The regression forecast is based on the assumption of a model
consisting of a constant and a linear trend.
For the purposes of a forecast where the parameters of the
model may change, it is more convenient to express the model
as a function of ,
is the positive displacement from a reference time T.
The forecast is based on estimated parameters.
The parameters at time T are computed from the observation
at time T and the previous m-1 observations:
Using these m observations, we find the linear equation
that minimize thes sum of squares of the difference of the observations
from the fitted line. The values of the indices, -k,
are the independent variables for the simple regression. The
values of the observations, ,
are the dependent variables. The following parameter estimates
are based on the least squares normal equations
for fitting a linear equation.
The forecast for the expected value for future periods is a
constant plus a linear term that depends on the number of periods
into the future.
With a trend estimate as part of the forecast, this method
will track changes in trend. We use the same data as for the
other forecasting methods. We repeat the data below. Recall
that the simulated data begins with a constant mean of 10. At
time 11 the mean increases with a trend of 1 until time 20 when
the mean becomes a constant again with value 20. The noise is
simulated using a normal distribution with mean 0 and standard