#  Expectations Formation with Fat-tailed Processes: Evidence from Sales
Forecasts ††thanks: We thank seminar participants at CFM and Columbia GSB for
their constructive feedback. Thesmar is a consultant for CFM.

Eugene Larsen-Hallock111Columbia & CFM Adam Rej 222CFM David Thesmar 333MIT,
CEPR & NBER

###### Abstract

We empirically analyze a large sample of firm sales growth expectations. We
find that the relationship between forecast errors and lagged revision is non-
linear. Forecasters underreact to typical (positive or negative) news about
future sales, but overreact to very significant news. To account for this non-
linearity, we propose a simple framework, where (1) sales growth dynamics have
a fat-tailed high frequency component and (2) forecasters use a simple linear
rule. This framework qualitatively fits several additional features of data on
sales growth dynamics, forecast errors, and stock returns.

##  1 Introduction

Expectations formation is a core question in economics. In recent years, a
strain of literature in macroeconomics and finance has been collecting
empirical regularities using survey data on subjective forecasts. It finds
that forecasts largely deviate from the full information model that
predominates in economic modelling: forecast errors are biased and predictable
using past errors and past revisions. Two types of explanations for this have
been put forward. The first one is that the data-generating process (DGP) is
simple and known to forecasters, but forecasting rules are irrational but
linear, featuring for instance under-reaction (Bouchaud et al., 2019) or
overreaction (Bordalo et al., 2019; Bordalo et al., 2020; Afrouzi et al.,
2020). The second approach to explaining observed biases is the tenet that the
data-generating process is too complex to be known by forecasters. Thus, they
use a misspecified model calibrated on the data they observe. This may come
from the fact that the DGP is hard to learn (for recent contributions along
these lines see Kozlowski et al., 2020; Farmer et al., 2022), or alternatively
from bounded rationality of the forecasters. They can only use simple
forecasting rules (Fuster et al., 2010; Gabaix, 2018). In any case, forecast
errors are predictable because forecasters use an imperfect model. In this
paper, we find evidence consistent with the second view, i.e. that, facing
complex (non-Gaussian) processes, forecasters use simple rules.

We use data on some 63,601 analyst forecasts of corporate revenue growth and
their realizations. An advantage of focusing on revenue growth (instead of EPS
as the literature typically does) is that revenue is always positive so that
growth rate is always well defined. We first show that the relationship
between forecast revisions and future forecast error is non-linear, a feature
that is not reported in the existing literature. In some settings, revisions
linearly and _positively_ predict forecast errors, a feature commonly
interpreted as evidence of under-reaction (Coibion and Gorodnichenko, 2015a).
In others, revisions linearly and _negatively_ predict forecast errors, which
is considered as evidence of overreaction (Bordalo et al., 2019; Bordalo et
al., 2020). In our sample, which is much larger than the existing studies, and
which focuses on a rather new object, sales growth, we find evidence of both.
For intermediate values of revisions, forecasters underreact to news (an
increasing relation between revisions and errors). For large values of
revisions, forecaster overreact (a decreasing relation between revisions and
errors). This non-linearity is robust. It holds in U.S. data and international
data. It holds across most industry groups.

The remainder of the paper is dedicated to explaining this fact. Our framework
is based on the simple assumption that forecasters use a linear rule to
forecast sales growth, but that this rule is misspecified because the true DGP
is more complex. Taking inspiration from the literature on firm size
distribution (in particular, Axtell, 2001; Bottazzi and Secchi, 2006), we
posit that sales growth distribution may be modelled by the sum of a low-
frequency and high-frequency shock. The low frequency shock is Gaussian, while
the high-frequency shock is non-Gaussian. It may have a very large (positive
or negative) realizations. With such a model, the optimal forecast of future
growth, conditional on current growth, is non-linear. A perfectly rational
forecaster anticipates more reversion to the mean when realizations are
extreme and more persistence when realizations are intermediate. We assume,
however, that agents stick to a linear rule to make their forecasts. The fact
that agents use a misspecified model may be grounded in bounded rationality
(i.e., agents use a simple rule even if the process is complex, as in Fuster
et al., 2010) or the difficulty of learning about complex processes (shocks
with multiple frequencies are hard to learn Farmer et al., 2022; shocks with
fat tails also Kozlowski et al., 2020).

Combined, these two assumptions (linear forecasting rule but short-term non-
Gaussian shocks) are enough to generate the non-linear relation between
forecast errors and past revisions that observe empirically. The mechanism is
intuitive. When revisions are large, the rational forecaster should anticipate
mean reversion, but the linear forecaster won’t. She overreacts to big
positive (or negative) news. When fitting her forecasting rule to the data,
she does, however, take this overreaction into account, and optimally
attenuates the sensitivity of her forecast to recent observations in the bulk
of the distribution. As a result, she underreacts to news of lesser
significance.

We then qualitatively test four additional predictions of the model. We start
with two natural predictions of the data-generating process. The first such
prediction is that the distribution of sales growths has fat tails, a fact
that holds strongly in the data (and previously shown by Bottazzi and Secchi,
2006). In particular, we check that this fact is not driven by an alternative
model of firm dynamics, where firms have heterogeneous volatility, but
Gaussian dynamics. In such a setting, large growth shocks could be generated
by the subset of firms who are more volatile than average (Wyart and Bouchaud,
2003). We thus rescale sales growths by estimates of firm-level standard
deviation and find that the resulting distribution still has very fat tails,
suggesting that growth shocks occur within firms, not across firms.

The second prediction from our DGP is that, conditional on past growth, future
growth should follow a S-shaped pattern as discussed above. We show that this
holds in the data, whether we normalize sales growth by firm-level standard
deviation or not.

The third prediction is on forecast errors. A natural prediction of our
forecasting model is that the autocorrelation of forecast errors should have
the same non-linear relation as the relation between errors and lagged
revision. In our model, where the forecasting rule is linear, they are the
same. Large past errors are equivalent to big shocks and therefore transient
ones: This leads to overreaction, as in the error-revision relation. We find
that this pattern holds in the data: forecast error are positively correlated
for intermediate values and negatively for large absolute values.

Our fourth and last prediction is on stock returns. Assuming risk-neutral
pricing and that equity cash-flows follow a dynamic similar to revenues, it is
easy to show that our model predicts that the autocorrelation of returns
should have a shape similar to the autocorrelation of forecast errors. For
intermediate values of past returns, momentum should dominate, but for extreme
values of returns, stock returns should mean revert. We find this pattern to
hold in the data. Our findings line up with recent research from Schmidhuber
(2021), who also finds evidence of momentum for “normal past returns” and
mean-reversion for extreme values of returns. We conclude from this analysis
that the risk-adjusted performance of momentum strategies would be
considerably improved by excluding stocks whose past returns have been large
in absolute value.

