### The seasonal model - underlying theory

- What is the underlying seasonal model?
- What is a seasonally adjusted series?
- What is the seasonal adjustment process?

### Trend cycle

- What is the trend?
- Why "trend cycle"?
- Why do trends get revised?
- Why do seasonally adjusted series get revised?
- What are the desirable characteristics of a trend estimator?

#### Seasonal component

#### Irregular component

- What is the irregular component?
- How does the irregular component arise?
- What information does the irregular component give about the quality of the model?

#### Calendar effects

- What are calendar effects?
- What are trading day effects and trading day adjustments?
- Why does Statistics New Zealand make so few trading day adjustments?
- Are trading day adjustments made in quarterly series?

#### Choice of model

#### Direct and indirect adjustment

- What are direct and indirect adjustments?
- How do you choose between a direct and indirect adjustment?
- When is seasonal adjustment not appropriate?

### The Seasonal Model - underlying theory

#### What is the underlying seasonal model?

A seasonal model gives a simplified description of the data. It is used to assist in analysis. At Statistics New Zealand, we use the model to help us understand what is happening now, and what has happened in the past. We do not use it for forecasting future values.

The standard seasonal model assumes the actual (observed) series (A) is composed of factors or components:

C - the trend cycle

S - the seasonal component - this includes calendar effects

I - the irregular component

We assume that some relationship exists between them. Generally it is either

multiplicative: A = C x S x I

or additive: A = C + S + I

How do you decide whether to apply an additive or a multiplicative model?

#### What is a seasonally adjusted series?

This is the actual series with the seasonal component removed. Generally it is either

- Multiplicative: Seasadj = C x I = A / S
- Additive: Seasadj = C + I = A - S

#### What is the seasonal adjustment process?

The mechanics of seasonal adjustment involve breaking down a series into trend cycle, seasonal, and irregular components.

### Trend

Trend, seasonal component, irregular component and calendar effects

#### What is the trend?

Trend estimates reveal the smooth, relatively slowly changing features in a time series. They are usually estimated by applying repeated moving averages.

How to use the trend

The trend cycle component shows the fundamental movement of the series, reflecting the prevailing economic conditions in an economic series. It merges any cyclical movements present with the underlying trend:

##### Underlying trend

Most economic time series have a long-term underlying trend present. It is often associated with some basic characteristic of the economy, such as population growth. In some series the trend may be steadily upwards, while in others it may show considerable variability. In general the trend has little effect on the short-term month-to-month movements of the series, but it is the most important component for determining the general level and broader movement of the activity as measured by the series.

##### Cyclical movements

These are composed of cumulative, reversible, short-run movements. They are characterised by alternating periods of expansion and contraction as they reflect general economic activity. In some large overseas economies these accelerations and recessions are self-generating. In New Zealand, however, they are often the direct or indirect result of fluctuation in the terms of trade, as New Zealand is heavily dependent on overseas trade. Typically the frequency of these cycles is less than once a year and hence are not confounded with the seasonal component whose frequencies are faster than once a year.

Whenever cyclical variations are present in a New Zealand time series, they merge with the underlying trend to form a component that is usually referred to as trend, though more correctly it is the trend cycle. This component is therefore abbreviated C, not T.

#### Why do trends get revised?

Important caveat on trend estimates:

Trend estimates at the end of a series are not as reliable and are subject to greater revisions than estimates that are more central.

If the actual observed data itself is revised, this will result in changes to the trend and seasonally adjusted series.

Even if the actual data is not revised, trend estimates towards the end of a series are subject to revision. This is because the moving average used in the body of the series is symmetric, making use of the same number of points before and after the time point. But at the end of the series, where there is no "future" data, the moving average used is asymmetric. As new data becomes available data points which are not yet in the body of the series have their trend re-estimated using different, and decreasingly asymmetric moving averages. The number of periods where the revisions can be of consequence depends on the length of the moving average, but usually only affects the most recent trend estimates. There are likely to be substantial revisions for many recent periods in the vicinity of turning points.

