The purpose of seasonal adjustment is to remove systematic calendar-related variation associated with the time of the year, that is, seasonal effects. This facilitates comparisons between consecutive time periods. Data that are collected over time form a time series. Many of the most well-known statistics published by the Office for National Statistics are regular time series, including: the claimant count, the Retail Prices Index (RPI), balance of payments and gross domestic product (GDP).

Those analysing time series typically seek to establish the general pattern of the data, the long-term movements and whether any unusual occurrences have had major effects on the series. This type of analysis is not straightforward when one is reliant on raw time series data, because there will normally be short-term effects, associated with the time of the year, which obscure or confound other movements.

For example, retail sales rise each December due to Christmas. The purpose of seasonal adjustment is to remove systematic calendar-related variation associated with the time of the year, that is, seasonal effects. This facilitates comparisons between consecutive time periods.

### Components of a time series

Time series can be thought of as combinations of three broad and distinctly different types of behaviour, each representing the impact of certain types of real world events on the data. These three components are: systematic calendar-related effects, irregular fluctuations and trend behaviour.

#### Systematic calendar-related effects

Systematic calendar-related effects comprise seasonal effects and calendar effects. Seasonal effects are cyclical patterns that may evolve as the result of changes associated with the seasons. They may be caused by various factors, such as:

- weather patterns: for example, the increase in energy consumption with the onset of winter
- administrative measures: for example, the start and end dates of the school year
- social, cultural and religious events: for example, retail sales increasing in the run up to Christmas
- variation in the length of months and quarters due to the nature of the calendar

Other calendar effects relate to factors that do not necessarily occur in the same month (or quarter) each year. They include:

- trading day effects, which are caused by months having differing numbers of each day of the week from year to year: for example, spending in hardware stores is likely to be higher in a month with five, rather than four, weekends
- moving holidays, which may fall in different months from year to year: for example Easter, which can occur in either March or April

Taken together these effects make up the seasonal component.

#### Irregular fluctuations

Irregular fluctuations may occur due to a combination of unpredictable or unexpected factors, such as: sampling error, non-sampling error, unseasonable weather, natural disasters, or strikes. While every member of the population is affected by general economic or social conditions, each is affected somewhat differently, so there will always be some degree of random variation in a time series. The contribution of the irregular fluctuations will generally change in direction and/or magnitude from period to period. This is in marked contrast with the regular behaviour of the seasonal effects.

#### The trend (or trend cycle)

The trend (or trend cycle) represents the underlying behaviour and direction of the series. It captures the long-term behaviour of the series as well as the various medium-term business cycles.

### The seasonal adjustment process

Although there are many ways in which these components could fit together in a time series, we select one of two models:

- additive model: Y equals C plus S plus I
- multiplicative model: Y equals C multiplied by S multiplied by I

where Y is the original series, C is the trend-cycle, S is the seasonal component and I is the irregular component.

The seasonally adjusted series is formed by estimating and removing the seasonal component.

- for the additive model: seasonally adjusted series equals Y minus S equals C plus I
- for the multiplicative model: seasonally adjusted series equals Y divided by S equals C multiplied by I

In a multiplicative decomposition, the seasonal effects change proportionately with the trend. If the trend rises, so do the seasonal effects, while if the trend moves downward the seasonal effects diminish too. In an additive decomposition the seasonal effects remain broadly constant, no matter which direction the trend is moving in.

In practice, most economic time series exhibit a multiplicative relationship and hence the multiplicative decomposition usually provides the best fit. However, a multiplicative decomposition cannot be implemented if any zero or negative values appear in the time series.

### Other factors that affect seasonal adjustment

There are a variety of issues that can impact on the quality of the seasonal adjustment. These include:

- outliers, which are extreme values; these usually have identifiable causes, such as strikes, war, or extreme weather conditions, which can distort the seasonal adjustment – they are normally considered to be part of the irregular component
- trend breaks (also known as level shifts), where the trend component suddenly increases or decreases sharply; possible causes include changes in definitions relating to the series that is being measured, to take account of, say, a reclassification of products or a change in the rate of taxation
- seasonal breaks, where there are abrupt changes in the seasonal pattern

These issues need to be addressed before the seasonal adjustment process begins, in order to obtain the most reliable estimate of the seasonal component.