Statistics for Environmental Engineers

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It is also true for long lead times that:


Var[e,(€)] — Var(z,)


which says that the forecast error variance equals the variance of the z, about the mean value. This result is valid for all stationary ARMA time series processes.


Example 53.2


Compute the variance and approximate 95% confidence intervals for the lead one and lead five forecast errors for the AR(1) process in Example 53.1. The variance of the lead one forecast is dl = 0.89. The approximate 95% confidence interval is ±2 da = ±2(0.94)= ±1.88. The variance of the lead five forecast error is:


The approximate 95% confidence interval is ±2(1.33) = ±2.66. These confidence intervals are shown in Figure 53.1.

Forecasting an AR(2) Process


The forecasts for an AR(2) model are a damped sine that decays exponentially to the mean. This is illustrated with the AR(2) model:


z, = фЛ1 02Z,-2 + a,


The series of и = 100 observations in Figure 53.2 (top) is described by the fitted model zt = 0.797z,-i — 0.527zt-2 + a,. The forecasted values from origin t = 25 are shown as open circles connected by a line to define the damped sine. The forecasts converge to the mean, which is zero here because the Zt are deviations from the long-term process mean.



Forecasts from ’ origin t = 25


…….


20    40


Time


FIGURE 53.2 The top panel shows an AR(2) process fitted to a time series of size n = 100. The bottom panel shows the forecasts from origin , = 25 for this series. The forecasts converge to the mean value as a damped sine.

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