Illuminating ARIMA model-based seasonal adjustment with three fundamental seasonal models
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Findley, David F.; Lytras, Demetra P.; Maravall Herrero, Agustin Article Illuminating ARIMA model-based seasonal adjustment with three fundamental seasonal models SERIEs - Journal of the Spanish Economic Association Provided in Cooperation with: Spanish Economic Association Suggested Citation: Findley, David F.; Lytras, Demetra P.; Maravall Herrero, Agustin (2016) : Illuminating ARIMA model-based seasonal adjustment with three fundamental seasonal models, SERIEs - Journal of the Spanish Economic Association, ISSN 1869-4195, Springer, Heidelberg, Vol. 7, Iss. 1, pp. 11-52, https://doi.org/10.1007/s13209-016-0139-4 This Version is available at: https://hdl.handle.net/10419/158550 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. http://creativecommons.org/licenses/by/4.0/
SERIEs (2016) 7:11–52 DOI 10.1007/s13209-016-0139-4 ORIGINAL ARTICLE Illuminating ARIMA model-based seasonal adjustment with three fundamental seasonal models David F. Findley1·Demetra P. Lytras1· Agustin Maravall2 Received: 9 December 2014 / Accepted: 1 February 2016 / Published online: 25 February 2016 © The Author(s) 2016. This article is published with open access at Springerlink.com Abstract Our starting place is the first order seasonal autoregressive model. Its series are shown to have canonical model-based decompositions whose finite-sample estimates, filters, and error covariances have simple revealing formulas from basic linear regression. We obtain analogous formulas for seasonal random walks, extending some of the results of Maravall and Pierce (J Time Series Anal, 8:177–293, 1987). The seasonal decomposition filters of the biannual seasonal random walk have formulas that explicitly reveal which deterministic functions they annihilate and which they reproduce, directly illustrating very general results of Bell (J Off Stat, 28:441–461, 2012; Center for Statistical Research and Methodology, Research Report Series, Statistics #2015-03, U.S. Census Bureau, Washington, D.C. https://www.census.gov/srd/ papers/pdf/RRS2015-03,2015). Other formulas express phenomena heretofore lacking such concrete expression, such as the much discussed negative autocorrelation at the first seasonal lag quite often observed in differenced seasonally adjusted series. An innovation that is also applied to airline model seasonal decompositions is the effective use of signs of lag one and first-seasonal-lag autocorrelations (after differencing) to indicate, in a formal way, where smoothness is increased by seasonal adjustment and where its effect is opposite. Keywords ARIMA models ·Signal extraction smoothness ·Timeseries BDavid F. Findley david.findle[email protected] Demetra P. Lytras [email protected] Agustin Maravall amarav[email protected] 1U.S. Census Bureau, Suitland, USA 2Bank of Spain, Madrid, Spain 123
12 SERIEs (2016) 7:11–52 JEL Classification C4 ·C8 1Overview Much of this document radiates from the stationary first order seasonal autoregressive model or SAR(1), Zt=Zt−q+at,−1<<1,(1) with uncorrelated (white noise or w.n.) at, whose variance Ea2 tis denoted σ2 a.The autocovariances of Ztare γj=EZt+jZt=σ2 a1−2−1k,|j|=kq,k=0,1,... 0,otherwise. (2) See Chapter 9 of Box and Jenkins (1976) for example. Hence the autocorrelations are ρj=k,|j|=qk,k=0,1,... 0,otherwise. .(3) We only consider 0 <<1 in order to have positive correlation at the seasonal lags q,2q,.... For large enough ,(3) shows that Zthas the fundamental characteristics of a strongly seasonal time series, namely a strong tendency for year-to-year movements in the same direction, with magnitudes (relative to the underlying level, e.g. its mean zero) that change gradually more often than not. For the monthly case q=12, Fig. 1 shows that when =0.95, then even after 12 years the correlation is greater than 0.5. By contrast, when =0.70, after 5 years the correlation is negligible. Graphs of such a Zt(not shown) do not clearly indicate seasonality. Figure 3in Sect. 3.2 shows a simulated =0.95 monthly SAR(1) series Zt of length 144 displaying quite seasonal features. It also shows the residual series ˆ Nt=Zt−ˆ Stresulting from removal of the estimate ˆ Stof the unobserved signal component Stof a signal plus noise decomposition, Zt=St+Nt,(4) with uncorrelated components, ES tNt−j=0,−∞ <j<∞. The signal Stis specified to have the smallest variance γS 0<γ 0compatible with having the same nonzero-lag autocovariances as Zt,γS j=γj,j= 0. This is equivalent to specifying Ntas white noise with the largest variance possible for a w.n. component in an uncorrelated decomposition (4)ofZt. This variance, γ0−γS 0, is the minimum value of the spectral density (s.d.) of Zt. It has a simple formula in the SAR(1) case, as does the s.d., see (10) and (11) in Sect. 3.2. The graph of ˆ Ntin Fig. 3appears less smooth than Zt, and this will be established in a formal way in Sect. 13.2. The signal estimate ˆ Stis graphed by calendar month in Fig. 4.The ˆ Stvisibly smooth each of the 12 annual calendar month series of Zt,a 123
SERIEs (2016) 7:11–52 13 Fig. 1 The nonzero monthly (q=12)SAR(1) autocorrelations for seasonal lags 12, 24,…, 144 and two values of .For=0.95, the autocorrelations are still greater than 0.5 at a lag of twelve years, indicative of well defined and similar seasonal movements for a number of years, as Fig. 3confirms. For =0.70, they are negligible after 5 years, indicating substantially weaker, perhaps negligible “seasonality” property connected to the fact that the lag 12k,k≥1, autocorrelations of ˆ Stare larger than those of Zt, see Sect. 13. For any stationary Ztwith known autocovariances γj, typically from an ARMA model for Zt, the first step toward obtaining linear estimates of an uncorrelated component decomposition (4) is the determination or specification of an appropriate autocovariance decomposition,γj=γS j+γN j,j=0,1,.... The SAR(1) estimated decomposition is detailed in Sect. 3.2. For a vector of observations Z=(Z1,...,Zn), the autocovariances at lags 0 to n−1 furnish a corresponding n×nautocovariance matrix decomposition, ZZ =SS +NN [γ|j−k|]=[γS |j−k|]+[γN |j−k|].(5) This decomposition enables simplified linear regression formulas (reviewed in Sect. 4) to yield a decomposition Z=ˆ S+ˆ N, with minimum mean square error (MMSE) linear estimates (or estimators), ˆ S=βSZand ˆ N=βNZ, of the unobserved components. Such estimates are also called minimum variance estimates. Another standard formula provides the variance matrix of the estimation errors. Everything is illustrated for the two-component SAR(1) decomposition. Our most extensive analyses are for the simplest seasonal ARIMA model, the seasonal random walk or SRW, obtained by setting =1in(1), Zt=Zt−q+at.(6) 123
