Describe the properties of the ols

Question

Discuss the below:

Q1. Consider the model Y1=β0+β1X_1i+β2X_2i+ε1 where Var(ε1)=X_3iσ² where a2 is a constant and X 31 is another independent variable in the system.

(a) Describe the properties of the OLS estimates of the coefficients of this model. Are the OLS estimates of 13's biased or unbiased? Explain your answer.

(b) Describe step by step how you will overcome the problem
of heteroscedasticity in the errors and obtain efficient parameter estimates.

2 The following regression was fitted by OLS , using 32 annual observations on time series data:

Y_t = 4.52 - 0.62 X_1t +0.93 X_2t +0.61 X3t +0.16 X_lt ; Adjusted R² = 0.638

S.E (1.23) (0.28) (0.38) (0.21) (0.12)                      DW= 0.61

Note that Y = quantity of wheat exports from Canada

X1 = Price of Canadian wheat in international market

X2 = Quantity of Canadian wheat harvested

X3 = GNP per capita in countries importing Canadian wheat

X4 = price of barley in world market.

(a) Test at 5% level of significance the hypothesis that, all else equal, the coefficient on GNP per capita is less than 1.

(b) Test at 5% level of significance the hypothesis that there is no serial correlation in the residuals.

(c) Assuming that you cannot reject the hypothesis of a positive serial correlation in the residuals in the preceding case, describe in detail the various steps in finding consistent and efficient estimates of the parameters in the above regression.

(d) Explain how would you test the hypothesis that β1 + β2 = 0. What additional information is required to calculate the test statistic?