How one thinks about probability affects how one thinks about economics. Consider the use of the word “probability” in each of the following sentences:1. What is the probability that when a fair coin is flipped it will come up heads?

2. What is the probability that exactly two number-one seeds will make it to the final four in the March Madness basketball tournament?

3. What is the probability that New York City will rank higher relative to other cities five years from now in terms of college graduates?

We would answer the first question by saying that the probability is 50 percent, based on the very definition of a fair coin. This is an *axiomatic* interpretation of probability. The axiomatic view treats probability as a matter of pure logic, with statements that do not require any empirical testing.

We would answer the second question by looking up historical records for the NCAA basketball tournament. This is the *frequentist* account of probability, which treats probability as counting outcomes from repeated trials. A frequentist would claim that the only way we can know that a coin has a 50 percent probability of coming up heads is by actually flipping a coin enough times to verify this empirically.

The third question cannot be answered on the basis of axioms or observed frequencies. The probability estimate is purely *subjective*. The subjective account of probability is that it reflects an individual belief that cannot be proven either logically or empirically.

Sometimes, these three accounts or probability are presented as mutually exclusive. Supposedly, there is a “debate” among adherents of the three views. Instead, I prefer to look at this issue as one that is caused by the same word (“probability”) being used with multiple meanings. Asking me which theory of probability I believe is asking me to face a false choice. I believe in axiomatic probability, frequentist probability, *and* subjective probability. Each account of probability is true in its own context. My resolution of the issue is to argue that the “debate” arises only because of linguistic imprecision. If we used three words instead of one, then the question of which is the proper account of probability would never arise. We would use word A to refer to axiomatic probability, word B to refer to frequentist probability, and word C to refer to subjective probability.

John Maynard Keynes’ first book was *A Treatise on Probability*. I have never read it. What I know about it is that Keynes is regarded as a leading exponent of the subjectivist account of probability. I also know that Robert Skidelsky, Keynes’ biographer, argues that Keynes’ outlook on probability is linked to his views of economics. I think this is true, and I think I am naturally inclined to follow Keynes in his views of probability and economics.

The way I see it, the “rational expectations” school of economic theory is linked to the axiomatic treatment of probability. The economist derives predictions about a model from a set of axioms, one of which is that the people whose behavior is modeled are assumed to fully understand the model.

This axiomatic approach is, in my view, absurd. Economist X has one model of the economy. Economist Y has another model of the economy. In X’s model, people believe in X’s theory. In Y’s model, people believe in Y’s theory. It is logically impossible for economist X and economist Y to inhabit the same universe! Yet they do. This tells me that the axiom of rational expectations is too strong.

The frequentist approach is, in my view, too hubristic. When I think of frequentists, I think of Bob Shiller, who is confident that he knows the true prices of assets and is certain that the variation of stock prices and house prices around their true values is unjustifiably high. Shiller’s first paper on asset prices, published in 1979, made the claim that one can compute bounds for the volatility of, say, stock prices. Violation of such variance bounds, he argued, proves that markets are not rational.

When Shiller’s paper came out in the *Journal of Political Economy* in 1979, I submitted a critical comment, which he rejected, so it was never published. In retrospect, I think that our dispute was over his frequentist interpretation of probability. I was arguing, in effect, that under a subjective interpretation of probability there are no variance bounds on asset prices.

Consider the value of stock prices today. Typical Shiller calculations would suggest that stock prices may be too low relative to fundamentals. However, subjectively, it might be rational to believe that stock prices are reasonable. By the time the overall economy recovers, many of today’s companies will have failed or shrunk. Entirely new industries may emerge. If this is the case, it is possible that owning a representative portfolio of current companies will be a losing strategy. (I am not saying that I personally believe in this pessimistic stock market scenario, but it is rational to regard it as plausible.)

More generally, at any point in time, structural change may be under way that affects the entire future path of share prices. The past is one path out of many possible paths that could have occurred had structural change evolved differently. It is necessarily much more tightly bounded than the future, where the path of structural change is unknown. Shiller inevitably confuses the tight variance bounds that look backwards with a forward-looking variance that need not be so bounded.

