Box-Jenkins. (1) Identification ( ) (2) Estimation ( ) (3) Diagnostic Checking ( ) (1) Identification: ARMA(p,q) p, q. (2) Estimation: ARMA(p,q)

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1 4 A RMA Box-Jenkins () Identification ( ) (2) Estimation ( ) (3) Diagnostic Checking ( ) () Identification: ARMA(p,q) p, q (2) Estimation: ARMA(p,q) φ(l)y t = m + θ(l)ε t φ = (φ,,φ p ) θ = (θ,,θ q ) m = σ 2 = ε t 2 NLSE) ε 2 t = { } 2 φ(l)y t m θ(l) Note. AR(p) 2 σ 2 MLE T 2 log(2πσ2 ) 2 log Σ 2σ 2 (y µ) Σ (y µ) ARMA φ, θ, m NLSE MLE T T ( ) N(, W ) W Fisher W = lim T E [ T 2 { β β 2σ 2 }] T ε 2 t t=, β =(φ,θ,m) 24

2 AR() y t = m + φ y t + ε t T ( ˆβ β) N(,W ), β =(φ,m) W = lim T E = lim T = lim T [ 2 { }] T (y T β β 2σ 2 t m φ y t ) 2 t= ( y Tσ E 2 ) t yt 2 yt T ( T (γ + µ 2 ) ) Tµ Tσ 2 Tµ T = σ 2 ( γ + µ 2 µ µ ) γ = V (y t )= W = σ2 φ 2, µ = E(y t )= m φ ( φ 2 ( φ 2 )µ ) ( φ 2 )µ σ 2 +( φ 2 )µ 2 MA() y t = m + ε t θ ε t T ( ˆβ β) N(,W ), β =(θ,m) [ 2 { }] T ε 2 T β β 2σ 2 t t= ( ( ) ) 2 εt = lim T Tσ E θ + 2 ε εt t ( 2 ε θ 2 εt t + ) εt ε t m θ m 2 ( ε 2 ε t t + ) εt ε t ( ( ) ) 2 εt m θ m m + 2 ε εt t m 2 ( ( ) ) 2 εt = lim T Tσ E θ 2 ( ( ) ) 2 εt m = V ( ) ε t θ L σ 2 = θ 2 ( θ ) 2 σ 2 ( θ ) 2 W = lim T E W = ( θ 2 ) σ 2 ( θ ) 2 (3) Diagnostic Checking: ARMA 25

3 ˆε t = ˆφ(L)y t ˆm ˆθ(L) ˆr h = Tt=h+ ˆε t hˆε t Tt= ˆε 2 t h ( ˆr h N, ) T 5% [.96,.96 ] T T Portmanteau tests Q = K T ˆr h 2 : h= Q = K T (T +2) h= Box-Pierce T h ˆr2 h : Ljung-Box Q Q K p q χ 2 AIC (Akaike s Information Criterion): AIC(p, q) = 2 +2(p + q) p q ARMA(p,q) Note. AIC(p,q) p, q {y t } DGP (Data Generating Process: ) AR(3) y t = m + φ y t + φ 2 y t 2 + φ 3 y t 3 + ε t m =.5, φ =.55, φ 2 =., φ 3 =.4, σ 2 = DGP 26

4 y t =.5+.55y t.y t 2 +.4y t 3 + ε t (T=) Y T ARIMA Procedure Name of variable = Y. Mean of working series = Standard deviation = Number of observations = Autocorrelations Lag Covar Corr ******************** 27

5 **************** ********* *** * * * * * * ** *** **** **** *** * * * ** ** **. "." marks two standard errors Inverse Autocorrelations Lag Correlation **************** ********* ***** ***** **** *** ** *** **..86. ** * * * ** ** * ** *** ****. 28

6 ***** ***** ***** *** *. Partial Autocorrelations Lag Correlation **************** ************ *** ** * * * ** ** * ** * * * ** ** ****. Autocorrelation Check for White Noise To Chi Autocorrelations Lag Square DF Prob AR(2) Conditional Least Squares Estimation Approx. 29

7 Parameter Estimate Std Error T Ratio Lag MU AR, AR, Constant Estimate = Variance Estimate = Std Error Estimate = AIC = * SBC = * Number of Residuals= * Does not include log determinant. Correlations of the Estimates Parameter MU AR, AR,2 MU AR, AR, Autocorrelation Check of Residuals To Chi Autocorrelations Lag Square DF Prob Model for variable Y Estimated Mean = Autoregressive Factors Factor : -.35 B**() B**(2) AR(3) Conditional Least Squares Estimation Approx. Parameter Estimate Std Error T Ratio Lag MU AR, AR, AR,

8 Constant Estimate = Variance Estimate = Std Error Estimate = AIC = * SBC = * Number of Residuals= * Does not include log determinant. Correlations of the Estimates Parameter MU AR, AR,2 AR,3 MU AR, AR, AR, Autocorrelation Check of Residuals To Chi Autocorrelations Lag Square DF Prob Model for variable Y Estimated Mean = Autoregressive Factors Factor : B**() B**(2) B**(3) ( AR(3) SAS filename out chap4-.dat ; data ar3; do i= to 2; 2 e=rannor( ); lags=sum(.55*lagy,-.*lag2y,.4*lag3y,e); y=.5+lags; lag3y=lag2y;lag2y=lagy;lagy=y; if i> then output; end; data b;set ar3; 3

9 file out;put y f7.3; chap4-.dat ( ARMA SAS options ls=65 ps=5; (ls= ps= ) filename in chap4-.dat ; filename graph chap4-.ps ; goptions device=pslepsfc gsfmode=replace gsfname=graph; goptions cback=yellow colors=(rose cyan yellow lilg rose); data ar3;infile in;input y; t=_n_; symbol i=join l= c=blue v=none; proc gplot;plot y*t; GPLOT title f=simplex h=.5 Data from AR(3) ; proc arima; ARIMA identify var=y; estimate p=2; AR(2) estimate p=3; AR(3) ( ARMA SHAZAM sample read(chap4-.dat) y chap4-.dat y genr t=time() t=,2,... arima y/plotac plotpac nowide (identification) arima y/nar=3 coef=beta resid=ry nowide (estimation) gen s=sqrt($sig2) arima y/nar=3 coef=beta fbeg= fend=2 sigma=s predict=py resid=res plot ry t /gnu lineonly device=postscript output=ar3.ps stop 32