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

Transcription

1 CONFERENCE ON SEASONALITY, SEASONAL ADJUSTMENT AND THEIR IMPLICATIONS FOR SHORT-TERM ANALYSIS AND FORECASTING MAY 2006 Seasonal heteroskedasticity in time series data: modeling, estimation, and testing Thomas M. Trimbur

2 Seasonal heteroskedasticity in time series data : modeling, estimation, and testing Thomas M. Trimbur Division of Research and Statistics Board of Directors of the Federal Reserve System Washington DC USA April 30, 2006 Abstract Seasonal heteroskedasticity exists in a number of monthly time series from major statistical agencies. Accounting for such systematic variation in calendar month effects can be important in estimating seasonal effectsandmovementsin underlying trend. In the context of seasonal adjustment, the standard approach to this phenomenon has been to use nonparametric (X-11) filters of different lengths in the signal extraction routine of the X-12-ARIMA program. This serves as a simple, pragmatic approach that is, however, limited in its ability to adapt to different datasets, and furthermore, the nonparametric framework gives no clear way to test for the presence of seasonal heteroskedasticity or to evaluate the plausibility of the heteroskedastic effects as a model for the time series. Model-based methods such as those introduced recently by Proietti (2004) and Bell (2004) allow for a more flexible and informative approach to handling time varying seasonal influences. In this paper, we give extensions of the modelling approach and develop statistical tests for seasonal heteroskedasticity, applying them to a number of U.S. Census Bureau time series. We also discuss the treatment of periodic volatility linked to calendar month, which gives rise to seasonal noise in a time series, showing its implications for seasonal 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 not necessarily those of the Federal Reserve Board. 1

3 1 Introduction Seasonal heteroskedasticity is defined by regular changes in variability over the calendar year. In this paper we examine approaches to modeling two different forms of seasonal heteroskedasticity: the seasonal specific models introduced recently by Proietti (2004), and an extension of the airline model proposed by Bell (2004). We examine estimation of these models and their use in signal extraction estimation of time 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 presence of seasonal heteroskedasticity. We then examine application of the models and tests to 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 behavior, and thus there are connections here with the periodic autoregressive-moving average models studied in Tiao and Grupe (1980) and the form-free seasonal effects models of West and Harrison (1989). A key difference between these two general models is that in the latter a complete set of processes, one for each month, is defined at all time points, though only the process corresponding to the calendar month of observation is observed at each time point. The required hidden components are easily 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), but augments it with an additional white noise component whose variance changes seasonally. 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 s model is more related to those of Tiao and Grupe (1980), and could be thought of as a simple extension to ARIMA component models of their approach. While Bell s model differs fundamentally from Proietti s model, it suggests a natural modification of Proietti s model to allow for a seasonally heteroskedastic irregular component, 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 here the concern would be about having too many variances to estimate. Therefore, for all three models considered we focus on the case where there are only two distinct variances, leading to what can be called high variance months and low variance months. Determination of which months fall in each of these two groups can be based partly on a priori knowledge (bad winter weather could adversely affect construction activity 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 selecting different seasonal moving averages for different calendar months. The models consid- 2

4 ered here have two potential advantages over this approach: they offer more flexibility in dealing with seasonal heteroskedasticity, and they provide a framework for formal statistical inference about seasonal heteroskedasticity (e.g., via the likelihood ratio tests 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 ARIMA plus seasonal noise model of Bell (2004), and the modification of Proietti s model to have a seasonally heteroskedastic irregular component. Section 2 also presents a modification of Proietti s (2004) approach to estimation of trend and seasonal components, noting that this modification is needed to avoid violating a standard requirement for these components. Section 3 considers estimation of the three models, including the issue of determining which months should be regarded as high variance and which as low variance. Section 4 examines likelihood ratio tests for the presence of seasonal heteroskedasticity in the contexts of the three models. Section 5 then applies the three models, and their associated likelihood ratio tests, to time series of regional housing starts and building permits from the U.S. Census Bureau. Section 6 considers treatment of the seasonal noise from Bell s model in signal extraction for model-based seasonal adjustment. Finally, Section 7 offers conclusions. 2 Models for seasonal heteroskedasticity This section presents the three time series models designed to capture seasonal heteroskedasticity: those of Proietti (2004) and Bell (2004), and a modification of Proietti s model. 2.1 Seasonal specific levels model (Proietti 2004) Here we review the model proposed in Proietti (2004), where heteroskedastic movements are specified in a set of level equations. Following presentation and discussion ofthemodel,wenoteamodification needed of Proietti s definition of the trend componenttoachieveadefinition 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 January observations is assumed to follow a basic specification with certain parameter values. The same equation also applies to the February series but with distinct disturbances and possibly different parameter values, and similarly for the other months. The assumed covariance structure of the disturbances across the equations for the different seasons determines the form of the heteroskedasticity. Given a seasonal time series y t =(y 1,..., y T ) 0, the model is 3

