Question

Assume that you will run the following regression: yt = α + βxt + ut. You believe that the errors might be serially correlated in the following fashion: ut = γ0 + γ1ut−1 + εt

a) Will OLS estimation be inconsistent? Why?

Answer 1

If errors are serially correlated in any regression, then it is said to have the problem of Autocorrelation.

Autocorrelation is nothing but the degree of similarity of correlation between members of observations ordered in time ( as in time series data) or space (as in cross-sectional data).

Since the error term(ut) in above regression is correlated with the error term of one previous period(ut-1), it is the case of autocorrelation.

Consequences of autocorrelation:-

1) OLS estimators are linear.

2) OLS estimators are unbiased and hence consistent.

3)OLS estimators are not efficient since the variance is not minimum.

4)The formula to estimate the variance of OLS estimator is generally biased.  

Hence, OLS estimators under autocorrelation are consistent, meaning as the sample size is increased the estimator can be made to lie arbitrarily close to its true value with prob close to 1. Moreover, it has been seen the rate of convergence becomes faster when autocorrelation is negative while it becomes slower when it is positive.