Intrinsic Value of House Price by jolinmilioncherie

VIEWS: 2 PAGES: 67

									    Dynamic Model: House Price Returns, Mortgage rates and Mortgage

                                          Default rates

                                Xiangjing Wei and Shaun Wang1

                                      (Revised in Nov 2008)

                                             Abstract

     We apply vector auto regression models (VAR) and simultaneous equations models (SEM) to

estimate the dynamic relations among house price returns, mortgage rates and mortgage default

rates, using historical data during the time period of 1979 till 2007.

     We estimate that, holding all the other factors constant, two consecutive one-percent

increases of default rates can drive OFHEO’s house price returns down by about 5 percent and

Case-Shiller’s current house price return down by about 12 percent. Conversely, two consecutive

1-percent decreases of OFHEO’s or Case-Shiller’s house price returns can drag the current

default rate up by 0.08 percent or 0.05 percent, respectively.

     We apply our models in making predictions using data up to the second quarter of 2008. Not

surprisingly, the OFHEO’s and Case-Shiller’s indices exhibit different patterns and thus they

yield different predictions as well. On an expected value basis, the future level of OFHEO’s

house price returns will remain negative and reach the lowest value in 2010; it may take quite a

few years for the house price returns to become positive. However, we get more optimistic

forecasts using the Case-Shiller’s index, whereas the future house price returns would become

positive since 2010, and mortgage default rates will peak by 2010 and decrease thereafter.




1
 Shaun Wang
Professor, Department of Risk Management and Insurance, Robinson College of Business, Georgia State
University
Xiangjing Wei (Correspondent)
Department of Risk Management and Insurance, Robinson College of Business, Georgia State University
Email: insxxwx@langate.gsu.edu


                                                  1
    Dynamic Model: House Price Return, Mortgage rate and

                             Mortgage Default rate

                          Xiangjing Wei and Shaun Wang

1. Introduction

   Since late 2006 up to the time of writing this paper, the mortgage default rates and

foreclosure rates have been increasing steeply. According to the National Delinquency

Survey from Mortgage Banker's Association (MBA), in the fourth quarter of 2007 among

all the mortgage loans in the USA, the percentage of loans past due rose by 17.58 percent

to 5.82, compared with the fourth quarter of 2006; similarly, the percentage of loans past

due 30 days increased by 3.90 percent to 3.20; the percentage of loans past due 60 days

climbed 27.78 percent to 1.15; the percentage of loans past due 90 days was up 54.17

percent to 1.48; and the percentage of loans in foreclosure jumped 53.70 percent to 0.83

(Table 1).

   Although it is agreed that this mortgage meltdown originated in subprime adjustable

rate mortgages (ARM), the default rates of all types of mortgages have increased in

different degrees based on the data from MBA. Although some relatively new mortgage

products (such as interest only mortgage and negative amortization mortgage) and loose

underwriting may be blamed for the mortgage meltdown, there indeed exist fundamental

economic drives: the changes in house prices and in mortgage rates.




                                            2
                                 Table 1: Mortgage Delinquency Rates

The default rates of all types of mortgages have increased in different degrees, especially the adjustable
rate mortgages (ARM).
                                       2005Q4 Increased 2006Q4 Increased 2007Q4 Increased
            Loans Past Due              4.70%        7.31%    4.95%         5.32%     5.82%       17.58%
            Loans Past Due 30 Days      2.85%        2.89%    3.08%         8.07%     3.20%         3.90%
All loans   Loans Past Due 60 Days      0.83%      10.67%     0.90%         8.43%     1.15%       27.78%
            Loans Past Due 90 Days      1.02%      18.60%     0.96%        -5.88%     1.48%       54.17%
            Loans in Foreclosure        0.42%       -8.70%    0.54%        28.57%     0.83%       53.70%

           Loans Past Due               2.21%        8.33%     2.27%        2.71%     2.56%       12.78%
Prime      Loans Past Due 30 Days       1.49%        0.00%     1.64%       10.07%     1.72%        4.88%
FRM        Loans Past Due 60 Days       0.35%       12.90%     0.34%       -2.86%     0.44%       29.41%
Loans      Loans Past Due 90 Days       0.37%       48.00%     0.29%      -21.62%     0.40%       37.93%
           Loans in Foreclosure         0.15%      -11.76%     0.16%        6.67%     0.22%       37.50%

           Loans Past Due               2.54%       20.38%     3.39%       33.46%     5.51%       62.54%
Prime      Loans Past Due 30 Days       1.76%       13.55%     2.30%       30.68%     2.89%       25.65%
ARM        Loans Past Due 60 Days       0.44%       33.33%     0.63%       43.18%     1.20%       90.48%
Loans      Loans Past Due 90 Days       0.34%       47.83%     0.47%       38.24%     1.41%      200.00%
           Loans in Foreclosure         0.20%        5.26%     0.41%      105.00%     1.06%      158.54%

           Loans Past Due               9.70%       -0.21%    10.09%        4.02%    13.99%       38.65%
Subprime   Loans Past Due 30 Days       5.06%        1.00%     5.57%       10.08%     7.17%       28.73%
FRM        Loans Past Due 60 Days       1.60%        1.27%     1.73%        8.12%     2.54%       46.82%
Loans      Loans Past Due 90 Days       3.04%       -2.88%     2.78%       -8.55%     4.29%       54.32%
           Loans in Foreclosure         1.05%      -23.36%     1.09%        3.81%     1.52%       39.45%

            Loans Past Due            11.61%       18.11% 14.44%           24.38%    20.02%       38.64%
Subprime Loans Past Due 30 Days        6.74%       13.66%    7.93%         17.66%     8.80%       10.97%
ARM         Loans Past Due 60 Days     2.35%       23.68%    3.13%         33.19%     4.58%       46.33%
Loans       Loans Past Due 90 Days     2.53%       25.87%    3.38%         33.60%     6.64%       96.45%
            Loans in Foreclosure       1.55%        3.33%    2.70%         74.19%     5.29%       95.93%
Sources: Mortgage Bankers Association: National Delinquency Survey
*: Increased percentage compared with one year ago




                                                    3
       As for the mechanism of mortgage default, intuitively speaking, default will occur if,

(i) compared with family income, payment is unbearable and (ii) there is insufficient

equity to enable a refinance or sale. Only when the two situations occur simultaneously,

will the default rate increase sharply. Therefore default rate is influenced by both housing

equity and affordability. Since the house prices and the mortgage rates are the

determinants of housing equity and affordability, the mortgage default rates heavily

depend on the house prices and the mortgage rates.

       On the one hand, as house prices decrease or mortgage rates increase, we expect the

default rates to rise. On the other hand, as foreclosures mounted, unsold homes piled up,

slowing the pace of home sales. This pushed home prices even lower. According to the

House Price Index (HPI)2 released by the Office of Federal Housing Enterprise Oversight

(OFHEO), in the third quarter of 2007, the national quarter-over-quarter house price

return fell to -0.24 percent, first negative since 1995. Based on OFHEO’s News Release,

the states with the lowest rates of appreciation between the fourth quarter of 2006 and the

fourth quarter of 2007 were California (-6.6%), Nevada (-5.9%), Florida (-4.7%) and

Michigan (-4.3%).

       During the past six months from September 18, 2007 till March 18, 2008, the central

bank has slashed its federal funds rate, a key overnight bank lending rate, to 2.25% from

5.25%, by six interest rate cuts including two 75-basis-point cuts in January 2008 and

March 2008 respectively. The aim is to lower borrowing costs for consumers and to keep

the economy from slipping into recession. The effects of lowering interest rates on

mortgage rates are complicated. On one side, lower interest rates tend to pull down the



2
    Includes data from home sales and appraisals for refinancings.


                                                       4
mortgage rates. On the other side, due to the banks’ reluctance to provide credit, they

tend to raise the margin (interest spread) in the mortgage rates.

    In this paper, we analyze the dynamic relations among house prices, mortgage rates

and default rates. Figure 1 shows the changes in these variables over time.

    We will investigate both OFHEO’s House Price Index and S&P/ Case-Shiller’s Home

Price Index for house prices.3 These two indices are most widely accepted nowadays.

Both are repeat sales indexes. S&P/ Case-Shiller index is value-weighted, based on 10 or

20 metropolitan areas4, available from 1987. OFHEO’s index is unit-weighted, based on

the fifty states and Washington D.C., available from 1975. Moreover OFHEO’s House

Price Index only uses the data based on Fannie Mae and Freddie Mac mortgages. Case-

Shiller’s House Price Index obtains data from county assessor and recorder offices and

therefore covers more houses in the specific areas.

    Figure 1 displays the differences between the two indices. Case-Shiller’s Index shows

larger fluctuations than OFHEO’s Index. For the third and the fourth quarter of 2007,

OFHEO’s Index shows quarter-over-quarter house price returns of -0.24 percent and 0.09

percent respectively, while Case-Shiller’s Index shows quarter-over-quarter house price

returns of -1.79 percent and -5.51 percent respectively.

    As for the mortgage rate, we use data from the Freddie Mac’s website, where the 30-

year fixed rates, 15-year fixed rates, 5-year adjustable rates, and 1-year adjustable rates



3
  Additionally, the current house price series (or indexes) used to measure national trends include the
median price of existing homes sold (published by the National Association of Realtors) and the median
price of new homes sold (published by the Bureau of the Census of the U.S. Department of Commerce).
These two indices are not seasonally adjusted and reflect only recent sales, so they are volatile in the short
run.
4
  10 metropolitan areas include Boston, Chicago, Denver, Las Vegas, Los Angeles, Miami, New York, San
Diego, San Francisco, Washington DC. 20 metropolitan areas also include Atlanta, Charlotte, Cleveland,
Dallas, Detroit, Minneapolis, Phoenix, Portland (Oregon), Seattle, Tampa


                                                      5
are released. In selecting the most representative mortgage rates, we use 30-year fixed

mortgage rate.

   Before determining the measure of default rate, it is necessary to specify delinquency

and default. Practically, delinquency is differentiated from default based on the number

of days of missed installments. Delinquency refers to the non-payment of a mortgage

payment due, so it may be defined as a 30-days-and-over delinquency, a 60-days-and-

over delinquency or a 90-days-and-over delinquency. Default happens when a borrower

fails to pay back 90-days’ installment due and the fourth payment is due. So here we

utilize percent of all loans past due 90 days.

   In this paper, we work through simultaneous equations models of house price returns,

mortgage rates and default rates to investigate their dynamic relations. Three models are

defined. The first is vector auto regression model to analyze only relations among the

three variables. The second is extended based on Case and Shiller (1990) and examines

the effects of both the lagged terms and other related variables. The third model follows

Abraham and Hendershott (1996) and incorporates the cumulative fundamental-actual

differences.

   The major contribution of this paper is to extend from single house price return

models to simultaneous equations models. And, by working on a structural model, the

dynamic relations among house price returns, mortgage rates and default rates could be

investigated better. With all the other factors constant, two consecutive one-percent

increases of default rates can drive OFHEO’s house price returns down by about 5

percent and Case-Shiller’s current house price return down by about 11 percent.

Conversely, two consecutive 1-percent decreases of OFHEO’s or Case-Shiller’s house




                                                 6
price returns can drag the current default rate up by 0.08 percent or 0.05 percent,

respectively. However, the relative high standard errors may render the above estimates

fluctuating in a wide range. We apply our models in making predictions using data up to

the second quarter of 2008. Not surprisingly, the OFHEO’s and Case-Shiller’s indices

exhibit different patterns and thus they yield different predictions as well. On an expected

value basis, the future level of OFHEO’s house price returns will remain negative and

reach the lowest value in 2010; it may take quite a few years for the house price returns to

become positive. However, we get more optimistic forecasts using the Case-Shiller’s

index, whereas the future house price returns would become positive since 2010, and

mortgage default rates will peak by 2010 and decrease thereafter.

   The structure of the rest of this paper is as follows: Section 2 gives a literature review.

Section 3 presents the three models. Section 4 discusses the model specification and

empirical results. Section 5 makes predictions based on the models. Section 6

summarizes our conclusions.




                                             7
                                                                                                                                                                                                                                                                                                                                                             %




                                                                                                                                                                                                               -8
                                                                                                                                                                                                                    -6
                                                                                                                                                                                                                                                                                                                                    -4
                                                                                                                                                                                                                                                                                                                                         -2
                                                                                                                                                                                                                                                                                                                                                 0
                                                                                                                                                                                                                                                                                                                                                     2
                                                                                                                                                                                                                                                                                                                                                         4
                                                                                                                                                                                                                                                                                                                                                             6




            0.20
                   0.40
                          0.60
                                 0.80
                                        1.00
                                                       1.20
                                                                                            1.40
                                                                                                       4.00
                                                                                                              6.00
                                                                                                                     8.00
                                                                                                                            10.00
                                                                                                                                    12.00
                                                                                                                                            14.00
                                                                                                                                                    16.00
                                                                                                                                                                                               18.00
                                                                                                                                                                                                       20.00
                                                                                                                                                                                                                                                                                                                                              u -7
                                                                                                                                                                                                                                                                                                                                              Jn 5
     a 9
    M r-7
                                                                                                    a 5
                                                                                                   M r-7
                                                                                                                                                                                                                                                                                                                                              u -7
                                                                                                                                                                                                                                                                                                                                              Jn 6
     a 0
    M r-8                                                                                           a 6
                                                                                                   M r-7
                                                                                                                                                                                                                                                                                                                                              u -7
                                                                                                                                                                                                                                                                                                                                              Jn 7
     a 1
    M r-8                                                                                           a 7
                                                                                                   M r-7
                                                                                                                                                                                                                                                                                                                                              u -7
                                                                                                                                                                                                                                                                                                                                              Jn 8
    M r-8
     a 2                                                                                            a 8
                                                                                                   M r-7
                                                                                                                                                                                                                                                                                                                                              u -7
                                                                                                                                                                                                                                                                                                                                              Jn 9
                                                                                                    a 9
                                                                                                   M r-7
     a 3
    M r-8
                                                                                                                                                                                                                                                                                                                                              u -8
                                                                                                                                                                                                                                                                                                                                              Jn 0
                                                                                                    a 0
                                                                                                   M r-8
     a 4
    M r-8
                                                                                                                                                                                                                                                                                                                                              u -8
                                                                                                                                                                                                                                                                                                                                              Jn 1
                                                                                                    a 1
                                                                                                   M r-8
     a 5
    M r-8
                                                                                                                                                                                                                                                                                                                                              u -8
                                                                                                                                                                                                                                                                                                                                              Jn 2
                                                                                                    a 2
                                                                                                   M r-8
     a 6
    M r-8                                                                                                                                                                                                                                                                                                                                     u -8
                                                                                                                                                                                                                                                                                                                                              Jn 3
                                                                                                    a 3
                                                                                                   M r-8
     a 7
    M r-8                                                                                                                                                                                                                                                                                                                                     u -8
                                                                                                                                                                                                                                                                                                                                              Jn 4
                                                                                                    a 4
                                                                                                   M r-8

     a 8
    M r-8                                                                                          M r-8
                                                                                                    a 5                                                                                                                                                                                                                                       u -8
                                                                                                                                                                                                                                                                                                                                              Jn 5

