In Introductory Econometrics: A Modern Approach, Wooldridge writes the following regarding the omitted variable bias and its effect on the variance of the OLS estimator (x1 and x2 are correlated):
This intutivley makes sense given that by defintion of the omitted variable bias x1 and x2 are correlated. Thus including x2 into the regression should inflate the variance due to multicollinearity.
However when I run a simulation in R I consitently get the exact opposite of what Wooldridge suggest. Consider the data generating process:
x1 <- rnorm(10000) x2 <- rnorm(10000) + 0.2*x1 y <- 0.5 -2*x1 -2.5*x2 + rnorm(10000) summary(lm(y ~ x1 + x2)) summary(lm(y ~ x1))
No matter how many times I run this simulation the standard error of beta1 in the omitted variable case is always larger than in the unbaised specification.
How is this possible?