Stationary Time Series

Transcription

1 3-Jul-3 Time Series Analysis Assoc. Prof. Dr. Sevap Kesel July 03 Saionary Time Series Sricly saionary process: If he oin dis. of is he same as he oin dis. of ( X,... X n) ( X h,... X nh) Weakly Saionary :{ X } is weakly saionary if i. ) E( X ) is independen of. x ( Cov( X, X ) ( r, s) E[( X ( r))( X ( s))] ii. r s x does no depend on. r x s x

2 3-Jul-3 Auocorrelaion Auocovariance Funcion (ACVF) ( h) Cov( X, X ) x h Auocorrelaion Funcion Properies:. ACF is an even funcion of he lag. ACF lies beween -, ( h) ( h) (0) ( h) ( h) ( h) Sample Auocovariance Funcion: Le X,X,..,X n be observaions of a series. Given he sample mean is X and n The sample auocovariance funcion h ˆ( h) ( X X )( X h n and he sample auocorrelaion funcion ˆ( h) ˆ( h) ˆ(0) he following is rue: Lemma: If he series is random, hen for large n, he sample ACF is approximaely normally disribued wih mean zero and variance (/n). ˆ( h) ~ N(0, ) n X )

3 3-Jul-3 Correlogram Correlogram is an aid o inerpre a se of ACF where, sample auocorrelaionsare ploed agains lag h. Remarks: For daa conaining rend will exhibi slow decay as h increases. For daa wih a periodic componen will exhibi similar behavior wih he same periodiciy. ˆ ~ N(0, ) k 3. Random Series: n 3.. If a ime series compleely random, for large n, 3.. For a random ime series If 95% of ˆ( h) he values lie wihin ime series is random. n ˆ( h) 0 Correlogram Shor-erm correlaion: Fairly large value of ˆ() is followed by or more coefficiens which is significanly smaller han zero, end o ge successively smaller. ˆ( h) ges o zero for large h. Alernaing series: Correlogram also ends o alernae. Non-saionary series: If he series conains a rend, ˆ( h) values will no come down o zero excep very large h. Trend should be removed firs. Seasonal flucuaions: Correlogram exhibi an oscillaion a he same frequency. If follows a sinusoidal paern, hen so does ˆ( h) 3

4 3-Jul Auocorrelaion Auocorrelaion Funcion for reacion im Auocorrelaion LBQ T Corr Lag LBQ T Corr Lag LBQ T Corr Lag Auocorrelaion Funcion for sales Examples o correlograms for differen series Auocorrelaion LBQ T Corr Lag LBQ T Corr Lag LBQ T Corr Lag LBQ T Corr Lag Auocorrelaion Funcion for raffic faa

5 3-Jul-3 Tesing Serial Correlaion DW -ρ When here is no serial correlaion, ρ=0 and DW saisic akes a value close o. Posiive serial correlaion produces a DW<, while negaive serial correlaion produces a DW>. DW n ( ) n 5

6 3-Jul-3 Example Tesing Serial Correlaion Pormaneau Tes: An imporan source of informaion in deecing he presence and form of serial correlaion is he correlogram. Qualiaive examinaion of he correlogram is an imporan diagnosic ool bu i does no consiue a formal saisical es. The Box-Pierce and is relaed es he Lung-Box es are boh pormaneu ess which allow us o es he hypohesis ha he firs h poins in he correlogram are random wih a rue value of zero. Box-Pierce es is defined as h Q n ˆ i i A beer sample saisics is Lung-Box saisics is h * Q n( n ) ˆ i i i ( ni) Q and Q* are disribued Chi-square wih degrees of freedom of h. 6

7 3-Jul-3 MA(q) Model Linear Process: X Whie Noise Process (WN): {X } is a sequence of i.i.d random variables wih zero mean and finie variance σ. The series is saionary wih γ(+h,)= σ for h=0. Whie Noise process is a purely random process where all auocorrelaion funcions for every h are close o zero. X i0 i, i 0... k k X The plo of index numbers having Whie Noise model 4 7

8 3-Jul-3 Random Walk Random Walk Model: Le {S, =,..,n} be a process wih S =Σ where is WN. Then, E[S ]=0, Var[S ]=.σ and γ(+h,)=.σ Since γ(+h,)=.σ depends on, he series is no saionary. However, is saionary. X 5 6 8

9 3-Jul-3 Backward shif operaor Backward Shif Operaor, B, is anoher form of expressing he series BX X B( BX ) BX X B X B X X B 0 MA(q) Model Le { }~WN(0,σ ) MA(q) is X q 0 q For q= MA() E[ ]=0 V[ ]= σ (+θ ) X ( h, ) x x ( h) 0 ( ) 0 h 0 h h h 0 h h 8 9

10 3-Jul-3 9 AR(p) Model and is auocorrelaion Le { }~WN(0,σ ) AR(p) is For p= AR() E[ ]=0 V[ ]= σ /(-Φ ) X X X X... p p X X h ( h x ) h ( h ) x 0 0

11 3-Jul-3 AR() Model From he oupu, he behavior of ACF for AR() is ails off; he behavior of PACF for AR() is cus off afer lag.

