ACCOUNTING WORKSHOP

“The Remarkable Multidimensionality in the Cross-Section of Expected U.S. Stock Returns”

By

Jeremiah Green Penn State University

John R. M. Hand* UNC Chapel Hill

X. Frank Zhang Yale University

Thursday, October 24th, 2013 1:20 – 2:50 p.m.

Room C06 *Speaker Paper Available in Room 447

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The Remarkable Multidimensionality in the

Cross-Section of Expected U.S. Stock Returns

Jeremiah Green John R. M. Hand X. Frank Zhang

Penn State University UNC Chapel Hill Yale University

jrg28@psu.edu hand@unc.edu frank.zhang@yale.edu

Abstract

20+ years after Fama & French (1992), we re-measure the dimensionality of the cross-section of

expected U.S. stock returns using 93 academically documented firm-specific return predictive signals

(RPS). We find that 26 RPS are reliably multidimensionally priced as defined by their mean

coefficient estimates having an absolute t-statistic 3.0 in Fama-MacBeth regressions where all 93

RPS are simultaneously projected onto 1-month ahead returns during 1980-2012. Such remarkable

dimensionality in part reflects the small mean absolute cross-correlation amongst RPS of just 0.03.

We also observe that the multidimensional pricing of several famous RPS is strikingly different to

their unidimensional pricing, and that the hedge returns earned by multidimensioned RPS are on

average two-thirds smaller than those of unidimensioned RPS. We conjecture that RPS that are

either priced in non-near-term returns or persistently priced in near-term returns are more likely to

capture the rational pricing of economic risks, while RPS that are inconsistently priced in near-term

returns more likely reflect mispricing. Based on this view, and employing an expanded set of

multidimensional regressions that project our 93 RPS onto 2-12 months ahead and 13-36 months

ahead returns, we estimate that up to 19 RPS likely reflect the rational pricing of economic risks,

while an equal number likely reflect mispricing.

This version: Oct.21, 2013

Our paper has benefitted from helpful comments received from Jeff Abarbanell, Sanjeev Bhojraj,

Joshua Coyne, John Gallemore, Oleg Grudin, Ed Maydew, Jacob Sagi, Doug Shackelford, Eric Yeung,

and participants at Cornell University, UNC Chapel Hill, and the Fall 2013 Chicago Quantitative

Alliance conference. All errors are entirely our responsibility.

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1. Introduction

In a seminal and highly influential study, Fama and French (1992, hereafter FF92) measured

the dimensionality of the cross-section of expected monthly U.S. stock returns over the period 1963-

1990. After jointly evaluating the roles of beta, firm size, book-to-market, earnings-to-price and

leverage, they determined that while beta was not priced in expected returns, firm size and book-to-

market were, and in a manner that absorbed the unidimensional explanatory power of earnings-to-

price and leverage. FF92 concluded that the cross-section of monthly U.S. stock returns was two-

dimensional, though neither dimension was consistent with the CAPM. A third dimension in the

form of Jegadeesh (1990) and Jegadeesh and Titman‘s (1993) 12-month return momentum was

added in Fama and French (1996) and Carhart (1997) to create what has for two decades been seen in

academia and investment practice as the dominant and default set of priced firm-specific risks in

equities (Royal Swedish Academy of Sciences, 2013, p.43).

The chief goal of our paper is to re-measure the dimensionality of the cross-section of

expected U.S. stock returns in light of the 300+ firm-level return predictive signals (RPS) that have

been identified by business academics since 1990 (Green, Hand and Zhang, 2013; Harvey, Liu and

Zhu, 2013). Over the period 1980-2012, we find that the dimensionality of monthly U.S. stock

returns is at least an order of magnitude larger than that originally estimated by FF92. In our re-

estimation, we find that 26 RPS derived only from CRSP, Compustat and I/B/E/S data are reliably

multidimensionally priced as defined by their mean coefficient estimates having an absolute t-

statistic of 3.0 in Fama-MacBeth regressions where the monthly recalculated scaled decile ranks of

93 documented RPS are simultaneously projected onto 1-month ahead returns.1,2 In updating the

empirical measured dimensionality of the cross-section of expected monthly returns, our study makes

a start on executing both Cochrane‘s 2010 AFA Presidential Address in which he issues a

‗multidimensional challenge‘ and calls for Fama and French‘s ‗anomaly digestion exercise‘ to be

repeated, and Goyal‘s (2012) call for researchers to ‗synthesize the huge amount of collected [RPS]

evidence.‘ Via our regressions, we do so and provide initial answers to two questions posed by

Cochrane, namely: ―Which characteristics really provide independent information about mean

returns?‖ and ―Which characteristics are subsumed by other RPS?‖ (Cochrane 2011, p.1060).

A second goal of our study is to compare and contrast important economic aspects between

unidimensionally and multidimensionally priced RPS. Among several results, we find that [1] the

multidimensional pricing of several heavily cited RPS is quite different to their unidimensional

pricing; [2] none of firm size, book-to-market, and 12-month momentum have t-statistics large

enough to put them in the top 10 largest t-statistics in 1-month ahead returns multidimensional

regressions; [3] the hedge returns earned by multidimensioned RPS are two thirds smaller than those

earned by unidimensioned RPS, while the t-statistics on the hedge returns of multidimensioned RPS

are only between one third and two-thirds the magnitude of those on the hedge returns of

unidimensioned RPS; [4] high dimensionality is present in each decade of our data period; and [5]

large firms have far fewer multidimensionally priced RPS than do small firms, but their RPS explain

three times as much cross-sectional return variance.

1 For simplicity, throughout the paper we use the present tense when describing the dimensionality of returns,

recognizing that our analysis is in-sample and historical, and so may not necessarily describe the future

dimensionality of returns. We also use the term ―priced‖ recognizing that RPS may have reliably in-sample

predictive correlations with returns because they reflect firm-specific exposures to common risk factors. 2 We use 3.0 as the absolute t-statistic cutoff for inferring statistical significance based on the insights of Harvey,

Liu, and Zhu (HLZ, 2013). HLZ seek to answer the related but different question of whether the documented hedge

returns to RPS are real or whether they are statistical artifacts stemming from multiple testing of the same

underlying data, and conclude that an absolute t-statistic of 3.0 offers sufficient protection against data-snooping.

