I want to model the relationship between two (time-series) variables by using a Vector Autoregressive model (VAR). Since I am not entirely familiar with time series analysis yet, the following question came up in my mind: Is it a good idea to simply model the relationship of the variables by using OLS? I thought it might be a good starting point when writing a text in which I analyze relationships of certain variables, before diving into the more serious analysis. Do you guys think it is a good idea? Or is it rather a no-go (I am asking this since I once heared that OLS is not sophisticated enough for time series analysis)? Do I have to carefully consider something before applying OLS? Thanks in advance!
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$\begingroup$ Estimating VAR models with ordinary least squares is a commonplace, perfectly acceptable practice in finance and economics. If you're willing to declare that your error terms follow a particular distribution, you can also go the route of maximum likelihood estimation. $\endgroup$– Matthew GunnCommented May 24, 2017 at 22:50
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$\begingroup$ @MatthewGunn: Thanks for your comment, it is really helpful! $\endgroup$– KumaCommented May 25, 2017 at 10:38
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$\begingroup$ Even more than that, if typical VAR assumptions are met (i.e., each equation has the same regressors, the errors are mean independent of the lagged variables - i.e., you got the dynamics right), OLS is even the efficient systems estimator. $\endgroup$– Christoph HanckCommented May 26, 2017 at 10:47
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$\begingroup$ Okay I didn't know this is true. Thanks for clarifying that! I will apply OLS then. $\endgroup$– KumaCommented May 26, 2017 at 14:08
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1 Answer
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There are time series models (such as VAR, ARIMA, etc.) and there are estimation techniques (such as OLS, maximum likelihood (ML), etc.). Different models can be estimated by different techniques (sometimes more than one). E.g. a VAR can be estimated by OLS or ML while ARIMA (with a nonempty MA part) cannot be estimated by OLS but can be estimated by ML.
When modelling some data, you need to choose a sensible model and then estimate it. The choice of a sensible model may be hard, I would say, much harder than the estimation of the chosen model. But once the choice is done, you proceed to estimation. If you choose a VAR, then you can estimate it by OLS. Indeed, as Matthew Gunn says, Estimating VAR models with ordinary least squares is a commonplace, perfectly acceptable practice in finance and economics. And as Christoph Hanck correctly adds, if typical VAR assumptions are met (i.e., each equation has the same regressors, the errors are mean independent of the lagged variables - i.e., you got the dynamics right), OLS is even the efficient systems estimator.
Thus the statement OLS is not sophisticated enough for time series analysis is simply not true in general.
answered Jun 25, 2017 at 19:45
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1$\begingroup$ @Kuma, thanks! A lot of credit goes to Matthew Gunn and Christoph Hanck, of course. $\endgroup$ Commented Jun 26, 2017 at 18:55
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$\begingroup$ But isn't the assumption of strict exogenity violated in, say, AR model and doesn't this then mean that OLS estimate is biased? Thank you $\endgroup$ Commented Dec 18, 2019 at 19:25
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$\begingroup$ @Confounded, why don't you post it as a separate question? $\endgroup$ Commented Dec 18, 2019 at 19:51
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$\begingroup$ Because my question (or rather, point) is directly related to the OP, in my opinion. I believe, the bias of OLS in AR is a well-know issue. An demonstration of the phenomenon is given, for example, here: Asatoshi Maeshiro (2000) "An Illustration of the Bias of OLS for $Y_t = λY_{t-1} + U_t$", The Journal of Economic Education, 31:1, 76-80. $\endgroup$ Commented Dec 19, 2019 at 12:32
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$\begingroup$ OK, if you only want a brief answer, then yes, the estimator is biased. (It is still consistent, i.e. asymptotically unbiased, and asymptotically normal.) A follow up question would be, is there a better estimator? Shrinkage estimators are usually good in terms of mean squared error and similar metrics, but they are typically even more biased in small samples. $\endgroup$ Commented Dec 19, 2019 at 12:49