Masking of volatility by seasonal adjustment methods

Economic Modelling 33 (2013) 676–688

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Masking of volatility by seasonal adjustment methods Aziz Hayat a,⁎, M. Ishaq Bhatti b a b

School of Accounting, Economics and Finance, Deakin University, VIC 3125, Australia Department of Finance, La Trobe Business School, La Trobe University, VIC 3086, Australia

a r t i c l e

i n f o

Article history: Accepted 21 May 2013 JEL classification: C22 C82 Keywords: Seasonality TRAMO–SEATS X-12 ARIMA Variability Under-estimation Seasonal adjustments

a b s t r a c t We report that the X-12 ARIMA and TRAMO–SEATS seasonal adjustment methods consistently underestimate the variability of the differenced seasonally adjusted series. We show that underestimation is due to a non-zero estimation error in estimating the seasonal component at each time period, which is the result of the use of low order seasonal filter in X12-ARIMA for estimating the seasonal component. Hence, we propose the use of high order seasonal filter for estimating the seasonal component, which helps reducing the estimation error noticeably, helps amending the underestimation problem, and helps improving the forecasting accuracy of the series. In TRAMO–SEATS, Airline model is found to deliver the best seasonal filter among other ARIMA models. Crown Copyright © 2013 Published by Elsevier B.V. All rights reserved.

1. Introduction Seasonal adjustment of a time series means removing the seasonal component from a seasonal time series. The two seasonal adjustment methods used for estimating the seasonal component, X-12 ARIMA and TRAMO–SEATS, are based on different philosophies/methodologies. Thornton (2011) recently compares Butterworth low-pass filter for the UK new car registration with that of the X-12 ARIMA and the TRAMO– SEATS filters. The US Census Bureau's non-parametric seasonal adjustment method X-12 ARIMA (Findlay et al., 1998) uses moving average procedures (Shiskin et al., 1967) for estimating the seasonal component in each stage of its two stage iterative procedure. In the first stage, it uses S3 × 3 moving averages of SI differences or ratios, alternatively known as the seasonal filter. Following the Lothian procedure, the X-12 ARIMA chooses one of the following filters among S3 × 3, S3 × 5, S3 × 9, and S3 × 15 in its second stage. In almost all cases it ends up choosing the S3 × 5 filter. The TRAMO–SEATS1 procedure partitions the spectrum of the seasonal ARIMA model in order to estimate the seasonal ⁎ Corresponding author. E-mail addresses: [email protected] (A. Hayat), [email protected] (M.I. Bhatti). 1 TRAMO is an abbreviation of Time Series Regression with ARIMA Noise Missing Observations and Outliers, and SEATS is an abbreviation of Signal Extraction in ARIMA Time Series. TRAMO is a program for the estimation and forecasting of regression models with possibly ARIMA errors. SEATS is a program for the estimation of unobserved time series components following the ARIMA-model-based method. It uses a signal extraction technique with an ARIMA model to estimate the time series components. T–S (see Gomez, 1992; Gomez and Maravall, 1994, 1997) was developed from a program built by Burman (1980). Burman's program partitions the spectrum of the seasonal ARIMA model into trend, seasonal and irregular components. A nice application of the TRAMO–SEATS appears in somewhat different contexts of direct vs. indirect seasonal adjustment in Maravall (2006).

components. The default ARIMA model in TRAMO–SEATS is the Airline model of the form of (1 − LD)(1 − Ld)yt = (1 + θL)(1 + ΘmL)εt, where D, and d are seasonal and non-seasonal differences respectively, and θ and Θm are the non-seasonal and seasonal MA parts respectively, with m denoting the frequency of the data. Moreover, the Airline model is the model that TRAMO–SEATS selects from among other ARIMA models in many cases (Fischer and Planas, 2000). A quality seasonal adjustment requires the estimation of the seasonal component in such a way that the irregular component is not contaminated by it, in terms of under- or over-estimation of either the seasonal or irregular component variation (Burman, 1980). In fact, Miller and Williams (2004) documented the fact that the X-12 ARIMA's seasonal filter overestimates seasonal variation. Alternatively, X-12 ARIMA underestimates the variability of the log differenced seasonally adjusted series. Miller & Williams provide a solution to the problem of overestimation of the seasonal variation issue, but it works outside the X-12 ARIMA framework. The procedure is to dampen or smooth the seasonal variation by using one or other of the two types of estimators externally to the X-12 ARIMA estimated seasonal factors, namely the Global shrinkage estimator and the Local shrinkage estimator. The Global damping estimator is based on the shrinkage estimator of James and Stein (1961), whereas the Local damping is done by the method of Lemon and Krutchkoff (1969). Actually there have been a variety of works done on Stein-rule estimators including in the regression context, see for instance, Shalabh (1998), Chaturvedi and Shalabh (2004), and Shalabha et al. (2009) among others. The Local shrinkage estimator works well, especially for time series in which the random variation dominates the seasonal variation (Findley et al., 2004). This makes

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A. Hayat, M.I. Bhatti / Economic Modelling 33 (2013) 676–688

sense, as the aim of the Miller & Williams procedure is to smooth the seasonal variation, which is overestimated by the X-12 ARIMA default method. We find that this is because the X-12 ARIMA's default setting picks up the low order seasonal filter or moving average of SI (seasonal-irregular) ratios, S3 × 3 in the first stage and mostly S3 × 5 in the second stage of the seasonal adjustment. Findley et al. (2004) admitted the gains from Local shrinkage to the seasonal factors, but raised major concerns about the methodology of the Miller & Williams procedure, as it lacks theoretical justifications and practical implementations. These are the reasons why Miller & Williams' procedure is not adopted in X-12/13 ARIMA. Instead, the TRAMO–SEATS procedure is on the way to being implemented in X-13-ARIMA–SEATS, which aims to deliver gains similar to those of the Miller & Williams estimator for series in which the random variation dominates the seasonal variation (Findley et al., 2004). We contribute to the above discussed literature. We raise the question as to whether we can achieve Miller & Williams' (i.e., no underestimation of the variance of differenced seasonally adjusted series and better seasonal adjustments) within the X-12 ARIMA framework. The main advantage of this is that it is based on the X12-ARIMA methodology and offers easy implementation. We analyze the same question for the TRAMO–SEATS seasonal adjustment, as it is used globally second to the X-12 ARIMA method, and is gradually becoming the part of the X-12 ARIMA method. In addition, it is also interesting to analyze TRAMO–SEATS for such an issue, as TRAMO–SEATS's estimated seasonal component for the Airline model can be very close to that of X-12 ARIMA (Planas and Depoutot, 2002). Hence, TRAMO–SEATS may be no different to X-12 ARIMA on the issue of overestimation of the seasonal variation. However, it is not known whether the TRAMO–SEATS seasonal filters come close to the X-12 ARIMA filters when the data generating process (DGP) is not equivalent to the Airline model. In this case, X-12 ARIMA may produce better seasonal estimates for such series than TRAMO–SEATS, due to X-12 ARIMA's nonparametric nature and its tendency to match the seasonal filter of any DGP to the order of the moving averages of SI differences or ratios. This is the main advantage which X-12 ARIMA could have over TRAMO–SEATS. We report that the X-12 ARIMA and TRAMO–SEATS seasonal adjustment methods consistently under-estimate the variability of the differenced seasonally adjusted series. We show that this underestimation is due to a non-zero estimation error in estimating the seasonal component at each time period, which is the result of the use of a suboptimal (generally low) order seasonal filter in X12-ARIMA, suggested for estimating the seasonal component by its default method. Hence, the use of an optimal (generally high) order seasonal filter for estimating the seasonal component in X-12 ARIMA helps reduce the estimation error noticeably, helps amend the under-estimation problem, and helps improve the forecasting accuracy of the series. In TRAMO–SEATS, the Airline model is found to deliver the best seasonal filter of all ARIMA models. We find that the seasonal parameter estimate Θm comes close to its invertibility limit of − 1, which leaves almost no space for TRAMO–SEATS to produce a frequency transfer function which could be close to X-12 ARIMA's filter. This is the main advantage that X-12 ARIMA has over TRAMO–SEATS which we report. The paper is structured as follows: Section 2 provides the background for our research question of volatility under-estimation by the two seasonal adjustment methods. In Section 3, we show the causes of the under-estimation of volatility by the seasonal adjustment methods. In Section 4, we elaborate on the use of higher order seasonal filters in X-12 ARIMA for smoothing the SI (seasonalirregular) series. In Section 5, we show the masking of the volatility by the seasonal adjustment methods. In Section 6, we prescribe a remedial measure for the volatility under-estimation problem in X-12 ARIMA. In the second to the last section, we show the gains obtained

