A Basic Model
The most general expression of linear regression models that have both time and unit dimensions is seen in equation 15.1 below.
where i — 1,2,…,N and t — 1,2,…,T. If we have a full set of time observations for every individual then there will be NT total observations in the sample. The panel is said to be balanced in this case. It is not unusual to have some missing time observations for one or more individuals. When this happens, the total number of observation is less than NT and the panel is unbalanced.
The biggest problem with equation (15.1) is that even if the panel is complete (balanced), the model contains 3 times as many parameters as observations (NT)! To be able to estimate the model, some assumptions have to be made in order to reduce the number of parameters. One of the most common assumptions is that the slopes are constant for each individual and every time period; also, the intercepts vary only by individual. This model is shown in equation (15.2).
yit — ві i + @2X2 it + взХз it + eit (15.2)
Figure 15.3: This shows a portion of the series list window for the Barro and Lee data from the database server. From here you can display the values contained in a series, plot the series, or add a series to your dataset. Highlight the particular series you want and click on the appropriate icon at the top.
This specification, which includes N + 2 parameters, includes dummy variables that allow the intercept to shift for each individual. By using such a model you are saying that over short time periods there are no substantive changes in the regression function. Obviously, the longer your time dimension, the more likely this assumption will be false.
In equation (15.2) the parameters that vary by individual are called individual fixed effects and the model is referred to as one-way fixed effects. The model is suitable when the individuals in the sample differ from one another in a way that does not vary over time. It is a useful way to avoid unobserved differences among the individuals in your sample that would otherwise have to be omitted from consideration. Remember, omitting relevant variables may cause least squares to be biased and inconsistent; a one-way fixed effects model, which requires the use of panel data, can be very useful in mitigating the bias associated with time invariant, unobservable effects.
If you have a longer panel and are concerned that the regression function is shifting over time, you can add T — 1 time dummy variables to the model. The model becomes
Vit = ві i + віі + в2Х2 it + взхз it + eu (15.3)
where either e1i or e1t have to be omitted in order to avoid perfect collinearity. This model contains N + (T — 1) + 2 parameters which is generally fewer than the NT observations in the sample. Equation (15.3) is called the two-way fixed effects model because it contains parameters that will be estimated for each individual and each time period.