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1 Revisions in Seasonal Adjustment Methods: an Empirical Comparison of X-12-ARIMA & SEATS Jens Dossé* Christophe Planas* *Consultants to Eurostat, Bâtiment Jean Monnet C5/98, L – 2920 Luxembourg Revised Version, September 1996 1 Introduction Most of seasonal adjustment procedures involve moving average filters. The difference between moving average filters mainly lies in the way that the filters are constructed. With methods like X-12-ARIMA (using the X-11 decomposition), the filter is empirical in the sense that it does not depend on the statistical properties of the series under analysis. On the other hand, the ARIMA-model-based signal extraction method as implemented in the program SEATS uses the information obtained from the modelling of the series to estimate the components. In both cases, the filters used are symmetric, involving past and future realisations of the observed series. For large enough samples, the application of the filters around the central periods yields the so-called final estimators. For the most recent periods, no symmetric filter can be applied, and a preliminary estimator has to be considered instead. An optimal preliminary estimator can be obtained by replacing the unknown’s future observations by their forecasts (see Cleveland and Tiao 1976). The forecast errors imply that the preliminary estimates will be contaminated by an error, termed revision error. As new observations become available, forecasts are updated and eventually replaced by the observed values, and the preliminary estimate is revised. Since the seasonal adjusted series and the trend estimate for the concurrent period 2 is of most interest to short-term macroeconomic policy makers, the size of the revisions is then a critical point in a decomposition procedure. This note discusses and compares the revisions obtained with the use of an empirical filter as in X-12-ARIMA and with an ARIMA- model-based approach as in SEATS on the basis of an empirical investigation. 2 Revision in preliminary estimates Suppose that one wish to decompose a series into a seasonal t s plus non-seasonal component t n according to: t t t n s x + = (2.1) where additivity may be obtained after a log-transformation. The non- seasonal component t n can be further decomposed into a trend t p plus an irregular component t u . Let ( )B sν denote the filter used to compute the final estimator of the seasonal component: ( ) t s t x B s ν = ˆ (2.2) The filter ( )B sν is symmetric and can be written as ( ) + =L B sν L + + + − F B 1 0 1 ν ν ν . In the case of the model-based approach (cf. SEATS), ( )B sν corresponds to the Wiener-Kolmogorov filter, while in the case of X-12-ARIMA, ( )B sν may correspond to one of the so-called 3 by 3, 3 by 5 or 3 by 9 seasonal MA filter. Suppose that the correlation structure of the series t x is well described by the model: ( ) t t t t a B a a x ψ ψ = + + = − L 1 1 (2.3) where t a is a Gaussian variable and ( )B ψ is a polynomial which can be 3 infinite. Inserting (2.3) into (2.2), then the final estimator of t s is given as: ( ) ( ) ( ) t s t s t a B a B B s ξ ψ ν = = ˆ (2.4) where ( ) ( ) ( ) L L + + + + = = − F B B B B s s s s s 1 0 1 ξ ξ ξ ψ ν ξ . A preliminary estimate k t t s + / ˆ of t s obtained at time k t + is simply obtained by taking the expectation of the final estimator t ŝ conditional on the information available at time k t + : ( ) [ ] [ ] [ ] ( ) t k s t k sk s s s t k sk k sk s s s k t t s k t k t t a B a F F B a F F F B E a B E s ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ = = + + + + + = = + + + + + + + + = = = − + + − + + + L L L L L 1 0 1 1 1 1 0 1 / ˆ (2.5) where use has been made of the fact that the innovations can not be forecasted. The filter ( )B k sξ corresponds to the filter ( )B sξ truncated in k F . The revision k R in the preliminary estimate of t s obtained at time k t + is then simply obtained as: ∑ ∞ + = + + = − = 1 / ˆ ˆ k i i t si k t t t k a s s R ξ (2.6) The revisions follow thus a moving average process, and standard results (see Pierce 1980) show that the revisions are convergent. The update in the preliminary after one further observation is given by: 1 , , 0 , ˆ ˆ 1 1 / 1 / − = = − = + + + + + + T k a s s r k t sk k t t k t t k K ξ . (2.7) 4 It is clear that different forecasts and different filters imply different revision patterns. The result (2.5) also allows us to point out the following sources of revisions when new observations are added: • The update of the forecasts. • The re-estimation of the parameters of the observed series model yields an updated polynomial ( )B ψ , and thus a new polynomial ( )B ξ will be obtained (see (2.4)). Notice that in a model-based approach, an updated filter ( )B sν would also be obtained. • The new observations may affect the identification of the model for the observed series, giving another form of the polynomial ( )B ψ . • Since the model for the observed series and the optimal filter are related, it can be expected that a discrepancy between the properties of the model for forecasting and those of the filter in use may induce large revision. For example, if unstable seasonal patterns are predicted while the filter in use is well suited for a stable seasonality, then some large revisions may be produced. These considerations have led us to set up an experimental design that we present now. 3 Experimental Design Twenty-four monthly time series have been analysed. These series are French import and export series (nomenclature