The Nature of Stock Market Equilibrium
(Caution: This article described the stock market between 1968-1999. It does
not describe the stock market after the Financial Crisis of 2008 because
the economy’s intrinsic rate of growth is lower.)
The novelist, G.K. Chesterton wrote:
“The real trouble with this world of ours is
not that it is an unreasonable world, nor even
that it is a reasonable one. The commonest
kind of trouble is that it is nearly reasonable,
but not quite.” In 2003, Clive Granger won the
Nobel Prize for econometric research that
justified the use of real world data in theoretical
At the core of economics is the idea of equilibrium, that stable state of affairs to which an economy tends. Equilibrium makes possible the reasoning of textbooks. There is, however, a problem with this idea because the state of equilibrium requires no markets.
In real economies, people trade freely in markets to increase their welfare. Yet, as we have discussed in a previous article, a mathematical model of trading between value and momentum investors verges of chaos, not eliciting the efficiencies that are at the heart of equilibrium economics.
Econometrics is a branch of economics concerned with empirical studies. In 1987, Clive Granger solved a crucial econometric problem, also resolving this paradox. His research allows us to explain exactly why our cyclical modification of the Fed ratio works.
The econometric regression model assumes ideal Gaussian (bell-shaped) statistical distributions. Real world economic data, however, is often not Gaussian. Many economic time series share common trends due to general economic growth. Over time, these common trends can swamp the phenomena being studied, producing spurious regressions with very high statistical significances but with no grounding in fact. Some non Gaussian economic data, however, can be usefully analyzed.
discussion is necessarily technical. Those of our readers not acquainted with
econometrics might want to skip to the summary, where we will discuss the
practical consequences of dynamic error correction in the
The OLS regression model assumes a Gaussian normal data distribution, but our primary data is not Gaussian. We used the Lilliefors statistical test (Conover, 1980) which sets the model data against an ideal Gaussian normal distribution and measures the vertical distances. A Gaussian distribution is automatically stationary, having no serial correlation. Table I presents the resulting analyses of our model data for the periods (1968-1999). There are apparent problems.
The standard econometric technique is then to detrend the data by calculating first differences, that is the difference between a data point and its previous neighbor. The problem with this technique is that it removes long-term information, in this instance cyclical economic information, and the results are difficult to interpret because economic theories are usually formulated in terms of equilibrium levels rather than in rates of change. We also present the first differenced results in the same table.
Data Analysis Results
After First Differencing
Stock Earnings Yields
No Evidence Against
* We included this variable for theoretical reasons. Current research suggests that inflation is related more to Fed policy (Gauthier, 2001) and perhaps its statistical distribution could be modeled otherwise.
In our previous analysis, we used the primary data directly without first differencing. Why did we get statistically significant results? Here are the statistics: The overall R2 of the model explains 95% of the variation in the data, all variables are statistically significant at the 0% level, the regression almost perfectly fits the data mean, but the standard Durbin-Watson test for detecting distorting serial correlation in the residuals is inconclusive.
The Durbin-Watson statistic of our residual data is 1.49, below the preferred level for our sample exceeding 1.50. Since this test was inconclusive, we ran a next neighbor regression on the yearly residuals. There is no serial correlation in the residuals; the primary data is statistically significant. However, there are variations that are persistent in sign. In other words, investors overreact to major short-term changes. (Tversky and Kahneman, 1972 ). Our subsequent analysis will confirm this fact because we will be able to see exactly how a regression model operating on non-Gaussian data can error correct, but that’s getting ahead of the story.
In a landmark paper titled “Co-Integration and Error Correction: Representation, Estimation, and Testing,” Engle and Granger (1987) proved, even more generally than as follows, that economic data which is stationary at first differences (as our data substantially is) can be conventionally analyzed without differencing, provided the residuals of the model are not serially correlated. Furthermore, if this is so, there is a dynamic error-correcting model embedded within the data; that in the words of the authors, “allows long-run components of variables to obey equilibrium constraints while short-run components have a flexible dynamic specification.” These are the relevant consequences of the theory:
A stationary random variable has a stable mean and standard deviation.
A variable is integrated I(0) if it is stationary as is. A variable is integrated I(1) if it is stationary only after taking first differences (the difference between its value and its previous neighbor). The primary variables of our study are I(1), meaning they have large but undefined trend components (Granger, 1981) and large variances.
Regressing two I(1) variables will generally produce problematic I(1) residuals unless the variables are cointegrated. If they are cointegrated, they will produce the statistically random I(0) residual that regression model requires. The I(0) residual will be caused by an error correction model that ties the two variables together and which will cause the actual equilibrium values to occur at times.
We investigate the simplest error correction process for our cyclical modification of the Fed model:
D Bond Yields (t) / Stock Earnings Yields (t) = a + b ( residual (t-1) )
Where: a is a constant
b is a regression coefficient
What this model says is that a prior year’s residual error will cause a comparable correction in the Fed ratio. We ran this regression with our data; here are the results:
1. The sign of the residual coefficient, x^1, is negative as we expect.
2. This regression is significant at the 3% level. There is within the data an error correction model that keeps the willingness of investors to buy stocks related primarily to expected industrial capacity utilization.
