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I'm pretty sure that I'm missing something obvious here, but I'm rather confused with different terms in the time series field. If I understand it correctly, serially autocorrelated errors are a problem in regression models (see for example here). My question is now what exactly defines an autocorrelated error? I know the definition of autocorrelation and I can apply the formulas, but this is more a problem of comprehension with time series in regressions.

For example, let's take the time series of daily temperatures: If it is a hot day today (summer time!), it's probably hot tomorrow as well, and vice versa. I guess I have a problem to call this phenomenon a phenomenon of "serially autocorrelated errors" because it just doesn't strike me as an error, but as something expected.

More formally, let's assume a regression set-up with one dependent variable $y_t$ and one independent variable $x_t$ and the model.

$$ y_t = \alpha + \beta x_t + \epsilon_t $$

Is it possible that $x_t$ is autocorrelated, while $\epsilon_t$ is i.i.d? If so, what does that mean for all that methods that adjust standard errors for autocorrelation? Do you still have to do that or do they only apply to autocorrelated errors? Or would you always model the autocorrelation in such a setting in the error term, so it basically doesn't make a difference if $x_t$ is autocorrelated or $e_t$?

This is my first question here. I hope it's not too confusing and I hope I didn't miss anything obvious...I also tried to google it and found some interesting links (for instance, here on SA), but nothing really helped me.

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Is seems to me that you are getting hung up on the difference between autoregression (temperature today is influenced by temperature yesterday, or my consumption of heroin today depends on my previous drug use) and autocorrelated errors (which have to do with the off-diagonal terms in variance-covariance terms for $\epsilon$ being non-zero. Sticking with your weather example, suppose you model temperature as a function of time, but it is also influenced by things like volcanic eruptions, which you left out of your model. The volcano sends up clouds of dust, which block out the sun, lowering the temperature. This random disturbance will persist over more than one period. This will make your time trend appear less steep than it should be. To be fair, it is probably the case that both autoregression and autocorrelated errors are an issue with temperature.

Autocorrelated errors can also arise in cross-sectional spatial data, where a random shock that affects economic activity in one region will spill over to other areas because they have economic ties. A shock that kills grapes in California will also lower sales of beef from Montana. You can also induce autocorrelated disturbances if you omit a relevant and autocorrelated independent variable from your time-series model.

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  • $\begingroup$ Thank you very much, Dimitriy. You got it right: I got confused about the difference between autoregression and autocorrelated errors. Just to make sure, though: In my example, I would model $x_t$ as an autoregressive time series (abstracting from volcanic eruptions, etc.) because of summer and winter times and then wouldn't have to deal with autocorrelated errors? $\endgroup$ – Christoph_J Apr 10 '12 at 7:44
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    $\begingroup$ @Christoph_J Ideally you want to regress against one ore more time lags for the seasonal pattern and volcanic activity. If instead we were ignoring the cause for the autocorrelated errors a moving average model can help. In this case it would be an ARIMA model. $\endgroup$ – Robert Kubrick Apr 10 '12 at 11:40
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    $\begingroup$ @Christoph_J I am not sure I understand your question. Did you mean to write $y_{t}$ above? You should also tell us more about the actual problem you are dealing with. My temperature example was just a toy model to highlight the issues. There are several solutions to deal with AR, the easiest of which is the Koyck distributed lag specification, which boils down to estimating simple equation with an $MA(1)$ error term. However, you should still conduct some sort of autocorrelation test, like the Durbin-Watson, though that can give you a false positive if you don't get the specification right. $\endgroup$ – Dimitriy V. Masterov Apr 10 '12 at 13:23
  • $\begingroup$ Thanks to both of you. @DimitriyV.Masterov At this point, I don't have an actual problem. That's the reason I tried to frame my problem as general as possible. I think that I just struggle with time series on the one hand and regressions on the other hand. Sometimes they seem to be two completely different issues; if I get it right there are cases in which you just try to model a time series (how many lags does it have have? is it stationary? etc.). At the other extreme, sometimes you just seem to regress a time series on the other, without paying much attention on the fact that it's a TS. $\endgroup$ – Christoph_J Apr 10 '12 at 13:47
  • $\begingroup$ And I just have sometimes some problems what is the best way going forward: Do I have to model the autoregressive process first or can I just correct for autocorrelation in the error terms? However, as far as my question is concerned, your and Robert's answer helped a lot and I think that in my field (factor models in finance) should deal with serially autocorrelated errors, not with autoregression. If another question is coming up, I would ask a new question. $\endgroup$ – Christoph_J Apr 10 '12 at 13:51
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Just to add up to Dimitriy very good answer: error autocorrelation poses problems for the calculation of the coefficients standard error and thus the significance levels, or p-value, making the IVs selection less straightforward. $R^2$ and the F value are also affected.

Of all the assumptions in linear regression (homoscedasticity, independence of the residuals, linearity of the relationship IVs --> DV, normality of residuals) linearity and independence of the residuals are those that impact results more seriously if violated.

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