5 y t = z 0 t μ t + ε t, ε t WN(0,σε 2 ), t =1,..., T (1) μ t+1 = μ t + iβ t + iη t + η t, μ t =(μ 1t,..., μ st ) 0 η t =(η 1t,...,η st )0, η t WN(0,ση 2 ), η jt WN(0,σ2 η,j ) (2) β t+1 = β t + ζ t, ζ t WN(0,σζ 2 ) where i is an s 1 vector of ones, with s denoting the number of seasons in a year, that is s =12for monthly data. In the observation equation, z t is an s 1 selection vector that has a one in the position j =1+(t 1) mod s and zeroes elsewhere. The s elements of the vector η t are idiosyncratic level disturbances, which are assumed to be uncorrelated with each other and with η t. We refer to (1) as the seasonal specific levels model. Since most applications we consider involve monthly data, month and season are used interchangeably in what follows. The vector μ t contains the month-specific levels. At time t, eachofthes elements is subject to a shared level disturbance η t andisincrementedbythecommonslopeβ t. The disturbance ζ t that drives the time-varying slope process is assumed independent of the other disturbances in the model. The jth process μ jt is also subject to the idiosyncratic shock ηjt whose variance ση,j 2 depends, in general, on the season j. The observation at time t is the sum of the appropriate μ jt for the current season and the noise term ε t. The key property of the seasonal specific levels model is that the specification of the heteroskedasticity is embedded in the nonstationary level processes μ jt. Since in (1) the heteroskedastic part of the model is tied to the level equations, this suggests that the model might be useful for estimating trends in seasonally heteroskedastic series. 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 interpretable 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 the s heterogeneous levels that contribute to the observed process y t. Proietti (2004) suggests taking a weighted average of the level processes μ jt as the definition of the trend: μ t = w 0 μ t, w = N 1 η ι/(ι 0 N 1 η 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 seasonal heteroskedasticity are clearly visible in the graph, and model (1) was fitted to the series. Maximum likelihood estimation and smoothing methods are discussed in a later section; for now, we concentrate on the illustration of results. 4

6 The problem with this trend estimate is revealed by looking at the corresponding seasonal estimates. These are computed as γ t = μ jt μ t,thatis,thedifference between month-specific level and trend at each time point. Figure 2 shows the estimates in the form of a seasonal component by month plot. As they primarily track the winter dips, the seasonals center around negative values. Hence, not only do they absorb the volatility associated with winter, but they also have a negative offset, violating the standard condition of mean-zero seasonal sums. This negative offset represents a permanent contribution to the level of the series and should be assigned to the trend. In this example, a heteroskedastic seasonal component should concentrate on the winter volatility, but it should refrain from tracking the winter dips. The resulting bias in the trend is evident in figure 1, which also shows the trend estimated under the assumption of homoskedasticity, where ση,j 2 = ση 2 is constant across seasons, and the nonparametric trend produced by the X-12-ARIMA program using a log-additive adjustment. Relative to these two standard trends, there is a large discrepancy in the heteroskedastic trend. The biased estimates effectively ignore the movements in winter, and, as such, they fail to give a representative 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 figure remain well below zero for the entire sample period. This negative offset represents a permanent 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 higher variance in August. The trend effectively ignores the August drop in production. However, Proietti (2004) omits an investigation of seasonal components, and the bias in the level estimate goes unrecognized. It is unclear from his results whether un unbiased measure of trend for model (1) would successfully smooth out the extra variability 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 seasonal components and unbiased trends. One simple approach would be to demean the seasonal, by subtracting the average of the seasonal component over the full sample and adding the same quantity to the trend, but this would ignore the stochastic variation in the level. Another possibility is to use the correction mentioned in Proietti (2004) where the quantity (ι/s w) 0 μ 1 is added to the trend and subtracted from the seasonal. However, this would only ensure the calibration of the initial level 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, the adjustment (ι/s w) 0 μ t must be applied for all t. Therefore, the corrected definitions give 5

