I will be glad to help, but let me first advise you away from working on this topic until you have an academic position. This topic has been poison for me, but I am slogging on anyways. Before you use anything I do, get permission from your academic advisor.
I have an unpublished article on options pricing, and I am proposing a new branch of stochastic calculus. I will provide you the link to the stochastic calculus below. I am preparing a video series on how to price options because I have been desk rejected out the ears. Hence, the recommendation that you veer away from this topic since I would hate to poison your employment.
So, there are two answers concerning your question on options pricing and econometrics. They depend entirely on the loss function that you would face from choosing an unfortunate sample, and they will produce very different results. If your work is to be applied, your tools are different than if the search is for an academic tool. The applied tool minimizes average loss, the academic tool minimizes the maximum risk that you could be exposed to from an unfortunate sample.
For an applied purpose, we must back up to basic mathematical principles.
An option is a gamble; therefore, any methodology you use must satisfy de Finetti’s coherence principle. In Italian, it is called consistency rather than coherence, but consistency has a very defined meaning in English language statistics.
The Black-Scholes Options Pricing Model is not coherent in the statistical sense, even if all assumptions were correct. A contract that is not coherent will give rise to circumstances where you can create a Dutch Book and attack a market maker with their own quotes. No market manipulation is required.
I wrote an article to provide an example of a case very similar to Black-Scholes where this would happen. Instead of options, I used ghosts, cakes, engineers, elves, a king, and so forth. I also simplified it as it is really the principle that is the issue and not the specific case.
You can find it at
David Harris. Tool Induced Arbitrage Opportunities, also, how to cut cakes. Data Science Central. https://www.datasciencecentral.com/profiles/blogs/tool-induced-arbitrage-opportunities-also-how-to-cut-cakes?xg_source=activity . Nov 5, 2019.
You can go into it, link to my LinkedIn profile, and I will be glad to help with specifics with an advisor’s approval. You can get to the stochastic calculus by going to the first article in the series.
For an applied case, there cannot exist a non-Bayesian solution. All non-Bayesian solutions will, from time to time, give rise to circumstances where a hostile actor could attack a bank, and the bank would be unaware at the time. See my cake article to see an example of why. In it, a clever elf wins 93\% of the gambles on what is almost surely a fair coin toss. Almost surely is the measure-theoretic term for this type of outcome at the limit.
Returns depend on mergers, bankruptcy, dividends, and the state of a business continuing without interruption. The truncated Cauchy distribution depends on all of those items being factored out. It also doesn’t apply in a handful of cases, such as perpetual preferred stock. So an econometric solution will depend, in part, on the question you are asking. Nonetheless, in the generic case total return can be thought of as $$\Pr(R_{total}=r)=\Pr(R_G|G)\Pr(G)+\Pr(R_M|M)\Pr(M)+0\times\Pr(B)+\sum(\frac{\Pr(\delta_{t+\Delta{t}}|D)\Pr(D)}{P_t})^\frac{1}{\Delta{t}}$$ where G is going concern, M is the merged state, B is bankruptcy $\delta$ is a declared dividend, D is the state where a dividend is declared, and R is the return for the state. Note that I left liquidity out because the bid-ask spread or the stochastic budget constraint requires a little discussion.
If you are wondering where in all that is the truncated Cauchy distribution, it is at $\Pr(R_G|G)$. Imagine you were not pricing options, but only considering the case of firms that were not going bankrupt in the intermediate term and were not merger candidates. Let us also imagine you were going to do a regression against some factor and you have removed all other effects. Then your regression would be the distribution of $(\beta,\alpha,\gamma)\forall(\beta,\alpha,\gamma)\in\Re^2\times\Re^{++}$ for this one factor equation. $$\frac{1}{\pi}\frac{\gamma}{\gamma^2+(R_t-\beta{y_t}-\alpha)^2}.$$ Please note that this ignores the prior and the denominator. Without them, the distribution of posterior distribution will not sum to unity as required. In the above formula, $y$ is whatever you are regressing on, $\gamma$ is the scale parameter of your model and not of your data, $\alpha$ is an intercept.
