Last Update: Mar 26, 2011
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by Jan K. Brueckner, Paul S. Calem, and Leonard I. Nakamura
This paper explores the link between the house-price expectations of mortgage lenders and
the extent of subprime lending. It argues that bubble conditions in the housing market are
likely to spur subprime lending, with favorable price expectations easing the default concerns
of lenders and thus increasing their willingness to extend loans to risky borrowers. Since the
demand created by subprime lending feeds back onto house prices, such lending also helps to
fuel an emerging housing bubble. The paper, however, focuses on the reverse causal linkage,
where subprime lending is a consequence rather than a cause of bubble conditions. These ideas
are illustrated in a theoretical model, and empirical work tests for a connection between price
expectations and the extent of subprime lending.
WORKING PAPER NO. 11-12 SUBPRIME MORTGAGES AND THE HOUSING BUBBLE Jan K. Brueckner University of California–Irvine Paul S. Calem Board of Governors of the Federal Reserve System Leonard I. Nakamura Federal Reserve Bank of Philadelphia
Subprime Mortgages and the Housing Bubble
by† Jan K. Brueckner Department of Economics University of California, Irvine 3151 Social Science Plaza Irvine, CA 92697 e-mail: firstname.lastname@example.org Paul S. Calem Board of Governors of the Federal Reserve System Washington, D.C. 20551 e-mail: Paul.Calem@frb.gov and Leonard I. Nakamura Federal Reserve Bank of Philadelphia 10 Independence Mall Philadelphia PA 19106 e-mail: Leonard.Nakamura@phil.frb.org February 2011 Abstract This paper explores the link between the house-price expectations of mortgage lenders and the extent of subprime lending. It argues that bubble conditions in the housing market are likely to spur subprime lending, with favorable price expectations easing the default concerns of lenders and thus increasing their willingness to extend loans to risky borrowers. Since the demand created by subprime lending feeds back onto house prices, such lending also helps to fuel an emerging housing bubble. The paper, however, focuses on the reverse causal linkage, where subprime lending is a consequence rather than a cause of bubble conditions. These ideas are illustrated in a theoretical model, and empirical work tests for a connection between price expectations and the extent of subprime lending.
views expressed in this paper are those of the authors and do not necessarily reﬂect those of the Federal Reserve Bank of Philadelphia, the Federal Reserve Board of Governors, or the Federal Reserve System. This paper is available free of charge at http://www.philadelphiafed.org/ research-and-data/publications/working-papers./
Subprime Mortgages and the Housing Bubble
by Jan K. Brueckner, Paul S. Calem, and Leonard I. Nakamura*
The spectacular run-up in US housing prices over the ﬁrst years of the new century, along with the subsequent price collapse, are watershed events in real-estate history. No previous brief period witnessed such dramatic price escalation (on the order of 50 percent), and the rapid 30 percent price drop that ensued was also unprecedented. Researchers have expended considerable eﬀort in trying to understand these events, but the understanding is thus far incomplete. Investigators have asked whether expansionary monetary policy early in the decade, and the resulting sustained period of low mortgage interest rates, fueled the demand for housing, driving up prices. They have asked whether the growth in subprime lending, occurring partly in response to aﬀordable-housing goals, was the culprit, amplifying the eﬀect of low interest rates on demand. Researchers have also asked whether high loan-to-value ratios along with irresponsibly lax mortgage underwriting were the sources of price escalation. Finally, investigators have tried to link house-price patterns to market fundamentals such as population and income growth. A recent paper by Glaeser, Gottlieb and Gyourko (2010) conveniently tests a number of these competing explanations in the context of a single study, and the results are not particularly aﬃrmative. Relying on a multi-decade regression relating house prices to interest rates, the authors ﬁnd that low interest rates can explain at most half of the price escalation through 2008. To measure the eﬀect of lax underwriting standards (and the consequent easing of mortgage access), the authors use a regression of house prices on mortgage approval rates, ﬁnding little eﬀect on prices. They also show that the increase in loan-to-value ratios over the period cannot explain the price explosion. Similarly, although Coleman, LaCour-Little and Vandell (2008) show some role for market fundamentals in the price escalation through 2003, they cannot ﬁnd a link to the expansion in
subprime lending, low interest rates, or high LTV ratios.1 But they argue that price increases through 2003 laid the foundation for a “bubble” psychology, where market participants grew to expect continuing future price increases.2 Glaeser, Gottlieb and Gyourko (2010) reach a similar conclusion, arguing that, in the absence of measured links to the various factors they investigate, responsibility for the price explosion may lie in irrational expectations, as suggested by Case and Shiller (2003). The role of expectations is studied by Mian and Suﬁ (2009). But instead of asking whether optimistic expectations drove the escalation of house prices, they ask the narrower question of whether favorable price expectations helped spur the growth of subprime lending. In particular, the authors provide a test of what they call the “expectations hypothesis,” which asserts that expectations of future house-price growth, by easing default concerns, spurred subprime lending. They test the hypothesis in a highly indirect fashion. The approach compares price growth in zip codes with low credit scores at the beginning of the period (potential subprime areas) to price growth in non-subprime zip codes. The authors ﬁnd no diﬀerence in price escalation between potential subprime areas, where most subprime lending occurred, and nonsubprime zip codes, focusing on areas with elastic housing supply. They conclude that the faster expansion of mortgage credit in subprime areas could not have been driven by favorable price expectations, given that the actual price path was no diﬀerent in these areas than in nonsubprime zip codes where credit expanded less. Instead, Mian and Suﬁ conclude that subprime expansion was driven by supply factors, including relaxation of underwriting standards and the growth of mortgage securitization. As Glaeser et al. (2010) and Coleman et al. (2008) have argued, expectations likely played a signiﬁcant role in generating the housing “bubble.” Mian and Suﬁ (2009) explore this channel through their indirect test for an expectations eﬀect in subprime lending, implicitly viewing such lending as a driver of the bubble. However, given the likely importance of price expectations and the literature’s limited exploration of their eﬀects, additional research is clearly needed. The present paper ﬁlls this need by further exploring the role of lender price expectations in the operation of real-estate markets. The paper investigates a causal channel like that posited by Mian and Suﬁ, arguing that a favorable shift in lender expectations regarding
future prices spurs subprime lending by easing default concerns.3 The resulting increase in the demand for housing then feeds back into the market, driving up today’s housing prices. So expectations of future price growth generate current price escalation, feeding a housing bubble.4 Thus, the paper is built around the notion that subprime lending is both a consequence and a cause of bubble conditions in the housing market. Most of the eﬀort is devoted to exploring the ﬁrst channel, where the presence of a bubble spurs subprime lending, but the other causal direction is also kept in focus.5 These ideas are developed in a detailed theoretical model, and the paper then oﬀers a more direct test of the link between price expectations and subprime lending than the one oﬀered by Mian and Suﬁ (2009). In the model, subprime lending is portrayed as an extension of credit to borrowers with low “default costs.” These costs represent the penalty incurred by a borrower who defaults on a mortgage, which include the cost of credit impairment, moving costs, and possibly the costs associated with guilt. With their poor credit ratings, subprime borrowers presumably have low default costs since default will not greatly worsen a credit standing that is already bad. Although a previous paper by Brueckner (2000) analyzed mortgage lending when default costs are private information, the present analysis assumes that these costs (denoted C) are observable. The model generates a minimum value of C below which mortgage lenders cannot proﬁtably oﬀer a mortgage, and market-level changes that reduce this minimum will generate an expansion in subprime lending. One such change is a shift in lender expectations regarding future house prices. With a more optimist view of future prices, default is less of a concern for lenders, allowing them to extend mortgages to borrowers with even lower C’s, thus leading to an expansion of subprime lending. While the basic analysis in section 2 below makes this point, an extension of the model allows the expansion of lending to feed back onto current house prices, bidding them up. Section 3 of the paper goes further by embedding the model in an explicitly dynamic setting, where a shift in expectations generates an intertemporal adjustment process. To test for the eﬀect of price expectations on subprime lending, the paper uses the Consumer Credit Panel database from the Federal Reserve Bank of New York, aggregated up to
the state level over the 2001-2008 period. The borrower-quality measure is the credit “risk score,” which is analogous to the more familiar FICO score, being a measure of the borrower’s creditworthiness. The dependent variable in the regressions is a state-level risk-score measure, equal to the mean risk score for new borrowers in the state or, alternatively, the 10th or 25th percentile of the state’s risk-score distribution. The key explanatory variable, which is intended to capture price expectations, is the lagged annual rate of house-price appreciation, which relies on the state-level CoreLogic price index. The regressions include additional covariates as well as state and quarter ﬁxed eﬀects. While the regression is meant to test for the causal path running from price expectations to subprime lending, the reverse path (by which subprime lending causes price escalation) is a potential source of simultaneity bias. Appropriate steps are taken to deal with this bias. The empirical results, presented in section 4 of the paper, provide support for the main hypothesis by suggesting a causal link between prior price appreciation and subprime lending at the state level.
2. The Model
2.1. The setup The model, which draws on the framework of Brueckner (2000), is simple and stylized along a number of dimensions. The ﬁrst simplifying assumption is that the mortgage term consists of a single period. The borrower arranges a loan to purchase a standardized house of value P0 , and without loss of generality, the mortgage is assumed to be a 100 percent loan, so that the amount borrowed equals P0 . Repayment is required in the subsequent period. The balance-due, denoted B, includes the principal P0 as well as interest, with the interest rate r implicitly deﬁned by 1 + r = B/P0 . The value of the house in the next period is denoted P . Since P may be lower than P0 , with the house value falling over the period, the borrower may default on the mortgage. Default occurs when P is low relative to B, in which case the borrower is better oﬀ surrendering the house to the lender via foreclosure rather than paying oﬀ the mortgage. The resulting loss of the low-valued housing asset is more than oﬀset by cancellation of a larger liability.
