On some remarks about SEATS signal extraction
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Abstract
In seasonal adjustment a time series is considered as a juxtaposition of several components, the trendcycle, and the seasonal and irregular components. The Bureau of the Census X11 method, based on moving averages, correction of large errors and trading day adjustments, has long dominated. With the success of ARIMA modelling at the end of the 20th century, methods with better outlier detection and trading day corrections by regression with ARIMA errors have appeared, with the regARIMA module of Census X12ARIMA or Bank of Spain TRAMOSEATS. SEATS consists of extracting the components by an ARIMAmodelbased unobserved components approach. This means that models are used for each component such that the sum of the components is compatible with the ARIMA model for the corrected time series. The underlying theory of the SEATS program is studied in many papers but there is no complete and systematic description of its output. Our purpose is to examine SEATS text output and to explain the results in simple words and formulas. This is done on a simple example, a time series with a nonseasonal model so that the computations can be verified step by step. The principles behind SEATS are first described, including the admissible decompositions and the canonical decomposition, and the derivation of the WienerKolmogorov filter. Then the example is introduced: the interest rates of US certificates of deposits. The text output from SEATS is presented in edited form in several tables. Finally, the main results are checked on the example by means of a Microsoft Excel workbook and direct computations. In particular, the forecasts and backcasts are obtained; the admissible and canonical decompositions with two components are discussed; the filters are first derived using autocorrelations of two auxiliary ARMA processes, then applied on the prolonged time series; and the characteristics of the estimates, the revisions and the growth rates are analyzed.
Keywords
SEATS Seasonal adjustment Admissible decomposition Canonical decomposition WienerKolmogorov filterJEL Classification
C22 C82 C871 Introduction
Seasonal adjustment is fundamental for the analysis and interpretation of macroeconomic time series. The series is considered as a juxtaposition of several components: the trendcycle, the seasonal component and the irregular component. A seasonally adjusted series is obtained by removing the seasonal component.
During the second half of the 20th century, the Bureau of the Census X11 method (Shiskin and Eisenpress 1957) has dominated. It is based on a mixture of statistical techniques mainly moving averages, treatment of atypical observations and trading day adjustments. For a nice illustrated example showing the internals of X11, see Ladiray and Quenneville (2002). An alternative approach under the form of an Excel file is available (Online Resource 1).
Statistics Canada X11ARIMA (Dagum 1980) has introduced autoregressive integrated moving average (ARIMA) modelling in particular to extend the series in the past and in the future in order to avoid endadjustments in the moving averages. Census X12ARIMA (Findley et al. 1998) has improved on this by including a better outlier detection procedure and trading day corrections by regression with ARIMA errors, within a regARIMA module, while leaving nearly unchanged the old extraction of components by moving averages. Bank of Spain TRAMOSEATS (Gómez and Maravall 1994, 2001a, b) has used a more powerful automatic model selection (AMS) procedure called TRAMO (Time series Regression with ARIMA noise, Missing values and Outliers), see below, before a very different signal extractionbased procedure called SEATS (Signal Extraction in ARIMA Time Series). While keeping regARIMA and the available model selection procedures, versions of X12ARIMA after version 0.3 have added the automdl spec based on TRAMO. Now (Time Series Research Staff 2013), the Bureau of the Census X13ARIMASEATS has appeared, with few changes on regARIMA and automdl, but offering a choice between SEATS and the old X11 decomposition procedure, with some improvements for the latter. Therefore, the paper is valid for TRAMOSEATS but also for the SEATS part in X13ARIMASEATS. It should also be valid for JDemetra+ which intends to be a reimplementation of TRAMOSEATS and X13ARIMASEATS using the same concepts and algorithms. Before going into the details, let us mention the basic ingredients of these two modules, TRAMO (or regARIMA), on the one hand, and SEATS, on the other hand.
Although TRAMO is complex and contains features which are still not always standard in statistical software packages for time series, the use of ARIMA models is now well mastered. Besides the standard ARIMA modelling stage by maximum likelihood, generally in an automated way, the following techniques are involved: regression with autocorrelated errors; treatment of extreme observations using corrections for outliers (additive outliers, level shift, and transitory change); Easter and mobile holiday’s effect and calendar effect adjustments; and treatment of missing observations. The ARIMA model will be directly used by SEATS but it serves also to extend the finite series, by computing as many forecasts and backcasts (i.e. forecasts performed backwards, also called backforecasts by Box and Jenkins in 1970, e.g. Box et al. 2008) as needed. It should be stressed that these techniques are integrated in the ARIMA modelling, although the use of a concentrated likelihood approach allows to estimate the parameters more or less separately.
The seasonal adjustment procedure derived by SEATS on the basis of the prolonged series and the model fitted is more difficult to explain, except the basic objective: a decomposition of the series a little bit like in elementary seasonal decomposition methods. However, the signal extraction procedure in SEATS is based on engineering techniques which are generally much less mastered by economists and even statisticians, with some exceptions. The underlying theory of the SEATS program is studied in many papers but there is no complete and systematic description of its output. There are a few tutorial papers, like Kaiser and Maravall (2001), but they are perhaps still too complex for the interested audience. There does not appear to be a paper showing the main concepts in a simple way. Unfortunately, even for the airline model (or an \(\hbox {ARIMA}(0,1,1){(0,1,1)}_s\) model on logarithms of the data, with s \(=\) 12 for monthly observations), the simplest realistic ARIMA model at the TRAMO or regARIMA stage, things are still too complex.