I will provide an example further down.
I have a video series started, I am recording the second and third video now. The first video derives the distribution of returns more cleanly. You can find it at https://youtu.be/R3fcVUBgIZw. What you would do for mergers is precisely the same thing except you would move the bivariate normal distribution to the left to a point $p_t^*-\alpha$ and then perform the integration. That is because if you are buying another firm, you either believe it is undervalued or that you can shift the value to its long-run value.
For bankruptcy, it reduces the value to zero, so you would multiply zero times the probability of bankruptcy, which is part of the trinomial $(\Pr(G),\Pr(M),\Pr(B))$.
What you will end up with is a terrible, horrible, incredibly bad function. If you have not used a Bayesian method before, pick up Bolstad’s two books on it. One is an undergraduate level book, the other graduate.
You would solve the problem, most likely, using Markov Chain Monte Carlo methods. The purpose of that is to get your denominator for Bayes theorem. I can post specific tools that you could use to speed up your process, as this is a very slow tool. Depending on the computer language and the number of cores available, you could be several days awaiting an answer. Complexity hurts in this application.
Now for the Frequentist solution.
There are three possible solutions for regression, but only one is likely useful at this point, given the current state of the statistical literature.
The first is Thiel’s regression. Unfortunately, the assumption of independence is too strong, and so the results should not be considered trustworthy. The second is to take the log of prices, which works in a univariate model, but the problem comes in the multivariate case. The distribution in the multivariate case turns out to be the hyperbolic secant distribution. It lacks a covariance structure, so any squares minimizing tool will generate potentially spurious results. The third case is quantile regression. It is less precise than either of the two methods, but the results would not be suspect. Quantile regression produces an unbiased estimator, and I suspect it could be improved upon by imposing something more like Rothenberg’s estimator for the Cauchy distribution.
Rothenberg, Thomas J.; Fisher, Franklin, M.; Tilanus, C.B. (1964). “A note on estimation from a Cauchy sample”. Journal of the American Statistical Association. 59 (306): 460–463.
For the next two videos that I am producing, I include some content from regression. Top to bottom, the regressions are Bayesian, Quantile and OLS. From left to right are the X, Y and intercept for $$f(x,y)=2x+5y+5+\epsilon,\epsilon\sim\mathcal{C}(0,\gamma).$$ Each sample has ten thousand triplets as data points. That is quite large for a macroeconomic application but quite small if you have tick data.
The graph below is of the sampling distribution of the estimates and not the data. In other words, the same population was sampled from over and over again, and the slope and intercept estimates were plotted, and kernel density estimation was applied using the biweight algorithm.
Note that for the X slope, Quantile and Bayesian methods are quite comparable. The Quantile regression is a little less precise, but that is due to the information loss that is intrinsic to the technique. The sample size is so large that the loss is small. The results of least squares had to be zoomed in on as the range was enormous.
For the Y slope estimator, the Quantile estimates had to be zoomed in on. I purposely limited the range of the Y variable to make it small relative to the scale parameter of the independent variable. It had a pretty wide range because there wasn’t a lot of variability in Y. As before, the OLS estimator produces a spurious result. You can see it from the difference in the density.
The intercept suffers from quite a bit of information loss in the quantile estimator. If you are not doing predictive work, it does not matter. If your work is predictive, you may have some problems.
One thing to note is that most OLS estimates are outside the Bayesian credible range, essentially, the estimate has zero probability of even being possible. In that case, you can construct a functionally riskless arbitrage. The video, when it comes out, will show more specific content, including a case study.
What I have not touched is liquidity, which I consider a vital component. The reason is that there isn’t a unique discussion of what liquidity means. Is it depth? Is it ease of exit? Returns are, obviously, a function of liquidity and dividends. You can either directly model the bid-ask spread, or you can model a return as a function of the planetary stochastic budget constraint by noting that there is a 100% chance of selling 100 shares of IBM at \$0 per share and zero shares at an infinite price. A sigmoid function, such as the logistic function would work there. Liquidity skews the distributions because of this. A high return is less likely than a low return, given only the budget constrain as a factor.
