Description
This paper estimates the value added by Big
8/6/5 auditors after controlling for the
permanent and non-permanent impact of
earnings and cash flows using linear and nonlinear
(arctan) regression models. The linear
model shows significant value added for
industrial firms that utilise Big 8/6/5 auditors;
while an arctan model shows that large auditors
value-add by attesting to the permanence of
earnings for large firms. We demonstrate that
refinements to the audit research can be made
by using response coefficients to filter out the
different timing components inherent in
earnings and cash flows
Accounting Research Journal
The Effect of Returns History on the Current Period Relation Between Returns and Unexpected Earnings
Russell Calk Paul Haensly Mary J o Billiot
Article information:
To cite this document:
Russell Calk Paul Haensly Mary J o Billiot, (2007),"The Effect of Returns History on the Current Period Relation Between
Returns and Unexpected Earnings", Accounting Research J ournal, Vol. 20 Iss 1 pp. 5 - 20
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The Effect of Returns History on the Current Period Relation Between Returns and Unexpected Earnings
5
The Effect of Returns History on the Current
Period Relation Between Returns and
Unexpected Earnings
Russell Calk
Department of Accounting and Business Computer Systems
College of Business Administration and Economics
New Mexico State University
Paul Haensly
School of Business
University of Texas of the Permian Basin
and
Mary Jo Billiot
Department of Accounting and Business Computer Systems
College of Business Administration and Economics
New Mexico State University
Abstract
This study applies a model of systematic belief
revision to examine the effect of the relation
between current-period unexpected earnings
and prior-period security returns on the current-
period relation between those unexpected
earnings and returns. Cross-sectional analysis
blurs the effects of past information on current
returns in a manner that makes it easy to
overlook any dependence on historical patterns
in this information. We show that the market
responds to earnings innovations conditional on
these patterns but does not respond in the
manner predicted by the Hogarth and Einhorn
(1992) belief adjustment model. Nonetheless,
the results suggest that individual decision
processes are detectable in capital markets data.
1. Introduction
Over the past three decades, much of the
empirical financial accounting research has
focused on whether there is a discernable
Acknowledgments: The authors wish to express our
appreciation for the many helpful comments of the
workshop participants at Oklahoma State University and the
Annual Meeting of the American Accounting Association.
impact of accounting information in financial
markets. (For example, see reviews by Francis
and Schipper, 1999; Francis, Schipper, and
Vincent, 2002; Kross and Kim, 2000;
Landsman and Maydew, 2002.) This research
has centered on the relation between accounting
earnings and security returns with the general
focus on the effect of earnings information on
returns. However, cross-sectional analysis blurs
the effects of past information on current returns
in a manner that makes it easy to overlook any
dependence on historical patterns in this
information. We show that the market responds
to earnings innovations conditional on these
patterns but does not respond in the manner
predicted by the Hogarth and Einhorn (1992)
belief adjustment model.
We formulate and test hypotheses based on
the belief adjustment model proposed by Hogarth
and Einhorn (1992). Their decision model
describes belief adjustment in which decision-
makers evaluate new information based on its
relation to a decision anchor. We investigate how
investors (individual decision-makers) evaluate
accounting earnings (new information) and
incorporate the new information based on the
relation of earnings to prior-period returns
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ACCOUNTING RESEARCH JOURNAL VOLUME 20 NO 1 (2007)
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(decision anchor) to determine stock price and
hence, current period returns.
Experimental markets research provides
support for an individual decision model that
follows systematic belief adjustment. Such
research indicates that individual decision
biases can influence returns (Ackert, Church,
and Shehata, 1997; Camerer, 1992; Ganguly,
Kagel, and Moser, 1994; O’Brien and
Srivastava, 1991) but also cautions that these
decision biases may not be detectable in
efficient markets. By using empirical market
methods to analyze the systematic belief
adjustment model we pursue two goals. The
first goal is to determine whether systematic
belief adjustment is detectable in a market
setting. The second goal is to determine whether
systematic belief adjustment is based on the
relation between unexpected earnings (new
information) and prior period returns (decision
anchor).
Consistent with Hogarth and Einhorn’s
(1992) model, we find strong evidence that
current period returns respond to current
earnings innovations conditional on prior-period
returns. We also find that sensitivity of current
period returns to current earnings innovations
depends on whether innovations are positive or
negative, all else equal. However, we also find
results that are not consistent with Hogarth and
Einhorn’s claim that beliefs become less
sensitive to new information as information
accumulates. Our analysis supports this
proposition when the earnings innovation (new
information) is consistent with (i.e., has the
same sign as) prior-period returns; but the
reverse of this proposition is true when earnings
innovations are not consistent with prior-period
returns.
2. The Earnings/Return Relation
An extensive body of literature examines the
relation between accounting earnings and
security returns. An exhaustive review of this
literature is beyond the scope of this paper, but
it is reasonable to assert that an empirical
relationship between earnings and returns has
been well documented. While the specific
experimental design varies depending on the
nature and purpose of the research question,
most studies employ either a regression model
such as the one used by Kormendi and Lipe
(1987) or portfolio tests similar to those used by
Francis and Schipper (1999) in their
examination of the earnings/return relationship.
The regression tests focus on the earnings
coefficient estimate, commonly referred to as
the earnings response coefficient (ERC), of a
model relating unexpected earnings to changes
in stock price. The portfolio tests compare
returns for firms grouped according to
unexpected earnings. In either case, unexpected
earnings (UE) for firm j in period t are defined
as
t j
t j t j
E E UE ,
, ,
?
? =
(1)
where E
j,t
are the reported earnings and E
^
j,t
are
the forecasted earnings for firm j in period t.
These approaches have not fully explained
the relation between earnings and returns.
Studies have sometimes documented
unexpected results when examining ERCs. For
example, ERCs are sometimes negative which
implies that returns should increase as a
consequence of negative UE and vice versa.
Schroeder (1995) notes that firms with negative
ERCs had greater variability in UE over the
time period and proposes that returns may not
react in a predicted way to unexpected earnings
because UE interacts with other information. In
an effort to more fully understand this relation,
studies have included variables to capture the
effects of prior period prices (Donnelly and
Walker, 1995), analysts’ earnings forecasts
(Schroeder 1995), uncertainty in those analysts’
forecasts (Imhoff and Lobo, 1992), the
variability of UE caused by firm specific
characteristics (Teets and Wasley, 1996), the
time series pattern of earnings (Subramanyam
and Wild, 1996), the current level of earnings
(Ali and Zarowin, 1992), other financial
statement information (Penman, 1992), and firm
size (Shevlin and Shores, 1993) in the
experimental design. Other studies suggest that
the model itself is misspecified. Cheng,
Hopwood, and McKeown (1992), for example,
allow for fixed and random UE effects in the
model. Freeman and Tse (1992) and Cheng,
Hopwood, and McKeown (1992) also suggest
that the relationship between UE and returns is
nonlinear. Lipe, Bryant, and Widener (1998)
show that the variables included in the model
and the functional form of the model are both
significant factors in explaining the effect of
earnings on returns.
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The Effect of Returns History on the Current Period Relation Between Returns and Unexpected Earnings
7
We take a different approach to examine
the earnings/return relationship. Our principal
objective in this study is to determine whether or
not the Hogarth and Einhorn (1992) belief
adjustment model can be adapted to explain the
relation between returns and unexpected
earnings. In particular, we follow up on
Schroeder’s (1995) suggestion that trends in prior
period information may explain how current
period returns respond to current period UE. To
illustrate, consider the following examples.
Suppose that firm A reports negative returns in
the previous two reporting periods, and firm B
reports positive returns in each of the same
periods. Further suppose that both firms report
the same positive UE in the current period.
Would the price reaction to the current period UE
be the same for both firms? Conventional
efficient markets theory predicts that the response
should be the same, all else equal, because
information in past returns already is embedded
in price. Alternatively, suppose that firm A had
reported negative returns for three consecutive
prior periods instead of two. Would the
earnings/return relationship for Firm A be the
same in either case? To address questions such as
these, we formulate hypotheses based on the
belief adjustment model and test them with
models in which current period returns are a
function of prior period returns as well as current
period UE.
3. Hypotheses Development and
Experimental Design
Hogarth and Einhorn (1995, p. 8) propose a
structural model of systematic belief adjustment
based on the idea that, “People handle belief-
updating tasks by a general, sequential
anchoring-and-adjustment process in which
opinion, or the anchor, is adjusted by the impact
of succeeding pieces of evidence.” In particular,
the effect of new information on the anchor
depends on the direction of the new information.
If it is consistent with the anchor, then the new
information may be weighted differently than if it
is inconsistent with the anchor. In other words,
the weight placed on new information should
depend on the direction of the new information
relative to prior information.
For example, in the context of financial
markets, the weight placed on positive or
negative unexpected earnings (UE) in period t
should depend on the sign of returns in prior
periods. Positive UE reported in the period
following a sequence of positive returns would
be consistent with those returns and hence less
of a surprise than if the positive UE followed a
sequence of negative returns. According to the
Hogarth and Einhorn (1992) model, the positive
UE will be weighted differently and will have a
different effect in the two situations. Similarly,
negative UE reported in the period following a
sequence of negative returns would be
consistent with those returns and hence less of a
surprise than if the negative UE followed a
sequence of positive returns.
Calegari and Fargher (1997) provide
evidence to support a conditional response to
UE. Subjects in their experimental market
tended to underweight unexpected earnings that
were an extreme surprise. These results suggest
that individual investors are likely to
underweight reported UE that are inconsistent
with prior period returns. Thus, for example, if
UE > 0 follows negative prior returns, then the
current period returns would be lower than if
UE > 0 had followed positive prior returns.
Regardless of the individual weighting of
information, Hogarth and Einhorn (1992, p. 4)
assert that, “As information accumulates, beliefs
are expected to become less sensitive to the
impact of new information because this
represents an increasingly small proportion of
the evidence already processed.” This is a
manifestation of an escalation of commitment
documented by Staw (1976). In particular, the
effect of current period UE on returns should
decrease as the number of same sign, prior
period returns increases. The effect of UE
should diminish whether the sign of UE is
consistent or inconsistent with the sign of the
sequence of prior period returns. This idea is
similar to the notion of a decreasing marginal
effect of earnings persistence proposed by
Feltham and Ohlson (1995).
We reformulate Hogarth and Einhorn’s
(1992) two general hypotheses in the context of
accounting earnings and security returns. Our
hypotheses identify subsets of data in terms of
UE and the pattern of prior period returns. By
comparing returns conditional on these subsets,
we are able to examine the effects of decision-
making conditional on UE and prior period
returns. Let R
j,t|+,n+
be the return for firm j in
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ACCOUNTING RESEARCH JOURNAL VOLUME 20 NO 1 (2007)
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period t, conditional on UE
j,t
> 0 and the
sequence of n prior period returns such that
R
j,t–1
> 0, …, R
j,t–n
> 0. Let R
j,t|+,n-
be the return
for firm j in period t, conditional on UE
j,t
> 0
and the sequence of n prior period returns such
that R
j,t–1
< 0, …, R
j,t–n
< 0. Similarly, R
j,t|-,n+
represents the return for firm j in period t,
conditional on UE
j,t
< 0 and the sequence of n
prior period returns such that R
j,t-1
> 0, …,
R
j,t–n
> 0. R
j,t|-,n-
represents the return for firm j in
period t, conditional on UE
j,t
< 0 and the
sequence of n prior period returns such that
R
j,t–1
< 0, …, R
j,t–n
< 0.
Our first hypothesis expresses Hogarth and
Einhorn’s (1992) proposition that the effect of
new information on the anchor depends on the
direction of the new information.
H1: current period returns depend on both
current period UE and on the relation between
current period UE and prior period returns. In
particular, investor response to new information
contained in current period UE depends on the
direction of UE relative to the sign of the
sequence of n prior period returns:
a) if UE
j,t
> 0, then R
j,t|+,n+
? R
j,t|+,n-
, on average;
b) if UE
j,t
< 0, then R
j,t|-,n+
? R
j,t|-,n-
, on average.
We label this first version as weak-form H1.
The Calegari and Fargher (1997) experimental
results suggest the following version of the first
hypothesis, which we label as strong-form H1:
a) if UE
j,t
> 0, then R
j,t|+,n+
> R
j,t|+,n-
, on average;
b) if UE
j,t
< 0, then R
j,t|-,n-
< R
j,t|-,n+
, on average.
Our second hypothesis expresses Hogarth and
Einhorn’s (1992) proposition that as information
accumulates, belief becomes less sensitive to new
information. We follow Schroeder’s (1995)
suggestion when we translate Hogarth and
Einhorn’s proposition into the language of
earnings and returns. That is, we formulate the
hypotheses in terms of the length of the sequence
of same sign, prior period returns. In light of H1,
the second hypothesis has two parts
corresponding to UE > 0 and UE < 0.
H2: as the number of consecutive periods of
same sign, prior period returns increases, the
impact of current period UE on current period
returns will decrease.
a) If UE
j,t
> 0, then the positive effect on current
period returns will be smaller the longer the
sequence of same sign, prior period returns.
