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# Class Note for STAT 635 at OSU 12

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COURSE
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Class Notes
PAGES
14
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KARMA
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Date Created: 02/06/15
STATISTICS 635 SUMMER 2005 NONSTATIONARY AND SEASONAL TIME SERIES MODELS ARIMA MODELS FOR NONSTATIONARY TIME SERIES EH So far7 we have considered the ARMA family of models7 which rested on the crucial assumption of stationarity EH We now consider a more general family that allows the modeling of nonstationary time series through the application of differencing EH The simplest example of a nonstationary time series transformed into a stationary one after differencing is the random walk encountered in chapter 1 Recall that7 we de ned the random walk Xt as Xt Xt1 Zt whereZt N WNO 02 Xt so de ned turned out to be a nonstationary ARC process However7 VXt with VXt Xt Xt1 is a stationary process7 being just the white noise Zt STATISTICAL ANALYSIS OF TIME SERIES 1 STATISTICS 635 SUMMER 2005 EH Consider the following nonseasonal model with trend Xt mt Yt where mt is a polynomial of order k and Yt is a stationary process D Xt is nonstationary since it has trend polynomial D With mt 60 lt kt we have VkXt kl k VkYt D VkXt is a stationary process with mean Sig7 since VkYt is stationary D The operation VkXt removed the trend and yielded a stationary time series that can be analyzed with the ARMA machinery De nition 1 Hal is a nonnegative integer then Xt is an ARIMApdq process if Yt 1 BdXt is a causal ARMAp q process lt is clear from the de nition that ltBgtXt E Bgtlt1 BgtdXt 909 Zt N WN0 02 33 Note ARIMA stands for Integrated ARMA If E VdXt u y 07 then the model can be written as B1 BdXt oz 6BZt whereault1 1 pgt STATISTICAL ANALYSIS OF TIME SERIES 2 STATISTICS 635 SUMMER 2005 EH Since zd has a unit root With multiplicity d7 the process is still nonstationary even if the roots of are different from 1 However7 VdXt is stationary EXAMPLE 1 Simulate an ARIMA110 With gb 09 and 02 1 X lt arimasimlistorder c110 at 09 n 200 INTUITIVE UNDERSTANDING ARIMA PROCESSES EH The need for ARIMA arises from the fact that the series Xt in itself is nonstationary EH The key idea is that repeated applications of differencing operator V 1 B ultinIately yields a transformed series Yt VdXt that exhibits stationarity EH ln the ARIMA equation ltBgtXt E Bgtlt1 BgtdXt 909 Zt N WN0 02 1 the integer d 2 O is the number of applications of differencing STATISTICAL ANALYSIS OF TIME SERIES 3 STATISTICS 635 SUMMER 2005 oquot mum Wm M Mm Sanale Acr mdmuumed on mumm Sam pm md 39umcm on mumm EH ARMApq processes studied earlier are stationary7 and oorre spond to d 0 ie no need to apply differencing to get stationarity EH ARIMA models are useful for representing data with trend D Why is this the case EH The parameters 9b 0 and 02 are based on the observed differences 1 BdXt PARAMETER ESTIMATION WITH R EH Based on the original series X we use p 17 d 1 and q O7 M1lt arimaX orderc110 We nd 83 089 with segi5 00818 Also7 8 2 089247 and AIC 55033 EH Based on the differenoed Yt VXt7 we use p 17 d O and q O7 M2lt arimaY orderc100 We nd 83 088 with segi5 00322 Also7 8 2 089067 and AIC 55192 STATISTICAL ANALYSIS OF TIME SERIES 4 STATISTICS 635 SUMMER 2005 EH The diagnostics checks in both cases support the plausibility of the Chosen model 51 uuuuuuuuuuuuuuu Is 51 uuuuuuuuuuuuuuuu ls quot lll I lllllllll HM I ll if llll ll lllil l lll lllll l lllll llll if w IH I l l ll l Hl u 1m 1m mm D mu 15D mm W m Acr anesduds I rLlunuanx smusuc rnmlueslnr Llunuranx smusuc Note lrI light of its PACF7 it is plausible to t an ARlt2gt to this clata7 ie M3lt arimaX orderc 2 O 0 The result of which is gzgl 189 with segi51 003107 Q32 O89 with