To test what model you should use, test it through the Bayesian posterior density. I did a population test of the truncated normal versus the truncated Cauchy and the log-normal and the truncated Cauchy. I granted a 999,999:1 prior odds in favor of the normal distribution. The logic was that if a test can overcome such mathematical prejudice, then it makes no sense to continue to use it. The standard models were rejected with roughly 8.6 million leading zeros.
You can get away with regression because there is a co-scale parameter that collapses into the joint scale parameter in the multivariate case. It exists, but it vanishes. It isn’t the same thing as covariance. However, for the multivariate Cauchy distribution, you could think of the modal relationship between two variables as being represented by it. That, of course, is a very weak claim.
EDIT
I was trying to think about how to reply to your comments, or if to reply to your comments as they are not about the original question. Nonetheless, let us approach the problem by looking at the separate issues.
Please note, there is no obligation on my part to bring anyone over to my side. Marxist economics still operates as does Austrian economics, feminist economics, and Muslim economics. I see my duty as of an obligation to warn. If I am correct, there are up to six hundred trillion dollars in mispriced derivatives securities. Furthermore, based on well-known theorems in probability and statistics, I can take advantage of people using models such as the CAPM, Black-Scholes or the newer GARCH based models in such a way that I can transfer capital from the user to me. This is a deadly serious discussion
So let us back up a little. I will concede that I am a terrible writer. Neither Stephen King nor JK Rowling fears job loss because of my writing skills. If I had the option, I would rewrite the article.
Let us begin with some observations, and the problem with consensus and the strawman of all models are wrong.
Let us also assume that all of my conclusions are wrong. I will concede the position at the outset. That does not lessen the danger of the existing methodology to the banking system.
So, one of the assumptions in Ito and Stratonovich calculus is that the parameters are known with certainty by all actors. In rocketry, that is not a big issue because it is a manufactured object, and the parameters are built into the rocket. The excuse to use it in economics has always been that the market behaves as if the market knew the parameters.
The problem is that there was a proof in 1958 that showed that if the parameters are being estimated, then there is no Frequentist or Maximum Likelihood-based solution to models such as Black-Scholes or the CAPM. In the face of a non-existence proof, models like Black-Scholes or that use an adapted methodology are irrelevant to the discussion of options pricing. Likewise, Markowitz allocation models are just as invalid, unless all actors know the parameters. In that case, both models are valid. However, the posted question would be odd since no econometrics would be required.
That is why I built two new branches of stochastic calculus. Although economists mostly are unaware of it, the calculus in use assumes known parameters.
Assuming the article is totally in error, the bulk of financial econometrics must be dumped regardless. I have nothing involved with that. The proof also happens to form the basis of the Dickey-Fuller test. If it is wrong, then all unit root tests are wrong as they depend on this proof.
Now let us go at the problem from a different tack. The definition of a statistic is any function of data.
Returns are a statistic and not data. They are a transformation of data. A distribution for a statistic cannot be assumed into existence. It is the data generation process that matters. It is improper to say something like, “I am going to perform Student’s t-test and assume the sampling distribution is the Chi distribution.” When someone assumes returns are normally distributed, they are doing precisely the same thing.
Returns are the ratio of prices times the ratio of quantities minus one. It is here that we start getting into a sticky problem. Again, you can blame me for the quality of my writing and having less understanding of the problem when I wrote it.
If the problem is attacked directly as $Z=\frac{Y}{X}$ then the user ends up landing nowhere useful from an economist’s perspective. It isn’t wrong, but it pragmatically useless. For an economist, it either ends up requiring unobtainable knowledge or a negative number of degrees of freedom. On the other hand, in polar coordinates, the solution ends up looking like the data it models. The actual strength of the polar solution is that the real plane of prices is not an ordered space. Anywhere can be zero.
I chose to norm around the equilibrium, but there is no mathematical issue with choosing any point that someone would like. One could choose, arbitrarily, to always integrate around the point $(1,5)$ but would then have difficulty explaining why they wanted to do such a thing.