In particular:
i) If the sequence of prior period returns is
positive, then the sequence of conditional
current period returns R
j,t|+,n+
monotonically decreases as n increases,
on average.
ii) If the sequence of prior period returns is
negative, then the sequence of conditional
current period returns R
j,t|+,n-
monotonically decreases as n increases,
on average.
b) If UE
j,t
< 0, then the negative effect on
current period returns will be smaller the
longer the sequence of same sign, prior
period returns. In particular:
i) If the sequence of prior period returns is
positive, then the sequence of conditional
current period returns R
j,t|-,n+
monotonically increases as n increases, on
average; that is, the magnitude of these
negative current period returns decreases.
ii) If the sequence of prior period returns is
negative, then the sequence of conditional
current period returns R
j,t|-,n-
monotonically increases as n increases, on
average; that is, the magnitude of these
negative current period returns decreases.
Our primary objective in this study is to test
Hogarth and Einhorn’s (1992) belief adjustment
model — in particular, extensions suggested by
the work of Calegari and Fargher (1997) and
Schroeder (1995) — in the context of the
earnings/return relationship. To test these
hypotheses, we draw data for each 1994–2000
fiscal year from Research Insight (Compustat)
for firms listed as components of the S&P 1500
in that year. The S&P 1500, or Supercomposite,
is a large, representative list of mostly domestic
common stocks traded in U.S. markets. The
Supercomposite combines the S&P 500 (large
caps), S&P Mid-cap 400, and S&P Small-cap
600. S&P 1500 firms are widely followed by
analysts, and information about these firms
should have been widely available to investors.
The effect of earnings announcements on S&P
1500 stock prices should, therefore, represent
investor response to new information rather
than simply speculation. Furthermore, Buchheit
and Kohlbeck (2002) show that the increase
in the usefulness of earnings seems to be
concentrated among large firms. Our sample
should capture this effect.
We collected current period and three years
of prior period total returns for each firm. We
measured annual total return in two ways. First,
we measured each firm’s annual returns for
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The Effect of Returns History on the Current Period Relation Between Returns and Unexpected Earnings
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12-month periods that lag the corresponding
fiscal years by three months. The vast majority
of companies release their annual reports within
the mandatory 90-day period following their
fiscal year ends. By using lagged annual returns,
we allow the current earnings announcement
(and hence current period UE) to have an
informational effect on current annual returns.
We also measured each firm’s annual returns
for 12-month periods concurrent with fiscal
years (or, at least with fiscal year ends). In
principle, these annual returns should not be
directly affected by the final annual earnings
announcement for the concurrent fiscal year.
Thus, current period UE may not have a
significant effect on concurrent period annual
returns. However, many studies of the
earnings/return relation use concurrent annual
returns. So our use of concurrent annual returns
allows comparability with previous work. In
addition, quarterly earnings and preliminary
annual earnings estimates at the end of the fiscal
year may provide investors with much of the
information contained in the final version.
We collected current period operating
earnings per share for each firm. Use of
operating earnings per share controls for any
nonrecurring events. We also replicated our
analysis with basic EPS and diluted EPS. Our
results for these forms of EPS were similar to
those based on operating earnings per share but
tended to be slightly less significant. Hence, we
only report results based on operating earnings
per share.
We treat each fiscal year that a firm is in the
S&P 1500 as a potential current year
observation. We analyze pooled current years
from Research Insight fiscal years 1994–2000,
the period over which Research Insight
identifies S&P 1500 companies. (A Research
Insight fiscal year includes annual reports with a
fiscal year end that falls in June of that year
through May of the following calendar year. For
example, FY2000 includes reports with fiscal
year ends of June 2000 through May 2001.)
Operating earnings per share (OPEPS) are only
available back to FY1988 in Research Insight.
As a result, the methods we apply to forecast
earnings sometimes limit our choice of current
years. In particular, when we apply second
order autoregression methods to forecast
earnings, we require at least 10 prior years of
earnings for an annual report to be included in
our sample. Sample sizes in our reported results
vary from 1,762 observations (from FY1998–
2000 when we use second-order autoregressive
(AR(2)) earnings forecasts and limit fiscal year
ends to December) to 8,080 observations (from
FY1994–2000 when we use random walk
earnings forecasts and use all fiscal year end
months).
We report results based on the assumption
that annual earnings follow a random walk.
Under that assumption, we calculate UE,
defined in equation (1), from the forecast
Ê
j,t
= E
j,t-1.
Foster (1986, p. 240) concludes, “The
result that, on average, annual reported earnings
or EPS can be well-described by a random walk
model is one of the most robust empirical
findings in the financial statement literature.”
On the other hand, recent work, e.g., Ball and
Bartov (1996), suggests that the market acts as
if quarterly earnings are serially correlated.
Hence, for robustness, we also apply second-
order autoregression to forecast current annual
earnings E
j,t
for each observation. When
changes in annual earnings are assumed to
follow a second-order autoregressive process
(AR(2)), the forecast is Ê
j,t
= E
j,t-1
+ ?Ê
j,t
, where
change in earnings may be modeled as
?E
j,t
= ?
j1
?E
j,t-1
+ ?
j2
?E
j,t-2
+ a
j,t
. (2)
From equation (1) and the forecast for
current earnings,
UE
j,t
= E
j,t
– Ê
j,t
= E
j,t
– (E
j,t-1
+ ?Ê
j,t
)
= ?E
j,t
– ?Ê
j,t
. (3)
Thus, in a manner similar to Kormendi and
Lipe (1987), UE is the residual from the
estimated AR(2) process. That is,
?E
j,t
= f
j1
?E
j,t-1
+ f
j2
?E
j,t-2
+ UE
j,t
, (4)
where f
j1
and f
j2
are the estimated AR(2)
coefficients for equation (2). In our analysis,
neither earnings forecast model is superior in all
scenarios, and hypothesis testing with the two
alternatives produces similar results.
We analyze the data using two general types
of models: those with categorical independent
variables, and those with continuous
independent variables. The categorical models
can be used to analyze firm year observations
grouped based on the sign of UE
j,t
and the signs
of R
j,t-n
over the n periods of the returns
sequence. While we derive these results from
regression analysis of our categorical models,
the mean returns for these groups could have
been computed directly. Hence, these results are
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ACCOUNTING RESEARCH JOURNAL VOLUME 20 NO 1 (2007)
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robust in the sense that they depend only on the
independent variables selected and not on any
particular form of the regression model. The
general categorical model is
r
j,t
= ?
0
+ ?
1
DUE
j,t
+ ?
1
DR
j,t-1
+ ?
2
DR
j,t-2
+ · · ·
+ ?
n
DR
j,t-n
+ ?
1
DUE
j,t
DR
j,t-1
+ ?
2
DUE
j,t
DR
j,t-2
+ · · · + ?
n
DUE
j,t
DR
j,t-n
+ ?
j,t
, (5)
where
DUE
j ,t
=
1 if UE
j ,t
? 0
0 if UE
j ,t
< 0
?
?
?
?
?
?
,
DR
j, t ?k
=
1 if R
j, t ?k
? 0
0 if R
j, t? k
< 0
?
?
?
?
?
?
, k = 0, 1, 2,… , n,
and we use continuously compounded annual
returns r
j,t
= ln(1+R
j,t
).
The regression analysis of the categorical
model uses only the sign of UE
j,t
and the signs
of R
j,t-n
but not their magnitudes. To evaluate the
relation between current returns and prior period
returns for a given level of unexpected earnings,
we must use continuous variables. The general
continuous model is
r
j,t
= ?
0
+ ?
1
(UE
j,t
/P
j,t-1
) + ?
1
r
j,t-1
+ ?
2
r
j,t-2
+ · · ·
+ ?
n
r
j,t-n
+ ?
1
(UE
j,t
/P
j,t-1
)r
j,t-1
+ ?
2
(UE
j,t
/P
j,t-1
)r
j,t-2
+ · · · + ?
n
(UE
j,t
/P
j,t-1
)r
j,t-n
+ ?
j,t
(6)
where r
j,t-k
= ln(1+R
j,t-k
), k = 0, 1, 2,… , n, and
unexpected earnings are scaled in the same
manner that change of price is scaled in R
j,t
.
The primary objective of this paper is to
evaluate how new information in the form of
unexpected earnings affects investors’ behavior
as measured by current returns, conditional on
information in the form of prior returns. Hence,
all models that we examine include current
unexpected earnings and its interaction with any
prior returns in the model. Due to constraints in
availability of OPEPS and data requirements for
AR(2) modeling of earnings, we use at most
prior returns back to t-3. Thus, the full models
will be equations (5) and (6) with n = 3.
A secondary objective of this paper is to
evaluate a two-equation model proposed by
Hogarth and Einhorn (1992, p. 14) for
systematic belief revision. Their original model
is stated in terms of degree of belief:
S
k
= S
k?1
+?S
k?1
s(x
k
) ? R [ ] for s(x
k
) ? R,
(7a)
S
k
= S
k ?1
+ ? 1? S
k ?1
( ) s(x
k
) ? R [ ] for s(x
k
) > R,
(7b)
where S
k
is degree of belief in some hypothesis,
S
k-1
is prior belief or opinion, s(x
k
) is subjective
evaluation of new evidence, and R is the
reference point against which the new evidence
is evaluated. Degree of belief is a bounded
variable such that 0 ? S
k
? 1. Hogarth and
Einhorn define coefficients ? and ? as
sensitivity to positive and negative evidence,
respectively, and require that 0 ? ? ? 1 and
0 ? ? ? 1. Their model can be adapted to study
belief revision in the context of capital markets
by letting R
j,t
, the return for firm j in period t,
represent investors’ degree of belief in the
information content of earnings for firm j:
R
j ,t
= R
j ,t ?1
+ ?R
j ,t ?1
UE
j, t
/ P
j,t ?1
( ) for UE
j ,t
? 0,
(8a)
R
j ,t
= R
j ,t ?1
+ ? 1 ? R
j ,t ?1
( )UE
j ,t
/ P
j,t ?1
( ) for UE
j ,t
> 0,
(8b)
where UE
j,t
/P
j,t-1
measures the effect of new
evidence (earnings announcement) against the
forecasted earnings for the firm. Hogarth and
Einhorn posit that ? and ? in equation (7)
depend on the individual decision-maker.
However, prices and returns in the market are
determined by the collective decision-making of
investors. So we treat ? and ? as investors’
collective sensitivities to positive and negative
evidence (i.e., earnings innovations),
respectively.
We estimate the coefficients ? and ?
indirectly. Under the assumption that
sensitivities ? and ? are the same for all firms,
equation (8) implies that
? = (R
j, t
? R
j,t ?1
) R
j ,t ?1
( ) UE
j,t
/ P
j,t ?1
( ) for UE
j ,t
? 0,
(9a)
? = (R
j,t
? R
j ,t ?1
) (1? R
j ,t ?1
) ( ) UE
j ,t
/ P
j, t?1
( ) for UE
j ,t
> 0.
(9b)
Alternatively, the partial derivatives of
current period returns with respect to prior
period returns in equation (8) are
?R
j ,t
/ ?R
j ,t ?1
= 1+ ? UE
j,t
/ P
j ,t ?1
( ) for UE
j ,t
? 0,
(10a)
?R
j ,t
/ ?R
j ,t ?1
= 1? ? UE
j ,t
/ P
j, t ?1
( ) for UE
j,t
> 0.
(10b)
Thus, alternative approximations for ? and ? are
ˆ
? = ?R
j,t
/ ?R
j, t ?1
?1 ( ) UE
j ,t
/ P
j ,t ?1
( ) for UE
j,t
? 0,
(11a)
ˆ
? = 1? ?R
j ,t
/ ?R
j ,t ?1
( ) UE
j, t
/ P
j,t ?1
( ) for UE
j ,t
> 0.
(11b)
We first estimate the parameters in our
single-equation model in equation (6) and use it
to calculate fitted values for current period
returns, R
j,t-1
. We then estimate the sensitivity
coefficients, ? and ?, by substituting the fitted
current period returns and the corresponding
prior period returns and scaled unexpected
earnings into equations (9) or (11).
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The astute reader may wonder why we use
our single-equation model in (6) to test the
hypotheses in H1 and H2 rather than test the
two-equation model in (8) directly. One reason
is that Hogarth and Einhorn’s assumptions
unreasonably restrict current and prior period
returns to the range from zero to one. More
importantly, Hogarth and Einhorn’s (1992)
model (equation (7) above) is defined directly in
terms of degree of belief. Security returns, on
the other hand, include more information than
just degree of belief. Change in security price
due to an earnings innovation reflects not only
belief in persistence of the change in earnings
but also the magnitude of the expected change
in future cash flows.
An analogy from statistics may clarify this
issue. A point estimator may be evaluated in
terms of bias (or accuracy) and efficiency. Bias
refers to how much the expected value of the
estimator differs from the true value of the
parameter to be estimated, while efficiency
refers to the estimator’s variance relative to that
of other estimators. But degree of belief is
equivalent to variance of an estimator in the
sense that it measures efficiency but not
accuracy of belief. A belief may be strongly
held yet far from the truth. Change in security
price reflects both an estimate of the change in
value as well as degree of belief in that change.
