Class Note for STAT 635 at OSU 11
Class Note for STAT 635 at OSU 11
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Date Created: 02/06/15
STATISTICS 635 SUMMER 2005 FORECASTING ARMA PROCESSES As A Man Thinketh in His Heart So Is He Proverbs 23 7 ACF AND PACF OF ARMA PROCESSES EH What Is the pattern of ACF and PACF for MAltqgt processes W n2 nn n2 no nn nn H I I I I I I P am n2 nn n2 no W n2 nn n2 no nn nn H Pam A n n2 no Figure 2 ACF and PACF MA2 with X Z 0724 0122 EH Remark The ACF of an MAq tends to cut off after lag q7 while Its PACF tends to tail off STATISTICAL ANALYSIS OF TIME SERIES 1 STATISTICS 635 SUMMER 2005 ACF AND PACF OF ARMA PROCESSES EH What Is the pattern of ACF and PACF for ARltpgt processes W ACF EH Remark The ACF of an ARltpgt tends to tail off7 While Its PACF tends to cut off after lag p AM I11 In no mi U 00 02 04 06 GB in 402 ACF furAR1 m an o 5 7 lilhiiiliiilll i VVVVV quotH VVVVVVVVVVVVVVVVVVVVVVV 7 I JIC JLIL1IJiiiiiiiiil iiiiiiiiiiiwuiilj Paw AIM III III III I no as FACF AR1wth an o 5 Figure 3 ACF and PACF of an AR1 process X 05xt1 Z AR2 with phi 1 o7 and phi 2 01 AR2with phi 1 o7 and phi 2 01 liif hilw 777 39IIITCH 777 quotiii 777 Tilt WUt i D In 4D EEI an inn Lag Famai ACF Ga 04 02 an Figure 4 ACF and PACF of an AR2 process X 07xt1 7 01xt2 Zt Note The PACF of MA models tends to behave llhe the ACF of AR models whlle the PA CF of AR models tends to behave llhe the ACF of MA models STATISTICAL ANALYSIS OF TIME SERIES STATISTICS 635 SUMMER 2005 ACF AND PACF OF ARMA PROCESSES EH What Is the pattern of ACF and PACF for ARMAQ q processes nu nn n2 no Inn on PM no n2 nn n2 no nn I Anz Figure 5 ACF and PACF of an ARMA27 2 process xt708xt1701xt2 zt07zt101zt2 ACF of an ARMA22 PACF of an ARMA22 on ACF 05 PnIACF 02 an I nn n4 I Ans Lag Lag Figure 6 ACF and PACF of an ARMA27 2 process xt07xt101xt2 zt7082t1701ztg EH Remark The ACF of an ARMAQ q tends to tail off7 and Its PACF also tends to tail off STATISTICAL ANALYSIS OF TIME SERIES 3 STATISTICS 635 SUMMER 2005 Example A LOOK AT THE FAMOUS SUNSPOTS DATA Pim uIAnnuaI sunspms Imm 177EHEE9 ACF ofthe Annual SunSpots Data from 1770 1569 In ACF a 4 I U 2 I I I I n In 2D I An Lag Figure 7 Plot and ACF Of the yearly sunspots from 1770 1869 EH The ACF strongly suggests a lingering dependency EH Plausible tO assume underlying autoregressive mechanism PACF of the Annual SunSpots Data from 17701869 U 5 I Partial ACF an 405 I Lag Figure 8 PACF Of the yearly sunspots from 1770 1869 EH What does the PACF suggest STATISTICAL ANALYSIS OF TIME SERIES 4 STATISTICS 635 SUMMER 2005 FORECASTING AN ARMAp q PROCESS EH Given the series up to time point 717 ie XT Xn X714 Xl EH Predict a future value of Xnh for h 1 2 EH We developed the Best Linear Predictor BLP in Chapter 2 131th a0 aan aan1 anXl n a0 Z az39XnJrliz39 21 The two determining equations of BLP can be summarized as E Error gtlt PredictorVariable 0 EH We introduced the Durbin Levinson Algorithm D Naturally suitable for ARltpgt processes D Provides an estimate of the PACF as its byproduct D Computationally e icient Provides an 9012 solution where di