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MITOCW | watch?v=cDlbEQz1PQk The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation or view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. PROFESSOR: All right. Let's see. We're going to start today with a wrap up of our discussion of univariate time series analysis. And last time we went through the world representation theorem, which applies to covarient stationary processes, a very powerful theorem. And implementations of the covariant stationary processor with ARMA models. And we discussed estimation of those models with maximum likelihood. And here in this slide i just wanted to highlight how when we estimate models with maximum likelihood we need to have an assumption of a probability distribution for what's random, and in the ARMA structure we consider the simple case where the innovations, the eta t, are normally distributed white noise. So they're independent and identically distributed normal random variables. And the likelihood function can be maximized at the maximum likelihood parameters. And it's simple to implement the limited information maximum likelihood where one conditions on the first few observations in the time series. If you look at the likelihood structure for ARMA models, the density of an outcome at a given time point depends on lags of that dependent variable. So if those are unavailable, then that can be a problem. One can implement limited information maximum likelihood where you're just conditioning on those initial values, or there are full information maximum likelihood methods that you can apply as well. Generally though the limited information case is what's applied. Then the issue is model selection. And with model selection the issues that arise with time series are issues that arise in fitting any kind of statistical model. Ordinarily one will have multiple candidates for the model you want to fit to data. And the issue 1
is how do you judge which ones are better than others. Why would you prefer one over the other? And if we're considering a collection of different ARMA models then we could say, fit all ARMA models of order pq with p and q varying over some range. P from 0 up to p max, q from q up to q max. And evaluate those pq different models. And if we consider sigma tilda squared of pq being the mle of the error variance. Then there are these model selection criteria that are very popular. Akaike information criterion, and Bayes information criterion, and Hannan-Quinn. Now these criteria all have the same term, log of the mle of the error variance. So these criteria don't vary at all with that. They just vary with this second term, but let's focus first on the AIC criterion.

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• Spring '17
• Jim Angel

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