"Frontmatter". In: Analysis of Financial Time Series

40 LINEAR TIME SERIES ANALYSIS AND ITS APPLICATIONSˆr h (1) = E(r h+1 | r h , r h−1 ,...)= φ 0 +and the associated forecast error ise h (1) = r h+1 −ˆr h (1) = a h+1 .p∑φ i r h+1−iConsequently, the variance of the 1-step ahead forecast error is Var[e h (1)] =Var(a h+1 ) = σ 2 a .Ifa t is normally distributed, then a 95% 1-step ahead intervalforecast of r h+1 is ˆr h (1) ± 1.96 × σ a . For the linear model in Eq. (2.4), a t+1 is alsothe 1-step ahead forecast error at the forecast origin t. In the econometric literature,a t+1 is referred to as the shock to the series at time t + 1.In practice, estimated parameters are often used to compute point and intervalforecasts. This results in a conditional forecast because such a forecast does nottake into consideration the uncertainty in the parameter estimates. In theory, one canconsider parameter uncertainty in forecasting, but it is much more involved. Whenthe sample size used in estimation is sufficiently large, then the conditional forecastis close to the unconditional one.2-Step Ahead ForecastNext consider the forecast of r h+2 at the forecast origin h. From the AR(p) model,we havei=1r h+2 = φ 0 + φ 1 r h+1 +···+φ p r h+2−p + a h+2 .Taking conditional expectation, we haveˆr h (2) = E(r h+2 | r h , r h−1 ,...)= φ 0 + φ 1 ˆr h (1) + φ 2 r h +···+φ p r h+2−pand the associated forecast errore h (2) = r h+2 −ˆr h (2) = φ 1 [r h+1 −ˆr h (1)]+a h+2 = a h+2 + φ 1 a h+1 .The variance of the forecast error is Var[e h (2)] =(1 + φ1 2)σ a 2 . Interval forecastsof r h+2 can be computed in the same way as those for r h+1 . It is interesting to seethat Var[e h (2)] ≥ Var[e h (1)], meaning that as the forecast horizon increases theuncertainty in forecast also increases. This is in agreement with common sense thatwe are more uncertain about r h+2 than r h+1 at the time index h for a linear timeseries.Multistep Ahead ForecastIn general, we haver h+l = φ 0 + φ 1 r h+l−1 +···+φ p r h+l−p + a h+l .