ARMA Autocorrelation Analysis: Parameter Estimation and Goodness of Fit Test

The celebrated Ljung-Box residual analysis is a widely used method in time series for the parameter estimation and the goodness of fit test for the ARMA time series models. The question is whether the autocorrelation function of the fitted ARMA(p,q) model for an observed time series, at different lags, in the Ljung-Box estimation method, is close to the correlogram of observed series. The answer indeed is not affirmative. In this article, firstly, we present a new procedure in solving the Yule-Walker equations for the exact computation of the autocorrelation functions of ARMA(p,q) models. Secondly, we provided a goodness of fit procedure using the limiting distribution of the the sample correlation function. Thirdly, we establish a new parameters estimation method based on examining the model autocorrelation function against the series autocorrelation coefficients. The effectiveness of the procedure is brought into sight using simulated data.