Our single-equation model is designed to
capture important elements of the belief-
adjustment process represented by prior period
returns and the interaction between prior period
returns and unexpected earnings on the right-
hand side of the two-equation model. Hence,
our single-equation model is consistent with the
theoretical belief adjustment process described
by Hogarth and Einhorn (1992) and, at the same
time, with the ERC literature in accounting.
A potential drawback to our single-equation
model is that the estimate of the ERC will not be
conditional on the sign of UE, because the sample
includes observations with positive UE and
observations with negative UE. Hence,
examination of the ERC in our model does not
facilitate the above hypothesis tests, because each
test is conditional on the sign of UE. However, we
are primarily interested in testing the validity of
the belief adjustment model, not in estimation of
the ERC. The tests of the hypotheses do not
require estimation of the ERC. Should the
evidence support the above hypotheses and thus
the validity of the belief adjustment model, then
the natural next step would be to develop an
appropriate model for estimating the ERC
conditional on sign of the UE.
4. Results
The market data does not support key
predictions based on the belief adjustment
model. A natural extension of this model to
decision-making in financial markets states that
investors will tend to underweight their
response to earnings innovations when UE has
the opposite sign of prior period returns (strong-
form H1). Data from both the categorical and
continuous models generally do not support this
conclusion (Tables 1 and 5). A second natural
extension of the belief adjustment model states
that the market response to UE will diminish
with the length of the sequence of same sign,
prior period returns (hypotheses H2(a) for
UE > 0 and H2(b) for UE < 0). That is,
investors will behave as if they become
committed to their beliefs the longer the
sequence of consistent returns. Data from both
the categorical and continuous models generally
do not support this conclusion when UE has the
opposite sign from the sequence of prior period
returns (Tables 2 and 6). Instead, the market
behaves as if UE of opposite sign to the prior
period returns is a surprise containing
significant news that justifies a reversal of
valuation. Furthermore, the longer the sequence
of same sign, prior period returns, the stronger
the response, i.e., investors appear less
committed to their beliefs when UE is of
opposite sign as the prior period returns.
In this section, we discuss the above results
in detail for an experimental design in which
earnings are measured by operating earnings per
share, earnings forecasts are based on a random
walk, and annual returns lag the fiscal year end
by three months. Our results are robust to
choice of measure of earnings (OPEPS, basic
EPS, or diluted EPS), model for earnings
(random walk or AR(2)), and choice of period
for annual returns (concurrent with the fiscal
year end or measured with a three-month lag).
Tests of Hypothesis 1 with the Categorical
Model
The strong form H1 states that if UE is positive,
then R
j,t|+,n+
> R
j,t|+,n-
, on average, and if UE is
negative, then R
j,t|-,n-
< R
j,t|-,n+
, on average. That
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is, investors will tend to underweight their
response to earnings innovations that are a
surprise in the sense that sign of UE is opposite
to the sign of the prior period returns. The tests
show just the opposite is true. See Table 1. The
inequalities in current period returns are
significant but in the opposite direction from
that hypothesized. The difference is significant
at the 0.01 level for n = 1, 2, and 3 whether UE
is positive or negative. This result is robust in
the sense that we observe this level of
significance regardless of the experimental
design. Weak-form H1 is supported by our
results but not strong-form H1.
Tests of H1 are not consistent with the results
reported by Calegari and Fargher (1997).
Current returns do not exhibit underreaction to
earnings surprises. Market response is more
strongly positive when the good news (UE > 0)
is not consistent with prior returns and is more
strongly negative when the bad news (UE < 0)
is not consistent with prior returns. These results
hold regardless of the number of same sign,
prior period returns. While not consistent with
Calegari and Fargher, these results are
consistent with Skinner and Sloan (2001) who
document a particularly large reaction to
negative earnings surprises among growth
stocks. Calegari and Fargher used student
subjects in their experimental market study.
Mowen and Mowen (1986) suggest that student
subjects may not be appropriate when the
decision setting is very complex and when a
great deal of professional judgment or
experience is required. These characteristics of
the decision-makers in their study may explain
the discrepancy between Calegari and Fargher’s
experimental market results and the empirical
results reported here.
The magnitude of market response (i.e.,
investors in aggregate) is asymmetric and thus
is consistent with an alternative hypothesis: UE
that is inconsistent with prior returns has
informational value that overrides the anchoring
postulated by Hogarth and Einhorn (1992).
(Whether the market overreacts when new
information contradicts previous beliefs is an
issue that we do not explore in this paper.)
When prior returns have been positive, the
market is conditioned to expect continuing good
performance by the company. Thus, UE
j,t
> 0
provides little or no new information about
future performance. On the other hand, when
prior returns have been negative, the market
has been conditioned to expect continuing
bad performance. Thus, UE
j,t
> 0 provides
Table 1
Tests of H1: Mean Returns Conditional on Signs of UE and
Prior Period Returns
Mean current returns conditional on sequence
of prior period returns
No. of
same-sign
prior
returns
Unexpected
earnings r
t-1
? 0 r
t-1
< 0
Difference
of mean
returns
F-test for
difference of
means
1 UE
t
? 0 -0.0976 0.1172 -0.2147 92.90
***
1 UE
t
< 0 -0.4693 -0.1979 -0.2714 263.68
***
r
t-1
? 0, r
t-2
? 0 r
t-1
< 0, r
t-2
< 0
2 UE
t
? 0 -0.1577 0.1785 -0.3361 93.82
***
2 UE
t
< 0 -0.5334 -0.1259 -0.4074 261.70
***
r
t-1
? 0, r
t-2
? 0, r
t-3
? 0 r
t-1
< 0, r
t-2
< 0, r
t-3
< 0
3 UE
t
? 0 -0.1695 0.2027 -0.3722 70.21
***
3 UE
t
< 0 -0.5487 -0.0989 -0.4497 193.70
***
Notes. The sample is drawn from S&P 1500 firms for FY1998–2000 (fiscal years ending 6/98–5/01) in Standard &
Poor’s Research Insight/Compustat. Sample size is 3,826 firm-year observations. We use firms with fiscal year ends
from all calendar months. UE
t
= current period unexpected annual earnings, where earnings are forecast using a random
walk model. Returns are r
t-n
, the continuously compounded total annual return n periods before the current period t,
n = 0, 1, 2, 3. The predicted difference of mean returns in H1 is positive in all cases above.
*** All F-statistics reported above are significant at the 0.01 level.
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significant new information that performance
has changed for the better. Alternatively, the
good news in a positive UE following positive
prior returns is not very surprising (little new
information) but is a big surprise (substantial
new information) if it follows negative prior
returns.
Similarly, when prior returns have been
negative, the market is conditioned to expect
continuing bad performance by the company.
Thus, UE
j,t
< 0 provides little or no new
information about future performance. On the
other hand, when prior returns have been
positive, the market has been conditioned to
expect continuing good performance. Thus,
UE
j,t
< 0 provides significant new information
that performance has changed for the worse.
Alternatively, the bad news in a negative UE
following negative prior returns is not very
surprising (little new information) but is a big
surprise (substantial new information) if it
follows positive prior returns.
Tests of Hypothesis 2 with the Categorical
Model
Hogarth and Einhorn (1992) assert that all
decision-makers become committed to their
prior beliefs over time so the impact of
additional information diminishes. The results
in this study support that claim when new
information is consistent with prior information.
However, the results contradict that claim when
new information is inconsistent.
When UE
j,t
> 0, the market response to this
good news diminishes (i.e., becomes less
positive) as the length of the sequence of
positive prior returns increases. See Table 2,
Panel A. UE
j,t
> 0 carries less weight in
investors’ decisions as this sequence grows
longer.
Table 2
Tests of H2: Mean Returns Conditional on UE and
Sequence of Prior Period Returns
Sequence of positive prior period
returns
Sequence of negative prior period
returns
r
t-1
+ + + + – – – –
r
t-2
– – + + + + – –
r
t-3
– + – + + – + –
Panel A: UE > 0
Predicted ranking of
mean returns by H2(a)
High Middle Low High Middle Low
Observed mean returns -0.0329 -0.0642 -0.1382 -0.1695 0.0661 0.0974 0.1714 0.2027
F-test of difference in
means
1.68 5.61
**
1.68 1.68 5.61
**
1.68
Panel B: UE < 0
Predicted ranking of
mean returns by H2(b)
Low Middle High Low Middle High
Observed mean returns -0.3555 -0.4002 -0.5040 -0.5487 -0.2145 -0.1698 -0.0660 -0.0213
F-test of difference in
means
2.72
*
9.33
***
2.72
*
2.72
*
9.33
***
2.72
*
Notes. The sample is drawn from S&P 1500 firms for FY1998–2000 (fiscal years ending 6/98–5/01) in Standard &
Poor’s Research Insight/Compustat. Sample size is 3,826 firm-year observations. We use firms with fiscal year ends
from all calendar months. UE
t
= current period unexpected annual earnings, where earnings are forecast using a random
walk model. Returns are r
t-n
, the continuously compounded total annual return n periods before the current period t, n =
0, 1, 2, 3. Negative mean returns following UE
t
> 0 are due to the heavy influence of the market downturn that began in
March 2000. However, this does not affect interpretation of the above results, because H2 concerns relative order of the
means.
* significant at 0.10
** significant at 0.05
*** significant at 0.01
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Thus, the data support hypothesis H2(a)(i). The
results also indicate that the anchor extends
back about two years. As n increases from one
to two, the change is significant at the 0.05
level, but as n increases from two to three, the
change is not significant.
However, a different market response to
UE
j,t
> 0 occurs when the prior returns are
negative. See Table 2, Panel A. As the length of
the sequence of negative prior period returns
increases, the positive market response to the
good news becomes stronger. The good news of
UE
j,t
> 0 carries more weight in investors’
decisions as this sequence of negative prior
period returns grows longer. Thus, the data do
not support hypothesis H2(a)(ii).
The market response to the good news in
positive UE is better explained in terms of
information content conditional on prior
information than in terms of commitment to
prior beliefs. In Table 2, notice that as the length
of the sequence of positive prior returns
decreases (left side of Panel A) or as the length
the sequence of negative prior returns increases
(right side of Panel A), the market response to
UE
j,t
> 0 improves. This result is consistent with
an increase in information content as the
inconsistency between direction of UE and the
prior period returns increases.
When UE
j,t
< 0, the market response to
this bad news diminishes (i.e., becomes less
negative) as the sequence of negative prior
returns increases. See Table 2, Panel B. The bad
news in UE
j,t
< 0 carries less weight in
investors’ decisions as this sequence grows
longer. Thus, the data support hypothesis
H2(b)(ii). The results also indicate that the
anchor extends back about two years. As n
increases from one to two, the change is
significant at the 0.01 level, but as n increases
from two to three, the change is only significant
at the 0.10 level.
However, a different market response to
UE
j,t
< 0 occurs when the prior returns are
positive. See Table 2, Panel B. As the length of
the sequence of positive prior returns increases,
the negative market response to the bad news
becomes stronger. The bad news of UE
j,t
< 0
carries more weight in investors’ decisions as
this sequence grows longer. Thus, the data do
not support hypothesis H2(b)(i).
The market response to the bad news in
negative UE is better explained in terms of
information content conditional on prior
information than in terms of commitment to
prior beliefs. In Table 2, notice that as the length
of the sequence of negative prior returns
decreases (right side of Panel B) or as the
length of the sequence of positive prior returns
increases (left side of Panel A), the strength of
the negative response to UE
j,t
< 0 increases.
Table 3
Model Comparisons and Goodness of Fit — Continuous Model
Model # Independent variables in the model
Adj. R
2
(%)
F-statistic for test
of significant
difference between
adjusted R
2
Models
in
F-test
1 UE
t
/P
t-1
3.23 299.17
***
1 vs. 2
2 UE
t
/P
t-1
, r
t-1
, (UE
t
/P
t-1
)×r
t-1
16.29 71.73
***
2 vs. 3
3 UE
t
/P
t-1
, r
t-1
, (UE
t
/P
t-1
)×r
t-1
, r
t-2
, (UE
t
/P
t-1
)×r
t-2
19.28 39.50
***
3 vs. 4
4
UE
t
/P
t-1
, r
t-1
, (UE
t
/P
t-1
)×r
t-1
, r
t-2
, (UE
t
/P
t-1
)×r
t-2
, r
t-3
,
(UE
t
/P
t-1
)×r
t-3
20.87
5 UE
t
/P
t-1
, r
t-1
, r
t-2
, (UE
t
/P
t-1
) ×r
t-2
, r
t-3
, (UE
t
/P
t-1
)×r
t-3
5.47 372.77
***
5 vs. 4
6 UE
t
/P
t-1
, r
t-1
, r
t-3
, (UE
t
/P
t-1
)×r
t-3
3.84 33.91
***
6 vs. 5
Notes. The sample is drawn from S&P 1500 firms for FY1998–2000 (fiscal years ending 6/98–5/01) in Standard &
Poor’s Research Insight/Compustat. Sample size is 3,826 firm-year observations. We use firms with fiscal year ends
from all calendar months. UE
t
= current period unexpected annual earnings, where earnings are forecast using a random
walk model. P
t-1
is the share price at the end of the period preceeding the current period. Prior period returns are r
t-n
, the
continuously compounded total annual return n periods before the current period t.
