5.05 Making predictions using time series data

The purpose of smoothing time series data using a moving average or deseasonalising the data is to take out the 'peaks' and 'troughs' and view the underlying trend. Often, but not always, the smoothed data will appear to be linear in nature. If it is linear in nature we can calculate the least-squares regression line for the smoothed data and use this line to make future predictions. Making future predictions with time series data is also called forecasting.

Once we have a predicted value from the underlying trend line, we will need to factor back in the season it's from. If it's from a 'peak' season then our prediction should be adjusted upwards. If it's from a 'trough' season then it should be adjusted downwards. We multiply the predicted score from the regression line by the appropriate seasonal index to adjust the predicted value.

Note:

• Moving average data or deseasonalised data can be used to find the regression line. Read the question carefully!
• Correlation has little meaning with time series data as the smoothed line will almost always have a strong, linear correlation.
• Time series predictions are almost always future predictions so we don't tend to use the words 'extrapolation' or 'interpolation' with these problems. Instead, we consider how close the predicted value is to the original data set and use the words 'within one cycle'.
• Showing working is essential in these type of problems. You should always state the value of $t$t that you are using, the equation of the regression line and the seasonal index you are multiplying with.
• Time is always the explanatory variable (plotted on the horizontal axis).

Worked example

Example 1

An aluminium window company records quarterly figures on the number of windows manufactured. For planning purposes the company management wishes to predict the number of windows that will be manufactured in March 2020.

Think: Look at the table carefully. We can see that steps 1 and 2 have been completed already.

• Step 1–the data has been smoothed. The deseasonalised data can be fitted to a linear regression model.

• Step 2–the time periods have been allocated a number ( $t=1$t=1, $t=2$t=2, $t=3$t=3 etc).

Do: Step 3–calculate the least-squares regression line. Using our calculator, we enter the time values in list 1 and our deseasonalised data in list 2, then we fit a linear regression model. Then to complete the remaining steps, we can use the linear regression model to make a prediction for March 2020, and then adjust this value to include seasonality. Review how to fit a least squares regression line using your brand of calculator here.

Do: Write down the equation $y=-3.2519t+494.4746$y=−3.2519t+494.4746 as the equation of the least squares regression line. To predict for March 2020 we first need to find the $t$t value. March 2019 was $t=9$t=9 so one cycle or four more time periods ahead means March 2020 will be $t=13$t=13. We now substitute that value into our regression line to make our deseasonalised prediction.

Now we want to adjust for seasonality. We've just found the deseasonalised value for March 2020 which is the predicted value taken from the deseasonalised values. What we're interested in is the more realistic value that includes the seasonal trend in our actual data. So we must multiply our predicted value by the appropriate seasonal index. In this case we want to multiply by the seasonal index for March.

The seasonal indices for this data are given in the table below.

So the predicted value for March 2020 is $452.1999\times0.9310=420.9981$452.1999×0.9310=420.9981. In other words, the window company predicts that $421$421 windows will be manufactured in March 2020.

Reflect: The original data as shown below contains peaks and troughs that reflect the seasonality. We use the deseasonalised values to fit a least squares line, which is then used to to make a deseasonalised prediction. To include seasonality, this prediction is adjusted using the seasonal index.

How reliable is our prediction?

When using time series data our prediction will almost always be an extrapolation. As such, we know our predictions might be somewhat unreliable due to our inability to accurately predict future events. However, in general we say that as long as our prediction has been made within one cycle of the available data, we can consider our prediction reliable.

In the example above, one cycle was four quarters, and our last available piece of data was December 2019. So any prediction made for any of the quarters in 2020 would be considered reliable as they are within one cycle since December 2019. If we started predicting for 2021 and beyond, we'd run the risk of being too inaccurate.

Practice questions

Question 1

Question 2