User:Kerners/sandbox

EXPECTATIONS UNDER UNCERTAINTY AND PSYCHOLOGICAL TIME

How do

?

Allais’s “lost” theory of psychological time

in 1968 with the following words:

This work [the HRL formulation] introduces a very basic and important distinction between

Allais’s contribution has nevertheless been “lost”: it has been absent from the debate about expectations.[3]

Feedback loops in financial markets

Owing to periods during which

to fall. On the contrary, rising prices appear to stimulate the demand for risky assets, at least up to a certain point.

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

• The process whereby agents forget the past is akin and symmetrical to the one whereby they discount the future.
• Like discount factors, forgetting factors ${\displaystyle k(t,\tau )}$ decline exponentially with respect to time.

${\displaystyle k(t,\tau )=\mathrm {e} ^{-r(t-\tau )}}$

(1)

where ${\displaystyle r}$ is a constant between ${\displaystyle \tau }$ and ${\displaystyle t}$.

• Agents are responding to a coefficient of psychological expansion ${\displaystyle Z}$ , measuring the
  weighted sum
  of past growth rates ${\displaystyle x(\tau )}$ (hence the H in HRL)

${\displaystyle Z=\int _{-\infty }^{t}x(\tau )\mathrm {e} ^{-\int _{\tau }^{t}\chi (u)\mathrm {d} u}\mathrm {d} \tau }$

(2)

which, by

, implies

${\displaystyle {\dfrac {\mathrm {d} Z}{\mathrm {d} t}}=x-\chi Z}$

(3)

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

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

${\displaystyle {\dfrac {\mathrm {d} \chi }{\mathrm {d} Z}}=\alpha \left(\chi -{\dfrac {\chi _{0}}{1+b}}\right)}$

(4)

where ${\displaystyle \alpha }$, ${\displaystyle b}$ and ${\displaystyle \chi _{0}}$ are three psychological

and ${\displaystyle \chi _{0}/(1+b)}$ is the minimum value ${\displaystyle \chi _{min}}$ of ${\displaystyle \chi }$, which - by integration - implies

${\displaystyle \chi (Z)=\chi _{0}{\dfrac {1+b\mathrm {e} ^{\alpha Z}}{1+b}}}$

(5)

• The ratio of desired money balances ${\displaystyle M_{D}}$ to nominal spending ${\displaystyle D}$ (i.e
  money velocity
  owing to the assumption ${\displaystyle M_{D}\approx M}$ ) is a
  non-linear (logistic
  ) function of the coefficient of psychological expansion ${\displaystyle Z}$ (hence the L in HRL).

${\displaystyle {\dfrac {M_{D}}{D}}=\phi _{0}{\dfrac {1+b}{1+b\mathrm {e} ^{\alpha Z}}}}$

(6)

• There exist some limit-cycles.
• Collective
  human psychology
  is constant through time and space (although the parameters ${\displaystyle \alpha }$, ${\displaystyle b}$ and ${\displaystyle \chi _{0}}$ can vary from individual to individual, they are on average constant for a broad set of individuals; ${\displaystyle \alpha =b=1}$ and ${\displaystyle \chi _{0}=4.80\%}$ per annum).

Now, what is a

? 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

An hour[17] talking with a pretty girl sitting on a park bench passes like a minute^[18], but a minute^[19] sitting on a hot stove seems like an hour^[20].

Allais has conjectured that human psychology expands or compresses

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 ${\displaystyle \chi }$, 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 ${\displaystyle \chi _{0}}$ . Allais has estimated ${\displaystyle \chi _{0}}$ 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 ${\displaystyle f}$ , that is, the inverse of a time period ${\displaystyle T}$.

${\displaystyle f={\dfrac {1}{T}}}$

(7)

 

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

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.^[21] 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. ^[22] 

Beyond adaptive expectations

In standard adaptive expectations models, a type of model that Cagan^[23] and Allais^[24] 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 ${\displaystyle y}$ for a fraction ${\displaystyle k}$, comprised between 0 and 1, of the forecast error ${\displaystyle x-y}$ 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

.

