CONFERENCE ON SEASONALITY, SEASONAL ADJUSTMENT AND THEIR

IMPLICATIONS FOR SHORT-TERM ANALYSIS AND FORECASTING

10-12 MAY 2006

Seasonal heteroskedasticity in time series data: modeling, estimation, and testing

Thomas M. Trimbur

Seasonal heteroskedasticity in time series data :modeling, estimation, and testing

Thomas M. TrimburDivision of Research and Statistics

Board of Directors of the Federal Reserve SystemWashington DC 20551

USA

April 30, 2006

AbstractSeasonal heteroskedasticity exists in a number of monthly time series from

major statistical agencies. Accounting for such systematic variation in calendarmonth effects can be important in estimating seasonal effects and movements inunderlying trend. In the context of seasonal adjustment, the standard approachto this phenomenon has been to use nonparametric (X-11) filters of differentlengths in the signal extraction routine of the X-12-ARIMA program. Thisserves as a simple, pragmatic approach that is, however, limited in its abilityto adapt to different datasets, and furthermore, the nonparametric frameworkgives no clear way to test for the presence of seasonal heteroskedasticity orto evaluate the plausibility of the heteroskedastic effects as a model for thetime series. Model-based methods such as those introduced recently by Proietti(2004) and Bell (2004) allow for a more flexible and informative approach tohandling time varying seasonal influences. In this paper, we give extensions ofthe modelling approach and develop statistical tests for seasonal heteroskedas-ticity, applying them to a number of U.S. Census Bureau time series. We alsodiscuss the treatment of periodic volatility linked to calendar month, whichgives rise to seasonal noise in a time series, showing its implications for sea-sonal adjustment.KEYWORDS: seasonal adjustment, trends, unobserved components.JEL classification: C22, C51, C82

Disclaimer: The views expressed in this paper are those of the author and notnecessarily those of the Federal Reserve Board.

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1 IntroductionSeasonal heteroskedasticity is defined by regular changes in variability over the cal-endar year. In this paper we examine approaches to modeling two different formsof seasonal heteroskedasticity: the seasonal specific models introduced recently byProietti (2004), and an extension of the airline model proposed by Bell (2004). Weexamine estimation of these models and their use in signal extraction estimation oftime series components for model-based seasonal adjustment and trend estimation.We also examine use of likelihood ratio tests with the models to test for the presenceof seasonal heteroskedasticity. We then examine application of the models and teststo U.S. Census Bureau monthly time series of housing starts and building permits.For these time series there is a clear reason to expect seasonal heteroskedasticity —the potential effects of weather on activity surrounding new construction.Seasonal heteroskedasticity can be thought of as a particular form of periodic be-

havior, and thus there are connections here with the periodic autoregressive-movingaverage models studied in Tiao and Grupe (1980) and the form-free seasonal effectsmodels of West and Harrison (1989). A key difference between these two generalmodels is that in the latter a complete set of processes, one for each month, is definedat all time points, though only the process corresponding to the calendar month ofobservation is observed at each time point. The required hidden components are eas-ily handled in the state space model. Proietti’s model is of this type. Bell (2004),in contrast, starts with the popular “airline model” of Box and Jenkins (1976), butaugments it with an additional white noise component whose variance changes sea-sonally. This model can be rewritten in a form where additional “seasonal noise”appears only in some months, thus being zero in the remaining months. Thus, Bell’smodel is more related to those of Tiao and Grupe (1980), and could be thought of asa simple extension to ARIMA component models of their approach.While Bell’s model differs fundamentally from Proietti’s model, it suggests a nat-

ural modification of Proietti’s model to allow for a seasonally heteroskedastic irregularcomponent, so we consider this third model as well.In most economic applications, the limited length of available datasets rules out

the effective use of unrestricted periodic models. For the models considered herethe concern would be about having too many variances to estimate. Therefore, forall three models considered we focus on the case where there are only two distinctvariances, leading to what can be called “high variance months” and “low variancemonths.” Determination of which months fall in each of these two groups can be basedpartly on a priori knowledge (bad winter weather could adversely affect constructionactivity and lead to higher variance in the winter months for the series analyzed here)and partly on empirical evidence. This issue is taken up in Section 3.In nonparametric seasonal adjustment with X12-ARIMA (see Findley et. al,

1998), the analyst can attempt to account for seasonal heteroskedasticity by selectingdifferent seasonal moving averages for different calendar months. The models consid-

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ered here have two potential advantages over this approach: they offer more flexibilityin dealing with seasonal heteroskedasticity, and they provide a framework for formalstatistical inference about seasonal heteroskedasticity (e.g., via the likelihood ratiotests presented that test for the presence of seasonal heteroskedasticity).The paper proceeds as follows. Section 2 presents the three models for seasonal

heteroskedasticity: the seasonal specific levels model of Proietti (2004), the ARIMAplus seasonal noise model of Bell (2004), and the modification of Proietti’s model tohave a seasonally heteroskedastic irregular component. Section 2 also presents a mod-ification of Proietti’s (2004) approach to estimation of trend and seasonal components,noting that this modification is needed to avoid violating a standard requirement forthese components. Section 3 considers estimation of the three models, including theissue of determining which months should be regarded as high variance and which aslow variance. Section 4 examines likelihood ratio tests for the presence of seasonalheteroskedasticity in the contexts of the three models. Section 5 then applies the threemodels, and their associated likelihood ratio tests, to time series of regional housingstarts and building permits from the U.S. Census Bureau. Section 6 considers treat-ment of the seasonal noise from Bell’s model in signal extraction for model-basedseasonal adjustment. Finally, Section 7 offers conclusions.

2 Models for seasonal heteroskedasticityThis section presents the three time series models designed to capture seasonal het-eroskedasticity: those of Proietti (2004) and Bell (2004), and a modification of Proi-etti’s model.

