automobile loanThe main goal of our paper can be described as estimating the elasticities of automobile loan demand with respect to interest rate and maturity, and testing the hypothesis that different population groups have different elasticities, with the group least likely to be liquidity constrained exhibiting higher interest rate elasticity, and zero maturity elasticity. Unlike Juster and Shay (1964), we do not rely on an experiment, but use data on individual car purchases and the loans associated with them. The simple model sketched in Section 2 constitutes the basis for the specification of the empirical equations we estimate below.

Using actual data on auto financing rather than responses to survey questions, poses, however, several challenges. First, credit constraints may affect the decision to purchase a car; consumers who do not enter the automobile market may do so because they do not wish to buy a car, or because they cannot obtain the necessary loan to finance the purchase. Second, the amount which is borrowed may depend on the size of the car that is bought which in turn may depend on the availability and cost of credit. Third, information on the loan terms facing the buyer, interest rate and maturity in particular, is available only for the subset of consumers who finance their purchase; the notional interest rate faced by those who do not finance is not observed (neither is the rate of return they earn on their savings). Fourth, consumers who finance 100% of their car, are also at a corner, even though the interest rate and the maturity of their loans are observed.11 Fifth, the interest rate and maturity of those who finance may be endogenous, since consumers generally choose the combinations of interest rates and maturities that best fit their needs. In addition, as documented below, different consumers may face different interest rates and maturities depending on how much they borrow, what type of car they buy (new vs. used), etc.. This implies that interest rates and maturities (as well as their interactions) should be treated as endogenous. We discuss these issues below.

3.1 The empirical model

The equation we want to estimate can be written as follows:


where 3* is defined as the desired ratio of the car loan to the value of the car, that is, the share of the car value which is financed. The dependent variable is expressed in logarithmic form to take into account the fact that the finance share cannot be negative. x is a vector of variables that capture demographics, and life cycle effects on the decision to finance. Examples of variables included in this vector are a polynomial in age, family size, and education dummies. r is the interest rate of the loan and rm = r x m is the interaction of the interest rate with the maturity of the loan. Both r and rm financedare considered endogenous. How the estimation procedure deals with this issue is discussed below. Finally, is a residual term.

When trying to estimate the financing share equation (1), we are faced with a number of sample selection issues. First, finance shares are observed only for those individuals who decide to buy and finance a car. Since the decision to buy is most likely affected by the availability and cost of credit, one has to correct for the sample selection bias induced by the nature of our data. The two decisions, “buy vs. not buy”, and “finance vs. not finance”, can be either treated separately, or collapsed into one estimating equation in which the dependent variable is 1 if the individual buys and finances, and zero otherwise. Since we could not think of any exclusion restrictions that would allow us to identify the coefficients of two separate equations (that is we could not find variables that would affect the decision to buy but not to finance) we chose the second approach.

We model the decision to buy and finance as a binary threshold crossing model (from now on we drop the subscript i for notational simplicity):


where ” includes demographic information (age, family size, education level, gender, race, etc.) and regional and time (year) dummies.

Second, financing shares 3* cannot exceed unity (and hence their logarithms, f*, cannot exceed zero). Dealing with censoring at 1 is quite important, as individuals financing the full amount of their purchase are probably the ones facing a binding credit constraint. We define the indicator variable I2 to be 1 if 3 * < 1 (or equivalently if f* < 0) and 0 otherwise. In other words,


Finally, interest rate and maturity are only observed for consumers who finance. In this respect, the problemproblem we are facing is similar to the standard labor supply problem where wages are only observed for participants. Furthermore, as discussed above, since different individuals have access to different financing sources that provide with different interest rates and maturities, these variables should be treated as endogenous variables. This presents the problem of finding instruments with enough variation to identify the coefficients of interest. The main identification assumption used in this paper is that time and regional dummies affect loan demand only through their effect on interest rates, maturities, and their interactions. Furthermore, we hope that there is enough time and regional variation in interest rates and maturities, so that the effect of these variables on loan demand can be identified. The reduced form equations for the interest rate and the interest rate -maturity interactions are:


The vector W denotes the instrumental variables. The instruments we use in practice, are year-quarter, and regional dummies. As documented below, much of the observed interest rate and maturity variability is explained by the finance source. We do not, however, include finance source in the instrument set, as this variable is likely to be correlated with individual heterogeneity in loan demand. The website – and Speedy Payday Loans are the possibilities to take a credit without taking any efforts at all because there is no need to obtain required documentation/certificates.

