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					    Does it Pay Not to Pay? An Empirical Model of Subprime

                        Mortgage Default from 2000 to 2007


                       Patrick Bajari, University of Minnesota and NBER
                          Sean Chu, Federal Reserve Board of Governors
                               Minjung Park, University of Minnesota


                                                July 3, 2008




                                                  Abstract


          To understand the relative importance of various incentives for subprime borrowers to
      default on their mortgages, we build an econometric model that nests various potential
      drivers of borrower behavior.       We allow borrowers to default on their mortgages either
      because doing so increases their lifetime utility or because of the borrowers’ inability to
      pay, treating the decision as the outcome of a bivariate probit speci…cation with partial
      observability. We estimate our model using detailed loan-level data from LoanPerformance
      and the Case-Shiller home price index, and …nd that liquidity constraints are as empirically
      important an explanation as declining house prices for the increase in subprime defaults over
      recent years. Expectations about future home price movements and changes in the interest
      rate environment also contributed to the recent rise in defaults, but their actual e¤ects are
      not large.
    We thank Narayana Kocherlakota, Andreas Lehnert, Monika Piazzesi, Tom Sargent, and Dick Todd for helpful
conversations. Bajari would like to thank the National Science Foundation for generous research support. The
                                                                   ect
views expressed are those of the authors and do not necessarily re‡ the o¢ cial positions of the Federal Reserve
System. Correspondence: bajari@umn.edu; Sean.Chu@frb.gov; mpark@umn.edu.


                                                       1
1    Introduction


Subprime mortgages are made to borrowers with low credit quality or who have a higher prob-
ability of default due to risk factors associated with the loan itself, such as having a low down-
payment.   The subprime market experienced dramatic growth starting from the mid- to late
1990s, up until its recent implosion.    Fewer than 5% of mortgages originated in 1994 were
                                                                  s
subprime; by 2005 that …gure had risen to 20%, according to Moody’ Economy.com. Much of
this growth was made possible by an expansion in the market for private-issue mortgage-backed
securities (MBS). Securitization through MBS and related credit derivatives allowed for loans
that did not conform to the underwriting standards of Fannie Mae and Freddie Mac, the two
government-sponsored securitizers. Beginning in late 2006, the US subprime mortgage market
experienced a sharp increase in the number of delinquencies and foreclosures. In the third quar-
ter of 2005, 10.76% of all subprime mortgages were delinquent and 3.31% were in the formal
process of foreclosure. By contrast, in the fourth quarter of 2007, the corresponding …gures had
risen to 17.31% and 8.65%.


    The turmoil in mortgage and housing markets has generated broader …nancial instability.
Subprime lenders such as New Century Financial have been forced to declare bankruptcy. Banks
and investment banks experienced substantial losses from write-downs on the value of MBS and
collateralized debt obligations. Policymakers have initiated a number of responses or proposed
responses to the conditions in the mortgage and housing markets.        The Federal Reserve has
lowered the discount rate, and Federal Reserve Chairman Bernanke has advocated reducing
loan principal amounts in order to reduce the incentives of homeowners to default. There have
also been collaborative e¤orts by government and industry to freeze mortgage payments for
certain borrowers with adjustable-rate mortgages. Understanding the determinants of mortgage
defaults is clearly necessary for formulating appropriate policy in mortgage, housing and credit
markets. Also, understanding the determinants of default is an interesting positive economic
question in its own right.




                                                2
    In this paper, we explore four alternative explanations for the increase in mortgage defaults,
using a unique data set from LoanPerformance. An observation in the data set is a subprime
or Alt-A mortgage securitized between 1992 and 2007.         We observe information from the in-
         s
dividual’ loan application, including the mortgage term, the initial interest rate, interest rate
adjustments, the level of documentation, the appraised value of the property, the loan-to-value
                        s
ratio, and the borrower’ FICO score at the time of origination. We also have panel data on
the stream of payments made by the borrower and whether the mortgage goes into default.
We merge the LoanPerformance data with the Case-Shiller home price index for 20 major U.S.
cities.   This allows us to track the current value of the home by appropriately in‡ating the
original appraisal using this disaggregated price index.


    A …rst possible explanation for the rise in defaults is falling home prices.         Consider a
frictionless world in which there are no transaction costs from selling a home and no penalties
for defaulting on a mortgage. The buyer should compare the market value of the home to the
outstanding principal balance. If the current home value is less than the outstanding mortgage
balance, it is optimal to default. In the literature, this is referred to as the put option component
of the mortgage (see Crawford and Rosenblatt, 1995; Deng, Quigley, and van Order, 2000; Foster
and van Order, 1985; vandell, 1993).


    Second, increased defaults could result from borrowers’ inability to pay due to a lack of
income or access to credit. Subprime borrowers are likely to be liquidity-constrained. When
interest rates reset for adjustable-rate mortgages, monthly mortgage payments can rise by a
large amount. Buyers with low credit quality may simply lack the income or access to credit
necessary to make their mortgage payments.


    A third explanation is changes in expectations about home prices.           In a fully dynamic
model, the put option component of a mortgage is in‡uenced by expectations about future
home price appreciation.     If home prices are expected to appreciate rapidly, the incentive to
default decreases. This is because default would entail foregone capital gains from the increased
equity in the home. We use two measures of home price expectations. The …rst is a backward-


                                                 3
looking measure based on past trends. The second is a forward-looking measure based on the
ratio of rental to purchase prices for homes, following the approach proposed by Himmelberg,
Mayer, and Sinai (2005).


   Fourth, increased defaults could be due to an increase in contract interest rates relative to
market rates, particularly for adjustable-rate mortgages. When the contract interest rate is less
                                                  s
than the current market interest rate, a borrower’ incentive to default is lower ceteris paribus.
This is because the borrower would have to pay a higher interest rate than his current mortgage
rate. Conversely, when the interest rates on adjustable-rate mortgages increase, the incentives
for default increase.


   We build an econometric model that nests these four possibilities and therefore permits
us to quantify the relative importance of each factor.    The dependent variable in the model
is the decision to default.   Households act as utility maximizers and default if the expected
utility from continuing to make mortgage payments is less than the utility from defaulting on
the mortgage. We also include a second equation that re‡                s
                                                        ects a borrower’ ability to continue
paying the mortgage. If the buyer lacks adequate income or access to credit, this may also result
in default.   We demonstrate that our structural equations can be represented as a bivariate
probit with partial observability, a type of model …rst studied by Poirier (1980). We check the
robustness of our results by estimating a competing hazards model with unobserved borrower
heterogeneity similar to the speci…cation in Deng, Quigley, and van Order (2000).


   We …nd evidence for each of the hypothesized factors in explaining default by subprime
mortgage borrowers. In particular, our results suggest that declining house prices and borrower
and loan characteristics that a¤ect the borrowers’ ability to pay are the two most important
factors in predicting default. The …nding that liquidity constraints are as empirically important
an explanation as declining house prices suggests that the increase in subprime defaults over
recent years is partly linked to changes over time in the composition of mortgage recipients.
Higher numbers of borrowers with little or low documentation and low FICO scores, or who
only make small downpayments, contributed to the increase in foreclosures in the subprime


                                               4
mortgage market. The increasing prevalence of adjustable-rate mortgages also contributed to
rising foreclosures. The monthly payments for adjustable-rate mortgages come with periodic—
and sometimes very large— adjustments, forcing liquidity-constrained borrowers to default.


   There is a wealth of literature examining various aspects of mortgage borrowers’decision to
default. Existing research has typically focused on the put-option nature of default by studying
how net equity or home prices a¤ect default rates (Deng, Quigley, and van Order, 2000; Gerardi,
Shapiro, and Willen, 2008). Other studies have looked at the importance of borrowers’liquidity
constraints (Archer, Ling, and McGill, 1996; Carranza and Estrada, 2007), borrowers’ overall
ability to pay, as measured by their credit quality (Demyanyk and van Hemert, 2008), and the
role of rate resets for adjustable-rate mortgages (Pennington-Cross and Ho, 2006).


   We build on these earlier works by considering each of the factors proposed by other re-
searchers. However, our analysis di¤ers from the previous literature in at least four respects.
First, our econometric model nests the various potential incentives for default inside a uni…ed
framework, rather than studying these incentives individually.    In particular, we depart from
the previous literature by specifying two latent causes of default— …nancial incentives to raise
                                                                s
lifetime utility by defaulting and the violation of the borrower’ liquidity constraint— and using
the likelihood function that takes into account the fact that we do not observe which of the two
underlying causes was the trigger for default. Second, our data set includes recent observations
from a nationally representative sample of subprime mortgages, allowing us to focus on the
drivers behind the recent wave of mortgage defaults.     In contrast, a closely related paper by
Deng, Quigley, and van Order (2000) examines prime mortgage borrowers, for whom default
is much less common.    Third, since our data contain detailed information on loan terms and
borrower risk factors, we can control for these observables in our analysis, which some previous
work could not adequately address. Moreover, our paper systematically examines the e¤ects of
several variables that economic theory suggests ought to a¤ect borrower default, including ex-
pectations about home prices, the volatility of home prices, the amount of time remaining until
the next rate reset for ARMs, and the payment-to-income ratio. By using a more comprehensive



                                               5
list of potential drivers of default, we are better able to assess the relative importance of various
factors, compared to previous literature. Fourth, in contrast to more descriptive pieces such
as Demyanyk and van Hemert (2008), our model builds from the assumption that consumers
maximize their utility and face liquidity constraints, from which we then derive the equations
that we estimate.


    The rest of this paper proceeds as follows. In Section 2, we present a model of default by
mortgage borrowers. In Section 3 we describe the data. Section 4 presents model estimates
and other empirical …ndings. Section 5 concludes.




2    Model


Our model of housing default builds on Deng, Quigley, and van Order (2000), Archer, Ling,
and McGill (1996), and Crawford and Rosenblatt (1995).          We build on the stylized model of
optimal default described in this earlier research. In the model, borrowers maximize expected
discounted utility.   At each period in time, a borrower receives utility from housing services
and from consumption of a composite commodity. Consumption of the composite commodity
is equal to income less savings and the costs of housing services.       In this environment, it is
                                                                                    s
optimal for a homeowner to default if and only if defaulting increases the homeowner’ wealth.
We begin by considering the case of a frictionless environment in which default is optimal if and
only if the value of the home exceeds the expected discounted mortgage payments. We then
sequentially incorporate additional factors— expectations about home prices, interest rates and,
                                                         s
…nally, credit constraints. We demonstrate that an agent’ optimal decision rules take the form
of a system of inequalities.   This system of inequalities is naturally modeled using a discrete
choice framework.     We demonstrate that our system of inequalities can be modeled using a
bivariate probit with partial observability, …rst studied by Poirier (1980).




                                                 6
2.1    Optimal Default without Liquidity Constraints


                                                                                      s
Let i index borrowers and t index time periods. Denote by Vit the value of borrower i’ home
                                                                   s
at time period t, and denote by Lit the outstanding principal on i’ mortgage. Normalize the
time period in which i purchases her home to t = 0. Let git denote the rate of in‡ation in home
price between time period t    1 and t. Then
                                                  Q
                                                  t
                                      Vit = Vi0           (1 + git0 )                          (1)
                                                  t0 =1

That is, the current home value is the initial home value times the gross rate of in‡ation in
housing. In our empirical analysis, we shall let git correspond to the Case-Shiller price index for
                                                                              s
the city where i resides. Equation (1) tracks the evolution in the value of i’ home. Empirically,
we expect it generally to be the case that Vit > Vi0 for buyers who have held their homes for
many years.    However, for more recent buyers it may be the case that Vit < Vi0 because of
declining home prices over a shorter horizon.


   The evolution of the outstanding principal, Lit , is somewhat more complicated than the
evolution of Vit . In order to economize on notation, we will not write down an explicit formula
for the outstanding principal, since di¤erent households have di¤erent mortgage contracts. Lit
is a function of the original loan amount, previous mortgage payments by the borrower, the
mortgage term, and contract interest rates. Fortunately, in our empirical work, we have access
to the current principal and a complete speci…cation of the contract terms that determine how
Lit evolves.


   We begin by considering an environment without any frictions. In particular, we abstract
from penalties from default and transaction costs, and assume that markets are complete and
that there are no binding credit constraints. In this extremely stylized model, i will choose to
default if
                                          Vit      Lit < 0                                     (2)

The reason is that if the above inequality holds, the borrower is able to strictly increase her
lifetime utility by defaulting. She could default and then repurchase the same home, and thus

                                                   7
               ow
keep the same ‡ of housing services while increasing her wealth by Lit           Vit . On the other
hand, if Vit   Lit > 0 then default is suboptimal. For example, the borrower would be able to
sell her home at a price that strictly exceeds the outstanding principal. This would leave her
with net wealth Vit    Lit .


   The decision to default may be triggered by a fall in home prices git . For recent borrowers
who have made small downpayments, just a few su¢ ciently negative realizations of git may be
su¢ cient for (2) to hold. For borrowers who have made large downpayments or who have held
their homes for many periods in which git > 0, (2) is less likely to hold.



2.1.1   Expectations about Home Prices


In this subsection, we allow our default decision to become slightly more complicated and depend
on expectations about future home prices. This dependence would exist if, for example, there
is a lag between the decision to sell a home and the time period in which the sale actually takes
place. This assumption is quite realistic given that sale times for homes are typically three to
six months during normal housing markets, and may exceed a year during housing downturns.
As a result, there may be a gap between prices at the time during which the decision to sell was
made and the price that the seller was ultimately able to receive in the market.


                                 s
   Let Egit represent borrower i’ expectation in period t, given her current information, about
the future growth rate in home price. If the buyer is risk-neutral, the default decision depends
on the following condition:
                                       Vit (1 +   1 Egit )   Lit < 0                             (3)

We use the parameter     1   to scale the units of Egit : We shall describe our approach to measuring
Egit in the next subsection.