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The final goal of our paper is to explore implications of our results for academic practice, and

for the long-running question of whether RPS reflect the rational pricing of economic risks, or

mispricing, or a mixture of the two. With regard to academic practice, we do not suggest that prior

research has been fatally flawed because it did not control for all 26 RPS that we estimate were

multidimensionally priced over the past 30 years. Nor do we argue that future research will be

fatally flawed if it ignores the apparently very high multidimensionality in returns. Instead, we offer

four opinions that we expand on later in the text. First, the replicable but at times crude choices we

make in combining RPS across companies, time periods and databases, combined with our approach

to dealing with missing data, lead us to suspect that the true dimensionality in returns is larger than

we infer it to be. Second, future returns-based research that controls only for firm size, book-to-

market and 12-month momentum is likely to arrive at unnecessarily low-powered inferences, and at

times those inferences will be predictably biased. Third, the NPV of academic investment into

understanding the causes and consequences of multidimensionality is almost surely larger than the

NPV of academic investment into discovering yet more RPS. Finally, the very large degree to which

empirically-based inferences about the dimensionality of returns appear to have changed over two

decades suggests that future inferences about the dimensionality of returns may warrant more caution

in their production and consumption by academics and practitioners—including those we arrive at in

our study.

On the long-running question of market efficiency, we cannot confidently identify that any

particular RPS reflects the rational pricing of economic risks over mispricing, or vice-versa.

However, we propose that our results allow us to say which RPS may be more versus less likely to

reflect which kind of pricing. Specifically, we propose that RPS that are either priced in non-near-

term returns or persistently priced in near-term returns are more likely to capture the rational pricing

of economic risks, while RPS that are inconsistently priced in near-term returns are more likely to

reflect mispricing. We reason that the cumulative forces of information dissemination and arbitrage

make it unlikely that RPS that are multidimensionally priced in longer window returns that are

distant from the RPS measurement date reflect mispricing, leaving such RPS to likely reflect the

rational pricing of economic risks. Conversely, however, RPS that are multidimensionally priced in

short window returns that begin immediately after the RPS measurement date may reflect either

mispricing or the rational pricing of short-term economic risks or market frictions such as

transactions costs or market microstructure. We propose distinguishing between rational pricing and

mispricing in multidimensioned near-term short horizon future returns through the consistency of

pricing—thus, if multidimensional pricing is seen persistently over calendar time, then we propose

that such pricing likely captures rational risk pricing, leaving non-persistent multidimensional pricing

as likely reflecting mispricing.

We adopt this view and seek to identify which of our 93 RPS more likely reflect rational

pricing versus mispricing by estimating an expanded set of multidimensional regressions in which we

project the 93 RPS onto 2-12 months ahead and 13-36 months ahead returns. Based on our

hypothesis, we propose that there are up to 19 RPS out of our set of 93 RPS that reflect the rational

pricing of economic risks. We cautiously identify these as eight RPS that are multidimensionally

priced in 13-36 month ahead returns that reflect rational pricing, eight RPS that are

multidimensionally priced in only 2-12 month ahead returns or in both 1-month and 2-12 months

ahead returns, and three RPS that are consistently multidimensionally priced in 1-month ahead

returns in each of the 1980s, 1990s and 2000s. Insofar as mispricing is concerned, we cautiously

identify up to 19 different RPS in this category, based on their either being multidimensionally priced

based on the totality of the 1980-2012 window, or not being multidimensionally priced in each of the

1980s, 1990s and 2000s.

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The remainder of our paper proceeds as follows. In section 2 we review the prior literature

on dimensioning expected returns. In section 3 we describe the sample of RPS we employ and the

choices we make during the process of selecting, aligning and coding them. In section 4 we report

the results of a variety of analyses aimed at describing key aspects of multidimensionally priced RPS.

In section 5 we put forward and implement a perspective on using our results to try to differentiate

between RPS that likely reflect the rational pricing of economic risks versus those that likely reflect

mispricing. We emphasize certain limitations of our study in section 6, and conclude with

suggestions for future research in section 7.

2. Prior Literature on Dimensioning the Cross-Section of Expected Stock Returns

Since FF92, Jegadeesh and Titman (1993) and Fama and French (1996) established the

prevailing convention that firm size, book-to-market and 12-month momentum dimension the cross-

section of expected U.S stock returns, relatively few papers have directly revisited the dimensionality

of returns. This contrasts with the steady development of a vast literature that has identified

hundreds of firm-specific characteristics or RPS that predict the cross-section of future stock returns

in the sense that the signal is incrementally priced beyond one or more of firm size, book-to-market

and 12-month momentum. For example, Subrahmanyam (2010) identifies 50 RPS, McLean and

Pontiff (2013) identify 82 RPS, Harvey, Liu and Zhu (2013) identify 311 RPS and/or factors, and

Green, Hand and Zhang (2013) identify 330 RPS.

Papers that have studied the dimensionality of returns have either proposed a small

competing set of priced RPS to replace firm size, book-to-market and 12-month momentum, or have

put forward a larger set of priced RPS beyond firm size, book-to-market and 12-month momentum.

Hou, Xue and Zhang (2012) and Light, Maslov and Rytchkov (2013) exemplify the former approach,

while Haugen and Baker (1996), Fama and French (2008) and Lewellen (2013) illustrate the latter

method.

Motivated by q-theory, Hou, Xue and Zhang (2012) argue that a model consisting of the

excess market return, a small-minus-big firm size factor, a high-minus-low investment factor and a

high-minus-low return on equity factor performs similarly to firm size, book-to-market and 12-month

momentum but also captures many patterns that are anomalous to firm size, book-to-market and 12-

month momentum. As such, Hou, Xue and Zhang propose that their four factor model is ―a new

incarnation of Fama and French (1996)‖ (p.4) in that it is an alternative factor-based model for

estimating the cross-section of expected stock returns. Indeed, Hou, Xue and Zhang even go as far as

to propose that any new anomaly variable should be benchmarked against their q-factor model to see

if the variable provides any incremental information (p.35). Treating expected returns as latent

variables, Light, Maslov and Rytchkov (2013) develop a procedure that estimates them using 13 firm

characteristics as signals, and construct two new characteristics, one of which combines information

from all anomalies.