*** All F-statistics reported above are significant at the 0.01 level.
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This result is consistent with an increase in
information content as the inconsistency
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returns increases.
OLS Results for the Continuous Model
The categorical data leave us with a puzzle.
Predictions based on the belief revision model
are supported under some circumstances but not
others. The impact of new information
diminishes when it is consistent with prior
information but increases when it is
inconsistent. Investors, however, also know the
magnitudes of these variables. The continuous
model permits a more detailed picture based on
magnitude as well as sign and thus may resolve
this puzzle.
Table 3 reports goodness of fit for different
models. Goodness of fit of the continuous
model follows similar patterns in alternative
experimental designs. The full model, equation
(6) with n = 3, exhibits the best fit; adjusted R
2
is about 21%. The F-tests between consecutive
models in the nesting from n = 0 to 1 to 2 to 3
are significant at the 0.01 level, indicating that
each additional prior return and its interaction
term improve the model’s ability to explain the
variability of current return. The greatest
improvement in goodness of fit occurs when
r
j,t-1
and its interaction term are added to the
model with only UE
j,t
/P
j,t-1
. Successively adding
r
j,t-2
and then r
j,t-3
(along with their respective
interaction terms) also significantly improves
model fit, but the contribution to adjusted R
2
decreases as each additional prior return is
added to the model. This pattern is consistent
with a hypothesis that more recent information
has greater impact on investors’ beliefs.
Table 4 reports coefficient estimates for
different continuous models. Coefficient
estimates for the continuous model follow
similar patterns across all alternative
experimental designs. The coefficient estimate
for UE
j,t
/P
j,t-1
always is significant at the 0.01
level, and its sign always is positive. This result
is consistent with investors viewing current UE
as significant new information, interpreting
UE > 0 as good news, and interpreting UE < 0
as bad news.
We evaluate the continuous model for n = 3
at representative cases and present the results in
Tables 5 and 6. (The coefficient estimates for
equation (6) when n = 3 are listed in Table 4 as
model #4.) The behavior of current period
returns is generally inconsistent with the strong
Table 4
OLS Coefficient Estimates – Continuous Model
Model # Constant UE
t
/P
t-1
r
t-1
(UE
t
/P
t-1
)
×r
t-1
r
t-2
(UE
t
/P
t-1
)
×r
t-2
r
t-3
(UE
t
/P
t-1
)
×r
t-3
1
-0.13
***
(0.009)
1.21
***
(0.106)
2
-0.14
***
(0.008)
2.07
***
(0.127)
-0.39
***
(0.017)
1.40
***
(0.160)
3
-0.13
***
(0.008)
2.20
***
(0.137)
-0.43
***
(0.017)
1.32
***
(0.159)
-0.20
***
(0.019)
1.07
***
(0.216)
4
-0.12
***
(0.008)
2.36
***
(0.138)
-0.43
***
(0.016)
1.35
***
(0.156)
-0.23
***
(0.019)
1.39
***
(0.218)
-0.11
***
(0.021)
1.86
***
(0.275)
5
-0.11
***
(0.009)
1.49
***
(0.120)
-0.13
***
(0.020)
1.25
***
(0.237)
-0.10
***
(0.023)
1.45
***
(0.300)
6
-0.12
***
(0.009)
1.26
***
(0.107)
-0.07
***
(0.023)
1.18
***
(0.300)
Notes. The sample is drawn from S&P 1500 firms for FY1998–2000 (fiscal years ending 6/98–5/01) in Standard &
Poor’s Research Insight/Compustat. Sample size is 3,826 firm-year observations. We use firms with fiscal year ends
from all calendar months. UE
t
= current period unexpected annual earnings, where earnings are forecast using a random
walk model. P
t-1
is the share price at the end of the period preceeding the current period. Prior period returns are r
t-n
, the
continuously compounded total annual return n periods before the current period t. The value in parentheses is the
standard deviation of the coefficient estimate.
*** t-statistics for all coefficient estimates are significant at the 0.01 level.
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form H1. That is, the data generally do not
support the belief adjustment model’s prediction
that decision-makers (investors) underreact
when the sign of UE is not consistent with the
sign of prior period returns. See Table 5. The
exceptions are cases where UE
j,t
/P
j,t-1
> 0 is
extremely large. However, such cases are
unusual. For example, if the prior period share
price is $20 when UE
j,t
/P
j,t-1
= 0.35, then UE
j,t
=
$7 per share. If the belief adjustment model
only “works” in extreme cases, then it is not a
good model for behavior of decision-makers.
The belief adjustment model predicts that
decision-makers (investors) become committed
to their beliefs the longer the sequence of same
sign, prior period returns. That is, it predicts
that, when UE is positive, both R
j,t|+,n+
and
R
j,t|+,n-
should decrease as n increases. The
continuous model does not support this
conclusion in all cases. When UE
j,t
/P
j,t-1
> 0 has
relatively large values (e.g., 0.10 and greater,
which is well above the third quartile in our
sample), we observe cases where R
j,t|+,n+
does
not decrease as n increases, as predicted (top
half of Panel A in Table 6). Also, for most cases
where UE
j,t
/P
j,t-1
> 0 but is not extremely large
(e.g., the third quartile, 0.025, in our sample or
less), R
j,t|+,n-
increases as n increases, not
decreases as predicted (bottom half of Panel A
in Table 6). Thus, the data does not consistently
support hypothesis H2(a).
Nor does the data consistently support
hypothesis H2(b). When UE
j,t
/P
j,t-1
< 0, the
magnitude of the sequence R
j,t|-,n-
< 0 decreases
as n increases, as predicted by hypothesis
H2(b)(ii). On the other hand, the magnitude of
the sequence R
j,t|-,n+
< 0 increases as n increases,
thus violating hypothesis H2(b)(i). In summary,
the market does not behave in all situations as if
investors become committed to their beliefs the
longer the sequence of same sign, prior period
returns. In fact, it appears more likely that
investors regard UE of opposite sign from prior
period returns as significant new information
about a change in direction of company value.
Table 5
Tests of H1: Current Period Returns in the Continuous Model
Panel A. Positive UE.
Current period returns r
t
No. of
same-sign
prior
returns
Scaled
unexpected
earnings
r
t-1
= .1, r
t-2
= 0, r
t-3
= 0 r
t-1
= -.1, r
t-2
= 0, r
t-3
= 0
H1(a): hypothesized
direction of
difference is >.
Actual difference:
0.006 (Q1) -0.1480 -0.0637 <
0.011 (Q2) -0.1356 -0.0525 <
0.025 (Q3) -0.1006 -0.0214 <
0.200 0.3360 0.3680 <
1
0.350 0.7103 0.7018 >
Current period returns r
t
r
t-1
= .1, r
t-2
= .1, r
t-3
= 0 r
t-1
= -.1, r
t-2
= -.1, r
t-3
= 0
0.006 (Q1) -0.1702 -0.0415 <
0.011 (Q2) -0.1570 -0.0311 <
0.025 (Q3) -0.1202 -0.0018 <
0.200 0.3408 0.3632 <
2
0.350 0.7359 0.6761 >
Current period returns r
t
r
t-1
= .1, r
t-2
= .1, r
t-3
= .1 r
t-1
= -.1, r
t-2
= -.1, r
t-3
= -.1
0.006 (Q1) -0.1801 -0.0316 <
0.011 (Q2) -0.1660 -0.0221 <
0.025 (Q3) -0.1265 0.0045 <
0.150 0.2260 0.2420 <
3
0.200 0.3670 0.3370 >
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The Effect of Returns History on the Current Period Relation Between Returns and Unexpected Earnings
17
Panel B. Negative UE.
Current period returns r
t
No. of
same-sign
prior
returns
Scaled
unexpected
earnings
r
t-1
= .1, r
t-2
= 0, r
t-3
= 0 r
t-1
= -.1, r
t-2
= 0, r
t-3
= 0
H1(b): hypothesized
direction of difference
is >. Actual
difference:
-1.160 (min) -3.0572 -2.6580 <
-0.035 (Q1) -0.2503 -0.1549 <
-0.016 (Q2) -0.2029 -0.1126 <
-0.007 (Q3) -0.1805 -0.0926 <
1
-0.001 -0.1655 -0.0792 <
Current period returns r
t
r
t-1
= .1, r
t-2
= .1, r
t-3
= 0 r
t-1
= -.1, r
t-2
= -.1, r
t-3
= 0
-1.160 (min) -3.2414 -2.4738 <
-0.035 (Q1) -0.2782 -0.1270 <
-0.016 (Q2) -0.2281 -0.0874 <
-0.007 (Q3) -0.2044 -0.0686 <
2
-0.001 -0.1886 -0.0561 <
Current period returns r
t
r
t-1
= .1, r
t-2
= .1, r
t-3
= .1 r
t-1
= -.1, r
t-2
= -.1, r
t-3
= -.1
-1.160 (min) -3.4682 -2.2470 <
-0.035 (Q1) -0.2957 -0.1095 <
-0.016 (Q2) -0.2421 -0.0734 <
-0.007 (Q3) -0.2167 -0.0563 <
3
-0.001 -0.1998 -0.0449 <
Notes. Sample is drawn from S&P 1500 firms for FY1998–2000 (fiscal years ending 6/98–5/01). We use firms with
fiscal year ends from all calendar months. UE
t
= current period unexpected annual earnings; earnings are forecast as a
random walk. Unexpected earnings are scaled by P
t-1
, the share price at the end of period t-1. Returns are r
t-n
, the
continuously compounded total annual return n periods before the current period t, n = 0, 1, 2, 3. The current period
return is calculated using the continuous model #4 in Table 4. Q1, Q2, & Q3 are lower quartile, median, & upper
quartile, respectively, of UE
j,t
/P
j,t-1
, conditional on UE > 0 (Panel A) or UE < 0 (Panel B).
Indirect Tests of the Hogarth and Einhorn
Two-equation Model
We indirectly test the earnings-returns version
of the Hogarth and Einhorn (1992) two-
equation model by estimating the sensitivity
coefficients, ? and ?. In the Hogarth and
Einhorn model, these coefficients are bounded:
0 ? ? ? 1 and 0 ? ? ? 1.
We report results based on using model #2 in
Table 4 to calculate fitted values of the current
period returns. Estimates of ? and ? generally fall
well outside the assumed bounds and differ
substantially depending on the value of the prior
period return and current unexpected earnings.
For example, using equation (9a) to calculate the
estimate of ?, when UE
t
/P
t-1
= -0.035 (the first
quartile conditional on UE < 0), alpha is 101.8 if
R
t-1
= 0.10 and –19.6 if R
t-1
= -0.10. When
UE
t
/P
t-1
= -0.007 (the third quartile conditional on
UE < 0), then alpha is 420.7 if R
t-1
= 0.10 and
-20.7 if R
t–1
= -0.10. Using equation (9b) to
calculate the estimate of ?, when UE
t
/P
t-1
= 0.006
(the first quartile conditional on UE > 0), beta is
-49.2 if R
t-1
= 0.10 and 1.6 if R
t-1
= -0.10. When
UE
t
/P
t-1
= 0.025 (the third quartile conditional on
UE > 0), then beta is -9.9 if R
t-1
= 0.10 and 1.7 if
R
t–1
= -0.10. Estimates of ? and ? using equation
(11) generally are well above one and also vary
substantially for different values of prior period
returns and current unexpected earnings.
These results strongly suggest that the
earnings-returns version of the Hogarth and
Einhorn (1992) two-equation model is
misspecified. It may be that assuming ? and ?
are the same for all firms is too strong an
assumption. However, while this might explain
why estimates of ? and ? vary across prior
period returns and current unexpected earnings,
it does not explain why the estimates generally
are so far from the range [0, 1] specified by
Hogarth and Einhorn.
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ACCOUNTING RESEARCH JOURNAL VOLUME 20 NO 1 (2007)
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Table 6
Tests of H2: Current Period Returns in the Continuous Model
Panel A. Positive UE.