This paper contributes to the recent empirical literature on expectations
formation. Most papers in this space focus on linear and Gaussian data-
generating processes. Forecasting rules may, or may not, be optimal, but are
in general linear, so that the relationship between forecast errors and past
revisions (or past errors) is also linear. Our paper emphasizes the non-
linearity of such a relation, and as a result, suggests an different modelling
approach for the data-generating process to account for this non-linearity. We
emphasize non-Gaussian dynamics in firms’ growth (as Kozlowski et al., 2020,
have done in a different setting and in their case with a focus on Bayesian
learning).

In doing this we also connect the expectations formation literature with the
empirical literature on firm dynamics, which has since Axtell (2001)
emphasized the omnipresence of power laws in the distribution of firms sizes
(see Gabaix, 2009, for a survey of power laws in economics). That sales
_growths_ (rather than log sales) have fat tails is a less well-known fact,
although it was first uncovered by Bottazzi and Secchi (2006).

Last, our assumption that forecasters use a simple, linear, forecasting rule
that is misspecified is inspired by the literature on bounded rationality,
which assumes economic agents have a propensity to use oversimplified models
to minimize computation costs (Fuster et al., 2010, 2012; Gabaix, 2018). Such
models are correct on average, they are fitted on available data, but their
misspecification gives rise to predictability in forecast errors.

Section 2 describes the data we use: publicly available data on analyst
forecasts (IBES) and confidential data on international stock returns from
CFM. Section 3 documents the main fact: future errors are a S-shaped function
of past revision. Section 4.1 lays out the simple framework that we build in
order to explain this novel pattern. Section 5 tests four additional
predictions from this model. Section 6 concludes.

##  2 Data

###  2.1 Analyst Forecast data

This paper focuses on firm revenue (sales) forecasts made by analysts. Analyst
forecasts come from IBES Adjusted Summary Statistics files, which are
available both in the U.S. and internationally. Summary statistics files
contains “current” estimates as of the third Wednesday of each month. While
Earnings per Share forecasts have received greater attention in the
literature, sales forecasts are, in fact, better populated in the data than
EPS forecasts in recent years. Another advantage of revenue forecasts is that
they are never negative, so that we can easily calculate sales growth. A
downside of EPS is that it is frequently negative or small rendering the
calculation of EPS growth forecast impractical. Thus, the literature on EPS
forecasts studies a variable that is, in essence, non-stationary (typically,
future EPS normalized by current stock price).

For each firm $i$ and each year $t$, we denote sales by $R_{it}$, and
$F_{t}R_{it+1}$ the forecast made in year $t$ for the future realization of
$R_{it+1}$. We compute $F_{t}R_{it+1}$ as the consensus three months after the
end of fiscal year $t$ (i.e. nine months prior to the end of fiscal year
$t+1$) to ensure that sales results for fiscal year $t$ are available when the
forecast for $t+1$ is formed. Similarly, the two-year ahead forecast
$F_{t-1}R_{it+1}$ is measured three months after the end of fiscal year $t-1$.
Finally, we retrieve the realization of $R_{it+1}$ from the IBES actual files,
which is designed to recover the realization of the quantity actually forecast
by analysts.

In this paper, we focus on log sales growth and forecast of log sales growth.
We define $g_{it+1}=\log R_{it+1}-\log R_{it}$ the log-growth of this
quantity. The growth forecast is defined as $F_{t}g_{it+1}=\log
F_{t}R_{it+1}-\log R_{it}$ for the one-year ahead growth forecast, and
$F_{t-1}g_{it+1}=\log F_{t-1}R_{it+1}-\log F_{t-1}R_{it}$ for the two-year
ahead forecast of annual growth.

Finally, in the spirit of the expectations formation literature (Coibion and
Gorodnichenko, 2015a; Bouchaud et al., 2019), we construct two empirical
variables: the forecast error $ERR_{it+1}=g_{it+1}-F_{t}g_{it+1}$ and the
forecast _revision_ $R_{t}g_{it+1}=F_{t}g_{it+1}-F_{t-1}g_{it+1}$. These two
variables will be the main focus of our analysis.

To ensure forecast quality and improve sample consistency when we examine
returns, restrict our analysis of forecasts to firms that belong to one of the
major global stock indexes.444The list of stock markets used consists of: AEX,
AS5, CAC, DAX, HSC, HSI, IBE, IND, KOS, MID, NDX, NIF, NKY, OMX, SMI, SPT,
RAY, SX5, TOP, TPX, TWY, UKX. Further, we restrict ourselves to firm-year
observations for which both the forecast error $ERR_{it+1}$ and the revision
$R_{t}g_{it+1}$ are available. We give more details about the number of
observations and the start date in Table 1.

###  2.2 International Data on Stock Returns

In examining returns we restrict our sample to equities included in a major
national index. We rely on proprietary return data purchased and maintained by
CFM. The start of data availability differs by index and is shown in Table 2.
For all indexes data has been obtained through January, 2022. Each observation
is a ticker-month, and returns are log returns.

Table 1: Sample size by exchange (sales growth) Index | Total | 2000 | 2005 | 2010 | 2015 | 2020
---|---|---|---|---|---|---
AEX | 533 | 0 | 32 | 19 | 30 | 28
AS5 | 3228 | 48 | 122 | 167 | 196 | 161
CAC | 921 | 0 | 40 | 45 | 49 | 48
DAX | 680 | 0 | 29 | 38 | 37 | 35
HSC | 972 | 7 | 24 | 41 | 75 | 74
HSI | 572 | 15 | 24 | 28 | 29 | 29
IBE | 715 | 0 | 35 | 37 | 40 | 35
IND | 746 | 1 | 38 | 38 | 41 | 39
KOS | 1540 | 34 | 29 | 30 | 101 | 124
MID | 13016 | 10 | 586 | 818 | 782 | 646
NDX | 1174 | 1 | 47 | 67 | 72 | 61
NIF | 1037 | 13 | 24 | 47 | 66 | 64
NKY | 4959 | 207 | 206 | 226 | 224 | 233
OMX | 605 | 19 | 26 | 31 | 32 | 31
RAY | 15923 | 4 | 525 | 995 | 1057 | 881
SMI | 479 | 8 | 21 | 21 | 27 | 23
SPT | 998 | 0 | 34 | 56 | 67 | 63
SX5 | 215 | 0 | 10 | 11 | 13 | 11
TOP | 493 | 0 | 18 | 14 | 40 | 32
TPX | 10836 | 372 | 486 | 531 | 504 | 574
TWY | 1314 | 13 | 40 | 71 | 80 | 74
UKX | 2645 | 82 | 110 | 131 | 142 | 119

##  3 Motivating Facts

In this section we describe new evidence on expectations formation and
document a strong non-linearity in the link between forecast error and
revisions.

