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Identification of dynamic models with aggregate shocks with an application to mortgage default in Colombia Juan Esteban Carranza Salvador Navarro∗ May 23, 2011

Abstract We describe identification conditions for dynamic discrete choice models that include unobserved state variables that are correlated across individuals and across time periods. The proposed framework extends the standard literature on the structural estimation of dynamic models by incorporating unobserved serially correlated common shocks. The shocks affect all individuals’ static payoffs and the dynamic continuation payoffs associated with different decisions. We show that even in the simple binary choice optimal-stopping problem we study, the model is not identified when only market level data are used. That is, the unobserved aggregate states and their transition are only identified as long as there is cross-sectional variation in the observed states. We use the framework to estimate a model of mortgage default for a cohort of Colombian debtors between 1997 and 2004. Finally, we use the estimated model to study the effects on default of a class of policies and shocks that affected the evolution of mortgage balances in Colombia during the 1990’s. ∗

ICESI (Colombia) and University of Western Ontario, respectively. We thank Gadi Barlevy, Mariacristina De Nardi, Amit Gandhi, Jean-Fracois Houde, Alvin Murphy, Krishna Pendakur, Chris Taber, Ken Wolpin and seminar participants at the Federal Reserve Bank of Chicago, Penn, Simon Fraser, Wisconsin, WUSTL, Yale, the 2009 North American meetings of the Econometric Society, the 2010 meeting of the SED and the 2010 Econometric Society World Congress. Direct correspondence to Carranza: [email protected] and Navarro: [email protected]

1

1

Introduction

In this paper we develop a framework for identifying dynamic structural models, under the presence of unobservable states that are both correlated across individuals and over time. The correlation is caused by unobserved common states that are serially correlated.1 The early literature on the estimation of dynamic discrete choice models was based on the assumption that all the unobserved heterogeneity is independent across individuals and over time.2 More recent papers incorporate unobserved states that vary systematically across individuals but stay constant (as in Keane and Wolpin, 1994), are independent over time (as in Arcidiacono and Miller, 2008) or are correlated over time but independent across individuals (as in Erdem, Imai, and Keane, 2004 or Norets, 2009).3 The literature on the estimation of dynamic structural models with correlated unobserved common states (or shocks) is scarce. In the approach proposed by Altug and Miller (1998), the structure of the aggregate shocks is estimated separately and used as input in the dynamic model. Such approach is only practical when the aggregate shocks can be estimated from a separate model (e.g. a macroeconomic model or Euler equations). A closer paper to ours is Lee and Wolpin (2006), in which the aggregate shocks and their transition are computed throughout the estimation algorithm using a general equilibrium model, which is computationally demanding. We prove that, even in a simple binary choice optimal-stopping model, it is not possible to identify the unobserved common states separately from their transition from market level data alone. This is important because in the Industrial Organization literature static discrete choice models of demand are identified using only market level data. Our result implies that extending this approach to dynamic models will require either micro level data or additional restrictions. We show how to incorporate, identify and estimate unobserved common states that generate correlation both in the cross section and over time, using a standard micro data set. In contrast to Altug and Miller (1998) and Lee and Wolpin (2006), we show that estimating a dynamic model with aggregate unobserved shocks does not require the solution of an aggregate model. The identification and estimation of the common correlated states exploits the variation of the observed aggregate behavior implied by the microeconomic model, which is a piece of information that is not used directly by the existing literature. Our specification of the dynamic model is based on a Markovian decision problem with 1

For simplicity, we refer to these unobserved correlated common states as aggregate shocks. Early papers include Rust (1987), Wolpin (1984), Pakes (1986) and Hotz and Miller (1993). 3 For a comprehensive review of the literature see, for example, Aguirregabiria and Mira (2002). 2

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finite horizon. Each period’s payoff depends on observed and unobserved state variables that vary systematically across individuals. We show that in this particular formulation of the dynamic model, the micro data contain enough information to infer the aggregate shocks and their transition separately. The aggregate shocks and their transitions can be separately identified if and only if there is individual-level variation on the observables. We use the framework to estimate a dynamic mortgage default model using micro-level Colombian data spanning the years between 1997 and 2004. During this time mortgage default rates in Colombia were unusually high, due to an unprecedented economic downturn that was accompanied by a dramatic fall in home prices. The extent to which the fall in household incomes, the fall in home prices, and the fall in equity contributed separately to the unprecedented rates of default is a relevant policy question that can be answered with the proposed model. In addition, we use the estimated model to evaluate the impact of counterfactual policies, which cannot be evaluated with a model that does not account for the dynamic concerns of debtors. We show that, in the context of our data, the expectations of individuals regarding the evolution of equity had a substantial impact on default behavior. Moreover, we show that neither the level, nor the expected evolution of income contributed significantly to default. In the next section of the paper we describe our methodological framework. We formulate an optimal stopping problem with unobserved heterogeneity, describe our estimation approach, and discuss the identification of the different components of the model. In Section 3 of the paper we present the application of the model to the Colombian mortgage market. We describe the data, the estimation and the results. We then perform counterfactual simulations to illustrate the importance of income shocks, an evaluate the impact of the policies adopted by the Central Bank and the Colombian government in the mid-1990s. The paper concludes with a discussion of the limitations of the proposed framework.

