Sunday, December 15, 2013

Proxy Variables and Biased Estimation

Here's a problem from the exam. that one of my econometrics classes sat recently. It's to do with some of the consequences of mis-specifying a regression model, and then applying OLS estimation.

Specifically, let's suppose that data-generating process (the correct model specification) is actually of the form:

                       y = Xβ + ε     ;   ε ~ [0 , σ2In] .                         (1)

However, we can't observe the k variables in the X matrix, and instead we replace them with k "proxy variables" (substitutes) that we can observe. So, the model that we actually estimate is:

                      y = X*β + v .                                                     (2)

The students were asked to show that the usual (unbiased) estimator of σ2 is actually biased in this case; and they were asked if they could determine the "direction" of the bias.