2.1 Seasonal specific levels model (Proietti 2004)

Here we review the model proposed in Proietti (2004), where heteroskedastic move-ments are specified in a set of level equations. Following presentation and discussionof the model, we note a modification needed of Proietti’s definition of the trendcomponent to achieve a definition consistent with standard assumptions.The basic approach is to model seasonality by setting up separate processes for

the different calendar months. This means that the series of successive Januaryobservations is assumed to follow a basic specification with certain parameter values.The same equation also applies to the February series but with distinct disturbancesand possibly different parameter values, and similarly for the other months. Theassumed covariance structure of the disturbances across the equations for the differentseasons determines the form of the heteroskedasticity.Given a seasonal time series yt = (y1, ..., yT )0, the model is

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yt = z0tμt + εt, εt ∼WN(0, σ2ε), t = 1, ..., T (1)

μt+1 = μt + iβt + iηt + η∗t , μt = (μ1t, ..., μst)0

η∗t = (η∗1t, ..., η

∗st)

0, ηt ∼WN(0, σ2η), η∗jt ∼WN(0, σ2η∗,j) (2)

βt+1 = βt + ζt, ζt ∼WN(0, σ2ζ)

where i is an s× 1 vector of ones, with s denoting the number of seasons in a year,that is s = 12 for monthly data. In the observation equation, zt is an s× 1 selectionvector that has a one in the position j = 1+(t−1)mod s and zeroes elsewhere. Thes elements of the vector η∗t are idiosyncratic level disturbances, which are assumedto be uncorrelated with each other and with ηt.We refer to (1) as the seasonal specific levels model. Since most applications we

consider involve monthly data, “month” and “season” are used interchangeably inwhat follows.The vector μt contains the month-specific levels. At time t, each of the s elements

is subject to a shared level disturbance ηt and is incremented by the common slope βt.The disturbance ζt that drives the time-varying slope process is assumed independentof the other disturbances in the model. The jth process μjt is also subject to theidiosyncratic shock η∗jt whose variance σ

2η∗,j depends, in general, on the season j. The

observation at time t is the sum of the appropriate μjt for the current season and thenoise term εt.The key property of the seasonal specific levels model is that the specification of

the heteroskedasticity is embedded in the nonstationary level processes μjt. Since in(1) the heteroskedastic part of the model is tied to the level equations, this suggeststhat the model might be useful for estimating trends in seasonally heteroskedasticseries. Howevever, as written, (1) leaves the overall trend in the series unspecified.An explicit decomposition is thus needed to define trend and seasonal components.The principle of starting with an observed process and breaking it down into inter-pretable parts is the same as in the canonical decomposition of Hillmer and Tiao(1982). However, in the seasonal specific model, we can use the information in thes heterogeneous levels that contribute to the observed process yt. Proietti (2004)suggests taking a weighted average of the level processes μjt as the definition of thetrend:

μt = w0μt, w = N−1η ι/(ι0N−1η ι)

Notice that this definition gives more weight to the more stable (lower variance)seasons.An example of a trend computed with this weighting pattern is shown by the

dotted line in figure 1. For this series of building permits, the effects of seasonalheteroskedasticity are clearly visible in the graph, and model (1) was fitted to theseries. Maximum likelihood estimation and smoothing methods are discussed in alater section; for now, we concentrate on the illustration of results.

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The problem with this trend estimate is revealed by looking at the correspondingseasonal estimates. These are computed as γt = μjt − μt, that is, the differencebetween month-specific level and trend at each time point. Figure 2 shows theestimates in the form of a seasonal component by month plot. As they primarilytrack the winter dips, the seasonals center around negative values. Hence, not onlydo they absorb the volatility associated with winter, but they also have a negativeoffset, violating the standard condition of mean-zero seasonal sums.

This negative offset represents a permanent contribution to the level of the seriesand should be assigned to the trend. In this example, a heteroskedastic seasonalcomponent should concentrate on the winter volatility, but it should refrain fromtracking the winter dips. The resulting bias in the trend is evident in figure 1, whichalso shows the trend estimated under the assumption of homoskedasticity, whereσ2η∗,j = σ2η∗ is constant across seasons, and the nonparametric trend produced bythe X-12-ARIMA program using a log-additive adjustment. Relative to these twostandard trends, there is a large discrepancy in the heteroskedastic trend. The biasedestimates effectively ignore the movements in winter, and, as such, they fail to give arepresentative measure of the level over the full calendar year.The usual intuition is that seasonality arises from repetitive influences that tend

to cancel out over a complete seasonal cycle. However, the seasonal sums in the figureremain well below zero for the entire sample period. This negative offset represents apermanent contribution to the level of the series and should be assigned to the trend.A similar offset in the trend occurs in the illustration by Proietti (2004), where

model (1) is fitted to the Italian Industrial Production Index, assuming a highervariance in August. The trend effectively ignores the August drop in production.However, Proietti (2004) omits an investigation of seasonal components, and the biasin the level estimate goes unrecognized. It is unclear from his results whether ununbiased measure of trend for model (1) would successfully smooth out the extravariability in August.In what follows, a different definition of the trend and seasonal component will

be used from the one in Proietti (2004). The goal is to work with plausible seasonalcomponents and unbiased trends. One simple approach would be to demean theseasonal, by subtracting the average of the seasonal component over the full sampleand adding the same quantity to the trend, but this would ignore the stochasticvariation in the level. Another possibility is to use the correction mentioned inProietti (2004) where the quantity (ι/s−w)0μ1 is added to the trend and subtractedfrom the seasonal. However, this would only ensure the calibration of the initiallevel and would again fail to address the changes in the bias over the sample period.To yield representative estimates of the level of the series at each time point, theadjustment (ι/s−w)0μt must be applied for all t.Therefore, the corrected definitions give

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1985 1990 1995 2000 2005

9.00

9.25

9.50

9.75

10.00

10.25

10.50Trend, Seas. Het. Trend, Seas. Hom.

Series

Figure 1: Estimated trends in Total Midwest Building Permits (in logarithms) over thesample period January 1984 to March 2006. The trend for the heteroskedastic model iscomputed with the weights suggested by Proietti (2004). The trend for the homoskedasticmodel, as well as the trend produced by the X12-ARIMA program, are shown for compar-ison.

μt = w0μt, w = (1/s)ι

γt = z0t(Is − ιι0/s)μt

which is simply an equal weighted average of the μ0jts for the unbiased trend at eachtime point. Then the seasonal is given by the difference between the specific processμjt, for the season j at time t, and the trend. unbiased trend and seasonal componentderived from the heteroskedastic model in this manner.Figure 3 shows the corrected trend for the Midwest building permit series. The

corresponding seasonal, shown in figure 4, now centers around zero, and the aver-age level of the heteroskedastic trend matches the homoskedastic case. However,the estimates behave differently around the winter months; over these intervals, thesmoothness of the heteroskedastic trend contrasts with the responsiveness of thehomoskedastic level. From 1994 to 1996, for instance, the severity of the January

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−0.8

−0.6

−0.4

−0.2

0.0

0.2

Seasonal component by month plot

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

Heteroskedastic Homoskedastic

Figure 2: Estimated seasonals in Total Midwest Building Permits (in logarithms), shownseparately for each calendar month. The components for each full year from 1984 to 2005were computed with the heteroskedastic model using the weights suggested by Proietti(2004).

trough in the series induces temporary dips in the trend according to the homoskedas-tic model, but the heteroskedastic trend passes through these events with the growthrate more or less unaffected.After modifying the trend estimate for the heterokedastic model, to reflect an un-

biased measure of level, the differences with the homoskedastic model now concentrateon the variation in the winter months. Specifically, the seasonally heteroskedasticmodel gives a trend that is smoother around the turn of the year, as it responds lessto periodic increases in volatility.

