Fitting fully observed recursive mixed-process models with cmp
Abstract. At the heart of many econometric models are a linear function and a
normal error. Examples include the classical small-sample linear regression model
and the probit, ordered probit, multinomial probit, tobit, interval regression, and
truncated-distribution regression models. Because the normal distribution has a
natural multidimensional generalization, such models can be combined into multiequation
systems in which the errors share a multivariate normal distribution.
The literature has historically focused on multistage procedures for fitting mixed
models, which are more efficient computationally, if less so statistically, than maximum
likelihood. Direct maximum likelihood estimation has been made more practical
by faster computers and simulated likelihood methods for estimating higherdimensional
cumulative normal distributions. Such simulated likelihood methods
include the Geweke–Hajivassiliou–Keane algorithm (Geweke, 1989, Econometrica
57: 1317–1339; Hajivassiliou and McFadden, 1998, Econometrica 66: 863–896;
Keane, 1994, Econometrica 62: 95–116). Maximum likelihood also facilitates a
generalization to switching, selection, and other models in which the number and
types of equations vary by observation. The Stata command cmp fits seemingly unrelated
regressions models of this broad family. Its estimator is also consistent for
recursive systems in which all endogenous variables appear on the right-hand sides
as observed. If all the equations are structural, then estimation is full-information
maximum likelihood. If only the final stage or stages are structural, then estimation
is limited-information maximum likelihood. cmp can mimic a score of built-in
and user-written Stata commands. It is also appropriate for a panoply of models
that previously were hard to estimate. Heteroskedasticity, however, can render cmp
inconsistent. This article explains the theory and implementation of cmp and of a
related Mata function, ghk2(), that implements the Geweke–Hajivassiliou–Keane
algorithm.
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David Roodman
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cmp, ghk2, Geweke–Hajivassiliou–Keane algorithm, recursive mixed-process models, seemingly unrelated regression, conditional mixed-process models
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