In this research the simple linear regression (SLR) model with autocorrelated errors is considered. Traditionally, correlated errors are assumed to follow the autoregressive model of order one (AR (1)). Beside this model we will also study the SLR model with errors following the periodic autoregressive model of order one (PAR (1)). The later model is useful for modeling periodically autocorrelated errors. In particular, it is expected to be useful when the data are seasonal. We investigate the properties of the least squares estimators of the parameters of the simple regression model when the errors are autocorrelated and for various models. In particular, the relative efficiency of those estimates are obtained and compared for the white noise, AR(1) and PAR(1) models. Also, the generalized least squares estimates for the SLR with PAR(1) errors are derived. The relative efficiency of the intercept and slope estimates based on both methods is investigated via Monte-Carlo simulation. An application on real data set is also provided. It should be emphasized that once there are sufficient evidences that errors are autocorrelated then the type of this autocorrelation should be uncovered. Then estimates of model's parameters should be obtained accordingly, using some method like the generalized least squares but not the ordinary least squares.
Zusammenfassung
An important assumption of ordinary regression models is independence among errors. This research considers the case of periodically correlated errors following the periodic AR model of order 1 (PAR(1)). The remedial measure for correlated errors in regression known as the Cochran-Orcutt procedure is generalized to the case of periodically correlated errors. The motivation for making such generalizations is that the response data may inhibit some seasonality, which may not be captured by the traditional AR(1) autoregressive model. The proposed procedure is described and the bias and MSE of the resulting intercept and slope parameter estimates of the simple LR model with errors following PAR(1) are compared with those of ordinary least squares (OLS) estimates via simulation. An application of real data is provided.
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