From a subjective perspective, the future distribution of stock prices is much more uncertain than anything that is calculated based on past data. In my view, the question of what the stock market will be priced at five years from now is one that belongs in the realm of subjective probability, not frequentist probability. If one takes that position, then Shiller’s frequentist calculation of variance bounds is not valid.

I think that Keynes would have taken my side on this. His use of the term “animal spirits” reflected his view that future probabilities could not be known objectively. Instead, entrepreneurs’ subjective estimates would determine their behavior. Keynes was not alone in this view, of course. Frank Knight was famous for describing a notion of uncertainty that could not be reduced to an objective probability distribution.

In Shiller’s frequentist interpretation, I was merely concerned with what came to be called “the Peso problem,” or in more recent terminology, a Black Swan. In his view, the only way that asset prices can vary more than historical “fundamentals” is if investors were expecting a particular event that did not occur during the time frame. This is called “the Peso problem” because an example that is often used is a currency devaluation that was considered probable within a time frame but which in fact did not occur.

However, the “Peso problem” formulation is the frequentist’s attempt to recast pure Knightian uncertainty in terms of an objective, empirical distribution. In my view, the frequentists just don’t “get” how probability factors into asset prices.

Asset prices reflect subjective probability distributions about the future. The future differs fundamentally from the past in terms of uncertainty. The past followed one path. The future has many paths. Our ignorance of the future cannot be measured on the same scale as the variation of the past.

I guess there is something about the notion of subjective probability that makes it very hard for economists to accept. Many economists prefer the objective axiomatic approach of rational expectations. Still others, like Shiller, see the world through frequentist eyes. My own view, however, is that if you want to really understand the economy, make an effort to grasp the concept of subjective probability.

## READER COMMENTS

## David

## Mar 11 2009 at 6:09am

Have you looked at Weitzman (2007)? You might like it…

Weitzman, M.L., 2007. Subjective Expectations and Asset-Return Puzzles. The American Economic Review, 97(4): 1102-1130.

## Carl The EconGuy

## Mar 11 2009 at 8:31am

Speculative bubbles are by definition outside the historical variance, and then the “fundamentals” don’t apply. Bubbles, and the aftermath of their bursting, are driven by everyone guessing what everyone else is guessing, i.e., subjective forecasting. Keynes wrote extensively about this in the General Theory as a major reason for why depressions are so difficult to reverse. So the higher than historical variance during bubbles/implosions would, in Keynes’ GT version, be due to subjectivist elements overwhelming frequentist basics. That is, going back to the corridor theory, the two theories are not necessarily mutually exclusive. In normal times, frequency based expectations dominate, but outside-the-normal-range behaviors can occur. Irrational expectations do happen.

## shayne

## Mar 11 2009 at 9:25am

Thank you for this post, Dr. Kling.

## Greg Ransom

## Mar 11 2009 at 10:19am

What is the probability that academic economists will EVER have any incentive to use concepts like probability in accordance with their logic and meaning?

None.

Academics economists are paid to misuse formalisms and mathematical constructs contrary to their logic and meaning — and contrary to empirical reality.

That is there JOB.

## Pietro Poggi-Corradini

## Mar 11 2009 at 10:27am

Great post.

I’m halfway though this book that tries to challenge the subjective view-point:

http://www.math.washington.edu/~burdzy/Philosophy/

Haven’t made up my mind yet.

## Ryan

## Mar 11 2009 at 10:34am

This is one of your best posts Dr. Kling. This gets at the heart of why I feel so frustrated when my colleagues continually tell me that I just have to market equity investments because, over its entire history, the stock market has gained over 8% per year. There is no justifiable reason to invest based on past market returns, as though they are indicative of future market gains.

## kurt

## Mar 11 2009 at 11:01am

Is it a waste of time to read Keynes book if you have read Knight’s Risk, Uncertainty and Profit?

## Scott

## Mar 11 2009 at 11:27am

Arnold,

I have been a silent reader of this blog for many years and have learned a great deal from your posts. I wanted to specifically comment on this post because it is both succinct and profound.

I have been trying to work through some of these probability theories for some time and this post really helped me understand the fundamental issues more clearly.

Thank you and keep it up.

