Does it Pay Not to Pay? An Empirical Model of Subprime. Mortgage Default from 2000 to 2007

Transcription

1 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 oanperformance 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 ehnert, Monika Piazzesi, Tom Sargent, and Dick Todd for helpful conversations. Bajari would like to thank the National Science Foundation for generous research support. The views expressed are those of the authors and do not necessarily re ect the o cial positions of the Federal Reserve System. Correspondence: bajari@umn.edu; Sean.Chu@frb.gov; mpark@umn.edu. 1

2 1 Introduction Subprime mortgages are made to borrowers with low credit quality or who have a higher probability of default due to risk factors associated with the loan itself, such as having a low downpayment. 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 subprime; by 2005 that gure had risen to 20%, according to Moody s 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 quarter 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

3 In this paper, we explore four alternative explanations for the increase in mortgage defaults, using a unique data set from oanperformance. An observation in the data set is a subprime or Alt-A mortgage securitized between 1992 and We observe information from the individual s 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 ratio, and the borrower s 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 oanperformance 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

4 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 than the current market interest rate, a borrower s 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 ects a borrower s 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

5 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, ing, 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 researchers. 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 lifetime utility by defaulting and the violation of the borrower s 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 expectations 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

6 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, ing, 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 optimal for a homeowner to default if and only if defaulting increases the homeowner s 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, nally, credit constraints. We demonstrate that an agent s 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

7 2.1 Optimal Default without iquidity Constraints et i index borrowers and t index time periods. Denote by V it the value of borrower i s home at time period t, and denote by it the outstanding principal on i s mortgage. Normalize the time period in which i purchases her home to t = 0. et g it denote the rate of in ation in home price between time period t 1 and t. Then V it = V i0 tq (1 + g it 0) (1) t 0 =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 g it correspond to the Case-Shiller price index for the city where i resides. Equation (1) tracks the evolution in the value of i s home. Empirically, we expect it generally to be the case that V it > V i0 for buyers who have held their homes for many years. However, for more recent buyers it may be the case that V it < V i0 because of declining home prices over a shorter horizon. The evolution of the outstanding principal, it, is somewhat more complicated than the evolution of V it. 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. it 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 it 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 V it it < 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

8 keep the same ow of housing services while increasing her wealth by it V it. On the other hand, if V it it > 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 V it it. The decision to default may be triggered by a fall in home prices g it. For recent borrowers who have made small downpayments, just a few su ciently negative realizations of g it 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 g it > 0, (2) is less likely to hold 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. et Eg it represent borrower i s 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: V it (1 + 1 Eg it ) it < 0 (3) We use the parameter 1 to scale the units of Eg it : We shall describe our approach to measuring Eg it 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

9 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 V it 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 g it, 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 g it : V it (1 + 1 Eg it + 2 V g it ) it < 0 (4) Measuring Expectations about Home Prices We consider two di erent measures of Eg it : 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 = V it r rf it + V it! it V it it (rit c +! it) + V it it V it Eg it + V it it = R it (5) In this equation, V it is the house price and R it the annual rent. The term r rf it rate the homeowner i would have obtained in an alternative, risk-free investment. is the interest Therefore, V it r rf it captures the opportunity cost of the house relative to other potential investments. The term! it is the property tax rate, it the e ective tax rate on income, and r c it the contractual interest rate on the mortgage. The term V it it (r c it +! 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, Eg it the expected capital gain, and it the risk 9

10 premium. Using observed values of V it, R it, r rf it,! it, it, r c it, it, and it, obtained from data, we can deduce the expected capital gain Eg it that satis es (5). Himmelberg, Mayer, and Sinai (2005) recover Eg it from (5) and decompose it into two components. The rst is the expected growth due to fundamentals, Eg f it, which they proxy using the average annual home price growth rate between 1950 and The remainder, Eg b it, captures expected growth unexplained by their measure of fundamentals, and might be due to speculative bubbles. V it it (r c it +! it)+v it it For each MSA and quarter, they report the ratio of V it r rf it + V it! it V it Eg f it +V it it (the imputed rent ) to the actual rent R it. Comparing this expression to (5) shows that the ratio is greater than one if the market expects faster house price appreciation than warranted by fundamentals (Eg b it > 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 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

11 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, IR it = TPM it P i (1+rit m=1200)t t=1 TPM it t=1 TPM it P i (1+rit c t=1 =1200)t = P i (1+rit m=1200)t TPM it t=1 1 (1+r m it =1200)t TPM it TPM it 1 t=1 (1+rit c =1200)t (6) 1 (1+rit m t=1 =1200)t P i is the monthly payment for the mortgage, T M it is the number of remaining months until maturity, r m it is the market rate borrower i would get if he obtained a new loan in period t, and r c it is the contractual interest rate of the mortgage. For adjustable-rate mortgages, P i and r c it may vary over the course of the loan, but for simplicity, we assume that P i and r c it remain constant at the levels of the current month t. In practice, the available market rate of interest r m it varies across households because of differences 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 oanperformance data cover the universe of subprime mortgage originations, we can get a very precise estimate of the impact of the risk factors on contract rates. Our estimate of r m it also controls for unobserved household-level heterogeneity. Details behind the procedure for imputing r m it 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

12 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. etting MR it represent the number of months before the next rate reset for borrower i in period t (for xed-rate mortgages, we set MR it = 0 and then include a separate dummy for xed-rate mortgages), the default decision of the borrower depends on the following condition: V it (1 + 1 Eg it + 2 V g it ) it (1 + 3 IR it + 4 MR it ) < 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 it then yields: V it it (1 + 1 Eg it + 2 V g it ) (1 + 3 IR it + 4 MR it ) < 0 (8) 2.2 iquidity 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