The main purpose of this paper is therefore to examine the details of SEATS text output and to explain the results in simple words and formulas. It is done on the basis of an example, with a stepbystep description of a typical output from SEATS. To keep the explanation as simple as possible, the example will not use a seasonal decomposition but will use the simpler signal extraction of a trend. The example is for a time series with a nonseasonal model so that the computations can be easily verified. Therefore the models are much simpler and fewer numbers need to be interpreted while preserving the essential. We have designed a Microsoft Excel Workbook (Online Resource 2) which shows most of the output and a document with some instructions (Online Resource 3). Using Microsoft Excel for doing statistics is not generally recommended (see the references in Mélard 2014), but it is the right tool to describe simple computations. It would be difficult to use Excel to demonstrate SEATS in a more realistic example with a seasonal component. The regARIMA or TRAMO part of the treatment are not discussed, nor the graphical output.
We have found a monthly series called TICD (also used in Mélard 2007), the interest rates of U.S. certificates of deposit, between December 1974 and December 1979, which is a nonseasonal series. We have obtained a nonseasonal but otherwise very interesting decomposition, being able to compare the results for the filter weights and most of the output with those obtained using SEATS, see Tables and Online Resource 4.
The principles behind SEATS are described in Sect. 2, including the admissible decompositions and the canonical decomposition, and a procedure to implement the derivation of the WienerKolmogorov filter. In Sect. 3 the example is introduced: the time series and the text output from SEATS is presented in edited form in several tables. Finally, in Sect. 4, the main results are checked on the example by means of a Microsoft Excel workbook and direct computations. In particular, the forecasts and backcasts are obtained; the admissible and canonical decompositions with two components are discussed; the filters are first derived using autocorrelations of two auxiliary ARMA processes, then applied on the prolonged time series; and the characteristics of the estimates, the revisions and the growth rates are analyzed. We will conclude in Sect. 5. Appendix 1 will serve to introduce spectral analysis in a general approach. Appendix 2 will introduce the spectral analysis used in SEATS for the derivation of a canonical decomposition.
2 Principles behind SEATS
We will explain the principles first in informal terms and using simple models, before introducing SEATS more widely. In the next two sections we will use a simpler model by going into details first on the basis of SEATS output on the TICD series, then on the basis on an Excel file which describes the computations. As mentioned before, TICD is a nonseasonal series so that the decomposition will be simpler than with a seasonal series. We suppose we have obtained an ARIMA model for the original series or a corrected version of it, using either TRAMO or regARIMA. The corrected series will differ from the original series if the pretreatment by TRAMO or regARIMA has detected outliers or calendar effects. In the sequel, \(y_{t}\) denotes the corrected observation at t. SEATS is aimed at decomposing the obtained ARIMA model in a sum of components and extract them from the series. There can be up to 4 components: the permanent or trend (cycle) component,^{1} the transitory component, the seasonal component, and the irregular component or error. In the majority of cases of a multiplicative mode of composition, SEATS works with an additive decomposition of the data in logarithms, \(\hbox {log}(y_{t})\). Here we will restrain ourselves to two or three components and an additive decomposition, to simplify the presentation.
Moreover, even if we can determine the \(q_{P}+q_{S}\) of unknown coefficients, there is no reason why that solution would be unique. In other words, if there exists an admissible decomposition, we need to determine what is called a canonical decomposition. A detailed examination of these two problems of admissible and canonical decompositions are outside the scope of this paper and has been solved elsewhere (Hillmer and Tiao 1982; Maravall and Planas 1999; Fiorentini and Planas 2001). In Sect. 3 we will merely illustrate these problems on the example of a model for TICD. More precisely, after having determined a collection of admissible decompositions, we will select the canonical decomposition within that collection. We will not discuss other related problems such as the possible need to change the model for the corrected series in order to be able to perform the decomposition; the specification of the models for the components; the selection of the number of components; and the diagnostics surrounding the decomposition. Some of these problems will however be mentioned while looking at the example.
Since Burman (1980), several algorithms have been designed to improve the end effects. Several authors, like Maravall (1994) and McElroy (2008), have mentioned a link with the autocovariance function of some auxiliary processes based on the innovations of the corrected series, and on the polynomials of the ARIMA models of the corrected series and of the components. These autocovariances can be computed by a straightforward algorithm (McLeod 1975) of solving a linear system of equations or by fast algorithms due to Tunnicliffe Wilson (1979) or Demeure and Mullis (1989). These simple approaches do not appear often in the literature (e.g. Gómez 1999) although Burman (1980) already mentions developments by Tunnicliffe Wilson (1979).
