Dynamic unobserved effects model

A typical dynamic unobserved effects model is represented[1] as:

P(y_it = 1│y_i,t-1, ... , y_i,0 , z_i , c_i ) = G (z_it δ + ρ y_i,t-1 + c_i)

where c_i is an unobservable explanatory variable, z_it are explanatory variables which are exogenous conditional on the c_i, and G(∙) is a cumulative distribution function.

In this type of model, economists have a special interest in ρ, which is used to characterize the state dependence. For example, y_i,t can be a woman's choice whether to work or not, z_it includes the i-th individual's age, education level, number of children, and other factors. c_i can be some individual specific characteristic which cannot be observed by economists.[2] It is a reasonable conjecture that one's labor choice in period t should depend on his or her choice in period t − 1 due to habit formation or other reasons. This dependence is characterized by parameter ρ.

There are several MLE-based approaches to estimate δ and ρ consistently. The simplest way is to treat y_i,0 as non-stochastic and assume c_i is independent with z_i. Then by integrating P(y_i,t , y_i,t-1 , … , y_i,1 | y_i,0 , z_i , c_i) against the density of c_i, we can obtain the conditional density P(y_i,t , y_i,t-1 , ... , y_i,1 |y_i,0 , z_i). The objective function for the conditional MLE can be represented as: ${\displaystyle \sum _{i=1}^{N}}$ log (P (y_i,t , y_i,t-1, … , y_i,1 | y_i,0 , z_i)).

Treating y_i,0 as non-stochastic implicitly assumes the independence of y_i,0 on z_i. But in most cases in reality, y_i,0 depends on c_i and c_i also depends on z_i. An improvement on the approach above is to assume a density of y_i,0 conditional on (c_i, z_i) and conditional likelihood P(y_i,t , y_i,t-1 , … , y_t,1,y_i,0 | c_i, z_i) can be obtained. By integrating this likelihood against the density of c_i conditional on z_i, we can obtain the conditional density P(y_i,t , y_i,t-1 , … , y_i,1 , y_i,0 | z_i). The objective function for the conditional MLE[3] is ${\displaystyle \sum _{i=1}^{N}}$ log (P (y_i,t , y_i,t-1, … , y_i,1 | y_i,0 , z_i)).

Based on the estimates for (δ, ρ) and the corresponding variance, values of the coefficients can be tested[4] and the average partial effect can be calculated.[5]