Revisions can be particularly large if an observation is treated as an outlier in one quarter, but is found to be part of the underlying trend as further observations are added to the series.

There is no global solution to this problem. Statistics New Zealand has investigated the use of ARIMA models to forecast a year ahead and hence shift the most recent "real" data points into the body of the series.

#### Why do seasonally adjusted series get revised?

The seasonally adjusted series is the original series with the seasonal component removed. The seasonal component is estimated with moving averages and so it is subject to revision as new points are added to the series in the same way as the trend component. As with the trend series, when new data comes to hand, the assessment of what is an outlier can change. This occurs even for points earlier in the series. This will cause revisions in the seasonally adjusted series.

There is no global solution to this problem. SNZ has investigated the use of ARIMA models to forecast a year ahead and hence shift the most recent "real" data points into the body of the series.

The effect of ARIMA forecasting on revisions to seasonally adjusted time series

#### What are the desirable characteristics of a trend estimator?

The desirable characteristics of trend estimates in order of importance as assigned by most statistical institutions are:

- Speedy detection of turning points.
- Low proportion of false turning points.
- Minimal, unbiased revisions when a subsequent data point is added to the series.
- Robustness. The trend should not be affected by outliers.
- Smoothness.

No trend estimation method is able to satisfy all these criteria at once. There is generally a trade-off between the delay in recognising turning points and the frequency of false turning points.

### Seasonal component

#### What is the seasonal component?

It is the seasonal patterns found in many sub-annual (quarterly or monthly) economic series. It is reasonably stable in terms of annual timing, direction, and magnitude.

#### How does the seasonal component arise?

Possible causes include:

- Natural factors (eg seasonal weather patterns).
- Administrative measures (eg starting and ending dates of the school year).
- Social/cultural/religious traditions (eg fixed holidays such as Christmas).
- The length of the months (28, 29, 30 or 31 days) or quarters (90, 91 or 92 days).

Effects associated with the dates of moving holidays like Easter are not seasonal in this sense, because they occur in different calendar months and different quarters, depending on the date of the holiday.

The extent and nature of this seasonality can vary markedly between series. For example, it is especially prominent in the case of electricity generation, or agricultural production, but relatively insignificant for a series such as total New Zealand population.

### Irregular component

#### What is the irregular component?

This is the part of the observed value that is not included in the trend cycle or the seasonal effects (or in estimated trading day or holiday effects). Its values are unpredictable as regards timing, impact, and duration.

#### How does the irregular component arise?

It can arise from a combination of sampling error, non-sampling error, unseasonable weather, natural disasters, strikes, etc. Random fluctuations are the main cause. While every member of the population is affected by general economic or social conditions, each member is affected slightly differently. So there will always be some random variation in the series. This is not a problem, as long as it is small. If it is generally large, then it can become difficult to quantify the other components.

- Approximations or defects in the seasonal adjustment procedures. This is because the irregular component is measured as a residual in the process of seasonal adjustment.
- Sampling variability.
- Errors in measurement or statistical processing.

#### What information does the irregular component give about the quality of the model?

Much of the testing done in seasonal adjustment, as in any modelling procedure, is seeing if there is any structure (ie anything that can be modelled) left in the irregular component.

Many of the quality diagnostics are based on comparison of the variability of the other identified components with that of the irregular component.

### Calendar effects

#### What are calendar effects?

They are variations in the series that are attributable to calendar features such as the number of trading days in a month, holidays and Easter.

The difference between the lengths of the months (28, 29, 30 or 31 days) is incorporated in the seasonal factor.

#### What are trading day effects and trading day adjustments?

Monthly time series that are based on a daily flow of goods, services, or money can be influenced by each calendar month's weekday composition. This influence is revealed when monthly values consistently depend on which days of the week occur five times in the month. For example the month of January will have five Sundays in some years and four in others.

