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EXPECTATIONS UNDER UNCERTAINTY AND PSYCHOLOGICAL TIME
How do economic agents think about the future when they are confronted with uncertainty, that is, when they know that they do not even know the whole set of potential outcomes, nor the associated probabilities (unknown unknowns)? Is their thought process an endogenous source of financial instability?

Allais’s “lost” theory of psychological time
Allais’s Hereditary, Relativist and Logistic (HRL) theory of monetary dynamics was not meant to answer these two questions, but it does so by proposing an original theory of expectations formation that is a genuine alternative to both adaptive and rational expectations. Praised by Milton Friedman in 1968 with the following words: This work [the HRL formulation]'' introduces a very basic and important distinction between psychological time and chronological time. It is one of the most important and original paper that has been written for a long time … for its consideration of the problem of the formation of expectations.”

Allais’s contribution has nevertheless been “lost”: it has been absent from the debate about expectations.

Feedback loops in financial markets
Owing to periods during which asset prices are rising or falling fast and persistently, investors like George Soros, economic historians like Charles Kindleberger and some economists like Irving Fisher  , Hyman Minsky , George Akerlof and Robert Shiller  , Andrei Shleifer , to name but a few, have long hypothesized first, that financial behavior is influenced by past movements in prices, second, that in asset markets, rising prices do not necessarily cause demand to fall. On the contrary, rising prices appear to stimulate the demand for risky assets, at least up to a certain point.

Experimental economics has very much confirmed this intuition: while competitive markets operates surprisingly well for nondurable goods and services, the market mechanism works less efficiently in the case durable goods (including financial assets). In laboratory, asset markets are prone to generating large deviations of prices from fundamentals – bubbles - even when market participants benefit from perfect information about the fundamental value of the assets they trade. Vernon Smith explains this propensity to inflate bubbles by behavior uncertainty: a rational agent concerned by fundamental value cannot ignore the possibility that other less rational agents push prices away from the fundamental value. However, if it is true that there exist positive feedback loops in financial markets, how does it come that such cumulative price movements do not reinforce themselves forever? A theory of financial instability must answer this question, too.

Allais’s theory of money demand
Initially built to analyze the demand for money during hyperinflation, Allais’s theory of monetary dynamics provides a framework, which – as suggested by him - can easily be transposed to other fields – like financial markets - for it relies on very general psychological assumptions:
 * The process whereby agents forget the past is akin and symmetrical to the one whereby they discount the future.
 * Like discount factors, forgetting factors $$k(t,\tau)$$ decline exponentially with respect to time.

where $$r$$  is a constant between $$\tau$$ and $$t$$.
 * Agents are responding to a coefficient of psychological expansion $$Z$$, measuring the present value of the past and defined as a weighted sum of past growth rates $$x(\tau)$$ (hence the H in HRL)

which, by Leibniz's rule of differentiation under the integral sign, implies

where the variable $$z=\chi Z$$ is the expected value of $$x$$ and the difference $$x-z$$ the forecast error.


 * The (continuous) rate of memory decay $$\chi$$ varies through time in response to the coefficient $$Z$$ (hence the R in HRL)

where $$\alpha$$, $$b$$ and $$\chi_0$$ are three psychological parameters and $$\chi_0/(1+b)$$ is the minimum value $$\chi_{min}$$ of $$\chi$$, which - by integration - implies


 * The ratio of desired money balances $$M_D$$ to nominal spending $$D$$ (i.e liquidity preference or approximately the inverse of money velocity owing to the assumption $$M_D\approx M$$ ) is a non-linear (logistic) function of the coefficient of psychological expansion $$Z$$ (hence the L in HRL).


 * There exist some limit-cycles.
 * Collective human psychology is constant through time and space (although the parameters $$\alpha$$, $$b$$ and $$\chi_0$$ can vary from individual to individual, they are on average constant for a broad set of individuals; $$\alpha=b=1$$ and $$\chi_0=4.80\% $$ per annum).

