Estimating the Time Series Model Parameters

Estimating the model parameter values is in general performed using one of two methods the method of moments leading to the Yule-Walker equations or the maximum-likelihood method. Although the Yule-Walker equations are simpler, they only provide an efficient estimator for autoregressive models. Also, the Yule-Walker equations are useful for estimating the partial autocorrelation function. Least-squares estimates are also possible, but they are difficult to solve analytically due to the complex nature of the models. 

1 Yule-Walker Equations for Estimating an Autoregressive Model 

Multiply this equation by y, y, i. y, and take the expectation of the resulting p + equations to give 

Re-arranging this system of equations into matrix form gives 

5 Modelling Stochastic Processes with Time Series Analysis 

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