ISSN (Print) 0473-453X

Discussion Paper No. 1249

ISSN (Online) 2435-0982

SMOOTH DIAGNOSTIC EXPECTATIONSFrancesco Bianchi
Cosmin Ilut
Hikaru Saijo

July 2024

The Institute of Social and Economic Research
Osaka University
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan

Smooth Diagnostic Expectations∗Francesco Bianchi

Cosmin Ilut

Hikaru Saijo

Johns Hopkins University

Duke University

UC Santa Cruz

CEPR and NBER

NBER

July 4, 2024

Abstract
We show that in the formalization of representativeness (Kahneman and Tversky(1972)) developed by Gennaioli and Shleifer (2010), overreaction and confidence areaffected by uncertainty, as a news effect interacts with an uncertainty effect. In thetime series domain, this interaction emerges in a smooth version of Diagnostic Expectations (DE). Under smooth diagnosticity, agents overreact to new information.
Since new information typically changes not just the conditional mean, but also theconditional uncertainty, changes in uncertainty surrounding current and past beliefs
affect the severity of the DE distortion and confidence. Smooth DE implies a joint
and parsimonious micro-foundation for key properties of survey data: (1) overreactionof conditional mean to news, (2) stronger overreaction for weaker signals and longer
forecast horizons, and (3) overconfidence in subjective uncertainty. An analytical RBC
model featuring Smooth DE accounts for overreaction and overconfidence in surveys, aswell as three salient properties of the business cycle: (1) asymmetry, (2) countercyclicalmicro volatility, and (3) countercyclical macro volatility.
JEL Codes: D91, E32, E71.
Keywords: Diagnostic expectations, uncertainty, learning, overreaction, overconfidence∗

Emails: francesco.bianchi@jhu.edu, cosmin.ilut@duke.edu, and hsaijo@ucsc.edu. We thank Joel Flynn,
Spencer Kwon, Chen Lian, Karthik Sastry, Andrei Shleifer, Stephen Terry, Alonso Villacorta, as well as
conference participants at the ASSA 2024 Annual Meeting. Saijo gratefully acknowledges the financial
support of the Grants-in-Aid for Scientific Research (JP21H04397) from the Japan Society for the Promotionof Science.

1

Introduction

There has been a growing interest in psychological foundations that enrich models of belief
formation in economics. A prominent example is the widely-documented “representative
heuristic” of Kahneman and Tversky (1972), which serves as the underpinning of a recent
and expanding literature on the paradigm of Diagnostic Expectations (DE). According tothis heuristic, when new information arrives, as measured with respect to a reference distribution based on past data, memory selectively recalls more vividly past events that are
more associated with, or representative of, that current news. Models of DE formalize thedetails on how memory retrieval distorts the subjective probability of uncertain events awayfrom its objective, “kernel of truth,” frequency (see Bordalo et al. (2022) for an overview).
One immediate manifestation of the kernel of truth logic is that when the new information
completely eliminates uncertainty over the variable to be forecasted, there is objectively noroom for memory to distort conditional judgements (Gennaioli and Shleifer (2010)). While
in existing DE models this logic holds in its extreme version of no conditional uncertainty,
the literature has so far largely ignored the fact that representativeness, as formalized in
Gennaioli and Shleifer (2010) and Bordalo et al. (2016), also features a deeper and pervasive
relation between diagnosticity and uncertainty. Specifically, representativeness implies thatthe distance between the subjective distorted distribution and its objective counterpart depends on both the amount of information received and the precision of the revised objective
distribution. Uncertainty affects the strength of the representativeness distortion and leads
to variation in confidence, accommodating both cases of under- and overconfidence.
After illustrating the connection between representativeness and uncertainty in a simple,
static, two-state example, we argue that in the time series domain this connection manifests itself as a “smoothed” version of Diagnostic Expectations, in which the severity of
the DE distortion and confidence depend on conditional uncertainty. As a result, Smooth
Diagnostic Expectations (Smooth DE) end up connecting two vastly popular branches of
Economics that have largely proceeded in parallel: the Diagnostic Expectations literatureand the Uncertainty literature (Bloom (2009, 2014), Bloom et al. (2018), Baker et al. (2024)).
Representativeness and Uncertainty. We first study the relation between uncertainty
and representiveness in a simple static two-state categorical distribution. We build on theclassic example of inferring the probability of someone having red hair. In this example,
representativeness implies that if the person of interest is revealed to be Irish, agents overreact
and tend to overstate the probability that the person has red hair. This is because red hairis more representative of the Irish population with respect to the world population. We
use this example to emphasize that the strength of the overreaction is not monotonic, but

1rather hump-shaped. This is because two channels are at work: A news channel, measured
by the distance between the current and reference distribution, and an uncertainty channel,
measured by the entropy of the current distribution.
Suppose that there are only two categories - red and non-red hair color - and that,
before any information is revealed about the particular person, the hair color red has alow probability. As information is revealed that makes the red hair more likely (say, Irish
nationality), overreaction initially increases with the conditional probability assigned to thecolor red. This is because initially the news effect and the uncertainty effect work in the same
direction: the color red is more likely and there is more uncertainty on the hair color, as an
ex-ante low-probability hair color became more likely, but not certain. However, if the new
information makes the objective probability cross the point of maximum uncertainty (50:50
in a two-state categorical distribution), the uncertainty effect starts working in the opposite
direction. There is now progressively less uncertainty on the hair color and the distance
between subjective and objective distributions declines. In the limit, when the probabilityof red hair goes to 1, uncertainty is fully removed, the uncertainty channel dominates thenews channel, and the subjective and objective distribution coincide.
This interaction between the two channels also generate variation in the level of confidence. For small increases in the probability of the color red, the less likely hair color, thedistorted distribution features more uncertainty than the objective distribution because, byoverreacting, agents bring the perceived distorted probability closer to the point of maximum uncertainty (50:50). Eventually, overconfidence emerges, as agents’ overreaction leads
to attribute more than .5 probability to the color red even if the objective distribution has
not crossed the point of maximum uncertainty. As the probability of the hair color red
approaches 1, the severity of overreaction starts declining and eventually goes to zero.
Smooth Diagnostic Expectations. The natural adaptation of representativeness to thetime series domain involves a “smoothed” version of Diagnostic expectations. Under Smooth
DE the severity of the DE distortion and confidence depend on conditional uncertainty, in away similar to what we documented for categorical distributions. Agents overreact to new
information, defined as the difference between the current information set and a previous
information set. Since new information typically changes not just the conditional mean, butalso the conditional uncertainty, changes in uncertainty surrounding current and past beliefs
affect the extent of the DE distortion.
Smooth DE stems from a minimal, but conceptually important change to the baseline DE
paradigm developed by Bordalo et al. (2018) (BGS) and it aligns well with the original “representative heuristic” of Kahneman and Tversky (1972). In the BGS formulation of DE, thereference distribution is centered on the conditional mean under the true density formed at2some given past time, but shares the same uncertainty as the true distribution conditional on
current information. Instead, since we condition exclusively on the past information set, thereference distribution reflects the level of uncertainty at that past time in which expectations
were formed. This is in line with the original formalization of representativeness developedby Gennaioli and Shleifer (2010) and Bordalo et al. (2016) for categorical distributions, inwhich information sets between current and reference distributions are kept distinct.
Smooth DE is thus built on the key informational difference between conditional (orposterior) and unconditional (prior) distribution information. In this sense, our approach
relates to recent work in Bordalo et al. (2020), which features a sampling by similarity
framework that under conditional probability assessments yields a result reminiscent of DE.
In a similar spirit to Smooth DE, in that setting it is important to keep track of the entire
prior distribution, which plays a major role in memory interference. In fact, a version ofwhat we label Smooth DE appears in chapter 5 of Gennaioli and Shleifer (2018). However,despite this early appearance, the growing DE literature has focused on the simplified version
proposed in BGS. In this paper, we argue that representativeness features a fundamental
connection between memory recall and uncertainty. As a result, Smooth DE has distinctive
properties that render it an important point of contact with the uncertainty literature andhelp to make sense of several stylized facts.
When the current and reference distributions are Normal, the baseline BGS formulationdelivers a distorted distribution that is also Normal, but in which only the mean is affected by
DE. In comparison, under Smooth DE we uncover two key novel properties of the distorted
distribution. First, the severity of the Smooth DE distortion decreases as the current levelof uncertainty decreases compared to past uncertainty. Put differently, we obtain a smoothversion of the logic expressed by Gennaioli and Shleifer (2010), as now an agent is less prone
to overreact to the new information the more precise the current information is with respect
to past information. In the limit, as uncertainty is fully resolved by the new information,
the distortion vanishes, as in the baseline DE. However, with Smooth DE, the extent of the
distortion varies smoothly as current uncertainty increases with respect to past uncertainty,
while the baseline DE features a discontinuity once current uncertainty goes to zero.
Second, Smooth DE delivers a disconnect between the objective and subjective level of
uncertainty. This is because under Smooth DE, not only the mean, but also the variance of
the DE distribution is distorted. When agents experience a reduction in uncertainty withrespect to the reference distribution, agents over-state the precision of their forecasts, leading
to overconfidence. In other words, in that case the DE distribution features a variance lower
than under Rational Expectations (RE). Given that typically events close in the future areeasier to predict than events far into the future, agents’ beliefs will typically feature such3

overconfidence. However, the Smooth DE paradigm can also accommodate underconfidence
following an increase in uncertainty, like in response to an uncertainty shock (Bloom (2009)).
A parsimonious micro-foundation for survey evidence. As the traditional DE, Smooth
DE is characterized by a primitive stochastic environment and two parameters controlling(i) the severity of the distortion, θ > 0, and (ii) the lag of the reference distribution, J ≥ 1.
Thus, Smooth DE makes use of no additional degree of freedom. Instead, by allowing thereference distribution to be based only on the information set available at some given pasttime, the kernel of truth logic endogenously generates predictions for the effective distortion.
Under Smooth DE, the primitive parameter θ > 0 measures the severity of the DE distortion for a given level of relative uncertainty, while the effective severity changes with therelative uncertainty. These disciplined predictions allow Smooth DE to offer a parsimonious
micro-foundation for a wide range of stylized facts.
The novel property that the effective overreaction to news is stronger when relative uncertainty is higher helps to account for the stylized survey fact that overreaction increases withthe horizon of the survey forecast (see for example Bordalo et al. (2019), d’Arienzo (2020),
Bordalo et al. (2020), Augenblick et al. (2021), and Bordalo et al. (2023)). For standard
stationary processes the same piece of information is less informative about horizons further
in the future. Critically, under Smooth DE this relatively smaller reduction in conditional
uncertainty leads to a relatively stronger overreaction to news for longer horizons forecasts,
consistent with the stylized findings.
The property that Smooth DE implies a disconnect between subjective and measured
uncertainty makes the proposed framework relevant for a separate literature on overconfidence. Recent work documents that in survey data firms (i) overreact to news and (ii) areoverconfident in their subjective forecasts (see, Barrero (2022), Born et al. (2022), and thereviews in Altig et al. (2020) and Born et al. (2022)). While the baseline DE model canaccount for overreaction, it is silent on overconfidence. Smooth DE can instead account for
both these seemingly separate properties since it distorts both the mean and the variance ofagents’ expectations in a way to typically generate both overreaction and overconfidence.
More broadly, the overreaction and overconfidence properties have been typically studied as two distinct behavioral departures from full rationality (see Barberis (2018) for an
overview). While overreaction has been typically the focus on the standard DE literature,a separate literature (including for example De Bondt and Thaler (1995) and Daniel et al.
(1998, 2001)) is motivated by extensive psychological evidence for overconfidence and argues
that models based on this behavioral property are promising in accounting for asset market
puzzles. Our results indicate that Smooth DE can offer a joint micro-foundation, basedon the representativeness heuristic, of these two-widely documented and studied departures4

from standard Bayesian updating.
Business cycle implications. We leverage our theoretical insights to study a parsimonious
business cycle model with time-varying uncertainty to illustrate how state-dependent overreaction from Smooth DE generate important cyclical implications. We consider an island
economy subject to economy-wide and island-specific TFP shocks. Following Bloom et al.
(2018), we assume that the island-specific TFP shocks are subject to time-varying volatility that is negatively correlated with economy-wide TFP innovations. We show that thisparsimonious model can account for Barrero (2022)’s survey evidence on overreaction andoverconfidence, as well as three key empirical properties of the business cycle: (1) asymmetry(recessions are deeper than expansions), (2) countercyclical micro volatility (cross-sectionalvariances of microeconomic variables rise in recessions), and (3) countercyclical macro volatility (time-series variances of macroeconomic variables rise in recessions).1
First, consider the asymmetry property. A negative economy-wide TFP shock generates
higher uncertainty about the island-specific TFP shocks. Hence, agents overreact to theeconomy-wide TFP shock more than usual, leading to a sharper fall in hours, consumption, and output. In contrast, a positive TFP shock reduces agents’ uncertainty, and therise in economic activity is mild. Second, consider countercyclical micro volatility. In recessions, agents face higher uncertainty, so they overreact to the island-specific TFP and
as a result, the cross-sectional variances of island-level hours, output, and consumption
increase. Conversely, during expansions, agents’ overreactions are milder, and hence thecross-sectional dispersion decreases. Third, consider countercyclical macro volatility. The
state-dependent overreaction implies that in recessions, economic activity responds strongly
to an economy-wide shock to TFP, while in expansions the responses are more muted. As
a result, the aggregate volatility rises in recessions even when there is no change in thevolatility of economy-wide shocks. These mechanisms highlight that the micro-level uncertainty and macroeconomic volatility are tightly linked through the agent’s state-dependent
overreaction. As a result, a novel policy implication emerges: a redistributive policy that reduces idiosyncratic uncertainty could be beneficial for macroeconomic stabilization because
it dampens this state-dependent overreaction.
1

These properties have been extensively documented in the literature. For instance, Neftci (1984), Hamilton (1989), Sichel (1993), McKay and Reis (2008), and Morley and Piger (2012) show macroeconomic asymmetries using various econometric approaches. Bloom (2009), Fernández-Villaverde et al. (2011), Ilut et al.
(2018), Jurado et al. (2015), Basu and Bundick (2017), and Bloom et al. (2018) document that volatility or
uncertainty rise in recessions at the micro and macro levels.