7 10.50 Trend, Seas. Het. Trend, Seas. Hom. Series Figure 1: Estimated trends in Total Midwest Building Permits (in logarithms) over the sample period January 1984 to March The trend for the heteroskedastic model is computed with the weights suggested by Proietti (2004). The trend for the homoskedastic model, as well as the trend produced by the X12-ARIMA program, are shown for comparison. μ t = w 0 μ t, w =(1/s)ι γ t = z 0 t(i s ιι 0 /s)μ t which is simply an equal weighted average of the μ 0 jts for the unbiased trend at each time point. Then the seasonal is given by the difference between the specific process μ jt, for the season j at time t, and the trend. unbiased trend and seasonal component derived 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 average level of the heteroskedastic trend matches the homoskedastic case. However, the estimates behave differently around the winter months; over these intervals, the smoothness of the heteroskedastic trend contrasts with the responsiveness of the homoskedastic level. From 1994 to 1996, for instance, the severity of the January 6

8 Seasonal component by month plot Heteroskedastic Homoskedastic Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Figure 2: Estimated seasonals in Total Midwest Building Permits (in logarithms), shown separately for each calendar month. The components for each full year from 1984 to 2005 were 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 homoskedastic model, but the heteroskedastic trend passes through these events with the growth rate more or less unaffected. After modifying the trend estimate for the heterokedastic model, to reflect an unbiased measure of level, the differences with the homoskedastic model now concentrate on the variation in the winter months. Specifically, the seasonally heteroskedastic model gives a trend that is smoother around the turn of the year, as it responds less to 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, extending it to include seasonally heteroskedastic noise. In particular, the airline model can be generalized as follows : 7

9 10.50 Trend, Seas. Het. Trend, Seas. Hom. Series Figure 3: Trend in Total Midwest Building Permits, computed for the heteroskedastic model using the weights suggested in the text. The trend for the homoskedastic model is shown for comparison. (1 B)(1 B 12 )y t =(1 θb)(1 θ s B 12 )u t + ε jt (3) u t WN(0,σu), 2 ε jt WN(0,σεj) 2 It is assumed that the parameters θ, θ s,σu 2 admit a well-defined canonical decomposition into seasonal, trend, and irregular. The separate irregular ε jt has season-dependent variance, but to identify the model, the minimum σεj 2 are set to zero. Then the ε 0 jts have the interpretation of additional noise present in the higher variability seasons. In the application in Bell (2004), for instance, the effects are concentrated in January and February in a model for a housing start series. When the months are divided into two groups, then there is a single idiosyncratic variance parameter, denoted by σε, 2, that represents the extra variance in the high versus low variability seasons. In constrast to the seasonal specific levels model discussed in the last section, the effect of the seasonal noise is temporary and is kept separate from other components of the model. 8

10 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), shown separately for each calendar month. The components for each full year from 1984 to 2005 were 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 used in the seasonal specific approach. Thus, instead of starting with a seasonal ARIMA model for the observations, we can start with a homoskedastic case of (1), and add a generalized noise term to capture changes in variability across calendar months. Therefore, the model becomes y t = z 0 tμ t + ε jt, var(ε jt) =σε,j, 2 t =1,...,T (4) μ t+1 = μ t + iβ t + iη t + η t, var(η t )=ση, 2 η t NID(0,ση I) 2 β t+1 = β t + ζ t, ζ t NID(0,σζ 2 ) The random shocks ε jt are uncorrelated across seasons, and their variance depends only on the season index j =1+(t 1) mod s. The model is homoskedastic in the level disturbances so that all elements of η t havethesamevarianceση. 2 The trend and seasonal component are extracted in the same way as for (1). 9

