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1. (6 points) Suppose we compute the regression through the origin of y on x. (That is, we use OLS to estimate y = β1x + u.) Assume that, in our sample,
(Note: By the Cauchy-Schwartz inequality, we know the expression on the left-hand side can’t exceed unity). Assume also that x = 0. Show R2 ≡ SSE/SST will exceed unity if the absolute value of the sample mean of y exceeds the sample standard deviation of y. (Hints: use SSE = Pn i=1(ybi − yb) 2 ), and the fact that the OLS estimate of β1 will be βˆ P 1 = n i=1 P xiyi n i=1 x 2 i
2. (5 points) Suppose that assumptions MLR.1-MLR.3 hold but we replace MLR.4 with MLR.4’: E(u|x1…xk) = 5. Show that the OLS estimators of the slope coefficients are unbiase
3. (6 points) Suppose Y ∼ t(m). Use the Law of Large Numbers and the Continuous Mapping Theorem to show that Y converges in distribution to a N(0, 1) random variable as m → ∞. (Hint: Start with the representation of a t and a chi-square random variable from the Statistics Review
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