Bootstrap Approximations to Prediction Intervals for Explosive Ar(1)-Processes

Let X₀, X₁,…, X_n be observed values from some time series. An important issue then is to predict future values X_n+s from the observables. Usually, the quality of the predictor depends on how well a parametric or semiparametric model may be fitted to the data. E.g., if there is strong evidence for an AR(p)-model $$ {X_i} = {\beta_1}{X_{{i - 1}}} + \ldots + {\beta_p}{X_{{i - p}}} + {\varepsilon_i} $$ in which the errors (ε_i)_i are i.i.d. with d.f. F, zero means and finite variance, then the optimal predictor for X_n+1 under L2-loss equals $$ {\hat{X}_{{n + 1}}} = {\beta_1}{X_n} + \ldots + {\beta_p}{X_{{n + 1 - p}}} $$.