677

using the prescribed remedial measure in real life data. The last section concludes the paper. 2. Overestimation of seasonal variation The aim of the Global and Local shrinkage estimators of Miller & Williams' procedure is to dampen the seasonal variation of the X-12 ARIMA estimated seasonal factors using shrinkage estimators. We look this issue from the perspective of the (moving) variance of the log differenced seasonally adjusted series. The plot of the variance of the log differenced seasonally adjusted data is plotted in Fig. 1. TRAMOS–SEATS and X-12 ARIMA are compared and analyzed similarly below. We assume that the time series data are in logarithmic (log) form and that the seasonal component is therefore additive (rather than multiplicative) in X-12 ARIMA. In Section 7, we allow seasonal adjustments to be conducted in X-12 ARIMA in additive and multiplicative modes, with a log transformation and no transformation to the data, respectively. The findings of this paper are invariant to the seasonal adjustment mode. The models, estimation method and simulation design behind Fig. 1 are discussed in detail in the upcoming sections. The main observations from Fig. 1 which we will highlight here are the following: • the variance of the log differenced seasonally adjusted series is consistently under-estimated by X-12 ARIMA (default) and TRAMO– SEATS relative to the true variance; • the X-12 ARIMA seasonal filter tends to match the non-seasonal part of any DGP, comparing the variance from X-12 ARIMA's S3 × 15 filter to the true variance; and • the improvement of seasonal adjustment in TRAMO–SEATS seasonal filter is limited when the DGP does not match the Airline model. These points imply that: • the default seasonal filters in X-12 ARIMA and TRAMO–SEATS are not optimal for an unknown DGP; • the X-12 ARIMA seasonal filter has an advantage over the TRAMO– SEATS due to its non-parametric nature, which can help in matching the seasonal filter of any seasonal DGP, meaning that we do not need to go out of the X-12 ARIMA framework, as was suggested by Miller & Williams; and • the TRAMO–SEATS Airline model cannot do any better than this for matching the seasonal filter of the true seasonal DGP. These findings are robust to various DGPs and parameter values, as we actually repeated the exercise for different parameter values, with results which are not shown here, and the story remains the same. The true variance of the non-seasonal part of the DGP is shown by the black line. We have used seven different DGPs, which are referred to in the figure as models. The X-12 ARIMA (default) procedure underestimates the variance of the corresponding nonseasonal part of the series, as is shown by the red line. The green line indicates that TRAMO–SEATS performs similarly. The tendency of the X-12 ARIMA to match the seasonal filter of the DGP can be traced by the variance of the non-seasonal part, which increases with the length of the seasonal filter (from default: S3 × 3 and S3 × 5 in the first and second iterations to the highest seasonal filter, S3 × 15), as is depicted by the blue line. Miller & Williams presented the same idea of damping the seasonal variation by using the shrinkage estimators to X-12 ARIMA estimated seasonal factors out of X-12 ARIMA. We do the same, but employ a higher order seasonal filter of X-12 ARIMA within X-12 ARIMA to dampen the seasonal variation, and analyze the same issue for TRAMO–SEATS. TRAMO–SEATS (default) does a far better job than the X-12 ARIMA (default) procedure; however, its improvement gets constrained by the seasonal parameter Θm, as its estimate comes close to the invertibility limit of − 1 in almost all cases. Our results complement Fischer & Planas' finding

A. Hayat, M.I. Bhatti / Economic Modelling 33 (2013) 676–688

5.2

5.6

Model 2

4.4

⋅ ) var (m t

3.6 3.8 4.0 4.2 4.4

⋅ ) var(m t

Model 1

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True (simulated) X−12 ARIMA (default) TRAMO−SEATS X−12 ARIMA (S3x15)

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Model 7

0

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Fig. 1. Variance of the growth from simulated Australian GDP, with a moving window of 50 observations. The under-estimation of the variability in the growth can be seen as the vertical difference between the black line with that of the green or red lines. Note that vertical scales are different for each model, in order to highlight the masking of the volatility by the seasonal adjustments in the difference of the simulated GDP series.

that the Airline model offers the best seasonal adjustment to the series of all ARIMA models. The matter is not limited to the volatility estimate of the seasonally adjusted series. Its influence is much wider in scope. As is shown in Section 3, the over-estimation of the seasonal variation by the seasonal adjustment methods is proportional to the estimation error in the seasonal component. Thus, an improvement in the volatility measure will ensure an improvement in the seasonal adjustments as a whole. Later, we show that this can help in achieving better forecasts. The most intriguing part of the story is that X-12 ARIMA can match the seasonal filter of any DGP, whereas TRAMO–SEATS can only do so when the DGP matches its own model of estimation, for instance the Airline model.2 It is clear from Models 6 and 7 that opting for the correct (higher order than default) seasonal filter could lead to a dramatic improvement in the volatility estimate, and thus in the seasonal adjustments in X-12 ARIMA (as shown in Section 6). On the other hand, TRAMO–SEATS cannot do any better than what has already been achieved, due to what we see is the invertibility constraint on the seasonal MA parameter in the Airline model.

3. Reason for the masking of the volatility The reason why the seasonal adjustment methods could overestimate the seasonal variation or under-estimate the variability of the difference of the non-seasonal part of the series lies in the estimation error of the seasonal component. Below, we establish the link between the variability of the first difference of the seasonally adjusted series and the estimation error in the seasonal factors. Let us consider the decomposition of yt as follows: yt ¼ xt þ zt ¼ x^t þ z^t ;

ð1Þ

where xt and zt are the true seasonally adjusted and true seasonal factors, respectively. Similarly, let x^t and z^t be the corresponding estimated seasonally adjusted and seasonal factors, respectively (for instance from X-12 ARIMA or TRAMO–SEATS). We can write Eq. (1) as: xt ¼ x^t −ϱt

ð2Þ

where ϱt ¼ zt −z^t is the estimation error in zt. From Eq. (2), we have 2 It would be an interesting exercise to conduct a similar study of the newly developed data-driven method of seasonal adjustment by Heiler and Feng (2000).

2

2

2

σ x ¼ σ x^ þ σ ϱ −2σ x^ϱ ;

ð3Þ

A. Hayat, M.I. Bhatti / Economic Modelling 33 (2013) 676–688

where σ 2x ; σ 2x^ and σ 2ϱ are the variances of Δxt, Δx^t and Δϱt, respectively. Here, σ x^ϱ is the covariance between Δx^t and Δϱt. From Eq. (3) under no (or constant) estimation error in the seasonally adjusted or seasonal factors, i.e., ϱt is either zero or constant ∀ t, then σ 2 ¼ σ x^ ¼ 0, and as a result of this, σ 2x ¼ σ 2x^ . This conveys the message that the over-variation in the estimates of the seasonal factors reported by Miller & Williams or the under-estimation of the variability of the difference of the seasonally adjusted series is undesirably proportional to the estimation error ϱt.

see the complete list of the moving average coefficients of the S3t × 3 filter. Following Eq. (4) with n = 1 for quarterly series, we note 33

St

¼

3

St−4 ¼

St ¼

Proof. If ϱt = constant (including zero), σ 2 , and σ x^ will be zero. Eq. (3) will then reduce to σ 2x ¼ σ 2x^ .

St−4 ¼

In contrast to Miller & Williams, we recommend the use of higher order seasonal moving averages for dampening the seasonal variation within X-12 ARIMA. Below, we explain how the higher order seasonal filter in X-12 ARIMA relates to the mitigation of the seasonal variation while having no repercussions for the irregular component. The default X-12 ARIMA symmetric seasonal filter is a simple 3-term moving average, of simple average of odd length, 2n + 1, of SI (seasonalirregular or detrended series) values from the same calender quarter as quarter t, i.e., 3ð2nþ1Þ

¼

 1
ð2nþ1Þ 2nþ1 2nþ1 S þ St þ Stþ4 ; 3 t−4

ð4Þ

2nþ1

¼

n X 1 SI : 2n þ 1 j¼−n tþ4j

1 1X 1 SI ¼ ðSI þ SIt−4 þ SIt Þ; 3 j¼−1 t−4þ4j 3 t−8

ð7Þ

3

1  1X 1 SI ¼ SI þ SIt þ SItþ4 ; 3 j¼−1 t−4j 3 t−4

3

1  1X 1 SI ¼ SI þ SItþ4 þ SItþ8 : 3 j¼−1 tþ4þ4j 3 t

ð8Þ

ð9Þ

Substituting Eqs. (7), (8), and (9) into Eq. (6), we get: 33

St

¼

 1 SI þ 2SIt−4 þ 3SIt þ 2SItþ4 þ SItþ8 : 9 t−8

The weights attached to the SI differences above are the list of the moving average coefficients below. 33

SIt

ðSI Þ :

1 f1; 2; 3; 2; 1g: 9

Similarly, we can write these weights to SI differences for the other seasonal moving averages/filters as follows: 1 f1; 2; 3; 3; 3; 2; 1g; 15 1 39 St ðSI Þ : f1; 2; 3; 3; 3; 3; 3; 3; 3; 2; 1g; 27 1 315 St ðSI Þ : f1; 2; 3; 3; 3; 3; 3; 3; 3; 3; 3; 3; 3; 3; 3; 2; 1g: 45 It is obvious that the weights to the SI series decline with the increase in the order of the seasonal moving average. This is the basis for controlling the seasonal variation with the order of the seasonal filter within X-12 ARIMA. To explain this issue in more detail, take   πt St ¼ sin k as the first estimate of the SI series obtained by X-12 ARIMA, where k is the order of each of the 3-term moving averages and t is the time variable. We can write various X-12 ARIMA seasonal moving averages or seasonal filters for St above as follows:

with

St

ð6Þ

ðSI Þ : S35 t

4. Damping seasonal variation within X-12 ARIMA

St

 1
3 3 3 S þ St þ Stþ4 ; 3 t−4

where

Preposition. The variance of the difference of the estimated seasonal part can never be the same as the variance of the difference of the seasonal part of the DGP unless the estimation error in estimating the seasonal component is either constant or zero.