An essential part of SEATS is the distinction between different estimators of component i at time \(t, y_{it}\). Again assume the data are \(y_{t}, t=1, ..., n\). These estimators are conditional expectations \(E(y_{it}{\vert }y_{1}, ...y_{n})\), denoted \(\hat{y}_{itn}\). For a large enough series and values of t not close to 1 or n, they are the final or historical estimators, denoted \(\hat{y}_{it}\). In practice it occurs when \(k=nt\) is large enough. For \(t=n\), the concurrent estimator is obtained. When \(nk<t<n\), we have a preliminary estimator, and of course for \(t>n\), a forecast. A part of the output is devoted to the three differences, \(y_{it}\hat{y}_{it}, y_{it}\hat{y}_{itn}\), and \(\hat{y}_{it}\hat{y}_{itn}\), which are called, respectively, the final estimation error, the preliminary estimation error and the revision error in the preliminary estimator.
3 Example of a simple case of signal extraction
3.1 ARIMA modelling
 1.
This point shows information about the program (SEATS), about the series (TICD) and the estimation method used (exact maximum likelihood). Then the data is shown.
 2.
The input parameters are recalled. We have used the following input parameters in TRAMO: LAM = 1 (to avoid a logarithmic transformation), RSA = 3 (for automatic model identification^{9}). Note that RSA appears as being 0 in SEATS output.
 3.
The recommended specification is to use a regular difference and no seasonal difference. The differenced series is however omitted here.
 4.
The first twelve autocorrelations of the differenced series are displayed. They show a truncation pattern after lag 1, which confirms the model (3.1).
Edited SEATS output, part 1 (see text for details)

Edited SEATS output, part 2 (see text for details)

 5.
The final value of the (unique) estimated parameter of the ARMA model is equal to 0.4995. A more accurate value 0.499479 shown only in TRAMO output will be used in the computations of Sect. 4. The standard errors and Student statistics are not shown by SEATS but by TRAMO, where the corresponding parameter is denoted “TH1”.
 6.
The residuals of the fitted model are displayed. They will be used in Sect. 4.1 for the computation of the forecasts.
 7.
Some of the tests statistics on the residuals are shown, including the mean (which is not significantly different from 0, according to the t statistic). We will use the residual standard deviation, 0.4829, and its square 0.2332, the residual variance, denoted V in Sect. 2.
 8.
Some of the additional tests are shown, such as a test for residual seasonality and the LjungBox test based on the 24 first residual autocorrelations.
 9.
The backward residuals of the fitted model are displayed. They are the residuals when time is reversed, going from the future to the past. They will be used in Sect. 4.1 for the computation of the backcasts.
3.2 Admissible and canonical decompositions
Edited SEATS output, part 3 (see text for details)

Edited SEATS output, part 4 (see text for details)

 10.
The second part of the output is entitled “Derivation of the models for the components”. Each polynomial of the model for the corrected series is first recalled. Here, we have here only one non trivial polynomial, the regular moving average polynomial denoted Theta: \(\theta (B)=1+0.50B\). It is given with more significant digits, like in (3.1), a few lines below.
 11.
The new section is about the factorization of the (generalized) autoregressive polynomial, i.e. \(\varphi (B)\). Here it is \(\nabla =1B\) and thus the factorization is trivial. We have only given the total autoregressive polynomial subsection.
 12.
The model allows a decomposition which would have not been possible in some cases.
 13.
The derivation of the model for each component is detailed, starting with the trendcycle component, referred to here as the permanent component. We see that the numerator of the model for that component is \(1+B\), confirming that \(\theta _{P}=1\) was retained for the canonical decomposition, see (3.5).
 14.
It is followed by the irregular component.
 15.
In our case, the seasonally adjusted component corresponds to the original series. Logically its model should be identical to the global model.
 16.
Note that all variances are expressed in proportion to the global model innovation variance V.^{10} The effective variances of the two component models are respectively \(V_{P}=0.1311\) and \(V_{I}=0.01461\).
3.3 Moving average representation and WienerKolmogorov filters
 17.
An explanation that will be commented at the end of Sect. 4.4.
 18.
The infinite moving average representation of the component estimators, also called \(\psi \)weights, are given here in terms of the innovations (here only for lags \(\)10 to 2, the remaining ones being uninformative). We will give more details in Sect. 4.4.
 19.
It is first confirmed that the filters could be derived.
 20.
The weights of the WienerKolmogorov filters for each of the components are given but only for one side since they are symmetric, and only the lags 0 to 12, the remaining entries being 0 to 4 decimal places. This does not mean that they are negligible, as will be seen in Sect. 4.4.
3.4 Distinction between theoretical components, their estimators and estimates
 21.
The contribution of the original series and of its innovations to the estimator of the components for the present period. In the column “observation”, it is the weight of the WienerKolmogorov filters (for lags 0 and 1) given in item 18. In the column “innovation”, it is the \(\psi \)weight (for lags 0 and \(\)1) given in item 20.
 22.
In a subsection entitled “Distribution of component, theoretical estimator and empirical estimate”, the autocorrelation characteristics of the theoretical trendcycle component, its estimator and its estimate, after transformation for achieving stationarity, are given. These results will be discussed and checked in Sect. 4.5. We have omitted the seasonally adjusted component which has no meaning here.
 23.
It is followed by the autocorrelation characteristics of the theoretical irregular component, its estimator and its estimate. These results will be commented in Sect. 4.5. The error on Vis repeated here.
