review and the simplest forecasting model: the sample mean (pdf)
Notes on the random walk model (pdf)
Mean (constant) model
Linear trend model
Random walk model
Geometric random walk model
Three types of forecasts: estimation, validation, and the future
Most naturally-occurring time series in business and economics are not at all stationary (at least when plotted in their original units). Instead they exhibit various kinds of trends, cycles, and seasonal patterns. For example, here is a time series (Series #2) which exhibits steady, if somewhat irregular, linear growth:
The mean model described above would obviously be inappropriate here. Many persons, upon seeing this time series, would naturally think of fitting a simple linear trend model--i.e., a sloping line rather than horizontal line. The forecasting equation for the linear trend model is:
where t is the time index. The parameters alpha and beta (the "intercept" and "slope" of the trend line) are usually estimated via a simple regression in which Y is the dependent variable and the time index t is the independent variable. (The equations for this are given on the introduction-to-regression page.) Here is a plot of the forecasts produced by the "Linear Trend" option in Statgraphics:
If you are plotting the data in Excel, you can just right-click on the graph and select "Add Trendline" from the pop-up menu to slap a trend line on it. (You can also display R-squared and the estimated slope and intercept, but no other numerical output.) Despite the fact that it is more appropriate than the mean model, the linear trend model is still a terrible forecasting model for Series #2. The most obvious problem is that it flunks the "eyeball test": it appears to the eye as though the next few values of the series will be slightly above the last observed value, but this is outside the 95% confidence limits for the predictions!
If we study the residual plots and diagnostic tests more carefully, we notice that the errors are severely autocorrelated: there are long runs of negative errors alternating with long runs of positive errors. The autocorrelation plot of the errors looks like this:
If the model has succeeded in extracting all the "signal" from the data, there should be no pattern at all in the errors: the error in the next period should not be correlated with any previous errors, and the bars on the autocorrelation plot therefore should all be close to the zero line. The linear trend model obviously fails the autocorrelation test in this case.
Although linear trend models have their uses, they are often inappropriate for business and economic data. Most naturally occurring business time series do not behave as though there are straight lines fixed in space that they are trying to follow: real trends change their slopes and/or their intercepts over time. The linear trend model tries to find the slope and intercept that give the best average fit to all the past data, and unfortunately its deviation from the data is often greatest near the end of the time series, where the forecasting action is! (I call this the "business end" of the time series.)
When trying to project an assumed linear trend into the future, we would like to know the current values of the slope and intercept--i.e., the values that will give the best fit to the next few periods' data. We will see that other forecasting models often do a better job of this than the simple linear trend model. (Return to top of page.)