    M r-8
     a 9                                                                                           M r-8
                                                                                                    a 6                                                                                                                                                                                                                                       u -8
                                                                                                                                                                                                                                                                                                                                              Jn 6


    M r-9
     a 0                                                                                            a 7
                                                                                                   M r-8                                                                                                                                                                                                                                      u -8
                                                                                                                                                                                                                                                                                                                                              Jn 7

                                                                                                   M r-8
                                                                                                    a 8                                                                                                                                                                                                                                       u -8
                                                                                                                                                                                                                                                                                                                                              Jn 8
     a 1
    M r-9
                                                                                                    a 9
                                                                                                   M r-8                                                                                                                                                                                                                                      u -8
                                                                                                                                                                                                                                                                                                                                              Jn 9
     a 2
    M r-9
                                                                                                    a 0
                                                                                                   M r-9                                                                                                                                                                                                                                      u -9
                                                                                                                                                                                                                                                                                                                                              Jn 0
     a 3
    M r-9
                                                                                                    a 1
                                                                                                   M r-9                                                                                                                                                                                                                                      u -9
                                                                                                                                                                                                                                                                                                                                              Jn 1




8
     a 4
    M r-9
                                                                                                    a 2
                                                                                                   M r-9                                                                                                                                                                                                                                      u -9
                                                                                                                                                                                                                                                                                                                                              Jn 2
     a 5
    M r-9
                                                                                                    a 3
                                                                                                   M r-9
                                                                                                                                                                                                                                                                                                                                              u -9
                                                                                                                                                                                                                                                                                                                                              Jn 3
     a 6
    M r-9                                                                                           a 4
                                                                                                   M r-9
                                                                                                                                                                                                                                                                                                                                              u -9
                                                                                                                                                                                                                                                                                                                                              Jn 4
     a 7
    M r-9                                                                                           a 5
                                                                                                   M r-9
                                                                                                                                                                                                                                                                                  Quarter-over-quarter OFHEO's House Price Return




                                                                                                                                                                                                                                                                                                                                              u -9
                                                                                                                                                                                                                                                                                                                                              Jn 5
    M r-9
     a 8                                                                                            a 6
                                                                                                   M r-9
                                                                                                                                                                                                                         Quarter-over-quarter Case-Shiller's House Price Return




                                                                                                                                                                                                                                                                                                                                              u -9
                                                                                                                                                                                                                                                                                                                                              Jn 6

    M r-9
     a 9                                                                                            a 7
                                                                                                   M r-9
                                                                                                                                                                                                                                                                                                                                              u -9
                                                                                                                                                                                                                                                                                                                                              Jn 7
                                                                                                    a 8
                                                                                                   M r-9
     a 0
    M r-0                                                                                                                                                                                                                                                                                                                                     u -9
                                                                                                                                                                                                                                                                                                                                              Jn 8
                                                                                                    a 9
                                                                                                   M r-9
     a 1
    M r-0                                                                                                                                                                                                                                                                                                                                     u -9
                                                                                                                                                                                                                                                                                                                                              Jn 9
                                                                                                    a 0
                                                                                                   M r-0
     a 2
    M r-0                                                                                                                                                                                                                                                                                                                                     u -0
                                                                                                                                                                                                                                                                                                                                              Jn 0
                                                                                                    a 1
                                                                                                   M r-0
    M r-0
     a 3                                                                                                                                                                                                                                                                                                                                      u -0
                                                                                                                                                                                                                                                                                                                                              Jn 1
                                                                                                    a 2
                                                                                                   M r-0




                                               90 days and over default rate of all loans
    M r-0
     a 4                                                                                                                                                                                                                                                                                                                                      u -0
                                                                                                                                                                                                                                                                                                                                              Jn 2
                                                                                                    a 3
                                                                                                   M r-0
                                                                                                                                                                                                                                                                                                                                              u -0
                                                                                                                                                                                                                                                                                                                                              Jn 3
                                                                                                                                                            30-year fixed rate Mortgage rate




     a 5
    M r-0
                                                                                                                                                                                                                                                                                                                                                                                 Figure 1: house price returns, mortgage rates and default rates




                                                                                                    a 4
                                                                                                   M r-0
                                                                                                                                                                                                                                                                                                                                              u -0
                                                                                                                                                                                                                                                                                                                                              Jn 4
    M r-0
     a 6                                                                                            a 5
                                                                                                   M r-0

                                                                                                                                                                                                                                                                                                                                              u -0
                                                                                                                                                                                                                                                                                                                                              Jn 5
    M r-0
     a 7                                                                                            a 6
                                                                                                   M r-0

                                                                                                   M r-0
                                                                                                    a 7                                                                                                                                                                                                                                       u -0
                                                                                                                                                                                                                                                                                                                                              Jn 6

                                                                                                                                                                                                                                                                                                                                              u -0
                                                                                                                                                                                                                                                                                                                                              Jn 7
                                                                                                                                                                                                                                                                                                                                                                 (This figure shows the changes of house price returns, mortgage rates and default rates over time.)
2. Literature Review

2.1. House Market Models

   A substantial literature exists for economic models of house price returns. Capozza

and Helsley (1989, 1990) find that house value is the sum of construction cost and the

land value. The real land value is the sum of three components: the real value of

agricultural land rent, the cost of developing the land for urban use, and the value of

“accessibility” to the central business district. Abraham and Hendershott (1993, 1996)

express the equilibrium house price return as a function of the growth in real construction

costs, the growth in real income, the employment growth and the change in real after-tax

interest rates. Case and Shiller (2003) model the house price mainly from the demand

side, including the following fundamentals: personal income per capita, population,

employment, unemployment rate, housing starts and mortgage rate. They also list the

relation between the demand for house and the stock market fluctuations.

   Some literature identifies the house price model as a system of two equations

(demand and supply). McCarthy and Peach (2002, 2004) express the long-term house

price of demand side as a function of housing stock, consumption and housing user cost

and the long term house price of supply side as a function of investment rate and structure

cost. Edelstein and Tsang (2007) relate rent, property values and capitalization rates with

demand fundamentals and relate housing investment and property values with supply

fundamentals.

   So previous literature on house price (or return) model basically focuses on the

relations of house price (or return) with income, construction cost, mortgage rate or



                                            9
interest rate, and other economic variables. However, few consider the impacts of default

rates on house prices (or returns). These impacts will be emphasized in our paper.

    Another strand of the real estate literature focuses on house price cycles. Case and

Shiller (1989) report that the lagged house price appreciation rate should be included

when doing price regressions. Shiller (1990) obtains from his survey that house

appreciation is related with the backward-looking expectation of the market participants.

Abraham and Hendershott (1996) specify the house price error term as two parts: one-

period lagged hours price return and serial correlated adjustment of lagged values of

house price return. Capozza et al. (2002) estimate equations relating the extent of serial

correlation and mean reversion to possible determinants. In this paper, we will apply this

cycle analysis to the dynamic process of house price returns, mortgage rates and default

rates.



2.2. Mortgage Default Models

    The literature on mortgage defaults mainly has two methods: option theoretical

approaches and empirical studies.

    Option theoretical approaches, which are based on Black and Scholes (1972) and Cox,

Ingersoll, and Ross (1985), provide an explicit theoretical framework to understanding

the default risks or prepayment risks inherent in home mortgages (e.g. Kau, Keenan,

Muller, and Epperson, 1990, 1992, 1993, 1995). According to these models, mortgages

can be viewed as ordinary debt instruments embedded with two basic options: a put

option, which reflects the ability to default on the mortgage, and a call option, which

reflects the ability to refinance the mortgage. So, when the present value of the mortgage




                                           10
including the inherent options is less than the present value of the remaining payment

stream, the borrower will choose to hold the mortgage, no default or prepayment.

However, because of transaction costs and reputation costs, the borrowers may not

exercise their options (default or prepayment) when the options are in-the-money.

    Most empirical studies have fitted the Cox Proportional hazard model5 on the account

level data to evaluate mortgage default risk or prepayment risk (e.g. Epperson, Kau,

Keenan, and Muller, 1985; Schwartz and Torous, 1993; Quigley and Van Order, 1995;

Deng, 1997; Deng, Quigley and Van Order, 2000; Ambrose, Capone and Deng, 2001).

The basic idea is that it is difficult to observe the critical house price and mortgage rate

that trigger the option exercise. What may be estimated is the probability that either

option is in-the-money. Hence the default rate or prepayment rate would be a function of

these probabilities.

    In this paper, we will estimate the mortgage default rates as a function of house price

returns, mortgage rates, loan-to-value ratio and other related variables at the macro level.



2.3. Recent Papers after Mortgage Meltdown

    After the mortgage meltdown in 2007, quite a few papers discuss its reasons and

impacts. Cagan (2007) projects the amount of default mortgages, including prime

mortgages and subprime mortgages, due to mortgage payment reset. Weaver and Reeves

(2007) states the impacts on default of Subprime Adjustable Rate Mortgages (ARMs) at

the fully indexed rate, instead of the low introductory rate.




5
 See David R Cox (1972) “Regression models and life tables”, Journal of the Royal Statistical Society
Series B 34:187-220


                                                   11
    As things went on, people began to think what the crucial problems are in the

mortgage market. For example, Foote, et al (2008) state that interest-rate resets may not

be the main problem in the mortgage market; and higher foreclosure rates stem from

falling house prices. They use the data from a private firm and focus on the situations in

Massachusetts and New England. One of their concerns is that home prices have a bigger

impact on foreclosures than foreclosures have on home prices. Greenlaw, et al (2008) and

Hatzius (2008) put emphasis on modeling mortgage credit losses, based on the effects of

home price declines on foreclosure and mortgage credit losses.

    In this paper we build dynamic models to estimate the interactions among house price

returns, mortgage rates and default rates. We find that mortgage default rates also have

huge impacts on house price returns.



3. Our Models

3.1. Model 1

    Due to the dynamic interactions between the house price returns, the mortgage rates

and the default rates, we first utilize a vector autoregressive process VARMA(p,d,q) to

interpret them.

           p                   q
    Yt = ∑ Φ i Yt −i + ε t + ∑ Θ j ε t − j ,
          i =1                j =1


                            ′                                                                  ′
where Yt = (HRt , Dt , MRt ) refers to the endogenous variables and ε t = (ε 1t , ε 2t , ε 3t )

refers to a vector white noise process. Φ i and Θ j are 3 × 3 matrices. HRt is the quarter-

over-quarter house price return at time t; MRt means the mortgage rate; Dt refers to the




                                               12
default rate. d means the differencing times due to non-stationarity of some variables.

This process is the reduced form of vector autoregressive model.



3.2. Model 2

    Case and Shiller (1990) build a forecasting model for house price return, including

both lagged house price return and other exogenous variables. The lagged house price

returns reflect a momentum part.

    In our paper, since we have more than one endogenous variable, a simultaneous

equations model is introduced and some exogenous variables are included. Our model

can be represented as6

                ⎛ p2            p3
                                          ⎞ p1
     HRt = f 1 ⎜ ∑ MRt − s , ∑ Dt − s , X ⎟ + ∑ a s HRt − s + ε 1 ,
                ⎜                         ⎟
                ⎝ s =0         s =0       ⎠ s =1
                 ⎛ p1            p3
                                          ⎞ p2
     MRt = f 2 ⎜ ∑ HRt − s , ∑ Dt − s , Y ⎟ + ∑ bs MRt − s + ε 2 ,
                 ⎜                        ⎟                                                 (2)
                 ⎝ s =0         s =0      ⎠ s =1
              ⎛ p1           p2
                                          ⎞ p3
     Dt = f 3 ⎜ ∑ HRt − s , ∑ MRt − s , Z ⎟ + ∑ c s Dt − s + ε 3 ,
              ⎜                           ⎟
              ⎝ s =0        s =0          ⎠ s =1

where X, Y and Z refer to vectors of economic variables. We will define these three

vectors specifically in the next section.

    The first part of each equation can be regarded as a fundamental value or an intrinsic

value of the endogenous variable. The serial correlation part represents the momentum.

This model is under the structural framework, so that the relationships among the

variables could be more clearly examined.



6
  If mortgage rates are modeled as an exogenous variable, the simultaneous equations model will only
contain the first and third equation in the system. The following equations are the same, so we will not
identify the two-equation system anymore in this section.


                                                     13
3.3. Model 3

   Assume that in each time period t the house price return, the mortgage rate and the

default rate have their fundamental values determined by economic conditions.

                ⎛ p2            p3
                                          ⎞
    HR = f 1 ⎜ ∑ MRt − s , ∑ Dt − s , X ⎟
        *
        t       ⎜                         ⎟
                ⎝ s =0         s =0       ⎠
                 ⎛ p1            p3
                                          ⎞
    MRt* = f 2 ⎜ ∑ HRt − s , ∑ Dt − s , Y ⎟
                 ⎜                        ⎟                                                                     (3)
                 ⎝ s =0         s =0      ⎠
              ⎛ p1           p2
                                          ⎞
    Dt* = f 3 ⎜ ∑ HRt − s , ∑ MRt − s , Z ⎟
              ⎜                           ⎟
              ⎝ s =0        s =0          ⎠

where HRt* , MRt* and Dt* represent the fundamental values determined by equation (3).

So we have

    HRt = HRt* + υ t1
    MRt = MRt* + υ t2
    Dt = Dt* + υ t3

where υ t1 , υ t2 and υ t3 are the error terms. Following Abraham and Hendershott (1996),

the error terms could be described as an adjustment dynamics:


   υ t1 = a0 + ∑ a s HRt − s + α ∑ (HRi* − HRi ) + η t1
                   p1                            t −1


                   s =1                          i =1


   υ t2 = b0 + ∑ bs MRt − s + β ∑ (MRi* − MRi ) + η t2
                   p2                            t −1


                   s =1                          i =1


   υ t3 = c0 + ∑ c s Dt − s + γ ∑ (Di* − Di ) + η t3
                   p3                     t −1


                   s =1                   i =1




        ∑ (HR                     )         ∑ (MR                   )          ∑ (D              )
            t −1                             t −1                              t −1
where                *
                     i    − HRi       ,                 *
                                                        i   − MRi        and          *
                                                                                      i   − Di       are the cumulative
            i =1                             i =1                              i =1



fundamental-actual differences. η t1 , η t2 and η t3 are the error terms. So, putting these

equations together, we may model the dynamics as



                                                                    14
               ⎛ p2                      ⎞
                                                                   (           )
                               p3                   p1                   t −1
    HRt = f1 ⎜ ∑ MRt − s , ∑ Dt − s , X ⎟ + a0 + ∑ a s HRt − s + α ∑ HRi* − HRi + δ 1 ,
               ⎜                         ⎟
               ⎝ s =0         s =0       ⎠         s =1                  i =1

                ⎛ p1                     ⎞
                                                                   (           )
                                p3                 p2                   t −1
    MRt = f 2 ⎜ ∑ HRt − s , ∑ Dt − s , Y ⎟ + b0 + ∑ bs MRt − s + β ∑ MRi* − MRi + δ 2 , (4)
                ⎜                        ⎟
                ⎝ s =0         s =0      ⎠        s =1                  i =1

             ⎛ p1                        ⎞
                                                               (        )
                            p2                     p3              t −1
    Dt = f 3 ⎜ ∑ HRt − s , ∑ MRt − s , Z ⎟ + c0 + ∑ c s Dt − s + γ ∑ Di* − Di + δ 3 ,
             ⎜                           ⎟
             ⎝ s =0        s =0          ⎠        s =1             i =1


   Here we make a development by estimating the variables in a simultaneous equations

model.