 24.
The crosscorrelations, without any lag, between the estimators and the estimates of the two components are given here. See Sect. 4.5.
 25.
Tests of comparisons between the estimators and the estimates according to several criteria: the variance, autocorrelation of order 1 and 12, and crosscorrelation. See Maravall (2003).
 26.
Under the heading “Weights”, the weights for the asymmetric filter for the trend given a semiinfinite realization are given. They are identical to the \(\psi \)weights for the trend, already shown in item 18.
Edited SEATS output, part 5 (see text for details)

Edited SEATS output, part 6 (see text for details)

Edited SEATS output, part 7 (see text for details)

Several pieces of output have been deleted (phase diagram, seasonal diagnostics, the conclusions of the spectral diagnostics, the residual stochastic seasonality spectral evidence, and the trading day effect).
3.5 Error analysis
 27.
It is started with the final estimation error, the only one that counts for an observation near the middle of the series, and the revision error, which is due on the arrival of a new observation.
 28.
Under the heading “Total estimation error (concurrent estimator)” the output contains the sum of the two preceding errors. The variance is thus the sum of the two variances. For the autocorrelations, see Sect. 4.6.
 29.
The item called “Variance of the revision error” contains the variance of the revision error after additional time has passed.
 30.
Under the heading “Percentage reduction in the standard error of the revision”, the output contains the percentage of reduction of the variance of the revision error after additional time has passed.
Edited SEATS output, part 8 (see text for details)

Edited SEATS output, part 9 (see text for details)

Edited SEATS output, part 10 (see text for details)

Edited SEATS output, part 11 (see text for details)

 31.
The item entitled “Decomposition of the series: recent estimates”, contains the values of the components (which are also repeated further) but also their standard deviations.
 32.
Under the heading “Decomposition of the series: forecasts”, the output contains the forecasts that can be made for each component with their standard deviations.
3.6 Estimates of the components
3.7 Rates of growth
 35.
Periodtoperiod growth estimation error variance for the concurrent estimator,
 36.
Periodtoperiod growth for the most recent periods for the original series and the trend (with standard error of revision for the latter),
 37.
Accumulated growth during the present year for the original series and the trend (with standard error of revision for the latter),
 38.
Annual growth estimation error variance for the concurrent estimator,
 39.
Annual growth for the most recent periods for the original series and the trend (with standard error of revision for the latter),
 40.
Annual centred growth with 6 observed periods and 6 forecasts,
 41.
Growth forecasts for the original series and the trend, with standard error of revision.
4 Verification of the results in the example
It is clear that TRAMOSEATS is a complex piece of software. If the basic theory is given in a few papers (Gómez and Maravall 1994, 2001a, b, and others), and can be checked empirically, some of the more sophisticated features like the selection of the canonical decomposition, the derivation of the WienerKolmogorov filters, revisions, and the various diagnostics are more difficult to grasp. We will therefore describe their use on the same example as in Sect. 3, but this time with the help of an Excel file (Online Resource 2).^{11} These results can then be compared with the edited output from the program in Tables 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 or with the full output (Online Resource 4). At some places simple algebraic computations are also used.
4.1 Forecasts and backcasts of the series
Before looking at decompositions, let us look at the forecasts and backcasts of the TICD series that will be needed. In the Excel file, worksheet Main, we have used the example based on the model (3.1) for TICD.
Let us first show how to evaluate the forecasts for the future months, with December 1979 as forecast origin. (3.1) implies that \(\hbox {TICD}_{t}=\hbox {TICD}_{t1}+e_{t}+0.4995e_{t1}\), hence the forecast at time \(T+1\) equals: \(\hbox {TICD}^{*}_{T+1}=\hbox {TICD}_{T}+0.4995e_{T}\). The next forecasts are \(\hbox {TICD}^{*}_{T+2}=\hbox {TICD}^{*}_{T+1}, \hbox {TICD}^{*}_{T+3} =\hbox {TICD}^{*}_{T+2}\), and so on. The residuals are obtained starting from the same model \(\nabla \hbox {TICD}_{t}=e_{t}+0.4995e_{t1}\), by writing \(e_{t}=\nabla \hbox {TICD}_{t}0.4995e_{t1}\) where the cell V49 contains the moving average coefficient 0.4995. The data of November 1979 and December 1979 are equal to 13.97 and 13.42, respectively, and the residual in December 1979 equals \(\)0.2863 (cell U113 which corresponds to the last number of item 6 in Table 2). Then \(\hbox {TICD}^{*}_{T+1}=13.42+0.4995\times (0.2863)=13.28\), hence \(\hbox {TICD}^{*}_{T+2} =\hbox {TICD}^{*}_{T+3}=13.28\), and so on, see cells V114 to V131. This is shown in item 32, column “Series Forecast”, in Table 8. The computation of the forecast standard errors for the series is classic using the socalled \(\psi \) weights of the pure moving average representation of the process. Given that these weights are \(\psi _{j}=1+{\theta }\), these standard errors for horizons 1, 2 and 3 are the square roots of V, \(V(1+(1+ {\theta })^{2}), V(1+2(1+ {\theta })^{2})\), and so on. This gives the values shown in the range X114 to X125 of the worksheet and in column “Series S.E.” of item 32 in Table 8. Note that the derivation of the other columns of item 32 is differed to Sect. 4.6.
