I'am comparing two 14 years time series, one being a diversity index and the other annual temperature. I suspect annual temperature to be a strong driver of annual diversity. Accordingly, I found a significant granger causality index when testing for granger causality between temperature and diversity with a lag of three years. However when running a cross correlation analysis between the two time series I found really low correlation between temperature at time t-3 and diversity at time t. Does anyone ever experience such a situation ? In other work, is it possible to observe causality but no correlation ? Thanks in advance !
Thank you Samuel for your answer. I must admit that I still find it difficult to understand how a causal relationship can exist without correlation, even with a time lag... But it's good to know that such situations exist. Would you have an example of a situation where a Granger test is significant but the correlation is not (including when time lags are taken into account)?*
What is the frequency of your data. You speak of 14 years of data and a lag of 3 years. If your data are weekly or monthly, I suppose you have also some seasonality. If your data are yearly, I am not surprised. Correlation analysis requires relatively long series. Moreover, it is not straightforward in autocorrelated time series because there is a severe bias. I am more surprised by the result of the Granger causality test.
Guy Mélard Thank you Guy for your answer. They are yearly data so yes, the length of my times series are quiet short (14years). Is there a "rule" concerning the length of time serie ? I agree that 14 points is short for correlation test ..
As far as I know, there is no strict rule since it can depend on many factors (including the multivariate distribution of the random variables, which is not under control). Surely, the large sample distribution is not applicable for such a short series. I remember of a paper by Provost and Rudiuk (1995). The sampling distribution of the serial correlation coefficient. American Journal of Mathematical and Management Sciences., 15, No. 1&2, 57-81. My former student Abdelghafour Ayadi and me, we had in 2004 a working paper on ResearchGate on a similar subject but I don't think it covers cross-correlations, only autocorrelations. Please contact me directly for more information.
There are 2 points you must consider:
1. Granger causality simply tests if lagged values of explanatory are relevant to explain the actual levels of the dependent. Correlation may be weak even when it its not null
2. Granger causality tests the effect of 3 lagged periods, while correlation only takes into account one lagged period (in you case, just t-3). Remember that in Granger causality the effects of t-1 and t-2 are also included in the F-test.
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