rect methods would require 9013 D Useful for estimating the order of an ARp EH We introduced the Innovations Algorithm D Strength More general7 does not require stationarity D Naturally suitable for MAltqgt processes B Good for ARMA models With q gt O D Can be made even better See textbook STATISTICAL ANALYSIS OF TIME SERIES 5 STATISTICS 635 SUMMER 2005 GENERAL APPROACH To FITTING ARMA PROCESSES EH Obtain the observations EH Do the observations come from a stationary process D Does the ACF decay to zero rapidly D lf need be7 perform transformation to obtain stationarity EH What is the order of the process D What is the autoregressive order p D What is the moving average order q EH Assuming p and q known and xed7 we have to D Estimate Q51Q52quot39 ng7 then 61 62 g and 02 D Total of p q 1 parameters EH How do we estimate the parameters of the ARMAQ q E Maximum Likelihood Estimation D Numerically requires good preliminary estimates b Durbin Levinson for pure autoregressive processes b Innovations vvhen q gt O in ARMAQ q EH Remember to perform goodness of t Remark Since the methods assume zero mean processes7 it is good practice to subtract the sample mean out prior to using the technique STATISTICAL ANALYSIS OF TIME SERIES 6 STATISTICS 635 SUMMER 2005 REVISITING THE DURBIN LEVINSON ALGORITHM Step 1 Initialize 2500 O and 1310 70 Step 2 For n 2 17 calculate p01 3 9257mm if Cbnn mil 1 1 Z Cbnil puw k1 Where7 for n 2 2 71k Cbnil Cbnngbnil ik Step 3 For n 2 17 calculate 1321 P 1lt1 in lt3gt WHAT IS THE ALGORITHM REALLY DOING Goal Solve an ARltpgt problem D L proceeds as follows EH Solve ARC problem rst EH Then solve the larger ARlt2gt problem7 using results from AR1 EH Then solve the larger ARlt3gt problem7 using results from AR2 EH So on EH Solve the original ARltpgt problem7 using results from ARltp I STATISTICAL ANALYSIS OF TIME SERIES 7 STATISTICS 635 SUMMER 2005 Exercise FINDING SOME USEFUL ACF EH Consider the causal ARlt2gt process Xt 1Xt71 2Xt72 Zt EH Find Its ACF using the method of difference equation STATISTICAL ANALYSIS OF TIME SERIES 8 STATISTICS 635 SUMMER 2005 Exercise PREDICTING AN AR2 PROCESS EH Consider the causal ARlt2gt process Xt 1Xt71 2Xt72 Zt EH Expressing in matrix vector fornI gives 71 70 71 71 2 71 70 72 1 vlt0 vlt1gt 71 117 2 l p 17p A Pltf3 gtgt EH The ACF derived earlier to simplify the above solution EH What do you conclude EH Apply the Durbin Levinson algorithm and compare STATISTICAL ANALYSIS OF TIME SERIES 9 STATISTICS 635 SUMMER 2005 PREDICTING h STEP AHEAD EH Assume that we still have our 71 observations Xm X714 Xl EH The best linear prediction h step ahead is given by X72731 hl an71quot39 Where a all ax aw and 779 Numb 1gt Wt n 1gtgtT and all satisfy anbff 75 EB The mean square error in this case is given by P531 E Xnm X512 70 lt7 MgtTF7217 M EH What is the key difference between the h step ahead and the one step ahead Each one has to find his peace from within And peace to be real must be unaffected by outside circumstances Mahatma Gandhi STATISTICAL ANALYSIS OF TIME SERIES 10
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