Let us begin with the normality assumption and weaken it a bit and also explore it a bit.
Normality entered economics because the Office of Naval Research needed an inexpensive way to test nuclear weapons without all the expense associated with fission or fusion. They needed someone to study Brownian motion and Brownian motion with drift. The University of Chicago’s economics department used to have a classified section to it. Wall Street did not fund finance in those days. It was hoped at the time that stock prices could be modeled by Brownian motion.
The very first test in the literature by Osborne showed the method to be very successful, but they had to trim the data because the tails were too broad. The assumption of normality was not a convenience and was not based on data. It had to do with the applicability of the math to weapons research.
When the Navy stopped paying, Wall Street started paying. Wall Street started paying for specific projects, and normality and log-normality survived despite mathematical arguments in the 1960s that it could not be.
Let us assume the article and the video are incorrect. The result will end up being the same as the assumption of normality is a stronger assumption than is required.
At a deep level, the issue is the symmetry of errors. Symmetry and support on the reals are sufficient to arrive at a Cauchy distribution. The ratio of any elliptically distributed variables is the Cauchy distribution. The ratio of two normals, two Student’s distributions, two Cauchy distributions will result in a Cauchy distribution under the construction used in the article and video.
For any other outcome to hold, the argument has to be that the bid or the ask is unique and special and that people appraise the value of a firm in such a way that it biases someone to buy or sell without respect to the underlying value. The article makes that argument for English style open outcry auctions. They are subject to the winner’s curse, and so returns follow the ratio of two Gumbel distributions. The error terms of the bids are asymmetric with respect to the appraisals. The stock market uses a double auction, so the Gumbel distribution does not apply.
If the article is wrong but the appraisals are symmetric as time goes to infinity about each local equilibrium, then finance must arrive at the same conclusion. The problem with the symmetry proof is that it would be a book and not an article. It would require multiple authors because there would be so much base content to cover, and nobody has all those skills in one person.
My argument for normality is simple, however. It follows from the central limit theorem. I did cheat by assuming many buyers and sellers, but that was for brevity.
Consider the joint distribution of $\Pr(R_G|G)\Pr(G)$ from above. Focusing on $R_G|G$ reads as the return given that survival is true. The implication, without the second half of the joint distribution, is that of an infinitely long life. With infinite draws, the Central Limit Theorem holds. The distribution of $p_t-p_t^*$ is intrinsically normal at the limit if everyone bids their expectation, which is the rational behavior from auction theory.
If people take expectations as economists so love, then the distribution of returns for going concerns is the truncated Cauchy distribution.
There are a couple of other attacks I think I could make, but let us now move on to the unfortunate comment by Box on all models being wrong. The AIC and the BIC make no sense if that is a useful statement.
All model selection methods are Bayesian methods. The information criterion, such as the AIC or the BIC, can be shown to map from a Bayesian posterior to a real number. They differ in their prior probability. The BIC gives equal weight to all models before having seen the data. The AIC has a complicated prior that converges to the BIC as the sample size becomes large relative to the model complexity. The information criteria are all stylized Bayesian tools that permit fast computation. The AIC was derived from information theory, while the BIC was derived from Bayesian theory. Subsequent researchers showed the tight linkages between information theory and Bayesian theory.
One advantage of Bayesian theory over information theory for model selection is the simplicity of developing a problem specific tool. As I mentioned above, I gave prior odds in favor of the normal or log-normal of 999,999:1 for the standard models.
If you have never used a Bayesian method, that may not mean much, but it implies two things. First, the data would only have to be vaguely normal to win over the Cauchy distribution. Those are highly bigoted odds. A Bayesian test of a model is $\Pr(Model=Gaussian|Data)$ versus $\Pr(Model=Cauchy|Data).$ Actually, I tested for their truncated forms. It was also a population study, so there was no data left to test. It could not be due to a bad sample.
What a Bayesian analysis does is consider the likelihood of seeing a specific data point under every possible combination of parameter values and every model specification. It is the intrinsically optimal solution in that no methodology can create a less risky test. Its optimality is related to the concept of Pareto optimality.
