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I know that for panel data RE probit model we should average out the individual effects and in order to do that we do the following: $$f(y_1,...,y_T|x_i, \theta) = \int[\prod_{t=1}^T f(y_t|x_{it},c;\beta)\frac{1}{\sigma_c}\phi(\frac{c}{\sigma_c})]dc $$ I would be very grateful if someone could explain how an assumption that $c$ is independent of $x_i, \epsilon_i$ allows me to conclude that $$ f(y_1,...,y_T|x_i, \beta) = \int[f(y_1,...,y_T|x_{it},c;\beta)\frac{1}{\sigma_c}\phi(\frac{c}{\sigma_c})]dc $$

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The conditional exogeneity assumption does not provide the link

$$ \prod_{t=1}^T f_{Y\mid \boldsymbol{X}, \alpha}(Y_{it} \mid \boldsymbol{X}_{it}, \alpha_i; \boldsymbol{\theta}) = f_{\boldsymbol{Y}_{\cdot} \mid\mathbf{X}_{\cdot}, \alpha}(\boldsymbol{Y}_{i\cdot}\mid \mathbf{X}_{i\cdot}, \alpha_i; \boldsymbol{\theta}) $$

That follows from the static specification of the model. The reason the use of the conditional exogeneity of the individual effects is not clear is because you have already used that assumption in writing your first expression.


  • The model that you have is the following $$ Y_{it} = \boldsymbol{X}_{it}'\boldsymbol{\beta} + \alpha_i + \varepsilon_{it};\, t=1, \ldots, T, \, i=1, \ldots, n $$ together with the assumption that $\mathbb{E}(\varepsilon_{it} \mid \boldsymbol{X}_{it}) = 0$ for all $i,t$ (I introduce the conditional exogeneity assumption on the individual effects later).

  • The correct factorization of the joint density of the outcome variables conditional on the regressors is $$ \begin{align} f_{\boldsymbol{Y}_{\cdot} \mid\mathbf{X}_{\cdot}}(\boldsymbol{Y}_{i\cdot}\mid \mathbf{X}_{i\cdot}; \boldsymbol{\theta}) &= \int f_{\boldsymbol{Y}_{\cdot} \mid\mathbf{X}_{\cdot}, \alpha}(\boldsymbol{Y}_{i\cdot}\mid \mathbf{X}_{i\cdot}, \alpha_i; \boldsymbol{\theta}) f_{\alpha \mid \mathbf{X}_{\cdot}}(\alpha_i \mid \mathbf{X}_{i\cdot}; \boldsymbol{\theta}) \, d\alpha_i\\ &= \int \left(\prod_{t=1}^Tf_{Y\mid \boldsymbol{X}, \alpha}(Y_{it} \mid \boldsymbol{X}_{it}, \alpha_i; \boldsymbol{\theta})\right)f_{\alpha \mid \mathbf{X}_{\cdot}}(\alpha_i \mid \mathbf{X}_{i\cdot}; \boldsymbol{\theta}) \, d\alpha_i \end{align} $$ where, $\mathbf{X}_{i\cdot} = [\boldsymbol{X}_{i1}, \ldots, \boldsymbol{X}_{iT}]'$, $\boldsymbol{Y}_{i\cdot} = [Y_{i1}, \ldots, Y_{iT}]'$, and $\boldsymbol{\theta}$ collects all the parameters in the model. So far, this follows from the specification of the regression equation.

  • The conditional exogeneity of the individual effects says that $\alpha_i \mid \mathbf{X}_{i\cdot} \sim \alpha_i$, that is, the individual effects are conditionally exogenous of the included regressors. In particular, this means that $f_{\alpha \mid \mathbf{X}_{\cdot}} = f_{\alpha}$, from which you get the expression you have $$ f_{\boldsymbol{Y}_{\cdot} \mid\mathbf{X}_{\cdot}}(\boldsymbol{Y}_{i\cdot}\mid \mathbf{X}_{i\cdot}; \boldsymbol{\theta}) = \int \left(\prod_{t=1}^Tf_{Y\mid \boldsymbol{X}, \alpha}(Y_{it} \mid \boldsymbol{X}_{it}, \alpha_i; \boldsymbol{\theta})\right)f_{\alpha}(\alpha_i; \boldsymbol{\theta}) \, d\alpha_i $$
    which is the required expression.

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