Estimation of Coefficient on "observable" Time-Invariant Variable in Panel Data Analysis

[Necip Bulut]

Hello everyone,

Basically, I want to estimate the coefficient on an observable time-invariant variable in a panel data analysis. This time-invariant variable varies for each but does not change over time (please see table below). Let's say there are several agricultural firms and we want to estimate the effects of soil quality on agricultural production (along with the effects of other explanatory variables). In the textbooks I read, this soil quality is assumed as an "unobserved" time-invariant variable. Now, I wonder what if we can observe this soil quality variable? As you can see below, I assume that this soil quality is not the same for all firms but is constant for each firm.

By the way, I know that such a time-invariant variable does not make sense in a time series analysis performed for just one firm. But it can make sense for the difference between the agricultural production levels of these firms. Therefore, I just estimate the effects of this soil quality variable, which does not change over time for each firm, on the difference between agricultural production. Some resources advise performing "fixed effect estimation" or "random effect estimation" by using Stata command like "xtreg , fe". But I am not sure whether this process is proper for an "unobserved" or an "observed" time-invariant variable.

Perhaps, we can consider this variable as a dummy variable but I am not sure. Because this variable is not made up of values like 0 or 1. It consists of values 3.15, 4.29, 5,12, etc and the number of these numbers is equal to the firm number (n). For example, the soil quality of firm 1 is equal to 3.15 over the sample period, and that of firm 2 is equal to 4,29, etc.

Could you advise lecture notes, a textbook or an article that deals with this issue and provide comprehensive knowledge about the estimation method (such as GMM, OLS..) and which tests should be conducted before and after estimation in order to achieve unbiased and consistent estimators?

i=1,2...n
t= 1,2...T

Firm (i)	Year (t)	Dependent Var. (Yi,t) (production)	Independent Var. (Xi,t)	Observable Time-Invariant Var. (Zi) (soil quality)
1	1	y1,1	x1,1	z1
1	2	y1,2	x1,2	z1
1	...	...	...	z1
1	T	y1,T	x1,T	z1
2	1	y2,1	x2,1	z2
2	2	y2,2	x2,2	z2
2	...	...	...	z2
2	T	y2,T	x2,T	z2
3	1	y3,1	x3,1	z3
3	2	y3,2	x3,2	z3
3	...			z3
3	T	y3,T	x3,T	z3
n	1	yn,1	xn,1	zn
n	2	yn,2	xn,2	zn
n	...	...	...	zn
n	T	yn,T	xn,T	zn

[Clyde Schechter]

In a fixed-effects analysis it is mathematically impossible to estimate the effect of any variable, observed or not, that does not vary over time within panels. You can adjust for their confounding effects on other variable relationships in the model (indeed, the fixed-effect estimator does so automatically). But you cannot estimate their effects.

If estimating the effect of this soil quality variable is one of your research goals, then you simply cannot use a fixed-effects analysis for the purpose. Using -xtreg, re- to do a random effects analysis will serve the purpose, assuming you have other variables that do vary within time that you wish to include in the regression. Another possibility is to do a correlated random effects model, using the -xthybrid- command (by Reinhard Schunk, available from SSC). This will give results equivalent to fixed-effects for the time-varying variables, but will still allow you to estimate the effect of soil quality, which is a between-units effect.