Introduction to Forecasting Methods

The FORECAST Procedure

Introduction to Forecasting Methods

This section briefly introduces the forecasting methods used by the FORECAST procedure. Refer to textbooks on forecasting and see "Forecasting Methods" later in this chapter for more detailed discussions of forecasting methods.

The FORECAST procedure combines three basic models to fit time series:

time trend models for long-term, deterministic change
autoregressive models for short-term fluctuations
seasonal models for regular seasonal fluctuations

Two approaches to time series modeling and forecasting are time trend models and time series methods.

Time Trend Models

Time trend models assume that there is some permanent deterministic pattern across time. These models are best suited to data that are not dominated by random fluctuations.

Examining a graphical plot of the time series you want to forecast is often very useful in choosing an appropriate model. The simplest case of a time trend model is one in which you assume the series is a constant plus purely random fluctuations that are independent from one time period to the next. Figure 12.8 shows how such a time series might look.

Figure 12.8: Time Series without Trend

The x_t values are generated according to the equation

$x_{t}=b_{0}+{\epsilon}_{t}$

where ${\epsilon}$ _t is an independent, zero-mean, random error, and b₀ is the true series mean.

Suppose that the series exhibits growth over time, as shown in Figure 12.9.

Figure 12.9: Time Series with Linear Trend

A linear model is appropriate for this data. For the linear model, assume the x_t values are generated according to the equation

$x_{t}=b_{0}+b_{1}t+{\epsilon}_{t}$

The linear model has two parameters. The predicted values for the future are the points on the estimated line. The extension of the polynomial model to three parameters is the quadratic (which forms a parabola). This allows for a constantly changing slope, where the x_t values are generated according to the equation

$x_{t}=b_{0}+b_{1}t+b_{2}t^2+{\epsilon}_{t}$

PROC FORECAST can fit three types of time trend models: constant, linear, and quadratic. For other kinds of trend models, other SAS procedures can be used.

Exponential smoothing fits a time trend model using a smoothing scheme in which the weights decline geometrically as you go backward in time. The forecasts from exponential smoothing are a time trend, but the trend is based mostly on the recent observations instead of on all the observations equally. How well exponential smoothing works as a forecasting method depends on choosing a good smoothing weight for the series.

To specify the exponential smoothing method, use the METHOD=EXPO option. Single exponential smoothing produces forecasts with a constant trend (that is, no trend). Double exponential smoothing produces forecasts with a linear trend, and triple exponential smoothing produces a quadratic trend. Use the TREND= option with the METHOD=EXPO option to select single, double, or triple exponential smoothing.

The time trend model can be modified to account for regular seasonal fluctuations of the series about the trend. To capture seasonality, the trend model includes a seasonal parameter for each season. Seasonal models can be additive or multiplicative.

$x_{t}=b_{0}+b_{1}t+s(t)+{\epsilon}_{t} \hspace*{2in}\rm{(Additive)}$

$x_{t}=(b_{0}+b_{1}t) s(t)+{\epsilon}_{t} \hspace*{2in}\rm{(Multiplicative)}$

where s(t) is the seasonal parameter for the season corresponding to time t.

The Winters method is similar to exponential smoothing, but includes seasonal factors. The Winters method can use either additive or multiplicative seasonal factors. Like exponential smoothing, good results with the Winters method depend on choosing good smoothing weights for the series to be forecast.

To specify the multiplicative or additive versions of the Winters method, use the METHOD=WINTERS or METHOD=ADDWINTERS options, respectively. To specify seasonal factors to include in the model, use the SEASONS= option.

Many observed time series do not behave like constant, linear, or quadratic time trends. However, you can partially compensate for the inadequacies of the trend models by fitting time series models to the departures from the time trend, as described in the following sections.

Time Series Methods

Time series models assume the future value of a variable to be a linear function of past values. If the model is a function of past values for a finite number of periods, it is an autoregressive model and is written as follows:

$x_{t}=a_{0}+a_{1}x_{t-1} +a_{2}x_{t-2}+ ... +a_{p}x_{t-p}+{\epsilon}_{t}$

The coefficients a_i are autoregressive parameters. One of the simplest cases of this model is the random walk, where the series dances around in purely random jumps. This is illustrated in Figure 12.10.

Figure 12.10: Random Walk Series

The x_t values are generated by the equation

$x_{t}=x_{t-1}+{\epsilon}_{t}$

In this type of model, the best forecast of a future value is the present value. However, with other autoregressive models, the best forecast is a weighted sum of recent values. Pure autoregressive forecasts always damp down to a constant (assuming the process is stationary).

Autoregressive time series models can also be used to predict seasonal fluctuations.

Combining Time Trend with Autoregressive Models

Trend models are suitable for capturing long-term behavior, whereas autoregressive models are more appropriate for capturing short-term fluctuations. One approach to forecasting is to combine a deterministic time trend model with an autoregressive model.

The stepwise autoregressive method (STEPAR method) combines a time-trend regression with an autoregressive model for departures from trend. The combined time-trend and autoregressive model is written as follows:

x_t=b₀+b₁t+b₂t²+u_t

$u_{t}=a_{1}u_{t-1}+a_{2}u_{t-2}+ ... +a_{p}u_{t-p}+{\epsilon}_{t}$

The autoregressive parameters included in the model for each series are selected by a stepwise regression procedure, so that autoregressive parameters are only included at those lags at which they are statistically significant.

The stepwise autoregressive method is fully automatic and, unlike the exponential smoothing and Winters methods, does not depend on choosing smoothing weights. However, the STEPAR method assumes that the long-term trend is stable; that is, the time trend regression is fit to the whole series with equal weights for the observations.

The stepwise autoregressive model is used when you specify the METHOD=STEPAR option or do not specify any METHOD= option. To select a constant, linear, or quadratic trend for the time-trend part of the model, use the TREND= option.

Chapter Contents
Previous
Next
Top