3. The R2 coefficient of determination that measures the overall fit of the yearly error model with the data is only .15. The R2 measures the strength of the linear relationship, and it would be 1 if market corrected perfectly in one year. Large persistent deviations in the market from the levels predicted in our model relate to actual events such as the OPEC crisis and 9/11. The yearly error correction process operates only to a degree because investors tend to overreact to large changes in the economy.
4. The non-cyclical Fed model tracked the market well between (1982-1997), years that excluded the OPEC crisis and the later Internet exuberance. We ran the cyclical error correction model for these years. The R2 was a higher .43; the regression was significant at the 1% level. The results are as expected; error correction exists.
5. On the full data set, the error correction model operates only to a degree. However, because the mean of our cyclical Fed Model is nearly equal to the mean of the data, the slow but cumulative effect of error correction eventually dominates over a number of economic cycles as positive and negative events, or more precisely as investors’ reactions to these events, cancel out. With error correction cointegrating the independent and dependent variables, the primary regression equation is the best estimate of the equilibrium relationship.
In 1935 the British economist John Maynard Keynes, an accomplished investor, published The General Theory. The theory can be stated in the mathematical form found in the economics texts. Keynes also argued that, at equilibrium, investors equate the marginal returns of all investments, whether stocks or bonds.
Nevertheless, in the oft quoted Chapter 12 titled “Long-Term Expectation,” Keynes likened stock selection to a beauty contest, an investor being rewarded if his choice corresponds to the average preferences of all investors.
Keynes left this major contradiction unresolved, other than to say that the chapter was at a different level of abstraction. In 1987, Clive Granger published the theory of cointegration, the theory that non-Gaussian economic variables will be tied together by an error correction model that defines equilibrium, if the primary regression residuals are statistically random.
We have shown that
The fact that the
1. The short-term error correcting process will return the actual Fed ratio to its equilibrium value, but only if there are no further events. Events, both positive and negative, cause the actual ratio to cross its equilibrium value. Major events prolong the error correction process because investors overreact, sometimes for a period of several years. Furthermore, good times and bad times alternate; but not with mathematical precision.
Considered over the long-term, several
business cycles, the
2. As a practical matter, value investors look to short-term catalysts as well – to positive changes in the economy and company events.
3. There is a statistical basis for tactical asset allocation, the marginal adjustment of balanced portfolios according to relative values.
4. The idea of dynamic error correction resolves the contradiction between freedom and necessity. The stock market allows for short-term innovations and also drives companies towards profitable operating efficiencies over the long-term.
5. Simple rules of thumb can contain remarkable logic.
We have stated our argument in the empirical form. More concisely, Warren Buffet said, “…in the short run, (the market is) a voting machine; in the long run, it’s a weighing machine.”
Answering our readers’ questions, here are some additional observations:
The practical application of this quantitative analysis requires judgment. These are some of the shorter term considerations of our analysis:
1) Under substantially balanced macroeconomic conditions, error correction will cause the stock market to trade around its equilibrium value.
2) Under most macroeconomic conditions, more than a quantitative formula is necessary to describe short-term stock market behavior. The stock market is also moved by catalytic economic events; but the calculated equilibrium is still important because the stock market will cross this estimate, at some time in the future.
3) Excessive stock markets will overcorrect, actual events also matter.
Considering the long-term, and the importance of appropriate institutions, the formula describes an open economy and universe. Future equilibrium stock market prices depend both upon realized earnings and future interest rates. The future depends upon what you do today. The future also depends upon the patterned facts and circumstances at the time. What are the consequences of this?
Error correction makes unnecessary the logical distinction between Gaussian risk, a state where the probabilities can be quantified, and uncertainty, where the probabilities are unknown.
Equilibrium economics is assumed to be always true. In fact equilibrium economics, and the associated math, is true only in a formal sense:
Economic theory based on utilitarian premises, which is to say all “economic” theory in the proper sense of the word, is purely abstract and formal… . Any question as to what resources, technology, etc., are met with at a given time and place must be answered in terms of institutional (our emphasis) history, since all such things, in common with the impersonal system of market relations itself, are obviously culture-history facts and products…the first step, toward a practical comprehension of the social system is to isolate and follow out to their logical conclusion...(the) fundamental tendencies discoverable in it.
Frank H. Knight
Frank H. Knight
We have shown that a cyclical modification of the Fed ratio is the best long-run estimate of the relationship between stock and bond prices.
Using a cointegration methodology, Dupuis and Tessier (2003) find that changes in inflation-adjusted dividends (earnings) account for 76% of the changes in long-term U.S. stock prices and that changes in long-term interest rates account for 24%. This is a significant empirical result for investors. The authors, however, do not use the main regression to directly calculate the level of the stock market for methodological reasons. They directly analyze interest rates and stock prices, trending variables whose causes must be further explained if possible Wickens (1996).
By analyzing the ratio between bond and stock earnings yields, the common long-term trends simply cancel out. We can then use the regression to estimate the long-term equilibrium level of the stock market. The Fed’s countercyclical monetary policy is likely the main source of statistical error correction.