Similarly, the backcasts are obtained in reversed time, with December 1974 as forecast origin. (3.1) is an invertible model which implies that we have the following model in reverse time: \(\hbox {TICD}_{t}\hbox {TICD}_{t+1} =u_{t}+0.4995u_{t+1}\), where we will denote \(u_{t}\) the innovations in reverse time. This implies that \(\hbox {TICD}_{t}= \hbox {TICD}_{t+1}+u_{t}+0.4995u_{t+1}\), thus the forecast at time 0 equals: \(\hbox {TICD}^{*}_{0}=\hbox {TICD}_{1}+0.4995u_{1}\). The preceding forecasts are \(\hbox {TICD}^{*}_{2}=\hbox {TICD}^{*}_{1}=\hbox {TICD}^{*}_{0}\), and so on. The residuals in reverse time are obtained from \(\hbox {TICD}_{t}\hbox {TICD}_{t+1}=u_{t}+0.4995u_{t+1}\), by writing \(u_{t}=\hbox {TICD}_{t}\hbox {TICD}_{t+1}0.4995u_{t+1}\), where cell V49 contains the moving average coefficient 0.4995. The data of December 1974 and January 1975 are equal to 8.82 and 7.42, respectively, and the residual in December 1974 is equal to 0.9990 (cell Y53 which corresponds to the first number of item 9 in Table 2). Then \(\hbox {TICD}^{*}_{0} =8.82+0.4995\times (0.9990) =9.32\), hence \(\hbox {TICD}^{*}_{1}=\hbox {TICD}^{*}_{2}=9.32\), and so on, see cells AA41 to AA52. The whole series of backward residuals is shown in SEATS output, see Table 2, item 9.
4.2 Admissible and canonical decompositions
In Sect. 3.2, we have seen a general form for the difference of the permanent or trend component given by \(\nabla P_{t}=(1+\theta _{P}B)e_{t}^{P}\). We have obtained a system of two equations in the variances of the innovations of the two components \(V_{P}\) and \(V_{I}\). This system can be solved for any value of \(\theta _{P}\) between \(\)1 and 1. In the Excel file, worksheet Second, we have used the context of the example based on the TICD model. We have entered in cells B9 and B10 the value of \(\theta \) and V, respectively. In cells A15 to A35, we have placed potential values of \(\theta _{P}\) between \(\)1 and 1. In the next two columns formulas give the solutions of \(V_{P}\) and \(V_{I}\). in function of \(\theta _{P}\). The resulting table is copied in Table 12. We can see that not only \(\theta _{P}=0\) is not admissible, because one of the two variances is negative, but also that only values of \(\theta _{P}\) between 0.6 and 1 lead to admissible decompositions. There remains to justify why the canonical decomposition should be the one corresponding to \(\theta _{P}=1\).
4.3 The canonical decomposition
Admissible and nonadmissible decompositions
\(\theta _{P}\)  \(V_{P}\)  \(V_{I}\) 

1.0  0.1311  0.0146 
0.9  0.1452  0.0142 
0.8  0.1618  0.0130 
0.7  0.1814  0.0105 
0.6  0.2048  0.0064 
0.5  0.2330  0.0000 
0.4  0.2675  \(\)0.0095 
0.3  0.3103  \(\)0.0234 
0.2  0.3641  \(\)0.0437 
0.1  0.4333  \(\)0.0731 
0.0  0.5244  \(\)0.1165 
\(\)0.1  0.6473  \(\)0.1812 
\(\)0.2  0.8193  \(\)0.2803 
\(\)0.3  1.0701  \(\)0.4375 
\(\)0.4  1.4565  \(\)0.6991 
\(\)0.5  2.0974  \(\)1.1652 
\(\)0.6  3.2772  \(\)2.0828 
4.4 Moving average representation and WienerKolmogorov filters
It is easier to present the derivation of the components using the WienerKolmogorov filters before discussing the moving average representation.
We can now use the two filters to extract the components. To see the calculation of the components for the permanent and for the irregular component, we have applied the weighted moving averages in Fig. 3 to the series prolonged with forecasts and backcasts. This is done in the Excel file, worksheet Main in the range AB53 to AB113 for the permanent component and in the range AC53 to AC113 for the irregular component. To achieve the accuracy for the components shown in Table 9 of the SEATS output, we have made a correction to take care of a few more weights. Nevertheless, there are some differences for the irregular component.^{12}
4.5 Autocorrelations of the components, the estimators and the estimates
It is specified in the output (see Table 5) that what is shown is the autocorrelation function of a “stationary transformation of components and their estimators”. We first check the components and the estimates, before the estimators which are more complex to handle.