   In a richer model, we would also expect the variance of home prices to in‡uence the default
decision. Buyers may be risk-averse and therefore demand a risk premium when home prices



                                                     8
‡uctuate.     Moreover, option pricing theory suggests that the variance in home prices should
in‡uence the optimal default decision. For example, homeowners may be willing to take on a
mortgage even when the expected change in Vit is negative, so long as the variance is su¢ ciently
large. The reason is that the borrower would still gain in the event that the house price does
appreciate, while at the same time the option to default mitigates the downside risk if the home
value instead falls.


      In practice, measuring expectations about the variance, V git , is even more di¢ cult than
measuring the expected growth rate. Also, modeling the impact of variance on consumer utility
in a structural way is beyond the scope of this paper. As a compromise, we add to (3) a term
that captures the reduced-form impact of the variance of git :




                                      Vit (1 +     1 Egit   +       2 V git )   Lit < 0                                (4)



Measuring Expectations about Home Prices                                  We consider two di¤erent measures of Egit :
one based on user costs, and another measure based on price trends in the recent past.                                 For
the former, we follow Himmelberg, Mayer, and Sinai (2005) and exploit a no-arbitrage condition
between renting and purchasing a house. In a given housing market, the annual user cost of
ownership must equal the annual rent:

                 Cost of ownership at time t                =
                                                                                                                       (5)
                      rf                          c
                 Vit rit + Vit ! it    Vit   it (rit   + ! it ) + Vit     it    Vit Egit + Vit   it   = Rit

                                                                            rf
In this equation, Vit is the house price and Rit the annual rent. The term rit is the interest
rate the homeowner i would have obtained in an alternative, risk-free investment. Therefore,
     rf
Vit rit captures the opportunity cost of the house relative to other potential investments. The
term ! it is the property tax rate,                                                      c
                                                   the e¤ective tax rate on income, and rit the contractual
                                              it

interest rate on the mortgage.           The term Vit                c
                                                                it (rit   + ! it ) therefore represents savings to the
homeowner due to the tax-deductibility of mortgage payments and property taxes. The term

 it   represents the depreciation rate of the house, Egit the expected capital gain, and                      it   the risk

                                                                9
                                               rf                              c
premium. Using observed values of Vit , Rit , rit , ! it ,             it ,   rit ,   it ,   and   it ,   obtained from data,
we can deduce the expected capital gain Egit that satis…es (5).


      Himmelberg, Mayer, and Sinai (2005) recover Egit from (5) and decompose it into two
                                                                     f
components. The …rst is the expected growth due to “fundamentals,” Egit , which they proxy
                                                                                        b
using the average annual home price growth rate between 1950 and 2000. The remainder, Egit ,
captures expected growth unexplained by their measure of “fundamentals,” and might be due
                                                                                rf
to speculative bubbles. For each MSA and quarter, they report the ratio of Vit rit + Vit ! it
           c                            f
Vit   it (rit + ! it ) + Vit it   Vit Egit + Vit   it                      )
                                                        (the “imputed rent” to the actual rent Rit . Comparing
this expression to (5) shows that the ratio is greater than one if the market expects faster house
                                                     b
price appreciation than warranted by fundamentals (Egit > 0), with a higher ratio indicating
a larger bubble.          Conversely, a ratio less than one indicates that the market expects slower
growth than implied by fundamentals.                      We include this ratio of imputed rent to actual rent
as one measure of borrowers’ expectations about home prices.                                 This measure is denoted by
Exp_HM S in our empirical section.


      In addition to looking at the e¤ects of speculative home price appreciation, we also use
backward-looking measures to form expectations about home prices. Speci…cally, we allow the
default decision to depend on home price appreciation in the previous period, based on the idea
that borrowers may be extrapolating from the recent past in forecasting future growth. This
measure is denoted by Exp_Bwd in our empirical section. The econometric model that we
estimate will allow us to separately identify the impacts of these two alternative measures of
expectations.



2.1.2      Interest Rates


Finally, we allow the optimal default decision to depend on interest rates. Theory predicts that
when market interest rates are high relative to the contractual rate, the incentive to default
is lower. Below-market-rate contractual rates imply that borrowers will lose the future value


                                                              10
of the discount if they default. To operationalize this idea, we follow Deng, Quigley, and van
Order (2000) and compute the normalized di¤erence between the present value of the payment
stream discounted at the mortgage note rate and the present value discounted at the current
market interest rate. For borrower i in period t,
                TP
                 Mit                     TP
                                          Mit                        TP
                                                                      Mit                     TP
                                                                                               Mit
                             Pi                       Pi                           1                        1
                           m
                       (1+rit =1200)t               c
                                                (1+rit =1200)t                  m
                                                                            (1+rit =1200)t               c
                                                                                                     (1+rit =1200)t
                t=1                       t=1                        t=1                       t=1
       IRit =               TP
                             Mit
                                                                 =               TP
                                                                                  Mit
                                                                                                                      (6)
                                         Pi                                                    1
                                       m
                                   (1+rit =1200)t                                           m
                                                                                        (1+rit =1200)t
                             t=1                                                  t=1

Pi is the monthly payment for the mortgage, T Mit is the number of remaining months until
           m
maturity, rit is the market rate borrower i would get if he obtained a new loan in period t,
     c
and rit is the contractual interest rate of the mortgage. For adjustable-rate mortgages, Pi and
 c                                                                                   c
rit may vary over the course of the loan, but for simplicity, we assume that Pi and rit remain
constant at the levels of the current month t.

                                                       m
   In practice, the available market rate of interest rit varies across households because of dif-
ferences in credit histories and other risk factors. Some of these risk factors are unobservable to
us as econometricians. Therefore, in order to determine the market rate of interest available to
a household i at time t, we …rst compute the predicted rates based on observable borrower and
loan characteristics (FICO, loan-to-value, etc.), where the prediction parameters are estimated
using actual originations of all subprime mortgages observed in the data. Since the LoanPer-
formance data cover the universe of subprime mortgage originations, we can get a very precise
                                                                               m
estimate of the impact of the risk factors on contract rates. Our estimate of rit also controls
for unobserved household-level heterogeneity.                                                          m
                                                            Details behind the procedure for imputing rit
are described in Appendix A.


   Just as we measure expectations about future house prices, we would also ideally like to
control for household expectations about future interest rates. We have not yet done so, and
leave this extension to future work.            Although we currently abstract away from expectations
about market rates, we do incorporate one prominent source of interest rate changes: rate resets
for adjustable-rate mortgages.           Borrowers with ARMs presumably are able to anticipate—
at least to a limited degree— future interest rate resets, which a¤ect the option value of not

                                                          11
defaulting. If a borrower expects that her contractual interest rates will reset to a higher level
in the near future, the borrower will have a stronger incentive to default at any given level of
net equity.       In the data, we observe the number of months until the next rate reset of each
ARM, and we can use this measure to investigate how expectations of future rate changes a¤ect
default decisions. Letting M Rit represent the number of months before the next rate reset for
borrower i in period t (for …xed-rate mortgages, we set M Rit = 0 and then include a separate
dummy for …xed-rate mortgages), the default decision of the borrower depends on the following
condition:
                        Vit (1 +    1 Egit   +       2 V git )     Lit (1 +    3 IRit   +    4 M Rit )   <0      (7)

Similar to    1   and    2,   the terms      3   and       4     are necessary to properly scale the units. Dividing
both sides of the equation by Lit then yields:

                         Vit
                             (1 +     1 Egit     +     2 V git )     (1 +     3 IRit   +    4 M Rit )   <0       (8)
                         Lit



2.2   Liquidity Constraints


So far, we have considered the optimal default decision of borrowers in a frictionless world
without any liquidity constraints or penalties from default. In such a world, a borrower would
default on her mortgage whenever equation (8) is satis…ed. This type of default rule is sometimes
referred to as “ruthless”default in the mortgage literature (vandell, 1995), which has found that
although the ruthless default rule does explain borrowers’ default behavior to some extent, a
signi…cant portion of default behavior remains unexplained. Researchers have conjectured and
also empirically investigated the additional role played by liquidity constraints, reputational
costs, and trigger events such as divorce in explaining default (Deng, Quigley, and van Order,
2000; Kau, Keenan, and Kim, 1993).                        In particular, for subprime mortgage borrowers, who
tend to have poor credit quality and limited credit lines, liquidity constraints are likely to be a
signi…cant factor for their default decisions.


   To capture the idea that a mortgage borrower may default simply because she cannot meet


                                                                   12
the monthly payments, and not for the purpose of increasing lifetime wealth, we introduce a
second equation that captures frictions associated with household illiquidity and inability to
pay. Key determinants of whether a household has su¢ cient liquidity to meet its contractual
                                                                                                Pit 1
obligations are its monthly principal and interest payments relative to income,                 Yit ,   and the
          s
household’ overall credit quality, Zit . The latter matters because it has an e¤ect on whether the
household has the ability to borrow from other sources in order to meet its mortgage payments.


       We start by making an assumption that subprime borrowers cannot save and that no addi-
tional borrowing is available to the mortgage holders because they cannot tap into the capital
market to borrow against future income.             As a result, borrowers must meet their period-by-
period budget constraints in every single period.             The period-by-period budget constraint of
household i can be written as follows.

                                                Pit + Cit     Yit                                           (9)

Cit denotes the consumption of the household i in period t.                   We further assume that the
                                                                                 s
household must have a minimum level of consumption in each period. The household’ budget
constraint then takes the following form.

                                                                    Pit
                               Budget Constraint Binds , 1                 cit < 0                         (10)
                                                                    Yit

                                                                               s
where cit is the minimum required consumption as a proportion of the household’ income.
                 s
When a household’ monthly payment Pit increases relative to its income, the budget constraint
is more likely to bind, forcing the household to default.


       The budget constraint (10) is appropriate only for those who have no access to any form of
credit. Most borrowers, however, have at least limited access to certain forms of credit, with the
level of access varying by their credit quality. For households that are able to borrow from the
                                                                                              Pit
capital market in order to meet their monthly payments, the relationship between              Yit   and default
is less stark. Only for those with low credit quality and limited borrowing ability do we expect
   1
       Since the imputed income for each household remains constant over time, the variation in the payment-to-
income ratio comes from across households as well as rate resets for a given household.


                                                       13
such a rigid relationship between the payment-to-income ratio and default. Under the extreme
assumption of complete capital markets, the relevant budget constraint for a household would be
                                                          s
its lifetime budget constraint, which pools the household’ period-by-period budget constraints
over all time periods. To capture the notion that the relevance of the period-by-period budget
                                                                                    Pit
constraint is weaker for borrowers with high credit quality, we interact            Yit   with measures of
borrowers’credit quality. We categorize each borrower into one of three credit quality groups—
low credit, medium credit, and high credit— and allow the impact of the payment-to-income
ratio on default to vary across these groups. We also allow for the possibility that the measures
of credit quality, Zit , may a¤ect the budget constraint independently of their e¤ects through
interactions with the payment-to-income ratio. After making appropriate normalizations, these
considerations yield the following condition.
                                                                      Pit
                 Budget Constraint Binds ,      1 Zit   +   2 Zit (       )   cit + 1 < 0             (11)
                                                                      Yit


2.3   Bivariate Probit with Partial Observability


The structural equations (8) and (11), derived from our model, represent two drivers of default:
borrowers are utility maximizers and will exercise an option to default either because doing
so increases their wealth or because credit constraints prevent them from continuing to make
payments. Thus, at a given point in time, the household can be in one of four possible situations:
                                    s                         s
(a) default increases the household’ wealth and the household’ budget constraint is binding, (b)
                                s                        s
default increases the household’ wealth and the household’ budget constraint does not bind,
                                    s                         s
(c) default decreases the household’ wealth and the household’ budget constraint is binding,
                                       s                          s
and (d) default decreases the household’ wealth and the household’ budget constraint does
not bind. (a), (b), and (c) lead to default, while (d) leads to no default. As econometricians,
all that we observe in the data is whether a given household defaults or not in a given period t.
When we observe no default, we know that (d) holds. However when we observe default, we
cannot distinguish whether it is due to (a), (b), or (c).


   If the agents’latent utilities have a bivariate-normally distributed error, the data generating

                                                14
process for the observed outcome corresponds to a bivariate probit model with partial observ-
ability, which was …rst studied by Poirier (1980). By modeling default as the outcome of two
separate (but potentially correlated) underlying propensities, our approach contrasts with the
existing literature, in which researchers have typically included in a single equation both the
determinants of …nancial incentives as well as measures of liquidity (Archer, Ling, and McGill,
1996; Demyanyk and van Hemert, 2008). A single-equation model leads to misspeci…cation be-
cause it fails to account for the fact that the …nancial incentives are relevant for default decisions
only if the liquidity constraint does not bind, and vice versa. Such a fallacy may lead to bias
in the estimated empirical signi…cance of one or the other type of incentive.


       Our econometric model is formulated by simply adding stochastic errors to the structural
equations (8) and (11). For household i at time t:



                                        Vit
                   U1;it =     0i   +   Lit (1   +   1 Egit   +     2 V git )      (1 +      3 IRit   +     4 M Rit )   + "1;it
                                                                                                                                       (12)
                                                                                          Pit
                                          U2;it =      0i +       1 Zit +       2 Zit (   Yit )   + "2;it


       U1;it represents the latent utility associated with not defaulting, and is equal to the nor-
malized di¤erence between the market value of the house and the option-adjusted value of the
mortgage. The option value stems from either anticipated changes in home prices or interest
rates, or from deviations of the contractual interest rate from the market rate. The term "1;it
is an iid shock, and represents idiosyncratic di¤erences across borrowers in their utility from
not defaulting.            The term U2;it represents the budget constraint of household i, and "2;it is
an idiosyncratic shock to the tightness of the household budget constraint.                                               The terms U1;it
and U2;it are correlated with each other through the observable covariates Vit , Lit , Egit , V git ,
                   Pit
IRit , M Rit ,     Yit ,   and Zit , as well as through the distribution of the unobservables "1;it and "2;it ,
which we assume are jointly normal with a variance of 1 and a covariance of                                                  .    The terms

 0i    and    0i   capture the unobserved borrower heterogeneity in U1;it and U2;it .2                                       Our data are
   2
                                                                                              s
       We assume that cit , the minimum required consumption as a proportion of the household’ income, remains
constant over time for a given individual. Hence, the term is now subsumed in                               0i .




