I want to estimate the parameters of $a_t = a_{t-1}+\theta+\epsilon_t$ using MLE. Assume errors $\epsilon_t$ are serially correlated, then how would I choose the likelihood function?

  • $\begingroup$ Do you know anything about how they are serially correlated? $\endgroup$
    – cc7768
    Feb 3, 2016 at 17:40
  • $\begingroup$ No, but let's assume a first order serial correlation $\epsilon_t = \rho\epsilon_{t-1}+e_t$ $\endgroup$
    – london
    Feb 3, 2016 at 17:49
  • $\begingroup$ What paramters are you estimating? Presumably the coefficeint on the lagged value.. $\endgroup$
    – ChinG
    Feb 3, 2016 at 22:57
  • $\begingroup$ yes, it is the parameter on $da_t$ $\endgroup$
    – london
    Feb 5, 2016 at 18:11

1 Answer 1


This is model is known an GARCH(general autoregressive conditional heteroskedasticity) where the AR component of the observable series $\{a_t\}$ is equal to unity.

The likelihood is determined by your model assumptions on the innovation terms $\varepsilon_t$ and $e_t$.

If you are looking for a canned function to estimate an GARCH see https://www.kevinsheppard.com/MFE_Toolbox. If you are looking for some more theory on how to form the likelihood see http://www.math.elte.hu/probability/markus/AlkmatTS1/ARCH_GARCH_Eload.html.

  • $\begingroup$ This is not an ARCH model. $\endgroup$
    – luchonacho
    Apr 25, 2017 at 19:41
  • $\begingroup$ Updated to reflect that it is really GARCH. My mistake. $\endgroup$ Apr 25, 2017 at 21:54

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