We would like to obtain an estimator of β1 from the following regression model with only one independent regressor:

yi =β0 +β1x1i +ui. (1) However, there is another variable x2i, which is missing from the model and poten-

tially correlated with x1i. That is, the true model would be
yi =β0 +β1x1i +β2x2i +vi (2)

where vi is an observation error, which satisfies E (vi|x1i, x2i) = 0.

(a) Show that the OLS estimator of β1 obtained from model (1) is biased.

(b) When is this bias equal to 0?

(c) Derive the OLS estimators of β1 and β2 from model (2). (Hint : You don’t need to solve it all the way through : Write down a systems of equations in terms of

β and β ] is enough.)12

(d) Show that, when the sample covariance between x1i and x2i is equal to 0, then the OLS estimator of β1 derived in (c) is the same as the OLS estimator of β1derived in (a) (Hint : You can use the system equations derived in (c)).