14 SERIEs (2016) 7:11–52 The MMSE two-component decomposition filter formulas for such nonstationary Ztcan be obtained simply by setting =1 in the stationary SAR(1) formulas. This follows from results of Bell (1984) which expand to difference-stationary series the Wiener-Kolmogorov (W-K) filter formulas presented in Sect. 6. The original W- K formulas provide MMSE component estimates from bi-infinite stationary data Zt,−∞ <t<∞and immediately reproduce the SAR(1) formulas of Sect. 5.1 for the intermediate times between the first and last years, q+1≤t≤n−q. In Sect. 7, after formally defining the pseudo-spectral density (pseudo-s.d.) of an ARIMA model, we illustrate the kinds of non-stationary W-K calculations that are done with pseudo-s.d.’s in TRAMO-SEATS (Gómez and Maravall 1996), hereafter T-S, and in its implementations in TSW (Caporello and Maravall 2004), X-13ARIMA-SEATS (U.S. Census Bureau 2015), hereafter X-13A-S, and JDemetra+ (Seasonal Adjustment Centre of Competence 2015), hereafter JD+. We derive the simple formulas of all of the filters associated with the three-component seasonal, trend, and irregular decomposition of the q=2 SRW. We proceed a little more directly than the tutorial article Maravall and Pierce (1987), which develops fundamental properties of this model’s decomposition estimates with somewhat different goals. In Sect. 7.2, we obtain the forecast and backcast results which are required to derive the asymmetric filters for initial and final years of a finite sample as well as the error variances of their estimates. These illustrate in a simple way the fundamental role of Bell’s Assumption A. In Sect. 10, we provide an extended and corrected version of Maravall and Pierce’s Table 1 giving variances and autocorrelations of the stationary transforms of the estimates. For the Box-Jenkins airline model, Sect. 12 provides graphs of the MMSE filters determined by small, medium and large values of the seasonal moving average parameter . Graphs also display the quite different visual smoothing effects of filters from such on the monthly International Airline Passenger totals series for which the model is named. This is a prelude to Sect. 13, which has the most experimental material. In a formal way based on autocorrelations of the component estimates of the different types of models considered (fully differenced in the nonstationary case), it shows where smoothness is enhanced, and where an opposite result occurs, among the seasonal decomposition components. Same-calendar-month subseries are the main setting and are considered separately from the monthly series. Complete results are presented, first for the two-component SAR(1) decomposition, next for the q=2 SRW’s threecomponent decomposition. Thereafter, smoothness results are presented for airline model series over an illustrative set of coefficient pairs. There are many formulas. In most cases, more useful for readers than their derivations or details will be to study how the formulas are used. 2 Some conventions and terminology A generic primary time series Xt, stationary or not, will be assumed to have q≥2 observations per year, with the j-th observation for the k-th year having the time index t=j+(k−1)q,1≤j≤q. For simplicity, the series of j-th values from all available years of is called the j-th calendar month subseries of Xteven when q= 12. When 123
SERIEs (2016) 7:11–52 15 q=12, these are the series of January values, the series of February values, etc., 12 series in all. Some seasonal adjustment properties, especially those of seasonal component estimates, are best revealed by the calendar month subseries. When Xtis stationary (mean EXt=0 assumed), the lag kautocorrelation of a calendar month subseries is the lag kq,ork-th seasonal autocorrelation, of Xt. Because some formulas simplify when q/2 is an integer, we only consider even q. In our examples q=2or 12. (In practice, q=3,4,6 also occur.) Some basic features of canonical ARIMA- model-based seasonal adjustment (AMBSA for short) will be related to smoothing of the calendar month subseries or detrended versions thereof, see Sect. 11. The definition of canonical is given in Sect. 3.2. Features of SEATS referred to are also features of the implementations of SEATS in X-13A-S and JD+. 3 The general stationary setting Seasonal adjustment is an important example of a time series signal extraction procedure. In the simplest setting, the observed series Ztis treated as the sum (4)of two not directly observable components, the “signal” Stof interest and an obscuring component, the “noise” Nt. In the case of stationary Ztwith known autocovariances, γj,j=0,±1,..., typically from an ARMA model, estimates of both components can be obtained from an autocovariance decomposition γj=γS j+γN j,j=0,±1,... , (7) when γS jand γN jhave properties suitable for Stand Nt. Effectively, the additive decomposition (7) implies uncorrelatedness of the signal and noise, ES t+jNt=0,j=0,±1,..., (8) see Findley (2012), which we always assume. As a consequence, for a given finite sample Zt,1≤t≤n, simplified linear regression formulas (21) summarized in Sect. 4provide a decomposition Zt=ˆ St+ˆ Nt,1≤t≤nwith MMSE estimates. 3.1 Autocovariance and spectral density decompositions The information in an autocovariance sequence γj,j=0,±1,...can be reexpressed, often both compactly and revealingly, by its spectral density function (s.d.), g(λ)=∞ j=−∞ γjei2πjλ=γ0+∞ j=1 γjei2πjλ+e−i2πjλ =γ0+2∞ j=1 γjcos 2πjλ, −1/2≤λ≤1/2. 123
16 SERIEs (2016) 7:11–52 Fig. 2 The q=12 SAR(1) spectral densities for =0.70 (darker line)and=0.95 with σ2 a= 1−2, which results in γ0=1. So the area under each graph is 1/2 (in the units of the graph). The peaks are at the trend frequency λ=0 and at each seasonal frequency, λ=k/12 cycles per year, 1 ≤k≤6, always with amplitude σ2 a(1−)−2=σ2 a(1+)( 1−)−1. The peaks for =0.70 are broader and much lower. The minimum value σ2 a(1+)−2occurs midway between each pair of peaks The second and third formulas arise from γ−j=γjand cos 2πjλ=1 2ei2πjλ+ e−i2πjλ. White noise is characterized by having a constant s.d. equal to its variance. In the AMBSA paradigm of Hillmer and Tiao (1982) implemented in SEATS, spectral densities play an essential role in specifying the canonical components, as will be demonstrated shortly. An s.d. is nonnegative, g(λ)≥0 always, see (50) for the ARMA formula. It is an even function, g(−λ)=g(λ), so it is graphed only for 0 ≤λ≤1/2. See the SAR(1) example in Fig. 2. It is integrable and for any jthe autocovariance γjcan be recovered from g(λ)as γj=1/2 −1/2 e−i2πjλg(λ)dλ=21/2 0 cos 2πjλg(λ)dλ, j=0,±1,±2,... . An autocovariance decomposition (7) is equivalent to the s.d. decomposition g(λ)=gS(λ)+gN(λ),−1/2≤λ≤1/2,(9) with gS(λ)=γS 0+2∞ j=1γS jcos 2πjλand gN(λ)=γN 0+2∞ j=1γN jcos 2πjλ, a key fact. 3.2 The SAR(1) canonical signal + white noise decomposition Conceptually attractive and unique decompositions result from the following restriction, introduced by Tiao and Hillmer (1978). An s.d. decomposition with two or more 123
