z c j = N 1 N t= j1 [ ( z t z ) ( )] z t j z
q 2 1 2 r j /N j=1 1/ N J Q = N(N 2) 1 N j j=1 r j 2 2 χ J B d z t = z t d (1 B) 2 z t = (z t z t 1 ) (z t 1 z t 2 ) (1 B s )z t = z t z t s
_ARIMA CONSUME / NLAG=8 NLAGP=8 PLOTAC PLOTPAC IDENTIFICATION SECTION - VARIABLE=CONSUME NUMBER OF AUTOCORRELATIONS = 8 NUMBER OF PARTIAL AUTOCORRELATIONS = 8 0 0 0 SERIES (1-B) (1-B ) CONSUME NET NUMBER OF OBSERVATIONS = 17 MEAN= 134.51 VARIANCE= 555.89 STANDARD DEV.= 23.577 LAGS AUTOCORRELATIONS STD ERR 1-8.72.53.42.21.16.04 -.18 -.29.24 MODIFIED BOX-PIERCE (LJUNG-BOX-PIERCE) STATISTICS (CHI-SQUARE) LAG Q DF P-VALUE 1 10.59 1.001 LAG Q DF P-VALUE 5 22.43 5.000 2 16.64 2.000 6 22.48 6.001 3 4 20.64 21.74 3.000 4.000 7 8 23.53 26.51 7.001 8.001 LAGS PARTIAL AUTOCORRELATIONS STD ERR 1-8.72.01.06 -.24.17 -.22 -.26 -.12.24 AUTOCORRELATION FUNCTION OF THE SERIES 0 0 0 (1-B) (1-B ) CONSUME 1.72. RRRRRRRRRRRRRRRRRRRRRRRRRR 2.53. 3.42. RRRRRRRRRRRRRRRRRRR RRRRRRRRRRRRRRR 4.21. RRRRRRRR 5.16. 6.04. RRRRRR RR 7 -.18. 8 -.29. RRRRRRR RRRRRRRRRRR 0 0 0 PARTIAL AUTOCORRELATION FUNCTION OF THE SERIES (1-B) (1-B ) CONSUME 1.72. RRRRRRRRRRRRRRRRRRRRRRRRRR
2.01. R 3.06. RRR 4 -.24. RRRRRRRRR 5.17. RRRRRRR 6 -.22. RRRRRRRRR 7 -.26. RRRRRRRRRR 8 -.12. RRRRR. SAMPLE 1 462 ARIMA TBILL / NLAGP=6 PACF=PACREC DIM PACOLS 6 BETA 6 PACDOLS 6 DO #=1,6 *?OLS?OLS TBILL TBILL(1.#) / COEF=BETA GEN1 PACOLS:#=BETA:# ENDO * * ARIMA TBILL / NLAGP=6 PACF=PACDREC NDIFF=1 SAMPLE 2 462 GENR TBILLD=TBILL-LAG(TBILL) DO #=1,6?OLS TBILLD TBILLD(1.#) / COEF=BETA GEN1 PACDOLS:#=BETA:# ENDO * SAMPLE 1 6 PRINT PACOLS PACREC PACDOLS PACDREC
_PRINT PACOLS PACREC PACDOLS PACDREC PACOLS PACREC PACDOLS PACDREC.9853103.9849570.3211617.3210678 -.3292023 -.2853001 -.2370160 -.2369215.2268389.1769929 -.1310004E-02 -.1195282E-02 -.6588286E-02.1690130E-01 -.2410940E-01 -.2402594E-01.1629671E-01.5984359E-02.2512564E-01.2511990E-01 -.3295992E-01 -.2934608E-01 -.2968466 -.2964889 z t φ 1 z t 1 = u t θ 1 u t 1 δ φ θ δ φ (B) = 1 φ 1 B... φ p B p θ(b) = 1 θ 1 B... θ q B q φ(b)(1 B) d z t = θ(b) u t δ Γ(B s ) = 1 Γ 1 B s... Γ P B sp (B s ) = 1 1 B s... Q B sq
