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The Probability Approach to Econometrics .

Sample Mean To start with the simplest setting, we Örst consider the intercept-only model yi =  + ei E (ei) = 0: which is equivalent to the regression model with k = 1 and xi = 1: In the intercept model,  = E (yi) is the mean of yi : (See Exercise 2.15.) The least-squares estimator b = y equals the sample mean as shown in equation (3.9). We now calculate the mean and variance of the estimator y. Since the sample mean is a linear function of the observations, its expectation is simple to calculate E (y) = E 1 n Xn i=1 yi ! = 1 n Xn i=1 E (yi) = : This shows that the expected value of the least-squares estimator (the sample mean) equals the projection coe¢ cient (the population mean). An estimator with the property that its expectation equals the parameter it is estimating is called unbiased. DeÖnition 4.1 An estimator b for  is unbiased if E  b  = . We next calculate the variance of the estimator y under Assumption 4.1. Making the substitution yi =  + ei we Önd y  = 1 n Xn i=1 ei : Then var (y) = E (y ) 2 = E 0 @ 1 n Xn i=1 ei ! 0 @ 1 n Xn j=1 ej 1 A 1 A = 1 n2 Xn i=1 Xn j=1 E (eiej ) = 1 n2 Xn i=1  2 = 1 n  2 : The second-to-last inequality is because E (eiej ) =  2 for i = j yet E (eiej ) = 0 for i 6= j due to independence. We have shown that var (y) = 1 n  2 . This is the familiar formula for the variance of the sample me