Since (Coibion and Gorodnichenko, 2015b) many papers in the expectations
formation literature estimate the following linear relationship between
forecast errors and revision:

$ERR_{it+1}=\alpha+\beta R_{t}g_{it+1}+\epsilon_{it+1}$ (1)

which is intuitive to interpret. Full information rationality predicts
$\beta=0$ for consensus forecasts (Coibion and Gorodnichenko, 2015b). Plain
rationality predicts $\beta=0$ for individual forecasts. $\beta>0$ is
typically interpreted as evidence of information frictions (Coibion and
Gorodnichenko, 2015a), or, if run at the forecaster level, plain under-
reaction (Bouchaud et al., 2019, study EPS forecasts; Ma et al., 2020, study
the revenue forecasts of managers). In contrast, $\beta<0$ is interpreted as
evidence of overreaction (Bordalo et al., 2019, study long-term EPS growth
forecasts; Bordalo et al. (2018) focus on macroeconomic expectations). All
these papers restrict their analyses to linear functional forms, as in
equation (1).

Figure 1: Revenue Forecast Error as a Function of Past Revision

Note: In this figure we use international sample of firm revenue expectations
to report the binned scatter plot of future log forecast errors
$g_{t+1}-F_{t}g_{t+1}$ as a function of past revision
$F_{t}g_{t+1}-F_{t-1}g_{t+1}$. The blue line is a local polynomial
approximation, centered in the middle of each centile.

In this section we show that for revenue growth forecasts this relationship is
actually non-linear. In Figure 1, we represent the relationship in a non-
parametric way through a binned scatter plot where the x-axis are the
revisions $R_{t}g_{it+1}$ and the y-axis is the average forecast error
$FE_{it+1}$. Each black dot corresponds to a centile of the distribution of
revisions, with the x coordinate being the average revision in this centile
and the y coordinate being the average forecast error. The grey shaded area
shows a bootstrapped 95% confidence interval. The blue line shows the
predicted error from a local polynomial regression (or LOESS) model estimated
at the center of each percentile of lagged revision. The kernel for this local
regression model is Gaussian with the bandwidth set equal to the average of
the distances between the centers of the 1st and 2nd, and the 99th and 100th,
percentiles.

For revisions of relatively small to moderate magnitude we find that errors
are increasing in revision. Thus, forecasters are under-reacting in response
to moderately-sized news shocks. This consistent with evidence from Bouchaud
et al. (2019) on EPS forecasts in United states. Ma et al. (2020) find similar
evidence on revenue forecasts from managers’ expectations in the U.S. (using
guidance data) and Italy (using a survey from the Bank of Italy). Their
samples are, however, much smaller than ours (a few 10,000 observations at
most), which precludes observing the tails of the distribution of revisions.

The key difference is in the tails of the distribution of revisions, for which
this relationship is reversed. In the face of exceptionally bad news,
forecasters are over-reacting: a larger, positive revision leads to more
negative surprises. A similar non-linearity is marginally observable in U.S.
EPS forecasts in Bouchaud et al. (2019), but the S shape in not complete
there.

We then explore the robustness of this relationship across sub-categories in
Figure 2. This figure has two panels: one that splits between U.S. and
non-U.S. firms (Panel A) and one that splits the sample into industries (Panel
B). In both cases we only show the prediction from flexible polynomial
approximation. In both subcategories the S-shaped function emerges. In
particular, it is visible both in US and non-US firms, although more
pronounced among U.S. firms.

Figure 2: The Error-Revision relationship: Sample Splits

Panel A: US v RoW Panel B: By Industry

Note: In this figure we use our international sample of firm revenue
expectations to report the binned scatter plot of future log forecast errors
$g_{t+1}-F_{t}g_{t+1}$ as a function of past revision
$F_{t}g_{t+1}-F_{t-1}g_{t+1}$. The blue line is a local polynomial
approximation, centered in the middle of each centile. In Panel A, we split
the sample between U.S. and non U.S. observations. In Panel B, we split the
sample into 1 digit GICS industries.

Overall, the evidence on log forecast errors and revisions points towards a
different treatment of large v. smaller shocks. Such evidence is hard to
square with established models of expectations formations, which feature
linear DGPs (typically, AR1 models) and linear expectations models. In what
follows, we set up a simple model that features extreme (i.e. non-Gaussian)
shocks and linear expectations formation in order to captures the above non-
linearity.

##  4 Model

In this section we develop a parsimonious model that features extreme shocks
and linear expectations in order to capture the non-linear behavior of
expectation errors of Figure 1.

###  4.1 Modeling Sales Growth

The first piece of the model is the data-generating process. We will omit the
firm index $i$ for clarity’s sake and assume that log sales growth,
$g_{it+1}$, evolves according to:

$\displaystyle g_{t+1}$ $\displaystyle=\underline{g}_{t+1}+\epsilon_{t+1}$ (2)
$\displaystyle\underline{g}_{t+1}$
$\displaystyle=\overline{\underline{g}}+\phi(\underline{g}_{t}-\overline{\underline{g}})+u_{t+1}$
(3)

where $\underline{g}_{t+1}$ is the unobservable latent state that follows the
classic linear-Gaussian AR1 dynamics. The key difference with most existing
models of expectations formation is that $\epsilon_{t+1}$ follows a
probability distribution density that has heavy tails. Because it fits the
data quite well (as we document below), we assume that $\epsilon_{t+1}$
follows a Student’s t distribution with $\nu$ degrees of freedom. Thus:

$\displaystyle\epsilon_{t+1}$ $\displaystyle\sim\text{Student-t}(0,1,\nu)$
$\displaystyle u_{t+1}$ $\displaystyle\sim\text{Normal}(0,1)$

Note that, although we analyze the cross-section of firms, we assume a single
process for all firms. In this paper, we do not explore the consequences of
firm heterogeneity for forecasting biases. For instance, such biases could
arise from forecasters using one single forecasting model for firms following
different processes. We believe such an avenue is interesting, but beyond the
scope of this paper, which focuses on one single deviation from the classical
model, i.e. that temporary shocks have fat tails. In order to bring the data
closer to the model however, we will conduct all of our analysis with
“normalized growth” data, thereby ensuring that all firms have the same growth
volatility. We discuss this adjustment extensively in Section 5.

