# Testing Significance of Correlation

Lets say I have the returns of two stocks(stock1 and stock2). Now without running a regression, I lag one of the variables, calculate the correlation between the two stocks and repeat this process as I keep stock1's returns fixed as I continually lag stock2's returns forward. What are some valid statistics that one could use to determine if the correlation (corr(stock1 return, stock2 return(-n)) at any given lag is statistically significant? Ljung-Box is pretty much used for looked at autocorrelation for a variable and its past lags, not for another variable.

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The correlation tests you ran with differents lags are themselves 'significance' tests (???). The higher the Pearson (Spearman, ...) correlation the more 'significant'. Besides, you might wish to ensure correlations you found are stable over time. – edouard Apr 7 '13 at 12:36

## 1 Answer

Simple rule: correlation coefficient R of N samples is statistically significant if: $|R| > 2 / \sqrt{N}$ http://capone.mtsu.edu/dwalsh/436/CORRSIG.pdf But watch out for spurious correlations. It is possible to find statistically significant correlation for non stationary data series even though there is no correlation. http://www.investopedia.com/terms/s/spurious_correlation.asp

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Thanks for your response Wisent. Though the whole point of my question is to find a way to test whether a correlation is significant and indeed not spurious. Assuming you have an economic/financial reason as to why the correlation holds, do you think testing the significance of the correlation is enough to preclude that the relationship between the two variables is not spurious? – jessica Apr 7 '13 at 22:42
Pearson correlation will be significant, assuming no assumptions are violated, when the bivariate time series is fitted well by a linear relationship // straight line. This makes sense because the square of the pearson correlation is the R^2 from the linear model y = a + bx + e. You find out whether it's spurious by seeing if the estimator's assumptions are violated, meaning the finite sample properties of the test statistic will not approximate the asymptotic case well. The Pearson correlation is highly susceptible; bivariate non-normality,hetero,serial corr,etc. Kendall/Spearman is robust. – Jase Apr 8 '13 at 11:30