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8 out of sample forecasting the linear regression

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8. Out-of-sample forecasting The linear regression model was introduced as a forecasting scheme. The question we now address is: How reliable is an out-of-sample forecast? Consider the linear regression model (2), and suppose we observe Then the forecast X n % 1 . of is where the OLS estimators are computed on the basis of Y n % 1 ˆ Y n % 1 ' ˆ α % ˆ β . X n % 1 , ˆ α and ˆ β the observations for j = 1,2,..., n . The actual but unknown value of is Y n % 1 = α + Y n % 1 β . X n % 1 % U n % 1 , so that the forecast error is: Y n % 1 & $ Y n % 1 ' U n % 1 & ( $ " & " ) & ( $ $ & $ ). X n % 1 ' U n % 1 & j n j ' 1 1 n % ( X n % 1 & ¯ X )( X j & ¯ X ) ' n i ' 1 ( X i & ¯ X ) 2 . U j . (28) See the Appendix for the latter equality. It follows now from Lemma 3 that under Assumptions I through V, where Y n % 1 & ˆ Y n % 1 - N [0, σ 2 Y n % 1 & ˆ Y n % 1 ], F 2 Y n % 1 & $ Y n % 1 ' F 2 n % 1 n % ( X n % 1 & ¯ X ) 2 ' n j ' 1 ( X j & ¯ X ) 2 . (29) See the Appendix. Denoting, $ F 2 Y n % 1 & $ Y n % 1 ' $ F 2 n % 1 n % ( X n % 1 & ¯ X ) 2 ' n j ' 1 ( X j & ¯ X ) 2 , (30) it follows now similar to Proposition 6 that
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16 Proposition 8 . Under assumptions I - V, ( Y n % 1 & ˆ Y n % 1 )/ ˆ σ Y n % 1 & ˆ Y n % 1 - t n & 2 . This result can be used to construct a 95% confidence interval, say, of Look up in Y n % 1 . the table of the t distribution the critical value of the two-sided t-test with n ! 2 degrees of t ( freedom. Then it follows from Proposition 7 that 0.95 ' P [ & t ( # ( Y n % 1 & $ Y n % 1 )/ $ F Y n % 1 & $ Y n % 1 # t ( ] ' P [ & t ( $ F Y n % 1 & $ Y n % 1 # Y n % 1 & $ Y n % 1 # t ( $ F Y n % 1 & $ Y n % 1 ] ' P [ $ Y n % 1 & t ( $ F Y n % 1 & $ Y n % 1 # Y n % 1 # $ Y n % 1 % t ( $ F Y n % 1 & $ Y n % 1 ] (31) Thus, the 95% confidence interval of is Y n % 1 [ ˆ Y n % 1 & t ( ˆ σ Y n % 1 & ˆ Y n % 1 , ˆ Y n % 1 % t ( ˆ σ Y n % 1 & ˆ Y n % 1 ]. Observe from (30) that increases with and so does the width of the ˆ σ Y n % 1 & ˆ Y n % 1 ( X n % 1 & ¯ X ) 2 , confidence interval. Thus, the father is away from the more unreliable the forecast X n % 1 ¯ X , ˆ Y n % 1 of becomes. Also observe from (30) that and that gets close to Y n % 1 ˆ σ Y n % 1 & ˆ Y n % 1 $ ˆ σ , ˆ σ Y n % 1 & ˆ Y n % 1 ˆ σ if n is large because lim n 64 ' n j ' 1 ( X j & ¯ X ) 2 ' 4 . 9. Relaxing the non-random regressor assumption As said before, the assumption that the regressors X j are non-random is too strong an assumption in economics. Therefore, we now assume that the X j ‘s are random variables. This requires the following modifications of the Assumptions I-V: Assumption I * : The pairs are independent and identically distributed . ( X j , Y j ), j ' 1,2,3, .... , n , Assumption II * : The conditional expectations are equal to zero : E [ U j | X j ] E [ U j | X j ] / 0. Assumption III * : The conditional expectations do not depend on the X j 's and are E [ U 2 j | X j ] finite, constant and equal : ( This is called the homoscedasticity E [ U 2 j | X j ] / σ 2 < 4 . assumption .)
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17 Assumption IV * : Conditional on X j , U j is N (0, σ 2 ) distributed . The Assumptions I * and II * imply that for j = 1,..., n , E [ U j | X 1 , X 2 ,..., X n ] / 0, (32) and similarly the Assumptions I * and III * imply that for j = 1,..., n , E [ U 2 j | X 1 , X 2 ,..., X n ] / F 2 . (33) Because (loosely speaking) conditioning on is effectively the same as X 1 , X 2 ,..., X n treating them as given constants, most of the previous propositions carry over: Proposition 9. Under Assumptions I * -IV * , Propositions 1 and 4 through 7 carry over, and the results in Propositions 2 and 3 now hold conditional on X 1 , X 2 ,..., X n .
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