4. Model Specification and Empirical Results

4.1. Data Description

   We consider additional exogenous variables that include:

   1) Inflation rate ( Inf ). The measure of inflation rate comes from the Consumption

Price Index from the U.S. Bureau of Labor Statistics (BLS), which is available monthly

and converted into quarterly data.

   2) Disposable personal income ( Inc ).The disposable personal income is available

from the Bureau of Economic Analysis’ website.

   3) Unemployment rate ( Unem ). The unemployment rate is from the BLS Household

Survey.

   4) Construction cost (CC). We use Construction Price Index as the measure of

construction cost. This index is the price deflator index of new one-family houses under

construction from U.S. Census Bureau.

   5) Gross domestic product (GDP). GDP is from the Bureau of Economic Analysis

(BEA).



                                            15
   6) 10-year treasury bond rate (TB). The 10-year treasury bond rate is available on

the Federal Reserve Board’s website.

   7) 3-month treasury bill rate (TB3m). The 3-month treasury bill rate is available on

the Federal Reserve Board’s website.

   8) Composite loan-to-price ratio (CLTV). The composite loan-to-price ratio of all

loans comes from U.S. Federal Housing Finance Board.

   9) Homeownership Rate (HO). The Homeownership Rates are obtained from US

Census Bureau.

   For the models dealing with OFHEO’s house price returns, we use quarterly data

from the first quarter of 1979 till the fourth quarter of 2007, with 126 observations in

total, due to the data source restrictions of mortgage default rate. For the models dealing

with Case-Shiller’s house price returns, we use quarterly data from the first quarter of

1987 till the fourth quarter of 2007, with 84 observations in total. In order to eliminate the

confusion caused by the constant term, we utilize the de-meaned data here.

   We checked the stationarity of all the variables and the results are presented in Table

4. Only house price return and inflation rate reject the non-stationary null at 1%

significance level. For all other variables, the nonstationary null hypothesis cannot be

rejected, while the null is easily rejected for the first differences, showing that these

variables are integrated of order 1. In order to avoid the spurious regression, we may

correspondingly add the lagged or differenced terms.




                                             16
    To detect multicollinearity, as a first indicator, we may look at the correlation of the

possible independent variables (Table 57). With the variables left in Table 5, there is no

extremely large positive or negative correlation.

    The most commonly used approaches to assessing collinearity are tolerance, variance

inflation factor and condition indexes. Basically, tolerance measures the correlation

between one independent variable and all the other independent variables. If we define

  2
R X , X as the correlation between one dependent variable X and all the other independent
      ~



variables, then the tolerance (TOL) would be TOLX = 1 − R X , X . A small value of
                                                          2
                                                              ~



tolerance means that the variable X is highly correlated with the other variables. The

variance inflation factor (VIF) is the inverse of tolerance, VIFX = 1 / TOL X , showing the

degree by which the standard error of the estimator is inflated by multicollinearity. So a

VIF of 16 shows that the standard error of the estimator is 4 times inflated due to

multicollineary.     Practically,     TOL < 0.1      and    equivalently VIF > 10          indicate      a

multicollinearity problem. Table 6 exhibits the tolerances and variance inflation factors of

the possible independent variables, which shows no serious multicollinearity.

    Conditional index is the ratio of a specific eigenvalue over the maximum of all

eigenvalues of the model matrix. As an informal rule, conditional index over 30 may

show multicollinearity. Table 7 shows all the conditional indices for the possible

independent variables. The maximum conditional index is around 10, reflecting no

serious multicollinearity.




7
 Table 5 shows the correlations with OFHEO’s house price returns. The results for Case-Shiller’s house
price returns are similar and we do not list in this paper.


                                                   17
                                    Table 3: data sources.
                   Data                                              Sources
House Price Index                               OFHEO
30-year fixed-rate mortgage contract rate       Federal Reserve Board /Freddie Mac
10 year treasury bond rate                      Federal Reserve Board
3-month treasury bill rate                      Federal Reserve Board
Mortgage default rate                           Mortgage Bankers Association (MBA)
(Disposable) Personal Income                    Bureau of Economic Analysis (BEA)
Gross domestic product                          Bureau of Economic Analysis (BEA)
Consumer Price Index (CPI)                      U.S. Bureau of Labor Statistics (BLS)
Unemployment rate                               U.S. Bureau of Labor Statistics (BLS)
Composite loan-to-price ratio                   U.S. Federal Housing Finance Board (FHFB)
Construction Price Indexes                      U.S. Census Bureau
Homeownership Rates                             U.S. Census Bureau


                   Table 4: Augmented Dickey-Fuller Unit Root Test
    This table exhibits the stationarity of the variables in our model. Only house price return
    and inflation rate reject the non-stationary null at 1% significance level. For all other
    variables, the non-stationary null hypothesis can not be rejected, while the null is easily
    rejected for the first differences, showing that these variables are integrated of order 1.
                                          Original Data         First Difference
           House Price Return                    -5.24***
             Mortgage Rate                            -0.8                  -8.6***
          Mortgage Default Rate                      -0.68                 -9.03***
              Inflation Rate                     -4.42***
                 Income                               6.29               -11.43***
           Unemployment rate                         -1.36                 -6.19***
            Construction Cost                         2.55                  -7.1***
                   GDP                               10.31                 -5.87***
       10-year Treasury Bond Rate                    -0.88                 -8.87***
       3-month Treasury Bill Rate                    -1.34                 -9.32***
     Composite Loan-To-Value Ratio                   -2.09               -10.86***
          Homeownership Rates                        -0.41               -13.97***
    Note: *10%, **5%, ***1% indicate the corresponding significance levels.
          The above data are checked based on Single Mean.
         After checking Zero Mean or Trend, we got the similar results.




                                                18
Table 5: Correlations of the possible independent variables
With the variables left in Table 5, there is no extremely large positive or negative correlation.
                                                                                                                                         1-
                      1-lagged     2-lagged    3-lagged     1-Lagged     1-lagged Change      2-lagged Change       3-lagged Change      lagged     1-lagged
                      HR           HR          HR           MR           of MR                of MR                 of MR                D          Change of D
 2-lagged HR               0.64
 3-lagged HR               0.54         0.64
 1-Lagged MR              -0.14        -0.08       -0.04
 1-lagged Change          -0.02         0.30        0.23          0.12
 of MR Change
 2-lagged                  0.05        -0.02        0.30          0.17                 0.27
 of MR Change
 3-lagged                  0.10         0.05       -0.02          0.18                 0.10                 0.27
 of MR
 1-lagged D               -0.10        -0.14       -0.18         -0.23                -0.19                -0.23                -0.19
 1-lagged Change          -0.07        -0.10       -0.09         -0.05                -0.14                -0.08                 0.10        0.52
 of D
 2-lagged Change          -0.01        -0.07       -0.10         -0.03                -0.05                -0.16                -0.06        0.48                0.36
 of D
 3-lagged Change           0.02        -0.01       -0.07         -0.02                -0.04                -0.06                -0.15        0.43                0.22
 of D
 Change of GDP             0.33         0.29        0.22          0.18                 0.04                -0.09                 0.06       -0.08                0.00
 Change of 10-             0.24         0.18        0.21         -0.07                 0.13                 0.07                 0.08       -0.13               -0.11
  ear TB
 Change of 3-              0.31         0.21        0.25         -0.12                -0.05                 0.02                 0.08       -0.14               -0.18
 month TB
 Inf rate                  0.36         0.39        0.37          0.36                 0.37                 0.27                 0.31       -0.28                0.03
 1-lagged Inf Rate         0.21         0.36        0.39          0.45                 0.45                 0.37                 0.27       -0.26                0.04
 change of CC              0.55         0.51        0.39         -0.03                 0.12                 0.20                 0.07       -0.25               -0.07
 change of Inc             0.24         0.26        0.25          0.28                 0.13                 0.19                 0.17       -0.18                0.06
 Unem Rate                -0.15        -0.16       -0.18          0.69                -0.16                -0.13                -0.08        0.05                0.03
 change of Unem           -0.17        -0.04       -0.04          0.19                 0.08                 0.17                 0.27       -0.10                0.23
 Rate
 CLTV                     -0.33        -0.34       -0.33         -0.46                 0.01                -0.04                -0.06       -0.04               -0.08
 change of CLTV           -0.18        -0.08       -0.01         -0.03                 0.06                -0.13                -0.04        0.14               -0.04
  change of HO             0.09         0.03       -0.02         -0.17                -0.13                 0.05                 0.00       -0.12               -0.03
HR=house price return; MR= 30-year fixed mortgage rate; D=default rate; TB=treasury bill rate; Inf=inflation; CC=construction cost; Inc=Income; Unem Rate=Unemployment
rate; CLTV=composite loan-to-value ratio; HO=homeownership rates




                                                                                 19
Table 5 (Cont.): Correlations of the possible variables
With the variables left in Table 5, there is no extremely large positive or negative correlation.
                             2-                                           Change                                                         change
                             lagged     3-lagged               Change     of 3-                1-                                        of                 change
                             Change     Change      Change     of 10-     month                lagged      change    change     Unem     Unem               of
                             of D       of D        of GDP     year TB    TB          Inf rate Inf Rate    of CC     of Inc     Rate     Rate         CLTV  CLTV
  2-lagged HR
  3-lagged HR
  1-Lagged MR
  1-lagged Change of MR
  2-lagged Change of MR
  3-lagged Change of MR
  1-lagged D
  1-lagged Change of D
  2-lagged Change of D
  3-lagged Change of D           0.29
  Change of GDP                 -0.03        0.00
  Change of 10-year TB          -0.03       -0.07       0.43
  Change of 3-month TB          -0.16       -0.10       0.44       0.65
  Inf rate                       0.10        0.07       0.47       0.32       0.19
  1-lagged Inf Rate              0.03        0.10       0.31       0.12       0.00        0.73
  change of CC                  -0.09       -0.04       0.36       0.23       0.25        0.40         0.34
  change of Inc                 -0.10        0.09       0.51       0.38       0.33        0.43         0.33      0.36
  Unem Rate                      0.08        0.15       0.27     -0.11       -0.15        0.17         0.23      0.04       0.23
  change of Unem Rate            0.18        0.19     -0.48      -0.36       -0.50        0.13         0.24     -0.17      -0.12       0.07
  CLTV                          -0.09       -0.17     -0.21      -0.01        0.09       -0.44        -0.47     -0.29      -0.24 -0.55         -0.29
  change of CLTV                 0.12       -0.09       0.11       0.26       0.09       -0.06        -0.08      0.02       0.15       0.15    -0.14 0.18
  change of HO                   0.02        0.01       0.04       0.15       0.10        0.00        -0.02      0.12       0.08 -0.14          0.03 0.05       -0.11
HR=house price return; MR= 30-year fixed mortgage rate; D=default rate; TB=treasury bill rate; Inf=inflation; CC=construction cost; Inc=Income; Unem Rate=Unemployment
rate; CLTV=composite loan-to-value ratio; HO=homeownership rates




                                                                                 20
                  Table 6: Tolerances and Variance Inflation Factors
                         of the Possible Independent Variables
  Practically, TOL < 0.1 and equivalently VIF > 10 indicate a multicollinearity problem. The
   results in this table show no serious multicollinearity for the possible independent variables.
                            Variable              TOL      VIF
                            1-lagged HR              0.426     2.349
                            2-lagged HR              0.353     2.829
                            3-lagged HR              0.363     2.756
                            1-Lagged MR              0.210     4.763
                            1-lagged Change of MR    0.475     2.104
                            2-lagged Change of MR    0.458     2.183
                            3-lagged Change of MR    0.613     1.631
                            1-lagged D               0.503     1.989
                            1-lagged Change of D     0.721     1.388
                            2-lagged Change of D     0.815     1.227
                            3-lagged Change of D     0.760     1.316
                            Change of GDP            0.363     2.753
                            Change of 10-year TB     0.405     2.470
                            Change of 3-month TB     0.391     2.554
                            Inf rate                 0.380     2.634
                            1-lagged Inf Rate        0.341     2.934
                            change of CC             0.573     1.745
                            change of Inc            0.558     1.792
                            Unem Rate                0.164     6.115
                            change of Unem Rate      0.264     3.788
                            CLTV                     0.258     3.878
                            change of CLTV           0.635     1.575
                            Change of HO             0.743     1.346
HR=house price return; MR= 30-year fixed mortgage rate; D=default rate; TB=treasury bill rate; Inf=inflation;
CC=construction cost; Inc=Income; Unem Rate=Unemployment rate; CLTV=composite loan-to-value ratio;
HO=homeownership rates




                                                    21
Table 7: the Conditional Indices for the Possible Independent Variables.
Conditional index is the ratio of a specific eigenvalue over the maximum of all eigenvalues of the model
matrix. As an informal rule, conditional index over 30 may show multicollinearity.
This table reports the eigenvalues and condition index for all the variables in Table 6 and an “one” vector.
The maximum conditional index here is around 10, reflecting no serious multicollinearity.
 Number                  Eigenvalue Condition Index
                     1        3.942                1.000
                     2        3.277                1.097
                     3        2.870                1.172
                     4        2.161                1.351
                     5        1.898                1.441
                     6        1.420                1.666
                     7        1.273                1.760
                     8        1.043                1.945
                     9        0.856                2.147
                   10         0.813                2.202
                   11         0.749                2.294
                   12         0.714                2.349
                   13         0.565                2.641
                   14         0.423                3.052
                   15         0.382                3.213
                   16         0.336                3.425
                   17         0.274                3.796
                   18         0.252                3.952
                   19         0.199                4.454
                   20         0.197                4.474
                   21         0.132                5.455
                   22         0.101                6.247
                   23         0.090                6.614
                   24         0.033               10.954




                                                    22
4.2 Model Specification

Model 1

    The mortgage rates and default rates are first-differenced due to their non-stationarity.

Table 8 displays the results for different Vector Auto Regression models, based on model

selection criteria.

    Appendix A shows the descriptions for the five information criteria listed in Table 8.

Basically, according to all the five criteria, model with the smallest values is preferred.

Here for OFHEO’s house price return, we choose VAR(7). Based on the significance

level of the coefficients, 1-period, 3-period, and 7-period house price returns, first-

differenced mortgage rates, and first-differenced default rates are chosen. Table 9

exhibits the model diagnostic check, showing that this model is homogenous.

    Similarly, for Case-Shiller’s house price return, we choose VAR(12). Based on the

significance level of the coefficients, 1-period, 3-period, 4-period, 5-period, 8-period, 10-

period, 11-period, and 12-period house price returns, first-differenced mortgage rates, and

first-differenced default rates are chosen.