5

2

Representativeness and Uncertainty

The building block for our Smooth DE model is the concept of representativeness. Here
we directly use the definition of representativeness introduced in recent work on selective
memory by Gennaioli and Shleifer (2010), Bordalo et al. (2016), Bordalo et al. (2020). Thisdefinition, detailed below, captures formally Tversky and Kahneman (1975) own definitionof representativeness: “an attribute is representative of a class if it is very diagnostic; thatis, the relative frequency of this attribute is much higher in that class than in a relevant
reference class” (p. 296).
To understand how we later build on this definition in a time series dimension, we firstconsider some simple examples, within a very standard probability environment that is staticand discrete. These examples illustrate how representativeness implies that overreaction to
the relative frequency is shaped by the entire distribution of attributes, both in the class ofinterest and the relevant reference class, consistently with the above definition by Tversky
and Kahneman (1975). We highlight how the amount of uncertainty characterizing these
distributions is of particular importance in shaping the effective manifestation of representativeness. This focus on the role of uncertainty will allow us to draw an explicit connection
between the building block of representativeness and Smooth DE as developed in Section 3.
Representativeness. Consider a simple environment where (Ω, P ) is a discrete probability
space, with two random variables defined on this space: X, the trait that the agent seeks toassess, and D, the available data. The agent looks to form the conditional probability of agiven trait xb ∈ X, given a particular realized data d ∈ D. A Bayesian agent would simply
use the conditional probability P (bx|d) = P (bx ∩ d)/P (d). The environment here is static,
without a sense of repeated accumulation of information.
In assessing conditional probabilities, the agent subject to the representativeness heuristic
distorts the Bayesian belief. Here we follow the formalization of representativeness in Gennaioli and Shleifer (2010), Bordalo et al. (2016), Bordalo et al. (2020). This work explainsin detail how representativeness can be intuitively understood as the tendency to overweight
representative traits, arising due to limited memory and the fact that representative traits
are easier to recall. Formally, the representativeness of a given trait x
b conditional on a particular data d compared to another conditioning data realization, dref (referred to as thereference data) is defined as the relative frequency of that trait across the two data groups:
rep(bx|d, dref ) ≡

P (b
x|d)
P (b
x|dref )(1)

The conditional belief under representativeness is modelled as a weight that distorts the6underlying absolute frequency of xb conditional on the data dP θ (bx|d, dref ) = P (bx|d)weight(bx|d, dref ).

(2)

The weight reflects the effect of the relative frequency in equation (1), and is given byθ
weight(bx|d, dref ) = rep(bx|d, dref )1.
Z(d, dref )(3)

where Z(d, dref ) is a constant of integration, ensuring that the distorted probability integrates
to one:XθZ(d, dref ) =P (bx|d) rep(bx|d, dref )
(4)
x
b∈X

In this formal representation representativeness, the parameter θ ≥ 0 measures the extent
to which the relative frequency across two conditioning data groups affects judgements.
When θ = 0, the agent’s memory retrieval is perfect and beliefs collapse to the standard
frictionless model. When θ > 0, memory is limited and the agent’s judgments are shaped byrepresentativeness.
Illustrations. Our simple illustrations below let Ω be the universe of people, X be different
hair colors, and d ∈ D take on specific Nationalities values, with one of them being a reference
dref = {W orld}.2 We evaluate the effects of representativeness of the event x
b conditional onrefa specific Nationality group, compared to the reference group d .

2.1

Two-state distribution

Consider the case where X assume only two values, “red” or “non-red/other”, which wedenote as xb = R and x
b = N R, respectively. The agent observes the nationality of an
unknown individual and assess the probability of the hair color red based on P θ (bx =R|N ationality, W orld) as defined in equation (2). In particular, the agent twists the underlying probability P (bx = R|N ationality) by the weight weight (bx = R|N ationality, W orld)
defined by equations (3) and (4).
We are interested in how the distorted probability distribution and its distance (appropriately defined) from the actual conditional distribution depend on the underlying probability
distributions. We emphasize the following three qualitative properties.
2

The particular numbers we will use are simply chosen to illustrate a series of key features of representativeness and are not necessarily reflecting the actual incidence of the hair colors in the world population. In
fact, we will implicitly assume that there are nationalities for which the probability of the hair color red is 1.

7

Figure 1: News and uncertainty in a two-states discrete distribution

Notes: The figure displays how, as we vary the actual P (bx = R|N ationality), the following objects change:(a.1) the distorted probability of red hair, (a.2) the distorting weight on the red hair probability, (b.1) thenews component, measured using the KL defined in equation (5), (b.2) the uncertainty component,
measured using the normalized entropy of the actual distribution in equation (6), (c.1) the overall
overreaction, measured by the KL divergence of the distorted probability distribution from the actual one,defined in equation (7), and (c.2) the entropy of the distorted distribution relative to the entropy of theactual distribution. We fix the reference probability of a red hair color at P (bx = R|W orld)=0.2.

1. Overreaction. The distorted probability amplifies the relative change in frequency.
This first and most immediate characteristic of representativeness can be immediately seen
by equation (2). We show it graphically in Figure 1 (a.1), where we plot the distortedP θ (bx = R|N ationality, W orld) for different values of P (bx = R|N ationality) and diagnosticity parameter values θ = 1, 2, as well as θ = 0 corresponding to no memory distortion. The
frequency under the reference data is P (bx = R|W orld) = 0.2 marked by a vertical red dottedline. When the frequency of red hair for a given Nationality is higher (lower) than under thereference data, the distorted probability is higher (lower) than the actual frequency for that
Nationality. The magnitude of the difference is increasing in θ.
2. News and uncertainty. Representativeness is characterized by an effective degreeof overreaction that is a function of the entire shape of the distributions for the two datagroups that get compared. This property has typically been outside the main interest of8

recent applied work on representativeness and diagnostic expectations. However, it will play
an important role in our development of Smooth DE.
We separate two channels that create this relation between overreaction and the amountof information present in the conditional distributions that get compared. On the one hand,equation (3) features a news effect. When the knowledge of the Nationality implies a larger
revision in the frequency of red hair, the representativeness rep(bx = R|N ationality, W orld)changes by more. Holding fixed the constant of integration Z(N ationality, W orld) in equation (3), the distorting weight is then larger in absolute terms, and therefore there is more
overreaction to the news. On the other hand, there is an uncertainty effect: the more theknowledge of a given Nationality decreases the uncertainty over the hair color, the less room
there is for overreaction to effectively manifest itself. This intuitive force mathematicallyappears in equation (3) through the constant of integration of equation (4). The news anduncertainty effects are jointly determined by the shapes of distributions getting comparedthrough representativeness. We dissect these effects through a series of plots in Figure 1.
First, Panel (a.1) already shows that the difference between the distorted and underlying
probability is hump-shaped. This difference is largest for intermediate values between theprior of 0.2 and the lower and upper bounds (0 and 1). By equation (2) the reason for thehump shape is the behavior of the distorting weight (bx = R|N ationality, W orld), which getsplotted in Figure 1 (a.2). Indeed, the weight is non-monotonic in the actual frequency: theweight is increasing from an underlying P (bx = R|N ationality) = 0 to some intermediate
value between the reference frequency of 0.2 and the upper bound of 1 and then starts
decreasing. Formally, this is because the constant of integration in equation (4) becomeslarger as the frequency P (bx = R|N ationality) approaches 1. Intuitively, as the actual
frequency for a given Nationality becomes more certain towards one hair color, the room for
distortion becomes smaller as the total probability has to sum to 1.
Second, we dissect the news and uncertainty channels by using information-theoretic
measures. In Panel (b.1), we measure the size of the news component. This is capturedby the change in the distribution induced by conditioning on a Nationality as opposed toconditioning on World, as measured by the Kullback-Leibler (KL) divergence of the two
distributions:


X
P (b
x = i|N ationality).
(5)
KL(Nationality||World) =P (bx = i|N ationality) ln
P (bx = i|W orld)i
The KL divergence is U-shaped, with KL = 0 when the knowing the Nationality does not
imply any revision in the distribution.
To measure uncertainty for the objective distribution of hair colors for a given Nationality,9Panel (b.2) in Figure 1 plots the normalized entropyPΩ(P (N ationality)) = −x = i|N ationality) ln P (bx = i|N ationality)i P (b
ln(2)

,

(6)

where the denominator ensures that the normalized entropy is between 0 and 1. This entropy
measure of uncertainty is particularly suitable to capture uncertainty in this environment of
categorical random variables. The entropy measure shows an inverted U-shape and peaks
when P (bx = R|N ationality) = 0.5, the point of maximum uncertainty in this two-state case.
To measure the extent to which overreaction distorts the perceived frequency with respect to the objective frequency, Panel (c.1) reports the KL divergence between the two
distributions

 θ
X
P (b
x = i|N ationality, W orld)θθ(7)
KL(P ||P ) =P (bx = i|N ationality, W orld) ln
P (bx = i|N ationality)i
We can now see how the degree of overreaction shows two humps: a smaller one that peaks
at the intermediate value between 0 and reference frequency P (bx = R|W orld) = 0.2 and alarger one that peaks at the intermediate value between that reference of 0.2 and the upperbound of 1. The two humps are the result of the interaction between news and uncertainty
effects, which affect overreaction, but display opposing patterns (U- and inverted U-shape).
The particular sizes of the two humps are different because in panels (b.1) and (b.2) thetrough of the news component and the peak of the uncertainty effect do not coincide.
3. Over/under confidence. The overreaction to information also affects the amountof uncertainty perceived by the agent compared to the actual frequency. Panel (c.2) reports
the entropy Ω(P θ (N ationality)) of the distorted distribution normalized with respect tothe entropy Ω(P (N ationality)) of the actual frequency. A relative entropy smaller than
0 implies overconfidence; the agent is too certain about the hair color compared to theobjective distribution. The interaction between the news effect and uncertainty effect leads
to an interesting pattern.
First, there is no over- or underconfidence if there is no news (i.e. the distributions
conditional on Nationality or World coincide) or the news completely removes uncertainty(P (bx = R|N ationality) = {0, 1}). Second, when the frequency of red hair is smaller for agiven Nationality than for the World, news and uncertainty effects work in the same direction,and only over-confidence is possible. Third, when instead the relative frequency is larger, thetwo effects can work in opposite directions. On the one hand, uncertainty keeps increasing
until the frequency is .5, at which point uncertainty starts decreasing again. As a result, for10

small increases in the relative frequency of red hair, over-reaction to that relative frequency
can lead to underconfidence. Eventually, the news effects starts dominating, leading tooverconfidence. However, the strength of overconfidence declines as the news starts to fully
remove uncertainty. The degree of overconfidence is more severe for an actual frequency on
the right tail (P (bx = R|N ationality) > 0.8) than on the left tail (P (bx = R|N ationality) <0.2). This is because the news effect is larger for a frequency on the right of a small probability
event such as the one considered in this example.

2.2

Three-state distribution

Consider now an extension of the event space to three hair colors - “red,” “blond,” and“dark,” denoted by R, B, and D, respectively. We consider a symmetric reference frequency
for R and B so that conditional on World, their frequencies are equal to 0.2, while for D it
is equal to 0.6. This three-state example is instructive because the actual frequencies of R
and B for a given Nationality jointly determine the news and uncertainty component, thus
underscoring the importance of the shape of the joint probability distribution in affecting
the degree of overreaction.
In Figure 2, we use contour plots to examine how news and uncertainty affect overreaction
as we vary P (bx = R|N ationality) and P (bx = B|N ationality) when θ = 2. To ease interpretation, we mark the coordinates corresponding to the reference frequencies with white
pluses, and the coordinates representing the equal probabilities of 1/3 for all three outcomeswith red diamonds. We can recover the three properties discussed for the two-state example:
1. Overreaction. Panels (a.1) and (a.2) report the distorted probability and the weightassociated with the red hair color. As in the two-states example, the distorted probabilities
overreact to the relative frequency of red hair for the given Nationality compared to theWorld. The distorting weight of the red probability peaks when P (bx = R|N ationality) takes
an intermediate value between 0.2 and 1 and P (bx = B|N ationality) takes an intermediate
value between 0 and 0.2. Similarly, Panel (b.1) and (b.2) report the distorting probability
and the weight associated with a blond hair color.
2. News and Uncertainty. The three state example is particularly useful here to see
how the shape of the distributions matter. Panel (c.1) displays the news component, measured again as the KL(Nationality||World) of equation (5). The news component increases
as the two distributions differ and peaks when either of the hair color outcomes reaches
certainty (P (bx = R|N ationality) = 1 or P (bx = B|N ationality) = 1). The uncertainty
component, Ω(Nationality), reported in Panel (c.2) peaks when the distribution for a given
Nationality assigns equal probabilities for all three outcomes.
11Figure 2: News and uncertainty in a three-states discrete distribution

Notes: The figure displays how, as we vary the probability of red and blond hair colors for a given
Nationality, the following objects change: (a.1) the distorted probability of red hair, (a.2) the distorting
weight on the red hair probability, (b.1) the distorted probability of blond hair, (b.2) the distorting weighton the blond hair probability, (c.1) the news component, measured using the KL defined in equation (5),(c.2) the uncertainty component, measured using the normalized entropy of the actual distribution inequation (6), (d.1) the overall overreaction, measured using the KL divergence of the distorted probability
distribution from the actual one, defined in equation (7), and (d.2) the entropy of the distorted probability
distribution relative to the entropy of the actual distribution. We mark the coordinates corresponding to
the reference probabilities with white pluses, and those representing equal probabilities for all threeoutcomes with red diamonds.

