OBVey
HD28
.M414
ALFRED
P.
WORKING PAPER
SLOAN SCHOOL OF MANAGEMENT
ANALYSTS* FORECASTS AS EARNINGS EXPECTATIONS
Patricia C. O'Brien
WP# 1743-86
revised March 1987
MASSACHUSETTS
INSTITUTE OF TECHNOLOGY
50 MEMORIAL DRIVE
CAMBRIDGE, MASSACHUSETTS 02139
ANALYSTS' FORECASTS AS EARNINGS EXPECTATIONS
Patricia
WP# 1743-86
C.
O'Brien
revised March 1987
ANALYSTS'
FORECASTS AS EARNINGS EXPECTATIONS
Patricia
C.
O'Brien
Massachusetts Institute of Technology
examine three composite analyst forecasts of earnings per
share as proxies for expected earnings.
The most current
forecast weakly dominates the mean and median forecasts in
accuracy.
This is evidence that forecast dates are more relevant
for determining accuracy than individual error.
Consistent with previous research, I find analysts more
accurate than time-series models.
However, prior knowledge of
forecast errors from a quarterly autoregressive model predicts
excess stock returns better than prior knowledge of analysts'
errors.
This is inconsistent with previous research, and is
anomalous given analysts' greater accuracy.
I
Sloan School of Management
Massachusetts Institute of Technology
50 Memorial Drive
Cambridge, MA
02139
M.I.T.
LIBRARIES
1
Introduction
.
Analysts'
forecasts of earnings are increasingly used in
accounting and finance research as proxies for the unobservable
"market" expectation of
diverse set of forecasts
firm's earnings,
into
a
future earnings realization.
a
is
available at any time for
Since
a
given
composites are used to distill the diverse set
single expectation.
This paper considers the relative
merits of several composite forecasts as expectations data.
primary result
is
a
that
the most current
forecast available
The
is
more accurate than either the mean or the median of all available
forecasts.
This suggests forecast timeliness as
for distinguishing better forecasts.
is
that,
A
characteristic
a
second and related result
conditional on only relatively recent forecasts being
included, means or medians increase accuracy by aggregating
across idiosyncratic individual error.
A
second contribution of this paper is
analysts with time series models,
expectations.
a
a
comparison of
competing source of earnings
Consistent with previous research,
I
generally more accurate than time-series models.
that prior knowledge of the forecast errors from
find analysts
However,
a
I
find
simple
autoregressive model on the univariate series of quarterly
earnings provides better predictions of excess stock returns than
prior knowledge of analysts'
forecast errors.
This result is
inconsistent with prior research, and somewhat anomalous given
analysts'
greater accuracy.
A
third contribution is
a
methodological refinement of
techniques used to evaluate forecasts.
existence of significant
errors.
t
I
me-pe r i od -spec i
demonstrate the
f i c
effects in forecast
time-series and cross-section data are pooled without
If
taking these effects into account,
statistical results may be
overstated, and results are subject to an aggregation bias.
In
section
forecasts,
2
,
well
as
the empirical
I
as
tests.
describe proxies for consensus
in
two quarterly time-series models used
In
section
3
I
describe the data.
statistical tests and results are discussed in section
section
5
is
a
analysts'
in
The
4
,
and
summary with some concluding remarks.
.El2^i§5_I°r_l5D£Cted_Earnings
2.1
Defining Consensus for Analysts'
The motivations
for
seeking
a
Forecasts
consensus expectation when
many forecasts are available are primarily practical.
In
contexts earnings expectations are not the central issue,
necessary data.
For example,
to
many
but are
remove the effects of
simultaneous earnings releases on non-earnings events,
unanticipated earnings are necessary.
in
anticipated earnings, and therefore
Reducing measurement error
in
unanticipated earnings,
increases the power of tests of the non-earnings event.
individual
If
forecasts contain idiosyncratic error which can be
diminished by aggregation, more accurate forecasts can be
obtained by combining forecasts.!
Academic researchers have used
variety of methods to
a
aggregate analysts' earnings forecasts into
expec ta t i on
.
use
and Brown,
(1978)
a
(1979)
Griffin,
Hagerman, and Zmijewski
select the "most active" forecaster for each firm from
and Gultekin
Earnings Forecaster.
Foster and Noreen
Elton,
Gruber
(1984)
consider both means and medians published
.
Summary database.
I
the ValueLine
Brown,
(1981)
the mean,
and Brown,
in
the
I/B/E/S
Richardson, and Schwager (1986) use
forecasts and the I/B/E/S Summary forecasts.
compare three composites from
An
(1986)
Givoly and Lakonishok
those available in Standard and Poors'
both
Brown and
set of forecasts.
single forecast from ValueLine.
a
single
Barefield and Comiskey (1975), and Fried and
2
Givoly (1982) use the mean of
Rozeff
a
the median,
a
set
of
available forecasts:
and the most current forecast.
implicit assumption behind the use of either the mean or
the median forecast to represent consensus
are current,
so
is
that all forecasts
that cross-sectional differences
in
forecasts are
attributable to differential use of the same global set of
information.
Gains from combining forecasts arise either from
the employment of more
by
any individual,
information
in
the aggregate than
or from diversification across
is
used
individuals'
idiosyncratic errors.
In
fact,
however,
the analysts'
forecasts available at any
time have been produced at varying dates.
If
analysts
incorporate new information into their forecasts as the year
progresses as previous research suggests (Crichfield, Dyckman and
Lakonishok (1978),
Gultekin (1982)),
Collins and Hopwood (1980), Elton, Gruber. and
then more current forecasts are expected to be
However,
more accurate than older ones.
individual
i
d
a
diversifying across
osync rac i es is more important than eliminating out-
dated forecasts,
aggregations of many forecasts,
may be more accurate than
date,
if
a
regardless of
single current one.
I
provide
comparison of the relative importance of forecast age versus
diversification, by examining the most current forecast as an
alternate to the mean and median definition of consensus.
My results
indicate that the single most current forecast
dominates aggregations which ignore forecast dates.
relatively recent forecasts are included
is
in
When only
the aggregations,
possible to increase accuracy by aggregating forecasts.
Since the aggregate forecasts
in
it
3
published databases (e.g.
the
I/B/E/S Summary data and the Zacks Investment Research data)
ignore forecast dates,
my
results are relevant for researchers
using these sources.
2.2
Quarterly Time-series Models of Earnings
I
use quarterly time-series models of earnings as
benchmarks, against which analysts'
forecasts are compared.
Time-series models have been used frequently
to
provide earnings expectations.
in
previous research
Analysts, however,
have the
advantage of
a
broader information set,
firm sales and production figures,
information, and other analysts'
historical series of earnings.
including industry and
general macroeconomi c
forecasts,
Analysts'
in
addition to the
forecasts,
therefore,
seem likely to be more accurate than forecasts from univariate
models
.
Several studies
(Brown and Rozeff
(1978),
Collins and
Fried and Givoly (1982)) demonstrate that
Hopwood (1980),
analysts are more accurate than univariate models, presumably
because of the broader information set they can incorporate.
Fried and Givoly
(1982)
also find that analysts'
are more closely associated with excess
those of univariate models.
remain
forecast errors
stock returns than are
Nonetheless, univariate models
common means of generating earnings expectations.
a
An advantage of univariate time-series models
is
the
relative ease with which earnings data can be obtained for
moderate samples of firms.
This advantage is tempered by the
caveat that the data requirements of the models impart
"survivorship" bias to samples.
models
is
the relative
Another advantage to time-series
simplicity of the models used to generate
Parsimonious models with
expectations.
a
a
single,
simple ARIMA
structure applied to all firms have been shown to predict at
least as well as univariate models with individually-specified
structures, when one-step-ahead forecast errors are compared
(Foster (1977),
I
Foster
use
(
the
1977
)
:
Watts and Leftwich
(1977)).
following two quarterly time-series models,
from
E[a
and
..
J tq
where
a
j t
quarter
q
jtq ]
.
