Distributional Assumptions for Raw and Log-Return Betas

# Finance beta: normally distributed?


# Finance beta: normally distributed?












If we assume normally distributed return (or normally distributed log Returns) for an asset and the market, can be then also say that the betas derived by this are also normally distributed?

How about time-varying betas calculcated by a rolling window OLS? Are they also normally distributed?

## Answer by Dave Harris (score -1)

https://quant.stackexchange.com/a/31585

So you actually asked two questions, one regarding $\beta$ in raw form and one involving $\beta$ in log form. For purposes of notation, I will distinguish $\beta_{raw}$ and $\beta_{log}$ as these are two very different questions. If I am discussing a population parameter, then I will use $\beta$.

As a general note, there are two possible answers and the answer for finance, and also for cancer research, is different than the answer for something such as a prediction of warranty costs for new cars. The reason has to do with the central limit theorem. You can assume that $\beta_{raw}$ converges to a Student's t distribution for most things except stock and cancer, and the infrared divergence in quantum mechanics.

In 1801 Laplace wrote a letter to Poisson with a proof for what used to be called the law of errors and is now called the central limit theorem. He wanted Poisson to review his proof. Poisson found a counter-example to the central limit theorem, but wrote that a mere footnote is required "as we will, without a doubt, not encounter it in practice." This language is unfortunate for two reasons. First, this is correspondence between two of histories greatest mathematicians and a footnote for the greatest might need an entire book for the rest of us. The second is that the statistics textbooks often omitted this exception, even in graduate works, because why add a footnote no one will need in practice. Stocks sit inside this exception.

The next encounter with this problem was in 1851, 52, or 53, I no longer remember which year. The great Augustin Cauchy had just derived a method of regression using the median. Just to be clear, I should have written the Great Augustin Cauchy, not only because he was great, but because he knew he was great. The lesser mathematician Bienayme' about a month later published proofs that OLS was the best method and Cauchy took this as an insult. Most of the practical things we teach undergraduates come from Bienayme', including the misnamed Chebychev inequality.

Cauchy found a specific case where OLS and OLS type methods would always produce an incorrect answer, even with an infinite amount of data. In fact it would not even be an approximation. It was a variant on Poisson's original issue. $\beta_{raw}$ is a form of an arithmetic mean, but for slopes. Cauchy found a case, usually ignored now, where the distribution had no mean, only the zeroth moment exists. As such, $\beta_{raw}$ is the estimator of a non-existent thing. It would behave in a strange way, for time data, it would be time-varying. For non-time data, it would simply vary from sample to sample, but not converge. It would appear to have heteroskedasticity, or even time-varying heteroskedasticity, but in fact is askedastic. You would even get runs that look like volatility clusters.

The next appearance is in 1958 when John White solved the test statistic for the explosive root case of AR(1) problems. It was generalized in 1962 by Rao for stochastic polynomials of any order with explosive roots. Mann and Wald showed in 1943 that the estimator was the OLS estimator. White showed that the distribution of the error term did not matter, provided it had finite variance greater than zero and a mean of zero. This is important because it can have heteroskedasticity, or not. It could be Wishart distributed errors or normal, it didn't matter. Pick ANY error structure and the test statistic was the same and was solved.

The problem was, it was Cauchy's distribution that had no mean. So the sampling distribution of $\beta_{raw}$ for equity securities is the Cauchy distribution. The reason is can be found at https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2828744 where you can get an extended discussion.

$\beta_{log}$ is a different creature however. The likelihood function is the hyperbolic secant distribution. You can read why in the above paper. It has a mean and a variance of a sort. Returns in logs will converge to a hyperbolic secant distribution, in a Markowitz style world,with no liquidity costs and no bankruptcy or mergers. It is heavy tailed, but not fat tailed. The sampling distribution of a regression for the slopes will converge to normality as the sample becomes very large, but, you have to ignore the covariance matrix that OLS provides you. The hyperbolic secant distribution has no covariance matrix implying that stocks can comove, but cannot covary. Indeed, they are not independent and cannot be independent, yet they do not covary.

The covariance matrix describes the uncertainty in the sample, but it does not point to a population parameter. In this type of math $\sigma_{ij}$ does not exist as a population parameter. The logarithmic form of regression points to Theil's median based regression for the raw data.

It has been the assumption, since 1952, that some form of normality was present if we could just find the correct restrictions to put on a model. Maybe if finance added ARCH or GARCH, or time-varying means, we could get to where we are going, but there was a warning in 1953 by John von Neumann that this branch of mathematics was as yet unsolved and that many things that looked like proofs at that time may turn out not to be proofs. This branch of mathematics has been solved. If you are getting weird effects in your $\beta_{raw}$ measurements, take a look at your tails. Do they sit inside six $\sigma$?