Does it really matter if a correlation is spurious? Let’s say you are trying to find if there is a correlation between two stock prices, where both are likely non stationary series. You have no concern as it relates to a potentially causal relationship...
You run a simple correlation analysis against all the rules. Both our series are autocorrelated and non stationary. You find there is a 98% correlation so you conclude they depend on each other.
This is the conversation I just had with a colleague... but I think they are 100% wrong and I’d like some confirmation.
If you find two autocorrelated and non stationary series to be 98% correlated, then the correlation is likely spurious. What this means to me is that the correlation we observe is likely due to complete chance (and their correlation is likely a result of their mutual dependence on something else outside of the two series themselves). So if our goal is to identify the extent to which these two series “depend” on each other, finding a valid correlation coefficient is necessary. Is this correct?

There are two concerns with correlations of time series

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*Correlation when causal relationship is absent. Correlation does not imply causation. An example is the correlation between ice cream sales and the death rate due to drawing. These two are both high in summer and low in winter and they correlate in time, but this is not due to a direct causal relationship between the two. In such case, if a causal relationship between two variables is inferred based on a correlation between two variables then people use the term 'spurious relationship' (the inference is not correct).


*Correlation in a sample when statistical relationship for the population is absent.  Another concern is that the correlation might be likely found in data, even in the absence of an underlying statistical relationship. Time series with autocorrelation have a tendency to go up/down for short periods of time and so they tend to correlate with each other within short windows of time. But, this correlation is not significant. Yes, if you would compute the significance assuming that the datapoints are independently distributed according to a bivariate normal distribution (for which you can compute the exact sample distribution for the correlation coefficient), then it will turn out to be significant, but that assumption of independence is not correct when the time series follow trends or are autocorrelated.