2 2.1

The methodological framework A generic optimal stopping problem

Consider the standard optimal stopping problem of an individual i ∈ {1, ..., N } at period t who has to choose between two actions di,t = j : “stopping” (j = 0), which is an absorbing state, or “continuing” (j = 1) to face the same decision next period. For simplicity, we assume a finite horizon Ti which may be different across individuals. For example, in our aplication Ti is different across individuals, because mortgages have different term lengths 3

across debtors. Each choice generates a static payoff ui,j,t = uj (Xi,t ) + εi,j,t . The payoff consists of a component uj (Xi,t ) that depends on a vector of observable (to the econometrician) states Xi,t , and of an additive unobserved (to the econometrician) state variable εi,j,t that may be correlated across individuals and time periods. We assume that the vector of observed states Xi,t follows a first order Markov process. This process is assumed to be independent of the contemporaneous unobserved states, a common assumption in the literature. That is, if we let Λ(Xi,t |.) be the conditional cdf of X, then Λ(Xi,t |Xi,t−1 , εi,0,t , εi,1,t ) = Λ(Xi,t |Xi,t−1 ). The unobserved states {εi,0,t , εi,1,t } are also assumed to be Markovian as described below. Hence, S˜i,t = {Xi,t , εi,0,t , εi,1,t } is the set of decision-relevant state variables for individual i at time t. Consider the Bellman representation of the problem of individual i who, as of time t − 1 < Ti − 1, has not chosen j = 0 in the past: h i V˜t (S˜i,t ) = max{u1 (Xi,t ) + εi,1,t + βE V˜t+1 (S˜i,j,t+1 )|S˜i,j,t , u0 (Xi,t ) + εi,0,t },

(1)

where β is an assumed known exogenous discount rate. To simplify notation we assume that once an individual stops, he gets the static payoff of stopping and nothing else.4 At t = Ti the continuation payoff of the problem is zero, so that: E[V˜Ti +1 (S˜i,j,Ti +1 )|S˜i,j,Ti ] = 0.

(2)

It has been shown before that this model is generically not identified non-parametrically even with uncorrelated unobserved states.5 Therefore, the mapping of the model into data has to be based partly on assumptions on the net observable utility, u (Xi,t ) = u1 (Xi,t ) − u0 (Xi,t ) , and/or on the stochastic properties of the unobserved states ε. In order to allow for a rich pattern of unobserved correlation, we decompose the unobserved states as follows: εi,1,t − εi,0,t = ξt + µi + i,t ,

(3)

where i,t is an idiosyncratic random variable distributed iid across individuals, choices and 4

Since j = 0 is an absorbing state, this is equivalent to simply redefining the reward function to be the expected present value of continuing in state j = 0 until Ti . 5 Rust (1994); see also Taber (2000) and Heckman and Navarro (2007) for conditions under which these models are semiparametrically identified.