segi52 003137 Also7 amp2 087457 arid AIC 55982 Note lrI factorized form7 this ARlt2gt is very similar to the ARIMA1 1 0 EH Clearly7 ARIMA1 1 O can be written as 1 08931 BXt Z Z N WNO 08924 EH The ARlt2gt can be written as 1 1893 08932Xt 1 08931 BXt Z with Z N WNO 08745 STATISTICAL ANALYSIS OF TIME SERIES 5 STATISTICS 635 SUMMER 2005 Problem The ARlt2gt tting assumes stationarity7 yet the resulting coef cients are very similar to those of the nonstationary generating process Fact From a sample of nite length7 it Will be extremely di icult to distinguish between a nonstationary process for which Which O and a process Which has very similar coe icients bur for Which gbquot has all of its zeros outside the unit circle Lesson Applying differencing until the resulting series has a sample ACF that decays rapidly EH The di erenced data can be tted by a lovv order ARMA process EH The autoregressive polynomial gbquot Will have zeros that are comfort ably outside the unit circle as desired IDENTIFICATION TECHNIQUES EH Preliminary transformations These are mainly aimed at forming sta tionary series7 since the techniques described assume the stationarity of the process being studied Typical easily detectable indicators of nonstationarity are D Unstable variance D Trend D Seasonal variations First7 here are some variance stabilizing transformations STATISTICAL ANALYSIS OF TIME SERIES 6 STATISTICS 635 SUMMER 2005 D Logarithmic transformation This is one of the more common transformations used whenever Ut is a series whose standard deviation tends to increase linearly with the mean Vt ln Ut D General Box Cox transformation WT Ut 2 O gt 07 ant Ut gt O O fAltUtgt Transformations for elimination of trend and seasonal variations through D Classical decomposition D Differencing D Fitting of a sum of harmonics and a polynomial EH Model Identification and Estimation Let Xt be the mean corrected transformed time series obtained from preliminary transformations E Main guiding question What is the most satisfactory ARMAQ q for Xt 9 If p and q are known7 then simply estimate 9b 0 and 02 D Model size determination In general however7 we will need to nd p and q from the data by minimizing 2p q 1n W Note Some software packages allow the speci cation of a range AICC 21nL p 9g snip Hqn for p and q STATISTICAL ANALYSIS OF TIME SERIES 7 STATISTICS 635 SUMMER 2005 EH Identification and estimation in practice 1 0051 Plot your data and inspect to determine the type of preliminary transformation that is appropriate e g What is a reasonable value of d Transform the data is necessary Plot ACF and PACF to get rough estimates of p and or q Note You may need to repeat and several times before nding an ACF andor PACF that exhibits a rapidly decaying behavior which is our indicator of the reasonahility of the assumption of stationarity Choose a range for p and q if necessary Estimate the parameters 7 0 and 02 for each plausible model Perform a residual analysis for each plausible model Narrow everything down to models with good diagnostic check Select the model with the lowest AIC Perform prediction based on this best model Caution When performing differencing7 one should be careful as some correlation may be unnecessarily introduced as a result of overdifferenc ing For example7 from an uncorrelated th7 we get VZt Zt Zt1 that is correlated STATISTICAL ANALYSIS OF TIME SERIES 8 STATISTICS 635 SUMMER 2005 SEASONAL ARMA MODELS EH Consider the following seasonal model Without trend Xt 5t Yt Where is a stationary process St has the rst property 5t Stirs Where 5 denotes the length of the seasonal period Also St is i Sj O j1 EH Recall the lag 5 differencing operator V5 We have Vth Xt XFS Xt BSXt 1 BSXt EH Applying V5 to Xt de ned above7 we get Vth 5t Yt Stirs K75 VsYt EH The seasonal ARMA Inodel denoted by ARMAltP Q57 With BSXt BSZt Where 132 1 ltIgt1z ltIgt2z2 