Sample selection models, pioneered in [1] - [3] , are indispensable to researchers who use observational data for statistical inference. Among the many variants of these types of models, there is a growing interest in multiva- riate sample selection models. These are used to model a system of two or more seemingly unrelated equations, where the outcome variable for each equation may be non-randomly missing or censored according to its own stochastic selection variable. Applications range from modeling systems of demand equations [4] [5] to house- hold vehicle usage [6] - [8] . A common specification is to assume a correlated multivariate normal distribution underlying both the outcomes of interest and the latent variables in the system.

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. [9] discusses the ML estimation of these models and proposes to use the popular Geweke, Hajivassiliou, and Keane (GHK) algorithm to approximate these integrals in a simulated ML frame- work. While this strategy works reasonably well, the GHK algorithm can be difficult to implement. Another popular approach is to use two-step estimation (see [10] for a survey). In general, there is a tradeoff in the statistical properties and the computational simplicity for these estimators. If efficiency and consistency are of pri- mary concern, then ML estimation should be preferred over two-step estimation.

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 [9] in a few important ways. First, the PX-MCEM algo- rithm does not use the observed-data likelihood directly, so it avoids the aforementioned integrations. Second, the proposed iterative algorithm does not require the evaluations of gradients or Hessians, which become increa- singly difficult to evaluate with more parameters and equations. Third, the algorithm is straightforward to implement. It only depends on quantities that are either easy to simulate or have closed form expressions. This last point is especially appealing when estimating the covariance matrix parameter since there are non-standard restrictions imposed onto it for identification.

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, and equations. In the previous expressions, is the continuous outcome of interest for observation and equation. Using similar indexing notation, is the latent

variable underlying the binary selection variable, where denotes an indicator function

that equals if event is true and 0 otherwise. Sample selection is incorporated by assuming that is

missing when. Otherwise, is observed and equal to. For later use, define and, where the prime symbol in, , and in the rest of this paper is used to denote matrix

transpose.

Furthermore, and are column vectors of exogenous covariates, and and are conforming

vectors of parameters. Define and. For identification, must

contain at least one exogenous covariate that does not overlap with (refer to [11] for these exclusion res-

trictions). The unobserved errors and are jointly distributed as a

-dimensional multivariate normal with a mean vector of zeros and an unknown covariance matrix of. Formally, with

The submatrix is restricted to be in correlation form to identify the parameters corresponding to the latent variables [9] . The other elements of are restricted such that the matrix is symmetric and positive definite.

The covariates and binary selection variables are always observed. Without loss of generality, assume that the outcomes for any observation are only missing for the first equations, where. Define

, and let denote the observed data. The observed-data likelihood derived from (1) through (5) is denoted as. See [9] for an exact expression of this likelihood.

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.