CHAPTER 7
SEASONAL AND CALENDAR ADJUSTMENT
Seasonal and calendar adjustment
Introduction
1.1.Due to the periodicity at which they are recorded, quarterly time series quite often show short-termmovements that tend to repeat themselves in the same period(month or quarter) each year,and which are caused by the weather, habits, legislation, etc. These effects are usually defined as seasonal variations. In addition, there are calendar variations, such as Easter, that affect different periods in different years.
1.2.Although seasonal and calendar variation is an integral part of quarterly data, it is generally considered an impediment toeffective analysis of recent economic developments or the business cycle. Therefore, the derivation of series with the seasonal and calendar variations removed, and which are generally referred to simply as seasonally adjusted series, has become commonplace. Seasonally adjusted series are much more heavily used and much more prominently reported in the media than the non-seasonally adjusted series[1].
1.3.Seasonal adjustment of the quarterly national accounts should cover at least Table 1 of the ESA transmissionprogramme. All series should be tested for the existence of seasonal and calendar variation and adjusted accordingly if they are present.Adjusted results should be produced for data in current prices, volumes and deflators byadjusting any two of these and deriving the third from the other two series after they have been seasonally adjusted andbenchmarked. When the deflator is the derived one, care should be takento prevent its path being affected by arbitrary differences in the separate adjustment ofcurrent price and volume data.
1.4.Seasonal adjustment is a complex process, which is comprehensively addressed in the forthcoming Handbook on Seasonal Adjustment[2]. This chapter briefly discusses the causes and nature of seasonality, and gives a summary description of the most commonly used seasonal adjustment methods and their application. The chapter also addresses issues that are of special importance to the national accounts. Further guidelines specific to the QNA are provided in the Final Report of the Task Force on Seasonal Adjustment of Quarterly National Accounts[3], which was endorsed by the European Committee on Monetary, Financial and Balance of Payments Statistics (CMFB) in 2008. Further, more general, guidelines on seasonal adjustment are provided in the 2009 edition of ESS Guidelines on Seasonal Adjustment[4].
Causes of seasonal variation[5]
1.5.There are at least four, not totally distinct, classes of causes of seasonal and calendar variation ineconomic data:
A. Calendar
The timing of certain public holidays, such as Christmas and Easter, clearly affect the quarter-on-quarter, or month-on-month, movements of some time series, particularly those related to production and consumption. Also, the number and distribution of days in a month/quarter varies from one month/quarter to another and contributes to period-on-period variations.
B. Timing decisions
The timing of school vacations, the ending of university sessions, the payment ofcompany dividends, the choice of the end of a tax-year or accounting period are allexamples of decisions made by individuals or institutions that cause importantseasonal effects, as these events are inclined to occur at similar times each year. Theyare generally deterministic, or pre-announced, and are decisions that produce verypronounced seasonal components in series such as employment rates. These timingdecisions are generally not necessarily tied to any particular time in the year but bytradition have become so.
C. Weather
Changes in temperature, rainfall and other weather variables have direct effectson various economic series, such as those concerned with agricultural production,construction and transportation, and consequent indirect effects on other series. Itcould be argued that these weather-related factors are the true seasonal effect, being a consequence of the annual movement of the earth's axis relative to the sun which leads to the seasons.Weather can lead to major random effects, too.
D. Expectations
The expectation of a seasonal pattern in a variable can cause an actual seasonal effectin that or some other variable, since expectations can lead to plans that then ensureseasonality. An example is toy production in the expectation of a sales peak during theChristmas period. Without the expectation-planning aspect, the seasonal pattern maystill occur but might be of a different shape or nature. Expectations may arise becauseit has been noted that the series being considered has in the past contained a seasonalpattern, or because it is observed that acknowledged causal series have a seasonalcomponent.
1.6.These four groups may be thought of as basic causes of seasonal variation, but there may be others. They are not always easilydistinguishable and may oftenmerge together.Some series may have seasonal components that are only indirectly due to these basiccauses. Weather may cause a seasonal pattern in grape production which then causes aseasonal distribution in grape prices, for example. Formany series, the actual causation of aseasonal effect may be due to a complicated mix of many factors or reasons, due to thedirect impact of basic causes and many indirect impacts via other economic variables. Evenif only a single basic cause is operating, the causal function need not be a simple one andcould involve both a variety of lags and non-linear terms.
1.7.Two important conclusions can be reached from such considerations:
a.The causes ofthe seasonal components can be expected to have differing properties;
b.Theseasonal components cannot be assumed to be deterministic. It is common to observe the seasonality of a series changing over time, and often one can only hypothesise what the cause may be.
1.8.There are no hard and fast “rules” about which series are seasonal and which are not, but some types of series are commonly seasonal and others are not. In general, flows, such as production and consumption, are seasonal but there may be exceptions (purchases of toothpaste?). Stock variables, such as unemployment, employment and inventories, are commonly seasonal by virtue of the seasonality of their inflows and outflows but, since the inflows and outflows are small relative to the stock level, they display a proportionately lower degree of seasonality. By contrast, many prices, including interest rates and exchange rates, are not seasonal but there are exceptions, such as the prices of agricultural products and holiday accommodation.