Opposite to Hou, Xue and Zhang‘s focus on a small-in-number competitor set of RPS,

Haugen and Baker (1996) and Lewellen (2013) examine whether a larger set of RPS than firm size,

book-to-market and 12-month momentum are multidimensionally priced. Thus Haugen and Baker

report that out of 40 interrelated RPS they choose in an ad hoc manner that is only partially based on

prior published research, a total of 11 RPS are reliably multidimensionally priced, while Fama and

French (2008) and Lewellen (2013) show that from a more rigorously prescribed set of RPS taken

from prior academic research, six out of seven RPS, and nine out 15 RPS, respectively, are reliably

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multidimensionally priced using the Harvey, Liu and Zhu (2013) absolute t-statistic 3.0 cutoff that

we adopt in our paper.3

These studies notwithstanding, the thesis of our paper is that prior research has not yet

sufficiently responded to the multidimensional challenges issued by Cochrane (2011) and Goyal

(2012). Prior dimensional research has studied only a fraction of the 300+ RPS that have been

documented by academics. It is the existence of a ―veritable zoo‖ of RPS—particularly those that

have not been highly cited yet in their originating papers exhibit mean hedge returns and Sharpe

ratios that are far larger than those of highly cited RPS—that leads us to propose that FF92‘s original

very useful data reduction now warrants repeating, and executing this task is the focus of our paper.

3. Data and Methodology

3.1 RPS dataset of integrated CRSP, Compustat and I/B/E/S data aligned in calendar time

Since the goal of our paper is to simultaneously project a large number of RPS onto future

returns, we face choices about how many RPS to include, how to combine RPS across companies,

time periods and databases, and how to address missing data. We therefore document the choices we

made during the process of selecting, aligning and coding our RPS. Some choices unavoidably

distance us from the exact research design used in the original papers, the exact definitions of RPS,

and the exact sample periods used in the originating papers. We expect this will make it less likely

that we will observe multidimensional statistical significance for some RPS and therefore make it

more likely that we will inferentially underestimate the true degree of multidimensionality in U.S.

stock returns.

The most powerful way to measure the degree of dimensionality in expected stock returns

would be to use the entire population of known RPS. We judged this to be infeasible, since Green,

Hand and Zhang (2013) catalogue 330 different firm-specific signals spread across many different

data sources in their approximation of the population of RPS publicly identified by accounting,

finance and other business faculty as of December 2010. In Figure 1 we highlight both the dramatic

growth in, and the total number of, RPS that were publicly documented between 1970-2010 (panel

A), as well as the breakdown of these 330 RPS into those that are accounting-based, finance-based,

and other-based signals (panel B).

To balance the benefits of analyzing the largest number of RPS with the costs of gathering,

programming and analyzing the relevant data, and also seeking to not erect barriers to the

replicability of our work, we selected 100 RPS primarily from the Green, Hand and Zhang database,

requiring only that each RPS be based entirely on CRSP, Compustat and I/B/E/S data items.4 Our

dataset spans the period Jan. 1980 - Dec. 2012. We begin in 1980 because 1980 represents a point at

which most of the RPS data items are robustly available. We end in 2012 because 2012 was the date

of the most recently available data as of the writing of the initial draft of our paper.

The full set of our 100 RPS are reported in Table 1, listed in the order in which the RPS were

first published, or where not yet published, appeared as a working paper. We also provide the

3 Many of the 40 RPS used by Haugen and Baker (1996) are highly correlated variants of few constructs, with the

likely result that the analysis in Haugen and Baker is based on fewer than 40 independent RPS. Fama and French

(2008) orient their analysis around the question of whether RPS pricing is robust across firm size. Lewellen (2013)

focuses his study on the cross-sectional dispersion and out-of-sample predictive ability of the stock return forecasts

that he extracts from the particular set of 15 RPS that he employs. 4 We also restricted the RPS to main effect signals. We do not include RPS that are interactions between other RPS.

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acronyms we use, and the authors, journal and year of publication. Inspection of Table 1 shows that

the RPS we select span both famous studies and sparsely cited papers, published and working papers,

and publication dates that spread out between 1977-2013. On some occasions we identify several

RPS from one paper.

Table 2 defines from a programming point of view each RPS implemented in our study.

Monthly stock returns are collected for the month following the month end at which the RPS data is

available. Missing Compustat and I/B/E/S data are the main reason that few RPS can be computed

for every firm at every point in calendar time. However, deleting observations with missing

Compustat and I/B/E/S data would greatly reduce both the number of observations and/or firms

included in our analysis and the representativeness of our results. To avoid this, Table 3 details our

data retention strategy, relative to our baseline of starting with all firms with common stock traded on

the NYSE, AMEX or NASDAQ exchanges. We proceed as follows:

First, following FF92 we exclude approximately 46 observations per month where market

cap or book value of equity was unavailable (panel A). Second, we deleted a few observations due

them having implausibly extreme or impossible monthly returns (panel B).5 Third, we reset 19

Compustat missing data items such as R&D expense, intangible assets and total inventory to zero if

they are reported missing by Compustat (panel C).6 We also reset one I/B/E/S data item with

missing values to zero, namely analyst following nanalyst. I/B/E/S is the most restrictive of our

databases in terms of its coverage of companies and the sample time period available, so we only use

I/B/E/S-based RPS starting in January 1989 when more expansive coverage begins.7 We note that

while resetting a large number of missing observations across several data items to zero preserves the

large number of RPS values, it also likely reduces the quality of the RPS in that it injects into their

measurement largely uninformative zero values. This will make it more likely that our statistical

analysis will yield underestimates of the dimensionality of returns.

