You have an i.i.d. sample {(, X, X2i): i = 1,...,n). You want to estimate the causal effect of X, on Y. a) You first run a regression Y = a + aX +u, and get the following result: = 0.5 +1.2X (0.2) (0.6) where the numbers in parentheses are standard errors. Construct a 90% confidence interval for a. (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, into the regression. The only condition you can check is that whether X, and X are correlated. Propose a test with the null Ho: Px,x = 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 find X and X are indeed correlated with covariance equal to 0.7. You decide to include X in the regression: Y = Bo + BX + B2X2 + e. The results are as follows: 0.6 + - 4Xzi 1.4X1 (0.3) (0.875) (2.67) Construct 90% confidence intervals for B, and for B. Is B, significantly different from 0 at 10% level? How about B? (1) d) You want to further conduct a Wald test for Ho: B B = 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) Does parte) yield the same result as part 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 a method to compute (estimate) the OVB based on the information given so far and the regression you run in b).