${\displaystyle y_{n}=y_{n-1}+k(x_{n}-y_{n-1})\quad {\text{with}}\quad 0<k\leq 1}$

(8)

or, in continuous time,

${\displaystyle {\dfrac {\mathrm {d} y}{\mathrm {d} t}}=k(x-y)}$

(9)

By recurrence, it is easy to show that

${\displaystyle y_{n}={\dfrac {x_{n}+(1-k)x_{n-1}+(1-k)^{2}x_{n-2}+\cdots +(1-k)^{n}x_{0}}{\dfrac {1}{k}}}}$

(10)

where

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

In

) becomes

${\displaystyle y=k\int _{-\infty }^{t}x(\tau )(1-k)^{(t-\tau )}\mathrm {d} \tau }$

(11)

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

${\displaystyle S=\int _{-\infty }^{t}(1-k)^{t-\tau }\mathrm {d} \tau =\left[{\dfrac {-(1-k)^{t-\tau }}{\ln(1-k)}}\right]_{-\infty }^{t}={\dfrac {-1}{ln(1-k)}}\approx {\dfrac {1}{k}}}$

(12)

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

${\displaystyle \Theta =\int _{-\infty }^{t}(t-\tau )(1-k)^{t-\tau }\mathrm {d} \tau =\underbrace {\left[{\dfrac {-(t-\tau )(1-k)^{t-\tau }}{\ln(1-k)}}\right]_{-\infty }^{t}} _{=0}-{\dfrac {1}{ln(1-k)}}\int _{-\infty }^{t}(1-k)^{t-\tau }\mathrm {d} \tau =\left[{\dfrac {-1}{ln(1-k)}}\right]^{2}\approx {\dfrac {1}{k^{2}}}}$

(13)

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

${\displaystyle {\dfrac {\Theta }{S}}\approx {\dfrac {1}{k}}}$

(14)

or writing

${\displaystyle 1-k=\mathrm {e} ^{-r\mathrm {d} t}\Longleftrightarrow \ln(1-k)=-r\mathrm {d} t}$

(15)

where ${\displaystyle r}$ is a constant continuous rate of memory decay.

we get

${\displaystyle {\dfrac {\Theta }{S}}={\dfrac {1}{r}}}$

(16)

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

${\displaystyle E(y;x)={\dfrac {\dfrac {\mathrm {d} y}{y}}{\dfrac {x-y}{y}}}=k\mathrm {d} t}$

(17)

where ${\displaystyle E(y;x)}$ is the elasticity of ${\displaystyle y}$ with respect to ${\displaystyle x}$, 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 ${\displaystyle k}$ ? Koyck’s transformation gives a rigorous answer to this question when the variable ${\displaystyle y}$ intervenes in a

This being said, a value close to 0 ensures that noise will be smoothed out (as it implies ${\displaystyle y_{n}\approx y_{n-1}}$) 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 ${\displaystyle y_{n}\approx x_{n}}$) , 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.^[26]

The

), 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 ${\displaystyle r}$ - 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 ${\displaystyle \alpha }$, to the forecasting error. By differentiation, relationship (4) implies indeed

${\displaystyle {\dfrac {\chi \prime \prime (Z)}{\chi \prime (Z)}}=\alpha }$

(18)

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

${\displaystyle {\dfrac {\mathrm {d} \ln(\chi \prime (Z))}{\mathrm {d} Z}}=\alpha }$

(19)

Since, by relationship (4), ${\displaystyle \chi \prime =\alpha (\chi -\chi _{min})}$, we get

${\displaystyle {\dfrac {\mathrm {d} \ln(\chi (Z)-\chi _{min})}{\mathrm {d} Z}}=\alpha }$

(20)

Since, by relationship (3), ${\displaystyle \mathrm {d} Z=(x-z)\mathrm {d} t}$, relationship (20) is equivalent to

${\displaystyle {\dfrac {\mathrm {d} \ln(\chi (Z)-\chi _{min})}{\mathrm {d} t}}=\alpha (x-z)}$

(21)

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

${\displaystyle \chi =\underbrace {\chi _{min}} _{\text{fixed part}}+\underbrace {\chi -\chi _{min}} _{\text{variable part}}}$

(22)

so that finally

${\displaystyle {\dfrac {\mathrm {d} \chi }{\mathrm {d} t}}=(\chi -\chi _{min})\left(\mathrm {e} ^{\alpha (x-z)}-1\right)}$

(23)

The relative variation of the variable part of the rate of memory decay being linear in ${\displaystyle Z}$ 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.^[27]. If Allais's formulation is not rational in the sense of Muth, it can be considered as an ecological form of rationality.^[28]

Updating equation

Now, from ${\displaystyle z=\chi Z}$ , we get

${\displaystyle {\dfrac {\mathrm {d} z}{\mathrm {d} t}}=\chi {\dfrac {\mathrm {d} Z}{\mathrm {d} t}}+Z{\dfrac {\mathrm {d} \chi }{\mathrm {d} t}}}$

