Limit theorems on a linear explosive stochastic model for time series with moving average error

Summary LetX(t) be a linear autoregressively generated explosive time series, with autoregressive coefficientsb ₁,…,b_q, and a constant termb ₀, and an error term ; a₀=1. Where ε(t),t≧1 are independent, Eε(t)=0, and Eε ²(t)=σ² is positive and finite. In this paper two categories of -consisent and asymptotically singularly normal estimators are proposed for (b ₁,…,b_q, b₀) thus settling an open problem since the publication of the paper (Venkataraman [5]). Based on these estimators several additional limit theorems based on estimated error residuals are proved. The parameter-free limit theorems of Spectral and Quenouille types of this paper serve as asymptotic goodness of fit tests for the model generatingX(t).