In this paper we consider time series models belonging to the AR(autoregressive) familiy and deal with the estimation of the residual variance. This is important because estimates of the variance enter, for example, into confidence sets for the parameters of the model, in the estimation of the spectrum , in expressions for the estimated error of prediction and in sample quantities used to make inferences about the order of the model. We consider the asymptotic biases for moment and least squares estimators of the residual variance, and compare them with known results when available and with those for maximum likelihood estimators under normality. For finite samples, simulation results are presented. Key words: AR models, bias, least squares estimator, maximum likelihood estimator, moment estimator, residual variance, time series. 1. Introduction. We consider time series models belonging to the AR(p) family in which the observable stationary process fX t g has EfX t g = ¯ and finite...

We consider time series models of the MA (moving average) family, and deal with the estimation of the residual variance. Results are known for maximum likelihood estimates under normality, both for known or unknown mean, in which case the asymptotic biases depend on the number of parameters (including the mean), and do not depend on the values of the parameters. For moment estimates the situation is different, because we find that the asymptotic biases depend on the values of the parameters, and become large as they approach the boundary of the region of invertibility. Our approach is to use Taylor series expansions, and the objective is to obtain asymptotic biases with error of o(1=T ), where T is the sample size. Simulation results are presented, and corrections for bias suggested. 1. INTRODUCTION The moving average time series models of order q, denoted MA(q), is de- fined by X t \Gamma ¯ = q X k=0 ff k a t\Gammak ; t = : : : ; \Gamma1; 0; 1; : : : ; (1:1) where ff 0 = 1, X t...

We consider the ARMA(1,1) model and deal with the estimation of the residual variance. Results are known for the maximum likelihood(ML) estimators under normality, both for known and unknowm mean, in which case the asymptotic biases depend on the number of parameters(including the mean) and on the true residual variance, but not on the values of the remaining parameters. For moment and least squares estimators the situation is different: the asymptotic biases depend on the values of the parameters, besides the true variance. Some simulation results are also presented. Key words: Time series, autoregressive moving average models,residual variance, bias, maximum likelihood,method of moments, least squares. Address for Correspondence: Pedro A. Morettin University of S~ao Paulo C.P 66281 05315-970-S~ao Paulo,SP Brazil e-mail: pam@ime.usp.br 1. Introduction We consider the stationary ARMA(1,1) model (X t \Gamma ¯) + fi(X t\Gamma1 \Gamma ¯) = a t + ffa t\Gamma1 ; t = 0; \Sigma1; : : : ; (...