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2. Question 2: Heteros kedastic-Robust Standard Errors Consider the following model Y = Bo+B,X + (1+x) where * 0, E[U|X] = 0, and E[42]x] = 0?. In this case, B1,0Ls will be consistent. Although E[42]x] = 0, the model is not homos kedastic because, in a regression of Y, a constant, and X, the error term is 11 Y = fo+BX+V V= (1+x) Here E[v+]x] = (1+yX)2o2. Therefore, the model exhibits heteroskedasticity. Recall tha{] BioLNB en (hu 1 E [(X E[x])245] Var (X2) We defined the asymptotic variance of g to be Avar (10) 1E [(x - E[x]){v}] Var(x1) For this exercise, you will run S = 100 regressions with sample size n = 1,000, assuming that Bo = 0, B1 = 1, and 1 = 1. Also, that X ~ N(0, 1) and U~ N(0,1). (a) Save the values of Blous, and of Aar(B1.OLS) . n11'-,(X-X) AVar(108) 12-[(X; )207) [I L-:(X 8)+] where V; = Y; - BOOLS - B1.OLX. Here AVar(Bus) is an estimator of AVar(81.us) under the incorrect homoskedasticity assumption. Here Aar(1.0Ls) is the correct) heteroskedastic- robust estimator of the asymptotic variance. 1 11 (b) Construct 95% confidence intervals for $1 using each of the estimators of the asymptotic vari- ance Blous 61.96 AVar(Blous) B1.OLS 1.9 x Aar(BLOLS) We refer to AVar(B)x as robust standard errors. Plot the confidence intervals and compute the proportion of confidence intervals that contain the true value B1 = 1 in each case. This is called the empirical coverage of the confidence intervals. Comment on your findings. (c) In the case where 1 = 0, then the assumption of homoskedasticity is correct. However, still Allar (B1os) + AVar (10)R. Which one should you use? Base your answer in the empirical coverage of the confidence intervals now computed for 1 = 0. (Before we had assumed that 7=1