Panel (d.1) reports the KL(P θ ||P) divergence of equation (7). The overreaction displays
twin peaks resembling the double humps in the two-states example. However, in the current
three-state example how the actual frequencies are split matters for the overreaction. Indeed,
the overreaction is mild as we increase both the actual probabilities of red and blond hairsimultaneously in equal proportions. This is because both hair colors are becoming more
representative for the given Nationality compared to the World.
3. Over/under confidence. Finally, Panel (d.2) reports the entropy Ω(P θ (N ationality))
with respect to the underlying entropy Ω(P (N ationality)). The former is smaller in mostregions, indicating overconfidence, except for the neighborhood of the reference frequency
12and the coordinate representing equal probabilities for all three outcomes. Overconfidence
is most severe around the three corners of the triangle, when the actual frequency puts mostprobability weights on one hair color. Intuitively, as one specific hair color becomes more
representative, the agent overestimates the probability of that hair color and becomes too
certain. As before, overconfidence eventually disappears as uncertainty is fully removed.
To summarize, both the two-and three-states example underscore a nuanced but tightconnection between news/uncertainty effects and overreaction/overconfidence.

3

Smooth Diagnostic Expectations

In this section, we show that the key principles uncovered for categorical distributions naturally find their counterparts under Smooth DE, which allows for continuous distributions andaccumulation of information. In particular, in order to model the representativeness heuristic into time-series, we mirror the logic of equation (1), and interpret groups as different
information sets that become available over time.

3.1

Smooth DE density: representativeness and information sets

We introduce the definition of representativeness and information sets with some generality.
Consider the filtered probability space (Ω, F, (Ft )t≥0 , P ) where (Ft )t≥0 is an increasing familyof sub-σ-algebras of F, with Ft ⊂ Ft+1 . Here the filtration (Ft )t≥0 represents the evolution of
information over time. A stochastic process (Xt )(t≥0) on (Ω, F, P ) is adapted to the filtration(Ft )t≥0 if, for each t ≥ 0, Xt is Ft -measurable. Let the conditional density function of Xt+h ,for some horizon h ≥ 1, given the filtration Ft , be denoted by f (xt+h |Ft ). This function
describes the probability density of Xt+h , given all the information available up to time t.
In our approach, the counterpart of equation (1) is obtained by defining the referencegroup as a past available information set. As information flows and is accumulated, representativeness is then defined as follows:
Definition 1 (Representativeness and information sets) Given the current information set Ft and a reference information set Ft−J , formed J ≥ 1 periods ago, the representativeness of a random event x
bt+h for some future horizon h ≥ 1 is defined asrep (bxt+h |Ft , Ft−J ) ≡f (bxt+h |Ft )f (bxt+h |Ft−J )(8)Mirroring the language of Tversky and Kahneman (1975) in Section 2, in the time-series
domain an event is representative for the current information set if the relative frequency of13this event is higher conditional on the current information set than conditional on some pastreference information set.3
Based on this definition of the representativeness of an event, we then follow the same
construction of the distorted belief as introduced early in equation (2) in Section 2. In
particular, we build the following distorted conditional density, which we refer to as Smooth
Diagnostic Expectations, or in a more abbreviated form as Smooth DE.
Definition 2 (Smooth Diagnostic Expectations). In the time series domain, the conditional density f θ (bxt+h |Ft , Ft−J ), distorted by representativeness as defined in Definition 1,
is constructed as followsf θ (bxt+h |Ft , Ft−J ) = f (bxt+h |Ft ) [rep (bxt+h |Ft , Ft−J )]θ

1
Z

(9)

where Z is a constant of integration and the parameter θ ≥ 0 measures the distortion severity.
Our formalization of Smooth DE strongly connects to that of Diagnostic Expectations(DE) introduced in Bordalo et al. (2018) (BGS) to incorporate representativeness in timeseries. As we detail further below, the only, but consequential difference is in the constructionof the reference group in the definition of representativeness. Given such a definition, thefunctional form for the distorted density in equation (9) is identical to DE.

3.2

Smooth DE with Normal densities

In the rest of this paper we follow BGS and focus on normal densities. As we show below, thisfocus leads to significant gains in tractability and in the range of possible applications. Thisis because when the true conditional density is Normal, expression (9) delivers a distorted
distribution that is also Normal. However, it is important to emphasize that while this anatural point of departure, Smooth DE can easily accommodate alternative distributions, as
illustrated in Section 2.
With Normal densities, two moments, the conditional mean µt+h|s and conditional vari2
ance σt+h|s, summarize the whole conditioning information:
2f (bxt+h |Fs ) = N x
bt+h ; µt+h|s , σt+h|s
,

for

s≤t

It follows that given Definition 1, the representativeness of the event x
bt+h is evaluated asrep (bxt+h |Ft , Ft−J ) =
3

2
N (b
xt+h ; µt+h|t , σt+h|t)2
N (bxt+h ; µt+h|t−J , σt+h|t−J
)

,

(10)

The approach is easily extended when the available information is incomplete or noisy - see Section E.

14

capturing its relative frequency under the current information set Ft , compared to some past
information set Ft−J , with J ≥ 1.
Under the representativeness characterizing normal densities in equation (10), the Smooth
DE density in equation (9) has a closed-form solution given by Proposition 1 below.
Proposition 1 Denote the ratio of variances for the current and reference groups as2
2
Rt+h|t,t−J ≡ σt+h|t/σt+h|t−J(11)

If Rt+h|t,t−J < (1 + θ) /θ, the Smooth DE density f θ (bxt+h |Ft , Ft−J ) in equation (9) is Normal
with conditional meanEθt (xt+h ) = µt+h|t + θ
Rt+h|t,t−J µt+h|t − µt+h|t−J
1 + θ 1 − Rt+h|t,t−J(12)and conditional varianceVθt (xt+h ) =2
σt+h|t1 + θ 1 − Rt+h|t,t−J



(13)

Proof. See Appendix.
The condition Rt+h|t,t−J < (1 + θ) /θ guarantees that the variance of the resulting distorted Normal distribution is finite and positive. As the ratio of conditional variances between the current and reference distribution approaches this limiting value, the variance of
the Smooth DE distribution approaches infinity and the corresponding Normal distribution
approaches a degenerate, flat distribution. Thus, the condition requires that the current
2) is not too high with respect to the past
uncertainty with respect to a future event (σt+h|t2). The condition typically holds in stationary
uncertainty about the same event (σt+h|t−J
environments with homoskedastic innovations in which events closer into the future are easier to predict than events far into the future. However, the condition also allows for the
possibility of an increase in uncertainty, for example as a result of heteroskedasticity, as longas the increase is not too large with respect to the DE distortion.4
4

In Appendix B we discuss an approach that deals with this threshold condition by implementing an
upper bound on the Smooth DE overreaction in the conditional mean. The approach guarantees that boththe mean and variance distortions remain finite and non-decreasing as the ratio Rt+h|t,t−J goes to infinity.

15

3.3

Three key properties

Smooth DE is characterized by three important properties. To understand them, it is helpful
to define the effective overreaction of the conditional mean to news in equation (12) as
θet,t−J ≡ θ

Rt+h|t,t−J.
1 + θ 1 − Rt+h|t,t−J(14)Corollary 1 Assume the presence of residual uncertainty with respect to a future event:2
σt+h|t> 0. Compared to the RE forecast (θ = 0), the conditional forecast under Smooth DE(θ > 0), characterized in Proposition 1, exhibits
1. overreaction of the conditional mean to new information, sinceθet,t−J > 0(15)

2. an effective overreaction of the conditional mean to new information that is monotonically increasing in the ratio Rt+h|t,t−J between current and past uncertainty∂ θet,t−J>0∂Rt+h|t,t−J(16)

3. overconfidence when Rt+h|t,t−J < 1, since then by equation (13)2Vθt (xt+h ) < σt+h|t(17)

or underconfidence when Rt+h|t,t−J > 1, since then by equation (13)2Vθt (xt+h ) > σt+h|t.

(18)

On the one hand, these properties mirror the qualitative ones for the static, discrete
state example introduced and discussed in Section 2.1 - properties labelled there ’overreaction’, ’news and uncertainty’, and ’over/under confidence’, respectively. The formalism
and closed-form solution of Normal densities make these properties particularly transparent.
This connection is useful, indicating that Smooth DE maintains similar insights when extending the concept or representativeness and over-reaction to new information to dynamic
models. Additionally, the tractability of the Normal distribution in the time series domain
will allow us to connect these three properties to stylized survey facts in Section 4.
The first property in Corollary 1, overreaction of the conditional mean, is an immediate16

manifestation of the amplification of the relative frequency of an event that becomes more
likely. This property is shared with the standard DE approach, as we detail further below.
The other two properties are novel and reflect the tight connection between uncertainty andthe degree of over-reaction, in line with the analysis presented in Section 2.
In particular, like in the static, discrete case, the shapes of the current and reference
densities matter for the effective degree of overreaction. In the Normal case, the shape issummarized only by the conditional uncertainty. Indeed, Smooth DE captures formally,by equation (16), how the severity of the mean distortion increases as the ratio Rt+h|t,t−J
of today’s uncertainty to past uncertainty increases. Smooth DE thus micro-founds in atime-series domain an inverse smooth link between overreaction of conditional mean to news
and the precision of the news compared to the reference distribution. This is captured by2compared to
the ratio Rt+h|t,t−J : The more the new information reduces uncertainty σt+h|t2
σt+h|t−J , the lower is the role of memory in distorting probability judgements, and thus thelower is the effective overreaction to news. Conversely, everything else equal, the larger
today’s uncertainty, the larger the Smooth DE distortion.
Similar to the basic intuition of the static, discrete case, Smooth DE also has important implications for the level of subjective confidence that agents express with respect totheir expectations. As summarized in Corollary 1, if agents experience a reduction of uncertainty with respect to the reference distribution, so that Rt+h|t,t−J < 1, Smooth DE implies
overconfidence, i.e. agents overstate the precision of their expectations. Under this scenario,
independently of the direction and size of the mean distortion, agents are overconfident about
the precision of their expectations. If agents do not experience a change in uncertainty, and2. Finally, if
Rt+h|t,t−J = 1, we do not observe a change in confidence with respect to σt+h|tagents experience an increase in uncertainty, so that Rt+h|t,t−J > 1, they will be less confident
than under the true density.

3.4Standard DE

The standard approach to apply representativeness in the time-series domain follows the
BGS formulation, in which the representativeness of x
bt+h is given by the relative frequency
rep

BGS

(b
xt+h |Ft , Ft−J ) ≡2
N (bxt+h ; µt+h|t , σt+h|t)2
N (bxt+h ; µt+h|t−J , σt+h|t
)

,

(19)

In equation (19), the reference density uses the mean conditional on the information Ft−J
available J periods ago, µt+h|t−J , but the uncertainty conditional on the current information
set Ft , σt+h|t . The reference density can thus be understood as a mixture of information sets,meant to keep the conditional uncertainty the same for the current and reference density.
17Instead, under Smooth DE, the reference density captures entirely and solely the role of
new information, as the difference between information sets Ft and Ft−J . Under Normality,
comparing the definition of representativeness in equations (10) and (19), this difference
formally manifests itself in the variance of the reference distribution.
This BGS assumption delivers the following closed form standard DE density:
Proposition 2 (BGS implementation for DE). Consider the BGS definition of represen2
tativeness in equation (19). When σt+h|t> 0, the resulting DE density f θ (bxt+h |Ft , Ft−J )defined by equation (9) has a Normal distribution with mean:Eθt (xt+h ) = µt+h|t + θ µt+h|t − µt+h|t−J .

(20)2Vθt (xt+h ) = σt+h|t.

(21)

and variance:2= 0, the DE conditional mean Eθt (xt+h ) collapses to µt+h|t .
When σt+h|t

Proof. See Bordalo et al. (2018).
Smooth DE and the standard BGS formulation of DE coincide in two cases. First, in thelimit of no conditional uncertainty. In particular, in both approaches, with θ > 0, there is a22
distortion if and only if the conditional variance σt+h|t> 0. Intuitively, when σt+h|t= 0, theconditional likelihood of observing any other scenario for xt+h than the one the agent is now
fully informed on has become equal to zero. As noted by Gennaioli and Shleifer (2010), thelack of such conditional (or “residual”) uncertainty leaves no room for memory to distortconditional forecasts. According to Proposition 1, Smooth DE formally nests that limiting
possibility, which would amount to Rt+h|t,t−J = 0 and thus effectively no distortion even if
θ > 0. In the BGS formulation that limit is instead imposed through a discontinuity at2= 0: in the language developed in Bordalo et al. (2018), to compute Eθt (xt+h ) the
σt+h|trealization xt+h constitutes its infinitely representative state (see appendix in Bordalo et al.
(2018) on Corollary 1), and the result is Eθt (xt+h ) = µt+h|t . Under Smooth DE the effective
overreaction θet,t−J in equation (14) smoothly goes to zero as current uncertainty goes to zero.
Second, and more importantly, away from the zero conditional uncertainty case, in theoriginal BGS formulation the ratio Rt+h|t,t−J is always 1 by assumption. Thus, DE coincides
with Smooth DE if and only if the stochastic process is characterized by information sets2
2
Ft and Ft−J that happen to deliver σt+h|t= σt+h|t−J, ∀t and for any given J. Indeed, if
Rt+h|t,t−J = 1 in Proposition 1 the effective overreaction θet,t−J = θ, and formulas (12) and(13) collapse to their respective counterparts in equations (20) and (21).
In Figure 3 we use a series of illustrative examples to show the different effects at work
18

Figure 3: Smooth Diagnostic Expectations and standard Diagnostic Expectations densities

Notes: The figure is obtained using the following parameter values for the mean and variances of the2current and reference distributions: µt+h|t−J = [0, 0, 0, 0, 0], µt+h|t = [1, 0, 0, 1, 1], σt+h|t−J= [1, 1, 1, 1, 1],
2and σt+h|t−J = [1, .5, 1.3, .5, 1.3].

