Average partial effects in multivariate probit models with latent heterogeneity: Monte Carlo experiments and an application to immigrants' ethnic identity and economic performance

We extend the univariate results in Wooldridge (2005) to multivariate probit models, proving the following. 1) Average partial effects (APEs) based on joint probabilities are consistently estimated by conventional multivariate probit models under general forms of conditionally independent latent heterogeneity (LH) as long as the only constraints beyond normalization, if any, are within-equation homogenous restrictions. The normalization of choice is not neutral to consistency in models with cross-equation parameter restrictions beyond normalization, such as those implemented by Stata's asmprobit command or in the panel probit model: if the normalization is through an error covariance matrix in correlation form, consistency breaks down unless the LH components are truly homoskedastic. This is substantial because an error covariance matrix in correlation form is the only normalization permitted by Stata's biprobit and mvprobit commands or Limdep's BIVARIATE PROBIT and MPROBIT. Covariance restrictions beyond normalizations generally conflict with an arbitrary covariance matrix for the LH components. The multinomial probit model with i.i.d. errors, implemented by Stata's mprobit, is a case in point. 2) Conditional independence of the LH components is not generally sufficient for consistent estimation of APEs on conditional probabilities. Consistency is restored by maintaining an additional independence assumption. This holds true whether or not the response variables are used as regressors. 3) The dimensionality benefit observed by Mullahy (2011) in the estimation of partial effects extends to APEs. We exploit this feature in the design of a simple procedure estimating APEs, which is both faster and more accurate than simulation-based codes, such as Stata's mvprobit and cmp. To demonstrate the finite-sample implications of our results, we carry out extensive Monte Carlo experiments with bivariate and trivariate probit models. Finally, we apply our procedure in (3) to Italian survey data of immigrants in order to estimate the APEs of a trivariate probit model of ethnic identity formation and economic performance.