The critical value of P , below which default occurs, depends on the magnitude of default costs. These costs, denoted C, include the cost of credit impairment, moving costs, and any psychic costs from failing to honor the mortgage contract, as noted above. The role of default costs, which are sometimes called transaction costs, has been analyzed previously by Kau, Keenan and Kim (1993, 1994), Riddiough and Thompson (1993), Brueckner (2000), Foote, Gerardi, and Willen (2008) and others. In the model, default costs are heterogeneous across borrowers, ranging from a minimum value C to a maximum value of C. The magnitude of some elements of C, and thus the outcome of the default decision, can be aﬀected by “trigger events,” as discussed by Foote, Gerardi, and Willen (2008).6 Thus, default costs provide a way in which trigger events can play a role in a model that retains the spirit of the home-equity approach to default decisions. Taking account of default costs, the borrower defaults on the mortgage when P satisﬁes
P − B < − C.
P − B in (1) equals housing equity, which is retained by the borrower if he repays the mortgage and thus represents the (possibly negative) increment to wealth in the absence of default. Conversely, −C in (1) gives the (negative) increment to wealth if default occurs. Therefore, wealth is larger with default when housing equity is less than −C < 0. Note that the presence of default costs implies that equity must fall well below zero before default is desirable. Rearranging, (1) shows that default is optimal when
< B − C.
Thus, default occurs when the house price is less than the mortgage balance-due less default costs. In contrast to the analysis of Brueckner (2000), where default costs are private information, the current model assumes that C is observable. In reality, the borrower’s credit rating appears to give a good picture of the default propensity, with C small (and default more likely) when
the credit score is low. The model assumes that there are no other unobserved inﬂuences on C, which is perfectly proxied by the credit score. With default costs thus observable, the lender tailors the mortgage contract oﬀered to the borrower to reﬂect the observed value of C. Competition among lenders ensures that the resulting contract generates zero lender proﬁt. Computation of proﬁt involves a key element: lender expectations regarding the next period’s house prices, which help determine the perceived likelihood of default. Expectations are represented by the continuous density function f(P ), which is the same for all lenders. Generally, the density f will depend on the entire past history of house prices, which will aﬀect both the mean and variance of the distribution of anticipated prices, perhaps through a Bayesian updating process. The basic analysis that follows, however, does not require a description of exactly how expectations are formed. Instead, the main focus is on the eﬀect of an exogenous shift in expectations, conditional on a general form for f. The formation of expectations must be speciﬁed, however, when the goal is to analyze the intertemporal adjustment process generated by an expectations shift, as seen in section 3 below. Suppose the lender is risk neutral, focusing on expected proﬁt, and that he incurs a mortgage origination cost of k for each loan issued. Then, incorporating f and letting η < 1 denote the lender’s discount factor, the expected present value of proﬁt from oﬀering a loan with a balance-due of B to a borrower with default costs C is given by
π ≡ − k − P0 + η
P f(P )dP + η
Note that the lender transfers P0 to the borrower when the loan is made. In the next period, he receives the loan balance B from the borrower over the range of house values where P ≥ B − C holds and default does not occur. He takes possession of the house through foreclosure, earning P from resale of the property, over the range of values where P < B − C and default does occur. Without loss of generality, the foreclosure costs incurred by the lender are assumed to be zero.7 Although this formulation portrays the lender as using his own money to extend the loan, the analysis would be unaﬀected if he relied instead on borrowed funds (bank deposits). Then
the outlay of P0 would disappear from (3), being replaced by the discounted next-period interest payment −ηzP0 , where z is the lender’s cost of funds. Note that under either formulation, the supply of mortgage funds is viewed as perfectly elastic, unaﬀected by the total volume of lending. The zero-proﬁt condition for lenders requires that π in (3) equals zero. The resulting equation then implicitly deﬁnes a value of B that yields zero expected proﬁt for a given value of C. Recalling that 1 + r = B/P0 , the equation also implicitly deﬁnes the interest rate that generates zero proﬁt from lending to a borrower with the given default costs. 2.2. The relation between B and C Setting (3) equal to zero and totally diﬀerentiating, the eﬀect of C on B is given by ∂B ∂C πC , πB
where πC and πB are the derivatives of π with respect to these variables. Using Leibniz’s rule,
= η[Bf(B − C) − (B − C)f(B − C)] = ηCf(B − C) > 0.
By making default less likely for a given B, higher default costs thus raise proﬁt. Similarly, the eﬀect of B on proﬁt is given by
f(P )dP − Cf(B − C) ,
where the ﬁrst term comes from diﬀerentiating under the integral in (1). The sign of (6) is ambiguous as a result of two opposing eﬀects. First, an increase in B raises the lender’s revenue in the event that default does not occur, an eﬀect captured by the ﬁrst term in brackets. However, an increase in B makes default more likely, which has the opposite eﬀect and leads to the negative second term in (6). One would expect the positive eﬀect to dominate the second one, so that a larger balance-due raises expected proﬁt. This outcome, for example, is
guaranteed in the case where f is uniform and given by 1/(P − P ) over the support [P , P ] , provided that an additional assumption holds. In this case, P −B , P −P
which is positive under the assumption that the lender does not ask the borrower to pay back more than the maximal anticipated next-period house price (B < P ). More generally, with a unimodal f, it can be shown that πB is positive provided that the density is not too steep over its descending range. Assuming that πB > 0, (4) is then negative, implying ∂B/∂C < 0. Thus, an increase in C allows a reduction in B, with the lender setting the balance-due lower, or equivalently charging a lower interest rate, for a better quality borrower (one with a higher C). Since such a borrower is less likely to default, the amount B that the bank recovers when default does not occur can be set lower. Therefore, a curve relating B to C is downward sloping over the relevant range, as shown in Figure 1.8 That range is limited at the upper end by C, the maximal value of default costs. A key additional assumption puts a lower limit on C. This assumption realistically imposes an upper bound on B via a stylized form of the mortgage payment-to-income constraint. Such a constraint has been a key feature of mortgage underwriting for decades, although income requirements were substantially loosened for subprime borrowers in recent years. To see the eﬀect of the payment-to-income constraint, note that the mortgage payment is a component of B, being equal to rP0 = B − P0 . Assuming that borrower incomes are uniform and equal to y despite heterogeneous default costs, the payment-to-income constraint is written rP0 /y ≤ α for some constant α < 1. Substituting, the constraint becomes (B − P0 )/y ≤ α, or
B ≤ αy + P0 ≡ B.
Let the value of C associated with B in Figure 1 be denoted C. Then, referring to the ﬁgure, the B ≤ B requirement implies C ≥ C, so that an upper limit on the balance-due puts
a ﬂoor of C under allowable default costs. The reason is that, in lending to a borrower with default costs below C, zero proﬁt would require a balance-due above the limit ﬁxed by the payment-to-income constraint. Thus, mortgages cannot be oﬀered to borrowers of less than a certain quality, those with C values lying below C. As seen in Figure 1, C is assumed to be less than C, so that low-quality potential borrowers in the range [C, C] cannot get a mortgage. The discussion so far has implicitly assumed that borrowers will always accept an oﬀered mortgage, regardless of the magnitude of the balance-due. In eﬀect, (owner-occupied) housing is assumed to be essential, so that a mortgage will be accepted regardless of its cost. Thus, the demand more mortgage funds is perfectly inelastic. 2.3. Subprime lending In the model, subprime lending corresponds to a reduction in C. In other words, subprime lending gives some low-quality potential borrowers, who previously could not get a loan, access to mortgage funds. Parametric changes in the model can generate a reduction in C and thus the emergence of subprime lending. Three changes in particular are of interest. The ﬁrst is a relaxation of the payment-to-income constraint, as has occurred in recent years. The second change is a decline in the cost of lending, perhaps reﬂecting eﬃciency gains from greater reliance on the Internet. The third change, which is the main focus of the analysis, is a favorable shift in the lender’s house-price expectations. To analyze the eﬀects of these changes, the explicit condition that determines C is needed. This condition comes from setting π from (3) equal to zero, and then substituting B in place of B. The resulting condition determines the associated C value, namely C, and it is written
− k − P0 + η
P f(P, δ)dP + η
In (9), δ is a shift parameter for the house-price density. It is assumed that an increase in δ shifts the density toward higher values in the sense of stochastic dominance. In other words, with an increase in δ, the cumulative distribution function shifts downward at each value of P within the support of f. Thus, the δ derivative of F (P, δ) ≡
f(z, δ)dz satisﬁes Fδ (P, δ) < 0
over this support, where the subscript denotes partial derivative.
