6 Consider the equation y = B. + Bix + Bax? + u E(ulx) = 0 where the explanatory variable x has a standard normal distribution in the population. In particular, E(x) = 0, E(x) = Var(x) = 1, and E(X) = 0. This last condition holds because the standard normal distribution is symmetric about zero. We want to study what we can say about the OLS estimator of B, we omit x and compute the simple regression estimator of the intercept and slope. (i) Show that we can write y = 0 + Bix + v where E(v) = 0. In particular, find v and the new intercept, Qo. (ii) Show that E(vlx) depends on x unless B2 = 0. (iii) Show that Cov(x, v) = 0. (iv) If , is the slope coefficient from regression y; on x, is , consistent for B.? Is it unbiased? Explain. (v) Argue that being able to estimate B, has some value in the following sense: B, is the partial effect of x on y evaluated at x = 0, the average value of x. (vi) Explain why being able to consistently estimate B, and B, is more valuable than just estimating B. 6 Consider the equation y = B. + Bix + Bax? + u E(ulx) = 0 where the explanatory variable x has a standard normal distribution in the population. In particular, E(x) = 0, E(x) = Var(x) = 1, and E(X) = 0. This last condition holds because the standard normal distribution is symmetric about zero. We want to study what we can say about the OLS estimator of B, we omit x and compute the simple regression estimator of the intercept and slope. (i) Show that we can write y = 0 + Bix + v where E(v) = 0. In particular, find v and the new intercept, Qo. (ii) Show that E(vlx) depends on x unless B2 = 0. (iii) Show that Cov(x, v) = 0. (iv) If , is the slope coefficient from regression y; on x, is , consistent for B.? Is it unbiased? Explain. (v) Argue that being able to estimate B, has some value in the following sense: B, is the partial effect of x on y evaluated at x = 0, the average value of x. (vi) Explain why being able to consistently estimate B, and B, is more valuable than just estimating B