13 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 obligations are its monthly principal and interest payments relative to income, P it Y it, 1 and the household s overall credit quality, Z it. 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 additional 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-byperiod budget constraints in every single period. The period-by-period budget constraint of household i can be written as follows. P it + C it Y it (9) C it denotes the consumption of the household i in period t. We further assume that the household must have a minimum level of consumption in each period. The household s budget constraint then takes the following form. Budget Constraint Binds, 1 P it Y it c it < 0 (10) where c it is the minimum required consumption as a proportion of the household s income. When a household s monthly payment P it 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 capital market in order to meet their monthly payments, the relationship between P it Y it is less stark. and default 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-toincome ratio comes from across households as well as rate resets for a given household. 13

14 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 its lifetime budget constraint, which pools the household s period-by-period budget constraints over all time periods. To capture the notion that the relevance of the period-by-period budget constraint is weaker for borrowers with high credit quality, we interact P it Y it 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, Z it, 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. Budget Constraint Binds, 1 Z it + 2 Z it ( P it Y it ) c it + 1 < 0 (11) 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: (a) default increases the household s wealth and the household s budget constraint is binding, (b) default increases the household s wealth and the household s budget constraint does not bind, (c) default decreases the household s wealth and the household s budget constraint is binding, and (d) default decreases the household s wealth and the household s 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

15 process for the observed outcome corresponds to a bivariate probit model with partial observability, 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, ing, and McGill, 1996; Demyanyk and van Hemert, 2008). A single-equation model leads to misspeci cation because 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: U 1;it = 0i + V it it (1 + 1 Eg it + 2 V g it ) (1 + 3 IR it + 4 MR it ) + " 1;it (12) P U 2;it = 0i + 1 Z it + 2 Z it ( it Y it ) + " 2;it U 1;it represents the latent utility associated with not defaulting, and is equal to the normalized 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 U 2;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 U 1;it and U 2;it are correlated with each other through the observable covariates V it, it, Eg it, V g it, IR it, MR it, P it Y it, and Z it, 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 U 1;it and U 2;it. 2 Our data are 2 We assume that c it, the minimum required consumption as a proportion of the household s income, remains constant over time for a given individual. Hence, the term is now subsumed in 0i. 15

16 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 Z it entering the liquidity equation, we include the most obvious measures of borrowers credit quality, such as FICO scores. We also include observable loan characteristics 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 origination. 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 ND it = 1 if household i does NOT default in period t and as 0 otherwise. The condition for default is as follows: ND it = U 1;it U 2;it = 0 (default), U 1;it < 0 or U 2;it < 0 (13) where the outside options for both U 1;it and U 2;it are normalized to zero. Given the available data, when a default occurs we cannot observe whether it is because U 1;it < 0, because U 2;it < 0, or for both reasons. Two points are worth mentioning. First, in principle we could specify the borrower s decision 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

17 checks, we estimate an alternative model in which prepayment and default are dependent competing 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 speci cation of an agent s 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 oanperformance on subprime and Alt-A mortgages that were originated between 1992 and 2007 and securitized in the private-label market. The oan- 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 oanperformance 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

18 interest rate, the level of documentation (full, low, or nonexistent 4 ), 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 s FICO score, 5 and the borrower s debt-to-income ratio. One limitation of the oanperformance 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 oanperformance data, see Chomsisengphet and Pennington-Cross (2006), Demyanyk and van Hemert (2007), and Keys, Mukherjee, Seru, and Vig (2007). The oanperformance 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 that could a ect a borrower s liquidity constraints, we use monthly unemployment rates reported at the county level by the Bureau of abor and Statistics (BS). 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 Full documentation indicates that the borrower s income and assets have been veri ed. ow 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

19 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. oans that default tend to be adjustablerate 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

20 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 =) is on average over the course of each loan and 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 over the course of the loan and in the nal period. This ratio tends to be highest among loans that default (on average 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 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 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

21 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 at the point of default (versus an overall average of 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 ) 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 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, as Vegas, os 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. ower weights are also assigned to houses for which the change in transaction price is an outlier within a geographic 21

22 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 continued 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 U 1;it in equation (12). The column eq2 includes the parameter estimates and covariates that determine U 2;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-standarddeviation 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 U 1;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 (U 1 > 0; U 2 > 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 (U 1 > 0) and P (U 2 > 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 U j on P (U 1 > 0; U 2 > 0) is virtually equal to its marginal e ect on P (U j > 0) for j = 1, 2. Therefore, we do not need to discuss both e ects. 22

23 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 valueto-loan ratio are less likely to default. Our estimates of the marginal e ects imply that a one-standard-deviation increase in log( V ) is associated with a 47.8% reduction in the hazard of default in a given month. The sharp decline in home prices played an important role in the recent increase in foreclosures. Consider a hypothetical household in Phoenix that purchases a home in February 2007 with a 30-year xed-rate mortgage and no downpayments. The household s log( V ) is then 0 at the time of purchase. Further assume that the household makes monthly payments such that the outstanding balance on the mortgage in February 2008 is of the original loan amount. If there is no change in home price between February 2007 and February 2008, the household s log( V ) in February 2008 would be During this time period, however, home prices in Phoenix fell by 21.7%. If this household s property value experienced the average home price change in Phoenix, its log( V ) at the end of this time period would be Thus, the decline 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 U 2 are important drivers of default as well. A low-documentation loan has a 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 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-standarddeviation 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 payments to monthly income also predicts an increase in the probability of default. A one-standarddeviation increase in this ratio (0.12) generates an 18.6% increase in the hazard of default. For 23