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time periods. The term µi , which we call individual heterogeneity, is an individual-specific unobserved state that stays constant over time. It is assumed to be distributed among the population of individuals according to a distribution Υ(µ). The term ξt , which we will refer to as the aggregate shock, is a common unobserved state assumed to follow a first order Markov process. Both the distribution of the individual heterogeneity and the transition of the aggregate shocks have to be estimated simultaneously with the whole model. Under this specification individual choices are correlated over time and across individuals, even after conditioning on the observed states. In addition, the unobserved heterogeneity can be allowed to depend on Xi,t , which would be equivalent to a parametric model with heterogeneous coefficients. The model is similar to standard dynamic discrete choice models, except for the presence of the shock ξt which is allowed to be serially correlated. The importance of including this form of heterogeneity is that it permits individual choices to be correlated (in unobservable ways) in a given cross-section (since all individuals face the same shock), allowing this correlation to persist over time. We will refer to these shocks as aggregate shocks, but they can be understood more generally as the common component of the unobserved heterogeneity. The model we specify nests the standard models in the literature. Specifically, if we set µi = ξt = 0, all the unobserved heterogeneity in the model is iid and the model is similar to the models in Rust (1987), Wolpin (1987) and Hotz and Miller (1993). If we assume away the aggregate shocks so that ξt = 0, but account for a correlated individual shock µi 6= 0 the model is similar to Keane and Wolpin (1994). In addition, if we allow µi to depend on X, the model is equivalent to a model with random coefficients. In contrast to the models by Altug and Miller (1998) and Lee and Wolpin (2006), we do not specify where the aggregate shocks come from. In Section 2.3 we show that it is not necessary to do so; micro data alone is enough to identify the aggregate shocks and their transition separately. In a general equilibrium setup, the specification of a model for the determination of the aggregate shocks ξ and their transition would be necessary for the computation of counterfactual equilibria, but not for the estimation of the model. Let Si,t = {Xi,t , µi , ξt } be the the set of state variables excluding the idiosyncratic iid error. Define the expected value function as the expectation of the value function in (1) with respect to the idiosyncratic iid shock: Vt (Si,t ) = E



5

 ˜ Vt (Si,t , i,t ) .

(4)

Conditional on survival, the predicted probability that individual i chooses j = 1 at time t is given by: P ri,t (Si,t ) = P rob [i,t < u(Xi,t ) + µi + ξt + βE [Vt+1 (Si,t+1 )|Si,t ]] ,

(5)

where the continuation payoffs correspond to the conditonal expectation of (4). Notice that this probability is not observed by the econometrician, since it depends on the realization of both the individual heterogeneity µi and the aggregate shock ξt . Given (5), let P˜ri denote the probability of an individual history, which can be computed as the product of probabilities over the given sequence of choices, conditional on the realization of the individual heterogeneity and the aggregate shocks: P˜ri (Si ) =

T¯i Y

P ri,t (Si,t )di,t [1 − P ri,t (Si,t )] (1−di,t ) ,

(6)

t=1

where Si = {Si,t=1,...,T¯i } is the matrix of states and T¯i is the last time period at which an individual is observed making choices. In other words, T¯i is either the time when individual i first chooses di,t = 0, or the final period Ti if he always chooses di,t = 1.

2.2

Recovering the aggregate shocks

Before discussing identification, we first consider the problem of recovering the aggregate shocks, taking as given all remaining aspects of the model. That is, we provide conditions under which the aggregate shocks can be inferred from data on individual choices without the specification of an aggregate model. In Section 2.3 we discuss which of the remaining aspects of the model can potentially be recovered non- or semi-parametrically. Assume that we have a population of i = 1, ..., N individuals. For notational convenience, we assume that all individuals start to solve the problem simultaneously but then have potentially different non-random problem horizons Ti . The individuals are observed solving the described optimal stopping problem during a sequence of T¯ = max{T¯1 , ..., T¯N } time periods, where T¯i ≤ Ti . In addition to the time horizons, the econometrician observes states Xi = {Xi,1 , ..., Xi,T¯i } and decisions di = {di,1 , ..., di,T¯i }. Given the assumed exogeneity of the observed states, their transition can be recovered directly from the data.

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The likelihood, as a function of the aggregate shocks, is thus given by:

`(ξ) =

=

N ˆ Y i=1 N ˆ Y

"

T¯i Y

# P ri,τ (Si,τ )di,τ (1 − P ri,τ (Si,τ ))1−di,τ dΥ (µ)

τ =1

P˜ri (µ, Xi , ξ)dΥ(µ),

(7)

i=1

where the choice probabilities are integrated with respect to the initial distribution of the individual heterogeneity Υ(µ). Consider recovering the vector of aggregate shocks, ξ, using (7) and taking all the other elements of the model as given. Depending on the case maximizing the likelihood can be difficult, especially if the number of periods T¯ is large. Moreover, because the aggregate shocks enter both the static payoffs and the continuation payoffs, it is not clear whether the maximization problem has a unique solution. The key insight for the identification results we derive is that the properties of the aggregate shocks can be inferred from the observed aggregate behavior. Let the survival probQ −1 P ri,t (Si,t ), where ξ (u(Xi,t+1 ) + µi + h(ξt0 ) + υt+1 + i,t+1 ). Therefore, (A7) holds. We will show now that if ξt − ξt0 > 0, (A7) holds for t < Ti − 1. Abusing notation, let Ψt (ξt ) = E[V (., ξt+1 )|ξt ]. At t = Ti − 2: ∆max(ξt , ξt0 ) =βmax{u(Xi,t+1 ) + µi + h(ξt0 ) + υt+1 + i,t+1 + βΨt+1 (h(ξt0 ) + υt+1 ), 0} − βmax{u(Xi,t+1 ) + µi + h(ξt ) + υt+1 + i,t+1 + βΨt+1 (h(ξt ) + υt+1 ), 0}. Again, four cases arise: • ∆max(ξt , ξt0 ) = 0: In this case (A7) holds again trivially, because ξt − ξt0 > 0 by assumption. • ∆max(ξt , ξt0 ) = −β(u(Xi,t+1 ) + µi + h(ξt ) + υt+1 + i,t+1 + Ψ(h(ξt ) + υt+1 )): In this case ∆max(ξt , ξt0 ) < 0 < ξt − ξt0 , so that (A7) holds.