Nec02, codes 01-12) starting in January 1980 and ending in December 1994, that is along a sample of 180 observations. Prior outlier removing and correction 5 for special effects such that trading days and Easter effect have been performed before the series were sent to X-12-ARIMA and to SEATS. For every series, the last three years have been removed and a shorter sample is first considered. That is, a number 36 = T of observations are initially suppressed. The analysis concentrates on the revisions in the preliminary estimates in the last period in the short sample. The preliminary estimates considered are the seasonally adjusted series and the trend estimates. Every time a new observation is added, new estimates are computed and the revisions in the previous estimates are straightforwardly available. In particular, the value of k r is obtained for every estimate. The estimators obtained after the adding of the 36th observation is taken as final, yielding k R . Three different ways of dealing with the forecasting model and its parameters are undertaken: first, the model and its parameter is fixed; second, the model is fixed but the parameters re-estimated; three, the model is re-identified/selected and re-estimated every time a new observation is added. The parameters sent to SEATS and X-12-ARIMA are given in the appendix. The two programs have been made running as much as possible in an automatic way. The statistics used to analyse the results are: • Absolute Revision Variation (ARV): % 100 ˆ 36 / 2 0 1 ⋅ − = + − = + ∑ t t T k k k s r r ARV (3.8) • Smoothness of Revisions (SMR): ( ) % 100 ˆ 2 2 36 / 2 0 2 1 ⋅ − = + − = + ∑ t t T k k k s r r SMR (3.9) 6 • Sum of Squared Revisions (SQR): % 100 ˆ 2 2 36 / 1 0 2 ⋅ = + − = ∑ t t T k k s r SQR (3.10) • Mean Convergence (MC): Write: 1 , , 0 %, 100 0 2 − = ⋅ = ∑ = T k SQR r C k i i k K (3.11) Then, ∑ − = = 1 0 1 T k k C T MC (3.12) • Smoothness of Convergence (SC): ( ) ∑ − = + − = 2 0 2 1 T k k k C C SC (3.13) The first three measures have been standardised so as to yield results interpretable in proportion of the level of the final estimators. This standardisation was introduced because the use of different approaches may lead to different final estimators. Although the differences have been found often negligible, the correction facilitates the interpretation of the results. 7 4 Revision in Seasonal Adjusted Series: Results of the Comparison (i) For the 24 series, the re-estimation of the models does not significantly modify the results: when the models are taken as fixed, whether the parameter are set or re-estimated does not much affect the results. On the contrary, proceeding to a re-identification/selection of the model before re-estimating the parameters have a significant impact. The SQR statistics indicates that larger revisions are obtained. (ii) When the model is fixed and for the majority of the cases seen, the model-based approach (SEATS) yields smaller revisions than with an empirical filter approach (X-12-ARIMA). This can be seen on the SQR tables. (iii) As read from the ARV and SMR figures, X-12-ARIMA often implies erratic revisions while the model-based approach yields much smoother revision. This holds whether the model is set or not. (iv) The convergence may be faster with X-12-ARIMA than with SEATS (see the MC plot). However, this effect was often related to relatively large revisions when the first new observations were added. The plot of the smoothness of the convergence, which shows an as smooth convergence of the SEATS preliminary estimates as this of the X-12-ARIMA ones, confirms this. 5 Revision in Trends: Some Theoretical Consid- erations The comparison of the revisions in the trend preliminary estimates has come out to be more difficult for the reason that X-12-ARIMA estimates trends with a Henderson 13-term moving average filter, and then suggests the users to select another filter if needed. Furthermore, depending on the series under analysis, the Henderson 13-term moving average filter may be more concentrated around the low frequencies 8 than the model-based trend. In that case, the SEATS trend corresponds to a shorter-term trend and the X-12-ARIMA trend shows longer terms movements. The comparison of the revisions is thus biased, the two programs estimating much different trends. Some points out about the size of the revisions in the trend preliminary estimators may however be worked. In particular, a result derived in Maravall and Planas (1995) shows that when a time series displays a stable (alternatively unstable) pattern, then the component catching this feature must be made as stable (unstable) as possible to minimise the variance of the estimation error. In the case of the trend component, if a time series shows a stable long-run path, then the trend estimator must be made as stable as possible to be more accurate. In the model-based approach, the filter for estimating the trend is derived from the model for the observed series. An empirical filter is instead selected not strictly in relation with the properties of the series under analysis, and thus may take benefit from more freedom in its design. In particular, it may perfectly be that the empirical filter would be more concentrated around the low frequencies that the filter obtained in the model-based approach. In that case, lower revisions may be expected. To check if investigate that point, we have focused the attention on the time series displaying the most stable long-term pattern, namely the series Frexp04. The empirical