J
.
e
j0
jl
U jtq-l
^tq-S
or egr e
=
a
.
+
.
.
Jtq-4
year
of
s s
and 6j
t,
The models are,
i
ve process
have proven to be at
models.
)
in
.
©jl
j
in
and 6j2 are estimated
respectively,
a
first order
fourth differences with
a
drift.
a
I
drift,
and
a
chose these
and because they
least as good as other mechanical quarterly
The data used and the estimation of these models are
described in section
3.1
1
(2)
j2
models because of their relative simplicity,
.
(
e ..
random walk in fourth differences with
3
1
denotes quarterly reported earnings for firm
q
parameters.
t
6
Jtq-4
:
E[a
au
3
]
3.
Data
Sources
The forecast data are from the
Estimate System,
brokerage house.
or
I/B/E/S,
Institutional Brokers
developed by the Lynch, Jones
The database consists of
&
Ryan
individual analysts'
forecasts of earnings per share (EPS) 4 made between early 1975
and mid-1982,
by
analysts at between 50 and 130 brokerage houses.
EPS are forecast for approximately
on
the month and year
in
question.
1000 to 2500 firms,
The
depending
individual forecasts are
used by I/B/E/S to compute summary data,
such as means,
The summary statistics are sold to
and standard deviations.
primarily institutional investors.
clients,
medians
The I/B/E/S Summary
data have been analyzed extensively by Brown,
Foster and Noreen
among others.
(1984),
Each brokerage house in the database employs many analysts,
but at most one forecast
any time,
for
a
reported from each brokerage house at
is
given firm and year.
houses are identified
Analysts and brokerage
the database by code numbers.
in
The
I/B/E/S data are updated once per month with new forecasts.
use primarily two pieces of
forecasts,
information:
individual analysts'
and their associated forecast dates.
selecting the sample of available forecasts for
year,
described more fully
by Lynch,
Jones
&
I/B/E/S publication dates,
a
COMPUSTAT
a
given firm and
differs from that used
I
to
use analysts'
The most
forecast dates, not
define which forecasts are
given day.
is
the
announcement dates.
the Wall
My method of
Ryan to produce the Summary data.
important difference is that
available on
section 3.2,
in
I
source of earnings data and most earnings
The remaining announcement dates are from
Street Journal
(WSJ)
and its
Index.
Stock return data
and the trading day calendar are from the CRSP Daily Returns
file.
Data on stock splits and stock dividends are from the CRSP
Monthly Master file.
Some analysts occasionally forecast fully-diluted EPS,
rather than primary EPS.
This
is
indicated in the I/B/E/S detail
8
data.
I
convert forecasts of fully-diluted EPS to primary EPS,
using the reported ratio of fully-diluted to primary EPS for that
firm and year from COMPUSTAT.
time series model
I
also adjust both analysts'
and
forecasts for any stock splits and stock
dividends announced between the forecast date and the annual
earnings announcement date.
Sample Selection
3.2
The sample selection criteria and effects on the sample size
are summarized
in
Table
1
.
The
initial sample comprises the set
of
firms in the I/B/E/S database with December yearends,
at
least one forecast available in each year from 1975 through
1981.
This set contains 508 firms,
year is excluded
if
the annual
COMPUSTAT.
if
all
on
or
and 3556 firm-years.
and with
A
firm-
earnings number is not available
four quarterly earnings announcement
dates are not available from COMPUSTAT or the WSJ.
This
criterion reduces the sample to 497 firms, with 3440 firm-years.
The requirement
that returns data be available on CRSP reduces
the sample to 410 firms.
The estimation requirements of the
quarterly models. 30 continuous quarters of data prior to 1975IV,
impose the most drastic reduction in the sample,
to
184
firms
with 1260 firm-years.
Forecasts for each firm and year are selected at five fixed
horizons of less than one year.
120.
60 and
5
The horizons are:
240,
180.
trading days prior to the announcement of annual
9
The first four horizons correspond roughly to dates
earnings.
following each of the year's quarterly earnings announcements;
the fifth is
immediately prior to the annual earnings
announcement.
For example,
corresponds to
a
a
horizon of 240 trading days usually
date after the previous year's annual
announcement, but before the current year's first quarter
announcement.
A
horizon of 180 trading days typically
corresponds to
a
date between the first quarter announcement and
the second quarter announcement;
I
define the set of available analysts'
horizon,
T
and so on.
The horizon date for horizon
firm and year as follows.
for firm
j
in
year
t
is
the date
announcement of firm j's year
t
forecasts for each
T
EPS.
trading days prior to the
Given
a
horizon date,
I
select the most recent forecast available from each brokerage
house forecasting firm j's EPS for year
t.
The number of
available forecasts increases as the horizon shrinks, as reported
in
Panel
B
of
Table
1
.
Some brokerage houses issue forecasts
before the start of the year,
the year.
and update them periodically during
Many others add forecasts as the annual EPS
announcement approaches.
To
determine this set of available forecasts,
I
use the
dates assigned to forecasts by the analysts,
not the dates of
first publication of the forecasts.
The publication
I/B/E/S'
lag,
or
time between the analyst's forecast date and the date of
first appearance on I/B/E/S,
the forecast's
days,
and has
a
averages 34 trading
standard deviation of 44.5 trading days.
Thus,
10
some recently-updated forecasts are omitted from each monthly
listing by I/B/E/S.
From the set of available forecasts for each firm
and horizon x,
current.
consensus.
the analysts'
year
t,
compute the mean and median, and find the most
I
use the mean,
I
j,
median and most current as proxies for
Since published means and medians are
computed from the monthly lists,
reflect some recent updates.
probably results
in
the
I/B/E/S Summary data fail to
Eliminating the publication lag
my consensus analyst
forecasts being somewhat
more accurate than those published in the Summary data.
Most previous studies of analysts'
forecasts have used
publication dates, not analysts' dates, to select forecasts.
example,
For
Fried and Givoly (1982) and Givoly and Lakonishok (1979)
select their samples based on the publication date of the
Standard
&
Poors'
Earnings Forecaster,
their source of forecast
Givoly go on to use analysts'
dates within that
sample to distinguish new and old forecasts.
Brown and Rozeff
Fried
data.
(1978),
&
Brown.
and Zmijewski
Foster and Noreen
(1986)
and
Brown,
(1984),
Brown.
Richardson and Schwager (1986)
use datasets for which individual analysts'
not available.
Griffin, Hagerman
forecast dates are
Using publication dates instead of forecast dates
probably biases results against analysts, by failing to include
some recent updates of forecasts.
In
spite of eliminating the publication lag,
for any given
horizon date many of the forecasts available were made prior to
the
last quarterly earnings announcement.
This may indicate
1
analysts'
failure to incorporate new information,
but need not.
announced quarterly EPS may be close to the
For example,
analyst's expectation,
so
little new information is conveyed by
the quarterly announcement and
forecast is unnecessary.
a
revision of the annual EPS
investigate
I
a
subsample consisting of
only those forecasts which have been updated since the most
recent announcement of quarterly EPS.
in
section 4.4.
reported
3.3
The subsample is described
where the results of this investigation are
.
Measuring Forecast Errors
Forecast errors are the elementary data
The forecast error
forecasts.
between
Aj t
i,
at
a
j
in
year
horizon
x
and
t,
prior to
6
e,
.„
UtT
The source of
following:
use to evaluate
defined as the difference
is
tT
actual earnings per share of firm
,
the realization:
analysts'
i j
forecast of EPS from source
tne
fi tt
j
e
I
=
A
.
Jt
the forecast,
the mean,
forecasts:
the median,
or
one of
i
denoted by
or
(3)
jtT
i,
is
one of the
the most current of available
the two benchmark quarterly models
described in section 2.2.
The fundamental difference between the most current analyst
forecast as
median,
is
a
consensus definition and either the mean or the
that the former is constructed using the forecast
12
date,
while the latter two are not.
distributions of forecast ages
forecast
is
in
Table
In
2,
this sample.
defined as the difference,
compare
I
The age of
trading days,
in
a
between
the forecast date and the horizon date chosen for this study.
