Averaging and smoothing models

Notes on forecasting with moving averages (pdf)
Averaging and exponential smoothing models
Slides on inflation and seasonal adjustment and Winters seasonal exponential smoothing
Spreadsheet implementation of seasonal adjustment and exponential smoothing
Equations for the smoothing models (SAS web site)


Spreadsheet implementation of seasonal adjustment and exponential smoothing

It is straightforward to perform seasonal adjustment and fit exponential smoothing models using a spreadsheet program such as Excel. The screen images and charts below are taken from a spreadsheet which has been set up to illustrate multiplicative seasonal adjustment and linear exponential smoothing on the Outboard Marine data. To obtain a copy of the spreadsheet file itself, click here.  The version of linear exponential smoothing that will be used here for purposes of demonstration is Brown’s version, merely because it can be implemented with a single column of formulas and there is only one smoothing constant to optimize.  Usually it is better to use Holt’s version that has separate smoothing constants for level and trend.

The forecasting process proceeds as follows: (i) first the data are seasonally adjusted; (ii) then forecasts are generated for the seasonally adjusted data via linear exponential smoothing; and (iii) finally the seasonally adjusted forecasts are "reseasonalized" to obtain forecasts for the original series. The seasonal adjustment process is carried out in the first six columns.

The first step in seasonal adjustment is to compute a centered moving average (performed here in column C). This can be done by simply taking the average of two one-year-wide averages that are offset by one period relative to each other. The next step is to compute the ratio to moving average--i.e., the original data divided by the moving average in each period--which is performed here in column D. The estimated seasonal index for each season is computed by first averaging all the ratios for that particular season, and then renormalizing the ratios if necessary so that they sum to exactly 100% times the number of periods in a season, or 400% in this case. This is done in the first few rows of columns E and F.  In this case the number of rows is small, and for purposes of illustration I just manually typed the row numbers for all the "June" ratios inside the AVERAGE( ) formula in cell E3, and then I copied this formula into cells E4-E6.  (There are better, more systematic ways to average the ratios, which really ought to be used in practice to avoid formula errors.  In particular, you could add a column that contains numeric codes for seasons and then use an AVERAGEIF formula to average only the values for a particular season number.)  Cells E13-E16 contain cell references pointing to the appropriate seasonal indices computed in cells F3-F6.  (Careful:  you must be sure to line them up correctly.  In this case, the formulas in cells E13 to E16 are "=F5", "=F6", "=F3", "=F4", respectively.)  Cells E17 and below contain cell references pointing to the value in the same column 4 quarters earlier. Note that you do NOT copy the seasonal index formula into ALL the other cells in the same column--you just replicate it enough times to obtain the seasonal index once for each season in the year. Then, in the remainder of the cells in the column, you insert cell references pointing to the seasonal index used in the same season in the preceding year. The seasonally adjusted data is obtained by dividing the original data by the column of seasonal indices (here, by dividing column B by column E). The centered moving average and the seasonally adjusted data end up looking like this:

Note that the moving average typically looks like a smoothed version of the seasonally adjusted series.

The same spreadsheet also shows the application of the linear exponential smoothing model to the seasonally adjusted data, beginning in column G.

A value for the smoothing constant (alpha) is entered above the forecast column (here, in cell G9) and for convenience it is assigned the range name "Alpha." (The name is assigned using the "Insert/Name/Create" command.) The LES model is initialized by setting the first two forecasts equal to the first actual value of the seasonally adjusted series. The formula used here for the LES forecast is the single-equation recursive form of Brown’s model:


This formula is entered in the cell corresponding to the third period (here, cell G15) and copied down from there. Notice that the LES forecast for the current period refers to the two preceding observations and the two preceding forecast errors, as well as to the value of alpha. Thus, the forecasting formula in row 15 refers only to data which were available in row 14 and earlier. (Of course, if we wished to use simple instead of linear exponential smoothing, we could substitute the SES formula here instead.  We could also use Holt’s rather than Brown’s LES model, which would require two more columns of formulas to calculate the level and trend that are used in the forecast.)