In our simple model the conditional expectation
$E\left(g_{it+1}|g_{it}\right)$ is non-linear. We show this numerically in
Figure 3. For different values of $\nu$, we numerically simulate the process
and compute the conditional expectation $E\left(g_{it+1}|g_{it}\right)$ on
simulated data. As shown in Figure 3 this relationship is indeed quite linear
in the body of the distribution, but experiences “reversals” in the tails.
While not visible in Figure 3, all finite values of $\nu$ leads to such
reversals in the tail, but as $\nu$ gets larger (and $\epsilon$ is closer to
being Gaussian) they get pushed out farther into the tails and are very sharp
and localized.

Figure 3: Actual vs. lagged actual in model simulation.

Note: We simulate the model over T periods with $u$ following a normal
distribution $N(0,1)$, and $\epsilon$ following $\text{Student-t}(0,1,\nu)$.
We then show local polynomial regressions of $g_{t}$ on $g_{t-1}$ estimated at
the center of each percentile of lagged realization $g_{t-1}$. We explore
values of $\nu$ from 1.6 (fat tailed) to $\infty$ (Gaussian).

The economic intuition is simple. Since the underlying state variable is
Gaussian, extreme negative or positive realizations are more likely to come
from the transitory process $\epsilon$ than the persistent one
$\underline{g}$, since it features more extreme shocks. As a result, a large
sales growth realization today is unlikely to translate into future large
sales growth tomorrow: This suggests the presence of “reversals” in the tails,
as we see in Figure 3.

The above process has several predictions about the distribution of growth
rates, one of them being that the cross-sectional distribution of growth rates
should have fat tails. We will explore these predictions in Section 5.

###  4.2 Expectations Formation

The second building block of the model is the formation of expectations. Our
core assumption is that forecasters fail to perceive the non-linearity of true
expectations $E\left(g_{it+1}|g_{it}\right)$ and use a linear rule. This
assumption is based on the idea that economic agents use simplified, “sparse”,
model of reality to formulate expectations (Fuster et al., 2012; Gabaix,
2018). Agents assume $g_{it}$ follows a linear AR(p) model, estimate it on
data and use this model to form forecasts. One advantage of this
representation is that the term structure of forecasts is naturally defined,
as agents calculate mathematical expectations under the AR(p) model. Hence, we
assume that the forecaster believes growth follows the following AR(p) model:

$g_{t+1}=\underline{g}+\sum_{s=0}^{p-1}{\beta_{k}\left(g_{t-k}-\underline{g}\right)}+u_{t+1}$

We denote the subjective expectation operator by $F_{t}g_{t+k}\equiv
EL(g_{t+k}|\underline{g_{t}})$.

We assume that this prior is dogmatic. The forecaster is willing to re-
estimate the model’s parameters as new data comes in, but does not explore
models outside of the AR(p) set-up. As a result, the agent does not really
formulate rational expectations since she does not estimate the right DGP, as
in Fuster et al. (2010). One foundation for such dogmatism is that learning is
extremely slow in non-Gaussian, non-linear environments, so that it takes many
periods to modify the prior about the model (in recent literature, see
Kozlowski et al. (2020) and Farmer et al. (2022)).

Thus the agent estimates the parameters of the misspecified model using OLS on
expanding windows – using all information until date $t$. Let
$\widehat{\underline{g}}$ and $\widehat{\beta_{k}}$ be these estimates. The
one-period ahead forecast and the revision are given by:

$\displaystyle F_{t}g_{t+1}$
$\displaystyle=\widehat{\underline{g}}+\sum_{s=0}^{p-1}{\widehat{\beta_{k}}\left(g_{t-k}-\widehat{\underline{g}}\right)}$

###  4.3 Predictions of the Model: Errors on Revisions

We now check that our model indeed generates the non-linear relation between
revenue forecast errors and revenue forecast revisions shown in Figure 1.

In Figure 4, we report results from simulations, assuming that forecasts are
based on a fitted AR(2) model. We vary the thickness of the tail of the
temporary shock $\epsilon$, which is governed by $\nu$. $\nu=+\infty$
corresponds to a normal distribution, while $\nu=1.6$ is the thickest tail
possible.

Figure 4: Error as a function of lagged revision

Note: We simulate the model over T periods with $u$ following a normal
distribution $N(0,1)$, and $\epsilon$ following $\text{Student-t}(0,1,\nu)$.
We then show local polynomial regressions of error
$ERR_{t+1}=g_{t+1}-F_{t}g_{t+1}$ on revision
$R_{t}g_{t+1}=F_{t}g_{t+1}-F_{t-1}g_{t+1}$ estimated at the center of each
percentile of revision. We explore values of $\nu$ from 1.6 (fat tailed) to
$\infty$ (Gaussian). Forecasters are assumed to employee an AR(3) model when
predicting dividend growth rates.

Figure 4 shows that, as long as the temporary shock has sufficiently fat
tails, the linear expectations model generates predictable forecast error that
display a non linear pattern similar to Figure 1. This is quite intuitive. As
shown previously, the true conditional expectation is non-linear (see Figure
3). When the past realization of revenue growth is large, it is likely that it
was driven by the temporary fat-tailed process. As a result, the rational
forecaster would expect some mean-reversion, but the linear forecaster does
not. This creates overreaction to large shocks. In contrast, when past
realizations are moderate, there is underreaction. This comes from the fact
that the linear forecaster is on average rational: She fits a linear relation
on the S-shaped data of Figure 3. The slope of forecasts for smaller
realizations incorporates some of the overreaction in the tails.

To gain further insights in Figure 3 we vary the fatness of the tail $\nu$.
The less thick-tailed the innovation process, the less predictable errors are.
When $\nu=+\infty$, the temporary shock $\epsilon$ is Gaussian and forecast
errors are very close to zero for all lagged realizations (the black dots line
up on the x-axis). This is because in this case the linear AR2 forecasting
rule is nearly rational. Indeed, in this case, the rational expectation is a
Kalman filter:

$K_{t}g_{t+h}=\underline{g}+\phi^{h}G\sum_{s=0}^{+\infty}{\left(1-G\right)^{s}\left(g_{t-s}-\underline{g}\right)}$

where $G$ is the Kalman “gain”. The AR2 process is close enough to the above
equation that forecast errors are nearly zero in our simulations.