filter for estimating a long-term trend as been taken as the standard 23-term Henderson filter. The revision in the estimates of the trend from SEATS, 13-term and 23-term Henderson filter (X-12-ARIMA) obtained in the last period when up to 3 years of observations are added are presented in appendix. The most striking figure is that the model-based filter always yields lower revisions. That can be seen simply from the spectra of the filters designed to estimate the trend: the SEATS one appears to be much more concentrated at the low frequency, yielding a more stable estimator and, as a consequence, less revisions. This illustrates a result in Maravall and Planas: the narrowest filter more accurately estimates a stable trend. France export, part 1 10 100 1000 10000 100000 8001 8101 8201 8301 8401 8501 8601 8701 8801 8901 9001 9101 9201 9301 9401 Date Logarithmic scaleFrExp01/3 FrExp03/3 FrExp05 FrExp06 FrExp10 FrExp12 9 France export, part 2 10 100 1000 10000 100000 8001 8101 8201 8301 8401 8501 8601 8701 8801 8901 9001 9101 9201 9301 9401 Date Logarithmic scaleFrExp02 FrExp04 FrExp07 FrExp08 FrExp09 FrExp11 10 France import, part 1 100 1000 10000 100000 8001 8101 8201 8301 8401 8501 8601 8701 8801 8901 9001 9101 9201 9301 9401 Date Logarithmic scaleFrImp01 FrImp02 FrImp03 FrImp07 FrImp08 FrImp11 11 France import, part 2 100 1000 10000 100000 8001 8101 8201 8301 8401 8501 8601 8701 8801 8901 9001 9101 9201 9301 9401 Date Logarithmic scaleFrImp04 FrImp05 FrImp06 FrImp09 FrImp10 FrImp12 12 SA revisions of Seats with re-identified models (all time series) -600 -400 -200 0 200 400 600 r0 r1 r2 r3 r4 r5 r6 r7 r8 r9 r10 r11 r12 r13 r14 r15 r16 r17 r18 r19 r20 r21 r22 r23 r24 r25 r26 r27 r28 r29 r30 r31 r32 r33 r34 r35 Revisions rk (updates in preliminary) during the last three years 13 SA revisions of X-12-Arima with re-selected models (all time series) -600 -400 -200 0 200 400 600 r0 r1 r2 r3 r4 r5 r6 r7 r8 r9 r10 r11 r12 r13 r14 r15 r16 r17 r18 r19 r20 r21 r22 r23 r24 r25 r26 r27 r28 r29 r30 r31 r32 r33 r34 r35 Revisions rk (updates in preliminary) during the last three years 14 Trend revisions of Seats with re-identified models (all time series) -600 -400 -200 0 200 400 600 r0 r1 r2 r3 r4 r5 r6 r7 r8 r9 r10 r11 r12 r13 r14 r15 r16 r17 r18 r19 r20 r21 r22 r23 r24 r25 r26 r27 r28 r29 r30 r31 r32 r33 r34 r35 Revisions rk (updates in preliminary) during the last three years 15 Trend revisions of X-12-Arima with re-selected models (all time series) -600 -400 -200 0 200 400 600 r0 r1 r2 r3 r4 r5 r6 r7 r8 r9 r10 r11 r12 r13 r14 r15 r16 r17 r18 r19 r20 r21 r22 r23 r24 r25 r26 r27 r28 r29 r30 r31 r32 r33 r34 r35 Revisions rk (updates in preliminary) during the last three years 16 SA revisions of Seats with re-estimated model parameters (all time series) -600 -400 -200 0 200 400 600 r0 r1 r2 r3 r4 r5 r6 r7 r8 r9 r10 r11 r12 r13 r14 r15 r16 r17 r18 r19 r20 r21 r22 r23 r24 r25 r26 r27 r28 r29 r30 r31 r32 r33 r34 r35 Revisions rk (updates in preliminary) during the last three years 17 SA revisions of X-12-Arima with re-estimated model parameters (all time series) -600 -400 -200 0 200 400 600 r0 r1 r2 r3 r4 r5 r6 r7 r8 r9 r10 r11 r12 r13 r14 r15 r16 r17 r18 r19 r20 r21 r22 r23 r24 r25 r26 r27 r28 r29 r30 r31 r32 r33 r34 r35 Revisions rk (updates in preliminary) during the last three years 18 Trend revisions of Seats with re-estimated model parameters (all time series) -600 -400 -200 0 200 400 600 r0 r1 r2 r3 r4 r5 r6 r7 r8 r9 r10 r11 r12 r13 r14 r15 r16 r17 r18 r19 r20 r21 r22 r23 r24 r25 r26 r27 r28 r29 r30 r31 r32 r33 r34 r35 Revisions rk (updates in preliminary) during the last three years 19 Trend revisions of X-12-Arima with re-estimated model parameters (all time series) -600 -400 -200 0 200 400 600 r0 r1 r2 r3 r4 r5 r6 r7 r8 r9 r10 r11 r12 r13 r14 r15 r16 r17 r18 r19 r20 r21 r22 r23 r24 r25 r26 r27 r28 r29 r30 r31 r32 r33 r34 r35 Revisions rk (updates in preliminary) during the last three years 20 SA revisions of Seats with fixed model parameters (all time series) -600 -400 -200 0 200 400 600 r0 r1 r2 r3 r4 r5 r6 r7 r8 r9 r10 r11 r12 r13 r14 r15 r16 r17 r18 r19 r20 r21 r22 r23 r24 r25 r26 r27 r28 r29 r30 r31 r32 r33 r34 r35 Revisions rk (updates in preliminary) during the last three years 21 SA revisions of X-12-Arima with fixed model parameters (all time series) -600 -400 -200 0 200 400 600 r0 r1 r2 r3 r4 r5 r6 r7 r8 r9 r10 r11 r12 r13 r14 r15 r16 r17 r18 r19 r20 r21 r22 r23 r24 r25 r26 r27 r28 r29 r30 r31 r32 r33 r34 r35 Revisions rk (updates in preliminary) during the last three years 22 Trend revisions of Seats with fixed model parameters (all time series) -600 -400 -200 0 200 400 600 r0 r1 r2 r3 r4 r5 r6 r7 r8 r9 r10 r11 r12 r13 r14 r15 r16 r17 r18 r19 r20 r21 r22 r23 r24 r25 r26 r27 r28 r29 r30 r31 r32 r33 r34 r35 Revisions rk (updates in preliminary) during the last three years 23 Trend revisions of X-12-Arima with fixed model parameters (all time series) -600 -400 -200 0 200 400 600 r0 r1 r2 r3 r4 r5 r6 r7 r8 r9 r10 r11 r12 r13 r14 r15 r16 r17 r18 r19 r20 r21 r22 r23 r24 r25 r26 r27 r28 r29 r30 r31 r32 r33 r34 r35 Revisions rk (updates in preliminary) during the last three years 24 Absolute Revision Variation for all seasonally adjusted and trend series 0 5 10 15 20 25 30 35 0 5 10 15 20 25 30 35 Seats X-12-ArimaSA Model free Trend Model free SA Model order fixed Trend Model order fixed SA Model param. fixed Trend Model param. fixed N° of series for which ARV(Seats) < ARV(X-11): 131/142 25 Smoothness of Revisions for all seasonally adjusted and trend series 0 10 