## fundamentalist

## Mar 11 2009 at 11:40am

Mises agrees with Kling and Keynes in his book “Human Action” in the chapter on uncertainty. What Kling calls frequency probability Mises calls class probability, and Kling’s subjective probability becomes Mises’ case probability. But Mises goes further. Subjective probability can be improved through understanding how humans act. You can never apply class probabilities to a specific case, but you can predict the future better if you understand how things work, just as an engineer can predict how a machine will behave in the future because he understands how it works.

Carl: “Speculative bubbles are by definition outside the historical variance, and then the “fundamentals” don’t apply. Bubbles, and the aftermath of their bursting, are driven by everyone guessing what everyone else is guessing, i.e., subjective forecasting.”

That depends upon the range of history your data includes. If you include enough years, say a century, and then bubbles may not be outside historical variance. Also, a better understanding of economics will show that bubbles and their popping are more than the results of just subjective forecasting and have a clear cause in manipulation of credit. Such manipulation causes greater errors to occur in subjective forecasting than would otherwise occur because it distorts prices, which are the main data for subjective forecasting.

## SheetWise

## Mar 11 2009 at 11:56am

But subjective probability only makes sense with a good narrative that describes the process and estimates the bounds of confidence at each step. With that knowledge, other people can recast the probability for themselves. Math doesn’t work very well when everything is a variable. Even those who believe they can lock a few of them down are chasing a moving target.

That’s why I like the Austrians.

## fundamentalist

## Mar 11 2009 at 12:00pm

Kling: “Many economists prefer the objective axiomatic approach of rational expectations.”

I guess you might classify Austrian econ as an axiomatic approach with rational expectations but imperfect knowledge. Austrians don’t expect people to even know about their theories, let alone agree with them. As knowledge changes, expectations change, too, but people generally act rationally according to the knowledge they possess. People may not know economics, but they think they know what is best for them and will act accordingly. As a result, they will generally buy more of something at a lower price, all other things held constant. The general quantitative theory of money (not the math equation) is built on similar principles. Austrian econ builds on axioms such as these.

## dearieme

## Mar 11 2009 at 12:40pm

The teaching of probability and statistics is, or was, routinely blemished with silliness. You’ve put your finger on one of the most important, but I can remember as a freshman also being driven first to anger, and then to contempt, by the multiple meanings of “random” and even by the stupid little problems that were posed to us. Remember them? A bag contains 3 white balls and 6 black balls. You have picked out one ball at random, what is the probabilty that it is black? The answer, I felt like shouting, is that probability has nothing to do with it – you’ve picked the bloody ball out; just look at it and say which colour it is! Argh. The teaching of elementary probability called for higher standards of clear and precise explanation, or definition, than mathematicians seemed inclined to meet. All very odd.

## David

## Mar 11 2009 at 1:32pm

Excellent post.

I’ve been thinking about this recently given all the nonsense about Knightian uncertainty and how the credit markets are “frozen” and “not working” and therefore the government must intervene.

I would have thought that it would be obvious that any valid forward-looking estimate of probabilibity cannot (at least in the world of economics) be exclusively backward-looking (i.e., frequentist). Clearly, therefore, they must have a subjective element. It may be a small adjustment to a probability derived from past data or it may be a big adjustment, depending on the circumstances, or it may be a completely subjective estimate. The subjective assessment of confidence associated with the subjective estimate of probability will obviously also vary so that Knightian uncertainty is really a special case, at one end of the continuum.

I would suggest that Shiller’s heavy reliance frequentist calculation of variance bounds is not far different from the type of very complex, but ultimately substantively empty, statistical analysis practiced by the “quants” in securitizing mortgages. How did that work out?

## jeff

## Mar 11 2009 at 2:39pm

If you have a model, you’re saying that “This is how an important part of the world works.” If your model has a role for agents’ expectations, and those expectations are not rational, (i.e. they are not what the model itself would lead you to predict), what you are in effect saying is that people are too stupid to understand how this part of the world works. But you, the modeler, are not.