SERIEs (2016) 7:11–52 17 component s.d.’s is called canonical if at most one of the components, usually a constant (white noise) s.d., has a non-zero minimum. A nonconstant s.d. (or pseudo-s.d. as defined in Sect. 7) is called canonical if its minimum value is zero. The two-component SAR(1) case provides the simplest seasonal example. By calculation from (2) or from the general ARMA formula (50) below, for a series Ztwith model (1), the s.d. g(λ)=σ2 a1−2−1∞ j=−∞ |j|ei2πjqλhas the formula g(λ)=σ2 a1−ei2πqλ−2=σ2 a1+2−2cos 2πqλ−1 ,−1/2≤λ≤1/2. (10) For q=12, Fig. 2shows an overlay plot of g(λ)for the cases =0.70 and 0.95, each with σ2 a=1−2, which results in γ0=1 for both SAR(1) processes, showing their prominent peaks at the seasonal frequencies λ=k/12,1≤k≤6 cycles per year. A canonical two-component s.d. decomposition of (10) is achieved by separating g(λ)from its minimum value, min −1/2≤λ≤1/2g(λ)=σ2 a(1+)−2,(11) which occurs at frequencies in −1/2≤λ≤1/2 where ei2πqλ=cos 2πqλ=−1, such as λ=±(2q)−1. The resulting decomposition g(λ)=g(λ)−σ2 a(1+)−2+σ2 a(1+)−2=gS(λ)+gN(λ)(12) prescribes a matrix decomposition (5) for any sample size n≥1: With ZZ = 1−2−1σ2 a|j−k|j,k=1,...,nand Ithe identity matrix of order n, ZZ =ZZ −σ2 a(1+)−2I+σ2 a(1+)−2I =SS +NN,(13) where SS and NN have the formulas indicated. Substitution into the regression formulas (21) yields estimated signal factors ˆ Stand noise factors ˆ Ntthat are exemplified in Figs. 3and 4for the simulated SAR(1) Z1,...,Z144 shown. The function gS(λ)=g(λ)−σ2 a(1+)−2, being nonnegative, even and integrable, is the s.d. of a stationary process St, a SARMA(1,1)qas a more informative formula (19) will reveal. Since gS(λ)differs from g(λ)by a constant, it too has a peak at λ=0. This is the trend frequency, the lower limit of the long-term cycle frequencies. This peak is identical in shape, and therefore contributes as much to the integral of gS(λ)over [0,1/2], as one-half of each seasonal peak at λ=k/12,1≤k≤6 and the same as the seasonal peak at λ=1/2.Hence Sthas a trend component that is not negligible compared to its seasonal component. So Stis not the seasonal component 123
18 SERIEs (2016) 7:11–52 Fig. 3 A length 144 simulated monthly =0.95 SAR(1) series and its estimated noise component ˆ Nt (darker line) from (21). The series Ztshows the consistent prominent variations by calendar month seen with quite seasonal time series. The oscillations of ˆ Ntare considerably smaller yet ˆ Ntcan be considered somewhat less smooth than Ztalso after the difference of scale is taken into account, see Sect. 13.2 Fig. 4 The 12 calendar month subseries of Fig. 3overlaid with their estimated signal component ˆ Stvalues (darker line) from Sect. 5. For each month, the horizontal line shows the calendar month average of the ˆ St. The ˆ Stclosely follow all but the most rapid movements of the series, but with fewer changes of direction over the 12 years. Autocorrelation properties help to explain why they evolve somewhat more smoothly than the Ztsubseries. Their slightly reduced standard deviation explains why they have slightly reduced extremes, see Sect. 13.3 of Zt. The decomposition we will obtain from (13) can be regarded as a smoothnonsmooth decomposition, see Sect. 13. (SEATS and its implementations calculate an algebraically more complex canonical seasonal, trend, and irregular decomposition for an SAR(1) model to estimate its seasonal and nonseasonal components, see Findley et al. 2015 for the derivation.) Further insight into the properties of Stcome from a formula for gS(λ)that displays the autocovariances of Stexplicitly: 123
SERIEs (2016) 7:11–52 25 filters, (1+)−2{−Bq+1}and (1+)−2{Bq+(+2)}respectively, could be applied to all Ztafter the first year, q+1≤t≤n, but are only MMSE in the final year. 5.2 The error covariances of the SAR(1) estimates For the SAR(1) model, the formulas (27), (18) and (21) yield =NN −ˆ Nˆ N =σ2 NI−σ2 N−1 ZZ=σ2 N(I−βN)=σ2 NβS.(43) Hence, for q=2 and n=7, from (18) and (32), =σ2 a (1+)4 ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 2+000 0 0 02+000 0 02000 0020 0 0002 0 00002+0 000002+ ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ,(44) which reveals the general pattern. The error variances of the initial and final years are larger than the error variance 2σ2 a(1+)−4at intermediate times by the amount σ2 a2(1+)−4. This is the mean square error2of using (1+)−2{Zt}to backcast/forecast (1+)−2Zt±qin (34) and (37), since from (2)wehave EZt±q−Zt2=1+2γ0−2γq=1−2γ0=σ2 a.(45) The intermediate-time mean square error is γ 0=ESt−ˆ St2=ENt−ˆ Nt2 =2σ2 N(1+)−2=2(1+)−4σ2 a. On the scale of the variance σ2 Nof Nt,thisis γ 0/σ2 N=2(1+)−2, which is approximately 0.4997 for =0.95 and therefore quite substantial. By contrast, for Stwe have γ 0/γ S 0=(1−)( 1+)−2from (15). This is approximately 0.013 for =0.95. The fact that the intermediate-time mean square error has the same positive value for all n≥5 reminds us that the error does not become negligible with large n. The two-component SAR(1) decomposition derived above has exceptional pedagogical value because of the simplicity of its filter and error variance formulas derived above. With the aid of results of Bell (1984), a rederivation of the filter formulas via the Wiener-Kolmogorov formulas, presented next, makes possible a quick transition 2With more general models for Zt, more forecasts and backcasts are needed, and their error covariances at nonzero leads and lags occur in the more complex mean square error formulas. 123
26 SERIEs (2016) 7:11–52 to the nonstationary SRW model (6) and its canonical MMSE two-component decomposition filter and error autocovariance formulas, all obtained by setting =1inthe SAR(1) formulas above. 6 Wiener–Kolmogorov formulas applied to SAR(1) and SRW models Initially for the two-component case (4) with bi-infinite data, we consider a fundamental and relatively simple approach to obtaining MMSE decomposition estimates. It also applies to the ARIMA case under a productive assumption of Bell (1984) discussed and applied in Sect. 7. 6.1 Filter transfer functions and the input–output spectral density formula Not only finite filter formulas but also general bi-infinite filter formulas Yt= ∞ j=−∞ βjXt−jare usefully reexpressed as Yt=β(B)Xtwith filter β(B)= ∞ j=−∞ βjBj. The s.d. of the filter output series Ytis related to the input series s.d. gX(λ)by the fundamental formula gY(λ)=βei2πλ2 gX(λ),(46) see (4.4.3) of Brockwell and Davis (1991). The function βei2πλis called the transfer function of the filter β(B)and βei2πλ2is its squared gain. When a filter’s transfer function βei2πλis known, then the filter coefficients can be obtained from it, in general by integration βj=1/2 −1/2 e−i2πjλβei2πλdλ, j=0,±1,... , but in practice, for ARMA or ARIMA related transfer functions, by algebraic/numerical algorithms such as those encoded in SEATS. For example, the transfer function of ˆ Stin (42)is βSei2πλ=(1+)−2ei2πqλ+2+e−i2πλ =(1+)−21+ei2πqλ2 . Hence from (46), the spectral density of ˆ Stis gˆ S(λ)=2(1+)−41+ei2πqλ4 g(λ) =2(1+)−4σ2 a1+ei2πqλ4 1−ei2πqλ2.(47) 123