Current period returns r
t
Scaled
unexpected
earnings
r
t-1
= .1, r
t-2
= -.1,
r
t-3
= -.1
r
t-1
= .1, r
t-2
= .1,
r
t-3
= -.1
r
t-1
= .1, r
t-2
= .1,
r
t-3
= .1
H2(a)(i): hypothesized
direction of differences
is >, >. Actual:
0.006 (Q1) -0.1160 -0.1603 -0.1801 >, >
0.011 (Q2) -0.1051 -0.1481 -0.1660 >, >
0.025 (Q3) -0.0748 -0.1138 -0.1265 >, >
0.100 0.0880 0.0698 0.0850 >, <
0.200 0.3050 0.3146 0.3670 , >. Actual:
0.006 (Q1) -0.0957 -0.0514 -0.0316 , >
-0.016 (Q2) -0.1637 -0.2142 -0.2421 >, >
-0.007 (Q3) -0.1442 -0.1921 -0.2167 >, >
-0.001 -0.1312 -0.1774 -0.1998 >, >
Current period returns r
t
Scaled
unexpected
earnings
r
t-1
= -.1, r
t-2
= .1,
r
t-3
= .1
r
t-1
= -.1, r
t-2
= -.1,
r
t-3
= .1
r
t-1
= -.1, r
t-2
= -.1,
r
t-3
= -.1
H2(b)(ii): hypothesized
direction of differences
is
This paper estimates the value added by Big
8/6/5 auditors after controlling for the
permanent and non-permanent impact of
earnings and cash flows using linear and nonlinear
(arctan) regression models. The linear
model shows significant value added for
industrial firms that utilise Big 8/6/5 auditors;
while an arctan model shows that large auditors
value-add by attesting to the permanence of
earnings for large firms. We demonstrate that
refinements to the audit research can be made
by using response coefficients to filter out the
different timing components inherent in
earnings and cash flows
Accounting Research Journal
The Effect of Returns History on the Current Period Relation Between Returns and Unexpected Earnings
Russell Calk Paul Haensly Mary J o Billiot
Article information:
To cite this document:
Russell Calk Paul Haensly Mary J o Billiot, (2007),"The Effect of Returns History on the Current Period Relation Between
Returns and Unexpected Earnings", Accounting Research J ournal, Vol. 20 Iss 1 pp. 5 - 20
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The Effect of Returns History on the Current Period Relation Between Returns and Unexpected Earnings
5
The Effect of Returns History on the Current
Period Relation Between Returns and
Unexpected Earnings
Russell Calk
Department of Accounting and Business Computer Systems
College of Business Administration and Economics
New Mexico State University
Paul Haensly
School of Business
University of Texas of the Permian Basin
and
Mary Jo Billiot
Department of Accounting and Business Computer Systems
College of Business Administration and Economics
New Mexico State University
Abstract
This study applies a model of systematic belief
revision to examine the effect of the relation
between current-period unexpected earnings
and prior-period security returns on the current-
period relation between those unexpected
earnings and returns. Cross-sectional analysis
blurs the effects of past information on current
returns in a manner that makes it easy to
overlook any dependence on historical patterns
in this information. We show that the market
responds to earnings innovations conditional on
these patterns but does not respond in the
manner predicted by the Hogarth and Einhorn
(1992) belief adjustment model. Nonetheless,
the results suggest that individual decision
processes are detectable in capital markets data.
1. Introduction
Over the past three decades, much of the
empirical financial accounting research has
focused on whether there is a discernable
Acknowledgments: The authors wish to express our
appreciation for the many helpful comments of the
workshop participants at Oklahoma State University and the
Annual Meeting of the American Accounting Association.
impact of accounting information in financial
markets. (For example, see reviews by Francis
and Schipper, 1999; Francis, Schipper, and
Vincent, 2002; Kross and Kim, 2000;
Landsman and Maydew, 2002.) This research
has centered on the relation between accounting
earnings and security returns with the general
focus on the effect of earnings information on
returns. However, cross-sectional analysis blurs
the effects of past information on current returns
in a manner that makes it easy to overlook any
dependence on historical patterns in this
information. We show that the market responds
to earnings innovations conditional on these
patterns but does not respond in the manner
predicted by the Hogarth and Einhorn (1992)
belief adjustment model.
We formulate and test hypotheses based on
the belief adjustment model proposed by Hogarth
and Einhorn (1992). Their decision model
describes belief adjustment in which decision-
makers evaluate new information based on its
relation to a decision anchor. We investigate how
investors (individual decision-makers) evaluate
accounting earnings (new information) and
incorporate the new information based on the
relation of earnings to prior-period returns
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ACCOUNTING RESEARCH JOURNAL VOLUME 20 NO 1 (2007)
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(decision anchor) to determine stock price and
hence, current period returns.
Experimental markets research provides
support for an individual decision model that
follows systematic belief adjustment. Such
research indicates that individual decision
biases can influence returns (Ackert, Church,
and Shehata, 1997; Camerer, 1992; Ganguly,
Kagel, and Moser, 1994; O’Brien and
Srivastava, 1991) but also cautions that these
decision biases may not be detectable in
efficient markets. By using empirical market
methods to analyze the systematic belief
adjustment model we pursue two goals. The
first goal is to determine whether systematic
belief adjustment is detectable in a market
setting. The second goal is to determine whether
systematic belief adjustment is based on the
relation between unexpected earnings (new
information) and prior period returns (decision
anchor).
Consistent with Hogarth and Einhorn’s
(1992) model, we find strong evidence that
current period returns respond to current
earnings innovations conditional on prior-period
returns. We also find that sensitivity of current
period returns to current earnings innovations
depends on whether innovations are positive or
negative, all else equal. However, we also find
results that are not consistent with Hogarth and
Einhorn’s claim that beliefs become less
sensitive to new information as information
accumulates. Our analysis supports this
proposition when the earnings innovation (new
information) is consistent with (i.e., has the
same sign as) prior-period returns; but the
reverse of this proposition is true when earnings
innovations are not consistent with prior-period
returns.
2. The Earnings/Return Relation
An extensive body of literature examines the
relation between accounting earnings and
security returns. An exhaustive review of this
literature is beyond the scope of this paper, but
it is reasonable to assert that an empirical
relationship between earnings and returns has
been well documented. While the specific
experimental design varies depending on the
nature and purpose of the research question,
most studies employ either a regression model
such as the one used by Kormendi and Lipe
(1987) or portfolio tests similar to those used by
Francis and Schipper (1999) in their
examination of the earnings/return relationship.
The regression tests focus on the earnings
coefficient estimate, commonly referred to as
the earnings response coefficient (ERC), of a
model relating unexpected earnings to changes
in stock price. The portfolio tests compare
returns for firms grouped according to
unexpected earnings. In either case, unexpected
earnings (UE) for firm j in period t are defined
as
t j
t j t j
E E UE ,
, ,
?
? =
(1)
where E
j,t
are the reported earnings and E
^
j,t
are
the forecasted earnings for firm j in period t.
These approaches have not fully explained
the relation between earnings and returns.
Studies have sometimes documented
unexpected results when examining ERCs. For
example, ERCs are sometimes negative which
implies that returns should increase as a
consequence of negative UE and vice versa.
Schroeder (1995) notes that firms with negative
ERCs had greater variability in UE over the
time period and proposes that returns may not
react in a predicted way to unexpected earnings
because UE interacts with other information. In
an effort to more fully understand this relation,
studies have included variables to capture the
effects of prior period prices (Donnelly and
Walker, 1995), analysts’ earnings forecasts
(Schroeder 1995), uncertainty in those analysts’
forecasts (Imhoff and Lobo, 1992), the
variability of UE caused by firm specific
characteristics (Teets and Wasley, 1996), the
time series pattern of earnings (Subramanyam
and Wild, 1996), the current level of earnings
(Ali and Zarowin, 1992), other financial
statement information (Penman, 1992), and firm
size (Shevlin and Shores, 1993) in the
experimental design. Other studies suggest that
the model itself is misspecified. Cheng,
Hopwood, and McKeown (1992), for example,
allow for fixed and random UE effects in the
model. Freeman and Tse (1992) and Cheng,
Hopwood, and McKeown (1992) also suggest
that the relationship between UE and returns is
nonlinear. Lipe, Bryant, and Widener (1998)
show that the variables included in the model
and the functional form of the model are both
significant factors in explaining the effect of
earnings on returns.
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7
We take a different approach to examine
the earnings/return relationship. Our principal
objective in this study is to determine whether or
not the Hogarth and Einhorn (1992) belief
adjustment model can be adapted to explain the
relation between returns and unexpected
earnings. In particular, we follow up on
Schroeder’s (1995) suggestion that trends in prior
period information may explain how current
period returns respond to current period UE. To
illustrate, consider the following examples.
Suppose that firm A reports negative returns in
the previous two reporting periods, and firm B
reports positive returns in each of the same
periods. Further suppose that both firms report
the same positive UE in the current period.
Would the price reaction to the current period UE
be the same for both firms? Conventional
efficient markets theory predicts that the response
should be the same, all else equal, because
information in past returns already is embedded
in price. Alternatively, suppose that firm A had
reported negative returns for three consecutive
prior periods instead of two. Would the
earnings/return relationship for Firm A be the
same in either case? To address questions such as
these, we formulate hypotheses based on the
belief adjustment model and test them with
models in which current period returns are a
function of prior period returns as well as current
period UE.
3. Hypotheses Development and
Experimental Design
Hogarth and Einhorn (1995, p. 8) propose a
structural model of systematic belief adjustment
based on the idea that, “People handle belief-
updating tasks by a general, sequential
anchoring-and-adjustment process in which
opinion, or the anchor, is adjusted by the impact
of succeeding pieces of evidence.” In particular,
the effect of new information on the anchor
depends on the direction of the new information.
If it is consistent with the anchor, then the new
information may be weighted differently than if it
is inconsistent with the anchor. In other words,
the weight placed on new information should
depend on the direction of the new information
relative to prior information.
For example, in the context of financial
markets, the weight placed on positive or
negative unexpected earnings (UE) in period t
should depend on the sign of returns in prior
periods. Positive UE reported in the period
following a sequence of positive returns would
be consistent with those returns and hence less
of a surprise than if the positive UE followed a
sequence of negative returns. According to the
Hogarth and Einhorn (1992) model, the positive
UE will be weighted differently and will have a
different effect in the two situations. Similarly,
negative UE reported in the period following a
sequence of negative returns would be
consistent with those returns and hence less of a
surprise than if the negative UE followed a
sequence of positive returns.
Calegari and Fargher (1997) provide
evidence to support a conditional response to
UE. Subjects in their experimental market
tended to underweight unexpected earnings that
were an extreme surprise. These results suggest
that individual investors are likely to
underweight reported UE that are inconsistent
with prior period returns. Thus, for example, if
UE > 0 follows negative prior returns, then the
current period returns would be lower than if
UE > 0 had followed positive prior returns.
Regardless of the individual weighting of
information, Hogarth and Einhorn (1992, p. 4)
assert that, “As information accumulates, beliefs
are expected to become less sensitive to the
impact of new information because this
represents an increasingly small proportion of
the evidence already processed.” This is a
manifestation of an escalation of commitment
documented by Staw (1976). In particular, the
effect of current period UE on returns should
decrease as the number of same sign, prior
period returns increases. The effect of UE
should diminish whether the sign of UE is
consistent or inconsistent with the sign of the
sequence of prior period returns. This idea is
similar to the notion of a decreasing marginal
effect of earnings persistence proposed by
Feltham and Ohlson (1995).
We reformulate Hogarth and Einhorn’s
(1992) two general hypotheses in the context of
accounting earnings and security returns. Our
hypotheses identify subsets of data in terms of
UE and the pattern of prior period returns. By
comparing returns conditional on these subsets,
we are able to examine the effects of decision-
making conditional on UE and prior period
returns. Let R
j,t|+,n+
be the return for firm j in
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period t, conditional on UE
j,t
> 0 and the
sequence of n prior period returns such that
R
j,t–1
> 0, …, R
j,t–n
> 0. Let R
j,t|+,n-
be the return
for firm j in period t, conditional on UE
j,t
> 0
and the sequence of n prior period returns such
that R
j,t–1
< 0, …, R
j,t–n
< 0. Similarly, R
j,t|-,n+
represents the return for firm j in period t,
conditional on UE
j,t
< 0 and the sequence of n
prior period returns such that R
j,t-1
> 0, …,
R
j,t–n
> 0. R
j,t|-,n-
represents the return for firm j in
period t, conditional on UE
j,t
< 0 and the
sequence of n prior period returns such that
R
j,t–1
< 0, …, R
j,t–n
< 0.
Our first hypothesis expresses Hogarth and
Einhorn’s (1992) proposition that the effect of
new information on the anchor depends on the
direction of the new information.
H1: current period returns depend on both
current period UE and on the relation between
current period UE and prior period returns. In
particular, investor response to new information
contained in current period UE depends on the
direction of UE relative to the sign of the
sequence of n prior period returns:
a) if UE
j,t
> 0, then R
j,t|+,n+
? R
j,t|+,n-
, on average;
b) if UE
j,t
< 0, then R
j,t|-,n+
? R
j,t|-,n-
, on average.
We label this first version as weak-form H1.
The Calegari and Fargher (1997) experimental
results suggest the following version of the first
hypothesis, which we label as strong-form H1:
a) if UE
j,t
> 0, then R
j,t|+,n+
> R
j,t|+,n-
, on average;
b) if UE
j,t
< 0, then R
j,t|-,n-
< R
j,t|-,n+
, on average.
Our second hypothesis expresses Hogarth and
Einhorn’s (1992) proposition that as information
accumulates, belief becomes less sensitive to new
information. We follow Schroeder’s (1995)
suggestion when we translate Hogarth and
Einhorn’s proposition into the language of
earnings and returns. That is, we formulate the
hypotheses in terms of the length of the sequence
of same sign, prior period returns. In light of H1,
the second hypothesis has two parts
corresponding to UE > 0 and UE < 0.
H2: as the number of consecutive periods of
same sign, prior period returns increases, the
impact of current period UE on current period
returns will decrease.
a) If UE
j,t
> 0, then the positive effect on current
period returns will be smaller the longer the
sequence of same sign, prior period returns.