More generally,
this might correspond to
a
lag between the
forecast date and an event date of interest to the researcher.
For Table
as,
2,
define the ages of the mean and median forecasts
I
respectively,
the mean and the median of
the ages of
the
forecasts in the set of available forecasts for each firm, year
The distribution described in Table
and horizon.
firms and years,
As
has
a
over all
the most current
forecast
for each horizon.
expected from its definition,
distribution of ages much closer to zero than either the
mean or median.
60
is
2
trading days),
For the four
longer horizons
(240,
180.
and
120.
over fifty percent of the most current
forecasts are less than five trading days old.
By contrast,
over
fifty percent of the mean or median ages at all horizons are more
than forty trading days old.
While some of these older forecasts
may represent circumstances where little new information has
arrived,
so
there was no need to update,
the accuracy results
which follow suggest that this is not always the case.
Quarterly models
(1)
and
for each quarter from 1975-1
(2)
are estimated for each firm,
through 1981-IV.
estimates are updated each quarter,
quarters'
observations.
the number of
Parameter
using the previous thirty
Observations are adjusted for changes
outstanding shares.
Annual forecasts are
in
13
constructed from quarterly forecasts by summing the appropriate
realizations and forecasts.
For example,
in quarter
there
3,
have been two realizations of quarterly earnings for the year,
and two quarters remain to be forecast.
The annual forecast from
quarterly model during the third fiscal quarter
a
actual earnings for quarters
and
3
4
A
and
1
is
the sum of
and forecasts for quarters
2,
.
small number of
influential observations altered the
regression results reported below.
Since analysts and brokerage
7
houses are identified only by code numbers
in
the database,
was no way to trace these observations to other sources.
this reason.
I
there
For
imposed an arbitrary censoring rule on the data
forecast errors larger
for errors which could not be traced:
than $10.00 per share in absolute value were deleted from the
Since typical values for EPS numbers are in the range of
sample.
$1.00 to $5.00 per share,
errors of sufficient magnitude to be
deleted are rare, and suggest
error
3
.
3
data entry or transcription
a
.
Stock Returns
I
measure the new information impounded in stock returns by
the cumulated prediction errors from a market model
in
logarithmic form:
E[ln(l
where Rj
s
is
+
the return
R. )]
js
to
=
a
.
+
B
j
security
.
1
n
(
1
+
J
j
on
day
RM
Ms
s,
(4)
)
Rm s
is
the return
14
on
the CRSP equally-weighted market portfolio of securities on
day
and
s,
denotes the natural logarithm
In
The parameters of
(4)
rans f orma t i on
.
are estimated for each firm in the
study using 200 trading days of data at
time,
a
beginning
in
July
Estimated parameters are used to predict ahead 100 trading
1974.
and excess returns are the difference between the
days,
realization
each
t
1
n
(
1
R
+
and tne prediction based on
js>
iteration of estimation and prediction,
period
After
the estimation
rolled forward by 100 trading days,
is
(4).
and new parameters
are estimated.
The estimation procedure produces
daily excess returns,
forecast horizon,
4.1
u
1
form
to
arriving over horizon
s
.
The
are cumulated over each
e j s
from the horizon date through the announcement
date of annual EPS,
4^__Re
e j s
stream of predicted
a
in
x
U
j t
t
year
the measure of new information
•
for firm
t
j.
t
Aggregating Forecast Errors
If
a
forecast incorporates all the information available on
the forecast date
the usual
forecast
in
unbiased manner,
an
it
statistical sense of the word.
is
an expectation
Let f* denote such
in
a
:
f
•
JtT
=
E[A
.
Jt
I
A
y
tT
]
(5)
15
where
4>
represents the information available at
tT
prior to the realization,
and
E
[
•
|
horizon
x
the conditional
is
]
a
expectation operator.
Between the forecast date and the realization date, new
information may arrive.
ideal
Even
forecast like (5), which may be
a
the sense of employing all
in
forecast date,
omits unanticipated information which arrives
Forecast errors consist,
later.
information available on the
in
part or entirely,
information revealed over the forecast horizon,
i.e.
of
new
between
forecast and realization.
Two closely -related implications of unanticipated
information reflected
in
forecast errors are important for the
specification of statistical tests.
within
to
forecast errors
year which are aggregated cross-sectionally may appear
a
"biased" because of the common new information reflected in
be
Second,
them.
this common information is not accounted for,
if
induce contemporaneous cross-sectional correlation
will
it
First,
in
forecast errors.
An example of
errors
has
a
an unanticipated macroeconomic
is
firms in
a
similar manner.
non-zero mean,
forecast errors,
non-zero mean.
t
information which may be reflected in forecast
then
If
a
shock affecting many
the effect of the shock on firms
cross-sectional aggregation of
even from unbiased forecasts,
This non-zero mean is not bias,
me-per i od -s pec i f i c new information.
If
will also have
but rather is
time-period-specific
a
16
they induce correlation in forecast errors
effects are ignored,
across firms,
within
for
a
given year and horizon,
and across horizons
year.
a
forecast errors are positively correlated across firms
If
within years,
statistical comparisons based on pooled time-series
and cross-section forecast error data which assume cross-
sectional independence will overstate the statistical validity of
the results.
Several studies
Gruber and Gultekin
(1981).
(Brown and Rozeff
(1978),
Malkiel and Cragg (1982).
Elton,
for
example) have compared forecasts using criteria such as the
number or proportion of times that one forecasting method
outpredicts another.
This criterion,
independent observations and
is
or any other
applied to
a
that assumes
cross-section, could
obtain the appearance of statistically significant superiority in
forecasting ability from an anecdotal difference.
tests developed in this paper adjust for time-period-
The
specific shocks using
and its
A
s
simple fixed effects model.
This model,
importance for the results, are described below.
Evaluating Forecasts
4.2
1
a
-
Bias
simple model of time-period effects in forecast errors
9
e
where
u tT
is
.
jtx
=
U
M
+
tT
P
(6)
jtT
the average forecast
error for year
t
and horizon
x
17
and 'HjtT
i
random error term,
a
s
representing the deviation of
firm j's forecast error from the common annual mean.
There may also be
f
rm- spec
i f i
c
information effects which
persist through time,
but the unanticipated information argument
does not apply. 10
a
If
forecast fully impounds information from
previous mistakes in an unbiased manner,
systematically recurring
events will not remain unanticipated year after year.
recurrent
on
f
i
r
m-
s
pec
i
f
i
Thus
forecast errors are not expected to arise
c
information that was unavailable at the time the
the basis of
forecast was made.
estimate (6) using least squares with
I
each year.
The test for bias
is
a
dummy variable for
based on the grand mean of the
estimated annual averages:
1
(7)
t?/ tT
where
T
is
the number of
years in the sample,
The average
x.
combination of
1
ea
uT
defined in (7)
-squa r es coefficients,
s
u tT
are the
estimated separately for
year-specific average forecast errors,
each horizon
and the
is
a
linear
with estimated
standard error:
T1
s-
=
T
.
[\' (X'X)
_1
i]
1/2
(8)
T
where
\
is
variables,
a
vector of ones of length
and s^ is
standard error (8)
is
the
a
T,
X
is
the matrix of dummy
regression residual standard error.
The
weighted version of the residual standard
18
constructed from the
error s^,
nj
t t
•
The residuals n j
tt
are
deviations from the annual averages, and so are purged of average
time-period-specific information which induces cross-sectional
correlation
Table
In
the
in
e-j
tT
.
the bias results are presented.
3.
