TS_Fall_2011_Lecture_4

TS_Fall_2011_Lecture_4 - FIN 271 FINANCIAL MODELING AND...

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FIN 271 FINANCIAL MODELING AND ECONOMETRICS LECTURE SET 4 TIME SERIES MODELING REFIK SOYER THE GEORGE WASHINGTON UNIVERSITY
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2 MOVING AVERAGE (MA) PROCESSES ]  oe] oe  â t t t 1 t 1 2 t 2 q t q . % ) % ) % ) % . is a MA(q). MA process of order q Ê Note: E( ) and E( ) 0 . ] oe ] oe t t . The MA process is a linear combination of the current and the past values of the white noise process; thus, the MA process is always stationary. The back shift operator representation : ] oe â . ( 1 B B B ) t 1 2 q t 2 q ðóóóóóóóóóóóóóóóóñóóóóóóóóóóóóóóóóò ) ) ) % (B) ) Ê ] oe (B) . t t ) % ) (B) transfer function of the MA process. ´ The MA process is a linear combination of the current and the past values of the white noise process; recall the MA representation of AR(1) process. 1
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3 • Special Case: First order MA process: MA(1) ] oe   t t 1 t 1 . % ) % (Conventional representation) Variable of interest is written as a "moving average" of random disturbance terms. ] t For AR and MA models (also for ARMA models), it can be shown that the random disturbance terms are one step ahead forecast errors (1) ^ % t t t 1 oe]  ] where (1) is the forecast for at time (t 1). ^ ] ] t 1 t Ê ] % t t is the one step ahead forecast error for forecasting based on the information at time (t 1). We can write the MA(1) process ] oe oe . (1 B) t t 1 t 1 1 t % ) % ) % where ( ) 0 for all t, since ( ) 0 for all t. . . ] oe]  ÊI ] oe I oe t t t t . % 2
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4 Z+< ] oeZ+< oeZ+< Z+< ( ) ( ) ( ) ( ) . t t 1 t 1 t t 1 1 2 % ) % % ) % (1 ). Ê oe # 5 ) 0 2 2 1 % The ACF of MA(1) Process: # . . k t t k t t k oeIÒ ]  ]  ÓoeI ] ] ( ) ( ) ( ) . . First, consider . We can multiply both sides by and take expectations: . # 1 t 1 ] ] ] oe ] oe . . . ( ) ( ) ( ) t t 1 t 1 t 1 t 1 t 1 t 1 t 1 1 t 2 % ) % % ) % % ) % I ] ] oeI  ( ) ( ) ( ) . . t t 1 t 1 t 1 t 1 1 t 2 % ) % % ) % I ] ] oeI  I  I  I ( ) ( ) ( ) ( ) ( ) . . t t 1 t t 1 1 t t 2 1 t 1 t 2 2 2 t 1 1 %% ) %% ) % ) % % I oe Ê oe ] ] oe  oe  ( ) for all t E( ) E( ) . . % 5 # ) % ) 5 t 2 2 2 2 1 t t 1 1 1 t 1 % % Thus, we have (1 ) (1 ) 3 # ) 5 ) # 5 ) ) 1 1 1 1 0 2 2 1 1 2 2 oe oe oe % % What about higher order autocorrelations ? # % ) % % ) % 2 t t 2 t 1 t 1 t 2 1 t 3 oe ] ] oe oe E( ) E( ) ( ) 0 . . 3
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5 In general for the MA(1) process # 3 k k oe Ê oe 0 for k 1 0 for k 1. Recall that a stationary AR process has an infinite MA process representation. Similarly an MA process can be represented as infinite AR process under certain conditions.
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