Marginal effects in multivariate probit models

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.

Appendices

Appendix 1: Detailed derivations for the general case

For an intuition for (2), note that in the $M{=}2$ case the partial derivative w.r.t. $u_1$ of the function $g\left( {u_1 ,u_2 } \right) \equiv {\partial F\left( {u_1 ,u_2} \right) }big/{\partial u_2 }$ evaluated at u $=$ v must in light of (1) yield the joint density $f\left( {v_1 ,v_2 } \right)$. One function $g\left( {v_1 ,v_2 } \right)$ satisfying this is $g\left( {v_1 ,v_2 } \right) =f_2 \left( {v_2 } \right) \times F\left( {v_1 \left| {v_2 } \right. } \right)$, which is of the form (2); this follows since, at u $=$ v,

$$\begin{aligned} \frac{\partial f_2 \left( {v_2 } \right) \times F\left( {v_1 \left| {v_2 } \right. } \right) }{\partial v_1 }=f_2 \left( {v_2 } \right) \times \frac{\partial F\left( {v_1 \left| {v_2 } \right. } \right) }{\partial v_1 }=f_2 \left( {v_2 } \right) \times f\left( {v_1 \left| {v_2 } \right. } \right) =f\left( {v_1 ,v_2 } \right) .\nonumber \\ \end{aligned}$$

(8)

By recursion, this result generalizes to M>2 by working backwards from the Mth cross-partial derivative. The general sequence of partial derivatives of $F\left( {\ldots } \right)$ is (differentiating w.o.l.o.g. in the order $j=1,2,\ldots ,M$):

$$\begin{aligned} \frac{\partial F\left( {\mathbf{v}} \right) }{\partial v_1 }= & {} f_1 \left( {v_1 } \right) \times F_{-1} \left( {v_2 ,\ldots ,v_{M} \left| {v_1 } \right. } \right) \\ \frac{\partial ^{{r}}F\left( {\mathbf{v}} \right) }{\partial v_1 \cdots \partial v_{r} }= & {} f_1 \left( {v_1 } \right) \times \left\{ {\prod \limits _{{{k}}=2}^{r} {f\left( {v_{k} \left| {v_1 ,\ldots ,v_{{k}-1} } \right. } \right) } } \right\} \\ \quad\times & {} F_{-\left\{ {1,\ldots ,{r}} \right\} } \left( {v_{{r}+1} ,\ldots ,v_{M} \left| {v_1 ,\ldots ,v_{r} } \right. } \right) ,~ r=2,\ldots ,M-1 \\ \frac{\partial ^{{M}}F\left( {\mathbf{v}} \right) }{\partial v_1 \cdots \partial {v}_{M} }= & {} f\left( {\mathbf{v}} \right) . \end{aligned}$$

This result is trivial when the $v_{{j}}$ are mutually independent, in which case all the conditioning arguments are irrelevant.

Alternatively, (2) can be obtained directly using Leibniz’s rule for differentiation of integrals whose limits depend on the variable of differentiation. Since $F\left( {\mathbf{v}} \right) =\int _{-\infty }^{v_{M} } \cdots \int _{-\infty }^{v_1 } {f\left( {\mathbf{u}} \right) \hbox {d}u_1 \cdots \hbox {d}u_{M} }$, then one can obtain ${\partial F\left( {\mathbf{v}} \right) }\big /{\partial v_{j} }$ by noting that $v_{j}$ appears in this expression only once, as the upper limit of one integration, so that passing Leibniz’s rule into the integral yields