in Corollary 1.5 In the first row, we report the reference, current, DE, and Smooth DE
distributions. In the second row, we report the weights that capture the belief distortion
under DE and under Smooth DE. These are computed as:

weight (bxt+h ) ≡2)
N (bxt+h ; µt+h|t , σt+h|t

!θ

2
N (b
xt+h ; µt+h|t−J , σt+h|t
)

1
,
Z

for the standard DE andweight (bxt+h ) =2
N (bxt+h ; µt+h|t , σt+h|t)2
N (bxt+h ; µt+h|t−J , σt+h|t−J
)

!θ

1
,
Z

for the Smooth DE, where the constant of integration accounts for the respective formulation.
We organize the examples to follow the three key properties presented in Section 2 andformally characterized for the Normal distribution in Corollary 1.6
5

We use the following parameter values for the mean and variances of the current and reference distribu2
2
tions: µt+h|t−J = [0, 0, 0, 0, 0], µt+h|t = [1, 0, 0, 1, 1], σt+h|t−J= [1, 1, 1, 1, 1], and σt+h|t−J= [1, .5, 1.3, .5, 1.3].
6
Appendix D uses these examples to further and more generally discuss how the news and uncertainty

19

Overreaction of the conditional mean to new information. Consider first a casein which the reference and current distributions only differ in terms of the mean: µt+h|t >µt+h|t−J . The new information does not induce any change in uncertainty and Rt+h|t,t−J = 1.
Smooth DE and DE lead to the same normal distribution, in which only the conditionalmean is distorted. Both the SDE and DE means are shifted to the right with respect tothe current distribution by θ(µt+h|t − µt+h|t−J ). The lower panel in the first column showsthat the weights increase moving from left to right: The weights shift probability mass to
the right of the current density, lowering the probability assigned to events that became lesslikely, and inflating the probability of events that became more likely. The weights keepincreasing as elements in the right tail of the current density became much more likely tooccur in relative terms. However, the true probability of these events goes to zero fasterthan the weights increase, preserving the normality of the SDE and DE distributions.
Over/underconfidence. In the second and third columns, the reference and current
distributions only differ in terms of variance, while µt+h|t = µt+h|t−J . In the second col22
umn the current true distribution features a lower variance, i.e σt+h|t< σt+h|t−J
and thus
Rt+h|t,t−J < 1. This case is typical for standard stochastic processes when new information
leads to a reduction of uncertainty and more precise forecasts. Under the standard BGS
implementation of DE, the fact that events close to the mean, and the mean itself, became
more likely does not have any effect. Instead, under Smooth DE, the agent revises her beliefs in light of the new information. She inflates the probability of the mean and the otherevents that became more representative, while further downplaying the probability of events
that became less likely. The result is an even narrower distribution than the current true
distribution. Under Smooth DE, the new information leads to overreaction in terms of thedecline in uncertainty and, as a result, to a novel implication: overconfidence.
In the third case the current distribution has a larger variance than the reference distribu2
2
tion, i.e. σt+h|t> σt+h|t−J, and thus Rt+h|t,t−J > 1. This situation could arise, for example, in
response to a positive uncertainty shock. Now tail events become more representative under
the revised density and receive a magnified weight under Smooth DE. The probability mass
is moved from the center to the tails, but preserving normality.7 Under DE, the weights areonce again uniformly equal to 1, and the DE density coincides with the true density.
Changes in uncertainty affect the overreaction of the conditional mean. The
fourth and fifth columns combine a revision in mean with a revision in variance. The key
effects introduced in Section 2 determine the effective overreaction of the Smooth DE density.
7
This example also allows us to illustrate the role of the upper bound on Rt+h|t,t−J : As this ratio
increases, more and more probability mass is moved to the tails, flattening the normal distribution. As
Rt+h|t,t−J → (1 + θ)/θ , the variance of the Smooth DE density goes to infinity, as an increasing probability
mass is moved to the tails.

20observation here is illustrated by Corollary 1: under Smooth DE, a change in uncertainty
affects the degree to which the distorted revision responds to the true revision. Since changes
in information sets typically involve actual changes in both conditional moments, this is aparticularly novel and important aspect of Smooth DE.
In particular, the fourth case combines the first two, with an increase in the conditionalmean and a reduction of conditional uncertainty. As in the first case, under DE we observea shift of the probability mass to the right. Accordingly, the DE density moves to the right,
but with no change in shape with respect to the true density. Under Smooth DE, instead,the agent recognizes that, despite the increase in the mean, the new information made tailevents to the right less representative. Thus, for a given θ, the Smooth DE density still shifts
to the right, but by a smaller amount, and becomes visibly narrower, as the weights take
into account the change in uncertainty.8
Finally, new information can also bring a shift in the mean, but now with more uncertainty. The last column considers this case, where the shift in the mean is positive like inthe fourth column. The agent’s overreaction in terms of revisions to the conditional mean
is now stronger than under standard DE because tail events have become more likely under
the current distribution. Thus, the weights determine an even more significant shift of probability mass to the right (the scale for the SDE weights is on the left). In terms of distorted
conditional uncertainty, in this case, we observe underconfidence. This is again in itself aform of overreaction, as the agent magnifies the increase in uncertainty as this appears large
compared to the reference uncertainty.

3.5

AR(1) process

The Smooth DE density can be easily applied for a standard AR(1) process. For a simpler
exposition we focus on the one-step-ahead horizon (h = 1) and recent past (J = 1). We
extend this to the more general h and J in Section 4.1. Consider
xt+1 = ρxt + εt+1 ,εt+1 ∼ N (0, σ 2 )where ρ ≤ 1 and σ 2 > 0. The true conditional density is simplyf (bxt+1 |Ft ) = N x
bt+1 ; ρxt , σ 2 .
8As the current conditional uncertainty becomes continuously smaller and converges to zero, the shift
in the distorted conditional mean also smoothly converges to zero. In contrast, under standard DE, there
2
would be a discrete jump in the extreme case of σt+h|t=0.

21

while the reference density is 
f (bxt+1 |Ft−1 ) = N x
bt+1 ; ρ2 xt−1 , 1 + ρ2 σ 2As a result, Rt+1|t,t−1 defined in Proposition 2 takes the form
Rt+1|t,t−1 =

σ2
1
=
2
2
(1 + ρ ) σ
1 + ρ2(22)

The Smooth DE density f θ (bxt+1 ) in Proposition 1 is then Normal, with a conditional meanEθt (xt+1 ) = ρxt +θ
1 + ρ2 (1 + θ)

ρxt − ρ2 xt−1



(23)

and conditional distorted varianceVθt (xt+1 ) =σ22ρ
1 + 1+ρ2θ

(24)This AR(1) example further illustrates some of the general principles behind Smooth DE.
First, equation (23) implies that, as long as θ > 0, Smooth DE exhibits effective overreactionof the conditional mean to news. Second, given σ 2 , a higher persistence parameter ρ implies
that the new information determines a larger reduction in current uncertainty about thevariable of interest xt+1 compared to the reference density. This lower variance ratio Rt+1|t,t−1
leaves less room for memory to distort probability judgements which makes the effective
overreaction in equation (23) decrease in the persistence parameter ρ. Third, since new
information at time t lowers the conditional uncertainty from (1 + ρ2 ) σ 2 to σ 2 , the ratio
Rt+1|t,t−1 < 1 in equation (22). Thus, the distorted density is characterized by overconfidence,
so that Vθt (xt+1 ) < σ 2 , as seen in equation (24).

4

A parsimonious micro-foundation for survey evidence

Like in existing work on DE, we take the underlying θ ≥ 0 and J ≥ 1 as primitive parameterscharacterizing the decision’s maker limited memory and the effect that the representativenessheuristic has on agent’s judgments. As discussed in Section 3, given θ and J, the Smooth
DE density does not introduce any further degrees of freedom. Nevertheless, by allowing
the density for the representative group to reflect the time t − J conditional uncertainty, we
find that Smooth DE can offer a joint and parsimonious micro-foundation for a range of
observable implications consistent with survey data. These implications refer to the broad
22

properties of Smooth DE emphasized and collected by Corollary 1.
At its core, Smooth DE captures the intuition that new information that significantly
reduces current uncertainty over the variable of interest leaves less room for memory andrepresentativeness to distort judgements. As we discuss below, this implication of a stronger(weaker) effective overreaction of the conditional mean to new information that reduces less(more) current uncertainty helps to account for two sets of stylized survey facts.

4.1

Stronger overreaction for longer forecast horizons

The first set of over-identifying restrictions on our theory of overreaction relates to themodel’s implications for short- versus long-horizon forecasts. A strand of literature using
survey data argues that overreaction appears to be increasing with the horizon of the forecast. For example, Bordalo et al. (2019) and Bordalo et al. (2023) point to such stronger
overreaction for equity analysts’ forecasts of long-term earnings growth and emphasize thepotential for this type of overreaction to account for stock market volatility. Using professional forecasters’ forecasts of interest rates, other contributions, including for example
Bordalo et al. (2020), d’Arienzo (2020), find evidence of significant overreaction for expectations of long-term interest rates, but not for expectations of short-term interest rates.
Augenblick et al. (2021) use field data from betting and financial markets to argue thatcompared to the Bayesian forecast there is relatively stronger overreaction to signals with alonger (shorter) time-to-resolution, conceptually similar to longer (shorter) forecast horizons.
Smooth DE is consistent with such evidence as it predicts that overreaction increases withthe horizon of the forecast. The basic intuition appears in Section 3. Smooth DE formalizes
an inverse relation between the informativeness of the new piece of information obtainedby the decision-maker and the overreaction of her conditional forecasts (see for example
equation (16) in Corollary 1). In the context of forecasting at different horizons, the same
piece of information is less informative about horizons further in the future, leading to asmaller reduction in uncertainty and a stronger overreaction. Thus, Smooth DE naturally
predicts that overreaction is relatively stronger for long-horizon forecasts.
The simplest environment to showcase this basic insight is the AR(1) process introducedwith equation (3.5) in Section 3. For an horizon h ≥ 1 and a J− lagged reference distribution(J ≥ 1), the conditional mean for the Smooth DE density f θ (bxt+h ) isEθt (xt+h ) = ρh xt + θet,t−J ρh xt − ρh+J xt−J



(25)

where the effective severity θet,t−J of DE distortion is given in equation (14).
Given the AR(1) process in equation (3.5), the ratio Rt+h|t,t−J of conditional variances,23

defined in Proposition 1, takes the particular formVt [xt+h ]=Rt+h|t,t−J =Vt−J [xt+h ]

"

1−ρ2h
when ρ2 < 1
1−ρ2(h+J)hwhen ρ2 = 1
h+J#

Proposition 3 The ratio Rt+h|t,t−J increases in the forecast horizon h. Thus, the effective
overreaction θet,t−J of Eθt (xt+h ) in equation (25) is stronger for longer forecast horizons.
For a given lag J in the reference distribution, as the forecast horizon h increases, theeffective horizon of the current RE forecast (h), and the effective horizon of the reference
RE forecast (h + J) become increasingly similar. As a result, the levels of uncertainty
associated with the two forecasts also become increasingly similar because the information
set is implicitly more similar. Intuitively, the uncertainty around the two forecasts reflects
a larger and larger number of the same shocks. In relative terms, the current information
set is less and less informative for the variable that the agent is trying to predict. Given
that under Smooth DE overreaction is increasing in the level of relative uncertainty, as hincreases, so does the amount of overreaction to a given revision of the RE forecasts.

4.2

Overreaction and overconfidence

A recent literature studying the properties of survey responses, including Barrero (2022),
Born et al. (2022), and the reviews in Altig et al. (2020) and Born et al. (2022), documentsthat while firms’ forecasts are unconditionally unbiased, i.e forecast errors are on average
not significantly different from zero, firms make conditionally predictable forecast errors. In
particular, firms overreact to news and are overconfident in their subjective forecasts.
These stylized facts provide a challenge for models featuring standard rational belief
updating. As a result, the overreaction and overconfidence empirical properties have been
typically addressed in existing models through two distinct behavioral primitive assumptions
that do not distort unconditional forecasts. Overreaction of conditional forecasts has been
explained as an outcome of DE, modelled according to the original BGS formulation. Under
DE, agents overreact only in presence of new information and in a symmetric way, preserving
unbiased unconditional forecasts, but failing to account for overconfidence. Thus, the findingof overconfidence has been typically addressed with an additional overconfidence bias. An
example of this approach is Barrero (2022), who uses both distinct features to account for
the three survey facts.
Smooth DE can instead account for all three stylized facts. Consider for example theGaussian environment of Section 3 where a firm’s fundamental (eg. productivity) follows a
simple AR(1) process. Or, the arguably more empirically plausible extension, where those24

fundamentals are not directly observed, but firms can learn about their realizations fromnoisy signals, like in the simple state-space model described in Section E. In either case,
Corollary 1 and Proposition 10, respectively, describe how forecasts made under Smooth
DE are characterized by precisely these two key properties: overreaction to news and overconfidence. Moreover, as in the RE case (θ = 0), forecasts under DE are neverthelessunconditionally unbiased, being on average driven by the underlying rational forecasts.
The discussion and formalism in Section 3 indicate that, in contrast to the generalityof the overreaction of the conditional mean, overconfidence is not a universal property of
Smooth DE. However, we view it as a ’typical’ property, because the necessary condition
for overconfidence is simply that new information reduces uncertainty. This condition is
ubiquitous as it holds in stationary, homoskedastic environments, where events closer into
the future are naturally easier to predict than events far into the future. At the same time,
the condition might not hold if new information entails a sufficiently large and unexpected
increase in uncertainty, as indicated by some of our examples in Figure 3.
Finally, we further note the broader context of a large literature on overconfidence (eg.
De Bondt and Thaler (1995) and Daniel et al. (1998, 2001)). This work has been motivatedby extensive psychological evidence for overconfidence and argues that models based on thisbehavioral property are promising in accounting for asset market puzzles. Our key insighthere is that Smooth DE emerges as a potential parsimonious micro-foundation, based on the
representativeness heuristic, for overreaction and overconfidence, two behavioral features
argued as important in understanding a variety of economic outcomes.9

5

Business cycle implications

We illustrate the business cycle implications of Smooth DE in a parsimonious RBC modelwith time-varying uncertainty. We argue that Smooth DE emerges as a novel behavioral
propagation and amplification mechanism for time-varying uncertainty. We first show thatthe model replicates, without relying on additional frictions, several salient features of thedata thanks to the state-dependent overreaction: (1) asymmetry (recessions are deeper than
expansions), (2) countercyclical micro volatility (cross-sectional variances of microeconomic
variables rise in recessions), and (3) countercyclical macro volatility (time-series variances ofmacroeconomic variables rise in recessions).10 We also show that, the perceived increase of
9

See further Barberis (2018) for a review of these two-widely documented, but typically studied separately,
departures from standard Bayesian updating.
10
As described in the Introduction, these properties have strong empirical support in the literature. The
concept of asymmetries have a long tradition in macroeconomics, including Neftci (1984), Hamilton (1989),
Sichel (1993), and more recently McKay and Reis (2008) and Morley and Piger (2012). The extensive

25

uncertainty in recessions is more than three times larger than the actual uncertainty increase.
We then discuss a novel policy implication: a redistributive policy that reduces idiosyncratic
uncertainty could be beneficial for macroeconomic stabilization because it dampens the statedependent overreaction.

