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Law of Iterated Expectations

Mean-Square Forecast Error One use of an estimated regression is to predict out-of-sample values. Consider an out-of-sample observation (yn+1; xn+1) where xn+1 is observed but not yn+1. Given the coe¢ cient estimate b the standard point estimate of E (yn+1 j xn+1) = x 0 n+1 is yen+1 = x 0 n+1 b: The forecast error is the di§erence between the actual value yn+1 and the point forecast yen+1. This is the forecast error een+1 = yn+1 yen+1: The mean-squared forecast error (MSFE) is its expected squared value MSF En = E ee 2 n+1 : In the linear regression model, een+1 = en+1 x 0 n+1  b  ; so MSF En = E e 2 n+1 2E  en+1x 0 n+1  b  (4.28) + E  x 0 n+1  b   CHAPTER 4. LEAST SQUARES REGRESSION 116 The Örst term in (4.28) is  2 : The second term in (4.28) is zero since en+1x 0 n+1 is independent of b and both are mean zero. Using the properties of the trace operator, the third term in (4.28) is tr  E xn+1x 0 n+1 E  b   b 0  = tr  E xn+1x 0 n+1 E  E  b   b 0 j X  = tr  E xn+1x 0 n+1 E  V b  = E tr  xn+1x 0 n+1 V b  = E  x 0 n+1V bxn+1 (4.29) where we use the fact that xn+1 is independent of b, the deÖnition V b = E  b   b 0 j X  and the fact that xn+1 is independent of V b. Thus MSF En =  2 + E  x 0 n+1V bxn+1 : Under conditional homoskedasticity, this simpliÖes to MSF En =  2  1 + E  x 0 n+1 X0X 1 xn+1 : A simple estimator for the MSFE is obtained by averaging the squared prediction errors (3.47) e 2 = 1 n Xn i=1 ee 2 i where eei = yi x 0 i b (i) = ebi(1 hii) 1 : Indeed, we can calculate that E e 2  = E ee 2 i  = E  ei x 0 i  b (i) 2 =  2 + E  x 0 i  b (i)   b (i) 0 xi  : By a similar calculation as in (4.29) we Önd E e 2  =  2 + E  x 0 iV b (i) xi  = MSF En1: This is the MSFE based on a sample of size n 1; rather than size n: The di§erence arises because the in-sample prediction errors eei for i  n are calculated using an e§ective sample size of n1, while the out-of sample prediction error een+1 is calculated from a sample with the full n observations. Unless n is very small we should expect MSF En1 (the MSFE based on n 1 observations) to be close to MSF En (the MSFE based on n observations). Thus e 2 is a CHAPTER 4. LEAST SQUARES REGRESSION 117 Theorem 4.6 MSFE In the linear regression model (Assumption 4.2) and i.i.d. sampling (Assumption 4.1) MSF En = E ee 2 n+1 =  2 + E  x 0 n+1V bxn+1 where V b = var  b j X  : Furthermore, e 2 deÖned in (3.47) is an unbiased estimator of MSF En1 : E e 2  = MSF En