On X-12-ARIMA2000 Abstract : This is a tentative Japanese translation of the first part Chapter 1 Chapter 5 of the X-12-ARIMA Manual with two short ap

Transcription

1 CIRJE-J-47 X-12-ARIMA

2 On X-12-ARIMA2000 Abstract : This is a tentative Japanese translation of the first part Chapter 1 Chapter 5 of the X-12-ARIMA Manual with two short appendices on the seasonal adjustment programs. Because the X-12-ARIMA program developed by the time series research group of the U.S. Census Bureau uses the statistical time series analysis extensively, it may be helpful for practitioners in Japan to use or understand the X -12-ARIMA program although the evaluation of the X-12-ARIMA program is still under discussion. We shall translate the remaining parts of the X -12-ARIMA Manual in the near future hopefully.

3 X-12-ARIMA ( ) U.S.Census Bureau X-12-ARIMA Reference Manual Version 0.2.7) X-12-ARIMA X-12-ARIMA X-12-ARIMA Reference Manual (Version TEX ) Dr. David Findley Dr. Brian Monsell 1

4 X-12-ARIMA (Input) (Output) X-12-ARIMA (Log Files) (Flags) (Program limits) X-12-ARIMA RegARIMA ARIMA (Forecasting) RegARIMA MA AR AR MA

5 A. X-12-ARIMA(2000) B

6 1. X 12 ARIMA (1967) II-X11 X-11 Findley=Hood (1999) RegARIMA RegARIMA ARIMA( ) RegARIMA ARIMA RegARIMA ARIMA X-12-ARIMA X11 X-12 RegARIMA RegARIMA RegARIMA X-12-ARIMA X11 X-11-ARIMA Geweke (1978), Dagum (1988), Bobbitt=Otto (1990) X-12-ARIMA 386 DOS (3.0 ) (Sun 4UNIX) (VAX/VMS) X-12- ARIMA (FORTRAN) URL http : // ftp pub/ts/x12a ftp.census.gov anonymous (1967) Dagun(1988) X11 4

7 X-12-ARIMA X11 X11 X11 X11 (a) (sliding spans) Findley=Monsell=Shulman=Pugh (1990) (b) (c) (d) (e) (f) (g) X-12-ARIMA RegARIMA ( ) RegARIMA X-12-ARIMA X-12-ARIMA X-12-ARIMA X-12-ARIMA ARIMA (p dq) (P DQ) s (p dq) (AR) (MA) (P DQ) s X-12-ARIMA RegARIMA ARIMA (p dq) (P DQ) s ARIMA RegARIMA (IGLS) X-12-ARIMA 5

8 AIC AIC AIC 5 X-12-ARIMA 2 X-12-ARIMA 3 X-12-ARIMA RegARIMA RegARIMA X-12-ARIMA 6 X-12-ARIMA X-12-ARIMA 2. X-12-ARIMA X-12-ARIMA ftp.census.gov /pub/ts/x12a/readme DOS X-12- ARIMA > path\x12a path\filename path\filename.spc X-12-ARIMA path\filename.out path ( ) X-12-ARIMA filename.spc filename.out filename.err xuu1.spc DOS > x12a xuu1 <return> ( <enter>) xuu1.out xuu1.err 6

9 UNIX DOS UNIX VAX/VMS X-12-ARIMA 2.1 (Input) X-12-ARIMA.spc path\filename X-12-ARIMA 2.2 (Output) path\filename.out (print) (save) (save) 2.3 (Input errors) path\filename.err (^) ERROR WARNING X-12-ARIMA (series) 7

10 WARNING 2.4 filename.spc X-12- ARIMA filename.out filename.err RegARIMA > path\x12a path\filename path\outname (2.1) filename.spc outname.out outname.err 2.5 X-12-ARIMA X-12-ARIMA X-12-ARIMA (a) (multi-spec mode) : (b) (single spec mode) : (metafile) X-12-ARIMA X-12-ARIMA 2 X-12-ARIMA X-12-ARIMA (input metafile) X-12-ARIMA 1 2 8

