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This paper develops a parameter-expanded Monte Carlo EM (PX-MCEM) algorithm to perform maximum likelihood estimation in a multivariate sample selection model. In contrast to the current methods of estimation, the proposed algorithm does not directly depend on the observed-data likelihood, the evaluation of which requires intractable multivariate integrations over normal densities. Moreover, the algorithm is simple to implement and involves only quantities that are easy to simulate or have closed form expressions.

Sample selection models, pioneered in [

There are two dominant approaches in the current literature to estimate these models. One approach is to use maximum likelihood (ML) estimation. However, as noted in the literature, a major hurdle in evaluating the like- lihood is that it requires computations of multivariate integrals over normal densities, which do not generally have closed form solutions. [

The objective of this paper is to develop a simple ML estimation algorithm for a commonly used multivariate sample selection model. In particular, this paper develops a parameter-expanded Monte Carlo expectation maximization (PX-MCEM) algorithm that differs from [

This paper is organized as follows. The multivariate sample selection model (MSSM) is formulated in Section 2. Section 3 begins with a brief overview of the EM algorithm for the MSSM and continues with the develop- ment of the PX-MCEM algorithm. Methods to obtain the standard errors are discussed. Section 4 offers some concluding remarks.

The MSSM is

for observations

variable underlying the binary selection variable

that equals

missing when

transpose.

Furthermore,

vectors of parameters. Define

contain at least one exogenous covariate that does not overlap with

trictions). The unobserved errors

The submatrix

The covariates and binary selection variables are always observed. Without loss of generality, assume that the outcomes for any observation

The PX-MCEM algorithm is based on the EM algorithm of [

where

The EM algorithm then proceeds iteratively between an expectation step (E-step) and a maximization step (M-step) as follows. In iteration

where the expectation is taken with respect to the conditional predictive distribution for the missing data,

Denote the maximal values as

For the MSSM,

of complete data,

with

normal with mean

Equation (10) is a degenerate density since conditioning on

The standard EM algorithm using (7) and (8) is difficult to implement for the MSSM as the E-step and M-step are intractable. The PX-MCEM algorithm addresses this issue by modifying the E-step in two ways and leads to an M-step that can be evaluated with closed form quantities. Stated succinctly, the PX-MCEM algorithm is as follows.

1. Initialize

At iteration

2. Draw

3. PX-MC E-step: Estimate

4. PX-MC M-step: Maximize

obtain the maximizing parameters

5. Reduction step: Apply reduction functions to

6. Repeat Steps 2 through 5 until convergence. The converged values are the ML estimates

Each step is described in more detail in the subsequent sections.

Following [

with

are defined analogously to

Second, instead of computing

where

for

Similarly, for the latent variables,

for

The Gibbs sampler recursively samples from the full conditional distributions in (14) and (15) in the usual way. After a sufficient burn-in period, the last

By recognizing that (11) is proportional to the log-likelihood function of a seemingly unrelated regression model with

and

where

and (17) recursively until convergence. Denote the converged values as

In the reduction step, set

maining

The observed information matrix is

where

estimated by taking additional draws from the Gibbs sampler and constructing their Monte Carlo analogs. The standard errors are the square roots of the diagonals of the inverse estimated quantity in (18).

A new and simple ML estimation algorithm is developed for multivariate sample selection models. Roughly speaking, the implementation of this algorithm only involves iteratively drawing sets of missing data from well- known distributions and using IGLS on the complete data, both of which are inexpensive to perform. By using parameter expansion and Monte Carlo methods, the algorithm only depends on quantities with closed form expressions, even when estimating the covariance matrix parameter with correlation restrictions. This algorithm is readily extendable to other types of selection models, including extensions to various types of outcome and selection variables with an underlying normal structure, and modifications to time-series or panel data.

I would like to thank the referee, Alicia Lloro, Andrew Chang, Jonathan Cook, and Sibel Sirakaya for their helpful comments.