Consider the standard binary logit regression model (see Section 14.3).

(a) Write down the log-likelihood function.

(b) Introduce a random intercept assumption in which the intercept is drawn from a suitable distribution with finite mean and variance. What justification can you offer for introducing an unobserved heterogeneity term in this way? If the logit model is derived from the random utility model with extreme value errors, how does the random intercept affect that interpretation and/or derivation? [See Revelt and Train, 1998.]

(c) Suggest a suitable distributional assumption for the random intercept; rewrite the likelihood function conditional on unobserved heterogeneity. Next write down the likelihood function with unobserved heterogeneity integrated out.

(d) Describe in a step-by-step manner how to use the maximum simulated likelihood estimation procedure to estimate this model. Explain, with details, how to calculate the variance matrix of unknown parameters. How would you decide how many simulations you will use?

(e) Consider the method of simulated moments as an alternative to the MSL procedure for the random parameter logit. Write down the moment condition(s) conditional on the unobserved heterogeneity term. Then outline an MSM estimation procedure for this model.

14.3. Logit and Probit Models We now provide more formal theory for these models. We present binary outcomes as a direct extension of the coin-toss example of introductory statistics to situations where the probability of success is modeled to depend on regressors. Two commonly used parameterizations lead to the logit and probit models. Motivation for these pa- rameterizations, using latent variables, is deferred to Section 14.4.