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
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-
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
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
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
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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