The reported
numbers are the forecast bias, estimated jointly for all forecast
sources by stacking equation
(7)
to
evaluated as
is
(8)
hypothesis of no bias,
and most current,
and
a
(7)
across sources.
t-statistic,
The ratio of
against
for each analyst composite,
null
a
mean,
the quarterly models.
for
forecast errors exhibit statistically significant
Generally,
negative bias.
the
Of
three analyst consensus measures,
median uniformly exhibits the smallest bias,
indistinguishable from zero.
bias in analysts'
forecasts
11
is
management.
however,
12
in
to
Negative
consistent with some conventional
that analysts prefer optimistic predictions and
recommendations,
the
usually
Negative bias corresponds to overestimates of EPS.
wisdom,
median
"buy"
help maintain good relations with
The evidence supporting this story is weak,
two
respects.
appears to be unbiased.
First,
Second,
the median analyst
forecast
and more importantly,
analyst estimates are significantly negative,
when the
they are
statistically indistinguishable from those of mechanical timeseries models.
The motive of maintaining good relations with
management cannot be ascribed to these models.
the contention
forecasts
is
at
Thus,
support for
that analysts preferentially issue optimistic
best weak.
19
An alternative explanation which
also consistent with
is
these results is that analysts issue unbiased forecasts,
seven-year period,
1975 through 1981,
negative unanticipated EPS.
to
Unfortunately,
the most obvious way
distinguish between the hypothesis of deliberate optimistic
years.
-
accuracy.
section for evaluating relative forecast
Accuracy
is
defined as absolute or squared forecast
For the absolute error case,
I
1
e
.
jtT
1=6.
ljT
tT )2
as
the
the model
a
similar model
left-hand-side variable.
In
*ljT measure average accuracy for each firm
measure average accuracy for each year
deviations from the average accuracy
and for year
arise,
t.
is:
+6 2tT +£.JtT
1
For the squared error case,
(e-j
Accuracy
use an approach similar to the bias evaluation described
the previous
error.
longer span of
a
This is not possible using the I/B/E/S detail data.
Evaluating Forecasts
I
in
in
t.
(9)
is
estimated, with
equation (9) the
j,
The
and the
£
j t
x
are
j
Differences in accuracy across firms could
for example,
if
there are persistent differences in the
accuracy across years could arise
larger,
6 2 t t
this sample for firm
amount of information available for different firms.
in
this
one with primarily
is
bias and this alternative is to collect data for
4.3
but
unanticipated events
in
if
Differences
there are more,
or
some years than in others.
20
Equation
estimated using least squares on
is
(9)
Average accuracy
dummy variables for firms and years.
computed as
a
set of
a
is
linear combination of the estimated effects;13
1
£
7
1
JT
t=
l
(10)
2JT
Equation (10) defines the average absolute error accuracy.
Average squared error accuracy
defined similarly using the
is
coefficients from the squared error version of (9).
estimated standard error of the average accuracy
l
s
equation (10)
:
-
[W' (Z 'Z
In
in
The
(11).
Z
)
1/2
'
1
u>]
(
11
)
represents the matrix of dummy variables used to
estimate equation (9) or its squared error analogue,
U)
is
the
vector of weights that transform the estimated parameters into
the average defined
(10),
in
and
s
error from the regression equation
The estimates
in
^
is
the residual
standard
(9).
equations (9) through (11) are computed
jointly for all five forecast sources (mean, median and most
current analyst; and two quarterly models),
across sources.
by stacking equations
Pairwise differences in accuracy are compared
across forecast sources using
a
t-statistic constructed from the
average accuracies from (10) or its squared error analogue, and
the standard
Tables
accuracy.
error from (11).
4
through
6
summarize the results on forecast
The average absolute errors and average squared
21
errors,
computed as described in equation (10) for each forecast
source,
appear in Table
4.
Table
5
contains t-statistics testing
pairwise differences in accuracy among analysts.
Table
6
contains t-statistics testing pairwise differences in accuracy
between analysts and the quarterly models.
Table
4
displays the expected pattern of increasing forecast
accuracy as the earnings announcement date approaches,
for all
Both average absolute error and average
forecast sources.
squared error decline uniformly as the year progresses,
analysts and for quarterly models.
For example,
for
the average
absolute error of the most current forecaster declines from
$0,742 per share at
a
horizon of 240 days (almost
prior to the announcement)
to
$0,292 per share at
year
a
full
a
horizon of
days.
This pattern of convergence toward the announced EPS
number
is
5
consistent with forecasts incorporating some new
information relevant to the prediction of EPS over the course of
the year
.
From Table
it
4
appears that the most current analyst
worse than the other sources,
quarterly models
in
is
results are highly correlated,
the differences
Table
4
In
in
5
important caveat to
that the relative performance
both across horizons and across
accuracy which are suggested by
appear in Tables
Tables
An
The results of statistical
definitions of accuracy.
and 6,
5
a
no
and that analysts dominate
the longer horizons.
this qualitative statement
is
a
tests for
perusal of
and 6.
positive difference means that the
22
Table
first of the pair is less accurate.
of
5
contains the results
pairwise comparisons among the three analyst consensus
definitions.
In
terms of absolute error, which is reported in
when the differences are significant they favor the mean
Panel A,
over the median and the most current forecaster over either the
mean or the median.
For example,
at
the 60-trading-day horizon,
the t-statistic on the difference between the mean and median
forecasts in average absolute accuracy is -2.14, which favors the
and is significant at the
mean,
the difference between
for the same horizon
is
is
4.97,
.01
contrast,
Panel
in
The t-statistic on
the median and the most
significant at the
In
level.
.05
accuracy are presented,
current forecasts
which favors the most current,
and
level.
B,
where differences in squared error
there are no measurable differences in
accuracy among the three analyst consensus definitions.
The
signs of differences in this panel generally are the same as
those in Panel A,
so
the squared error evidence does not
contradict the absolute error results.
However,
the very small
t-statistics lend no additional support to the conclusions,
either
.
The results reported in Table
6
indicate that analyst
forecasts generally dominate the time-series models at the longer
horizons.
For the 240,
180 and
1
20- rad i ng-day horizons,
wherever differences are statistically significant,
This evidence is
favor analysts over the quarterly models.
consistent with the explanation that analysts use
information set than can be exploited by
the results
a
a
broader
univariate model.
23
At the 60-t rading-day horizon,
however,
the quarterly time-
series models dominate the mean and the median analyst forecast
in
comparisons where significant differences exist.
all
current forecast is never dominated to
a
The most
statistically
significant extent by the quarterly models, but generally
is
indistinguishable from them.
In
through
summary,
6
the accuracy results reported in Tables
generally support the conjectures that
presumably incorporating
analyst forecast,
set,
is
model,
at
and
least as accurate as
is
at
a
a
a
4
current
broader information
forecast from
a
time-series
least as accurate as aggregations which ignore
the forecast date.
Although the statistical significance of
results varies across forecasting horizons and accuracy criteria,
wherever differences
in
accuracy between forecast sources are
statistically significant, the results conform with expectations.
The results reported here probably understate the difference
between the most current forecast and the mean and median
definitions which appear
is
in
most other published work.
My sample
selected to eliminate the publication lag, and so the mean and
median forecasts in my sample are probably more accurate than,
e.g.,
those in the I/B/E/S Summary Data.
The comparisons between analysts and quarterly models may
understate differences because of the sample selection process.
Since the sample of firms is weighted toward stabler and longerlived firms by the data requirements of the time-series models,
the selection process may exclude firms where analysts'
24
information advantage
is
largest:
newer,
less stable firms,
or
where time-series models that assume stationarity are less
suitable.
However,
stability,
this may be mitigated.
(1986)
if
firm size is
a
proxy for longevity or
Brown,
Richardson and Schwager
find that the superiority of ValueLine forecasts to
a
random walk model increases with firm size.
Accuracy Within
4.4
a
Subsample of Timely Forecasts
The results reported
timeliness
no worse
in
in
section 4.3 suggest an advantage to
forecast selection.
single current forecast is
A
and sometimes dominates,
than,
aggregations which
include both recently-updated and out-of-date forecasts.
this,
the question arises whether,
Given
conditional on timeliness,
there is an advantage to diversifying idiosyncratic error.
this section,
I
address this question using
a
In
subsample which
includes only forecasts made after the firm's most recent
announcement of quarterly EPS.