12 3-Jul-3 ARMA(p,q) Model Le { }~WN(0,σ ) ARMA(p,q) is X X X... X.. p p q q ( B)( X ) ( B) where B B B B ( )... ( B) B.. B q q p p ARMA(,) X ( X ) ( B)( X ) ( B) where ( B) B ( B) B 3 Parial Auocorrelaion PACF For an AR(p) process PACF, Φ hh is he correlaion beween X and X -h conrolling he effec of X -h- AR(): Φ = Φ=ρ() AR(): Φ = Φ=ρ() hh () () () 0 h 4

13 3-Jul-3 PACF AR(p) (); h h, () () ; (), hh h, h,,.., h hh ( h) h h h, h, ( ) ( ), h 3,4,... Yule Walker Equaions ( h) ( h ) ( h ) ( h p) ( h) ( h ) ( h ) p p ( h p) h 0 h 0 5 h ( ) for h 0 hh ( h) Parial Auocorrelaion for MA() process ( ) ( ) ( ) Asympoic disribuion of Parial Auocorrelaions For a causal AR(p) process, he asympoic disribuion n ˆ d N(0,) kk 6 3

14 X 3-Jul-3 Properies of he ACF and PACF for various ARMA Models Model ACF PACF AR() Exponenial or =0 for h> kk oscillaory decay AR() Exponenial or sine kk =0 for h> wave decay AR(p) Exponenial or sine kk =0 for h>p wave decay MA() h 0 =0 for h> Dominaed by damped exponenial MA() =0 for h> h 0 Dominaed by damped exponenial or sine wave MA(q) =0 for h>q h 0 Dominaed by linear combinaion of damped exponenial and/or sine waves ARMA(,) Tails off. Exponenial Tails off. Dominaed by exponenial decay from lag decay from lag ARMA(p,q) Tails off afer (q-p) Tails off afer (p-q) lags. Dominaed by lags. Exponenial damped exponenials and or sine waves and/or sine wave decay afer (p-q) lags afer (q-p) lags 7 Example: Yield Daa Yield Morgages-Yield on Gov. Loan Monhly) ACF year monhs h 7 9 lag Parial Auocorrelaion Funcion for lag h=,,3,..,0 h kk h kk PACF,00 0,80 0,60 0,40 0,0 0,00-0,0-0,

15 3-Jul-3 Table 6: Oupu of he daa se Dependen Variable: YIELD_DATA Mehod: Leas Squares Dae: 0//08 Time: 00:4 Variable Coefficien Sd. Error -Saisic Prob. C AR() R-squared Mean dependen var Adused 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) Yˆ Y 9 Saionary Condiions: Causaliy The esimaed values of parameers have cerain condiions. X X where X ( X ) afer subsiuions X ( B B...) X ( ) ( B) B In general, AR(p) is causal if he roos of Φ(B) lie in uni circle p p (.. p ) 0 X B B B B X ( ) ( B) B B 30 5

16 3-Jul-3 Inveribiliy The esimaed values of parameers have cerain condiions. X B ( B) X ( ) X ( B) B B In general, MA(q) is inverible if he roos of θ(b) lie in uni circle X... q q X B B.. B q q q q (.. q ) 0 X B B B B X ( ) ( B) B B 3 Exercise Dependen Variable: SERIES0 Variable Coefficien Sd. Error -Saisic Prob. AR() MA() MA() R-squared Mean dependen var Adused R-squared S.D. dependen var S.E. of regression.7754 Akaike info crierion Sum squared resid Schwarz crierion Log likelihood Durbin-Wason sa.9979 Invered AR Roos.85 Invered MA Roos

17 3-Jul-3 Dependen Variable: SERIES0 Variable Coefficien Sd. Error -Saisic Prob. AR() AR() MA() R-squared Mean dependen var Adused R-squared S.D. dependen var S.E. of regression.8997 Akaike info crierion Sum squared resid 3.76 Schwarz crierion Log likelihood Durbin-Wason sa Invered AR Roos.85. Invered MA Roos