Expectations Formation with Fat-tailed Processes Evidence from Sales Forecasts Eugene Larsen-HallockAdam RejDavid Thesmar

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Expectations Formation with Fat-tailed

Processes: Evidence from Sales Forecasts *

Eugene Larsen-Hallock†Adam Rej ‡David Thesmar §

October 20, 2022

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.

*We thank seminar participants at CFM and Columbia GSB for their constructive feedback. Thesmar is a consultant for

CFM.

†Columbia & CFM

‡CFM

§MIT, CEPR & NBER

arXiv:2210.10169v1 [q-fin.ST] 18 Oct 2022

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 regulari-

ties 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 irra-

tional but linear, featuring for instance under-reaction (Bouchaud et al., 2019) or

overreaction (Bordalo et al., 2019, 2020; Afrouzi et al., 2020). The second ap-

proach 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 cali-

brated 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 exist-

ing 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, 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 decreas-

ing 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 fre-

quency 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 ex-

treme and more persistence when realizations are intermediate. We assume, how-

ever, 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 er-

rors 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 observa-

tions 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, fu-

ture 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 simi-

lar 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 re-

cent research from Schmidhuber (2021), who also ﬁnds evidence of momentum for

“normal past returns” and mean-reversion for extreme values of returns. We con-

clude 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 for-

mation. 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 em-

pirical 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 com-

putation 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.

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ExpectationsFormationwithFat-tailedProcesses:EvidencefromSalesForecasts*EugeneLarsen-HallockAdamRejDavidThesmar§October20,2022AbstractWeempiricallyanalyzealargesampleofrmsalesgrowthexpectations.Wendthattherelationshipbetweenforecasterrorsandlaggedrevisionisnon-linear.Forecastersunderreacttotypic...

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Expectations Formation with Fat-tailed Processes Evidence from Sales Forecasts Eugene Larsen-HallockAdam RejDavid Thesmar

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