                                                                       15
in the form of a panel and we will treat              0i   and   0i   as random e¤ects. In principle, we could
potentially estimate        0i   and   0i   using …xed e¤ects techniques for discrete choice models in panel
data settings. However, the computational burden of these techniques is prohibitive because of
the large size of our sample.


      Among the covariates Zit entering the liquidity equation, we include the most obvious mea-
sures of borrowers’credit quality, such as FICO scores. We also include observable loan charac-
teristics and the monthly unemployment rate at the county level. Among loan characteristics,
we focus on the age of the loan, the level of documentation, and the loan-to-value ratio at orig-
ination. For reasons other than actual …nancial incentives, holders of older loans are less likely
to be liquidity-constrained simply because mortgages held by liquid borrowers are more likely
to survive. Borrowers with low documentation on income or wealth are also more likely to have
low credit and liquidity problems. Finally, after controlling for the current loan-to-value ratio,
loans with higher loan-to-value ratios at origination are more likely to attract illiquid borrowers,
many of whom probably cannot obtain mortgages under tighter terms.


      We de…ne the random variable N Dit = 1 if household i does NOT default in period t and
as 0 otherwise. The condition for default is as follows:

                      N Dit = U1;it         U2;it = 0 (default) ,        U1;it < 0 or U2;it < 0                 (13)

where the outside options for both U1;it and U2;it are normalized to zero. Given the available
data, when a default occurs we cannot observe whether it is because U1;it < 0, because U2;it < 0,
or for both reasons.


      Two points are worth mentioning.                                                                s
                                                    First, in principle we could specify the borrower’ de-
cision as a choice among three options, instead of a binary choice, by distinguishing between
prepayment and continued payment according to schedule. In the above baseline speci…cation,
the choice of no default includes both prepayment as well as the decision to continue making
only scheduled payments. However, we do not believe that such an extension would signi…cantly
change our key …ndings with regard to the drivers behind default.3 Nevertheless, as robustness
  3
      Ceteris paribus, declining house prices increase the incentive to default and decrease the incentive to prepay.


                                                            16
checks, we estimate an alternative model in which prepayment and default are dependent com-
peting hazards, and as a separate exercise also try dropping from the estimation sample all loans
ending in prepayment (leaving only loans that end in default, censoring, or scheduled payment
to maturity).


    Second, note that the above speci…cation is basically a static discrete choice model.                     A
natural alternative would be to incorporate future-looking behavior using a dynamic discrete
choice framework, in the spirit of Rust (1987). However, these types of models require a full
                        s
speci…cation of an agent’ optimization problem and constraints. We believe that our current
results are useful for determining our modeling strategy in such a framework.                   For example,
our results will be informative about whether we should include credit constraints in this model
and how we should model price expectations.                We hope to pursue a fully speci…ed model in
upcoming work.




3    Data


Our estimation exploits data from LoanPerformance on subprime and Alt-A mortgages that
were originated between 1992 and 2007 and securitized in the private-label market. The Loan-
Performance data set covers more than 85% of all securitized subprime and Alt-A mortgages.
According to the Mortgage Market Statistical Annual, 55%-75% of all subprime mortgages were
securitized in the early- to mid- 2000s. Because sample selection is based on securitization, the
loans covered by LoanPerformance may di¤er from the subprime mortgage market as a whole.


    For each loan, we observe the terms and borrower characteristics reported at the time of loan
origination, including the identity of the originator, the type of mortgage (…xed rate, adjustable
rate, interest-only, etc.), the frequency of rate resets (in the case of ARMs), the initial contract
Therefore, the e¤ects on the choice between “default” and “no default” are unambiguous. On the other hand,
declining interest rates increase the value of the mortgage and therefore increase the propensity both to prepay
as well as to default (See Foster and van Order, 1984 and Quigley and van Order, 1995).



                                                      17
interest rate, the level of documentation (full, low, or nonexistent4 ), the appraisal value of the
property, the loan-to-value ratio, whether the loan is a …rst-lien loan, the existence of prepayment
penalties, the location of the property (by zip code), the borrower’ FICO score,5 and the
                                                                   s
         s
borrower’ debt-to-income ratio. One limitation of the LoanPerformance data is that they do
not report the number of mortgage points purchased by the borrower at the time of origination,
so we are only able to observe the interest rate before any adjustments for points.6


       In addition to loan and borrower characteristics at the date of origination, the data also track
each loan over the course of its life, reporting the outstanding balance, delinquency status, and
the current interest rate in each month. For more detailed discussions of the LoanPerformance
data, see Chomsisengphet and Pennington-Cross (2006), Demyanyk and van Hemert (2007), and
Keys, Mukherjee, Seru, and Vig (2007).


       The LoanPerformance data contain detailed information on the credit quality of borrowers,
but do not report their demographic characteristics. Therefore, we match the loan-level data
to 2000-Census data on demographic characteristics at the zip-code level (per-capita income,
average household size and education, racial composition, etc.).                In addition, as one measure
                            s
that could a¤ect a borrower’ liquidity constraints, we use monthly unemployment rates reported
at the county level by the Bureau of Labor and Statistics (BLS). These variables are a proxy
for individual-level demographics.         Because our proxies are measured with error, we will not
be able to consistently estimate the e¤ect of individual-level demographics on mortgage default.
   4
                                                      s
       Full documentation indicates that the borrower’ income and assets have been veri…ed. Low documentation
refers to loans for which some information about only assets has been veri…ed. No documentation indicates there
has been no veri…cation of information about either income or assets.
   5
     According to Keys, Mukherjee, Seru, and Vig (2007), FICO scores represent the credit quality of a potential
borrower based on the probability that the borrower will experience a negative credit event (default, delinquency,
etc.) in the next two years.     FICO scores fall between 300 and 850, with higher scores indicating a lower
probability of a negative event.
   6
     Borrowers may purchase “points” at the time of origination, in return for a reduction in interest rates.
(Negative points are also obtainable in exchange for an increase in interest rates.) Because the lumpsum is not
returned if the borrower prepays, buying points is a better deal for borrowers the longer they plan to keep the
mortgage before prepaying.



                                                       18
However, since we expect the proxies to be correlated— and in many cases strongly correlated—
with actual demographics, including these variables will provide some evidence about the impact
of demographics on mortgage default.


       Another important variable that enters the equation determining the budget constraint is
the payment-to-income ratio. While we do not observe income at the household level, we can
obtain a noisy imputation of household income based on the reported debt-to-income ratio.7
De…nitions and summary statistics for key variables are reported in Tables 1 and 2.


       In Table 2, we also report separate summary statistics according to the termination mode
of each loan— that is, whether a loan prepays (a category comprising 67,056 loans), defaults
(20,060 loans), or is either paid to maturity or censored by the data (111,179 loans).                    In the
last category, virtually all of the loans are censored, while only 4 loans are observed paying
to maturity, so in the following discussion, we shall simply refer to the third category as the
“censored” observations.


       The relationships between the termination mode and the measures of borrowers’ability to
pay are generally consistent with our hypotheses.              Loans that default tend to be adjustable-
rate mortgages, are associated with higher initial loan-to-value ratios, and tend to be issued to
borrowers with lower credit scores.           For instance, …xed-rate mortgages comprise 26.2% of all
loans, 24.6% among loans that prepay, and 32.1% among the censored loans, while comprising
only 15.4% of loans that default.           The average FICO score in the sample is 631 and is lower
conditional on default (596), higher conditional on prepayment (627), and higher still among
censored loans (647).


       Table 2 also summarizes the time-varying variables, both as an average over the course of
each loan (the second panel) as well as for the last period in which we observe each loan (the
   7
       Speci…cally, we assume that household income stays constant over time, and approximate it by the scheduled
monthly payment divided by the “front-end” debt-to-income ratio, both reported as of the time of origination.
The front-end ratio measures housing-related principal and interest payments, taxes, and insurance as a percentage
of monthly income.



                                                        19
third panel). Relative to the overall average, borrowers that default tend to have less equity
at the point in time when they default, as well as higher payment-to-income ratios and higher
contractual interest rates. Conditional on being an ARM, loans also tend to default at times
when fewer periods remain until the next rate reset, though the e¤ect is weak.


   To be more precise about the magnitudes of these e¤ects, log(V =L) is on average 0.512
over the course of each loan and 0.515 in the last observed period.        The average is higher
conditional on prepayment (0.572 in the last period), much lower for loans that default (0.365),
and intermediate for the remaining loans (0.466).      The average monthly payment-to-income
ratio is 0.294 over the course of the loan and 0.307 in the …nal period. This ratio tends to be
highest among loans that default (on average 0.339 in the …nal period), somewhat lower among
loans that prepay (on average 0.312), and lowest among the censored loans (on average 0.290).
The data are also suggestive of ARM holders tending to default when fewer periods remain until
the next reset, but the di¤erence is small, which suggests to some extent that borrowers do not
so much default in anticipation of rate resets as much as they wait until after the resets have
actually occurred, when the higher payments have come due.


   Consistent with theory, default tends to occur at points in time when the trend in housing
prices is low, as measured by the change on the previous month or as realized ex post over
the course of the following month. Default is also associated with lower volatility in housing
prices, though of course, our measure of volatility (i.e., the normalized standard error of housing
prices over the previous twelve months) is highly correlated with the trend.        Upon default,
the annualized rates of appreciation over the previous and subsequent months are on average
2.0% and 1.3%, respectively, while the recent volatility is on average 1.053. By contrast, upon
prepayment, the average annualized rates of appreciation in the previous and subsequent months
are 8.7% and 8.1%, respectively, while the recent volatility is on average 1.919. Because the data
are censored at October 2007, when housing markets were falling in many areas, the censored
loans tend to end at a point in time when recent housing appreciation has been negative. Finally,
user costs tell largely the same story as the actual house price trends, though the implied rate



                                                20
of appreciation tends to be much higher— at an average annualized rate of 11.7% at the point in
time when loans default, 14.1% at the point of time when loans prepay, and 22.9% at the …nal
observation for all remaining loans.


       Furthermore, as we would expect, the data indicate that conditional on default, borrowers
tend to be paying higher interest rates than the market rate.                  For loans that end in default,
IR has an average value of 0.0347 at the point of default (versus an overall average of 0.0235
for the …nal observation across all loans). Somewhat surprisingly, at the time of prepayment
for loans that prepay, the average IR is actually somewhat lower (at 0.0208) than the overall
average. However, this is consistent with the fact that market interest rates were quite low at
the censoring date of October 2007, which brings down the overall average.


       The demographic data indicate that both default and prepayment tend to occur in zip codes
with higher-than-average unemployment (5.10% and 5.16%, respectively, versus 4.73% for all
other loans). Default is also more prevalent in lower-income zip codes (with the zip-code–level
income averaging $20,880 for loans that default, versus an overall average of $22,340 and an
average of $22,610 among loans that prepay).


       To track movements in home prices, we use housing price indices at the MSA level, from
Case-Shiller.8 The HPI for each MSA is normalized to 100 for January 2000. The home price
indices are reported at a monthly frequency, and are determined using the transaction prices of
those properties that undergo repeat sales at di¤erent points in time in a given geographic area.
Since the index is designed to measure price changes for homes whose quality remains unchanged
over time, homes are assigned di¤erent weights depending on the length of time between the two
transactions, along with other rules of thumb indicating that the home has undergone major
renovations.9
   8
       Cities covered by Case-Shiller are Atlanta, Boston, Charlotte, Chicago, Cleveland, Dallas, Denver, Detroit,
Las Vegas, Los Angeles, Miami, Minneapolis, New York, Phoenix, Portland, San Diego, San Francisco, Seattle,
Tampa, and Washington D.C.
  9
    The index assigns zero weight to houses that have undergone repeat transactions within a span of six months.
Lower weights are also assigned to houses for which the change in transaction price is an outlier within a geographic



                                                         21
4       Results


We begin by discussing estimates from our baseline model, i.e., the bivariate probit with partial
observability. In these speci…cations, the dependent variable “no default”includes both contin-
ued payments and prepayments. We consider a wide range of alternative speci…cations in order
to assess the robustness of our results to alternative modeling assumptions.


    The …rst set of speci…cations is described in Table 3a.              For a particular speci…cation, the
column eq1 includes the covariates and parameter estimates that determine U1;it in equation
(12).    The column eq2 includes the parameter estimates and covariates that determine U2;it .
Each cell in this table contains the parameter estimate, the standard error and the marginal
e¤ect of the covariate.10          In Table 5, we display estimates of the impact of a one-standard-
deviation increase in the independent variables on the probability of default. A particular cell
reports the change in the default probability due to the increase in the independent variable
by one standard deviation, divided by the baseline default probability.                   The baseline default
probability is de…ned by setting all explanatory variables equal to their sample means.


    In Speci…cation 1, we start with a parsimonious model in which U1;it is determined by the
ratio of the home value to the outstanding loan balance. The discussion of Section 2.1 suggests
that the incentives to default decrease as the ratio of the value to the loan increases. In our
empirical analysis, we choose to use the natural logarithm of the ratio of the value to the loan
instead of this ratio directly, as discussed in Section 2.1. In the data, as the term of the loan
ends, the denominator of this ratio can become quite small. These observations have a smaller
e¤ect on our estimates when we use the natural log.
area. Finally, houses with a higher initial sales price are assigned a higher weight.
  10
                                                                                                           ,
     In the tables, we express all marginal e¤ects in terms of the e¤ect on the probability of “no default” P (U1 >
0; U2 > 0), with all independent variables set at their sample means. For the sake of brevity, we shall not always
explicitly state this assumption. Furthermore, because P (U1 > 0) and P (U2 > 0) are each individually very close
to one, and because none of the covariates is included in both equations, the marginal e¤ect of any covariate of
Uj on P (U1 > 0; U2 > 0) is virtually equal to its marginal e¤ect on P (Uj > 0) for j = 1, 2. Therefore, we do
not need to discuss both e¤ects.


