Regarding "financial" data:  I have a high frequency (1 minute) measure of
supply/demand and I'd like to know if it has any influence on short term
price changes (also 1 minute).

Question: How do I calculate the correlation between this supply/demand
measure and price changes (correctly)?

Some facts about that data:
The price changes and supply/demand measure are non-normal. An assumption of
stationarity in either measure is certainly questionable. There is
non-homogeneity in variance in both measures.

In R there are 3 methods used with the cor() function, "pearson", "kendall",
"spearman". Can any of these be used without a gross violation of the
assumptions?

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