Now, what is a financial bubble if not a kind of hyperinflation? What is money velocity if not what market participants call market liquidity? Considering such questions, it is natural to transpose Allais’s framework to the analysis of financial instability.

Psychological time
Psychological time is the keystone of Allais’s theory of monetary dynamics. It is the fundamental time scale along which clocks’ time flows more or less rapidly depending on how it is filled by the phenomenon we observe. Every day, we experience psychological time. Stuck in a traffic jam, it seems that clocks’ time flows slowly; captivated by a page-turner, we do not see time going by. Or, as Einstein reportedly put it:

An hour talking with a pretty girl sitting on a park bench passes like a minute, but a minute sitting on a hot stove seems like an hour .

Allais has conjectured that human psychology expands or compresses physical time in response to how it is filled and further worked out that while economic growth and inflation fill time, recession and deflation empty it. If clocks’ time can flow more or less quickly, then the pace at which we forget the past, the rate of memory decay $$\chi$$, must in turn be variable, elevated when time is going by quickly, low when time flows slowly, as if our attention was limited. But, along the psychological time scale, the rate of memory decay is constant and equal to $$\chi_0$$. Allais has estimated $$\chi_0$$ to be close to 5% a year.

Allais was led to psychological time by his analysis of money velocity during hyperinflation. This is not really surprising: money velocity is what mathematicians and physicists call a frequency $$f$$, that is, the inverse of a time period $$T$$.

Money velocity measures indeed how frequently money changes hands during a given period of time. Now, during an hyperinflation episode, money velocity increases; as a result, the duration of the elementary planning period shrinks; time flows faster and faster; the inflation rate observed one year ago or even only three months ago is of little help to assess the current situation. Not only can we forget it; we must actually forget it.

Allais has always declined to use the word expectations to designate a vision of the future that is actually rooted in the memory of the past. His terminological qualms have not helped his contribution to be recognized. However, if one interprets Allais’s theory of memory in terms of expectations, it is clear that Allais has constructed a theory worthy of attention, for it goes well beyond adaptive expectations, and this at a time when rational expectations theorists are coming back to adaptive expectations (as in rational learning with forgetting) after having much criticized them. That prominent theorists recently called for a theory of expectations having the attributes of Allais’s “lost” theory of psychological time and memory decay is evidence of its relevance and timeliness.

Beyond adaptive expectations
In standard adaptive expectations models, a type of model that Cagan and Allais used independently in the early 1950’s, agents are assumed to recursively update their knowledge of a given phenomenon, say inflation, by adjusting their prior knowledge $$y$$ for a fraction $$k$$, comprised between 0 and 1, of the forecast error $$x-y$$ they have just made (i.e. of the surprise, positive or negative, they have just experienced). Furthermore, agents are assumed to use the same updating coefficient irrespective of the magnitude of the forecast error. Mathematically, this is the very definition of an exponential average.

or, in continuous time,

By recurrence, it is easy to show that

where

$$\dfrac{1}{k}=\lim_{n\rightarrow\infty}[1+(1-k)+(1-k)^2+\cdots+(1-k)^n]$$

In continuous time, relationship ($$) becomes

For $$k\approx 0$$, the sum $$S$$ of the weights is indeed

and the weighted sum $$\Theta$$ of the distances in time is

Hence, the average distance in time (or characteristic length of an exponential average) is given by the ratio

or writing

where $$r$$ is a constant continuous rate of memory decay.

we get

The updating coefficient $$k$$ can be interpreted either as the responsiveness of expectations to fresh data, since

where $$E(y;x)$$ is the elasticity of $$y$$ with respect to $$x$$, or as a forgetting coefficient. Hence, to a low (resp. high) elasticity of expectations corresponds a slow (resp. fast) forgetting process, or a long (resp. short) duration of memory.