Using (9), the eﬀect on C of an increase in α, the maximum payment-to-income ratio, is easily derived. Totally diﬀerentiating (9) with respect to α and C, recognizing that α determines B via (8), yields ∂C ∂α πB (∂ B/∂α) πC
where the inequality follows because πC > 0, πB is assumed positive, and ∂ B/∂α > 0 from (8). Thus, by raising the maximum possible B, relaxation of the payment-to-income constraint lowers the ﬂoor on C, allowing C to fall and spurring subprime lending. In other words, by allowing lenders to recoup a larger balance-due when default does not occur, a higher α allows lenders to extend mortgages to borrowers of even lower quality without incurring a loss.9 Similarly, since πk = −1, it follows that ∂ C/∂k = 1/πC > 0. Thus, by raising proﬁt, a decline in mortgage-origination costs allows lenders to extend loans to lower-quality borrowers. The eﬀect on C of a shift in the lender’s house-price expectations requires more extensive computation. The appendix shows that an increase in δ raises π holding C ﬁxed, with
= −η CFδ (B − C, δ) +
Fδ (P, δ)dP
Therefore, diﬀerentiating (9), ∂C ∂δ = − πδ πC < 0. (12)
A favorable shift in the house-price density thus leads to a reduction in C, with more optimistic price expectations spurring subprime lending. With better expectations, the perceived likelihood of default declines, other things equal, allowing loans to be extended to even lower-quality borrowers without generating a loss for the lender. Recognizing that a shift in expectations is characteristic of a housing bubble, the results in (10) and (12) are summarized as follows: Proposition 1. A favorable shift in the density of anticipated future house prices (as occurs under a housing bubble) spurs subprime lending, with the caliber of the
lowest-quality borrower falling. This eﬀect is reinforced by a relaxation of the paymentto-income constraint or a decline in mortgage-origination costs. 2.4. Endogenizing P0 So far, the initial house price P0 has been viewed as ﬁxed despite the increase in the number of home buyers that follows from a decline in C. A full analysis, however, should allow P0 to be endogenous, so that it can change along with C in response to changes in the model’s parameters. The size of houses, however, remains ﬁxed at some standardized value. To carry out this extension, the ﬁrst step is to relate the number of demanders of housing to C. Let G(C) denote the cumulative distribution function for default costs over the [C, C] range, with G(C) = 0 and G(C) = N, the number of potential mortgage borrowers. Then, the number of housing demanders as a function of C is simply N − G(C), which gives the number of individuals with C values large enough to get a mortgage. On the supply side, let S(P0 ) denote the supply function for new houses, which gives the number of houses supplied as an increasing function of price. Note that housing production is assumed to be instantaneous, so that producers have no need to consider prices outside of the current period. Taking account of the supply function, the number of houses available would then equal the stock passed on from the previous period, denoted H, plus S(P0 ). The condition that equates housing supply and demand is then
Φ ≡ H + S(P0 ) − [N − G(C)] = 0.
While a high price generates an increase in the number of houses, so that S(P0 ) > 0 holds for large values of P0 , units are removed from the housing stock when P0 is low. This outcome reﬂects the existence of some alternate use for the housing land input, which is superior when housing commands a low price. Thus, the supply function satisﬁes S(P0 ) > (<) 0 when P0 > (<) P ∗ , with P ∗ giving the price where supply is zero. The second equilibrium condition, which is based on (9), links P0 and C from the lender’s side. The required modiﬁcation comes from (8), which makes B a function of the nowendogenous P0 instead of simply a constant. This function is B(P0 ) ≡ αy + P0 , and it
shows a positive relationship between B and P0 .10 Substituting B(P0 ) in place of B in (9), the condition is rewritten as
B(P0 )−C ∞
− k − P0 + η
P f(P, δ)dP + η
B(P0 )f(P, δ)dP
Conditions (13) and (14) jointly determine equilibrium values of C and P0 . The eﬀect on these equilibrium values of a shift in house-price expectations can be analyzed diagrammatically. The equilibrium is the intersection of two locii in (C, P0 ) space, with one given by (13) and the other by (14). The locus given by (13), denoted the “supply-demand (s-d) locus,” is downward sloping. Totally diﬀerentiating (13), the slope of the locus is given by ∂P0 ∂C
G (C) S (P0 )
where the sign follows from G , S > 0. The locus given by (14), denoted the “π locus,” is upward sloping. Its slope is given by ∂P0 ∂C
πC πP 0
−πC πB (∂ B/∂P0 ) − 1
Given πC > 0, the inequality in (16) is a consequence of the negativity of the last denominator expression, which is easily demonstrated.11 Figure 2 illustrates the s-d and π locii and their intersection. When house-price expectations shift, with δ increasing, the π locus shifts, while the s-d locus (which is independent of δ) remains ﬁxed. The shift in the π locus is leftward in the C direction, as shown in the ﬁgure. This conclusion follows given that ∂ C/∂δ|π < 0 holds from (12), which indicates that C falls as δ increases, holding P0 ﬁxed. As seen in Figure 2, this shift in the π locus reduces the equilibrium value of C while raising P0 . Since (10) shows that the π locus again shifts to the left when α increases, a relaxation of the payment-to-income constraint has the same impacts on C and P0 as a shift in expectations. These same eﬀects also arise when origination costs k decline. Summarizing yields.
Proposition 2. When the current house price P0 is endogenous, a rightward shift in the density of anticipated future prices (as occurs under a housing bubble) raises P0 while again spurring subprime lending, with the caliber of the lowest-quality borrower falling. A relaxation of the payment-to-income constraint or a decline in mortgage origination costs reinforces these eﬀects. Thus, growth in subprime lending continues to be a consequence of a shift in future houseprice expectations, a relaxation of the payment-to-income constraint or a decline in mortgageorigination costs. In addition, all three changes feed back onto current prices, making houses more expensive. 2.5. Eventual default The actual default decisions of borrowers occur in the subsequent period, and they depend on the house price that emerges in that period. To analyze default, the realized value of P must be inserted into the default condition (2) along with the balance due B owed by a given borrower, which is shown by the curve in Figure 1. To write the balance-due as a function of the relevant variables, let (3) be rewritten to include the shift parameter δ. Then, setting (3) equal to zero, the condition determines B as a function of C and the other variables P0 and δ, written as B = φ(C, P0 , δ). This function is decreasing in C, as seen above, and it is also increasing in P0 and decreasing in δ.12 Let C denote the critical value of C that makes a borrower indiﬀerent in the subsequent period between defaulting and not doing so. Referring to (2), C satisﬁes
= φ(C, P0 , δ) − C.
Since φ is decreasing in C, it follows that φ(C, P0 , δ) − C is also decreasing in C, which means that P < φ(C, P0 , δ) − C holds for C < C. Thus, after taking account of B’s dependence on C, actual defaulters are those borrowers with the lowest default costs. For future reference, note that (17) determines the critical value C as function of the remaining variables in the equation. This function, which is written C(P, P0 , δ), is decreasing in P given that a larger realized house value makes a borrower less prone to default, requiring C to fall below a lower critical value to make default desirable. In addition, since a larger P0
raises B, higher default costs are required to forestall default, so that C is increasing in P0 . Finally, since a higher δ reduces the required B, lower default costs are required to induce default, making C a decreasing function of δ.13 While this discussion is silent about how the realized P value is determined, this omission is remedied below when the model is embedded in a dynamic setting.
3. Making the Model Dynamic: An Example
While the paper’s empirical work relies only on Proposition 2, further insight can be gained by using the model to carry out a dynamic analysis. In a dynamic setting, a one-time shift in house-price expectations leads to an intemporal adjustment process. Analysis of this process can give insight into the evolution of housing and mortgage markets in the presence of a housing bubble.
t The ﬁrst step is to attach a time index t to P0 and C, which are now written P0 and C t.
The realized house price in the period following t then equals the P0 value for that period, or
t+1 P0 . The next step is to specify exactly how the lender’s house-price expectations are formed,
a question that could be skirted in the analysis up to this point. While the expectations density f will depend on the entire past history of house prices, as mentioned above, this dependence is assumed to take a very simple form in order to permit a tractable analysis. Given this limitation, the ensuing discussion should be viewed as only providing an example of how the model might behave in a dynamic setting. The simplifying assumption is that the position of the density of anticipated prices depends on the previous period’s house price, with a change in that price shifting the density without changing its shape. Thus, the time-t density for the next period’s anticipated price is written
t−1 t−1 f(P − P0 ), so that a given increase in P0 shifts the density to the right by the amount
of the increase.14 This assumption could correspond, for example, to a situation where f is
t−1 t−1 symmetric and centered at P0 , so that a change in P0 alters the mean of the distribution
while its variance is independent of the past history. In addition, a change in expectations, as reﬂected in an increase in δ, is assumed to shift the density in the same way as the past price,
t−1 so that when δ is nonzero, the density is written as f(P − P0 − δ).
Incorporating these changes, the zero-proﬁt condition (14) at time t is rewritten as
t B(P0 )−C t
t−1 − P0 − δ)dP
t B(P0 )−C t
t t−1 B(P0 )f(P − P0 − δ)dP = 0.
(18) To adapt the supply-demand condition in (13) to a dynamic setting, recall that H in (13) is the number of housing units inherited from the previous period. Assuming that housing is perfectly durable (unless intentionally removed from the stock), H simply equals the number of mortgage borrowers in the previous period. Therefore, (13) at time t becomes
t N − G(C t−1 ) + S(P0 ) − [N − G(C t )] = 0.
Finally, the dynamic setup involves a particular assumption on the fate of borrowers who default on their mortgages.15
t−1 t−1 t Since P0 enters (18), the position of the π locus relating P0 and C t depends on P0 . An t−1 increase in P0 shifts that locus to the left, just like the increase in δ analyzed above. Similarly,
the position of the s-d locus depends on C t−1 , with an increase in C t−1 shifting the locus to the right.16 Note that, with a higher C t−1, a smaller housing stock is handed forward from time t − 1, so that supply-demand equilibrium at t requires fewer current mortgage borrowers
t (a higher C t) for any given P0 .
