# Fama-French three-factor model vs four-factor (Carhart) and five-factor model

I'm performing a study where I compare the Fama-French three factor model to the CAPM on the Swedish industrials industry. I do this to compare which of the models is the best performer, but also if FF3FM better explains stock returns industry-wise than just country-wise.

In my study I want to motivate why I did not choose to compare the four- or five-factor model with CAPM, but I have problems with finding good explanations. One would be that the four-factor model is commonly used with mutual funds (while I measure single stocks), why is that?

Any other suggestions why the three-factor would be more interesting or better to perform a study on?

• It is true that Mr. Carhart developed the Carhart model in conjunction with his thesis on mutual fund performance. May 9 '16 at 17:13
• This might be interesting for you: quant.stackexchange.com/a/18187/12 Jul 12 '17 at 16:25

Why are you using only stock returns? Do you Mean you're using portfolios or individual stocks? The latter is a tough sell, and I wouldn't recommend it. Running regressions on portfolios is far more standard.

So, now, motivating the models: If you want to take a theoretical stance, recent(ish) work in asset pricing has focused on grounding the Fama-French factors. To the best of my knowledge, less work has gone into theoretically founding the fourth factor. In particular, I would look into the work of Kogan (at MIT) and production based asset pricing if you want a theoretical justification.

A good argument for not using Carhart's momentum factor is that it's more based on behavioural finance arguments whereas the size and value factor are more rooted in the efficient market hypothesies. I.e, value and smaller companies are fundamentlly riskier than growth and big companies.

There is a theoretical answer to your question. Actually, there are two of them. The first is that if there is a grounded, theoretical reason for one to be superior to the other, then that is the answer. Of course, if there were, this post wouldn't be sought.

When you have multiple models, the only solution lies in Bayesian statistics. The standard implementation for either of these in most financial literature is through the AIC or the BIC, which is closer to a summary statistic that a complete answer. It is uncommon in finance to search for the full-blown Bayesian solution as it will tend to match the AIC or BIC. The AIC and BIC could be thought of as stylized Bayesian solutions with restrictive assumptions.

I am going to extend the discussion because a good grounding in Bayesian methods is a fundamental skill for those who use statistical methods and this tends to be lacking in finance and economics.

First, it is important to understand that Pearson and Neyman's Frequentist school of thought is grounded in a different set of axioms than Bayesian methods. They tend to be constructed inside "measure theory." Measure theory is the mathematical study of measurement where you allow infinitesimals in your math. If you do not, then you cannot use the real numbers. You are seeking several things in the Frequentist school.

First, you are seeking a parameter estimation algorithm that has good qualities such as consistency and unbiasedness. You are also trying to minimize the variance in the estimate among the set of all possible estimators that meet your other property requirements. This statistic, or set of statistics as in the case of multiple regression, is paired with a hypothesis, the null hypothesis.

The null hypothesis is used to generate the sample space. This is the set of all possible samples that could happen, given the null is true. Then, at least conceptually, the set of all possible sample statistics is created from the set of samples. This is the sampling distribution of the statistic and is the statistical test you use.

This contrasts with Bayesian methods. Bayesian methods can be constructed from a variety of axiom systems. The three biggest are de Finetti's, Savage's and Cox's. Each is important to finance in their own right. The method of de Finetti could be described as the defining axioms of the entire field of finance.

He begins with a bookie, an event, and a gamble. He defines a coherent statistic as a statistic such that the bookie cannot be forced to take a sure loss in all states of the world. This is a weaker constraint than the no arbitrage constraint. It simply says the market maker cannot be gamed, but it does not say a market maker or financial institution cannot game a naive investor.

Probability and utility do not exist in his thinking. Only a bookie, an event, and a gamble exist. Probability and utility are derived from this. His argument is that probability is a mental calculation that we do to makes sense of reality, but which is not part of reality itself. The same is true for utility. It may reflect internal brain processes that we then define as preferences, but the argument is that the brain processes are real, but utility is not. Utility is us trying to make sense of the world.

Because of these axioms, anyone using them can be sure that their stated probabilities can be gambled on. This is never true of a Frequentist gamble. Frequentist statistics can never be gambled on because you can always game a Frequentist bookie. Hence, any Frequentist mean-variance finance model is always stochastically dominated by Bayesian decision theory.

If mean-variance finance were valid, and I have argued in papers that it cannot be, then it would only be valid for scientific purposes. You could never bet money on the results.

Savage does not take the tack of de Finetti. Instead, he derives utility and probability from cognitive preferences alone. This creates a purely personalistic set of statistics. You have yours. I have mine. They are incredibly powerful. They force you to acknowledge in a disciplined manner that we have different knowledge, data, experiences, and preferences and put them into the open so we can explore the scientific space.

Cox's axioms are derived from set theory and Venn diagrams with function theory. His method is built around beliefs about logical statements. This is powerful. Do you believe the CAPM is better than the Fama-French model? How much? You assert logical claims, and they are tested with data. A probability is a statement of how much belief you should have in an idea.

If this seems long and convoluted for an answer, then you understand why people say "I'll just use the AIC."