11 The notation ε jt is used to distinguish it from the heteroskedastic noise process ε jt in (3). While the minimum variance of the ε 0 jts must be zero, the form of the model in (4) means that the variance parameters σε,j 2 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-specific levels. By shifting the heteroskedasticity into the irregular, we obtain (4), the analogue of (3) in the seasonal specific framework. For seasonal variation linked to weather and other temporary factors, models (3) and (4) seem natural since the heteroskedastic structure is built into the stationary part of the model. Model (3) involves a simple extension of a well-known model. Further, generalizing from the airline model to the class of seasonal ARIMA (SARIMA) models is straightforward. 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 seasonally heteroskedastic irregular that accounts for total variation, so it could be used as an unobserved 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 properties of tests based on the models. 3 Estimation of seasonally heteroskedastic models In this section, we discuss the estimation of seasonally heteroskedastic effects in monthly time series. Estimated trend and seasonal components were illustrated for the series of total MidWest building permits. In this example, there is a clear indication of changing variability around the winter troughs. Though the seasonal heteroskedasticity in the series is prominent in the graph, this notion could be made more precise through quantitative analysis. In the general situation, the visual evidence is usually murkier for the existence of seasonal heteroskedasticity, much less for the structure that underlies it. In a given setting, previous experience or the substance of the problem will usually suggest a probable pattern of seasonal heteroskedasticity. The researcher may have information about the timing of regular events known to increase variability, or about monthly patterns that influence uncertainty. This prior knowledge can help in designing a parsimonious model. Such knowledge is usually vague, however, while the assumptions on model structure are precise. Therefore, we suggest the use of prior estimation in setting up the model. In this way, a sample of historical data is used to determine the details of the heteroskedastic structure in the series. This gives a definitive basis for design, and when applied to a separate sample of recent data, the properties of the estimation and testing process are preserved. 10

12 3.1 Determination of heteroskedastic structure In (1) the relative variances of the ηjt s determines the precise form of the heteroskedasticity. In the most general case, we could consider the unrestricted form of the model. However, a completely flexible approach leads to an overparametrized specification - the availability of data limits the scope for estimating a large number of idiosyncratic parameters. In practice, therefore, it is necessary to restrict attention to at most two or three categories. This still allows for flexibility in the choice of grouping, and further, in a typical application, the main effect of the heteroskedasticity is often 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 of a 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, the effect of winter weather may differ between December and January, and we might want to set up the model to reflect this difference. Thus, in certain cases the use of a third group could provide incremental information, though this must be weighed against 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 and single-unit building permits for the four Census regions of the US. In the Series column, Total Permits, MidWest stands for the number of Building Permits issued in the MidWest region for all structures, Single-unit Starts, NorthEast stands for the number of Housing Starts in the NorthEast region for single-unit structures, and so forth. For models of the form (1), for each individual time series, the month-specific variances ση,j 2 are divided into two groups. For each month j of low variation, ση,j 2 = ση,l 2, and similarly for each month j of high variation, σ2 η,j = ση,h 2. The heteroskedastic 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 preselected groupings. If, in contrast, the test were applied following a within-sample optimization routine for choosing the set of high variance months, then the resulting testing procedure would involve a bias. We now discuss two basic approaches for selection of groups prior to model estimation and testing for a given sample. The first approach is based on experience with seasonal adjustment. The seasonally adjusted estimates published by the U.S. Census Bureau are produced by staff with the X-12-ARIMA program. The signal extraction routine allows for a limited set of nonparametric filters. The default involves the use of a fixed filter, but to handle seasonally heteroskedastic series, the analyst has the possibility to use different filters 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 an example, in running X-12-ARIMA, monthly I/S ratios are shown as output; these 11