14
described in Table
The selection of the subsample,
follows.
For each horizon t,
set of available
firm
j,
and year
I
,
is
as
select the
forecasts which were made after the most recent
previous quarterly earnings announcement.
in
t,
7
That is,
all
forecasts
this subsample have forecast dates indicating they have been
revised since the last quarterly earnings release.
This
criterion reduces the number of forecasts available by about onethird at the 5-trading-day horizon,
more at the other horizons.
and by 70 to 80 percent or
25
The dramatic difference between the 5-trading-day and the
other horizons in sample reduction,
evident in Table
largely due to the definition of forecast horizons.
by construction,
trading-day horizon date is,
months after the third quarter announcement.
also by construction,
dates,
quarterly announcement
7,
is
The 5-
approximately three
All
other horizon
tend to be closer to the previous
.
Both the timeliness and the number of forecasts included in
this subsample depend,
among other things,
on
the number of days
between the quarterly EPS announcement date and the horizon date.
This fact is evident in Table
the distribution of
of
where summary information about
forecast ages in this subsample is reported.
240 trading days,
For horizons between 60 and
the age of
8,
differences between
the most current forecast and the mean or the median
forecast ages are much less pronounced than those reported in
Table
2
for the full
Table
7
sample.
illustrates that up to 27% of the firm-years
in
the
sample are eliminated entirely by this timeliness criterion.
That is,
for some firms
in
some years,
none of the analysts'
forecasts available on the fixed horizon date had been updated
since the last quarterly EPS announcement.
the 5-trading-day horizon,
eliminated.
A
This is true even at
where one percent of firm-years are
related feature of the subsample is an increase in
the number of firm-years with only forecast available.
These
observations, where by definition the mean, median and most
current forecast are the same,
reduce the power of statistical
26
tests to distinguish between the three consensus definitions.
However,
in
the full
observations.
In
sample they are
this subsample,
a
trivial proportion of
they are
4
percent of
observations at the 5-trading-day horizon, but
of
2
to
30 percent
observations at other horizons.
Tests of pairwise differences in accuracy among the three
consensus definitions
this subsample are reported in Table
in
the most current forecast
In
contrast to the results in Table
no
longer dominates when all forecasts are reasonably current.
5,
9.
There are no statistically significant differences in accuracy in
the
longer horizons,
in
part due to the sample selection
raised in the previous paragraph.
issues
the 5-trading-day horizon,
At
where sufficient forecasts are available to distinguish the
different consensus measures,
the most current forecast
significantly worse than either the mean or the median.
is
That is,
conditional on timeliness and on the availability of sufficient
numbers of forecasts,
there are gains in accuracy from
aggregating to reduce idiosyncratic error.
4.5
Evaluating Forecasts
-
The criteria developed
Market Association
in
relative accuracy, are common
evaluation.
They do not,
forecasts are used.
the previous sections,
in
the
however,
bias and
literature on forecast
address the context in which
Both researchers'
and
investors'
use of
forecast data in contexts related to securities markets suggests
27
that association with stock returns may provide
relevant
a
empirical comparison.
If
forecast errors reflect information relevant to the
firm's prospects arriving after the forecast date,
to
two important qualifications discussed below,
will
be
this
forecast errors
information relevant
The first qualification
is
not precisely the same as the
current-year earnings.
to
There are errors
both variables with respect to the measured association
Non-recurring events, whether they are treated as
between them.
extraordinary items or not. may affect earnings in
year,
particular
events that influence longer-term prospects,
Conversely,
such as changes
a
inconsequential to the long-term value of the
but may be
firm.
of
in
implied association is that the information relevant to
valuing the firm's common stock
in
subject
positively correlated with new information impounded
stock returns over the forecast horizon.
to
then,
investment opportunities,
in
may affect the value
the firm without altering current earnings.
The second qualification
is
that excess returns are
constructed to exclude one source of unanticipated information.
The excess return
is
purged of its systematic relation with
market returns, which includes both anticipated and unanticipated
market returns.
It
is
desirable to purge the stock returns of
the anticipated component of
i
n f
or ma
t i
ona
1 1
y
however,
since
inefficient forecasts will be correlated with
anticipated information.
return,
the market return,
Eliminating the unanticipated market
may reduce the measurable association between
28
excess returns and forecast errors.
the other hand,
On
it
is
important to note that while the power of tests for positive
association
the reduction
reduced,
is
across forecast sources,
the same excess
in
power does not vary
since they are all evaluated relative to
returns.
other words,
In
the relative degree of
association across sources will be unaffected.
Both qualifications noted above will have the effect of
reducing the measurable association between forecast errors and
excess returns.
Nevertheless,
previous studies facing the same
inherent difficulties have found statistically significant
positive associations between unexpected earnings and excess
stock returns
(197 9),
and Brown
(Ball
and Fried and Givoly
Beaver,
(1968),
(1982),
Clarke and Wright
for example).
The regression model used to estimate the association
between cumulated excess returns,
forecast errors.
e
In
i
(12),
at
f3i
.
,.
T U-jtT
horizon
returns.
.
lJtT
x
e
M
=
a
is
t
x
,
.
represented by Uj
.
llJT
+
a„
.
+8.IT U_
Jtl
.
2ltT
and
+
v
.
(
.
ljtl
12)
the portion of the forecast error from source
which is systematically related to excess
The slope coefficient
in
,
is:
•
8
T
excess returns and forecast errors,
effects,
tT
is
the covariance between
adjusted for firm and year
units of the variance of excess returns.
Using
excess returns as the independent variable and forecast errors as
dependent has the desirable feature that
8j T
and
t-statistic have the same scale for all sources
of
its associated
i.
If
the roles
these two variables were reversed in the regression equation,
29
the estimated regression slope coefficient would depend
explicitly on the forecast error variance from source
The constants oti;j
and
y ea r
-spec
i
f i c
and 0t2itT measure,
T
The
play an important role in equation
capture the
t
respectively,
firm-
average forecast errors, conditional on the
systematic relation with excess returns.
effects,
i.
me-per i od -spec
i
f i c
cx
2
jtT-
(12),
the year
since they
information in forecast errors
which is not captured by excess returns.
I
5
Among other things,
they include the average effect of omitting the unanticipated
component of the market return.
the model,
they are
omitted variable.
the residuals,
impounded
This
If
in
the
the
ot
2 i t T
are
n
°t
included in
regression residuals as an
induces cross-sectional correlation in
which if ignored leads to incorrect statistical
inferences, as was discussed above for the bias computation. 16
Equation
(12)
is
estimated by stacking the regressions for
the five forecast sources,
and estimating them jointly.
forecast sources are the mean,
The
median and most current analyst
Estimations are performed
forecast and the two quarterly models.
jointly for the five forecast sources, and separately for each
forecast horizon.
Since the matrix of independent variables,
which consists of
cumulated excess returns over the forecast horizon and dummy
variables indicating firms and years,
forecast sources,
is
the same for each of the
there is no efficiency gain over equation-by-
equation least squares (see Zellner (1962)).
stacking the equations
is
for
The advantage of
joint estimation of the firm and
30
year effects,
so
that observations from each of the five forecast
sources are adjusted for the same firm and year effects.
Tables 10 through 12 contain the regression results from
Table 10,
I
slope coefficients and their associated
t
estimation of model
(12).
In
report the estimated
-s ta
t i s t i
c s
,
testing the
statistical significance of the relation between forecast errors
and excess
Table
returns over the forecasting horizon.
11
contains t-statistics which test for differences in the slope
Table 12 contains
coefficients across forecast sources.
regression summary statistics,
including adjusted
,
f -
and sample sizes.
and numbers of parameters,
statistics,
R2
According to results reported in Table 12,
equation (12)
explains between 9% and 16 % of the variation in forecast errors,
with slight variations across horizons.
appears at the 5-trading-day horizon,
explanatory power are not large.
