Maximum-Likelihood Parameter Estimates for ARMA Models

2 Maximum-Likelihood Parameter Estimates for ARMA Models 

Assume that the ARMA process of interest can be written as an infinite-order autoregressive process, that is. 

5 Modelling Stochastic Processes with Time Series Analysis 

Assuming that the residuals (or e,) are normally distributed, then y = (yj, 2, , will have a multivariate normal distribution with a probability density 

The parameter estimates are then obtained by maximising either Eqs. (5.88) or (5.89). The result given by Eq. (5.89) is called the exact log-likelihood function. However, when dealing with a long time series with many data points, it will be computationally expensive to compute the determinant and inverse required by the exact log-likelihood function at every iteration. Instead, the exact probability can be 

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