13 reflect estimates of an inverse signal-noise ratio for seasonality and so give a rough measure of how the seasonal variability evolves over the calendar year. The more variable months tend to be assigned a shorter filter length. Since the use of three or more distinct filters would usually lead to an erratic seasonal pattern, the signal extraction routine in X-12-ARIMA is nearly always implemented with at most two different filter lengths. This suggests that the details of seasonal adjustment practice 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 the filter lengths are chosen with a focus on the empirical properties of the seasonal adjustment. One criterion, for instance, pertains to the magnitude of subsequent revisions to estimates. The selection is experimental and is only indirectly related to modeling the statistical properties of the data. Nonetheless, the approach may be useful 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 is reserved for determining the high variance months. This can be done by choosing the set that maximizes the likelihood function. The test for seasonal heteroskedasticity is then applied to the remaining part of the sample. Since the prior sample does not overlap with the testing sample, the pre-selection of groups is not influenced by the testing sample. We emphasize this approach in what follows. The set of high variance months for each series, determined in this way for the three different models, areshownintables1to3. Series High Variance Months log L Permits, NorthEast Dec, Jan, Feb 0.00 Permits, South Dec 1.07 Permits, West Dec, Jan 0.94 Permits, MidWest Dec, Jan, Feb 0.00 Total Starts, NorthEast Dec, Jan, Feb 0.00 Single-unit Starts, NorthEast Dec, Jan, Feb 0.00 Total Starts, South Jan 1.05 Single-unit Starts, South Dec, Jan, Feb 0.00 Total Starts, West Dec, Jan 1.91 Single-unit Starts, West Jan 2.43 Total Starts, MidWest Dec, Jan, Feb 0.00 Single-unit Starts, MidWest Dec, Jan, Feb 0.00 Table 1 : Calendar month groups for seasonal specific levels model for Census Bureau construction series. The second column shows, for each series, the months assigned the 12

14 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 log L Permits, NorthEast Dec, Jan, Feb, Mar 0.05 Permits, South Dec 2.45 Permits, West Dec 0.73 Permits, MidWest Dec, Jan, Feb, Mar 1.69 Total Starts, NorthEast Dec, Jan, Feb 0.00 Single-unit Starts, NorthEast Dec, Jan, Feb 0.00 Total Starts, South Dec, Jan, Feb, Mar 0.09 Single-unit Starts, South Dec, Jan, Feb 0.00 Total Starts, West Dec, Jan, Feb 0.00 Single-unit Starts, West Dec, Jan, Feb, Jul, Aug 0.10 Total Starts, MidWest Dec, Jan, Feb 0.00 Single-unit Starts, MidWest Dec, Jan, Feb 0.00 Table 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 the 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. Series High Variance Months log L Permits, NorthEast Jan, Feb, Dec 0.00 Permits, South Dec 2.33 Permits, West Feb 1.01 Permits, MidWest Jan, Feb, Mar, Dec 0.64 Total Starts, NorthEast Jan, Feb, Dec 0.00 Single-unit Starts, NorthEast Jan, Feb, Dec 0.00 Total Starts, South Jan, Feb, Mar, Dec 0.08 Single-unit Starts, South Jan, Feb, Dec 0.00 Total Starts, West Jan, Feb, Dec 0.00 Single-unit Starts, West Jan, Feb, Jul, Aug, Dec 0.09 Total Starts, MidWest Jan, Feb, Dec 0.00 Single-unit Starts, MidWest Jan, Feb, Dec 0.00 Table 3 : Calendar month groups for seasonal specific irregular model for Census Bureau construction series. The second column shows, for each series, the months assigned the higher variance parameter in the optimal grouping (based on likelihood comparisons). 13

15 The third column shows the difference between the maximized log-likelihood using the optimal grouping and the maximized log-likelihood using the winter grouping. For each series listed in the tables, the available sample period is divided into prior and testing subperiods. The regional series on total building permits are split into the two samples, 1959:1 to 1984:12 (prior) and 1985:1 to 2006:3 (testing). Since the history of the single-unit permit series is limited, they are assigned the same grouping as the total permit series, and the same testing sample is used. Likewise, the housing start series are split into the two samples, 1964:1 to 1984:12 (prior) and 1985:1 to 2006:3 (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 classification across series. For closely related series, for example single-unit and total for the same region and variable type, the differences might simply reflect random variation in the prior sample period, but they might also reflect differences in the seasonal properties of the series. To leave a sufficiently long series for testing, it is necessary to limit the sample size for the pre-selection of months. Nonetheless, the results indicate a focus on the winter months, especially for the Northeast and Midwest regions. The use of two categories for the idiosyncratic variance parameters gives a parsimonious framework for studying the effects of heteroskedasticity on seasonal and trend components, and for constructing a test for the property. Selection of the groups based on prior estimation is feasible for sufficiently long series; otherwise, an alternative based on expert judgment or diagnostic criteria could be used. Of course, introducing an additional category could increase the flexiblity of the model and might give some incremental improvement in fit, but only at the cost of added complexity. 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 winter; otherwise, knowledge about the heteroskedastic effects is unclear. To justify the inclusion of a third group would require some special information about a particular series. 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 seasonality encountered in practice; frequent switches in variability over a single seasonal cycle are 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 idiosyncracies of the sample period and partly from the design criteria, as the diagnostics examined to calibrate the filter lengths relate not to modeling performance but to practical seasonal 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 is Building Permits in the NorthEast, for which estimation with the airline-seasonal 14