Conditions (18) and (19) constitute a system of diﬀerence equations that governs the evolution of the C and P0 variables over time. Suppose the economy is in a steady state up to time τ − 1. The steady-state price equals the zero-supply price P ∗ , and the steady-state C is denoted C ∗ .17 Then, suppose that the equilibrium is perturbed at time τ by a shift in house-price expectations, with δ becoming positive. The π locus shifts as in Figure 2, with P rising and C falling at time τ , as shown again in Figure 3. The equilibrium moves from the steady-state values P ∗ and C ∗ , which lie at the intersection of the steady-state π ∗ and s-d∗ locii, to the values at the intersection of π τ and s-d∗ locii. At time τ + 1, the expectations shock is passed, with δ returning to zero. But the time-τ changes in P and C aﬀect the positions of the π and s-d locii at τ + 1, following the above
discussion. Since C τ +1 < C ∗ , the s-d locus at time τ + 1 lies to the left of the steady-state locus, as seen in Figure 3. To locate the π τ +1 locus, it can be shown that the house price at
τ time τ does not rise by the full amount of the expectations shift, so that P0 < P ∗ + δ.18 As τ a result, the π locus at τ + 1 (whose position depends on P0 ) is not as far to the left as the
π τ locus (whose position depends on P ∗ + δ), as shown in Figure 3. Given these two shifts, it
τ τ follows that the house price at τ + 1 must be lower than at τ , with P0 +1 < P0 . Depending on
the exact positions of the curves, C could either rise or fall between times τ and τ + 1, with Figure 3 showing the latter case.19 This information can be used to investigate the time path of defaults. Letting C t denote, for time t, the critical C value below which default occurs, the default rate at t is given by G(C t ) − G(C t−1 ) N − G(C t−1)
The denominator equals the number of consumers getting mortgages at t − 1 (whose C values lie above C t−1) and the numerator equals the number of these individuals who default at t (whose C values lies above C t−1 and below C t). At time τ − 1, the G arguments in (20) equal the starred, steady-state values, yielding the steady-state default rate D∗ . The increase in the house price at τ reduces the incentive for default, pushing the critical C value below the steady-state level, so that C τ < C ∗ .20 Since C τ −1 equals the steady value C ∗ , it follows from (20) that the default rate declines at time τ , with Dτ < D∗ . In moving from time τ to τ + 1, the default rate is aﬀected by changes in both C and the lagged C. Diﬀerentiation of (20) shows Dt rises when C t−1 decreases, so that an expansion in subprime lending raises the subsequent default rate, holding C ﬁxed.21 Thus, the decline in the lagged C from C τ −1 = C ∗ to C τ < C ∗ tends to increase the default rate between times τ and τ + 1. However, the change in C between these periods is ambiguous.22 Although the drop in P0 raises the incentive to default, which tends to increase C, the higher past price and the previous period’s expectation shift have opposing eﬀects on the balance-due owed at time τ + 1, which makes the direction of change in C unclear. However, if the C eﬀect dominates, the default rate will rise. Summarizing yields
Proposition 3. Under the maintained assumptions, a house-price expectations shift at time τ raises the current house price P0 above the steady-state level while reducing the default rate D at τ relative to its own steady-state level. In the subsequent period (τ + 1), the house price falls relative to the level at time τ . The change in the default rate is ambiguous, but a large expansion in subprime lending at τ will tend to make it positive. The ambiguity regarding Dτ +1 as well as the evolution beyond time τ + 1 can be explored numerically. The calculations assume that the density f at time t is uniform, with height 1/2µ
t−1 t−1 over the interval [P , P ] = [P0 + δ − µ, P0 + δ + µ]. In addition, default costs are uniformly
distributed, and the supply function is linear. Under the parameter values listed in the appendix, P ∗ = 5.0 and C ∗ = 2.2. At time τ , P0 rises to 5.36 and C falls dramatically to 0.67, as seen in Figure 4. From a steady-state value of 1.7 percent, the default rate at time τ drops to zero, as seen in Figure 5. At time τ + 1, P0 falls almost all the way back to the steady-state value, dropping to 5.01, while C declines further to 0.57 (matching Figure 3). The previous period’s expansion of subprime lending dominates in determining the change in the default rate, which surges to 40 percent at time τ + 1, a number that approximates actual experience.23 At time τ + 2, the house price drops below the steady-state level while C rises almost back to C ∗ , indicating a drop in subprime lending. The default rate declines only slightly.24 The subsequent evolution of P0 and C is depicted in Figure 4, which shows convergence back to the steady state.25 Close inspection of the adjustment paths shows that they exhibit the cyclical convergence seen in Figure 3, an outcome that is robust to variations in the underlying parameter values. Figure 5 shows that the default rate also convergences cyclically back to its steady-state value, spending several periods at zero before returning to D∗ . Given the analytical and simulation results, several observations can be made about the intertemporal path generated by the shock to house-price expectations. In response to the initial shock, subprime lending surges as C declines, and the resulting increase in demand pushes up the current house price and enlarges the housing stock. While this price increase partly sustains favorable price expectations for the subsequent period, the optimism is weaker than under the initial shock. This fact (captured in the π locus’s rightward shift), together with
the larger housing stock handed forward from time τ (reﬂected in the s-d locus’s leftward shift), pushes down the housing price at time τ + 1. This drop, along with the presence of new low-C borrowers, leads to a spike in defaults. As the adjustment process continues, subprime lending eventually abates, with P0 , C and the default rate ultimately returning to their steady-state values.26
4. Empirical Evidence
4.1. Data The discussion now turns to a description of the empirical work designed to test the predictions contained in Proposition 2. Credit risk scores, drawn from the New York Fed’s Consumer Credit Panel database, are used to measure the extent of subprime lending in a state. The database is a 5 percent sample of the entire US credit-bureau population (individuals with credit information), taken as a panel. The same set of social security numbers is drawn each quarter. The risk-score measures are scaled like the FICO scores that are also commonly used as a criterion for mortgage lending, and they are similarly based on credit-bureau data (the sample score range is from 280 to 850). In the data, the borrower’s risk score is tabulated when a new mortgage is originated in a given quarter. Risk scores are then aggregated by state for each quarter, with the state’s mean score, the 10th percentile score, and the 25th percentile score computed. The sample covers the 32 quarters running from 2001Q1 to 2008Q4. The sample of borrowers is split into repeat home buyers, reﬁnancers, investors and ﬁrsttime buyers. A repeat buyer is a borrower who had a previously recorded mortgage and changed address upon receiving the new mortgage. A reﬁnancer is a borrower with a previously recorded mortgage who did not change address in the quarter before or the two quarters after receiving the new mortgage. An investor is a borrower who has two additional ﬁrst-lien mortgages with positive balances when the new mortgage was received. Identiﬁcation of ﬁrst-time home buyers is less straightforward than for these other categories. A ﬁrst-time buyer is a borrower who did not have a recorded mortgage in the prior four quarters, who was 40 years old or younger, and whose oldest account still active (as recorded by the credit bureau) was less than or equal to 240 months from its time of origination. Although
some households could have paid oﬀ an unobserved previous mortgage well before borrowing again, thus looking like ﬁrst-time buyers, the restriction to households no older than 40 with credit histories shorter than 20 years lessens the chance of such misidentiﬁcation since few borrowers with these characteristics could have paid oﬀ a previous mortgage. Another potential problem is that some prior mortgages may not be recorded in credit-bureau records, although in recent years this oversight is likely to be uncommon. Additional variables include quarterly state house price indexes from Core Logic, quarterly state unemployment rates from the Bureau of Labor Statistics, and the Conference Board’s quarterly regional Consumer Conﬁdence Index for the nine U.S. Census regions. Another variable is real quarterly state personal income, constructed by Haver Analytics from state personal income data deﬂated using the PCE chained price index, both from the Bureau of Economic Analysis. These latter three variables, which were downloaded from Haver DLX databases, control for demand-side eﬀects, including the possibility that the risk composition of borrowers varies with the economic cycle.27 4.2. Empirical model The empirical model portrays lender expectations somewhat diﬀerently from the theoretical model of section 3. For tractability, that model assumed that the location of the density of anticipated house prices depended on the level of the previous period’s price, with an expectation shock having the same eﬀect on the density as an exogenous increase in the previous price. A theoretically less tractable but more realistic approach for empirical purposes is to assume that rapid past price appreciation, rather than a high past price level, produces favorable expectations for the future price. In this setting, an expectations shock would be equivalent to an exogenous increase in the past appreciation rate. The empirical work does not provide an internal test of this assumption that lenders extrapolate past appreciation forward in predicting future house prices. But such behavior seems reasonable, and it is supported by indirect evidence.28 Any empirical speciﬁcation relating borrower risk scores to past price appreciation must, however, confront a simultaneity problem. Past appreciation may encourage subprime lending, but the feedback eﬀect from this lending will raise current house prices and thus the rate of
appreciation through the current period. In other words, since housing demand at time t will depend on the volume of new subprime mortgages originated at t, it follows that price appreciation over the period ending at t will be aﬀected by this lending volume and hence by the borrower risk-score measure at time t. To isolate the reverse causal path, where fast price appreciation signals high future prices and thus spurs subprime lending (depressing risk scores), the appreciation rate must be lagged, unlinking it from the current price. Accordingly, the empirical model relates the aggregate risk-score measure in state j in quarter t, denoted RISKSCOREjt , to the one-year (4-quarter) lag of annual house-price appreciation, denoted HP ICHGjt−4 . This variable is computed as (HP Ijt−4 − HP Ijt−8 )/HP Ijt−8 , where HP I is house-price index. A second version of the model instead uses the sixth-month lag of annual house-price appreciation, HP ICHGjt−2 , computed as (HP Ijt−2 − HP Ijt−6 )/HP Ijt−6 . RISKSCOREjt is assumed to also depend on a vector of additional covariates, denoted Xjt . These variables include the state unemployment rate for the given quarter (UNRjt ), per capita income (P Cjt), and the regional consumer conﬁdence index (CCjt). Also included in Xjt are state ﬁxed eﬀects, captured by a vector S of state dummy variables that takes the value Sj for state j, and quarterly ﬁxed eﬀects, captured by a vector T of quarter dummy variables that takes the value Tt for quarter t. By capturing economic well-being at the household level, UNRjt and P Cjt help determine borrower creditworthiness and thus individual risk scores, so that their presumed eﬀects on the aggregate measure are respectively negative and positive.29 The consumer-conﬁdence variable may help determine lender optimism about the future, while also capturing economic well-being. Under the ﬁrst interpretation, an increase in CCjt should lead to a reduction in the aggregate risk-score measure as lenders serve riskier borrowers, but under the second interpretation, the eﬀect of CCjt could be positive. State and quarterly ﬁxed eﬀects are also elements of Xjt . The equation determining RISKSCOREjt is thus written
RISKSCOREjt = β0 + β1HP ICHGjt−4 + β2 Xjt +
is an error term. Under the maintained hypothesis, β1 is negative.
