# Variance of a rational forecast

In Chapter 24 of Richard Thaler's book Misbehaving, he writes:

An important property of rational forecasts---as a stock price is supposed to be---is that the predictions cannot vary more than the thing being forecast.

He then goes on to explain this using an example of weather forecast. If the temperature in a region typically varies between 85°F and 95°F, then it would be wrong to predict 50°F one day and 115°F the next day (although this would be approximately right on average).

What exactly is underlying mathematical principle for when he says "the predictions can't vary more than the thing being forecast"? Do forecasts have to get both the mean and the variance right to qualify as rational?

• I have not read the book, but from your quotation it seems that making forecasts outside the plausible range of outcomes would be irrational so perhaps any forecasts inside the plausible range would be rational even if some rational forecasts could be better than others. Flipping a coin, predicting heads all the time might be rational but predicting it remains spinning in the air forever would not. – Henry Nov 14 '17 at 8:05

A colleague of mine pointed me to Steven M. Sheffrin's Rational Expectations, where I found the explanation (on p.122):

[L]et $P_t^*$ be the variable to be forecast. Then the optimal forecast is given by \begin{equation}P_t=E[P_t^*|I_{t-1}].\end{equation} Let $u_t$ denote the forecast error, which, by the orthogonality principle, must be uncorrelated with the forecast $P_t$. Thus $P_t^*=P_t+u_t$ and, taking the variance of both sides, \begin{equation} \mathrm{var}(P_t^*)=\mathrm{var}(P_t)+\mathrm{var}(u_t) \end{equation} or \begin{equation} \mathrm{var}(P_t^*)\ge \mathrm{var}(P_t),\quad\text{since $\mathrm{var}(u_t)\ge0$}. \end{equation}

• optional forecast? – Richard Hardy Nov 14 '17 at 20:35
• I do not know these things myself. I thought perhaps it was a typo and optimal was the intended word, but pardon if I am wrong. – Richard Hardy Nov 15 '17 at 6:25

I do not know what Thaler was trying to say, but I can guess at what the context was. Regardless, until quite recently forecasts were problematic in finance because the underlying distributions that were involved were unknown. Indeed, the validity of estimators was unknown.

Mean-variance finance models, if both literally true in their assumptions and correctly derived, do not give rise to estimators. If the CAPM $\beta$ exists, then the disappointing problem is that a proof exists to show that no stronger statement than it is in the real numbers can be found that is both consistent with Frequentist theory and mean-variance finance.

One of the assumptions of mean-variance finance models is that their parameters are known with probability one. If that is not the case, then you will not be able to find the parameters. There are several reasons for this, but the simplest is to note that if $w_{t+1}=Rw_t+\epsilon_{t+1}, R>1$(people want a profit), then the correct Frequentist estimator is the least squares estimator for any distribution of $\epsilon$ centered on zero with finite variance greater than zero. White, however, shows that the distribution of $\hat{R}-R$ is the Cauchy distribution. As $\hat{\beta}$ is a form of the sample mean and the mean of the Cauchy distribution does not exist, the power of one billion observations is no greater than the power of one observation.

If the parameters are unknown, then no person could ever find the mean or the variance to solve the model to get to a $\beta.$ Further, because of the distributions involved, $\hat{\beta}$ is meaningless.

A recent paper derived the various families of distributions that must be present in financial returns, as well as returns for antiques sold at auction, seeds, tractors and so forth. It isn't a single distribution and cannot be.

The paper noted that returns are not data. Prices are data. Volumes are data. Cash flows are data. Returns are not observed, they are computed. Any function of actual data is a statistic. It inherits its properties from the properties in the data. You could no more assume returns are normally distributed than you could assume Student's t-test follows a Weibull distribution.

Under the assumptions used in CAPM models, the distribution of returns would be the Cauchy distribution as above. In the real world, there is bankruptcy, mergers and a finite but unknown planetary budget constraint. Because most of the distributions involved lack the first or higher moments, no computable non-Bayesian method exists that is also admissible.

Forecasts in Bayesian methods have the important property that they are coherent. A statistic is coherent if fair gambles could be based on them. Frequentist statistics are not coherent. Technically, the definition of coherence is that no market-maker or bookie could be gamed to take a sure loss in all states of nature by a crafty actor or set of actors.

You can find Mann and Wald's paper at:

Mann, H. and Wald, A. (1943) On the Statistical Treatment of Linear Stochastic Difference Equations. Econometrica, 11, 173-200

You can find White's paper at:

White, J.S. (1958) The Limiting Distribution of the Serial Correlation Coefficient in the Explosive Case. The Annals of Mathematical Statistics, 29, 1188-1197

You can find the derivation of the distributions involved at:

Harris, D.E. (2017) The Distribution of Returns. Journal of Mathematical Finance, 7, 769-804.

A Bayesian prediction is a distribution such that: $$\Pr(\tilde{x}|\mathbf{X})=\int_{\theta\in\Theta}\Pr(\tilde{x}|\theta)\Pr(\theta|\mathbf{X})\mathrm{d}\theta.$$

If you note, there is no parameter in $\Pr(\tilde{x}|\mathbf{X}).$ This is because the prediction does not depend upon knowing the true value of the parameter. This is an entire distribution of predictions, however, a point prediction can be formed by minimizing a cost function over the density.

Ignoring the coherence issue, the Frequentist prediction has quite a few problems. For starters, the distributions involved lack a sufficient point statistic. Any attempt to create a statistic would lose information. This is also true for someone minimizing a cost function over a Bayesian posterior density, but not the predictive density. The Bayesian posterior is sufficient, however. In addition, the limitation of liability truncates the distribution at -100% returns. This shifts the median away from the center of location and no Frequentist estimator exists for the mode that is admissible.

As a result, Frequentist statistics overestimate returns by 2% per year and understate risk by 4% per years for the period 1925-2013. Using the log returns retains this bias. Furthermore, in log space, no covariance style construction exists in these distributions.

The criticism of prior forecasts is not really relevant. If I forecast anything using a non-converging random number generator, which is what the method of least squares does with stock prices, then I should not be upset that I have bad forecasts.

Predictions will get either/both the center of location and the scale parameter correct with measure zero. Any countable set of points in a continuum of points has zero measure and therefore has zero probability. Forecasts should be judged on a system of scoring. There is an entire field that studies the proper scoring of forecasts. In this sense, Thaler is correct in that if the distribution of forecasts was stochastically dominated by the raw data alone, then the forecasts are worse than chance alone. As the entire set of raw data is a sufficient statistic for a parameter, having forecasts worse implies that they come from a forecast that is not sufficient for the parameter involved.

Do note that expectations cannot exist for distributions without a first moment, so rational expectations is a meaningless statement if it is over the raw distribution. They would exist for log utility, though. All risk-averse utility functions should have an expected utility.

Hopefully, a paper will be forthcoming soon that derives a pricing model for European, Asian, look back and American style options that are both distribution-free and assumes the absence of the first moment. It will, I hope, also provide a set of new operators for stochastic calculus to deal with the absence of expectations. Ito methods do not give rise to an admissible statistic, except in a handful of special cases, such cash-for-stock mergers. I am hoping to have it complete soon and I am looking for comment on the mathematics of certain components as they appear to be novel innovations and I want mathematicians to kick the tires.

• Thank you Dave for the in-depth explanation. I learned quite a bit from your exposition. – Herr K. Nov 17 '17 at 7:12