11 3 xuu1.spc, xuu2.spc, xuu3.spc xuu1 xuu2 xuu3 DOS c:\export\specs c:\export\specs\xuu1 c:\export\specs\xuu2 c:\export\specs\xuu3 X-12-ARIMA > x12a -m metafile (metafile.mta) -m X-12-ARIMA exports.mta > x12a -m exports.mta X-12-ARIMA exports.mta xuu1.out, xuu2.out, xuu3.out exports.out c:\export\specs\xuu1 c:\export\specs\xuu2 c:\export\specs\xuu3 c:\export\output\xuu1 c:\export\output\xuu2 c:\export\output\xuu X-12-ARIMA X-12-ARIMA (data metafile) 2 9

12 : X-12-ARIMA 6 (series) 3 xuu1.dat, xuu2.dat,xuu3.dat xuu1.dat xuu2.dat xuu3.dat c:\export\data c:\export\data\xuu1.dat c:\export\data\xuu2.dat c:\export\data\xuu3.dat X-12-ARIMA > x12a specfile -d metafile metafile.dta -d X-12- ARIMA specfile.spc 3 exports.dta > x12a default -d exports default.spc X- 12-ARIMA.dta X-12-ARIMA exports.dta xuu1.out, xuu2.out,xuu3.out exports.out c:\export\data\xuu1.dat c:\export\output\xuu1 c:\export\data\xuu2.dat c:\export\output\xuu2 c:\export\data\xuu3.dat c:\export\output\xuu3 10

13 2.6 (Log Files) X-12-ARIMA (log file) X-12-ARIMA.log > x12a -m exports exports.mta exports.log.log (series), (composite), (transform),x11, x11 (x11 regression), (regression), (automdl), (estimate), (check), (slidingspans), (history) (Flags) m d > path\x12a arg 1 arg 2 arg N x12a ( ) 2.1 X-12-ARIMA -m -d -i x12a test x12a -i test -i -m -i o (.out.err) (save) x12a test test2 x12a -i test -o test2 x12a -o test2 -i test 11

14 x12a -i test test2 i o m s RegARIMA s X-12-ARIMA (seasonal adjustment diagnostics file).xdg > x12a test -s test.xdg > x12a test -s -o testout testout.xdg s RegARIMA (model diagnostics file).mdg > x12a test -s test.mdg > x12a test -s -o testout testout.mdg ftp.census.gov Icon (Griswold=Griswold (1997) ) g (1) (.spc) (2) (3) (4) RegARIMA.gmt.xdg.mdg > x12a test -g c:\sagraph 12

15 c:\sagraph\test.gmt c:\sagraph\test.xdg c:\sagraph\test.mdg > x12a test -g c:\sagraph -o testout c:\sagraph\testout.gmt c:\sagraph\testout.xdg c:\sagraph\testout.mdg c:\sagraph g SAS/GRAPH(SAS) X-12-Graph ( Hood (1998a) Hood (1998b) ) X-12-ARIMA 2-2 g SAS/GRAPH s g n, w, p n X-12-ARIMA X-12-ARIMA n X-12-ARIMA w (series) p c (-m) X-12-ARIMA 6 (composite) c RegARIMA v X-12-ARIMA X-12-ARIMA 13

16 v s, c, n, w, p 2.8 X-12-ARIMA model.prm srslen.prm 2-3 model.prm srslen.prm 3. RegARIMA 3.1 X-12-ARIMA RegARIMA ARIMA X-12-ARIMA 6 RegARIMA X-12-ARIMA 3.1 (1976) ARIMA z t ARIMA φ(b)φ(b s )(1 B) d (1 B s ) D z t = θ(b)θ(b s )a t (1) B backshift operator Bz t = z t 1 ), s φ(b) = (1 φ 1 B φ p B p ) (AR), Φ(B s )=(1 Φ 1 B s Φ P B Ps ), θ(b) = (1 θ 1 B θ q B q ) (MA), Θ(B s )=(1 Θ 1 B s Θ Q B Qs ), a t σ 2 (1 B) d (1 B s ) D d D d = D =0( ) (1) z t z t µ µ = E[z t ] 14

