Asymptotic Efficiency in a Partly Autoregressive Model

Consider the model X_t = X_t-1β+g（U_t）+ ε_i for t≥ 1. Here g is an unknown function, β is an unknown parameter to be estimated and ε_j are i.i d. with mean 0 and variance σ² and U_t are i.i.d. random variables obeyed uniformly on 0, l The order of convergence of consistent estimators and the bound of asymptotic efficiency in sense of Takeuchi are given. Meanwhile we give a necessary and sufficient condition that the least squares of β is asymptotically efficient, and we also show that the MLE is asymptotically efficient.