Fourth, we integrate the missing-value-adjusted data across Compustat, I/B/E/S and CRSP

databases, and proceed to compute and align RPS in calendar time. Since Green, Hand and Zhang

(2013) report that 57% of the 330 RPS in their database study the RPS through the lens of monthly

returns, we re-measure and align RPS each month.8 While monthly updating is consistent with the

portfolio rebalancing approach used by many quantitative institutional investors, practitioners may

update the data aspects of their RPS as often as every minute or as infrequently as every 12 months.

We expect monthly updating to adequately tradeoff the lower transactions and trading costs at longer

frequencies with the greater timeliness from RPS that are updated at shorter frequencies.

Necessarily, though, our monthly RPS construction means that the RPS in our dataset that come from

studies that employ a shorter-than-monthly frequency will use signals that are less timely than in

5 We include delisting returns following Shumway and Warther (1999).

6 In doing so, we follow what is commonly done by quantitative practitioners. For balance sheet variables that are

missing in the quarterly Compustat database at the quarterly frequency but are available on an annual basis, we set

the quarterly values to the most recent annual values. 7 With the number in parenthesis being that shown in Tables 1 and 2, the RPS that use I/B/E/S are sue (#6), chfeps

(7), fgr5yr (#9), sfe (#49), nanalyst (#50), disp (#51) and chnanalyst (#75). 8 In aligning RPS in calendar month time we use the following conventions. At the end of each calendar month, the

most recent annual financial statement information is assumed to be available if the fiscal year ended at least 5

months prior to the month end. Quarterly financial statement information collected from Compustat is assumed to be

available with at least a 60-day lag, and I/B/E/S and CRSP information are aligned in calendar time using the

I/B/E/S statistical period date and the CRSP monthly or daily end date. We obtain similar results assuming a 90-day

lag. Our adopting a 60 day lag is intended to balance making sure financial statement data is available to the stock

market, and preventing certain RPS such as the presence of an earnings announcement from getting too stale.

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prior studies, while those RPS that come from studies using longer frequencies will use more timely

information. This slippage further degrades our ability to detect multidimensionally priced RPS.

Finally, once calculated using the data in place at the end of the four steps just described, we

further reset all missing values of RPS to the trimmed mean of the non-missing RPS values for that

calendar month.9 We do so to retain as many firm-month RPS observations as possible. Unlike our

resetting to zero of certain Compustat and I/B/E/S data items, the resetting of all missing RPS values

in a given month to the mean value of the RPS in that same month should not materially affect the

estimated slope coefficients in our regressions. It will, however, increase the standard errors on the

estimated coefficients.10 In panel D we report the number of firm-month observations in our full

dataset of 1,987,340 firm-month observations spanning Jan. 1980 - Dec. 2012 before resetting

missing RPS values to each RPS‘ monthly mean, and associated the percentage of firm-month

observations in which we then reset missing RPS values to each RPS‘ monthly mean. The mean

percentage of firm-month observations where we reset missing RPS values to each RPS‘ monthly

mean is 10%.11

3.2 Construction and limitations of scaled decile ranked RPS

We seek to dampen the potential impacts of economically genuine and data-error-based

outliers by using monthly cross-sectional scaled decile rankings of each continuous or non-indicator

RPS in our return prediction regressions. We implement the scaled decile ranked approach at the end

of every calendar month by ranking each non-indicator RPS into deciles where zero is the lowest

decile and nine is the highest decile, and then dividing by nine. However, the scaled decile ranked

RPS are only created after resetting missing RPS values to each RPS‘ monthly mean.

The scaled decile ranking approach directly follows work by Fama (1976) and Abarbanell

and Bushee (1998). In addition to minimizing the effects of outliers on our regression parameter

estimates, the method yields coefficient estimates that have a ready and powerful economic

interpretation. Specifically, Fama and Abarbanell and Bushee show that the coefficients estimated

from cross-sectional regressions of returns on RPS scaled to lie between 0 and 1 will be the returns to

optimal pre-transactions costs dollar-neutral in-sample RPS hedge portfolios that are orthogonal to

all the other RPS included in the regressions.

A key strength of scaled decile ranked RPS is that the entire distribution of RPS is used,

rather than only the top and bottom RPS deciles as has often been the case in prior RPS research. We

recognize that this benefit comes with two potential costs, however. First, the method imposes a

linear structure on ranked RPS, thereby ignoring non-linearities that may exist in the relations

between RPS and future returns. Second, to the extent that non-linearities exist, as with other data

and design choices we have made, we likely dilute our ability to detect truly multidimensional

relations conditional on the other statistical decision criteria we impose (such as our only accepting

absolute t-statistics 3.0 as indicative of statistical significance).

9 The trimmed mean is the mean calculated after extremes that are more than 3X the interquartile range (IQR) below

the first quartile Q1 or above the third quartile Q3 are reset to Q1 – [3 x IQR] or Q3 + [3 x IQR], respectively. 10

In unreported findings, our inferences about the general scale of multidimensionality in stock market returns, as

well as the inferences pertaining to most specific RPS, are unchanged if we use non-scaled decile ranked RPS,

standardized or normalized RPS, and winsorized RPS. 11

We recognize that there are many more sophisticated methods that could be used to infill missing observations

(e.g., modeling missing observations as a function of firm characteristics). We adopt a simple approach in order to

increase the replicability of our findings and decrease the likelihood of creating data snooping biases.

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3.3 Measuring and adjusting for high collinearity amongst a small subset of RPS

Lastly in terms of constructing our RPS dataset, we measure the degree of collinearity among

our 100 RPS and explain how we address the high collinearity that we find exists within a small

subset of signals. We do this because while high cross-correlations between independent variables in

OLS regressions do not per se create bias in the estimated slope coefficients, they greatly increase the

associated standard errors. To the extent that we can identify RPS that have large cross-correlations

because they are either definitionally or economically very closely related—for example, beta and

beta squared—we choose to include fewer RPS in exchange for more precise standard errors on the

RPS that are included.