Neither of the above options (estimation errors being either constant or zero) is practically attainable by any seasonal adjustment method. However, we can aim at reducing the gap between σ 2x^ and σx2 by minimizing ϱt. Below, we show that the application of the higher order seasonal filter in X-12 ARIMA could generate lower estimation errors in the difference of the non-seasonal part of the series, which therefore helps reduce the gap between σ 2x^ and σx2. We notice that a marginal decline in the estimation error of the seasonal component could substantially improve the under-estimation of the variance of the difference of estimated seasonally adjusted series, or, for that matter, the seasonal adjustment in general. The seasonal filter with the airline model in TRAMO–SEATS offers the minimum estimation error in the seasonal component among all ARIMA models.

679

ð5Þ

S3t × +1 is referred to as the 3 × (2n + 1) seasonal moving average or seasonal filter. In stage 1 of the seasonal filter, X-12 ARIMA uses a 3 × 3 moving average to obtain S3t × 3 as an initial estimate of the seasonal factors. In stage 2 of the seasonal adjustment, it uses a criterion from Lothian (1984) to select from among four filters, S3t × 3, S3t × 5, S3t × 9 and S3t × 15. A good description of these stages of X-12 ARIMA seasonal adjustment is documented in Appendix A of Findlay et al. (1998, pp. 148–149). We see that the symmetric moving average estimation of the seasonal component in X-12 ARIMA is strictly one of smoothing. More precisely, let us work through S3t × (2n + 1) for n = 1 so that we can

    
π  1 2π πt 2 cos þ 4 cos þ 3 sin ; 9 k k k       
π  1 3π 2π πt 35 2 cos þ 4 cos þ 6 cos þ 3 sin ; St ¼ 15 k k k k         3 2 5π 4π 3π 2π   1 6 2 cos k þ 4 cos k þ 6 cos k þ 6 cos k þ 7 πt 39 ; St ¼ 4 5 sin
π  27 k þ3 6 cos k         3 2 8π 7π 6π 5π   þ 4 cos þ 6 cos þ 6 cos þ7 2 cos 1 6 k k k k 315 6 7 sin πt :       ¼ St
π  4 5 4π 3π 3π 45 k þ 6 cos þ 6 cos þ 6 cos þ3 6 cos k k k k S33 ¼ t

  It is straightforward to see that the terms prior to sin πtk in each of the above equations are not dependent on t, and hence determine the amplitude of the series. The amplitude continues to decrease with the

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A. Hayat, M.I. Bhatti / Economic Modelling 33 (2013) 676–688

−1.0 −0.5 0.0

St3x(2n+1)

0.5

1.0

increase in the order of the seasonal filter, which allows us to see smoothing done elegantly by the moving averages. In the case of the S3t × 3 filter, moving averages offer us the lowest smoothing of the seasonal component (St). The largest smoothing to SI differences (or ratios, in the case of multiplicative seasonality) is done by the S3t × 15 filter. To get the visual feeling of the smoothing by the various orders of moving averages offered by the X-12 ARIMA, we plot S3t × 3, S3t × 5, S3t × 9 and S3t × 15 for k = 9 in Fig. 2. The seasonal variation or amplitude is dampened by the increase in the order of the seasonal filter or the moving average. This is the main idea which we attempt to deliver in the paper, namely of proposing that X-12 ARIMA uses higher order seasonal filters to smooth the seasonal variation, thus leaving more variation for the difference of the log seasonally adjusted series. The procedure of using higher seasonal filters works perfectly fine within the X-12 ARIMA, as opposed to Miller & Williams' procedure, which lacks theoretical justification and practical implementation. As was mentioned earlier, the default X-12 ARIMA is more conservative in choosing the seasonal filter in stage 2, meaning that it ends up choosing S3t × 5 in almost all cases, despite having the lowest seasonal filter S3t × 3 in stage 1. Thus, we recommend that X12-ARIMA's default procedure, which is almost universally adopted, be modified (so that it can select higher order seasonal filters). In this way, we can reap gains from a smoothed seasonal component in X-12 ARIMA as per Findley et al. (2004), but in a more orthodox way. The question now arises: if the aim is solely to curb the seasonal variation, why could not this be done using a lower order smoothing filter for the trend-cycle component (referred to hereafter as the trend component)? X-12 ARIMA uses a 2 × 4 moving average for quarterly series and a 2 × 12 moving average for monthly series in order to obtain the first estimate of the trend component. This could equally mean that the X-12 ARIMA uses a higher order trend filter to smooth the trend component, thereby leaving more variation for the seasonal component. A simple fix to the problem (of under-estimation of the variation of the differenced seasonally adjusted series) can be to use the lower order trend filter, so as to leave more variation for the irregular component—a part of the seasonally adjusted series, such as the 2 × 2 and 2 × 6 trend filters, respectively, in quarterly and monthly series, for instance. However, smoothing the trend with a lower order trend filter can do more harm than good relative to smoothing the seasonal component with a higher order seasonal filter. Smoothing with a low order trend filter aims to eliminate frequencies with long periods—an undesirable outcome. Smoothing the trend component with a lower order filter could eliminate the low frequencies with periods of more than a year, for example cyclical periods of 3–5 years; an undesirable outcome of smoothing the trend component with a low order filter. Fortunately, there are no such dire consequences in smoothing the seasonal component. The SI differences only contain seasonal components with high frequencies and irregular components with no periods. Smoothing the seasonal component eliminates high frequencies with

0

20 St

40 St3x3

60 St3x5

80 St3x9

100 St3x15

Fig. 2. The effect of various symmetric moving averages (seasonal filters) on SI series (St) in the X-12 ARIMA seasonal adjustment method.

periods which are associated with the seasonality, while leaving the irregular component unharmed for the periodicity. In X-12 ARIMA, smoothing the seasonal component (with a higher order seasonal filter) in order to fix the overestimation of its variation is a more viable option than smoothing the trend component (with lower order trend filter). To get a feel for it, the gain functions of the various seasonal filters are shown in Fig. 3. The point to highlight here is that the X-12 ARIMA seasonal filters including S3 × 15 preserve the annual seasonality, since they exactly restore the multiple frequencies of 30° = 2π/ 12 = π/6. 5. Masking of volatility by the seasonal adjustment methods In this section, we look at the percentage of under-estimation as per Eq. (15) and the seasonal adjustment accuracy based on some function of ϱ, which is defined in Eq. (2), comparing the results (i) when we use the default seasonal filters, and (ii) when we use our suggested higher order seasonal filter in X-12 ARIMA and the best ARIMA seasonal filter in TRAMO–SEATS. Our methodology is based on Monte Carlo simulations, which are discussed in detail in Section 2, following the subsection on the data generating processes which are considered in the study. The Monte Carlo results are discussed in Section 3. At the end of this section, we suggest a simple solution to the problem of under-estimation in X-12 ARIMA, which works fine within the seasonal adjustment method. 5.1. Data generating processes To assess the above objectives for X-12 ARIMA and TRAMO– SEATS, we employed seven models (DGPs). For each DGP, we used three different sets of parameter values, in the hope that this range of models will compensate by covering a number of possible DGPs, since we do not know the actual underlying DGP. Miller & Williams conducted their study reporting the X-12 ARIMA's consistent over-estimation of the seasonal variation using a variety of parameter values but only one DGP. We therefore ensure that our findings are not limited by the choice of parameter values, but it would be a redundant exercise to replicate the study for other parameter values. However, we allow a range of models which model a variety of time series components, for instance, varying from fixed seasonal factors to time varying seasonal factors, and similarly with the trend and cycle components. For the model descriptions, see Table 4. We used Gross Domestic Product (GDP) data for estimating the parameter values of the model. We opted to use the GDP data due to its importance in policy making at the individual, business and government levels. The release of the seasonally adjusted GDP by the respective statistical agency therefore receives full media attention. The parameter values for the models were obtained as estimates associated with the best fit of each of the corresponding models to the quarterly Australian, UK and US seasonally unadjusted real GDP data from September 1959 to December 1983. We ended the estimation period in December 1983 to avoid a structural break in the mid-1980s (McConnell and Perez-Quiros, 2000), which complicates seasonal adjustment. Using three different data sets means that we have three different sets of parameter estimates for each model, referred to as Sets 1, 2 and 3, associated with the best fit to Australian, UK and US seasonally unadjusted real GDP data, respectively. The parameter values (estimates) of the models are presented in Table 5. In this way, we are able to note any parameter value or model specific effects in the Monte Carlo simulation results. Model 1 is an example of modeling the trend, seasonal and cyclical components deterministically, with the components being modeled as non-latent variables. We estimate Model 1 by non-linear least squares. Models 2 to 4 are known as “single source of error models”, because each of the time series components has only a single source