20 30 40 50 60 70 0 10 20 30 40 50 60 70 Seats X-12-ArimaSA Model free Trend Model free SA Model order fixed Trend Model order fixed SA Model param. fixed Trend Model param. fixed N° of series for which SMR(Seats) < SMR(X-11): 127/142 26 Sum of Squared Revisions for all seasonally adjusted and trend series 0 10 20 30 40 50 60 0 10 20 30 40 50 60 Seats X-12-ArimaSA Model free Trend Model free SA Model order fixed Trend Model order fixed SA Model param. fixed Trend Model param. fixed N° of series for which SQR(Seats) < SQR(X-11): 126/142 27 Mean Convergence for all seasonally adjusted and trend series 0500000100000015000002000000250000030000000 500000 1000000 1500000 2000000 2500000 3000000 Seats X-12-ArimaSA Model free Trend Model free SA Model order fixed Trend Model order fixed SA Model param. fixed Trend Model param. fixed N° of series for which MC(Seats) < MC(X-11): 62/142 28 Smoothness of Convergence for all seasonally adjusted and trend series 100100001000000100000000100000000001E+121E+14100 10000 1000000 100000000 10000000000 1E+12 1E+14 Seats X-12-ArimaSA Model free Trend Model free SA Model order fixed Trend Model order fixed SA Model param. fixed Trend Model param. fixed N° of series for which SC(Seats) < SC(X-11): 70/142 29 40 60 80 100 120 140 160 180 200 0 20 40 60 80 100 120 140 160 180 EXP04: Original Series Trend revisions with fixed model parameters for series FrExp04 -1 -0.5 0 0.5 1 1.5 2 2.5 r0 r1 r2 r3 r4 r5 r6 r7 r8 r9 r10 r11 r12 r13 r14 r15 r16 r17 r18 r19 r20 r21 r22 r23 r24 r25 r26 r27 r28 r29 r30 r31 r32 r33 r34 r35 Revisions rk (updates in preliminary) during the last three years FrExp04 Seats trend FrExp04 X-12-Arima trend (Henderson-13) FrExp04 X-12-Arima trend (Henderson-23) 31 32 Parameters sent to Seats and X-12-Arima: Model parameters fixed: FrExp01 Seats input: D=1,BD=1,P=0,BP=0,Q=1,BQ=1, LAM=0,IMEAN=0,PG=1,MQ=12,MODEL=0,INIT=2, TH(1)=-0.458662, BTH(1)=-0.646929, TRAMO=-1,BLQT=15.573744,HS=1.500000,CRMEAN=0,EPSPHI=2.000000, OUT=0,RSA=0,QMAX=50,NOADMISS=1,BIAS=0, SMTR=0,THTR=-0.400000,RMOD=0.500000,MAXBIAS=0.500000,IQM=24 X-12-Arima input: transform{function=log} estimate{save=(model)} arima{model=(2 1 2)(0 1 1) ar=(-0.6146149903E+00f 0.2810814997E+00f) ma=(-0.1942233453E+00f 0.6335310999E+00f 0.6486130414E+00f)} x11{mode=mult trendma=13 seasonalma=s3x5 save=(d11 d12)} check{save=(acf)} FrExp02 Seats input: D=1,BD=1,P=0,BP=0,Q=1,BQ=1, LAM=0,IMEAN=0,PG=1,MQ=12,MODEL=0,INIT=2, TH(1)=-0.368447, BTH(1)=-0.601682, TRAMO=-1,BLQT=18.250837,HS=1.500000,CRMEAN=0,EPSPHI=2.000000, OUT=0,RSA=0,QMAX=50,NOADMISS=1,BIAS=0, SMTR=0,THTR=-0.400000,RMOD=0.500000,MAXBIAS=0.500000,IQM=24 X-12-Arima input: transform{function=log} estimate{save=(model)} arima{model=(0 1 2)(0 1 1) ma=( 0.3132496774E+00f 0.1165852982E+00f 0.6061667409E+00f)} x11{mode=mult trendma=13 seasonalma=s3x5 save=(d11 d12)} check{save=(acf)} FrExp03 Seats input: D=1,BD=0,P=0,BP=1,Q=1,BQ=0, LAM=0,IMEAN=0,PG=1,MQ=12,MODEL=0,INIT=2, BPHI(1)=-0.201888, TH(1)=-0.706600, TRAMO=-1,BLQT=21.535236,HS=1.500000,CRMEAN=0,EPSPHI=2.000000, OUT=0,RSA=0,QMAX=50,NOADMISS=1,BIAS=0, SMTR=0,THTR=-0.400000,RMOD=0.500000,MAXBIAS=0.500000,IQM=24 X-12-Arima input: transform{function=log} estimate{save=(model)} arima{model=(0 1 1) (0 1 1) ma=( 0.7203044275E+00f 0.7243152389E+00f)} x11{mode=mult trendma=23 seasonalma=s3x5 save=(d11 d12)} check{save=(acf)} 33 FrExp04 Seats input: D=1,BD=1,P=0,BP=0,Q=1,BQ=1, LAM=0,IMEAN=0,PG=1,MQ=12,MODEL=0,INIT=2, TH(1)=-0.774064, BTH(1)=-0.923241, TRAMO=-1,BLQT=27.665483,HS=1.500000,CRMEAN=0,EPSPHI=2.000000, OUT=0,RSA=0,QMAX=50,NOADMISS=1,BIAS=0, SMTR=0,THTR=-0.400000,RMOD=0.500000,MAXBIAS=0.500000,IQM=24 X-12-Arima input: transform{function=log} estimate{save=(model)} arima{model=(2 1 0)(0 1 1) ar=(-0.7038447493E+00f -0.4339817706E+00f) ma=( 0.8899731440E+00f)} x11{mode=mult trendma=23 seasonalma=s3x5 save=(d11 d12)} check{save=(acf)} FrExp05 Seats input: D=1,BD=1,P=0,BP=0,Q=1,BQ=1, LAM=0,IMEAN=0,PG=1,MQ=12,MODEL=0,INIT=2, TH(1)=-0.589691, BTH(1)=-0.652684, TRAMO=-1,BLQT=15.123951,HS=1.500000,CRMEAN=0,EPSPHI=2.000000, OUT=0,RSA=0,QMAX=50,NOADMISS=1,BIAS=0, SMTR=0,THTR=-0.400000,RMOD=0.500000,MAXBIAS=0.500000,IQM=24 X-12-Arima input: transform{function=log} estimate{save=(model)} arima{model=(2 1 2)(0 1 1) ar=( 0.2011812314E+00f -0.2173513383E+00f) ma=( 0.9082461034E+00f -0.4553338938E+00f 0.6273839741E+00f)} x11{mode=mult trendma=13 seasonalma=s3x5 save=(d11 d12)} check{save=(acf)} FrExp06 Seats input: D=1,BD=1,P=2,BP=0,Q=0,BQ=0, LAM=0,IMEAN=0,PG=1,MQ=12,MODEL=0,INIT=2, PHI(1)=0.560617, PHI(2)=0.304828, TRAMO=-1,BLQT=26.825666,HS=1.500000,CRMEAN=0,EPSPHI=2.000000, OUT=0,RSA=0,QMAX=50,NOADMISS=1,BIAS=0, SMTR=0,THTR=-0.400000,RMOD=0.500000,MAXBIAS=0.500000,IQM=24 X-12-Arima input: transform{function=log} estimate{save=(model)} arima{model=(0 1 2)(0 1 1) ma=( 0.6531437910E+00f -0.2369674316E+00f 0.2524437642E+00f)} x11{mode=mult trendma=13 seasonalma=s3x5 save=(d11 d12)} check{save=(acf)} FrExp07 Seats input: D=1,BD=1,P=0,BP=0,Q=1,BQ=1, LAM=0,IMEAN=0,PG=1,MQ=12,MODEL=0,INIT=2, TH(1)=-0.706218, 34 BTH(1)=-0.648650, TRAMO=-1,BLQT=8.929244,HS=1.500000,CRMEAN=0,EPSPHI=2.000000, OUT=0,RSA=0,QMAX=50,NOADMISS=1,BIAS=0, SMTR=0,THTR=-0.400000,RMOD=0.500000,MAXBIAS=0.500000,IQM=24 X-12-Arima input: transform{function=log} estimate{save=(model)} arima{model=(0 1 1)(0 1 1) ma=( 0.7073814950E+00f 0.6534618659E+00f)} x11{mode=mult trendma=23 seasonalma=s3x5 save=(d11 d12)} check{save=(acf)} FrExp08 Seats input: D=1,BD=1,P=0,BP=0,Q=2,BQ=1, LAM=0,IMEAN=0,PG=1,MQ=12,MODEL=0,INIT=2, TH(1)=-0.933254, TH(2)=0.372568, BTH(1)=-0.381034, TRAMO=-1,BLQT=11.891964,HS=1.500000,CRMEAN=0,EPSPHI=2.000000, OUT=0,RSA=0,QMAX=50,NOADMISS=1,BIAS=0, SMTR=0,THTR=-0.400000,RMOD=0.500000,MAXBIAS=0.500000,IQM=24 X-12-Arima input: transform{function=log} estimate{save=(model)} arima{model=(0 1 1)(0 1 1) ma=( 0.7630920875E+00f 0.4253977234E+00f)} x11{mode=mult trendma=13 seasonalma=s3x5 save=(d11 d12)} check{save=(acf)} FrExp09 Seats input: D=1,BD=1,P=0,BP=0,Q=1,BQ=1, LAM=0,IMEAN=0,PG=1,MQ=12,MODEL=0,INIT=2, TH(1)=-0.743045, BTH(1)=-0.593829, TRAMO=-1,BLQT=18.527110,HS=1.500000,CRMEAN=0,EPSPHI=2.000000, OUT=0,RSA=0,QMAX=50,NOADMISS=1,BIAS=0, SMTR=0,THTR=-0.400000,RMOD=0.500000,MAXBIAS=0.500000,IQM=24 X-12-Arima input: transform{function=log} estimate{save=(model)} arima{model=(0 1 1)(0 1 1) ma=( 0.7449801036E+00f 0.5928487864E+00f)} x11{mode=mult trendma=23 seasonalma=s3x5 save=(d11 d12)} check{save=(acf)} FrExp10 Seats input: D=1,BD=1,P=0,BP=0,Q=1,BQ=1, LAM=0,IMEAN=0,PG=1,MQ=12,MODEL=0,INIT=2, TH(1)=-0.696577, BTH(1)=-0.534302, TRAMO=-1,BLQT=7.808433,HS=1.500000,CRMEAN=0,EPSPHI=2.000000, OUT=0,RSA=0,QMAX=50,NOADMISS=1,BIAS=0, SMTR=0,THTR=-0.400000,RMOD=0.500000,MAXBIAS=0.500000,IQM=24 X-12-Arima input: transform{function=log} estimate{save=(model)} arima{model=(0 1 2)(0 1 1) ma=( 0.6885028428E+00f 35 0.1484695287E-01f 0.5331679452E+00f)} x11{mode=mult trendma=13 seasonalma=s3x5 save=(d11 d12)} check{save=(acf)} FrExp11 Seats input: D=1,BD=1,P=0,BP=0,Q=1,BQ=1, LAM=0,IMEAN=0,PG=1,MQ=12,MODEL=0,INIT=2, TH(1)=-0.543608, BTH(1)=-0.762261, TRAMO=-1,BLQT=11.599968,HS=1.500000,CRMEAN=0,EPSPHI=2.000000, OUT=0,RSA=0,QMAX=50,NOADMISS=1,BIAS=0, SMTR=0,THTR=-0.400000,RMOD=0.500000,MAXBIAS=0.500000,IQM=24 X-12-Arima input: transform{function=log} estimate{save=(model)} arima{model=(0 1 1)(0 1 1) ma=( 0.5438274305E+00f 0.7756838981E+00f)} x11{mode=mult trendma=13 seasonalma=s3x5 save=(d11 d12)} check{save=(acf)} FrExp12 Seats input: D=1,BD=1,P=1,BP=0,Q=0,BQ=1, LAM=0,IMEAN=0,PG=1,MQ=12,MODEL=0,INIT=2, PHI(1)=0.477178, BTH(1)=-0.899710, TRAMO=-1,BLQT=35.070401,HS=1.500000,CRMEAN=0,EPSPHI=2.000000, OUT=0,RSA=0,QMAX=50,NOADMISS=1,BIAS=0, SMTR=0,THTR=-0.400000,RMOD=0.500000,MAXBIAS=0.500000,IQM=24 X-12-Arima input: transform{function=log} estimate{save=(model)} arima{model=(0 1 1) (0 1 1) ma=( 0.4141482226E+00f 0.8635620383E+00f)} x11{mode=mult trendma=13 seasonalma=s3x5 save=(d11 d12)} check{save=(acf)} FrImp01 Seats input: D=1,BD=1,P=0,BP=0,Q=1,BQ=1, LAM=0,IMEAN=0,PG=1,MQ=12,MODEL=0,INIT=2, TH(1)=-0.469437, BTH(1)=-0.742854, TRAMO=-1,BLQT=26.281439,HS=1.500000,CRMEAN=0,EPSPHI=2.000000, OUT=0,RSA=0,QMAX=50,NOADMISS=1,BIAS=0, SMTR=0,THTR=-0.400000,RMOD=0.500000,MAXBIAS=0.500000,IQM=24 X-12-Arima input: transform{function=log} estimate{save=(model)} arima{model=(2 1 2)(0 1 1) ar=(-0.3790070744E-01f 0.2054644806E+00f) ma=( 0.4158746446E+00f 0.2081458431E+00f 0.7489245721E+00f)} x11{mode=mult trendma=13 seasonalma=s3x5 save=(d11 d12)} check{save=(acf)} 36 FrImp02 Seats input: D=1,BD=1,P=0,BP=0,Q=1,BQ=1, LAM=0,IMEAN=0,PG=1,MQ=12,MODEL=0,INIT=2, TH(1)=-0.519174, BTH(1)=-0.814682, TRAMO=-1,BLQT=26.025157,HS=1.500000,CRMEAN=0,EPSPHI=2.000000, OUT=0,RSA=0,QMAX=50,NOADMISS=1,BIAS=0, SMTR=0,THTR=-0.400000,RMOD=0.500000,MAXBIAS=0.500000,IQM=24 X-12-Arima input: transform{function=log} estimate{save=(model)} arima{model=(0 1 2)(0 1 1) ma=( 0.5107649806E+00f 0.1405083626E-01f 0.8219543211E+00f)} x11{mode=mult trendma=13 seasonalma=s3x5 save=(d11 d12)} check{save=(acf)} FrImp03 Seats input: D=1,BD=1,P=0,BP=0,Q=1,BQ=1, LAM=0,IMEAN=0,PG=1,MQ=12,MODEL=0,INIT=2, TH(1)=-0.358081, BTH(1)=-0.937180, TRAMO=-1,BLQT=18.781023,HS=1.500000,CRMEAN=0,EPSPHI=2.000000, OUT=0,RSA=0,QMAX=50,NOADMISS=1,BIAS=0, SMTR=0,THTR=-0.400000,RMOD=0.500000,MAXBIAS=0.500000,IQM=24 X-12-Arima input: transform{function=log} estimate{save=(model)} arima{model=(0 1 1) (0 1 1) ma=( 0.3603913654E+00f 0.9991202971E+00f)} x11{mode=mult trendma=13 seasonalma=s3x5 save=(d11 d12)} check{save=(acf)} FrImp04 Seats input: D=1,BD=1,P=0,BP=0,Q=1,BQ=1, LAM=0,IMEAN=0,PG=1,MQ=12,MODEL=0,INIT=2, TH(1)=-0.597559, BTH(1)=-0.833972, TRAMO=-1,BLQT=25.785357,HS=1.500000,CRMEAN=0,EPSPHI=2.000000, OUT=0,RSA=0,QMAX=50,NOADMISS=1,BIAS=0, SMTR=0,THTR=-0.400000,RMOD=0.500000,MAXBIAS=0.500000,IQM=24 X-12-Arima input: transform{function=log} estimate{save=(model)} arima{model=(2 1 0)(0 1 1) ar=(-0.6170016810E+00f -0.2754927696E+00f) ma=( 0.8087940547E+00f)} x11{mode=mult trendma=23 seasonalma=s3x5 save=(d11 d12)} check{save=(acf)} FrImp05 Seats input: D=1,BD=1,P=1,BP=0,Q=0,BQ=1, LAM=0,IMEAN=0,PG=1,MQ=12,MODEL=0,INIT=2, PHI(1)=0.580016, BTH(1)=-0.819570, TRAMO=-1,BLQT=16.770111,HS=1.500000,CRMEAN=0,EPSPHI=2.000000, OUT=0,RSA=0,QMAX=50,NOADMISS=1,BIAS=0, 37 SMTR=0,THTR=-0.400000,RMOD=0.500000,MAXBIAS=0.500000,IQM=24 X-12-Arima input: transform{function=log} estimate{save=(model)} arima{model=(2 1 0)(0 1 1) ar=(-0.7015689375E+00f -0.2035230056E+00f) ma=( 0.8312610341E+00f)} x11{mode=mult trendma=13 seasonalma=s3x5 save=(d11 d12)} check{save=(acf)} FrImp06 Seats input: D=1,BD=1,P=0,BP=0,Q=1,BQ=1, LAM=0,IMEAN=0,PG=1,MQ=12,MODEL=0,INIT=2, TH(1)=-0.496396, BTH(1)=-0.712429, TRAMO=-1,BLQT=21.718406,HS=1.500000,CRMEAN=0,EPSPHI=2.000000, OUT=0,RSA=0,QMAX=50,NOADMISS=1,BIAS=0, SMTR=0,THTR=-0.400000,RMOD=0.500000,MAXBIAS=0.500000,IQM=24 X-12-Arima input: transform{function=log} estimate{save=(model)} arima{model=(2 1 0)(0 1 1) ar=(-0.5464865572E+00f -0.2236194447E+00f) ma=( 0.7096894858E+00f)} x11{mode=mult trendma=13 seasonalma=s3x5 save=(d11 d12)} check{save=(acf)} FrImp07 Seats