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• ∆max(ξt , ξt0 ) = β(h(ξt0 )−h(ξt )+β(Ψt+1 (h(ξt0 )+υt+1 )−Ψt+1 (h(ξt )+υt+1 ))) ≥ −β(h(ξt )+ υt+1 − h(ξt0 ) − υt+1 + ∆max(h(ξt ) + υt+1 , h(ξt0 ) + υt+1 ))). From (A9) ξt − ξt0 > β(h(ξt ) − h(ξt0 )). Since we proved that (A6) is true for t = T −1, then each element of the integral β(Ψt+1 (h(ξt0 ) + υt+1 ) − Ψt+1 (h(ξt ) + υt+1 )) is negative. Therefore, ξt − ξt0 > ∆max(ξt , ξt0 ) and (A7) is true. • ∆max(ξt , ξt0 ) = β(u(Xi,t+1 ) + µi + h(ξt0 ) + υt+1 + i,t+1 + βΨt+1 (h(ξt0 ) + υt+1 )): In this case, β(u(Xi,t+1 ) + µi + h(ξt ) + βυt+1 + i,t+1 + βΨt+1 (h(ξt ) + υt+1 )) < 0. Using again (A9): ξt − ξt0 > β(h(ξt0 ) − h(ξt )) = β(u(Xi,t+1 ) + µi + h(ξt0 ) + υt+1 + i,t+1 + βΨt+1 (h(ξt0 ) + υt+1 )) − β(u(Xi,t+1 ) + µi + h(ξt0 ) + υt+1 + i,t+1 + βΨt+1 (h(ξt0 ) + υt+1 )) > ∆max(ξt , ξt0 ), which is what we wanted to show. Proceeding recursively, we can show that (A7) holds for any t. Therefore s˜(ξt ) is monotone and the proof is complete.

Proof of Proposition 1 To show necessity, we show first that whenever Xi,t = Xj,t ∀i, j, t the model is not identified by showing that the likelihood function of the sample is flat. For simplicity we assume that ρ is a scalar. The proof can easily be generalized to the vector case by changing derivatives for jacobians and determinants of these jacobians (to equate to zero) where needed. We can rewrite the log-likelihood function (7) of the sample as follows:

ln(`) =

T¯i N X X

ˆ ln

  P ri,t (Si,t ; ρ)di,t (1 − P ri,t (Si,t ; ρ))1−di,t dΥt (µ)

(A10)

i=1 t=1

where the integrals are taken with respect to the distribution of µ conditional on survival until time t, Υt (µ). The derivative of the log-likelihood w.r.t. ρ is: ´ ´  T¯i  N d P ri,T¯i (.)dΥT¯i (µ) d P ri,t (.)dΥt (µ) d ln(`) X X di,t 1 − di,t ´ = −´ dρ dρ dρ P ri,t (.)dΥt (µ) P ri,T¯i (.)dΥT¯i (µ) i=1 t=1 ´ T¯i N X X (2di,t − 1) d P ri,t (.)dΥt (µ) ´ = (A11) dρ P r (.)dΥ (µ) i,t t i=1 t=1

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Let Xi,t = Xj,t ∀i, j, t. Therefore: P ri,t (µ, ξt ; ρ) = P rj,t (µ, ξt ; ρ) ∀µ ˆ ˆ ⇒ P ri,t (.)dΥt (µ) = P rj,t (.)dΥt (µ) = st (ξ, ρ) ´ ´ ´ ∂ P ri,t (.)dΥt (µ) ∂ P ri,t (.)dΥt (µ) dξt d P ri,t (.)dΥt (µ) = + ∀i ⇒ dρ ∂ρ ∂ξt dρ

(A12)

