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.