# MLE estimation with serially correlated errors

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?

• Do you know anything about how they are serially correlated? – cc7768 Feb 3 '16 at 17:40
• No, but let's assume a first order serial correlation $\epsilon_t = \rho\epsilon_{t-1}+e_t$ – london Feb 3 '16 at 17:49
• What paramters are you estimating? Presumably the coefficeint on the lagged value.. – ChinG Feb 3 '16 at 22:57
• yes, it is the parameter on $da_t$ – london Feb 5 '16 at 18:11

## 1 Answer

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.

• This is not an ARCH model. – luchonacho Apr 25 '17 at 19:41
• Updated to reflect that it is really GARCH. My mistake. – hipHopMetropolisHastings Apr 25 '17 at 21:54