2.2 Airline model with seasonal noise

In Bell (2004), the approach is to start with a standard time series model, extendingit to include seasonally heteroskedastic noise. In particular, the airline model can begeneralized as follows :

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1985 1990 1995 2000 2005

9.00

9.25

9.50

9.75

10.00

10.25

10.50 Trend, Seas. Het. Trend, Seas. Hom.

Series

Figure 3: Trend in Total Midwest Building Permits, computed for the heteroskedasticmodel using the weights suggested in the text. The trend for the homoskedastic model isshown for comparison.

(1−B)(1−B12)yt = (1− θB)(1− θsB12)ut + εjt (3)

ut ∼WN(0, σ2u), εjt ∼WN(0, σ2εj)

It is assumed that the parameters θ, θs, σ2u admit a well-defined canonical decompo-sition into seasonal, trend, and irregular.The separate irregular εjt has season-dependent variance, but to identify the

model, the minimum σ2εj are set to zero. Then the ε0jts have the interpretationof additional noise present in the higher variability seasons. In the application inBell (2004), for instance, the effects are concentrated in January and February in amodel for a housing start series.When the months are divided into two groups, then there is a single idiosyncratic

variance parameter, denoted by σ2ε,∆, that represents the extra variance in the highversus low variability seasons. In constrast to the seasonal specific levels model dis-cussed in the last section, the effect of the seasonal noise is temporary and is keptseparate from other components of the model.

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−0.6

−0.4

−0.2

0.0

0.2

0.4 Seasonal component by month plot

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

Figure 4: Estimated seasonals in Total Midwest Building Permits (in logarithms), shownseparately for each calendar month. The components for each full year from 1984 to 2005were computed with the heteroskedastic model using the weights suggested in the text.

2.3 Seasonal specific irregular model

The same device of additive seasonal heteroskedasticity, as in Bell (2004), can be usedin the seasonal specific approach. Thus, instead of starting with a seasonal ARIMAmodel for the observations, we can start with a homoskedastic case of (1), and add ageneralized noise term to capture changes in variability across calendar months.Therefore, the model becomes

yt = z0tμt + ε∗jt, var(ε∗jt) = σ2ε∗,j, t = 1, ..., T (4)

μt+1 = μt + iβt + iηt + η∗t , var(ηt) = σ2η, η∗t ∼ NID(0, σ2η∗I)

βt+1 = βt + ζt, ζt ∼ NID(0, σ2ζ )

The random shocks ε∗jt are uncorrelated across seasons, and their variance dependsonly on the season index j = 1 + (t− 1)mod s. The model is homoskedastic in thelevel disturbances so that all elements of η∗t have the same variance σ

2η∗. The trend

and seasonal component are extracted in the same way as for (1).

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The notation ε∗jt is used to distinguish it from the heteroskedastic noise processεjt in (3). While the minimum variance of the ε0jts must be zero, the form of themodel in (4) means that the variance parameters σ2ε∗,j are unrestricted.For now, we focus attention on the three models discussed above. In (1), the

heteroskedastic structure is built directly into the specification of the month-specificlevels. By shifting the heteroskedasticity into the irregular, we obtain (4), the ana-logue of (3) in the seasonal specific framework. For seasonal variation linked toweather and other temporary factors, models (3) and (4) seem natural since theheteroskedastic structure is built into the stationary part of the model.Model (3) involves a simple extension of a well-known model. Further, general-

izing from the airline model to the class of seasonal ARIMA (SARIMA) models isstraightforward. The heteroskedastic noise process differs between (4) and (3). In(3), it focusses on the extra variation in certain months. In (4), ε∗jt is a seasonallyheteroskedastic irregular that accounts for total variation, so it could be used as anunobserved component in a more general framework.Given the differences between the three model structures, in following sections we

look at results of estimation for a set of series and make comparisons in the propertiesof tests based on the models.

3 Estimation of seasonally heteroskedastic modelsIn this section, we discuss the estimation of seasonally heteroskedastic effects inmonthly time series. Estimated trend and seasonal components were illustratedfor the series of total MidWest building permits. In this example, there is a clearindication of changing variability around the winter troughs. Though the seasonalheteroskedasticity in the series is prominent in the graph, this notion could be mademore precise through quantitative analysis. In the general situation, the visual ev-idence is usually murkier for the existence of seasonal heteroskedasticity, much lessfor the structure that underlies it.In a given setting, previous experience or the substance of the problem will usu-

ally suggest a probable pattern of seasonal heteroskedasticity. The researcher mayhave information about the timing of regular events known to increase variability, orabout monthly patterns that influence uncertainty. This prior knowledge can helpin designing a parsimonious model.Such knowledge is usually vague, however, while the assumptions on model struc-

ture are precise. Therefore, we suggest the use of prior estimation in setting up themodel. In this way, a sample of historical data is used to determine the details of theheteroskedastic structure in the series. This gives a definitive basis for design, andwhen applied to a separate sample of recent data, the properties of the estimationand testing process are preserved.

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3.1 Determination of heteroskedastic structure

In (1) the relative variances of the η∗jt’s determines the precise form of the het-eroskedasticity. In the most general case, we could consider the unrestricted formof the model. However, a completely flexible approach leads to an overparametrizedspecification - the availability of data limits the scope for estimating a large number ofidiosyncratic parameters. In practice, therefore, it is necessary to restrict attention toat most two or three categories. This still allows for flexibility in the choice of group-ing, and further, in a typical application, the main effect of the heteroskedasticity isoften captured by a simple classification of the months.For most of the construction series we investigate, the excess variation in winter

tends to dominate. In what follows, therefore, we develop the analysis on the basis ofa simple division of the months into high and low variance groups. For some series,there could be reason to investigate a more refined classification. For instance, theeffect of winter weather may differ between December and January, and we mightwant to set up the model to reflect this difference. Thus, in certain cases the use ofa third group could provide incremental information, though this must be weighedagainst the increase in model complexity.We consider the sixteen construction time series listed in table 1. Together these

