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TS_Fall_2011_Lecture_7

# TS_Fall_2011_Lecture_7 - FIN 271 FINANCIAL MODELING AND...

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FIN 271 FINANCIAL MODELING AND ECONOMETRICS LECTURE SET 7 TIME SERIES MODELING REFIK SOYER THE GEORGE WASHINGTON UNIVERSITY

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ARCH and GARCH Models Using dependence of as motivation, Engle (1982) suggested the ARCH(q) model % 2 t % ttt œ2 a È â t01 q 22 t1 tq αα % α %  , where a 's independent and identically distributed random variables with mean 0 and t variance 1. There are certain conditions on 's that need to be satisfied for 0, that is, 0 for it i 2 i 0, ,q. œá Note that in the above, since , the conditional variance is a function of the past %. œ]  values of series as well. ] t Thus, given history H , the volatility equation is an exact function for h . In the above the t mean corrected process is stationary. % t 1
• Special case q 1; the ARCH(1) model is given by œ % ttt œ a h È h , , 0. t01 0 1 2 t1 œ αα % α α It is easy to show that E ( )0 V a r ( )E ( ) . %% % tt 2 t œÊ œ Also, we can write a α % 0 1 2 É E( ) E( ) α % 22 1 01 and stationarity of implies that E( ) E( ), % t t œ E( ) . 1 % α α 2 t 0 1 œ For the variance to exist we need 0 and 0 1. If we need higher order moments  to exist then we need to impose other restrictions. 2

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Note that we can have any form for the conditional mean in the model . .. % tt t t ]œ  For example, if C then .9 t1 t 1 œ] 2œÐ  Ñœ  Ð]   ] Ñ t0 1 0 1 t 1 1 t 2 22 αα % 9  C If we have a regression type model where ."" t01 t œ X then  Ð]   Ñ t 01 t 101 t 1 2 "" X Thus, the conditional variance depends on the past values.
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TS_Fall_2011_Lecture_7 - FIN 271 FINANCIAL MODELING AND...

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