In particular:
i) If the sequence of prior period returns is
positive, then the sequence of conditional
current period returns R
j,t|+,n+
monotonically decreases as n increases,
on average.
ii) If the sequence of prior period returns is
negative, then the sequence of conditional
current period returns R
j,t|+,n-
monotonically decreases as n increases,
on average.
b) If UE
j,t
< 0, then the negative effect on
current period returns will be smaller the
longer the sequence of same sign, prior
period returns. In particular:
i) If the sequence of prior period returns is
positive, then the sequence of conditional
current period returns R
j,t|-,n+
monotonically increases as n increases, on
average; that is, the magnitude of these
negative current period returns decreases.
ii) If the sequence of prior period returns is
negative, then the sequence of conditional
current period returns R
j,t|-,n-
monotonically increases as n increases, on
average; that is, the magnitude of these
negative current period returns decreases.
Our primary objective in this study is to test
Hogarth and Einhorn’s (1992) belief adjustment
model — in particular, extensions suggested by
the work of Calegari and Fargher (1997) and
Schroeder (1995) — in the context of the
earnings/return relationship. To test these
hypotheses, we draw data for each 1994–2000
fiscal year from Research Insight (Compustat)
for firms listed as components of the S&P 1500
in that year. The S&P 1500, or Supercomposite,
is a large, representative list of mostly domestic
common stocks traded in U.S. markets. The
Supercomposite combines the S&P 500 (large
caps), S&P Mid-cap 400, and S&P Small-cap
600. S&P 1500 firms are widely followed by
analysts, and information about these firms
should have been widely available to investors.
The effect of earnings announcements on S&P
1500 stock prices should, therefore, represent
investor response to new information rather
than simply speculation. Furthermore, Buchheit
and Kohlbeck (2002) show that the increase
in the usefulness of earnings seems to be
concentrated among large firms. Our sample
should capture this effect.
We collected current period and three years
of prior period total returns for each firm. We
measured annual total return in two ways. First,
we measured each firm’s annual returns for
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12-month periods that lag the corresponding
fiscal years by three months. The vast majority
of companies release their annual reports within
the mandatory 90-day period following their
fiscal year ends. By using lagged annual returns,
we allow the current earnings announcement
(and hence current period UE) to have an
informational effect on current annual returns.
We also measured each firm’s annual returns
for 12-month periods concurrent with fiscal
years (or, at least with fiscal year ends). In
principle, these annual returns should not be
directly affected by the final annual earnings
announcement for the concurrent fiscal year.
Thus, current period UE may not have a
significant effect on concurrent period annual
returns. However, many studies of the
earnings/return relation use concurrent annual
returns. So our use of concurrent annual returns
allows comparability with previous work. In
addition, quarterly earnings and preliminary
annual earnings estimates at the end of the fiscal
year may provide investors with much of the
information contained in the final version.
We collected current period operating
earnings per share for each firm. Use of
operating earnings per share controls for any
nonrecurring events. We also replicated our
analysis with basic EPS and diluted EPS. Our
results for these forms of EPS were similar to
those based on operating earnings per share but
tended to be slightly less significant. Hence, we
only report results based on operating earnings
per share.
We treat each fiscal year that a firm is in the
S&P 1500 as a potential current year
observation. We analyze pooled current years
from Research Insight fiscal years 1994–2000,
the period over which Research Insight
identifies S&P 1500 companies. (A Research
Insight fiscal year includes annual reports with a
fiscal year end that falls in June of that year
through May of the following calendar year. For
example, FY2000 includes reports with fiscal
year ends of June 2000 through May 2001.)
Operating earnings per share (OPEPS) are only
available back to FY1988 in Research Insight.
As a result, the methods we apply to forecast
earnings sometimes limit our choice of current
years. In particular, when we apply second
order autoregression methods to forecast
earnings, we require at least 10 prior years of
earnings for an annual report to be included in
our sample. Sample sizes in our reported results
vary from 1,762 observations (from FY1998–
2000 when we use second-order autoregressive
(AR(2)) earnings forecasts and limit fiscal year
ends to December) to 8,080 observations (from
FY1994–2000 when we use random walk
earnings forecasts and use all fiscal year end
months).
We report results based on the assumption
that annual earnings follow a random walk.
Under that assumption, we calculate UE,
defined in equation (1), from the forecast
Ê
j,t
= E
j,t-1.
Foster (1986, p. 240) concludes, “The
result that, on average, annual reported earnings
or EPS can be well-described by a random walk
model is one of the most robust empirical
findings in the financial statement literature.”
On the other hand, recent work, e.g., Ball and
Bartov (1996), suggests that the market acts as
if quarterly earnings are serially correlated.
Hence, for robustness, we also apply second-
order autoregression to forecast current annual
earnings E
j,t
for each observation. When
changes in annual earnings are assumed to
follow a second-order autoregressive process
(AR(2)), the forecast is Ê
j,t
= E
j,t-1
+ ?Ê
j,t
, where
change in earnings may be modeled as
?E
j,t
= ?
j1
?E
j,t-1
+ ?
j2
?E
j,t-2
+ a
j,t
. (2)
From equation (1) and the forecast for
current earnings,
UE
j,t
= E
j,t
– Ê
j,t
= E
j,t
– (E
j,t-1
+ ?Ê
j,t
)
= ?E
j,t
– ?Ê
j,t
. (3)
Thus, in a manner similar to Kormendi and
Lipe (1987), UE is the residual from the
estimated AR(2) process. That is,
?E
j,t
= f
j1
?E
j,t-1
+ f
j2
?E
j,t-2
+ UE
j,t
, (4)
where f
j1
and f
j2
are the estimated AR(2)
coefficients for equation (2). In our analysis,
neither earnings forecast model is superior in all
scenarios, and hypothesis testing with the two
alternatives produces similar results.
We analyze the data using two general types
of models: those with categorical independent
variables, and those with continuous
independent variables. The categorical models
can be used to analyze firm year observations
grouped based on the sign of UE
j,t
and the signs
of R
j,t-n
over the n periods of the returns
sequence. While we derive these results from
regression analysis of our categorical models,
the mean returns for these groups could have
been computed directly. Hence, these results are
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robust in the sense that they depend only on the
independent variables selected and not on any
particular form of the regression model. The
general categorical model is
r
j,t
= ?
0
+ ?
1
DUE
j,t
+ ?
1
DR
j,t-1
+ ?
2
DR
j,t-2
+ · · ·
+ ?
n
DR
j,t-n
+ ?
1
DUE
j,t
DR
j,t-1
+ ?
2
DUE
j,t
DR
j,t-2
+ · · · + ?
n
DUE
j,t
DR
j,t-n
+ ?
j,t
, (5)
where
DUE
j ,t
=
1 if UE
j ,t
? 0
0 if UE
j ,t
< 0
?
?
?
?
?
?
,
DR
j, t ?k
=
1 if R
j, t ?k
? 0
0 if R
j, t? k
< 0
?
?
?
?
?
?
, k = 0, 1, 2,… , n,
and we use continuously compounded annual
returns r
j,t
= ln(1+R
j,t
).
The regression analysis of the categorical
model uses only the sign of UE
j,t
and the signs
of R
j,t-n
but not their magnitudes. To evaluate the
relation between current returns and prior period
returns for a given level of unexpected earnings,
we must use continuous variables. The general
continuous model is
r
j,t
= ?
0
+ ?
1
(UE
j,t
/P
j,t-1
) + ?
1
r
j,t-1
+ ?
2
r
j,t-2
+ · · ·
+ ?
n
r
j,t-n
+ ?
1
(UE
j,t
/P
j,t-1
)r
j,t-1
+ ?
2
(UE
j,t
/P
j,t-1
)r
j,t-2
+ · · · + ?
n
(UE
j,t
/P
j,t-1
)r
j,t-n
+ ?
j,t
(6)
where r
j,t-k
= ln(1+R
j,t-k
), k = 0, 1, 2,… , n, and
unexpected earnings are scaled in the same
manner that change of price is scaled in R
j,t
.
The primary objective of this paper is to
evaluate how new information in the form of
unexpected earnings affects investors’ behavior
as measured by current returns, conditional on
information in the form of prior returns. Hence,
all models that we examine include current
unexpected earnings and its interaction with any
prior returns in the model. Due to constraints in
availability of OPEPS and data requirements for
AR(2) modeling of earnings, we use at most
prior returns back to t-3. Thus, the full models
will be equations (5) and (6) with n = 3.
A secondary objective of this paper is to
evaluate a two-equation model proposed by
Hogarth and Einhorn (1992, p. 14) for
systematic belief revision. Their original model
is stated in terms of degree of belief:
S
k
= S
k?1
+?S
k?1
s(x
k
) ? R [ ] for s(x
k
) ? R,
(7a)
S
k
= S
k ?1
+ ? 1? S
k ?1
( ) s(x
k
) ? R [ ] for s(x
k
) > R,
(7b)
where S
k
is degree of belief in some hypothesis,
S
k-1
is prior belief or opinion, s(x
k
) is subjective
evaluation of new evidence, and R is the
reference point against which the new evidence
is evaluated. Degree of belief is a bounded
variable such that 0 ? S
k
? 1. Hogarth and
Einhorn define coefficients ? and ? as
sensitivity to positive and negative evidence,
respectively, and require that 0 ? ? ? 1 and
0 ? ? ? 1. Their model can be adapted to study
belief revision in the context of capital markets
by letting R
j,t
, the return for firm j in period t,
represent investors’ degree of belief in the
information content of earnings for firm j:
R
j ,t
= R
j ,t ?1
+ ?R
j ,t ?1
UE
j, t
/ P
j,t ?1
( ) for UE
j ,t
? 0,
(8a)
R
j ,t
= R
j ,t ?1
+ ? 1 ? R
j ,t ?1
( )UE
j ,t
/ P
j,t ?1
( ) for UE
j ,t
> 0,
(8b)
where UE
j,t
/P
j,t-1
measures the effect of new
evidence (earnings announcement) against the
forecasted earnings for the firm. Hogarth and
Einhorn posit that ? and ? in equation (7)
depend on the individual decision-maker.
However, prices and returns in the market are
determined by the collective decision-making of
investors. So we treat ? and ? as investors’
collective sensitivities to positive and negative
evidence (i.e., earnings innovations),
respectively.
We estimate the coefficients ? and ?
indirectly. Under the assumption that
sensitivities ? and ? are the same for all firms,
equation (8) implies that
? = (R
j, t
? R
j,t ?1
) R
j ,t ?1
( ) UE
j,t
/ P
j,t ?1
( ) for UE
j ,t
? 0,
(9a)
? = (R
j,t
? R
j ,t ?1
) (1? R
j ,t ?1
) ( ) UE
j ,t
/ P
j, t?1
( ) for UE
j ,t
> 0.
(9b)
Alternatively, the partial derivatives of
current period returns with respect to prior
period returns in equation (8) are
?R
j ,t
/ ?R
j ,t ?1
= 1+ ? UE
j,t
/ P
j ,t ?1
( ) for UE
j ,t
? 0,
(10a)
?R
j ,t
/ ?R
j ,t ?1
= 1? ? UE
j ,t
/ P
j, t ?1
( ) for UE
j,t
> 0.
(10b)
Thus, alternative approximations for ? and ? are
ˆ
? = ?R
j,t
/ ?R
j, t ?1
?1 ( ) UE
j ,t
/ P
j ,t ?1
( ) for UE
j,t
? 0,
(11a)
ˆ
? = 1? ?R
j ,t
/ ?R
j ,t ?1
( ) UE
j, t
/ P
j,t ?1
( ) for UE
j ,t
> 0.
(11b)
We first estimate the parameters in our
single-equation model in equation (6) and use it
to calculate fitted values for current period
returns, R
j,t-1
. We then estimate the sensitivity
coefficients, ? and ?, by substituting the fitted
current period returns and the corresponding
prior period returns and scaled unexpected
earnings into equations (9) or (11).
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The astute reader may wonder why we use
our single-equation model in (6) to test the
hypotheses in H1 and H2 rather than test the
two-equation model in (8) directly. One reason
is that Hogarth and Einhorn’s assumptions
unreasonably restrict current and prior period
returns to the range from zero to one. More
importantly, Hogarth and Einhorn’s (1992)
model (equation (7) above) is defined directly in
terms of degree of belief. Security returns, on
the other hand, include more information than
just degree of belief. Change in security price
due to an earnings innovation reflects not only
belief in persistence of the change in earnings
but also the magnitude of the expected change
in future cash flows.
An analogy from statistics may clarify this
issue. A point estimator may be evaluated in
terms of bias (or accuracy) and efficiency. Bias
refers to how much the expected value of the
estimator differs from the true value of the
parameter to be estimated, while efficiency
refers to the estimator’s variance relative to that
of other estimators. But degree of belief is
equivalent to variance of an estimator in the
sense that it measures efficiency but not
accuracy of belief. A belief may be strongly
held yet far from the truth. Change in security
price reflects both an estimate of the change in
value as well as degree of belief in that change.
Our single-equation model is designed to
capture important elements of the belief-
adjustment process represented by prior period
returns and the interaction between prior period
returns and unexpected earnings on the right-
hand side of the two-equation model. Hence,
our single-equation model is consistent with the
theoretical belief adjustment process described
by Hogarth and Einhorn (1992) and, at the same
time, with the ERC literature in accounting.