Simple as it is, this approach leaves one important question unanswered: which value to assign to the forgetting coefficient $$k$$ ? Koyck’s transformation gives a rigorous answer to this question when the variable $$y$$ intervenes in a linear relationship. This being said, a value close to 0 ensures that noise will be smoothed out (as it implies $$y_n \approx y_{n-1}$$) but exposes to the risk of being systematically behind the curve of a persistently accelerating process, if such a phenomenon occurs. No doubt, one should assume that rational agents would recognize such a systematic pattern and learn something from it. Conversely, a value close to 1 ensures that one stays on top of the most recent developments (as it implies $$y_n \approx x_n$$), but at the risk of too extreme a versatility. In 1960, Muth has demonstrated that the use of exponential averages should be restricted to time series exhibiting certain attributes (stationarity and randomness) rarely encountered in the real world.

The volatility of most economic and financial time series being volatile through time (heteroscedasticity), common sense suggests that an ideal expectations model would be one in which the elasticity of expectations would vary between a minimum close to 0 and 1, in response to the behavior of the phenomenon under consideration and without making any assumption as regards the distribution of outcomes. For this to happen, the rate of memory decay - instead of being a constant $$r$$ - must vary between a minimum close to 0 and infinity. This is precisely what Allais’s model of expectations under uncertainty assumes.

Law of variation of the rate of memory decay
As we shall now see, in Allais’s model, the relative variation of the difference between the rate of memory decay and its minimum value is assumed to be equal, up to the constant $$\alpha$$, to the forecasting error. By differentiation, relationship ($$) implies indeed

in which we recognize the logarithmic derivative of $$\chi \prime$$

Since, by relationship ($$), $$\chi \prime=\alpha (\chi-\chi_{min})$$, we get

Since, by relationship ($$), $$\mathrm d Z= (x-z) \mathrm d t$$, relationship ($$) is equivalent to

In other words, a surprise - depending on it being minor or major - instantaneously triggers, up to the constant $$\alpha$$, a relative variation, itself small or large, of the variable part of the rate of memory decay, since

so that finally

The relative variation of the variable part of the rate of memory decay being linear in $$Z$$ and a straight line minimizing the path between two points, the law of variation of the rate of memory decay can be said to be an optimal one from an economic point of view. . If Allais's formulation is not rational in the sense of Muth, it can be considered as an ecological form of rationality.

Updating equation
Now, from $$z=\chi Z$$, we get

which, by relationships ($$) and ($$), yields the following updating equation

In contrast to relationship ($$), Allais's updating equation is not linear in the forecast error, since the term $$x-z$$ appears twice, one of which as an exponent. Furthermore, unlike $$k$$ in relationship ($$), the coefficient $$\chi$$ is variable.

Elasticity of expectations varying between almost 0 and 1
The variable $$z$$ is a function of $$Z$$

The variable $$Z$$ is itself a function of $$x$$.

Hence, the variable $$z$$ is ultimately a function of $$x$$

The following general chain rule

gives the elasticity of the composition of two functions.

Hence, we have

Elasticity of $$Z$$ with respect to $$x$$'

By relationship ($$)

Elasticity of $$z$$ with respect to $$Z$$

Again, by relationship ($$)

Continuous elasticity of $$z$$ with respect to $$x$$

By relationships ($$), ($$) and ($$)

Discrete elasticity of $$z$$ with respect to $$x$$

$$\chi + Z \chi \prime$$ being a continuous rate of interest, there exists a number $$k(t)$$ such as

from which we get $$k(t)$$, the elasticity of $$z$$ with respect to $$x$$ over the period $$p$$

It immediately follows that

and

From relationships ($$), we get

which implies

$$z + \mathrm d z \approx x$$

In other words, when $$z \rightarrow +\infty$$ (as is the case during hyperinflation), the expected rate of inflation converges toward the observed rate of inflation.