    Table 8: Model Selection Criteria of difference Vector Auto Regression
 Information Criteria                        p=(1)        p=(1,2)    p=(1,2,3) p=(1,3)        p=(1,3,7)   p=(1,3,7,8)
 AICC(Corrected AIC)                          -8.33175 -8.21573 -8.25565 -8.38624 -8.62784                  -8.63024
 HQC(Hannan-Quinn Criterion)                  -8.22251     -8.0346 -8.01273 -8.20443 -8.38244               -8.33612
 AIC(Akaike Information Criterion)              -8.3394 -8.24028 -8.30818 -8.41124 -8.68453                 -8.73105
 SBC(Schwarz Bayesian Criterion)              -8.05138 -7.73342 -7.58001 -7.90152 -7.93949                  -7.75684
 FPEC(Final Prediction Error Criterion)       0.000239 0.000264 0.000247 0.000222 0.000169                 0.000162
    The descriptions for all the above information criteria are in Appendix A.
    p=(1 3 7) refers to the 1-period, 3-period, and 7-period lagged variables are chosen in the model.




                                                    23
    Table 9: VAR(7) Model Diagnostic Check
     The VAR model with 1-period, 3-period, and 7-period house price returns, first-differenced mortgage
rates, and first-differenced default rates is homogenous.
          Variable                                         DW(1)       ARCH F Value                 Prob>F
          House Price Return                                  1.89                      0.16          0.6889
          First-differenced Default Rate                      1.87                      0.05          0.8223
          First-differenced Mortgage Rate                     2.07                      1.64          0.2027



Model 2

    In the house model, we use inflation rate ( Inf ), disposable personal income ( Inc ),

unemployment rate ( Unem ), construction cost (CC), and 3-month Treasury bill rate

(TB3m) as the exogenous variables. The variables (inflation rate, disposable personal

income, unemployment rate, construction cost) are chosen as the exogenous variables of

house price (or return) model as in many literature. We choose 3-month Treasury bill rate

as the indicator of market interest rate.

    So the house price return equation in Equation (2) would be

                                           p2                                                p3
HR t = α 0 + α 1 MR t + α 2 MR t −1 +     ∑b
                                          s =1
                                                 1
                                                 s   ∆ MR t − p 3 + α 3 D t + α 4 D t −1 +   ∑c
                                                                                             s =1
                                                                                                    1
                                                                                                    s   ∆ D t − p 3 (5a)


      + α 5 Inf t + α 6 Inf t −1 + α 7 ∆CC t −1 + α 8 ∆Inct + α 9Unemt + α 10 ∆Unemt + α 11∆Tb3mt

           p1
        + ∑ a 1 HRt − s + ε 1
              s
          s =1


    The mortgage rate equation in Equation (2) includes additional variables, such as

inflation rate (Inf), gross domestic product (GDP), 10-year treasury bond rate (TB), 3-

month Treasury bill rate (TB3m) and has the form

                   p1                                    p3
MRt = β 0 + ∑ a s2 HRt − s + β1 Dt + β 2 Dt −1 + ∑ c s2 ∆Dt − s + β 3 Inf t −1 + β 4 ∆GDPt
                  s =0                                   s =1


                                                                p2
                 + β 5 ∆TBt + β 6 ∆TB3mt + β 7 MRt −1 + ∑ bs2 ∆MRt − s + ε 2                                        (5b)
                                                                s =1




                                                          24
   The default rate equation in Equation (2) may contain inflation rate (Inf), composite

loan-to-price ratio (CLTV), disposable personal income ( Inc ), 3-month Treasury bill

rate (TB3m) and Homeownership Rate (HO).

   So the equation could be expressed as

                p1                                           p2
Dt = γ 0 + ∑ a s3 HRt − s + γ 1 MRt + γ 2 MRt −1 + ∑ bs3 ∆MRt − s + γ 3 Inf t −1 + γ 4 CLTVt
            s =0                                           s =1


                                                                                     p3
    + γ 5 ∆CLTVt + γ 6 ∆Inct + γ 7 ∆TB3mt + γ 8 ∆HOt + γ 9 Dt −1 + ∑ c s3 ∆Dt − s + ε 3                           (5c)
                                                                                 s =1




Model 3

   Assuming the linear model with the fundamentals, equation (4) can be specified as:

                                          p2                                              p3
HR t = α 0 + α 1 MR t + α 2 MR t −1 +     ∑ b s1 ∆ MR t − p 3 + α 3 D t + α 4 D t −1 +
                                          s =1
                                                                                          ∑c
                                                                                          s =1
                                                                                                 1
                                                                                                 s   ∆ D t − p 3 (6a)


     + α 5 Inf t + α 6 Inf t −1 + α 7 ∆CCt −1 + α 8 ∆Inct + α 9Unemt + α 10 ∆Unemt + α 11 ∆TB3mt


                                      (           )
           p1                 t −1
        + ∑ a 1 HRt − s + α 12 ∑ HRi* − HRi + ε 1
              s
          s =1                 i =1


                      p1                                p3
MRt = β 0 + ∑ a s2 HRt − s + β1 Dt + β 2 Dt −1 + ∑ c s2 ∆Dt − s + β 3 Inf t −1 + β 4 ∆GDPt
                     s =0                               s =1




                                                                                     (               )
                                                      p2                      t −1
     + β 5 ∆TBt + β 6 ∆TB3mt + β 7 MRt −1 + ∑ b ∆MRt − s + β 8 ∑ MRi* − MRi + ε 2
                                                                  2
                                                                  s                                                (6b)
                                                      s =1                    i =1


                p1                                           p2
Dt = γ 0 + ∑ a s3 HRt − s + γ 1 MRt + γ 2 MRt −1 + ∑ bs3 ∆MRt − s + γ 3 Inf t −1 + γ 4 CLTVt                           (6c)
            s =0                                           s =1




                                                                                                            (      )
                                                                         p3                          t −1
+ γ 5 ∆CLTVt + γ 6 ∆Inct + γ 7 ∆TB3mt + γ 8 ∆HOt + γ 9 Dt −1 + ∑ c s3 ∆Dt − s + γ 10 ∑ Di* − Di + ε 3
                                                                        s =1                         i =1




                                                           25
   For such a simultaneous equations model, the typical estimation method is three-

stage-least-square (3SLS). And another difficulty is that HRt* , MRt* and Dt* themselves

depend on the estimation results from the model. So here we basically follow the

estimation method of Abraham and Hendershott (1996). The steps are as follows:

   (1)       estimating equation (6a, 6b, 6c) without the cumulative fundamental-actual

             difference terms;

   (2)       calculating HRi* , MRi* and Di* (i=1,2,…,t-1), so that we can obtain the


                                                                                                    ∑ (HR                  )
                                                                                                    t −1
             cumulative          fundamental-actual                   difference          terms                *
                                                                                                               i   − HRi           ,
                                                                                                    i =1




             ∑ (MR                )           ∑ (D               )
             t −1                             t −1
                      *
                      i   − MRi and                    *
                                                       i    − Di .
             i =1                             i =1




                      ∑ (HR                     ) ∑ (MR                   )        ∑ (D             )
                          t −1                       t −1                          t −1
   (3)       Adding              *
                                 i    − HRi ,                   *
                                                                i    − MRi and              *
                                                                                            i   − Di and re-estimating
                          i =1                       i =1                          i =1


             equation (6a, 6b, 6c).


                                                                                                        ∑ (HR                  )
                                                                                                        t −1
   (4)       Recalculating HRi* , MRi* , Di* (i=1,2,…,t-1) and                                                     *
                                                                                                                   i   − HRi       ,
                                                                                                        i =1




             ∑ (MR                ) ∑ (D                    )
             t −1                      t −1
                      *
                      i   − MRi ,                *
                                                 i   − Di .
             i =1                      i =1


   (5)       Re-estimating equation (6a, 6b, 6c) till the coefficients converge.



4.3. Estimation Results

1, Model 1

   The coefficients of the VAR models for OFHEO’s and Case-Shiller’s house price

returns are exhibited in Table 10 and 11 respectively. The columns (1), (2) and (3)




                                                                26
represent the separate equations in the vector auto regression model. The results reflect

that most of coefficients on the lagged terms have swinging signs on the cross-sectional

variables, displaying complex dynamic relationships.

   Since the model is a reduced form and the variables for default rate and mortgage rate

are first-differenced, the coefficients may not explain better the dynamic relationships

among the three variables. We will analyze the estimates in Model 2.




                                           27
            Table 10: VAR Coefficient Estimates for OFHEO’s HPI Return

     In this VAR(7) model, only the 1-period-lagged, 3-period-lagged, and 7-period-
     lagged variables are selected based on the Akaike Information Criterion (AIC).
     The columns (1), (2) and (3) represent the separate equations in the vector auto
     regression model.
     Panel A exhibits the estimated coefficients and Panel B shows the sums of each
     variable in each equation.
                                        House Price     First-Differenced       First-Differenced
                                           Return          Default Rate          Mortgage Rate
       Lag               Variable            (1)                 (2)                    (3)
                         Intercept       0.299***               0.007               -0.216***
                                          (0.128)             (0.011)                (0.109)
                   House Price Return   0.504****              -0.008              0.194****
                                          (0.082)             (0.007)                (0.069)
                    First-Differenced      -0.134            0.211***                 -0.086
        1              Default Rate       (1.249)             (0.105)                (1.059)
                    First-Differenced     -0.175*              -0.005              0.396****
                     Mortgage Rate        (0.109)             (0.009)                (0.092)
                   House Price Return   0.387****              -0.001                 -0.073
                                          (0.088)             (0.007)                (0.075)
                    First-Differenced      -1.780           0.321****                 -0.679
        3              Default Rate       (1.339)             (0.113)                (1.135)
                    First-Differenced      0.098                0.002                 -0.055
                     Mortgage Rate        (0.105)             (0.009)                (0.089)
                   House Price Return     -0.130*               0.006                  0.012
                                          (0.081)             (0.007)                (0.069)
                    First-Differenced      -1.445            0.282***                  0.438
        7              Default Rate       (1.360)             (0.115)                (1.154)
                    First-Differenced      0.015                0.002                 -0.008
                     Mortgage Rate        (0.095)             (0.008)                (0.080)
Note: *15%, **10%, ***5%, ****1% indicate the corresponding significance levels. The numbers in
brackets refers to the standard errors.




                                                 28
                     Table 11: VAR Coefficient Estimates for Case-Shiller’s HPI Return
        In this model, house price returns are from Case-Shiller’s Index. The columns (1), (2) and (3)
        represent the separate equations in the vector auto regression model.
        Panel A exhibits the estimated coefficients and Panel B shows the sums of each variable in each
        equation.
                                        Case-Shiller House Price   First-Differenced Default   First-Differenced
                                        Return (1)                 Rate (2)                    Mortgage Rate (3)
                                        Esti-      Std     P       Esti-       Std    P        Esti-       Std   P
Lag   Variable                          mate       Error Value     mate        Error Value     mate        Error Value
      Intercept                           0.170 0.173 0.332           0.000 0.013 0.989          -0.215 0.087 0.017
  1   House Price Return                  0.879 0.096 0.000           0.000 0.007 0.981          -0.009 0.048 0.849
      First-Differenced Default Rate     -3.037 2.127 0.160           0.204 0.161 0.212          -0.356 1.063 0.739
      First-Differenced Mortgage Rate    -0.759 0.266 0.006           0.012 0.020 0.571           0.099 0.133 0.462
  3   House Price Return                  0.028 0.111 0.804           0.010 0.008 0.256          -0.025 0.055 0.654
      First-Differenced Default Rate     -1.845 2.230 0.412           0.140 0.169 0.413          -1.305 1.115 0.248
      First-Differenced Mortgage Rate     0.205 0.275 0.460           0.000 0.021 0.984           0.120 0.137 0.388
  4   House Price Return                  0.636 0.174 0.001          -0.025 0.013 0.069           0.153 0.087 0.086
      First-Differenced Default Rate      5.215 2.055 0.015           0.055 0.156 0.726          -1.093 1.027 0.293
      First-Differenced Mortgage Rate     0.031 0.314 0.922           0.034 0.024 0.165          -0.605 0.157 0.000
  5   House Price Return                 -0.634 0.143 0.000          -0.004 0.011 0.731          -0.131 0.071 0.072
      First-Differenced Default Rate     -0.516 2.051 0.802          -0.058 0.156 0.711          -0.626 1.026 0.545
      First-Differenced Mortgage Rate    -0.007 0.307 0.982           0.018 0.023 0.448          -0.174 0.153 0.262
  8   House Price Return                  0.098 0.139 0.482           0.011 0.011 0.295           0.125 0.069 0.078
      First-Differenced Default Rate     -7.145 2.214 0.002           0.076 0.168 0.652          -1.199 1.107 0.284
      First-Differenced Mortgage Rate    -0.143 0.331 0.669           0.007 0.025 0.777          -0.156 0.166 0.351
 10   House Price Return                 -0.253 0.099 0.014          -0.001 0.008 0.908           0.020 0.050 0.684
      First-Differenced Default Rate      3.610 2.605 0.172           0.543 0.198 0.008          -0.637 1.302 0.627
      First-Differenced Mortgage Rate     0.375 0.265 0.162           0.010 0.020 0.620          -0.087 0.132 0.515
 12   House Price Return                  0.103 0.121 0.398           0.017 0.009 0.070          -0.054 0.061 0.375
      First-Differenced Default Rate      5.456 2.532 0.036          -0.317 0.192 0.105          -0.196 1.266 0.878
      First-Differenced Mortgage Rate     0.148 0.285 0.607          -0.029 0.022 0.187          -0.071 0.143 0.622




                                                         29
2, Model 2

   As a structural model, Model 2 could explain the dynamic relationship among the

endogenous variables better. Due to the high correlations between the residuals of the

models, Model 2 is estimated by three-stage-least-square method. We carry out two

regressions for both OFHEO’s and Case-Shiller’s house price returns. One contains only

one-period-lagged house price returns, mortgage rates and default rates, while the other

one includes multi-period-lagged or multi-period-changed house price returns, mortgage

rates and default rates. The results are listed in Table 12 and 13.



Serial Correlation Term

   All the three endogenous variables have highly significantly positive serial correlation

coefficients. Obviously, they have the strong tendency to keep their original values.

   For OFHEO’s house price returns, the estimate of one-period-lagged house price

return in Regression 1 of the house equation is 0.54. The sum of estimates of lagged

house price returns in Regression 2 of the house equation is 0.76. For Case-Shiller’s

house price returns, the two estimates are 0.51 and 0.80 respectively. They are roughly

consistent with the previous literature, such as Case and Shiller (1989) and Abraham and

Hendershott (1993). The national house price index data we used here are supposed to

show higher momentum effects than the local house price index data. Following the

explanation of Abraham and Hendershott (1996) and Capozza (2002), this coefficient

shows the degree for house bubble to build up.

   In the mortgage rate equation, the current mortgage rate reflects 90-95 percent of the

lagged one.



                                             30
   And the current default rate in the default rate equation reflects around 100 percent of

the lagged one, showing a strong character of unit root.



