Bootstrapping Some Statistics Useful in Identifying ARMA Models

Consider a zero mean weakly stationary stochastic process with a continuous and nonzero spectral density function, which satisfies the stochastic difference equation (1)$$ {X_t} = \sum\limits_{{j = 1}}^{\infty} {{a_j}{X_{{t - j}}} + {\varepsilon_t}} $$ for tεZ. We assume that the associated power series $$ A(z) = 1 - \sum\nolimits_{{j = 1}}^{\infty} {{a_j}{z^j}} $$ converges and is nonzero for |z| ≤ 1. The random variables ε_t are assumed to be independently and identically distributed according to an unknown distribution function F with Eε_t = 0 and $$ E\varepsilon_t^2 = {\sigma^2} > 0 $$.