5.1

The model

To isolate the role of Smooth DE as a propagation mechanism, we keep the model simple andabstract from conventional frictions in the uncertainty shock literature, such as adjustment
costs (Bloom (2009), Bloom et al. (2018)) and sticky prices (Basu and Bundick (2017),
Fernández-Villaverde et al. (2015), and Bianchi et al. (2023a)). The economy consists of acontinuum of islands i ∈ [0, 1]. In each island i, an agent has the per-period utility function
1−γ
h1+η
ci,t
i,t
−β
.
U (ci,t , hi,t ) =1−γ1+η

where ci,t is consumption, hi,t is the amount of hours worked, γ is the coefficient of relativerisk aversion, and η is the inverse of the Frisch labor elasticity. We simplify the algebra
below by multiplying the disutility of labor by the discount factor β.
Output in each island is produced according toyi,t = zi,t hi,t−1 .

(26)

The t − 1 subscript on hours reflects the assumption that the labor input is chosen before
the random realization of productivity zi,t is known. The island resource constraint is
ci,t = yi,t .

(27)

We obtain aggregate variables by simply adding up variables of all islands:
Z 1
Ht =

Z 1
hi,t di,

0

Yt =

Z 1
yi,t di,Ct =0ci,t di0

The island productivity zi,t+1 is the sum of aggregate and idiosyncratic components:
ln zi,t+1 = At+1 + ai,t+1 ,literature of the macroeconomics of time-varying uncertainty, including Bloom (2009), Fernández-Villaverde
et al. (2011), Ilut et al. (2018), Jurado et al. (2015), Basu and Bundick (2017), and Bloom et al. (2018)confirm that volatility or uncertainty is countercyclical at both micro and macro levels.

26

The economy-wide TFP shock At+1 is common across all islands and follows the process
At+1 = ρA At + uA,t+1 ,uA,t+1 ∼ i.i.d.N (0, σA2 ).

The idiosyncratic TFP ai,t+1 is instead specific to island i, and it is composed of a predictablecomponent si,t ∼ i.i.d.N (0, σs2 ), known one-period-in-advance, and an unpredictable compo2) realized at t + 1: ai,t+1 = si,t + ua,i,t+1 . Following Bloom
nent ua,i,t+1 ∼ i.i.d.N (0, σa,tet al. (2018), we assume the volatility σa,t is time-varying and negatively correlated with theeconomy-wide TFP. In particular, as we describe in Section 5.4, σa,t increases when there isa negative innovation to the economy-wide TFP, and vice versa. We use σa,t to denote thevolatility of the period t+1 innovation to reflect the assumption that the volatility of the next
period’s innovation is known one-period-in-advance. We also assume that the volatility ofthe predictable component, si,t , is constant. This implies that the cross-sectional dispersionof labor is driven only by the news effect of the uncertainty shock. If we were to relax theassumption of constant volatility of si,t , the cross-sectional dispersion would also depend on
its realized volatility, but none of the main qualitative properties of the model would change.
The mean and variance of ai,t+1 conditional on the predictable component si,t are
Ei,t [ai,t+1 ] = si,t ,2
Vi,t [ai,t+1 ] = σa,t.

We can define the residual uncertainty (posterior variance relative to ex-ante uncertainty)
222
as in David et al. (2016) as σa,t/(σs2 + σa,t), which is increasing in σa,t
. Intuitively, in times
2of low aggregate TFP and higher uncertainty σa,t , the predictable component si,t serves asa weaker signal in forecasting ai,t+1 relative to times of lower uncertainty.

5.2

Rational Expectations solution

We first characterize the equilibrium under RE. The island i agent’s Bellman equation isV(hi,t−1 , zi,t ) = max {U (ci,t , hi,t ) + βEi,t [V(hi,t , zi,t+1 )]} .
hi,tCombining the first order condition for labor with the envelope condition, we obtain(hi,t )η = Ei,t (ci,t+1 )−γ zi,t+1 .

(28)

The optimality condition equates the current marginal disutility of working with its expected
benefit. The latter is given by the marginal product of labor weighted by the marginalutility of consumption. We log-linearize the condition and use the method of undetermined27coefficients to obtain the RE solution. Hats denote log-deviations from steady state.
Proposition 4 The equilibrium under RE is given as follows:
1. Individual hours worked are given byb
hi,t = ε [ρA At + si,t ] ,1−γ. Equilibrium output and consumption follow immediately as
where ε = η+γ

ybi,t = At + ai,t + b
hi,t−1 = b
ci,t .
2. Equilibrium aggregate variables are given byb t = ερA At ,Hb t−1 = Cbt
Ybt = At + H

Proof. See Appendix.
The response of individual and aggregate hours to news about expected economy-wide
productivity ρA At and island-specific productivity si,t is affected by the intertemporal elasticity of consumption (IES), which here also equals the inverse of the coefficient of relativerisk aversion. When the IES is large enough, so that γ −1 > 1 and thus ε > 0, an increasein expected productivity raises hours. In that case the intertemporal substitution effectdominates the wealth effect that would lower hours through the effect on marginal utility.
The next proposition characterizes the cross-sectional variance under RE.
Proposition 5 The cross-sectional variance of hours worked is given by

2
Z 1
2
1−γ
b
b
σs2 ,
hi,t − Ht di =η+γ0
and is constant over the business cycle. The cross-sectional variances of output yi,t and2
consumption ci,t are increasing in the volatility σa,t−1
of the idiosyncratic TFP.
Proof. See Appendix.
Under RE, the cross-sectional variance of hours stays constant over the business cycle.
This is because once the model is linearized, the news effect of changes in uncertainty is muted
under RE. The cross-sectional variances of output and consumption are instead mechanically2affected by σa,t−1
because of the change in realized volatility.

28

5.3

Smooth DE solution

We now solve the model under Smooth DE. We consider the case of distant memory, meaningthat agents’ memory recall is based on a more distant past, rather than just the immediate
past. This means that the reference group is based on the information set available J > 1
periods ago. Bianchi et al. (2024) find that in standard models, distant memory can accountfor salient features of data, such as persistence and repeated boom-bust cycles. Under
distant memory, a time-inconsistency problem arises due to the failure of the law of iterated
expectations. Bianchi et al. (2024) address this issue by adopting the naı̈veté approach(O’Donoghue and Rabin (1999)), which we follow here. Under this approach, the agent fails
to take into account that her preferences are time-inconsistent and thinks that in the future
she will make choices under perfect memory recall, or RE. However, when the future arrives,
the agent ends up changing behavior and be again subject to her imperfect memory recall.11
Let θ-superscripts and RE-superscripts denote equilibrium Smooth DE choices and choices
under a RE policy function, respectively. The island i agent’s Bellman equation ismax U (cθi,t , hθi,t ) + βEθi,t V(hθi,t , zi,t+1 ) ,
hθi,t

θ
θ
. The continuation value is given by= zi,t hθi,t−1 and cθi,t = yi,tsubject to yi,t




RE
RE
V(hθi,t−1 , zi,t ) = max U (cRE,i,t , hi,t ) + βEi,t V(hi,t , zi,t+1 )
hRE
i,t

RE
RE
subject to yi,t= zi,t hθi,t−1 and cRE
i,t = yi,t .
Similar to the RE problem, the agent optimally equates the marginal disutility of laborwith its expected benefit, except that the benefit is evaluated under Smooth DE:

hθi,t

η

= Eθi,t

h

cRE
i,t+1

−γ

i
zi,t+1 .

(29)

Proposition 6 The equilibrium under Smooth DE is given as follows:
1. Individual hours worked are given bybhθi,t = ε [ρA At + si,t ]+θeη t,t−J ε [ρA Nt−J,t [At ] + si,t ] ,η + 1 + θet,t−J γ

11

(30)

Bianchi et al. (2024) further argue that the naı̈veté approach is psychologically coherent and consistent
with the underlying foundation of diagnostic beliefs as a heuristic and a mental short-cut.

29where ε is given in Proposition 4 and Nt−J,t [At ] ≡ At − Et−J [At ] represents the news
in At , compared to past expectation. Equilibrium output and consumption follow asθcθi,t .
= At + ai,t + b
hθi,t−1 = b
ybi,t

(31)

2. The effective diagnosticity parameter θet,t−J is given byθet,t−J =where
Rt+1|t,t−J =Rt+1|t,t−J θ ,1 + 1 − Rt+1|t,t−J θ

Vi,t −γbcREi,t+1 + At+1 + ai,t+1

(32)


.

(33)

θe
η
 t,t−J ερA Nt−J,t [At ]eη + 1 + θt,t−J γ(34)Vi,t−J −γbcREi,t+1 + At+1 + ai,t+1

3. Equilibrium aggregate variables are given byb tθ = ερA At +Hbθ = Cbθ
Ybtθ = At + H
t−1tProof. See Appendix.
First, consider the policy function for individual hours hθi,t . The first line of (30) is identical to the RE policy function. The second line captures the overreaction to news, i.e.
surprises.12 Consider, for instance, a positive surprise to an economy-wide TFP At . Smoothdiagnostic agents are over-influenced by this surprise and become over-optimistic about the
future benefit of working, and hence work more (if ε > 0). The coefficient on this overreθet,t−J ηaction, η+ 1+, is increasing in the effective diagnosticity θet,t−J . From (31) individual( θet,t−J )γoutput and consumption also overreact when individual hours overreact. Second, the effective diagnosticity θet,t−J is positively related to Rt+1|t−J , given by (33): the ratio between
the current uncertainty about the marginal benefit of labor and the uncertainty perceived
at period t − J. Third, aggregate hours, output, and consumption also feature overreaction,
controlled by θet,t−J , to news about economy-wide shocks.
The expressions (32) and (33) in Proposition 6 suggest that an increase in uncertainty
about future idiosyncratic productivity could raise θet,t−J and, in turn, the overreaction to
news.13 The Proposition below indeed confirms that this is the case.
12

Note that, for si,t , since it is i.i.d., the surprise is si,t itself.
In the current model, we must take a stance on how agents deal with time-varying volatility whenforming expectations. There are two approaches to compute the conditional variance at t − J in (33)that preserve normality of the Smooth DE density. The first approach consists of making an “anticipated
13

30

2
Proposition 7 An increase in the volatility σa,tof idosyncratic TFP raises the effective
diagnosticity parameter θet,t−J .

Proof. See Appendix.
There are two important implications of this proposition. First, the business cycle isasymmetric, even if the underlying shocks are symmetric. A positive TFP shock would
lower uncertainty and, as result, overreaction. In contrast, a negative TFP shock would
raise uncertainty and, as result, overreaction. Thus, a drop in economic activity in response
to a negative shock would be sharper, while an expansion in response to a symmetric positive shock would be milder, generating asymmetric fluctuations. Second, macroeconomic
volatility is countercyclical. During expansions uncertainty and overreaction are low while
in recessions agents overreact more to economy-wide shocks.
State-dependent overreaction also implies that micro-level volatility is countercyclical:
Proposition 8 The cross-sectional variance of hours worked is given by
#2
"
Z 1
2
et,t−J )(1 − γ)(1+θbb θ di =σs2 ,hθi,t − Hte
η + (1 + θt,t−J )γ02of the idiosyncratic TFP. The
and is increasing in θet,t−J and, thus, in the volatility σa,tcross-sectional variances of output yi,t and consumption ci,t are similarly increasing in the2volatility σa,t−1
of the idiosyncratic TFP.

Proof. See Appendix.
As uncertainty increases, the overreactions to the predictable component of idiosyncratic
h ei2(1+θt,t−J )(1−γ)
TFP and the future benefit of labor, captured in the η+(1+θe )γterm, rise. Hence,
t,t−J
an increase in uncertainty about idiosyncratic TFP raises the cross-sectional variances ofindividual actions.
Our theory has an important policy implication. As we saw above, the micro-level volatility and macroeconomic volatility are tightly linked through the state-dependent overreaction
controlled by θet,t−J . Thus, a policy that reduces microeconomic uncertainty through, for instance, a redistributive tax policy can also be effective in stabilizing the macroeconomy. To
fix ideas, consider a progressive income tax and subsidy scheme where the individual rate
τi,t is increasing in the realized idiosyncratic productivity level τi,t = τ ai,t , where τ ≥ 0 is autility” assumption (Kreps (1998)). In this case, agents’ uncertainty depends on the volatility at the timeof the forecast, disregarding the possibility of volatility changes. The second approach consists of assumingthat agents take into account the possibility of volatility changes, but that memory retrieves a Normal
approximation of the resulting mixture of Normal’s. We adopt the first approach, as it is arguably moreconsistent with the naı̈veté assumption and the general motivation of DE as a mental heuristics.