681

0.6 0.8 1.0

A. Hayat, M.I. Bhatti / Economic Modelling 33 (2013) 676–688

St3x3

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0

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100

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Quarterly data

0

50

100

150

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0.6 0.8 1.0

Monthly data

0

50

100

150

Fig. 3. The gain functions of the various seasonal filters in X-12 ARIMA. The gain function over each quarter/month is displayed at the top, and those over the whole series are given in the bottom two plots.

of variation (the error term). They incorporate time series components latently, assuming the innovations of each of the latent variables (trend, seasonal and cycle) to be perfectly correlated with the innovation of the observational equation (yt). Following Archibald and Koehler (2003), Model 3 employs the Roberts/McKenzie renormalization of the seasonal factors, which produce unbiased state vectors of the level, trend and seasonality. Model 4 assumes a constant growth (g), which is a feature of most economic time series with a cycle. Models 2–3 do not include a cycle equation explicitly, but instead model trend (bt) stochastically, which splits up the level (łt) and observation (yt) equations. Models 2–4 are formed and estimated within the state space framework using the exponential smoothing technique by maximizing the log-likelihood of the model. For detailed descriptions of these models, see Archibald and Koehler (2003), Ord et al. (1997), Snyder (1985) and Hyndman et al. (2002, 2005). All of the time series components in Models 2 to 4 are stochastic. Models 5 to 7 are referred as “multiple source of error models” (MSOEM), because there are multiple sources of variation for each of the time series components in the models (different errors for each latent variable), which are modeled latently. In contrast to the SSOE models, the estimation of the MSOEM must rely on the Kalman filtering technique (Kalman and Bucy, 1961), assuming that all of the Table 1 Selection criteria for the seasonal filter using the default option. A

B

C

D

E

MSR b 2.5 3×3

2.5 b MSR b 3.5

3.5 b MSR b 5.5 3×5

5.5 b MSR b 6.5

6.5 b MSR 3×9

innovations (of measurement and latent variable equations) are independent of each other in the model. Model 5, like Model 4, assumes a constant growth b for yt in level lt, but with no cycle. Model 6 assumes a deterministic trend tt with cycle. Model 7 assumes a constant growth b in level lt, with stochastic seasonality and the cyclical equations. For further details on these models, see Harvey (1989). 5.2. Monte Carlo simulations Below we explain how we assess whether X-12 ARIMA or TRAMO–SEATS smoothes the seasonally adjusted series more than what is needed, using a Monte Carlo simulation exercise. The steps taken for the ith simulation are as follows: • Simulate the seasonally unadjusted series yit from each of the above models for t = 1,2, so that yit ¼ xit þ zit ;

ð10Þ

where xit and zit are the true seasonally adjusted series and the true seasonal component, respectively. While generating yit, we restricted it to positively sloped series by imposing the constraint y1t b ynt.3 When this condition was not met, we generated another y1t before we went on to the next step.4

3 This is because the GDP data, and most of the (flow) economic data, have a strong tendency to grow over long periods of time, which means that they will be positively sloped. 4 We have noticed that the relaxation of the y1t b ynt condition in simulations does not change the findings of the study qualitatively, this condition is already met as in most cases without imposing it explicitly in simulations.

682

A. Hayat, M.I. Bhatti / Economic Modelling 33 (2013) 676–688

Table 2 The variance of the fourth order difference and the first order of the fourth order differences of the series. Series

AU-GDPa

UK-GDPa

USA-GDPa

AU-ComFin

AU-GenFin

(1 − L4)yt
 1−L4 x^ X12−Def t
 315 S 1−L4 x^ t t
 4 ^ TS−Def 1−L x t
 1−L4 x^ Bestmodel t

517.4

505.1

656.4

448.1

2809.3

514.8

502.6

654.5

443.2

2807.6

516.1

504.7

656.0

446.2

2799.8

504.9

491.3

648.2

445.6

2698.9

504.9

490.7

650.4

313.3

2697.3

388.6

367.7

141.8

43840.4

298433.4

384.8

360.4

136.8

42014.6

291764.8

385.4

366.4γ

140.7

43136.8

296173.9

362.0

332.7

127.5

432.5

2718.1

361.9

330.9

330.9

5889.4

271435.3

4

(1 − L)(1 − L )yt
 ð1−LÞ 1−L4 x^ tX12Def
 315 S ð1−LÞ 1−L4 x^ t t
 4 ^ TSDef ð1−LÞ 1−L x t
 ð1−LÞ 1−L4 x^ Bestmodel t 315

S yt ; x^ X12Def ; x^ t t ; x^ tTSDef ; and x^ Bestmodel are respectively the seasonally unadjusted series, the estimated seasonally adjusted series by default X-12 ARIMA, the estimated seasonally t t adjusted series by X-12 ARIMA's S3t × 15 filter, the estimated seasonally adjusted series by TRAMO–SEATS's default model, and the estimated seasonally adjusted series by TRAMO– SEATS' best ARIMA model. L stands for the lag operator. a The variances for the AU-GDP, UK-GDP, and USA-GDP are multiplied by 100.

• For y1t, estimate its corresponding seasonally adjusted series ( x^it ) by using X-12 ARIMA or RAMR0–SEATS, i.e.: x^it ¼ yit −z^it

ð11Þ

where z^it is the X-12 ARIMA or TRAMO–SEATS estimated seasonal component.5 • Compute the difference (growth rate6) of xit or x^it as:
 _ it_ ¼ mit_ −mi;t_ ¼ 1  100; m

ð12Þ

_ it_ runs for t_ ¼ 2. _ it_ could be xit_ or x^it_ , and m where m _ it_ with a (moving) window of 50 • Compute moving variances of m observations until t_ ¼ 165 as follows:

vij ¼

jþ50 ∑ __ t ¼jþ1


2 _ it_ −m _ ij m 49

• Repeat the above steps a total of r times, and compute an average

 moving variance v j over the v j replications for each r as follows. vj ¼

r

∑i¼1 vij : r

ð14Þ

These average moving variances for the DGP with Set 1 parameter values are shown in Fig. 1. • An average (across t_ ) of vj is obtained for analyzing the masking of the volatility by the seasonal adjustment methods. For purposes of convenience and clarity, we refer to this as the AAV (Average of Averaged moving Variance). Lastly, we compare the percentage under-estimation by the AAV of the non-seasonal part estimated by using X-12 ARIMA or TRAMO–SEATS with the AAV of the non-seasonal part of the DGP, i.e., Percentage of under  estimation ¼ ðAAV of the estimated non  seasonal part  AAV of the non  seasonal part of DGPÞ

;

ð13Þ

_ ij is the mean of where j = 1, 2, …, 115 for t_ ¼ 2; 3; …; 165 and m _ it_ . We computed moving variances as to allow us to notice the m how the variance varies with time.7

5 For quarterly data, the default X-12 ARIMA procedure forecasts yit using ARIMA modeling (Box and Jenkins, 1970) for one year before the seasonal adjustments are done on the modified series. The forecasts of the series minimize the revisions when the new data arrive (Dagum, 1975). The reason for this is that the forecast values make possible the use of a two-sided symmetrical seasonal filter, which is not applied to the beginning of the series unless the series is backcasted. The default procedure of the seasonal adjustment has no backcast and the forecast from the end of the series for one year. We present here the results from this default option but removing the initial and last eight observations in Eq. (11), so that x^ it_ ½9 : 173 is 165 observations long. This is because we look for the under-estimation of volatility of the difference between seasonally adjusted series under the best scenario of no use of one-sided filter and that of no forecast contamination for the seasonal adjustment. Moreover, the findings of the study do not change qualitatively when (i) only the default settings are used, that is, the use of one-year forecasts and a one-sided moving average filter at the beginning of the series, and (ii) the forecast option in X-12 ARIMA is muted, so that the onesided moving average seasonal filter is applied at both ends of the series, with comprising 181 observations. These findings are with the authors and can be provided upon request. 6 Since the DGP is in the log form. 7 The window size of 50 observations was good enough to see smooth changes in the moving variance.