16 noise model leads to the high variance group including March. However, on further inspection, the maximized log-likelihood declines only by a small amount when March is switched to the low variance group. In particular, the difference between the loglikelihoods for {Dec, Jan, Feb, March} and {Dec, Jan, Feb} is about This is 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 variability of the month lies between the two extremes so that their assignment to one of the groups 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 = 5.2. Also, the selections based on the seasonal specific levels and airline-seasonal noise models match for this series. These results support the idea of making a distinction between the winter and the remaining seasons in modeling the series. This finding is consistent with 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 shows the decline in the maximized log-likelihood occurring when the high variance group is automatically set to {Dec, Jan, Feb}. The effect is neglible in fiveofthetwelve cases for the season specific levels model and in seven of the twelve cases for the airlineseasonal noise model, suggesting the use of a rough approximation where winter is treated separately for all series. Yet in some cases, the differences exceed 1.5, so there may 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 values is considered large without getting into a discussion of statistical significance in the context of estimation across different groupings, but for our purposes, interest centers on the use of appropriate groupings for subsequent analysis. In what follows, we set up 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 likelihood function for the testing sample is evaluated for any permissible set of parameter values based on the prediction error decomposition, as described in Harvey (1989) for example. The parameter estimates are computed by optimizing over the likelihood surface in each case. Programs to perform the necessary calculations were written in the Ox language [Doornik (1999)] and included the Ssfpack library of C++ routines for state space calculations [Koopman et. al (1999)]. For optimization, the BFGS algorithm, a Newton-type method, was applied. For each series, the trading day effects estimated with X-12-ARIMA were removed, 15

17 and logarithms were then taken. Thus, the three models for seasonal heteroskedasticity correspond to additive decompositions of the logged series. As such, the model-based estimates may be compared with the output of a log-additive seasonal adjustment from X-12-ARIMA. 3.2 Estimation of seasonal specific model A homoskedastic model assumes constant variability in the seasonal component for different calendar months. In model (1), setting the high and low variance parameters equal gives a single variance parameter ση 2 = ση,h 2 = σ2 η,l. This amounts to a single restriction on the heteroskedastic model in our problem. Parameter estimates are shown in table 4 for the sixteen series. The estimates of σζ 2 (not reported) were less than 10 8 for all series, indicating a nearly fixed slope. The two signal-noise ratios defined separately for the level, q η = ση 2/σ2 ε, and for the seasonal, q η = ση /σ 2 ε, 2 indicate the variation in each stochastic component relative to irregular. They play a role similar to the seasonal and nonseasonal MA coefficients in the airline model, summarizing the scale-invariant dynamics of the model. However, q η and q η relate directly to the trend and seasonal components. The estimates of q η in table 4 range from 0.03, for total NorthEast housing starts, to 0.50, for single-unit NorthEast buliding permits. The series with low q η appear noisy relative to level. The estimated q η range from 0 (fixed seasonal) to for the series of single-unit MidWest building permits, whose seasonal has a strong stochastic part. Series ση 2 ( 10 3 ) ση 2 ( 10 3 ) σε 2 ( 10 3 ) q η q η Total Permits, NorthEast Single-unit Permits, NorthEast Total Permits, South Single-unit Permits, South Total Permits, West Single-unit Permits, West Total Permits, MidWest Single-unit Permits, MidWest Total Starts, NorthEast Single-unit Starts, NorthEast Total Starts, South Single-unit Starts, South Total Starts, West Single-unit Starts, West Total Starts, MidWest Single-unit Starts, MidWest Table 4 : Parameter estimates for homoskedastic form of model (1). The sample period is Jan 1985 to March 2006 for all series. The parameter ση 2 denotes the variance of the idiosyncratic level disturbance, which is the same for all months. 16