(24)

which, by relationships (3) and (23), yields the following updating equation

${\displaystyle {\dfrac {\mathrm {d} z}{\mathrm {d} t}}=\chi (x-z)+z\left(1-{\dfrac {\chi _{min}}{\chi }}\right)\left(\mathrm {e} ^{\alpha (x-z)}-1\right)}$

(25)

In contrast to relationship (9), Allais's updating equation is not linear in the forecast error, since the term ${\displaystyle x-z}$ appears twice, one of which as an exponent. Furthermore, unlike ${\displaystyle k}$ in relationship (9), the coefficient ${\displaystyle \chi }$ is variable.

Elasticity of expectations varying between almost 0 and 1

The variable ${\displaystyle z}$ is a function of ${\displaystyle Z}$

${\displaystyle z=z(Z)=\chi (Z)Z}$

(26)

The variable ${\displaystyle Z}$ is itself a function of ${\displaystyle x}$.

Hence, the variable ${\displaystyle z}$ is ultimately a function of ${\displaystyle x}$

${\displaystyle z=z(Z(x))=\chi (Z(x))Z(x)}$

(27)

The following general chain rule

${\displaystyle E(g[f(x)];x)=E(g[f(x)];f(x))\times E[f(x);x]}$

(28)

gives the elasticity of the composition of two functions.

Hence, we have

${\displaystyle E(z[Z(x)];x)=E(z(Z),Z)\times E[Z(x);x]}$

(29)

Elasticity of ${\displaystyle Z}$ with respect to ${\displaystyle x}$'

By relationship (3)

${\displaystyle E(Z(x);x)={\dfrac {\dfrac {\mathrm {d} Z}{Z}}{\dfrac {x-z}{z}}}={\dfrac {\dfrac {(x-z)\mathrm {d} t}{Z}}{\dfrac {x-z}{\chi Z}}}=\chi \mathrm {d} t}$

(30)

Elasticity of ${\displaystyle z}$ with respect to ${\displaystyle Z}$

Again, by relationship (3)

${\displaystyle E(z(Z);Z)=Z{\dfrac {z\prime (Z)}{z(Z)}}=Z{\dfrac {\chi +Z\chi \prime }{\chi Z}}={\dfrac {\chi +Z\chi \prime }{\chi }}}$

(31)

Continuous elasticity of ${\displaystyle z}$ with respect to ${\displaystyle x}$

By relationships (29), (30) and (31)

${\displaystyle E(z[Z(x)];x)={\dfrac {\chi +Z\chi \prime }{\chi }}\times \chi \mathrm {d} t}$

(32)

${\displaystyle E(z[Z(x)];x)=(\chi +Z\chi \prime )\mathrm {d} t}$

(33)

Discrete elasticity of ${\displaystyle z}$ with respect to ${\displaystyle x}$

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

${\displaystyle \ln(1-k(t))=-[\chi (t)+Z(t)\chi \prime (t)]p\quad {\text{with}}\quad p=\Delta t}$

(34)

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

${\displaystyle k(t)=1-\mathrm {e} ^{-p[\chi (t)+Z(t)\chi \prime (t)]}}$

(35)

It immediately follows that

${\displaystyle \lim _{Z\rightarrow +\infty }k(t)=1}$

(36)

and

${\displaystyle \lim _{Z\rightarrow -\infty }k(t)\approx 0}$

(37)

From relationships (36), we get

${\displaystyle \lim _{z\rightarrow +\infty }{\dfrac {\dfrac {\mathrm {d} z}{z}}{\dfrac {x-z}{z}}}=\lim _{z\rightarrow +\infty }{\dfrac {\mathrm {d} z}{x-z}}=1}$

(38)

which implies

${\displaystyle z+\mathrm {d} z\approx x}$

In other words, when ${\displaystyle z\rightarrow +\infty }$ (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

, 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 ${\displaystyle x-z}$ is not correlated with the expectation ${\displaystyle z}$.

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”

risks

are nothing but the present value ${\displaystyle Z}$ of past returns (${\displaystyle x(\tau )}$) or losses (${\displaystyle x(\tau )<0}$).

Bull markets have a short memory and hence are inherently unstable

This analytical framework implies that the memory of market participants shrinks during a

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

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.

Allais’s HRL formulation and other approaches

Behavioral finance

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

.

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).

---