Coefficient sign changes in fixed effect and first-difference estimation

Question

I have a large empirical panel, where I basically want to regress the standard deviation of (equity) returns ($y_{it}$) of firm i at time t, on leverage (equity/debt) of firm i at time t ($x_{it}$). Further I would like to control for the equity size of the firm ($z_{it}$).

Unfortunately the $\beta$ coefficient sign changes (significant in both specifications) if I run the following two models:

Fixed Effects Model: $$ y_{it} = a_{i} + \beta * x_{it} + c * z_{it} + u_{it}$$

First-Difference Model: $$ \Delta y_{it} = \beta * \Delta x_{it} + c * \Delta z_{it} + \Delta u_{it}$$

Since I only have basic knowledge about econometrics and my colleagues are also puzzled, I was hoping that some statistic guys might help me.

I identified some potentially problems (but there might be more):

• $y_{it}$ might be autocorrelated
• Since leverage is defined as Debt/Equity, controlling for equity brings some potential problems

The true question now is: Which model (if any) is econometrically correct?

Answer 1

Another thing that could go awry is an unbalanced panel where you have "gaps" in the middle of the time series. The FD estimator will lose two observations if there a single period missing. The dummy approach will loose only one. Are the sample sizes wildly different between the two regressions by any chance? What happens if you use only observations where you have all the periods?

If that's not's what causing it, we have to think harder. You may have contemporaneous correlation between $x_{it}$ and $u_{it}$. In that case, both the FD and FE estimators will be inconsistent and have different probability limits (Adult Wooldridge, p.321-322). It's hard to know which one should be preferred ex ante and or what to do about it.

If you have non-contemporaneous correlation, it will have similar effects, but there may be a solution. When $x_{it}$ and $u_{is}$ for $t < s$ are correlated, you can include lags of $x$. I think this is likely the culprit given your comment above. If there's feedback from $u_{is}$ to $x_{it}$ for $t > s$, a more complicated solution is described in chapter 11 of Wooldridge.

If you maintain that you have contemporaneous exogeneity, then the inconsistency of the FE estimator from the failure of strict exogeneity goes to zero at the rate $\frac{1}{T}$, while the FD's is independent of $T$. But that's only true if $x_{it}$ and $y_{it}$ are cointegrated (in the time series sense). In fact, FE may be worse than FD in that case for fixed $N$ as $T$ grows. So FD may deal better with spurious regression.