gt-ln04

gt-ln04 - LN04: FORECASTING YASUTOMO MURASAWA Contents 1....

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Unformatted text preview: LN04: FORECASTING YASUTOMO MURASAWA Contents 1. Forecasting 1 1.1. Statistical Decision 1 1.2. Best Prediction 1 1.3. Linear Projection 2 1.4. OLS 3 2. Forecasts with Infinite Observations 3 2.1. MA Sequences 3 2.2. ARMA Sequences 4 3. Forecasts with Finite Observations 4 Appendix A. LDU Decomposition 4 1. Forecasting 1.1. Statistical Decision. Let ( y t , x t ) be a (1 + k )-variate random vector. Let y e t be a prediction of y t , g ( . ) be a predictor of y t . Definition 1. A loss function is L ( ., . ) that gives the loss of using y e t instead of y t . Definition 2. A quadratic loss function is L ( ., . ) s.th. ( y e , y ) R 2 , L ( y e , y ) ( y e- y ) 2 . Definition 3. The risk of g ( . ) is R ( g ( . )) := E( L ( g ( x t ) , y t )) . Definition 4. The mean squared error (MSE) of g ( . ) is MSE( g ( . )) := E ( ( y t- g ( x t )) 2 ) . Definition 5. The MSE is the risk under a quadratic loss. 1.2. Best Prediction. Let G be the set of all predictors of y t . Definition 6. The best predictor of y t is g ( . ) := arg min g ( . ) G R ( g ( . )) . Theorem 1. Suppose that (1) L ( ., . ) is quadratic, (2) ( y t , g ( x t )) is square-integrable. Then g ( x t ) = E( y t | x t ) . Date : October 28, 2009. 1 2 YASUTOMO MURASAWA Proof. By the LIE, g ( . ) G , E ( ( y t- g ( x t )) 2 ) = E ( [( y t- E( y t | x t )) + (E( y t | x t )- g ( x t ))] 2 ) = E ( ( y t- E( y t | x t )) 2 ) + 2 E(( y t- E( y t | x t ))(E( y t | x t )- g ( x t ))) + E ( (E( y t | x t )- g ( x t )) 2 ) E ( ( y t- E( y t | x t )) 2 ) + 2 E(E(( y t- E( y t | x t ))(E( y t | x t )- g ( x t )) | x t...
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This note was uploaded on 02/22/2012 for the course EE 441 taught by Professor Neely during the Spring '08 term at USC.

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gt-ln04 - LN04: FORECASTING YASUTOMO MURASAWA Contents 1....

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