Question 1
Consider the simple linear regression model:
Setup the minimisation problem to derive the ordinary least squares (OLS) estimator for and use that to find the formula of OLS estimator. [20 marks]
State the Gauss Markov Theorem including all the assumptions and the meaning of the acronym BLUE. [15 marks]
What are the implications for OLS estimation if heteroskedasticity exists? [5 marks]
State the formula to calculate the standard error of the OLS estimate of when the error terms are homoskedastic. State the formula to calculate the standard error of the OLS estimate of when the error terms are heteroskedastic. [10 marks]Question 2
Given the model the estimated regression based on 20 observations yielded (standard errors in parentheses):
(3.41) (0.06) (0.50)
=0.87
Test against . Explain your method in doing this test. State any required assumption to use this regression analysis for statistical inference.[10 marks]
Test the overall significance of the model. If you think that there is not enough information to answer please state which information is needed in order to do so.[15 marks]
Explain how to use the Goldfeld-Quandt test to detect heteroskedasticity. [10 marks]
Suppose that, by mistake, you consider the following model Y_t=β_0+〖β_1 X〗_1t+u_t and you omit the variable X_2t. State the formula which shows the expected value of β ̂_1 when there is an omitted variable. Discuss when the omitted variable bias is different from zero. Let the variance of X_1 be 5.15, the variance of X_2 be 1.47, and the covariance between X_1 and X_2 be 2.38. Determine whether the regression with the omitted variable will overestimate or underestimate the real value of the parameter β_1. [15 marks]Sample Solution