The errors are computed in the next column (here, column H) by subtracting the forecasts from the actual values. The root mean squared error is computed as the square root of the variance of the errors plus the square of the mean. (This follows from the mathematical identity: MSE= VARIANCE(errors)+ (AVERAGE(errors))^2.) In calculating the mean and variance of the errors in this formula, the first two periods are excluded because the model does not actually begin forecasting until the third period (row 15 on the spreadsheet). The optimal value of alpha can be found either by manually changing alpha until the minimum RMSE is found, or else you can use the "Solver" to perform an exact minimization. The value of alpha that the Solver found is shown here (alpha=0.471154).

It is usually a good idea to plot the errors of the model (in transformed units) and also to compute and plot their autocorrelations at lags up to one season. Here is a time series plot of the (seasonally adjusted) errors:

The error autocorrelations are computed by using the CORREL( ) function to compute the correlations of the errors with themselves lagged by one or more periods--details are shown in the spreadsheet model. Here is a plot of the autocorrelations of the errors at the first five lags:

The spike at lag 4 is slightly troublesome--it suggests that the seasonal adjustment process has not been completely successful. However, it is actually only marginally significant. The 95% significant limits for autocorrelations are roughly plus-or-minus-2-over-the-square-root-of-n, where n is the sample size. Here the sample size for the autocorrelations varies from 34 to 38 depending on the lag, so the square-root-of-n is about 6, and hence the confidence limits are roughly plus-or-minus 2/6, or 0.33. If you vary the value of alpha by hand, you can observe the effect on the time series and autocorrelation plots of the errors, as well as on the mean squared error.

At the bottom of the spreadsheet, the forecasting formula is "bootstrapped" into the future by merely substituting forecasts for actual values at the point where the actual data runs out--i.e., where "the future" begins. (In other words, in each cell where a future data value would occur, a cell reference is inserted which points to the forecast made for that period.) All the other formulas are simply copied down from above:

Notice that the errors for forecasts of the future are all computed to be zero. This does not mean the actual errors will be zero, but rather it merely reflects the fact that for purposes of extrapolation we are assuming that the future data will equal the forecasts. The resulting LES forecasts for the seasonally adjusted data look like this:

Note that with this particular value of alpha--the "optimal" value for one-step-ahead prediction--the projected trend is slightly upward, reflecting the local trend that was observed over the last 2 years or so. For other values of alpha, a different trend projection might be obtained.

The final step in building the forecasting model is to "reasonalize" the LES forecasts by multiplying them by the appropriate seasonal indices. Thus, the reseasonalized forecasts in column I are simply the product of the seasonal indices in column E and the seasonally adjusted LES forecasts in column G.

It is relatively easy to compute confidence intervals for one-step-ahead forecasts made by this model: first compute the RMSE (root-mean-squared error, which is just the square root of the MSE) and then compute a confidence interval for the seasonally adjusted forecast by adding and subtracting two times the RMSE. (Remember that a 95% confidence interval is roughly equal to the point forecast "plus-or-minus-two-standard deviations," assuming the error distribution is approximately normal and the sample size is 20 or more.  Here, the RMSE is the best estimate of the error standard deviation.) The confidence limits for the seasonally adjusted forecast are then reseasonalized, along with the forecast, by multiplying them by the appropriate seasonal indices. In this case the RMSE is equal to 27.4 and the seasonally adjusted forecast for the first future period (Dec-93) is 273.2, so the seasonally adjusted 95% confidence interval is from 273.2-2*27.4 = 218.4 to 273.2+2*27.4 = 328.0. Multiplying these limits by December's seasonal index of 68.61%, we obtain lower and upper confidence limits of 149.8 and 225.0 around the Dec-93 point forecast of 187.4.

Confidence limits for forecasts more than one period ahead will generally widen as the forecast horizon increases, due to uncertainty about the level and trend as well as the seasonal factors, but it is difficult to compute them in general by analytic methods.  (The appropriate way to compute confidence limits for the LES forecast is by using ARIMA theory, but the uncertainty in the seasonal indices is another matter.)  If you want a realistic confidence interval for a forecast more than one period ahead, taking all sources of error into account, your best bet is to use empirical methods: for example, to obtain a confidence interval for a 2-step ahead forecast, you could create another column on the spreadsheet to compute a 2-step-ahead forecast for every period (by bootstrapping the one-step-ahead forecast). Then compute the RMSE of the 2-step-ahead forecast errors and use this as the basis for a 2-step-ahead confidence interval.