SERIEs (2016) 7:11–52 27 A stationary ARMA series Zthas a representation ϕ(B)Zt=ϑ(B)at, (48) with AR and MA polynomials ϕ(B)=1−φ1B−···−φrBrand ϑ(B)=1−θ1B− ···−θmBmsatisfying ϑ(0)=ϕ(0)=1,ϕ(z)= 0for |z|≤1,ϑ(z)= 0for |z|<1,(49) where atis white noise with variance denoted σ2 a.(ϑis script θ,ϕ is script φ.) The general ARMA s.d. formula, g(λ)=σ2 aϑei2πλ2 ϕei2πλ2,(50) follows from two applications of (46) as in Brockwell and Davis (1991, p. 123). Conversely, if (50) and (49) hold, then so does (48) for some w.n. process atwith variance σ2 a. This fact can be used to identify ARMA models for bi-infinite data component estimates. For example, from (47) and (50), ˆ Sthas the noninvertible SARMA(1,2)q model 1−Bqˆ St=1+Bq2bt,σ 2 b=2(1+)−4σ2 a.(51) Note from (50) that an ARMA model is noninvertible, i.e. ϑei2πλ=0forsomeλ, if and only if its spectral density is zero at some λ. 6.2 The W–K formulas For a stationary series Ztwith a spectral density decomposition (9) specifying a two-uncorrelated-component decomposition Zt=St+Nt,Kolmogorov (1939) and Wiener (1949) independently derived the formulas βSei2πλ=gS(λ) g(λ),β Nei2πλ=gN(λ) g(λ),(52) of the transfer functions of each component’s MMSE linear estimate ˆ St=∞ j=−∞ βS jZt−j=⎛ ⎝∞ j=−∞ βS jBj⎞ ⎠Zt, ˆ Nt=∞ j=−∞ βN jZt−j=⎛ ⎝∞ j=−∞ βN jBj⎞ ⎠Zt,(53) 123
28 SERIEs (2016) 7:11–52 from bi-infinite data, Zτ,−∞ <τ<∞. For decompositions with more components, the same ratio form applies: each component estimate’s transfer function is the ratio of its spectral density to g(λ). The filters are finite only when Zthas an AR model but at most one component also does. The W–K formulas are implemented in SEATS using an algorithm of Tunnicliffe Wilson, presented in Burman (1980), which yields finite-sample MMSE estimates. From a moderate number of forecasts and backcasts, the algorithm efficiently calculates the result of using backcasts and forecasts for the infinitely many missing past and future ARMA or ARIMA data. This has led to all finite-sample MMSE component estimates being confusingly called Y–K estimates in some of the literature. The bi-infinite error processes t=St−ˆ Stand −t=Nt−ˆ Ntare stationary with spectral density g(λ)=gS(λ)gN(λ) g(λ),(54) see Whittle (1963) or Appendix A of the research report Findley et al. (2015)for derivations of (52) and (54). The formulas (52) and (54) are analogues of the matrix formulas of (21) and (29). Finite-sample mean square errors variances and covariances cannot be obtained from g(λ). In their place, the measures of uncertainty output by SEATS for its finite-sample estimates are measures for semi-infinite data estimates, used as approximations. JD+ obtains exact finite-sample measures from state space algorithms. 6.3 SAR(1) intermediate-time filters again and an alternate model form As a simple W-K application, for the SAR(1), from (10) and (17), the intermediate-time ˆ Ntfilter has the transfer function βNei2πλ=gN(λ) g(λ)=σ2 N σ2 a1−ei2πqλ2 =(1+)−2−ei2πqλ+1+2−e−i2πqλ.(55) Substituting Bjfor ei2πjλand B−jfor e−i2πjλyields (41), and (42) follows similarly using βS(B)=1−βN(B). We enter new territory by substituting Zt=(1−Bq)−1atinto (41). This yields the model formula ˆ Nt=(1+)−21−B−qat=(1+)−2{at−at+2},(56) the forward-time form of a seasonal MA(1). This form is used by SEATS for revision variance calculations illustrated in Maravall and Pierce (1987). It will also be the form provided by the most direct derivations of seasonal random walk component models in later sections. 123
SERIEs (2016) 7:11–52 29 Remark A formula of the usual backward-time form, ˆ Nt=(1+)−2(ct−ct−2), can be obtained for (56) with ct=1−B2−11−B−2at. Complex conjugation preserves magnitude, so 1−ei2π2λ−21−e−i2π2λ2=1 for all λ. Thus, from (46), the spectral densities satisfy gc(λ)=ga(λ)=σ2 ashowing that ctis white noise with variance σ2 a. The expanded formula ct=1−B−2∞ j=0jat−2j, shows that ctis a function of future and past at, specifically of at+2−2j,0≤j<∞. Analogues of these results hold for all forward-time moving average models derived below. 6.4 Going nonstationary with the seasonal random walk The ARIMA W-K formula generalization of Bell (1984), discussed in detail in the next section, shows that setting =1in(41)–(42) yields the MMSE estimates of the two-component decomposition Zt=ˆ St+ˆ Ntof the SRW model (6) with ˆ St=1 41+Bq1+B−qZt,ˆ Nt=1 41−Bq1−B−qZt.(57) This is the result obtained with =1in(42) and (41). From (57), the forecasting and backcasting results of Sect. 7.2 will yield that the initial year and final year estimates of ˆ Ntand ˆ Stare obtained by setting =1in(34) and (36) and in (39) and (40), respectively. Also, error variances and covariances of the estimates are obtained by setting =1in(44) and in the general matrix result described below (44). The formulas (57) start from the directly verifiable decomposition of g(λ)=σ2 a 1−ei2πqλ2,(58) given by g(λ)=1 4σ2 a1+ei2πqλ2 1−ei2πqλ2+1 4σ2 a=gS(λ)+gN(λ),(59) which also follows from setting =1 in the formulas associated with the component s.d.’s of (12). The functions g(λ)and gS(λ)are pseudo-spectral densities, see (61), of the model (6) and the SARIMA (1,1)qmodel of ˆstobtained by setting =1in (51). In Sect. 8, we will detail the seasonal, trend, irregular decompositions of (58) and the SRW with q=2. 7 ARIMA component filters from pseudo-spectral density decompositions For an ARIMA Ztwith degree d≥1 differencing operator δ(B)=1−δ1B−···− δdBdand model 123