$$\begin{aligned}&\frac{\partial }{\partial v_{{j}} }\left( {\int _{-\infty }^{v_{{M}} } \cdots \int _{-\infty }^{v_1 } {f\left( {\mathbf{u}} \right) \hbox {d}u_1 \cdots \hbox {d}u_{{M}} } } \right) \\&\quad =\int _{-\infty }^{v_{{M}} } \cdots \int _{-\infty }^{v_{{j}+1} } {\int _{-\infty }^{v_{{j}-1} } \cdots } \int _{-\infty }^{v_1 }\\&\qquad \times \, {\left( {\frac{\partial }{\partial v_{{j}} }\int _{-\infty }^{v_{{j}} } {f\left( {u_1 ,\ldots ,u_{{M}} } \right) \hbox {d}u_{{j}} } } \right) } \hbox {d}u_1 \cdots \hbox {d}u_{{j}-1} \hbox {d}u_{{j}+1} \cdots \hbox {d}u_{{M}} \\&\quad =\int _{-\infty }^{v_{{M}} } \cdots \int _{-\infty }^{v_{{j}+1} } {\int _{-\infty }^{v_{{j}-1} } \cdots } \int _{-\infty }^{v_1 }\\&\qquad \times \,{\left( {f\left( {u_1 ,\ldots ,u_{{j}-1} ,v_{{j}} ,u_{{j}+1} ,\ldots ,u_{{M}} } \right) } \right) } \hbox {d}u_1 \cdots \hbox {d}u_{{j}-1} \hbox {d}u_{{j}+1} \cdots \hbox {d}u_{{M}} \\&\quad =\int _{-\infty }^{v_{{M}} } \cdots \int _{-\infty }^{v_{{j}+1} } {\int _{-\infty }^{v_{{j}-1} } \cdots } \int _{-\infty }^{v_1 }\\&\qquad \times \,{\left( {f\left( {u_1 ,\ldots ,u_{{j}-1} ,u_{{j}+1} ,\ldots ,u_{{M}} \left| {v_{{j}} } \right. } \right) \times f\left( {v_{{j}} } \right) } \right) } \hbox {d}u_1 \cdots \hbox {d}u_{{j}-1} \hbox {d}u_{{j}+1} \cdots \hbox {d}u_{{M}} \\&\quad =f\left( {v_{{j}} } \right) \times \int _{-\infty }^{v_{{M}} } \cdots \int _{-\infty }^{v_{{j}+1} } {\int _{-\infty }^{v_{{j}-1} } \cdots } \int _{-\infty }^{v_1 }\\&\qquad \times \,{\left( {f\left( {u_1 ,\ldots ,u_{{j}-1} ,u_{{j}+1} ,\ldots ,u_{{M}} \left| {v_{{j}} } \right. } \right) } \right) } \hbox {d}u_1 \cdots \hbox {d}u_{{j}-1} \hbox {d}u_{{j}+1} \cdots \hbox {d}u_{{M}} \\&\quad =f\left( {v_{{j}} } \right) \times F\left( {v_1 ,\ldots ,v_{{j}-1} ,v_{{j}+1} ,\ldots ,v_{{M}} \left| {v_{{j}} } \right. } \right) \\ \end{aligned}$$

Analogous results appear in the literature on copula joint distribution functions (Frees and Valdez 1998; Trivedi and Zimmer 2005) in which the joint distribution of y is represented in copula form as

$$\begin{aligned} C\left( {F_1 \left( {y_1 } \right) ,\ldots ,F_{{M}} \left( {y_{{M}} } \right) } \right) =C\left( {u_1 ,\ldots ,u_{{M}} } \right) =F\left( {\mathbf{u}} \right) \end{aligned}$$

with $F_{{j}}$ denoting the marginal distribution function of $y_{{j}}$, with the $u_{{j}}$ being marginally uniform variates. A familiar result in the bivariate copula literature is that ${\partial C\left( {u_1 ,u_2 } \right) }\big /{\partial u_1 =}F\left( {u_2 \left| {u_1 } \right. } \right)$. This is essentially equivalent to (8) since uniform marginal densities satisfy $f_{{j}} \left( {u_{{j}} } \right) =1$. Note, however, that there are instances in the copula literature in which results like ${\partial F\left( {u_1 ,u_2 } \right) }\big /{\partial u_1 =}F\left( {u_2 \left| {u_1 } \right. } \right)$ are stated. In light of (8), this result in general does not hold unless $f_{{j}} \left( {u_{{j}} } \right) =1$.

