(a) Suppoe (y, Xi.Xu) smtisfy the a (a) Suppose (y,Xu X2.) satisfy the assumptio
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Question
(a) Suppoe (y, Xi.Xu) smtisfy the a (a) Suppose (y,Xu X2.) satisfy the assumptions in Key Concept 6.4 (The Least Squares Assumptions in the Mulitple Regression Model), you're intersted in 1, i e. the causal effect of Xi on Y, and the two variables Xi and X2 are uncorrelated. You just estimate by regressing Y onto Xi (so that X2 is not included in the regression). Does this estimator suffer from omitted variable bias? Explain. (b) Now, you have other assumptions: A random sample of size n = 401 is drawn frorn the population. The conditional sample variance of residals d, 2-4 and sample variance Var(X6. Using the following formula, calculate the variance of B when Xii and X2 are uncorrelated: Var() = where Ri is the coefficient of determination from a regression of Xi on X2. (c) Assume that corr(Xu, X2i) = 0.5, in other words, R = 0.25. Compute the variance of 1 again. (d) Comment on the following statements: "When Xi and X2 are correlated, the variance of , is larger than it would be if X1 and X2 were uncorrelated. Thus, if you are interested in 1, it is best to leave X2 out of the regression if it is correlated withX" Hint: Use your answers to (b) and (c).Explanation / Answer
In any linear regression, for omitted-variable bias to exist following two conditions must hold true:
1) The omitted variable must be a determinant of the dependent variable (i.e., its true regression coefficient is not zero) and
2) the omitted variable must be correlated with an independent variable specified in the regression (i.e., cov(z,x), is not equal to zero).
In the given question we have two varibales X1 and X2. We are trying to ommit X2 and see the regression of Y on X1. Now it is mentioned that X1 and X2 are uncorrelated i.e., cov(X1 and X2)is equal to zero(condition 2 in above list is violated). So even though X2 is a determinant of Y(condition 1 holds), there will not be any omitted variable bias in this problem.
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