I did try and test some of the more modern models built around conditional heteroskedasticity, but I found that it was impossible due to the state of standard economic theory. It turns out that there is no way to defend and ARCH/GARCH model with leads/lags without having an economic theory that drives the test.
The issue has to do with the nature of the Bayesian/Frequentist split. Usually, most economists ignore the difference in the math as they also do with the method of maximum likelihood or the method of moments. The math implies far more than economists generally attend to.
Frequentist models are built on a theory of sampling. The solutions are independent of the problem. Any problem where the assumptions are met, whether the domain is physics or psychology, can use a particular test. The actual problem does not matter. If weighted least squares is used in economics or in particle physics, it will produce exactly the same sampling properties of the estimates. Of course, the specific values may vary, but the sampling properties are tied to the estimator and not the problem.
So the above regression, while preposterous for economics in terms of parameter values, would be the properties of any bivariate regression using those tools under those circumstances. The problem domain does not matter. That is a strength of Frequentist statistics. The tool to find the average mass of a pile of sand is the same tool to determine the average length of bird wings. That does not hold true for a Bayesian method. If you read my article on the cakes and the follow-on article, you will see a good example of why.
An alternative way to understand the difference is by considering the famous example by Brad Effron.
Efron, B.; Morris, C. (1977), “Stein’s paradox in statistics” (PDF), Scientific American, 236 (5): 119–127e difference, consider this example by Brad Effron.
In the example, Effron compares the sample mean with Stein’s shrinkage estimator. Stein’s estimator maps to a particular Bayesian procedure so I will discuss it from a Bayesian perspective. I am hoping it will help in understanding the nature of the difference.
In the article, Effron tries to predict the final batting average of a group of major league players based on the first forty-five at-bats. That is about the first sixth of the season.
The logic from a Frequentist perspective is that each at-bat is independent of all other batters but depends on the batter alone. A Yankees batter in Boston at noon on July 3rd does not influence a Pittsburgh Pirate’s batter at the plate at noon on July 3rd. Each at-bat is independent. Effron used the normal approximation instead of the binomial.
The maximum likelihood estimator and the minimum variance unbiased estimator for the batting average is the vector of sample means. That is driven by the assumption of normality and the independence assumption.
The Bayesian estimator that maps to the Stein estimator would use an empirical prior and a joint variance. What that would imply is that while the at-bats are independent, the batters are not. They are part of a system called Major League Baseball. It implies that there is information about the parameters of all batters in the batting history of all the others. It also implies that the variability in the system is driven by the system and its incentives.
The Bayesian estimator had 2/7ths of the variance of the prediction versus the final batting average. The Bayesian estimator also assumed independence of the at-bats but did not assume independence of the system’s variability or that the batters were chosen independently to play in major league baseball.
That difference creates a testing problem without a theory of leads and lags. The reason that leads and lags are considered at all is that the OLS estimator is no longer the MVUE. The concern of unbiasedness and minimizing variance requires someone using a Frequentist method to adapt to the dependence structure implicit in some time series by collecting additional information lost in OLS.
Likewise, Frequentist methods concern themselves with heteroskedasticity because it messes up the test statistics. The only concern, however, is that it exists. There is no concern with why it exists.
A Bayesian estimator does not lose information. It cannot lose information. It also needs a reason for heteroskedasticity and a mechanism by which it is propagated. In Bayesian testing, it is not enough to say it exists. There must be an economics-based theoretical reason for the form that it takes. The same thing would be true of lags. There should be some economic theory that says fifteen-year mortgage loans should always have three monthly lags because (fill in reason here). It would never be because the economist tested it to be so. If OLS didn’t lose information, the lags would not be used.
The only way to test a more modern method would be combinatorically; however, the number of combinations vastly exceeds the CRSP data set. Any test would be under-identified.
That creates a problem that is independent of me and my work. There is no defense for Frequentist model selection that is possible using strict testing standards. If asked, “why do you do that,” the answer is “because that is what we do.”
Consensus gets you to Ptolemaic models of planetary motion, not Copernican.
