Volatility of x and y variables in linear regression? 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?
Edit:
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?
 A: 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.      