17 ARIMA y t y t = i β i x it + z t (2) y t x it y t β i z t = y t β i x it (1) ARIMA z t ARIMA (2) z t ) (2) z t X-12-ARIMA RegARIMA (1) (2) φ(b)φ(b s )(1 B) d (1 B s ) D( y t i β i x it ) = θ(b)θ(b s )a t. (3) RegARIMA (1) ARIMA ( β i x it ) (2) z t (1) ARIMA RegARIMA y t z t z t w t w t ARMA φ(b)φ(b s )w t = θ(b)θ(b s )a t (3) RegARIMA (1 B) d (1 B s ) D y t = i β i (1 B) d (1 B s ) D x it + w t. (4) w t ARMA (4) RegARIMA x it y t (1 B) d (1 B s ) D ARIMA (3) RegARIMA x it (3) βx i,t 1 X-12-ARIMA X-12-ARIMA RegARIMA ARIMA (i) ARIMA, (ii) AR MA, (iii) AR MA, (iv) ((1 B) d (1 B s ) D y t ) RegARIMA ARIMA (1976) Abraham=Ledolter (1983), Vandaele (1983) Bell (1999) RegARIMA X-12-ARIMA 15

18 3.2 (series) (series) (span) (modelspan) (series) ( ) transform (Box-Cox) 0 1 (length of month) ( (length of quarter) 3.3 RgARIMA ((2) x it ) z t (1) ARIMA (regression) (arima) 3.4 X-12-ARIMA 3.1 (regression) 3.5 X-12-ARIMA t χ 2 - ARIMA ARIMA (4) ARIMA X-12-ARIMA (d >0) (D =0) t d t j 0 j<d) (1 B) d (3)( (4)) (D >0) d + D d + D 1 (Fixed seasonal effects) 16

19 3-1 X-12-ARIMA X-12-ARIMA (Trading-day effects) ( ) ( ),..., ( ) ( ) (lom) (lpyear) X-12-ARIMA tdnolpyear m Feb Y t /m t Y t m t t (28 29) m Feb = Y t RegARIMA tdnolpyear X-12-ARIMA (td) (regression) (flow) (stock) 3-1 X-12-ARIMA X-12-ARIMA X-12-ARIMA T t =( ) 5 2 ( ) X-12-ARIMA td td1coef 17

20 tdnolpyear td1nolpyear (Holiday effects) (i) (ii) (Easter effects) (Labor Day) (Thanksgiving) X-12-ARIMA w w t 3-1 X-12-ARIMA X-12-ARIMA 4 (additive outliers) (AO), (level shifts) (LS), (temporary changes) (TC), (ramps) AO LS TC ramp 3-1 LS -1 ramp (regression) AO, LS, TC, ramps (outlier) (outlier) AO, TC, LS ( ramp AO, LS, TC, ramp (1975) (interventions) X-12-ARIMA AO, LS, TC, ramp ( )RegARIMA (1976) 18

21 RegARIMA (1976) ARIMA RegARIMA ARIMA (1) (p dq), (P DQ),s ARIMA ARIMA y t (ACF) (PACF) RegARIMA ACF PACF y t ACF PACF RegARIMA (AR) (MA) RegARIMA Bell=Hillmer(1983) Bell(1999) RegARIMA (identify) y t,(1 B)y t,(1 B 12 )y t,(1 B)(1 B 12 )y t ACF (identify) (identify) (regression) (i) y t, (ii) ACF PACF 1 1 (d =1 D =1), (identify) (regression) ARIMA (1 B)(1 B 12 )y t = i β i (1 B)(1 B 12 )x it + w t (5) (OLS) (5) ACF PACF X-12-ARIMA d =1 D =1 (identify) (regression) (i) (5) OLS β i, (ii) z t = y t i β i x it, (iii) z t, (1 B) z t,(1 B 12 ) z t,(1 B)(1 B 12 ) z t ACF PACF (regression) (5) β 1 x 1t z t = y t Σ i 2 βi x it (5) 3.3 (5) (1 B)(1 B 12 ) t 0 1 t 19