Appendix 2: Detailed derivations for the multivariate probit model

Let $s_{{jp}} =2k_{{jp}} -1$ so that $s_{{jp}} \in \left\{ {-1,1} \right\}$, and define correspondingly the $M\times M$ diagonal transformation matrixes ${\mathbf{T}}_{p} =\hbox {diag}\left[ {s_{{jp}} } \right] , p=1,\ldots ,2^{{M}},j=1,\ldots ,M$. Define for each p the transformation $\mathbf{Q}_{p} =\mathbf{T}_{p} \mathbf{RT}_{p}$ of the original covariance (i.e., correlation) matrix R, so that $\mathbf{Q}_{{p}}$ is of the form

$$\begin{aligned} \mathbf{Q}_{p} =\left[ {{\begin{array}{cccc} 1&{}\quad {s_{1{p}} s_{2{p}} \rho _{12} }&{}\quad \cdots &{}\quad {s_{1{p}} s_{{Mp}} \rho _{1{M}} } \\ {s_{1{p}} s_{2{p}} \rho _{12} }&{}\quad 1&{}\quad &{}\quad \vdots \\ \vdots &{}\quad &{}\quad \ddots &{}\quad \\ {s_{1{p}} s_{{Mp}} \rho _{1{M}} }&{}\quad \cdots &{}\quad &{}\quad 1 \\ \end{array} }} \right] =\left[ {{\begin{array}{cccc} 1&{}\quad {\tau _{12{p}} }&{}\quad \cdots &{}\quad {\tau _{1{Mp}} } \\ {\tau _{12{p}} }&{}\quad 1&{}\quad &{}\quad \vdots \\ \vdots &{}\quad &{}\quad \ddots &{}\quad \\ {\tau _{1{Mp}} }&{}\quad \cdots &{}\quad &{} 1\quad \\ \end{array} }} \right] . \end{aligned}$$

The conditional-on-x probability of any particular outcome configuration $\mathbf{k}_{p}$ is thus given by

$$\begin{aligned}&\Pr \left( {y_1 =k_{1{p}} ,\ldots ,y_{{M}} =k_{{Mp}} \left| \mathbf{x} \right. } \right) =\Phi _{\mathbf{Q}_{p} } \left( {s_{1{p}} \mathbf{x}{\varvec{\upbeta }}_1 ,\ldots ,s_{{Mp}} \mathbf{x}{\varvec{\upbeta }}_{{M}} } \right) \\&\qquad =\Phi _{\mathbf{Q}_{p} } \left( {\alpha _{1{p}} ,\ldots ,\alpha _{{Mp}}} \right) , \end{aligned}$$

where $\Phi _\mathbf{Q}$ is the cumulative of an $\hbox {MVN}\left( {\mathbf{0,Q}} \right)$ distribution with density ${\phi }_{\mathbf{Q}} \left( {\ldots } \right)$ and $\alpha _{{jp}} =s_{{jp}} \mathbf{x}{\varvec{\upbeta }}_{{j}}$. Using the transformed matrixes Q in place of the original correlation matrixes R streamlines the exposition since for each configuration p, the outcome orthant probability can be described by a joint cumulative rather than by a notationally messy mix of cumulatives and survivor functions. This amounts to a linear change-of-variables operation on ${\varvec{\varepsilon }} =\left[ {\varepsilon _1 ,\ldots ,\varepsilon _{m} } \right]$ of the form $\mathbf{T}_{p} {\varvec{\varepsilon }}$ which becomes the effective error structure of model at each p; this transformation works due to the symmetry of the distribution of ${{\varvec{\varepsilon }}}$ around the origin.

To obtain the MVP’s marginal effects, it thus suffices to obtain the particular expressions corresponding to the second line in (3). $f_{{j}} \left( {c_{{j}} \left( {\varvec{\theta }} \right) } \right)$ is a univariate $N\left( {0,1} \right)$ density and $F_{-{j}} \left( {c_1 \left( {\varvec{\theta }} \right) ,\ldots ,c_{{j}-1} \left( {\varvec{\theta }} \right) ,c_{{j}+1} \left( {\varvec{\theta }} \right) ,\ldots ,c_{m} \left( {\varvec{\theta }} \right) \left| {c_{{j}} \left( {\varvec{\theta }} \right) } \right. } \right)$ is the cumulative of a conditional (M-1)-variate multivariate normal distribution. The $c_{{j}} \left( {\varvec{\theta }} \right)$ in (3) are equal to $s_{{j}} \mathbf{x}{\varvec{\upbeta }}_{{j}}$ in the MVP context, with x playing the role of the “parameter” that is common across outcomes, so that ${\hbox {d}c}_{{j}} \left( {\varvec{\theta }} \right) \big /{\hbox {d}{\varvec{\theta }} }$ is ${\hbox {d}\left( {s_{{j}} \mathbf{x}{\varvec{\upbeta }} _{{j}} } \right) }\big /{\hbox {d}{} \mathbf{x}}=s_{{j}} {\varvec{\upbeta }}_{{j}}$. Substituting into (14) ${\phi }\left( {\ldots } \right)$ for $f\left( {\ldots } \right)$, $\Phi \left( {\ldots } \right)$ for $F\left( {\ldots } \right)$, and $\alpha _{{jp}}$ for $c_{{j}} \left( {\varvec{\theta }} \right)$ gives:

