# 14 - I n t r o d u c t i o n t o T i m e S e r i e s A n a...

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Unformatted text preview: I n t r o d u c t i o n t o T i m e S e r i e s A n a l y s i s . L e c t u r e 1 4 . L a s t l e c t u r e : M a x i m u m l i k e l i h o o d e s t i m a t i o n 1 . R e v i e w : M a x i m u m l i k e l i h o o d e s t i m a t i o n 2 . M o d e l s e l e c t i o n 3 . I n t e g r a t e d A R M A m o d e l s 4 . S e a s o n a l A R M A 5 . S e a s o n a l A R I M A m o d e l s 1 R e c a l l : M a x i m u m l i k e l i h o o d e s t i m a t i o n T h e M L E ( ˆ φ , ˆ θ , ˆ σ 2 w ) s a t i s fi e s ˆ σ 2 w = S ( ˆ φ , ˆ θ ) n , a n d ˆ φ , ˆ θ m i n i m i z e l o g parenleftBigg S ( ˆ φ , ˆ θ ) n parenrightBigg + 1 n n summationdisplay i = 1 l o g r i − 1 i , w h e r e r i − 1 i = P i − 1 i / σ 2 w a n d S ( φ , θ ) = n summationdisplay i = 1 ( X i − X i − 1 i ) 2 r i − 1 i . 2 R e c a l l : M a x i m u m l i k e l i h o o d e s t i m a t i o n W e c a n e x p r e s s t h e l i k e l i h o o d i n t e r m s o f t h e i n n o v a t i o n s . S i n c e t h e i n n o v a t i o n s a r e l i n e a r i n p r e v i o u s a n d c u r r e n t v a l u e s , w e c a n w r i t e X 1 . . . X n bracehtipupleft bracehtipdownright bracehtipdownleft bracehtipupright X = C X 1 − X 1 . . . X n − X n − 1 n bracehtipupleft bracehtipdownright bracehtipdownleft bracehtipupright U w h e r e C i s a l o w e r t r i a n g u l a r m a t r i x w i t h o n e s o n t h e d i a g o n a l . T a k e t h e v a r i a n c e / c o v a r i a n c e o f b o t h s i d e s t o s e e t h a t Γ n = C D C ′ w h e r e D = d i a g ( P 1 , . . . , P n − 1 n ) . 3 R e c a l l : M a x i m u m l i k e l i h o o d e s t i m a t i o n | Γ n | = | C | 2 P 1 · · · P n − 1 n = P 1 · · · P n − 1 n a n d X ′ Γ − 1 n X = U ′ C ′ Γ − 1 n C U = U ′ C ′ C − T D − 1 C − 1 C U = U ′ D − 1 U . W e r e w r i t e t h e l i k e l i h o o d a s L ( φ , θ , σ 2 w ) = 1 ( ( 2 π ) n P 1 · · · P n − 1 n ) 1 / 2 e x p parenleftBigg − 1 2 n summationdisplay i = 1 ( X i − X i − 1 i ) 2 / P i − 1 i parenrightBigg = 1 ( ( 2 π σ 2 w ) n r 1 · · · r n − 1 n ) 1 / 2 e x p parenleftbigg − S ( φ , θ ) 2 σ 2 w parenrightbigg , w h e r e r i − 1 i = P i − 1 i / σ 2 w a n d S ( φ , θ ) = n summationdisplay i = 1 ( X i − X i − 1 i ) 2 r i − 1 i ....
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• Fall '08
• Staff
• Maximum likelihood, Autoregressive moving average model, maximum likelihood estimation, Xt, ARMA Models, seasonal arma

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14 - I n t r o d u c t i o n t o T i m e S e r i e s A n a...

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