I bring the axioms up because they will ground you for what you want to do, at least at a deeper level. The statement, "the CAPM is better than the Four Factor Model," is a logical assertion that can be tested. Further, as the method would be identical to de Finetti's, you can also gamble money on the outcome. It is more than just saying "the AIC is better for the CAPM." It is a tested logical assertion that a gamble can be placed on. However, as per Savage, you have to state your beliefs in an open and public forum about how you are constructing your problem so that it can be torn apart by all who disagree with you and in the best of cases, described in a way your opponents could test their own beliefs using your data. This can be operationalized through Bayes factors as a table.

Now as to the operational aspects. In Bayesian methods data are not random. It cannot be random. You saw it. It is a fact. A fact cannot be random. An observation is a fixed thing. The explanation of the fact is what you are uncertain about. Randomness is not chance in Bayesian understanding; it is uncertainty. In Bayesian thinking, parameters are random variables meaning there is uncertainty as to their true value. Since there is uncertainty regarding the explanation of the facts, the explanation is random. Let that sink it a second. It goes against your training.

Indeed, there was a debate in Bayesian statistics about taking random samples, as is required in Frequentist thought because it imposes a mathematical function on nature that is not found in the data generating function itself. It was later proved this was a harmless imposition on the data.

What you need to do is construct a probability statement regarding each model. That is to say, you need to solve $$\Pr(model=CAPM|data)$$ versus $$\Pr(model=Fama-French|data.$$

This is done through Bayes theorem. You would begin by assigning prior probabilities over each parameter and model. To do this, you would research prior articles in other countries on likely values and create a statistical distribution of these observed parameters. It won't be exact, of course, but neither are sampling statistics. You would create some functional form for each parameter and model. Models are treated as parameters here.

Then you would assign a likelihood function to the data generation process. The likelihood looks like a density function, but it isn't the same. For i.i.d. variables, if $$\frac{1}{\pi}\frac{1}{1+(x-\mu)^2},\forall{x}\in\chi$$ is your density function, then $$\frac{1}{\pi}\frac{1}{1+(x-\mu)^2},\forall{\mu}\in\Theta$$ is your likelihood function, where $\chi$ is your sample space and $\Theta$ is your parameter space. In a likelihood function, $x$ is not random, $\mu$ is.

Then over the entire model space and parameter space, you would multiply the prior by the likelihood. This would give you the numerator of Bayes theorem. Summing over the models and integrating over the parameters would give you the denominator. This gives you the posterior density function. For the CAPM it would be: $$\Pr(model=CAPM;\beta;\alpha;\sigma|data),$$ where $\sigma$ is the uncertainty of the model over the data.

Unfortunately, this is not what you asked for. You want to know $$\Pr(model=CAPM|data).$$ $\beta,\alpha$ and $\sigma$ are called "nuisance parameters." A nuisance parameter is any parameter that you do not care about, but which you have to calculate to solve your problem. They are removed through a process called marginalization. The $$\Pr(model=CAPM|data)=\int\int\int\Pr(model=CAPM;\beta;\alpha;\sigma|data)\mathrm{d}\beta\mathrm{d}\alpha\mathrm{d}\sigma.$$

While this long process may not seem worth it, Bayesian methods do something that Frequentist methods cannot do, they can make predictions about future events. Once the events have happened, you can the "score" the predictions based on the future data. So, even if you find that the CAPM was the best model, it may score terribly. It may be the best of the three models over the data set, but it may be useless out of sample implying there is another model that you have not considered.

Generically, where $x$ is data and $\tilde{x}$ is future data, the Bayesian predictive distribution is $$\Pr(\tilde{x}|x)=\int_\theta\Pr(\tilde{x}|\theta)\Pr(\theta|x)\mathrm{d}\theta.$$

Now if you noticed I didn't tell you how to calculate this, it is because that takes you over to the world of textbooks. I strongly recommend William Bolstad's Introduction to Bayesian Statistics. Its ISBN is 1118091566. He also has a book on computational methods that is a graduate text that follows on this text which is mostly restricted to problems with analytic solutions. You should also get a copy of Parmigiani's Decision Theory: Principles and Practices. Its ISBN is 047149657X.

Frequentist methods cannot solve this because the null hypothesis is that your model is the true model. You cannot test among possible true models, because for the null to work, you have to start with it. The Bayesian method tests how likely a model is near the model in nature. It does not make the rather arrogant assertion, "I know the true model."

The hidden advantage of using a Bayesian method is exactly when your model is wrong. Imagine you had a model where people were hypothesized to behave in a certain way. What you do not know is that they behave one way when it rains and another when it is sunny. The Bayesian posterior probability when have two modes because it would pick up the lurking variable. You would end up with, in practice, two covariance matrices and two sets of parameters. The Frequentist would average them out.

Now it can be the case that a multimodal parameter is due to a statistical run and you cannot test the difference between these two possibilities, but it gives you a warning to investigate. The posterior density warns you when things are wrong with your model because it will look wrong. Of course it takes skill and practice to know what it "should" look like.

The AIC and the BIC say "choose this model." Bayesian scoring says "wow, that was the best model, but boy is it bad."

• Great answer! Can you elaborate on the statement (or provide a link for) "you can always game a frequentist bookie?"
– jd8
Aug 12 '17 at 5:41