My independent variable is the monthly returns of Stock A. My dependent variable is the monthly returns of Stock B. Returns are calculated using [(price at end of month) / (price at beginning of month)] - 1.

If I run a simple linear regression, I get a regression coefficient of 2 for Stock A. Since my data are returns, I can interpret this as saying “for every 1% marginal rise in Stock A’s monthly return from month a to month a+b, we get a 2% marginal rise in Stock B’s monthly return from month a to month a+b” where a 1% rise in Stock A returns is interpreted as going from 2% to 3%, or 8% to 9%, etc.

Question A: Am I limited to interpreting this relationship considering one month or more (and not considering days, or other smaller chunks of time)? For example, the model can predict what happens if Stock A returns rise by 1% from month 1 to month 1000, just as much as if they rose by 1% from month 5 to month 6 - Stock B would rise by 2% considering both time frames, right? What if we calculate 30 day returns but making it more random (maybe beginning price at day 5 and ending price 30 days after so that the period of 30 days). If we observe a 1% increase in Stock A using this return formula, then we can’t say our model will predict a 2% rise in Stock A’s return because it’s not the same formula? So in short, our interpretations must stick with the way we calculated our returns in the first place?

Question B: if we use one month lagged Stock A returns, how does the interpretation change? Do we say “if Stock A’s monthly return rises by 1% from month a to month a+b, then Stock B’s monthly return will rise by 2% from month a+b to month a+b+1?”


2 Answers 2


The criticisms made about OLS regressing financial securities against each other are partially fair.

It doesn't help that the cornerstone theory of markets that every CFA student must learn (and recite), the CAPM, holds that expected returns are a linear function of such an OLS regression (versus the market). Empirically, that's total nonsense!

And yes, the dangers of spurious correlations and associated regressions are a famous problem. Granger and Newbolt (1973) famously showed that random walks will increasingly correlate over time!

But there's a "throwing with the baby out with the bathwater" to taking all this too far. Suppose that A and B were Ford and GM. Are we really saying that a correlation between their stock returns is spurious? The analyst should indeed be cautious. The movement in one is not the cause of the move in the other. As such, the model is probably indeed wrong at the outset. However, the returns of either represent a reasonable and intuitive imperfect proxy for unobserved drivers that affect the other; and intuitively are good or bad for both in unison. As the model will have "omitted variable bias". However, this is an expedient reality if share prices are observable, while the underlying mutual drivers themselves, the "confounding variables" are not. No magic Dutch auctioneer is necessary here.

If this inherent problem in model specification is present (and it usually is in real-world financial market data), then it will be a problem for all regression methodologies. There's no special problem for OLS here.

It is also a stretch to argue that returns have no mean or variance. That argument hinges on the assumption that it is the price that are the random variable. If however, price is the function of returns that are the true random variable, then the latter can then easily have a mean and variance. Why the Bayesian confidence that it is the prices and not the returns are the random variable here? How is that a given (except in statistics exam courses)?

In the classic econometrics approach, analysts should attempt to correct for "simultaneous equations bias", given both A and B driven by C. Ideally, you'd use a third auto company's stock returns, D, as instrument to ensure statistically consistency. But that process is just way too painful for all but the most diehard academic econometricians ;-)


Let me list my concerns with this method before I answer your questions.

The first is about what any statistical model does. For example, if you test $y=\beta{x}+\alpha$ and the true model is something else, then your regression will be spurious. You are assuming there is a relationship between $X$ and $Y$.

My finance colleagues will jump in about here and start posting about financial theory. That isn't my intent with the last paragraph. What I am saying is that you cannot just assume some relationship exists simply because you ran a regression and found it. It may be spurious. For example, there is a correlation between the number of donkeys in the United States and the number of doctorates granted. There is no true relationship, but there is a relationship between the population size and the number of donkeys and the population size and the number of degrees granted. It is never enough to find a relationship.

It is important that you have a reason for your model other than that you may feel it is obvious. There is a tendency to forget this in quantitative finance and a lot of spurious relationships get found due to the naturally high level of correlation in our data sets.

The second is that you are not specifying what you mean when you say regression. I am assuming that you mean ordinary least squares, (OLS). There are several problems with that. The first of which is that OLS violates the Dutch Book Theorem.

The Dutch Book Theorem is predicated on two separate ideas. The first is that there is a market maker or a bookie clearing all bets. The market maker will refuse to play any game of "heads you win, tails I lose." To prevent that, the market maker will accept any order at their stated price. You can be long or short. The second is that every decision is a marginal decision by the market maker. In standard terminology, they are trying to maximize profit.

For highly technical reasons, OLS leads to sure losses against a clever opponent. That has been settled math since 1955. Of course, since most people don't know that, there haven't been opponents actively trying to make others lose money, but it is becoming better known.

The only techniques that are not subject to sure losses are Bayesian methods with proper prior distributions. So if you were running a Bayesian regression, it would be less of a big deal.

The Dutch Book Theorem does not apply to academic work. It only applies when using a statistical method to trade. If you were doing academic work, then the range of tools opens up quite a bit.