The bottom line of this analysis is that the non-linear structure of
expectations errors can easily arise when forecasters use linear models when
the data generating process has temporary shocks that have fat tails. Indeed
in this case, the optimal forecasting rule is non-linear, even though the
process is itself linear.

###  4.4 An Additional Prediction: Error on Lagged Error

The empirical expectations literature also investigates a different moment:
The autocorrelation of expectation errors (for instance, Ma et al. (2020) and
Farmer et al. (2022) among many others).

In our model the autocorrelation of errors is equivalent to the error-revision
coefficient. This happens because revisions are directly proportional to
current forecast errors:

$\underbrace{F_{t}g_{t+1}-F_{t-1}g_{t+1}}_{\equiv
R_{t}g_{t+1}}=\widehat{\beta_{0}}\cdot\left(g_{t}-F_{t-1}g_{t}\right)$ (4)

which means that a positive surprise translates into a positive revision about
future growth. The fact that the prior is linear makes this relationship
linear, whatever the number of lags $p$.

As a result, we expect the non-linear relation between errors and lagged
revision of Figure 1 to also hold between error and lagged _errors_. We test
this additional prediction in Section 5.

###  4.5 Model Prediction on Returns: Building Intuition

We also derive predictions on stock returns. Our simple model, as we will see,
predicts that momentum occurs for intermediate returns and mean-reversion
occurs for extreme returns.

In the spirit of Bouchaud et al. (2019), we assume stock prices are given by:

$P_{t}=\sum_{s\geq 1}{\frac{F_{t}D_{t+s}}{\left(1+r\right)^{s}}}$ (5)

where $F_{t}D_{t+s}$ is based on the forecasting rule described above in
Section 4.2. Hence, the stock is priced by investors who form expectations
based on a linear AR2 fitted on past realizations, while we assume dividends
to follow the process described in Section 4.1. We also assume for simplicity
that investors are risk-neutral, so that the discount rate is fixed at $r$.

In this very simple asset pricing model we expect returns to be a non-linear
function of past returns, similar to what we documented for the error-revision
relation in Figure 1. Before we discuss simulation results and economic
intuition, it is worth showing the algebra. The standard first order Campbell-
Shiller approximation writes as:

$r_{t+1}-F_{t}r_{t+1}\approx\left(F_{t+1}-F_{t}\right)\sum_{s=0}^{\infty}\rho^{s}g_{t+1+s}-\left(F_{t+1}-F_{t}\right)\sum_{s=1}^{\infty}\rho^{s}r_{t+1+s}$

where we denote log dividend growth as $g$ with a slight abuse of notation
($g$ stands for log revenue growth in the rest of the paper). Equation (5)
assumes constant expected returns $F_{t}r_{t+k}=r$ (investors may be biased
but are risk neutral), so that the CS decomposition simplifies into:

$\displaystyle r_{t+1}-r$
$\displaystyle=\left(F_{t+1}-F_{t}\right)\sum_{s=0}^{\infty}\rho^{s}g_{t+1+s}$
$\displaystyle=g_{t+1}-F_{t}g_{t+1}+\rho\left(F_{t+1}-F_{t}\right)\sum_{s=0}^{\infty}\rho^{s}g_{t+2+s}$

It then remains to describe the infinite sum of discounted dividend growth. In
this paper, we assume that forecasters (mistakenly) estimate dividend growth
as an AR(p) process:

$g_{t}-\underline{g}=\sum_{s=1}^{p}\beta_{s}\left(g_{t-s}-\underline{g}\right)+\epsilon_{t}$

We can then stack the estimated AR(p) coefficients $\beta_{1},\dots,\beta_{p}$
into “companion” form:

$\begin{bmatrix}g_{t}-\underline{g}\\\ g_{t-1}-\underline{g}\\\ \vdots\\\
g_{t-p}-\underline{g}\end{bmatrix}=\begin{bmatrix}\beta_{1}&\cdots&\beta_{p-1}&\beta_{p}\\\
1&\cdots&0&0\\\ \vdots&\ddots&0&0\\\
0&\cdots&1&0\end{bmatrix}\begin{bmatrix}g_{t-1}-\underline{g}\\\
g_{t-2}-\underline{g}\\\ \vdots\\\
g_{t-p-1}-\underline{g}\end{bmatrix}+\begin{bmatrix}\epsilon_{t}\\\ 0\\\
\vdots\\\ 0\end{bmatrix}$

or more compactly:

$\mathcal{G}_{t}=\mathbf{B}\mathcal{G}_{t-1}+\mathbf{\epsilon}_{t}$

Time $t$ forecasts for $g_{t+s}-\underline{g}$ are then given by:

$F_{t}(g_{t+s}-\underline{g})=\mathbf{e}_{1}^{\prime}\mathbf{B}^{s}\mathcal{G}_{t}$

where $\mathbf{e}_{1}$ is a “selector” vector picking out the first element of
the following vector. The infinite sum of discounted forecast dividend growth
is given by:

$\displaystyle F_{t}\sum_{s=0}^{\infty}\rho^{s}(g_{t+1+s}-\underline{g})$
$\displaystyle=\sum_{s=0}^{\infty}\rho^{s}\mathbf{e}_{1}^{\prime}\mathbf{B}^{s+1}\mathcal{G}_{t}$
$\displaystyle=\mathbf{e}_{1}^{\prime}\mathbf{B}(\mathbf{I}-\mathbf{B})^{-1}\mathcal{G}_{t}$

We then plug this formula into the CS decomposition and obtain:

$\displaystyle r_{t+1}-r$
$\displaystyle=\mathbf{e}_{1}^{\prime}\left(\mathcal{G}_{t+1}-\mathbf{B}\mathcal{G}_{t}\right)+\rho\mathbf{e}_{1}^{\prime}\mathbf{B}(\mathbf{I}-\mathbf{B})^{-1}\left(\mathcal{G}_{t+1}-\mathbf{B}\mathcal{G}_{t}\right)$
(6)
$\displaystyle=\mathbf{e}_{1}^{\prime}\left(\mathbf{I}+\rho\mathbf{B}(\mathbf{I}-\mathbf{B})^{-1}\right)\underbrace{\left(\mathcal{G}_{t+1}-\mathbf{B}\mathcal{G}_{t}\right)}_{=ERR_{t+1}\mathcal{G}_{t+1}}$
(7)

The above expression shows that under the AR(p) assumption, returns are a
linear function of past forecast errors ($ERR_{t+1}\mathcal{G}_{t+1}$ is the
vector of past $p$ forecast error). In this simple asset-pricing model,
returns are only predictable if dividend growth forecast errors are
predictable. Under rational expectations (i.e. if the true DGP for dividends
is an AR(p)), they are not. But if dividends are driven by a thick-tailed
state variable, the true DGP is far from AR(p) as we have documented. Thus,
expected forecast errors are non-linear functions of past errors, and the same
should hold for returns and past returns. So our model predicts that returns
should be a non-linear function of past returns, in other words, momentum
should only be present for intermediate values of past returns.