22 y t.x-12-arima z t 2 t 2 (4) ARIMA (identify) (4) i 2 β i 3.5 (regression) (arima) RegARIMA (estimate) (1976) AR) MA) (AR) MA) ( 4.1 ) AR MA ARIMA (More=Garbow=Hillstrom(1980) MINPACK) RegARIMA Burman=Otto=Bell(1987) (IGLS) (i) AR MA (GLS) ARIMA (ii) β i ARIMA z t = y t β i x it IGLS (estimate) Box=Jenkins(1976, 7 ), Ljung=Box(1979), Hillmer=Tiao(1979), Wilson(1983) 4 RegARIMA ARIMA RegARIMA (Box=Jenkins(1976) 7, Brockwell=Davis(1987) 8 Pierce(1971), Bell(1999)) X-12-ARIMA ARMA ARMA 20

23 ARMA t χ 2 - X-12-ARIMA X-12-ARIMA σ 2 ˆσ 2 =SS/(n d s D) SS n d s D AR n d s D n p d s P s D X-12-ARIMA ARIMA(0 0 0) ˆσ 2 t χ 2 X-12-ARIMA AIC, AICC(F- AIC), Hannan-Quinn, BIC t χ 2 - (nonnested) AR(1) MA(1) 4.5 RegARIMA 3.6 RegARIMA (3) a t ) N(0,σ 2 ) (check) ACF PACF (Ljung=Box(1978)) Q- X-12-ARIMA (outlier) AO TC LS (AO, TC, LS) 3.3 X-12-ARIMA Chang=Tiao(1983) Chang=Tiao=Chen(1988) Bell(1983, 1999) Otto=Bell(1990) GLS 21

24 AO,LS, TC n AO, LS, TC n 6 (outlier) (DETAILS) t AO, LS, TC AR MA t X-12-ARIMA addone addall 6 (outlier) Findley=Monsell=Bell=Otto=Chen(1997) 1.48 Hampel et al. (1986) (regression) X-12-ARIMA 2 3 (temporary level shift) 3.7 X-12-ARIMA RegARIMA (forecast) y t MMSE RegARIMA ARIMA RegARIMA 22

25 AR MA (1976, AR MA Y t y t = log(y t ) (3) y t y t MMSE X-12-ARIMA X-12-ARIMA 4. RegARIMA X-12-ARIMA IGLS RegARIMA X-12-ARIMA 4.1 AR MA X-12-ARIMA AR MA 0.1 GLS X-12-ARIMA X-12-ARIMA 23

26 MA MA θ(b) =1 θ 1 B θ q B q θ(b) =0 G 1,...,G q ( G j > 1 Brockwell=Davis (1991, ) ARIMA MA MA MA ( G j =1) MA X-12-ARIMA MA X- 12-ARIMA MLE MA X- 12-ARIMA X-12-ARIMA 4.4 AR MA MA Θ MLE 1 1 ΘB s ARIMA AR AR φ(b) =1 φ 1 B φ p B p φ(b) =0 (1) φ(b)φ(b s )w t = θ(b)θ(b s )a t w t =(1 B) d (1 B s ) D z t AR AR AR AR AR AR X-12-ARIMA 24

27 AR MA X-12-ARIMA AR X-12-ARIMA 3.5 AR Fuller(1976) AR MA ARMA AR MA (1) (3) p>0 q>0 P>0 Q>0 ARMA p>0 Q>0 P>0 q>0 ARMA AR MA ARMA(1,1) (1 φb)z t =(1 θb)a t φ = θ (1 φb) z t = a t φ = θ ARMA(1,1) MLE ˆφ = ˆθ AR MA (1976, ) X-12-ARIMA ARMA AR MA AR MA (estimate) print=roots AR MA AR MA MA MA (1 B) MA (1 B s ) (1 B)(1 B s )z t =(1 θb)(1 ΘB s )a t θ Θ MA ˆθ =1 MA (1 θb) ˆθ =1 MA (1 ΘB s ) Abraham=Box(1978) Bell(1987) MA 25

28 MA ARMA AR MA AR MA 4.5 X-12-ARIMA AIC ( Akaike(1973), Findley(1985),(1999) ), AICC (Hurvich=Tsai(1989) ) Hannan=Quinn(1979), BIC (Schwarz(1978)) n p N L N AIC N = 2L N +2n p AICC N = 2L N +2n p { N N np+1 N Hannan Quinn N = 2L N +2n p log log N BIC N = 2L N + n p log N. } AR MA X-12-ARIMA X-12-ARIMA AO, LS, TS, ramp RegARIMA AO t Gomez=Maravall=Pena(1999) ) AO 26

