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From basic statistics and hearing "correlation is not causation" all the time, I tend to think it's fine to say that "X and Y are correlated" even if X and Y aren't in a causal relationship. For example, I'd normally think it's perfectly okay to say that ice cream sales and swimsuit sales are correlated, since high swimsuit sales probably means high ice cream sales (even though increases in swimsuit sales don't cause an increase an ice cream sales).

However, when studying time analysis, I get a little confused about this terminology. It seems like a time series analyst would not say that ice cream sales are correlated with swimsuit sales, but rather that ice cream sales are spuriously correlated with swimsuit sales. An unmodified "X is correlated with Y" seems to be reserved for the case where X actually causes Y, so it's fine to say temperature (but not ice cream) is correlated with swimsuit sales.

Is this correct? My problem is that there seem to be two meanings to spurious correlation:

  1. Regress two independent random walks against each other, and ordinary statistical tests will say that they're correlated, even though the two random walks are obviously unrelated in any fashion. (I'm fine with this meaning of spurious correlation, since there really is no relationship.)
  2. Regress ice cream sales against swimsuit sales. It confuses me that this correlation is called spurious, since there really is a relationship between ice cream sales and swimsuit sales, even though this relationship isn't causal.

So I guess my question is: do time series analysts reserve the term "(non-spurious) correlation" for causal relationships -- so that for time series analysts, correlation is meant to suggest causation! -- while statisticians in general are fine with using "correlation" to indicate any kind of (possibly non-causal) relationship?

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    $\begingroup$ Firstly notice that, in time series context you actually are dealing with spurious relationships (if the latter are linear than spurious regressions) rather than spurious correlations. What you will typically see in time-series analysis are auto-correlations, that are obviously not spurious. Well all your questions are actually covered by 3 responses below. By the way even more sound Granger (non)causality concept does not imply theory based causation. $\endgroup$ – Dmitrij Celov Mar 12 '11 at 16:17
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In order to avoid the spurious correlation problem, you should regress two stationary time series against one another. This can (potentially) provide a causal story. It is non-stationary series that lead to spurious correlation. See the reasoning given by my answer to this question (As a footnote, you may not need stationary series if they are integrated series, but I'd point you to any of the applied time series books to learn more about that.)

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    $\begingroup$ two stationary time series may be spuriously correlated as well. Just take simple textbook example, generate 100 independent realisations of Gaussian white noise process (very stationary process :) ), on average about 5 per cent of them will be correlated significantly with 5 per cent significance level. What is important - the theoretical background or deeper understanding of what's going on rather than a theoretical data-mining approach. Co-integration, Granger causality all concepts are useful, but sound model have to add some story behind the statistics it uses. $\endgroup$ – Dmitrij Celov Mar 12 '11 at 11:09
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    $\begingroup$ Thanks, Dmitrij, of course some of the series will be falsely correlated by random chance, but that's not how I typically use the "spurious correlation" term; I define this term as correlation resulting from the series each containing a time trend. And I also distinguish the cross-section variant as "omitted variables bias," though I do see spurious correlation (as I define it) as a special case of omitted variables bias. But I guess that's how the terms are used in economics. $\endgroup$ – Charlie Mar 12 '11 at 15:45
  • $\begingroup$ I just re-read your answer and cross-reference to other question and think that all 3 answers here are jointly right (+1) for this one. $\endgroup$ – Dmitrij Celov Mar 12 '11 at 16:38
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There is a good definition of spurious relationship in wikipedia. Spurious means that there is some hidden variable or feature which causes both of the variables. In both time-series and in usual regression then terminology means the same, the relationship between two variables is spurious when something else causes both variables. In time-series context this something else is inherent property of random walks, in usual regression analysis some other variable.

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  • $\begingroup$ I agree to the definition from wikie (+1) for this, by the way trends may be deterministic as well (not only random walks), for instance linear, so two linear trends will be indeed very correlated. Another source of spuriousness in time series (from the dirty tricks set) is the seasonal components, that are also very likely to be correlated wrongly. "Correlation does not imply causation article" in wikipedia is also useful. en.wikipedia.org/wiki/Correlation_does_not_imply_causation $\endgroup$ – Dmitrij Celov Mar 12 '11 at 11:15
  • $\begingroup$ @Dmitrij, all the things you mention fall into the same category, something else is causing the relationship. That is why I cited the wikipedia. $\endgroup$ – mpiktas Mar 12 '11 at 11:45
  • $\begingroup$ So would you say that "how sweet a dessert tastes" has a spurious relationship with "number of calories in the dessert" (assuming for the sake of example that sugar is the hidden variable influencing both sweetness and calories)? Also: we all know that correlation does not imply causation. But what about non-spurious correlation? Is non-spurious correlation meant to imply causation (assuming in theory that you've considered every possible hidden variable -- obviously you couldn't really check every possible hidden variable in real life)? $\endgroup$ – raegtin Mar 12 '11 at 18:39
  • $\begingroup$ @raegtin, yes in your example the relationship is spurious. But that does not mean that it is not useful. If the sugar is the true hidden variable and the relationship between sweetness and number of calories is linear, you can fit the regression and use it to determine the sweetness depending on the calories (note that this though takes this example too far), but you cannot say that the cause of the sweetness is the number of calories. In the time series context though the spurious relationship has worse consequences, since the usual regression does not make sense. $\endgroup$ – mpiktas Mar 12 '11 at 18:55
  • $\begingroup$ @raegtin, the non-spurious correlation with your given definition might imply the causation, but it does not matter, since there is no way you can calculate it. See this question and the answers below it, I think it might be of interest. $\endgroup$ – mpiktas Mar 12 '11 at 19:01
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As to your main question, my answer is No. If you've seen such a distinction between the way the terms are used in time series contexts vs. cross-sectional contexts, it must be due to one or two idiosyncratic authors you've read. Rigorous authors would never be correct in using "correlation" to mean "causation." You've gotten ahold of a spurious terminology distinction there. Interesting question, though.

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  • $\begingroup$ Then what exactly is spurious about the spurious correlation between ice cream sales and swimsuit sales? $\endgroup$ – raegtin Mar 12 '11 at 7:37
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    $\begingroup$ @raegtin, seasonal component that could be described by the weather conditions, temperature for instance. $\endgroup$ – Dmitrij Celov Mar 12 '11 at 16:19

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