31

parameter that controls the progressivity. The island resource constraint is
ci,t + τi,t yi,t = yi,t ,so the agent pays a tax (τi,t > 0) if the realized TFP shock is positive (ai,t > 0) and receives
a transfer (τi,t < 0) otherwise. The scheme is budget neutral.
Proposition 9 A higher progressivity τ is associated with a smaller increase in the effective
2of idiosyncratic TFP rises.
diagnosticity parameter θet,t−J when the volatility σa,tProof. See Appendix.
Intuitively, redistribution dampens the state-dependent overreaction by reducing crosssectional uncertainty about the future benefit of labor. Thus, the government can stabilize
the macroeconomy by using the tax policy to reduce the increase in uncertainty and overreaction. For instance, in times of low aggregate TFP and high uncertainty, the government can
implement the tax policy or increase its progressivity. These interventions would dampen
the overreaction and make the downturn less severe.

5.4

Calibrated example

We illustrate the quantitative potential of the Smooth DE mechanism in the context of theparsimonious RBC model presented above by examining its dynamics.
Calibration. We calibrate the model to a quarterly frequency. We set the discountfactor β = 0.99, the IES γ −1 = 0.25−1 , and η = 0.4, which implies a Frisch elasticity of labor
supply of 2.5.14 For the economy-wide TFP shock, we set ρA = 0.95 and σA = 0.7/100. The
calibration satisfies the condition γ −1 > 1, so labor increases in response to an increase in
expected TFP.
Consider the time-varying standard deviation σa,t of the idiosyncratic TFP shocks. Using
Census micro data, Bloom et al. (2018) and Ilut et al. (2018) find that, during recessions,the dispersion of TFP shocks increases by 13% and 7%, respectively. Motivated by thesefindings, we assume that a negative innovation to economy-wide TFP larger or equal than
one standard deviation is associated with a 10% increase in the standard deviation σa,t of
the idiosyncratic TFP shocks relative to the steady-state standard deviation σa . Conversely,
a positive economy-wide TFP innovation of the same magnitude is associated with a 10%
decrease in the standard deviation σa,t of the idiosyncratic TFP shocks.
14

These values of IES and Frisch elasticity allow us to generate realistic labor volatility. Our calibrated
model generates the time-series standard deviation of aggregate hours worked of 1.67%. In the data, thestandard deviation of total hours worked in the nonfarm business sector (1983:Q1–2019:Q4) is 1.66%.

32

Table 1: Internally calibrated parameters and targeted moments
Parameters
σa
σs
θe

0.022
0.027
1.547

Targeted moments
Data

Model

Realized absolute forecast error 0.143
Residual uncertainty
0.41
Skewness of aggregate hours
−0.21

0.143
0.41
−0.21

Notes: The table reports the parameters and their calibrated values as well as the targeted moments. σa isthe steady-state standard deviation of unpredictable component of the idiosyncratic TFP shock, σs is thestandard deviation of the predictable component of the idiosyncratic TFP shock, and θe is the long-run
average effective diagnosticity implied by the calibrated value of θ. The realized absolute forecast error isreported in Barrero (2022) using survey data on US managers, calculated from realized forecast errors ofsales growth between t to t + 4, with observations employment-weighted. The residual uncertainty fromDavid et al. (2016) captures the amount of posterior uncertainty relative to the ex-ante uncertainty. The
skewness of aggregate hours is calculated using total hours worked in the nonfarm business sector(1983:Q1–2019:Q4). The model moments are calculated using simulated data from the Smooth DE model.

We assume that the agent’s comparison group is the expectation formed J = 5 periodsago. The parameter J mainly determines the persistence of overreaction and the value isconsistent with Bianchi et al. (2024), who find that in an estimated structural model thememory weights center around five- and six-quarters-ago expectations.
There are three remaining parameters: the steady-state standard deviation of the unpredictable component of the idiosyncratic TFP shock σa , the standard deviation of thepredictable component of the idiosyncratic TFP shock σs , and the diagnosticity parameterθ. We calibrate these parameters so that the model matches the three empirical momentssummarized in Table 1.15 While multiple model parameters jointly affect these moments, weselect the moments so that each moment is informative about a parameter of interest.
The first empirical moment, the mean of realized absolute forecast errors, is from Barrero (2022) who uses survey data (Atlanta Fed/Chicago-Booth/Stanford Survey of Business
Uncertainty (SBU)) on US managers. Forecast errors are computed by subtracting realized
sales growth between t to t + 4 from managers’ forecasts. The model counterpart is obtainedby calculating the mean absolute forecast error on the simulated distribution of the real RE θized forecast error Eθi,t ybi,t+4− ybi,t+4.16 This moment is informative about the steady-state15

We choose the parameters so that the squared-sum of distance between the data moments and themodel-implied moments is minimized.
16
Specifically, we generate 100 replications of T = 200 time series with n = 500 islands. The number ofislands roughly matches the the number of firms surveyed in the SBU data in Barrero (2022).

33

standard deviation of the unpredictable component of idiosyncratic TFP.
The second moment, residual uncertainty, captures the amount of posterior uncertainty
relative to the ex-ante uncertainty. David et al. (2016) estimate the residual uncertainty to
be 41%. The model counterpart is given by σa2 /(σs2 + σa2 ). This moment is useful to pin down
the standard deviation σs of the predictable component of idiosyncratic TFP.
Finally, the third moment is the skewness of total hours worked in the nonfarm business sector (1983:Q1–2019:Q4).17 The negative skewness (−0.21) reflects macroeconomic
asymmetry: drops in hours worked are steeper than increases. In our model, a negative2
economy-wide TFP innovation increases uncertainty σa,tand, in turn, the effective overreaction θet,t−J . A positive TFP innovation, in contrast, reduces overreaction. Under Smooth
DE, the diagnosticity parameter θ governs the strength of this mechanism to generate asymmetry. Under RE model and the standard DE model where the overreaction is constant,
there is no asymmetry, and the skewness is zero.
The model moments match the empirical moments perfectly. The calibrated σa and σs
imply the predictable and unpredictable components’ variances are about the same in steadye implied by the calibrated
state. The long-run average effective diagnosticity parameter θ,value of θ, is 1.54. This value is somewhat larger than Bordalo et al. (2018), Bordalo
et al. (2019), and d’Arienzo (2020), which tend to estimate the standard DE diagnosticity
parameter around 1, but smaller than the estimate of 1.97 in Bianchi et al. (2024).18
Implications for untargeted survey moments. We examine to what extent ourtheory can explain untargeted survey evidence on overreaction and overconfidence. We use
Barrero (2022)’s survey moments as an external validation because the study shows both
overreaction and overconfidence based on a single dataset (SBU). The first column of Table
2 reports the coefficient from a panel regression where managers’ time t forecast of t + 4 sales
growth minus the realization is regressed on the sales growth between quarter t − 1 to t. The
coefficient is positive, meaning that managers’ forecasts tend to be excessively optimistic
during high growth period. The second column is the realized mean absolute forecast error,
reported in Table 1, and is shown here again to facilitate comparison. The third column isthe subjective mean absolute forecast error, where the hypothetical realizations are drawn
from the managers’ subjective probability distributions. The subjective absolute forecast
error is only 16% the size of the empirical errors (fourth column), indicating overconfidence:
17

The empirical skewness of hours increases significantly to −1.75 when we extend the sample until 2022:Q1
to include the 2020 Covid-19 recession. We use the simulated data to compute the skewness of aggregate
hours worked in the model. Both simulated and actual time series are HP-filtered with λ = 1600.
18
Like Bianchi et al. (2024), our current model features distant memory (J > 1). Bianchi et al. (2024) notesthat existing estimates are based primarily on models where imperfect memory is assumed to be driven onlyby the immediate past (J = 1), and this assumption changes inference about the diagnosticity parameter.

34

Table 2: Untargeted survey moments: overreaction and overconfidence

Data
Model

(1)
Ft (∆yi,t+4|t ) − ∆yi,t+4|t

(2)

(3)
(4)
Absolute forecast error

on ∆yi,t|t−1
0.173

Realized
0.143

Subjective
0.023(0.059)(0.012)(0.002)

0.095

0.143

0.017

(Subjective)/(Realized)
0.16
0.12

Notes: The table reports the coefficient on overextrapolation and realized and subjective mean absoluteforecast errors. The data moments are computed by Barrero (2022) using survey data on US managers,
with observations employment-weighted and standard errors in parentheses. The first column reports thecoefficient from a panel regression where managers’ time t forecast of t + 4 sales growth minus therealization is regressed on the sales growth between quarter t − 1 to t. The second column is the realized
mean absolute forecast error, calculated using realized forecast errors of sales growth between t to t + 4.
The realized mean absolute forecast error is used in the calibration as a target, but is included in this tableto facilitate comparison. The third column is the subjective mean absolute forecast error, where thehypothetical realizations are drawn from managers’ subjective probability distributions. The fourth column
is the ratio of the subjective absolute forecast error to the realized error. The model moments arecalculated using the simulated data from the Smooth DE model.

managers overestimate the precision of their forecasts.
The model moments are computed by simulating the model under Smooth DE. First,
consider the overextrapolation regression coefficient. The model counterpart is the coefficient on pooled OLS where we regress the Smooth DE four-quarters-ahead forecast error, θ RE 
θθ− ybi,t−1, which proxies for news. The coefficient, on output growth, ybi,tEθi,t ybi,t+4− ybi,t+4
is positive, but smaller than in the data. The reason why the calibrated model understates
this coefficient relative to the data is as follows. In our model, economy-wide shocks are
persistent while island-specific shocks are i.i.d. In contrast to persistent shocks, when shocks
are i.i.d., the Smooth DE forecasts are orthogonal to news, so the idiosyncratic shocks push
the coefficient toward zero.19 While we specified idiosyncratic TFP shocks to be i.i.d. fortractability, allowing for persistence would increase the overextrapolation coefficient. Thus,
our model provides a conservative lower bound on the macroeconomic effects of Smooth DE.
Next, consider the mean absolute forecast errors. The subjective error (third column) is
19This need not be the case when agents forecast endogenous variables in models with slow-moving endogenous states, such as capital. See Bianchi et al. (2024) for details.

35

Table 3: Countercyclical cross-sectional standard deviation of labor growth(1)
Data(Recessions)/(Expansions) 1.16

(2)
Smooth DE
1.12

(3)
DE
1

(4)
RE
1

Notes: The table reports the ratio of the cross-sectional standard deviation of labor growth during
recessions to the cross-sectional standard deviation during expansions. The first column shows the ratio in
the data, reported by Ilut et al. (2018), where recessions and expansions are defined as NBER recessions
and NBER expansions, respectively. The second, third, and fourth columns report the model-implied ratios
for the Smooth DE, DE, and RE models, respectively.

obtained first by calculating the Smooth DE variance of output growthVθi,t RE 
ybi,t+4 = RE 
Vi,t ybi,t+4 ,1 + 1 − Rt+1|t,t−J θ(35)

and then leverage the normality of the RE output growth so that the subjective absolutep RE  21
forecast error is given by 2/π Vθi,t ybi,t+4. The model closely matches the subjective
forecast error. The size of the absolute subjective error is 12% of the size of the realized forecast error (fourth column), in line with the survey data’s 16%. According to (35), the Smooth RE  θ DE variance Vθi,t ybi,t+4
would be lower than the econometrician’s variance Vi,t ybi,t+4
due
to two factors. The first factor is that, under naı̈veté, (Smooth) DE agents perceive futureθRE.
instead of the equilibrium law of motion ybi,t+4
output to follow the RE law of motion ybi,t+4
The second factor is the Smooth DE effect (the denominator in (35)) on uncertainty, according to which a reduction of uncertainty contributes to overconfidence about the precision of
expectations. To disentangle these two factors, we calculate the subjective mean absolutep RE  12= 0.053, which
forecast error without the Smooth DE effect. We obtain 2/π Vi,t ybi,t+4
is 37% of the size of the realized errors. This ratio is more than double the values recoveredby the data (16%) and implied by the baseline model (12%). We conclude that the Smooth
DE effect is important to account for overconfidence as observed in survey data.
Countercyclical micro and macro volatility. We now study the model’s ability togenerate countercyclical micro and macro volatility. First, consider the micro volatility. In
Table 3, we report the ratio of the cross-sectional standard deviation of labor growth during
recessions to the cross-sectional standard deviation during expansions. The first column
shows this ratio from the data, as reported by Ilut et al. (2018), where recessions andexpansions are defined as NBER recessions and NBER expansions, respectively. The crosssectional dispersion is countercyclical: in recessions, it is 16% higher than during expansions.
36

Table 4: Countercyclical volatility of aggregate labor growth(1)
Data(Recessions)/(Expansions) 1.23

(2)
Smooth DE
1.22

(3)
DE
1

(4)
RE
1

Notes: The table reports the ratio of the rolling standard deviation of aggregate labor growth during
recessions to the rolling standard deviation during expansions. The first column shows the ratio in the data
for the period 1983:Q1-2019:Q4, where recessions and expansions are defined as NBER recessions and
NBER expansions, respectively. The second, third, and fourth columns report the model-implied ratios for
the Smooth DE, DE, and RE models, respectively.

