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

If the variable of interest is a time series, then naturally
it is important to identify and fit any systematic time patterns which may be
present. Consider again the
variable X1 that was analyzed on the page for the mean model, and suppose
that it is a time series. Its graph
looks like this:

(The file containing this data and the models below can be found here.) There is indeed a suggestion of a time
pattern, namely that the local mean value appears somewhat higher at the end of
the series than at the beginning.
There are several ways in which a change in the mean over time could be
modeled. Possibly it underwent a
“step change” at some point.
In fact, the sample mean of the first 15 values of X1 is 32.3 with a
standard error of 2.6, and the sample mean of the last 15 values is 44.7 with a
standard error of 2.8. If 95%
confidence intervals for these two means are calculated (approximately) by
adding or subtracting two standard errors, the intervals do not overlap, so the
difference in means is statistically very significant. If there is independent evidence
for a sudden change in the mean in the middle of the sample, then it might make
sense to break up the data into subsets or else fit a regression model with a
dummy variable whose value is equal to zero up the point at which the change
occurred and equal to 1 afterward.
The estimated coefficient of such a variable would measure the magnitude
of the change.

Another possibility is that the local mean is increasing gradually over
time, i.e., that there is a constant trend. If that is the case, then it might be
appropriate to fit a sloping line rather than a horizontal line to the entire
series. This is a** linear trend model**, also known as a **trend-line model**. It is a special case of a simple
regression model in which the independent variable is just a time index
variable, i.e., 1, 2, 3, ... or some other equally spaced sequence of
numbers. When it is estimated by
regression, the trend line is the unique line that minimizes the sum of squared
deviations from the data, measured in the vertical direction. (More information about this and other
properties of regression models is provided in the regression pages on this
web site.)

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 use the trendline options to display R-squared and the estimated slope and
intercept, but no other numerical output, as shown here:

The
intercept of the trend line (the point at which the line crosses the y-axis) is
30.5 and its slope (the increase per period) is 0.516. More detail can be
obtained by fitting the regression model using statistical software such as RegressIt. Here is some of the standard output that
is provided by RegressIt, including 50% confidence bands around the regression
line:

(The time
index variable was named T in this data set.) R-squared for this model is 0.143,
which means that the variance of the regression model's errors is 14.3% less
than the variance of the mean model's errors, i.e., the model has
“explained” 14.3% of the variance in X1. *Adjusted* R-squared, which is 0.112, is the fraction by which *the square of the standard error of the
regression *is less than the variance of the mean model's errors, and it is
an *unbiased* measure of the fraction
of variance that has been explained.
(See this page
for a more thorough discussion of R-squared and adjusted R-squared.)

So, the
linear trend model does improve a bit on the mean model for this time
series. Is the improvement
statistically significant? To help
answer that question, we can look at the t-statistic of the slope coefficient,
whose value is 2.16, and its associated P-value, which is 0.039. These statistics indicate that the
estimated slope is different from zero at (better than) the 0.05 level of
significance, so the model passes that conventional test, but not by a lot.

If the
objective of the analysis is to forecast what will happen next, the most
important issue in comparing the models is the extent to which they make
different predictions. Here is a
table and chart of the forecast that the linear trend model produces for X1 in
period 31, with 50% confidence limits:

And here
is the corresponding forecast produced by the mean model:

Notice
that the mean model’s point forecast for period 31 (38.5) is almost the
same as the lower 50% limit (38.2) for the linear trend model’s
forecast. Roughly speaking, the
mean model predicts that there is a 50% chance of observing a value less than
38.5 in period 31, while the linear trend model predicts that there is only a
25% chance of this happening.

Which
model should be chosen? The data
argues in favor of the linear trend model, although consideration should also
be given to the question of whether it is *logical* to assume that this
series has a steady upward trend (as opposed, say, to no trend or a randomly
changing trend), based on everything else that is known about it. The trend that has been estimated from
this sample of data is statistically significant but not overwhelmingly so.

Here
is a graph of another variable, X2, which exhibits a much stronger upward
trend:

If a
linear trend model is fitted, the following results are obtained, with 95%
confidence limits shown:

R-squared
is 92% for this model! That means
it is very good, right? Well,
no. The straight line does not
actually do a very good job of capturing the fine detail in the time
pattern. Here is a plot of the
errors (“residuals”) of the model versus time:

It is seen
here (and was also evident on the regression line plot, if you look closely)
that **the linear trend model for X2 has a
tendency to make an error of the same sign for many periods in a row**. This
tendency is measured in statistical terms by the **lag-1 autocorrelation** and **Durbin-Watson
statistic**. If there is no time
pattern, the lag-1 autocorrelation should be very close to zero, and the
Durbin-Watson statistic ought to be very close to 2, which is not the case
here. 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. The linear trend model obviously fails the
autocorrelation test in this case.

If we are
interested in using the model to predict the *future*, the fact that 8 out its last 9 errors have been positive and
they appear to be getting worse is cause for concern. Here is a chart of the predictions,
together with the forecast and 95% confidence interval for period 31. The forecast clearly appears to be too
low, given what X2 has been doing lately and given that in the past it did not
show a tendency to quickly return to the regression line after wandering away
from it.

For this
time series, a better model would be a **random-walk-with-drift
model**, which merely predicts that the next period’s value will be the
same as the current period’s value, plus a constant. The standard deviation of the errors
made by the random-walk-with-drift model is simply the standard deviation of
the period-to-period change (the so-called “first difference”) of
the variable, which is 1.75 for X2.
This is significantly less than the standard error of the regression for
the linear trend model, which is 2.28. The random-walk-with-drift model
would predict the value of X2 in period 31 to be slightly above its observed
value in period 30, which seems more realistic here.

Although
trend lines have their uses as visual aids, they are often poor for purposes of
*forecasting* outside the historical range of the data. Most time series
that arise in nature and economics do not behave as though there are straight
lines fixed in space to which they want to return some day. Rather, their
levels and trends undergo evolution. 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 at the very end of
the time series (the “business end” as I like to call it), where
the forecasting action is! 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 more
discussion of the linear trend model, and its comparison to the mean model for
another data sample, see pages 12-16 of the handout: “Review
of basic statistics and the simplest forecasting model: the mean model.” For complete details of how the slope
and intercept are estimated and how confidence limits for forecasts are
computed, see the
mathematics of simple regression page.