Asymptotic normality in partial linear models based on dependent errors

In this paper we are concerned with the regression model y i = x i β + g ( t i ) + V i ( 1 ⩽ i ⩽ n ) under correlated errors V i = σ i e i and V i = ∑ j = - ∞ ∞ c j e i - j , where the design points ( x i , t i ) are known and nonrandom, the slope parameter β and the nonparametric component g are unknown, { e i , F i } are martingale differences. For the first case, it is assumed that σ i 2 = f ( u i ) , u i are known and nonrandom, f is unknown function, we study the issue of asymptotic normality for two different slope estimators: the least squares estimator and the weighted least squares estimator. For the second case, we consider the asymptotic normality of the least squares estimator of β . Also, the asymptotic normality of the nonparametric estimators of g ( · ) under the two cases are considered.