Ly Y Uq 9y LqQ 0Q

will also be strongly consistent for 8y and 0Q.

proof: The proof follows almost directly from Hartley and Mallela (1977). Details of the proof are given in section 13.7.

It would be desirable to derive an expression for the asymptotic distribution of the second-stage estimators 9y and 8Q. Unfortunately an examination of a Taylor expansion of the normal equations indicates that, unless relatively restrictive conditions are met, an expression for the asymptotic distribution of 8y and 90 will involve terms from the reduced forms as well as the second stage. We can, however, derive easily estimable expressions for the asymptotic distribution of the maximum likelihood estimators 8y and 0G. Using 0y as an example, these can be estimated by the inverse of the information matrix of (13.27) through (13.28) as follows:

theorem 13.2: If conditions 13.1 through 13.4 are met, then where

Furthermore, if conditions for theorem 13.1 hold, then 2. GY(nQ, ôy) converges to GY(nQ, 8y) a.e. the same holding for proof: Part 1 follows directly from Hartley-Mallela (1977). Part 2 follows from Hartley-Mallela lemma 10(showing uniform convergence for Gy( - )) and lemma 13.1 of Section 13.7 (Amemiya 1973, lemma 4).

Although the asymptotic distribution of the maximum likelihood estimators is unlikely to be the same as that of {0y, 0O}, we could interpret estimates of their standard errors as lower bounds of the standard errors of 0y and 0Q.

The major advantage of our proposed procedure is computational simplicity and feasibility. Each of the four equation systems—the two reduced forms and the two second-stage models—can be estimated separately, using computer programs for standard switching regression models. An alternative procedure, full information ML, might require estimation of a prohibitively large number of parameters. However, by not using full information ML, we are likely to be losing efficiency as we are ignoring cross-equation restrictions and common system switching points. If cross-equation restrictions are necesssary for full identification of the model, then our procedure, which ignores such restrictions, will be incapable of producing consistent structural parameter estimates.

Our proposed procedure requires the estimation of four switching regression model systems. Since these systems are relatively expensive to estimate, as a final modification we show how one of these runs can be eliminated.

The expectation of the observed dependent variable of a switching regression model, given the exogenous variables, can be computed by a straightforward method. This formula can be used to estimate an instrument for observed debt, Q, from the debt-reduced form. This instrument can then be plugged into (13.11) in place of Q and structural parameters of the y equation estimated by OLS. This step avoids having to estimate the second-stage Y switching regressions. Formally the instrument is derived from the debt-reduced forms as follows.

Let â*2, â*2, â*d be the variance and covariance estimates of the errors of the reduced forms (13.24) and (13.25), respectively, and let â — ^Jô*2 + ô*z — 2â*d. Defining/( ■ ) as the standard normal density function, and F( ■ ) as the cumulative normal, a consistent estimate of the expectation of Q„ given XlB, X2n, and X3n is where {55„, D*} are estimates of {S*, Ddn) calculated using the estimated reduced form parameters. It can then be shown that under very general conditions, consistent estimates of the structural parameters and «j can be obtained by substituting E(Qn | Xln, X2„, X3n) for Q„ in the original linear structural equation (13.11) and running OLS.

We propose to estimate the parameters of the model we derived in section 13.3, using a procedure based on the methods just described. We discuss our specific application and present empirical estimates in section 13.5.

13.5 The Evidence

The estimation procedure used for our model follows directly from the procedure outlined in the previous section. We can briefly summarize the process. The first step is the estimation of switching regression reduced form equations for each of the three endogenous variables—autos, other durables, and consumer debt. Parameters of the reduced forms are estimated by maximum likelihood assuming a Hartley-Mallela switching regression model with correlated equation errors. A complication of the reduced form estimation is the requirement that the sign of the debt variable in both the auto and durable structural equations be known.

The second step of the procedure is to construct instrumental variables for the three endogenous variables. Although all are not used, potentially three different instruments could be computed for each reduced form: (1) an instrument for the observed dependent variable computed using the formula given in equation (13.32), (2) an instrument for the unobserved demand dependent variable formed from the demand equaton, and (3) an instrument for the unobserved supply dependent variable.

The final step of the procedure is the estimation of structural equation parameters. Both the auto and other durable equations are estimated by OLS, using instruments for the observed dependent variables (instrument form 1). Although OLS yields consistent parameter estimates, we note that the form of equation (13.32) implies that OLS standard errors are unlikely to be consistent.

The equations of particular interest to this chapter are the structural equations for debt. Unfortunately, since debt demand is assumed to be a function of the same exogenous variables as durable and auto demand, the structural equation for debt demand is not fully identified. Thus we are forced to estimate a reduced form for the second-stage debt demand equation. The debt supply equation, however, is identified.

Estimates of the debt supply and demand equations are computed by maximizing a switching regression likelihood function of the exogenous variables and supply instruments for autos and other durables. As shown in section 13.4, these estimates will be strongly consistent.

Maximum likelihood estimates (or quasi-maximum likelihood in case of the second-stage equations) of the switching regression parameters are computed using a Davidon-Fletcher-Powell iterative procedure with two criteria for convergence. Each element of the vector of first derivatives and the change in the log likelihood function value is required to be within a preset tolerance of zero. As a check on the quality of convergence the negative of the matrix of log likelihood second derivatives is computed and, if invertible, used as an estimate of the information matrix and for standard errors. We note, however, that for the second stage, the matrix of second derivatives will not yield estimates of our actual coefficient standard errors. A correct interpretation is that they yield consistent estimates of the standard errors of the maximum likelihood estimators and hence can be considered lower bounds.

Before we report the results of our estimation, it may be useful to show a few sample statistics. Sample means of several key variables broken down by race are given in table 13.2. One statistic that stands out is the substantially higher average debt holdings for whites than for blacks or Hispanics. This difference, which supports Bell's (1974) evidence, is mirrored in auto and durable holdings but is not nearly as apparent in income.

The first step of the estimation was the three switching regression reduced forms, parameter estimates of which are given in section 13.8. It

Table 13.2

Sample statistics (means broken down by race)

Table 13.2

Sample statistics (means broken down by race)

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