Γ(B s ) φ (B)(1 B) d (1 B s ) D z t = (B s ) θ (B) u t δ β = [ φ θ Γ δ] N S(β) = u t (β) 2 t=1 ˆ (i β ) ut ( ˆ ( i) ) [ u ( ˆ ( i) i h e ) u ( ˆ ( ) β β β )]/ h k = 1,..., K β k t k k t k h k = hβ ˆ (i) k ˆ ( i) ˆ ( i 1) ˆ ( i) β β / β <. 0001 k k k S( β ˆ (i) ) S( β ˆ (i 1) )/ S( β ˆ (i 1) )<.000001
SIGMA**2 σ ˆ 2 1 u = N K N u t (ˆ β ) 2 t=1 σ ˆ 2 u X β ˆ Xˆ 1 β x kt = u t β k (ˆ β ) 2 χ J K1 2 χ J K AIC(K) = log σ ˆ 2 () u 2 K/N SC(K) = log σ ˆ 2 u ( ) Klog(N )/N y t = (1 B)d (1 B s ) D z t u ˆ t j c (y ˆ u )j = 1 N N (y t t= j1 y ) ( ˆ u t j u )
N 1 y= N t= 1 y t 1 u = N N t= 1 uˆ t r (yˆ u )j = c (yˆ u )j / c (yy)0 c ( u ˆ u ˆ )0
ARIMA IINV / NAR=5 START RESTRICT BEG=1950.1 END=1985.4.5 0 0.1.1 5
ARIMA CONSUME / NAR=1 NMA=1 START 0.5-0.2 100 DIM ALPHA 3 GEN1 ALPHA:1= 0.5 GEN1 ALPHA:2= -.2 GEN1 ALPHA:3= 100 ARIMA CONSUME / NAR=1 NMA=1 START=ALPHA _ARIMA CONSUME / NAR=1 NMA=1 START=ALPHA ESTIMATION PROCEDURE STARTING VALUES OF PARAMETERS ARE:.50000 -.20000 100.00 MEAN OF SERIES = 134.5
VARIANCE OF SERIES = 555.9 STANDARD DEVIATION OF SERIES = 23.58 INITIAL SUM OF SQUARES = 17958.972 ITERATION STOPS - RELATIVE CHANGE IN EACH PARAMETER LESS THAN.1E-03 NET NUMBER OF OBS IS 17 DIFFERENCING: 0 CONSECUTIVE, 0 SEASONAL WITH SPAN 0 CONVERGENCE AFTER 30 ITERATIONS INITIAL SUM OF SQS= 17958.972 FINAL SUM OF SQS= 2527.1076 R-SQUARE =.7159 R-SQUARE ADJUSTED =.6753 VARIANCE OF THE ESTIMATE-SIGMA**2 = 166.02 STANDARD ERROR OF THE ESTIMATE-SIGMA = 12.885 AKAIKE INFORMATION CRITERIA -AIC(K) = 5.4650 SCHWARZ CRITERIA- SC(K) = 5.6121 PARAMETER ESTIMATES AR( 1).59682 STD ERROR.2130 T-STAT 2.802 MA( 1) -.94992.9190E-01-10.34 CONSTANT 55.041 28.77 1.913 RESIDUALS LAGS AUTOCORRELATIONS STD ERR 1-12 -.37.04.23 -.12.12 -.04.00 -.16.16 -.06 -.06 -.06.24 13-16 -.05.00 -.08 -.03.30 MODIFIED BOX-PIERCE (LJUNG-BOX-PIERCE) STATISTICS (CHI-SQUARE) LAG Q