###  4.6 Model Prediction on Returns: Simulations

In order to check that this prediction also holds without CS approximation, we
proceed to simulate our model. On simulated data, we build returns as
$R_{t+1}=(P_{t+1}+D_{t}-P_{t})/P_{t}$. We then plot average future returns by
bins of past returns in Figure 5. In this very simple asset pricing model,
returns are predictable as soon as the dividend process

Figure 5: Momentum for Intermediate Past Returns; Reversal in the Tails

Note: We simulate the model over many periods with $u$ following a normal
distribution $N(0,1)$, and $\epsilon$ following $\text{Student-t}(0,1,\nu)$.
We then show local polynomial regressions of $R_{t+1}$ on $R_{t}$ estimated at
the center of each percentile of lagged realization $R_{t}g_{t+1}$. We explore
values of $\nu$ from 1.6 (fat tailed) to $\infty$ (Gaussian). Forecasters are
assumed to employee an AR(3) model when predicting dividend growth rates.

The intuition is the same one as before. Very high past returns likely emerge
from surprises due to a large thick-tailed temporary shock. Linear forecasters
overreact: large past return events are likely to be situations where dividend
realization a one-off boon. As a result, linear forecasters overestimate the
level of future dividends: The stock price rises too much and future returns
are lower. Intermediate past returns, however, are likely generated by
standard shocks. There, the linear forecaster under-reacts to small dividend
news and the price does not respond enough to these news. Future returns are
thus positively correlated with past returns for “smaller” absolute values of
returns. Note that when $\nu=+\infty$, temporary shocks are Gaussian and, as
discussed previously, linear expectations are quasi optimal – a true Kalman
filter would be perfectly rational – and past returns do _not_ predict future
returns.

##  5 Testing the Model’s Predictions

###  5.1 Predictions of Growth Rate Dynamics

In this section we discuss two key predictions of our data-generating process
(2)-(3). The first one is that the distribution of firms _revenue growths_
should have fat tails. The second one is that the conditional expectation of
$g_{t+1}$ on $g_{t}$ should be non-linear as in Figure 3.

First, a key prediction of our DGP (2)-(3) is that the distribution of firm
size growth has fat tails. Many variables relevant for finance and economics
are not normally distributed Gabaix (2009). It is for instance well-known that
the distribution of firm sizes follows a Zipf law (Axtell, 2001). Less well-
known is also the fact that the distribution of firm _growth rates_ has fat
tails. Bottazzi and Secchi (2006) show that the distribution of Compustat
firms follows a Laplace distribution. Here, we provide similar evidence from
our sample.

In Figure 6 we show the QQ plot of the sales growth distribution in our
sample, along some other textbook distributions. This QQ plot focuses on
observations above the 90th percentile distribution of $|g_{it}|$, the
_absolute_ value of sales growth (so both negative and positive shocks). Each
point of this chart corresponds to one quantile of the data distribution. For
a given quantile $q$ and a c.d.f. $F$, the y coordinate of the point is the
average value of absolute growth at quantile $q$ of the data distribution.
Since we focus on the top 10% of absolute sales growth, this number is
positive (so the y axis does not start at zero). The x coordinate of that
point is the average value of the _same quantile_ of the chosen distribution
$F$, or $F^{-1}(q)$. The closer is $F$ to the data distribution, the more the
chart will look like a 45 degree line (the black line on the Figure).

Figure 6: Tail of log sales growth distribution: Fit of various distributions

Note: This is a Q-Q plot of log sales growth vs. some textbook distributions
(Laplace, Normal and Student). This plot shows the tail above the 90th-
percentile of the distribution of $|g_{it}|$. On the x-axis, we report the
value of the quantile ($F^{-1}(q)$) of the comparison distribution. On the
y-axis, we report the value of the same quantile of the data distribution
($F_{\text{data}}^{-1}(q)$). The data distribution is normalized so that its
variance is one. The comparison distribution also have unit variance. By
design, the “data” line is the 45 degree line.

Looking at Figure 6 it is clear that the distribution of growth rates is very
different from normal in the tail. The green line increases faster than the 45
degree line meaning that the sales growth distribution has much heavier tails
than normal distribution. The best fit is obtained by fitting a Student
distribution.

Our model crucially assumes that this distribution comes from temporary shocks
occurring _within_ firms. Wyart and Bouchaud (2003) suggest an alternative
explanation for such thick tails: sales growth has a normal distribution at
the firm-level, but that the standard deviation of this process varies across
firms. In this case, extreme growth rates would typically occur among firms
that have very volatile growth rates (for instance, smaller firms). This
alternative interpretation does not explain our findings, but it is worthwhile
to analyze its validity.

In order to do that we normalize growth rates by a measure of firm-level
“volatility”. To do this we compute the mean absolute deviation of log sales
for each firm. This measure of volatility has the advantage of being more
immune to fat tails in the growth distribution (since variance may not exist
in such cases). For firm $i$, we thus compute:

$MAD_{i}=\frac{1}{T_{i}}\sum_{t=0}^{T_{i}}|g_{it}-\overline{X}_{i}|$

where $T_{i}$ is the number of observations for the firm, and
$\overline{X}_{i}$ is the average sales growth at the firm level.

In Figure 7 we show the QQ plot of the distribution of normalized
$\frac{g_{it}}{MAD_{i}}$. If heavy tails are driven by firms with larger
growth variance, this adjustment should significantly reduce the fat tails of
the data. The QQ plot shows that the distribution is still strongly non-
normal, though now the fit of the Laplace distribution is much better
(consistent with Bottazzi and Secchi, 2006) and the one of the student
distribution is nearly perfect.