 AAV of the non seasonal part of DGP  100:

ð15Þ This percentage of under-estimation is presented in Table 6. 5.3. Findings The results of the Monte Carlo experiment are presented in Table 6. A glance at it reveals that the direction of under-estimation of AAV is invariant to the model and parameter values used for the DGP, as well as to the seasonal adjustment method used for estimating the non-seasonal part of the DGP (see the top panel of Table 6). However, the percentage of under-estimation varies with the change in model, parameter values used for the DGP and seasonal adjustment method employed for estimating the non-seasonal part of the DGP. The lowest average under-estimation of AAV was 10.80% with Model 7 as the DGP, and the highest was 27.84% with Model 2 as the DGP and the non-seasonal part of the DGP was estimated by X-12 ARIMA. Similarly, the lowest average under-estimation of AAV across the DGPs was 16.64% with the Set 1 parameter values and the highest was 21.80% with the Set 2 parameter values for X-12 ARIMA estimated seasonally adjusted series. The average underestimation of AAV from TRAMO–SEATS is virtually the same. The most interesting observation is that the average (across models) of the average (across three parameter sets) under-estimation of AAV with X-12 ARIMA estimating the non-seasonal part of the DGP is

A. Hayat, M.I. Bhatti / Economic Modelling 33 (2013) 676–688

683

Table 3 The estimation error (in the seasonal component) as a sum of the square of the differenced series, as per Eqs. (16) and (17). Differenced series

 ð1−LÞ 1−L4 yt −ð1−LÞ 1−L4 x^ tX12Def

 315 S ð1−LÞ 1−L4 yt −ð1−LÞ 1−L4 x^ t t

 4 4 ^ TSDef ð1−LÞ 1−L yt −ð1−LÞ 1−L x t

 ð1−LÞ 1−L4 yt −ð1−LÞ 1−L4 x^ tTSDef

AU-GDP

UK-GDP

AU-ComFin

AU-GenFin

9.8

12.3

USA-GDP 4.8

1454.2

17048.7

3.6

3.0

1.3

202.3

2307.9

8.8

17.0

4.6

Large number

Large number

8.8

17.7

23.3

Large number

Large number

See the notes in Table 2.

about 19%, which is greater than that for its TRAMO–SEATS counterpart, which turned out to be 15%. Thus, we can assert that TRAMO–SEATS performs better than X-12 ARIMA on average for estimating the non-seasonal part of the DGP. This observation would have been masked had we not used a range of models, as well as different sets of parameter values for the DGP. For instance, had we used only either Model 1 or Model 6, there would have been a bias in favor of one seasonal adjustment method over the other, since, on average, Model 1 favors TRAMO–SEATS for the lesser amount of under-estimation of the AAV, whereas Model 6 favors X-12 ARIMA. However, there is no room for any further improvement of the AAV estimate for TRAMO–SEATS, as we find that the seasonal MA parameter Θ estimate comes just below the invertibility limit of −1 in almost all cases. In contrast, X-12 ARIMA has the tendency to match the seasonal filter of any DGP, due to its non-parametric nature, so that the under-estimation in AAV estimates can be virtually zero in Table 6. In this sense, X-12 ARIMA improves on TRAMO–SEATS for seasonal adjustment with an unknown DGP, as is generally the case in practice. Below, we show how changing the seasonal filter can improve the AAV estimate in X-12 ARIMA; unfortunately, there is no room for such maneuvering of the seasonal filter in TRAMO–SEATS. 6. A simple remedy for masking the volatility in X-12 ARIMA We propose a simple idea for fixing the masking of the volatility of the seasonally adjusted series in X-12 ARIMA, and hence improving the seasonal adjustments overall. We do not claim here that the higher order seasonal filter S3 × 15 for each quarter/month is the ideal filter for every data series (of course, this would depend partly on the seasonal variation to irregular variation ratio); instead, this exercise is an attempt to show that X-12 ARIMA is fully capable of matching the seasonal filter of any DGP if we moderate and enhance the default procedure for choosing the right seasonal filter. Theoretically, X-12 ARIMA can have seasonal filters of the type of Skt × (2n + 1), where k = 3, 5, … and n = 1, 2 …. Currently, X-12 ARIMA offers Skt × (2n + 1) for n = 1, 2, 4, 7, and this can easily be extended to n = 3, 5, 6 as well. 6.1. Higher order seasonal filter and improvement to the seasonal adjustments Following the findings that X-12 ARIMA tends to under-estimate the volatility of the seasonally adjusted series, we propose that the default X-12 ARIMA should use a higher order seasonal filter.8 This is because it smooths the seasonal variation, and thus leaves more variation for the seasonally adjusted series. For instance, we employed an S3t × 15 seasonal filter for each quarter in the Monte Carlo study and observed a dramatic improvement (19.35% vs. 4.20%, comparing the averaged under-estimation in the bottom panel with the averaged

under-estimation in the top panel of Table 6) in the volatility estimate of the differenced seasonally adjusted series. As Eq. (3) indicated, an improvement in the volatility estimate of the difference of seasonally adjusted series is linked to the improvement in the difference of estimation error in the seasonal component. We report the sum of the absolute difference of estimation errors   ∑Δ2 and the sum of the squared difference of estimation errors ð∑jΔjÞ in the seasonal factors in Table 7. The seasonal filter used in the top panel of Table 7 was the default X-12 ARIMA and that used in the bottom panel was the highest order seasonal filter available in X-12 ARIMA, 3 × 15. We observed an improvement, as the sum of jΔj with the default seasonal filter was 2.45, compared to 2.14 with the 3 × 15 seasonal filter. Similarly, the sum of Δ2t came to about 0.064 with the default seasonal filter and about 0.050 with the 3 × 15 seasonal filter, on average. We note that, under the Set 2 and Set 3 parameter values, Models 5, 6 and 7 over-estimate the variability, on average, of the difference of the seasonally adjusted series with the 3 × 15 seasonal filter (see the bottom panel of Table 6),which suggests that we have used a higher order seasonal filter than was required. Had the choice of other seasonal filters, for instance 3 × 11 or 3 × 13, and the appropriate method for choosing the correct seasonal filter for each quarter been available in the X-12 ARIMA method, we would have chosen the better (optimal) seasonal filter, and hence reduced the estimation error further, even for the cases where we still observe under-estimation. Thus, we can reduce the volatility under-estimation problem by increasing the length of the seasonal filter. The nice thing about our idea is that it works within the X-12 ARIMA method and can easily be implemented for its default seasonal adjustment method. We find the default seasonal filter in TRAMO–SEATS to be the best filter, in offering the lowest estimation error in the seasonal component when the true DGP of the underlying series is unknown. These findings are in line with the findings of Fischer and Planas (2000) paper for the best seasonal adjustment. Repeating our strategy for X-12 ARIMA, we tried to lengthen the seasonal filter in TRAMO–SEAT by choosing the best ARIMA model among the available ARIMA models based on the AIC model selection criterion, and observed that the under-estimation of AAV got worst (see Table 6), as did the estimation error 9 of the seasonal filter. The under-estimation of the AAV, on average, increased from 14.53% to 16.54% under the best ARIMA model, compared to the basic Airline model. As discussed above, the number of as of seasonal filters can be increased from currently 4 seasonal filters to create more room within X-12 ARIMA to reach to the optimal seasonal filter. In contrary, the TRAMO–SEATS will remain to underestimate the variability of the seasonally adjusted series if the DGP does not match the Airline model, in which seasonal MA parameter stretches to its maximum value just below the invertibility limit of − 1. 6.2. Proposed selection criterion for the seasonal filter

8

The appropriateness of a longer seasonal filter has been suggested by researchers investigating ARIMA model based signal-extraction seasonal adjustments (Bell and Hilmer, 1984, pp.308–309).

The current procedure for selecting the seasonal filter in the second stage of X-12 ARIMA is based on the method of Lothian (1984).

684

A. Hayat, M.I. Bhatti / Economic Modelling 33 (2013) 676–688

Table 4 Models employed as the DGP for the time series data. Time series components Model

DGP

1

yt ¼ t t þ st þ ct ða; b; α 1 ; β 1 ; α 2 ; ϕ1 ; ϕ2 ; ϕ3 ; σ e ; c0 ; c−1 ; c−2 Þ