30 SERIEs (2016) 7:11–52 ϕ(B)δ(B)Zt=ϑ(B)at,(60) the pseudo-spectral density (pseudo-s.d.) is defined by g(λ)=σ2 aϑei2πλ2 δei2πλϕei2πλ2=σ2 aϑei2πλ2 δei2πλ2ϕei2πλ2.(61) Its integral is infinite because of the λat which δei2πλ=0. In the nonstationary signal plus nonstationary noise case of interest, δ(B)= δS(B)δN(B), and δSei2πλand δNei2πλhave no common zero. In the seasonal plus nonseasonal case, δSei2πλhas zeroes only at seasonal frequencies λ=k/q,k=±1,...,q/2, and δNei2πλ=0 only for λ=0, as with δS(B)=1+B+···+Bq−1and δN(B)=(1−B)2for δ(B)=(1−B)( 1−Bq) of the airline model. The pseudo-s.d. g(λ)must be decomposed into a sum of seasonal and nonseasonal pseudo-s.d.’s associated with δS(B)and δN(B), respectively. Under mild assumptions described below, Bell (1984) established the MMSE optimality of the pseudo-spectral generalization of the W-K transfer function formulas for ARIMA component signal extraction. Tiao and Hillmer (1978), Burman (1980), and Hillmer and Tiao (1982) developed the canonical approach used with extensions and refinements in SEATS and its implementations. The last reference provides a number of examples of canonical seasonal, trend and irregular pseudo-s.d. decompositions g(λ)=gs(λ)+gp(λ)+gu(λ),(62) as well as examples of ARIMA models whose pseudo-s.d. does not admit such a decomposition (e.g. airline models with certain parameter values), the inadmissible case. Stationary components in addition to the irregular occur with s.d.’s for cyclical or other “transitory” components, see Gómez and Maravall (1996) and Kaiser and Maravall (2001). SEATS has options for several. Generalizing the stationary case definition, a pseudo-s.d. decomposition is canonical if, with at most one exception, its component pseudos.d.’s and s.d.’s have minimum value zero, as in (59). 7.1 Bell’s assumptions In addition to the requirements on δS(B)and δN(B),Bell (1984) also requires that the series δS(B)Stand δN(B)Ntbe uncorrelated. This can be obtained “automatically” from the implied s.d. decomposition gδ(B)Z(λ)=gδ(B)S(λ)+gδ(B)N(λ),seeFindley (2012). From here, Bell’s Assumption A provides MMSE optimality for finite-sample component estimates from the matrix formulas of McElroy (2008), for bi-infinite data estimates from pseudo-spectral W-K formulas, and for semi-infinite data estimates like those considered in Bell and Martin (2004). Bell’s Assumption A: For δ(B)of degree d,the d initial values, say Z1,...,Zd, are uncorrelated with the bi-infinite ARMA series wt=δ(B)Zt,−∞ <t<∞, which generates the bi-infinite Ztvia 123
SERIEs (2016) 7:11–52 31 Zt=wt+δ1Zt−1+···+δdZt−d,t>d, Zt−d=δ−1 d(wt+δd−1Zt−d+1+···+δ1Zt),t<1.(63) An assumption is necessary. Inference about the initial values is impossible because they are nonstationary and there is one observation of each. Assumption A provides both signal extraction optimality and, as we demonstrate next, a more fundamental MMSE optimality, that of ARIMA forecasts as commonly produced. 7.2 Assumption A yields MMSE forecasts and backcasts Forecasts of ARIMA Ztare traditionally obtained as in Box and Jenkins (1976,Ch. 5), namely by generating future Zt,t>dfrom initial Z1,...,Zdand subsequent wt=δ(B)Zt,t>d, replacing unobserved wtby their MMSE ARMA forecasts to obtain desired forecasts of Zt. The MMSE optimality of such forecasts follows somewhat straightforwardly from the forecasting results obtained under Assumption A in the third text paragraph on p. 652 of Bell (1984). For backcasts the reverse recursion in (63)isused. We illustrate this procedure by deriving results we require for Ztfrom the biannual SRW model, 1−B2Zt=at,(64) which is analyzed in detail in the next section. The white noise variates atare all uncorrelated with one another and, under Assumption A, also with the initial values Z1,Z2of any n≥2 observations Z1,...,Zn.Fort>2, repeated application of the recursion Zt=Zt−2+atyields all Zt,t≥3 from the left hand formulas of (65) and (66) below: odd “months” Z2(m+k)−1and even “months” Z2(m+k)m≥1,k≥1are generated independently. The right most white noise sum in (65) shows the forecast error when Z2m−1is the forecast of Z2(m+k)−1from data Z1,Z2,...,Z2m−1or from Z1,Z2,...,Z2m−1,Z2m. The right most white noise sum in (66) shows the error of the forecast Z2mof Z2(m+k),k≥1fromZ1,Z2,...,Z2m−1,Z2m: Z2(m+k)−1=Z1+ m+k i=1 a2i−1=Z2m−1+ m+k i=m+1 a2i−1,(65) Z2(m+k)=Z2+ m+k j=1 a2j=Z2m+ m+k i=m+1 a2i.(66) For any k≥1,Z2m−1is the MMSE forecast of Z2(m+k)−1because each term of the error m+k i=m+1a2i−1is uncorrelated with the data, as explained above. Similarly Z2mis the MMSE forecast of Z2(m+k),k≥1, because its error m+k i=m+1a2i−1is uncorrelated with Z1,Z2,...,Z2m−1,Z2m. In both cases the mean square error is kσ2 a. For backcasts the result that, for all k≥1,Z1and Z2are the MMSE optimal backcasts of Z−1−2(k−1)and Z−2(k−1)respectively, follows from Assumption A and 123
32 SERIEs (2016) 7:11–52 the analogues of (65) and (66) yielded by the backward form Zt−2=Zt−atof the recursion used above. 8 The 3-component q=2 SRW decomposition For (64), paralleling what the software does with more complex models, the differencing operator δ(B)=1−B2is factored into the product of the seasonal sum operator δs(B)=1+Band the differencing operator for the trend δp(B)=1−B. The next step is a partial fraction calculation (see Wikipedia (2011)) to decompose the pseudos.d. g(λ)of Ztinto a sum of two ratio functions, one with δsei2πλ2=1+ei2πλ2 as the denominator and the other with δpei2πλ=1−ei2πλ2as denominator. For σ−2 ag(λ)=1 1−ei2π2λ2=1 1+ei2πλ21−ei2πλ2, we seek α1,α 2such that for all λ, α1 1+ei2πλ2+α2 1−ei2πλ2=α11−ei2πλ2+α21+ei2πλ2 1+ei2πλ21−ei2πλ2 =1 1−ei2π2λ2. This requires α11−ei2πλ2+α21+ei2πλ2=α12−e−i2πλ −ei2πλ +α22+e−i2πλ +ei2πλ=1, for all λ, which is equivalent to 2 (α1+α2)=1,−α1+α2=0, whose unique solution is α1=α2=1/4. Thus we have an initial decomposition, σ−2 ag(λ)=1 41+ei2πλ−2+1 41−ei2πλ−2 . Both pseudo-s.d.’s on the right have a nonzero minimum value of 1/16. So the canonical 3-component pseudo-s.d. decomposition is g(λ)=gs(λ)+gp(λ)+gu(λ),(67) with σ−2 ags(λ)=1 41+ei2πλ−2−1/16,σ−2 agp(λ)=1 41−ei2πλ−2−1/16, and σ−2 agu(λ)=1/8. (Note our use of lower case letters, e.g. sinstead of S,for components of three-component decompositions.) It is easily verified that 123
SERIEs (2016) 7:11–52 33 gs(λ)=σ2 a 16 1−ei2πλ2 1+ei2πλ2,gp(λ)=σ2 a 16 1+ei2πλ2 1−ei2πλ2,gu(λ)=σ2 a 8.(68) Thus stand ptare nonstationary, with differencing polynomials δs(B)=1+Band δn(B)=1−B, with δn(B)also the differencing polynomial of sat=pt+ut.Using the W-K formulas as before, e.g. βsei2πλ=gs(λ)/g(λ), one obtains from (68)the decomposition’s transfer functions βsei2πλ=1 16 1−ei2πλ4 ,β pei2πλ=1 16 1+ei2πλ4 , βuei2πλ=1 81−ei2π2λ2 .(69) For the seasonal adjustment filter, we have βsa ei2πλ=1−βsei2πλ. These provide the estimates ˆst=βs(B)Zt,ˆpt=βp(B)Zt,ˆut=βu(B)Ztand sat= (1−βs(B)) Ztfrom bi-infinite data, and also for intermediate times 3 ≤t≤n−2 from n≥3 observations. 8.1 The estimates’ symmetric filters and ARIMA models For translations from squared gains jαjei2πqλ2to filters, we adopt a useful convention of Maravall and Pierce (1987) which uses replacement of e±i2πjλby B±jto obtain the symmetric filter formula, jαjBj2=jαjBjjαjB−j.(70) For example, |1+B|2=(1+B)1+B−1=B+2+B−1, so from (69), the trend filter is βp(B)=1 16 |1+B|4=1 16 (1+B)21+B−12 =1 16 B2+4B+6+4B−1+B−2. To obtain the (forward-time) moving average polynomial of each estimates’ model, we use the result that if β(B)=δ(B)˜ β(B), then the substitution Zt=δ(B)−1atis allowed in β(B)Ztand results in β(B)Zt=˜ β(B)at,see Theorem 4.10.1 of Brockwell and Davis (1991). Applying this to (1−B)ˆpt= 1 16 (1−B)( 1+B)21+B−12Zt=1 16 1−B2(1+B)1+B−12Zt,we have (1−B)ˆpt=1 16 (1+B)1+B−12 at=1 16 B+3+3B−1+B−2at. 123