The
largest adjusted
R2
though differences in
The model has statistically
significant explanatory power, according to the regression
F -
statistics, which reject the null of no explanatory power at the
.001
level.
The
incremental F-statistics in Table 12 confirm that the
year-specific effects are important
statistic on the year -s pec
H
That is,
the
estimation
to
of
the model.
:
a
i
=
2ilT
f i c
a
in
equation (12).
?
The F-
effects tests the null hypothesis:
2i2T
•
=
a„
m
2iTx
.
=0
F-statistic tests the null hypothesis that
year -specific
intercepts adds no explanatory power
This hypothesis is rejected at the
.05
level at
31
all
horizons,
longer than
and at the
level or better at the horizons
The importance of year effects in the model
days.
5
.001
increases with the length of the horizon.
This is consistent
with the information-based explanation for their inclusion in the
model,
namely that forecast errors impound time-period-specific
unanticipated information.
speaking,
Over longer horizons,
loosely
"quantity" of unanticipated information
the
greater.
is
The strength of this result also confirms the assertions made
earlier that
period is not
cross-section of forecast errors for
a
a
set
of
a
single time
independent observations.
The F-statistics on firm-specific effects reported
12
in
Table
also reject the null hypothesis, which is:
H
:
"lilt
1
a,
i2T
.
.
=
There are measurable firm-specific differences
forecast error at all horizons,
in
average
even after the adjustment for
firm-specific information impounded
in
excess returns.
The
strength of this result varies little across forecast horizons
and across transformations of the dependent variable.
The importance of the slope coefficients in model
indicated by the F-statisic reported in Table 12.
(12),
varies across
forecast horizons and across transformations of the dependent
variable.
10,
The individual
however,
Table 10 show
are of greater relevance.
a
first,
Generally,
the results
in
pattern of positive association between forecast
errors and excess returns.
if,
slope coefficients reported in Table
A
positive association is expected
there is some overlap between information relevant to
32
firm value and information relevant to current-year earnings,
second,
and
some of this overlapping information is unanticipated
both by investors and by the predictor of EPS.
significance of the positive association
The statistical
varies somewhat across forecast horizons,
across forecast sources.
and more importantly
The strongest results are for the 120-
Among the analyst consensus forecasts,
day horizon.
the
strongest results are generally for the most current forecaster,
which is consistent with the most current forecaster acting as
reasonable composite,
expectation.
or
The strongest results,
the only ones which are consistently positive and
a
and
statistically
significant, however, are for the quarterly autoregressive model,
equation
(1).
This pattern of relative performance does not vary
substantially across horizons or across transformations of the
dependent variable.
This result is anomalous,
especially
light of the quarterly model's relative inaccuracy.
that prior knowledge of the forecast error from
autoregressive model
is
a
a
It
in
indicates
quarterly
better predictor of excess returns than
prior knowledge of the forecast error from analysts'
forecasts. 18
The greater association of excess returns with forecast
errors from
a
time-series model than with those from analysts
not consistent with
use quarterly data
the
in
results of Fried and Givoly (1982).
the time series models of annual
while Fried and Givoly (1982) use annual data.
is
I
earnings,
Since models
using quarterly data produce more accurate forecasts of annual
earnings than models using only annual data (see Hopwood, McKeown
33
and Newbold
presumably my tests are more demanding of
(1982)),
analysts than those used by Fried and Givoly.
However,
the
result remains anomalous since analysts are more accurate and can
employ more information than quarterly time series models.
This anomaly is further investigated in Table 11,
for the statistical
coefficients.
by
testing
significance of differences in the slope
further advantage of stacking equations across
A
forecast sources to estimate them is that statistical testing is
simplified,
since
a
set
slope coefficients Bj
T
of
linear constraints on the estimated
generates direct tests of differences
For example,
slope across forecast sources.
of
is
the vector
difference
in
8
-r
is
the vector
one for each forecast source,
five slope coefficients,
Q22'
if
(1,-1,0.0,0),
then
c
1
2
'
in
and
if
estimates the
£T
slopes between the first and second sources, with
estimated standard error:
s
In
(13),
sv
is
C
,
„
e
=
s„
.
[c
1
- 1
1/2
(X'X)/c]
(
13)
6
the residual
estimation of equation (12).
standard error from the joint
X
(
'
X
~
)
1
g
the
is
lower-right
submatrix of five rows and five columns from the (X'X)~1 matrix
of
equation (12).
This submatrix determines the variance-
covariance relations among the five slope coefficients.
Linear constraints of the form of
differences
i.e.
in
the slope coefficients
c
j
2
are used to evaluate
that appear in Table 10,
differences across forecast sources
in
between excess returns and forecast errors.
the association
Table
11
contains
34
the results of
these statistical
Results are shown for
tests.
tests of pairwise differences between the quarterly
au oregres
s
i
ve
model and all other forecast sources,
and for
differences between the most current analyst and the mean and
median analyst forecasts.
anomalous result,
The statistical
that errors from
tests confirm the
mechanical quarterly model
a
often are significantly more closely related to excess returns
than errors from analysts.
In
addition,
the tests indicate that
while the most current forecast typically shows the strongest
result among analyst consensus forecasts,
statistically significant,
5^
the difference
not
is
general.
in
§y!D!D3iy._§Qd_Conclusi_ons
Analysts'
predictions of EPS are
"market expectations"
information.
different composite forecasts,
span approximately
the most current
a
year.
on
I
a
potential source of
have examined properties of
arbitrarily-chosen dates which
Results reported here indicate that
forecast available from an analyst dominates the
mean or median of the available forecasts.
Five alternate sources of earnings expectations are
examined:
the mean,
analysts'
forecasts;
median,
an
and most current of available
au tor egress i ve model
in
fourth
differences of the univariate series of quarterly EPS, and
fourth-differenced random walk using quarterly EPS.
a
The two
35
quarterly time-series models are included primarily as
benchmarks
.
The most current forecast is at least as accurate as either
the mean or median forecast,
and generally dominates them in
absolute error terms, when all available forecasts are
This result indicates that the date of the forecast
considered.
is
relevant for determining its accuracy, and dominates the
diversification obtained by aggregating forecasts from different
sources as
"first cut" criterion.
a
Since most published
aggregations of forecasts and much previous research treat all
forecasts for
a
given firm as if they are equally current,
they
ignore this relevant piece of information.
When forecasts are censored in
a
way that eliminates the
most out-of-date forecasts from the sample,
it
improve accuracy by aggregating forecasts.
When only forecasts
from the fourth fiscal quarter are included,
forecast at
a
announcement
horizon of
is
5
is
possible to
the mean or median
trading days before the annual EPS
more accurate than the single most current
forecast.
In
this sample the forecast error from the most current
forecast
is
more closely associated with excess returns over the
forecast horizon than the error from the mean or the median
forecast,
but
the difference
in
association
is
not,
in
general,
statistically significant.
Analysts generally are significantly more accurate than
time-series models.
Errors from the quarterly autoregressive
36
model,
however, appear to be more closely related with excess
returns over the forecasting horizon than those of analysts.
Because of this anomalous result,
provide
a
it
is
unclear that analysts
better model of the "market expectation" than
mechanical models.
It
should be noted,
though,
that the sample of firms was
reduced sharply by the data requirements of the time-series
models.
This sample,
with its selection bias toward longer-lived
firms with continuous data available,
does not clearly isolate
cases where analysts might be expected to have the most advantage
over mechanical models,
cases.
and perhaps eliminated many of these
These firms, where there is
a
substantial amount of non-
earnings information expected to have an impact on earnings, may
be a
fruitful area for future investigation.
37
ACKNOWLEDGEMENT
This paper is based on work from my doctoral dissertation at the
I
am grateful to the members of my
Univesity of Chicago.
committee--John Abowd, Craig Ansley, Nicholas Dopuch, Richard
Leftwich, Victor Zarnowitz, Mark Zmijewski, and especially to
Robert Holt hausen --for their helpful comments.
Others to whom I
an indebted for their comments on previous drafts are Linda
DeAngelo. Paul Healy. Rick Ruback, and an anonymous reviewer.