represent the available data on total and single unit housing starts and total andsingle-unit building permits for the four Census regions of the US. In the Seriescolumn, ‘Total Permits, MidWest’ stands for the number of Building Permits issuedin the MidWest region for all structures, ‘Single-unit Starts, NorthEast’ stands for thenumber of Housing Starts in the NorthEast region for single-unit structures, and soforth. For models of the form (1), for each individual time series, the month-specificvariances σ2η∗,j are divided into two groups. For each month j of low variation,σ2η∗,j = σ2η∗,l, and similarly for each month j of high variation, σ2η∗,j = σ2η∗,h. Theheteroskedastic variance parameters for models (3) and (4) are similarly assigned.In using prior information to determine the set of high variance months for each

series, we ensure the validity of the statistical test properties, based on preselectedgroupings. If, in contrast, the test were applied following a within-sample optimiza-tion routine for choosing the set of high variance months, then the resulting testingprocedure would involve a bias. We now discuss two basic approaches for selectionof groups prior to model estimation and testing for a given sample.The first approach is based on experience with seasonal adjustment. The season-

ally adjusted estimates published by the U.S. Census Bureau are produced by staffwith the X-12-ARIMA program. The signal extraction routine allows for a limitedset of nonparametric filters. The default involves the use of a fixed filter, but to han-dle seasonally heteroskedastic series, the analyst has the possibility to use differentfilters for different calendar months.The profile of filters used to produce the official estimates is guided by diagnostics

indicating the quality of the seasonal adjustment, for instance its stability. As anexample, in running X-12-ARIMA, monthly I/S ratios are shown as output; these

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reflect estimates of an inverse signal-noise ratio for seasonality and so give a roughmeasure of how the seasonal variability evolves over the calendar year. The morevariable months tend to be assigned a shorter filter length. Since the use of threeor more distinct filters would usually lead to an erratic seasonal pattern, the signalextraction routine in X-12-ARIMA is nearly always implemented with at most twodifferent filter lengths. This suggests that the details of seasonal adjustment prac-tice can be informative for classifying months into groups of high and low variance,specifically by examining the filter lengths chosen for adjusting the different series.The prior knowledge in this approach comes from previous experience of Census

Bureau staff in time series analysis and seasonal adjustment. However, note thefilter lengths are chosen with a focus on the empirical properties of the seasonaladjustment. One criterion, for instance, pertains to the magnitude of subsequentrevisions to estimates. The selection is experimental and is only indirectly related tomodeling the statistical properties of the data. Nonetheless, the approach may beuseful when available data is limited.A second approach to the pre-selection of month groupings is based on model

estimation with a prior sample. In this approach, an initial part of the sample isreserved for determining the high variance months. This can be done by choosing theset that maximizes the likelihood function. The test for seasonal heteroskedasticityis then applied to the remaining part of the sample. Since the prior sample doesnot overlap with the testing sample, the pre-selection of groups is not influenced bythe testing sample. We emphasize this approach in what follows. The set of highvariance months for each series, determined in this way for the three different models,are shown in tables 1 to 3.

Series High Variance Months ∆ logLPermits, NorthEast Dec, Jan, Feb 0.00Permits, South Dec 1.07Permits, West Dec, Jan 0.94Permits, MidWest Dec, Jan, Feb 0.00Total Starts, NorthEast Dec, Jan, Feb 0.00Single-unit Starts, NorthEast Dec, Jan, Feb 0.00Total Starts, South Jan 1.05Single-unit Starts, South Dec, Jan, Feb 0.00Total Starts, West Dec, Jan 1.91Single-unit Starts, West Jan 2.43Total Starts, MidWest Dec, Jan, Feb 0.00Single-unit Starts, MidWest Dec, Jan, Feb 0.00

Table 1 : Calendar month groups for seasonal specific levels model for Census Bureauconstruction series. The second column shows, for each series, the months assigned the

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higher variance parameter in the optimal grouping (based on likelihood comparisons).The third column shows the difference between the maximized log-likelihood using the

optimal grouping and the maximized log-likelihood using the winter grouping, that is with{Jan, Feb, Dec} as the set of high variance months.

Series High Variance Months ∆ logLPermits, NorthEast Dec, Jan, Feb, Mar 0.05Permits, South Dec 2.45Permits, West Dec 0.73Permits, MidWest Dec, Jan, Feb, Mar 1.69Total Starts, NorthEast Dec, Jan, Feb 0.00Single-unit Starts, NorthEast Dec, Jan, Feb 0.00Total Starts, South Dec, Jan, Feb, Mar 0.09Single-unit Starts, South Dec, Jan, Feb 0.00Total Starts, West Dec, Jan, Feb 0.00Single-unit Starts, West Dec, Jan, Feb, Jul, Aug 0.10Total Starts, MidWest Dec, Jan, Feb 0.00Single-unit Starts, MidWest Dec, Jan, Feb 0.00Table 2 : Calendar month groups for the airline-seasonal noise model for Census Bureau

construction series. The second column shows, for each series, the months assigned thehigher variance parameter in the optimal grouping (based on likelihood comparisons). Thethird column shows the difference between the maximized log-likelihood using the optimalgrouping and the maximized log-likelihood using the winter grouping.

Series High Variance Months ∆ logLPermits, NorthEast Jan, Feb, Dec 0.00Permits, South Dec 2.33Permits, West Feb 1.01Permits, MidWest Jan, Feb, Mar, Dec 0.64Total Starts, NorthEast Jan, Feb, Dec 0.00Single-unit Starts, NorthEast Jan, Feb, Dec 0.00Total Starts, South Jan, Feb, Mar, Dec 0.08Single-unit Starts, South Jan, Feb, Dec 0.00Total Starts, West Jan, Feb, Dec 0.00Single-unit Starts, West Jan, Feb, Jul, Aug, Dec 0.09Total Starts, MidWest Jan, Feb, Dec 0.00Single-unit Starts, MidWest Jan, Feb, Dec 0.00Table 3 : Calendar month groups for seasonal specific irregular model for Census

Bureau construction series. The second column shows, for each series, the months assignedthe higher variance parameter in the optimal grouping (based on likelihood comparisons).

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The third column shows the difference between the maximized log-likelihood using theoptimal grouping and the maximized log-likelihood using the winter grouping.