34 SERIEs (2016) 7:11–52 So ˆpthas an ARIMA (0,1,3) model, noninvertible due to the factors (1+B)and 1+B−12=B−2(1+B)2. Similar calculations from (69) provide the symmetric filters and ARIMA or ARMA models of the other estimates of three-component q=2 SRW decomposition shown below. The factored filter formulas reveal the differencing factor(s) latent in each filter, causing each model to be noninvertible. Further consequences are described in Sect. 9. Also, we difference each estimate appropriately to obtain its model’s stationary moving average component, for calculation of autocorrelations of its stationary transform for tables in later sections. The MA order is noted after each model. βs(B)=1 16 (1−B)21−B−12 =1 16 B2−4B+6−4B−1+B−2,(71) (1+B)ˆst=1 16 1−B21−B−12 at =1 16 {−at−1+3at−3at+1+at+2}MA(3) (72) βsa (B)=1−βs(B)=1 16 −B2+4B+10 +4B−1−B−2 =1 16 −1+6B−1−B−2(1+B)2 (1−B)sat=1 16 −1+6B−1−B−2(1+B)at =1 16 {−at−1+5at+5at+1−at+2}MA(3) (73) βu(B)=1 81−B21−B−2=1 8−B2+2−B−2(74) ˆut=1 81−B21−B−21−B2−1 at =1 81−B−2atMA(2) (75) Apart from the seasonal adjustment formulas, the preceding formulas are equivalent to canonical filter and model formulas in Maravall and Pierce (1987), who in addition consider non-canonical MMSE estimates. Maravall (1994, p. 169ff) shows how the forward-time form of the MA polynomials can facilitate economic analyses. 8.2 The estimates’ asymmetric filter formulas As in Sect. 5.1, we can use MMSE forecasts and backcasts, now from Sect. 7.2,to obtain the filters for the MMSE estimates from the initial and final years. We give finalyear examples for a series of odd length n=2m+1. (Their time reverses provide the initial year filters). The factored formulas reveal the differencing operator factor, often of lower degree than in the symmetric filter. Starting with the concurrent filters, the asymmetric seasonal and seasonal adjustment filters for the last year are: 123
SERIEs (2016) 7:11–52 41 Fig. 9 When θ=0.9and=0.9, the effective length of the filter is the length of the data span (also for somewhat larger n>97 ). The filter resists domination by rapid changes of Ztvalues. However, revisions from Zn+12 ,Zn+24,...need not diminish in size over time and could cumulatively be large Fig. 10 Having =0.0 results in erratic movements in the seasonal factors estimated from the Airline Passenger series data due to close tracking of detrended series movements by a filter similar to Fig. 7.This leads to strong smoothing, see Fig. 12, and a potential for large revisions estimates for the International Airline Passenger data from the filters determined by small, intermediate and large values of , always with θ=0.6. The coefficient values were specified as fixed in X-13ARIMA-SEATS and thus are not data-dependent. (For the Airline Passenger series the estimates are ˆ θ, ˆ =(0.4,0.6)). The factors from =0.0 change rapidly, resulting in excessive smoothing of the seasonal adjustment, see Fig. 12, and in large revisions (not shown). For =0.9 the seasonal factors are effectively fixed and not locally adaptive. They thus have small revisions (whose cumulative effect over time can be large). 123
42 SERIEs (2016) 7:11–52 Fig. 11 With =0.9, the calendar month seasonal factor estimates from the Airline Passenger series are almost fixed across the 12 years, as in Fig. 9. For a series with changing seasonality, such factors can leave residual seasonality, see Fig. 12 Fig. 12 Two extremes. The very smooth, trend-like seasonal adjustment of the Airline Passenger series shown is obtained from division by the volatile =0.0 seasonal factors of Fig. 10, whose text explains why revisions from future data are likely to be large. By contrast, the nearly stable =0.9 same-calendar- month factors of Fig. 11 produce a much less smooth adjustment, with residual seasonality visible in the later years against the background of the original series 13 Smoothness properties of the models’ estimates 13.1 Simplistic autocorrelation comparison criteria for smoothness and nonsmoothness We begin our autocorrelation-based consideration of the smoothing properties of estimates. The simplistic definitions used will support a systematic analysis that provides insight regarding seasonal decompositions. 123
SERIEs (2016) 7:11–52 43 Given two stationary series Xtand Yt, we say that Xtis smooth if ρX 1>0. If also ρX 1>ρ Y 1, then Xtis smoother than Yt.IfρY 1<0, the series Ytis nonsmooth. A smooth series is therefore smoother than all white noise series and all nonsmooth series. If Ytis nonsmooth and ρY 1<ρ X 1holds, then Ytis more nonsmooth than Xt.A nonsmooth series is thus more nonsmooth than all white noise series and all smooth series. To examine if visual impressions of smoothness or nonsmoothness align with the conclusions of these formal criteria, differences of scale must be accounted for, see Sect. 13.2 and 13.3 and associated figures for illustrations. We make smoothness/nonsmoothness comparisons between the series Ztand its intermediate-time/bi-infinite-data component estimates. This is done for the two SAR(1) component estimates in Sect. 13.2 and 13.3. In the nonstationary cases, the comparisons are made for the stationary decompositions δ(B)Zt=δ(B)ˆst+δ(B)ˆpt+δ(B)ˆut=δ(B)ˆst+δ(B) sat.(80) These represent MMSE optimal three- and two-component decompositions of δ(B)Zt. For example, the MMSE property of ˆst, that ˆst−stis uncorrelated with Zτ for all tand τ, immediately yields that δ(B)ˆst−stis uncorrelated with δ(B)Zτ for all tand τ. When a series considered is a calendar month subseries, the autocorrelations considered are the seasonal autocorrelations in the time scale of the original Zt.We sometimes say that the conclusion of smoother is strengthened if the relevant autocorrelation at lag 2 (and perhaps consecutive higher lags) is also positive. This calls attention to the stronger smoothness properties of monthly trends and calendar month seasonal factors or their stationary transforms. Other intensifiers are only suggestive and will not be formally defined. 13.2 SAR(1): increased nonsmoothness of ˆ Nt By direct calculation from (33) or from its seasonal MA(1) model (56), the autocorrelations of intermediate-time ˆ Ntare ρˆ N j=−1+2−1,j=q, 0j= q.(81) Whereas calendar month series of Ztare smooth because ρq=>0,ρ ˆ N q<0 so ˆ Nthas more nonsmooth calendar month series than Zt. Of course, the calendar month subseries of ˆ Ntare less variable than those of Ztin the sense that, from (56), γˆ N 0=(1+)−41+2σ2 a=(1−)1+2(1+)−3γ0is less than γ0. Consequently, the scale of the ˆ Ntis related to that of the Ztthrough !γˆ N 0=!(1−)1+2(1+)−3√γ0.(82) 123