´ We know from lemmas 1 and 2 that ξt is uniquely defined by st (ρ, ξ) ≡ P ri,t (.)dΥt (µ) = s0t where s0t is the observed time t share. We can obtain its derivative using the implicit function theorem: ´ ∂ P ri,t (.)dΥt (µ)/∂ξt dξt ∂st /∂ξt =− =− ´ dρ ∂st /∂ρ ∂ P ri,t (.)dΥt (µ)/∂ρ ´ ´ ´ ´ d P ri,t (.)dΥt (µ) ∂ P ri,t (.)dΥt (µ) ∂ P ri,t (.)dΥt (µ) ∂ P ri,t (.)dΥt (µ)/∂ρ ´ ⇒ = − dρ ∂ρ ∂ξt ∂ P ri,t (.)dΥt (µ)/∂ξt ´ d P ri,t (.)dΥt (µ) ⇒ = 0 ∀ρ, i dρ which implies that (A11) is zero for all ρ and therefore the model is not identified. To show sufficiency, we show that whenever Xi,t 6= Xj,t the model is generically identified. t will not vary with X (since we are integrating against First, notice that for any given t dξ dρ the distribution of X). Conditional on a particular value X = x we have that ´ ´ ´ d P ri,t (x, .)dΥt (µ) ∂ P ri,t (x, .)dΥt (µ) ∂ P ri,t (x, .)dΥt (µ) dξt = + . dρ ∂ρ ∂ξt dρ

(29)

t does not vary with X this derivative cannot, generically, be zero for all values of Since dξ dρ ρ. This in turns implies that (A11), which consists of a weighted sum of terms like the ones above, will generically not be zero. Therefore, the model can only be identified when there is cross sectional variation in X. Now consider the “true” value of ρ∗ such that d ln(`(ρ∗)) = 0. dρ From (A11) we can show that the second derivative of the likelihood function evaluated at ρ∗ is: ´ T¯i N d2 P ri,t (Xi,t , .)dΥt (µ) d2 ln(`(ρ∗)) X X (2di,t − 1) ´ = (A13) 2 dρ2 dρ P r (X , .)dΥ (µ) i,t i,t t i=1 t=1

From (A12) we can show that

d2

´

P ri,t (Xi,t ,.)dΥt (µ) dρ2

t 2 ( dξ ) ) is the same for all individuals, Xi,t 6= dρ

∂2

´

P ri,t (Xi,t ,.)dΥt (µ) t 2 (1 − ( dξ ) ). Since (1 − ∂ρ2 dρ ´ ´ 2 d P ri,t dΥt (µ) d2 P rj,t dΥt (µ) Xj,t implies that = 6 . dρ2 dρ2

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=

2

Therefore, if Xi,t 6= Xj,t , d ln(`(ρ∗)) can only be zero around ρ∗ if the terms in (A13) add up dρ2 to zero which is generically not true. Therefore, the model is (locally) identified.

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Appendix 2 In this appendix we lay down a very simple 2-period model with income maximizing agents to illustrate the point that negative equity is, at best (i.e., when default is costless and agents are income maximizers) a necessary condition for default. Consider an income maximizing agent who lives for two periods and who (for simplicity) does not discount the future. The agent owns an asset (i.e., a house) that has a price of p today. The next period the house can be worth either pL or pH > pL with probability 0.5. Each period t = 1, 2 the individual has to pay the amount ct if he lives in the house (i.e., the mortgage payment). We further assume that the price of the asset is below the cost in the bad state pL ≤ c2 but above cost in the good state pH ≥ c2 . In equilibrium, the price of the house (i.e. the value of the asset) today will be given by p = d + 0.5pL + 0.5pH

(A15)

where d is the “dividend” payed by the asset today (which can be understood as the flow “utility” the agent gets from living in the house today). We further assume that the house has negative equity today so p < c1 + c2 where c1 + c2 is the mortgage balance. We assume that default is costless, so if an individual defaults, he simply walks out of the house and gets nothing (i.e. he does not sell the house) and pays nothing.20 In this case, an individual will choose not to default today if the expected value of holding the house is larger than the value of defaulting. That is, if d − c1 + 0.5 (pH − c2 ) > 0.

(A16)

Notice that, because the individual can (and will) default tomorrow in the bad state, he simply gets zero in that case and so the bad state does not directly affect the decision of whether to default today. A simple way of illustrating the importance of accounting for the dynamic incentives 20

Notice that default is costless in the sense that there are no direct expenses associated with default. Default is costly because we are ruling out the possibility that the debtor walks out and purchases again a similar home at the lower market price. If this was the case, the equilibrium price of homes would not be (A15) and the analysis would be more complicated. The general point remains valid in the sense that negative equity is not sufficient for default. We thank an anonymous referee for pointing this out.

