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I know it has something to do with the errors being correlated with the variable, but I'm not sure exactly what that means. Could someone please give me a quick simple explanation about why you must use MLE over OLS?

Thank you

edit: I'm also slightly confused about the details of the LM test used in the MLE procedure. I don't really need anything too in-depth, just some details of what actually goes on in this test.

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    $\begingroup$ Exactly which time series model do you have in mind? Although OLS is out of the question, due to the hypothesized lack of independence of observations, plenty of time series models still can be fit with generalized least squares. There is no mathematical obligation to use maximum likelihood in any statistical problem whatsoever: it's just one of many procedures one could select. $\endgroup$ – whuber Apr 20 '15 at 21:47
  • $\begingroup$ Can you be more specific? What is the "LM test used in the MLE procedure" that you want to know about? I'm not sure this question is answerable in its current form. $\endgroup$ – gung - Reinstate Monica Apr 20 '15 at 22:08
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    $\begingroup$ Well, in one sense you don't have to do anything; the issue is what properties you're prepared to accept. In some circumstances it can make sense to do a form of OLS, but it depends on what time series model you mean, and OLS estimating which model in particular. For example, with a reasonably long time series, and an AR(p) model, one might condition on the first p observations and apply linear regression on lagged values of the series (i.e. OLS could be reasonable). For comments on what you're actually dealing with ("time series" is much too broad), you need to give specific details. $\endgroup$ – Glen_b -Reinstate Monica Apr 20 '15 at 22:35
  • $\begingroup$ Sorry yeah you're right I was pretty vague. I'm writing a paper for an introductory time series analysis class and we have to explain pretty much everything and these are 2 areas I was a bit confused about. My model is an AR2 and I know you can't use regular OLS because it's biased but I'm having a hard time explaining why. And MLE is the only other method we've learned. And I was talking about the Lagrange multiplier test that MLE procedure uses for it's estimates $\endgroup$ – Nick Apr 21 '15 at 0:18
  • $\begingroup$ But one is not forced to use ML instead of OLS. Take the U-MIDAS model, for example. That's a mixed-frequency time-series model and can be estimated using OLS. There are other cases, for sure. $\endgroup$ – Graeme Walsh Apr 23 '15 at 18:51
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Despite a number of good comments, one point seems to have been overlooked in the discussion. When there are unobserved or missing data, the likelihood of the model can be specified without problems, so the ML estimation is feasible. Meanwhile, regular OLS is not feasible, i.e. it does not work with unobserved or missing data (perhaps some modification of OLS could work, though).

One example is the MA(1) model:

$$x_t=\varepsilon_t+\theta \varepsilon_{t-1}$$.

The $\varepsilon_t$ is unobserved for any $t$. Thus there is no way you can run a regular OLS regression; the right-hand-side variables needed in it "are not there"; OLS is not feasible. Meanwhile, you can specify the likelihood function for this model, and maximize it (so you get the maximum likelihood). That is why ML may be more natural than OLS for some time series models.

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