For each series listed in the tables, the available sample period is divided into priorand testing subperiods. The regional series on total building permits are split intothe two samples, 1959:1 to 1984:12 (prior) and 1985:1 to 2006:3 (testing). Since thehistory of the single-unit permit series is limited, they are assigned the same groupingas the total permit series, and the same testing sample is used. Likewise, the housingstart series are split into the two samples, 1964:1 to 1984:12 (prior) and 1985:1 to2006:3 (testing).The set of high variance months nearly always includes January and tends to

cluster around the winter season. There is slight variation in the precise classificationacross series. For closely related series, for example single-unit and total for the sameregion and variable type, the differences might simply reflect random variation in theprior sample period, but they might also reflect differences in the seasonal propertiesof the series. To leave a sufficiently long series for testing, it is necessary to limit thesample size for the pre-selection of months. Nonetheless, the results indicate a focuson the winter months, especially for the Northeast and Midwest regions.The use of two categories for the idiosyncratic variance parameters gives a par-

simonious framework for studying the effects of heteroskedasticity on seasonal andtrend components, and for constructing a test for the property. Selection of thegroups based on prior estimation is feasible for sufficiently long series; otherwise,an alternative based on expert judgment or diagnostic criteria could be used. Ofcourse, introducing an additional category could increase the flexiblity of the modeland might give some incremental improvement in fit, but only at the cost of addedcomplexity. There is a clear rationale for a two-group structure in our application.For the construction series, there is a strong case for specifying extra variance in win-ter; otherwise, knowledge about the heteroskedastic effects is unclear. To justify theinclusion of a third group would require some special information about a particularseries.Note that, according to the selections in tables 1 to 3, adjacent months tend to

cluster within a group. This pattern is consistent with the usual form of seasonalityencountered in practice; frequent switches in variability over a single seasonal cycleare rare. Correspondingly, in the official seasonal adjustment with X-12-ARIMA,the calendar months assigned the same length of filter are often consecutive ones,though there are frequent exceptions. These may arise partly from the idiosyncraciesof the sample period and partly from the design criteria, as the diagnostics examinedto calibrate the filter lengths relate not to modeling performance but to practicalseasonal adjustment.For the results in the tables, based on model fitting over the prior sample, the

groups {Dec, Jan} and {Dec, Jan, Feb} are especially common. An exception isBuilding Permits in the NorthEast, for which estimation with the airline-seasonal

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noise model leads to the high variance group including March. However, on furtherinspection, the maximized log-likelihood declines only by a small amount when Marchis switched to the low variance group. In particular, the difference between the log-likelihoods for {Dec, Jan, Feb, March} and {Dec, Jan, Feb} is about 0.05. Thisis reported in the last column of table 2. For a borderline month such as March,the classification is expected to be unstable, but this just means that the variabilityof the month lies between the two extremes so that their assignment to one of thegroups is unimportant for the model’s performance.In contrast to the above example, the high variance group {Dec, Jan, Feb} is

decisive, or robust for single-unit MidWest housing starts. The best alternative is{Jan, Feb}, but this reduces the maximized log-likelihood by 70.1− 64.9 = 5.2. Also,the selections based on the seasonal specific levels and airline-seasonal noise modelsmatch for this series. These results support the idea of making a distinction betweenthe winter and the remaining seasons in modeling the series. This finding is consistentwith expectations about the effects of winter on construction in the MidWest.Each of tables 1 to 3 shows how the optimal grouping compares with a simple

separation into winter and non-winter months. The last columns on the right showsthe decline in the maximized log-likelihood occurring when the high variance group isautomatically set to {Dec, Jan, Feb}. The effect is neglible in five of the twelve casesfor the season specific levels model and in seven of the twelve cases for the airline-seasonal noise model, suggesting the use of a rough approximation where winter istreated separately for all series. Yet in some cases, the differences exceed 1.5, so theremay be additional information in the grouping established through prior estimation.For the airline-seasonal noise model, for three of the series (MW and NE regions),

the difference in the optimal grouping from the winter grouping seems important.Thus, it is better to specify NE housing starts as having a single high variance month,February, with December and January moved to the low variance group.

Of course, it is difficult to say exactly what difference in log-likelihood valuesis considered large without getting into a discussion of statistical significance in thecontext of estimation across different groupings, but for our purposes, interest centerson the use of appropriate groupings for subsequent analysis. In what follows, we setup each heteroskedastic model based on the divison of months listed in the tables.The maximum likelihood estimation routine, used for the prior estimation and for

the results given below, is based on state space modeling. In particular, the likelihoodfunction for the testing sample is evaluated for any permissible set of parametervalues based on the prediction error decomposition, as described in Harvey (1989) forexample. The parameter estimates are computed by optimizing over the likelihoodsurface in each case. Programs to perform the necessary calculations were written inthe Ox language [Doornik (1999)] and included the Ssfpack library of C++ routinesfor state space calculations [Koopman et. al (1999)]. For optimization, the BFGSalgorithm, a Newton-type method, was applied.For each series, the trading day effects estimated with X-12-ARIMAwere removed,

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and logarithms were then taken. Thus, the three models for seasonal heteroskedas-ticity correspond to additive decompositions of the logged series. As such, themodel-based estimates may be compared with the output of a log-additive seasonaladjustment from X-12-ARIMA.

3.2 Estimation of seasonal specific model

A homoskedastic model assumes constant variability in the seasonal component fordifferent calendar months. In model (1), setting the high and low variance parametersequal gives a single variance parameter σ2η∗ = σ2η∗,h = σ2η∗,l. This amounts to a singlerestriction on the heteroskedastic model in our problem. Parameter estimates areshown in table 4 for the sixteen series. The estimates of σ2ζ (not reported) were lessthan 10−8 for all series, indicating a nearly fixed slope.The two signal-noise ratios defined separately for the level, qη = σ2η/σ

2ε , and for the

seasonal, qη∗ = σ2η∗/σ2ε , indicate the variation in each stochastic component relative to

irregular. They play a role similar to the seasonal and nonseasonal MA coefficients inthe airline model, summarizing the scale-invariant dynamics of the model. However,qη and qη∗ relate directly to the trend and seasonal components. The estimates of qηin table 4 range from 0.03, for total NorthEast housing starts, to 0.50, for single-unitNorthEast buliding permits. The series with low qη appear noisy relative to level.The estimated qη∗ range from 0 (fixed seasonal) to 0.016 for the series of single-unitMidWest building permits, whose seasonal has a strong stochastic part.