1.9.While seasonality always has to be determined empirically, one should always try to understand what the causal factors may be. If the seasonality is contrary to what might be expected one should investigate the possibility of it being created as an artefact of how it is reported. A special case is where there is no apparent seasonality for a series where one would expect to find it.
1.10.Any stationary time series can be represented as the sum of sine and cosine waves of different frequencies, phase and amplitude. The seasonal frequencies for quarterly data correspond to waves of length two and four quarters, as these are the only waves that recur in the same quarter(s) each year. Similarly, the seasonal frequencies for monthly data correspond to waves of length two, three, four and six months.
1.11.In simple terms, monthly/quarterly time series can be considered to have seasonality if the amplitudes of the sinusoidal waves with these wavelengths are large relative to those of other wavelengths, and seasonal adjustment can be thought of as filtering out these waves.
Calendar variation
1.12.The calendar effect is the impact of working/trading-days, fixed and moving holidays, leapyear and other calendar related phenomena (e.g. bridging days[6]) on a time series. These calendar effects can be divided into a seasonal and a non-seasonal component. The formercorresponds to the average calendar situation that repeats each year in the same month or quarter,and thelatter corresponds to the deviation of the calendar variables (such as numbers oftrading/working days, moving holidays, leapyear days) from the long-term month-orquarter-specific average. The seasonal component of the calendar effect is part of theseasonal component of the time series and is removed by seasonal adjustment. The non-seasonalpart of the calendar effect is commonly referred to as “calendar variation”, and it has essentially three different elements:
- effects related to moving holidays;
- effects related to working-days/trading-days;
- leap years
Moving holidays
1.13.Easter, as well as other moving holidays, may concern different months or quarters according to the year (Catholic Easter can affect March or April, that is the first or the second quarter, and Orthodox Easter can affect April or May). Different dates of such holidays (mainly Easter) from one year to another imply instability of the seasonal pattern related to the corresponding quarter or month. For this reason, Easter and moving holiday effects require a special statistical treatment.
1.14.The impact of moving holidays varies between products and industries. For example, while Easter and the associated holidays may reduce production in some industries, they can increase production in others because of increased consumption. Also, the extent and duration of the effects can vary between products and industries. This implies that each time series needs to be carefully analysed for calendar effects individually.
Working-days
1.15.Many industries undertake a lot more production on normal (days that are not public holidays) weekdays than on weekend days, and given that the number of normal weekdays and weekend days in a particular month, or quarter, varies from year to year, then one would expect to see a working-day variation. An adjustment for the effect is sometimes based on the proportion of normal weekdays and weekend days in a month or a quarter. However, proportional adjustment has proved to be not very accurate, and it is not recommended. The preferred approach to calendar adjustment is based on regression methods, possibly assuming an autoregressive integrated moving average dynamic (ARIMA) process for the error term (as currently implemented in TRAMO/SEATS and X-12-ARIMA).
Trading-days
1.16.Whilst the working-day effect highlights the differences of business activity between weekdays and weekend days, the trading-day effect catches the differences in economic activity between every day of the week. The word “trading” suggests that it relates only to distributive industries, but in practice it is applied to all industries and products where activity varies according to the day of the week. Therefore, working-day variation can be considered a subset of trading-day variation.
Leap year/length of the month
1.17.Some time series may be affected by the fact that every four years February, and therefore the first quarter, has one extra day.
Seasonal adjustment models
1.18.Seasonality is intrinsically an unobserved component in timeseries. Consequently, all seasonal adjustment methods are based on a model that decomposes the observed non-seasonally adjusted data into a seasonal component, calendar component and non-seasonal unobserved components on the basis of a set of assumptions of their characteristics. The simplest decomposition model to apply to a time series is the additive model:
Xt= Tt + Ct + St + Kt + Ut,
where:
- Ttis the trend component;
- Ct is the cyclical component;
- Stis the seasonal component;
- Kt is the calendar component;
- Ut is the irregular component.
1.19.These components are usually defined in the following way:
- Trend is the underlying level of the time series, and changes in trend reflect the long-term growth of the phenomenon being considered.
- Cycle comprises short tomedium-term fluctuations characterised by alternate periods of expansion and contraction, whichare commonly related to fluctuations in economic activity, such as the business cycle.
- Seasonalreflects the effects of weather-related or institutional events, decisions or expectations, which repeat themselves more or less regularly each year in the same period.
- Calendar captures the effects related to the calendar that do not repeat themselves each year, e.g. the number of working-days per month or special situations like the dating of Easter.
- Irregular fluctuations represent unexpected movements related to events other than those previously considered.
1.20.The trend and cycle components are often combined together as asingle(trend-cycle) component because for seasonal adjustment purposes there is no need to separate them, and so the additive model simplifies to:
Xt= TCt + St + Kt + Ut,
whereTCt denotes the combined trend-cycle. St,KtandUtall describe fluctuations around the trend-cycle and take positive and negative values in the additive model.