ltIgtpzp and 921 1z 2z2 QZQ are respectively7 the seasonal AR operator and the seasonal MA operator7 With period 5 STATISTICAL ANALYSIS OF TIME SERIES 9 STATISTICS 635 SUMMER 2005 CAUSALITY AND INVERTIBILITY OF SEASONAL ARMA EH Just like with ARMAQ q7 the ARMAP Q5 model is causal only if the roots of zs lie outside the unit circle EH ln the same way7 the ARMAP Q5 model is invertible only if the roots of zs lie outside the unit circle EXAMPLE OF SEASONAL ARMA EH Example 1 Consider the seasonal ARMA1 D12 We can write 1 B Xt 1 B Zt or equivalently Xt DXtilQ Z Zt712 EH For our seasonal ARMAltL 112 to be causal7 we require ltlgt lt 17 and for invertibility7 we require lt 1 EH When we studied the nonseasonal ARMA1 1 process we had Xt CbXtil Zt 0Zt71 Note The difference here is that lag 1 is now replaced by lag 12 EB A seasonal ARMAO Q5 reduces to a seasonal lIAQS7 and A seasonal ARMAP OS reduces to a seasonal ARltPgtS STATISTICAL ANALYSIS OF TIME SERIES 10 STATISTICS 635 SUMMER 2005 MORE EXAMPLES OF NONSEASONAL ARMA EH Example 2 Compute the ACVF and the ACF of the following seasonal MAlt1gt12 With simple calculations we get for 739 0 77 for 739 i otherwise and mi EH Example 3 Compute the ACVF and the ACF of the following seasonal ARlt1gtS Xt Xtis Zt Calculations similar to those used for nonseasonal ARlt1gt7 yield for 739 0 77 for 739 i otherwise and therefore for 739 0 97 for 739 i otherwise STATISTICAL ANALYSIS OF TIME SERIES 11 STATISTICS 635 SUMMER 2005 SUMMARY OF ACF AND PACF BEHAVIORS FOR SEASONAL ARMA ARPs MAQS ARMMR QL ACF Tails off at lags k5 Cuts off after lags Q5 Tails off at lags k5 PACF Cuts off after lags P5 Tails off at lags k5 Tails off at lags k5 EH The values of ACF and PACF are zero at non seasonal lagS 739 y k5 EH Here7 5 is the length of the period7 and k 12 MIXED SEASONAL ARMA MODELS EH The seasonal and non seasonal ARMA models can be combined The resulting model has equation ltIgtltBSgt ltBgtXI 9ltBsgt6ltBgtz EH This resulting model is called the mixed seasonal ARlVIA7 and is denoted by ARMAm 1 X P Qgts EH The behavior of a mixed seasonal ARMA model is a combina tion of the the behaviors of its seasonal and nonseasonal constituents EH EXAMPLE Consider the model ARMAO1 gtlt 1012 D Write down the equation of this model E Find the expression of the ACF When ltlgt lt l and 6 lt l STATISTICAL ANALYSIS OF TIME SERIES 12 STATISTICS 635 SUMMER 2005 SEASONALAEUAAABACDELS De nition 2 If d and D are nonnegative integers then Xt is a seasonal ARIMAp d q x P D Q process With period 5 if the di erenced time series Yt 1 Bd1 BSDXt is a causal ARMA process de ned by BltIgtBSYt 6BBSZt Z N WNO 02 2 Where gt1 2 W 132 1 ltIgt1z ltIgt2z2 ltIgtpzp 621 61z 62z2 6qu 921 1z 222 QZQ NOTEWORTHY ASPECTS OF SARIMA PROCESSES EH The process is causal If and only if y O and z y O for z g 1 EH ln practice7 the value of D is rarely more than 17 and P and Q are typically less than 3 EH Seasonal ARIMA models allow for for randomness in the seasonal pattern from one cycle to the next7 making it more general than the decomposition seen earlier STATISTICAL ANALYSIS OF TIME SERIES 13 STATISTICS 635 SUMMER 2005 SARIMA MODELS FOR MONTHLY DATA OVER YEARS YearMonth 1 2 12 1 Y1 Y2 YI2 2 Y13 Y14 Y24 3 325 Y26 Y36 7quot lfl12r71 B12r71 quot YI2r EH Each column of the table may itself be Viewed a realization of a time series7 so that we have a total of 12 such realizations EH Suppose that each of the 12 time series is generated by the same ARMAltP model7 ie for t O 7quot 17 Yj12t 1lg12t71 39 39 39 DPYj12t7P Uj12t 1Uj12t71 39 39 39 QUj12tiQ 3 where Uj12t71gt t gt1gtOgt1gtquot39 EH Since the same ARMAltP model is assumed for j 1 127 we can rewrite 3 as Bmm enemm 5 Uj112t1 t 10 1 N WNO02 for each j with j 1 2 12 5 will be referred to as the between year model STATISTICAL ANALYSIS OF TIME SERIES 14

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