In modelling and estimating the conditional variance of stock returns, I understand that most empirical studies outline that stock returns are leptokurtic and are asymmetric/have negative skewness, but I still see studies employing the normal distribution to model and forecast volatility, in addition to the t-distribution and GED.

Is there any justification for ever using the normal distribution outright on its own? Which papers employ the distribution on its own, and what is the main reason for doing so?


Is there any justification for ever using the normal distribution outright on its own?

When designing a GARCH model, we are making an assumption on standardized errors of stock returns, not the stock returns themselves. GARCH as a structure generates heavy-tailed outputs (even) from Normal inputs. Thus leptokurtic stock returns are compatible with Normal standardized errors. Nevertheless, a stylized fact from the stock markets is that even the standardized errors tend to be heavy-tailed, although less so than the stock returns.

<...> what is the main reason for doing so?

If a normal distribution is assumed, the MLEs can (under some not-too-stringent conditions) be treated as QMLEs; the estimators are consistent but have higher variances than under the correct distribution. I guess the normal distribution is computationally convenient and thus is often used as a quick-fix solution.

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  • $\begingroup$ This answer, along with @Aksakal combined seems to make the most sense. But I am accepting this as you've described it in the context of GARCH. From what I got from this answer, is that in the context of GARCH and what it outputs (heavy tailed outputs regardless of the input), I can use a normal distribution although it is probably not the most ideal/accurate. It is however the most convenient. I assume I can apply the reasoning to daily log returns as well, but perhaps not intraday log returns? $\endgroup$ – Albe Apr 16 '17 at 11:11
  • $\begingroup$ @Albe, Roughly, yes. This is a rather common approach. $\endgroup$ – Richard Hardy Apr 16 '17 at 14:40

When the raw data, conditional on time and other covariates, do not follow a normal distribution, predicted means can sometimes be OK (though confidence intervals will not be) but predicted individual realizations will not be. In general it is not a good idea to rely on the central limit theorem for predicted means, and it is more fruitful and robust to use more flexible models such as semiparametric ones or ones that use distributions having more than two parameters to control their center and shape.

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    $\begingroup$ Does that mean I pretty much have to redo the results using a student t or GED distribution? I see some papers acknowledge the fact that returns are leptokurtic, asymmetric etc, but still use a normal distribution when using GARCH to model volatility which confuses me a little as to what logic they are using. Seeing as I have little time, would it be somewhat acceptable if I acknowledged that suggestions for further research could include other distributions? $\endgroup$ – Albe Apr 14 '17 at 13:49
  • $\begingroup$ Please elaborate on what GED is. Perhaps it's like Tukey's generalized lambda distribution, which is quite flexible. $\endgroup$ – Frank Harrell Apr 14 '17 at 16:09
  • $\begingroup$ Your arguments apply for linear models but may not be helpful for GARCH models. A relevant idea is that GARCH as a structure generates heavy-tailed outputs from Normal inputs. Thus the inputs (the standardized errors) may well be normal. Nevertheless, in financial data even the inputs tend to be heavy-tailed, somewhere between Student's $t(2)$ and Student's $t(4)$. $\endgroup$ – Richard Hardy Apr 15 '17 at 17:30
  • $\begingroup$ Also, a GARCH model often assumes the conditional mean is just a constant, so predicting it is not very interesting and typically not the goal. However, predicting conditional variance or the (constant-mean) distribution of returns is of central interest. $\endgroup$ – Richard Hardy Apr 15 '17 at 17:39
  • $\begingroup$ Yes. Thanks for clearing this up, I am referring to GARCH models, not linear models. Whereby obviously the inputs are heavy tailed, leptokurtic, daily returns. $\endgroup$ – Albe Apr 16 '17 at 10:53

The fat tails and a general non-normality is are usually encountered on short horizon returns such as daily or intraday. Unless you believe that the daily returns are autocorrelated or from stable distributions it is hard to argue why the longer holding period (horizon) would have non-normal distribution or fat tails due to the central limit theorem (assuming the variance is stationary). So, empirically monthly returns are not necessarily leptokurtic (have fat tails) or obviously non-normal otherwise. Hence, it would help if you referred to particular research when making statements in your question. Are they talking about monthly returns vs. daily? etc.

The autocorrelation obviously flies in the face of efficient market hypothesis. It doesn't mean it's not there, but in short term there's no autocorrelation detectable usually. Otherwise, you'd be making money on this, right?

The stable distributions are not very popular. People tried them on stock returns but they don't seem to work that well.

