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Empirical Economics

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11.06.2016

Marginal effects in multivariate probit models

verfasst von: John Mullahy

Erschienen in: Empirical Economics | Ausgabe 2/2017

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Abstract

Estimation of marginal or partial effects of covariates x on various conditional parameters or functionals is often a main target of applied microeconometric analysis. In the specific context of probit models, estimation of partial effects involving outcome probabilities will often be of interest. Such estimation is straightforward in univariate models, and results covering the case of quadrant probability marginal effects in bivariate probit models for jointly distributed outcomes y have previously been described in the literature. This paper’s goals are to extend Greene’s results to encompass the general \(M\ge 2\) multivariate probit context for arbitrary orthant probabilities and to extended these results to models that condition on subvectors of y and to multivariate ordered probit data structures. It is suggested that such partial effects are broadly useful in situations, wherein multivariate outcomes are of concern.

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One obvious example is that of conditional product moments \(E\left[ {\prod _{{ j}=1}^{M} {y_{j}^{{b}_{j} } } \left| \mathbf{x} \right. } \right] \) of which conditional covariances may be the most familiar example. In such cases, how \(\sigma _{{i,j}} \left( \mathbf{x} \right) \) varies with conditioning sets x may be of interest in applications (consider, for example, GARCH and related literature).

See also Christofides et al. (1997, 1998).

To streamline the analysis and notation, the x’s will be treated as continuous so that “\(\partial \mathbf{x}\)” calculus can be used. Discrete x’s (e.g., dummy variables, count measures) can be accommodated straightforwardly with the understanding that discrete differences in \(\Pr \left( {y_1 =k_1 ,\ldots ,y_{M} =k_{M} \left| \mathbf{x} \right. } \right) \) due to \(\Delta x_{j} =1\) will be of interest; these can be computed by evaluating \(\Pr \left( {y_1 =k_1 ,\ldots ,y_{M} =k_{M} \left| \mathbf{x} \right. } \right) \) at two different values of \(x_{j}\) and then differencing.

Somewhat informally, the paper uses the term “orthant probability” in reference to the vector of binary outcomes y to refer to the probabilities that the underlying latent random variables that map into the observed binary y (see (4) below) occupy any of the \(2^{{M}}\) orthants in \({\mathbb {R}}^{{M}}\) defined implicitly by k. Some additional notation will also prove useful. Let K be the \(2^{{M}} \times M\) matrix whose rows (arranged arbitrarily) are the \(2^{{M}}\) possible outcome configurations k. Let \({\mathbb {P}}\) be a \(2^{{M}}\)-element set indexing rows of K having typical indexing element p, so that \(\mathbf{k}_\mathbf{p} =\mathbf{K}_{\mathbf{p}{\bullet }} \) will denote a particular (pth) outcome configuration.

This stochastic structure allows for but does not appeal specifically to a common factor error structure for \({\varvec{\varepsilon }}\) in (4). It may be that such an assumption would simplify estimation and, ultimately, computation of the marginal effects.

In applied studies, an explicit formulation of the model of interest as \(\Pr \left( {\mathbf{y}_\mathrm{a} ={\mathbf{k}}_{{p,a}} \left| {\mathbf{y}_{b} =\mathbf{k}_{{p,b}} ,\mathbf{x}} \right. } \right) \) is often absent, and this conditional probability may or may not be the parameter whose marginal effects are of interest. See Greene (1996) for conceptual discussion.

Allowing the \(y_{j} \) to have different numbers of outcomes is straightforward; the assumption of equal numbers of categories across j is made solely to keep notation from becoming unwieldy.

Estimation of the M-variate multivariate ordered probit model can be approached using the methods spelled out in Mullahy (2016).

Of course, for each covariate the sum of the marginal effects across all 32 patterns must be zero.

See Huguenin et al. (2009) for a discussion of other considerations that arise in estimation of MVP models, wherein dimension reduction is a primary consideration.

Cappellari L, Jenkins SP (2003) Multivariate probit regression using simulated maximum likelihood. Stata J 3(3):278–294

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Christofides LN, Stengos T, Swidinsky R (2000) Corrigendum. Econ Lett 68(3):339CrossRef

Frees EW, Valdez EA (1998) Understanding relationships using copulas. N Am Actuar J 2(1):1–25CrossRef

Greene WH (1996) Marginal effects in the bivariate probit model. NYU Stern School of Business working paper EC-96-11

Greene WH (1998) Gender economics courses in liberal arts colleges: further results. J Econo Educ 29(4):291–300CrossRef

Greene WH (2004) Convenient estimators for the panel probit model: further results. Empir Econ 29(1):21–47CrossRef

Greene WH, Hensher DA (2010) Modeling ordered choices: a primer. Cambridge University Press, CambridgeCrossRef

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Huguenin J, Pelgrin F, Holly A (2009) Estimation of multivariate probit models by exact maximum likelihood. University of Lausanne, IEMS working paper 09-02

Mullahy J (2016) Estimation of multivariate probit models via bivariate probit. Stata J 16(1):37–51

Rao CR (1973) Linear statistical inference and its applications, 2nd edn. Wiley, New YorkCrossRef

Trivedi PK, Zimmer DM (2005) Copula modeling: an introduction for practitioners. Found Trends Econom 1(1):1–111CrossRef

Titel: Marginal effects in multivariate probit models
verfasst von: John Mullahy
Publikationsdatum: 11.06.2016
Verlag: Springer Berlin Heidelberg
Erschienen in: Empirical Economics / Ausgabe 2/2017
Print ISSN: 0377-7332
Elektronische ISSN: 1435-8921
DOI: https://doi.org/10.1007/s00181-016-1090-8

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