ARMA – Fitting a Non-Normal ARMA Process with OLS or MLE

armaleast squaresmaximum likelihoodnormality-assumption

If I want to fit an ARMA on data whose white noise part is non-normal is it better to estimate the model with OLS or with MLE? In other words, is OLS or MLE better for non-normal white noise in time series?

Best Answer

ARMA($p,q$) with $q>0$

ARMA($p,q$) model cannot be estimated by OLS if $q>0$, because the design matrix cannot be constructed (it includes lagged errors which are unobservable). However, MLE or conditional MLE are feasible options.

ARMA($p,q$) with $q=0$

When $q=0$, both OLS and (conditional or unconditional) MLE are feasible.
OLS estimation will be equivalent to conditional MLE with the assumption that the errors are normally distributed.

For nonnormal error distributions, OLS will be consistent, and

  • if the error distribution is correctly specified, MLE will be more efficient than OLS (however, specifying a correct error distribution might not be easy);
  • if the error distribution is misspecified, MLE might be inconsistent.

Summary

  • If $q>0$, forget OLS.
  • If $q=0$, OLS is a relatively safe choice due to consistency regardless of the error distribution.
    MLE or conditional MLE could be preferred to OLS when the error distribution can be guessed with good accuracy as then MLE would be more efficient. Otherwise OLS seems safer.
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