Multi-Step Prediction

In many cases one wants to predict for several days (hours, weeks, months, e.t.c) ahead. We call that as "Multi-Step Prediction (MSP)".

A simple way to do that is by using some type of Monte Carlo simulation. The residuals $\epsilon_t$ (see expression (1.44)) are determined up to the simulation starting moment $t(s)$ using the observed data. The rest of residuals $\epsilon_t,\ t \ge t(s)$ are generated by a Gaussian distribution with zero mean and variance $\sigma^2$. The simulation is repeated $K$ times Considering external factors we just replace them by the nearest previous values (see section 1.4.1). The illustration is in Figure 1.2

The line denoted as $call.rate.actual$ shows the observed call rate. The lines $call.rate.min$, $call.rate.mean$, and $call.rate.max$ show the minimal, the average, and the maximal results of MSP predictions.

The "min" and "max" lines denote the lower and the upper values of simulation. Therefore, these lines are referred to as "MSP- confidence intervals", meaning that if the model is true, one may expect those "intervals" to cover the real data with some "MSP-confidence level" $\alpha(MSP)$. It is very difficult to define $\alpha(MSP)$ exactly. Assuming that "interval deviations" may be regarded as independent and uniformly distributed random variables, we obtain $\alpha(MSP)=1-K$. Here $K$ is the number of Monte-Carlo repetitions. In the example, $K=10$, thus $\alpha(MSP)=0.9$

Unfortunately, this assumption "over-simplifies" the statistical model. Therefore, one may regard "MSP-confidence levels" $\alpha(MSP)$ merely as a Monte-Carlo approximation.