Probit model for panel data with heterogeneity and endogenous explanatory variables

In many cases, there is an unobservable heterogeneity in the probit model. For instance, when modelling the consumption choice of a certain brand, consumers’ personal preference is unobserved but needs to be considered in the model. Owing to omitted variable or measurement error, endogeneity issue also could arise. A probit model including both of these two issues can be represented as:

y_it = 1[y_it^*>0]

y_it^* = x_it⁽¹⁾β + z_itδ + c_i + u_it

z_it = x_it⁽¹⁾γ₁ + x_it⁽²⁾γ₂ + v_it

where c_i is the unobservable heterogeneity effect and u_it ∣ x_i N(0,1), v_it|x_i N(0,σ²). If v_it and u_it are independent, this model will degenerate to a probit model with unobservable heterogeneity. In this case, we can just integrate P(y_iT,…,y_i0∣x_i,c_i) against the density of c_i conditional on x_i, then P(y_iT,…,y_i0|x_i) can be obtained and the objective for the conditional Maximum Likelihood Estimation is

$\sum_{i=1}^N \log [P(y_{iT},\ldots,y_{i0} |x_i)]$

If v_it and u_it are correlated, under the normality assumption, it can be assumed that v_it =ρu_it + ϵ_it, where ϵ_it∼_iidN(0,σ²−ρ²) and ϵ_i is independent with v_i and u_i. Then the model can be rewritten as:

y_it = 1[x_it⁽¹⁾(β+δγ₁)+x_it⁽²⁾δγ₂+c_i+ω_it>0]

where ω_it = (1+ρδ)u_it + δϵ_it, ω_it/simN(0,(1+ρδ)²+δ²(σ²−ρ²)) and $corr(\omega_{it} , \omega _{(it-s)} ) = \frac {(1 + \rho\delta)^2 corr (u_{it}, u_{(it-s)} )} { ((1 + \rho\delta)^2 + \delta^2 (\sigma^2 - \rho^2 ) )}.$

Based on this, following the same Maximum Likelihood Estimation procedure and the scaled parameter $(\beta + \delta\gamma_1 , \delta\gamma_2) / \sqrt {(1+\rho\delta)^2 + \delta^2 (\sigma^2 - \rho^2 )}$ can be consistently estimated, then the APE can be consistently estimated correspondingly.