# Under which circumstances can conditional distribution of asset returns be less Gaussian than the unconditional distribution?

I looked into the unconditional and the conditional distribution of a return series, where the unconditional distribution is simply the marginal distribution of the returns, and the conditional distribution is the distribution of the residuals of a GARCH model (i.e., the return series adjusted for GARCH effects).

Intuitively, I would think that a good part of the deviations from Gaussianity (e.g., a high Jarque-Bera or Kolmogorov-Smirnov statistic) are due to the GARCH effects, so I would expect the conditional distribution to be closer to a Gaussian (i.e., lower JB or KS statistics). For a reference, see Cont (2000).

My question is, how it is possible that the conditional distribution (e.g., the distribution of GARCH residuals) is further from a Gaussian than the unconditional distribution? How can the GARCH model aggravate the deviations from normality? What does this say about the appropriateness of the GARCH model and how can I diagnose this? What are the underlying assumptions that could be violated here?

It cannot be the case if the Gaussian model is the true model. If you see that then your observations are consistent with Mandelbrot's observations in 1963. See in particular:

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

I am going to get downvoted for this, but the reason it is not Gaussian is that it is already proven that it can only be Gaussian under one circumstance which is when you believe you will take a loss in every period ignoring the impact of chance effects. It is possible to get some positive returns if the mean is a small persisting loss and a high enough relative standard deviation. You can find the literature for this under:

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

and

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

This last article was extended by Rao to all finite order AR cases, but I cannot remember where the citation is.

These are both for the Fisher's likelihoodist perspective on probability. For Pearson and Neyman's interpretation of probability, the CAPM and related models cannot exist. The discussion is way too long for here. The short version would be this, however. The White article shows that no estimator exists for mean-variance finance in Fisher's understanding of probability for models like the CAPM or Black-Scholes. This isn't directly obvious, unfortunately. Pearson and Neyman's method would allow something like Theil's regression as a valid tool, but as it is a median and interquartile range based tool, you won't have mean-variance finance you will have median-interquartile range finance.

I wrote an article on the Bayesian case because Bayesian statistics are always admissible statistics and non-Bayesian statistics are admissible only if they match the Bayesian at ever sample or at the limit. The original Wald article on this,* An Essentially Complete Class of Decision Functions* is less readable than, say

Parmigiani G, Inoue L (2009) Decision Theory: Principles and Approaches. Wiley Series in Probability and Statistics, Wiley, Chichester, West Sussex

For my article, you can read Harris, David E., The Distribution of Returns (August 24, 2016). Available at SSRN: https://ssrn.com/abstract=2828744

I would point out that the original GARCH article was tested on stocks and they discovered that stocks strongly violated the underlying assumptions of the tool. The irony is that no one seems to read that part and GARCH has been around ever since. I am about to revise the above article for final submission for publication. My article arrives at the Bayesian solution and the solution for regression models, including non-Bayesian cases. I would reproduce it here with citations, but it took three page, and I don't care to retype and recite three pages.

I did a population test on the major portion of the paper for all end of day trades in the CRSP universe from 1925-2013 minus some things like shell companies and so forth. As all model selection processes are either Bayesian processes or limiting forms of Bayesian processes, I granted only one chance in one million that the above article was correct. The assumption of Gaussianity was granted 999999/100000 prior probability of truth. The Gaussian model was rejected overwhelmingly despite the prior bias for it. People can argue theory all they want, but it's crazy to argue with the population of the data.

Go back to the very first GARCH paper, I have it in a file somewhere, but I don't have the citation on me because I cannot find it in them. It warns that stocks don't have the properties necessary for GARCH to work. You are getting the response because you are using algorithms whose distance from nature is so great that the Kullback–Leibler divergence between what you are estimating and how you are estimating is very large.

I recommend all of the above papers and you can quit using GARCH now. Also, note this is not a self-promotion posting, but there are few peer-reviewed arguments since the '60s on this, and they don't provide a first principles reason for the distributions that are observed. While there are articles on various stable distributions, they are basically articles on fitting parameters rather than why the parameters exist in the first place.