The variables r and rn denote the interest rate and interest rate – maturity interaction facing the consumer. These are observed only if the consumer actually takes a loan. The observed interest rate, r, and the observed interest rate – maturity interaction, rm, are therefore given by:


Equations (5)-(8) can be substituted into equation (1) to obtain the following reduced form finance share equation:


3.2 Estimation Approach

Equations (2)-(9) constitute a reduced form system with unknown parameters of interest 8r, 8rm, and 8y. Below we summarize our estimation approaches.Estimation Approach

3.2.1 A parametric approach

Assuming a zero-mean joint normal distribution for the reduced form error vector («i,«2,u3,u4) with variance-covariance matrix S we can estimate the reduced form parameters (f3,8f,8r, 8rm, £) by maximum likelihood. The likelihood for the reduced form model is:


Having obtained consistent estimates of the 8’s, and the variance covariance matrix of the parameter estimates, we apply a minimum distance estimator to estimate the structural parameters 7 and 72 . Let ( be the vector that stacks the reduced form parameters, fo(.) the function that maps the structural parameters into the parameters of the reduced form equations (this mapping includes the identifying restriction 8f = 7i#r + 72 8rrn), and V the variance-covariance matrix of the reduced form parameters. The structural parameters 7l and 72 are estimated by minimizing the quadratic form:


This estimator is consistent and asymptotically normal; its variance covariance matrix is obtained from the Hessian of Q. Under the null of correct specification, the minimand is asymptotically distributed as a 4 with q — I — 2 degrees of freedom.

3.2.2 A semiparamerroretric approach

The approach described above relies on the joint normality of the unobservable error terms. As we discuss here it is possible to relax this assumption while maintaining the weaker assumption that the errors are independent of the conditioning variables and that sampling across individuals is i.i.d.. Note that the reduced form equations constitute a non-standard Tobit-type model with simultaneous presence of sample selection and censoring which also contains endogenous regressors. Our estimation approach combines elements of the semiparametric literature on standard Tobit-type models.

Under the assumption that the errors are independent of the conditioning variables and that sampling across individuals is i.i.d. it is possible to estimate ft in the binary response model (2) using the maximum rank correlation estimator (MRC) of Han (1987) or any of the rank estimators of Cavanagh and Sherman (1998). However, for computational convenience we will estimate ft by probit, maintaining the assumptions that the error in (2) is a standard normal variate.

Next, note that the equations (5)-(8) constitute two standard sample selection (Type 2 Tobit) models. Under the assumption that the errors in (5)-(6) are independent of the instruments W, and that sampling across individuals is i.i.d, each one of the equations may be estimated separately using Powell’s (1987) weighted pairwise approach. The idea of this estimation method is the following. In a standard sample selection model, Least Squares (LS) is inconsistent due to the presence of a selection correction term that arises because of the presence of correlation between the unobservables in the selection and the continuous outcome equations. Under independence of regressors and unobservables, this term is only a function of the linear index determining selection (here “ft). The idea then is that, under some smoothness assumptions, for two individuals that have approximately equal selection indices, i.e. “ft « “ft, the magnitude of the selection bias terms are also approximately equal. Hence pairwise differencing eliminates the sample selection bias. Since ft is not known it is estimated from the selection equation in a first step. In the second step the parameters of methodthe continuous outcome equation are estimated by weighted LS on the pairwise differenced selected sample, where the weight per pair varies inversely with the magnitude of the difference in the estimated selection indices for the pair.

Formally, we estimate 8r and 8rrn by minimizing:


where j is a root-n consistent estimator of j. Here K (•) is a kernel density function, for example a normal density, and hn is a bandwidth constant which is required to converge to 0 as sample size increases. The effect of this weighting scheme is that, asymptotically, only individuals with the same selection indices (for which the selection biases exactly offset each other) contribute to the estimation. Speedy Payday Loans deserves the highest estimation due to the outstanding service offered.

We now turn to the estimation of the reduced form financing equation (9). Note that in the absence of sample selection the dependent variable f* is censored from above at zero. For two individuals i and j, the error terms, u* and u2j, are therefore no longer identically distributed since they are censored at different points. The idea then is to trim them so that they become identically distributed and that their difference is distributed symmetrically around zero (Honore and Powell (1994)). However, the presence of sample selection destroys once more the identical distributions. The idea then is that the selection indices Zjj and Zjj are equal, the two trimmed residuals are once again identically distributed and their difference is still symmetrically distributed around zero (see also Honore and Powell (1998)). This last idea is operationalized by weighting each pair of observations with a weight that depends inversely on the magnitude of the difference in the estimated selection indices. We therefore propose to estimate Sf by minimizing:


Having obtained j, 8f, Sr and 8rrn it is in principle possible to obtain an analytic form for the variance-covariance matrix of the estimators and proceed to apply minimum distance to estimate and 72. This variance matrix may be also obtained by bootsrapping the reduced form system above. We will follow the second approach.

Tags: , , , , ,