Measures of Fit Two complementary measures of how well the regression line \"fits\" the data: a regression R2 measures the fraction of the variance of Y that is explained by X a standard error of the regression {SER) measures the magnitude of a typical regression residual (in the units of Y) Econometrics The regression R2 Y,- => 5-3, => Total Sum of Squares = o 0 E R2 :1 1; R2 =0 means \"no fit\The larger is the variance of X, the smaller is the variance of B1 FIGURE 4.6 ) The Variance of #, and the Variance of X The colored dots represent a set of X/'s with a small 206 The math: variance. The black dots represent a set of X"'s with 03 = 1 x var(X;-Hx) u:] a large variance. The 204 (ox ) 2 regression line can be esti- mated more accurately with the black dots than with the colored dots. The intuition: Variation in X helps fitting the regression line, as in the figure: 194 09 100 101 102 103 The number of black and blue dots is the same. Econometrics 33 / 60 What is the sampling distribution of 31? The exact sampling distribution is complicated, but when n is large we get some simple (and good) approximations: (1) Since o2, ox 1 and E[B1] = B1, B1 is consistent: B1 7p B1 (2) When n is large, the sampling distribution of B1 is well approximated by the Normal distribution (CLT). Recall the CLT: suppose {v;}, i = 1, . .., n is i.i.d., with E[v;] = 0 and var[vi] = 0?. Then, when n is large, _ _ v; is approximately distributed as N(0, o?). Econometrics 34 / 60Problem 1. This problem asks you to retrace the steps we have made looking at the properties of the standard OLS estimator in class, and to apply the same logic to the model of this problem. (The counterpart to (a) are p.21-22 of the slides for OLS with a Single Regressor; the counterpart to (b) are p. 33-34). Consider the population regression model without an intercept Yi= aXi+ Ui. (1) (a) Obtain the least squares estimator of the parameter o in this model. You will first need to form the least squares criterion that is the sum of squares of the differences between Y; and aXi. You will then need to minimize it by differentiating the criterion function with respect to the parameter a. You will then equate the derivative to zero, and solve the resulting equation to obtain the estimator. If you do everything correctly you will obtain the estimator a = (b) Assume that the Least Squares Assumptions #1-#3 hold. Show that the estimator a is an unbiased estimator of the population parameter o. As in the slides, you may want to first rearrage the expression for a, then compute E [a|X] showing that a is conditionally unbiased, and then conclude by using the Law of Iterated Expectations