Nonstationarity-Integrated Models. Time Series Analysis Dr. Sevtap Kestel 1

Transcription

1 Nonsaionariy-Inegraed Models Time Series Analysis Dr. Sevap Kesel 1

2 Diagnosic Checking Residual Analysis: Whie noise. P-P or Q-Q plos of he residuals follow a normal disribuion, he series is called a Gaussian process. 2. Overfiing : To conclude which model explains he series beer we Akaike s Informaion Crierion (AIC) and Schwarz s Informaion Crierion (SIC) are compared for each model. The model having smaller value of AIC or SIC proposes a beer fi.

3 Expeced Normal Value Normaliy es by graph (page 7) 9 Normal Q-Q Plo of Wai ime in minues Observed Value

4 AIC and SIC AIC 2 n 2k lnˆ k n Beer for small samples SIC kln n lnˆ 2 k n Beer for large samples

5 Exercise 12 Dependen Variable: SERIES01 Dependen Variable: Difference(SERIES01) Variable Coefficien Sd. Error -Saisic Prob. Variable Coefficie n Sd. Error -Saisic Prob. C AR(1) AR(2) AR(1) MA(1) R-squared Mean dependen var R-squared Mean dependen var Adjused R-squared S.D. dependen var S.E. of regression Akaike info crierion Sum squared resid 6.99E+08 Schwarz crierion Log likelihood F-saisic Durbin-Wason sa Prob(F-saisic) Invered AR Roos Adjused R-squared S.D. dependen var S.E. of regression Akaike info crierion Sum squared resid 6.86E+08 Schwarz crierion Log likelihood Durbin-Wason sa Invered AR Roos.77 Invered MA Roos.27

6 Difference funcion Used as a filer o randomize he series X = X - X -1 =(1-B) X 2 X = (X - X -1 )-(X -1 - X -2 )=(1-B) 2 X d X = (1-B) d X

7 ARIMA(p,d,q) Seps: 1. Take he d-difference of he series 2. Fi an appropriae sochasic model ARIMA(p,0,q) ARMA(p,q) ARIMA(0,d,q) IMA(d,q) ARIMA(p,d,0) ARI(p,d)

8 Example

9 Example The daa are Real U.S. Gross Naional Produc in billions of chained 1996 dollars and hey have been seasonally adjused. The daa were obained from he Federal Reserve Bank of S. Louis.

10 The sample ACF and PACF of he quarerly growh rae are ploed in he upper figure. Inspecing he sample ACF and PACF, we migh feel ha he ACF is cuing off a lag 3 and he PACF is ailing off.

11 Coefficiens: MA(1) MA(2) MA(3) inercep s.e The variance =8.853e-05: log likelihood = AIC = Coefficiens: AR(1) inercep s.e Variance =9.03e-05: log likelihood = , AIC =

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13 UNIT ROOT 1 Y = 0,8y -1 1 Y = -0,8y -1 0,9 0,8 0,8 0,6 0,7 0,4 0,6 0,2 0,5 0 0,4-0,2 0,3-0,4 0,2-0,6 0,1-0, j j 18 Y = 1,1y Y = -1,1y j j

14 NON-STATIONARY TIME SERIES MODELS Non-consan in mean Non-consan in variance Boh 14

15 NON-STATIONARITY IN MEAN Deerminisic rend Derending Sochasic rend Differencing 15

16 DETERMINISTIC TREND A deerminisic rend is when we say ha he series is rending because i is an explici funcion of ime. Using a simple linear rend model, he deerminisic (global) rend can be esimaed. This way o proceed is very simple and assumes he paern represened by linear rend remains fixed over he observed ime span of he series. A simple linear rend model: Z Z 16

17 DETERMINISTIC TREND The parameer measure he average change in X from one period o he anoher: X X X X 1 1 Z Z 1 E The sequence {X } will exhibi only emporary deparures from he rend line +. This ype of model is called a rend saionary (TS) model. 17

18 TREND STATIONARY If a series has a deerminisic ime rend, hen we simply regress X on an inercep and a ime rend (=1,2,,n) and save he residuals. The residuals are derended series. If X is sochasic, we do no necessarily ge saionary series. 18

19 DETERMINISTIC TREND Many economic series exhibi exponenial rend/growh. They grow over ime like an exponenial funcion over ime insead of a linear funcion. For such series, we wan o work wih he log of he series: ln So E X Z he average growh lnx rae is : 19

20 DETERMINISTIC TREND Sandard regression model can be used o describe he phenomenon. If he deerminisic rend can be described by a k-h order polynomial of ime, he model of he process 2 X k where ~ 2 Z WN 0,. z k Z 20

21 DETERMINISTIC TREND This model has a shor memory. If a shock his a series, i goes back o rend level in shor ime. Hence, he bes forecass are no affeced. Rarely model like his is useful in pracice. A more realisic model involves sochasic (local) rend. 21