Even though use of the lagged appreciation rate eliminates a main source of potential simultaneity in the empirical model, a residual eﬀect may remain as a consequence of serial correlation in the error terms in (21). To understand this point, observe that the feedback eﬀect from subprime lending to prices can be captured by an equation relating HP ICHGjt−4 to RISKSCOREjt−4 and additional covariates Zjt−4 , which is written HP ICHGjt−4 = γ0 + γ1 RISKSCOREjt−4 + γ2 Zjt−4 + ηjt−4 . (22)
Note that since an expansion of subprime lending (a lower RISKSCOREjt−4 ) raises price appreciation, γ1 < 0 holds. The error term η in (22) is assumed to be uncorrelated with from (21), with ηjr and
being uncorrelated for any r, s, j and k. The Zjt−4 vector would
include the Xjt−4 variables UNRjt−4 , P CIjt−4 and CCjt−4 , all of which are shifters of housing demand, as well as additional demand shifters such as the annual rate of population change for the state. To see that serial correlation in the ’s poses a threat to consistent estimation, let (21) be lagged four quarters to generate RISKSCOREjt−4 , with the result substituted into (22). After simplifying, the resulting equation can be written HP ICHGjt−4 = θ0 + θ1 HP ICHGjt−8 + θ2 Zjt−4 + γ1 Because HP ICHGjt−4 depends on with the error term
+ ηjt−4 .
from (23), HP ICHGjt−4 in (21) will be correlated
if the ’s are themselves serially correlated. This correlation will in turn
bias the ordinary least-squares estimates of (21). This source of bias can be eliminated by applying an autoregressive transformation to the model. Suppose that the follows an annual AR process, so that
jt−4 + vjt ,
where ρ is
the autoregressive parameter and the vjt are i.i.d. error terms.30 Then, lagging (21) by four quarters, multiplying by ρ and subtracting the result from (21) yields RISKSCOREjt = (1 − ρ)β0 + ρRISKSCOREjt−4 + β1 HP ICHGjt−4 − ρβ1HP ICHGjt−8 + β2 Xjt − ρβ2 Xjt−4 + vjt
Since HP ICHG depends on the contemporaneous
from (23) and thus on the contempora-
neous v, it is independent of later v’s.31 As a result, both HP ICHGjt−4 and HP ICHGjt−8 in (24) are independent of vjt, eliminating the correlation between the right-hand variable and the error term that prevents consistent estimation of (21). To estimate the model, the usual approach would rely on the OLS residuals from (21) to estimate ρ, with the result substituted in (24) and the β coeﬃcients then estimated by OLS. However, since OLS estimation of (21) is inconsistent given the correlation between HP ICHGjt−4 and the error term, the resulting estimate of ρ is inconsistent as well. A diﬀerent approach that circumvents this problem is to estimate (24) by nonlinear least squares. The next section presents the results of this approach along with the inconsistent OLS estimates for comparison purposes. Nonlinear squares estimates are also presented for the alternative case of where (21) uses a two-quarter rather than four-quarter lag of annual houseprice appreciation (HP ICHGjt−2 rather than HP ICHGjt−4 ).32 In order to give bigger states (with their larger number of mortgage borrowers) more weight, (24) is estimated in weighted fashion, with quarterly observations for each state weighted by the average annual borrower count for the state. Signiﬁcance tests are based on robust standard errors.33 Table 1 shows the summary statistics for the variables used in the model and gives their deﬁnitions. One noteworthy comparison is that each of the three RISKSCORE measures for ﬁrst-time buyers is lower than the corresponding measure for the other borrower groups. Before presenting the empirical results, it is important to recognize that borrower risk scores and prior house-price appreciation may be linked for a reason diﬀerent from the one envisioned in the model. In particular, since rapid prior appreciation will raise the home equity of repeat buyers and reﬁnancers, a lower loan-to-value ratio becomes feasible on a new mortgage. The resulting reduction in default risk may in turn make lenders more willing to extend mortgages to borrowers with lower risk scores, generating an inverse relationship between prior appreciation and risk scores like that predicted by the model. This linkage, however, is not operative in the case of ﬁrst-time buyers, for whom prior appreciation does not generate a wealth gain. Therefore, the regression results for ﬁrst-time buyers provide a crucial way of distinguishing between these two sources of correlation between prior appreciation and
borrower risk scores.34 4.3. Results Table 2 shows the results for the subsample of repeat buyers. In the table’s ﬁrst threecolumn block, the aggregate risk score measure is the mean state risk score for this borrower group. The two remaining three-column blocks show results when the aggregate measures are the 10th and 25th percentile state risk scores, respectively. Within the ﬁrst column block, the ﬁrst regression shows the results of estimating (21) by OLS using the mean RISKSCORE as dependent variable along with a four-quarter HP ICHG lag. The HP ICHGt−4 coeﬃcient (β1 in (21)) is negative and statistically signiﬁcant, suggesting that more rapid past price appreciation lowers the mean risk score among current repeat buyers, reﬂecting an expansion of subprime lending. The estimates are thus consistent with the empirical hypothesis that subprime lending is spurred by favorable price expectations. However, the OLS estimates are inconsistent, as explained above, and the next two regressions show the results from consistent nonlinear estimation of the transformed model. The second column, which is again based on a four-quarter lag, yields a signiﬁcantly negative HP ICHGt−4 coeﬃcient and a signiﬁcant ρ estimate of about 0.2. The β1 coeﬃcient is smaller in absolute value than in the OLS regression, suggesting that the β1 estimate under OLS is downward biased, a conclusion that also follows from the model. To see this point, note that since HP ICHGjt−4 is inversely related to
by (24) (recall γ1 < 0), a negative correlation
exists between HP ICHGjt−4 and the error term bias in the OLS estimate of β1 .
in (21) when ρ > 0, leading to downward
These more reliable estimates again support the hypothesis that subprime lending is spurred by favorable price expectations. But to see whether this conclusion is robust to the period over which past price appreciation is measured, the last regression in the ﬁrst column block shows the nonlinear estimation results when the lag in (21) is two quarters rather than four, with HP ICHGjt−4 replaced by HP ICHGjt−2 . The β1 estimate is again signiﬁcantly negative and similar in size to the previous values, and the ρ estimate is now larger at 0.26, a diﬀerence that makes sense given the shorter lag. Among the other covariates, only the consumer conﬁdence index has a signiﬁcant eﬀect on
the mean risk score. The positive direction of the eﬀect suggests that CCt is capturing current economic well-being, which generates high borrower risk scores, rather than proxying favorable lender expectations about the future, which would encourage lending to riskier borrowers. The regressions shown in the remaining column blocks of Table 2 use the 10th and 25th percentile state risk scores as dependent variables, and the estimates are similar to those based on the mean risk score. The β1 coeﬃcients are again signﬁcantly negative, and an increase in the consumer conﬁdence index again leads to higher risk scores. Note also that the positive expected eﬀect of state per capita income emerges in the ﬁrst two 25th-percentile regressions. Since the model predicts that more-favorable price expectations should spur lending to the riskiest borrowers, one might expect lagged HP ICHG to have a stronger eﬀect at lower percentiles of the risk-score distribution. A comparison of the nonlinear four-quarter-lag results for the 10th and 25th percentiles conﬁrms this expectation. The elasticity of the 10th percentile risk score with respect to the lagged HP ICHG is more negative than the 25th percentile elasticity, an outcome that can be seen by noting that the larger absolute value of the 25th percentile β1 is more than oﬀset by the larger 25th percentile riskscore in the ratio used to compute the elasticity (see Table 1). In contrast, the nonlinear results with a two-quarter lag yield elasticities that are about equal. This outcome suggests that the ﬁrst elasticity pattern may not be robust, a conclusion that is reinforced below in the case of ﬁrst-time buyers. The explanation could be that the 25th percentile risk score already embraces the bulk of subprime borrowers, with impacts at the 10th percentile governed by other considerations. To gauge the quantitative implications of the results, note that the mean regressions in Table 2 imply that a 10 percentage-point increase in past annual house-price appreciation reduces the mean risk score among repeat buyers by about 4 points. According to the model, this reduction comes from a decrease in the minimum risk score (C) due to an expansion in subprime lending. Further analysis shows this drop in the lowest risk score will equal some multiple of the 4-point decrease in the mean score.35 Thus, the results are consistent with an appreciable reduction in the risk score of the worst borrowers receiving mortgages in response to higher past price appreciation price. Table 3 shows the results for the case of reﬁnancers. The main patterns established in the
repeat-buyer case again emerge. All the β1 coeﬃcients are signiﬁcantly negative, with the OLS estimates larger in absolute value than the nonlinear estimates for the four-quarter-lag case. HP ICHG elasticities are larger for the 10th percentile case than at the 25th percentile, now for both the four and two-quarter-lag speciﬁcations. As for the other covariates, the positive consumer conﬁdence eﬀect disappears in the nonlinear regressions, while the positive per capita income eﬀect now emerges more consistently. Although the estimates in Tables 2 and 3 are consistent with the main hypothesis, they could also reﬂect favorable wealth eﬀects from past price appreciation, which would allow riskier borrowers among the repeat-buyer and reﬁnancer groups to secure mortgages. As explained above, ﬁrst-time buyers experience no such wealth eﬀect, which means that they oﬀer a more stringent test of the main hypothesis. The ﬁrst-time buyer results are shown in Table 4, and they mostly conform to previous patterns. When the dependent variable is the mean risk score or the 25th percentile score, the β1 coeﬃcients are again signiﬁcantly negative, although the OLS and nonlinear 4-quarter-lag coeﬃcients are now similar in size. In contrast, the OLS and nonlinear β1 estimates are insigniﬁcant when the dependent variable is the 10th percentile risk score. While this result is unexpected, the fact that the anticipated negative eﬀect of the lagged HP ICHG still emerges for ﬁrst-time buyers near the bottom of the risk score distribution (at the 25th percentile), as well as emerging at the mean, supports the main hypothesis. Among the other covariates, the positive per capita income eﬀect is again present, while the consumer conﬁdence eﬀect is more consistently positive than in the reﬁnancer case, mirroring the repeat-buyer results. Table 5 shows the regression for the investor case, and the results once again show negative β1 coeﬃcients. As in the case of ﬁrst-time borrowers, the OLS and nonlinear four-quarter-lag estimates are similar in size. In contrast to previous results, the coeﬃcients of the other covariates are mostly insigniﬁcant, with the few signiﬁcant cases showing negative rather than positive signs. Overall, the empirical results provide considerable support for the empirical hypothesis that faster past price appreciation, by raising lender optimism about future prices, spurs lending to riskier borrowers. Although a wealth eﬀect could contribute to the observed association
between past appreciation and borrower riskiness for previous homeowners, the evidence that this association is also present for ﬁrst-time buyers lends substantial credence to the maintained hypothesis.
