A COMPANION TO Theoretical Econometrics

Appendix B Existence of Joint Posterior Distribution

Let ю denote the number of missing wage observations, and let Q = [Aq, BQ]“ £ ^++ be the domain of unobserved wages, where 0 < AQ < BQ < Also, let A = [Ад, BA]3NT £ ^3NT be the domain of latent relative utilities, where -^ < Aa < BA < ro. We want to show that:

Here, we have subsumed a and Л into n and ¥, respectively (as we did in Step 5 of Section 3) and defined

g(V, p, E-1) = IE-1 |—exp{-±(V - Yp)'(E-1 ® INT)(V - Yp)}, (22.B2)

where

(22.B3)

We have also defined

h(Z, W, n, En1) = |En1 |~exp{-£(Z - W - Wn)'(Ej ® N)(Z - W - Wk)}

■ I(Zijt > 0, Zm(k Ф j) < 0 if dit = j and j £ {1, 2, 3}, {Zjj 1,2,3 < 0 otherwise),

(22.B4)

(0).

Since the only arguments common to both g and h are those that include func­tions of wages, we can express (22.B1) as

We first observe that for any configuration of unobserved wages in Q,

To see this, note that we can express g(V, p, Eg1) as

g(V, в, Ee) = IE-1 |^exp{-^tr(S(P)E-1) - l(P - P),Y,(E-1 ® INT)Y(p - p)},

(22.B7)

where

S = (Y'(E-1 ® INt)Y)-1Y'(E-1 ® Int)V

S(S) = (V1 - Y1S1, V2 - Y2S2)'(V - Y1S1, V2 - Y2S2) (22.B8)

Hence, g(V, p, Eg) is proportional to a normal-Wishart density (Bernardo and Smith, 1994, p. 140), hence finitely integrable to a value g(V) that, in general, will depend on the configuration of unobserved wages. Since g(V) is finite over the compact set Q, it follows that g(V) is bounded over Q.

Turn next to h(Z, W, n, E,-1). Since (22.B4) has the same form as (22.B2), just as (22.B7) we can write:

h(Z, W, n, Xn1) = |ХЛ1 |^exp{- ^1tr(S(l)En1) - i(n - n)'W'(En1 ® Im№ - I)}

■ I(Zijt > 0, ZiRt(k Ф j) < 0 if du = j and j Є {1, 2, 3}, {Zijt}j=1Z3 < 0 otherwise) (22.B9)

where I and S(I) are defined in a way that is exactly analogous to (22.B8). Hence,

for any configuration of unobserved wages in Q and latent utilities in A. It fol­lows that h(Z, W) is bounded over Q x A. Thus, the integral (B1) reduces to

which is finite since each element of the integrand is bounded over the compact domain of integration.

Notes

0 Currently, approaches to numerical integration such as quadrature and series expan­sion are not useful if the dimension of the integration is greater than three or four.

1 Restrictions of this type can be tested easily by estimating versions of the model with different but nested future components.

2 These findings are related to those of Lancaster (1997), who considered Bayesian infer­ence in the stationary job search model. He found that if the future component is treated as a free parameter (rather than being set "optimally" as dictated by the offer wage function, offer arrival rate, unemployment benefit and discount rate) there is little loss of information about the structural parameters of the offer wage functions. (As in our example, however, identification of the discount factor is lost.) The station­ary job search model considered by Lancaster (1997) has the feature that the future component is a constant (i. e. it is not a function of state variables). Our procedure of treating the future component as a polynomial in state variables can be viewed as extending Lancaster's approach to a much more general class of models.

3 As noted earlier, the future component's arguments reflect restrictions implied by the model. For instance, because the productivity and preference shocks are serially inde­pendent, they contain no information useful for forecasting future payoffs and do not appear in the future component's arguments. Also, given total experience in each occupation, the order in which occupations one and two were chosen in the past does not bear on current or future payoffs. Accordingly, only total experience in each occupation enters the future component.

4 To begin the Gibbs algorithm we needed an initial guess for the model's parameters (although the asymptotic behavior of the Gibbs sampler as the number of cycles grows large is independent of starting values). We chose to set the log-wage equation Ps equal to the value from an OLS regression on observed wages. The diagonal elements of £e were set to the variance of observed log-wages, while the off-diagonal elements were set to zero. The school payoff parameters were all initialized at zero. All of the future component's n values were also started at zero, with the exception of the alternative-specific intercepts. The intercepts for alternatives one, two, and three were initialized at -5,000, 10,000, and 20,000, respectively. These values were chosen with an eye towards matching aggregate choice frequencies in each alternative. We initialized the covariance matrix by setting all off-diagonal elements to zero, and each diagonal element to 5 x 108. We used large initial variances because doing so increases the size of the initial Gibbs steps, and seems to improve the rate of con­vergence of the algorithm.

5 Space considerations prevent us from reporting results for individual expectations parameters. Instead, below we will graphically compare the form of the estimated future component to that which was used to generate the data.

6 We also ran OLS accepted log-wage regressions for the 1-EMAX through 5-EMAX data sets. The results are very similar to those in Table 22.4, so we do not report them here. The estimates again show substantial biases for all the wage equation parameters. Thus, the Gibbs sampling algorithm continues to do an impressive job of implement­ing a dynamic selection correction despite the fact that the agents' decision rules are misspecified.

7 The mean state vectors were derived from the choices in data set 5-EMAX, and the coefficients of the polynomial were valued at the posterior means derived from the analysis of data set 5-EMAX.

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