Student t distribution and its skewed version is used in practice on daily returns. However, again, when you roll them up to monthly the returns will look a lot like Gaussian, not much like Student due to the central limit theorem. You can observe this all yourself easily by downloading daily returns from Yahoo Finance, for instance

Also, please, distinguish the variance of returns from the variance of the variance. There are models where the variance is assumed stochastic. Some of these models are under GARCH/ARCH family, some are stochastic variance. In all of them there will some constant part, such as the variance of the variance in GARCH. These models may fatten the tails, although you assume a constant variance of variance, for instance.

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  • $\begingroup$ The non-normality is scale-invariant. The people who worked in stable distributions attempted to use Frequentist methods. For truncated data, there is no admissible Frequentist solution, so they cannot work well. Stable distributions shouldn't be popular, they have terrible properties, excluding the normal. $\endgroup$ – Dave Harris Apr 15 '17 at 18:40
  • $\begingroup$ To clear it up: My returns are daily log returns. Brooks (1998) & McMillan et al (2000) use daily log returns under the the normal distribution. The latter uses daily, monthly, weekly frequencies as well. I'm aware monthly returns look a lot more gaussian, but it seems from your description that daily log returns are really only compatible with non-gaussian distributions. This makes sense but not all literature does this. But @Richard_Hardy also describes it in the context of GARCH, and that explains why some literature uses a normal distribution for daily log returns despite your points above $\endgroup$ – Albe Apr 16 '17 at 11:16
  • $\begingroup$ Normal returns simplify a lot of math that's why academic literature tends to use them, especially old literature. Another reason in the central limit theorem and the continuous time finance. The latter is intellectually compelling, but the implication is that when you integrate to finite time intervals, it's hard to see from where the Student t would come up in daily. The CLT appears to suggest that any finite interval should lead to Gaussian. That's why academics also like stochastic volatility models - they seem to be compatible with CLT and continuous time analysis. $\endgroup$ – Aksakal Apr 16 '17 at 15:09

I have written a proof that I am preparing for publication that the normal distribution can never occur, even in approximation, and the use of it has already been shown to have perfect relative asymptotic inefficiency in a paper by Sen in 1968.

The simplified version of the argument would go like this. Either you could view returns as an autoregressive processive such as $x_{t+1}=Rx_t+\epsilon_{t+1}$, or you could view a return as a future price divided by a purchase price minus one. A proof by Mann and Wald in 1943 shows that $R<1$ when $R$ is normal and a proof by White in 1958 that that is the only case where normality holds. It can only hold if the marginal investor is planning on losing money in every period.

Contrary to the expectation of some, the posterior density is very well behaved and as Bayesian methods are admissible and the likelihood function is a minimal sufficient statistic, it has all the nice properties that are desired except unbiasedness.

Let us assume that you anticipate making money when you invest. It turns out that it is sufficient that you have that subjective belief. This does not mean that you did not buy Yahoo at its peak of $235 per share at the tech bubble, it just means you didn't believe that was what you were doing. There is a nice proof commented on by Keynes for a similar proof in 1943 (whose author I forget) on the mathematical linkage between subjective probabilities and objective frequencies. In any case, subjective belief is sufficient.

If you have the AR(1) equation of $x_{t+1}=Rx_t+\epsilon_{t+1}$ with many buyers and many sellers, with no liquidity constraints and no bankruptcy limitation, as in Markowitz with the additional assertion that markets are in equilibrium, then the marginal distribution of R will be $$\frac{1}{\pi}\frac{\sigma}{\sigma^2+(r-\mu)^2},$$ with a flat prior.

This, of course, is the Cauchy distribution. It isn't just leptokurtic, it has no mean. You have to use Bayesian methods because White proved that no non-Bayesian solution existed.

If you view it as a ratio then under the same assumptions you again get the Cauchy distribution. This is because prices should be normally distributed. In equilibrium and in a double auction, there is no winner's curse so actors should bid their expectation and, from the central limit theorem, the distribution of bids should tend toward normality as the number of buyers and sellers become large. So the denominator should have a normal error and the numerator should have a normal error. The ratio of two normal distributions centered on zero should be the Cauchy distribution, by well-known proofs.

The normal distribution was assumed to be present because they were working with punch card systems, because the math was understood, because the Navy was financing it because stock quotes are cheaper than chemical processes if stocks were a natural Brownian motion, and because no one had any better ideas. This idea then morphed into a ton of different variants trying to explain the natural behavior of the Cauchy distribution by using normal distributions as mixtures, or explaining it by adding jumps or by using a varying variance. There are a few other explanations out there as well.