2

yt ¼ lt−1 þ bt−1 þ st−4 þ et ðα; β; γ; σ e ; l0 ; b0 ; s−3 ; s−2 ; s−1 ; s0 Þ

3

yt ¼ lt−1 þ bt−1 þ skt þ et ðα; β; γ; σ e ; l0 ; b0 ; s−3 ; s−2 ; s−1 ; s0 Þ

4

yt ¼ lt þ st þ ct ðα; β; γ; ϕ; σ e ; l0 ; g; s−3 ; s−2 ; s−1 ; s0 ; c0 Þ

5

6

7

Trend

Seasonal

Cyclical

Irregular

st ¼ α 1 cos

ct = ϕ1ct − 1 + ϕ2ct − 2 + ϕ3ct − 3 + et

et

lt = lt − 1 + bt − 1 + αet bt = bt − 1 + βet lt = lt − 1 + bt − 1 + αet + rt r t ¼ 0 þ γ4 et bt = bt − 1 + βet lt = lt − 1 + g + αet

st = st − 4

–

et

–

et

st ¼ −  ðst−1 þ st−2 þ st−3 Þ þ ωt ωt ∼N 0; σ 2ω

ct = ϕ1ct − 1 + ϕ2ct − 2 + ϕ3ct − 3 + et –

et

lt ¼ b
þ lt−1þ ηt ηt ∼N 0; σ 2η



tt = a + bt

yt ¼ t t þ st þ ct  b; σ e ; σ η ; σ w ; l0 ; s−1 ; s−2 ; s0 yt ¼ t t þ ct þ st−4 þ et ða; b; γ1 ; γ 2 ; λc ; σ e ; σ w ; s−3 ; s−2 ; s−1 ; s0 Þ

s4t



1 1 πt þ β1 sin πt 2 2 þα 1 cosðπt Þ + γet



s1t − 1

= + γet − rt r t ¼ 0 þ γ4 et +1 stk = skt − 1 − rt st = st − 4 + βet

tt = a + bt

yt ¼ lt þ t t þ st þ ct þ et   b; ρ; λc ; σ e ; σ η ; σ w ; σ h ; σ h ; l0 ; s−3 ; s−2 ; s−1 ; s0



ct ¼ γ 1 cos þ ðγc t Þþ γ2 sinðγ c t Þ

st ¼ st−1  þω t ωt ∼N 0; σ 2ω st ¼ − ð s t−1  þ st−2 þ st−3 Þ þ ωt ωt ∼N 0; σ 2ω

lt ¼ b
þ lt−1þ ηt ηt ∼N 0; σ 2η

ct ¼ ρ cosðλc Þ þ ct−1 þ ρ sinðλc Þct−1 þ ht

et

et

et

ht ∼ N(0,σ2ω) ct ¼ −ρ sinðλc Þ þ ct−1 þ ρ cosðλc Þct−1 þ ht    ht ∼N 0; σ 2ω Notes: et in Models 1–7 are niid. ct in Model 1 is an autoregressive process of order at most 3. We have listed parameters of the models to be estimated in parentheses in column 2. In Model 7, ht and ht⁎ are assumed to have same variance.

Table 5 Parameter values used in the twenty-one DGPs. Model

Parameter values

1

Set 1 Set 2 Set 3

2

Set 1 Set 2 Set 3

3

Set 1 Set 2 Set 3

4

Set 1 Set 2 Set 3

5

Set 1 Set 2 Set 3

a

b

10.458 11.221 12.809

0.007 0.005 0.006

α

β

0.550 0.551 0.950 α 0.550 0.551 0.950 α 0.000 0.000 0.193 σe

6

7

Set 1 Set 2 Set 3

Set 1 Set 2 Set 3

0.007 0.008 0.000

ϕ1 0.814 0.557 0.874 γ

0.05 0.05 0.05 β

0.050 0.204 0.181 γ

0.05 0.05 0.05 β

0.050 0.204 0.181 γ

0.000 0.207 0.147 ση

0.816 0.597 0.660 σω

0.017 0.012 0.013

0.002 0.002 0.001

ϕ2

ϕ3

α1

β1

α2

– 0.334 0.544

0.146 – −0.462

−0.014 −0.016 −0.016

−0.064 −0.021 −0.016

−0.034 −0.012 −0.019

σe

l0

b0

s−3

s−2

s−1

s0

0.022 0.018 0.014

10.567 11.380 12.973

0.010 0.008 0.010

−0.022 0.002 −0.005

0.098 0.023 0.033

−0.044 −0.034 −0.036

−0.032 0.009 0.008

σe

l0

b0

s−3

s−2

s−1

s0

0.022 0.018 0.014

10.567 11.380 12.973

0.010 0.008 0.010

−0.022 0.002 −0.005

0.098 0.023 0.033

−0.044 −0.034 −0.036

−0.032 0.009 0.008

σe

l0

b0

g

s−3

s−2

0.961 0.997 1.000

0.0205 0.0177 0.0141

10.739 11.152 11.967

l0

b

s−2

10.574 11.340 12.982

0.009 0.006 0.008

−0.033 −0.033 −0.006

α

b

γ1

γ2

γc

σe

10.617 11.441 13.016

0.010 0.004 0.006

−0.061 −0.069 −0.055

−0.028 0.079 0.210

0.068 0.034 0.029

0.023 0.019 0.024

σe

ση

σω

σh

λc

ρ

0.003 0.000 0.003

0.656 0.3349 0.256

0.958 1.000 0.983

0.008 0.009 0.000

0.014 0.010 0.011

0.002 0.002 0.002

0.008 0.006 0.008 s−1

−0.020 −0.001 −0.005

σe 0.021 0.019 0.014

0.098 0.025 0.034

c0

c−1

c−2

* * *

* * *

* * *

s−1

s0

c0

−0.047 −0.033 −0.036

−0.030 0.008 0.008

−0.182 0.215 1.007

s−1

s0

−0.020 −0.006 −0.013

−0.030 −0.001 −0.001

s0

0.086 0.004 0.034 σω

−0.054 4.6 × 10−4 −0.035 s−3

0.003 0.007 0.009 l0 10.579 11.34 12.977

−0.050 −0.050 −0.004

s−2 0.100 0.032 0.018

b

s−2

s−1

s0

c0

c0*

0.099 0.006 0.008

−0.012 −0.033 −0.006

0.099 0.004 0.034

−0.042 0.000 −0.035

−0.006 2 × 10−4 0.006

−0.006 0.002 0.006


 σ 2e Notes: * The initial values for ct, c0, c−1 and c−2 in Model 1 are assumed to follow N 0; 1−ϕ2 −ϕ , where σ2e is the variance of et. In Models 2–4, the initial parameters such as 3 −ϕ2 1

3

3

l0, b0, s− 3,… are estimated by optimizing their respective optimum criteria. In Models 5–7, the initial parameters are estimates from the respective estimated state, for example, l0 and b0 in Model 5 are the first observations of estimated l0 and bt.

A. Hayat, M.I. Bhatti / Economic Modelling 33 (2013) 676–688 Table 6 The percentage under-estimation of AAV by the seasonal adjustment methods. Models

Model 1 Model 2 Model 3 Model 4 Model 5 Model 6 Model 7 Average

Table 7 The sum of the absolute estimation error (jj) and the sum of the squared estimation error (ϱ2) in the difference of X-12 ARIMA estimated seasonally adjusted series ( x^_ t ) under the default and 3 × 15 seasonal filters respectively.

X-12 ARIMA (seas. filter: default)a

T–S (default Airline model)c

Set 1

Average

Set 1

Set 2

Set 3

Average

Seas. filter: default

Seas. filter: default

19.07 27.84 26.70 25.76 11.56 13.71 10.80 19.35

3.67 13.04 12.21 3.25 4.42 11.51 3.09 7.31

3.60 33.71 31.19 30.73 5.88 19.53 6.54 18.74

23.48 24.24 24.06 20.91 4.98 20.97 4.18 17.55

10.25 23.66 22.48 18.30 5.10 17.34 4.60 14.53

Sum of jϱj

Sum ϱ2

16.66 21.49 20.74 15.26 12.95 17.01 12.35 16.64

Set 2 16.74 34.50 32.01 34.49 10.98 12.44 11.57 21.82

Set 3 23.81 27.53 27.35 27.52 10.75 11.69 8.47 19.59

X-12 ARIMA (seas. filter: 3 × 15)b

T–S (best ARIMA model)d

Models

Set 1

Set 2

Set 3

Average

Set 1

Set 2

Set 3

Average

Model 1 Model 2 Model 3 Model 4 Model 5 Model 6 Model 7 Average

6.17 11.52 10.67 5.72 −1.63 4.40 −2.27 4.94

6.37 19.54 16.49 19.49 −9.97 −11.04 −9.80 4.44

8.15 13.66 13.45 15.44 −6.19 −13.60 −8.40 3.22

6.90 14.91 13.54 13.55 −5.93 −6.75 −6.82 4.20

6.34 14.33 13.51 3.80 5.13 14.24 4.89 8.89

7.24 34.71 32.23 31.27 7.33 22.65 8.85 20.61

34.09 24.73 24.55 21.45 5.76 24.18 6.12 20.13

15.89 24.59 23.43 18.84 6.07 20.36 6.62 16.54

a In 2 of the 3 stages, the default X-12 ARIMA uses a 3 × 3 moving average procedure to obtain an s3t × 3 as the initial estimates of the seasonal factors in its stage 1. In stage 2, it uses a criterion due (Lothian, 1984) to select from among the four seasonal filters, s3t × 3, s3t × 5, s3t × 9 and s3t × 15, to obtain the refined seasonal estimates. The moving average representation of the seasonal filters is discussed in Section 6. b s3t × 15 seasonal filter is used for each quarter in both stages 1 and 2 of the X12-ARIMA. c For the default seasonal adjustment, T–S uses the Airline ARIMA model in (??) prior to partitioning it into time series components via frequency domain. d The TRAMO part of T–S is invoked to select the best ARIMA model (of order at most (p, d, q)(P, D, Q) = (3, 2, 1)(3, 1, 1)) prior to partitioning it into time series components via the frequency domain, where (p, q) are the orders of AR and MA polynomials of non-seasonal terms of the differenced (d) series, and likewise, (P, D, Q) are the seasonal counterparts of an ARIMA model.