Series σ2η (×103) σ2η∗ (×103) σ2ε (×103) qη qη∗Total Permits, NorthEast 1.758 0.023 5.597 0.314 0.0041Single-unit Permits, NorthEast 2.052 0.014 4.111 0.499 0.0034Total Permits, South 0.839 0 3.857 0.217 0Single-unit Permits, South 0.721 0.004 2.733 0.263 0.0014Total Permits, West 1.667 0 4.887 0.341 0Single-unit Permits, West 1.609 0 3.399 0.473 0Total Permits, MidWest 1.528 0.029 7.039 0.217 0.0042Single-unit Permits, MidWest 1.662 0.061 3.913 0.424 0.0155Total Starts, NorthEast 0.460 0.058 14.23 0.032 0.0049Single-unit Starts, NorthEast 1.057 0.047 11.77 0.089 0.0039Total Starts, South 0.770 0.015 4.117 0.187 0.0035Single-unit Starts, South 0.809 0.008 3.809 0.213 0.0021Total Starts, West 1.331 0.001 6.811 0.196 0.0002Single-unit Starts, West 1.394 0.003 6.364 0.219 0.0005Total Starts, MidWest 0.951 0.052 11.87 0.080 0.0043Single-unit Starts, MidWest 1.136 0.073 8.472 0.134 0.0086Table 4 : Parameter estimates for homoskedastic form of model (1). The sample

period is Jan 1985 to March 2006 for all series. The parameter σ2η∗ denotes the varianceof the idiosyncratic level disturbance, which is the same for all months.

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The series of monthly Midwest building permits was shown in figure 3. Thehomoskedastic trend is based on the estimates qη = 0.22, qη∗ = 0.004. Note thattrend estimates for (1) and (4) are computed using the state space model. First,we use the Kalman smoother algorithm to get the smoothed vector of monthly levelseμt = (eμ1t, ..., eμst)0. Second, we take the product w0eμt, which gives the MMSEestimate of the level at each time t.The homoskedastic model is able to capture stochastic variation in the level and

seasonal. Yet the winter months are assigned the same degree of uncertainty as theother months, so an unusually high or low value is treated the same whether it occursin January or in July. In separating out components, the homoskedastic model isforced to accomodate the seasonal increases in volatility, partly through short-termadjustment of the trend. Figure 3 shows that this has the effect of inducing temporarydips in the trend around the turn of the year.

Series σ2η (×103) σ2η∗,h (×103) σ2η∗,l (×103) σ2ε (×103)Total Permits, NorthEast 1.234 0.755 0.0030 4.269Single-unit Permits, NorthEast 1.7075 0.2339 0 3.6786Total Permits, South 0.83566 0 0 3.845Single-unit Permits, South 0.7054 0.0024 0.00243 2.7079Total Permits, West 1.651 0.01422 0 4.7303Single-unit Permits, West 1.609 0 0 3.399Total Permits, MidWest 0.985 2.632 0.00852 2.912Single-unit Permits, MidWest 1.224 1.2956 0.01437 2.227Total Starts, NorthEast 0.3190 1.245 0.0151 11.455Single-unit Starts, NorthEast 0.7250 1.2196 0.01686 9.1977Total Starts, South 0.73264 0.1874 0 3.5098Single-unit Starts, South 0.7552 0.11073 0 3.3978Total Starts, West 1.336 0 0.001868 6.7937Single-unit Starts, West 1.4054 0.000208 0.00398 6.3508Total Starts, MidWest 0.76484 3.004 0.01775 6.939Single-unit Starts, MidWest 0.94721 2.134 0.01265 5.041Table 5 : Parameter estimates for heteroskedastic form of model (1) with idiosyncratic

variance parameters σ2η∗,l (low variance) and σ2η∗,h (high variance) corresponding to thegrouping of months in table 1. The sample period is Jan 1985 to March 2006 for all series.

For heteroskedastic models, the months are distinguished by variance parameter.For MidWest building permits, according to table 1, the level disturbance in model(1) has high variance σ2η∗,h in December, January, and February, and low varianceσ2η∗,l from March to November. From table 4, parameter estimates indicate that thevariability in season-specific level, which includes contributions from both ηt and η∗t ,more than doubles during the winter months. In figure 3, the smoothness of the

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heteroskedastic trend around the turn of the year complements the extra variationin the seasonal component evident in figure 4. The irregular variance falls by morethan half once the heteroskedasticity is attributed to the seasonal part of the model.Note that in the homoskedastic model, the low estimates of bσ2η∗ reported in table

4 indicate that most of the monthly variation in the level arises from the shareddisturbance ηt. However, nonzero η∗t is required for describing stochastic seasonalitysince it allows the monthly level processes to evolve apart from one another. Settingη∗t = 0 gives identical local linear trend equations for the different μj,t so that fixedseasonality is produced that depends only on the initialization of μt.In extending (1) to the heteroskedastic case, the estimates of the high idiosyncratic

variance σ2η∗,h are comparable to the estimates of σ2η. Thus, the effects of seasonality

become more apparent once the homoskedastic constraint is removed.

3.3 Estimation of airline-seasonal noise model

Parameter estimates for the standard airline model, that is with σ2ε,∆ = 0, are shown intable 6. The MA coefficient values are consistent with prior experience in estimatingthe airline model for monthly time series. The nonseasonal MA coefficient rangesfrom 0.5 to 0.75. The seasonal MA coefficient ranges from 0.74 to 1.00; as thisparameter increases, the model approaches the case of deterministic, or fixed calendar-month effects. For the two permit series in the West, the estimate is approximatelyequal to unity, so a fixed seasonal could be used in the model.

Series θ θs σ2u (×103)Total Permits, NorthEast 0.571 0.821 11.78Single-unit Permits, NorthEast 0.501 0.846 9.615Total Permits, South 0.593 0.945 6.601Single-unit Permits, South 0.596 0.877 5.138Total Permits, West 0.555 1 8.735Single-unit Permits, West 0.509 1 6.676Total Permits, MidWest 0.622 0.830 13.45Single-unit Permits, MidWest 0.519 0.738 10.24Total Starts, NorthEast 0.711 0.803 23.29Single-unit Starts, NorthEast 0.666 0.814 20.55Total Starts, South 0.635 0.818 7.739Single-unit Starts, South 0.626 0.861 6.940Total Starts, West 0.637 0.933 11.25Single-unit Starts, West 0.629 0.927 10.85Total Starts, MidWest 0.751 0.813 19.36Single-unit Starts, MidWest 0.680 0.757 16.06Table 6 : Parameter estimates for the standard airline model, sample period is January

1985 to March 2006.

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The heteroskedastic term has positive variance σ2ε,∆ only in the high variancemonths, and zero variance in the remaining months. Table 7 shows that the estimatesare large and have the same order of magnitude as σ2u in many cases for the NorthEastand MidWest regions. The inclusion of σ2ε,∆ has the effect of squeezing some of thevariation out of the airline model part, so that estimates of σ2u decline.

