# How to determine the confidence interval or significance of a covariance estimate

I was wondering if there is a way to determine the significance of a covariance? So, if have two vectors of returns, and then calculate the covariance, how do I determine if the sample covariance is significantly different from zero? A professor at the university told me to look for the confidence interval? Can use the Wald test for this?

John's answer is correct. I will just try and explain it a bit more clearly.

The covariance, $\sigma(x, y)$, is given as: $$\sigma(x, y)=E[(x-\mu_x)(y-\mu_y)]$$ to consider if a value f the covariance is significantly different from zero first consider what are the limits on the covariance. The range of covariance possible is $$-\sigma(x)\sigma(y)\leq\sigma(x, y)\leq\sigma(x)\sigma(y)$$ $\sigma(x)$ & $\sigma(y)$ are the standard deviations of your datasets. You can prove this result using the Cauchy-Schwarz inequality.

In this case you can see it is hard to define what covariance is significant as the range of values depends on the variance of your datasets. The logical thing to do here is to normalise the covariance to remove this effect. If we define the correlation as $$\rho(x, y) = \frac{\sigma(x, y)}{\sigma(x)\sigma(y))}$$ this has a range $-1\leq\rho\leq1$. This makes it much easier to determine when the value is close to zero.

Obviously the exact value that you consider significant will depend on your exact setup and how much correlation you think you can ignore.

• thank you for your response. I talked to a professor on my school and he suggested me to look for a way of calculating the confidence interval of your co-variance.. I ran into this talkstats.com/showthread.php/….. If I understand it correctly, co-variance= \hat{s}, the confidence interval for \hat{s} is, [\frac {\hat{s}} {\chi^2_n(1-\frac {\alpha} {2})}, \frac {\hat{s}} {\chi^2_n(\frac {\alpha} {2})}] Commented May 21, 2014 at 8:56

Correlation can be calculated as the covariance divided by the geometric mean variance. It's the proportion of covariance given the variance. Therefore, the correlation includes the covariance you want to assess in a standardized way and has known solutions for your questions. If there's an issue with the scaling of the correlation then another useful way of examining your relationship would be the regression coefficient which is equivalent to the correlation in the scale of whatever you make the response variable.

It's important to correct for the variance because the covariance is the sum of the products of how much each variable deviates from it's mean. Therefore, if either or both of your returns has high variance then the covariance will be a larger number even if the relationship between the numbers is unchanged. Therefore, you need to scale the covariance by the variance and that's what the correlation does for you.

• I have to calculate the variance of a portfolio, therefore I need a co-variance matrix.. In order to reduce the numbers in the matrix, I need to know if the estimated co-variance is significantly different from zero Commented May 9, 2014 at 14:06
• As I tried to describe above, just consider the correlation standardized covariance. What would the covariance mean without considering the variance in the first place? A covariance significantly different from 0 would mean very little if the variance was 10x larger woudn't it? The correlation is derived from all of the values in the covariance matrix so you get to make an assessment that considers their relationship.
– John
Commented May 9, 2014 at 14:12
• John thank you, but I do not get your point Commented May 9, 2014 at 14:17
• @John, your answer would be improved with a reference or two, and possibly a formula or two. On the edge of my seat to +1 that answer here. Commented May 9, 2014 at 14:58