A potential drawback to our single-equation
model is that the estimate of the ERC will not be
conditional on the sign of UE, because the sample
includes observations with positive UE and
observations with negative UE. Hence,
examination of the ERC in our model does not
facilitate the above hypothesis tests, because each
test is conditional on the sign of UE. However, we
are primarily interested in testing the validity of
the belief adjustment model, not in estimation of
the ERC. The tests of the hypotheses do not
require estimation of the ERC. Should the
evidence support the above hypotheses and thus
the validity of the belief adjustment model, then
the natural next step would be to develop an
appropriate model for estimating the ERC
conditional on sign of the UE.
4. Results
The market data does not support key
predictions based on the belief adjustment
model. A natural extension of this model to
decision-making in financial markets states that
investors will tend to underweight their
response to earnings innovations when UE has
the opposite sign of prior period returns (strong-
form H1). Data from both the categorical and
continuous models generally do not support this
conclusion (Tables 1 and 5). A second natural
extension of the belief adjustment model states
that the market response to UE will diminish
with the length of the sequence of same sign,
prior period returns (hypotheses H2(a) for
UE > 0 and H2(b) for UE < 0). That is,
investors will behave as if they become
committed to their beliefs the longer the
sequence of consistent returns. Data from both
the categorical and continuous models generally
do not support this conclusion when UE has the
opposite sign from the sequence of prior period
returns (Tables 2 and 6). Instead, the market
behaves as if UE of opposite sign to the prior
period returns is a surprise containing
significant news that justifies a reversal of
valuation. Furthermore, the longer the sequence
of same sign, prior period returns, the stronger
the response, i.e., investors appear less
committed to their beliefs when UE is of
opposite sign as the prior period returns.
In this section, we discuss the above results
in detail for an experimental design in which
earnings are measured by operating earnings per
share, earnings forecasts are based on a random
walk, and annual returns lag the fiscal year end
by three months. Our results are robust to
choice of measure of earnings (OPEPS, basic
EPS, or diluted EPS), model for earnings
(random walk or AR(2)), and choice of period
for annual returns (concurrent with the fiscal
year end or measured with a three-month lag).
Tests of Hypothesis 1 with the Categorical
Model
The strong form H1 states that if UE is positive,
then R
j,t|+,n+
> R
j,t|+,n-
, on average, and if UE is
negative, then R
j,t|-,n-
< R
j,t|-,n+
, on average. That
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is, investors will tend to underweight their
response to earnings innovations that are a
surprise in the sense that sign of UE is opposite
to the sign of the prior period returns. The tests
show just the opposite is true. See Table 1. The
inequalities in current period returns are
significant but in the opposite direction from
that hypothesized. The difference is significant
at the 0.01 level for n = 1, 2, and 3 whether UE
is positive or negative. This result is robust in
the sense that we observe this level of
significance regardless of the experimental
design. Weak-form H1 is supported by our
results but not strong-form H1.
Tests of H1 are not consistent with the results
reported by Calegari and Fargher (1997).
Current returns do not exhibit underreaction to
earnings surprises. Market response is more
strongly positive when the good news (UE > 0)
is not consistent with prior returns and is more
strongly negative when the bad news (UE < 0)
is not consistent with prior returns. These results
hold regardless of the number of same sign,
prior period returns. While not consistent with
Calegari and Fargher, these results are
consistent with Skinner and Sloan (2001) who
document a particularly large reaction to
negative earnings surprises among growth
stocks. Calegari and Fargher used student
subjects in their experimental market study.
Mowen and Mowen (1986) suggest that student
subjects may not be appropriate when the
decision setting is very complex and when a
great deal of professional judgment or
experience is required. These characteristics of
the decision-makers in their study may explain
the discrepancy between Calegari and Fargher’s
experimental market results and the empirical
results reported here.
The magnitude of market response (i.e.,
investors in aggregate) is asymmetric and thus
is consistent with an alternative hypothesis: UE
that is inconsistent with prior returns has
informational value that overrides the anchoring
postulated by Hogarth and Einhorn (1992).
(Whether the market overreacts when new
information contradicts previous beliefs is an
issue that we do not explore in this paper.)
When prior returns have been positive, the
market is conditioned to expect continuing good
performance by the company. Thus, UE
j,t
> 0
provides little or no new information about
future performance. On the other hand, when
prior returns have been negative, the market
has been conditioned to expect continuing
bad performance. Thus, UE
j,t
> 0 provides
Table 1
Tests of H1: Mean Returns Conditional on Signs of UE and
Prior Period Returns
Mean current returns conditional on sequence
of prior period returns
No. of
same-sign
prior
returns
Unexpected
earnings r
t-1
? 0 r
t-1
< 0
Difference
of mean
returns
F-test for
difference of
means
1 UE
t
? 0 -0.0976 0.1172 -0.2147 92.90
***
1 UE
t
< 0 -0.4693 -0.1979 -0.2714 263.68
***
r
t-1
? 0, r
t-2
? 0 r
t-1
< 0, r
t-2
< 0
2 UE
t
? 0 -0.1577 0.1785 -0.3361 93.82
***
2 UE
t
< 0 -0.5334 -0.1259 -0.4074 261.70
***
r
t-1
? 0, r
t-2
? 0, r
t-3
? 0 r
t-1
< 0, r
t-2
< 0, r
t-3
< 0
3 UE
t
? 0 -0.1695 0.2027 -0.3722 70.21
***
3 UE
t
< 0 -0.5487 -0.0989 -0.4497 193.70
***
Notes. The sample is drawn from S&P 1500 firms for FY1998–2000 (fiscal years ending 6/98–5/01) in Standard &
Poor’s Research Insight/Compustat. Sample size is 3,826 firm-year observations. We use firms with fiscal year ends
from all calendar months. UE
t
= current period unexpected annual earnings, where earnings are forecast using a random
walk model. Returns are r
t-n
, the continuously compounded total annual return n periods before the current period t,
n = 0, 1, 2, 3. The predicted difference of mean returns in H1 is positive in all cases above.
*** All F-statistics reported above are significant at the 0.01 level.
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significant new information that performance
has changed for the better. Alternatively, the
good news in a positive UE following positive
prior returns is not very surprising (little new
information) but is a big surprise (substantial
new information) if it follows negative prior
returns.
Similarly, when prior returns have been
negative, the market is conditioned to expect
continuing bad performance by the company.
Thus, UE
j,t
< 0 provides little or no new
information about future performance. On the
other hand, when prior returns have been
positive, the market has been conditioned to
expect continuing good performance. Thus,
UE
j,t
< 0 provides significant new information
that performance has changed for the worse.
Alternatively, the bad news in a negative UE
following negative prior returns is not very
surprising (little new information) but is a big
surprise (substantial new information) if it
follows positive prior returns.
Tests of Hypothesis 2 with the Categorical
Model
Hogarth and Einhorn (1992) assert that all
decision-makers become committed to their
prior beliefs over time so the impact of
additional information diminishes. The results
in this study support that claim when new
information is consistent with prior information.
However, the results contradict that claim when
new information is inconsistent.
When UE
j,t
> 0, the market response to this
good news diminishes (i.e., becomes less
positive) as the length of the sequence of
positive prior returns increases. See Table 2,
Panel A. UE
j,t
> 0 carries less weight in
investors’ decisions as this sequence grows
longer.
Table 2
Tests of H2: Mean Returns Conditional on UE and
Sequence of Prior Period Returns
Sequence of positive prior period
returns
Sequence of negative prior period
returns
r
t-1
+ + + + – – – –
r
t-2
– – + + + + – –
r
t-3
– + – + + – + –
Panel A: UE > 0
Predicted ranking of
mean returns by H2(a)
High Middle Low High Middle Low
Observed mean returns -0.0329 -0.0642 -0.1382 -0.1695 0.0661 0.0974 0.1714 0.2027
F-test of difference in
means
1.68 5.61
**
1.68 1.68 5.61
**
1.68
Panel B: UE < 0
Predicted ranking of
mean returns by H2(b)
Low Middle High Low Middle High
Observed mean returns -0.3555 -0.4002 -0.5040 -0.5487 -0.2145 -0.1698 -0.0660 -0.0213
F-test of difference in
means
2.72
*
9.33
***
2.72
*
2.72
*
9.33
***
2.72
*
Notes. The sample is drawn from S&P 1500 firms for FY1998–2000 (fiscal years ending 6/98–5/01) in Standard &
Poor’s Research Insight/Compustat. Sample size is 3,826 firm-year observations. We use firms with fiscal year ends
from all calendar months. UE
t
= current period unexpected annual earnings, where earnings are forecast using a random
walk model. Returns are r
t-n
, the continuously compounded total annual return n periods before the current period t, n =
0, 1, 2, 3. Negative mean returns following UE
t
> 0 are due to the heavy influence of the market downturn that began in
March 2000. However, this does not affect interpretation of the above results, because H2 concerns relative order of the
means.
* significant at 0.10
** significant at 0.05
*** significant at 0.01
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ACCOUNTING RESEARCH JOURNAL VOLUME 20 NO 1 (2007)
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Thus, the data support hypothesis H2(a)(i). The
results also indicate that the anchor extends
back about two years. As n increases from one
to two, the change is significant at the 0.05
level, but as n increases from two to three, the
change is not significant.
However, a different market response to
UE
j,t
> 0 occurs when the prior returns are
negative. See Table 2, Panel A. As the length of
the sequence of negative prior period returns
increases, the positive market response to the
good news becomes stronger. The good news of
UE
j,t
> 0 carries more weight in investors’
decisions as this sequence of negative prior
period returns grows longer. Thus, the data do
not support hypothesis H2(a)(ii).
The market response to the good news in
positive UE is better explained in terms of
information content conditional on prior
information than in terms of commitment to
prior beliefs. In Table 2, notice that as the length
of the sequence of positive prior returns
decreases (left side of Panel A) or as the length
the sequence of negative prior returns increases
(right side of Panel A), the market response to
UE
j,t
> 0 improves. This result is consistent with
an increase in information content as the
inconsistency between direction of UE and the
prior period returns increases.
When UE
j,t
< 0, the market response to
this bad news diminishes (i.e., becomes less
negative) as the sequence of negative prior
returns increases. See Table 2, Panel B. The bad
news in UE
j,t
< 0 carries less weight in
investors’ decisions as this sequence grows
longer. Thus, the data support hypothesis
H2(b)(ii). The results also indicate that the
anchor extends back about two years. As n
increases from one to two, the change is
significant at the 0.01 level, but as n increases
from two to three, the change is only significant
at the 0.10 level.
However, a different market response to
UE
j,t
< 0 occurs when the prior returns are
positive. See Table 2, Panel B. As the length of
the sequence of positive prior returns increases,
the negative market response to the bad news
becomes stronger. The bad news of UE
j,t
< 0
carries more weight in investors’ decisions as
this sequence grows longer. Thus, the data do
not support hypothesis H2(b)(i).
The market response to the bad news in
negative UE is better explained in terms of
information content conditional on prior
information than in terms of commitment to
prior beliefs. In Table 2, notice that as the length
of the sequence of negative prior returns
decreases (right side of Panel B) or as the
length of the sequence of positive prior returns
increases (left side of Panel A), the strength of
the negative response to UE
j,t
< 0 increases.
Table 3
Model Comparisons and Goodness of Fit — Continuous Model
Model # Independent variables in the model
Adj. R
2
(%)
F-statistic for test
of significant
difference between
adjusted R
2
Models
in
F-test
1 UE
t
/P
t-1
3.23 299.17
***
1 vs. 2
2 UE
t
/P
t-1
, r
t-1
, (UE
t
/P
t-1
)×r
t-1
16.29 71.73
***
2 vs. 3
3 UE
t
/P
t-1
, r
t-1
, (UE
t
/P
t-1
)×r
t-1
, r
t-2
, (UE
t
/P
t-1
)×r
t-2
19.28 39.50
***
3 vs. 4
4
UE
t
/P
t-1
, r
t-1
, (UE
t
/P
t-1
)×r
t-1
, r
t-2
, (UE
t
/P
t-1
)×r
t-2
, r
t-3
,
(UE
t
/P
t-1
)×r
t-3
20.87
5 UE
t
/P
t-1
, r
t-1
, r
t-2
, (UE
t
/P
t-1
) ×r
t-2
, r
t-3
, (UE
t
/P
t-1
)×r
t-3
5.47 372.77
***
5 vs. 4
6 UE
t
/P
t-1
, r
t-1
, r
t-3
, (UE
t
/P
t-1
)×r
t-3
3.84 33.91
***
6 vs. 5
Notes. The sample is drawn from S&P 1500 firms for FY1998–2000 (fiscal years ending 6/98–5/01) in Standard &
Poor’s Research Insight/Compustat. Sample size is 3,826 firm-year observations. We use firms with fiscal year ends
from all calendar months. UE
t
= current period unexpected annual earnings, where earnings are forecast using a random
walk model. P
t-1
is the share price at the end of the period preceeding the current period. Prior period returns are r
t-n
, the
continuously compounded total annual return n periods before the current period t.
*** All F-statistics reported above are significant at the 0.01 level.
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This result is consistent with an increase in
information content as the inconsistency
between direction of UE and the prior period
returns increases.
OLS Results for the Continuous Model
The categorical data leave us with a puzzle.