Economic models usually have agents in them who act in what they perceive is their own self-interest. If the quality of the their predictions makes a difference to their own outcomes, optimizing implies that they will invest some effort to figure out how the world works. Voila, you have rational expectations. It seems to me that the only way to do economics with non-rational expectations is to have a situation where agents don’t spend the time and effort to be better informed because it isn’t worth the cost. Bryan Caplan’s

Myth of the Rational Voterhas this flavor. Not many macro models do.## Barkley Rosser

## Mar 11 2009 at 5:19pm

Arnold,

I strongly recommend reading Keynes’s TOP, and I suspect that kurt has not read it, given his remarks. Keynes’s book is far more sophisticated than Knight’s although it is fully reasonable to refer to “true uncertainty” as either “Keynes-Knight” or “Knight-Keynes” rather than to just one or the other, which tends to show someone ideolgically squawking off in an unintellectual manner, given that their respective books came out in 1921, although Keynes’s was based on his undergraduate thesis written quite some time earlier.

While Knight simply distinguishes measurable (either axiomatically or frequentistically) risk from unmeasurable uncertainty, Keynes recognizes a variety of intermediate cases, counting up to more than the three you propose here. Also, while Skidelsky fits in with the conventional view, and one can certainly find Keynes quite often taking subjectivist positions on things, often quite radically so, in TOP he clearly allows for classical axiomatic frequentism, and has no problem at all with assigning a true probability of one half to a “fair coin” showing heads each time (although there is the awuful problem of how to determine if one has a “fair coin” in actual practice, see the opening segment of Tom Stoppard’s Rosencrantz and Guildenstern are Dead”).

BTW, I would agree that you were right and that probably Mises and Hayek and Knight and Keynes (and some other folks as well) would have agreed with you, although the response by Merton had to do with the practice of “dividend smoothing” by companies, which simply throws his whole effort to estimate bubbles econometrically into the old “misspecified fundamental” one.

And, the peso problem is no big deal and perfectly consistent with frequentism. One is dealing with an underlying skewed distribution and simply suffers from small sample bias in not observing periods with those asymmetric tail events. Not a problem. No big deal.

## Brian Bochenek

## Mar 11 2009 at 10:17pm

This post reminds me strongly of a book that was assigned reading in my undergrad course on inductive reasoning, “Probability a Philosophical Introduction” by D.H. Mellor. In it he lays out the three essentially different types of probabilistic reasoning we have: Physical Probabilities; degress of belief/ credences; and epistemic probabilities. Later on he goes over how philosphers and statisticians attempt to estimate the chances in these various categories, and in doing so really points out how different each of them are from one and other.

I personally think that book is a great read for introducing student to the distinct natures of these three methods different meanings of “chance.”

## The Sheep Nazi

## Mar 12 2009 at 12:34am

The UnknownAs we know,

There are known knowns.

There are things we know we know.

We also know

There are known unknowns.

That is to say

We know there are some things

We do not know.

But there are also unknown unknowns,

The ones we don’t know

We don’t know.

—Feb. 12, 2002, Department of Defense news briefing

Hart Seeley, “The Poetry of D.H. Rumsfeld”, Slate Magazine, April 2 2003

The second and third stanzas strike me as frequentist and subjective, respectively. So I guess the question for Arnold is, how badly do you want to see subjectivism drive policy?

## SheetWise

## Mar 12 2009 at 12:44am

Brian —

In finance, we’re certainly safer if we concentrate on our individual ignorance and submit to the collective intelligence. It’s a bit dicier when we recognize our individual intelligence, and try to leverage it against a consensus. Life in the fast lane is actually wagering that the collective is ignorant.

That represents another three ways of looking at things. We could overlay everything in this thread and come up with at least 9^3 different ways of looking at the world.

I would suggest “Judgment under uncertainty: Heuristics and Biases” — a careful reading might get you to wager that the collective is insane.

## Yosef Javed

## Mar 12 2009 at 12:47am

This is my first time posting and I just have to say thank you as this was a great post! First time I have ever came across an examination of stat fundamentals. This makes me very interested to examine this a lot further in stats. I wonder what I can do with this…

## Jon

## Mar 12 2009 at 4:18am

I guess there is something about the notion of subjective probability that makes it very hard for economists to accept.The basic requirement for any candidate theory of probability is that it adequately interprets the probability calculus. The problem with subjectivism, at least as you have sketched it here, is that it singularly fails to do that.