Panel A of Table 4 reports the distribution of the variance inflation factors (VIFs) from an

initial pooled time-series cross-sectional regression of all 100 RPS onto 1-month ahead returns.

Although the mean VIF of 3.4 is smaller than the level of 5.0 that econometricians often see as

indicating excessive collinearity, 14 individual VIFs exceed 5.0. We therefore examine the RPS

associated with these 14 extreme VIFs in order to reduce the number of RPS to where the maximum

VIF is below 5.0. We find that after removing the seven RPS shown on the right hand side of panel

A, the VIFs of the remaining 93 RPS in a new pooled time-series cross-sectional regression of just

the 93 RPS onto 1-month ahead stock returns are all 4.0 or less.

In panel B we report key percentiles in the distribution of the absolute values of the 4,200+

cross-correlations between the scaled decile ranked RPS, as estimated from pooled time-series cross-

sectional data. We report these percentiles for two reasons. First, a comparison of the 99th percentile

and maximum absolute cross-correlations across the pre-VIF-outlier-trimmed set of 100 RPS and the

post-VIF-outlier-trimmed set of 93 RPS shows that eliminating seven highly cross-correlated RPS

based on their VIFs eliminates the most extreme RPS cross-correlations. Second, we highlight that

both the mean absolute cross-correlation of 0.03 and the 99th percentile absolute cross-correlation of

0.27 in the VIF-outlier-trimmed set of 93 RPS are surprisingly small. This suggests that a large

fraction of unidimensionally priced RPS will be multidimensionally priced, which is what we indeed

find to be the case.

In panel C we provide a different but related histogram of absolute cross-correlations among

our scaled decile ranked RPS. In each month Jan. 1980 - Dec. 2012 we compute the mean absolute

cross-correlation between all 100 of the initial set of RPS, yielding a vector of 396 mean absolute

cross-correlations. Panel C visually shows that the surprisingly small mean absolute cross-

correlation of 0.03 that we calculate and report in panel B using pooled data varies little over time,

and has a maximum value of just 0.047. Untabulated results show that the mean absolute cross-

correlation amongst RPS is similarly small if cross-correlations are calculated before missing RPS

values are reset to each RPS‘ monthly mean.

3.4 Fama-MacBeth cross-sectional regressions of RPS on future returns

We estimate Fama and MacBeth (1973) regressions over the period 1980-2012 to determine

how many and which RPS are priced and not priced when they are unidimensionally and

multidimensionally projected onto future stock returns. We calculate the mean estimated slope

coefficients and their associated t-statistics from the time-series of cross-sectional regressions. When

calculating t-statistics we employ Newey-West adjustments for 12 lags for 1-month ahead returns,

and one lag for 2-12 and 13-36 months ahead returns.

We measure the dimensionality of returns beyond only the 1-month ahead horizon for two

reasons. First, while monthly returns are the most popular horizon used in finance research, there is

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nothing special per se about a 1-month horizon. We thus see value in more broadly describing the

dimensionality of returns over longer and more distant (or shorter intra-month) horizons. Second, in

section 5 we use the differences in the degree of and overlap between the dimensionality of 1, 2-12

and 13-36 months ahead returns to approximately discriminate between those RPS that capture or are

proxies for the rational pricing of fundamental firm-specific economic risks, and those RPS that are

or are proxies for mispricing. With this later goal in mind, we choose future return horizons that get

longer the further away from the monthly RPS construction date they begin. We define future return

windows in non-overlapping calendar time, so that 2-12 months ahead returns cumulate from Aug. of

year t thru June of year t+1, while 13-36 months ahead returns cumulate in two-year periods starting

Jan. 1981 - Dec. 1982.

As highlighted by Fama (1976) and Abarbanell and Bushee (1998), using scaled decile

ranked independent variables yields mean RPS coefficients that are the hedge return to a zero-

investment portfolio that optimally exploits the information in the RPS and that are orthogonal to the

information in every other RPS included in the regression.12 We denote the annualized value of these

hedge returns as the MALSRets (= mean annualized long/short hedge returns) of the RPS. We do

not view MALSRets as implementable by long/short practitioners over our data window 1980-2012

for many reasons. Even setting aside transactions costs, realizing the in-sample MALSRets we

document would have required knowing about each and every RPS in real time, knowing the 1980-

2012 multidimensional relations between every RPS and expected returns both before the RPS were

discovered and as of 1980, and having the necessary computer power, real-time data feeds, and

specialized human capital. We do, however, argue that the MALSRet on a given RPS provides an

estimate of the economic significance of the RPS in the cross-section of expected returns.

4. Main RPS Dimensioning Results

4.1 Estimating the conventional 3-dimensional pricing of firm size, book-to-market, and 12-

month momentum in our 1980-2012 data period

We begin remeasuring of the dimensionality of the cross-section of U.S. stock returns by

estimating the empirical pricing associated with the dominant and default view that firm size mve,

book-to-market bm, and 12-month momentum mom12m robustly describe the cross-section of

expected stock U.S. stock returns. Using the data window 1980-2012, which overlaps only partially

with the 1963-1990 window used by FF92, we estimate Fama-MacBeth regressions in which the

dependent variables are 1-month ahead, 2-12 months ahead and 13-36 months ahead returns, and the

independent variables are the scaled decile ranks of mve, bm and mom12m.13 In doing so, our

objectives are limited to [1] determining whether the signs and statistical significance of the relations

between 1-month ahead returns and mve, bm and mom12m during 1980-2012 do or do not parallel

those documented in the original FF92, Jegadeesh (1990) and Jegadeesh and Titman (1993) papers,

and [2] documenting the extent to which mve, bm and mom12m explain the cross-section of 2-12

months ahead and 13-36 months ahead returns.

12

Technically, the zero investment portfolio interpretation relies on the regression including an intercept, which is

always the case in our analyses. Inclusion of an intercept ensures that the weighted average of the focal RPS,

measured in scaled decile ranks, is one, and that the weighted averages of the other RPS included in the regression

are zero. The return on the zero-investment portfolio is of course only optimal if the assumption that the RPS is

linearly related to the scaled decile ranks is correct. 13

Following Fama and French (1996) and others, we define 12-month momentum as the cumulative returns

calculated over the 11 months consisting of the prior 12 months except for the immediately preceding month.

