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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. References [1] Aguirregabiria, Victor, and Pedro Mira. 2007. “Swapping the Nested Fixed-Point Algorithm: a Class of Estimators for Discrete Markov Decision Models.” Econometrica, 70(4): 1519-1543. 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[30] vandell, Kerry. 1993. “Handing Over the Keys: A Perspective on Mortgage Default Re- search.” Real Estate Economics, 21(3): 211-246. [31] vandell, Kerry. 1995. “How Ruthless is Mortgage Default? A Review and Synthesis of the Evidence.” Journal of Housing Research, 6(2): 245-264. [32] Von Furstenberg, George. 1969. “Default Risk on FHA-Insured Home Mortgages as a Function of the Terms of Financing: A Quantitative Analysis.” Journal of Finance, 24(3): 459-477. 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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