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Q1. You have an i.i.d. sample {(Yi, X if, X 2i): i = 1, , n}. You want to estimate the causal effect of X1 on Y. a) You rst run a regression Yi = a0 + alei + ui and get the following result: Y. = 0.5 + 1.2x\"- (0.2) (0.6) where the numbers in parentheses are standard errors. Construct a 90% confidence interval for al. (The critical value for a t-test at 1%, 5% and 10% levels are 2.58, 1.96 and 1.64, respectively). b) You worry about omitted variable bias (OVB), so you are considering whether to put X 21- into the regression. The only condition you can check is that whether X1 and X 2 are correlated. Propose a test with the null Ho: pX1X2 = 0 based on regression. Describe what regression you would run and what the test procedure would be. Explain why the proposed test works. c) Now you nd X1 and X 2 are indeed correlated with covariance equal to 0.7. You decide to include X2 in the regression: Yi = )60 + lei + ,BzXzi + ei. The results are as follows: 17i=0.6 + 1.4-X1i + 4X2i (1) (0.3) (0.875) (2.67) Construct 90% condence intervals for 51 and for 32. Is 51 signicantly different from 0 at 10% level? How about [32? d) You want to further conduct a Wald test for H0: ,81 = B2 = 0. Based on the information in c), can you calculate the Wald statistic? If so, provide the derivation and the test statistic value. If not, explain why. e) Suppose the Wald statistic is equal to 4.76. Is the null in d) rejected at 10% level? f) What can you say about the conclusion in part c) given e)? g) Suppose the difference between the estimates in a) and c) is only due to OVB. If the model in c) is the true model, provide two ways to compute (estimate) the OVB based on the information given so far and the regression you run in b)