4
$\begingroup$

I read the following statement in the book python for data analysis, chapter 11, and I was wondering if someone could give me intuition about why regression has this effect? The purpose of the exercise was to compare a basic correlation between microsoft and apple versus a dynamic regression.

One issue with correlation between two assets is that it does not capture differences in
volatility. Least-squares regression provides another means for modeling the dynamic
relationship between a variable and one or more other predictor variables.
$\endgroup$
3
  • $\begingroup$ This is more of a basic statistics question than a mathematical/computational finance question. $\endgroup$ – Joshua Ulrich May 9 '14 at 1:16
  • $\begingroup$ Please consider registering on the site. $\endgroup$ – SRKX May 9 '14 at 15:07
  • $\begingroup$ you mean the stats stackoverflow? will do $\endgroup$ – user3314418 May 9 '14 at 18:57
6
$\begingroup$

I guess what they are trying to say here is that, assume you have two time series $X$ and $Y$ which are exactly the same i.e. $X=Y$, the correlation is :

$$\rho_{X,Y}= \frac{Cov(X,Y)}{\sigma_X \sigma_Y}\overset{X=Y}{=}\frac{Cov(X,X)}{\sigma_X \sigma_X}=\frac{\sigma_X^2}{\sigma_X^2}=1$$

Now assume a time series $Z=2 \cdot X$, you have:

$$\sigma_Z=2 \sigma_X$$

and

$$Cov(X,Z)=Cov(X,2X)=2 Cov(X,X) = 2 \sigma_X^2$$

So,

$$\rho_{X,Z}= \frac{Cov(X,Z)}{\sigma_X \sigma_Z}=\frac{2 \sigma_X^2}{2\sigma_X^2}=1$$

The fact that $Z$ is twice as volatile as $Y$ does not appear in the correlation measure.

In regression you would fit:

$$Y_t=\alpha_Y + \beta_Y X_t + \epsilon_t ~ \text{and} ~ Z_t=\alpha_Z + \beta_Z X_t + \epsilon_t$$

Your regression would give you $\alpha_Y=\alpha_Z=0$, $\beta_Y=1$ and $\beta_Z=2$, which shows that $Z$ and $Y$ have a different relationship with $X$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.