Hi! This might seem like a dumb question, and it probably is. I´ve read a little time series econometrics and have come to learn that you can´t compare non-stationary series (unless there is cointegration). Then I came a across a paper where the authors correlated two series in level, and I became confused. The answer from the author was that "We can do this since we don´t try to establish Granger causality". Is this allowed?
The answer depends upon the meaning of the word "compare" in " you can´t compare non-stationary series".
If you compute simple correlation between two non-stationary time-series and on the basis of non-zero correlation try to conclude that there is a kind of cause-and-effect relation between them, then you'll make a mistake: the correlation will be a consequence of existing linear trends in both series.
The simplest workaround is to subtract the linear trends from both time-series and to correlate the remainders. If the remainders are correlated (possibly with a time-lag - that is the "Granger model"), it is a stronger signal of possible cause-effect relation. Note that no evidence of statistical relation can be a 100% proof of cause-effect relation.
But if I don't want to make inferences of causality? What if the linear trends says something about the relatoinship, without delving into causality?
If you have some additional information that permits you to prove and interpret the "relationship-but-not-causality" somehow, correlating the time-series is an OK instrument to find such pairs of "co-trended" time-series (and to determine positive or negative sign of their "interaction").
But in the real world, there are too many time-trended variables, and it is unreasonable to bring any conclusion of "relationship" solely on the basis that both variables have time-trends.
Besides that, you will have problems with non-linear trends (because simple Pearson's correlation measures only linear interaction). The stationarity condition pretty much prohibits the non-linear effects, and so simplifies the situation a lot.
Ok. what do you mean my "additional information"? An example? Do you mean theory?
An example of additional information:
Imagine, you are performing a biological or physical experiment and measure the time-series of several parameters that describe the state of your test object. Thus, you know, that the parameters are related to the same object and have a good chance to be connected somehow. And if you see correlation between two time-series (especially if only a small fraction of possible pairs of parameters in your experiment are correlated), it may be interpreted as a possible interaction. After it you will have to carry out additional experiments to determine if it is a real cause-effect relationship or, for example, just simultaneous consequences of some third (observable or unobservable) parameter.
One can compare anything with anything, having both a trend, such as, for example, the consumption of cigarettes and energy consumption. To avoid spurious correlation, we must work in differentiating the two series.
Lars, it is not generally possible to infer the "direction" or "kind" of cause-effect relationship of two variables from their statistical relationship only. The additional experiments (or additional a priori "expert" knowledge) may help you to establish the direction (including the dialectic one) and the very existence of cause-effect relationship.
If the two series are integrated of order one (i.e. both contain a unit root), the distribution of the sample correlation is a semi-ellipse (see e.g. Banerjee et at. (1993): Co-Integration, Error-Correction, and the Econometric Analysis of Non-Stationary Data, Oxford University Press, pp. 77-78). You may compute the correlation but it will not tell you anything because all values between -1 and 1 are 'almost equally likely'.
On the other hand, if the two series are trend-stationary (stationary around a linear trend e.g.), then you may de-trend the series by regressing them on a constant and linear trend and compute the correlation between the de-trended series (i.e. the residuals).
As a follow-up to the issue of the analysis of non-stationary series:
- correlation is not causality and Granger-causality is a special case of an F test comparing "predictability" or forecasting quality;
- analysis of non-stationary series and not cointegrate series would most likely lead to spurious results, even if you use non-parametric methods;
- yes, the logical start is the cointegration test - better to use the Johansen cointegration test - even if you do not find the cointegration with the standard Johansen (1995) test, try using the fractional cointegration test for long-memory processes - it may be the case that there is fractional cointegration if your variables are not I(1) but are not I(0) as well;
- if the variables appear to be cointegrated, you can use VECM or in case of fractional cointegration - FCVAR - in both cases Granger causality test is not appropriate (this fact is often debated) because of the presence of the cointegrating vector - usually the "long-run exogeneity" test with restrictions is conducted.
Now a remark on the authors, who "compared time series in levels" - is it a published work? is it possible that the series were I(0) or stationary already from the beginning? Not everything in levels is apriori not stationary ...
I don´t think it's published in a peer-reviewed article. The series are most certainly non-stationary. Here is the paper: http://www.uueconomics.se/danielw/Research_files/BengtssonWaldenstrom_Capitalshares_long.pdf
Dear Lars,
it is indeed an interesting research question these authors try to pursue! I take it, you were referring to Table 1 "pearson correlation" coefficients: it is good that the authors report significance, but the correlation of these variables is most likely spurious (providing we are talking about the variables displayed on Figure 1). I can not understand why the authors do not follow the standard procedures (like adjusting for level shift, structural breaks and outliers) and analyse the growth rates or at least changes, which are stationary.
It is also unclear where are the interpolated observations - if there are any. In general, this paper has potential, but the econometric side there is somewhat far from perfect.
But from what I observe on Figure 1, it can be the case of fractional cointegration, at least for Germany, Japan, The Netherlands and New Zealand. There are few observations, but it may be the case.
So if you have any datasets of this kind with roughly 100 observations I am open for cooperation.
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