The second column reports the ratio in our model, where we define recessions and expansions
as periods when there are one-standard-deviation negative and positive innovations to theeconomy-wide TFP, respectively. Under Smooth DE the cross-sectional standard deviationof labor growth is 12% higher during recessions than in expansions, so the model explains
75% of the empirical countercyclicality of micro volatility. In the model, a negative aggregate
TFP innovation triggers an increase in idiosyncratic TFP uncertainty σa,t . As a result, theoverreaction θet,t−J to the predictable component si,t of idiosyncratic TFP rises, so the crosssectional dispersion of actions such as labor increases. As shown in the third and the fourthcolumns, the cross-sectional dispersion is constant over the business cycle under the standard
DE model, where the overreaction is constant, and the RE model, where we have θ = 0.
Next, consider macro volatility. In our model, in times of low TFP and high idiosyncratic
uncertainty σa,t , aggregate labor responds more to economy-wide shocks because the overreaction is stronger. Table 4 examines to what extent this countercyclical macro volatility is
consistent with the data. To measure time-varying volatility of aggregate hours worked in
the data, similar to Ilut et al. (2018), we compute the rolling window standard deviation asvuu
σH,t = t1
nw − 1(nw −1)/2

2∆ ln Ht+k − ∆ ln Ht ,

X

(36)

k=−(nw −1)/2

where ∆ ln Ht+k is the log change of total hours worked in the nonfarm business sector fromP(nw −1)/2
a quarter t + k − 1 to t + k and ∆ ln Ht ≡ (1/nw ) k=−(n∆ ln Ht+k . We set the windoww −1)/2
size nw = 3 and consider the sample 1983:Q1-2019:Q4. The first column of Table 4 reports
the measured σH,t during NBER recessions relative to σH,t during NBER expansions. The
measured volatility of aggregate labor growth is 23% higher in recessions than in expansions.
We then compute the same rolling standard deviation (36) on the simulated data from37

the model. We define recessions and expansions as periods when there are larger-than-orequal-to one-standard-deviation negative and positive innovations to the economy-wide TFP,
respectively. The second column shows that in the Smooth DE model, the aggregate labor
growth volatility σH,t is 22% higher in recessions than in expansions. Thus, the Smooth DEmodel generates the countercyclical macro volatility that is quantitatively in line with thedata, even though the volatility of economy-wide shocks is constant. DE and RE models, incontrast, do not generate such countercyclical volatility (third and fourth columns).
Perceived vs. actual increase in uncertainty. Discussions of Corollary 1 and Figure
3 indicate that, in our model, agents would overestimate the increase in uncertainty inrecessions. This is because tail events become more representative when uncertainty rises. To
quantify how much the agent’s perceived uncertainty rises relative to the actual uncertainty,
we compute the Smooth DE variance of the future marginal benefit of labor,
Vθi,t

−γb
cRE
i,t+1 + At+1 + ai,t+1
Vi,t −γbcREi,t+1 + At+1 + ai,t+1.
=
1 + θ(1 − Rt+1|t,t−J )(37)

We are interested in (37) because it controls the labor-supply decision in response to an uncertainty increase. We find that, under our calibration, a 10% rise in σa,t raises the perceived uncertainty (37) by 69%. In contrast, actual uncertainty, given by Vi,t −γbcθi,t+1 + At+1 + ai,t+1
rises only by 19%. Thus, the perceived rise in uncertainty is more than three times larger
than the actual uncertainty increase in recessions.20

6Conclusions

We developed a tractable and structural bridge from the representativeness heuristic ofKahneman and Tversky (1972) to the time-series domain. We built on the formalizationof representativeness by Gennaioli and Shleifer (2010) and of diagnostic expectations (DE)by Bordalo et al. (2018) to allow for what we call “smooth diagnosticity.” Under Smooth
DE new information is defined as the difference between the current information set and aprevious information set. A critical consequence of this basic approach is that current andpast uncertainty interact to determine the intensity of the DE overreaction, but also create
the preconditions for novel properties such as over- and under- confidence.
After formally characterizing Smooth DE and its key properties, we leveraged its insights
20

Since we linearize our model, the increase in uncertainty affects first-order economic outcomes throughthe state-dependent overreaction and not through the conventional risk channel. The risk adjusted loglinearization method as in Bianchi et al. (2023b) would allow us to capture the impact of perceived increasein uncertainty while preserving tractability.

38

along two substantive directions. First, we embedded the Smooth DE framework in a standard signal extraction problem and showed that Smooth DE can account for recent evidence
indicating that overreaction is stronger for weaker signals and for longer horizon forecasts.
Second, we embedded Smooth DE in a parsimonious RBC model with time-varying uncertainty. This model can account for survey data on overreaction and overconfidence as wellas three salient properties of the business cycle: (1) asymmetry, (2) countercyclical micro
volatility, and (3) countercyclical macro volatility. We uncovered a novel policy implication: a redistributive policy that reduces idiosyncratic uncertainty could be beneficial for
macroeconomic stabilization because it dampens the state-dependent overreaction.

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AppendixA

Proof of Proposition 1

Expression (9) can be written as:

 θ

2
1
#
exp − 2σ2
xt+1 − µt+1|t
 1
2 
t+1|t




xt+1 − µt+1|t
2  Z1
exp − 2σ2
xt+1 − µt+1|t−J"ftθ (bxt+1 ) ∝ exp−

1
2
2σt+h|tt+1|t−J

Collecting the terms in the exponents, we get:"ftθ (bxt+1 ) ∝ exp −1
2
2σt+1|t"(1 + θ) xt+1 − µt+1|t

2

−

2
σt+1|t2
σt+1|t−J##θ xt+1 − µt+1|t−J

2

1
Z

Developing the squared terms and keeping track of the terms involving xt+1 , we obtain:





2
σt+1|t− 2σ2
1 + θ − σ2
θ


t+1|t
t+1|t−J
 "
# 1
θ




ft (b
xt+1 ) ∝ exp 
−1

σ2
σ2

Z
x2t+1 − 2xt+1 1 + θ − σ2 t+1|t θµt+1|t (1 + θ) − σ2 t+1|t θµt+1|t−J
1

t+1|t−J42

t+1|t−Jwhere the remaining terms are absorbed in the constant of integration.
σ2
Define: Rt+1|t,t−J ≡ σ2 t+1|t . If Rt+1|t,t−J > (1 + θ)/θ, the expression above corresponds
t+1|t−Jto the kernel of a normal with mean:Eθt (xt+1 ) =1+θ−2
σt+1|t2
σt+1|t−J!−1 "µt+1|t (1 + θ) −θ2
σt+1|t2
σt+1|t−J"#θµt+1|t−J
Rt+1|t,t−J θ µt+1|t − µt+1|t−J= µt+1|t +1 + 1 − Rt+1|t,t−J θ#and variance:2Vθt (xt+1 ) = σt+1|t
1+θ−

=

2
σt+1|t2
σt+1|t−J!−1
θ2
σt+1|t .
1 + 1 − Rt+1|t,t−J θThis gives us the result stated in Proposition 2.

B

Upper bound on DE distortion

Suppose that we are interested in imposing an upper bound on the Smooth DE distortion.
Imposing such upper bound on the overreaction in the mean guarantees that both distortions
remain finite and non-decreasing as the ratio Rt+h|t,t−J goes to infinity. Thus, we propose
the following approach.
Let θe be the desired upper bound of effective overreaction in conditional mean. By
effective overreaction we refer to the object defined in equation (14). As a first step, we
exploit the fact that the size of the distortion is increasing in Rt+h|t,t−J to find the thresholdvalue R, such that, for a given θ, for each Rt+h|t,t−J > R, the overreaction in the mean wouldebe larger than θ:Rθ = θe1+θ 1−R
It follows that the upper threshold in terms of Rt+h|t,t−J isR=θe 1 + θ
1 + θe θ

43Whenever Rt+h|t,t−J > R, we thus replace θ with θR , the value of θ such that the overree Thus, we solve:
action in the mean is equal to θ.
Rt+h|t,t−J θR = θe1 + θR 1 − Rt+h|t,t−J
and obtain:
θe

θR =

Rt+h|t,t−J − θe 1 − Rt+h|t,t−JPlugging in θR in the formulas for the overreaction in mean and variance, we obtain:

Rt+h|t,t−J µt+h|t − µt+h|t−J
1 + θR 1 − Rt+h|t,t−JEθt (xt+h ) = µt+h|t + θe µt+h|t − µt+h|t−J
Eθt (xt+h ) = µt+h|t + θRand12
 σt+h|t
1 + θR 1 − Rt+h|t,t−J



1
θ
2
e
Vt (xt+h ) = 1 + θ 1 −σt+h|tRt+h|t,t−JVθt (xt+h ) =Note that while the overreaction in the mean remains constant once Rt+h|t,t−J > R, theoverreaction in the variance keeps growing as relative uncertainty increases, but it converges
to a finite value:
i
h
2
lim
Vθt (xt+h ) = 1 + θe σt+h|t
Rt+h|t,t−J →∞C

Overreaction to new information and Smooth DE

To further understand Smooth DE, we rewrite the distorted conditional mean and variancein Proposition 1 as a function of the revised information, as follows.
Corollary 2 (A revision representation). The Smooth DE density of Proposition 1 can be44

represented as distorting the RE revisions in conditional mean and variance, as follows:−1Eθt (xt+h ) − µt+h|t−J = µt+h|t − µt+h|t−J (1 + θ) 1 + θ 1 − Rt+h|t,t−J
|
{z
} |
{z
} | {z } |{z}Smooth DE revisionVθt (xt+h )2
σt+h|t−J| {z }

Smooth DE revision

RE revision

BGS effect

Smooth DE effect2
σt+h|t
−1= 2
1 + θ 1 − Rt+h|t,t−J
σt+h|t−J |{z}| {z }
Smooth DE effectRE revisionThis representation indicates how the revision in conditional moments under Smooth DE
can be decomposed as having three parts: (1) the RE revision, (2) an overreaction effectfrom representativeness as assumed in the standard BGS implementation of DE, and (3) aseparate and novel effect stemming from Smooth DE.

DNews and uncertainty effects for the Normal density

In Figure 4, we study how news and uncertainty effects determine the overreaction of the
Smooth DE density relative to the true current distribution. To facilitate comparison with
Figure 3, we mark the coordinates corresponding to the reference mean and variance withwhite pluses, and the coordinates corresponding to the current means and variances of eachscenario in Figure 3 with red circles. Panels (a) and (b) visually illustrate how the Smooth
DE mean and variance change as the posterior mean and variance change and confirm ourresults in Corollary 1: Panel (a) shows larger overreactions for higher current variances andPanel (b) highlights overconfidence for current variances lower than the reference varianceand underconfidence for current variances higher than the reference variance. Panel (c)shows the news effect, measured in terms of the KL divergence of the current distribution
from the reference distribution. Interestingly, the KL divergence is large when the current
variance is low and there is a large shift in the mean. The news effect is large in that casebecause we are moving probability masses of tail events under the reference distribution. In
contrast, the uncertainty effect is large when the current variance is high (Panel (d)). Panel(e) shows that the uncertainty effect of higher current variance dominates the news effect,
so the overreaction of the Smooth DE density (measured in terms of the KL divergence of
the Smooth DE density to current density) is largest when there is a large shift in the mean
and an increase in variance.

45

Figure 4: News and uncertainty in a Normal distribution

Notes: The figure displays how, as we vary the posterior mean and variance, the following objects change:(a) the Smooth DE mean, (b) the Smooth DE variance, (c) the news component, measured using the KL
divergence of the posterior distribution from the prior distribution, (d) the uncertainty component,
measured using the posterior variance, and (e) the overall overreaction, measured using the KL divergenceof the Smooth DE distribution from the posterior distribution. We mark the coordinates corresponding to
the prior mean and variance with white pluses, and the coordinates corresponding to the posterior meansand variances of each scenario in Figure 3 with red circles.

E

Signal extraction under Smooth DE

An important class of models emphasizing changes in subjective uncertainty belongs to thelarge literature on Bayesian learning. In what follows, we show that Smooth DE can beeasily extended to this class of models.
Let us start with some more general notation, which connects to the one used in Definition
1. Consider the same probability space (Ω, F, (Ft )t≥0 , P ) introduced in Section 3.1. But
now allow for noisy information. In particular, let the information available up to time t
be represented by another filtration (Gt )t≥0 , generated by an observed process st . Then let
f (xt+h |Gt ) be the conditional density function of Xt+h based on the information available inGt from the imperfect observations.
Like in Definitions 1 and 2, the representativeness of an event would then becomerep (bxt+h |Gt , Gt−J ) ≡

46

f (b
xt+h |Gt )f (bxt+h |Gt−J )(38)while the conditional density distorted by representativeness isf θ (bxt+h |Gt , Gt−J ) = f (bxt+h |Gt ) [rep (bxt+h |Gt , Gt−J )]θ Z −1(39)where Z is a constant of integration and the parameter θ ≥ 0.
Smooth Diagnostic Kalman filter. To derive closed form solutions, we focus on astandard Gaussian case of noisy information and maintain J = 1. In particular, consider astandard state-space representation. The observation equation is:εt ∼ N (0, σε2 )s t = xt + εt ,and the state transition equation for the unobserved xt is
ut ∼ N (0, σu2 )xt = ρxt−1 + ut ,The Kalman Filter gives the Bayesian forecast, and its derivation is standard. The onestep-ahead prediction from the period t−1 estimate x̃t−1|t−1 and its associated error varianceΣt−1|t−1 are given byx̃t|t−1 = ρx̃t−1|t−1 ; Σt|t−1 = ρ2 Σt−1|t−1 + σu2 .
Then, the estimates are updated according tox̃t|t = x̃t|t−1 + Kt (st − x̃t|t−1 ),Kt =Σt|t−1,
Σt|t−1 + σε2

where Kt is the Kalman gain and the updating rule for the variance isσε2
Σt|t =
Σt|t−1 .
Σt|t−1 + σε2(40)

In this environment, we now derive the version of Kalman filter used by agents subject
to Smooth DE. This connects and extends the diagnostic Kalman Filter derived within thestandard BGS formulation in earlier work like Bordalo et al. (2019) and Bordalo et al. (2020).
Let f (xt |Gt ) be the probability density of the rational, or Bayesian, period t estimate of
the current underlying state xt based on the Kalman Filter derived above. Intuitively, by
equation (38), a state xt is more representative if it becomes more likely relative to the t − 1
forecast. As in our discussion of equation (9), the key feature with respect to the original
BGS formulation is to condition on the whole past information set, and as a result, to take47

into account the associated uncertainty.
Let the ratio of current to prior estimation uncertainty under RE be denoted as
Rt|t,t−1 ≡ Σt|t /Σt|t−1 .
We derive the following result.
Proposition 10 (Smooth DE Kalman Filter.) The density f θ (bxt |Gt , Gt−1 ) in equation (39)
has a Normal distribution with meanEθt (xt ) = x̃t|t +
Rt|t,t−1 θ x̃t|t − x̃t|t−1 ,1 + 1 − Rt|t,t−1 θand varianceVθt (xt ) =

(41)

Σt|t

1 + 1 − Rt|t,t−1 θ(42)Proof. Re-writing the expression (39):h2 i θ
1


exp
−
x
−
x̃
t
t|t
2
2Σt|t
1
1

h
i
xt − x̃t|tftθ (xt ) ∝ exp −

2
1
2Σt|t
Z
exp − 2Σt|t−1
xt − x̃t|t−1
Collecting the terms in the exponents, we get:ftθ (xt ) ∝ exp




2 1
2
Σt|t
1
−
θ xt − x̃t|t−1(1 + θ) xt − x̃t|t −2Σt|tΣt|t−1Z

Developing the squared terms and keeping track of the terms involving xt , we obtain:− 2Σ1t|t

Σt|t
1 + θ − Σt|t−1
θ
−1 

 

ftθ (xt ) ∝ exp Σt|t
x2t − 2xt 1 + θ − Σt|t−1
θ



Σt|t
(1 + θ) x̃t|t − Σt|t−1
θx̃t|t−1


1
 

Z

where the remaining terms are absorbed in the constant of integration. The one above is48

the kernel of a normal with mean:
−1 


Σt|t
Σt|t
θ
(1 + θ) x̃t|t −θx̃t|t−11+θ−Σt|t−1
Σt|t−1

Rt|t,t−1 θ x̃t|t − x̃t|t−1= x̃t|t +1 + 1 − Rt|t,t−1 θ!