We have already specified the processes for the two components. The process for the permanent component \(P_{t}\) is nonstationary. Therefore what is shown is the autocorrelation function of the difference of the process, i.e. \(\nabla y_{t}\). Since that process is MA(1) with coefficient 1, the firstorder autocorrelation is equal to 0.5 and the other autocorrelations are equal to 0. This is shown in the second column of item 22 in Table 5. The Excel file, sheet Main (range AF51 to AM121), shows the computation of the first three autocorrelations for the estimate Open image in new window of the permanent component \(P_{t}\) detailed in Sect. 4.4: 0.539, 0.023, \(\)0.051. See cells AI121, AK121 and AM121. They agree with item 22 in Table 5, column “Estimate”. Computing the autocorrelations of the estimator \(p_{t}\) defined by (4.7) in function of the innovations of the process defined by (3.1) is more complex. The coefficients of the moving average representation for \(\nabla p_{t}\) have already been found in (4.9) up to a factor \(V_{P}/V\). To evaluate the variance, we have to take the sum of the squares of these coefficients given by \(1+(2\theta )^{2}+(1 \theta )^{4}/(1\theta ^{2})\), and multiply it by \((V_{P}/V)^{2}\). This yields 1.054, as indicated by item 22 in Table 5, on the line “VAR” and in cell AZ22, or 0.24575 in original units, see cell BA22. The autocovariances are computed in worksheet Main, in the range BS1 to DC19. Dividing them by the variance located in cell AZ21, we obtain the autocorrelations displayed in cells DD4 to DD19, e.g. 0.5501 for lag 1. They correspond to the contents of SEATS output, item 22 in Table 5, column “Estimator”. The variance in units of V is in cell AZ22.
The process for the irregular component \(I_{t}\) is stationary. Therefore a difference is not needed to make it stationary and all autocorrelations are equal to 0 for strictly positive lags. This is shown in the second column of item 23 in Table 5. The Excel file, sheet Main, range AP51 to AW121, shows the computation of the first three autocorrelations for the difference of the estimate \(\hat{\imath }_{t}\) of the irregular component \(I_{t}\) detailed in Sect. 4.4: \(\)0.766, 0.421, \(\)0.229, in cells AS121, AU121 and AW121. They agree nearly perfectly with item 23 in Table 5, column “Estimate”, where the respective results are \(\)0.765, 0.421, \(\)0.229. This discrepancy can be explained by the relatively small number of exact significant digits for the irregular estimates. Even if the autocorrelations for the estimates are not too far from the autocorrelations for the estimator, they are not comparable with the autocorrelations for the component. Computing the autocorrelations of the estimator \(i_{t}\) defined by (4.8) in function of the innovations of the process defined by (3.1) is again more complex. The coefficients for the moving average representation for \(i_{t}\) has already been found in (4.12). To evaluate the variance, we have to take the sum of the squares of these coefficients and multiply it by (\(V_{I}/V)^{2}\), giving 0.016, as indicated by item 23 in Table 5, on the line “VAR”. The autocovariances are computed in worksheet Main, in the range BS24 to DC41. Dividing them with the variance located in cell AZ43, we obtain the autocorrelations displayed in cells DD26 to DD41. They correspond to the contents of SEATS output, item 23 in Table 5, column “Estimator”. The variance in units of V is in cell AZ44.
Like the computation of the autocorrelations between the estimates of the two components, Open image in new window and \(\hat{\imath }_{t}\), we have tried to compute the contemporary (i.e. without lag) crosscorrelation between the estimates of the two components. Similarly with the previous results for the irregular component, we had some difficulties to recover exactly the crosscorrelation between the two components 0.279 shown in item 24 in Table 6, column “Estimate”. We have also tried to find the crosscorrelation between the estimators \(p_{t}\) and \(i_{t}\) using the coefficients of the moving average representation of the two estimators. The results shown in worksheet Main, cell BT47, coincides with the value 0.274 shown in item 24 in Table 6, column “Estimator”.
4.6 Revisions
This is related to the fact that occurrence of new observations will have a direct impact on the forecasts, and thus on the estimates near the end of the series and, hence, to the computation of onesided filters, also called concurrent filters. This problem is the subject of several papers (Maravall 1986; Gómez and Maravall 2001b; Bell and Martin 2002). There is no impact on the derivation of the estimates of the component, but on the interpretation side.
The total estimation error of the concurrent estimator \(t_{t}\) defined by (4.13) is the sum of the final estimation error \(f_{t}\) and of the revision error \(r_{t}\), which are uncorrelated. Hence, the covariance at lag j of \(t_{t}\) is the sum of the covariance at lag j of \(f_{t}\) and of the covariance at lag j of \(r_{t}\). Thus the variance of \(t_{t}\) is the sum of the variances of \(f_{t}\) and of \(r_{t}\): \(V_{t}=V_{f}+V_{r}\). It is computed in cell AN36 of worksheet Main and is equal to \(V_{t}=0.01369\) or 0.05871 in units of V. The correlations are deduced in the range from AN1 to AN37 as weighted averages of the autocorrelations of the final estimation errors and the revisions with the respective variances as weights. This corresponds to the contents of item 28 in Table 7. For example, the firstorder autocorrelation equals 0.1002 and the autocorrelation for lag 12 is 0 with 4 decimals.