Model

Set 1

Set 2

Set 3

Average

Set 1

Set 2

Set 3

Average

Model 1 Model 2 Model 3 Model 4 Model 5 Model 6 Model 7 Average

2.78 2.30 2.27 2.68 2.24 3.58 2.24 2.58

2.67 2.14 2.06 3.12 1.89 3.50 1.99 2.48

3.33 1.56 1.52 2.13 1.48 4.54 1.42 2.28

2.92 2.00 1.95 2.64 1.87 3.88 1.88 2.45

0.077 0.055 0.054 0.072 0.045 0.113 0.054 0.067

0.071 0.047 0.044 0.092 0.033 0.118 0.043 0.064

0.101 0.023 0.022 0.043 0.020 0.201 0.019 0.061

0.08 0.04 0.04 0.07 0.03 0.14 0.04 0.064

The method depends on the value of the moving seasonality ratio (MSR), given by the global irregular-seasonal ratio (I/S). In X-12 ARIMA, the automatic selection of a moving average in stage 2 is based on the following strategy (refer to Table 1): if the overall MSR occurs within zone A (MSR b 2.5), a 3 × 3 moving average is used; if it occurs within zone C (3.5 b MSR b 5.5), a 3 × 5 moving average is selected; if it occurs within zone E (MSR > 6.5), a 39 moving average is selected. In the X-12 ARIMA manual, we did not find any option for selecting a 3 × 15 seasonal filter in its default (automatic) selection scheme. If the MSR occurs within zones B or D, one year's observations are removed from the end of the series, and the MSR is re-calculated. If the ratio again occurs within zones B or D, we repeat the procedure, removing a maximum of five years of observations. If this does not work, i.e., if we are again within zones B or D, a 3 × 5 moving average is selected. Thus, we see from this approach that the seasonal filter selection in stage 2 of X-12 ARIMA's default method is biased towards the 3 × 5 seasonal filter, as the MSR limit is virtually from 3.5 to 6.5. This is the reason why X-12 ARIMA mostly ends up selecting a 3 × 5 seasonal filter in stage 2 of the seasonal estimate. Below, we suggest an alternative criterion for selecting an appropriate seasonal filter in X-12 ARIMA, which is free from any biases. We attempt to measure the seasonal adjustment accuracy for the real data as follows. 6.2.1. No trend Following Eq. (1), where yt is the seasonal data with no stochastic trend (i.e., no non-seasonal unit root), 1−L

m

  m m yt ¼ 1−L x^ðsf Þt þ 1−L ^z t ;

where x^ðsf Þt is the estimated non-seasonal part as a function of sf (seasonal filter), ^ z t is as defined earlier, and m stands for the seasonal

Seas. filter: 3 × 15

Seas. filter: 3 × 15

Sum of jϱj

Sum of ϱ2

Model

Set 1

Set 2

Set 3

Average

Set 1

Set 2

Set 3

Average

Model 1 Model 2 Model 3 Model 4 Model 5 Model 6 Model 7 Average

2.00 1.35 1.35 2.07 1.78 2.51 2.03 1.87

2.18 1.98 1.96 2.89 1.73 3.65 2.15 2.36

2.79 1.54 1.52 1.83 1.28 4.85 1.43 2.18

2.32 1.62 1.61 2.26 1.59 3.67 1.87 2.14

0.043 0.018 0.018 0.042 0.029 0.056 0.041 0.035

0.044 0.038 0.036 0.085 0.028 0.127 0.044 0.058

0.075 0.021 0.021 0.034 0.015 0.225 0.019 0.059

0.05 0.03 0.03 0.05 0.02 0.14 0.03 0.050

See the footnotes in Table 6.

frequency. If z^t ¼ zt and seasonality is deterministic, this will mean       that 1−Lm ^z t ¼ 0;x^ðsf Þt ¼ xt ; and 1−Lm yt ¼ 1−Lm x^ðsf Þt . Howevm m er, if ^z t ≠zt ; 1−L yt ≠ 1−L xðsf Þt . Thus, we can aim at minimizing min sf



685

n  X

m

1−L

2  m yt − 1−L x^t ðsf Þ

ð16Þ

i¼1

s.t. sf, where sf = X-12 ARIMA default, 3 × 3, 3 × 5, 3 × 9, and 3 × 15. The quantity within squared brackets (roughly) measures the estimation error in estimating the seasonal component embedded into the seasonally adjusted series, provided that the other parameters of the seasonal adjustments remain the same. 6.2.2. With trend Let yt be the seasonal data with a stochastic trend (i.e., with a non-seasonal unit root),    m m m ð1−LÞ 1−L yt ¼ ð1−LÞ 1−L x^ðsf Þt ð1−LÞ 1−L z^t ; and we can minimize min sf

n  X

2   m m ð1−LÞ 1−L yt −ð1−LÞ 1−L x^t ðsf Þ

ð17Þ

i¼1

s.t. sf, where the interpretation of the term within squared brackets is the same as for Eq. (16) above. 7. Application to real life data We employ five different data sets. As was mentioned earlier, it is unnecessary to replicate the (Miller and Williams, 2004) exercise which was done on the M3 competition data (Makridakis and Hibon, 2000), whereby it was established that the shrinkage estimators for the seasonal damping help to improve the forecasting accuracy, following which (Findley et al., 2004) showed that they help in improving the seasonal adjustment. Instead, we take five time series and elaborate in detail as to where we see the gains for the seasonal adjustment from using a higher order seasonal filter. This (higher order seasonal filters) is a

A. Hayat, M.I. Bhatti / Economic Modelling 33 (2013) 676–688

15.0

1e+07

16.0

3e+07

17.0

5e+07

18.0

686

1985

1990

1995

2005

2000

2010

1985

1990

1995

2005

2010

Additive

1.3

Multiplicative

2000

default

0.2

1.2

default

St3x15

0.8

−0.2

0.9

1.0

0.0

1.1

0.1

St3x15

2

4

6

8

10

12

2

4

6

8

10

12

Fig. 4. The X-12 ARIMA estimated seasonal components of the Commercial Finance in Australia using default and S3 × 15 seasonal filters. The bottom plots are of the X-12 ARIMA estimated seasonal factors in the year 1994.

better approach to seasonal adjustment than Miller & William's (Findley et al., 2004), in that it works with the X-12 ARIMA seasonal adjustments. Moreover, its superiority to the TRAMO–SEATS filter, due its tendency to match the seasonal filter of any DGP, was seen clearly in Fig. 1 earlier. The details of the data series are as follows: • Australian seasonally unadjusted GDP from September 1959 to March 2011. The series can be obtained from http://www.abs.gov.au/. • UK seasonally unadjusted GDP from September 1959 to March 2011. The series can be accessed at http://www.statistics.gov.uk/default.asp. • USA seasonally unadjusted GDP from September 1959 to December 2004. The series can be downloaded at http://www.bea.gov. The BEA no longer prepares seasonally unadjusted estimates of GDP, as a result of its budget cuts (private email correspondence). • Australian Commercial Finance and Australian General Lease Finance from January 1985 to February 2011. The data can be obtained from the Australian Bureau of Statistics website given above. We first show that the higher order seasonal filter in X-12 ARIMA helps improve the seasonal adjustment.9 We show later that using higher order seasonal filters for series with a higher random variation than seasonal improves the forecasting accuracy. The last two finance series plotted in Fig. 4 are shown to have an improved forecasting accuracy when the seasonal filter is S3 × 15. This is due mainly to the fact that the default X-12 ARIMA seasonal adjustment overestimates the variation in the seasonal component, as is shown by the lower panels for the year 1994 in Fig. 4. These findings are invariant to the seasonal adjustment question of whether to seasonally adjust the series at its original scale in multiplicative mode or to perform additive seasonal adjustment on the log scale of the series. The 9 We again admit that the order of the seasonal filter that we employ here, S3 × 15, may not be the optimal filter for each series. The aim of this exercise is to show that the deterioration which we find in seasonal adjustments and which was reported by Findley et al. (2004) is due to the fact that X-12 ARIMA uses lower order seasonal filters for the seasonal adjustment. This means that the optimal filter for the X-12 ARIMA seasonal adjustment for most series would be higher than what is offered by its default method. Our simulations and application findings support this view.