input: D=1,BD=1,P=0,BP=0,Q=1,BQ=1, LAM=0,IMEAN=0,PG=1,MQ=12,MODEL=0,INIT=2, TH(1)=-0.626725, BTH(1)=-0.927222, TRAMO=-1,BLQT=10.685485,HS=1.500000,CRMEAN=0,EPSPHI=2.000000, OUT=0,RSA=0,QMAX=50,NOADMISS=1,BIAS=0, SMTR=0,THTR=-0.400000,RMOD=0.500000,MAXBIAS=0.500000,IQM=24 X-12-Arima input: transform{function=log} estimate{save=(model)} arima{model=(0 1 1) (0 1 1) ma=( 0.6266837511E+00f 0.9260471749E+00f)} x11{mode=mult trendma=13 seasonalma=s3x9 save=(d11 d12)} check{save=(acf)} FrImp08 Seats input: D=1,BD=1,P=0,BP=0,Q=1,BQ=1, LAM=0,IMEAN=0,PG=1,MQ=12,MODEL=0,INIT=2, TH(1)=-0.473246, BTH(1)=-0.701354, TRAMO=-1,BLQT=23.196692,HS=1.500000,CRMEAN=0,EPSPHI=2.000000, OUT=0,RSA=0,QMAX=50,NOADMISS=1,BIAS=0, SMTR=0,THTR=-0.400000,RMOD=0.500000,MAXBIAS=0.500000,IQM=24 X-12-Arima input: transform{function=log} estimate{save=(model)} arima{model=(2 1 0)(0 1 1) ar=(-0.4651910337E+00f -0.9500174616E-01f) ma=( 0.7022830855E+00f)} 38 x11{mode=mult trendma=13 seasonalma=s3x5 save=(d11 d12)} check{save=(acf)} FrImp09 Seats input: D=1,BD=1,P=0,BP=0,Q=1,BQ=1, LAM=0,IMEAN=0,PG=1,MQ=12,MODEL=0,INIT=2, TH(1)=-0.623627, BTH(1)=-0.654008, TRAMO=-1,BLQT=27.272252,HS=1.500000,CRMEAN=0,EPSPHI=2.000000, OUT=0,RSA=0,QMAX=50,NOADMISS=1,BIAS=0, SMTR=0,THTR=-0.400000,RMOD=0.500000,MAXBIAS=0.500000,IQM=24 X-12-Arima input: transform{function=log} estimate{save=(model)} arima{model=(2 1 0)(0 1 1) ar=(-0.5352737975E+00f -0.2403309022E+00f) ma=( 0.6387507205E+00f)} x11{mode=mult trendma=13 seasonalma=s3x5 save=(d11 d12)} check{save=(acf)} FrImp10 Seats input: D=1,BD=1,P=0,BP=0,Q=1,BQ=1, LAM=0,IMEAN=0,PG=1,MQ=12,MODEL=0,INIT=2, TH(1)=-0.516122, BTH(1)=-0.548226, TRAMO=-1,BLQT=15.851545,HS=1.500000,CRMEAN=0,EPSPHI=2.000000, OUT=0,RSA=0,QMAX=50,NOADMISS=1,BIAS=0, SMTR=0,THTR=-0.400000,RMOD=0.500000,MAXBIAS=0.500000,IQM=24 X-12-Arima input: transform{function=log} estimate{save=(model)} arima{model=(0 1 2)(0 1 1) ma=( 0.4877054534E+00f 0.5116658498E-01f 0.5445323535E+00f)} x11{mode=mult trendma=13 seasonalma=s3x5 save=(d11 d12)} check{save=(acf)} FrImp11 Seats input: D=1,BD=1,P=0,BP=0,Q=1,BQ=1, LAM=0,IMEAN=0,PG=1,MQ=12,MODEL=0,INIT=2, TH(1)=-0.550019, BTH(1)=-0.562474, TRAMO=-1,BLQT=17.402828,HS=1.500000,CRMEAN=0,EPSPHI=2.000000, OUT=0,RSA=0,QMAX=50,NOADMISS=1,BIAS=0, SMTR=0,THTR=-0.400000,RMOD=0.500000,MAXBIAS=0.500000,IQM=24 X-12-Arima input: transform{function=log} estimate{save=(model)} arima{model=(0 1 1)(0 1 1) ma=( 0.5503492125E+00f 0.5615805274E+00f)} x11{mode=mult trendma=13 seasonalma=s3x5 save=(d11 d12)} check{save=(acf)} FrImp12 Seats input: D=1,BD=1,P=3,BP=0,Q=0,BQ=1, LAM=0,IMEAN=0,PG=1,MQ=12,MODEL=0,INIT=2, PHI(1)=0.381493, 39 PHI(2)=0.280700, PHI(3)=0.308912, BTH(1)=-0.980000, TRAMO=-1,BLQT=33.808182,HS=1.500000,CRMEAN=0,EPSPHI=2.000000, OUT=0,RSA=0,QMAX=50,NOADMISS=1,BIAS=0, SMTR=0,THTR=-0.400000,RMOD=0.500000,MAXBIAS=0.500000,IQM=24 X-12-Arima input: transform{function=log} estimate{save=(model)} arima{model=(0 1 1) (0 1 1) ma=( 0.4771239134E+00f 0.9992253600E+00f)} x11{mode=mult trendma=13 seasonalma=s3x5 save=(d11 d12)} check{save=(acf)} Model orders fixed, model parameters free: FrExp01 Seats input: D=1,BD=1,P=0,BP=0,Q=1,BQ=1, LAM=0,IMEAN=0,PG=1,MQ=12,MODEL=0,INIT=1, TH(1)=-0.458662, BTH(1)=-0.646929, TRAMO=-1,BLQT=15.573744,HS=1.500000,CRMEAN=0,EPSPHI=2.000000, OUT=0,RSA=0,QMAX=50,NOADMISS=1,BIAS=0, SMTR=0,THTR=-0.400000,RMOD=0.500000,MAXBIAS=0.500000,IQM=24 X-12-Arima input: transform{function=log} estimate{save=(model)} arima{model=(2 1 2)(0 1 1) ar=(-0.6146149903E+00 0.2810814997E+00) ma=(-0.1942233453E+00 0.6335310999E+00 0.6486130414E+00)} x11{mode=mult save=(d11 d12)} check{save=(acf)} FrExp02 Seats input: D=1,BD=1,P=0,BP=0,Q=1,BQ=1, LAM=0,IMEAN=0,PG=1,MQ=12,MODEL=0,INIT=1, TH(1)=-0.368447, BTH(1)=-0.601682, TRAMO=-1,BLQT=18.250837,HS=1.500000,CRMEAN=0,EPSPHI=2.000000, OUT=0,RSA=0,QMAX=50,NOADMISS=1,BIAS=0, SMTR=0,THTR=-0.400000,RMOD=0.500000,MAXBIAS=0.500000,IQM=24 X-12-Arima input: transform{function=log} estimate{save=(model)} arima{model=(0 1 2)(0 1 1) ma=( 0.3132496774E+00 0.1165852982E+00 0.6061667409E+00)} x11{mode=mult save=(d11 d12)} check{save=(acf)} FrExp03 Seats input: D=1,BD=0,P=0,BP=1,Q=1,BQ=0, LAM=0,IMEAN=0,PG=1,MQ=12,MODEL=0,INIT=1, BPHI(1)=-0.201888, TH(1)=-0.706600, 40 TRAMO=-1,BLQT=21.535236,HS=1.500000,CRMEAN=0,EPSPHI=2.000000, OUT=0,RSA=0,QMAX=50,NOADMISS=1,BIAS=0, SMTR=0,THTR=-0.400000,RMOD=0.500000,MAXBIAS=0.500000,IQM=24 X-12-Arima input: transform{function=log} estimate{save=(model)} arima{model=(0 1 1) (0 1 1) x11{mode=mult save=(d11 d12)} check{save=(acf)} FrExp04 Seats input: D=1,BD=1,P=0,BP=0,Q=1,BQ=1, LAM=0,IMEAN=0,PG=1,MQ=12,MODEL=0,INIT=1, TH(1)=-0.774064, BTH(1)=-0.923241, TRAMO=-1,BLQT=27.665483,HS=1.500000,CRMEAN=0,EPSPHI=2.000000, OUT=0,RSA=0,QMAX=50,NOADMISS=1,BIAS=0, SMTR=0,THTR=-0.400000,RMOD=0.500000,MAXBIAS=0.500000,IQM=24 X-12-Arima input: transform{function=log} estimate{save=(model)} arima{model=(2 1 0)(0 1 1) ar=(-0.7038447493E+00 -0.4339817706E+00) ma=( 0.8899731440E+00)} x11{mode=mult save=(d11 d12)} check{save=(acf)} FrExp05 Seats input: D=1,BD=1,P=0,BP=0,Q=1,BQ=1, LAM=0,IMEAN=0,PG=1,MQ=12,MODEL=0,INIT=1, TH(1)=-0.589691, BTH(1)=-0.652684, TRAMO=-1,BLQT=15.123951,HS=1.500000,CRMEAN=0,EPSPHI=2.000000, OUT=0,RSA=0,QMAX=50,NOADMISS=1,BIAS=0, SMTR=0,THTR=-0.400000,RMOD=0.500000,MAXBIAS=0.500000,IQM=24 X-12-Arima input: transform{function=log} estimate{save=(model)} arima{model=(2 1 2)(0 1 1) ar=( 0.2011812314E+00 -0.2173513383E+00) ma=( 0.9082461034E+00 -0.4553338938E+00 0.6273839741E+00)} x11{mode=mult save=(d11 