I
All remaining errors and omissions are my responsibility.
acknowledge the generous support of Deloitte, Haskins and Sells
Foundation
.
38
FOOTNOTES
^ee Bates and Granger (1969)
Figlewski and Urich (1982).
Granger and Newbold (1977) and
1
use words like "aggregate", "composite" and "consensus" in the
general sense, to include weighting schemes which put all weight
on a single forecast, and none on others.
2
3 Dimson
and Marsh (1984) find similar results for forecasts of
Their data are from a designed experiment in
stock returns.
which forecasts were collected at regular intervals, so they were
assured a sample of nearly simultaneous forecasts.
In analysts'
EPS forecast data such as those used here, the availability of
recent forecasts depends, in part, on analysts' private decisions
about when to update.
This point is discussed further in section
4.4.
"Earnings" and "earnings per share" (or "EPS") are used
interchangeably in this paper.
The data are forecasts of EPS.
4
^Since firms' annual earnings announcement dates are remarkably
consistent year after year, choosing fixed lengths of time prior
to the announcement date is a fairly accurate means of finding
dates that differ by one quarter.
For the 6218 horizons in this
paper, 13 horizon dates did not fall between the quarterly
announcements as intended.
The results are not affected by
deletion of these observations.
^Results reported in this paper are for unsealed forecast errors
defined in equation (3).
Other forecast error metrics, or
scales, may be appropriate, for example to control for
he ter oscedas t i c i t y
However, the qualitative conclusions of the
paper, other than the bias results, are not affected by the
choice of scale.
The scales I have investigated are:
(1)
standardized forecast errors, where the denominator is (a) the
average, over the previous five years, of absolute changes in
EPS, or (b) the standard deviation of EPS changes; and (2)
percent forecast errors, where the denominator is (a) the
absolute value of the prior year's EPS, or (b) where the sample
is censored to exclude negative denominators, or (c) to exclude
denominators less than $0.20.
These results are available upon
request from the author.
as
.
7
The numbers of observations deleted are:
1,
2, 2. 5,
the 5-. 60-, 120-. 180-. and 24 - t rad i ng-day horizons,
respectively
8M
of
y
and
5,
at
.
results do not differ if the value-weighted market portfolio
securities is used as a proxy for the market.
^The
subscript
i.
which indexes the source of the forecast,
is
39
suppressed in the following discussion for readability.
10 The estimation was also done with firm-specific effects in the
model.
The bias results do not differ qualitatively from those
reported here.
1:1
Fried and Givoly (1982) report negative statistically
significant bias in analysts' April forecasts for December
yearends for the period 1969 through 1979, but none in timeseries forecasts.
I
replicate the bias test for percent forecast
errors (see footnote 6), which is approximately their definition.
find no bias in 120-day through 240-day analyst forecasts, and
I
significant positive bias in the time-series model forecasts.
Thus, I conclude that their bias result is not replicated in this
later (1975-1981) sample.
12 For example, see Dirks and Gross (1974), especially
pp. 252257.
Also see "Bank Analysts Try to Balance their Ratings", WSJ,
May 29, 1984, p. 33; and
"Picking a Loser", WSJ, September 28,
1
1983
p
,
.
13 The
estimated effects are not the same as the estimated
coefficients, because model (9) has two sets of effects, firms
and years. A simple example will illustrate this.
If there were
two years and three firms in the sample, model (9) could be
estimated with no intercept, using two year dummy variables (DY1
and DY2) and two firm dummy variables (DF1 and DF2):
=
JtT
d
11T
DF1
d
12T
DF2
d
2lT
DYl
d
22T
DY2
£
JtT'
year 1 effect" is the average |e-j tT
This effect
for t = l
not estimated by d 2 i T
Rather, d 2 it is the average
e j ^t
for
year 1 for the omitted (third) firm,
year 1 effect" is
The
estimated in this formulation by:
The
|
is
.
|
=
21T
The
firm
6
The
firm
&
1
21T
effect"
+
(1/3
13T
"
(1/2)
is
d
12T
estimated by:
is
1/2
y*'*>
+
*
effect"
(1/3)
+
lit
=d lit +(l/2)d
J/ ^' u
2lT
11T
3
d
I
(
)
d
u
22t
estimated by:
21T
+
(
1/2
)
d
22T
Since any non-redundant spanning set of dummy variables can be
used, the particular linear combinations of coefficients to
estimate the firm and year effects depend on the model used.
^This analysis
was suggested to me by the referee.
15 Dropping the
O-hjx, the firm-specific effects, does not alter
the estimates of the slope coefficients 6j T or their statistical
Omitting the a 2 ^ t T
significance by a substantial amount.
their statistical
and
however, alters both the estimates
,
significance.
40
lf>An
alternate way to model the problem is to include the year or
firm effects as "random effects", contributing off-diagonal
elements to the covariance matrix of the vjj tT
Mundlak (1978)
shows that the estimate of the slope coefficient 6j T obtained
from a model like (12), is identical to the GLS estimate which
would be obtained if the firm- and year-specific effects, aiijx
and 0C2itT- were modeled as random effects and included in the
covariance matrix.
.
17 The
statistical significance of year -spec i f i c effects as a
determinant of forecast errors, although not reported in the
previous tables, is similar in the estimates of bias and
accuracy
.
8An
alternate method of measuring the association between
forecast errors and excess returns, similar to that of Ball and
Brown (1968), is to construct portfolios based on foreknowledge
of the sign of EPS forecast error.
That is, a long position is
taken in each of the securities for which the forecast error is
positive, and a short position is taken in those with negative
This procedure, applied to these data, produces results
errors.
qualitatively identical to those reported here.
1
41
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Fried. D. and D. Givoly (1982).
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Their Ratings.
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Bank Analysts Try to Balance
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The Wall Street Journal.
p. 33.
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Financial Analysts' Forecasts of Earnings.
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348Journal of the American Statistical Association 57:
368
.
Table
1
Sample Selection Criteria and Their Effects on the Sample
Panel A:
Cr
Elimination of Firms and Firm-Years
Table
2
Selected Characteristics of the Distribution of Forecast Ages
for the Mean, Median and Most Current Analyst Forecasts,
Measured Across Firms and Years
[forecast ages are measured in trading days]
Table
Forecast Bias,
3
for Five Forecast Sources,
[bias numbers are denominated in
Forecast Horizon,
Source
q
q
.
.
a
r
.
.
r
-0.15
(-3.82)
-0.11
(-3.15)
(-5.18)
(-4.11
-0.11
(-4.24)
(-3.25)
)
-0.12
(-4.70)
(
(-1.52)
(-2.25)
-0.10
(-2.93)
(-3.27)
-0
.
.
-0
.
-0
.
03
08
.
05
(-1.49)
-0.12
08
(-1.91)
Notes
-0
09
(-0.72)
current
Trading Days
120
(-2.09)
median
in
per share]
180
(-2.17)
mean
$
240
-0
w
Five Forecast Horizons
at
(
-3 .57
-0
.
05
-0
-0
-0
.
.
.
60
-0
14
-0
-0
-4
-0
06
.
.
.
.
.
5
08
)
06
09
96
-0
)
02
(
-0
-2
.
.
04
34
05
02
.00)
.
(-1.19)
09
.
(-2.74)
(
1
-0
)
.
02
(-1.42)
:
The computation of forecast bias and its associated t-statistic (in
parentheses) are described in equations (6) through (8) in the text.
The forecast sources are:
q.a.r. - a quarterly auto regressive model in fourth differences;
equation (1) in the text,
q.r.w. - a random walk model with drift in fourth differences;
equation (2) in the text,
mean - the mean of the available analysts' forecasts,
median - the median of the available analysts' forecasts,
current - the most recent forecast from an analyst.
The forecast horizons are measured in trading days prior to the annual
earnings announcement.
The degrees of freedom for all reported t-statistics are over 1,000,
so they are approximately normal.