44 SERIEs (2016) 7:11–52 Fig. 13 Calendar month plots of the SAR(1) intermediate-time white noise factor estimates ˆ Nt(darker line) and the rescaled Zt, downscaled to have the same standard deviation as the ˆ Ntfor =0.95. The horizontal lines are the calendar month averages of the rescaled Zt.The ˆ Ntcalendar month subseries are visually less smooth than the rescaled Zt, in alignment with result of the lag 12 autocorrelation analysis The scale reduction factor !(1−)1+2(1+)−3is approximately 0.113 for =0.95. This factor quantifies the diminished scale of oscillations about the level value 0 seen for the intermediate years in Fig. 3. Figure 13 shows calendar month graphs of the nonsmooth component ˆ Ntand the scale-reduced Zt, the latter downscaled to have ˆ Nt’s standard error. ˆ Ntis visibly less smooth, in alignment with the formal conclusion. 13.3 SAR(1): greater calendar month smoothness of ˆ St Directly calculating the seasonal lag autocovariances of Zt−q+2Zt+Zt+qand (3), we obtain from (38) that the variance and the nonzero autocovariances γˆ S kq,k≥1of intermediate-time ˆ Stare γˆ S 0=Eˆ S2 t=γ0 2 (1+)42(+3)( 1+) γˆ S q=Eˆ Stˆ St+q=γ0 2 (1+)44+3+2(1+) γˆ S 2q=Eˆ Stˆ St+2q=γ0 2 (1+)4(1+)4 γˆ S kq =k−2γˆ S 2q,k≥3.(83) Division by γˆ S 0yields the intermediate-time autocorrelations (84). 123
SERIEs (2016) 7:11–52 45 Table 2 Autocorrelations of fully differenced estimates Estimate γ0/σ2 aρ1ρ2ρ3ρ4 1−B2ˆst70/162−0.80 0.40 −4 35 . =−0.114 1 70 . =0.014 1−B2ˆpt70/1620.80 0.40 4 35 . =0.114 1 70 . =0.014 1−B2 sat134/16272 134 . =0.537 −2 67 . =−.030 0 1 134 . =0.007 1−B2ˆut24/1620−2 3 . =−.667 0 1 6 . =0.167 ρˆ S j=⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ 1 2(4+(3+)) (+3)−1,|j|=q 1 2(1+)3(+3)−1|j|=2q, k−2ρˆ S 2q,|j|=kq,k≥3, 0,|j|= 0,kq. (84) Using (3+)−1>1/4 and 1 2(1+)>for 0 <<1, one readily obtains from (84), (16) and (3) that the intermediate-time calendar month autocorrelations of ˆ Stdominate those of Zt, ρˆ S kq >ρ S kq > k>0,k≥1.(85) By calendar month, ˆ Stis smoother than Ztin alignment with the visual impression from Fig. 4.(Thescaleof ˆ Stis only slightly smaller than the scale of Zt:From(83), we have !γˆ S 0/γ0. =0.981 for =0.95). 13.4 Smoothness properties of q=2 SRW component estimates For q=2 SRW smoothness results, we require autocorrelations of the fully differenced components of (80): 1−B2ˆst=1 16 B−2(1−B)4at=1 16 {at−2−4at−1+6at−4at+1+at+2} 1−B2ˆpt=1 16 B−2(1+B)4at=1 16 {at−2+4at−1+6at+4at+1+at+2} 1−B2ˆut=2 16 B−21−B22 at=−2B2+4−2B−2at =1 16 {−2at−2+4at−2at+2} 1−B2 sat=1 16 {−at−2+4at−1+10at+4at+1−at+2} These formulas yield the autocorrelations of Table 2. 123
46 SERIEs (2016) 7:11–52 δ(B)Zt=atis white noise, so ρδ(B)Z j=0,j>0. Thus in Table 2,ρ2>0 indicates a differenced estimate with smoother calendar month series than δ(B)Zt, whereas ρ1<0 indicates a differenced estimate whose monthly series is more nonsmooth than monthly δ(B)Zt, etc. Regarding the monthly series: as determined by ρ1,δ(B) satand δ(B)ˆptare smoother than δ(B)Ztwith strengthened smoothness. The seasonal’s δ(B)ˆstis more nonsmooth than δ(B)Zt. Regarding calendar month series, the seasonal adjustment’s, δ(B) satand especially the irregular’s δ(B)ˆut’s are more nonsmooth than δ(B)Zt. The seasonal’s δ(B)ˆstand the trend’s δ(B)ˆpt are smoother than δ(B)Zt, with strengthened smoothness. 13.5 Airline model component estimates: autocorrelations after full differencing 13.5.1 Empirical lag 12 autocorrelation results of McElroy (2012) for seasonal adjustments With a set of 88 U.S. Census Bureau economic indicator series for which the airline model was selected over alternatives, McElroy (2012) found that all but one had negative lag 12 sample autocorrelation in the fully differenced seasonally adjusted series, δ(B) satin our notation. This negative autocorrelation is statistically significant at the 0.05 level for 46 of the series. From the perspective of detrended calendar month series, which seem always to be visually smoothed by seasonal factor estimates (in logs when appropriate for AMBSA), this should not a be surprising result–removal of a smooth component causes loss of smoothness, which negative autocorrelation can formally identify. 13.5.2 Correct model results for various θ, and component estimates For monthly series from (79), with δ(B)=(1−B)1−B12, we examine the estimates in the corresponding version of (80). The autocorrelations of δ(B)Zt= at−θat−1−at−12 +θat−13 of interest are ρδ(B)Z 1(,θ)=−θ1+θ2−1 ,ρ δ(B)Z 12 (,θ)=−1+2−1 .(86) For any choices of −1<θ<1 and 0 ≤<1, SEATS outputs the coefficients of the ARIMA or ARMA models of ˆst, sat=Zt−ˆst,ˆpt, and ˆut, with innovation variances given in units of σ2 a(so σ2 a=1). With this information, W-K formulas can be used to obtain models for δ(B)ˆst,δ(B)ˆptand δ(B)ˆut. From these models, the autocorrelations needed for smoothness analysis like those presented below can be calculated. The simplest differenced estimate’s model, that of δ(B)ˆut, is derived in Appendix 1 as an illustration, after that of ˆut. We start with the calendar month series. 13.5.3 Seasonal autocorrelations and calendar month smoothness Results are presented in the Tables 4,5,6,7,8for comparison with the autocorrelations of δ(B)Ztin Table 3from (86). Our formal relative smoothness criterion only 123
SERIEs (2016) 7:11–52 47 Table 3 Lag 12 autocorrelations ρδ(B)Z 12 (,θ)=−1+2−1of δ(B)Z \θ−0.3 0.0 0.3 0.6 0.9 0.000000 0.3 −0.275 −0.275 −0.275 −0.275 −0.275 0.6 −0.442 −0.442 −0.442 −0.442 −0.442 0.9 −0.497 −0.497 −0.497 −0.497 −0.497 From (86), ρδ(B)Z 12 <0for>0,so δ(B)Ztwill have nonsmooth calendar month subseries Table 4 Lag 12 Autocorrelations of δ(B)ˆs \θ−0.3 0.0 0.3 0.6 0.9 0.0 0.347 0.467 0.589 0.622 0.222 0.3 0.568 0.644 0.714 0.731 0.481 0.6 0.763 0.803 0.836 0.844 0.715 0.9 0.943 0.952 0.959 0.960 0.931 ρδ(B)ˆs 12 >ρδ(B)Z 12 =−ρδ(B)Z 12 always. δ(B)ˆsthas substantially smoother calendar month subseries than δ(B)Z Table 5 Lag 24 Autocorrelations of δ(B)ˆs \θ−0.3 0.0 0.3 0.6 0.9 0.0 0.035 0.072 0.121 0.131 0.013 0.3 0.197 0.244 0.294 0.305 0.154 0.6 0.474 0.510 0.545 0.552 0.435 0.9 0.852 0.864 0.873 0.875 0.840 ρδ(B)ˆs 24 >0=ρδ(B)Z 24 . The calendar month. smoothing indicated in Table 4is reinforced, moderately to strongly at 2 years remove involves first seasonal lag autocorrelations. Some higher lag results are presented for a broader perspective. Here is a summary of the tabled seasonal lag results. Tables 4,5,6 show that, in contrast to δ(B)Ztfrom Table 3, the series δ(B)ˆstfrom the seasonal estimates ˆstis positively correlated at all seasonal lags considered, 12, 24 and 36, often strongly, indicating that the calendar month subseries of δ(B)ˆstwill often be substantially smoother than δ(B)Zt. Table 7shows that the opposite holds for the seasonally adjusted series δ(B) sat=δ(B)Zt−δ(B)ˆst, and Table 8’s results for δ(B)ˆutare similar. Both have more negative autocorrelations at lag 12 than δ(B)Zt and positive autocorrelations at lag 24 (and negligible autocorrelations at lag 36–not shown), increasing the tendency for more changes of direction than δ(B)Zt. 123