Series θ θs σ2u (×103) σ2ε,∆ (×103)Total Permits, NorthEast 0.557 0.820 7.989 11.35Single-unit Permits, NorthEast 0.507 0.873 7.348 6.502Total Permits, South 0.593 0.945 6.597 0.014Single-unit Permits, South 0.592 0.877 4.952 0.805Total Permits, West 0.5367 1 7.768 4.395Single-unit Permits, West 0.499 1 6.144 2.360Total Permits, MidWest 0.552 0.817 5.992 22.20Single-unit Permits, MidWest 0.468 0.706 5.815 11.85Total Starts, NorthEast 0.701 0.808 16.61 20.79Single-unit Starts, NorthEast 0.653 0.814 14.22 19.18Total Starts, South 0.612 0.857 6.136 4.540Single-unit Starts, South 0.602 0.894 5.146 5.224Total Starts, West 0.642 0.905 9.678 5.409Single-unit Starts, West 0.629 0.917 10.18 2.210Total Starts, MidWest 0.699 0.802 11.06 26.06Single-unit Starts, MidWest 0.615 0.744 9.535 19.21Table 7 : Parameter estimates for airline model with seasonal noise, sample period is

January 1985 to March 2006.

3.4 Estimation of seasonal specific irregular model

Table 8 shows parameter estimates for model (4).

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Series σ2η (×103) σ2η∗ (×103) σ2ε∗,l (×103) σ2ε∗,h (×103)Total Permits, NorthEast 0.83 0.0120 3.93 16.5Single-unit Permits, NorthEast 1.58 0.0038 3.28 10.5Total Permits, South 0.83 0 3.84 3.94Single-unit Permits, South 0.70 0.0036 2.61 3.47Total Permits, West 1.60 0 4.20 8.60Single-unit Permits, West 1.54 0 3.06 5.43Total Permits, MidWest 0.94 0.0134 2.76 25.7Single-unit Permits, MidWest 1.10 0.0510 1.78 14.1Total Starts, NorthEast 0.04 0.0350 10.3 32.2Single-unit Starts, NorthEast 0.33 0.0311 8.24 28.3Total Starts, South 0.64 0.0019 3.38 8.86Single-unit Starts, South 0.67 0 2.85 8.74Total Starts, West 1.06 0.0035 5.72 11.22Single-unit Starts, West 1.29 0.0043 5.88 8.12Total Starts, MidWest 0.77 0.0333 6.20 32.5Single-unit Starts, MidWest 0.97 0.0482 4.43 24.4Table 8 : Parameter estimates for season-specific irregular model, sample period

January 1985 to March 2006.

Relative to the homoskedastic model, the inclusion of the extra irregular leads to adecline in σ2η∗ for many of the series. Typically, the estimates satisfy σ

2ε∗,l < σ2ε < σ2ε∗,h

where σ2ε is the homoskedastic variance in the constrained model. The estimate ofσ2ε∗,h is more than double that of σ

2ε for the NorthEast and MidWest series, and for

Housing starts in the South. This indicates that part of the stochastic movementsin the seasonal now show up as a switch in variance from low to high variabilitymonths.In comparing models (1) and (4), the heteroskedastic effects are easier to interpret

in (4. They are clearly concentrated in the high variance months as the ε∗0jts areone-period shocks. This gives a more natural starting point for applications wherethe seasonal changes in variability are due to temporary developments like weatherpatterns.In (1), a surprising shock in a high variance month j at time t is reflected in

a large value of η∗j,t. From section 2, the assumptions of the model state that(η∗j,t, η

∗j,t+1, η

∗j,t+2, ..., η

∗j,t+12) are uncorrelated. Since it operates through the level of

the process, the effect of the shock tends to spill over into adjacent months. Incontrast, the heteroskedastic noise in (3) or (4) is specifically attached to individualmonths.Finally, note that there is a loose relation between the extra variance parameter

σ2ε,∆ and the difference in variances, σ2ε∗,h − σ2ε∗,l. This depends on how well thecanonical white noise variation, that can be removed from the airline model, can bematched with the baseline variance σ2ε∗,l in the seasonal specific model.

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4 Finite sample test for seasonal heteroskedastic-ity

This section develops a test for seasonal heteroskedasticity that can be applied in finitesamples. The null hypothesis of homoskedasticity implies restrictions on the varianceparameters in models (1), (3), and (4). Maximum likelihood estimation was discussedin the previous section, and results were shown for heteroskedastic and homoskedasticmodels. Denoting the maximized log-likelihoods for estimation of the restricted andunrestricted models as eL∗ and eL, respectively, we construct the likelihood-ratio teststatistic LR = −2( eL∗− eL) for the presence of seasonal heteroskedasticity.In each of the three cases, the asymptotic distribution of LR under the null of

homskedasticity is related to a chi-squared density. However, it is unclear whetherthe large sample results give useful approximations in our application without furtherinvestigation. The construction series consist of about 250 observations. Further,the performance of the test for shorter-length series is of interest for other possibleapplications. We therefore present results on test statistics for each model, followedby simulation studies of the test properties.

4.1 LR test for Proietti model

In (1), under the null the idiosyncratic level disturbances are assigned equal variances,that is σ2η∗,h = σ2η∗,l = σ2η∗. As there is a single restriction, the distribution of LRunder the null of homskedasticity is asymptotically a χ2(1) (chi-squared with onedegree of freedom).Note that, when the variance σ2η∗ is small, there is a finite probability of estimating

both σ2η∗,l and σ2η∗,h in the heteroskedastic model to be zero. When this happens,the unrestricted model reduces to the restricted one with σ2η∗ = 0, giving a likelihoodratio test statistic of exactly zero. Therefore, the distribution of LR inherits thepile-up problem in the likelihood function. The probability mass at LR = 0 increasesas σ2η∗ approaches zero.More generally, the shape of the sampling distribution will depend on the set

of parameters appropriate for each series, as well as the series length. The MLEsfrom table 5 give a range of parameter values, and on this basis we can choose aset of parameter combinations that yield plausible models for the data generatingproces. In future work, we will present results for a simulation study of the LR testfor Proietti’s model.Note that, in the cases we have investigated so far, the empirical distribution of

LR under the null deviates from a χ2(1). The positive probability mass at LR = 0distinguishes it from a purely continuous distribution. The discrete probability ofLR = 0 increases as the true value of σ2η∗ gets closer to zero. Further, the nonzero,or continuous, part of the distribution has a thinner tail than the χ2(1).

