7.4.1 Forecasting Models with Seasonal Components

7.4.1 Forecasting Models with Seasonal Components

Seasonal patterns can be modeled either as deterministic or stochastic variations, or some combination thereof. The stochastic modeling approach uses differencing to incorporate seasonal components. Intuition for this is that if a variable varies strongly at the seasonal frequency, one way to model its dynamics is to consider year-onyear changes in the variable. For example, corporate earnings or dividends are often compared with the figure in the same quarter during the previous year.

Box and Jenkins (1970) considered seasonal ARIMA, or SARIMA, models of the form

$$\varphi (L)(1-L^S)y_t=\theta (L){\epsilon }_t.\text{\hspace{1em}(7.32)}$$

Here S denotes the seasonal frequency. With quarterly data, $S=4,$ with monthly data $S=12$ , with weekly data $S=52$ , and so forth, so that applying the seasonal differencing operator $(1-L^S)$ means that the dynamics in year-on-year changes are

being modeled in (7.32). Other than this, the mechanics of computing forecasts does not change.

Another way to write seasonal models is to convert the linear time indicator $t=1,\dots ,T$ into a seasonal indicator such as 2000:01, 2000:02, . . . , 2000:04, 2001:01, etc., in the quarterly case:

$$\varphi (L)(1-L^S){\gamma }_{S\tau +s}=\theta (L){\epsilon }_{S\tau +s}.\text{\hspace{1em}(7.33)}$$

Here $\tau =0,\dots ,T-1$ counts years, while $s=1,\dots ,s$ counts seasons, and $t=$ $S\tau +s$ . For example, if $S=4_{:}$ , corresponding to quarterly data, and $s=1$ , then as τ varies from $0,1,2,\dots ,T-1$ , first-quarter observations $y_1,y_5,y_9,...,y_{(T-1)S+1}$ are selected.

The model in (7.32) has a unit root, but the year-on-year changes will be stationary provided that the roots of $\varphi (L)$ lie outside the unit circle and the sum of squared coefficients of $\theta (L)$ is bounded. The autoregressive polynomial $\varphi (L)$ could have additional unit roots, although this is usually found only empirically in strongly trending series. For example, the so-called airline model of Box and Jenkins (1970) takes the form

$$(1-L)(1-L^S)y_t=(1-{\theta }_{1}L)(1-{\theta }_S{L}^S){\epsilon }_t.\text{\hspace{1em}(7.34)}$$

This model allows for seasonal variation in both the AR and the MA part.

Deterministic seasonal components can easily be incorporated by assuming that once a seasonal deterministic component, ${\mu }_{S\tau +s}$ , has been taken out of $y_{S\tau +s}$ , the remaining stochastic component ${\tilde{y}}_{S\tau +s}=y_{S\tau +s}-{\mu }_{S\tau +s}$ , follows an ARMA process:

$$\begin{array}{r}{y}_t={\mu }_t+{\tilde{y}}_t,\\ \\ \varphi (L){\tilde{y}}_t=\theta (L){\epsilon }_t,\end{array}\text{\hspace{1em}(7.35)}$$

where ${\epsilon }_t\sim \mathrm{WN}(0,{\sigma }^2)$ . The simplest approach for modeling seasonal variation in the mean, ${\mu }_t$ , is to apply S seasonal dummies, each of which equals 1 in season s and is 0 otherwise:

$$D_{1t}=\{\begin{array}{ll}1& \mathrm{if}s=1\mathrm{at}\mathrm{time}t,\\ 0& \mathrm{otherwise}.\end{array}$$

For example, with quarterly data there will be four seasonal dummies of the form (assuming the initial point occurs in the first quarter)

$$\begin{array}{rl}& D_{1t}=\left(10000100010001000\right),\\ & D_{2t}=\left(010001001000100\right),\\ & D_{3t}=\left(001000100010010\right),\\ & D_{4t}=\left(00001000010001\right).\end{array}$$

The ARMA model with seasonal dummies takes the form

$$\varphi (L)(y_t-\sum _{s=1}^S{\delta }_s{D}_{st})=\theta (L){\epsilon }_t.\text{\hspace{1em}(7.36)}$$

Application of such seasonal dummies can sometimes yield large improvements in predictive accuracy.

Seasonal patterns are examples of regularly occurring patterns linked to the calendar. These can induce patterns in the mean or volatility of the distribution of economic variables such as volume on the stock exchange during the day (volume tends to be highest during the first and last few minutes of the day), credit card transactions, road congestion, and energy usage.

Example 7.4.1 (Modeling seasonal components). Day-of-the-week dummies, monthof-the-year dummies, and dummies for holidays tend to be very important when modeling electricity demand. Following Diebold (2007), this suggests a deterministic component of the form

$${\mu }_t=\sum _{\mathrm{day}=1}^7{\beta }_{\mathrm{day}}{D}_{\mathrm{day},t}+\sum _{\mathrm{holiday}=1}^H{\beta }_{\mathrm{holiday}}{D}_{\mathrm{holiday},t}+\sum _{\mathrm{month}=1}^{12}{\beta }_{\mathrm{month}}{D}_{\mathrm{month},t},\text{\hspace{1em}(7.37)}$$

where H is the number of holidays. Holiday dummies could include a Christmas dummy that equals 1 on December 25, otherwise is 0, a Thanksgiving dummy that equals 1 on Thanksgiving, otherwise equals 0, etc. Such dummies can be used to capture the typical (average) electricity consumption on a given day. We have included daily and monthly dummies in addition to a holiday dummy since electricity consumption typically is very different on public holidays such as Christmas or Thanksgiving. Interaction terms such as $\begin{array}{r}{\sum }_{\mathrm{day}=1}^7{\beta }_{\mathrm{day},\mathrm{holiday}}{D}_{\mathrm{day},t}{D}_{\mathrm{holiday},t}\end{array}$ can also be included to account for differences in electricity consumption patterns for holidays occurring on different weekdays. Subtracting the deterministic seasonal term from the ARMA model, future demand at time $t+h$ can be predicted using a model of the form

$$\varphi (L)(y_{t+h}-{\mu }_{t+h})=\theta (L){\epsilon }_{t+h},\text{\hspace{1em}(7.38)}$$

so that the deseasonalized component $y_{t+h}-{\mu }_{t+h}$ follows an ARMA process. Assuming that we have estimates of $\{{\beta }_{\mathrm{day}},{\beta }_{\mathrm{holiday}},{\beta }_{\mathrm{month}}{\}}_{\mathrm{\Omega }}$ , it is straightforward to compute an estimate of the deterministic component of $y_{t+h}$ . This can then be added to the ARMA forecast of ${\tilde{y}}_{t+h}$ given the available information at time t. See Diebold (2007) and Ghysels, Osborn, and Rodrigues (2006) for further discussion of forecasting models with seasonal components.