d12)} check{save=(acf)} FrExp06 Seats input: D=1,BD=1,P=2,BP=0,Q=0,BQ=0, LAM=0,IMEAN=0,PG=1,MQ=12,MODEL=0,INIT=1, PHI(1)=0.560617, PHI(2)=0.304828, TRAMO=-1,BLQT=26.825666,HS=1.500000,CRMEAN=0,EPSPHI=2.000000, OUT=0,RSA=0,QMAX=50,NOADMISS=1,BIAS=0, SMTR=0,THTR=-0.400000,RMOD=0.500000,MAXBIAS=0.500000,IQM=24 X-12-Arima input: transform{function=log} estimate{save=(model)} arima{model=(0 1 2)(0 1 1) ma=( 0.6531437910E+00 41 -0.2369674316E+00 0.2524437642E+00)} x11{mode=mult save=(d11 d12)} check{save=(acf)} FrExp07 Seats input: D=1,BD=1,P=0,BP=0,Q=1,BQ=1, LAM=0,IMEAN=0,PG=1,MQ=12,MODEL=0,INIT=1, TH(1)=-0.706218, BTH(1)=-0.648650, TRAMO=-1,BLQT=8.929244,HS=1.500000,CRMEAN=0,EPSPHI=2.000000, OUT=0,RSA=0,QMAX=50,NOADMISS=1,BIAS=0, SMTR=0,THTR=-0.400000,RMOD=0.500000,MAXBIAS=0.500000,IQM=24 X-12-Arima input: transform{function=log} estimate{save=(model)} arima{model=(0 1 1)(0 1 1) ma=( 0.7073814950E+00 0.6534618659E+00)} x11{mode=mult save=(d11 d12)} check{save=(acf)} FrExp08 Seats input: D=1,BD=1,P=0,BP=0,Q=2,BQ=1, LAM=0,IMEAN=0,PG=1,MQ=12,MODEL=0,INIT=1, TH(1)=-0.933254, TH(2)=0.372568, BTH(1)=-0.381034, TRAMO=-1,BLQT=11.891964,HS=1.500000,CRMEAN=0,EPSPHI=2.000000, OUT=0,RSA=0,QMAX=50,NOADMISS=1,BIAS=0, SMTR=0,THTR=-0.400000,RMOD=0.500000,MAXBIAS=0.500000,IQM=24 X-12-Arima input: transform{function=log} estimate{save=(model)} arima{model=(0 1 1)(0 1 1) ma=( 0.7630920875E+00 0.4253977234E+00)} x11{mode=mult save=(d11 d12)} check{save=(acf)} FrExp09 Seats input: D=1,BD=1,P=0,BP=0,Q=1,BQ=1, LAM=0,IMEAN=0,PG=1,MQ=12,MODEL=0,INIT=1, TH(1)=-0.743045, BTH(1)=-0.593829, TRAMO=-1,BLQT=18.527110,HS=1.500000,CRMEAN=0,EPSPHI=2.000000, OUT=0,RSA=0,QMAX=50,NOADMISS=1,BIAS=0, SMTR=0,THTR=-0.400000,RMOD=0.500000,MAXBIAS=0.500000,IQM=24 X-12-Arima input: transform{function=log} estimate{save=(model)} arima{model=(0 1 1)(0 1 1) ma=( 0.7449801036E+00 0.5928487864E+00)} x11{mode=mult save=(d11 d12)} check{save=(acf)} FrExp10 Seats input: D=1,BD=1,P=0,BP=0,Q=1,BQ=1, LAM=0,IMEAN=0,PG=1,MQ=12,MODEL=0,INIT=1, 42 TH(1)=-0.696577, BTH(1)=-0.534302, TRAMO=-1,BLQT=7.808433,HS=1.500000,CRMEAN=0,EPSPHI=2.000000, OUT=0,RSA=0,QMAX=50,NOADMISS=1,BIAS=0, SMTR=0,THTR=-0.400000,RMOD=0.500000,MAXBIAS=0.500000,IQM=24 X-12-Arima input: transform{function=log} estimate{save=(model)} arima{model=(0 1 2)(0 1 1) ma=( 0.6885028428E+00 0.1484695287E-01 0.5331679452E+00)} x11{mode=mult save=(d11 d12)} check{save=(acf)} FrExp11 Seats input: D=1,BD=1,P=0,BP=0,Q=1,BQ=1, LAM=0,IMEAN=0,PG=1,MQ=12,MODEL=0,INIT=1, TH(1)=-0.543608, BTH(1)=-0.762261, TRAMO=-1,BLQT=11.599968,HS=1.500000,CRMEAN=0,EPSPHI=2.000000, OUT=0,RSA=0,QMAX=50,NOADMISS=1,BIAS=0, SMTR=0,THTR=-0.400000,RMOD=0.500000,MAXBIAS=0.500000,IQM=24 X-12-Arima input: transform{function=log} estimate{save=(model)} arima{model=(0 1 1)(0 1 1) ma=( 0.5438274305E+00 0.7756838981E+00)} x11{mode=mult save=(d11 d12)} check{save=(acf)} FrExp12 Seats input: D=1,BD=1,P=1,BP=0,Q=0,BQ=1, LAM=0,IMEAN=0,PG=1,MQ=12,MODEL=0,INIT=1, PHI(1)=0.477178, BTH(1)=-0.899710, TRAMO=-1,BLQT=35.070401,HS=1.500000,CRMEAN=0,EPSPHI=2.000000, OUT=0,RSA=0,QMAX=50,NOADMISS=1,BIAS=0, SMTR=0,THTR=-0.400000,RMOD=0.500000,MAXBIAS=0.500000,IQM=24 X-12-Arima input: transform{function=log} estimate{save=(model)} arima{model=(0 1 1) (0 1 1)} x11{mode=mult save=(d11 d12)} check{save=(acf)} FrImp01 Seats input: D=1,BD=1,P=0,BP=0,Q=1,BQ=1, LAM=0,IMEAN=0,PG=1,MQ=12,MODEL=0,INIT=1, TH(1)=-0.469437, BTH(1)=-0.742854, TRAMO=-1,BLQT=26.281439,HS=1.500000,CRMEAN=0,EPSPHI=2.000000, OUT=0,RSA=0,QMAX=50,NOADMISS=1,BIAS=0, SMTR=0,THTR=-0.400000,RMOD=0.500000,MAXBIAS=0.500000,IQM=24 X-12-Arima input: transform{function=log} estimate{save=(model)} arima{model=(2 1 2)(0 1 1) ar=(-0.3790070744E-01 0.2054644806E+00) 43 ma=( 0.4158746446E+00 0.2081458431E+00 0.7489245721E+00)} x11{mode=mult save=(d11 d12)} check{save=(acf)} FrImp02 Seats input: D=1,BD=1,P=0,BP=0,Q=1,BQ=1, LAM=0,IMEAN=0,PG=1,MQ=12,MODEL=0,INIT=1, TH(1)=-0.519174, BTH(1)=-0.814682, TRAMO=-1,BLQT=26.025157,HS=1.500000,CRMEAN=0,EPSPHI=2.000000, OUT=0,RSA=0,QMAX=50,NOADMISS=1,BIAS=0, SMTR=0,THTR=-0.400000,RMOD=0.500000,MAXBIAS=0.500000,IQM=24 X-12-Arima input: transform{function=log} estimate{save=(model)} arima{model=(0 1 2)(0 1 1) ma=( 0.5107649806E+00 0.1405083626E-01 0.8219543211E+00)} x11{mode=mult save=(d11 d12)} check{save=(acf)} FrImp03 Seats input: D=1,BD=1,P=0,BP=0,Q=1,BQ=1, LAM=0,IMEAN=0,PG=1,MQ=12,MODEL=0,INIT=1, TH(1)=-0.358081, BTH(1)=-0.937180, TRAMO=-1,BLQT=18.781023,HS=1.500000,CRMEAN=0,EPSPHI=2.000000, OUT=0,RSA=0,QMAX=50,NOADMISS=1,BIAS=0, SMTR=0,THTR=-0.400000,RMOD=0.500000,MAXBIAS=0.500000,IQM=24 X-12-Arima input: transform{function=log} estimate{save=(model)} arima{model=(0 1 1) (0 1 1)} x11{mode=mult save=(d11 d12)} check{save=(acf)} FrImp04 Seats input: D=1,BD=1,P=0,BP=0,Q=1,BQ=1, LAM=0,IMEAN=0,PG=1,MQ=12,MODEL=0,INIT=1, TH(1)=-0.597559, BTH(1)=-0.833972, TRAMO=-1,BLQT=25.785357,HS=1.500000,CRMEAN=0,EPSPHI=2.000000, OUT=0,RSA=0,QMAX=50,NOADMISS=1,BIAS=0, SMTR=0,THTR=-0.400000,RMOD=0.500000,MAXBIAS=0.500000,IQM=24 X-12-Arima input: transform{function=log} estimate{save=(model)} arima{model=(2 1 0)(0 1 1) ar=(-0.6170016810E+00 -0.2754927696E+00) ma=( 0.8087940547E+00)} x11{mode=mult save=(d11 d12)} check{save=(acf)} FrImp05 Seats input: D=1,BD=1,P=1,BP=0,Q=0,BQ=1, LAM=0,IMEAN=0,PG=1,MQ=12,MODEL=0,INIT=1, 44 PHI(1)=0.580016, BTH(1)=-0.819570, TRAMO=-1,BLQT=16.770111,HS=1.500000,CRMEAN=0,EPSPHI=2.000000, OUT=0,RSA=0,QMAX=50,NOADMISS=1,BIAS=0, SMTR=0,THTR=-0.400000,RMOD=0.500000,MAXBIAS=0.500000,IQM=24 X-12-Arima input: transform{function=log} estimate{save=(model)} arima{model=(2 