In my communications with Markowitz, he sent me his study on log returns. He is a wonderfully gracious gentleman and a true scientist. Unfortunately he made an error in the study. Although, in general, the math is correct, he interpreted the log approximation as being no different from the impact of the raw data. This gave him a student's t distribution of about three degrees of freedom. I assert that the appropriate distribution is the hyperbolic secant distribution, which looks like a t-distribution with a low number of degrees of freedom.

The cdf of the the Cauchy distribution is the same CDF of the hyperbolic secant distribution with log data. What was missed was that when you transform the data you transform the distribution as well. The difference between the hyperbolic secant distribution and the t-distribution is very important. The hyperbolic secant distribution has no covariance, so although no asset would be independent, they also couldn't covary either. This is why $\hat{\beta}$ seems to constantly be changing. The OLS algorithm creates a covariance matrix for data where no covariance exists in nature.

It can't settle down to a fixed point because there is no population parameter.

If it makes you feel better, there is no conditional variance because there is no mean. Now if you would like to estimate a conditional scale parameter, and the data is from a stock not going into bankruptcy or merger (inverse probability will require a different likelihood), then you will want to include liquidity constraints as well. You can model this as a reverse hazard function, independent of returns. You would model $\sigma=f()$. Fortunately, $\sigma$ appears to have a finite marginal variance as it is the ratio of two standard deviations. I haven't started to prove this though.

Sorry to have messed with your day, but I am about to mess with a lot of peoples' days. The good news is that graduate students will be able to make giant inroads because this sets aside all the standard models. I suggest reading as many textbooks on Bayesian methods as possible.

Citations as to why you cannot consider the normal distribution: Cramer H (1946) Mathematical Methods in Statistics. Princeton University Press

Curtiss JH (1941) On the distribution of the quotient of two chance variables. Annals of MathematicalStatistics 12:409

Mandelbrot B (1963) The variation of certain speculative prices. The Journal of Business 36(4):394–419

Mann H, Wald A (1943) On the statistical treatment of linear stochastic difference equations. Econometrica 11:173–200

White JS (1958) The limiting distribution of the serial correlation coefficient in the explosive case. The Annals of Mathematical Statistics 29(4):1188–1197

You will find that if you read the original article on GARCH and the stock market, the authors point out that the assumptions of GARCH were strongly violated by actual stock market data so GARCH, while a nice technique is not relevant to stocks.

No, there is nothing in the literature to support a normal distribution except desires.

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  • $\begingroup$ OP is asking about the published literature. So the answers must be in the context of the current state of art. Your answer is promoting rather unusual view point, hence it cannot answer the question in principle. You're basically saying that everyone's wrong, it's all messed up, and things should be this way. That puts it completely beyond the context of the question. $\endgroup$ – Aksakal Apr 16 '17 at 1:35
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    $\begingroup$ I have added published literature. It's outside the published literature because I get desk rejected from the finance journals, for years now. The first was from a top 3 journal. They said they had received ten such disproofs over ten years and rejected them all. Others said it would not be of interest to their readers. I sent it to the stats journals and it was rejected as obvious and not advancing the field of statistics because the content was so well known. Read the citations. Ask yourself if returns are a future value divided by a present value. $\endgroup$ – Dave Harris Apr 16 '17 at 19:15
  • $\begingroup$ I also did an empirical test using Bayes factors and posterior ratios, granting mean-variance finance 999,999:1 odds for mean-variance finance and mean-variance finance was excluded at approximately 10^(8,500,000):1 against mean-variance finance. $\endgroup$ – Dave Harris Apr 16 '17 at 19:19
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    $\begingroup$ @Aksakal I asked a mathematician how I could explain to economists that the CAPM and Black-Scholes had a math mistake in them. His response was "well that's obvious, they violate the laws of general summation." I told him that I cannot tell economists they can't add. His response was "then I don't know." You are right, I am promoting a view that is outside the mainstream, but fits wonderfully with the empirical data. I even replaced Black-Scholes with the corrected math and it works great. I found 3800 articles on just one anomaly within Black-Scholes. $\endgroup$ – Dave Harris Apr 16 '17 at 19:23
  • $\begingroup$ I'm not discouraging you from promoting an unusual point of view, but given that OP seems to be new to the field, it's helpful to disclose this to him. Claiming that BS or CAPM has a basic error which nobody noticed is quite extreme (out there), you must admit. It reminds me of this old joke about driving on the wrong side $\endgroup$ – Aksakal Apr 16 '17 at 19:55

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