Covariance greater than Variance?

Question

It is straightforward to verify that for two random variables $X$ and $Y$ with variances $\sigma^2_X \neq \sigma^2_Y$, we have that

$$\Big|{\rm Cov}(X, Y)\Big| \leq \max\{\sigma^2_X,\, \sigma^2_Y\}$$

On the other hand, is is not true in general that $\Big|{\rm Cov}(X, Y)\Big| \leq \min\{\sigma^2_X,\, \sigma^2_Y\}$

Assume $\sigma^2_X$ is the smaller variance. Then if we have

$$\sigma^2_X \leq \Big|{\rm Cov}(X, Y)\Big| \leq \sigma^2_Y \Rightarrow \sigma^2_X \leq |\rho| \sigma_X \sigma_Y \Rightarrow \frac {\sigma_X}{\sigma_Y}\leq |\rho|$$

Theoretically, this is perfectly feasible, the bi-variate normal case being the easiest example. But somehow it doesn't feel very "likely in practice", when $X$ and $Y$ are measured in the same units, to observe the ratio of standard deviations being smaller than the (absolute) correlation coefficient -but I may be wrong.

So I am looking for any kind of real-world-data-sets experience on such a phenomenon, in an attempt to informally assess whether it can be considered "likely to observe in practice" or not.

This relates to research in the following way: I am planning to use an approximating function which is piecewise, taking different functional forms depending on whether the argument of the approximated function is negative or not (which in turns depends on the above relations). And I need to obtain theoretical results prior to any empirical implementation, so it is not just a matter of "letting the data speak". Moreover this is not tied to any specific "part of the real world". In the interests of economy, it would be good to do all the theoretical work based on one hypothesis rather than derive all different scenarios from the beginning. And then it would be better if this hypothesis could be said to be the one "expected to hold" (informally).

Answer 1

I am not quite sure what the question is asking.

The absolute value of $\operatorname{cov}(X,Y)$, the covariance of $X$ and $Y$ is no larger than $\sigma_X\sigma_Y$ which is the geometric mean of the variances of $X$ and $Y$. Since $$\min\{\sigma^2_X,\, \sigma^2_Y\} \leq \sigma_X\sigma_Y \leq \max\{\sigma^2_X,\, \sigma^2_Y\},$$ it is certainly possible for the covariance to exceed $\min\{\sigma^2_X,\, \sigma^2_Y\}$. In other words, the question

Is it not true in general that $\big|\operatorname{cov}(X, Y)\big| \leq \min\{\sigma^2_X,\, \sigma^2_Y\}$?

has the answer that the desired relationship is not always feasible.

Consider, for example, $X$ and $Y$ having variances $6^2$ and $8^2$ respectively.

• Suppose that the correlation coefficient $\rho$ is $\displaystyle \frac 56$.
  Then, $\displaystyle\operatorname{cov}(X, Y) = \rho\sigma_X\sigma_Y = \frac 56 \times 6 \times 8 = 40 > \min \{\sigma^2_X,\, \sigma^2_Y\} = 36,$ and $\displaystyle \frac{\sigma_X}{\sigma_Y} = \frac{6}{8} < \rho = \frac 56.$

• With a smaller correlation coefficient $\displaystyle \frac 34$, we have that the covariance is $36$, same as $\sigma_X^2$, and of course $\displaystyle \frac{\sigma_X}{\sigma_Y} = \frac{6}{8} = \rho$.

• If $\rho$ were even smaller, say $\displaystyle\rho = \frac 12$, the covariance is $24$ which is smaller than $\sigma_X^2$.

So it would appear that the OP is asking whether most real-life data sets that people have encountered (with $\sigma_X < \sigma_Y$) happen to have correlation coefficients that do not exceed $\displaystyle\frac{\sigma_X}{\sigma_Y}$ in magnitude.

Answer 2

Unless I've made some mistake (which I may have, I'm not clear-headed right now):

$\text{Cov}(X,Y)/\text{Var}(X) = \rho \sigma_y/\sigma_x$ can be made smaller or larger by choosing different units for $x$ or $y$ (e.g. going from dollars to cents or meters to millimeters or vice-versa).

As a result, I think you can do it simply by changing units.

Answer 3

The trees data from the R datasets package:

This data set provides measurements of the girth, height and volume of timber in 31 felled black cherry trees. Note that girth is the diameter of the tree (in inches) measured at 4 ft 6 in above the ground.

The correlation between girth & height is $0.52$, while the ratio of the standard deviations of girth to height is only $\frac{3.14''}{76.4''}=0.04$.

It's rather common to measure correlations between measurements of large magnitude to measurements of small magnitude with roughly similar coefficients of variation, & there's no particular reason such correlations should be small.

Note that if girth were expressed as circumference rather than diameter the ratio of standard deviations would be 0.13; in fact some queer way of expressing the observations might always be found to raise the value of the ratio over that of the correlation. So the question's about not just what's likely to be observed, but what's likely to be written down.