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facing the individual is to consider the following direct implication of this model. Suppose that a policy is implemented that changes both c1 and c2 while keeping c1 + c2 constant. In this case, the individual’s incentive to default will change even though the equity the individual holds in the house is not affected (see condition (A16)). We now show that, even if there is negative equity (p < c1 + c2 ), an individual may still decide not to default. To see why notice that negative equity implies that d + 0.5pL + 0.5pH − c1 − c2 < 0 which can be rewritten as d − c1 + 0.5 (pH − c2 ) < 0.5 (c2 − pL ) . where c2 − pL ≥ 0 by assumption. If the left hand side of this condition is positive (i.e. if (A16) holds) the individual will not default today even though the house has negative equity. So, a necessary condition for the individual not to default is that 0.5 (c2 − pL ) is strictly positive. But 0.5 (c2 − pL ) is exactly the option value for the individual. To see why this is the case, consider what an individual gets today (in expected terms) if he is not allowed to default tomorrow max (d − c1 + 0.5pL + 0.5pH − c2 , 0) and compare it to what he gets when he has the option of defaulting tomorrow max (d − c1 + 0.5 (pH − c2 ) , 0) . How much will an individual be willing to pay to have the right to default tomorrow? He will be willing to pay up to the difference which is max (0, 0.5 (c2 − pL )) = 0.5 (c2 − pL ) . So, provided the value of the default option is positive, an individual may choose not to default even when the equity in the house is negative.

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Appendix 3 In this appendix we describe the Matzkin class of functions that allow for non-parametric identification of the binary choice model.21 We begin with her main result. Consider a binary choice model, D = 11 (U (X) > υ), where X is observed and υ is unobserved. Let U ∗ denote the “true” function U and let Fυ∗ denote the the true cdf of υ. Let Ω denote the set of monotone increasing functions from R into [0, 1]. Assume (i) X ∈ X ⊂ RK , U ∗ ∈ U where U is a set of functions mapping X into R that are continuous and strictly increasing in their K th coordinate. (ii) X ⊥⊥ υ (iii) The conditional distribution of the K th coordinate of X has a Lebesgue density that is everywhere positive conditional on the other coordinates of X. (iv) Fυ∗ is strictly increasing. (v) The support of the marginal distribution of X is included in X . Then (U ∗ , Fυ∗ ) is identified within U × Ω if and only if U is a set of functions such that no two functions in U are strictly increasing transformations of each other. The following functional forms are examples of functions satisfying her conditions for exact identification of U (X). 1. U (X) = Xγ, ||γ|| = 1 or γ1 = c for a known constant c. This is the same class considered by Manski (1988). 2. U (X) homogeneous of degree one such that U (˜ x) = α for a known α and some x˜ ∈ X . 3. Least concave functions that attain common values at two points in their domain. 4. Functions additively separable into a continuous and monotone increasing function and a continuous monotone increasing, concave and homogeneous of degree one function, e.g. U (X) = X1 + τ (X2 , ..., XK ).

21

See Matzkin (1992) and the appendix in Heckman and Navarro (2007) from where we borrow heavilly.

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Hendel, I., and A. Nevo (2006): “Measuring the Implications of Sales and Consumer Inventory Behavior,” Econometrica, 74(6), 1637–1673. Hotz, V. J., and R. A. Miller (1993): “Conditional Choice Probabilities and the Estimation of Dynamic Models,” Review of Economic Studies, 60(3), 497–529. Keane, M. P., and K. I. Wolpin (1994): “The Solution and Estimation of Discrete Choice Dynamic Programming Models by Simulation and Interpolation: Monte Carlo Evidence,” The Review of Economics and Statistics, 76(4), 648–672. Kotlarski, I. I. (1967): “On Characterizing the Gamma and Normal Distribution,” Pacific Journal of Mathematics, 20, 69–76. Lee, D., and K. I. Wolpin (2006): “Intersectoral Labor Mobility and the Growth of the Service Sector,” Econometrica, 74(1), 1–40. Manski, C. F. (1988): “Identification of Binary Response Models,” Journal of the American Statistical Association, 83(403), 729–738. Matzkin, R. L. (1992): “Nonparametric and Distribution-Free Estimation of the Binary Threshold Crossing and the Binary Choice Models,” Econometrica, 60(2), 239–270. Norets, A. (2009): “Inference in Dynamic Discrete Choice Models with Serially Correlated Unobserved State Variables,” Econometrica, 77(5), 1665–1682. Pakes, A. (1986): “Patents as Options: Some Estimates of the Value of Holding European Patent Stocks,” Econometrica, 54(4), 755–784. Rust, J. (1987): “Optimal Replacement of GMC Bus Engines: An Empirical Model of Harold Zurcher,” Econometrica, 55(5), 999–1033. (1994): “Structural Estimation of Markov Decision Processes,” in Handbook of Econometrics, Volume, ed. by R. Engle, and D. McFadden, pp. 3081–3143. North-Holland, New York. Taber, C. R. (2000): “Semiparametric Identification and Heterogeneity in Discrete Choice Dynamic Programming Models,” Journal of Econometrics, 96(2), 201–229. Wolpin, K. I. (1984): “An Estimable Dynamic Stochastic Model of Fertility and Child Mortality,” Journal of Political Economy, 92(5), 852–874. 40