$$\begin{aligned}&\frac{\partial \Phi _{\mathbf{Q}_{p} } \left( {\alpha _{1{p}} ,\ldots ,\alpha _{{Mp}} } \right) }{\partial \mathbf{x}}=\sum \limits _{{j}=1}^{M} {\left\{ {\left( {\frac{\partial \Phi _{\mathbf{Q}_{p} } \left( {\alpha _{1{p}} ,\ldots ,\alpha _{{Mp}} } \right) }{\partial \alpha _{{jp}} }} \right) \times \left( {\frac{\partial \alpha _{{jp}} }{\partial \mathbf{x}}} \right) } \right\} } \\&\quad =\sum \limits _{{j}=1}^{M} {\left\{ {\left( {{\phi }\left( {\alpha _{{jp}} } \right) \times \Phi _{\mathbf{Q}_{p} \left\{ {-{j}} \right\} } \left( {\alpha _{1{p}} ,\ldots ,\alpha _{\left( {{j}-1} \right) {p}} ,\alpha _{\left( {{j}+1} \right) {p}} ,\ldots ,\alpha _{{Mp}} \left| {\alpha _{{jp}} } \right. } \right) } \right) \times \left( {s_{{jp}} {\upbeta }_{{j}} } \right) ^{\mathrm{T}}} \right\} .} \\ \end{aligned}$$

Given consistent estimates $\widehat{\mathbf{B}}$ and $\widehat{\mathbf{Q}}$, estimation of ${\partial \Phi _{\mathbf{Q}_{p} } \left( {\alpha _{1{p}} ,\ldots ,\alpha _{{Mp}} } \right) }/{\partial \mathbf{x}}$ is complicated only by evaluation of the term $\Phi _{\mathbf{Q}_{p} \left\{ {-{j}} \right\} } \Big ( \alpha _{1{p}} ,\ldots ,\alpha _{\left( {{j}-1} \right) {p}} ,\alpha _{\left( {{j}+1} \right) {p}} ,\ldots ,\alpha _{{Mp}} \left| {\alpha _{{jp}} } \right. \Big )$. The following result provides a basis for this calculation:

Result: Joint Conditional Distribution of an MVN-Variate, Adapted from Rao (1973) (8a.2.11)

Suppose $\mathbf{z}=\left[ {z_1 ,\ldots ,z_{{M}} } \right] \sim \hbox {MVN}\left( {\mathbf{0},{\varvec{\Omega }} } \right)$. Partition ${\varvec{\Omega }}$ as $\left[ {{\begin{array}{ll} {\omega }_{11} &{} {{\varvec{\Omega }}_{12} } \\ {{\varvec{\Omega }}_{21} }&{} {{\varvec{\Omega }}_{22} } \\ \end{array} }} \right]$ with scalar. Then, ${\mathbf{z}}_{-1} =\left[ {z_2 ,\ldots ,z_{{M}} } \right]$ conditional on $z_1$ is ($M-1$)-variate $\hbox {MVN}\left( {{\varvec{\Omega }} _{21} {\omega }_{11}^{-1} z_1 ,\left( {\varvec{\Omega }}_{22} -{\omega }_{11}^{-1} {\varvec{\Omega }}_{21} {{\varvec{\Omega }}_{12} } \right) } \right)$. $\square$

