Candidate’s objectives:

After completing this reading you should be able to:

- Describe the sources of seasonality and how to deal with it in time series analysis.
- Explain how to use regression analysis to model seasonality.
- Explain how to construct an h-step-ahead point forecast.

## The Nature and Sources of Seasonality

One characteristic of seasonal patterns is their tendency of repeating themselves every year. In deterministic seasonality, there is an exact year to year repetition. In stochastic seasonality, the year to year repetition can be approximate. In business and economics seasonality is pervasive.

Seasonality in a series can be examined by removing it, then modeling and forecasting the seasonally adjusted time series. One application of this strategy is an instance when non-seasonal fluctuations are the explicit forecast of interest, a common scenario in macroeconomics.

## Modeling Seasonality

Regression on seasonal dummies is an important method of modeling seasonality. Assuming that there are \(s\) seasons in a year. Then the pure seasonal dummy model is:

$$ { y }_{ t }=\sum _{ i=1 }^{ s }{ { \gamma }_{ i }{ D }_{ it } } +{ \varepsilon }_{ t } $$

Where \({ D }_{ it }\) indicates whether we are in the \(i\)th quarter. The regression is on an intercept whereby in each season, a different intercept is permitted. The \(\gamma\)’s are referred to as the seasonal factors. Their role is to summarize the annual seasonal pattern. We will end up with similar \(\gamma\)’s, in the event that seasonality is absent. In that case, all the seasonal dummies can be dropped and an intercept included in the usual manner.

\(s – 1\) seasonal dummies and an intercept can be included rather than a full set of \(s\) seasonal dummies being included. The seasonal increment or decrement relative to the omitted season is determined by the coefficients on the seasonal dummies, while the omitted season’s intercept is given by the constant term.

We can also include a trend, and the model will change to:

$$ { y }_{ t }={ \beta }_{ 1 }{ TIME }_{ t }+\sum _{ i=1 }^{ s }{ { \gamma }_{ i }{ D }_{ it } } +{ \varepsilon }_{ t } $$

We can extend the concept of seasonality to accommodate more general calendar effects. One form of calendar effect is standard seasonality. Other forms are:

- Holiday variation
- Trading-day variation

In holiday variation, the idea is that over time, there can be changes in some holidays’ dates. The best example is the Easter holiday whose date differs despite arriving at approximately the same time each year. The timing of such holidays partly affects the behavior of most series and such our forecasting models should track them. Dummy variables are used to handle holiday effects, an Easter Dummy is a good example.

In trading-day variations, the number of trading days or business days in a month is not the same. This consideration should, therefore, be taken into account when some series are modeled or forecasted.

The complete model that takes into account the likelihood or holiday or trading day variation is written as:

$$ { y }_{ t }={ \beta }_{ 1 }{ TIME }_{ t }+\sum _{ i=1 }^{ s }{ { \gamma }_{ i }{ D }_{ it } }+\sum _{ i=1 }^{ { V }_{ 1 } }{ { \delta }_{ i }^{ HD }{ HDV }_{ it }+ } \sum _{ i=1 }^{ { V }_{ 2 } }{ { \delta }_{ i }^{ TD }{ TDV }_{ it }+ } { \varepsilon }_{ t } $$

In the above equation, there are \({ V }_{ 1 }\) holiday variables denoted by \(HDV\), and \({ V }_{ 2 }\) trading-day variables denoted as \(TDV\). The ordinary least square can be a good estimation of this standard regression equation.

## Forecasting Seasonal Series

Let’s construct an \(h\)-step-ahead point forecast, \({ y }_{ T+h,T }\) at time \(T\).

The full model is:

$$ { y }_{ t }={ \beta }_{ 1 }{ TIME }_{ t }+\sum _{ i=1 }^{ s }{ { \gamma }_{ i }{ D }_{ it } } +\sum _{ i=1 }^{ { V }_{ 1 } }{ { \delta }_{ i }^{ HD }{ HDV }_{ it }+ } \sum _{ i=1 }^{ { V }_{ 2 } }{ { \delta }_{ i }^{ TD }{ TDV }_{ it }+ } { \varepsilon }_{ t } $$

At time \(T + h\):

$$ { y }_{ T+h }={ \beta }_{ 1 }{ TIME }_{ T+h }+\sum _{ i=1 }^{ s }{ { \gamma }_{ i }{ D }_{ i,T+h } } +\sum _{ i=1 }^{ { V }_{ 1 } }{ { \delta }_{ i }^{ HD }{ HDV }_{ i,T+h }+ } \sum _{ i=1 }^{ { V }_{ 2 } }{ { \delta }_{ i }^{ TD }{ TDV }_{ i,T+h }+ } { \varepsilon }_{ T+h } $$

The right side of the equation is projected on what is provided at time \(T\), to obtain the forecast.

$$ { y }_{ T+h,T }={ \beta }_{ 1 }{ TIME }_{ T+h }+\sum _{ i=1 }^{ s }{ { \gamma }_{ i }{ D }_{ i,T+h } } +\sum _{ i=1 }^{ { V }_{ 1 } }{ { \delta }_{ i }^{ HD }{ HDV }_{ i,T+h }+ } \sum _{ i=1 }^{ { V }_{ 2 } }{ { \delta }_{ i }^{ TD }{ TDV }_{ i,T+h } } $$

The unknown parameters are then replaced by the estimates, to ensure this point forecast is operational:

$$ { \hat { y } }_{ T+h,T }={ \hat { \beta } }_{ 1 }{ TIME }_{ T+h }+\sum _{ i=1 }^{ s }{ { \hat { \gamma } }_{ i }{ D }_{ i,T+h } } +\sum _{ i=1 }^{ { V }_{ 1 } }{ { \hat { \delta } }_{ i }^{ HD }{ HDV }_{ i,T+h }+ } \sum _{ i=1 }^{ { V }_{ 2 } }{ { \hat { \delta } }_{ i }^{ TD }{ TDV }_{ i,T+h } } $$

The next concern is how to form an interval forecast. The regression disturbance is assumed to be normally distributed. An interval forecast of 95% ignoring parameter estimation uncertainty is:

$$ { y }_{ T+h,T }\pm 1.96\sigma $$

We have to apply the following for internal forecast to be operational:

$$ { \hat { y } }_{ T+h,T }\pm 1.96\hat { \sigma } $$

The density forecast is another concept that will need to be formed. The assumption here is that trend regression disturbance is a normal distribution. The density forecast is written as follows, with parameter estimation uncertainty being ignored:

$$ N\left( { y }_{ T+h,T },{ \sigma }^{ 2 } \right) $$

The standard deviation of the disturbance in the trend regression is given as \(\sigma\). Therefore, the operational density forecast is given as:

$$ N\left( { \hat { y } }_{ T+h,T },{ \hat { \sigma } }^{ 2 } \right) $$