between estimator regression

[tim hoeflinger]

Dear all,

I would like to collect some thoughts and opinions on when and why one should/could use the between estimator regression.
I have only recently learned about this method and find it to be quite an interesting alternative. It seems to be seldomly used and there is comparatively few information available about it.

Thanks for any thoughts.

Tim

[Chris Boudreaux]

Most will prefer the within estimator (I.e. fixed effects) because it removes the unobserved differences between groups. This is because they are time invariant. But, what if you are interested in a variable that is time invariant? These time invariant variables are omitted using a within estimator.

A between estimator will do the trick, but you have to assume no unobserved heterogeneity between individuals. In this case, most end up using a random effects estimator which combines both within and between group variation. Still, you should test whether your regressions are uncorrelated with the unobserved effects using a test like the Hausman test.

[Andrew Musau]

To expand on Chris's helpful reply, the between estimator is in general biased in the same way as pooled OLS. To see this, write out the general panel data model

$$y_{it}= \beta^{\prime}x_{it}+ \gamma^{\prime}z_{i}+\eta_{i}+u_{it}\;\;\;(i=1,... , N; t=1,..., T)$$

where the \(x\) variables are time-varying, the \(z\) variables are time invariant and \(\eta_{i}\) is the time-invariant individual effect. The between model is the cross-sectional equation

$$\bar{y}_{i}= \beta^{\prime}\bar{x}_{i}+ \gamma^{\prime}z_{i}+\eta_{i}+\bar{u}_{i}$$

where $$\bar{y}_{i}=\frac{1}{T}\sum_{t=1}^{T}y_{it}, \;\;\bar{x}_{i}=\frac{1}{T}\sum_{t=1}^{T}x_{it}, \;\;\bar{u}_{i}=\frac{1}{T}\sum_{t=1}^{T}u_{it}.$$

As you can see, after averaging and deriving the between estimator, the individual effect \(\eta_{i}\) does not drop out of the equation. For all intents and purposes, the between estimator is useful in considering the random effects model rather than an estimator in its own right.