29 (transform) (function) λ 1 (power) X-12-ARIMA X-12-ARIMA X-12-ARIMA X-12-ARIMA (series) (span) 5. X-12-ARIMA (specs) X-12-ARIMA (series) : (composite) : (series) (transform) : x11 : x11 (x11 regression) : (automdl) : (arima) : RegARIMA ARIMA 27

30 (regression) : RegARIMA (identify) (estimate) : (regression) (arima) (check) : (forecast) : (outlier) : (identify) : ARIMA (regression) (slidingspans) : (history) : RegARIMA (arguments)) (values) {} (argument) = (value) (argument) = ( 1, 2,...) (value) 5.1 (print) (save) (arguments) 5.2 (date) (series) (composite) X-12-ARIMA (series) (data) (file) X-12-ARIMA (series) data=( (data values)) x11 x11 (x11 regression), (slidingspans), (history) ( (estimates) X-12-ARIMA x11 x11 28

31 (identify) (arima), (regression), (estimate) (estimate) (outlier), (automdl), (check), (forecast), x11, (slidingspans), (history) X-12-ARIMA (estimate) estimate (arima) ARIMA(0 0 0) ( ) arima (series) (composite) X-11 RegARIMA ARIMA X-11 X : X-11 X-12-ARIMA series{title = "Monthly Retail Sales of Household Appliance Stores" data = ( ) start = 1972.jul} x11{} X-12-ARIMA ARIMA 29

32 RegARIMA (series) (identify) (transform) (regression) (series), (arima), (estimate) (transform), (regression), (outlier), (check), (forecast) X-12-ARIMA X-12-ARIMA 5.2 : RegARIMA X-12-ARIMA series{title = "Monthly Retail Sales of Household Appliance Stores" data = ( ) start = 1972.jul} transform{function = log} regression{variables = td} # Comment: Series has trading-day effects identify{diff=(0, 1) sdiff = (0, 1)} 5.2 (series), (transform), (regression), (identify) (series) ( (transform) ) (regression) (variables) = td (td6) 3.3 (regression) (identify) (1 B)(1 B 12 ) (identify) ARIMA (0 1 1)(0 1 1) 12 (identify) (arima) (estimate) : RegARIMA X-12-ARIMA series{title = "Monthly Retail Sales of Household Appliance Stores" data = ( ) 30

33 start = 1972.jul} transform{function = log} regression{variables = td} # Comment: Series has trading-day effects # identify{diff=(0, 1) sdiff = (0, 1)} arima{model = (0,1,1)(0,1,1)} estimate{print = iterations} (series), (transform), (regression), (arima), and (estimate) (regression) (arima) (estimate) (1 B)(1 B 12 ) ( 6 y t β i T it ) =(1 θb)(1 ΘB 12 )a t, i=1 T it y t x11 (forecast) : X-12-ARIMA series{title = "Monthly Retail Sales of Household Appliance Stores" data = ( ) start = 1972.jul} transform{function = log} regression{variables = td} # Comment: Series has trading-day effects # identify{diff=(0, 1) sdiff = (0, 1)} arima{model = (0,1,1)(0,1,1)} estimate{print = iterations} forecast{maxlead = 60} x11{seasonalma = s3x9} 3 9 (x11) ( (forecast)) 5.1 X-12-ARIMA (print) (save) (print) 31

34 (save) (print) (save) (Tables) ACF (print) (save) (print) (save) (save) (print) ( print=default ) print=() print=none print=none print=all print=alltables print=brief default, brief,none + (estimate) print =(+iterations + residuals) print =(default + iterations + residuals) default print=(none estimates) - print=brief -acf print=(all -iterations) (save) (estimate) save=(mdl estimates) (print) (save) (long) (short) (estimate) regcmatrix rcm none, all, alltables, default, brief (save) + - (save) (print) (save) DOS C:\TSERIES SALES.SPC X-12-ARIMA (estimate) save = (mdl estimates) C:\TSERIES\SALES.MDL C:\TSERIES\SALES.EST X-12-ARIMA.dat,.exe,.com,.for,.spc ) X-12-ARIMA 32