For item 29 in Table 7 and additional periods, we need to use \(m>0\) in (4.14). Additional period 12 corresponds to \(m=12\). The computations are done in worksheet Main in the range AQ1 to AV36. The number shown 0.683E09 can be seen in cell AU18. For item 31 in Table 8, the column entitled “TrendCycle standard error of revision” can be seen in the range AV6 to AV30 of the worksheet, going from bottom to top and with some approximations for long periods. For period 0, the number 0.05235 which can be seen in cell AV6 is the square root of the variance of revision error \(V_{r}=0.00274\) in cell AT6 already mentioned in cell AJ36. The percentage reduction in the standard error after one year, 99.98 %, shown in item 30 of Table 7, is obtained in cell AW6. Computation of the standard errors of the total error, obtained by adding \(V_{f}=0.01095\) to the variance of the revision error, can be seen in cells BS63 to CA71.
4.7 Growth rates
First of all, as already indicated in Sect. 3.7, the fifth part of the SEATS output, entitled “Rates of growth”, provides growths since the model is applied on the data, not on their logarithms. Hence we compute differences and standard errors are based on linear ARIMA models. In the presence of a logtransform, effective rates would be computed and standard error would be based on a linearized version of ARIMA models. Since the tables are numbered, we will use these table numbers in addition to the item indication.
SEATS Table 5.1 (item 35 in Table 9) shows the periodtoperiod growth estimation error variance for the concurrent estimator, so the first differences of the final estimation error, the revision error and the total estimation error. For example, the final estimation error, denoted \(f_{t}\), has been analysed in Sect. 4.6. We need to compute \(\hbox {var}(\nabla f_{t})\) which is equal to \(\hbox {var}(f_{t})+\hbox {var}(f_{t1})2\hbox {cov}(f_{t}, f_{t1}) =2\times 0.01095\times (10.2503) = 0.016\), see cell AF40.
SEATS Table 5.2 (item 36 in Table 10) shows the periodtoperiod growth for the most recent periods for the original series and the trend (with standard error of revision for the latter). The second column “Original series” coincides with the differenced series also shown in item 3 of Table 1, see also range BJ54 to BJ113 of worksheet Main, except that the rows are displayed in reverse order. The third column entitled “TrendCycle Estimate” contains \(\nabla P_{t}\), already shown in range AE54 to AE113, also in reverse order.
SEATS Table 5.3 (item 37 in Table 10) shows accumulated growth during the present year, thus at the last date T or Dec1979. For the original series, this is the seasonal difference \(\nabla _{12}y_{T}=y_{T}y_{T12}\), whereas for the trendcycle, this is the seasonal difference \(\nabla _{12}P_{T}=P_{T}P_{T12}\). These differences are computed in cells BN113 and BO113 of worksheet Main. The TrendCycle SER = 0.523E\(\)1 which is displayed is the standard error of revision for the trend shown in cell AV6 or AK36 and discussed in Sect. 4.6.
SEATS Table 5.4 (item 38 in Table 10) shows the annual growth estimation error variance for the concurrent estimator, which is a first seasonal difference of the final estimation error, the sum of the revision error and the total estimation error. As indicated, the numbers are to be multiplied by 10. For example, the first one, denoted \(f_{t}\), has been analysed in Sect. 4.6. We need to compute \(\hbox {var}(\nabla _{12}f_{t})\) which is equal to \(\hbox {var}(f_{t})+\hbox {var}(f_{t12})2\hbox {cov}(f_{t}, f_{t12})= 2\times 0.01095\times (1+0.0001) = 0.0219\), see cell AF41.
SEATS Table 5.5 (item 39 in Table 11) shows the annual growth for the most recent periods, thus not only at the last date T or Dec1979, as in SEATS Table 5.3, for the original series and the trendcycle (with standard error of revision for the latter). For the original series, this is the seasonal difference \(\nabla _{12}y_{t}=y_{t}y_{t12}\), whereas for the trendcycle, this is the seasonal difference \(\nabla _{12}P_{t}=P_{t}P_{t12}\). Hence the row for Dec1979 corresponds to SEATS Table 5.3. The other elements are computed in ranges BN65 to BN113 and BO65 to BO113 of worksheet Main but in reverse order. The standard error of revisions are those in cells AV6 to AV30, already discussed in Sect. 4.6.
SEATS Table 5.6 (item 40 in Table 11) shows annual centred growth rates, which are according to the documentation, base of a 6month forecast and the corresponding value 12 months before. Hence, the difference between the forecast in Jun1980 and the value in Jun1979. For the original series, it is \(13.28  10.44 = 3.30\) on row Dec1979. On row Nov1979, we have the difference between the forecast for May1980 and the value for May1979. The standard errors are the square roots of \(V(1+5(1+\theta )^{2})\) and \(V(1+4(1+\theta )^{2})\), respectively, since the horizons are 6 and 5, respectively, using the standard errors for future values presented in Sect. 4.1.
5 Conclusions
SEATS is now essential for seasonal adjustment since it is used in both Bureau of the Census X13ARIMASEATS and Bank of Spain TRAMOSEATS, the main two software packages for seasonal adjustment. There is a large amount of literature on SEATS but, with a few exceptions, it is not intended for the final user.