This paper has explored the link between the house-price expectations of mortgage lenders and the extent of subprime lending. It argues that bubble conditions in the housing market are likely to spur subprime lending, with favorable price expectations easing the default concerns of lenders and thus increasing their willingness to extend loans to risky borrowers. Since the demand created by this subprime lending feeds back onto house prices, the lending also helps to fuel an emerging housing bubble. The paper, however, focuses mostly on the reverse causal linkage, where subprime lending is a consequence rather than a cause of bubble conditions, further exploring what Mian and Suﬁ (2009) call the “expectations hypothesis.” The paper’s theoretical model portrays subprime lending as the extension of loans to borrowers with low observable “default costs,” and the analysis shows that a favorable shift in house-price expectations spurs such lending by encouraging loans to borrowers with even lower default costs (and thus higher default risks). The empirical work shows that rapid past appreciation, which is presumed to generate favorable expectations regarding future prices, does indeed lead lenders to extend mortgages to riskier borrowers (those with worse credit ratings). By providing a deeper theoretical grounding for the “expectation hypothesis” than prior work as well as a more-direct empirical test, the paper advances our understanding of the housing crisis. A better understanding of this watershed economic event is crucial in formulating policies to prevent its repetition, and this paper may add some of the required insights.
Derivation of (12) Integrating by parts, the proﬁt expression in (9) can be rewritten as
− k − P0 + η
(B − C)F (B − C, δ) −
F (P, δ)dP + B[1 − F (B − C, δ)]
k − P0 − η CF (B − C, δ) +
F (P, δ)dP − B .
Diﬀerentiating (a1) with respect to δ yields (11). Simulation assumptions The height of the uniform default-cost density is 1/g, where g = C −C, and the cumulative distribution function in (19) is then G(C) = (C − C)/g (the mass N of potential mortgage t t borrowers is normalized to unity). Furthermore, with linear housing supply, S(P0 ) ≡ nP0 − k, ∗ = k/n. Given these latter assumptions, (19) reduces to implying P
t C t − C t−1 + g(nP0 − k)
t t Letting a ≡ αy, so that B(P0 ) = a + P0 , and assuming η = 1, (18) reduces to t t−1 t t−1 (C t)2 − (a + P0 )2 + (P0 + δ − µ)[2(a + P0 ) − (P0 + δ − µ)] + 4aµ = 0.
t To carry out the simulation, (a2) is used to eliminate C t in (a3), so that P0 can be written t−1 t as a function of the prior values P0 and C t−1. With P0 then known given these values, t−1 substitution in (a2) determines C t as a function of P0 and C t−1. The time paths shown in Figure 4 reﬂect the following additional parameter values: a = 2.28, µ = 4.5, k = 10, n = 2 and g = 4. Under these parameter values, B < P holds at each point in time for all borrowers, so that πB in (7) is positive as assumed (note that B − C < P then also holds). In addition, assuming C = 0, the condition B − C > P is satisﬁed at each point in time for all borrowers. To compute default rates, the uniformity assumption is imposed in (18) (lagged one period), and the condition is solved for B to yield t−2 t−1 t−2 B = P0 + µ + δ − C 2 + 4δµ − 4µP0 + 4µP0 1/2
t This solution with C replaced by C is then substituted into the condition P0 = B − C to yield t−1 t−2 t C as a function of P0 , P0 , P0 and δ.
Figure 1: The relationship between B and C
Figure 2: The effect of a shift in house-price expectations
Figure 3: Adjustment to a house-price-expectation shock
Figure 4: Simulation Results
0.00000 τ−1 τ τ+1 τ+2 τ+3 τ+4 τ+5 τ+6 τ+7 τ+8 τ+9 τ+10 τ+11
Minimum borrower quality
Figure 5: Default rate
0.450 0.400 0.350 0.300 0.250 0.200 0.150 0.100 0.050 0.000 τ−1 τ τ+1 τ+2 τ+3 τ+4 τ+5 τ+6 τ+7 τ+8 τ+9 τ+10 τ+11
Table 1. Summary statistics N 1600 1600 1600 1600 1600 1600 1600 First time buyers Mean SD 689.0 15.47 585.4 636.4 4.92 95.52 5.23 35.25 20.52 20.26 1.17 21.90 8.07 11.54 Repeat buyers Mean SD 722.62 12.96 627.66 683.70 4.92 95.52 5.23 35.25 20.89 18.21 1.17 21.90 8.07 11.54 Refinancers Mean SD 715.6 16.36 610.4 669.8 4.92 95.52 5.23 35.25 25.61 23.90 1.17 21.90 8.07 11.54 Investors Mean 735.3 649.0 698.9 4.92 95.52 5.23 35.25 SD 17.44 29.39 25.52 1.17 21.90 8.07 11.54
Risk score – mean RISKSCOREM Risk score – 10th percentile RISKSCORE10 Risk score – 25th percentile RISKSCORE25 Unemployment rate UNR Consumer confidence ind. CC State house price inflation HPICHG Per capita income PCI
Table 2. RISKSCORE regressions for repeat buyers – 2001Q1 – 2008Q4 Mean OLS: 4Q lag Constant 467.779** (4.822) NL: 4Q lag 714.933** (7.551) 0.186** (0.033) -0.469** (0.047) -0.436** (0.046) -0.395** (0.048) 0.109** (0.018) 0.002 (0.332) 0.082** (0.020) 0.246 (0.346) 0.073** (0.021) 0.143 (0.377) 0.238** (0.035) -0.910 (0.657) 0.219** (0.037) -0.724 (0.672) NL: 2Q lag 716.652** (8.067) 0.264** (0.033) -0.647** (0.087) OLS: 4Q lag 407.462** (10.211) 10th Percentile NL: 4Q lag 611.468** (14.959) 0.104** (0.029) -0.625** (0.086) -0.501** (0.086) 0.196** (0.038) -0.461 (0.696) 0.176** (0.028) -0.345 (0.523) 0.145** (0.030) -0.002 (0.546) NL: 2Q lag 610.444** (15.635) 0.142** (0.032) -0.682** (0.072) OLS: 4Q lag 437.864 (8.029) 25th Percentile NL: 4Q lag 660.194** (12.205) 0.139** (0.032) -0.637** (0.071) -0.553** (0.074) 0.129** (0.033) -0.004 (0.585) NL: 2Q lag 662.026** (13.036) 0.211** (0.034)
HPICHGt-4 (β1) HPICHGt-2 (β1) CCt
UNRt (β2) PCIt (β2)
0.275 0.388 0.223 0.534 0.634 0.490 0.657* 0.822* 0.605 (0.171) (0.194) (0.200) (0.352) (0.382) (0.388) (0.283) (0.317) (0.323) Standard errors in parentheses. Significance: **1%, *5% N=1600 for each regression. The regressions use robust standard errors, and coefficients of the quarter and state dummy variables are not reported.