The second objection to OLS comes from your problem definition. Let us look at your definition again. If we define returns without dividends as $$r=\frac{p_{t+1}}{p_t}-1,$$ then $r$ is a statistic and not data. $r=r(p_t,p_{t+1}),$ so $r$ is a function. The random data are the prices. So $r$ is a statistic of a ratio distribution, that is one random variable divided by another.

Under very mild assumptions, it has been known since 1940 that such a distribution won't have a mean or a variance. It isn't trivial to show or prove, unfortunately. However, it is a first or second semester homework problem for statistic majors. They have to know how to add, subtract, multiply or divide random variables. If you are not a statistics major, there is no reason that you would need to know how to do that. It is not trivial math. The background math is actually quite difficult.

If there is no mean or variance, the results of OLS have been known to be spurious since Augustin Cauchy proved it in 1851. So you cannot use OLS. That is the reason equity returns have heavy tails.

Nonetheless, you can still run several regressions, most prominently Theil's regression or quantile regression. Theil's regression is a resampling method and is very robust. Quantile regression is fast but is a bit more fragile to extreme points.

So, if you were using a Bayesian method, Theil's or quantile regression as your tool we can talk about regression.

Let us work with Theil's regression first, the results would tend to be the same for quantile regression at the 50th percentile.

Because there can be no average return, as such a thing does not exist, we will map through the median return. The most you can say when the coefficient is 2 is that if stock A rises 1% then 50% of the time stock B will rise 2 or more percent and will produce a return less than 2% half the time including losses.

As a separate issue, and your actual question A, can you change the time frame? No, returns on equity securities are not scale invariant and logically cannot be. If you want another time frame, such as annual returns, that will be a separate calculation.

As to your second question, if your regression is $_Br_{t+1}=\beta_Ar_t+\epsilon,$ where $\epsilon$ is a random variable, then you would be predicting that a one percent increase in the return of security $A$ would result in a two percent or more return in $B$ in the following period for fifty percent of the time, with it being less the other half.

Now, for the Bayesian regression, it is important to note that returns cannot be below -100%. The center of location will be the mode, not the median, so the interpretation would be that the most common relationship between the returns when $A$ went up 1% is that $B$ would go up 2%.

Nonetheless, the regression would be of the modal slope passing through the modal point.

In summary, equity securities are not scale-invariant. If you are using a correct statistical method on a valid model, then you have to do the work over for each time frame. Secondly, if the current return on security A is a leading indicator for the return on security B, then if you are using a proper method then you can certainly find it with a regression model. Note again, that assumes it really is a leading variable and that the relationship is valid. I recommend reading a history book on what was known as Dow Theory to look at that idea.

It is a bit more complex than that because Bayesian methods produce an entire probability distribution and have a separate predictive distribution. Getting a point estimate also would depend on your choice of utility function.

EDIT As to the Dutch Book Theorem, it isn't uniquely OLS that you can cause institutions to be forced to take losses. It is any non-Bayesian procedure and some Bayesian procedures.

The Dutch Book Theorem is the result of a mathematical observation by Bruno de Finetti. He observed that bookie or market maker would not play a game of "heads you win, tails I lose." So the very first axiomatization of probability follows from that. Such a book is called a Dutch Book by 19th-century gamblers. It is unclear if the term is British and refers to the Dutch, with whom they have had multiple wars, or American, and referring to the Deutch immigrants from Germany which Americans mistook as Dutch. In either case, it is a racist term because only the Dutch or Germans could be dumb enough to force themselves to lose a game.

The Dutch Book Theorem and its converse both hold. The key element that impacts OLS has to do with the addition of sets. Under the Dutch Book Theorem, sets are finitely additive but not countably additive. Under Kolmogorov's axioms, sets are countably additive.

I believe it was Leonard Jimmie Savage that used the following analogy as to why Frequentist probabilities give rise to sure losses.

Imagine an urn with a finite number of balls in it. Even if the balls did not have equal probability to be drawn, with enough information it would be possible to construct gambling odds for each ball. The number of balls does not matter. It does matter that the number is finite.

Now imagine an urn that contained the integers. Assume, momentarily, that you could somehow draw one ball in a sensible manner. For simplicity, let us assume they are equiprobable. Also assume that you could short sell a bet or go long a bet. The odds are infinite against drawing any one ball, so how could you assign prices, remembering that your stated odds could be short sold.

Now it is not true that 100% of all bets using Frequentist models will result in a pure arbitrage opportunity, just a percentage of them, although there are games where it happens with 100% of all bets. I have a demonstration game where 48% of the gambles result in sure wins and 75% of all gambles are wins based on the MVUE passing a line through an unknown point. It is too long to post here. In the demonstration, the Frequentist would give even odds around the line as the uncertainty is symmetric by force of math.

What gets lost in that example is that frequencies are not probabilities. For example, if you take any normally distributed population and draw a very large number of samples without replacement and have two people perform estimates from each sample, one using the mean and the other using the median, their confidence intervals will be of different size.