                                                        22
   The estimates of Speci…cation 1, and all other speci…cations used in Table 3a, are consistent
with the predictions of Section 2.1. As the theory predicts, borrowers that have a high value-
to-loan ratio are less likely to default. Our estimates of the marginal e¤ects imply that a
                                        V
one-standard-deviation increase in log( ) is associated with a 47.8% reduction in the hazard of
                                        L
default in a given month.


   The sharp decline in home prices played an important role in the recent increase in foreclo-
sures. Consider a hypothetical household in Phoenix that purchases a home in February 2007
                                                                            V
                                                                       s
with a 30-year …xed-rate mortgage and no downpayments. The household’ log( ) is then 0 at
                                                                             L
the time of purchase. Further assume that the household makes monthly payments such that
                                                                29
the outstanding balance on the mortgage in February 2008 is     30   of the original loan amount.
If there is no change in home price between February 2007 and February 2008, the household’   s
     V
log( ) in February 2008 would be 0.034. During this time period, however, home prices in
     L
                                          s
Phoenix fell by 21.7%. If this household’ property value experienced the average home price
                           V
change in Phoenix, its log( ) at the end of this time period would be -0.211. Thus, the decline
                           L
in home price makes the household 25.4% more likely to default in February 2008 compared to
the hypothetical case of no change in home price.


   In Speci…cation 1, we see that the variables that enter U2 are important drivers of default
as well. A low-documentation loan has a 0.161 percentage point higher chance of default in a
given month, or equivalently, a 53.1% increase in the default hazard computed at the sample
means. The marginal e¤ect of a one-standard-deviation increase in the FICO score— about 71
points— corresponds to a decrease in default probability of 0.223 percentage points, or 73.7%
of the hazard computed at the sample means. Similarly, a one-standard-deviation increase in
the original loan-to-value (0.14) is associated with a 16.5% greater hazard, and a one-standard-
deviation increase in local unemployment rate (1.36%) is associated with a 9.4% greater hazard.


   As we would expect from equation (10), an increase in the ratio of monthly mortgage pay-
ments to monthly income also predicts an increase in the probability of default. A one-standard-
deviation increase in this ratio (0.12) generates an 18.6% increase in the hazard of default. For


                                               23
                4                                                      s
Speci…cations 2– of Table 3a, we interact this ratio with the borrower’ credit score, and …nd
that the e¤ect is stronger for borrowers with low or medium credit than for those with high
credit. This is consistent with the idea that liquidity constraints are less severe for high-credit
households because they have greater access to the capital market.


   We also include additional terms representing …nancial incentives to default in U1 , and report
the estimates in Speci…cations 2 and 3 of Table 3a. Speci…cation 4 is the most comprehensive
speci…cation, with the loan age, local demographics, MSA dummies, and year …xed e¤ects all
                                                           4
included as regressors. The estimates from Speci…cations 2– indicate that higher house price
                                       )
growth in the previous month (“Exp_Bwd” reduces the …nancial incentive to default, but that
the e¤ect is not large. The estimate from Speci…cation 4, for instance, implies that in markets
where housing prices have been appreciating at an annual rate 10% above the sample average,
                                                                   V
the hazard of default (for a borrower with an average value of log( )) is 2.30% lower than for
                                                                   L
an otherwise identical borrower in an average housing market.


   Besides the expected trend, expectations about price volatility also a¤ect default behavior,
but again the e¤ect is small.   When we include the volatility of housing prices over the past
twelve months (P ast V olatility) among the independent variables, along with its interaction
         V
with log( ), the uninteracted term has almost no e¤ect, while the interaction decreases the
         L
                                                                    V
propensity to default. Speci…cally, at the average level of log( ), an increase of 1.47 (one
                                                                    L
standard deviation) in the volatility measure is associated with a 3.06% lower hazard of default,
according to our results in Speci…cation 4. Therefore, our …ndings suggest that volatile home
price movements raise the option value of holding on to the mortgage, and that this e¤ect is
larger for those borrowers with higher net equity in the property. It is a bit unclear why the
e¤ect is larger for borrowers with higher net equity.    One possibility is that households with
higher net equity have lower risk aversion. This would be the case if risk aversion decreases with
wealth, because borrowers with higher net equity have greater housing wealth, by de…nition.

                                 V
   The estimated impact of log(     ) declines as we add Exp_Bwd and P ast V olatility to
                                 L
our model (Speci…cations 2 and 3), and declines further as we add MSA- and year …xed e¤ects


                                                24
                                                                             V
(Speci…cation 4). This is not surprising given the positive correlation among log(
                                                                                ), Exp_Bwd,
                                                                              L
and P ast V olatility. Adding MSA- and year …xed e¤ects also soaks up some of the variation in
home price changes. However, we still …nd that net equity in the property plays an important
role in default decisions. Speci…cation 4 in Table 5 shows that a one-standard-deviation increase
        V
in log( ) is associated with a 8.77% lower hazard of default even after we control for expectations
        L
about home price appreciation, expectations about house price volatility, and MSA- and year
…xed e¤ects.


   The estimates on the e¤ect of interest rates are somewhat weak but consistent with model
predictions.   A one-standard-deviation increase in the value of IR (a measure of how “over-
priced” contractual interest rates are, relative to the market rate) predicts a 0.19% greater
hazard compared to the sample average (Table 3a, Speci…cation 4). The small magnitude of
this e¤ect is likely due to the fact that high contractual interest rates increase both prepayment
and default and that prepayment is classi…ed under the category of no default in our baseline
speci…cations. As expected, borrowers with ARMs are also somewhat riskier. Conditional on
everything else being equal, they have an 8.9% higher hazard of default. Among ARM-holders,
default is also more likely when rate resets are imminent: adding an extra 12 months between
the present period and the next reset results in a lowering of the hazard of default by about
1.4%.


   Finally, the parameter estimates for Loan Age and (Loan Age)2 indicate that there is an
initial increase in the probability of default, but that after approximately the …rst three years,
older loans are much less likely to default, conditional on survival. This “hump-shaped”hazard
pro…le is consistent with the …ndings of other researchers (Gerardi, Shapiro, and Willen, 2008;
von Furstenberg, 1969). Part of this e¤ect is due to unobserved heterogeneity: loans that survive
are, by de…nition, more likely to be held by borrowers with a lower unobserved propensity to
default. However, the estimated e¤ect of loan age is only slightly weaker after controlling for
random e¤ects (Table 4), suggesting that the hazard of default for a given individual indeed
varies over the life of the loan.



                                                25
       We also re-run Speci…cations 1-4 using Exp_HM S instead of Exp_Bwd.                           The results,
reported in Table 3b, suggest that the relationship between default and expectations about
future house prices depends on how we measure expectations. In contrast to our earlier …nding
                                                         )
that higher price growth in the previous month (“Exp_Bwd” reduces the …nancial incentive
to default, the propensity to default is actually higher in markets where the user-cost approach
implies stronger house price appreciation.11 For a hypothetical borrower who is average in all
observable respects other than living in a market where the user-cost-based expectation of house
price appreciation is 10% above the sample average, the hazard of default is 0.91% higher than
would otherwise be the case (Table 3b, Speci…cation 4). It thus appears that borrower behavior
is more consistent with beliefs that are based on extrapolation, but not with beliefs imputed
from the price-to-rent ratio. Alternatively, it could be the case that price to rent ratios are not
particularly good measures of buyers’expectations or that housing and rental prices are related
by a more complicated mechanism than the one proposed by the standard user cost theory (see
equation (5)).


       The two expectation measures have a raw correlation of                     0:22.   We ought to see this
sort of negative correlation if market participants believe that housing price growth is mean-
reverting, in which case above-average growth in the recent past would lead to below-average
expectations that get capitalized into the price-to-rent ratio.                   But if this is the case, it is
unclear why— assuming that default and rent-or-buy decisions are optimal given expectations—
borrowers base their expectations on past trends, while the housing market as a whole anticipates
mean-reversion.12


       Table 4 adds random e¤ects to the model—                0i   and   0i   in (12)— in order to control for
  11
       Estimated coe¢ cients for other …nancial and credit quality variables are very similar in Tables 3a and 3b.
  12
       We also run the same speci…cations using a measure of expectation based on perfect foresight, an extreme form
                                                              s
of rational expectations. Speci…cally, we use the next period’ home price growth rate, Exp_F wd, as a measure
of borrowers’ expectations about future home price, and investigate the relationship between this expectation
measure and default behavior. The results are reported in Table A2. The results are very similar to those from
a speci…cation that use Exp_Bwd (Table 3a), which is not surprising given that there is a very high correlation
between home price growth rates in two adjacent months.



                                                          26
unobserved borrower heterogeneity.            In principle, we might be able to include …xed e¤ects in
our model.        However, given the number of observations in our sample, this does not appear
to be computationally feasible.           In our random e¤ects speci…cation, we see that the results
are very similar to those in Table 3, both qualitatively and quantitatively. As before, higher
net equity, higher expectations about future home prices (measured using Exp_Bwd), higher
volatility in home prices, and lower contractual rates all lead to a smaller hazard of default.
Similarly, we still …nd that variables representing higher credit quality and less severe liquidity
constraint predict a lower probability of default.             The only noticeable change compared to
Table 3 is that now low-credit borrowers do not appear any more sensitive to high payment-to-
income ratios than high-credit borrowers. The random e¤ects scale parameters in U1;it and U2;it
are signi…cantly di¤erent from zero, suggesting that there is a substantial degree of unobserved
borrower heterogeneity in‡uencing the …nancial incentives to default and the tightness of budget
constraint. We …nd it reassuring that most results carry over to the random e¤ects speci…cations
despite the large degree of unobserved borrower heterogeneity.


       Table 5, which reports estimates of the impact of a one-standard-deviation increase in the
independent variables on the probability of default, is informative in conveying the relative
signi…cance of each regressor in default decisions.             Alternatively, we could ask the following
question to determine the impact of each regressor: We know that 2006 vintage loans have
much worse performance than 2004 vintage loans. The empirical probability of default within
the …rst 12 months is 1.50% and 8.28% for mortgages originated in 2004 and 2006, respectively.
Then, how much of this increase in defaults could be explained by the observed change in
each regressor?         Table 6 provides an answer to this question.             Table 6 reports the mean
values of each regressor among 2004 vintage loans and 2006 vintage loans (the …rst and second
columns), the change in the mean for each regressor (the third column), and multiplies it by the
marginal e¤ect to obtain the contribution of each regressor to the high default probability of
2006 vintage loans compared to 2004 vintage loans (the fourth column).13 Table 6 con…rms our
  13
       We use the marginal e¤ects from Speci…cation 3 in Table 3a, not Speci…cation 4. Since our objective is to
compare loans of di¤erent vintages, it makes more sense to use a speci…cation that does not include year …xed
e¤ects.


                                                        27
prior …ndings: The biggest contributors to the high probability of default among 2006 vintage
loans are declining home prices and deteriorations in the credit quality and liquidity conditions
of mortgage borrowers. Our results indicate that declining home equity led to a 5.26% higher
hazard of default for 2006 vintage loans compared to 2004 ones.                    A decrease in house price
                                   ect
volatility, which could largely re‡ the slowdown in home price appreciation, made the holders
of 2006 vintage loans 5.65% more likely to default than otherwise identical holders of 2004
vintage loans. Lower credit quality, as measured by FICO scores, is responsible for an almost
50% larger hazard of default among 2006 vintage mortgage holders. Finally, low downpayments
and high payment-to-income ratios among low- to medium-credit borrowers are also signi…cant
contributors to the high incidence of defaults among 2006 vintage loans.


       Our …ndings can be summarized as follows:


       (1) The estimation results provide evidence for each of the hypothesized factors discussed in
Section 2 in explaining default by subprime mortgage borrowers.


       (2) Declining home prices are an important driver of subprime mortgage default.                        For
a borrower who purchased a home a year earlier with a 30-year …xed-rate mortgage and no
downpayment, a 20% decline in home price makes the borrower 23.2% more likely to default
than an otherwise identical borrower whose home price remained stable.14


       (3) Borrower and loan characteristics a¤ecting borrowers’ ability to pay are as empirically
important in predicting default as declining house prices, as evidenced by the magnitudes of the
marginal e¤ects in Table 5.15 Our results suggest that the increase in defaults in recent years is
partly linked to changes over time in the composition of mortgage recipients. Higher numbers of
  14
       Based on Speci…cation 1 in Table 3a. If we use the marginal e¤ects from Speci…cation 4, a 20% decline in
home price would make the borrower 4.26% more likely to default, which is much smaller than 23.2%, but still
economically signi…cant.
  15
     We will also compare loglikelihoods from two speci…cations: a univariate probit model with …nancial covariates
only and a univariate probit model with measures of credit quality and liquidity constraints only (corresponding to
Speci…cations 1 and 2 of Table 7). The comparison, which we will discuss in the next section, provides additional
support for this claim that liquidity constraints are as important as declining home prices in explaining default.


                                                        28
borrowers with little or no documentation and low FICO scores, or who only make small down-
payments, contributed to the increase in foreclosures in the subprime mortgage market. The
increasing prevalence of adjustable-rate mortgages also contributed to rising foreclosures, be-
cause the monthly payments for adjustable-rate mortgages come with periodic— and sometimes
very large— adjustments, forcing liquidity-constrained borrowers to default.


   (4) Other option-value-based indicators of whether it makes …nancial sense to default—
expected housing price appreciation, home price volatility, the gap between the market rate and
contract interest rate, and an expectation of future rate resets for ARMs— have e¤ects that are
consistent with economic theory. However, they do not appear to be quantitatively important
factors in default decisions. These results therefore suggest, albeit not strongly, that subprime
mortgage borrowers are not very forward-looking in making their default decisions. The main
drivers behind subprime mortgage holders’default decisions are realized home price movements
up to the time of decision-making and whether the budget constraint is binding in this period,
while the option value of waiting— due to expectations about future home price movements or
interest rate changes— does not seem to matter as much.



