## 3.5 Prediction intervals

As discussed in Section 1.7, a prediction interval gives an interval within which we expect $$y_{t}$$ to lie with a specified probability. For example, assuming that the forecast errors are uncorrelated and normally distributed, a simple 95% prediction interval for the next observation in a time series is $\hat{y}_{t} \pm 1.96 \hat\sigma,$ where $$\hat\sigma$$ is an estimate of the standard deviation of the forecast distribution. In this book we usually calculate 80% intervals and 95% intervals, although any percentage may be used.

When forecasting one step ahead, the standard deviation of the forecast distribution is almost the same as the standard deviation of the residuals. (In fact, the two standard deviations are identical if there are no parameters to be estimated, as is the case with the naïve method. For forecasting methods involving parameters to be estimated, the standard deviation of the forecast distribution is slightly larger than the residual standard deviation, although this difference is often ignored.)

For example, consider a naïve forecast for the Dow-Jones Index. The last value of the observed series is 3830, so the forecast of the next value of the DJI is 3830. The standard deviation of the residuals from the naïve method is 22.00 . Hence, a 95% prediction interval for the next value of the DJI is $3830 \pm 1.96(22.00) = [3787, 3873].$ Similarly, an 80% prediction interval is given by $3830 \pm 1.28(22.00) = [3802, 3858].$

The value of the multiplier (1.96 or 1.28) determines the percentage of the prediction interval. The following table gives the values to be used for different percentages.

Table 3.1: Multipliers to be used for prediction intervals.
Percentage Multiplier
50 0.67
55 0.76
60 0.84
65 0.93
70 1.04
75 1.15
80 1.28
85 1.44
90 1.64
95 1.96
96 2.05
97 2.17
98 2.33
99 2.58

The use of this table and the formula $$\hat{y}_{t} \pm k \hat\sigma$$ (where $$k$$ is the multiplier) assumes that the residuals are normally distributed and uncorrelated. If either of these conditions does not hold, then this method of producing a prediction interval cannot be used.

The value of prediction intervals is that they express the uncertainty in the forecasts. If we only produce point forecasts, there is no way of telling how accurate the forecasts are. However, if we also produce prediction intervals, then it is clear how much uncertainty is associated with each forecast. For this reason, point forecasts can be of almost no value without the accompanying prediction intervals.

To produce a prediction interval, it is necessary to have an estimate of the standard deviation of the forecast distribution. For one-step forecasts, the residual standard deviation provides a good estimate of the forecast standard deviation. But multi-step forecasts, a more complicated method of calculation is required. These calculations are usually done with standard forecasting software and need not trouble the forecaster (unless he or she is writing the software!).

A common feature of prediction intervals is that they increase in length as the forecast horizon increases. The further ahead we forecast, the more uncertainty is associated with the forecast, and thus the wider the prediction intervals. However, there are some (non-linear) forecasting methods that do not have this property.

If a transformation has been used, then the prediction interval should be computed on the transformed scale, and the end points back-transformed to give a prediction interval on the original scale. This approach preserves the probability coverage of the prediction interval, although it will no longer be symmetric around the point forecast.

Prediction intervals are computed for you when using any of the benchmark forecasting methods. For example, here is the output when using the naïve method for the Dow-Jones index.

naive(dj2)
#>     Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
#> 251           3830  3802  3858  3787  3873
#> 252           3830  3790  3870  3769  3891
#> 253           3830  3781  3879  3755  3905
#> 254           3830  3774  3886  3744  3916
#> 255           3830  3767  3893  3734  3926
#> 256           3830  3761  3899  3724  3936
#> 257           3830  3755  3905  3716  3944
#> 258           3830  3750  3910  3708  3952
#> 259           3830  3745  3915  3701  3959
#> 260           3830  3741  3919  3694  3966

When plotted, the prediction intervals are shown as shaded region, with the strength of colour indicating the probability associated with the interval.

autoplot(naive(dj2))