Time Series Analysis: The Basics
WHAT IS A TIME SERIES?
It also includes calendar related systematic effects that are not stable in their annual timing or are caused by variations in the calendar from year to year, such as:
HOW DO WE IDENTIFY SEASONALITY?
Seasonality in a time series can be identified by regularly spaced peaks and troughs which have a consistent direction and approximately the same magnitude every year, relative to the trend. The following diagram depicts a strongly seasonal series. There is an obvious large seasonal increase in December retail sales in New South Wales due to Christmas shopping. In this example, the magnitude of the seasonal component increases over time, as does the trend.
Figure 1: Monthly Retail Sales in New South Wales (NSW) Retail Department Stores
WHAT IS AN IRREGULAR?
The irregular component (sometimes also known as the residual) is what remains after the seasonal and trend components of a time series have been estimated and removed. It results from short term fluctuations in the series which are neither systematic nor predictable. In a highly irregular series, these fluctuations can dominate movements, which will mask the trend and seasonality. The following graph is of a highly irregular time series:
Figure 2: Monthly Value of Building Approvals, Australian Capital Territory (ACT)WHAT IS THE TREND?
The ABS trend is defined as the 'long term' movement in a time series without calendar related and irregular effects, and is a reflection of the underlying level. It is the result of influences such as population growth, price inflation and general economic changes. The following graph depicts a series in which there is an obvious upward trend over time:
Figure 3: Quarterly Gross Domestic Product
WHAT ARE THE UNDERLYING MODELS USED TO DECOMPOSE THE OBSERVED TIME SERIES?
Decomposition models are typically additive or multiplicative, but can also take other forms such as pseudo-additive.
In some time series, the amplitude of both the seasonal and irregular variations do not change as the level of the trend rises or falls. In such cases, an additive model is appropriate.
In the additive model, the observed time series (Ot) is considered to be the sum of three independent components: the seasonal St, the trend Tt and the irregular
Each of the three components has the same units as the original series. The seasonally adjusted series is obtained by estimating and removing the seasonal effects from the original time series. The estimated seasonal component is denoted by The seasonally adjusted estimates can be expressed by:
The following figure depicts a typically additive series. The underlying level of the series fluctuates but the magnitude of the seasonal spikes remains approximately stable.
Figure 4: General Government and Other Current Transfers to Other Sectors
In many time series, the amplitude of both the seasonal and irregular variations increase as the level of the trend rises. In this situation, a multiplicative model is usually appropriate.
In the multiplicative model, the original time series is expressed as the product of trend, seasonal and irregular components.
The seasonally adjusted data then becomes:
Under this model, the trend has the same units as the original series, but the seasonal and irregular components are unitless factors, distributed around 1.
Most of the series analysed by the ABS show characteristics of a multiplicative model. As the underlying level of the series changes, the magnitude of the seasonal fluctuations varies as well.
Figure 5: Monthly NSW ANZ Job AdvertisementsPseudo-Additive Decomposition
The multiplicative model cannot be used when the original time series contains very small or zero values. This is because it is not possible to divide a number by zero. In these cases, a pseudo additive model combining the elements of both the additive and multiplicative models is used. This model assumes that seasonal and irregular variations are both dependent on the level of the trend but independent of each other.
The original data can be expressed in the following form:
The pseudo-additive model continues the convention of the multiplicative model to have both the seasonal factor St and the irregular factor It centred around one. Therefore we need to subtract one from St and It to ensure that the terms Tt x (St - 1) and Tt x (It - 1) are centred around zero. These terms can be interpreted as the additive seasonal and additive irregular components respectively and because they are centred around zero the original data Ot will be centred around the trend values Tt .
The seasonally adjusted estimate is defined to be:
where and are the trend and seasonal component estimates. In the pseudo-additive model, the trend has the same units as the original series, but the seasonal and irregular components are unitless factors, distributed around 1.
An example of series that requires a pseudo-additive decomposition model is shown below. This model is used as cereal crops are only produced during certain months, with crop production being virtually zero for one quarter each year.
Figure 6: Quarterly Gross Value for the Production of Cereal Crops
Example: Shiskin Decomposition
The Shiskin decomposition gives graphs of the original series, seasonally adjusted series, trend series, residual (irregular) factors and the between month (seasonal) and within month (trading day) factors that are combined to form the combined adjustment factors. The residual (irregular) factors are found by dividing the seasonally adjusted series by the trend series. Figure 7 shows a Shiskin decomposition for the Australian Retail series.
Figure 7: Shiskin decomposition for Australian Total Retail Turnover, May 1990 to May 2000
HOW DO I KNOW WHICH DECOMPOSITION MODEL TO USE?
To choose an appropriate decomposition model, the time series analyst will examine a graph of the original series and try a range of models, selecting the one which yields the most stable seasonal component. If the magnitude of the seasonal component is relatively constant regardless of changes in the trend, an additive model is suitable. If it varies with changes in the trend, a multiplicative model is the most likely candidate. However if the series contains values close or equal to zero, and the magnitude of seasonal component appears to be dependent upon the trend level, then pseudo-additive model is most appropriate.
WHAT IS A SEASONAL AND IRREGULAR (SI) CHART?
Once the trend component is estimated, it can be removed from the original data, leaving behind the combined seasonal and irregular components or SIs. A seasonal and irregular or SI chart graphically presents the SI's for particular months or quarters in the series span.
The following graph is an SI chart for a monthly series, using a multiplicative decomposition model.
Figure 8: Seasonal and Irregular (SI) Chart - Value of Building Approvals, ACT
The points represent the SIs obtained from the time series, while the solid line shows the seasonal component. The seasonal component is calculated by smoothing the SI's, to remove irregular influences.
SI charts are useful in determining whether short-term movements are caused by seasonal or irregular influences. In the graph above, the SIs can be seen to fluctuate erratically, which indicates the time series under analysis is dominated by its irregular component.
SI charts are also used to identify seasonal breaks, moving holiday patterns and extreme values in a time series.