Dynamic unobserved effects model

The “dynamic” here means the dependence of the dependent variable on its past history, this is usually used to model the “state dependence” in economics. For instance, a person who cannot find a job this year, it will be hard for her to find a job next year because the fact that she doesn’t have a job this year will be a very negative signal for the potential employers. The “unobserved effects” means that one or some of the explanatory variables are unobservable. For example, one’s preference affects quite a lot her consumption choice of the ice cream with a certain taste, but preference is unobservable. A typical dynamic unobserved effects model is represented 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 is 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 work or not, z_it includes the i-th individual’s age, education level, numbers of kids and so on. c_i can be some individual specific characteristic which cannot be observed by economists. 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 is 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 integrate 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: ∑ 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 of the 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. Integrate this likelihood against the density of c_i conditional on z_i and 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 is ∑ 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, test about the coefficients can be implemented and the average partial effect can be calculated.