The point of this exercise is to show that tests for functional form cannot be relied on as a general test for omitted variables. Suppose that, conditional on the explanatory variables x1 and x2, a linear model relating y to x1 and x2 satisfies the Gauss-Markov assumptions: y = B + BX + Bxy + u
E(ulx, x) = 0
Varulx, x) = 0
To make the question interesting, assume B, = 0.
Suppose further that x, has a simple linear relationship with x1:
X2 = d + dx, + r
E(rix) = 0
Var(rix) = 0
1) Show that
Evrị) = (Bo + B-50) + 8 + Bồ) XI.
1) Under random sampling, what is the probability limit of the OLS estimator from the sim- ple regression of y on X? Is the simple regression estimator generally consistent for Bi?
Regression Analysis with Cross-Sectional Data
2) If you run the regression of y on x1, x1^2, what will be the probability limit of the OLS estimator of the coefficient on x1^2? Explain.
3) Using substitution, show that we can write
y = (beta0 + beta2 delta0) + (beta1 + beta2 delta1) x1 + u + beta2r
It can be shown that, if we define v = u + beta2r then E(vlx1) = 0. Var(vlx1) = sigma^2 + beta2^2 tau^2. What consequences does this have for the t statistic on x1^2 from the regression in part (ii)?
4) What do you conclude about adding a nonlinear function of x1- in particular, x1^2- in an attempt to detect omission of x2?