A dynamic unobserved effects model is a statistical model used in econometrics for panel analysis. It is characterized by the influence of previous values of the dependent variable on its present value, and by the presence of unobservable explanatory variables.
The term “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, for a person who cannot find a job this year, it will be harder to find a job next year because her present lack of a job will be a negative signal for the potential employers. “Unobserved effects” means that one or some of the explanatory variables are unobservable: for example, consumption choice of one flavor of ice cream over another is a function of personal preference, but preference is unobservable.
In a panel data tobit model,[1] [2] if the outcome
Yi,t
Yi,0,\ldots,Yt-1
Yi,t
| 1[Y | |
| =Y | |
| i,t |
>0];
Yi,t=zi,t\delta+\rhoyi,t-1+ci+ui,t;
ci\midyi,0,\ldots,yi,t-1\simF(yi,0xi);
ui,t\midzi,t,yi,0,\ldots,yi,t-1\simN(0,1).
In this specific model,
\rhoyi,t-1
ci
Based on this setup, the likelihood function conditional on
\{yi,0
| N | |
| \} | |
| i-1 |
| N | |
| \prod | |
| i=1 |
\intf\theta(ci\midyi,0,xi)\left[
| Tl(1[y | |
| \prod | |
| i,t |
=0][1-\Phi(zi,t\delta+\rhoyi,t-1>0]
| \varphi(zi,t\delta+\rhoyi,t-1+ci) | |
| \Phi(zi,t\delta+\rhoyi,t-1+ci) |
r)\right]dci
For the initial values
\{yi,0
| N | |
| \} | |
| i-1 |
A typical dynamic unobserved effects model with a binary dependent variable is represented[6] as:
P(yit=1\midyi,t-1,...,yi,0,zi,ci)=G(zit\delta+\rhoyi,t-1+ci)
where ci is an unobservable explanatory variable, zit are explanatory variables which are exogenous conditional on the ci, 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, yi,t can be a woman's choice whether to work or not, zit includes the i-th individual's age, education level, number of children, and other factors. ci can be some individual specific characteristic which cannot be observed by economists.[7] 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 yi,0 as non-stochastic and assume ci is independent with zi. Then by integrating P(yi,t, yi,t-1, …, yi,1 | yi,0, zi, ci) against the density of ci, we can obtain the conditional density P(yi,t, yi,t-1, ..., yi,1 |yi,0, zi). The objective function for the conditional MLE can be represented as:
| N | |
| \sum | |
| i=1 |
Treating yi,0 as non-stochastic implicitly assumes the independence of yi,0 on zi. But in most cases in reality, yi,0 depends on ci and ci also depends on zi. An improvement on the approach above is to assume a density of yi,0 conditional on (ci, zi) and conditional likelihood P(yi,t, yi,t-1, …, yt,1,yi,0 | ci, zi) can be obtained. By integrating this likelihood against the density of ci conditional on zi, we can obtain the conditional density P(yi,t, yi,t-1, …, yi,1, yi,0 | zi). The objective function for the conditional MLE[8] is
| N | |
| \sum | |
| i=1 |
Based on the estimates for (δ, ρ) and the corresponding variance, values of the coefficients can be tested[9] and the average partial effect can be calculated.[10]