Difference between Fixed effects and Mean-Difference Technique

[Yasir Shahab]

Hello Everyone.

I am using Chinese firms data for eight years and intending to use panel data regression for analysis. In one of the paper published in "Corporate Governance: An International Review", the authors have specified fixed-effects regression model as CSRDit= α0 + β1CGit + sum of βi CONTROLSit +γi + εit
with "γ" referring to the company-specific fixed-effects, consisting of a vector of the mean differences of all time variant variables. In the notes the authors have given that "We also follow Guest (2009) and Ntim et al. (2012a) in implementing the mean-difference technique, which is more robust in the presence of hetereoskedasticity (Gujarati, 2003; Wooldridge, 2010). However, we get essentially similar results if we run our fixed-effects models by employing the year dummy alternative instead of the mean-difference method."

I am wondering how to apply "mean difference technique" in panel data and how it differs from Fixed Effects model we normally use.

Thanks
Yasir

[David Dauria]

Suppose you only have a couple of firms. One way to run a fixed effects model would be to just include a dummy variable for each firm (and no constant). So you'd have a different intercept for each firm.

Now suppose you have many firms. Then computing the ols estimator will require you to invert a large matrix. That can be time consuming, or maybe impossible.

The mean difference technique may be described as a residual based regression. Regress the dependent variables on the firm ids. So this would be like including one dummy variable for each firm, and no intercept. If you look at the residuals of each of those regressions, you're effectively seeing the true value y minus the group mean, y_hat. Call that value y*. You can do the same for each of the dependent variables, and call the resulting matrix of residuals X* = X - X_hat. We are subtracting the means, hence "mean differencing technique."

If you then run a regression of y* on X* you would have a residual based regression. By the Frisch Waugh Lovell Theorem, the coefficient you get from the residual based regression will be identical to the one you get from the dummy variable technique. The difference is that you do not actually compute the fixed effect for each firm. But often researchers aren't interested in the fixed effects themselves, they just want to control for them in the model.

Implement the mean differencing technique using the areg command in Stata. Then compare with what you get from the dummy variable method. The coefficients should be the same.

[Yasir Shahab]

David Dauria Thanks a lot for your detailed explanation. So now I have to check the results with "areg" command. I will check the model with this technique as per your specifications.

Thanks
Yasir

[David Dauria]

You are most welcome. It took me a long time to understand the FWL Theorem and its applications, and I wish to become skilled at explaining it. So please follow up if you have more questions.

[Yasir Shahab]

David Dauria yes sure. Thanks a lot I will definitely ask you more about it in case of issues.