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4.2 LR Test for airline-seasonal noise model

The parameter estimates in table 5 suggest a range of combinations for plausible nullmodels. Thus, we study the distribution of LR for several models with differentθ, θS. The results of the simulation study are set out in Trimbur and Bell (2006).For now, we give a summary discussion.The grouping of the months has a definite impact on the distribution of the test

statistic. In the case of a six-six division, for instance winter and summer in thehigh variance group, then the symmetry of the likelihood function implies that thedistribution is close to a mixture of point mass at zero (Pr = 1/2) and a χ2(1) (Pr =1/2).However, for other groupings, this result is less accurate. The deviations are

most substantial for the case of a single high variance month. One must make useof Monte Carlo simulation to get accurate estimates.At first, it may seem that, under the null, we are setting a variance parameter in

(3) to zero and the distribution should be similar to the result for the Proietti model.However, there is a distinction worth noting. The test based on model (3) could beconsidered from a different perspective. Assuming that the parameters of the airlinemodel part are such that the canonical decomposition exists, or is admissible - seeHillmer and Tiao (1982) for instance - then one can rewrite yt as a sum of trend,seasonal, and (nonzero) irregular term. Therefore, the null model is equivalent to acomponents model with additive irregular.That is, (3) can be rewritten as a model with a heteroskedastic irregular, whose

variance alternates between a high and low value depending on the season. Theparameter then represents an extra variance. As long as the canonical irregular fromthe airline model has positive variance, there is no reason why σ2ε,∆ cannot be negative.By setting up the model as we are, assuming correct specification of the high variancemonths, there is an implicit restriction on . This leads to the probability of gettingexactly zero, which is one-half for the symmetric case of a six-six grouping. Small-sample critical values for different series lengths and different groupings are given inTrimbur and Bell (2006).

4.3 Test for seasonal specific irregular model

In this case, one expects the sampling distribution of LR to resemble the χ2(1). Thepile-up situation at zero is absent because, in the context of the construction series,the irregular variation is usually large. There is no boundary question in the modelparametrization, as the test is equivalent to setting two unconstrained variances equal.This situation corresponds to testing under standard conditions.We expect the critical values to correspond well to standard chi-squared percentiles

for large enough series. The finite sample properties are reported in Trimbur andBell (2006).

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4.4 Test results

Table 5 shows the likelihood ratio test statistics for the sixteen series under study.

Series Levels Airline IrregularTotal Permits, NorthEast 23.6 27.0 28.6Single-unit Permits, NorthEast 16.1 13.1 13.8Total Permits, South 0.053 0.000 0.007Single-unit Permits, South 1.21 0.727 0.795Total Permits, West 1.79 5.92 5.94Single-unit Permits, West 0.000 3.23 3.24Total Permits, MidWest 68.5 85.1 84.6Single-unit Permits, MidWest 47.6 51.7 51.1Total Starts, NorthEast 16.5 21.7 24.7Single-unit Starts, NorthEast 15.7 25.7 28.4Total Starts, South 20.1 11.0 12.4Single-unit Starts, South 13.1 16.8 18.4Total Starts, West 0.03 6.48 6.53Single-unit Starts, West 0.04 1.38 1.39Total Starts, MidWest 29.3 49.5 49.6Single-unit Starts, MidWest 41.4 44.6 44.7Table 9 : Results of likelihood-ratio tests for different models of seasonal heteroskedas-

ticity. The "Levels" column refers to model (1), "Airline" to model (3), and "Irregular" tomodel (4).

There is a good deal of agreement in the implications of the tests. In many cases,the results for models (4) and (3) differ only slightly. The test for model (1) differsnoticeably for some series, for instance housing starts in the South, but even here, allthree results are clearly signicant at the 5% level.For the South and the West, evidence is mixed. For building permits in the

South, the null hypothesis of homoskedasticity cannot be rejected at the 5% level ofsignicance. The results are decisive for the NorthEast and MidWest regions, wherein all cases the null is strongly rejected.

5 ConclusionsThis paper introduces a test for seasonal heteroskedasticity. The strategy of seasonalspecific models, as introduced in Proietti (2004), is to set up a collection of processes,one for each season. Such models are well-designed to handle seasonal heteroskedas-ticity, and we have illustrated their use in forming trend estimates that are robust tochanging variability in winter. An alternative model was introduced by Bell (2004)by extending the well-known airline model.

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The models form the basis for likelihood ratio tests. In examining a set of CensusBureau construction series, the seasonal increase in variance tends to be concentratedin the winter months. A prior estimation routine was used to set up the individualheteroskedastic pattern for each series.The results of the likelihood ratio test gave consistent support for the role of sea-

sonal heteroskedasticity in regional construction series for the NorthEast and Mid-West. The models also detected its presence in some series in the West and South,though evidence was generally less conclusive. In preliminary work with simulationstudies, we find that when the heteroskedasticity is concentrated in a single month,then the critical values for the test should be modified. Otherwise, the standardlikelihood-ratio tests can be applied with simple adjustment for series length. Infuture work, we plan to investigate the properties of the finite-sample test in moredetail, and to make further theoretical and empirical comparisons of the alternativemodels.AcknowledgementsSpecial thanks go to David Findley for extensive discussion on this project. Tomasso

Proietti is acknowledged for providing Ox code where the seasonal specific levelsmodel is implemented for the Italian industrial production series; we developed pro-grams to handle the seasonal specific model applications in this paper by extendingand modifying this code as needed. We also thank Tucker McElroy, for commentsand for providing Ox code for estimation of SARIMA models, and John Hisnanick,for his comments on an earlier draft. Richard Gagnon is acknowledged for his as-sistance in using the RegComponent program, and we are grateful to Brian Monsellfor discussion and information on using X-12-ARIMA. We would also like to thankKathleen McDonald-Johnson, who provided the construction dataset used in the em-pirical analysis and gave important practical information on characteristics of theseries and details of seasonal adjustment. Trimbur is grateful to the Statistical Re-search Division of the Census Bureau for hospitality and financial support during hisresidence as a PostDoctoral Researcher.References

Bell, W., 2004. On RegComponent time series models and their applications. In A.Harvey et al. (eds), State space and unobserved components models : Theoryand applications, Cambridge : Cambridge University Press.

Bell, W., Chu R., and Tiao G., 2005. A model-based criterion for choosing filters inthe X-12-ARIMA seasonal adjustment program, unpublished manuscript.

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Findley, D., Monsell B., Bell W., Otto M., and Chen B.-C. 1998. New Capabilitiesand Methods of the X-12-ARIMA Seasonal Adjustment Program, Journal ofBusiness and Economic Statistics 16 (2), 127-77.

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Harvey A.C., 1989. Forecasting, structural time series models and the Kalman filter(Cambridge: Cambridge University Press).

Hillmer, S., and Tiao, G., 1982. An ARIMA-Model-Based Approach to SeasonalAdjustment, Journal of the American Statistical Association 77, 63-70.

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