35 * 5.2 year.month year.quarter) 67 AD( CE)67 AD1967 (jan, feb, mar, apr, may, jun, jul, aug, sep, oct, nov, and dec) dec regression { variables=(ao1978.apr ls1982.sep) } (series) ao1972.jan 5.3 (Allowable input characters),,,, (newline) :=., {}()[]+- ASCII! % ˆ (Braces), (parentheses), (brackets) {},(),[] {} ( ) [ ] (i) (regression) { variables = (td Easter[14])}] (ii) ARIMA arima { model = (0 1 [1,3]) } (Case sensitivity) none all td seasonal ) X-12-ARIMA TD td (regression) x11 (x11regression) (variables) 33

36 (Comments) (Equals sign) = print = none title = "Monthly Retail Sales of Household Appliance Stores" (file) FORTRAN variables=(td seasonal const) variables=td, variables=(td), variables = (td seasonal), start=1967.4, start=(1967.4) (Null list) outlier{} (Numerical values) 400, 400.0, e+2 (Ordering) (series) (composite) ( (series), (transform), (regression) ), (model) ( (arima) ARIMA (span) ( (series) (outlier) (Separators) data=(0, 1, 2, 3, 4, 5) (span) span=(1967.4, ) (span) ( title or "title") 34

37 35

38 Abraham, B., and G.E.P. Box (1978), Deterministic and Forecast-Adaptive Time-Dependent Models, Applied Statistics, 27, Abraham, B., and J. Ledolter (1983), Statistical Methods for Forecasting, New York: John Wiley and Sons. 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, Baxter, M.A. (1994) A Guide to Seasonal Adjustment of Monthly Data With X-11 (Third Edition), Central Statistical Office, United Kingdom. Bell, W.R. (1983), A Computer Program for Detecting Outliers in Time Series, Proceedings of the American Statistical Association, Business a nd Economic Statistics Section, Bell, W.R. (1987), A Note on Overdifferencing and the Equivalence of Seasonal Time Series Models With Monthly Means and Models With (0,1,1)12 Seasonal Parts When Θ = 1, Journal of Business and Economic Statistics, 5, Bell, W.R. (1999), An Overview of regarima Modeling, forthcoming Research Report, Statistical Research Division, U. S. Census Bureau. Bell, W.R., and S.C. Hillmer (1983), Modeling Time Series with Calendar Variation, Journal of the American Statistical Association, 78, Bobbitt, L. and Otto, M. C. (1990), Effects of Forcasts on the Revisions of Seasonally Adjusted Values Using the X-11 Seasonal Adjustment Procedure, Proceedings of the American Statistical Association, Business and Economic Statistics Section, Box, G.E.P., and D.R. Cox (1964), An Analysis of Transformations, Journal of the Royal Statistical Society, B, 26, Box, G.E.P., and G.M. Jenkins (1976), Time Series Analysis: Forecasting and Control, San Francisco: Holden Day. Box, G.E.P., and G.C. Tiao (1975), Intervention Analysis with Applications to Economic and Environmental Problems, Journal of the American Statistical Association, 70, Brockwell, P.J., and R.A. Davis (1991), Time Series: Theory and Methods, New York: Springer-Verlag. Burman, J.P., M.C. Otto, and W. R. Bell (1987), An Iterative GLS Approach to Maximum Likelihood Estimation of Regression Models with ARIMA Errors, Research Report No. 87/34, U. S. Census Bureau. Chang, I., and G.C. Tiao (1983), Estimation of Time Series Parameters in the Presence of Outliers, Technical Report No. 8, Statistics Research Center, University of Chicago. Chang, I., G.C. Tiao, and C. Chen (1988), Estimation of Time Series Parameters in the Presence of Outliers, Technometrics, 30, Chen, B., and D.F. Findley (1998) Comparison of X-11 and regarima Easter Holiday Adjustments, Statistical Research Division, U. S. Census Bureau, manuscript in preparation. 36

