# A question about Fixed effects estimation

I have a fixed effects model as follows:

$Y = x_1, x_2, x_3$, (fixed effect), (error term)

Is there any way I can check whether Cov($x_1$, (fixed effect)) is larger or smaller than 0?

The panel data I have consists of 7858 individuals and 5 years.

Thanks!

• How about calculating Cov(X1, (fixed effect)? Or is there a reason why this is difficult? – Giskard Sep 15 '16 at 11:28
• I think the fixed effect is unobservable. How can I calculate the covariance? Do you mean calculating intercept estimates for each individual and then covariance btw estimated intercepts and X1? – YBHan Sep 15 '16 at 23:20
• Sorry, my bad, I was thinking of another type of model. – Giskard Sep 16 '16 at 4:52

Having read up on your question it seems the fixed effect is fixed. If this is indeed the case it will have zero variance and hence zero covariance with any variable.

• While the true parameter ($\gamma_i$) might be constant and therefore have a zero covariance, it may well be that the estimate of the parameter ($\hat{\gamma}_i$) co-varies with other variables. – BKay Oct 20 '16 at 13:56

Assuming that you mean "fixed effects" of econometricians, not of statisticians, you can check it as follows. You have $v_{it} = u_i + e_{it}$ consistently estimated (as $\beta$ is consistently estimated), where $u_i$ are the fixed effects and $e_{it}$ are the idiosyncratic errors. Under the assumption that $x_{it}$ is strictly exogenous to $e_{it}$ (which is assumed whenever you do FE regression), the covariance between $x_{it}$ and $v_{it}$ is the same as the covariance between $x_{it}$ and $u_{i}$. Thus, this is what you can do:

1. Obtain the fixed effects estimation residuals (if using Stata, xtreg y x1 x2 x3, fe, and then predict v, ue).
2. Regress x1 on v either for each $t$ (reg x1 v if year==1991 etc) or pooled. The sign of the estimate is what you want.

By the way, let me clarify that "fixed effects" are not nonrandom in econometrics, so we can't say that fixed effects are uncorrelated with other random variables. By "fixed effects" we mean individual effects that are possibly correlated with explanatory variables.

In statistics (mixed effects models), "fixed effects" mean variables whose coefficients are the same for all $i$, while "random effects" are those with individually different coefficients.

The answer is, that this is not possible. A fixed effect by itself is nothing that is measured. It just means that you allow for different intercepts for different individuals.

It seems that what you want to do is checking the sensitivity of your specification. Do so by going to the "econometrics kitchen". Run y just on x1 without fixed effects, then with fixed effects. Then y on x2, x3 and so on. Than you will get a feeling for whats going on in the data.

• thank you very much. Now I see some picture, with your help! I will try. – YBHan Oct 17 '16 at 23:45