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Could someone explain that the sign of coefficient estimates and their corresponding marginal effects in the Multivariate Probit Model is the same or they could be different? IF they are different, are they because the marginal effects take into account the means of other covariates or it has nothing to do with it OR Is there some other mechanics behind it?

Thanks

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  • $\begingroup$ Do you have interactions or polynomial terms in the model? $\endgroup$
    – dimitriy
    Aug 23, 2021 at 18:39
  • $\begingroup$ Hello Dimitriy. No, I don't have any interactions or polynomial variables. $\endgroup$
    – Jamal Shah
    Aug 24, 2021 at 9:17

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In a probit model, $\Pr(y_i=1 \vert x_i,z_i,t_i)=\Phi(\alpha +\beta x_i+\gamma z_i + \psi t_i),$ where $\Phi()$ is the standard normal cdf. The marginal effect of a continuous variable is the derivative of that function (using the chain rule): \begin{equation} \frac{\partial \Pr(y_i=1 \vert x_i,z_i,t_i)}{\partial x}=\varphi(\alpha +\beta x_i+\gamma z_i + \psi t_i)\cdot\beta, \end{equation} where $\varphi()$ is the standard normal pdf. This means then first term is always positive, regardless of its arguments.

Unless there are interactions or polynomial terms, the index function coefficients' significance and their sign will agree with various types of marginal effects. This will be the case since $\Phi(.)$ is monotonic since it is entirely non-decreasing. This applies to many other GLMs as well.

For categorical variables, you could calculate finite differences instead of derivatives:

$$Pr(y=1 \vert, x, z, t=1)-Pr(y=1 \vert x, z, t=0)$$

The same monotonicity logic applies and a difference of two probabilities will inherit the sign of the coefficient.

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