Rt|t,t−1 θ= x̃t|t−1 + 1 +x̃t|t − x̃t|t−1
1 + 1 − Rt|t,t−1 θ!
Rt|t,t−1 θ= x̃t|t−1 + 1 +Kt (st − x̃t|t−1 ),1 + 1 − Rt|t,t−1 θEθt (xt ) =where Rt|t,t−1 ≡ Σt|t /Σt|t−1 and in the fourth line we used (E), and variance:
−1

Σt|t
θ
1+θ−
Σt|t−1Σt|t .
=1 + 1 − Rt|t,t−1 θVθt (xt ) = Σt|tThis gives us the result stated in Proposition 10.
Like in our earlier general discussion, we observe overreaction of the conditional mean
when θ > 0 and the new information does not fully resolve uncertainty, i.e. when σε2 > 0.
Furthermore, similarly to the earlier AR(1) example, this environment is also characterized
by a conditional reduction in uncertainty, and therefore by overconfidence. Indeed, as longas σε2 is finite, by equation (40) estimation uncertainty decreases over time, as the new signal
is at least partly informative. It follows that the ratio Rt|t,t−1 < 1, ∀t and that given equation(42), subjective uncertainty is lower than Bayesian estimation uncertainty, i.e. Vθt (xt ) < Σt|t .

F

Proofs for the business cycle model

F.1

Proof of Proposition 4First, consider the equilibrium individual policy functions. To characterize dynamics we use
a log-linear approximation of decision rules around the steady state. We take logs of the
optimality condition with respect to hours in (28) and constraints (26) and (27):ηbhi,t = Ei,t [−γbci,t+1 + zbi,t+1 ] ,ybi,t = zbi,t + b
hi,t−1 = b
ci,t .

49

Substitute the constraints into the labor supply condition:
ηbhi,t = Ei,t [−γbci,t+1 + zbi,t+1 ]hi= Ei,t −γ zbi,t+1 + b
hi,t + zbi,t+11−γbhi,t =
Ei,t [bzi,t+1 ]
η+γ

1−γ 
ρA At + e
ai,t+1|t
=
η+γ
1−γ
=
[ρA At + si,t ] .
η+γEquating the coefficients we obtain the equilibrium elasticities.
Next, consider aggregate variables. Note we have
Z 1

Z 1
si,t di = 0,

0

0

Z 1
zbi,t di = At +ai,t di = At ,0

by law of large numbers. Then
Z 1
bt =
H

Z 1
b
hi,t di = ερA At + ε0si,t di0= ερA At
Z 1
Z 1
Z 1
b
Ybt =
ybi,t di =zbi,t di +hi,t−1 di
0

0

0

b t−1
= At + H
bt .
=C

F.2

Proof of Proposition 5

Note we have
Z 1
s2i,t di = σs2 ,
0

Z 1
0

2
u2a,i,t di = σa,t−1,

Z 1

a2i,t di =

0

Z 1

2
(si,t−1 + ua,i,t )2 di = σs2 + σa,t−1
.

0

Then
Z 1
Z 1
Z 1
2
2
2
b
b t di =hi,t − H(εsi,t ) di = (ε)
s2i,t di
0

0

0
2

= (ε)

σs2 ,

50

which is constant. Next consider the cross-sectional variance of output:
Z 1
0

ybi,t − Ybt

2

di =

Z 1 
Z0 1

=

 
2
b
b
At + ai,t + hi,t−1 − At + Ht−1
di(ai,t + εsi,t−1 )2 di

Z0 1(si,t−1 + ua,i,t + εsi,t−1 )2 di
0
Z 1
Z 1
2
2
u2a,i,t disi,t−1 di += (1 + ε)== (1 + ε)202
σs2 + σa,t−1,

02. It follows that the cross-sectional variance of consumption:
which is increasing in σa,t−1

Z 1
2
2
bt di = (1 + ε)2 σs2 + σa,t−1bci,t − C,02
is increasing in σa,t−1
as well.

F.3

Proof of Proposition 6

First, consider the equilibrium individual policy functions. As in the RE solution, to characterize dynamics we use a log-linear approximation of decision rules around the steady state.
We take logs of the optimality condition with respect to hours in (29) and constraints (26)and (27):ηbhθi,t = Eθi,t −γb
cRE
+
z
b
i,t+1
i,t+1
θ
θ
b
yb = zbi,t + h
=b
cθ .
i,t

i,t−1

51

i,t

Substitute the constraints into the labor supply condition:
bi,t+1bi,t+1 − θet,t−J Ei,t−J −γb
cRE
cRE
ηb
hθi,t = (1 + θet,t−J )Ei,t −γbi,t+1 + zi,t+1 + z

i
h

ηb
hθi,t = (1 + θet,t−J )Ei,t −γ zbi,t+1 + b
hθi,t + zbi,t+1hi− θet,t−J Ei,t−J −γ zbi,t+1 + b
hRE
+
z
b
i,t+1
i,t
i
h
h

 i
zi,t+1 ] − θet,t−J (1 − γ) Ei,t−J [bzi,t+1 ] + θet,t−J γEi,t−J bhRE
η + 1 + θet,t−J γ bhθi,t = (1 + θet,t−J ) (1 − γ) Ei,t [b
i,t
h

 i
η + 1 + θet,t−J γ bhθi,t = (1 + θet,t−J ) (1 − γ) [ρA At + si,t ]
 J+1

e
− θet,t−J (1 − γ) ρJ+1
A At−J + θt,t−J γ ερA At−J
(1 + θet,t−J ) (1 − γ)
(1 + θet,t−J ) (1 − γ)b ρA At + si,thθi,t =
η + 1 + θet,t−J γ
η + 1 + θet,t−J γθet,t−J η
1 − γ J+1
ρA At−J ,−η + 1 + θet,t−J γ η + γ
where the effective diagnosticity parameter θet,t−J is given by (32). Equating the coefficients
we obtain the equilibrium elasticities. As in the RE economy, we obtain equilibrium aggregate
variables by simply aggregating the individual policy functions.

F.4

Proof of Proposition 7

Consider Rt+1|t,t−J :Rt+1|t,t−J =Vi,t −γbcREbi,t+1i,t+1 + zVi,t−J −γbcREbi,t+1i,t+1 + z




θ
b
Vi,t −γ zbi,t+1 + hi,t + zbi,t+1




=
Vi,t−J −γ zbi,t+1 + b
hRE
+
z
b
i,t+1
i,t


Vi,t (1 − γ) zbi,t+1 − γ b
hθi,t

,
=
Vi,t−J (1 − γ) zbi,t+1 − γ bhREi,twhere the numerator is



θ
2
b
Vi,t (1 − γ) [ρA At + uA,t+1 + si,t + ua,i,t+1 ] − γ hi,t = (1 − γ)2 σA2 + σa,t,22which is increasing in σa,t
. Thus Rt+1|t,t−J and in turn θet,t−J are increasing in σa,t
.

52

F.5

Proof of Proposition 8(1+θe)(1−γ), we havet,t−J )γt,t−JFirst consider the cross-sectional variance of hours. Defining εθs,t ≡ η+(1+
θe

Z 1
Z 1
Z
2
2
2 1 2θθθθbb t di =hi,t − Hεs,t si,t di = εs,tsi,t di
0
0
0
2

e
(1 + θt,t−J ) (1 − γ)  2


σs ,
=
η + 1 + θet,t−J γ
which is increasing in θet,t−J . Next consider the cross-sectional variance of output:
Z 1
0

=

θ
ybi,t
− Ybtθ

Z 1 

2

di

 
2
bθ
At + ai,t + b
hθi,t−1 − At + H
di
t−1

Z0 1

2
si,t−1 + ua,i,t + εθs,t−1 si,t−1 di
0


2
e
(1 + θt,t−J ) (1 − γ)  22
σs + σa,t−1= 1 + eη + 1 + θt,t−J γ=2which is increasing in θet,t−J and σa,t−1. It follows that the cross-sectional variance of consumption:

2


Z 1
2
et,t−J ) (1 − γ)(1+θ2bθ di 1 +   σs2 + σa,t−1bcθi,t − Ct0
η + 1 + θet,t−J γ
2is increasing in θet,t−J and σa,t−1
as well.

F.6

Proof of Proposition 9First, we solve for the log-linearized RE decision rules under the tax policy. The optimality
conditions areηbhi,t = Ei,t [−γbci,t+1 + At+1 + (1 − τ )ai,t+1 ]
ybi,t = zbi,t + b
hi,t−1bci,t + τ ai,t = ybi,t .

53

Substitute the constraints into labor supply conditions:ηbhi,t = Ei,t [−γbci,t+1 + At+1 + (1 − τ )ai,t+1 ]hi= Ei,t −γ At+1 + (1 − τ )ai,t+1 + b
hi,t + At+1 + (1 − τ )ai,t+1
1−γ
1−γ
b
hi,t =
Ei,t [Ai,t+1 ] +(1 − τ )Ei,t [ai,t+1 ]
η+γ
η+γ

1−γ 
ρA At + (1 − τ )eai,t+1|t
=
η+γ
1−γ
=
[ρA At + (1 − τ )si,t ]η+γEquilibrium output and consumption follow immediately as
ybi,t = zbi,t + b
hi,t−1 = At + ai,t + b
hi,t−1 ,(43)bci,t = ybi,t − τ ai,t = At + (1 − τ )ai,t + b
hi,t−1 .

(44)

Next, consider the log-linearized SDE decision rules under the tax policy. To characterize
dynamics we use a log-linear approximation of decision rules around the steady state. The
optimality conditions areηbhθi,t = Eθi,t −γbcREi,t+1 + At+1 + (1 − τ )ai,t+1 ,ybθ = zbi,t + b
hθ ,
i,t

i,t−1

θ
b
cθi,t + τ ai,t = ybi,t.

Substitute the constraints into the labor supply condition:




e
ηb
hθi,t = (1 + θet,t−J )Ei,t −γbcREcREi,t+1 + At+1 + (1 − τ )ai,t+1 − θt,t−J Ei,t−J −γbi,t+1 + At+1 + (1 − τ )ai,t+1
h


i
ηb
hθi,t = (1 + θet,t−J )Ei,t −γ At+1 + (1 − τ )ai,t+1 + b
hθi,t + At+1 + (1 − τ )ai,t+1hi− θet,t−J Ei,t−J −γ At+1 + (1 − τ )ai,t+1 + b
hRE
+
A
+
(1
−
τ
)a
t+1
i,t+1
i,t
h

 i
η + 1 + θet,t−J γ bhθi,t = (1 + θet,t−J ) (1 − γ) Ei,t [At+1 + (1 − τ )ai,t+1 ]hi− θet,t−J (1 − γ) Ei,t−J [At+1 + (1 − τ )ai,t+1 ] + θet,t−J γEi,t−J b
hRE
i,t
h

 i
θ
η + 1 + θet,t−J γ bhi,t = (1 + θet,t−J ) (1 − γ) [ρA At + (1 − τ )si,t ]J+1e− θet,t−J (1 − γ) ρJ+1
A At−J + θt,t−J γερA At−J

(1 + θet,t−J ) (1 − γ)
(1 + θet,t−J ) (1 − γ)b ρ A At +
 (1 − τ )si,thθi,t =
η + 1 + θet,t−J γ
η + 1 + θet,t−J γθeη
1 − γ J+1
 t,t−J−ρA At−J ,η + 1 + θet,t−J γ η + γ

54

where the effective diagnosticity parameter θet,t−J is given by (32).
Consider Rt+1|t,t−J :Rt+1|t,t−J =Vi,t −γbcREi,t+1 + At+1 + (1 − τ )ai,t+1
Vi,t−J −γbcREi,t+1 + At+1 + (1 − τ )ai,t+1




θ
b
Vi,t −γ At+1 + (1 − τ )ai,t+1 + hi,t + At+1 + (1 − τ )ai,t+1




=
RE
b
Vi,t−J −γ At+1 + (1 − τ )ai,t+1 + hi,t + At+1 + (1 − τ )ai,t+1
Vi,t (1 − γ) (At+1 + (1 − τ )ai,t+1 ) − γ b
hθi,t


=
Vi,t−J (1 − γ) (At+1 + (1 − τ )ai,t+1 ) − γ bhREi,twhere the numerator isVi,t (1 − γ) [ρA At + uA,t+1 + (1 − τ )si,t + (1 − τ )ua,i,t+1 ] − γ b
hθi,t

2
2
2
= (1 − γ) σA+ (1 − τ )2 σa,t,22which is increasing in σa,tbut also a change in σa,thave a smaller impact when the progressivity τ is higher. Thus a higher τ is associated with a smaller increase in Rt+1|t,t−J andθet,t−J .

55