Relationship with other variables

1) House price return equation (Table 12(1) and Table 13(1))

   Default rates have consistent effects on house price returns for both regressions and

both indices. The current default rate has significantly negative coefficients on house

price returns, showing that the increased default rate will drive the house price returns

down immediately, due to shrunk demand or credit. Combining the current and one-

period-lagged default rates, we could get the current change of default rates, which has

the similar effects on house price returns as the current default rate. The significantly

positive estimates of the lagged default rate or lagged change of default rates would

reflect a complicated process. Take regression 1 for OFHEO’s house price return as an

example. A 1-percent increase of one-period-lagged default rate will incur a 10.77%

decrease   in   one-period-lagged     house        price   return   and   correspondingly   a

5.82%(=10.77%*0.54) decrease in current house price return. At the same time, the 1-

percent increase of one-period-lagged default rate will result in an 11.42% increase in

current house price return. Therefore, the net effects of a 1-percent increase of one-

period-lagged default rate on the current house price return would be a 5.60% (=11.42%-

5.82%) increase in current house price return. If the current default rate also increases by

1 percent, then the current house price return will decrease by 5.17% (=10.77%-5.60%)

finally. Similarly, the two consecutive 1-percent increases of default rates will cause

Case-Shiller’s current house price return down by 11.92%.




                                              31
   The estimates on mortgage rates are inconsistent for OFHEO’s and Case-Shiller’s

house price returns, which display that mortgage rates have complicated effects on house

price returns and it may be difficult to explain via a simple relationship. In the model

with OFHEO’s house price returns, the coefficients on the current and one-period-lagged

mortgage rates are statistically significant and the dominant effects is the current

mortgage rate, which is negatively correlated with the current house price return,

meaning that low mortgage rates will drive the housing demand up and so increase the

house price returns. In the model with Case-Shiller’s house price returns, the coefficients

on the one-period-lagged and three-period-lagged changes of mortgage rates are

significantly negative.

   The changes of 3-month Treasury bill rates are positively correlated with house price

returns, though insignificantly, especially after including the indirect effects on house

price returns through mortgage rates and default rates. From Figure 2, we could find that

the house price returns increases with the (change of) 3-month Treasury bill rates and

also decreases with the decrease of Treasury bill rates, due to the government

intervention effects.

   Figure 2: 3-month Treasury bill rates and change of 3-month Treasury bill rates
   The data are shown from the second quarter of 1979 till the second quarter of 2008
                                              3-month Treasury Bill rate
     %




     16
     14
     12
     10
      8
      6
      4
      2
      0
          Jun-79

                   Jun-81

                            Jun-83

                                     Jun-85

                                              Jun-87

                                                       Jun-89

                                                                Jun-91

                                                                         Jun-93

                                                                                  Jun-95

                                                                                           Jun-97

                                                                                                    Jun-99

                                                                                                             Jun-01

                                                                                                                      Jun-03

                                                                                                                               Jun-05

                                                                                                                                        Jun-07




                                                                                                              32
     %
                                             Change of 3-month TB Rate
     6

     4

     2

     0
         Jun-79

                  Jun-81

                           Jun-83

                                    Jun-85

                                             Jun-87

                                                      Jun-89

                                                               Jun-91

                                                                        Jun-93

                                                                                 Jun-95

                                                                                          Jun-97

                                                                                                   Jun-99

                                                                                                            Jun-01

                                                                                                                     Jun-03

                                                                                                                              Jun-05

                                                                                                                                       Jun-07
    -2

    -4

    -6




    The lagged inflation rate is much more significant than the current inflation rate.

Hence we may claim that the lagged inflation rate exhibits the dominant effects. The

lagged inflation rate is positively correlated with both house price returns.

    The increased income and decreased unemployment rate could enhance the earning

powers of the family and so drive up the house demand. It is not surprising to find the

positive coefficient for change of income and the negative coefficient for (change of)

unemployment in OFHEO’s house price return equation. In Case-Shiller’s house price

return equation, the estimates on change of income are negative, although highly

insignificant. It’s maybe because the changes of income do not always comply with the

income.

    The change of construction cost makes positive contributions to the housing price

return. It is because the increases of construction costs enhance house prices from the

view of housing supply. In the regressions, the coefficients on the change of

constructions costs cannot be rejected to be equal to zero at any reasonable significant

level,



2) Mortgage rate equation (Table 12(2) and Table 13(2))




                                                                                                             33
   After considering the indirect effects of lagged (change of) default rates on current

mortgage rates through the lagged mortgage rates, the effects of current default rates on

mortgage rates are dominant. The current default rates show significantly negative effects.

The impacts of default rate on mortgage rate are complicated. We know that the change

of mortgage rates may not be totally consistent with the market interest rates, because the

mortgage rate may be decomposed into two parts. One is market interest rate and the

other is margin, which reflects the willingness for the banks to provide credit. Generally,

when the lending standards are easy, the loan margins are expected to be low, and vice

versa. Due to the margin effect, although the market interest rates fell since September

2007, the mortgage rates did not change in the same direction or similar magnitude. The

increase of default rates commonly leads to declined market interest rates due to the

policymaker’s intervention and increased margin due to the unwillingness for the

mortgage providers to provide credit. The negative coefficient displays that here the

relation with market interest rate is dominant.

   As for the house price returns, although the lagged house price returns tend to

positively affect the mortgage rate, the dominant factor is the current house price return,

which has a significantly negative effect. It means that higher house price return will urge

the mortgage providers to provide more credit, and so ease the credit market and lower

mortgage rate.

   Compared with the change of 3-month Treasury bill rates, the 10-year Treasury bond

rates positively impacts the mortgage rate with stronger power. Change of GDP reflects

the performance of economy. Generally, the upward GDP exhibits the upswing of the




                                             34
economy and so increases the mortgage rate. It is totally intuitive that the mortgage rate

goes up with the 1-period lagged inflation.



3) Default rate equation (Table 12(3) and Table 13(3))

   In the model with OFHEO’s house price returns, the basically positive coefficients on

lagged mortgage rates (or changes of mortgage rates) and lagged house price returns

show positive correlations with the current default rate. We could regard the lagged

(changes of) mortgage rates or lagged house price returns as the values at origination of a

mortgage loan, then these increased origination values (mortgage contract rates or house

price returns) would augment the default rate later on, because the high mortgage contract

rates could lower the affordability of borrowers and high house price at origination may

make people borrow more than their affordability.

   The negative coefficient on the current house price return shows that the dropped

house price return lowers the housing equity and makes it more difficult to pay back the

mortgage by refinancing, which drives the default rate up.

   Again take Regression 1 as example. Two consecutive 1-percent decreases of

OFHEO’s house price returns will drag the current default rate up by 0.08 percent. And

two consecutive 1-percent decreases of Case-Shiller’s house price returns will cause the

current default rate up by 0.05 percent.

   By using the 30-year fixed-rate mortgage rate, we do not include the adjustment of

the mortgage rates in the current contracts. So as for the new mortgagors, the augmented

mortgage rate make the people who have lower affordability difficult to get a mortgage




                                              35
loan, which causes lower default rate. At the same time, the current mortgagors have

comparatively lower contract rates and tend to keep their contracts and not to default.

   Additionally, we use 3-month Treasury bill rates to reflect market interest rates and

adjustable mortgage rates. The positive coefficients could roughly show that, when

adjustable mortgage rates rise, the default rate would increase.

   The loan-to-value ratio used here is a composite one and may not reflect the exact

relation with each individual mortgage loan. The positive coefficients on the composite

loan-to-value ratio and the change of composite loan-to-value ratio show the intuition that

the higher loan-to-value ratio, defined as more loans on the same house value, endanger

the loan and enlarge the default rate.

   Homeownership rates are the ratio of housing units occupied by home owners over

total housing units. Decreased homeownership rates could reflect the raised default rates.

Our negative estimate is consistent with this relationship.

   In the model with Case-Shiller’s house price returns, the estimates of 30-year fixed

mortgage rates are not as neat. The significantly negative estimates on the one-period-

lagged and three-period-lagged change of 30-year mortgage rates are difficult to be

explained. It may display the complicated impacts of mortgage rates on default rates at

the aggregate level.



4) Dynamic interactions between house price returns and default rates

   We have observed that the relationships between house price returns and default rates

are consistent for both house price indices. Here we provide further discussions of their

dynamic relations.




                                             36
   With all other external variables constant, the impacts of default rates on house price

returns have two paths: one is a direct effect, and the other is an indirect effect through

mortgage rates. Similarly, the impacts of house price returns on default rates also have a

direct path and an indirect path through mortgage rates.

   Again take one-period-lagged SEM (regression 1) as example. With combined effects

of both direct and indirect paths, two consecutive 1-percent increases of default rates can

drive OFHEO’s current house price return down by 4.20% and Case-Shiller’s current

house price return down by 11.78%. Conversely, two consecutive 1-percent decreases of

OFHEO’s or Case-Shiller’s house price returns have similar effects and can drag the

current default rate up by 0.049 percent and 0.048 percent, respectively. However, the

relatively high standard errors may render the above estimates fluctuating in a wide range.




                                            37
Table 12 (1): regression results for Model 2 –OFHEO’s house price return equation

This table exhibits the regression results of three-stage least square for the OFHEO’s house price
return equation, based on Model 2. The data are demeaned.

 Equation: OFHEO's House Price Return
                                     Regression1                      Regression 2
                                     Para-      Std                   Para-    Std
 Variable                            meter      Error      Pr > |t|   meter    Error     Pr > |t|
 Intercept                              -0.057    0.145      0.697     -0.008    0.130     0.952
 1-period-lagged house price
 return                                  0.543     0.084   <.0001      0.481     0.123      0.000
 2-period-lagged house price
 return                                                                0.063     0.136      0.643
 3-period-lagged house price
 return                                                                0.223     0.138      0.108
 30-year fixed mortgage rate            -0.639     0.190     0.001    -0.656     0.204      0.002
 1-period-lagged 30-year fixed
 mortgage rate                           0.584     0.184     0.002     0.593     0.201      0.004
 1-period-lagged change of 30-
 year fixed mortgage rate                                              0.038     0.160      0.813
 2-period-lagged change of 30-
 year fixed mortgage rate                                              0.096     0.157      0.542
 3-period-lagged change of 30-
 year fixed mortgage rate                                              0.049     0.134      0.715
 default rate                          -10.773     4.515     0.019    -8.908     4.482      0.050
 1-period-lagged default rate           11.423     4.819     0.020     8.686     4.774      0.072
 1-period-lagged change of
 default rate                                                          2.449     1.882      0.196
 2-period-lagged change of
 default rate                                                          0.455     1.574      0.773
 3-period-lagged change of
 default rate                                                          1.342     2.275      0.557
 inflation rate                         -0.118     0.117     0.317    -0.074     0.105      0.482
 lagged inflation rate                   0.482     0.152     0.002     0.289     0.188      0.127
 change in income                        3.925     7.148     0.584     0.734     5.957      0.902
 change of 3-month Treasury bill
 rate                                    0.087     0.093     0.356     0.083     0.103      0.423
 change in construction cost             2.507     5.650     0.658    -0.698     5.247      0.895
 unemployment rate                      -0.020     0.052     0.706     0.048     0.058      0.415
 change in unemployment rate            -0.056     0.279     0.842    -0.224     0.246      0.364




                                              38
Table 12(2): regression results for Model 2—Mortgage Rate Equation

This table exhibits the regression results of three-stage least square for the mortgage rate
equation, based on Model 2. The data are demeaned.

 Equation: 30-year fixed Mortgage Rate
                                   Regression1                           Regression 2
                                   Para-   Std                           Para-    Std
 Variable                          meter   Error              Pr > |t|   meter    Error     Pr > |t|
 Intercept                              -0.030        0.095    0.755      -0.042    0.091    0.643
 1-period-lagged 30-year fixed                                <.000                         <.000
 mortgage rate                          0.960         0.014   1           0.957     0.017   1
 1-period-lagged change of 30-year
 fixed mortgage rate                                                      0.119     0.088    0.180
 2-period-lagged change of 30-year
 fixed mortgage rate                                                      0.029     0.090    0.749
 3-period-lagged change of 30-year
 fixed mortgage rate                                                      -0.030    0.072    0.674
 default rate                           -2.853        2.247    0.207      -6.047    2.536    0.019
 1-period-lagged default rate            2.964        2.390    0.218       5.992    2.631    0.025
 1-period-lagged change of default
 rate                                                                     1.924     1.048    0.070
 2-period-lagged change of default
 rate                                                                     -0.206    0.890    0.818
 3-period-lagged change of default
 rate                                                                      0.871    1.212    0.474
 house price return                     -0.522        0.149    0.001      -0.676    0.175    0.000
                                                              <.000                         <.000
 1-period-lagged house price return     0.351         0.087   1            0.424    0.091   1
 2-period-lagged house price return                                       -0.031    0.080    0.694
 3-period-lagged house price return                                        0.130    0.086    0.135
 lagged inflation rate                  0.242         0.077    0.002       0.225    0.096    0.022
 Change of nominal GDP                  1.857         5.172    0.720       3.034    4.986    0.544
 Change of 10-year Treasury bond                              <.000                         <.000
 rate                                   0.563         0.092   1           0.423     0.101   1
 change of 3-month Treasury bill
 rate                                   -0.013        0.055    0.811      0.022     0.063    0.734




                                                 39
Table 12 (3): regression results for Model 2—Default Rate Equation

This table exhibits the regression results of three-stage least square for the default rate
equation, based on Model 2. The data are demeaned.

 Equation: Default Rate
                                       Regression1                       Regression 2
                                       Para-   Std                       Para-    Std
 Variable                              meter   Error          Pr > |t|   meter    Error       Pr > |t|
 Intercept                              0.002    0.011         0.864      0.005     0.010      0.606
                                                              <.000                           <.000
 1-period-lagged default rate           1.062         0.037   1           1.014     0.041     1
 1-period-lagged change of default
 rate                                                                     0.249     0.117      0.036
 2-period-lagged change of default
 rate                                                                     0.023     0.117      0.843
 3-period-lagged change of default
 rate                                                                     0.258     0.138      0.064
 30-year fixed mortgage rate            -0.045        0.018    0.013     -0.043     0.020      0.036
 1-period-lagged 30-year fixed
 mortgage rate                           0.040        0.017    0.016      0.039     0.019      0.049
 1-period-lagged change of 30-
 year fixed mortgage rate                                                -0.009     0.013      0.493
 2-period-lagged change of 30-
 year fixed mortgage rate                                                 0.013     0.011      0.248
 3-period-lagged change of 30-
 year fixed mortgage rate                                                -0.001     0.010      0.908
 house price return                     -0.055        0.027    0.045     -0.051     0.030      0.090
 1-period-lagged house price
 return                                  0.030        0.015    0.047      0.025     0.016      0.132
 2-period-lagged house price
 return                                                                   0.006     0.010      0.533
 3-period-lagged house price
 return                                                                   0.003     0.013      0.797
 lagged inflation rate                   0.035        0.009    0.000      0.029     0.011      0.008
 composite loan-to-value ratio           0.165        0.218    0.452      0.147     0.180      0.416
 Change of composite loan-to-
 value ratio                            -0.198        0.335    0.557      0.112     0.353      0.752
 change in income                        0.333        0.598    0.578     -0.060     0.519      0.908
 change of 3-month Treasury bill
 rate                                    0.004        0.007    0.565      0.006     0.008      0.431
 change of home ownership rate          -0.699        1.026    0.497     -1.570     1.230      0.205




                                                 40
Table 13 (1): regression results for Model 2 –Case-Shiller’s house price return equation

This table exhibits the regression results of three-stage least square for the Case-Shiller’s
house price return equation, based on Model 2. The data are demeaned.