4.1   Univariate Probit Results


As a check for robustness, we also estimate a univariate probit model. Similar to the baseline
speci…cation, the outcome is default or no default in a given month. Here, however, we assume
that both the …nancial incentive to default and borrowers’ liquidity constraints enter into an
equation determining a single latent utility. Table 7 reports the estimates.


   Speci…cation 1 uses only the covariates included in the …nancial incentives equation (eq1)
for Speci…cation 3 of the bivariate probit (reported under Table 3a). Similarly, Speci…cation 2
includes only the covariates related to the liquidity equation (eq2) of the bivariate probit model
(except for loan age). Each of these speci…cations is equivalent to the bivariate probit model with
the constant term for one or the other equation constrained to equal in…nity and the covariance


                                                29
of the errors constrained to equal zero.     The model …t (as measured by the loglikelihood or
pseudo-R2 ) is very similar for Speci…cation 1 and Speci…cation 2, providing additional support
for the notion that illiquidity is an equally important driver behind default as …nancial incentives.


   From Speci…cations 3-4, we see that the qualitative results from the univariate probit model
are generally similar to those from the bivariate probit model, with a few exceptions. Similar
to before, default is more likely if the borrower has low net equity in the house. Moreover, the
probability of default also declines with Exp_Bwd, although the e¤ect is less robust.          How-
ever, unlike the case with the bivariate probit, the user-cost-based expectations have the same
qualitative e¤ect as the backward-looking measure: higher expectations of future home prices
are associated with less default, whether based on extrapolation or measured from user costs.
The implications are the same as before for the measures related to interest rates. Default is
less likely when the market interest rate is higher than the contract rate, because default entails
losing access to the discounted rate. Likewise, for ARMs, default is more likely as the next rate
reset gets closer in time.


   Parameter estimates for the measures that represent the liquidity constraint and the overall
credit quality of the borrowers also con…rm prior results. A high payment-to-income ratio, a
low FICO score, a low documentation level and a high loan-to-value ratio at origination all lead
to increased probability of default. The impact of a high payment-to-income ratio on default
is also larger for borrowers with low-credit quality than for borrowers with high-credit quality.


   Speci…cation 5 of Table 7 reports estimation results when we add loan-level random e¤ects
to the model. We see that the results are robust to the inclusion of loan-level random e¤ects, as
was the case with the bivariate probit. Finally, to partially address the concern that continued
payments and prepayments are rather distinct events, we re-run Speci…cation 4 after excluding
prepaid cases from the category of no default. The results are reported under Speci…cation 6
of Table 7, which shows that all of the coe¢ cients are qualitatively stable.


   Although the results are qualitatively very similar between the bivariate and univariate probit



                                                 30
                                                 V           V
models, the magnitudes of the coe¢ cients for log( ) and log( ) Exp_Bwd change a great
                                                 L            L
deal: their magnitudes are almost …ve times larger in the bivariate probit results than in the
univariate probit. On the other hand, the magnitudes of the coe¢ cients for the liquidity and
credit quality measures are similar between the two models. The …nancial incentives become
directly relevant for default only when the liquidity constraint does not bind.                    Because the
univariate probit model does not take this dependency into account, and because many defaults
are driven by the liquidity constraints of subprime mortgage borrowers, the univariate probit
model underestimates the e¤ects of …nancial incentives on default. Our bivariate probit model
with partial observability does not su¤er from this misspeci…cation, giving us better parameter
estimates for the …nancial incentive variables. This provides justi…cation for our baseline model.



4.2       Competing Hazards Model


As an alternative to the partial observability model, we also estimate a “competing hazards”
model, in which a mortgage can be terminated by either default or prepayment.                           Similar to
the univariate probit model, this model does not distinguish between covariates that a¤ect the
…nancial incentive to default and covariates that a¤ect household budget constraints: all of the
relevant observable characteristics simply a¤ect outcomes by shifting the hazards of default and
prepayment. The advantages of the hazards model, as compared to the bivariate probit baseline
model, include the fact that it treats prepayment and regularly scheduled payment as separate
outcomes and that we can allow default and prepayment to be correlated due to unobservables.
Moreover, the hazards model may be conceptually more appealing than the period-by-period
probit model, in the following sense. We essentially observe only one outcome for each loan— the
point in time when the loan defaults (if ever)— which the hazards model addresses by treating
the time to default (or prepayment) as the dependent variable. On the other hand, the period-
by-period probit model treats the status of the loan in each month as a separate observation,
arti…cially de‡ating the standard errors.16 Of course, the disadvantage of the hazards model,
  16
       The clustered standard errors that we report partially address this problem, but not entirely.    To further
investigate how treating each period as a separate observation might a¤ect our standard errors, we re-run var-


                                                         31
compared to the bivariate probit model, is the misspeci…cation resulting from treating default
as being determined by only a single equation instead of two equations.


    For household i, denote the time of default as Tdi and time of prepayment as Tpi , where Tdi
and Tpi are discrete random variables (Obviously, at least one of these stopping times must be
censored). The probabilities of survival past some time t in the future are:

                                                                         P
                                                                         t
                                          P [Tdi > t] = exp                    hd (k)
                                                                         k=1                                        (14)
                                                                         Pt
                                          P [Tpi > t] = exp                    hp (k)
                                                                         k=1



Suppose the instantaneous hazards of default and prepayment, hd (t) and hp (t), follow a propor-
tional hazards model as follows:

                                        hdi (t) = exp(                   0X
                                                             d (t)   +   d it    +      di )
                                                                                                                    (15)
                                        hpi (t) = exp(                   0X
                                                             p (t)   +   p it    +      pi )

In other words, the hazards depend on a time-dependent “baseline” hazard common across all
borrowers,     d (t)   and   p (t);   on (potentially) time-varying covariates, Xit ; and on unobserved,
borrower-speci…c random e¤ects             di   and   pi .     Changing the observed covariates Xit results in
a new hazard function that is proportional to the baseline hazard function, hence the name
“proportional hazards.”


    As is well known (Lunn and McNeil, 1995), if the unobserved heterogeneity terms                          di   and   pi

are independent, the two risks are independent conditional on observables, so separate estimation
of the two hazard functions yields consistent estimates. When estimating the hazard of default,
we would simply treat loans that end in prepayment as censored observations, similar to loans
that are censored by the end of the sample period. Similarly, when estimating the hazard of
prepayment, either a default or the end of the sample period would result in censoring. When
ious speci…cations of the univariate probit model using data aggregated by quarter, instead of using monthly
observations (and still clustering our standard errors).             The results are reported in Table A1.   Using quar-
terly observations increases our standard errors only very slightly, indicating that clustering largely mitigates the
problem of understated standard errors.



                                                              32
 di   and       pi   are not independent, estimation becomes more involved, but we can still estimate the
parameters using maximum likelihood, as in Deng, Quigley, and van Order (2000) and McCall
(1996).         The likelihood function and estimation details for the dependent competing hazards
model are provided in Appendix B.


      In Speci…cation 1 of Table 8, we report the estimation results for a particular case of inde-
pendent hazards.                Speci…cally, we assume that the hazards only depend on observables (i.e.,

 di   =    pi   = 0), while making no parametric assumptions about the underlying baseline hazards,

 d (t)    and        p (t).   This speci…cation is simply the standard Cox proportional hazards model (Cox,
1972; Cox and Oakes, 1984), and we can estimate the coe¢ cients                d   and   p   by minimizing the
“partial loglikelihood,” while essentially netting out the baseline hazards.


      We also estimate a speci…cation that allows for unobserved correlation in the hazards of
default and prepayment, following Deng, Quigley, and van Order (2000).                         Speci…cally, we
assume that there are two types of borrowers, where
                                     8
                                     < ( ;
                                        d1 p1 ) with probability
                       ( di ; pi ) =                                                                      (16)
                                     : ( ;
                                        d2 p2 ) with probability 1

Results for the model with correlated unobserved hazards are in Speci…cations 2-4 of Table 8.


      All of Speci…cations 1-4 generate implications for default behavior that are similar to what
we see in the bivariate probit and univariate probit models.                We …nd that higher net equity
decreases the probability of default, while its impact on the probability of prepayment is unclear.
The greater the contractual rate relative to the market interest rate, the more likely the borrower
is to default and prepay. In Speci…cation 2, the impact of IR on default is stronger than its
                                                                    4),
e¤ect on prepayment, but as we add more regressors (Speci…cations 3– the impact of IR
on prepayment is stronger than its e¤ect on default.                Borrowers are less likely to default or
prepay if they are farther away from the next rate reset for adjustable-rate mortgages. Again
not surprisingly, …xed-rate mortgages are less likely to default or prepay than adjustable-rate
mortgages.              The estimated hazard ratio indicates that adjustable-rate mortgages are about
three- to four times more likely to default and about three times more likely to prepay than

                                                           33
…xed-rate mortgages. The measures of liquidity constraints display similar patterns as before:
borrowers with low documentation, low FICO scores, high loan-to-value ratios at origination,
and high payment-to-income ratios are more likely to default.


    Finally, we …nd that there is a high degree of unobserved heterogeneity in both default
and prepayment risk, with the unobserved heterogeneity being greater for default. This result
contrasts with the …ndings of Deng, Quigley, and van Order (2000) who …nd substantial and
statistically signi…cant unobserved heterogeneity in exercising the prepayment option but not
in exercising the default option. The di¤erence between their …nding and ours may be due to
the di¤erent pools of borrowers in our respective data sets: while their sample is con…ned to
prime mortgage borrowers, who have a low probability of default in any case, we study subprime
mortgage borrowers, for whom the default risk is much higher. Thus, it makes intuitive sense
that our sample exhibits a much greater degree of unobserved heterogeneity in default behavior.




5    Conclusion


In this paper, we estimate a model of optimal default by subprime mortgage borrowers. Our
model nests four possible explanations for the recent increase in mortgage defaults: falling
home prices, lower expectations about future home prices, increases in borrowers’ contractual
interest rates relative to market rates, and borrowers’inability to pay due to a lack of income
or credit. The …rst three factors a¤ect borrowers’ …nancial gains from default, while the last
factor represents the possibility that liquidity constraints may force borrowers to default even
when defaulting is against their …nancial best interest. We account for the fact that long-run
…nancial incentives are relevant to default decisions only if the liquidity constraint does not bind,
and vice versa, thereby addressing a key misspeci…cation in previous studies.        The structural
equations of this model can be represented as a bivariate probit with partial observability, as
formulated by Poirier (1980).


    We estimate our model using unique data from LoanPerformance that track each loan over

                                                 34
the course of its life, and …nd evidence for each of the hypothesized factors in explaining default
by subprime mortgage borrowers.       In particular, our results suggest that borrower and loan
characteristics that a¤ect borrowers’ ability to pay are as important in predicting default as
the fundamental determinants of whether it makes …nancial sense to default. Declining home
prices are indeed an important driver behind the recent surge in defaults, but for the particular
segment of homeowners represented in our data, liquidity constraints are an equally important
factor. This …nding may provide some guidance on appropriate policy responses to the current
housing market turmoil. For example, the empirical importance of liquidity constraints suggests
that taking measures toward relaxing these constraints, such as providing income support to
struggling mortgage borrowers, could signi…cantly decrease the likelihood of default.


   The framework of this paper is essentially static: To capture the dynamic nature of borrow-
ers’ default decisions, we simply account for the reduced-form e¤ects of various option values
associated with holding a mortgage. In future work, we shall examine how our results change
when we explicitly model borrowers’ default decisions as an optimal stopping problem.         The
…ndings in this paper will be useful in informing us on whether to include credit constraints and
on how best to model price expectations in the fully dynamic model.




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                                             38
Appendix A


Imputation of counterfactual re…nancing interest rates for individual households


       We assume that in each time period t, household i is able to re…nance its mortgage at rate
 m
rit , the market rate adjusted by a household-speci…c risk premium. To impute this hypothetical
                                                                                      m
rate from the data, we make the following assumptions about the relationship between rit and
 c
ri0 , the initial contractual rate owed during the …rst month of the household’ actual loan. Let
                                                                               s
ti0 denote the time period corresponding to the initial month of the actual loan.                        j = 1:::J
index observable covariates other than time, with the covariates zij denoting the observable
                                                                                         m
household and loan characteristics upon which actual interest rates are determined, and zij
denoting the covariates that determine the re…nancing rate. Then,
                                                           J
                                                           X
                                        c
                                       ri0 = f (ti0 ) +          gj (ti0 )zij +       i
                                                           j=1
                                                            J                                                   (17)
                                                           X
                                          m                             m
                                         rit   = f (t) +         gj (t)zij   +    i
                                                           j=1

Crucially, among the covariates zit we include controls for the type of mortgage (ARM, hybrid,
…xed-rate, etc.) held by household i.17 We assume that zij = zij for all characteristics j except
                                                        m


for dummies related to the mortgage type: to preserve comparability across households, we
assume that all consumers re…nance into …xed-rate mortgages. By restricting the error,                          i,   to
                                                              s
be equal across equations, we are assuming that the household’ risk premium is constant over
time.


       The function f (t) captures the time-varying “baseline” market interest rate, and the func-
tions gj (t) capture the time-varying premia on characteristics j = 1 : : : J. For estimation, we
approximate f (t) and gj (t) by “natural”cubic spline functions. A natural cubic spline function
f (t) consists of piecewise cubic polynomials fn (t); n = 0; : : : ; N                    1 passing through nodes at
t0 ; t1 ; : : : ; tN , with the restriction that f (t) be twice-continuously di¤erentiable at each node and
  17
       Thus, we are also assuming that the risk premium does not vary with mortgage type. In principle, we could
allow the error to be nonadditive and to interact with the mortgage type.


