Part 1: Theoretical questions [30%]
Consider a linear regression model with i.i.d size n sample of dependent
variable yi and two regressors (x1i
, x2i), i.e., the sample is {(yi
, x1i
, x2i) : i =
1, . . . , n}.
yi = β0 + β1x1i + β2x2i + i
, E(i
|x1i
, x2i) = 0, i = 1, 2, . . . , n. (1)
(a) With suitably defining vectors and matrix, represent the n equations of
(1) as
y = Xβ + . (2)
In your answer, explain the constructions of y, X, β, and .
(b) Derive the ordinary least square (OLS) estimators βˆOLS for β in terms of
(y, X) you defined in part (a). In your answer, present the optimization
problem and the steps of derivation.
(c) Explain why βˆOLS is consistent and √
n(βˆOLS − β) is asymptotically normal. In addition, assuming the regression error i
is homoskedastic, derive
the asymptotic variance of √
n(βˆOLS − β).
(d) Suppose that your research assistant accidentally deleted the observations
of x2i from the data set so that you cannot run the regression equation of
(1) anymore. Consider instead estimating the following regression equation:
yi = γ0 + γ1x1i + νi
, E(νi
|x1i) = 0. (3)
We assume that x2i
is uncorrelated with x1i
, i.e., Cov(x1i
, x2i) = 0. In
that case, are the OLS estimator for (γ0, γ1) consistent to (β0, β1)? Explain.
(e) Under the homoskedasticity assumption imposed in part (c) and the zero
covariance assumption of part (d) which of the OLS estimators βˆ
1,OLS
for β1 or ˆγ1,OLS for γ1 has a smaller asymptotic variance? Provide your
reasoning.
Sample Solution