{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

LINREG2

8 out-of-sample forecasting the linear regression

This preview shows pages 15–17. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 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 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....
View Full Document

{[ snackBarMessage ]}

Page15 / 29

8 Out-of-sample forecasting The linear regression model was...

This preview shows document pages 15 - 17. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online