                                                           39
with the boundary conditions f 00 (t0 ) = f 00 (tN ) = 0. The boundary conditions— which impose
local linearity at the furthest endpoints— mitigate the tendency for cubic polynomials to take
on extreme values near the endpoints.


   We include the following variables among the covariates zit :



     FICO score

     “Low documentation” and “No documentation” dummies

     Dummy for …rst liens

     Dummies for mortgage type.       We categorize mortgages as …xed-rate mortgages, ARMs
     that have a …rst reset less than a year after origination (which tend to have much lower
     initial contractual rates), and other types of ARMs.

     The total loan-to-value ratio (for all liens) at origination

     The “front-end” debt-to-income ratio: the ratio of monthly housing-related principal and
     interest payments, taxes, and insurance to monthly income.

     The “back-end” debt-to-income ratio: similar to the front-end ratio, but also including
     in the numerator all payments for non–housing-related debts (e.g., car loans, credit card
     debt), as a percentage of monthly income.


                         m
   Note that by setting zij = zij for all characteristics j other than the mortgage type, we
abstract from the fact that re…nancing generally alters the debt-to-income ratio. Moreover, the
debt-to-income and loan-to-value ratios are endogenous, because the amount of debt borrowers
are willing to take on is presumably correlated with the interest rates they are able to obtain.
We ignore these issues, because our goal is not to obtain unbiased structural estimates for
the e¤ect of each covariate on interest rates, but merely to obtain adequate estimates for the
residual household risk premium. The operative assumption is that unobservable determinants
of borrowers’interest rates do not change over time.

                                                40
Appendix B


Estimation details for dependent competing risks model


   As a robustness check, we estimate a model of dependent competing risks. We assume that
at time t, borrower i is described by (potentially) time-dependent observable characteristics Xit
as well as a pair of unobservable characteristics (               di ; pi ),      which shift the hazards of default and
prepayment. We follow Han and Hausman (1990), Deng, Quigley, and van Order (2000), and
McCall (1996) in writing the likelihood function of this model.


   Denote the time to default as Td and time to prepayment as Tp , both being discrete random
variables. For economy of notation, we omit the subscript for individual i. The joint survival
function, conditional on observable characteristics X and unobservable type, is then as follows:

                                                                  P
                                                                  td
                                                                                                     0X
                    S(td ; tp j X;   d;      p ) = exp (               exp(       d (k)   +          d k   +   d)
                                                                k=1
                                                                  tp                                                         (18)
                                                                 P                                   0X
                                                                        exp(       p (k) +           p k   +   p ))
                                                                  k=1



We approximate the baseline hazards              d (t)    and      p (t)    using a third-order polynomial function
of time (t).
                                                                              2               3
                                     d (t)   =   0d   +    1d t   +    2d t       +    3d t
                                                                                                                             (19)
                                                                           2                3
                                     p (t) =     0p +      1p t +      2p t +          3p t

For the system to be identi…ed, we normalize                  0d      and         0p   to 0 (because these parameters are
not separately identi…ed from the population means of                         d   and         p ).    As a practical matter, using
a polynomial approximation does not seem to drive the results, in the sense that re-estimating
Speci…cation 1 of Table 8 but using nonparametric baseline hazard functions yields essentially
the same estimates for the remaining parameters.


   Default and prepayment are competing risks, so we only observe the duration associated
with the …rst terminating event. De…ne Fd (k j X;                      d; p)       as the probability that the mortgage
is terminated by default in period k, Fp (k j X;                      d; p)       as the probability of termination by


                                                           41
prepayment in period k, and Fc (k j X;                 d; p)   as the probability of censoring at period k by the
end of the sample. Following Deng, Quigley, and van Order (2000), and McCall (1996), we can
write the probabilities as follows:

   Fd (k j X;   d; p)   = S(k; k j X;          d; p)     S(k + 1; k j X;        d; p)   0:5(S(k; k j X;      d; p)

       +S(k + 1; k + 1 j X;            d; p)     S(k + 1; k j X;        d; p)     S(k; k + 1 j X;      d ; p ))

   Fp (k j X;   d; p)   = S(k; k j X;          d; p)     S(k; k + 1 j X;        d; p)   0:5(S(k; k j X;      d; p)
                                                                                                                       (20)

       +S(k + 1; k + 1 j X;            d; p)     S(k + 1; k j X;        d; p)     S(k; k + 1 j X;      d ; p ))

   Fc (k j X;   d; p)   = S(k; k j X;          d; p)


The term 0:5(S(k; k j        d ; p ) + S(k + 1; k + 1     j    d; p)    S(k + 1; k j    d; p)   S(k; k + 1 j      d ; p ))   is
an adjustment that is necessary because the durations are discrete random variables. Because
we do not observe       d   or   p   in the data, we must form the likelihood function using unconditional
probabilities, obtained by mixing over the type distribution:

                    Fd (k j X) = Fd (k j X;              d1 ; p1 )   + (1   )Fd (k j X;    d2 ; p2 )

                    Fp (k j X) = Fp (k j X;              d1 ; p1 )   + (1   )Fp (k j X;    d2 ; p2 )
                                                                                                                       (21)

                    Fc (k j X) = Fc (k j X;              d1 ; p1 )   + (1   )Fc (k j X;    d2 ; p2 )



The log likelihood function of this model is then given by:

                   P
                   N
         log L =         [(yi = d) log(Fd (Ki )) + (yi = p) log(Fp (Ki )) + (yi = c) log(Fc (Ki ))]                    (22)
                   i=1

                                             s
where (yi = j) is equal to one if borrower i’ mortgage ends by termination mode j, and equals
zero otherwise.




                                                               42
                               Table 1: Variable Definitions


Variable          Definition
                  = 1 if loan i does not default in period t, = 0 if defaults (in foreclosure or
NDit
                  Real Estate Owned)
V/L               Market value of the property / Outstanding principal balance
Exp_Bwd                         s
                  Previous month’ home price growth rate, multiplied by 12
                  Expected home price appreciation based on user costs. This is equal to
                  Imputed Rent / Actual Rent reported in Himmelberg, Mayer, and Sinai
Exp_HMS
                  (2005). This measure is normalized to an MSA-specific 24-yr average.
                  See the text for a detailed description
Exp_Fwd                     s
                  Next month’ home price growth rate, multiplied by 12
Past Volatility   Standard deviation of home prices for the past 12 months, divided by 10
                  Difference in the present values of the payment stream at the mortgage
IR
                  note rate and the current interest rate. Described in the text
MR                The number of months until the next reset for an ARM
FRM               = 1 if the mortgage is a fixed rate mortgage, = 0 otherwise
Low Doc           = 1 if the loan was done with no or low documentation, = 0 otherwise
                  FICO score, a credit score developed by Fair Isaac & Co.        Scores range
FICO
                  between 300 and 850, with higher scores indicating higher credit quality.
Low FICO          = 1 if FICO is less than 600, = 0 otherwise
Medium FICO       = 1 if FICO is between 600 and 700, = 0 otherwise
High FICO         = 1 if FICO is above 700, = 0 otherwise
                  Loan to value at origination. For second (third, fourth, … ) lien loans, this
Original LTV
                  will be the combined 1st and 2nd lien LTV
Unemployment      Monthly unemployment rate at the country level from BLS
                  Monthly Payments / Income.        We assume that Income stays constant
                  over time, and approximate it by the scheduled monthly payment divided
                  by the “front-end” debt-to-income ratio, both reported at the time of
PI ratio
                  origination.    (The front-end ratio measures housing-related principal
                  and interest payments, taxes, and insurance as a percentage of monthly
                  income). Monthly Payments may vary over time
Loan Age          The age of the loan in months
Local             Log population, mobility, per-capita income, % college educated, %
demographics      black, % Hispanic
                                            43
               Table 2: Summary Statistics for Estimation Sample

                          Prepaid        Defaulted       Censored or          All loans
                                                       paid to maturity

                                      Loan-level variables

                            Mean           Mean               Mean         Mean    Std dev
FRM                         .246            .154               .321        .262     .440
Original L=V                .780            .806               .782        .783     .135
(FICO score)/100            6.270          5.955              6.474        6.307    .709
Low FICO dummy              .372            .532               .244        .345     .475
Med. FICO dummy             .458            .407               .530        .477     .500
High FICO dummy             .170            .061               .226        .178     .382
Low documentation           .400            .385               .439        .412     .492

No. obs.                   67,056          20,060            111,179           198,295
                    Time-dependent loan-level variables over all periods

log(V =L)                   .519            .361               .537        .512     .460
Mo. payment/income          .306            .328               .272        .294     .120
Loan age in months          14.98          15.48              19.66        16.94    12.23
Mo. until next reset        14.64          13.81              17.06        15.41    13.45
Eg from user cost           1.084          1.081              1.167        1.118    .189
Eg from recent past         .103            .046              .0160        .062     .119
Eg from near future         .101            .043               .006        .057     .123
Recent volatility of V      1.827          1.046              1.401        1.581    1.474
IR                          .023            .030              -.0141       .008     .097

No. obs.                  1,827,840       337,612            1,497,954        3,663,406




                                               44
            Table 2: Summary Statistics for Estimation Sample, continued

                                Prepaid      Defaulted            Censored or            All loans
                                                             paid to maturity

                Time-dependent loan-level variables, as last observed for each loan

                                 Mean           Mean                Mean             Mean     Std dev
    log(V =L)                     .572           .365                .466            .515       .504
    Mo. payment/income            .312           .339                .290            .307       .129
    Loan age in months           19.02          19.36                25.42           21.22      13.27
    Mo. until next reset         11.71          11.18                11.47           11.57      11.16
    Exp_HM S                     1.141          1.117                1.23            1.168      .193
    Exp_Bwd                       .087           .020                -.120           .010       .144
    Exp_F wd                      .081           .013                -.183           -.015      .166
    Recent volatility of V       1.919          1.053                1.302           1.623      1.381
    IR                            .021           .035                .025            .024       .107

    No. obs.                     67,056        20,060               111,179               198,295
                              Zip-code–level demographics for each loan

    Unemployment (%)             5.163          5.102                4.729           5.010      1.359
    log(Population)              10.38          10.36                10.33           10.36       .70
    Mobility                      .006          -.259                .067           -.0002      2.312
    Per-cap. inc. ($K )          22.61          20.88                22.34           22.34      9.68
    Pct. college grad.            .352           .328                .348            .348       .130
    Pct. black                    .178           .270                .190            .191       .265
    Pct. Hispanic                 .224           .173                .210            .214       .227

    No. obs.                     67,056        20,060               111,179               198,295




     Conditional on being an ARM. “Mobility”is the year in which the average resident moved into her current

house, minus the nationwide average (1994.697, or August 1994).



                                                     45
              Table 3a: Bivariate Probit with Partial Observability using Exp_Bwd


                     Specification 1      Specification 2       Specification 3      Specification 4

                   Eqn 1       Eqn 2     Eqn 1       Eqn 2     Eqn 1      Eqn 2     Eqn 1      Eqn 2

                   1.939                 0.961                 0.937                1.008
       V
  Log(        )    (0.109)              (0.123)               (0.075)              (0.102)
       L
                   0.003                0.0004                0.0004               0.0001

        V                                6.796                 5.324                5.512
 log(        )×
        L                               (0.648)               (0.284)              (0.511)

  Exp_Bwd                                0.003                 0.002                0.001

                                         -0.058                -0.036               -0.065

Past Volatility                         (0.012)               (0.014)              (0.016)
                                        -0.00002              -0.00001             -0.00001

        V                                0.730                 0.642                0.626
 log(        )×
        L                               (0.059)               (0.069)              (0.087)
Past Volatility                         0.0003                0.0003               0.0001

                                                               -0.251               -0.245

        IR                                                    (0.050)              (0.069)

                                                              -0.0001              -0.00004

                                                               0.014                0.014
     MR                                                       (0.0004)             (0.0005)

                                                              6.65e-06             2.66e-06

                                                               0.757                1.167
    FRM                                                       (0.0311)             (0.156)
                                                              0.0002               0.0002

                               -0.187               -0.182                -0.191               -0.210

  Low Doc                     (0.007)               (0.008)              (0.008)               (0.008)

                               -0.001               -0.001                -0.001               -0.001

                               0.394                 0.431                0.431                 0.380
  FICO/100                    (0.007)               (0.017)              (0.016)               (0.015)
                               0.003                 0.002                0.002                 0.002

                               -0.462               -0.440                -0.459               -0.505
Original LTV                  (0.026)               (0.028)              (0.031)               (0.029)
                               -0.003               -0.003                -0.002               -0.003



                                                   46
                                                      Table 3a Continued

                        Specification 1             Specification 2             Specification 3        Specification 4

                       Eqn 1        Eqn 2          Eqn 1          Eqn 2        Eqn 1         Eqn 2    Eqn 1         Eqn 2

                                    -0.026                       -0.028                     -0.019               -0.028

Unemployment                       (0.002)                      (0.002)                     (0.002)              (0.004)
                                   -0.0002                      -0.0001                     -0.0001              -0.0001

                                    -0.589

    PI ratio                       (0.081)
                                    -0.004

                                                                 -0.563                     -0.536               -0.522
   PI ratio×
                                                                (0.097)                     (0.093)              (0.087)
  Low FICO
                                                                 -0.003                     -0.003               -0.003

                                                                 -0.721                     -0.691               -0.678
   PI ratio×
                                                                (0.076)                     (0.075)              (0.065)
Medium FICO
                                                                 -0.004                     -0.004               -0.004

                                                                 -0.361                     -0.313               -0.359
   PI ratio×
                                                                (0.106)                     (0.088)              (0.096)
  High FICO
                                                                 -0.002                     -0.001               -0.002

                                                                                            -0.029               -0.029
   Loan Age                                                                                 (0.001)             (0.0009)
                                                                                            -0.0001              -0.0002

                                                                                             0.039                  0.040
Loan Age2/100                                                                               (0.002)              (0.001)
                                                                                            0.0002               0.0002

     Local
                                                                                                                    Yes
Demographics

MSA Dummies                                                                                                         Yes

Year Dummies                                                                                                        Yes

   Corr. b/w                 -0.186                      -0.162                      -0.598                -0.494

   ε1 and ε2                 (0.032)                     (0.038)                     (0.037)              (0.031)

    No. Obs                 3649405                     3646195                     3444334               3390424

Log Likelihood            -118289.46                  -117308.79                   -106896.56             -104551



Dependent Variable = No Default. No Default includes both continued payments and prepayments.