1.21.Besides the additive model there is the multiplicative model:
Xt= TCt.St.Kt.Ut,
where the relationship among the components is multiplicative. As in the additive model,St,KtandUt all describe fluctuations around the trend-cycle, but in this case they take values around 1.
1.22.A variant of the multiplicative model is the log-additive model, which is obtained by taking logarithms of the multiplicative model:
logXt= log (TCt) + log (St) + log (Kt) + log (Ut)
This transformation allows the non-seasonally adjusted series to be decomposed using additive procedures. Note that while taking the logarithm of the multiplicative model does not change the model in concept it can lead to different outcomes in practice, depending on the methods used to estimate each of the components – see § 7.52 – 7.53.
1.23.A fourth model, the pseudo-additive, is a combination of the multiplicative and additive models:
Xt= Tt(St + Kt + Ut – 1)
1.24.Seasonal and calendar variations affect the level of the economic activity in specific periods in a predictable way, and it makes sense to remove such variations in order to get a clearer picture of the underlying growth of economic activity. In the additive model the calendar and seasonal components are subtracted from the non-seasonally adjusted data to yield calendar and seasonally adjusted data:
YtSKA = TCt+ Ut = Xt- St-Kt
1.25.The presence of calendar (and possibly other deterministic) effects can have an adverse effect on the estimation of the seasonal component, resulting in low quality seasonally adjusted data. Therefore, it is recommended calendar adjustments be made prior to the estimation of the seasonal component, or simultaneously if a model is to be estimated that includes both calendar and seasonal components.
1.26.As mentioned in §7.2, seasonally and calendar adjusted data are commonly referred to as seasonally adjusted data. In general, if the non-seasonally adjusted data show the effects of both calendar and seasonal variation, then data adjusted for only one of them are not published. However, non-seasonally adjusted data adjusted only for calendar variation are required for analytical purposes (see § 7.42) and to derive annual calendar adjusted data (see §7.75).
1.27.Having derived a seasonally and calendar adjusted time series, it is then possible to go one step further and remove the irregular component to obtain the trend-cycle component (see §7.65 – 7.73):
YTt = TCt
1.28.In a similar way, all the other models (such as the multiplicative and the log-additive) can be used to isolate seasonally adjusted and trend-cycle series.
Choice of seasonal adjustment method
1.29.At present, several techniques are used by EU Member States to obtain seasonallyadjusted data. Two main groups of methods can be distinguished:
- moving average-based methods;
- model-based methods.
1.30.Moving average-based methods are based on the use of different kinds of moving average filter[7]. They do not rely on an underlyingexplicit model and were developed mainly on an empirical basis.
1.31.The best known movingaverage-based method is the US Bureau of the Census’s X-11 (and its upgrades). It is also one of the most commonly used seasonal adjustment methodsworldwide. Currently, the latest upgrade in common use is X-12-ARIMA. All of the X-11 family involve the repeatedapplication of suitable moving average filters that leads to a decomposition of the non-seasonally adjusted data into its trend-cycle, seasonal and irregular components. The process is quite complex and involves many steps, but, leaving aside calendar variation, the basic steps are as follows when using the additive model:
a.derive an initial estimate of the trend-cycle by applying a moving average[8] to the non-seasonally adjusted data;
b.subtract this estimate from the non-seasonally adjusted data to obtain an initial estimate of the seasonal-irregular (SI) and apply a moving average to the SIs for each type of quarter separately to obtain initial estimates of the seasonal component;
c.subtract the initial seasonal factors from the non-seasonally adjusted data to obtain an initial estimate of the seasonally adjusted series (i.e. the trend-cycle/irregular) and apply a Henderson[9] moving average to obtain a second estimate of the trend-cycle;
d.subtract the second estimate of the trend-cycle from the non-seasonally adjusted data to obtain a second estimate of the SIs, and apply a moving average for each type of quarter separately to obtain final estimates of the seasonal component;
e.subtract the seasonal factors from the non-seasonally adjusted data to obtain a final estimate of the seasonally adjusted series and apply a Henderson moving average to obtain a final estimate of thetrend-cycle.
If the log-additive model is being used then the logarithm of the non-seasonally adjusted data is taken before step 1, and antilogarithms are taken of the final results. If the multiplicative model is being used then subtraction is replaced by division.
1.32.Model-based seasonal adjustment methods estimate the trend-cycle, seasonal and irregular components with signal extraction techniques applied to an ARIMA model fitted to the non-seasonally adjusted or transformed (e.g. logged) data. Each component is then represented by an ARIMA expression and some parameter restrictions are imposed to obtain orthogonal components. TRAMO/SEATS is one of the best known and most widely used methods of this type. In order to isolate a unique decomposition (i.e. the “canonical one”), TRAMO/SEATS imposes further constraints: the variance of the irregular component is maximized and, conversely, the other components are kept as stable as possible (compatible with the stochastic nature of the model used for their representations). There are other model-based methods that use different approaches to model estimation and decomposition[10].