<p>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. </p>

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

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

CC BY-SA 3.0
2017-04-25T19:41:27.380Z
This is not an ARCH model.
luchonacho
CC BY-SA 3.0
2017-04-25T21:54:58.300Z
Updated to reflect that it is really GARCH. My mistake.
hipHopMetropolisHastings
CC BY-SA 3.0
2017-04-25T18:52:37.353Z
2017-04-25T21:54:43.243Z
2017-04-25T21:54:43.243Z
hipHopMetropolisHastings
<p>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?</p>

CC BY-SA 3.0
2016-02-03T17:40:38.430Z
Do you know anything about how they are serially correlated?
cc7768
CC BY-SA 3.0
2016-02-03T17:49:03.780Z
No, but let's assume a first order serial correlation $\epsilon_t = \rho\epsilon_{t-1}+e_t$
london
CC BY-SA 3.0
2016-02-03T22:57:17.553Z
What paramters are you estimating? Presumably the coefficeint on the lagged value..
ChinG
CC BY-SA 3.0
2016-02-05T18:11:13.920Z
yes, it is the parameter on $da_t$
london
<macroeconomics><econometrics><statistics><applied-econometrics>
MLE estimation with serially correlated errors
london
# MLE estimation with serially correlated errors

(Asked by: username_0 on 2016-02-03)

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?

**Tags:** macroeconomics, econometrics, statistics, applied-econometrics

> **Comments:**
> **username_1**: Do you know anything about how they are serially correlated?  
> **username_0**: No, but let's assume a first order serial correlation $\epsilon_t = \rho\epsilon_{t-1}+e_t$  
> **username_2**: What paramters are you estimating? Presumably the coefficeint on the lagged value..  
> **username_0**: yes, it is the parameter on $da_t$  

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## 1 answers

### Answer by username_3 on 2017-04-25. Score: -2

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>.

> **Comments:**
> **username_4**: This is not an ARCH model.  
> **username_3**: Updated to reflect that it is really GARCH. My mistake.  

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