For a two-sided test, the .05 and
.01 critical points of the N(0.1) distribution are 1.96 and 2.58.
respectively
.
Table
4
Average Absolute or Squared Forecast Error
Forecast Accuracy:
for Five Forecast Sources, and Five Forecast Horizons
[forecast errors are denominated in
$
per share]
Table
5
Pairwise Differences in Forecast Accuracy
Among the Mean, Median and Most Current Analyst Forecasts
for Five Forecast Horizons
Panel A:
t-Statistics on Differences
Average Absolute Error
in
Forecast Horizon,
240
mean - median
mean - current
median - current
1.06
0.11
1.18
2
Trading Days
120
98
07
05
1.11
2
82
93
2
.
4
.
.
1
in
180
.
.
1
.
2
.
60
.
14
83
97
5
-1
Table
6
Pairwise Differences in Forecast Accuracy Between Analysts and
Quarterly Time-Series Models, for Five Forecast Horizons
Table
7
Selection of the Subsample, for Each Forecast Horizon,
of Forecasts Made Between the Last Announcement
of Quarterly Earnings and the Horizon Date
Forecast_Ho£2zon_L _in_Trading_Day_s
Full Sample
#
#
#
#
:
forecasts
firms
f i rm-year s
firm-years with
forecast
1
Subsample
240
180
120
16134
19294
20969
22078
184
184
184
184
184
1198
1235
1254
1259
1260
68
32
31
22
25
60
5
22864
:
#
#
forecasts
3017
5601
4425
3859
15352
firms
181
183
#
firm-years
firm-years witli
forecast
1
875
184
104
1039
183
961
1246
209
246
259
53
#
Notes
250
1
184
:
The full sample, whose selection is described in Table 1, includes,
for each firm and year, all available forecasts that have not been
revised or withdrawn prior to the horizon date.
The subsample includes, for each firm and year, only those forecasts
which have been revised or initiated between the firm's last
announcement of quarterly EPS and the horizon date.
Table
8
Selected Characteristics of the Distribution of Forecast Ages,
Measured Across Firms and Years, for the Subsample of Forecasts
Made Between the Last Announcement of Quarterly Earnings
and the Horizon Date
Table
9
Pairwise Differences in Forecast Accuracy Among the Mean,
Median and Most Current Analyst Forecasts, in the Subsample of
Forecasts Made Between the Last Announcement of Quarterly Earnings
and the Horizon Date, for Five Forecast Horizons
Panel A:
t-Statistics on Differences in Average Absolute Error
Forecast Horizon,
mean - median
mean - current
median - current
Panel
B:
.
.
0
.
240
180
034
240
275
0.210
1
.
258
1
.
.
.
048
.
60
059
169
228
Forecast Horizon,
Notes
Trading Days
.
1
1
.
.
067
106
039
5
.
388
-3 508
-3 896
.
.
t-Statistics on Differences in Average Squared Error
240
mean
mean
median
in
120
med i an
current
current
180
078
-0.318
0.489
-0 008
-0.411
0.310
-0
.
in
Trading Days
120
.
.
.
-0
.
101
704
603
60
-0 099
-0 370
-0 270
.
.
.
5
0.168
2
.
146
2.314
:
The subsample includes, for each firm and year, only those forecasts
which have been revised or initiated between the firm's last
announcement of quarterly EPS and the horizon date.
The computation of average absolute error is described in equations
The computation of average squared error is
(9) and (10) in the text.
analogous to the computation of average absolute error.
The forecast sources are:
q.a.r. - a quarterly au t or egres s i ve model in fourth differences;
equation (1) in the text,
q.r.w. - a random walk model with drift in fourth differences;
equation (2) in the text,
mean - the mean of the available analysts' forecasts,
median - the median of the available analysts' forecasts,
current - the most recent forecast from an analyst.
The forecast horizons are measured
earnings announcement.
in
trading days prior to the annual
Table 10
Slope Coefficients from the Regression of EPS Forecast Error
on Excess Return, for Five Forecast Horizons:
e
.
_tx =ot„..
1 1 jt
1 j
+
a
.
2
a
,_
tx
+8.ix
U
.
j
tx
ijtx
(12)
t
i
Table 11
of
Pairwise Differences in Slope Coefficients from the Regression
EPS Forecast Error on Excess Return, for Five Forecast Horizons
e
.
1
Panel
.
.
JtT
a„
=
.
1 1
.
JT
6
'2i tT
U
.
IT
1
(12)
JtT
t-Statistics on Differences between Quarterly Autoregressive Model [eqn. (1)] and Other Forecast Sources
A:
Quarter
Horizon:
1
2
3
240
180
120
q.a.r. - q.r.w.
q.a.r. - mean
q.a.r. - median
q.a.r. - current
2.69
3.28
3.55
2.59
:
Panel
..
J tT
3
4
3
2
.
.
.
.
08
08
82
85
1
.
3
.
2
.
68
09
74
2.16
4
60
1
.
1
.
1
.
.
47
68
37
56
t-Statistics on Differences between the Most Current
the Mean or Median Analyst Forecasts
B:
and
Horizon
mean
median
Notes
current
current
240
180
0.69
1.23
0
.
96
.
120
.
97
.
94
59
60
12
82
76
97
:
sources are:
a quarterly au oregres s i ve model in fourth
differences; equation (1) in the text,
q.r.w. - a random walk model with drift in fourth
differences; equation (2) in the text,
mean - the mean of the available analysts' forecasts,
median - the median of the available analysts' forecasts,
current - the most recent forecast from an analyst.
The forecast
q.a.r.
The forecast horizons are measured
in
trading days prior to the annual
earnings announcement.
In equation (12), ejj tT is the forecast error, in $ per share, from
source i for firm j in year t at horizon t, and Uj^ T is cumulated
Equation (12) is
excess returns for firm j in year t over horizon T.
estimated jointly for five forecast sources and separately for each
horizon.
The aijj T are f i rm- e f f ec t s
and the 0t2itT are year-effects.
,
slope B T is estimated for each forecast source i and horizon T.
The t-statistic on each slope coefficient is reported in parentheses.
For a
The t-statistics have degrees of freedom greater than 1,000.
one-sided test, the .05 and .01 critical points from the N(0,1)
distribution are 1.65 and 2.33, respectively.
A
Table 12
of
Regression Summary Statistics for the Regression
for Five Forecast Horizons
EPS Forecast Error on Excess Return,
1
JtT
1 i
a
jx
2
1
+
tx
6
it
U
j
tT
+
1
Forecast Horizon,
240
adjusted
R2
full model F(k-1 ,N-k)
year-effect
N-k
.
5
)
54
firm-effect F(k2,N-k)
3
excess returns F(k3,N-k)
6
d.f
#
.
for
F - s ta t
F
i
(
kl
s t
i
,
116
.
.
.
.
02
4
13
36
34
3
89
7
095
.
.
.
.
in
.
27
4
79
25
08
3
33
14
107
.
.
.
.
79
17
(12)
JtT
Trading Days
120
180
.
v
60
.
5
114
.
156
09
4
.
19.25
2
.
82
4
57
2.18
5
87
4
37
3
.
.
.
.
60
94
65
c s
sample size
(N)
(k)
of parameters
# of years (kl+1)
# of firms (k2+l
# of slopes
(k3)
5986
6171
6267
6293
3779
199
199
199
199
195
7
7
7
7
7
184
184
184
184
184
5
5
5
5
3
Notes
In equation (12), e^, tT is the forecast error, in $ per share, from
source i for firm j in year t at horizon T, and U j £ T is cumulated
Equation (12) is
excess returns for firm j in year t over horizon T.
estimated jointly for five forecast sources and separately for each
horizon.
The «jjj t estimate f rm- spec i f i c effects, and the a2itT
estimate year specific effects
model F-statistic tests the null hypothesis that the
The year, firm, and
regression model (12) has explanatory power.
excess return F-statistics test the incremental explanatory power of
Selected critical points
including groups of parameters in the model.
for the F distribution are:
The full
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