Because these fluctuations are most easily illustrated in retail trade it is commonly referred to as "trading day" effect. In reality the effect may have more to do with the way the data is collected, than with the actual daily flow.

Trading day effects can make it difficult to compare series values, or to compare movements in one series with movements in another. For this reason, when estimates of trading day effects are statistically significant, they are quantified and removed. The removal of such estimates is called trading day adjustment.

#### Why does Statistics New Zealand make so few trading day adjustments?

The task of quantifying a suspected effect accurately enough to be able to remove it makes trading day adjustments particularly difficult. A long series is required, much longer than for estimating seasonal factors, with a constant or nearly constant pattern. As a result very few of our series have trading day adjustments made in them.

Statistics New Zealand introduced trading day adjustments to monthly retail trade in Sept 1999, when we achieved a series that was seven years long.

#### Are trading day adjustments made in quarterly series?

Trading day effects are too small to quantify in quarterly series, and therefore adjustments cannot be made.

As the four quarters are almost all the same length (90, 91 or 92 days) and almost exact multiples of seven, whether the extra day is a Monday or Tuesday, etc, will only have a minuscule impact on the quarterly figure. Furthermore, it would be very difficult to estimate the weight of each day accurately, given the few occurrences of different extra days in quarterly data. Thus no trading day adjustment is made in the quarterly program.

#### Holidays and trading day effect

Unfortunately many holidays occur on a Monday or a Friday. This adds to the problems with quantifying Monday and Friday trading day factors. At present Statistics New Zealand does not make any adjustments for these effects.

### Choice of model

#### Multiplicative and additive models

##### How do you decide whether to apply an additive or a multiplicative model?

A multiplicative model cannot be implemented if there are zero or negative observed values in the series.

Most of Statistics New Zealand's economic series are multiplicatively adjusted, in line with "standard practice among producers of seasonally adjusted data, which is to assume that the components of an economic time series are multiplicatively adjusted.^{(1)}

In a multiplicative structure the seasonal effects change proportionately with the trend. If the trend increases, so do the seasonal effects and if the trend decreases the seasonal effects diminish too. In an additive structure the seasonal effects remain more or less the same no matter which direction the trend is moving in.

In practice we can inspect the graph of the series to decide what the structure is. Alternatively we can process the series through X-12-ARIMA, using first the multiplicative, then the additive decomposition. The decomposition that yields the higher stable seasonality F-value and the lower moving seasonality F-value is the preferable one.

^{(1)} Dagum, E B 1976, "X-11, the magic box and four golden rules of seasonal adjustment", Statistics Canada Research Paper 76-06-007E, p27, Ottawa.

### Direct and indirect adjustments

#### What are direct and indirect adjustments?

When seasonally adjusting a series which is a sum (or other composite) of a component series, then we have the option of:

a direct adjustment - seasonally adjusting the sum of the component series

SA direct(A1+A2+.....+An) = Seasadj(A1+A2+.....+An)

or

an indirect adjustment - summing the seasonally adjusted component series

SA indirect(A1+A2+.....+An) = Seasadj(A1) + Seasadj(A2) +.....+ Seasadj(An)

The two methods produce slightly different seasonal adjustments.

#### How do you choose between a direct and indirect adjustment?

When the component series have quite distinct seasonal patterns and have adjustments of good quality, indirect seasonal adjustment is usually of better quality. Indirect seasonal adjustments are preferred by many data users because they are consistent with the adjustments of the component series.

In the aggregate series direct seasonal adjustment removes any residual seasonality which may not be measurable in some of the individual components.

#### When is seasonal adjustment not appropriate?

A series that is dominated by either the trend or irregular component is not eligible for seasonal adjustment. Any seasonal component present is impossible to quantify accurately, and has minimal impact on the series.

Inappropriate seasonal adjustment of a non-seasonal series will often produce a series that has had a seasonal pattern introduced to it.