 # 2. Here's a question that I want you to answer intuitively (with pr...

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2. Here's a question that I want you to answer intuitively (with projection-type argu- ments), as opposed to algebraically. If you want to grind out the algebra, you may, but the point of the problem and the way I want you to solve it is to build your skill in using intuitive projection arguments. Consider the following true regression model in scalar notation (there is no constant here - do not worry about it, as I am ignoring it to make everything simpler in this question): Yi = Bxi + Swi + Azi + Ei Let's say that the regressors Wi and Zi are nearly collinear in the sense that we almost have Zi 22 CWi holding almost with equality where c is some constant. (a) Say that the researcher, aware of the collinearity issue, decides to estimate the 'short- ened' regression Yi = TTX + ywi + Ui Will IF be biased for the true parameter B? Describe the nature of the bias as the near equality of Zi 22 CWi becomes an exact equality. (b) In the general case where we have near collinearity of zi 22 cwi, what can we say about how the bias in the estimator V for the true parameter &? What does estimate in the case where the near equality holds with an exact equality? 4. In discussing empirical contexts and models which necessarily violate the strong form of exogeneity, namely 'strict exogeneity', I mentioned that the simple autocorrelation model of: Yt = at Byt-1 + Et is a prevalent example that violates strict exogeneity, and thus the conditional expectation 'tool' cannot be applied to establish that the estimator B obtained by applying OLS to this model (which is also the DGP) is unbiased. Nonetheless, as I mentioned in class - while leaving the details of the proof to you - you can establish that the OLS B is consistent, as the sample size T 80. excersise. This problem is that In other words, being careful to include the devatiations from means for the regressor, because of the presence of the estimated intercept a, write down the algebraic expression of B OLS, and then using the plim operator, establish the requisite conditions for this B to be consistent.

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