No random effects. Contents of each cell: estimated coefficient, standard error, marginal effects.

Standard errors are clustered by loan.                         47
             Table 3b: Bivariate Probit with Partial Observability using Exp_HMS


                    Specification 1      Specification 2       Specification 3      Specification 4

                   Eqn 1      Eqn 2     Eqn 1       Eqn 2     Eqn 1      Eqn 2     Eqn 1       Eqn 2

                   1.939                2.813                 2.300                1.775
       V
  Log(        )   (0.109)              (0.171)               (0.152)              (0.303)
       L
                   0.003                0.001                 0.001               0.0007

         V                              -1.934                -1.386               -0.928
  log(       )×
         L                             (0.104)               (0.094)              (0.202)

 Exp_HMS                                -0.001               -0.0009              -0.0004

                                        -0.061                -0.044               -0.053

Past Volatility                        (0.014)               (0.015)              (0.020)
                                       -0.00004              -0.00003             -0.00002

         V                              1.086                 0.869                0.712
  log(       )×
         L                             (0.073)               (0.072)              (0.105)
Past Volatility                        0.0007                0.0006               0.0003

                                                              -0.248               -0.325

      IR                                                     (0.052)              (0.075)

                                                             -0.0001              -0.0001

                                                              0.014                0.014
     MR                                                      (0.0004)             (0.0005)

                                                             0.00001              6.47e-06

                                                              0.735                1.114
    FRM                                                      (0.029)              (0.140)
                                                             0.0004               0.0004

                              -0.187               -0.188                -0.196               -0.214

  Low Doc                    (0.007)               (0.008)              (0.009)               (0.008)

                              -0.001               -0.001                -0.001               -0.001

                              0.394                 0.435                0.434                 0.384
  FICO/100                   (0.007)               (0.017)              (0.017)               (0.016)
                              0.003                 0.002                0.002                 0.002

                              -0.462               -0.441                -0.460               -0.521
Original LTV                 (0.026)               (0.029)              (0.033)               (0.031)
                              -0.003               -0.002                -0.002               -0.003



                                                  48
                                                      Table 3b Continued

                        Specification 1             Specification 2              Specification 3       Specification 4

                       Eqn 1        Eqn 2          Eqn 1          Eqn 2        Eqn 1         Eqn 2    Eqn 1         Eqn 2

                                    -0.026                      -0.022                      -0.014                  -0.031

Unemployment                       (0.002)                      (0.002)                     (0.003)              (0.004)
                                   -0.0002                      -0.0001                    -0.00008              -0.0002

                                    -0.589

    PI ratio                       (0.081)
                                    -0.004

                                                                -0.561                      -0.531                  -0.529
   PI ratio×
                                                                (0.102)                     (0.097)              (0.093)
  Low FICO
                                                                -0.003                      -0.003                  -0.003

                                                                -0.734                      -0.698                  -0.693
   PI ratio×
                                                                (0.084)                     (0.083)              (0.072)
Medium FICO
                                                                -0.004                      -0.004                  -0.004

                                                                -0.354                      -0.294                  -0.351
   PI ratio×
                                                                (0.109)                     (0.090)              (0.101)
  High FICO
                                                                -0.002                      -0.001                  -0.002

                                                                                            -0.030                  -0.029
   Loan Age                                                                                 (0.001)              (0.001)
                                                                                            -0.0001              -0.0001

                                                                                             0.039                  0.041
Loan Age2/100                                                                               (0.002)              (0.002)
                                                                                            0.0002                  0.0002

     Local
                                                                                                                     Yes
Demographics

MSA Dummies                                                                                                          Yes

Year Dummies                                                                                                         Yes

   Corr. b/w                 -0.186                      -0.246                        -0.554              -0.473
    ε1 and ε2                (0.032)                     (0.049)                     (0.031)              (0.029)

    No. Obs                 3649405                     3508428                     3314294               3269190

Log Likelihood            -118289.46                  -113156.95                   -103032.91           -100989.96



Dependent Variable = No Default. No Default includes both continued payments and prepayments.

No random effects. Contents of each cell: estimated coefficient, standard error, marginal effects.

Standard errors are clustered by loan.                         49
           Table 4: Bivariate Probit with Partial Observability (Random Effects)


                   Specification 1     Specification 2      Specification 3     Specification 4
                  Eqn 1     Eqn 2     Eqn 1      Eqn 2     Eqn 1      Eqn 2    Eqn 1     Eqn 2

          V       1.747               1.337                1.546               1.635
   log(     )
          L       (0.123)             (0.091)              (0.089)             (0.141)

        V
 log(     )×                          5.533                5.356               4.805
        L
                                      (0.462)              (0.443)             (0.710)
  Exp_Bwd
                                      0.042                0.043               0.030
Past Volatility
                                      (0.008)              (0.006)             (0.008)
                                                           -0.263              -0.233
      IR
                                                           (0.106)             (0.153)
                                                           0.013               0.014
     MR
                                                           (0.001)             (0.002)
                                                           0.646               0.892
    FRM
                                                           (0.046)             (0.129)
                            -0.205               -0.205              -0.225              -0.242
  Low Doc
                            (0.015)              (0.016)             (0.018)             (0.016)
                            0.406                0.493                0.539               0.459
    FICO
                            (0.015)              (0.022)             (0.023)             (0.021)
                            -0.429               -0.379              -0.391              -0.450
Original LTV
                            (0.061)              (0.059)             (0.062)             (0.061)
                            -0.025               -0.027              -0.016              -0.018
Unemployment
                            (0.004)              (0.005)             (0.005)             (0.008)
                            -0.710
   PI ratio
                            (0.021)
  PI ratio×                                      -0.627              -0.633              -0.593
 Low FICO                                        (0.025)             (0.029)             (0.031)
  PI ratio×                                      -0.905              -0.966              -0.882
Medium FICO                                      (0.062)             (0.066)             (0.063)
  PI ratio×                                      -0.792              -0.869              -0.912
 High FICO                                       (0.178)             (0.175)             (0.141)




                                                50
                                               Table 4 Continued

                      Specification 1         Specification 2         Specification 3         Specification 4
                    Eqn 1       Eqn 2       Eqn 1        Eqn 2       Eqn 1       Eqn 2      Eqn 1        Eqn 2
                                                                                 -0.030                 -0.027
   Loan Age
                                                                                (0.002)                 (0.001)
                                                                                 0.036                   0.036
Loan Age2/100
                                                                                (0.003)                 (0.002)
     Local
                                                                                                           Yes
Demographics
     MSA
                                                                                                           Yes
   Dummies
Year Dummies                                                                                               Yes
   Corr. b/w              -0.171                  -0.222                   -0.613                 -0.453
   ε1 and ε2              (9.507)                 (7.701)                (1453.45)               (73.469)
   RE Scale          0.936      0.784        0.027       0.009       0.059       0.289       0.061       0.215
  Parameter         (0.012)    (0.007)      (0.010)     (0.007)     (0.010)     (0.009)     (0.014)     (0.008)
   No. Obs                915936                  915936                  915936                  915936
      Log
                        -29143.58               -28910.24               -26329.06               -25658.72
  Likelihood


Dependent Variable = No Default. No Default includes both continued payments and prepayments. Contents of
each cell: estimated coefficient, standard error. Standard errors are clustered by loan. Due to the computational
burden, we use a 1/4 random sample of loans for estimation of random effects models.




                                                       51
                                 Table 5: Marginal Effects (based on Table 3a)


                                      Specification 1    Specification 2      Specification 3      Specification 4
                         1 std.          Marginal            Marginal            Marginal             Marginal
                          dev.            Effects             Effects              Effects             Effects

           V
    Log(     )           0.459            47.81%              24.16%              25.38%               8.77%
           L
    Exp_Bwd              0.118                                6.58%                6.02%                2.1%
  Past Volatility        1.473                                9.62%                10.4%               3.06%
        IR               0.096                                                    -0.58%               -0.19%
       MR               13.195                                                     4.46%               1.54%
      FRM                    1                                                    15.08%               8.93%
    Low Doc                  1           -53.10%             -58.43%              -61.84%             -70.38%
    FICO/100             0.709            73.66%              91.27%              91.73%               82.97%
  Original LTV           0.135           -16.46%             -17.75%              -18.62%             -21.03%
 Unemployment            1.358            -9.39%             -11.62%              -7.96%              -11.98%
     PI ratio            0.119           -18.56%
    PI ratio×
                         0.169                               -28.39%              -27.17%             -27.20%
   Low FICO
    PI ratio×
                         0.164                               -35.31%              -34.05%             -34.30%
 Medium FICO
    PI ratio×
                         0.113                               -12.18%              -10.62%             -12.49%
   High FICO
       Local
                                                                                                         Yes
  Demographics

  MSA Dummies                                                                                            Yes
  Year Dummies                                                                                           Yes


This table reports marginal effects (relative to the hazard of default computed at the sample means) associated with

a one-standard-deviation increase in each regressor. For binary variables, it is a unit change instead of a one-

standard-deviation change.




                                                        52
   Table 6: Comparison of 2004- and 2006 Vintage Loans (based on Table 3a Specification 3)


                         2004 Mean           2006 Mean          Δ in RHS Variable             Δ in Default
                              (1)                 (2)                 (2) –(1)                    Probability

           V
    Log(     )              0.512               0.402                  -0.109                      5.264%
           L
    Exp_Bwd                 0.090               -0.035                 -0.126                      0.016%
  Past Volatility           2.067               1.144                  -0.922                      5.653%
        IR                  -0.042              -0.006                 0.036                       0.192%
       MR*                 18.393               15.992                 -2.400                      0.705%
      FRM                   0.266               0.193                  -0.072                      0.955%
    Low Doc                 0.461               0.439                  -0.021                      -1.138%
    FICO/100                6.711               6.275                  -0.436                      49.002%
  Original LTV              0.769               0.802                  0.033                       3.993%
 Unemployment               4.806               4.387                  -0.418                      -2.132%
    PI ratio×
                            0.053               0.102                  0.049                       6.948%
   Low FICO
    PI ratio×
                            0.112               0.169                  0.056                       10.202%
 Medium FICO
    PI ratio×
                            0.079               0.031                  -0.047                      -3.861%
   High FICO


This table reports how the difference in each regressor between 2004- and 2006 vintage loans affects the
probability of default, relative to the hazard of default computed at the overall sample means.
* conditional on being an ARM.




                                                        53
                             Table 7: Univariate Probit


                   Spec 1     Spec 2      Spec 3      Spec 4    Spec 5      Spec 6
                   0.296                  0.234       -0.286     0.191      -0.692
       V
   log( )         (0.016)                 (0.017)     (0.047)   (0.014)    (0.049)
       L
                  0.0036                  0.002       -0.002    0.0012      -0.012

        V          0.526                  0.676                  -0.104
 log(     )×
        L         (0.070)                 (0.070)               (0.060)
  Exp_Bwd         0.0064                  0.006                 -0.0007

        V                                             0.432                 0.750
 log(     )×
        L                                             (0.044)              (0.049)
 Exp_HMS                                              0.004                 0.013
                   0.117                  0.089       0.059      0.050      0.017
Past Volatility   (0.005)                 (0.005)     (0.006)   (0.006)    (0.006)
                  0.0014                  0.0008      0.0005    0.0003     0.0002

        V          -0.057                 -0.035      -0.028    -0.0024     -0.030
 log(     )×
        L         (0.009)                 (0.010)     (0.009)   (0.0088)   (0.009)
Past Volatility   -0.00069                -0.0003    -0.0002    -0.00002   -0.00051
                   -0.571                 -0.113      -0.456     -0.564     -0.288
      IR          (0.026)                 (0.029)     (0.035)   (0.041)    (0.040)
                  -0.0069                 -0.001      -0.004    -0.0036    -0.0049
                   0.011                  0.004       0.003     0.0043     0.0062
     MR           (0.0003)               (0.0003)    (0.0003)   (0.0004)   (0.0003)
                  0.0001                 0.00004     0.00003    0.00003    0.0001
                   0.413                  0.283       0.255      0.290      0.404
    FRM           (0.007)                 (0.008)     (0.008)   (0.011)    (0.010)
                  0.0043                  0.002       0.002     0.0016     0.0063
                              -0.172      -0.175      -0.170     -0.193     -0.172
  Low Doc                     (0.005)     (0.006)     (0.006)   (0.008)    (0.007)
                              -0.002      -0.001      -0.001    -0.0013    -0.0031
                               0.316      0.295       0.296      0.343      0.309
  FICO/100                    (0.013)     (0.009)     (0.012)   (0.010)    (0.009)
                              0.0038      0.002       0.002     0.0022     0.0052




                                        54
                                             Table 7 Continued
                        Spec 1          Spec 2           Spec 3          Spec 4         Spec 5        Spec 6
                                        -0.812           -0.515          -0.536         -0.620        -0.667
  Original LTV                          (0.022)         (0.029)          (0.029)        (0.033)       (0.032)
                                        -0.009           -0.005          -0.005        -0.0039        -0.011
                                        -0.021           -0.004          -0.012         -0.026        -0.033
 Unemployment                            (0.001)        (0.002)          (0.003)        (0.004)       (0.004)
                                        -0.0002         -0.00004        -0.0001        -0.0002        -0.0005
                                        -0.487           -0.500          -0.479         -0.543        -0.639
    PI ratio×
                                        (0.061)         (0.039)          (0.048)        (0.030)       (0.035)
   Low FICO
                                        -0.0059          -0.005          -0.004        -0.0034        -0.011
                                        -0.632           -0.692          -0.638         -0.718        -0.863
PI ratio×Medium
                                        (0.049)         (0.039)          (0.045)        (0.029)       (0.030)
      FICO
                                        -0.0077          -0.006          -0.006        -0.0046        -0.015
                                        -0.533           -0.619          -0.586         -0.689        -0.813
    PI ratio×
                                        (0.096)         (0.079)          (0.101)        (0.047)       (0.054)
   High FICO
                                        -0.0064          -0.006          -0.005        -0.0044        -0.014
                                                         -0.023          -0.023         -0.028        -0.027
    Loan Age                                            (0.0007)        (0.0008)        (0.001)      (0.0009)
                                                        -0.0002         -0.0002        -0.00018       -0.0004
                                                           0.031          0.035         0.038          0.040
            2
 Loan Age /100                                          (0.001)          (0.001)        (0.001)       (0.002)
                                                         0.0003          0.0003        0.00024        0.0006
      Local
                                                                           Yes            Yes           Yes
 Demographics
YEAR Dummies                                                               Yes            Yes           Yes
     RE Rho             No RE           No RE            No RE           No RE          0.106         No RE
     No. Obs           3444360         3444334          3444334         3269190        3390424       1642033
                2
    Pseudo R            0.0395          0.0378           0.0669          0.0776         0.0772        0.0975
 Log Likelihood        -110150.6      -110339.6        -107004.4       -100540.7      -104208.4      -87451.9


Dependent Variable = No Default. Contents of each cell: estimated coefficient, standard error, marginal effects.