(1987): “Estimating a Structural Search Model: The Transition from School to Work,” Econometrica, 55(4), 801–817.

41

Table 1 Summary Statistics (Main Dataset) (1)

Quarter 1997:1 1997:2 1997:3 1997:4 1998:1 1998:2 1998:3 1998:4 1999:1 1999:2 1999:3 1999:4 2000:1 2000:2 2000:3 2000:4 2001:1 2001:2 2001:3 2001:4 2002:1 2002:2 2002:3 2002:4 2003:1 2003:2 2003:3 2003:4 2004:1 2004:2

(2)

(3)

Number of Loans 93 355 591 925 1,435 1,878 2,224 2,486 2,486 2,486 2,486 2,486 2,486 2,486 2,486 2,486 2,486 2,486 2,486 2,486 2,486 2,486 2,486 2,486 2,486 2,486 2,486 2,486 2,486 2,486

Outstanding Loans 93 351 575 892 1,366 1,775 2,078 2,267 2,153 2,022 1,946 1,837 1,738 1,699 1,616 1,565 1,532 1,496 1,458 1,425 1,404 1,389 1,369 1,338 1,321 1,307 1,296 1,289 1,279 1,270

*1997 Millions of Colombian Pesos

(4)

Default Rate 0.00% 1.14% 2.09% 1.91% 2.64% 1.92% 2.07% 3.22% 5.29% 6.48% 3.91% 5.93% 5.70% 2.30% 5.14% 3.26% 2.15% 2.41% 2.61% 2.32% 1.50% 1.08% 1.46% 2.32% 1.29% 1.07% 0.85% 0.54% 0.78% 0.71%

(5)

(6)

Mean House Price* Outstanding Loans

All Loans

167.98 85.69 87.28 85.12 73.97 71.17 65.57 62.81 65.05 63.09 55.94 52.37 50.72 55.54 52.90 53.63 60.29 54.85 59.13 62.29 56.83 64.03 60.11 68.76 63.46 66.76 65.94 67.34 65.94 73.50

167.98 85.35 86.32 84.02 73.04 70.41 64.83 61.68 63.07 62.87 59.31 63.56 60.75 66.33 65.53 65.94 73.96 67.14 71.87 75.04 68.17 76.52 71.51 81.35 74.80 78.52 77.41 79.43 78.06 86.83

Table 2 Estimation Results: Probability of Non-Default Model I Coefficient

Estimate

Model II

Marginal Effect % Points

Estimate

Model III

Marginal Effect % Points

Estimate

Marginal Effect % Points

Model IV Estimate

Marginal Effect % Points

Utility:

ζ1 (Price)

(Std. Error)

ζ2 (Balance)

(Std. Error)

ζ3 (Term)

0.061

0.266

0.061

0.269

0.049

0.418

0.049

0.428

(0.005)

(0.034)

(0.005)

(0.029)

(0.004)

(0.080)

(0.004)

(0.077)

-0.328

-0.673

-0.329

-0.714

-0.355

-1.090

-0.355

-1.181

(0.027)

(0.079)

(0.025)

(0.080)

(0.025)

(0.174)

(0.025)

(0.178)

-0.015

-0.102

-0.015

-0.102

-0.015

-0.228

-0.015

-0.230

(Std. Error)

(0.002)

(0.018)

(0.002)

(0.017)

(0.002)

(0.045)

(0.002)

(0.043)

ζ4 (Income)

-

-

-

-

0.001

0.061

0.001

0.062

(0.000)

(0.029)

(0.000)

(0.029)

-

-

-

-

(Std. Error)

ζ5 (Equity) (Std. Error)

-0.101 (0.059)

-

-0.168 (0.124)

-

Loan to Value:

α0 (Constant)

(Std. Error)

α1 (Heterogeneity)

(Std. Error)

α2 (Variance)

(0.005)

0.003 (0.002)

0.036

(Std. Error)