Figure 7: Tail of log sales growth distribution (Normalized)

Note: This is a Q-Q plot of _normalized_ log sales growth vs. some textbook
distributions (Laplace, Normal and Student). This plot shows the tail above
the 90th-percentile of the distribution of
$|g_{it}-\overline{g}_{i}|/MAD_{i}$. On the x-axis, we report the value of the
quantile ($F^{-1}(q)$) of the comparison distribution. On the y-axis, we
report the value of the same quantile of the data distribution
($F_{\text{data}}^{-1}(q)$). The data distribution is normalized so that its
variance is one. The comparison distribution also have unit variance. By
design, the “data” line is the 45 degree line.

From this analysis we draw the conclusion that sales growth distribution has
heavy tails and that the conditional expectation
$E\left(g_{it+1}|g_{it}\right)$ is non-linear. We will postulate below a model
of growth dynamics that fits these two facts, and show how it can explain the
non-linear relation between revisions and errors.

From now on we report results using the above normalization by the mean
absolute distance. This allows to account for fat-tails effects stemming from
heterogeneous growth variance. Also quite importantly our model (2)-(3)
assumes homoskedasticity, so it is important to rescale the data so that they
have the same property.

A second key prediction of our Data is that the conditional expectation
$E_{t}\left(g_{t+1}|g_{t}\right)$ should be non-linear, as shown in Figure 3.
As mentioned previously the intuition is that large revision presumably come
from large shocks to firm sales growth. Since in our model large shock are
transitory, the rational forecaster should not expect that large shocks should
persist going forward. Smaller shocks are, however, much more likely to stem
from the permanent component of revenue, and therefore the rational forecaster
should expect them to persist. We now check whether this relationship holds in
the data.

Figure 8: Future Growth as a Function of Past Growth (Normalized)

Note: This Figure reports the binned scatter plot of future growth
$|g_{it+1}-\overline{g}_{i}|/MAD_{i}$ by value of past growth
$|g_{it}-\overline{g}_{i}|/MAD_{i}$. Each bin corresponds to a centile of the
past growth distribution. The blue line is a local polynomial approximation
centered around each one of these centiles.

In Figure 8 we construct a binned scatter plot of sales growth against lagged
sales growth. To make sure all firms have the same growth volatility (as in
the model), we normalize growth by our estimate of the firm level standard
deviation $MAD_{i}$. Each black dot on this figure represents a centile of the
distribution of lagged log growth of sales. The x axis shows the average
lagged growth and the y axis measures the average current growth. This chart
shows that the relationship between current and lagged growth is far from
being linear and looks like the S curve shown in Figure 1. For intermediate
levels of growth (between 0 and 1 standard deviation), past growth translates
into higher future growth, with a coefficient of about 0.3. The relationship
does, however, become much flatter for high growth (with a slightly negative
slope in the tail). For negative growth the slope becomes strongly negative.
The lower the past growth, the higher the future growth will be, which is
consistent with the idea of a rebound. Conditional on survival, very poor past
performance predicts strong future growth, as in our model.

###  5.2 Predicting Forecast Errors

The model was designed to predict that forecast errors be a non-linear
function of past revisions. Given that the model assumes that all firms have
the same variance of shocks, it is natural to check that our main empirical
results holds after rescaling by firm-level variance. Another reason why it is
important to perform such a robustness check is discussed is discussed in the
previous Section. Assume, following Wyart and Bouchaud (2003), that large news
purely come from a separate group of firms (those with more volatile, but
still Gaussian, shocks). Then, if forecasters use a firm-level linear
forecasting rule, then their forecast errors should be close to unpredictable
(to the extent that the AR(p) model they use mimics the optimal Kalman
filter).555If, however, forecasters were to use a global forecasting rule (a
single rule estimated on all firms), the non-linear shape may be predicted.
Indeed, assume all shocks are Gaussian, but firms differ in the volatility of
their _temporary_ shock $\epsilon$. In this case, a unique forecasting rule
would overestimate the persistence of large shocks. We do not explore this
lead in this paper since we have documented in the previous Section that
normalized growth is far from being normally distributed.

Figure 9 shows that normalized error and normalized revisions follow the same
relationship as in our headline Figure 1. In this Figure, we simply show the
binned scatter plot of future log forecast errors
$\frac{g_{t+1}-F_{t}g_{t+1}}{MAD_{i}}$ as a function of past revision
$\frac{F_{t}g_{t+1}-F_{t-1}g_{t+1}}{MAD_{i}}$. This suggests that the non-
linear relationship does not stem from firm volatility heterogeneity.

Figure 9: Revenue Forecast Error as a Function of Past Revision (Normalized)

Note: In this figure we use our international sample of firm revenue
expectations to report the binned scatter plot of future log forecast errors
$\frac{g_{t+1}-F_{t}g_{t+1}}{MAD_{i}}$ as a function of past revision
$\frac{F_{t}g_{t+1}-F_{t-1}g_{t+1}}{MAD_{i}}$. The blue line is a local
polynomial approximation, centered in the middle of each centile.

Another natural prediction of our forecasting model is that current and past
forecast errors should follow a similar relationship. This comes from the fact
that in our linear forecasting model, errors and revisions are proportional
(equation 4). Thus, it mechanically follows that if error and lag revisions
are linked by the S-shaped curve of Figure 9, then error and lagged error
should follow the same relationship.

We look at the relation between error and lagged error in Figure 10. It shows
the binned scatter plot of future log forecast errors
$\frac{g_{t+1}-F_{t}g_{t+1}}{MAD_{i}}$ as a function of past error
$\frac{g_{t}-F_{t-1}g_{t}}{MAD_{i}}$, both of them normalized by firm-level
volatility. As can be seen from this figure, the forecast errors follow a
linear relationship for intermediate values (until about 1 unit of
volatility), but the relationship reverses for larger past errors.

Figure 10: Revenue Forecast Error as a Function of Past Error (Normalized)

Note: In this figure we use our international sample of firm revenue
expectations to report the binned scatter plot of future log forecast errors
$\frac{g_{t+1}-F_{t}g_{t+1}}{MAD_{i}}$ as a function of past error
$\frac{g_{t}-F_{t-1}g_{t}}{MAD_{i}}$. The blue line is a local polynomial
approximation, centered in the middle of each centile.

