350—Chapter 11. Series

3.Compute the seasonal indices. For monthly series, the seasonal index im for month m is the average of dt using observations only for month m . For quarterly series, the seasonal index iq for quarter q is the average of dt using observations only for quarter q .

4.We then adjust the seasonal indices so that they add up to zero. This is done by set-

ting sj = ij − i where i is the average of all seasonal indices. These s are the reported scaling factors. The interpretation is that the series y is sj higher in period

jrelative to the adjusted series.

5.The seasonally adjusted series is obtained by subtracting the seasonal factors sj from yt .

The main difference between X11 and the moving average methods is that the seasonal factors may change from year to year in X11. The seasonal factors are assumed to be constant for the moving average method.

Exponential Smoothing

Exponential smoothing is a simple method of adaptive forecasting. It is an effective way of forecasting when you have only a few observations on which to base your forecast. Unlike forecasts from regression models which use fixed coefficients, forecasts from exponential smoothing methods adjust based upon past forecast errors. For additional discussion, see Bowerman and O’Connell (1979).

To obtain forecasts based on exponential smoothing methods, choose Proc/Exponential Smoothing. The Exponential Smoothing dialog box appears:

You need to provide the following information:

•Smoothing Method. You have the option to choose one of the five methods listed.

•Smoothing Parameters. You can either specify the values of the smoothing parameters or let EViews estimate them.

To estimate the parameter, type the letter e (for estimate) in the edit field. EViews estimates the parameters by minimizing the sum of squared errors. Don't be surprised if the

352—Chapter 11. Series

(1979) suggest that values of α around 0.01 to 0.30 work quite well. You can also let EViews estimate α to minimize the sum of squares of one-step forecast errors.

Double Smoothing (one parameter)

This method applies the single smoothing method twice (using the same parameter) and is appropriate for series with a linear trend. Double smoothing of a series y is defined by the recursions:

Dt = αSt + ( 1 − α) Dt − 1

where S is the single smoothed series and D is the double smoothed series. Note that

double smoothing is a single parameter smoothing method with damping factor

0 < α ≤ 1 .

Forecasts from double smoothing are computed as:

The last expression shows that forecasts from double smoothing lie on a linear trend with intercept 2ST − DT and slope α( ST − DT) ⁄ ( 1 − α) .

Holt-Winters—Multiplicative (three parameters)

This method is appropriate for series with a linear time trend and multiplicative seasonal variation. The smoothed series yˆ t is given by,

where

apermanent component (intercept)

These three coefficients are defined by the following recursions:

354—Chapter 11. Series

where 0 < α, β, γ < 1 are the damping factors. This is an exponential smoothing method with two parameters.

Forecasts are computed by:

These forecasts lie on a linear trend with intercept a( T ) and slope b( T) .

It is worth noting that Holt-Winters—No Seasonal, is not the same as additive or multiplicative with γ = 0 . The condition γ = 0 only restricts the seasonal factors from changing over time so there are still (fixed) nonzero seasonal factors in the forecasts.

Illustration

As an illustration of forecasting using exponential smoothing we forecast data on monthly housing starts (HS) for the period 1985M01–1988M12 using the DRI Basics data for the period 1959M01–1984M12. These data are provided in the workfile HS.WF1. Load the workfile, highlight the HS series, double click, select Proc/Exponential Smoothing…. We use the Holt-Winters—multiplicative method to account for seasonality, name the smoothed forecasts as HS_SM, and estimate all parameters over the period 1959M1– 1984M12.

When you click OK, EViews displays the results of the smoothing procedure. The first part displays the estimated (or specified) parameter values, the sum of squared residuals, the root mean squared error of the forecast. The zero values for Beta and Gamma in this example mean that the trend and seasonal components are estimated as fixed and not changing.