18 The series of monthly Midwest building permits was shown in figure 3. The homoskedastic trend is based on the estimates q η =0.22,q η = Note that trend 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 levels eμ t = (eμ 1t,..., eμ st ) 0. Second, we take the product w 0 eμ t, which gives the MMSE estimate 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 the other months, so an unusually high or low value is treated the same whether it occurs in January or in July. In separating out components, the homoskedastic model is forced to accomodate the seasonal increases in volatility, partly through short-term adjustment of the trend. Figure 3 shows that this has the effect of inducing temporary dips in the trend around the turn of the year. Series ση 2 ( 10 3 ) ση,h 2 ( 103 ) ση,l 2 ( 103 ) σε 2 ( 10 3 ) Total Permits, NorthEast Single-unit Permits, NorthEast Total Permits, South Single-unit Permits, South Total Permits, West Single-unit Permits, West Total Permits, MidWest Single-unit Permits, MidWest Total Starts, NorthEast Single-unit Starts, NorthEast Total Starts, South Single-unit Starts, South Total Starts, West Single-unit Starts, West Total Starts, MidWest Single-unit Starts, MidWest Table 5 : Parameter estimates for heteroskedastic form of model (1) with idiosyncratic variance parameters ση,l 2 (low variance) and ση,h 2 (high variance) corresponding to the grouping 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 ση,h 2 in December, January, and February, and low variance ση,l 2 from March to November. From table 4, parameter estimates indicate that the variability in season-specific level, which includes contributions from both η t and ηt, more than doubles during the winter months. In figure 3, the smoothness of the 17

19 heteroskedastic trend around the turn of the year complements the extra variation in the seasonal component evident in figure 4. The irregular variance falls by more than half once the heteroskedasticity is attributed to the seasonal part of the model. Note that in the homoskedastic model, the low estimates of bσ η 2 reported in table 4 indicate that most of the monthly variation in the level arises from the shared disturbance η t. However, nonzero ηt is required for describing stochastic seasonality since it allows the monthly level processes to evolve apart from one another. Setting ηt =0gives identical local linear trend equations for the different μ j,t so that fixed seasonality is produced that depends only on the initialization of μ t. In extending (1) to the heteroskedastic case, the estimates of the high idiosyncratic variance ση,h 2 are comparable to the estimates of σ2 η. 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 σε, 2 =0,areshownin table 6. The MA coefficient values are consistent with prior experience in estimating the airline model for monthly time series. The nonseasonal MA coefficient ranges from 0.5 to The seasonal MA coefficient ranges from 0.74 to 1.00; as this parameter increases, the model approaches the case of deterministic, or fixed calendarmonth effects. For the two permit series in the West, the estimate is approximately equal to unity, so a fixed seasonal could be used in the model. Series θ θ s σu 2 ( 10 3 ) Total Permits, NorthEast Single-unit Permits, NorthEast Total Permits, South Single-unit Permits, South Total Permits, West Single-unit Permits, West Total Permits, MidWest Single-unit Permits, MidWest Total Starts, NorthEast Single-unit Starts, NorthEast Total Starts, South Single-unit Starts, South Total Starts, West Single-unit Starts, West Total Starts, MidWest Single-unit Starts, MidWest Table 6 : Parameter estimates for the standard airline model, sample period is January 1985 to March

20 The heteroskedastic term has positive variance σε, 2 only in the high variance months, and zero variance in the remaining months. Table 7 shows that the estimates are large and have the same order of magnitude as σu 2 in many cases for the NorthEast and MidWest regions. The inclusion of σε, 2 has the effect of squeezing some of the variation out of the airline model part, so that estimates of σu 2 decline. Series θ θ s σu 2 ( 10 3 ) σε, 2 ( 10 3 ) Total Permits, NorthEast Single-unit Permits, NorthEast Total Permits, South Single-unit Permits, South Total Permits, West Single-unit Permits, West Total Permits, MidWest Single-unit Permits, MidWest Total Starts, NorthEast Single-unit Starts, NorthEast Total Starts, South Single-unit Starts, South Total Starts, West Single-unit Starts, West Total Starts, MidWest Single-unit Starts, MidWest Table 7 : Parameter estimates for airline model with seasonal noise, sample period is January 1985 to March Estimation of seasonal specific irregular model Table 8 shows parameter estimates for model (4). 19