DF P-VALUE LAG Q DF P-VALUE 3 3.96 1.047 10 6.93 8.545 4 4.34 2.114 11 7.12 9.624 5 4.73 3.193 12 7.37 10.690 6 4.77 4.311 13 7.60 11.748 7 4.77 5.444 14 7.61 12.815 8 5.69 6.458 15 8.72 13.793 9 6.75 7.456 16 9.09 14.825 CROSS-CORRELATIONS BETWEEN RESIDUALS AND (DIFFERENCED) SERIES CROSS-CORRELATION AT ZERO LAG =.41 LAGS 1-12 CROSS CORRELATIONS Y(T),E(T-K).49.06.24.12 -.06.13 -.04 -.09 -.12 -.08 -.14 -.12 13-16 LEADS -.11 -.13 -.08 -.07 CROSS CORRELATIONS Y(T),E(TK) 1-12.04.26.10.08.06 -.07 -.11 -.04.01 -.15 -.03 -.21 13-16 -.11 -.04 -.15 -.05 zt.59682 z t 1 = ˆ u t.94992 ˆ u t 1 55.041
(1 Γ 1 B 4 )(1 φ 1 B) z t = u t δ z t Γ 1 z t 4 φ 1 z t 1 Γ 1 φ 1 z t 5 = u t δ ARIMA ZQ / NSPAN=4 NSA=1 NAR=1 z Tl l (l 1) ˆ z Tl ztl z Tl = φ 1 z Tl-1 u Tl θ 1 u Tl-1 δ z Tl = µ u Tl Ψ 1 u Tl-1 Ψ 2 u Tl-2... µ Ψ 1, Ψ 2,... z ˆ Tl = µ Ψ l u T Ψ l1 u T 1... e Tl = [ z Tl z ˆ Tl ]= u Tl Ψ 1 u Tl-1... Ψ l 1 u T1 2 [ ] = σ u 1 2 Ψ1 2...Ψl 1 Ve Tl ( )
β ˆ σ ˆ 2 u SIGMA**2 z ˆ T1 = φ ˆ 1 z T θ ˆ 1ˆ u T δ ˆ z ˆ T2 = φ ˆ 1ˆ z T1 δ ˆ z ˆ T3 = φ ˆ 1ˆ z T2 δ ˆ Ψ i PSI WT V ˆ [e T l ] z ˆ Tl 1.96 V ˆ e Tl [ ] 1/2
y ˆ Tl z ˆ Tl µ,σ2 µσ z Tl = exp(y Tl ) ˆ z Tl = exp y ˆ Tl 1 σ ˆ 2 2 u z ˆ Tl
GEN1 S=SQRT($SIG2) ARIMA CONSUME / NAR=1 NMA=1 COEF=BETA START=ALPHA GEN1 S=SQRT($SIG2) ARIMA CONSUME / NAR=1 NMA=1 COEF=BETA FBEG=14 FEND=19 SIGMA=S GNU SIGMA**2 _ARIMA CONSUME / NAR=1 NMA=1 COEF=BETA FBEG=14 FEND=19 SIGMA=S GNU ARIMA FORECAST PARAMETER VALUES ARE: AR( 1)=.59682 MA( 1)= -.94992 CONSTANT = 55.041 FROM ORIGIN DATE 14, FORECASTS ARE CALCULATED UP TO 5 STEPS AHEAD FUTURE DATE LOWER FORECAST UPPER ACTUAL ERROR 15 16 141.881 108.277 167.136 154.791 192.390 201.306 154.300 149.000-12.8356-5.79115 17 95.3940 147.424 199.453 165.500 18.0763 18 89.1687 143.027 196.885 19 85.9081 140.402 194.897 STEPS AHEAD STD ERROR PSI WT 1 12.88 1.0000 2 3 23.73 26.55 1.5467.9231 4 5 27.48 27.80.5509.3288 VARIANCE OF ONE-STEP-AHEAD ERRORS-SIGMA**2 = 166.0 STD.DEV. OF ONE-STEP-AHEAD ERRORS-SIGMA = 12.88