For my master thesis I estimated a linear regression. Durbin-Watson test shows, that the error terms are autocorrelated and thus, the estimator could be biased. So, what to do now? I found the Cochrane-Orcutt-Transformation but then I get new estimators.. What is the right next step?
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Cochrane-Orcutt assumes that the autocorrelation in your error term is due to the error term following an AR process. If this is your assumption, then just complete the estimation with the additional second-stage regression (with intercept) of $y_{t} - y_{t-1}\rho$ against $x_{t} - x_{t-1}\rho$ , where $\rho$ is the estimated coefficient in the AR of residuals (i.e. first-order autocorr in residuals of the first regression). For more info look here . This source is exhaustive enough.
However notice that the assumption of AR structure on residuals of the first regression may be very restrictive and maybe not the case of your data. So for a more general idea of how to solve the problem, you should likely use a ARIMAX model, where the regression allows for a more general ARIMA Error term (if error terms are linearly autocorrelated) or a regression with Garch Error term (if their squares are autocorrelated). For the first look here and here for the second look here. If you find it useful for clarifications also see this.
Clearly this will give you more flexibility in the choice of the assumed structure of dependencies between residuals. You should choose between the 3 (or more) alternatives by following the typical model specification rules: in this case, you have to choose the residual structure that most closely resembles the actual distribution of your first-stage regression residuals (i.e., to be more precise and statistically correct, the one that best removes the dependencies in the standardized innovations in your final MLE model if you use ARIMAX or GARCH regression).
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$\begingroup$ Note 1: ARIMAX and regression with ARIMA errors are not the same; see "The ARIMAX model muddle" by Rob J. Hyndman (also referred to in multiple earlier posts on CV). Note 2: Cochrane-Orcutt uses AR(1) specifically, not a more general AR(p). $\endgroup$ Commented Aug 21, 2019 at 20:34
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$\begingroup$ Note 3: Serial correlation in the residuals does not automatically imply that an estimator dealing with such serial correlation is the best fix. Serial correlation might also arise from, e.g., omitted variables such as quadratic terms. $\endgroup$ Commented Aug 22, 2019 at 11:10
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$\begingroup$ @RichardHardy Hi Richard! Great fan of you as I always say. On point one: yes, there is a difference but, look, I can’t write answers of million of characters. I just wanted to push the message that C-O might be too restrictive on the assumptions about the Error term and offer some links, that I judged sufficient and I had available very quickly.. just to make it clear that an AR is not necessarily the best fit. On point two: clearly AR1 yes, but I need to be short sometimes, again I just wanted to highlight that not necessarily it’s a AR (nor it is a AR1 necessarily, as you said). Ok $\endgroup$– Fr1Commented Aug 23, 2019 at 2:45
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$\begingroup$ Hey Fri, don't take the comment as a criticism, rather as a supplement. I would stress that precise language is pretty important in statistics, hence I added the notes to clarify a couple of points. Keep up the good work! $\endgroup$ Commented Aug 23, 2019 at 5:58
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$\begingroup$ No problem @RichardHardy , I just added a disclaimer to apologize for some things that could have been better expressed. Your comments are always welcome on my side. Yours is good work! $\endgroup$– Fr1Commented Aug 27, 2019 at 18:40