Question

Consider a simple linear regression model with time series data: yt=B0+ B1xt +ut t= 1;2,.....T

Suppose the error ut is strictly exogenous. That is E(utIx1;....xt,.....xT) = 0

Moreover, the error term follows an AR(1) serial correlation model. That is, ut= put-1 +et t= 1;2,.....T (3)

where et are uncorrelated, and have a zero mean and constant variance.

a. [2 points] Will the OLS estimator of B1 be unbiased? Why or why not?

b. [3 points] Will the conventional estimator of the variance of the OLS estimator be unbiased? Why or why not?

c. [5 points] Explain in detail how you will test for serial correlation in ut using a t-test. [Hint: Your null hypothesis is that p= 0 in equation 3.]

Answer 1

Solution:

Given

a simple linear regression model with time series data:

yt = B0 + B1xt + ut , t= 1;2,.....T

error ut is strictly exogenous.

That is E(ut I x1;....xt,.....xT) = 0

AR(1) serial correlation model.

That is, ut = put-1 + et , t= 1;2,.....T

a. OLS estimator of B1 be unbiased or not:

unbiasedness is a finite sample property, and if it held it would be expressed as

?(?̂ )=?E(β^)=β

(where the expected value is the first moment of the finite-sample distribution)

b. conventional estimator of the variance of the OLS estimator be unbiased or not:

while consistency is an asymptotic property expressed as:

plim?̂ =?plimβ^=β

The OP shows that even though OLS in this context is biased, it is still consistent.

c. test for serial correlation in ut using a t-test:

If r = 0,

then t t u = e and in that case the random errors t u satisfy

Assumption 4, i.e. there is no serial correlation.

Hence

a test for serial correlation is a test of : 0 H0 r =

?(?̂ )≠?butplim?̂ =?E(β^)≠βbutplimβ^=β

No contradiction here.