Equation: Case-Shiller's House Price Return
                                         Regression1                 Regression 2
                                         Para- Std                   Para-   Strd
Variable                                 meter Error        Pr > |t| meter   Error     Pr > |t|
Intercept                                    -0.245   0.517 0.638     -0.077   0.334 0.818
1-period-lagged house price return            0.511   0.210 0.017      0.824   0.138<.0001
2-period-lagged house price return                                    -0.751   0.192 0.000
3-period-lagged house price return                                     0.732   0.139<.0001
30-year fixed mortgage rate                   0.318   0.826 0.702     -0.084   0.585 0.886
1-period-lagged 30-year fixed mortgage
rate                                         -0.714   0.686 0.302     -0.149   0.533     0.781
1-period-lagged change of 30-year fixed
mortgage rate                                                         -1.258   0.447     0.007
2-period-lagged change of 30-year fixed
mortgage rate                                                          0.501   0.446     0.266
3-period-lagged change of 30-year fixed
mortgage rate                                                     -0.806       0.409     0.053
default rate                                -19.154 13.894 0.173 -13.185       6.928     0.062
1-period-lagged default rate                 17.022 14.527 0.245 10.788        7.314     0.146
1-period-lagged change of default rate                             2.429       3.655     0.509
2-period-lagged change of default rate                                -1.336 3.479       0.702
3-period-lagged change of default rate                                 0.050 3.806       0.990
inflation rate                               -0.058 0.348 0.868        0.033 0.197       0.868
lagged inflation rate                         0.744 0.847 0.383        0.514 0.539       0.344
change in income                             -1.916 24.481 0.938     -14.697 14.716      0.322
change of 3-month Treasury bill rate         -0.187 0.538 0.730        0.488 0.497       0.330


change in construction cost                  -0.758 16.315 0.963      -0.731   6.400     0.910
unemployment rate                             0.104 0.186 0.576        0.061   0.106     0.569
change in unemployment rate                  -0.476 1.037 0.648       -0.065   0.359     0.856




                                                41
Table 13(2): regression results for Model 2—Mortgage Rate Equation

This table exhibits the regression results of three-stage least square for the mortgage rate
equation, based on Model 2. The data are demeaned.

 Equation: 30-year fixed Mortgage Rate
                                          Regression1                     Regression 2
                                          Para-    Std                    Para-   Std
 Variable                                 meter    Error       Pr > |t|   meter   Error     Pr > |t|
 Intercept                                 -0.160    0.064      0.015     -0.152    0.064    0.021
 1-period-lagged 30-year fixed
 mortgage rate                              0.949     0.022    <.0001      0.922   0.028    <.0001
 1-period-lagged change of 30-year
 fixed mortgage rate                                                      -0.250   0.106       0.021
 2-period-lagged change of 30-year
 fixed mortgage rate                                                       0.115   0.076       0.136
 3-period-lagged change of 30-year
 fixed mortgage rate                                                      -0.115   0.082       0.167
 default rate                              -1.013     1.036     0.332     -3.014   1.300       0.024
 1-period-lagged default rate               0.670     1.040     0.522      2.388   1.235       0.058
 1-period-lagged change of default
 rate                                                                      1.150   0.571       0.049
 2-period-lagged change of default
 rate                                                                      0.172   0.551       0.756
 3-period-lagged change of default
 rate                                                                      0.587   0.575       0.312
 house price return                        -0.109     0.037     0.004     -0.195   0.066       0.005
 1-period-lagged house price return         0.060     0.026     0.026      0.166   0.059       0.007
 2-period-lagged house price return                                       -0.163   0.056       0.005
 3-period-lagged house price return                                        0.122   0.051       0.020
 lagged inflation rate                      0.110     0.066     0.099      0.237   0.084       0.007
 Change of nominal GDP                      7.695     4.025     0.060      5.640   4.061       0.170
 Change of 10-year Treasury bond
 rate                                       0.859     0.061    <.0001      0.798   0.073    <.0001
 change of 3-month Treasury bill rate       0.027     0.049     0.583      0.140   0.075     0.068




                                                42
Table 13 (3): regression results for Model 2—Default Rate Equation

This table exhibits the regression results of three-stage least square for the default rate
equation, based on Model 2. The data are demeaned.

 Equation: Default Rate
                                       Regression1                 Regression 2
                                       Para- Std                   Para-     Std
 Variable                              meter Error      Pr > |t|   meter     Error      Pr > |t|
 Intercept                             -0.015 0.014      0.287       -0.007     0.017    0.669
 1-period-lagged default rate           0.940 0.097     <.0001        0.833     0.113   <.0001
 1-period-lagged change of default
 rate                                                                 0.177     0.198     0.374
 2-period-lagged change of default
 rate                                                                -0.091     0.203     0.655
 3-period-lagged change of default
 rate                                                                 0.040     0.212     0.852
 30-year fixed mortgage rate            0.001   0.025     0.971      -0.009     0.030     0.760
 1-period-lagged 30-year fixed
 mortgage rate                         -0.020   0.026     0.446      -0.008     0.030     0.790
 1-period-lagged change of 30-
 year fixed mortgage rate                                            -0.091     0.041     0.032
 2-period-lagged change of 30-
 year fixed mortgage rate                                            0.038      0.026     0.139
 3-period-lagged change of 30-
 year fixed mortgage rate                                            -0.054     0.032     0.098
 house price return                    -0.033   0.020     0.095      -0.068     0.030     0.028
 1-period-lagged house price
 return                                 0.014   0.012     0.244       0.056     0.026     0.039
 2-period-lagged house price
 return                                                              -0.052     0.020     0.012
 3-period-lagged house price
 return                                                               0.048     0.021     0.027
 lagged inflation rate                  0.045   0.018     0.014       0.041     0.023     0.087
 composite loan-to-value ratio          0.074   0.439     0.867      -0.106     0.263     0.688
 Change of composite loan-to-
 value ratio                           -0.040   0.584     0.946       0.310     0.751     0.681
 change in income                       0.288   0.810     0.724      -1.011     1.030     0.330
 change of 3-month Treasury bill
 rate                                  -0.013   0.020     0.522       0.034     0.033     0.306
 change of home ownership rate         -1.219   1.257     0.336      -0.454     1.605     0.778




                                                43
3, Model 3

   Table 14 or 15 exhibits part of the three-stage-least-square regression results for

Model 3 with either house price index. We analyze both one-period-lag and multi-period-

lag (change) models.



Serial correlation term and fundamental variables term

   The signs and significant levels on the serial correlation terms and the fundamental

variables terms are basically similar with the results of Model 2. We do not report the

estimations on the fundamental variables terms in this paper.



Fundamental-actual difference term

   The fundamental-actual difference term presents the cumulative effects of the

fundamental-driven factors on the actual data. Since all the serial correlation coefficients

in our model are positive, a positive coefficient on the fundamental-actual difference term

displays the deviation from the lagged actual value, while a negative coefficient

strengthens the lagged actual value.

   For each kind of model in Table 14 (for OFHEO’s house price returns) and Table 15

(for Case-Shiller’s house price returns), three regressions are listed. In regression 1, we

only include the difference term in house price return equation. Regression 2 contains the

difference terms in house price return equation and the default rate equation. Regression

3 has the difference terms in all the three equations.




                                             44
    Basically, for both indices, the actual house price returns converge around 2-4 percent

each quarter of the difference. Under most cases, the difference term in house price

equation is statistically significant.

    The actual default rate deviates 0.5-2 percent. The actual mortgage rate converges

around 0.4 percent under the models with OFHEO’s house price returns and around 0.1

percent under the models with Case-Shiller’s house price returns.

    The low fundamental-actual convergence rates for all the three variables show a long-

term adjustment process toward the fundamental values.




                                            45
Table 14: Part of 3SLS Regression Results for Model 3 with OFHEO’s house price returns
                       Three-Equation SEM with 1-period lag terms
 House Price Return Equation
 Variable                         Regression 1       Regression 2        Regression 3
 1-period-lagged house price             0.563****           0.564****             0.611****
 return                                     (0.084)            (0.103)               (0.085)
 Lagged Deviation Term of                 0.038***             0.037**             0.048****
 House Price                                (0.019)            (0.021)               (0.017)
 Default Rate Equation
                                         1.043****           1.059****             1.079****
 1-period-lagged default rate               (0.035)            (0.036)               (0.037)
 Lagged Deviation Term of                                        0.004              0.009***
 Default Rate                                     --           (0.004)               (0.004)
 Mortgage Rate Equation
 1-period-lagged 30-year fixed           0.949****           0.962****             0.959****
 mortgage rate                              (0.017)            (0.016)               (0.020)
 Lagged Deviation Term of                                                          0.005****
 Mortgage Rate                                    --                  --             (0.001)
                 Three-Equation SEM with multi-period lag/change terms
 House Price Return Equation
 Variable                         Regression 1       Regression 2        Regression 3
 1-period-lagged house price             0.493****           0.468****             0.512****
 return                                     (0.113)            (0.114)               (0.105)
 2-period-lagged house price                  0.031              0.036                 0.006
 return                                     (0.131)            (0.131)               (0.121)
 3-period-lagged house price               0.270**           0.327****              0.284***
 return                                     (0.140)            (0.123)               (0.109)
 Lagged Deviation Term of                     0.014           0.041***              0.031***
 House Price                                (0.018)            (0.016)               (0.012)
 Default Rate Equation
 1-period-lagged default rate            1.013****           1.055****             1.056****
                                              (.038)           (0.060)               (0.065)
 1-period-lagged change of                0.231***               0.156                 0.090
 default rate                               (0.107)            (0.189)               (0.196)
 2-period-lagged change of                   -0.018             -0.075                -0.106
 default rate                               (0.108)            (0.160)               (0.178)
 3-period-lagged change of                0.267***               0.202                 0.035
 default rate                               (0.122)            (0.248)               (0.240)
 Lagged Deviation Term of                                        0.006                 0.010
 Default Rate                                     --           (0.009)               (0.008)
 Mortgage Rate Equation
 1-period-lagged 30-year fixed           0.934****           0.956****             0.958****
 mortgage rate                              (0.019)            (0.016)               (0.017)
 1-period-lagged change of 30-                0.009              0.097                 0.062
 year fixed mortgage rate                   (0.103)            (0.082)               (0.098)
 2-period-lagged change of 30-                0.061             -0.007                -0.031
 year fixed mortgage rate                   (0.104)            (0.083)               (0.092)
 3-period-lagged change of 30-               -0.019             -0.040                -0.065
 year fixed mortgage rate                   (0.085)            (0.067)               (0.078)
 Lagged Deviation Term of                                                            0.003**
 Mortgage Rate                                    --                  --             (0.002)
Note: *15%, **10%, ***5%, ****1% indicate the corresponding significance levels.
      The numbers in parentheses refers to the standard errors of the coefficients



                                             46
Table 15: Part of 3SLS Regression Results for Model 3 with Case-Shiller’s house price returns
                        Three-Equation SEM with 1-period lag terms
  House Price Return Equation
  Variable                         Regression 1       Regression 2        Regression 3
  1-period-lagged house price              0.455***             0.398**                0.398**
  return                                     (0.224)            (0.226)                (0.227)
  Lagged Deviation Term of                    0.029*           0.043***               0.043***
  House Price                                (0.020)            (0.020)                (0.020)
  Default Rate Equation
                                          0.943****           0.958****              0.957****
  1-period-lagged default rate               (0.095)            (0.092)                (0.092)
  Lagged Deviation Term of                                        0.004                  0.005
  Default Rate                                     --           (0.004)                (0.004)
  Mortgage Rate Equation
  1-period-lagged 30-year fixed           0.948****           0.949****              0.948****
  mortgage rate                              (0.022)            (0.022)                (0.023)
  Lagged Deviation Term of                                                             0.0001
  Mortgage Rate                                    --                  --              (0.001)
                  Three-Equation SEM with multi-period lag/change terms
  House Price Return Equation
  Variable                         Regression 1       Regression 2        Regression 3
  1-period-lagged house price             0.854****           0.882****              0.873****
  return                                     (0.130)            (0.118)                (0.118)
  2-period-lagged house price            -0.715****          -0.566****             -0.576****
  return                                     (0.179)            (0.157)                (0.152)
  3-period-lagged house price             0.750****           0.706****              0.705****
  return                                     (0.129)            (0.119)                (0.119)
  Lagged Deviation Term of                     0.014              0.019                  0.020
  House Price                                (0.022)            (0.021)                (0.021)
  Default Rate Equation
                                          0.957****           1.029****              1.014****
  1-period-lagged default rate               (0.073)            (0.068)                (0.067)
  1-period-lagged change of                    0.181              0.020                  0.054
  default rate                               (0.141)            (0.154)                (0.153)
  2-period-lagged change of                   -0.006             -0.141                 -0.112
  default rate                               (0.138)            (0.148)                (0.146)
  3-period-lagged change of                   0.219*              0.107                  0.128
  default rate                               (0.141)            (0.152)                (0.151)
  Lagged Deviation Term of                                     0.023***               0.022***
  Default Rate                                     --           (0.009)                (0.009)
  Mortgage Rate Equation
  1-period-lagged 30-year fixed           0.960****           0.967****              0.965****
  mortgage rate                              (0.019)            (0.017)                (0.023)
  1-period-lagged change of 30-               -0.078             -0.065                 -0.086
  year fixed mortgage rate                   (0.067)            (0.060)                (0.066)
  2-period-lagged change of 30-                0.048              0.035                  0.047
  year fixed mortgage rate                   (0.055)            (0.052)                (0.055)
  3-period-lagged change of 30-               -0.007             -0.001                 -0.013
  year fixed mortgage rate                   (0.056)            (0.052)                (0.057)
  Lagged Deviation Term of                                                               0.001
  Mortgage Rate                                    --                  --              (0.001)
 Note: *15%, **10%, ***5%, ****1% indicate the corresponding significance levels.
       The numbers in parentheses refers to the standard errors of the coefficients



                                               47
5. Prediction

    We use Model 1 and 2 to predict house price returns, mortgage rates and default rates,

using data up to the fourth quarter of 2007. Since the data for the first two quarters of

2008 are now available, we are able to compare the predicted values with the actual data.



5.1. Prediction via Model 1

    We first examine relationships between 3-month Treasury bill rates, 30-year fixed

mortgage rate, and mortgage rate spreads (as the differences between 30-year fixed

mortgage rates and 3-month Treasury bill rates). Although the 3-month Treasury bill

rates can come down to near zero during some time periods of Fed easing, the mortgage

rate spreads tend to move upward during these economic periods (See Figure 3). This

opposite movement of mortgage spreads will keep 30-year fixed mortgage rates above

certain level. In fact, the historical 30-year mortgage rates have never come below 5%.

Accordingly, in our econometric modeling of future mortgage rates, we put a constraint

on future 30-year fixed mortgage rates and they will be always no less that 4 percent.