48 SERIEs (2016) 7:11–52 Table 6 Lag 36 Autocorrelations of δ(B)ˆs \θ-0.3 0.0 0.3 0.6 0.9 0.0 . =0. =0. =0. =0. =0 0.3 0.059 0.073 0.088 0.092 0.046 0.6 0.284 0.306 0.327 0.331 0.261 0.9 0.767 0.777 0.786 0.788 0.756 ρδ(B)ˆs 36 >0=ρδ(B)Z 36 for ≥0.3 Calendar month smoothing is further reinforced at 3 years remove, not as strongly at as two Table 7 Lag 12 autocorrelations of δ(B) sa \θ-0.3 0.0 0.3 0.6 0.9 0.0 −0.297 −0.465 −0.590 −0.646 −0.659 0.3 −0.520 −0.548 −0.573 −0.586 −0.590 0.6 −0.520 −0.525 −0.529 −0.532 −0.533 0.9 −0.502 −0.502 −0.502 −0.502 −0.502 All ρδ(B) sa 12 <ρ δ(B)Z 12 <0.δ (B) satis more nonsmooth than δ(B)Z Table 8 Lag 12 autocorrelations of δ(B)ˆu \θ−0.3 0.0 0.3 0.6 0.9 0.0 −0.667 −0.667 −0.667 −0.667 −0.667 0.3 −0.591 −0.591 −0.591 −0.591 −0.591 0.6 −0.533 −0.533 −0.533 −0.533 −0.533 0.9 −0.502 −0.502 −0.502 −0.502 −0.502 ρδ(B)ˆu 12 is more negative than ρδ(B)Z 12 . The calendar month subseries of δ(B)ˆutare more nonsmooth 13.5.4 Lag 1 autocorrelation and monthly smoothness results Familiarly, an estimated trend visually smooths a seasonally adjusted monthly series. We examined the lag 1–12 autocorrelations (not shown) of the differenced trend estimates, δ(B)ˆptfor the ,θ under consideration. At lags 1–6 all are positive. At lags 7–11, some or all can have either sign. Thus δ(B)ˆptwill have at least a half-year of resistance to oscillation. At lag 12, all are negative. This is in strong contrast to δ(B)Zt, which, among lags 1–6, has a non-zero autocorrelation only at lag one, with a negative value indicating nonsmoothness (except when θ<0), see (86). For the differenced irregular component δ(B)ˆut, Tables 9and 10 below show that δ(B)ˆutis more nonsmooth than the nonsmooth δ(B)Zt. 123
SERIEs (2016) 7:11–52 49 Table 9 Lag 1 autocorrelations ρδ(B)Z 1(,θ)=−θ1+θ2−1of δ(B)Z \θ−0.30.00.30.60.9 all0.275 0.0−0.275 −0.441 −0.497 Monthly δ(B)Ztare nonsmooth for θ>0, smooth for θ<0 Table 10 Lag 1 Autocorrelations of δ(B)ˆu \θ−0.30.00.30.60.9 all −0.756 -2/3 −0.591 −0.533 −0.502 ρδ(B)ˆu 1<min 0,ρδ(B)Z 1.δ(B)ˆuis always more nonsmooth than δ(B)Zand ˆu 14 Concluding remarks The simple seasonal models considered have provided very informative formulas for two- and three-component decompositions of seasonal time series. The factored formulas for the seasonal random walk simply display the full range of differencing operators (in biannual form) of ARIMA model-based seasonal decomposition filters identified in Bell (2012,2015). The formulas for the estimates’ auto- and cross-correlation formulas have led to new insights and results. For example, the finding of negative sample autocorrelations at the seasonal lag of the differenced seasonally adjusted series now appears as an inevitable result of removing a seasonal component whose calendar month subseries are smooth. It is not a defect of the seasonal adjustment procedure, contrary to a view expressed in some of the literature motivating McElroy (2012). For the irregular component, there are the common empirical findings, with airline and similar models, of negative sample autocorrelations, often at the first lag (see Table 11 in Appendix 1) and at the first seasonal lag of the estimated irregular component ˆuor differenced ˆuas in Tables 1and 10. These can now be anticipated from the knowledge that ˆucan be regarded both as the detrended version of the seasonally adjusted series Z−ˆs, and also as the deseasonalized version of the detrended series Z−ˆp, in both cases resulting from removal of a smooth component. The capacity to provide illuminating precise answers to many questions is a very valuable feature of ARIMA-model-based seasonal adjustment, as is its conceptual simplicity relative to nonparametric procedures, at least for adjusters with sufficient modeling background and experience. (The challenge is always to find an adequate model for the data span to be adjusted, if one exists.) Also valuable are the error variance and autocovariance measures (not accounting for sampling or model error) that AMBSA easily provides (only) for additive direct seasonal adjustments and their period-to-period changes. The latter, with log-additive/multiplicative adjustments, the most common kind, yield approximate uncertainty intervals for growth rates, quantities of special interest for real-time economic analysis. 123
50 SERIEs (2016) 7:11–52 Disclaimer Results of ongoing research are provided to inform interested parties and stimulate discussion. Any opinions expressed are those of the authors and not necessarily those of the U.S. Census Bureau or the Bank of Spain. Acknowledgments The SAR(1) results were presented by the first author at the March 13-14, 2014 Bank of Spain conference “Celebrating 25 Years of TRAMO-SEATS and honoring the 70th Birthday of Agustin Maravall”. The authors are indebted Brian Monsell, Tucker McElroy and Bill Bell for beneficial comments on earlier drafts. Bell’s comments and suggestions were especially extensive and influential. We are also grateful to Domingo Pérez, Brian Monsell and especially Tucker McElroy for computational support. Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. 15 Appendix 1: Derivation of the SARMA models of ˆ utand δ(B)ˆ ut for the airline model The W–K estimate ˆutof the canonical airline model decomposition’s irregular component uthas the pseudo-s.d. gˆu(λ)=g2 u(λ) g(λ)=σ4 u σ2 aδei2πλ2 ϑei2πλ2, with gu(λ)=σ2 u.Thus,from(79 ), ˆuthas the stationary noninvertible seasonal ARMA(1,1)(1,1)12 model 1−θB−B12 +θB13ˆut=1−B−B12 +B13ct,(87) for white noise ctwith variance σ4 u/σ2 a, leading to Similarly, from (46), ˆ Jt=δ(B)ˆut, the fully differenced ˆuthas s.d. gˆ J(λ)=δei2πλ2 gˆu(λ)=σ4 u σ2 aδei2πλ4 ϑei2πλ2. So its model is the noninvertible seasonal ARMA(1,2)(1,2)12 model (1−θB)(1−Bq)ˆ Jt=(1−B)21−B122ct. Multiplied out, this model is 1−θB−B12 +θB13ˆ Jt =1−2B+B2−2B12 +4B13 −2B14 +B24 −2B25 +B26ct.(88) Expanded model formulas like (87) and (88) are what the algorithms for calculating autocovariances of Sect. 13.5.2 require and what SEATS outputs for Tables 1and 10. 123