The systematic difference comes from the fact that the sampling distribution of the median is not the same as for the mean. The frequencies are first conditioned on a loss function and the probabilities minimize the maximum level of risk under that loss function. The probabilities are not conditioned on the data, they are conditioned on the model.

Because they are partitioned in a manner that is based on your utility function they do not only depend on what you saw.

See: de Finetti, Bruno (1937), “Foresight: Its Logical Laws, Its Subjective Sources”, in Henry E. Kyburg and Howard E.K Smokler (eds.), Studies in Subjective Probability, Huntington, NY: Robert E. Kreiger Publishing Co.

Dubbins, Lester E and Savage, Leonard J. (1976) Inequalities for Stochastic Processes, How to Gamble If You Must. Dover Publications.

Kemeny, John (1955), “Fair Bets and Inductive Probabilities ”, Journal of Symbolic Logic, 20 (3): 263–273.

Lehman, R. Sherman (1955), “On Confirmation and Rational Betting”, Journal of Symbolic Logic, 20 (3): 251–262.

Savage, L.J. (1954), The Foundations of Statistics, New York: Wiley.

As to the ratio distributions, it is true that many ratio distributions have finite variance. It would depend on the long-run distribution of prices.

A sketch may go something like this.

Equity securities are sold in a double auction, so they are not subject to the winner's curse. A change in auction rules could change that. The rational behavior to bid for prices, in the absence of the winner's curse, is to bid one's expectation.

The sampling distribution, as time goes to infinity, around the equilibrium price, would be the normal distribution for the bids by extending the central limit theorem. Alternatively, it could be a truncated normal distribution as negative prices happen rarely.

Our concern for a distribution is generative and not sampling based, if you would change your model you would change the distribution. However for $p_{t+1}/p_t$ you would get two off-centered normal distributions. As it turns out there are several ways to model this. This version behaves badly but is the simplest and is sufficient to make the point.

If we follow Markowitz with infinite liquidity, no dividends in the period, no bankruptcy or mergers, then we are looking at going concerns. If we are closer to honest, there exists a positive probability that a firm will not survive. So there exists a probability, $\pi(g),$ and a case of returns given the firm survives. From Bayes rule, we can talk about $f(r|g)\pi(g)$, where $r$ is return and $f$ is a function that may be a probability or a likelihood depending on which system of thought we are walking around in.

Regardless of our system of thought, $f(r|g)$ is the case of returns where firms are known with certainty to survive. They cannot die, ever, inside that function. The remainder of the likelihood or probability would be $f(r|\tilde{}g)(1-\pi(g))$. In that segment, firms never survive.

Because $f(r|g)$ survives forever, the distribution of concern is the asymptotic one. There are an infinite number of limit book orders as $t\to\infty$ each is an expectation.

Now, for some obvious reasons, the simplest being that prices are not rationally stationary around a single point forever, that is too simple a discussion.

However, adding non-stationary distributions doesn't make the argument for a variance easier. It is also why I am restricting this as this could be book length.

The ratio of two non-centered normal distributions does not have a variance, although in some cases they may "nearly" have a mean and a variance.

For a more general case involving standard normals, see

Marsaglia, G. (2006). Ratios of Normal Variables. Journal of Statistical Software, 16(4), 1 - 10.

The article references an earlier article by the same author which covers the case in a better manner but I cannot find the cite for it.

You can find this as a homework problem in Freund's textbook on Mathematical Statistics, somewhere around the third or fourth edition. I don't know where I put my copy so I am not sure which one.

  • 4
    $\begingroup$ +1 for the donkey example!:D Quick question: ``*Under very mild assumptions, it has been known since 1940 that such a* [ratio] distribution won't have a mean or a variance.'' Aren't $t$- and $F$-distributions examples of ratio distributions? Their means are well-defined though (subject to some parameter restrictions). The Cauchy distribution as a ratio of normals has neither finite mean nor finite variance but no one would argue that prices are normally distributed. The ratio of log-normal prices has a finite mean, right? $\endgroup$
    – Kevin
    Aug 31, 2021 at 8:35
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    $\begingroup$ Would you have a source for the paper proving that OLS is loosing against a clever opponent? $\endgroup$
    – Mayeul sgc
    Aug 31, 2021 at 9:33
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    $\begingroup$ Instead of statements like “it has been known since 1940”, without any reference, it would be nice if you could provide at least a sketch of the proof of these arguments. $\endgroup$
    – fes
    Aug 31, 2021 at 10:44
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    $\begingroup$ @Mayeulsgc sources are provided. It has to do with countable versus finite additivity and is not specific to OLS but to any Frequentist procedure. $\endgroup$ Sep 5, 2021 at 23:41
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    $\begingroup$ @DaveHarris Thanks a lot for your response. I agree that some ratio distributions don't have a finite mean. But I hardly know any popular model in financial economics that suggests that asset prices are normally distributed (which would indeed cause problems for expected returns). However, it’s quite possible that asset prices are non-normal and that gross returns follow a ratio distribution that has finite moments? $\endgroup$
    – Kevin
    Sep 5, 2021 at 23:50

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