                                   Figure 3: 3-month Treasury bill rates vs. mortgage spreads
    Mortgage spreads are the differences between 30-year fixed mortgage rates and 3-month Treasury bill.
Although the 3-month Treasury bill rates can come down to near zero during some time periods of Fed
easing, the mortgage rate spreads tend to move upward during these economic periods, keeping 30-year
fixed mortgage rates above certain level .
                                           Comparison between 3-month Treasury Bill rate and mortgage spread
                16
       %
      ( )




                                                                                                                    TB3m
                14
                                                                                                                    spread



                12



                10



                     8



                     6



                     4



                     2



                     0
                 5




                              7




                                       9




                                                1




                                                         3




                                                                  5




                                                                           7




                                                                                    9




                                                                                              1




                                                                                                       3




                                                                                                                5




                                                                                                                          7




                                                                                                                                   9




                                                                                                                                            1




                                                                                                                                                     3




                                                                                                                                                              5




                                                                                                                                                                       7
                7




                             7




                                      7




                                               8




                                                        8




                                                                 8




                                                                          8




                                                                                   8




                                                                                             9




                                                                                                      9




                                                                                                               9




                                                                                                                         9




                                                                                                                                  9




                                                                                                                                           0




                                                                                                                                                    0




                                                                                                                                                             0




                                                                                                                                                                      0
              r-




                           r-




                                    r-




                                             r-




                                                      r-




                                                               r-




                                                                        r-




                                                                                 r-




                                                                                           r-




                                                                                                    r-




                                                                                                             r-




                                                                                                                       r-




                                                                                                                                r-




                                                                                                                                         r-




                                                                                                                                                  r-




                                                                                                                                                           r-




                                                                                                                                                                    r-
             a




                          a




                                   a




                                            a




                                                     a




                                                              a




                                                                       a




                                                                                a




                                                                                          a




                                                                                                   a




                                                                                                            a




                                                                                                                      a




                                                                                                                               a




                                                                                                                                        a




                                                                                                                                                 a




                                                                                                                                                          a




                                                                                                                                                                   a
            M




                         M




                                  M




                                           M




                                                    M




                                                             M




                                                                      M




                                                                               M




                                                                                         M




                                                                                                  M




                                                                                                           M




                                                                                                                     M




                                                                                                                              M




                                                                                                                                       M




                                                                                                                                                M




                                                                                                                                                         M




                                                                                                                                                                  M




                                                                                                                                                                           Time




                                                                                        48
   For both house price indices, we make 3-year forecasts via VAR, using historical data

up to the fourth quarter of 2007. By comparison, we also make forecasts via Auto-

Regressive models (AR). We employ Monte-Carlo simulations to estimate confidence

intervals for the predictions.

   The actual quarter-over-quarter OFHEO’s house price returns for the first two

quarters in 2008 have continuously deteriorated. The return is -0.23% in the first quarter

and -1.45% in the second quarter of 2008, which is the worst quarter-over-quarter return

since 1975. Figure 4 displays the predicted values of OFHEO’s house price returns for

the next three years from the first quarter of 2008 till the fourth quarter of 2010. The

mean values of predicted house price returns via AR model are always positive over time

since 2008, which obviously deviates from the actual data. On the contrary, the mean

values of predicted house price returns via VAR models are mainly negative. The

predictions based on VAR reach the lowest point in the third quarter of 2009. After that,

the predicted house price returns will improve gradually and should be back to positive in

2011. Additionally, although the confidence intervals via both AR and VAR fail to

exactly catch the huge deterioration in the second quarter of 2008, the 90% confidence

limit from VAR is relatively close to the actual data.

   The actual quarter-over-quarter Case-Shiller’s house price returns for the first two

quarters in 2008 show a different trend. The return is -6.99% in the first quarter, which is

lowest since 1987, and -2.36% in the second quarter of 2008, which is better than the

previous one. Figure 5 displays the predicted values of Case-Shiller’s house price returns

for the next three years from the first quarter of 2008 till the fourth quarter of 2010.

Again, the confidence intervals via both AR and VAR fail to exactly catch the huge




                                            49
deterioration in the first quarter of 2008, although the 90% confidence limits are

relatively close to the actual data. The confidence interval via VAR includes the actual

data in the second quarter of 2008. The mean values of predicted Case-Shiller’s house

price returns via VAR model are recovered a little bit quicker than the ones via AR model.

The expected house price returns from VAR model will be positive in 2010.

   The actual national default rates for the first two quarters in 2008 have deteriorated

further, with 1.63% in the first quarter of 2008 and 1.83% in the second quarter of 2008,

historically highest sine 1979. Figure 6 shows the expected predictions of national default

rates by AR model, VAR model with OFHEO’s house price returns, and VAR model

with Case-Shiller’s house price returns. For the default rates, the predictions via VAR

model with Case-Shiller’s house price returns (Figure 6c) show obvious improvements,

by comparing with the actual data of the first two quarters of 2008. And the expected

predictions via VAR model with Case-Shiller’s house price returns reach the highest in

2010 and display the slightly downward tendency afterwards.

   When investigating the predicted mortgage rate, the results from AR model and VAR

model with OFHEO’s house price returns seem more reasonable, compared with the

actual data.

   The differences between AR model and VAR models are that VAR models reflect the

interactions among house price returns, mortgage rates and default rates. These above

results clearly show the impacts of mortgage default on the housing market, no matter

which house price index we use.




                                            50
             Figure 4: Actual vs Predicted OFHEO’s House Price Returns
    The house price returns are from OFHEO’s index. The model use the data till the end of 2007
and there are 3-year predictions till the fourth quarter of 2010.
    Figure 4a: Predictions based on AR model.




    Figure 4b: Predictions based on VAR model.




                                            51
           Figure 5: Actual vs Predicted Case-Shiller’s House Price Returns
    The house price returns are from Case-Shiller’s index. The model use the data till the end of
2007 and there are 3-year predictions till the fourth quarter of 2010.
    Figure 5a: Predictions based on AR model.




    Figure 5b: Predictions based on VAR model.




                                             52
                   Figure 6: Actual vs Predicted National Default Rates
    The model use the data till the end of 2007 and there are 3-year predictions till the fourth quarter
of 2010.

    Figure 6a: Predictions based on AR model




    Figure 6b: Predictions based on VAR model with OFHEO’s House Price Returns




    Figure 6c: Predictions based on VAR model with Case-Shiller’s House Price Returns




                 Figure 7: Actual vs Predicted National Mortgage Rates


                                               53
    The model use the data till the end of 2007 and there are 3-year predictions till the fourth quarter
of 2010.

    Figure 7a: Predictions based on AR model




    Figure 7b: Predictions based on VAR model with OFHEO’s House Price Returns




    Figure 7c: Predictions based on VAR model with Case-Shiller’s House Price Returns




                                               54
Further Prediction: with Updated Data

   We re-estimate the VAR model using the data till the second quarter of 2008 and

make predictions. And the prediction results are graphed in Figure 8 with OFHEO’s

house price returns and in Figure 9 with Case-Shiller’s house price returns.

   The prediction results show great differences due to the different trends for the two

indices in the first two quarters of 2008. As we mentioned, OFHEO’s house price returns

reach the lowest value in the second quarter of 2008, while Case-Shiller’s house price

returns have the lowest one in the first quarter of 2008 and are somewhat better off in the

second quarter of 2008.

   On an expected value basis, the future level of OFHEO’s house price returns will

remain negative and reach the lowest value in 2010 and increase slowly thereafter,

although it may take quite a few years for the house price returns to become positive.

Based on the 90% confidence limits, if predicting optimistically, the house price returns

may go back to be positive after 2010.

   The expected Case-Shiller’s house price returns would become positive since 2010.

And, Figure 9 shows that default rates reach the highest value in 2010 and decrease

slowly thereafter.

   If we combine these two sets of results, we could say that, with only considering the

internal relationships among house price returns, mortgage rates and default rates and

without counting the effects of external factors, the year 2010 is an important turning

point for house price returns and default rates.




                                             55
       Figure 8: Predictions via VAR with OFHEO’s House Price Returns
    The model use the data till the second quarter of 2008 and there are 3-year predictions till the
second quarter of 2011.

    Figure 8a: Predictions of OFHEO’s house price returns




    Figure 8b: Predictions of Default Rate




    Figure 8c: Predictions of Mortgage Rate




                                              56
     Figure 9: Predictions via VAR with Case-Shiller’s House Price Returns
    The model use the data till the second quarter of 2008 and there are 3-year predictions till the
second quarter of 2011.

    Figure 9a: Predictions of Case-Shiller’s house price returns




    Figure 9b: Predictions of Default Rate




    Figure 9c: Predictions of Mortgage Rate




                                              57
5.2. Prediction via Model 2

   We first conduct conditional prediction via Model 2. Based on the known values of

the exogenous variables in the first two quarters of 2008, we predict the endogenous

variables. The expected results via Model with OFHEO’s house price returns are shown

in Table 16 and the results with Case-Shiller’s house price returns are in Table 17.

   For Model with OFHEO’s house price returns, the predictions, especially the multi-

period-lagged SEM, are more in line with the actual data, showing that the predictions of

the three variables rely on both the multi-period-lagged values and the other exogenous

variables.

   The multi-period-lagged SEM obtains the predicted house price returns with the

means of -0.24% and -0.83% (-0.89% if predicted dynamically) and with the confidence

intervals of [-1.31%, 0.81] and [-1.91%, 0.29%]            ([-2.07% 0.33%] if predicted

dynamically) in the first two quarters of 2008, close to the actual data -0.23% and -1.45%.

Similarly, for default rates and mortgage rates, the predicted means from the multi-

period-lagged SEM are pretty close to the actual values.

   For Model 2 with Case-Shiller’s house price returns, the main exception is the

predicted result for Case-Shiller’s house price return in the second quarter of 2008, which

deviates a lot from the actual value.

   The unconditional predictions need the predicted exogenous variables first, which

could be estimated via AR model or VAR model. The prediction results have not much

improvement, compared with the results from Model 1. We do not present the results

here.




                                            58
     Table 16: Conditional Predictions of OFHEO’s House Price Returns, Mortgage Rates and Default Rates via Three-
                                                  equation SEM
                                                        OFHEO’s House Price Returns (%)
            actual   VAR                                     1-period lag SEM                                   multi-period lag SEM
                             90% Conf       one step     90% Conf       dynamic 90% Conf        one step      90% Conf       dynamic 90% Conf
                             Interval       forecast     Interval       forecast Interval       forecast      Interval       forecast Interval
3/31/2008   -0.23     0.25   [-0.65 1.19]        0.24    [-0.79 1.34]                                 -0.24   [-1.31 0.81]
6/30/2008   -1.45    -0.04   [-1.02 0.96]        0.04    [-1.01 1.10]       0.27 [-0.87 1.43]         -0.83   [-1.91 0.29]      -0.89 [-2.07 0.33]
                                                                   Default Rate (%)
            actual   VAR                                     1-period lag SEM                                   multi-period lag SEM
                             90% Conf       one step     90% Conf       dynamic 90% Conf        one step      90% Conf       dynamic 90% Conf
                             Interval       forecast     Interval       forecast Interval       forecast      Interval       forecast Interval
3/31/2008    1.63    1.56    [1.48 1.63]         1.59    [1.51 1.67]                                   1.66   [1.57 1.74]
6/30/2008    1.83    1.64    [1.52 1.76]         1.73    [1.65 1.81]        1.68 [1.56 1.81]           1.79   [1.71 1.87]        1.82 [1.69 1.94]
                                                                  Mortgage Rate (%)
            actual   VAR                                     1-period lag SEM                                   multi-period lag SEM
                             90% Conf       one step     90% Conf       dynamic 90% Conf        one step      90% Conf       dynamic 90% Conf
                             Interval       forecast     Interval       forecast Interval       forecast      Interval       forecast Interval
3/31/2008    5.88    5.76    [5.01 6.53]         6.00    [5.45 6.57]                                   5.96   [5.33 6.57]
6/30/2008    6.09    5.33    [4.00 6.68]         6.11    [5.52 6.67]        6.28 [5.49 7.10]           6.12   [5.49 6.77]        6.19 [5.10 7.35]




                                                                     59
        Table 17: Conditional Predictions of Case-Shiller’s House Price Returns, Mortgage Rates and Default Rates via Three-
                                                           equation SEM
                                                        Case-Shiller’s House Price Returns (%)
            actual   VAR                                        1-period lag SEM                                      multi-period lag SEM
                             90% Conf        one step    90% Conf         dynamic    90% Conf        one step     90% Conf          dynamic    90% Conf
                             Interval        forecast    Interval         forecast   Interval        forecast     Interval          forecast   Interval
3/31/2008   -6.99    -5.69   [-6.70 -4.63]        -6.21 [-8.74 -3.65]                                     -6.46   [-8.30 -4.63]
6/30/2008   -2.36    -3.10   [-4.52 -1.68]        -7.28 [-9.73 -4.69]          -6.97 [-9.97 -3.81]        -7.00   [-8.94 -5.06]        -6.62   [-9.11 -4.18]

                                                                     Default Rate (%)
            actual   VAR                                        1-period lag SEM                                      multi-period lag SEM
                             90% Conf        one step    90% Conf         dynamic     90% Conf       one step     90% Conf          dynamic    90% Conf
                             Interval        forecast    Interval         forecast    Interval       forecast     Interval          forecast   Interval
3/31/2008    1.63    1.61    [1.52 1.69]          1.68   [1.60 1.77]                                      1.62    [1.52 1.73]
6/30/2008    1.83    1.86    [1.74 1.98]          1.82   [1.74 1.91]           1.87 [1.76 1.98]           1.82    [1.71 1.93]           1.81   [1.66 1.95]

                                                                    Mortgage Rate (%)
            actual   VAR                                        1-period lag SEM                                      multi-period lag SEM
                             90% Conf        one step    90% Conf         dynamic   90% Conf         one step     90% Conf          dynamic    90% Conf
                             Interval        forecast    Interval         forecast  Interval         forecast     Interval          forecast   Interval
3/31/2008    5.88    5.81    [5.30 6.31]          5.74   [5.50 5.99]                                      5.95    [5.65 6.25]
6/30/2008    6.09    4.98    [4.27 5.71]          6.16   [5.92 6.41]           6.00 [5.67 6.35]           6.42    [6.12 6.73]           6.48   [5.99 6.92]




                                                                          60
6. Conclusion

   In this paper, we present three models to describe the dynamic relations of house

price returns, mortgage rates and default rates. With their structural form, simultaneous

equation models can explain the relationships more clearly. By investigating both

OFHEO’s and Case-Shiller’s house price returns, we find the interactive negative

relationship between house price returns and default rates. For example, holding all the

other factors constant, two consecutive one-percent increases of default rates can drive

OFHEO’s house price returns down by about 5 percent and Case-Shiller’s current house

price return down by about 12 percent. Conversely, two consecutive 1-percent decreases

of OFHEO’s or Case-Shiller’s house price returns can drag the current default rate up by

0.08 percent or 0.05 percent, respectively. The effects of mortgage rates show different

results for models with the two different house price indices, reflecting complicated

relationships.

   In simultaneous equation models, the three level variables exhibit high serial

correlations, reflecting strong momentum effects, and relatively low fundamental-actual

convergence rates, showing a long-term adjustment process toward the fundamental

values.

   When making predictions using data up to the second quarter of 2008, we observe

that mortgage default rates have big impacts on house price returns, and vice versa. So

the situations on mortgage default could impact the recovery process of housing market.

