Bayesian procedures as a numerical tool for the estimation of an intertemporal discrete choice model

Abstract

Discrete choice models usually require a general specification of unobserved heterogeneity. In this paper, we apply Bayesian procedures as a numerical tool for the estimation of a female labor supply model based on a sample size that is typical for common household panels. We provide two important results for the practitioner: First, for a specification with a multivariate normal distribution for the unobserved heterogeneity, the Bayesian MCMC estimator yields almost identical results as a classical maximum simulated likelihood (MSL) estimator. Second, we show that when imposing distributional assumptions that are consistent with economic theory, e.g., log-normally distributed consumption preferences, the Bayesian method performs well and provides reasonable estimates, while the MSL estimator does not converge. These results indicate that Bayesian procedures can be a beneficial tool for the estimation of intertemporal discrete choice models.

Appendix

1.1 Additional details on the numerical implementation

We describe how the draws of $\beta _i$ have been taken by applying the Metropolis–Hastings algorithm. The draws of the fixed coefficients $\alpha$ have been taken analogously (see Train 2009 for more details). The algorithm is embedded into the Gibbs sampler. Starting with an arbitrary value for $\beta _{i}$, the following steps are implemented within each iteration of the Gibbs sampling conditional on the other subsets of the parameters:

• 1. Draw F independent values from a standard normal density to get a vector $\eta$.

• 2. This gives a new trial value $\beta _{i}^\mathrm{new}=\beta _{i}^\mathrm{old}+\rho L\eta$, where L is the Choleski factor of W and $\rho$ is the size of each jump that is adjusted iteratively to target an acceptance rate of 0.3. Hence, the proposal distribution is specified to be normal with mean zero and variance $\rho ^{2}W$.

• 3. Draw $\mu$ from a standard uniform distribution and calculate $R=\frac{L_{i\mathbf j }(\alpha ,\beta _{i}^\mathrm{new})\phi (\beta _{i}^\mathrm{new}|b,W)}{L_{i\mathbf j }(\alpha ,\beta _{i}^\mathrm{old})\phi (\beta _{i}^\mathrm{old}|b,W)}$.

• 4. If $\mu <=R$, accept $\beta _{i}^\mathrm{new}$, otherwise reject $\beta _{i}^\mathrm{new}$.

1.2 Estimated distributions of consumption preferences

The figure shows the marginal distributions of the consumption preferences $\beta _i^I$ for different distributional assumptions. We consider here the preferred specifications with four random coefficients (specifications 4–6). The density functions have been computed by taking a large number of draws from the respective distributions on the basis of the estimated moments and using a standard software package to obtain kernel density estimates. This shows how the different distributional assumptions translate into fairly different estimates for the unobserved heterogeneity (Fig. 1).

Fig. 1

Estimated distributions of consumption preferences for spec. 4–6