Our purpose was to explain the text output of SEATS on a simple example. The example is based on a nonseasonal series of U.S. interest rates on certificates of deposit. The modelbased decomposition in two components is simple and does not involve a seasonal model. Thanks to that, it is possible to check the computations rather easily using Microsoft Excel. In particular, we have verified the different admissible decompositions; the selection of the canonical decomposition using the criterion of a minimal variance for the irregular component and using a pseudospectrum criterion; the derivation of the WienerKolmogorov filter for the two components; and the difference between theoretical components, their estimators and their estimates. We could obtain the latter estimates but with a limited accuracy for the irregular component. We have also checked the autocorrelations of the estimates, with only two significant digits for some correlations involving the irregular component. On the contrary all the results relative to the estimators could be confirmed.
We were able to obtain most parts of the output. Note that we did not try to check the graphical output. This can be the subject of another paper. We have only found a real repeated mistake (items 16 in Table 3, 23 in Table 5, 28 and 30 in Table 7) in the text output, concerning the variance of the innovations of the series and a small mistake (item 17 in Table 4) which will be corrected with build 935. We have also pointed out some discordances (in certain standard errors for rates of growth not mentioned above) and a few misprints directly to the author. This does not affect the effective impact of SEATS which is a solid piece of software for seasonal adjustment.
One objection against the present paper is that the model is much too simple and without any seasonal component. A referee has made the following suggestion: to use a specific ARIMA model that should be as simple as possible, but should contain a seasonal component. For that purpose a pure seasonal model like \((0,0,0)(0,1,1)_{s}\), with \(s=2\) may be considered, as it admits three components (cycletrend, seasonal component and irregular component) but it requires the estimation of few parameters and few lags because \(s=2\). Of course another series instead of interest rates should be used, probably an artificial time series or a halfyearly version of a long real series. This would be a different paper, however, and can be the subject of a followup paper, possibly in collaboration. Also it is unclear whether Excel would be enough for the task.
Footnotes
 1.
The trendcycle can be further decomposed into trend and cycle using ARIMA models that reproduce the HodrickPrescott filter, with some improvements, see Kaiser and Maravall (2005).
 2.
 3.
When \(s=12\), a transitory component is needed in some cases: (i) if \(p+d+12(P+D)<q+12Q\); (ii) if the case of an AR(1) polynomial \((1+\phi \hbox {B})\) with a coefficient \(\phi >0.2\); (iii) series with peaks in the spectrum which are not associated to frequency 0 nor with the seasonal frequencies.
 4.
We assume \(q\le 4\), because otherwise \(I_{t}\) would be a moving average of order \(q4\), not a white noise.
 5.
The computations in SEATS are not precisely performed that way. The filter to estimate the component is derived in the frequency domain and does not explicitly require the models of the components. Given that the average user looks better in the time domain, we have preferred that equivalent approach.
 6.
 7.
Their treatment is based on a random walk signal, whereas an ARIMA(0,1,1) process is used here.
 8.
This is to save space and to concentrate on items that are commented on. For example, the partial autocorrelations are never shown. An unedited output is available (Online Resource 4).
 9.
No outlier was detected so the corrected series is identical to the original series.
 10.
Note a small mistake here and in some other places (in items 23, 28, 30) in builds up to 934 of the program, since the innovation standard deviation 0.4829 is printed instead of the innovation variance 0.2332. The mistake will be corrected in build 935 which was not publicly available at the time of writing.
 11.
It is based on the file CH13EX06.xls on the CDRom of Mélard (2007) except that there a model with a constant was used, see \(\hbox {footnote}^{1}\). A document (Online Resource 3) providing some instructions is also given but these instructions do not reflect the slight changes due to the absence of the constant; makes use of the BoxJenkins parameterization for AR and MA coefficients; and comments a former version of SEATS. It covers also some other aspects, such as generating artificial series from the models for the trendcycle and the irregular and showing that the autocorrelations of their sum behave like the series TICD.
 12.
Note that the sum of the two component series differs from the original series by 0.000004 on average, see cell BG116, showing only 5 correct significant digits. When the MA(1) parameter estimate 0.4995 is used, thus with four decimal places, the forecasts and backcasts are not very accurate, and the difference becomes 0.0002 on average, showing only 2 to 3 correct significant digits. To avoid numerical instability, Pollock (2006) recommends to compute the irregular component using the filter and to deduce the permanent component.
 13.
Based on Mélard (2006).
 14.
Notes
Acknowledgments
This project was started when I was involved in the development of a selflearning course for the National Bank of Belgium, see Cohen et al. (2002). The material included descriptions of the Census X11 procedure; multiple regression; ARIMA models and chapters on X12ARIMA and TRAMOSEATS; and using Microsoft Excel or free packages like Demetra. See the final product on the CDRom of Mélard (2007). For the present paper I received the help from several persons, including Stephen Pollock and Brian Monsell. I thank them all. I am particularly grateful to Agustín Maravall and the two referees and the editor Gabriele Fiorentini who made extremely useful suggestions for improving the text, especially suggesting Appendix 2. Agustín Maravall helped also with his course notes, recommendations and numerous comments on previous versions. I thank also Michelle Crowley for her editing contribution. Of course the remaining errors are mine. This paper is offered in the honour of Agustín Maravall. We associate his collaborators to our congratulations for his bright career.
Supplementary material
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