CIRJE-J-58 X-12-ARIMA 2000 : 2001 6
How to use X-12-ARIMA2000 when you must: A Case Study of Hojinkigyo-Tokei Naoto Kunitomo Faculty of Economics, The University of Tokyo Abstract: We illustrate how to use the X-12-ARIMA program developed by the U. S. Census Bureau when you have to make seasonal adjustment data at the statistical division of the central government. As an i llustration we use the Hojinkigyo -Toukei, which is one of the major statistics including sales and investments data by corporate firms in Japan. We shall discuss reasonable ways to use and/or not to use the procedures available in the X-12-ARIMA (2000) pr ogram.
X-12-ARIMA(2000) 2001 5 X-11 X-12-ARIMA DECOMP X-12-ARIMA(2000) X-11X-12-ARIMA(2000)DECOMP 1
1. X-11 X-12-ARIMA () X-12-ARIMA2000 X-12-ARIMA(2000) 2
X-12-ARIMA X-11, X-12-ARIMA, Decomp 3
2. X-12-ARIMA X-12-ARIMA 1 X-12-ARIMA(2000) X-12-ARIMA(2000) X-12 Experimental Version X-12-ARIMA(β Version) β Version 2 X-12-ARIMA X-12-ARIMA(1998) β Version X- 12-ARIMA Version 0.2.7 X-12-ARIMA(2000) (2001) U.S.Census Bureau (2000) (2001) X-12-ARIMA X-12-ARIMA(β Version) (1997) (1997) 1 (2001) 2 X-12-ARIMA Corrections() 4
X-12-ARIMA Findley et.al. (1998) (Technical Report) X-12-ARIMA statistical time series analysis spectral analysis periodogram NBERNational Bureau of Economic ResearchJulius Shiskin (Bureau of Labor Statistics) (moving average method) MITI 5
-III experimental methods Shiskin et.al. (1967) X-11 3 FORTRAN X-11 X-11 X-11 (Box=Jenkins (1976)) ARMA E.Dagun X-11-ARIMA X-11 ARIMA 4 X-11-ARIMA 3 (1983) 4 Autoregressive Integrated Moving Average Model 6
David FindleyX-12 X-11 X-11-ARIMA X-12-ARIMA (A.Zellner) BAYSEA BAYSEA DECOMP (1993) DECOMP Web-decomp Web http://www.ims.ac.jp/ sato 3. 4 3 =12 7
X-11 X-12-ARIMA DECOMP X-12-ARIMA 5 X-12-ARIMA 1-1, 2-1, 3-1, 4-1 - () X-12-ARIMA 5 (2001) 8
(statistical time series analysis) DECOMP 1-2(a) 6 X-12-ARIMA ARIMA (d + D>1) Web-Decomp 1-2 1-2(b) X-11 X-12-ARIMA 7 X-12-ARIMA 6 http : //www.ism.ac.jp/ sato Web-Decomp Web-Decomp Web-Decomp Decomp AR() 0 7 9
ARIMA 8 Web-Decomp 2-2(a) 2-2(b) ARIMA (d + D>1) 8 X-12-ARIMA transform() (2001) 10
Web-Decomp 3-2(a) 3-2(b) Web-Decomp 4-2 X-12-ARIMA ARIMA 4. X-12-ARIMA X-12-ARIMA X-12-ARIMA 9 X-12-ARIMA 9 X-12-ARIMA (2001) 11
X-12-ARIMA X-12-ARIMA Findley et. al. (1998) ( 5-1 ) X-12-ARIMA () {Y t, 1 t T } {y t } RegARIMA (forecasts) (backcasts) (regression) {Yt, H +1 t T + H} H ( 1) RegARIMA () {Yt } X-11 {Yt, 1 t T } X-11 X-11 X-11 () MPD 10 X-12-ARIMA X-11 RegARIMA 1 ( ) {y t,t =1, 2, } {y t } r {x it } r (4.1) φ p (B)Φ P (B s )(1 B) d (1 B s ) D (y t β i x it )=θ q (B)Θ Q (B s )a t. i=1 10 X-12-ARIMA X-12-ARIMA (U.S.Census Bureau (2000)) (2001) 12
y t () B (By t = y t 1 ) 11 s =(4 12), p, d, q, P, D, Q z φ p (z) =1 φ 1 z φ p z p, Φ p (z) =1 Φ 1 z Φ p z P,θ q (z) =1 θ 1 z θ q z q, Θ Q (z) =1 Θ 1 z Θ Q z Q Φ(B s ), Θ(B s ) B s (B s y t = y t s ) β i (i =1,,r), φ i (i =1,,p), Φ i (i =1,,P), θ i (i =1,,q), Θ i (i =1,,Q) {a t }, σ 2 (σ) t RegARIMA (linear regression) ARIMA() ARIMA (seasonal)arima (4.1) D =0, Φ P (z) =Θ Q (z) =1 ARIMA ARIMA ARIMA ARIMA (p, d, q) (P, D, Q) s (p, d, q)(p, D, Q)s Box=Jenkins(1976) ARIMA RegARIMA RegARIMA X-12-ARIMA (i)arima RegARIMA (identification) (4.2) (1 B) d (1 B s ) D y t = w t w r w t (r<t) t r 11 () B 2 y t = B(By t )=B(y t 1 )=y t 2 (1 B s )y t = y t y t s 13