Transition of

0.539

(0.001)

-1.321

(Std. Error)

(0.231)

ρ1 (Lagged ξ)

-0.896

ρ2 (Variance)

-

0.539 (0.005)

0.004 (0.004)

0.036 (0.001)

-

0.539 (0.005)

0.004 (0.005)

0.036 (0.001)

-

0.539 (0.005)

0.004 (0.002)

0.036 (0.001)

-

ξ:

ρ0 (Constant)

(Std. Error)

-

(0.156)

0.249

(Std. Error)

(0.065)

Variance(µ )

1.964

-

-1.309 (0.223)

-0.895 (0.154)

0.281 (0.067)

1.977

-

-0.035 (0.023)

0.123 (0.068)

0.001 (0.001)

1.862

-

-0.041 (0.024)

0.160 (0.075)

0.001 (0.001)

1.871

-

*The marginal Effects are computed as the average marginal effect across all debtors with a 15 year mortgage one year after the mortgage started. We evaluate the aggregate shock at its mean. We compute the marginal effect of a 10% increase in each variable and a 1 quarter increase in term left.

Table 3 Counterfactual Policy Simulations: Default Rate Quarter 1997:2 1997:3 1997:4 1998:1 1998:2 1998:3 1998:4 1999:1 1999:2 1999:3 1999:4 2000:1 2000:2 2000:3 2000:4 2001:1 2001:2 2001:3 2001:4 2002:1 2002:2 2002:3 2002:4 2003:1 2003:2 2003:3 2003:4 2004:1 2004:2

1

Baseline

3.64% 3.07% 2.29% 3.47% 2.16% 2.02% 2.96% 4.45% 5.97% 3.83% 6.50% 7.03% 3.72% 8.37% 6.42% 5.28% 5.69% 7.13% 9.22% 4.84% 4.65% 5.55% 10.77% 6.23% 5.85% 5.14% 4.14% 5.56% 6.25%

New Transition for Balances2 3.15% 2.50% 1.75% 2.52% 1.44% 1.28% 1.78% 2.58% 3.29% 1.89% 3.05% 2.98% 1.36% 2.93% 1.92% 1.41% 1.37% 1.54% 1.93% 0.65% 0.54% 0.57% 1.01% 0.43% 0.33% 0.24% 0.15% 0.18% 0.16%

Income Distribution Fixed for all t. Set Equal to Initial Distribution Original Transition3 Updated Transition4 3.64% 3.65% 3.07% 3.08% 2.30% 2.34% 3.45% 3.50% 2.15% 2.16% 2.02% 2.04% 2.95% 2.99% 4.42% 4.50% 5.96% 6.06% 3.82% 3.89% 6.51% 6.55% 7.04% 7.14% 3.72% 3.80% 8.38% 8.52% 6.44% 6.54% 5.28% 5.40% 5.68% 5.80% 7.13% 7.20% 9.20% 9.36% 4.84% 4.85% 4.65% 4.66% 5.54% 5.60% 10.74% 10.80% 6.20% 6.26% 5.85% 5.85% 5.15% 5.08% 4.15% 4.09% 5.55% 5.51% 6.26% 6.30%

1

Average default rate across debtors using the estimated transitions.

2

Let bt be the balance at time t and let Lt be the remaining periods until the mortgage ends. We show the average default rates across debtors when the transition for balances is replaced with the "standard" transition: bt=bt-1(1-1/Lt).

3

Average default rate across debtors when the distribution of income is fixed at the initial one for all periods but consumers use the original estimated transition to form expectations.

4

Average default rate across debtors when the distribution of income is fixed at the initial one for all periods, consumers know this and adjust their transition accordingly.

Figure 1 Default Rates: Baseline1 and Counterfactual Balances Transition2 12.00% Baseline Counterfactual

10.00%

Default Rate

8.00%

6.00%

4.00%

2.00%

0.00% 1

2

1Average

3

4

5

6

7

8

9

10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29

Quarter

default rate across debtors using the estimated transitions. bt be the balance at time t and let Lt be the remaining periods until the mortgage ends. We show the average default rates across debtors when the tran 2Let

Figure 2 Default Rates: Baseline1 and Counterfactual Balances Transition Announced in Current Period2 12.00% Baseline Counterfactual

10.00%

Default Rate

8.00%

6.00%

4.00%

2.00%

0.00% 1

2

1Average

3

4

5

6

7

8

9

10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29

Quarter

default rate across debtors using the estimated transitions. bt be the balance at time t and let Lt be the remaining periods until the mortgage ends. We show the average default rate across debtors when the estim 2Let