10

Table 5 reports the results from these regressions using all firms combined, and also per

Fama and French (1996) separately for large-cap, mid-cap, and small-cap firms. The predicted signs

in Table 5 are those that were empirically observed by FF92 and Jegadeesh and Titman. Large-cap

firms are defined as the largest 1,000 companies by market cap; mid-cap are the next largest 2,000;

and small-cap are all remaining companies.14 Our mean monthly number of between 5,018 and

5,140 firms is more than twice the 2,267 figure reported in FF92 because the number of firms in the

CRSP and Compustat databases has grown substantially since the 1990 endpoint of the data window

used by FF92. We report results using 1-month ahead returns in panel A, 2-12 months ahead returns

in panel B, and 13-36 months ahead returns in panel C.

The all-firm results reported in Panel A of Table 5 are partially comparable to the two-

dimensional regression in Table 3 of FF92. For firms as a whole, we observe estimated MALSRets

of –3.8% for mve (t-statistic = –0.9), 13.4% for bm (t-statistic = 3.5), and 16.1% for mom12m (t-

statistic = 5.6). The estimated Sharpe ratios of these MALSRets can be obtained by dividing the t-

statistics by 5.8.15 Panel A confirms some but not all of FF92 and Jegadeesh and Titman‘s findings.

Although we do observe reliably positive coefficient estimates on bm and mom12m, we do not

observe a reliably negative coefficient estimate on mve.16 Furthermore, panels B and C show that

increasing the length of and distance from the RPS construction date of the future return horizon onto

which mve, bm and mom12m are projected yields the somewhat surprising result that on average,

during the period 1980-2012 only bm was reliably priced in the cross-section of U.S. stock returns.

Results across large-cap, mid-cap and small-cap stocks reveal strong differences in pricing

based on firm size. Of the nine MALSRets estimated for large-cap stocks, only one is reliably

different from zero whereas five are reliably different from zero for small-cap stocks. This said, and

in contrast to the number of statistically significant MALSRets, in general mean regression adjusted

R2s increase as firm size increases, most likely because there is a greater degree of idiosyncratic

return volatility in small-cap stocks.

We conclude that the pricing of mve, bm and mom12m over the period 1980-2012 not only

differs at the aggregate all-firm level from that found by FF92 and Jegadeesh and Titman over the

period 1963-1990, but also varies across firm size and across the length of and distance from the RPS

construction date of the future return horizon.

4.2 Unidimensional vs. multidimensional pricing of RPS

4.2.1 Number of priced RPS

In Table 6 we report the main results of our paper where we extend the dimensioning of the

cross-section of expected stock returns from mve, bm and mom12m shown in Table 5 to the 30-fold

14

To create cutoffs for the largest 1,000 companies, we rank stocks by their month-end market cap. Since ties in the

market cap cannot be ordered within the tied values, we assign the average ranking of the next lowest and next

highest value of market cap to tied values. Thus, if there are 10 tied values and the next highest rank is 5 (so that the

next lowest rank is 16), then the 10 tied values are assigned a rank of 10.5, the average of ranks 6-15. The

assignment of ranks to tied values results in the cutoff for large firms, for example, being only approximately, not

exactly, the largest 1,000 firms. 15

Given monthly data over 1980-2012 and serially uncorrelated MALSRets, the Sharpe ratio of an RPS with a

Fama-MacBeth t-statistic of t* is obtained by multiplying t* by the square root of 12 (the number of months in a

year) and dividing it by the square root of 396 (the number of months in the 33-year period 1980-2012). 16

We are not the first to document that mve appears less robustly related to future returns than indicated in FF92.

See Chan, Karceski and Lakonishok (2000) and Horowitz, Loughran and Savin (2000).

11

larger set of 100 RPS listed in Tables 1 and 2. We focus on all firms combined, delaying analysis by

firm size until section 4.3. In this section we focus on the results from using 1-month ahead returns.

We return to discuss in detail the results from using 2-12 months ahead and 13-36 months ahead

returns in Section 5.

The top two rows of Table 6 report the number of RPS that are reliably priced in two sets of

Fama-MacBeth regressions: Unidimensional regressions in which each of the pre-VIF-outlier-

trimmed 100 RPS is singly projected onto future returns, and multidimensional regressions in which

all the post-VIF-outlier-trimmed 93 RPS are simultaneously projected onto future returns. For each,

we report the number of t-statistics that exceed in absolute value 1.96 and 3.0.

We employ two t-statistic cutoffs to speak to alternative ways of assessing the extent to

which dimensionality is present. On the one hand, a t-statistic cutoff of 1.96 yields the number of

significant RPS based on the conventional classical statistical hurdle, albeit before adjusting for the

number of t-statistics expected to be observed outside 1.96 when there are 100 randomly generated

independent variables—namely five.17 On the other hand, Harvey, Liu and Zhu (2013) justifiably

criticize a 1.96 cutoff. They argue that a t-statistic cutoff of 1.96 fails to take sufficient account of

multiple kinds of snooping biases that exist in the RPS research and publication processes. In place

of 1.96, they advocate that authors, editors and readers of papers that put forward a new RPS should

apply a t-statistic cutoff of 3.0. Although there are reasons to suppose that 3.0 may be too stringent,

in this paper we use 3.0 to try to avoid overstating the multidimensionality of returns.18 We note that

since the expected number of absolute t-statistics 3.0 for 100 randomly generated independent

variables is just 0.3, applying a t-statistic cutoff of 3.0 essentially means that no additional

adjustment need be made to the reported number of significant t-statistics. For purposes of visual

highlighting, we color in tan the cells of Table 6 that contain unidimensioned RPS with an absolute t-

statistic 3.0, and in light blue those RPS that have a multidimensioned RPS with an absolute t-

statistic 3.0.

