TS_Fall_2011_Lecture_6

TS_Fall_2011_Lecture_6 - FIN 271 FINANCIAL MODELING AND...

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FIN 271 FINANCIAL MODELING AND ECONOMETRICS LECTURE SET 6 TIME SERIES MODELING REFIK SOYER THE GEORGE WASHINGTON UNIVERSITY

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2 SEASONAL ARIMA (SARIMA) MODELS • Stochastic Seasonal Models: Seasonality may not follow a fixed pattern. We may have a seasonal TS with high correlations among observations from the same season, but ACF may still suggest a stationary behavior. For example: in monthly data high AC's at lags k 12, 24, . .. œ But in addition to high autocorrelations at seasonal lags, we may have high autocorrelations at nonseasonal lags as well. If we do not have a stationary TS, we can induce stationarity Ê seasonal and/or nonseasonal differencing. Consider a stationary series of quarterly observations. A very simple way of capturing a seasonal effect is by a seasonal AR model of the form ]œG ]  t1 t 4 t 1 F% F | | 1 The ACF for the model is 3 F k k/4 1 k 0, 4, 8, . ... 0 otherwise œ œ œ 1
3 Note that in the above model ACF is zero at nonseasonal lags. As long as | | 1, the seasonal pattern in ACF gradually dying away as lags getting F 1 larger stationarity. Ê The PACF will be chopped off after lag 4 (also zero at lags 1 through 3). The above is referred to as a first-order seasonal AR model (s 4 and P 1) AR(1) . œœ Ê 4 This model may be extended to allow for both AR and MA terms at other seasonal lags. Note that the ACF PACF of the above models will contain 'gaps' at nonseasonal lags Î and unless seasonal movements are the only feature of the series, these models are not appropriate. For example, in addition to high autocorrelations at seasonal lags, we can also expect correlations at nonseasonal lags need a more general model. Ê How do we deal with nonstationarity in seasonal models ? In nonseasonal TS differencing was needed to induce stationarity (1 B) . Ê d Seasonal differencing operator: B where s is the seasonal period, , e.g. s 12. s tt s ]œ] œ Seasonal TS can be made stationary by considering differences of the form (1 B ) . sD 2

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4 A more general model ARIMA (p, d, q) ( P, D, Q) model nonseasonal seasonal ï ðóñóò s where d denotes nonseasonal differencing and D denotes seasonal differencing. The above is known as as multiplicative SARIMA model and using backshift operator notation can be represented as 9F ) @% (B) (B ) (1 B) (1 B ) Y (B) (B ) sd s D s tt  œ Ex . (1 B) (1 B ) . ] œ % 11 t t 12 which is an (1, 0, 0) (1, 0, 0) multiplicative SARIMA model 12 ] œ G ] ] ] t 1 t-1 1 t-12 1 1 t-13 t 9 F % Ex . Airline model: (0, 1, 1) (0, 1, 1) [nonzero autocorrelations at lags 1, 11, 12, and 13.] 12 Ð Ñ Ñ ] œ 1 B 1 B 12 t 1 t 1 2 t 1 3 %) % @ % ) @ % 3
1 Modeling Johnson and Johnson Quarterly Earnings 1960-1980 0 5 10 15 JJ_EPS 0 10 20 30 40 50 60 70 80 90 Row -1 -0.5 0 0.5 1 1.5 2 2.5 3 LOG_JJ 0 10 20 30 40 50 60 70 80 90 Row 4

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2 data sales; infile 'c:\tsdata\JJ_Earnings.txt'; input eps; lgeps=log(eps); proc arima; i var=lgeps; i var=lgeps(1); i var=lgeps(1,4); run;
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TS_Fall_2011_Lecture_6 - FIN 271 FINANCIAL MODELING AND...

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