Question 2 You are given the following data sampling process yi = 1 + 2x2 + ; where: = X2i * Ui u~ N(0, 1) and 2, is a non-stochastic positive variable. (a) Show that this model is heteroskedastic {2 marks} (b) If the empirical information is Y = [4 2 5 7]' and 2i = [1 1 4 4]. Estimate BOLS {2 marks] (c) What are the characteristics of BOLS? {2 marks] (d) Discuss how you would transform the data so that you could remove the het- eroskedasticity {2 marks] (e) Now estimate the model with the empirical information given in section (b), but by GLS. {4 marks} 10 (f) Show that in this case var (GLS) = - [47] (2 marks) (g) Supply the robust standard errors that can be used to test the significance of 6 and B2 in 2(c) above {2 marks] (h) The figure below shows edited stata output detailing results for a heteroscedasticity test. Use it to test whether the reference estimation was heteroscedastic. {2 marks] hettest Breusch-Pagan/Cook-Weisberg test for heteroskedasticity Assumption: Normal error terms Variable: Fitted values of price HO: Constant variance chi2 (1) = 105.75 Prob chi2 = 0.0000 (i) Benji ran the following regression Price, = + lotsize + u. If this regression was heteroscedastic and price and lot-size are positively related, sketch the distribution of price around the line of best fit. {2 marks] [20 marks]

Question 2 You are given the following data sampling process yi=1+2x2i+i where: i=x2iuiuiiidN(0,1) and x2i is a non-stochastic positive variable. (a) Show that this model is heteroskedastic {2 marks } (b) If the empirical information is Y=[4257] and x2i=[1144]. Estimate ^OLS{2 marks } (c) What are the characteristics of ^OLS ? \{2 marks } (d) Discuss how you would transform the data so that you could remove the heteroskedasticity {2 marks } (e) Now estimate the model with the empirical information given in section (b), but by GLS. {4 marks } (f) Show that in this case var(^GLS)=[9109494185]{2 marks } (g) Supply the robust standard errors that can be used to test the significance of 1 and 2 in 2(c) above {2 marks } (h) The figure below shows edited stata output detailing results for a heteroscedasticity test. Use it to test whether the reference estimation was heteroscedastic. {2 marks } - hettest Breusch-Pagan/Cook-Weisberg test for heteroskedasticity Assumption: Normal error terms Variable: Fitted values of price H0: Constant variance chi2(1)=105.75 Prob > chi2 =0.0000 (i) Benji ran the following regression Price i=1+2 lotsize +ui. If this regression was heteroscedastic and price and lot-size are positively related, sketch the distribution of price around the line of best fit. {2 marks }