Table 3. RISKSCORE regressions for refinancers – 2001Q1 – 2008Q4 Mean OLS: 4Q lag Constant 839.825** (13.426) NL: 4Q lag 696.735** (12.117) 0.505** (0.056) -0.631** (0.084) -0.454** (0.051) -0.464** (0.064) 0.151** (0.022) 0.089 (0.468) 0.026 (0.036) 0.991* (0.500) -0.017 (0.034) 0.866 (0.483) 0.300** (0.039) 0.376 (0.898) 0.110 (0.065) 1.748 (1.060) NL: 2Q lag 671.574** (16.270) 0.612** (0.058) -0.891** (0.167) OLS: 4Q lag 654.739** (25.945) 10th Percentile NL: 4Q lag 521.663** (24.433) 0.412** (0.060) -0.687** (0.118) -0.679** (0.131) 0.033 (0.067) 1.644 (0.993) 0.268** (0.037) 0.171 (0.797) 0.049 (0.064) 1.411 (0.832) NL: 2Q lag 498.200** (30.094) 0.557** (0.079) -0.916** (0.146) OLS: 4Q lag 742.111** (22.600) 25th Percentile NL: 4Q lag 620.935** (20.992) 0.555** (0.068) -0.608** (0.082) -0.595** (0.102) -0.018 (0.058) 1.386 (0.829) NL: 2Q lag 581.994** (27.638) 0.663** (0.070)
HPICHGt-4 (β1) HPICHGt-2 (β1) CCt
UNRt (β2) PCIt (β2)
0.413 0.847** 0.840* 1.646** 2.406** 2.227** 1.246** 1.741** 1.519** (0.256) (0.309) (0.354) (0.502) (0.632) (0.660) (0.433) (0.492) (0.566) Standard errors in parentheses. Significance: **1%, *5% N=1600 for each regression. The regressions use robust standard errors, and coefficients of the quarter and state dummy variables are not reported.
Table 4. RISKSCORE regressions for first-time buyers – 2001Q1 – 2008Q4 Mean OLS: 4Q lag Constant 685.443** (7.656) NL: 4Q lag 663.657** (8.140) 0.264** (0.031) -0.152** (0.046) -0.149** (0.044) -0.162** (0.044) 0.048* (0.020) -0.487 (0.274) 0.041 (0.022) -0.315 (0.279) 0.034 (0.023) -0.339 (0.327) 0.0163** (0.025) 0.060 (0.383) 0.159** (0.027) 0.056 (0.389) NL: 2Q lag 658.563** (9.104) 0.382** (0.036) -0.085 (0.059) OLS: 4Q lag 589.630** (10.872) 10th Percentile NL: 4Q lag 564.056** (10.899) 0.450** (0.030) 0.100 (0.060) -0.082 (0.057) 0.143** (0.028) 0.074 (0.455) 0.122** (0.030) -0.611 (0.366) 0.118** (0.032) -0.407 (0.375) NL: 2Q lag 561.001** (12.103) 0.277** (0.035) -0.170* (0.067) OLS: 4Q lag 613.835** (10.632) 25th Percentile NL: 4Q lag 592.554** (11.176) 0.266** (0.033) -0.176** (0.065) -0.153* (0.065) 0.101** (0.035) -0.175 (0.431) NL: 2Q lag 583.233** (12.331) 0.380** (0.037)
HPICHGt-4 (β1) HPICHGt-2 (β1) CCt
UNRt (β2) PCIt (β2)
0.769** 0.790** 0.796** 0.508* 0.525* 0.533 1.245** 1.224** 1.283** (0.163) (0.194) (0.221) (0.237) (0.263) (0.292) (0.228) (0.269) (0.299) Standard errors in parentheses. Significance: **1%, *5% N=1600 for each regression. The regressions use robust standard errors, and coefficients of the quarter and state dummy variables are not reported.
Table 5. RISKSCORE regressions for investors – 2001Q1 – 2008Q4 Mean Constant OLS: 4Q lag 273.698** (3.495) NL: 4Q lag 765.342** (9.861) 0.236** (0.028) -0.283** (0.042) -0.292** (0.040) -0.140** (0.047) 0.031 (0.028) -0.669 (0.399) -0.516* (0.222) 0.003 (0.030) -0.637 (0.422) -0.382 (0.247) -0.019 (0.031) -0.078 (0.466) -0.297 (0.270) 0.095 (0.052) -2.754** (0.923) 0.033 (0.492) 0.089 (0.053) -2.649** (0.949) 0.094 (0.509) NL: 2Q lag 755.881** (10.987) 0.309** (0.031) -0.556** (0.101) OLS: 4Q lag 241.608** (7.664) 10th Percentile NL: 4Q lag 670.811** (20.458) 0.073* (0.029) -0.555** (0.100) -0.315** (0.107) 0.044 (0.057) -1.571 (1.023) 0.232 (0.547) 0.067 (0.042) -0.781 (0.630) -0.465 (0.333) 0.042 (0.044) -0.769 (0.657) -0.409 (0.362) NL: 2Q lag 657.788** (22.343) 0.169** (0.031) -0.464** (0.065) OLS: 4Q lag 263.383** (5.292) 25th Percentile NL: 4Q lag 734.584** (14.626) 0.155** (0.029) -0.466** (0.064) -0.250** (0.073) -0.001 (0.046) 0.075 (0.726) -0.262 (0.397) NL: 2Q lag 721.917** (16.191) 0.255** (0.032)
HPICHGt-4 (β1) HPICHGt-2 (β1) CCt
UNRt (β2) PCIt (β2)
Standard errors in parentheses. Significance: **1%, *5% N=1600 for each regression. The regressions use robust standard errors, and coefficients of the quarter and state dummy variables are not reported.
´ ´ Arce, O., Lopez-Salido D., 2011. Housing bubbles. American Economic Journal: Macroeconomics 3, 212-241. Brueckner, J.K., 2000. Mortgage default with asymmetric information. Journal of Real Estate Finance and Economics 20, 251-274. Case, K.E., Shiller, R.J., 2003. Is there a bubble in the housing market? Brookings Papers on Economic Activity, Fall, 299-342. Coleman, M., LaCour-Little, M., Vandell, K., 2008. Subprime lending and the housing bubble: Tail wags dog? Journal of Housing Economics 17, 272-290. Foote, C.L., Gerardi, K., Willen, P.S., 2008. Negative equity and foreclosure: Theory and evidence. Journal of Urban Economics 64, 234-245. Glaeser, E.L., Gottlieb J.D., Gyourko, J., Can cheap credit explain the housing boom? NBER working paper #16230. Himmelberg, C., Mayer, C., Sinai, T., 2005. Assessing high house prices: Bubbles, fundamentals and misperceptions. Journal of Economic Perspectives 19, 67-92. Iacoviello M., Neri, S., 2010. Housing market spillovers: Evidence from an estimated DSGE model. American Economic Journal: Macroeconomics 2, 125-164. Kau, J.B., Keenan, D.C., Kim, T., 1993. Transaction cost, suboptimal termination and default probabilities. Journal of the American Real Estate and Urban Economics Association 21, 247-263. Kau, J.B., Keenan, D.C., Kim, T., 1994. Default probabilities for mortgages. Journal of Urban Economics 35, 278-296. Kiyotaki, N., Moore, J., 1997. Credit cycles. Journal of Political Economy 105, 211-248. Lambertini, L., Mendicino, C., Punzi, M., 2010. Expectations-driven cycles in the housing market. Unpublished paper, Munich Personal RePEc Archive. Mian, A., Sufi A., 2009. The consequences of mortgage credit expansion: Evidence from the U.S. mortgage default crisis. Quarterly Journal of Economics 124, 1449-1496. Riddiough, T., Thompson, H.E.. 1993. Commercial mortgage pricing and unobservable
borrower default costs. Journal of the American Real Estate and Urban Economics Association 21, 265-291. Wheaton, W.C., Nechayev, G., 2008. The 1998-2005 housing “bubble” and the current “correction”: What’s diﬀerent this time? Journal of Real Estate Research 30, 1-26.
thank Rainald Borck, Bill Branch, Kangoh Lee, and Pierre Picard for helpful comments, and we are especially grateful to David Brownstone for econometric advice and to Jason Novak for extensive help with the data. Errors or shortcomings, however, are our responsibility. and Nechayev (2008), by contrast, show a connection between subprime lending and house-price escalation. They start by estimating a regression relating prices to economic fundamentals through 1998 and then show that the regression underpredicts prices during the subsequent 1998-2005 period. The authors then demonstrate that the forecast errors from this regression are larger in MSAs with substantial subprime lending activity, establishing a connection between such lending and price growth. Mayer and Sinai (2005) argue against the presence of bubble conditions as of 2004 by showing that the user-cost of owner-occupied housing (based on historical appreciation rates) was similar to rent levels across diﬀerent MSAs. Under bubble conditions, user-cost would be far below current rents. et al. (2010) acknowledge the existence of this channel, and they test for it by regressing subprime lending on prior price growth (reversing their main regression). However, despite the possible existence of two-way causation, they do not use simultaneous-equations methods. term “bubble” in this paper is used in its popular sense, not in the technical fashion seen in the theoretical macro literature. papers in macroeconomics explore the factors that generate price volatility and bubbles in the housing market. Lambertini, Mendicino and Punzi (2010) present an empirical analysis showing a link between expectations and boom-bust housing cycles. Arce and L´pez-Salido (2011) view a housing bubble as an equilibrium where houses are held only o for resale (not generating rent or utility) and show that a low downpayment requirement is required to sustain such an equilibrium. Iacoveillo and Neri (2010) use a calibrated DSGE model to explore the sources of volatility in house prices. All this work draws on the classic paper of Kiyotaki and Moore (1997), which demonstrated the eﬀect of collateral constraints on the dynamic behavior of the economy.
example, moving costs may be an element of C in the absence of a trigger event, while disappearing from C when a trigger event is present, as follows. Without a trigger event that requires a move, moving costs will be zero in the absence of default (in which case the
borrower stays in the house), but the cost will be positive with default, in which case a move is required. In this case, moving cost is an element of C. Suppose instead that a trigger event such as unemployment occurs. Then, assuming the house becomes unaﬀordable, the borrower must move regardless of whether default occurs, so that moving costs no longer represent part of default costs. Thus, the trigger event depresses C, making default more likely, as seen in the ensuing discussion.
density f(P ) is presumably positive only over a subinterval within [0, ∞], which can be denoted [P , P ]. For (3) to be relevant, B − C must lie within this interval. Otherwise, the probability of default is either zero or one. curve is drawn as linear even though other shapes are possible. The same point applies to Figure 2 below. that α and y are multiplied in (8), the eﬀect of an increase in income y is identical to the impact of a higher α.