1 0)(0 1 1) ar=(-0.7015689375E+00 -0.2035230056E+00) ma=( 0.8312610341E+00)} x11{mode=mult save=(d11 d12)} check{save=(acf)} FrImp06 Seats input: D=1,BD=1,P=0,BP=0,Q=1,BQ=1, LAM=0,IMEAN=0,PG=1,MQ=12,MODEL=0,INIT=1, TH(1)=-0.496396, BTH(1)=-0.712429, TRAMO=-1,BLQT=21.718406,HS=1.500000,CRMEAN=0,EPSPHI=2.000000, OUT=0,RSA=0,QMAX=50,NOADMISS=1,BIAS=0, SMTR=0,THTR=-0.400000,RMOD=0.500000,MAXBIAS=0.500000,IQM=24 X-12-Arima input: transform{function=log} estimate{save=(model)} arima{model=(2 1 0)(0 1 1) ar=(-0.5464865572E+00 -0.2236194447E+00) ma=( 0.7096894858E+00)} x11{mode=mult save=(d11 d12)} check{save=(acf)} FrImp07 Seats input: D=1,BD=1,P=0,BP=0,Q=1,BQ=1, LAM=0,IMEAN=0,PG=1,MQ=12,MODEL=0,INIT=1, TH(1)=-0.626725, BTH(1)=-0.927222, TRAMO=-1,BLQT=10.685485,HS=1.500000,CRMEAN=0,EPSPHI=2.000000, OUT=0,RSA=0,QMAX=50,NOADMISS=1,BIAS=0, SMTR=0,THTR=-0.400000,RMOD=0.500000,MAXBIAS=0.500000,IQM=24 X-12-Arima input: transform{function=log} estimate{save=(model)} arima{model=(0 1 1) (0 1 1)} x11{mode=mult save=(d11 d12)} check{save=(acf)} FrImp08 Seats input: D=1,BD=1,P=0,BP=0,Q=1,BQ=1, LAM=0,IMEAN=0,PG=1,MQ=12,MODEL=0,INIT=1, TH(1)=-0.473246, BTH(1)=-0.701354, TRAMO=-1,BLQT=23.196692,HS=1.500000,CRMEAN=0,EPSPHI=2.000000, OUT=0,RSA=0,QMAX=50,NOADMISS=1,BIAS=0, SMTR=0,THTR=-0.400000,RMOD=0.500000,MAXBIAS=0.500000,IQM=24 X-12-Arima input: transform{function=log} estimate{save=(model)} arima{model=(2 1 0)(0 1 1) ar=(-0.4651910337E+00 45 -0.9500174616E-01) ma=( 0.7022830855E+00)} x11{mode=mult save=(d11 d12)} check{save=(acf)} FrImp09 Seats input: D=1,BD=1,P=0,BP=0,Q=1,BQ=1, LAM=0,IMEAN=0,PG=1,MQ=12,MODEL=0,INIT=1, TH(1)=-0.623627, BTH(1)=-0.654008, TRAMO=-1,BLQT=27.272252,HS=1.500000,CRMEAN=0,EPSPHI=2.000000, OUT=0,RSA=0,QMAX=50,NOADMISS=1,BIAS=0, SMTR=0,THTR=-0.400000,RMOD=0.500000,MAXBIAS=0.500000,IQM=24 X-12-Arima input: transform{function=log} estimate{save=(model)} arima{model=(2 1 0)(0 1 1) ar=(-0.5352737975E+00 -0.2403309022E+00) ma=( 0.6387507205E+00)} x11{mode=mult save=(d11 d12)} check{save=(acf)} FrImp10 Seats input: D=1,BD=1,P=0,BP=0,Q=1,BQ=1, LAM=0,IMEAN=0,PG=1,MQ=12,MODEL=0,INIT=1, TH(1)=-0.516122, BTH(1)=-0.548226, TRAMO=-1,BLQT=15.851545,HS=1.500000,CRMEAN=0,EPSPHI=2.000000, OUT=0,RSA=0,QMAX=50,NOADMISS=1,BIAS=0, SMTR=0,THTR=-0.400000,RMOD=0.500000,MAXBIAS=0.500000,IQM=24 X-12-Arima input: transform{function=log} estimate{save=(model)} arima{model=(0 1 2)(0 1 1) ma=( 0.4877054534E+00 0.5116658498E-01 0.5445323535E+00)} x11{mode=mult save=(d11 d12)} check{save=(acf)} FrImp11 Seats input: D=1,BD=1,P=0,BP=0,Q=1,BQ=1, LAM=0,IMEAN=0,PG=1,MQ=12,MODEL=0,INIT=1, TH(1)=-0.550019, BTH(1)=-0.562474, TRAMO=-1,BLQT=17.402828,HS=1.500000,CRMEAN=0,EPSPHI=2.000000, OUT=0,RSA=0,QMAX=50,NOADMISS=1,BIAS=0, SMTR=0,THTR=-0.400000,RMOD=0.500000,MAXBIAS=0.500000,IQM=24 X-12-Arima input: transform{function=log} estimate{save=(model)} arima{model=(0 1 1)(0 1 1) ma=( 0.5503492125E+00 0.5615805274E+00)} x11{mode=mult save=(d11 d12)} check{save=(acf)} FrImp12 Seats input: D=1,BD=1,P=3,BP=0,Q=0,BQ=1, 46 LAM=0,IMEAN=0,PG=1,MQ=12,MODEL=0,INIT=1, PHI(1)=0.381493, PHI(2)=0.280700, PHI(3)=0.308912, BTH(1)=-0.980000, TRAMO=-1,BLQT=33.808182,HS=1.500000,CRMEAN=0,EPSPHI=2.000000, OUT=0,RSA=0,QMAX=50,NOADMISS=1,BIAS=0, SMTR=0,THTR=-0.400000,RMOD=0.500000,MAXBIAS=0.500000,IQM=24 X-12-Arima input: transform{function=log} estimate{save=(model)} arima{model=(0 1 1) (0 1 1)} x11{mode=mult save=(d11 d12)} check{save=(acf)} Model orders free: For all 24 time series (and all revisions) Seats input: lam=-1,interp=1,noadmiss=1,bias=0,imean=0,seats=2,inic=3,idif=3, X-12-Arima input: transform{function=log} estimate{save=(model)} automdl{method=best file=’x12a.mdl’} x11{mode=mult/add save=(d11 d12)} check{save=(acf)} Remark: mult/add decomposition depends on the result of the first automatic adjustment by Tramo Model parameters fixed, X-11: 13-term Henderson filter: FrExp04 Seats input: D=1,BD=1,P=0,BP=0,Q=1,BQ=1, LAM=0,IMEAN=0,PG=1,MQ=12,MODEL=0,INIT=2, TH(1)=-0.774068, BTH(1)=-0.923260, TRAMO=-1,BLQT=27.665645,HS=1.500000,CRMEAN=0,EPSPHI=2.000000, OUT=0,RSA=0,QMAX=50,NOADMISS=1,BIAS=0, SMTR=0,THTR=-0.400000,RMOD=0.500000,MAXBIAS=0.500000,IQM=24 X-12-Arima input: transform{function=log} estimate{save=(model)} arima{model=(2 1 0)(0 1 1) ar=(-0.7038447493E+00f -0.4339817706E+00f) ma=( 0.8899731440E+00f)} x11{mode=mult trendma=13 seasonalma=s3x5 save=(d11 d12)} check{save=(acf)} Model parameters fixed, X-11: 23-term Henderson filter: FrExp04 Seats input: D=1,BD=1,P=0,BP=0,Q=1,BQ=1, LAM=0,IMEAN=0,PG=1,MQ=12,MODEL=0,INIT=2, TH(1)=-0.774068, BTH(1)=-0.923260, TRAMO=-1,BLQT=27.665645,HS=1.500000,CRMEAN=0,EPSPHI=2.000000, OUT=0,RSA=0,QMAX=50,NOADMISS=1,BIAS=0, 47 SMTR=0,THTR=-0.400000,RMOD=0.500000,MAXBIAS=0.500000,IQM=24 X-12-Arima input: transform{function=log} estimate{save=(model)} arima{model=(2 1 0)(0 1 1) ar=(-0.7038447493E+00f -0.4339817706E+00f) ma=( 0.8899731440E+00f)} x11{mode=mult trendma=23 seasonalma=s3x5 save=(d11 d12)} check{save=(acf)} Explanation of the time series used: 24 series of the French external trade, nomenclature Nec 2, export and import, monthly data: Frexp##: French export Frimp##: French import ##: 01: Agriculture 07: Professional equipment 02: Food products 08: Electrical machinery and apparatus 03: Energy 09: Cars for households 04: Mineral goods 10: Car spare parts, commercial vehicles 05: Metallic products 11: Consumer goods 06: Non-metallic intermediate goods 12: Other goods starting date: January 1980 ending date: December 1994 length: 180 observations
