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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. 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