Inspection of the top two rows of Table 6 reveals several results that warrant emphasis. First

and foremost, we document that 26 of the post-VIF-outlier-trimmed 93 RPS are reliably

multidimensionally priced in 1-month ahead returns, a number that we propose is remarkably large.

Not only is 26 more than an order of magnitude larger than the 2 reported in panel A of Table 5‘s

dimensioning of the currently dominant trio of mve, bm and mom12m, but a case can be made for 26

being an underestimate of the true number of RPTS that are multidimensionally priced in 1-month

ahead returns. As we noted in section 3.1, several choices we made during the process of selecting,

aligning and coding our RPS unavoidably distance us from the RPS and methodologies used in the

originating papers, dampening down the estimated magnitudes of our estimated regression

coefficients noising up their standard errors. Consistent with this viewpoint, just 37 out of 100 RPS

are unidimensionally priced using an absolute t-statistic cutoff of 3.0—a figure that only increases to

48 if we instead apply an absolute t-statistic cutoff of 1.96. We argue that since with few exceptions

each RPS we include was found to be significant at the conventional t-statistic level of 1.96 in its

originating paper, it is not unreasonable to propose that as many as 50 (= 26 x 93/48) RPS were in

17

We acknowledge that there is a case to be made that a t-statistic cutoff of 1.96 is conservative in light of the fact

that we make one-sided sign predictions for every RPS. 18

One such reason is that that 39% of the 1980-2012 firm-months we use are post-publication, during which (as we

later show in section 4.5) mean RPS MALSRets decline relative to their pre-publication levels by between 28% and

37% and have commensurately smaller t-statistics. Since we use all firm-months equally in our Fama-MacBeth

regressions, the MALSRets and associated t-statistics that we estimate are in actuality weighted averages of pre- and

post-publication MALSRets and t-statistics.

12

reality multidimensionally priced during the period 1980-2012.19 The second result we highlight is

that as the distance of the future return from the RPS construction date the future return horizon

increases, the number of multidimensionally priced RPS decreases, and substantially so. We find

that 13 RPS are multidimensionally priced in 2-12 month ahead returns, and that 8 RPS are

multidimensionally priced in 13-36 month ahead returns. We return to this finding in section 5.

4.2.2 Multidimensional pricing of particular RPS

Rather than discuss the magnitude and significance of each of the 573 estimated MALSRets

coefficients in Table 6, we highlight certain key findings. Unless otherwise noted, we limit our

discussion to MALSRets obtained from 1-month ahead return regressions.

The first specific-RPS finding we call attention to in Table 6 is that the multidimensional

pricing in 1-month ahead returns of certain prominent RPS is markedly different to their

unidimensional pricing.20 We note the following examples, using the RPS acronym and RPS number

shown in Tables 1 and 2 as references:

beta (RPS #1, based on itemized listing shown in Tables 1 and 2) is neither unidimensionally nor

multidimensionally priced. This agrees with FF92 and much work since then.

mve (#4) is not unidimensionally priced but is positively multidimensionally priced, as are ep (#3)

and lev (#10). Our results contrast sharply with those of FF92 in two ways. First, whereas FF92

find mve to be reliably negatively priced, we find that mve is reliably multidimensionally

positively priced.21 Second, whereas FF92 find that neither ep and lev are not multidimensionally

priced, we find that both are positively multidimensionally priced.

mom12m (#21) and mom36m (#26) are positively unidimensionally priced but not

multidimensionally priced. Our results differ from those of the originating papers in that

Jegadeesh (1990) finds that mom12m is reliably positively unidimensionally priced, and De Bondt

and Thaler (1985) find that mom36m is reliably negatively unidimensionally priced.

mom1m (#24) is positively unidimensionally priced but negatively multidimensionally priced.

Our results are doubly different from Jegadeesh (1990) in that Jegadeesh finds that mom1m is

reliably unidimensionally negatively priced.

19

That just half of the 100 RPS are unidimensionally priced even at a t-statistic cutoff of 1.96 likely occurs for

multiple reasons, including that [1] we include a few variables that have been of historical interest to researchers but

have not always yielded statistical significance (e.g., lev, beta, dy), [2] our sample period differs from those used in

the paper that first identified the RPS, [3] our sample period typically includes both pre- and post-publication returns

(with evidence reported by McLean and Pontiff (2013) indicating that post-publications returns are materially

smaller than pre-publication returns), [4] the choices we make that aim to retain as many firm observations as

possible to study the full set of RPS, [5] the scaled decile ranks approach uses all firms, not just those in the extreme

deciles as is usually the case in the originating RPS paper, and assumes a linear RPS-to-returns relation, and [6] the

RPS were statistical artifacts that do not replicate with moderate changes in sample or methodology. 20

The multidimensional pricing in 1-month ahead returns that we document for certain prominent RPS also

sometimes markedly differs from that reported by Lewellen (2013, Table 2). For example, we find that mve is

positively multidimensionally priced, while Lewellen reports the opposite. Lewellen also reports that mom12m, acc

and beta are multidimensionally priced, while we do not. We attribute these differences to two key differences

between our two papers. First, our projecting 93 RPS onto 1-month ahead returns relative to Lewellen‘s 15 RPS.

Second, our not restricting our analysis to only those firms with non-missing data for all RPS. 21

This result parallels that of Brennan, Chordia and Subrahmanyam (1998) who find that the pricing of mve

becomes positive and significant when trading volume (turn, dolvol) is included.

13

sgr (Lakonishok, Shleifer and Vishny 1994, #27), acc (Sloan 1996, #33), chinv (Thomas and

Zhang 2002, #52), grltnoa (Fairfield, Whisenant and Yohn 2003, #54) and lgr (Richardson, Sloan,

Soliman and Tuna 2005, #60) are all negatively unidimensionally priced but are not

multidimensionally priced. Our results on all these RPS differ from the originating papers in that

we replicate the reliably negative unidimensional pricing reported in the originating papers, but

find no evidence of the RPS surviving the multidimensional challenge.

retvol (#68) is not unidimensionally priced but is negatively multidimensionally priced. Our

result here is the reverse of sgr, acc etc. in that we do not find the same negative unidimensional

pricing as the originating Ang, Hodrick, Xing and Zhang (2006) study, but we do observe strong

negative multidimensional pricing.

