4.05 Forecasts

Although it is not necessary to first deseasonalise time series data before fitting a least squares regression line, it is often performed in this order. The least squares line can then be used to make predictions being mindful of the issue of extrapolation.

Examine the smoothing effect that deseasonalising the data has on the original data set:

The deseasonalised data looks much smoother than the original data which will aid in fitting a least squares regression line.

In order to substitute values into a regression line, it is first necessary to give each time period a numerical value, as shown in the table below, where for example, March 2012 is represented by the number 1.

Using a CAS calculator, it is now possible to enter the time values as the independent variable and the deseasonalised data as the dependent variable, and fit to this a linear regression model.

The equation of the least squares line is now y=494.4746-3.2519t.

This line can now be used to predict a future value.

For example, consider the value for September 2015.

The time period associated with September 2015 will be t=15. This is obtained by counting on from the last piece of available data in the table.

Substituting t=15 into the regression line to make a prediction obtains y=445.6968.

It is important to remember however, that this is the deseasonalised value for September 2015, or the smoothed value. In the graph above, this would be the value taken from the green plotted data.

To find the more realistic value which contains the seasonality component, it is necessary to simply reverse the deseasonalising process by multiplying the predicted value by the appropriate seasonal index.

Recall the seasonal indices from before:

So to finalise the prediction for September 2015 we multiply by the seasonal index for September:

\begin{aligned} y&=445.6968\times 1.0135\\ y&=445.7137 \end{aligned}

When using time series data the prediction will almost always be an extrapolation. As such, the predictions might be somewhat unreliable due to the inability to accurately predict future events. However, in general so long as the prediction has been made within one cycle of the available data, it is considered reliable.

In the example above, one cycle was four quarters, and the last available piece of data was December 2014. So any prediction made for any of the quarters in 2015 would be considered reliable as they are within one cycle of December 2014. When predicting for 2016 and beyond, there is a much higher risk of being inaccurate.

Examples