###  5.3 Evidence from Returns

Table 2: Sample size by exchange (returns) Index | Total | 2000 | 2005 | 2010 | 2015 | 2020
---|---|---|---|---|---|---
AEX | 5892 | 0 | 290 | 300 | 278 | 283
AS5 | 47252 | 0 | 2127 | 2340 | 2164 | 2361
CAC | 9946 | 0 | 475 | 474 | 474 | 480
DAX | 6531 | 0 | 360 | 360 | 360 | 351
HSC | 5946 | 0 | 0 | 281 | 435 | 585
HSI | 8949 | 0 | 0 | 522 | 582 | 597
IBE | 8663 | 0 | 398 | 419 | 410 | 419
IND | 5377 | 0 | 0 | 360 | 360 | 348
KOS | 36595 | 0 | 0 | 2381 | 2345 | 2369
MID | 120017 | 4585 | 4773 | 4770 | 4583 | 4750
NDX | 21624 | 0 | 1171 | 1200 | 1234 | 1197
NIF | 6794 | 0 | 0 | 52 | 599 | 600
NKY | 60783 | 2642 | 2674 | 2698 | 2676 | 2698
OMX | 7565 | 0 | 348 | 360 | 360 | 360
RAY | 830673 | 29792 | 32936 | 33833 | 31915 | 33247
SMI | 3715 | 0 | 0 | 233 | 240 | 228
SPT | 10200 | 0 | 0 | 707 | 708 | 720
SX5 | 9744 | 0 | 0 | 600 | 600 | 587
TOP | 5229 | 0 | 0 | 5 | 504 | 452
TPX | 518734 | 16490 | 19074 | 19783 | 21722 | 25588
TWY | 13663 | 0 | 0 | 355 | 1168 | 1028
UKX | 29775 | 1086 | 1180 | 1196 | 1180 | 1187

Our model in Section 4.5 predicts that past returns should predict future
returns in a non-linear way. We now provide evidence on returns based on CFM’s
international monthly stock returns data described in the Data section of this
paper.

In Figure 11 we first show a smoothened binscatter plot of future returns on
past returns. Future returns are monthly and past returns are calculated over
the past 12 months excluding the last month of returns, as is common in the
literature on stock momentum. The only difference here with the standard
literature is that we take the log of returns (this is done because this
analysis tends to focus on extreme past returns).

Figure 11: Binscatter Plot of Returns by Past Returns

Panel A: Entire Sample Panel B: US vs International
Panel C: By holding Period Panel D: By S index

Note: These 4 panels represent smoothed binned scatter plots of future log
returns as a function of log past cumulative returns of the past 12 months
excluding the last month. Panel A is the entire sample, Panel B splits the
sample into US and International stocks, Panel C computes future returns using
different holding periods, and Panel D by size quintile.

Figure 11 shows the binned scatter plot for different splits of the data.
Panel A looks at the entire dataset and provides a picture consistent with our
prediction: There is momentum for most levels of past returns, but for extreme
values it is mean-reversion that prevails. Panel B shows that this pattern
holds both on US data and non-US returns. Panel C investigates the role of
various holding periods, i.e. looking for future returns over the following 1,
3, 6, 9 and 12 months. We find that the S-shaped curve emerges as soon as this
holding period is longer than one month. In panel D, we sort stocks into
market cap quintiles at the index-month level. Even when examining different
sizes of stocks, the S-shaped pattern is to be seen everywhere.

This finding suggests that the performance of traditional momentum strategies
could be “boosted” by allowing for a region of reversal in both tails. To test
this, we consider a self-financing strategy that goes long on momentum for
moderate values of the momentum signal, and goes short on a momentum (i.e.
long on reversal) for more extreme values of the signal. Let $s_{i,t-1}$ be a
momentum signal calculated from past returns. Specifically, we calculate the
momentum signal as the cross-sectional rank transform of cumulative returns
over 11 months from $t-12$ to $t-1$, normalized such that $s_{i,t-1}=0.5$ for
firm with the greatest past returns, and $s_{i,t-1}=-0.5$ for the firm with
the least. A portfolio with weights $w(s_{it})$ is then formed at time $t$ as
follows:

$w_{it}=\begin{cases}0.5-\frac{s_{i,t-1}}{a}&\text{if $s_{i,t-1}\leq a$}\\\
\frac{s_{i,t-1}-b}{b-a}-0.5&\text{if $a<s_{i,t-1}\leq b$}\\\
0.5-\frac{s_{i,t-1}-b}{1-b}&\text{if $b\leq s_{i,t-1}$}\end{cases}$

where $a$ and $b$ are constant “inflection” points at which our strategy flips
from reversal to momentum, and then from momentum to reversal.

Figure 12: The Error-Revision relationship: Sample Splits

Panel A: Portfolio weights Panel B: Sharpe ratio by inflection point

Note: Portfolio formed across all firms in our equity sample, both U.S. and
international. See text for details of momentum signal construction.

Our findings are depicted in figure 12. The inflection points of our strategy,
$a$ and $b$ are chosen to optimize the Sharpe ratio of our strategy over the
sample period. Panel A shows the Sharpe ratio maximizing weighting function,
while panel B shows how the Sharpe ratio varies for different upper and lower
inflection points. Note that a strategy with a low inflection point of 0 and
upper inflection point of 100 corresponds to a traditional momentum strategy,
with no tail reversal. Clearly, our approach contains some look ahead bias, as
the coefficients $a$ and $b$ are estimated on the entire sample. A more
systematic investigation of these returns is beyond the scope of this paper.

In line with our earlier empirical results, we find that the Sharpe ratio of
our strategy is maximized when the lower inflection point is set at the 13th
cross-sectional percentile of the normalized momentum signal, and the upper
inflection point is set at the 86th percentile. The maximum Sharpe ratio to
this strategy (0.61) is 1.27x the Sharpe ratio we observe for a pure momentum
strategy (0.48).

##  6 Conclusion

In this paper we emphasize that boundedly rational agents, when faced with
fat-tailed processes, will make predictable mistakes. In order to explore such
processes, we need large samples. Our empirical research here leverages the
international version of IBES which gives us a large panel of sales growth
forecasts. Consistently with the firm demographics literature, we find that
sales growth dynamics are well described by the sum of a short-run and and
long-run processes. The long-run process is a simple Gaussian, AR1 process,
but the short-run process has fat tails. As a result, a simple, linear
filtering rule will not be optimal. This simple model of expectations
formation matches a lot of the key features of the data.

Natural extensions of our work consists in exploring alternative forecasts
data. Macro forecasts are unlikely to provide us with non-Gaussian processes
and are in general too sparse to measure the tails of the DGP with enough
accuracy. Within IBES studying EPS forecasts is another natural research
direction, although it presents a scaling challenge. Growth cannot be computed
for a large number of firms. Internal sales forecast from large companies
could be another path.

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