d D (autocorrelation function acf) 12 (partial autocorrelation function pacf) acf pacf (stationary stochastic processes) d + D 1 d = D =1 I(2) I(2) overdifferencing 13 (ii) ARIMA (p) (q) (P) (Q) X-12-ARIMA automodel() ARIMA X-12-ARIMA 14 (Akaike s Information Criterion AIC) AIC AIC AIC 12 s γ(s) =E[(y t µ)(y t s µ)] s E( ) µ = E(y t ) 13 (1985) 14 Akaike(1973) (1989) 14
1ARIMA 1 (0,1,1) (0,1,1)4 2 (1,1,0) (0,1,1)4 3 (1,1,1) (0,1,1)4 4 (2,1,0) (0,1,1)4 5 (0,1,2) (0,1,1)4 6 (2,1,1) (0,1,1)4 7 (1,1,2) (0,1,1)4 8 (2,1,2) (0,1,1)4 (1) AIC AIC case1 1984.6227 case2 1984.3778 case3 1985.3950 case4 1985.2593 case5 1984.9030 case6 1986.3450 case7 1985.6934 case8 1981.2810 () AIC (common factor) 15 15 ARIMA (AR) (MA) 15
(2) AIC AIC case1 2411.9212 case2 2411.7487 case3 2412.3987 case4 2408.4527 case5 2409.1632 case6 2408.4143 case7 2411.1101 case8 2410.3991 (3) AIC AIC case1 1906.2882 case2 1905.8410 case3 1906.7123 case4 1902.7207 case5 1898.8119 case6 1902.0128 case7 1899.6596 case8 1897.9103 (4) AIC AIC case1 2146.5257 case2 2139.7715 case3 2141.4179 case4 2141.3743 case5 2142.3014 case6 2143.3436 case7 2143.2555 case8 2143.4295 () 16
1979 2 1983 1989 1 1996 2 1997 (iii) X-12-ARIMA X-11 X-12-ARIMA X-12-ARIMA regression() outlier() regression() X- 12-ARIMA (AO) (LS) AIC regression() arima( ) ARIMA AIC 17
AIC ARIMA RegARIMA X-12-ARIMA X-12-ARIMA outlier() X-12-ARIMA RegARIMA (non-stationary time series) (change point) 16 Findley et.al. (1998) (iv) X-12-ARIMA RegARIMA RegARIMA 17 X-12-ARIMA 1-3(a) 1-3(b) (v) X-12-ARIMA 16 17 (2000) 18
X-12-ARIMA slidingspans() history() (vi) X-12-ARIMA 18 5. X-11, X-12, DECOMP X-11, X-12-ARIMA, DECOMP X-11 X-12-ARIMA X-11 X-11 X-11 X-11 X-12-ARIMA RegARIMA 19 X-12-ARIMA X-11 X-11 X-11 X-12-ARIMA X-11 X-12-ARIMA DECOMP 18 (2000) 19 X-11 (1983) 19
Web-decomp 20 X-12-ARIMA 1-3(a) 1-3(b) X-11 Decomp 1-3(c) 1-3(d) 1-4(a) 1-4(b) X-12-ARIMA X-11 X-12-ARIMA X-11 RegARIMA X-12-ARIMA DECOMP X-11 X-11 20 X-12-ARIMA ARIMA d=d=1 DECOMP AIC X-12-ARIMA 20
X-12-ARIMA 2-3 X-12-ARIMA 2-4(a) 2-4(b) X-11 X-12-ARIMA DECOMP X-12-ARIMA DECOMP 3-3(a) 3-3(b) 3-4(a) 3-4(b) X-11 X-12-ARIMA DECOMP X-11 X-12-ARIMA X-12-ARIMA DECOMP 4-3(a) 4-3(b) 21
X-12-ARIMA X-11 4-3(c) 4-3(c) X-12-ARIMA RegARIMA DECOMP X-11 X-12-ARIMA X-12-ARIMA DECOMP X-12-ARIMA RegARIMA ARIMA AIC X-12-ARIMA composite() RegARIMA (direct method) (indirect method) 22
6. X-12-ARIMA(2000) X-11, X-12-ARIMA, DECOMP X-11 X-12-ARIMA X-11 X-12-ARIMA DECOMP X-11 X-11 X-11 X-12-ARIMA DECOMP X-11 X-12-ARIMA RegARIMA X-12-ARIMA 23
X-12-ARIMA X-12-ARIMA X-12-ARIMA X-12-ARIMA RegARIMA DECOMP X-12-ARIMA composite() 21 X-12-ARIMA RegARIMA 22 X-12-ARIMA X-12-ARIMA RegARIMA X-12-ARIMA DECOMP 21 22 24
7. Akaike, H. (1973), Information Theory and an Extension of the Likelihood Principle, in the Second International Symposium on Information Theory, eds. B.N. Petrov and F. Czaki, Budapest: Akademia Kiado, 267-287. Box, G.E.P., and G.M. Jenkins (1976), Time Series Analysis: Forecasting and Control, San Francisco: Holden Day. Findley, D.F., B.C. Monsell, W.R. Bell, M.C. Otto, B.C. Chen (1998), New Capabilities and Methods of the X-12-ARIMA Seasonal Adjustment Program, Journal of Business and Economic Statistics, 16, 127-176 (with Discussion). U.S. Bureau of Census (2000), X-12-ARIMA Reference Manual Version 0.2.7, Statistical Research Division, http : //www.census.gov/srd/www/x12a (1989), (), A.C. (1985) (1993) (1997) X-12-ARIMA, Discussion Paper No. J-97-1 Vol.25-1 http : //www.e.u tokyo.ac.jp (2001) X-12-ARIMA2000 Discussion Paper No. CIRJE-J-47, http : //www.e.u tokyo.ac.jp/cirje/research/03research02dp j/html (1983)() (2000) (1997) 25