I have a simple regression with price returns:

$r_{t+1} = \alpha + \beta r_t + \epsilon$

My question is: do I need to do anything if $r_{t+1}$ and $r_t$ are over different horizons? Suppose the x-variable is a one month return but the y-variable is a 1-day return: do I need to adjust the variables to account for the different volatilities?


In response to the current set of questions:

Imagine that we have two time series $r_{monthly}$ with rolling monthly returns (sampled daily) and $r_{daily}$ with daily returns. And then we have the simple regression:

$r_{daily, t+1} = \alpha + \beta r_{monthly, t} + \epsilon$

So we are predicting the next day return with the preceding monthly return. These returns are of different volatilities (assuming returns are gaussian, by the square-root of time rule). Should I do a transformation to normalize the volatilities before running the regression?

  • 4
    $\begingroup$ Normally one uses $t$ for time, which would be the "x-variable," so I'm wondering about the connection between your question and the equation you have provided. Where in that equation is the one month return? $\endgroup$ – whuber Sep 15 '11 at 5:55
  • $\begingroup$ @whuber Oops! Corrected. $\endgroup$ – Belmont Sep 15 '11 at 10:01
  • $\begingroup$ Thanks. I'm still puzzled, because now the notation suggests you have a sequence $(r_t)$ = $r_1, r_2, \ldots, r_n$. With such a situation it's simply impossible that every pair $(r_t, r_{t+1})$ consists of a (one day, one month) return (except in the trivial case $n=2$), because the case $t=1$, where $r_2$ is the y-variable, implies $r_2$ is a one-day return, but then $(r_2, r_3)$ has a one-day return for its x-variable, not a one-month return as stated. $\endgroup$ – whuber Sep 15 '11 at 16:18

No, it does not matter that $r_{daily}$ and $r_{monthly}$ have different volatilities, if you recognise this will largely be taken into account in the value of $\beta$.

I assume $r$ takes a value of $0.01$ for an increase of 1%, and a value of $-0.25$ for a fall of a quarter. I suppose you could turn $r_{monthly}$ into an average daily return rather than a cumulative return by using $\frac{r_{monthly}}{d}$ or $\sqrt[d]{1+r_{monthly}}-1$ instead, where $d$ is the number of (trading?) days in the month, but the main effect would be to multiply $\beta$ by about $d$. Better still would be to look at $\log_e(1+r_{daily})$ and either $\log_e(1+r_{monthly})$ or $\frac{\log_e(1+r_{monthly})}{d}$ although, for $x$ close to $0$, $\log_e(1+x)\approx x$ so it may not make much difference.

A much more serious issue is autocorrelation. Your model suggests there is autocorrelation (the change in a day is regressed on what has happened the previous month). But because your $r_{monthly}$ values are clearly autocorrelated with each other (they overlap), I would expect you to find autocorrelation of the residuals, and that may reduce the value of your model.


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