This generalizes straightforwardly to ${\mathbf{z}}_{-{j}} =\left[ {z_1 ,\ldots ,z_{{j}-1} ,z_{{j}+1} ,\ldots ,z_{{M}} } \right]$, j=2,...,M, by defining different partitions of ${\varvec{\Omega }}$. In the case of interest here, ${\varvec{\Omega }} =\mathbf{Q}_{p}$ so that $\omega _{11} =1$. It follows that the joint conditional distribution is

$$\begin{aligned} {\mathbf{z}}_{-1} \left| {z_1 \sim } \right. \hbox {MVN}\left( {\left[ {{\begin{array}{c} {z_1 \tau _{12{p}} } \\ \vdots \\ {z_1 \tau _{1{Mp}} } \\ \end{array} }} \right] ,\left[ {{\begin{array}{cccc} {1-\tau _{12{p}}^2 }&{} {\tau _{23{p}} -\tau _{12{p}} \tau _{13{p}} }&{} \ldots &{} {\tau _{2{Mp}} -\tau _{12{p}} \tau _{1{Mp}} } \\ {\tau _{23{p}} -\tau _{12{p}} \tau _{13{p}} }&{} {1-\tau _{13{p}}^2 }&{} &{} \vdots \\ \vdots &{} &{} \ddots &{} \\ {\tau _{2{Mp}} -\tau _{12{p}} \tau _{1{Mp}} }&{} \ldots &{} &{} {1-\tau _{1{Mp}}^2 } \\ \end{array} }} \right] } \right) ,\nonumber \\ \end{aligned}$$

(9)

again with obvious generalization to the distributions of ${\mathbf{z}}_{-{j}} \left| {z_{{j}} } \right.$, j=2,...,M.

To obtain $\Phi _{\mathbf{Q}_{p} \left\{ {-{j}} \right\} } \left( {\alpha _{1{p}} ,\ldots ,\alpha _{\left( {{j}-1} \right) {p}} ,\alpha _{\left( {{j}+1} \right) {p}} ,\ldots ,\alpha _{{Mp}} \left| {\alpha _{{jp}} } \right. } \right)$, define the ($M-1$)-vector of differences

$$\begin{aligned} \Delta _{-{j,p}}= & {} \left[ \left( {\alpha _{1{p}} -\alpha _{{jp}} \tau _{1{jp}} } \right) ,\ldots ,\left( {\alpha _{({j}-1){p}} -\alpha _{{jp}} \tau _{({j}-1){jp}} } \right) ,\right. \nonumber \\&\left. \times \left( {\alpha _{({j}+1){p}} -\alpha _{{jp}} \tau _{({j}+1){jp}} } \right) ,\ldots ,\left( {\alpha _{{Mp}} -\alpha _{{jp}} \tau _{\mathrm{Mjp}} } \right) \right] ^{\mathsf{T}}, \end{aligned}$$

(10)

and an $(M{-}1)\times (M{-}1)$ diagonal transformation matrix ${\mathbf{H}}_{{jp}} \,{=}\,\hbox {diag}_{{k}\ne {j}} \left[ {\left( {\sqrt{1-\tau _{\mathrm{jkp}}^2 }} \right) ^{-1}} \right]$. Let ${\mathbf{L}}_{{jp}} {=}{\mathbf{H}}_{{jp}} {\varvec{\Delta }} _{-{j,p}}$ be the corresponding (M-1)-vector of normalized differences. Then, $\Phi _{\mathbf{Q}_{{p}\left\{ {-{j}} \right\} } } \left( \alpha _{1{p}} ,\ldots ,\alpha _{\left( {{j}-1} \right) {p}} ,\alpha _{\left( {{j}+1} \right) {p}} ,\ldots ,\alpha _{{Mp}} \left| {\alpha _{{jp}} } \right. \right)$ can be computed by referring $\mathbf{L}_{{jp}}$ to $\Phi _{\mathbf{z},{\varvec{\Sigma }} } \left( {\ldots } \right)$, which is the cumulative of an (M-1)-variate $\hbox {MVN}\left( {\mathbf{0},{\varvec{\Sigma }} } \right)$ distribution in which the off-diagonals of ${\varvec{\Sigma }}$ may be nonzero. In this instance, ${\varvec{\Sigma }}$ is the variance-covariance matrix of $\mathbf{L}_{{jp}}$ which is in correlation matrix form having typical off-diagonal (r,c) element ${\left( {\tau _{{rcp}} -\tau _{{jrp}} \tau _{{jcp}} } \right) }\big /{\sqrt{\left( {1-\tau _{{jrp}}^2 } \right) \left( {1-\tau _{{jcp}}^2 } \right) }}$. Let this matrix be denoted ${\mathbf{V}}_{{jp}}$. The results derived in this appendix now provide the basis for computing the quantities of interest in (5).