The reason is that, since P0 is a component of B, with the remainder equal to the interest payment, a larger P0 would shrink that payment if B were held ﬁxed, pushing it below αy. For the interest payment to remain at its maximal level, B must then increase with P0 . follows because ∂ B/∂P0 = 1 and πB < 1. Referring to (6), this latter inequality follows because η ≤ 1 and the integral in (6) is less than unity. conclusions follow because the modiﬁed version of (3) is increasing in B, decreasing in P0 , and increasing in δ. properties follow from diﬀerentiation of (17) and use of the properties of the φ function.
t−1 property follows because, if P0 is increased by an amount , then P must also be increased by to keep the height of the density at its original value.
borrowers are assumed to reenter the pool of potential mortgage borrowers without penalty. The potential-borrower pool thus consists of the same individuals from period to period, who need a mortgage to repurchase the housing they just relinquished either through sale or default. In reality, borrowers who default enter the rental market for housing, which is suppressed under the current setup. This suppression is not completely unrealistic given that renter status for defaulters is typical temporary, with new mortgages available to them after a few years. Another point is that any capital gains earned upon sale of a house do not aﬀect a borrower’s ability to secure another mortgage. By aﬀecting wealth, not income,
these gains do not loosen the payment-to-income constraint, and they cannot be used for a downpayment given the assumption of 100 percent loans.
16 When t C t−1 increases, C t must increase holding P0 ﬁxed to maintain the equality in (19).
17 C t−1
= C t must hold in the steady state, so that the number of borrowers 1 − G(C) and t thus the number of houses is constant over time. From (19), this requirement implies P0 = ∗ , ensuring that housing supply equals zero and that the stock is thus constant. Then, P t−1 t substituting P0 = P0 = P ∗ into (18) (along with δ = 0), the condition yields a steady-state value for C, denoted C ∗. conclusion follows from computing ∂P0 /∂δ = −πδ /πP0 from (18) and showing that it is less than unity. Since the upward shift in the π locus is thus less than δ, it follows from Figure 2 that the increase in P0 must also be less than δ. In other words, given the negative slope of the s-d locus, the increase in P0 must be smaller than the vertical shift of the π locus, which is itself less than δ.
τ relation between C τ +1 and C τ depends on whether P0 +1 is above or below P ∗ . If τ P0 +1 > P ∗ holds, then the housing stock is growing, and the number of mortgage borrowers τ must be rising, implying C τ +1 < C τ , as in Figure 3. If P0 +1 < P ∗ , then the housing stock is shrinking, implying C τ +1 > C τ .
see this point, recall that the function C(P, P0 , δ) from above indicates that the critical C value below which default occurs depends on the realized price (P ), last period’s price (P0 ), and the position of last period’s anticipated house-price density, as captured by δ. In the dynamic setting, the critical value C τ at time τ is found by replacing P in this function τ by P0 , P0 by P ∗ (the price at τ − 1), and δ by P ∗ , recognizing that the density position in τ the previous period is now represented by the lagged price (in this case P0 −2 = P ∗ ) plus the τ expectations shift, which is zero at time τ − 1. Thus, C τ = C(P0 , P ∗ , P ∗ ). By contrast, the steady-state value C is given by C ∗ = C(P ∗ , P ∗ , P ∗ ). Since, from above, C is decreasing in its ﬁrst argument, it follows that C τ < C ∗ , so that the critical value declines at τ . derivative of Dt with respect to C t−1 has the sign of G (C t−1 )[G(C t) − N] < 0.
τ τ critical C value at time τ + 1 is given by C τ +1 = C(P0 +1 , P0 , P ∗ + δ), recognizing that τ the position of the density at τ is captured by the lagged price P0 −1 = P ∗ plus the time-τ τ τ τ expectations shift. Relative to C τ = C(P0 , P ∗ , P ∗ ), the ﬁrst argument of C(P0 +1 , P0 , P ∗ + δ) = C τ +1 is smaller while the second and third are larger. Since C is increasing in the second argument and decreasing in the others, it follows that the relationship between C τ +1
and C τ is ambiguous.
declines slightly between τ and τ + 1, tending to reduce the default rate, but the muchlarger change in C dominates. further slight decline in C is oﬀset by a continuing decline in C.
24 A 25 If
the lender’s price expectations are myopic, with f independent of the past price history, it can be shown analytically that convergence back to the steady state is guaranteed. We are indebted to Pierre Picard for this demonstration, which holds in the general version of the model. relaxation of the payment-to-income constraint (an increase in α) or a decline in origination costs k leads to a similar adjustment process. But if either of these changes is permanent, the steady-state is altered, in contrast to the case of a one-time expectations shock. Either change shifts the π locus upward in Figure 3, leading to an initial expansion of subprime lending and an increase in the house price. With the steady state altered in each case, however, convergence to new equilibrium values occurs. The steady-state house price remains at P ∗ given that neither change aﬀects the supply-demand condition (19), but the steady-state value of C falls when α increases or k declines. Therefore, the new steady-state equilibrium reﬂects a permanent increase in subprime lending and a larger housing stock. Note that since the adjustment process generated by either of these changes involves falling house prices, it will exhibit a temporary surge in defaults like that in Figure 5. these quarterly numbers are averages of monthly values.
evidence comes from regressions of building permits on past price appreciation. Presumably, a large volume of building permits signals optimism about future prices. In a regression of the permit volume in state j and quarter t on past annual price appreciation along with state and quarter ﬁxed eﬀects, the appreciation coeﬃcient is signiﬁcantly positive. This outcome emerges regardless of whether appreciation is measured through the current quarter or whether the annual appreciation rate is lagged by two or four quarters (being measured through quarter t − 2 or quarter t − 4). Thus, optimism about future prices, as reﬂected in building permits, appears to be linked to past appreciation.
This presumption could be reversed by other subtler eﬀects. For example, high unemployment could keep riskier borrowers from even entering the housing market, leading to a positive association between risk scores and this variable. As will be seen below, the unemployment coeﬃcient is almost always insigniﬁcant in the regressions, suggesting that such oﬀsetting eﬀects may be at work.
AR structure can be derived from an underlying process where the autoregressive lag is a single quarter, with jt = λ jt−1 + ujt, where the ujt are i.i.d. error terms with variance σ 2. Successive substitution yields jt = ρ jt−4 + vjt, where ρ = λ4 and vjt = ujt + λujt−1 + λ2 ujt−2 + λ3 ujt−3 . In this case, the vjt ’s are correlated, with E(vjt , vjt−1) = (λ + λ3 + λ5 )σ 2, E(vjt, vjt−2) = (λ2 + λ4 )σ 2, E(vjt, vjt−3) = λ3 σ 2 , and E(vjt, vjt−k ) = 0 for k > 3. In the case where the HP ICHG lag in (21) is two quarters rather than four, jt = ρ jt−2 + vjt, where vjt = ujt + λujt−1 , E(vjt, vjt−1 ) = λσ 2 , and E(vjt, vjt−k ) = 0 for k > 1. way to see this point is by successive substitution for the HP ICHGj ’s and j ’s on the right-hand side of (23), which shows that HP ICHGjt−4 depends on vjt−4 and all the previous vj values. state and quarter dummies are introduced directly into the nonlinear model rather than relying on the usual de-meaning of the data used to produce ﬁxed eﬀects estimates in OLS. Although ρ will be embedded in the dummy coeﬃcients in (24), the fact that the estimation of the underlying ﬁxed eﬀects is not a goal means that this presence can be ignored in the nonlinear procedure. In contrast, ρ and the remaining β coeﬃcients in (24) must be disentangled in order to measure the eﬀects of interest. the main sources of within-state error correlation are removed by the inclusion of state dummies and the use of an autoregressive transformation, the alternative approach of clustering at the state level does not appear necessary. Note that when the autoregressive structure is generated by an underlying process with a one-quarter lag, then the error terms vjt in (24) are correlated within each state but in a complex fashion. As seen in footnote 28, when the HP ICHG lag in (21) is four quarters, the error correlation is positive when the time index diﬀers by three or less and equals zero otherwise. When the HP ICHG lag is two quarters, the error correlation is positive when the time index diﬀers by one and equals zero otherwise. As a result, the error structure does not have the constant error correlation within states that justiﬁes clustering, making robust standard errors more appropriate. that reduced aﬀordability from faster house-price appreciation could actually mask an expectations-driven relationship between the risk score and prior appreciation for ﬁrst-time borrowers. The reason is that rapid prior appreciation would prompt households to seek loans with higher LTVs, which would require a better risk score to gain approval.
Letting g(c) denote the density of default costs, the mean C among borrowers getting mort∞ gages is Cm ≡ C [Cg(C)/(N − G(C))]dC. In the case of a uniform g with support [0, C], Cm = (C + C)/2, so that the mean drops at half the rate at which C declines, with C conversely dropping at double the rate of the mean. Generally, ∂Cm /∂ C equals (Cm − C) times the hazard-rate expression g(C)/[N − G(C)]. This hazard rate equals 1/(C − C) in the uniform case, yielding ∂Cm/∂ C < 1, a result that will be strengthened when the hazard
rate at C is smaller, as will happen with a unimodal density where C lies below the mode. In such cases, ∂Cm/∂ C will be well below one, implying that C drops by a multiple of any measured decline in Cm .