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I understand that the AR models can be estimated with both OLS and MLE, since all the values of the time series are known. However, I don't understand how the parameters for the MA parts can be estimated with MLE, as I found in literature. The error terms are unknown, so there aren't any data to maximise to find the most likely parameter. Any help to explain this point will be appreciated.

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    $\begingroup$ See this question ARIMA Estimation by Hand answers your question on MA component using MLE. $\endgroup$
    – forecaster
    Commented Jul 5, 2020 at 15:56
  • $\begingroup$ Did my comment help? $\endgroup$
    – forecaster
    Commented Jul 6, 2020 at 15:23
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    $\begingroup$ This is standard time-series textbook material. Try Hamilton "Time Series Analysis" or almost any other time series textbook at a sufficiently high technical level. There are also plenty of lecture notes available online discussing the question. $\endgroup$ Commented Jul 8, 2020 at 18:52

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