Predictions based on the belief revision model
are supported under some circumstances but not
others. The impact of new information
diminishes when it is consistent with prior
information but increases when it is
inconsistent. Investors, however, also know the
magnitudes of these variables. The continuous
model permits a more detailed picture based on
magnitude as well as sign and thus may resolve
this puzzle.
Table 3 reports goodness of fit for different
models. Goodness of fit of the continuous
model follows similar patterns in alternative
experimental designs. The full model, equation
(6) with n = 3, exhibits the best fit; adjusted R
2
is about 21%. The F-tests between consecutive
models in the nesting from n = 0 to 1 to 2 to 3
are significant at the 0.01 level, indicating that
each additional prior return and its interaction
term improve the model’s ability to explain the
variability of current return. The greatest
improvement in goodness of fit occurs when
r
j,t-1
and its interaction term are added to the
model with only UE
j,t
/P
j,t-1
. Successively adding
r
j,t-2
and then r
j,t-3
(along with their respective
interaction terms) also significantly improves
model fit, but the contribution to adjusted R
2
decreases as each additional prior return is
added to the model. This pattern is consistent
with a hypothesis that more recent information
has greater impact on investors’ beliefs.
Table 4 reports coefficient estimates for
different continuous models. Coefficient
estimates for the continuous model follow
similar patterns across all alternative
experimental designs. The coefficient estimate
for UE
j,t
/P
j,t-1
always is significant at the 0.01
level, and its sign always is positive. This result
is consistent with investors viewing current UE
as significant new information, interpreting
UE > 0 as good news, and interpreting UE < 0
as bad news.
We evaluate the continuous model for n = 3
at representative cases and present the results in
Tables 5 and 6. (The coefficient estimates for
equation (6) when n = 3 are listed in Table 4 as
model #4.) The behavior of current period
returns is generally inconsistent with the strong
Table 4
OLS Coefficient Estimates – Continuous Model
Model # Constant UE
t
/P
t-1
r
t-1
(UE
t
/P
t-1
)
×r
t-1
r
t-2
(UE
t
/P
t-1
)
×r
t-2
r
t-3
(UE
t
/P
t-1
)
×r
t-3
1
-0.13
***
(0.009)
1.21
***
(0.106)
2
-0.14
***
(0.008)
2.07
***
(0.127)
-0.39
***
(0.017)
1.40
***
(0.160)
3
-0.13
***
(0.008)
2.20
***
(0.137)
-0.43
***
(0.017)
1.32
***
(0.159)
-0.20
***
(0.019)
1.07
***
(0.216)
4
-0.12
***
(0.008)
2.36
***
(0.138)
-0.43
***
(0.016)
1.35
***
(0.156)
-0.23
***
(0.019)
1.39
***
(0.218)
-0.11
***
(0.021)
1.86
***
(0.275)
5
-0.11
***
(0.009)
1.49
***
(0.120)
-0.13
***
(0.020)
1.25
***
(0.237)
-0.10
***
(0.023)
1.45
***
(0.300)
6
-0.12
***
(0.009)
1.26
***
(0.107)
-0.07
***
(0.023)
1.18
***
(0.300)
Notes. The sample is drawn from S&P 1500 firms for FY1998–2000 (fiscal years ending 6/98–5/01) in Standard &
Poor’s Research Insight/Compustat. Sample size is 3,826 firm-year observations. We use firms with fiscal year ends
from all calendar months. UE
t
= current period unexpected annual earnings, where earnings are forecast using a random
walk model. P
t-1
is the share price at the end of the period preceeding the current period. Prior period returns are r
t-n
, the
continuously compounded total annual return n periods before the current period t. The value in parentheses is the
standard deviation of the coefficient estimate.
*** t-statistics for all coefficient estimates are significant at the 0.01 level.
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form H1. That is, the data generally do not
support the belief adjustment model’s prediction
that decision-makers (investors) underreact
when the sign of UE is not consistent with the
sign of prior period returns. See Table 5. The
exceptions are cases where UE
j,t
/P
j,t-1
> 0 is
extremely large. However, such cases are
unusual. For example, if the prior period share
price is $20 when UE
j,t
/P
j,t-1
= 0.35, then UE
j,t
=
$7 per share. If the belief adjustment model
only “works” in extreme cases, then it is not a
good model for behavior of decision-makers.
The belief adjustment model predicts that
decision-makers (investors) become committed
to their beliefs the longer the sequence of same
sign, prior period returns. That is, it predicts
that, when UE is positive, both R
j,t|+,n+
and
R
j,t|+,n-
should decrease as n increases. The
continuous model does not support this
conclusion in all cases. When UE
j,t
/P
j,t-1
> 0 has
relatively large values (e.g., 0.10 and greater,
which is well above the third quartile in our
sample), we observe cases where R
j,t|+,n+
does
not decrease as n increases, as predicted (top
half of Panel A in Table 6). Also, for most cases
where UE
j,t
/P
j,t-1
> 0 but is not extremely large
(e.g., the third quartile, 0.025, in our sample or
less), R
j,t|+,n-
increases as n increases, not
decreases as predicted (bottom half of Panel A
in Table 6). Thus, the data does not consistently
support hypothesis H2(a).
Nor does the data consistently support
hypothesis H2(b). When UE
j,t
/P
j,t-1
< 0, the
magnitude of the sequence R
j,t|-,n-
< 0 decreases
as n increases, as predicted by hypothesis
H2(b)(ii). On the other hand, the magnitude of
the sequence R
j,t|-,n+
< 0 increases as n increases,
thus violating hypothesis H2(b)(i). In summary,
the market does not behave in all situations as if
investors become committed to their beliefs the
longer the sequence of same sign, prior period
returns. In fact, it appears more likely that
investors regard UE of opposite sign from prior
period returns as significant new information
about a change in direction of company value.
Table 5
Tests of H1: Current Period Returns in the Continuous Model
Panel A. Positive UE.
Current period returns r
t
No. of
same-sign
prior
returns
Scaled
unexpected
earnings
r
t-1
= .1, r
t-2
= 0, r
t-3
= 0 r
t-1
= -.1, r
t-2
= 0, r
t-3
= 0
H1(a): hypothesized
direction of
difference is >.
Actual difference:
0.006 (Q1) -0.1480 -0.0637 <
0.011 (Q2) -0.1356 -0.0525 <
0.025 (Q3) -0.1006 -0.0214 <
0.200 0.3360 0.3680 <
1
0.350 0.7103 0.7018 >
Current period returns r
t
r
t-1
= .1, r
t-2
= .1, r
t-3
= 0 r
t-1
= -.1, r
t-2
= -.1, r
t-3
= 0
0.006 (Q1) -0.1702 -0.0415 <
0.011 (Q2) -0.1570 -0.0311 <
0.025 (Q3) -0.1202 -0.0018 <
0.200 0.3408 0.3632 <
2
0.350 0.7359 0.6761 >
Current period returns r
t
r
t-1
= .1, r
t-2
= .1, r
t-3
= .1 r
t-1
= -.1, r
t-2
= -.1, r
t-3
= -.1
0.006 (Q1) -0.1801 -0.0316 <
0.011 (Q2) -0.1660 -0.0221 <
0.025 (Q3) -0.1265 0.0045 <
0.150 0.2260 0.2420 <
3
0.200 0.3670 0.3370 >
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Panel B. Negative UE.
Current period returns r
t
No. of
same-sign
prior
returns
Scaled
unexpected
earnings
r
t-1
= .1, r
t-2
= 0, r
t-3
= 0 r
t-1
= -.1, r
t-2
= 0, r
t-3
= 0
H1(b): hypothesized
direction of difference
is >. Actual
difference:
-1.160 (min) -3.0572 -2.6580 <
-0.035 (Q1) -0.2503 -0.1549 <
-0.016 (Q2) -0.2029 -0.1126 <
-0.007 (Q3) -0.1805 -0.0926 <
1
-0.001 -0.1655 -0.0792 <
Current period returns r
t
r
t-1
= .1, r
t-2
= .1, r
t-3
= 0 r
t-1
= -.1, r
t-2
= -.1, r
t-3
= 0
-1.160 (min) -3.2414 -2.4738 <
-0.035 (Q1) -0.2782 -0.1270 <
-0.016 (Q2) -0.2281 -0.0874 <
-0.007 (Q3) -0.2044 -0.0686 <
2
-0.001 -0.1886 -0.0561 <
Current period returns r
t
r
t-1
= .1, r
t-2
= .1, r
t-3
= .1 r
t-1
= -.1, r
t-2
= -.1, r
t-3
= -.1
-1.160 (min) -3.4682 -2.2470 <
-0.035 (Q1) -0.2957 -0.1095 <
-0.016 (Q2) -0.2421 -0.0734 <
-0.007 (Q3) -0.2167 -0.0563 <
3
-0.001 -0.1998 -0.0449 <
Notes. Sample is drawn from S&P 1500 firms for FY1998–2000 (fiscal years ending 6/98–5/01). We use firms with
fiscal year ends from all calendar months. UE
t
= current period unexpected annual earnings; earnings are forecast as a
random walk. Unexpected earnings are scaled by P
t-1
, the share price at the end of period t-1. Returns are r
t-n
, the
continuously compounded total annual return n periods before the current period t, n = 0, 1, 2, 3. The current period
return is calculated using the continuous model #4 in Table 4. Q1, Q2, & Q3 are lower quartile, median, & upper
quartile, respectively, of UE
j,t
/P
j,t-1
, conditional on UE > 0 (Panel A) or UE < 0 (Panel B).
Indirect Tests of the Hogarth and Einhorn
Two-equation Model
We indirectly test the earnings-returns version
of the Hogarth and Einhorn (1992) two-
equation model by estimating the sensitivity
coefficients, ? and ?. In the Hogarth and
Einhorn model, these coefficients are bounded:
0 ? ? ? 1 and 0 ? ? ? 1.
We report results based on using model #2 in
Table 4 to calculate fitted values of the current
period returns. Estimates of ? and ? generally fall
well outside the assumed bounds and differ
substantially depending on the value of the prior
period return and current unexpected earnings.
For example, using equation (9a) to calculate the
estimate of ?, when UE
t
/P
t-1
= -0.035 (the first
quartile conditional on UE < 0), alpha is 101.8 if
R
t-1
= 0.10 and –19.6 if R
t-1
= -0.10. When
UE
t
/P
t-1
= -0.007 (the third quartile conditional on
UE < 0), then alpha is 420.7 if R
t-1
= 0.10 and
-20.7 if R
t–1
= -0.10. Using equation (9b) to
calculate the estimate of ?, when UE
t
/P
t-1
= 0.006
(the first quartile conditional on UE > 0), beta is
-49.2 if R
t-1
= 0.10 and 1.6 if R
t-1
= -0.10. When
UE
t
/P
t-1
= 0.025 (the third quartile conditional on
UE > 0), then beta is -9.9 if R
t-1
= 0.10 and 1.7 if
R
t–1
= -0.10. Estimates of ? and ? using equation
(11) generally are well above one and also vary
substantially for different values of prior period
returns and current unexpected earnings.
These results strongly suggest that the
earnings-returns version of the Hogarth and
Einhorn (1992) two-equation model is
misspecified. It may be that assuming ? and ?
are the same for all firms is too strong an
assumption. However, while this might explain
why estimates of ? and ? vary across prior
period returns and current unexpected earnings,
it does not explain why the estimates generally
are so far from the range [0, 1] specified by
Hogarth and Einhorn.
D
o
w
n
l
o
a
d
e
d
b
y
P
O
N
D
I
C
H
E
R
R
Y
U
N
I
V
E
R
S
I
T
Y
A
t
2
1
:
0
1
2
4
J
a
n
u
a
r
y
2
0
1
6
(
P
T
)
ACCOUNTING RESEARCH JOURNAL VOLUME 20 NO 1 (2007)
18
Table 6
Tests of H2: Current Period Returns in the Continuous Model
Panel A. Positive UE.
Current period returns r
t
Scaled
unexpected
earnings
r
t-1
= .1, r
t-2
= -.1,
r
t-3
= -.1
r
t-1
= .1, r
t-2
= .1,
r
t-3
= -.1
r
t-1
= .1, r
t-2
= .1,
r
t-3
= .1
H2(a)(i): hypothesized
direction of differences
is >, >. Actual:
0.006 (Q1) -0.1160 -0.1603 -0.1801 >, >
0.011 (Q2) -0.1051 -0.1481 -0.1660 >, >
0.025 (Q3) -0.0748 -0.1138 -0.1265 >, >
0.100 0.0880 0.0698 0.0850 >, <
0.200 0.3050 0.3146 0.3670 , >. Actual:
0.006 (Q1) -0.0957 -0.0514 -0.0316 , >
-0.016 (Q2) -0.1637 -0.2142 -0.2421 >, >
-0.007 (Q3) -0.1442 -0.1921 -0.2167 >, >
-0.001 -0.1312 -0.1774 -0.1998 >, >
Current period returns r
t
Scaled
unexpected
earnings
r
t-1
= -.1, r
t-2
= .1,
r
t-3
= .1
r
t-1
= -.1, r
t-2
= -.1,
r
t-3
= .1
r
t-1
= -.1, r
t-2
= -.1,
r
t-3
= -.1
H2(b)(ii): hypothesized
direction of differences
is