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TS_Fall_2011_Lecture_3

# TS_Fall_2011_Lecture_3 - FIN 271 FINANCIAL MODELING AND...

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FIN 271 FINANCIAL MODELING AND ECONOMETRICS LECTURE SET 3 TIME SERIES MODELING REFIK SOYER THE GEORGE WASHINGTON UNIVERSITY

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2 Forecasting from AR(1) Model Once the coefficients are estimated, we can forecast the future values of the TS . ] t Let (h) denote the forecast of at time n after observing , , . . ., . ] ] ] ] ] s n n h 1 2 n ] ´ s n (h) h-step ahead forecast at time n h lead time, n origin (of the forecast) ´ ´ For example, (1) is the one-step ahead forecast at time n. ] s n Consider the AR(1) process: ] œ G  ] t 1 t 1 t 9 % We are interested in forecasting ] n 1 ] œ G  ]  n 1 1 n n 1 9 % Given , , . . ., , one-step ahead forecast at time t ] ] ] 1 2 n ] Ð Ñ œ G  ] s n 1 n 1 9 1
3 Note that expected value of given past observations is 0 % n 1 Þ From the above we can write % n 1 n 1 n œ ]  ] Ð Ñ s 1 Ê is the one-step ahead forecast error ! % n 1 What about forecast for ? ] n 2 ] œ G  ] n 2 1 n 1 n 2 9 % ] Ð Ñ œ G  ] Ð Ñ s s n 1 n 2 1 9 ] Ð Ñ œ G  ÐG  ] Ñ œ Ð  ÑG  ] s n 1 1 n 1 n 2 1 2 1 9 9 9 9 In general for AR 1 Process: Ð Ñ ] Ð Ñ œ Ð   á  ÑG  ] s n 1 n 1 1 h 1 h h 1 9 9 9 2

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4 MEAN REVERSION OF FORECASTS What happens when h gets large ? In the above 1 is a geometric series thus it converges to Ð   á  Ñ 9 9 1 1 h 1 1 and thus ÎÐ"  Ñ 9 1 ] Ð Ñ Ä œ s G Ð  Ñ n 1 h . 1 9 . As the lead time increases the forecasts get closer to the mean and eventually the forecasts revert to the mean. This is a property of stationary time series models and will not apply to nonstationary series. See SAS example on price of the crude oil. 3
Analysis of the Price of Crude Oil Data * AVERAGE PRICE OF CRUDE OIL AT THE WELL JAN74-DEC82 INDEX=100 IN 1967; DATA OIL; INFILE 'C:\TSDATA\Crudeoil.txt'; INPUT PGAS PCRUDE STOCKGAS GASCONS STOCKOIL; PROC ARIMA DATA=OIL; IDENTIFY VAR=PCRUDE; RUN; Name of variable = PCRUDE. Mean of working series = 4173.5 Standard deviation = 2131.784 Number of observations = 108 Autocorrelations Lag Covariance Correlation -1 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 1 0 4544501 1.00000 | |********************| 1 4458346 0.98104 | . |********************| 2 4356540 0.95864 | . |******************* | 3 4244987 0.93409 | . |******************* | 4 4130834 0.90897 | . |****************** | 5 4014015 0.88327 | . |****************** | 6 3896140 0.85733 | . |***************** | 7 3780944 0.83198 | . |***************** | 8 3662695 0.80596 | . |**************** | 9 3541294 0.77925 | . |**************** | 10 3411041 0.75059 | . |*************** | 11 3267858 0.71908 | . |**************. | 12 3113620 0.68514 | . |************** . | 13 2957142 0.65071 | . |************* . | 14 2797785 0.61564 | . |************ . | 15 2638093 0.58050 | . |************ . | 16 2474790 0.54457 | . |*********** . | 17 2307640 0.50779 | . |********** . | 18 2139269 0.47074 | . |********* . | 19 1961795 0.43169 | . |********* . | 20 1776540 0.39092 | . |******** . | 1 4

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TS_Fall_2011_Lecture_3 - FIN 271 FINANCIAL MODELING AND...

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