22 STOCHASTIC TREND A more modern approach is o consider rends in ime series as a variable. A variable rend exiss when a rend changes in an unpredicable way. Therefore, i is considered as sochasic. 22

23 STOCHASTIC TREND Recall he AR(1) model: X = c + X 1 + Z. As long as < 1, everyhing is fine (OLS is consisen, -sas are asympoically normal, ec.). Now consider he exreme case where = 1, i.e. Y = c + Y 1 + Z. Where is he rend? No erm. 23

24 STOCHASTIC TREND Le us replace recursively he lag of Y on he righ-hand side: 24 i i Z X c Z Z X c c Z X c X Deerminisic rend This is wha we call a random walk wih drif. If c = 0, i is a random walk.

25 STOCHASTIC TREND Each Z i shock represens shif in he inercep. Since all values of {Z i } have a coefficien of uniy, he effec of each shock on he inercep erm is permanen. In he ime series lieraure, such a sequence is said o have a sochasic rend since each z i shock impars a permanen and random change in he condiional mean of he series. To be able o define his siuaion, we use Auoregressive Inegraed Moving Average (ARIMA) models. 25

26 DETERMINISTIC VS STOCHASTIC TREND They migh appear similar since hey boh lead o growh over ime bu hey are quie differen. To see why, suppose ha hrough any policies, you go a bigger X because he noise a is big. Wha will happen nex period? Wih a deerminisic rend, X +1 = c +(+1)+Z +1. The noise Z is no affecing X +1. Your supendous policy had a one period impac. Wih a sochasic rend, X +1 = c + X + Z +1 = c + (c + X 1 + Z ) + Z +1. The noise Z is affecing X +1. In fac, he policy will have a permanen impac. 26

27 DETERMINISTIC VS STOCHASTIC TREND Conclusions: When dealing wih rending series, we are always ineresed in knowing wheher he growh is a deerminisic or sochasic rend. There are also economic ime series ha do no grow over ime (e.g., ineres raes) bu we will need o check if hey have a behavior similar o sochasic rends ( = 1 insead of < a, while c = 0). A deerminisic rend refers o he long-erm rend ha is no affeced by shor erm flucuaions in he economy. Some of he occurrences in he economy are random and may have a permanen effec of he rend. Therefore he rend mus conain a deerminisic and a sochasic componen. 27

28 Trend saionary ime series are no mean saionary bu include a rend. Including a rend componen ino he regression model. Differencing his series increases variance of he error erm Y = a + b +βx +. Difference saionary ime series (which are mos of economic ime series) conain a sochasic rend, differencing resuls in a saionary ime series. Therefore: a linear process has a uni roo if 1 is a roo of he process's characerisic equaion which leads nonsaionariy.

29 RANDOM WALK PROCESS A random walk is defined as a process where he curren value of a variable is composed of he pas value plus an error erm defined as a whie noise (a normal variable wih zero mean and variance one). ARIMA(0,1,0) PROCESS X where X 1 Z Z X 2 ~ WN 0,. z 1 B X Z 29

30 RANDOM WALK PROCESS Behavior of sock marke. Brownian moion. Movemen of a drunken men. I is a limiing process of AR(1). 30

31 RANDOM WALK PROCESS The implicaion of a process of his ype is ha he bes predicion of Y for nex period is he curren value, or in oher words he process does no allow o predic he change (X -X -1 ). Tha is, he change of X is absoluely random. I can be shown ha he mean of a random walk process is consan bu is variance is no. Therefore a random walk process is nonsaionary, and is variance increases wih. In pracice, he presence of a random walk process makes he forecas process very simple since all he fuure values of X +s for s > 0, is simply X. 31

32 RANDOM WALK PROCESS 32

33 RANDOM WALK PROCESS 33

34 RANDOM WALK WITH DRIFT Change in X is parially deerminisic and parially sochasic. X X 0 1 X I can also be wrien as Z 0 X X 0 Z Pure model of a rend i deerminis i1 rend ic sochasic rend (no saionary componen) 34

35 RANDOM WALK WITH DRIFT Z Z 0 0 E Afer periods, he cumulaive change in X is 0. E X X X s fla s 0 no Each Z i shock has a permanen effec on he mean of X. 35

36 RANDOM WALK WITH DRIFT 36

37 Dickey Fuller Tes: Consider X X Z 1 Tes H 0 :φ=1 (here exiss uni roo) H a :φ<1 (he process is saionary) Augmened Dickey Fuller Tes X X X... X Z * p p1 Tes H 0 :φ*=0 (here exiss uni roo) H a :φ*<0 (he process is saionary) Augmened Dickey Fuller Tes Criical values are deermined by sandard -ables which may no be useful. MacKinnon (1991) developed a simulaion model o deermine he criical values fro arbirary sample sizes.