automatic mode selection in X-12 ARIMA chooses the multiplicative model since the seasonal variation increases with the series level (see the upper left panel of Fig. 4). We have allowed for the series to be adjusted for outliers automatically in X-12 ARIMA and TRAMO–SEATS. 7.1. Seasonal adjustment We measure the seasonal adjustment accuracy for the real data using Eqs. (16) and (17). These quantities are computed and compared for: (i) X-12 ARIMA with the default filter and the S3 × 15 filter; and (ii) TRAMO–SEATS with the default Airline filter and the best ARIMA filter. The seasonal adjustments are done on the log scale of the data mentioned above. In X-12 ARIMA, we used the additive model for the seasonal adjustment. We find that the findings are invariant to the scale of the data and the mode of the seasonal adjustment. We use the term ‘growth’ interchangeably for the fourth difference and the first difference of the fourth difference of the data series, since data is in the log scale and it simplifies our discussion below. Table 2 reveals an under-estimation, though not substantial, of the variability of the log differenced seasonally adjusted series by X-12 ARIMA, and improvements to the variability offered by its usage of higher order seasonal filters. For example, the actual variance of the US GDP growth was 141.8; the X-12 ARIMA's default estimated it be 136.8, whereas the application of the S3 × 15 filter brought the variance of the seasonally adjusted GDP growth up to 140.7—closer to the actual number of 141.8.10 The story is the same for the rest of the series, as the blue numbers come close to those in bold face. TRAMO–SEATS never performs better than X-12 ARIMA for the under-estimation of the variance of growth for these series. In fact, the variances of the growth of the TRAMO–SEATS adjusted series are quite a way off the actual variances of the growth series. For example, the variance of the US GDP growth of the TRAMO–SEATS 10 To show the effect of smoothing the seasonal component, we multiplied the variances of the various differences of the AU-GDP, UK-GDP, and USA-GDP by 100.

A. Hayat, M.I. Bhatti / Economic Modelling 33 (2013) 676–688

0.8

1.0

St3x15 St3x9

0.7

St

3x9

default

0.3

0.5

1.2

St3x15

Absolute Scaled Error

1.4

default

0.9

Additive seasoanl adjustment

0.6

Absolute Scaled Error

Multplicative seasonal adjustment

687

5

10

15

5

20

20

Additive seasonal adjustment

St3x15 St3x9 5

10

15

20

5 4 3

default St3x15

2

Absolute Scaled Error

6 5 4 3

default

2

Absolute Scaled Error

15

6

Multplicative seasonal adjustment

10

St3x9 5

10

15

20

Fig. 5. The absolute scaled error of the Commercial Finance (top two plots) and General Lease Finance (bottom two plots) data in Australia using various seasonal filters.

adjusted series (using the default Airline model) was found to be 127.5, compared to the actual number of 141.8—an even larger under-estimation. As was discussed earlier, the seasonal filter with the best ARIMA model in TRAMO–SEATS is no better than what is obtained from its Airline model—in fact, this time the best ARIMA model in TRAMO–SEATS provides the most unsatisfactory seasonal adjustments of all. The estimation error in the seasonal component, as measured by the sum of the square of the differenced series as per Eqs. (16) and (17), is reported in Table 3. As was shown earlier, the improvement in the volatility estimate is proportional to the decline in the estimation error of the seasonal component. We find a remarkable improvement in the estimation error of the X-12 ARIMA estimated seasonal component with the S3 × 15 seasonal filter. For instance, for the US GDP, the estimation error decreases from 4.8 with the default seasonal filter in X-12 ARIMA to only 1.3 with the S3 × 15 seasonal filter in X-12 ARIMA. The story is virtually the same for the other data sets. The estimation errors with the S3 × 15 seasonal filter in X-12 ARIMA are shown in blue. The TRAMO–SEATS did not do as well as X-12 ARIMA on average across series, as the estimation errors came out very large. The main observations from this exercise can be summarized as follows: • The default X-12 ARIMA seasonal adjustment underestimates the variability of the differenced seasonally adjusted series. • The higher order seasonal filter decreases the seasonal variation and increases the variability of the differenced seasonally adjusted series. • The estimation error in the seasonal component can get considerably lower with the higher order seasonal filter than with the default filter, and hence offers better seasonal adjustments. 7.2. Forecasting Miller and Williams (2004) have shown the gains to the forecasting accuracy from damping the X-12 ARIMA-estimated seasonal factors for

the huge and varied data set available from the M3 competition. In this sense, replicating Miller & Williams' finding using the new seasonal damping on the M3 competition data set would be a redundant exercise, given that we know that damping improves the forecasting accuracy. Instead, we illustrate that similar gains can be achieved within the X-12 ARIMA by opting to choose the correct (possibly higher order) seasonal filter. As in Section 5, we compute the seasonal adjustment accuracy measures, this time in terms of the forecasting error (a variant of the difference between the real data and its estimated counterpart) rather than the estimation error (a variant of the difference between the simulated (true) data and its estimated counterpart). We used the Mean Absolute Scaled Error (MASE) forecasting measure (Hyndman and Koehler, 2006). The most commonly adopted procedure when looking at the forecast accuracy is to deseasonalize the series, forecast it, reseasonalize the forecasts with the seasonal factors, and compare them with the withheld actual observations of the data. We used the exponential smoothing models of Hyndman et al. (2002) to forecast the deseasonalized series. The deseasonalization is done by the X-12 ARIMA default seasonal filter, and the higher order seasonal filters (S3 × 9 and S3 × 15). We used both of the seasonal adjustment modes: additive for the log of the data, and multiplicative for the data at its actual scale. We compute the MASE using data to time n, thereby leaving a window for the forecast horizon to period h = T − n = 20, where t = 1, 2, 3, … n, … T. We kept sliding the data size n forward by 1, thereby decreasing the forecast horizon h by 1, until we were left with forecast horizon h of 1. These MASEs are plotted in Fig. 5. The first observation is that the multiplicative model mentioned earlier offers a better seasonal adjustment for the forecast up to 20 months ahead for the two finance series. The improvements to the MASE for higher order seasonal filters (S3 × 9 and S3 × 15) are much clearer with the Commercial Finance series than with the General Lease Finance series. The S3 × 15 seasonal filter suits the Commercial Finance series better because the forecasts are slightly better with this filter than with the S3 × 9 filter. However, it is the other way

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A. Hayat, M.I. Bhatti / Economic Modelling 33 (2013) 676–688

round for General Lease Finance series, where the S3 × 9 filter appears to produce better forecasts for the initial 12 months. 7.3. Limitations of higher order seasonal filters We find that higher order seasonal filters improve the seasonal adjustment. However, higher order seasonal filters require large numbers of observations in order to seasonally adjust the data. Along with the forecasts, seasonal adjustments with higher order seasonal filters at the end of the data could be an issue. In particular, when new data come, the revised seasonally adjusted estimates could deliver large revisions. However, this is an issue which we leave unexplored for future work. 8. Concluding remarks In this paper, we have shown that the X-12 ARIMA and TRAMO– SEATS seasonal adjustment methods under-estimate the variability of the difference of seasonally adjusted series. We have demonstrated that this is due to the larger estimation errors in the estimated seasonal factors, which are the result of a low order (default) seasonal filter in the X-12 ARIMA seasonal adjustment method. Both are the X-12 ARIMA adjustment methods. We have proposed a simple remedy for fixing the under-estimation of the variability of the differenced estimated seasonally adjusted series within the X-12 ARIMA seasonal adjustment scheme. To fix the under-estimation problem, the default seasonal filter of X-12 ARIMA should be allowed to use a higher order seasonal filter in its stages 1 and 2, rather than the lower order seasonal filter. Ideally, X-12 ARIMA should have more options for using higher order seasonal filters Skt × (2n + 1), where k = 3, 5, … and n = 1, 2 … for the seasonal adjustment. In the TRAMO–SEATS method, the default procedure (the Airline model) seems to be the best (model), in that it offers the lowest under-estimation of the variability of the differenced seasonally adjusted series when the true DGP of the underlying series is unknown. This is because the use of the best ARIMA model among all available models in TRAMO worsens the under-estimation problem and results in a larger estimation error of the seasonally adjusted series. As was mentioned above, the X-12 ARIMA adjusts for seasonality twice during its iterative procedure of estimating time series component. In stage 1, it employs the S3t × 3 seasonal moving average or the seasonal filter. In stage 2 of the adjustment, it uses a criterion from Lothian (1984) to select a filter from among the four filters, S3t × 3, S3t × 5, S3t × 9, and S3t × 15, to estimate the seasonal component. In almost all cases, it ends up selecting S3t × 3. We find this to be the main reason for X-12 ARIMA over-estimating the seasonal variation or under-estimating the variability of the irregular component—part of the seasonally adjusted series. Allowing X-12 ARIMA to choose the higher order seasonal filter resolves this problem. Acknowledgment We would like to thank Professor Maxwell King (Monash University, Australia), anonymous referees, and Professor Joakim Westerlund (Deakin University, Australia) for their constructive and useful comments and suggestions. References Archibald, B.C., Koehler, A.B., 2003. Normalization of seasonal factors in Winters' method. International Journal of Forecasting 19, 143–148.

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