21 Series ση 2 ( 10 3 ) ση 2 ( 10 3 ) σε,l 2 ( 103 ) σε,h 2 ( 103 ) Total Permits, NorthEast Single-unit Permits, NorthEast Total Permits, South Single-unit Permits, South Total Permits, West Single-unit Permits, West Total Permits, MidWest Single-unit Permits, MidWest Total Starts, NorthEast Single-unit Starts, NorthEast Total Starts, South Single-unit Starts, South Total Starts, West Single-unit Starts, West Total Starts, MidWest Single-unit Starts, MidWest Table 8 : Parameter estimates for season-specific irregular model, sample period January 1985 to March Relative to the homoskedastic model, the inclusion of the extra irregular leads to a decline in ση 2 for many of the series. Typically, the estimates satisfy σε,l 2 <σ2 ε <σε,h 2 where σε 2 is the homoskedastic variance in the constrained model. The estimate of σε,h 2 is more than double that of σ2 ε for the NorthEast and MidWest series, and for Housing starts in the South. This indicates that part of the stochastic movements in the seasonal now show up as a switch in variance from low to high variability months. 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 0 jts are one-period shocks. This gives a more natural starting point for applications where the seasonal changes in variability are due to temporary developments like weather patterns. In (1), a surprising shock in a high variance month j at time t is reflected in a large value of ηj,t. From section 2, the assumptions of the model state that (ηj,t,η j,t+1,η j,t+2,..., ηj,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. In contrast, the heteroskedastic noise in (3) or (4) is specifically attached to individual months. Finally, note that there is a loose relation between the extra variance parameter σε, 2 and the difference in variances, σε,h 2 σ2 ε,l. This depends on how well the canonical white noise variation, that can be removed from the airline model, can be matched with the baseline variance σε,l 2 in the seasonal specific model. 20

22 4 Finite sample test for seasonal heteroskedasticity This section develops a test for seasonal heteroskedasticity that can be applied in finite samples. The null hypothesis of homoskedasticity implies restrictions on the variance parameters in models (1), (3), and (4). Maximum likelihood estimation was discussed in the previous section, and results were shown for heteroskedastic and homoskedastic models. Denoting the maximized log-likelihoods for estimation of the restricted and unrestricted models as L e and L, e respectively, we construct the likelihood-ratio test statistic LR = 2( L e L) e 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 whether the large sample results give useful approximations in our application without further investigation. The construction series consist of about 250 observations. Further, the performance of the test for shorter-length series is of interest for other possible applications. We therefore present results on test statistics for each model, followed by 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 ση,h 2 = σ2 η,l = ση. 2 As there is a single restriction, the distribution of LR under the null of homskedasticity is asymptotically a χ 2 (1) (chi-squared with one degree of freedom). Note that, when the variance ση 2 is small, there is a finite probability of estimating both ση,l 2 and ση,h 2 in the heteroskedastic model to be zero. When this happens, the unrestricted model reduces to the restricted one with ση 2 =0, giving a likelihood ratio test statistic of exactly zero. Therefore, the distribution of LR inherits the pile-up problem in the likelihood function. The probability mass at LR =0increases as ση 2 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 MLEs from table 5 give a range of parameter values, and on this basis we can choose a set of parameter combinations that yield plausible models for the data generating proces. In future work, we will present results for a simulation study of the LR test for Proietti s model. Note that, in the cases we have investigated so far, the empirical distribution of LR under the null deviates from a χ 2 (1). The positive probability mass at LR =0 distinguishes it from a purely continuous distribution. The discrete probability of LR =0increases as the true value of ση 2 gets closer to zero. Further, the nonzero, or continuous, part of the distribution has a thinner tail than the χ 2 (1). 21