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1 z c j = N 1 N t= j1 [ ( z t z ) ( )] z t j z

2 q 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

4 _ARIMA CONSUME / NLAG=8 NLAGP=8 PLOTAC PLOTPAC IDENTIFICATION SECTION - VARIABLE=CONSUME NUMBER OF AUTOCORRELATIONS = 8 NUMBER OF PARTIAL AUTOCORRELATIONS = SERIES (1-B) (1-B ) CONSUME NET NUMBER OF OBSERVATIONS = 17 MEAN= VARIANCE= STANDARD DEV.= LAGS AUTOCORRELATIONS STD ERR MODIFIED BOX-PIERCE (LJUNG-BOX-PIERCE) STATISTICS (CHI-SQUARE) LAG Q DF P-VALUE LAG Q DF P-VALUE LAGS PARTIAL AUTOCORRELATIONS STD ERR AUTOCORRELATION FUNCTION OF THE SERIES (1-B) (1-B ) CONSUME RRRRRRRRRRRRRRRRRRRRRRRRRR RRRRRRRRRRRRRRRRRRR RRRRRRRRRRRRRRR RRRRRRRR RRRRRR RR RRRRRRR RRRRRRRRRRR PARTIAL AUTOCORRELATION FUNCTION OF THE SERIES (1-B) (1-B ) CONSUME RRRRRRRRRRRRRRRRRRRRRRRRRR

5 2.01. R RRR RRRRRRRRR RRRRRRR RRRRRRRRR RRRRRRRRRR RRRRR. SAMPLE 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 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

6 _PRINT PACOLS PACREC PACDOLS PACDREC PACOLS PACREC PACDOLS PACDREC E E E E E E E E E E E E 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

7 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 < k k k S( β ˆ (i) ) S( β ˆ (i 1) )/ S( β ˆ (i 1) )<

8 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 )

9 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

10 ARIMA IINV / NAR=5 START RESTRICT BEG= END=

11 ARIMA CONSUME / NAR=1 NMA=1 START 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: MEAN OF SERIES = 134.5

12 VARIANCE OF SERIES = STANDARD DEVIATION OF SERIES = INITIAL SUM OF SQUARES = 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= FINAL SUM OF SQS= R-SQUARE =.7159 R-SQUARE ADJUSTED =.6753 VARIANCE OF THE ESTIMATE-SIGMA**2 = STANDARD ERROR OF THE ESTIMATE-SIGMA = AKAIKE INFORMATION CRITERIA -AIC(K) = SCHWARZ CRITERIA- SC(K) = PARAMETER ESTIMATES AR( 1) STD ERROR.2130 T-STAT MA( 1) E CONSTANT RESIDUALS LAGS AUTOCORRELATIONS STD ERR MODIFIED BOX-PIERCE (LJUNG-BOX-PIERCE) STATISTICS (CHI-SQUARE) LAG Q DF P-VALUE LAG Q DF P-VALUE CROSS-CORRELATIONS BETWEEN RESIDUALS AND (DIFFERENCED) SERIES CROSS-CORRELATION AT ZERO LAG =.41 LAGS 1-12 CROSS CORRELATIONS Y(T),E(T-K) LEADS CROSS CORRELATIONS Y(T),E(TK) zt z t 1 = ˆ u t ˆ u t

13 (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 ( )

14 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

15 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

16 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)= MA( 1)= CONSTANT = FROM ORIGIN DATE 14, FORECASTS ARE CALCULATED UP TO 5 STEPS AHEAD FUTURE DATE LOWER FORECAST UPPER ACTUAL ERROR STEPS AHEAD STD ERROR PSI WT VARIANCE OF ONE-STEP-AHEAD ERRORS-SIGMA**2 = STD.DEV. OF ONE-STEP-AHEAD ERRORS-SIGMA = 12.88

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y i = Z i δ i +ε i ε i δ X y i = X Z i δ i + X ε i [ ] 1 δ ˆ i = Z i X( X X) 1 X Z i [ ] 1 σ ˆ 2 Z i X( X X) 1 X Z i Z i X( X X) 1 X y i σ ˆ 2 ˆ σ 2 = [ ] y i Z ˆ [ i δ i ] 1 y N p i Z i δ ˆ i i RSTAT