Standard errors are clustered by loan. For Specifications 1-5, No Default includes both continued payments and

prepayments. For Specification 6, No Default includes continued payments only.



                                                      55
                                  Table 8: Competing Hazards Model


                      Specification 1
                                            Specification 2       Specification 3       Specification 4
                     No Unobserved
                                          2 Unobserved Types    2 Unobserved Types    2 Unobserved Types
                      Heterogeneity

                    Default     Prepay    Default     Prepay    Default     Prepay    Default     Prepay

                    -2.149      0.155     -1.838      0.477     -0.237      -0.204    -0.271      -0.164
     V
Log(        )       (0.037)     (0.011)   (0.042)     (0.014)   (0.117)     (0.037)   (0.119)     (0.038)
     L
                    0.116       1.168      0.159      1.611     0.789       0.815      0.763      0.849

                    0.280       0.015      0.268      -0.025    0.176       -0.060     0.179      -0.048
      V         2
(log(       ))      (0.004)     (0.003)   (0.005)     (0.003)   (0.006)     (0.004)   (0.006)     (0.004)
      L
                    1.323       1.015      1.307      0.975     1.193       0.942      1.196      0.953

       V                                                        -4.227      1.731     -4.148      1.651
log(     )×
       L                                                        (0.161)     (0.034)   (0.164)     (0.034)

Exp_Bwd                                                         0.015       5.648     (0.016)     5.212

       V                                                        -0.788      0.567     -0.772      0.503
log(     )×
       L                                                        (0.088)     (0.027)   (0.090)     (0.029)

Exp_HMS                                                         0.455       1.762      0.462      1.654

                                           1.579      0.948     0.496       0.841      0.599      0.784
       IR                                 (0.083)     (0.036)   (0.092)     (0.042)   (0.092)     (0.042)
                                           4.850      2.580     1.642       2.319      1.820      2.190

                                          -0.312      -0.189    -0.213      -0.150    -0.189      -0.146

   MR                                     (0.011)     (0.004)   (0.013)     (0.005)   (0.013)     (0.005)
                                           0.732      0.827     0.808       0.860      0.828      0.864

                                          -1.403      -0.965    -1.076      -0.815    -0.971      -0.821
  FRM                                     (0.028)     (0.011)   (0.030)     (0.012)   (0.029)     (0.012)
                                           0.245      0.381     0.341       0.443      0.379      0.440

                                                                0.553       0.156      0.541      0.128

Low Doc                                                         (0.019)     (0.008)   (0.019)     (0.009)
                                                                1.738       1.169      1.718      1.137

                                                                -0.828      -0.134    -0.898      -0.132

  FICO                                                          (0.015)     (0.006)   (0.016)     (0.006)
                                                                0.437       0.879      0.407      0.877




                                                     56
                                                       Table 8 Continued

                        Specification 1
                                                     Specification 2           Specification 3              Specification 4
                        No Unobserved
                                                  2 Unobserved Types        2 Unobserved Types           2 Unobserved Types
                        Heterogeneity

                     Default        Prepay        Default         Prepay     Default       Prepay        Default       Prepay

                                                                              1.414         0.672         0.848         0.861

Original LTV                                                                 (0.099)       (0.030)       (0.099)       (0.003)

                                                                              4.111         1.960         2.335         2.366

                                                                              2.205         1.143         2.179         1.095

   PI ratio                                                                  (0.069)       (0.031)       (0.071)       (0.033)
                                                                              9.061         4.108          8.84         2.988

                                                                                                         -0.525         0.126
  Refinance,
                                                                                                         (0.020)       (0.010)
     Cash
                                                                                                          0.592         1.134

                                                                                                         -0.512        -0.028
  Refinance,
                                                                                                         (0.028)       (0.013)
   No cash
                                                                                                          0.599         0.972

                                                                                                          1.478         0.949

  Loan Age                                                                                               (0.092)       (0.044)
                                                                                                          4.384         2.583

                                                                                                         -0.197        -0.153
     Loan
                                                                                                         (0.014)       (0.007)
  Age2/100
                                                                                                          0.822         0.858

     Local
                                                                                                           Yes             Yes
Demographics

                                                       (-2.9, -3.82,             (-2.96, -1.08,              (-4.22, 0.46,
   (ηp1, ηd1,                                         -5.38, -7.42)              -5.08, -4.33)               -6.55, -2.75)

   ηp2, ηd2)                                         SE (0.01, 0.03,           SE (0.05, 0.13,              SE (0.09, 0.19,
                                                       0.03, 0.06)                0.06, 0.14)                 0.10, 0.20)

 Pr. of Type 1                                        0.261 (0.002)              0.248(0.004)               0.256 (0.003)

  No. Loans                 177420                       177420                     177420                        177420

      Log
                         - 545233.004                 - 538260.594               -532946.781                 -531382.845
  Likelihood



Contents of each cell: estimated coefficient, standard error, hazard ratio. Hazard ratios are exponentiated coefficients and have

the interpretation of hazard ratios for a one-unit change in X.   57
 Table A1: Univariate Probit with Quarterly Observations


                   Specification 1   Specification 2   Specification 3

                       0.244             0.267             0.204
       V
   log( )             (0.011)            (0.018)          (0.017)
       L
                      0.0071            0.0070            0.0051

        V              1.409             0.555            -0.223
 log(     )×
        L             (0.067)           (0.074)           (0.077)
  Exp_Bwd              0.041             0.015            -0.0056
                                         0.121             0.084
Past Volatility                         (0.006)           (0.006)
                                        0.0032            0.0021

        V                               -0.052            -0.014
 log(     )×
        L                               (0.010)           (0.010)
Past Volatility                         -0.0014          -0.00034
                                        -0.080             -0.511
      IR                                (0.035)           (0.041)
                                        -0.0021           -0.013
                                        0.0048            0.0036
     MR                                (0.0004)          (0.0004)
                                        0.00013          0.000092
                                         0.311             0.279
    FRM                                 (0.009)           (0.010)
                                        0.0073            0.0063
                      -0.185            -0.198            -0.195
  Low Doc             (0.006)           (0.007)           (0.007)
                      -0.0057           -0.0055           -0.0052
                       0.359             0.338             0.335
  FICO/100            (0.010)           (0.011)           (0.012)
                       0.011            0.0088            0.0084




                               58
                                  Table A1 Continued
                                Specification 1     Specification 2     Specification 3

                                    -0.603              -0.598                -0.623
     Original LTV                   (0.029)             (0.032)              (0.032)
                                    -0.018              -0.016                -0.016
                                    -0.028              -0.0079               -0.025
    Unemployment                    (0.002)            (0.0024)              (0.004)
                                   -0.00083            -0.00021              -0.00063
                                    -0.627              -0.563                -0.537
        PI ratio×
                                    (0.030)             (0.042)               (0.048)
       Low FICO
                                    -0.018              -0.015                -0.013
                                    -0.797              -0.773                -0.724
        PI ratio×
                                    (0.039)             (0.046)              (0.051)
     Medium FICO
                                    -0.023              -0.020                -0.018
                                    -0.732              -0.726                -0.700
        PI ratio×
                                    (0.076)             (0.091)              (0.106)
       High FICO
                                    -0.021              -0.019                -0.018
                                                        -0.021                -0.022
        Loan Age                                       (0.0008)              (0.0009)
                                                       -0.00055              -0.00056
                                                         0.028                0.034
                2
     Loan Age /100                                      (0.002)              (0.002)
                                                       0.00073               0.00085
 Local Demographics                                                            Yes
   YEAR Dummies                                                                Yes
        No. Obs                    1284073             1212730               1193639
                    2
       Pseudo R                     0.0559              0.0757               0.0851
    Log Likelihood                -97064.66           -88143.61              -85911.14


Contents of each cell: estimated coefficient, standard error, marginal effects. Quarters

computed starting from month of initial observation for each loan (e.g., for a loan first
appearing in the data in 11/2007, the first quarter is 11/2007 – 01/2008.)    Right-hand-

side variables are averages over quarters.




                                             59
              Table A2: Bivariate Probit with Partial Observability using Exp_ Fwd


                      Specification 1       Specification 2       Specification 3       Specification 4

                    Eqn 1        Eqn 2     Eqn 1      Eqn 2      Eqn 1      Eqn 2      Eqn 1      Eqn 2

                    1.029                  0.974                 1.040                 1.011
       V
  Log(        )     (0.122)               (0.073)               (0.101)               (0.091)
       L
                   0.00043                0.0004                0.0002                0.0002

        V           6.462                  4.980                 5.100                 4.880
 log(        )×
        L           (0.554)               (0.243)               (0.419)               (0.434)

  Exp_Fwd           0.0027                0.0023                0.00097               0.0011

                    -0.0617               -0.0376               -0.0676               -0.0702

Past Volatility    (0.0113)               (0.0137)              (0.0162)              (0.0162)
                   -0.00002               -0.00001              -0.00001              -0.00001

        V           0.7906                 0.677                 0.670                 0.688
 log(        )×
        L          (0.0561)               (0.066)               (0.084)               (0.084)
Past Volatility    0.00033                0.0003                0.00013               0.00015

                                           -0.239                -0.227                0.145

        IR                                (0.051)               (0.070)               (0.116)

                                          -0.0001               -0.00004              0.00003

        V                                                                              -1.609
 log(        )×
        L                                                                             (0.460)

        IR                                                                            -0.0003

                                          0.0140                0.0145                0.0143
     MR                                   (0.0005)              (0.0006)              (0.0006)
                                           6.4e-6                2.8e-6                3.1e-6

                                           0.744                 1.105                 1.028

    FRM                                   (0.030)               (0.127)               (0.108)

                                          0.0002                0.00019               0.0002

                                -0.1830               -0.192                -0.211                -0.212
  Low Doc                      (0.0080)               (0.009)               (0.008)               (0.008)
                                -0.0013               -0.0012              -0.0016               -0.0016

                                0.4314                0.432                 0.381                 0.383
  FICO/100                     (0.0175)               (0.017)               (0.016)               (0.016)
                                 0.003                0.0025                0.0026                0.0026



                                                     60
                                                 Table A2 Continued

                        Specification 1          Specification 2           Specification 3            Specification 4

                      Eqn 1           Eqn 2     Eqn 1       Eqn 2        Eqn 1          Eqn 2        Eqn 1      Eqn 2

                                     -0.434                 -0.454                   -0.500                     -0.503

 Original LTV                     (0.028)                   (0.032)                  (0.029)                    (0.030)
                                     -0.003                -0.0027                   -0.0035                   -0.0035

                                  -0.0290                  -0.0200                   0.0285                    -0.0283

Unemployment                      (0.0025)                 (0.0029)                 (0.0042)                   (0.0042)
                                  -0.00020                 -0.0001                   -0.0002                   -0.00019

                                     -0.564                 -0.537                   -0.523                     -0.522
   PI ratio×
                                  (0.097)                   (0.094)                  (0.088)                    (0.088)
  Low FICO
                                  -0.0039                  -0.0032                   -0.0036                   -0.0036

                                     -0.719                 -0.692                   -0.679                     -0.678
   PI ratio×
                                  (0.076)                   (0.075)                  (0.066)                    (0.066)
Medium FICO
                                  -0.0049                  -0.0041                   -0.0047                   -0.0047

                                     -0.371                 -0.320                   -0.359                     -0.356
   PI ratio×
                                  (0.107)                   (0.089)                  (0.097)                    (0.097)
  High FICO
                                  -0.0025                  -0.0019                   -0.0025                   -0.0025

                                                           -0.0299                   -0.0293                   -0.0298
   Loan Age                                                (0.0011)                 (0.0010)                   (0.0010)
                                                           -0.0001                   -0.0002                   -0.00021

                                                            0.0396                   0.0408                     0.0420
Loan Age2/100                                              (0.0023)                 (0.0020)                   (0.0022)
                                                            0.0002                  0.00028                    0.00029

     Local
                                                                                         Yes                       Yes
 Demographics

MSA Dummies                                                                              Yes                       Yes

Year Dummies                                                                             Yes                       Yes

   Corr. b/w                -0.222                    -0.611                   -0.511                     -0.510

   ε1 and ε2                (0.033)                  (0.029)                   (0.035)                   (0.034)

    No. Obs                3646195                   3444334                  3390424                    3390424

Log Likelihood            -117246.9                 -106871.7                -104540.4                  -104535.0



Dependent Variable = No Default. No Default includes both continued payments and prepayments.

No random effects. Contents of each cell: estimated coefficient, standard error, marginal effects.
Standard errors are clustered by loan.                    61

				
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