Rational expectations in the sense of Muth
The rational expectations hypothesis (REH) and its twin - the efficient markets hypothesis (EMH) - spring from Muth's critique of adaptive expectations. Very influential until the 2007-2008 financial crisis, the REH leaves little room to uncertainty as it equates economic life to a game in which agents would know the complete set of potential outcomes and their associated probabilities (known unknowns or risk as opposed to uncertainty). According to this theory, since agents are supposed to play, say dices, and since there exists one and only one model, probability calculus for this matter, that forecasts the distribution of the outcomes of a large number of throws, it is rational to form expectations that are identical to the model's probabilistic forecasts. Such expectations are deemed to be rational in the sense of Muth.

There are many well-known theoretical and empirical reasons to reject the REH. Yet, its implication that the forecast error should not be correlated with the forecast itself should not be jettisoned, for if such a correlation existed, this systematic pattern would surely be exploited by agents. In the case of the HRL formulation, as a consequence of its variable elasticity, the forecast error $$x-z$$ is not correlated with the expectation $$z$$.

Allais’s HRL formulation and the dynamics of financial instability
Allais’s HRL formulation can be used to model financial behavior under the assumption that “expected” returns and risks are nothing but the present value $$Z$$ of past returns ($$x(\tau)$$) or losses ($$x(\tau)<0$$).

Bull markets have a short memory and hence are inherently unstable
This analytical framework implies that the memory of market participants shrinks during a bull market. Put differently, the elasticity of expected returns with respect to realized returns varies through time, potentially between almost 0 and 1, meaning that exuberant expectations, not to say irrational exuberance, are by nature unstable, while bearish expectations are inherently sticky.

The demand for risky assets rises when prices rise, but only up to a certain point
This framework also implies that, instead of falling, the demand for risky assets increases with the present value of past returns. But it does so in a non-linear way between two horizontal asymptotes: a lower bound and an upper limit. As the present value of past returns increases, the demand for risky assets rises less and less (the marginal demand falls): this would explain why the cumulative process whereby rising (resp. falling) prices beget rising (resp. falling) prices inevitably comes to an end, even though it is definitely self-reinforcing in its intermediate stage.

Policy rates are not apt at containing an incipient bubble
Given the order of magnitude of the expected returns during a bull market (double digit returns), this framework suggests that short-term (or policy) rates are not appropriate tools to contain an incipient financial bubble. It also suggests that a policy aiming at limiting drawdowns can only foster risk-taking and moral hazard by reducing the perceived risk of loss.

Behavioral finance
Behavioral finance has identified a number of cognitive biases (anchoring, conservatism, belief perseverance, overconfidence, availability biases, representativeness) showing that people do not update their knowledge according to Bayes’s theorem, in the sense that they either underweight or overweight fresh evidence relative to prior information. These observations are compatible with the HRL formulation’s key insight that people forget the past (i.e. prior information) at a pace which is determined by the sequence of fresh evidence. A low rate of memory decay (i.e. a long memory) implies anchoring, conservatism, belief perseverance and overconfidence, while an elevated rate of memory decay (i.e. a short memory) leads to availability biases and representative heuristic.

Rational inattention
According to Christopher Sims's rational inattention hypothesis, attention is a scarce resource: agents cannot be “on top of all things all the time”, as a result of which – as shown by linear models - their response to fresh data tends to be smoothed and delayed.

The HRL formulation provides an alternative interpretation of the observation that agents’ response to fresh data is smoothed and delayed: agents respond in a non-linear way to the present value of past data. Non-linearity implies smoothness of the response for some values of the present value of past data, while the latter implies a delay in the response.

Furthermore, the HRL formulation may explain how people allocate their attention. An elevated rate of memory decay can indeed be interpreted as a high degree of attention; conversely, a low rate of memory decay is akin to inattention to fresh data. Since the rate of memory decay increases in presence of a sequence of increasing rates of inflation, growth or return, the HRL formulation implies that agents allocate their attention to things that change the most. Once again, this seems to be a rather rational behavior (at least in the ecological sense).