Plus, there is no reason why your third question could not be answered on the basis of what you call the “axiomatic” (I prefer “logical”) account. Put very simplistically, you look at the factors that make a city attractive to graduates and mix in some weighting factors and some factors representing the potential for variation in the first set of factors.

## valter

## Mar 12 2009 at 7:50am

“We would answer the first question by saying that the probability is 50 percent, based on the very definition of a fair coin.”

Fairness of the coin is not enough. The coin thrower must also be fair (or inept) enough!

You might like to have a look at the section on coin throwing in Jaynes’s probability theory book.

## Milton Recht

## Mar 15 2009 at 3:09am

I believe that the black swan concept is misleading and gives a false impression about the ability of asset prices to incorporate accurately expectations about future low probability events. A black swan event tends to answer a question never asked. The black swan concept focuses on an individual event instead of the cumulative probability of the numerous events that will lead to the same outcome. The future many paths you discuss.

For example, the probability of me getting into a car accident and hitting a black swan is so low I can ignore it. However, I live in a somewhat rural-suburban area where the probability of me hitting a wild turkey, Canadian goose, duck, white swan, other birds, deer, moose, raccoon, opossum, squirrel, mole, weasel, fox, or feral cat is so high that everybody knows someone, if not themselves, who had a car accident hitting one of these creatures. In other words, the probability of an accident by hitting a non-human living creature is very high. It is so high in some areas that drivers drive expecting one of these creatures to appear suddenly on the road in the path of the car.

If black swans really exist, since there are white swans in the area, then someone has very likely hit a black swan.

Do we care? Should we protect ourselves from hitting a black swan? The answer is no because it is not the relevant question. The question that concerns us is the probability of having an accident. We might refine the question to ask the probabilities of hitting a living creature versus an inanimate object. We might further refine the question to ask the probability of hitting a person versus an animal.

It is the same with asset prices. We care about events that increase asset prices, decrease asset prices or leave them unchanged. We might refine the categories into more categories such as very large, moderate and very small increases and decreases. We might even create numerical categories, such as 1 percent, 2 percent increases etc.

What we do not care about is the individual probabilities of the causative events that will place an asset into one of the categories. Numerous events cause asset prices to increase or decrease substantially. The derivations of the equity premium and risk free rates show that there are classes of events that can cause great losses with significant likelihoods.

To focus on only one of the many events in a class of events is valueless. Black swan events only exist if we focus on causes instead of outcomes.

Future outcomes have ex ante (axiomatic) probabilities or ex post, historical (frequentist) probabilities. To use the word subjective to switch from probabilities of causes to outcome probabilities leaves an impression that it replaces the other two probabilities discussed. It does not. It just indicates that there are many paths to the same end.

## Matthew C Pollard

## Apr 2 2009 at 12:39am

One of the best posts on Econlog. Great work Arnold.

On Shiller’s Variance Bounds

What’s so surprising is that many very smart economists got tangled in Shiller’s confused thinking: West, Mankiw, Romer & Shapiro among others, and simply try to remedy it by better econometrics.

Its just an ex-ante/ex-post confusion.

Ex-ante, dividends and earnings are highly uncertain and there are many possible paths that both variables can take. Prices change because the probabilities of different paths change as information arises.

Ex-post, we the realization of one path, and one set of information. Shiller’s acolytes calculate P*. P* must be, by construction, smoother than P because its only one of many possible price paths. There is no uncertainty in P*, there is in P.

Example:

Two possible worlds. A company’s earnings are either 1 or 0. The state is realized at time T=10. No more earnings afterwards.

The price of stock will be between (0,1).

If state=1 is observed, the Shiller rational price would smoothly rise to 1 at the risk-free rate.

If state=0 is observed, the Shiller price will be constant equal to 0.

The actual price will jump around as investors are working out the probability of either state, conditional on their information. As time goes by, more information is gathered and prices will eventually converge to either 0 or 1.

This seems to be a far bigger issue with the Shiller bound than the non-stationarity issue. The whole model setup is defuct.

Comments are closed.