FORECAST INTERVALS

      Often, we will want more than just point forecasts f n , h of the future value xn +h . Instead, we

would like a forecast interval (A , B ) which will contain the future value with high probability. If the

time series is ARMA (p , q ) with known parameter values, we can in principle write the series in its

MA (∞) form

                                                    xt =   Σ c j εt −j
                                                           j =0

where co = 1 and {εt } is zero-mean white noise. Then

                                                  h −1                 ∞
                                       xn + h =   Σ c j εn +h −j + jΣ c j εn +h −j
                                                  j =0              =h

so the best forecast is

                                                   f n ,h =   Σ c j εn +h − j
                                                              j =h

and the forecast error is

                                                                      h −1
                                       en , h = xn +h − f n , h =     Σ c j εn +h −j
                                                                      j =0

The variance of the forecast error is

                                                                     h −1
                                              Var (en , h ) = σε2     Σ0 c j2

where σε2 = Var (εt ). Note that as the lead time h is increased, the variance of the forecast error

increases, and approaches Var (xt ): The longer the lead time, the less accurate the forecast.

      Assuming that en , h is normally distributed, a 95% forecast interval for xn +h is

                                            f n , h ± 1.96 √Var (en , h )            .

To prove this, note that

                     Prob {f n , h − 1.96 √Var (en , h ) < xn +h < f n , h + 1.96 √Var (en , h ) }

                          = Prob {−1.96 √Var (en , h ) < xn +h − f n , h < 1.96 √Var (en , h ) }

                                                          en , h
                                    = Prob {−1.96 <                    < 1.96 }
                                                      √Var (en , h )

                    = Prob {−1.96 < Standard Normal Random Variable < 1.96} = .95         .

      The correct interpretation of prediction intervals is rather tricky. First, we need to remember that

the realization of the time series that we actually got is just one of an infinite number of realizations

that we might have gotten but didn’t. Since the endpoints of the prediction interval would have been

different if we had seen a different realization, it is not correct to say that there is a 95% probability

that the given prediction interval will contain the future value. If we considered all possible realizations

and counted what proportion of the time the given, fixed prediction interval obtained from our given

data contained the future value, the answer would not be 95%. Instead, we could say that an interval of

this kind (not this one, but one obtained by this method) would contain the future value in 95% of all

possible realizations. This interpretation is very similar to the correct interpretation of confidence inter-

vals for the mean or for regression parameters.

      An easier interpretation of the prediction interval is possible if we are prepared to restrict our

attention to only those realizations which agree with the given observations. This is called the condi-

tional approach, and many forecasters find it reasonable. Then we can say that conditionally , there is a

95% probability that the future observation will fall in the given prediction interval. In other words, of

all realizations that agree with the given observations up to time n , 95% of them will produce a future

value xn +h which lies in the (fixed) prediction interval f n , h ± 1.96 √Var (en , h ).

      In practice the parameters must be estimated. We then pretend that the estimated parameters are

the actual parameters and proceed as above. The resulting prediction intervals tend to be too narrow

(optimistic). Thus, a (supposed) 95% interval based on estimated parameters will have an actual cover-

age rate which is less than 95%.

To top