Statistics
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:

Ŷ_{t} = α + βt

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.)

*For a more detailed
example of a linear trend model, and its comparison to the mean model for the
same data, see pages 14-16 of the handout:*** “Review
of basic statistics and the simplest forecasting model: the mean model.”**