According to our model, only considering the inter-relationships among house price

returns, mortgage rates and default rates, without counting the effects of other external




                                           61
factors, the year 2010 will probably be an important turning point for both house price

returns and mortgage default rates.




                                          62
Appendix A

1. Akaike.s information criterion (AIC)
The AIC procedure (Akaike, 1974) is used to evaluate how well the candidate model
approximates the true model. The criterion is
                  2 pq + q(q + 1)
 AIC = ln Σ p +
           ˆ
                         n
Where n is the number of observations; p refers the number of parameters including the
                                                  Σˆ
intercept; q is the number of dependent variables; p refers to the sum squared error for
a model with p parameters including the intercept. The model with the lowest AIC is
preferred, for a given data set.

2. The Corrected Form of Akaike.s information criterion (AICC)
The corrected version (AICC) of AIC is used for small sample sizes. The formula is
AICC = ln Σ p +
            ˆ       (n + p )q
                  n − p − q −1
Similarly, with this procedure, the model with the minimum AICC value is prefered.

3. Hannan-Quinn Criterion (HQC)
Hannan-Quinn Criterion introduced by Hannan and Quinn (1979) is
              2 ln[ln(n )] pq
HQ = ln Σ p +
         ˆ
                     n
The “best” model is the one with the minimum HQ criterion value.

4. The Corrected Form of Hannan and Quinn (HQC) Information Criterion
In order to applying to small samples, McQuarrie & Tsai (1998) proposed a corrected
version of Hannan and Quinn (HQ) criterion, which is
                 2 ln[ln(n )] pq
 HQC = ln Σ 2 +
            ˆ
                  n − p − q −1
              p


Similarly, the procedure identifies the “best” model that yields the smallest value.

5. Schwarz Bayesian Criterion (SBC)
Schwarz ’s Bayesian information is
         ˆ p ln(n ) p .
SBC = ln Σ 2 +
                 n
The model with the minimal value is prefered.

6. Final Prediction Error Criterion (FPEC)
The criterion of Akaike's Final Prediction Error (FPE) value [1] is adopted to represent
the fitness value of model order
          1+ p / n ˆ
 FPEC =            Σp
          1− p / n
Generally, the model with lower FPEC value indicates a better fit.



                                            63
References
Abraham, Jesse, and Bill Schauman. (1991). “Evidence on House Prices from FHLEC
   Repeat Sales”, Journal of the AREUEA 19, 333-352.
Abraham, Jesse, and Patric H. Hendershott. (1992). “Patterns and Determinants of
   Metropolitan House Prices, 1977-91”, NBER working paper.
Abraham, Jesse, and Patric H. Hendershott. (1996). “Bubbles in Metropolitan Housing
   Markets”, Journal of Housing Research, 7(2), 191-207
Akaike, Hirotugu (1974). "A new look at the statistical model identification". IEEE
   Transactions on Automatic Control 19 (6): 716–723.
Ambrose, Brent, Richard Buttimer, Jr. and Charles Capone. (1997). “Pricing Mortgage
   Default and Foreclosure Delay”, Journal of Money, Credit and Banking, 29, 314-325.
Ambrose, Brent and Michael LaCour-Little. (2001). “Prepayment Risk in Adjustable
   Rate Mortgages Subject to Initial Year Discounts: Some New Evidence”, Real Estate
   Economics, 29, 305-327
Ambrose, Brent, Charles A. Capone, JR. and Yongheng Deng. (2001). “Optimal Put
   Exercise: An Empirical Examination of Conditions for Mortgage Foreclosure”,
   Journal of Real Estate Finance and Economics, 23:2, 213-234.
Ang, Andrew and Monika Piazzesi (2003). “A No-arbitrage Vector Autoregression of
   Term Structure Dynamics with Macroeconomic and Latent Variables”, Journal of
   Monetary Economics, 50, 745-787.
Ayuso, Juan and Fernando Restoy (2006). “House Prices and Rents: An Equilibrium
   asset pricing approach”, Journal of Empirical Finance, 13, 371-388.
Black, Fischer and Myron S. Scholes (1972). “The valuation of option contracts and a
   test of market efficiency”, Journal of Finance, 27 (2), 399–418.
Black, Angela, Patrica Fraser and Martin Hoesli (2006). “House Prices, Fundamentals
   and Bubbles”, Journal of Business Finance and Accounting, 33, 1535-1555.
Bourassa, Steven C, Patric H. Hendershott and James Murphy (2001). “Further Evidence
   on the Existence of Housing Market Bubbles”, Journal of Property Research, 18:1, 1-
   19.
Cagan, Christopher. (2007). “Mortgage Payment Reset: the issue and the impact”,
   www.loanperformance.com.
Calhoun, Charles and Yongheng Deng. (2002) “A Dynamic Analysis of Fixed- and
   Adjustable-Rate Mortgage Terminations”, Journal of Real Estate Finance and
   Economics, 24, 9-33.
Capozza, Deninis, and Robert Helsley. (1989) “The Fundamentals of Land Prices and
   Urban Growth”, Journal of Urban Economics, 26, 295-306.
Capozza, Deninis, and Robert Helsley. (1990) “The Stochastic City”, Journal of Urban
   Economics, 28, 187-203.
Capozza, Dennis, Patric H. Hendershott, Charlotte Mack, and Christopher J. Mayer.
   (2002). “Determinants of Real House Price Dynamics”, NEBR working paper,
   available at http://www/nber.org/papers/w9262
Case, Karl E. and Robert J. Shiller. (1987). “Prices of Single-Family Homes Since 1970:
   New Indexes for Four Cities”, New England Economics Review, Sept. /Oct., 45-56.
Case, Karl E. and Robert J. Shiller. (1989). “The Efficiency of the Market for Single-
   Family Homes”, American Economic Review, 79, 125-137.




                                          64
Case, Karl E. and Robert J. Shiller. (1990). “Forecasting Prices and Excess Returns in the
   Housing Market”, AREUEA Journal, 18, 253-273.
Case, Karl E. and Robert J. Shiller. (2003a). “A Decade of Boom and Bust in the Prices
   of Single-Family Homes: Boston and Los Angeles, 1983 to 1993”, New England
   Economic Review, March.
Case, Karl E. and Robert J. Shiller. (2003b). “Is there a Bubble in the Housing Market”,
   Brookings Paper on Economic Activity, 2., 299-362.
Case, Karl E., Robert J. Shiller and Allan N. Weiss. (1995). “Mortgage Default Risk and
   Real Estate Prices: the Use of Index-based Futures and Options in Real Estate”,
   NBER Working.
Cox, John, Jonathan Ingersoll, and Stephen Ross. (1985). “A Theory of the Term
   Structure of Interest Rates”, Econometrica, 53, 358-407.
Dai, Qiang and Kenneth J. Singleton. (2000) “Specification Analysis of Affine Term
   Structure Models”, Journal of Finance, 55, 1943-1978.
Deng, Yongheng. (1997). “Mortgage Termination: An Empirical Hazard Model with a
   Stochastic Term Structure”, Journal of Real Estate Finance and Economics, 14, 309-
   331
Deng, Yongsheng, John M. Quigley, and Robert Van Order. (2000). “Mortgage
   Terminations, Heterogeneity and the Exercise of Mortgage Options”, Econometrica
   68, 275-307.
Dornbusch, Rudiger. (1976). “Expectations and Exchange Rate Dynamics”, Journal of
   Political Economics, 84, 1161-1176
Dunn, Kenneth and John McConnell. (1981). “Valuation of GNMA Mortgage-Backed
   Securities”, Journal of Finance, 36, 599-615
Edelstein, Robert and Desmond Tsang (2007). “Dynamic Residential Housing Cycles
   Analysis”, Journal of Real Estate Finance and Economics, 35, 295-313.
Engle, Robert and C. W. J. Granger (1987). “Co-Integration and Error Correction:
   Representation, Estimation, and Testing”, Econometica, 55, 251-276.
Emrath, Paul, (2005), “Interest Rates and House Prices: the “Priced Out” Effect”,
   National       Association         of      Home        Builders,      available      at
   http://www.nahb.org/generic.aspx?genericContentID=37153
Epperson, James, James Kau, Donald Keenan, and Walter Muller. (1985). “Pricing
   Default Risk in Mortgages”, Journal of the American Real Estate and Urban
   Economics Association, 13, 261-272
Fama, Eugene, and Kenneth French. (1988). “Permanent and Temporary Components of
   Stock Prices”, the Journal of Political Economy, 96, 246-273
Foote, Christopher L., Kristopher Gerardi, and Paul Willen, (2008a). “Subprime Facts:
   What (We Think) We Know about the Subprime Crisis and What We Don’t”,
   Federal Reserve Bank of Boston Public Policy Discussion Paper No. 08-2
Foote, Christopher L., Kristopher Gerardi, and Paul Willen, (2008b). “Negative Equity
   and Foreclosure: Theory and Evidence”, Federal Reserve Bank of Boston Public
   Policy Discussion Paper No. 08-3
Greenlaw, David, and Jan Hatzius, Anil Kashyap, and Hyun Song Shin, (2008),
   “Leveraged Losses: Lessons from the Mortgage Market Meltdown”, Proceedings of
   the U.S. Monetary Policy Forum 2008.
Hamilton, James (1994). “Time Series Analysis”, Princeton: Princeton University Press.



                                           65
Hannan, E. J. and Quinn, B. G. (1979), .The Determination of The Order of an
   Autoregression., Journal of the Royal Statistical Society, B 41, 190-195.
Harrington, Scott and Tong Yu. (2003). “Do Property-Casualty Insurance Underwriting
   Margins Have Unit Roots”, the Journal of Risk and Insurance, 70, 715-733.
Hatzius, Jan. (2008). “Beyond Leveraged Losses: the Balance Sheet Effects of the Home
   Price Downturn”, Brookings Papers on Economic Activity Fall 2008 Conference
   Draft.
Himmerlberg, Charles, Christopher Mayer, and Todd Sinai. (2005). “Assessing High
   House Prices: Bubbles, Fundamentals, and Misperceptions”, Federal Reserve Bank of
   New York Staff Reports.
Hott, Christian and Pierre Monnin. (2008). “Fundamental Real Estate Prices: An
   Empirical Estimation with International Data”, Journal of Real Estate Finance and
   Economics, 36, 427-450
Houston, Joel, J. Sa-Aadu, and James Shilling. (1991). “Teaser Rates in Conventional
   Adjustable-Rate Mortgage (ARM) Markets”, Journal of Real Estate Finance and
   Economics, 4, 19-31
Kau, James, Donald Keenan, Walter Muller and James Epperson. (1990). “The Valuation
   and Analysis of Adjustable Rate Mortgages”, Management Science, 36, 1417-1431.
Kau, James, Donald Keenan, Walter Muller and James Epperson. (1992). “A Generalized
   Valuation Model for Fixed-Rate Residential Mortgages”, Journal of Money, Credit
   and Banking 24, 279-299.
Kau, James, Donald Keenan, Walter Muller and James Epperson. (1993). “Optional
   Theory and Floating-Rate Securities with a Comparison of Adjustable- and Fixed-
   Rate Mortgages”, the Journal of Business, 66, 595-618.
Kau, James, Donald Keenan, Walter Muller and James Epperson. (1995). “The Valuation
   at Origination of Fixed Rate Mortgages with Default and Prepayment”, Journal of
   Real Estate Finance and Economics, 11, 5-36
Kelly, Austin. (2007). “Zero Down Payment Mortgage Default”, MPRA Paper, online at
   http://mpra.ub.uni-muenchen.de/4318
MaCarthy, Jonathan and Richard W. Peach. (2002). “Monetary Policy Transmission to
   Residential Investment”, Federal Reserve Bank of New York Economic Policy Review,
   8, No.1(May), 139-158
MaCarthy, Jonathan and Richard W. Peach. (2004). “Are Home Prices the Next
   ‘Bubble’?”, Federal Reserve Bank of New York Economic Policy Review, December,
   1-17
MaCarthy, Jonathan and Richard W. Peach. (2005). “Is There a ‘Bubble’ in the Housing
   Market Now?”, working paper.
McQuarrie A. D., and Tsai, C. (1998), .Regression and Time Series Model Selection.
   World Scientific Publishing Co. Pte. Ltd., River Edge, NJ.
Madsen, Chris and Hal Pedersen. (2002). “An Examination of Insurance Pricing and
   Underwriting Cycles”, working paper.
Meissner, Chris and Stephen Satchell. (2007). “A Comparision of the Case-Shiller House
   Price Index Methodology with the FT House Price Index Methodology”, available
   from www.acadametrics.co.uk.
Office of Federal Housing Enterprise Oversight. (2007) “News Release”, the second
   quarter, http://www.ofheo.gov/media/hpi/2q07hpi.pdf



                                         66
Ong, Seow Eng, Tien Foo Sing and Alan Hwee Loon Teo. (2007) “Delinquency and
    Default in ARMs: the Effects of Protected Equity and Loss Aversion”, Journal of
    Real Estate Finance and Economics, 35, 253-280.
Pearson, Neil D. and Tong-Sheng Sun. (1994). “Exploiting the Conditional Density in
    Estimating the Term Structure: an Application to the Cox, Ingersoll and Ross Model”,
    Journal of Finance, 54, 1279-1304.
Poterba, James M. and Lawrence H. Summers. (1988). “Mean Reversion in Stock Prices”,
    Journal of Financial Economics, 22, 27-59.
Quigley, John M. and Robert Van Order. (1995). “Explicit Tests of Contingent Claims
    Models of Mortgage Default”, Journal of Real Estate Finance and Economics 11, 99-
    117.
Risbjerg, Lars. (2006). “Money Growth, Inflation and the Business Cycle”, Denmarks
    Nationalbank: Monetary Review, 3rd Quarter, 23-35
Rogoff, Kenneth. (2002). “Dornbusch’s Overshooting Model After Twenty-Five Years”,
    Second Annual Research Conference, International Monetary Fund Mundell-Fleming
    Lecture
Schwartz, Eduardo, and Walter Torous (1993). “Mortgage Prepayment and Default
    Decisions: a Poisson Regression Approach”, Journal of the American Real Estate and
    Urban Economics Association, 21, 431-449.
Shiller, Robert J (1990) "Speculative Prices and Popular Models", Journal of Economic
    Perspectives, Spring, 55-65.
Shiller, Robert J (2007) "Low Long-term Interest Rates and High Asset Prices", for
    “Celebration of BPEA”.
Standard & Poor’s (2008). “S&P/Case-Shiller Home Price Indices Methodology”,
    available from www.standardandpoors.com.
Tian, Yisong. (1992) "A Simplified Binomial Lattice Approach to the Pricing of Interest-
    Rate Contingent Claims", Journal of Financial Engineering 1:1, 14-37.
Vandell, Kerry. (1993). “Handing Over the Keys: a Perspective on Mortgage Default
    Research”, Journal of the American Real Estate and Urban Economics Association,
    21, 211-246.
Weaver, Karen and Katie Reeves. (2007). “The Impact of Underwriting Subprime ARMs
    at the Fully Indexed Rate: An Analysis of Debt-to-Income Ratios”, Market Pulse,
    Mar.




                                          67

								
To top