38 Sequenial es procedure: Sar wih a relaively high number of lags, such as 10 Subsequenly, reduce he number of lags unil he las coefficien is significan differen from zero a 10 % level of significance. Compare he models (wihou drif and rend, wih drif, and wih drif and rend) by looking a he Akaike crierion. Then choose he model having he lowes Akaike crierion. If he value of he es saisic is greaer han (or in absolue values lesser han) he criical value, fail o rejec he exisence of uni roo. The Phillips-Peron Tes: A nonparameric mehod of conrolling for higher order serial correlaion in he series. The es saisic follows a - disribuion asympoically. Mac Kinnon able values are also used o es uni roo.

39 Example: Annual GDP of Counry XYZ GDP 2,000 1,600 DLG 1, LGDP

40 ACF of log(gdp) ACF of differenced log(gdp)

41 Uni roo es for log(gdp) Uni roo es for differenced log(gdp)

42 AR(1)??? log(gdp) ACF of Residuals and is Plo LGDP Residuals

43 ARIMA(1,2)?? for diff(log(gdp)) ACF of resiuals and plo DLGDP Residuals

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45 Mean VAR00003 Srucural Break- Chow Tes F ( ) 1 ( T 2 k)( ee) ( ee) k H 0 : No srucural break a ime Chow Breakpoin probabiliy Tes F-saisics Log Likelihood raio ime

46 Example: Yearly elecric use of a cerain plan has been recorded. ELECTRIC_USE DELECTRIC_USE

47 ACF of series ACF of differenced series

48 Uni roo Tes of he series Null Hypohesis: ELECTRIC_USE has a uni roo Exogenous: Consan Lag Lengh: 11 (Auomaic based on SIC, MAXLAG=12) -Saisic Prob.* Augmened Dickey-Fuller es saisic Tes criical values: 1% level % level % level ,1096>0.05 so we can no rejec he null hypohesis. The uni roo exiss. We ake he difference of he series.

49 Uni roo of he differenced daa Null Hypohesis: D(ELECTRIC_USE) has a uni roo Exogenous: Consan Lag Lengh: 10 (Auomaic based on SIC, MAXLAG=12) Augmened Dickey-Fuller es saisic -Saisic Prob.* Tes criical values: 1% level 5% level % level So he differenced series is saionary. To selec he model, we have wo opions: i)to look a he ACF and PACF graphs o see he cu-off ii) Compare he possible models by looking a he Akaike Info Crierion

50 MODEL 1: Dependen Variable: DIF_ELECTRIC_USE Convergence achieved afer 3 ieraions Variable Coefficien Sd. Error -Saisic Prob. C AR(1) AR(2) AR(3) AR(12) R-squared Mean dependen var Adjused R-squared S.D. dependen var S.E. of regression Akaike info crierion Sum squared resid Schwarz crierion Log likelihood F-saisic Durbin-Wason sa Prob(F-saisic) Invered AR Roos i i i i i i i i i i -.97

51 Model 2: Variable Coefficien Sd. Error -Saisic Prob. C AR(1) AR(2) AR(12) MA(1) R-squared Mean dependen var Adjused R-squared S.D. dependen var S.E. of regression Akaike info crierion Sum squared resid Schwarz crierion Log likelihood F-saisic Durbin-Wason sa Prob(F-saisic) All coefficiens are significan. We mosly ge rid of he serial correlaion. So his model is appropriae

52 Residual analysis of Model DELECTRIC_USE Residuals

53 Residual analysis of Model DELECTRIC_USE Residuals

54 NON-STATIONARITY IN VARIANCE Saionariy in mean Non-saionariy in mean variance Saionariy in variance Non-saionariy in If he mean funcion is ime dependen, 1. The variance, Var(X ) is ime dependen. 2. Var(X ) is unbounded as. 3. Auocovariance and auocorrelaion funcions are also ime dependen. 4. If is large wr X 0, hen k 1. 54

55 VARIANCE STABILIZING TRANSFORMATION The variance of a non-saionary process changes as is level changes Var X c. f for some posiive consan c and a funcion f. Find a funcion T so ha he ransformed series T(X ) has a consan variance. The Dela Mehod 55

56 VARIANCE STABILIZING TRANSFORMATION Generally, we use he power funcion T 1 X (Box and Cox, 1964) X Transformaion 1 1/X 0.5 1/(X ) ln X 0.5 (X ) X (no ransformaion) 56

57 VARIANCE STABILIZING TRANSFORMATION Variance sabilizing ransformaion is only for posiive series. If your series has negaive values, hen you need o add each value wih a posiive number so ha all he values in he series are posiive. Now, you can search for any need for ransformaion. I should be performed before any oher analysis such as differencing. No only sabilize he variance bu also improves he approximaion of he disribuion by Normal disribuion. 57