39 Cholette, P.A. (1978) A Comparison and Assessment of Various Adjustment Methods of Sub-Annual Series to Yearly Benchmarks, Research Paper, Seasonal Adjustment and Time Series Staff, Statistics Canada. Cholette, P.A. (1979) A Comparison of Various Trend Cycle Estimators, Research Paper, Seasonal Adjustment and Time Series Staff, Statistics Canada. Dagum, E.B. (1988) X-11-ARIMA/88 Seasonal Adjustment Method - Foundations and Users Manual, Statistics Canada. Doherty, M. (1991) Surrogate Henderson Filters in X-11, Technical Report, Statistics New Zealand, Wellington, New Zealand. Duffet-Smith, P (1981) Practical Astronomy With Your Calculator, 2nd Edition, Cambridge University Press. Findley, D.F. (1985) On the Unbiasedness Property of AIC for Exact or Approximating Linear Stochastic Time Series Models, Journal of Time Series Analysis, 6, Findley, D.F. (1999) Akaike s Information Criterion II in the Encyclopedia of Statistical Science, Update Volume 3, eds. S. Kotz, C. B. Read, and D. L. Banks. New York: John Wiley and Sons, 2 6. Findley, D.F. and C.C. Hood (1999). X-12-ARIMA and Its Application to Some Italian Indicator Series, to appear in Seasonal Adjustment Procedures Experiences and Perspectives, Istituto Nazionale di Statistica (ISTAT), Rome, Findley, D.F., B.C. Monsell, H.B. Shulman, and M.G. Pugh (1990), Sliding Spans Diagnostics for Seasonal and Related Adjustments, Journal of the American Statistical Association, 85, 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, (with Discussion). Fuller, W.A. (1976), Introduction to Statistical Time Series, New York: John Wiley. Geweke, J. (1978), Revision of Seasonally Adjusted Time Series, SSRI Report No. 7822, University of Wisconsin, Department of Statistics. Gomez, V., A. Maravall, and D. Peña (1999). Missing Observations in ARIMA Models: Skipping Approach versus Additive Outlier Approach, Journal of Econometrics, 88, Griswold, R.E., and M.T. Griswold (1997) The ICON Programming Language, 3rd Edition, San Jose: Peer-to-Peer Communications. Hampel, F.R., E.M. Ronchetti, P.J. Rousseeuw, and W.A. Stahel (1986), Robust Statistics: The Approach Based on Influence Functions, New York: John Wiley. Hannan, E.J., and B.G. Quinn (1979), The Determination of the Order of an Autoregression, Journal of the Royal Statistical Society, B, 41, Hillmer, S.C., and G.C. Tiao (1979), Likelihood Function of Stationary Multiple Autoregressive Moving Average Models, Journal of the American Statistical Association, 74, Hood, C.C. (1998a) X-12-Graph: A SAS/GRAPH Program for X-12-ARIMA Output, User s Guide for X-12-Graph Interactive for PC/Windows, Version 1.0, 37

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42 X-12-ARIMA. X-12-ARIMA(2000) X-12 Experimental Version X-12-ARIMA(β Version) β Version X-12-ARIMA URL Corrections( ) X-12-ARIMA X-12-ARIMA X-12-ARIMA(1998) β Version X-12-ARIMA Version X-12- ARIMA(2000) X-12-ARIMA URL X-12-ARIMA(2000) X-12-ARIMA X-12-ARIMA(β Version) (1997) X-12- ARIMA Discussion Paper No. J-97-1, ( URL ) (1997) X-12-ARIMA Findley et.al. (1998) (1996) X-12-ARIMA. 40

43 statistical time series analysis spectral analysis periodogram NBER National Bureau of Economic Research Julius Shiskin (Bureau of Labor Statistics) (Moving average method) MITI -III Experimental methods Shiskin et.al. (1967) X-11 FORTRAN X-11 X-11 X-11 X-11 ARMA E.Dagun X-11-ARIMA X-11 41

44 ARIMA X-11-ARIMA David Findley X-12 X-11 X-11-ARIMA X-11 A.Zellner X-12-ARIMA H.Akaike BAYSEA BAYSEA DECOMP (1993) Web-decomp Web sato 42