The second specific-RPS observation we note is that not every prominent RPS in Table 6 has

a MALSRet of an unexpected sign and/or statistical significance. For example, the signs of the

significant unidimensional and multidimensional pricings of unexpected quarterly earnings sue

(Rendleman, Jones and Latane 1982, #6), book-to-market bm (Rosenberg, Reid and Lanstein 1985,

#8) and asset growth agr (Cooper, Gulen and Schill 2008, #76) are all consistent with those reported

in the originating papers, particularly for 1-month ahead returns.

The last specific-RPS observation we highlight is that several relatively unknown RPS are

very strongly multidimensionally priced. For example, turn (#34) generates a MALSRet of 25.2% (t-

statistic = 11.1), dolvol (#46) generates a MALSRet of –10.0% (t-statistic = –10.1), sfe (#49)

generates a MALSRet of –16.1% (t- statistic = –8.6), ear (#80) generates a MALSRet of 9.4% (t-

statistic = 13.8), and roaq (#87) generates a MALSRet of 10.8% (t-statistic = 7.7). Of these, turn,

ear and roaq are also reliably unidimensionally priced, while dolvol and sfe are not.

4.2.3 Strength of firm size, book-to-market and 12-month momentum in multidimensionality

How strongly are firm size, book-to-market and 12-month momentum—the dominant and

default set of priced firm-specific risks in equities (Royal Swedish Academy of Sciences, 2013,

p.43)—multidimensionally priced relative to other RPS? In panel B of Table 6 we provide one

answer to this question by listing the largest 10 multidimensionally priced RPS at each future return

horizon. We find that none of firm size, book-to-market and 12-month momentum appear in the top

10 for 1-month ahead returns; bm places 7th in 2-12 months ahead returns; and mve places 1st and bm

places 3rd in 13-36 months ahead returns. This suggests that while prior research that has concluded

based on 1-month ahead returns that firm size, book-to-market and 12-month momentum are the

central determinants of the cross-section of returns is wrong, it is not wrong to conclude that firm

size, book-to-market and 12-month momentum are irrelevant. Our results suggest that they matter,

but only to an important degree in longer-horizon returns.

4.2.4 Explanatory power of multidimensionally priced RPS

Another result we note in Table 6 is that the adjusted R2 of multidimensional RPS regressions

are 15-20 times larger than those of the average unidimensional RPS regression, and are 3-4 times as

large as the adjusted R2s of partially multidimensional RPS regressions in which the RPS included

consist only of firm size, book-to-market and 12 month momentum. Per the third line from the

bottom of Table 6, the mean adjusted R2s of unidimensional projections of RPS onto 1, 2-12 and 13-

36 months ahead returns are 0.4%, 0.5% and 0.4%, respectively, and per Table 5 the mean adjusted

R2s of projections of firm size, book-to-market and 12 month momentum onto 1, 2-12 and 13-36

14

months ahead returns are 1.5%, 2.6% and 2.2%. In contrast, the mean adjusted R2s of

multidimensional projections of RPS onto 1, 2-12 and 13-36 months ahead returns are much larger at

6.5%, 9.0% and 7.4%, respectively.22

One interpretation of observing that the adjusted R2 of multidimensional RPS regressions are

15-20 times larger than those of the average unidimensional RPS regression may be that they

represent evidence that is inconsistent with the view that because financial data such as returns

contains a tremendous amount of uncertainty, predictable patterns will be at best modest and very

subtle (Hansen, 2013).

4.2.5 In-sample Sharpe ratios on portfolio of multidimensionally priced RPS

The final result we note in Table 6 is that large numbers of multidimensionally priced RPS

not surprisingly lead to portfolios of such RPS having large in-sample Sharpe ratios. We estimate the

magnitude of such Sharpe ratios by constructing equal-volatility-weighted portfolios consisting only

of the S RPS that are multidimensionally priced at a given future return horizon. Bailey and López

de Prado (2012) show that if the returns of S return predictive signals are described by

, (1)

and if an equal-volatility portfolio weighting of the RPS is constructed as

, (2)

then the Sharpe ratio of the equal-volatility-weighted RPS portfolio P is given by

, (3)

where

is the average Sharpe ratio of the individual RPS and

is the

mean absolute cross-correlation amongst RPS returns.

The last line of Table 6 indicates that this approach leads to very high estimated Sharpe

ratios, namely 3.0 for 1-month ahead returns, 1.9 for 2-12 months ahead returns, and 2.0 for 13-36

months ahead returns.23 It should be borne in mind that the Sharpe ratios we measure are by no

means implementable. Not only do they exclude implementation and other transactions costs, but

they are wholly in-sample. Nevertheless, we propose that they do provide an indication of the very

large economic footprint generated by, and intrinsically inherent in, the multidimensionality that we

document is present in the cross-section of expected returns.

4.3 Multidimensional pricing of RPS across decades and across firm size groups

In Table 7 we extend the analysis reported in Table 6 to sub-analyses by decade and by firm

size.24 Panel A shows that multidimensional RPS pricing is present in each decade of our 1980-2012

22

We note but do not investigate the reason why the multidimensional adjusted R2 is highest for mid-cap firms.

23 The very high values of the estimated Sharpe ratios stem from high average Sharpe ratios across the included

RPS, and very low cross-correlations between the returns generated by each individual RPS, averaging just 0.09. 24

Tables detailing the results by firm size as in Table 6 are available from the authors on request.

15

window, particularly for 1-month ahead returns. Projections of the 93 post-VIF-outlier-trimmed RPS

onto 1-month ahead returns for all firms yield 15, 19 and 22 t-statistics with an absolute value 3.0

in the 1980s, 1990s and 2000s, respectively, comparing favorably to 26 over the entire 1980-2012

period.

In panels B and C we contrast the number of unidimensional and multidimensional

MALSRets with significant t-statistics when regressions are estimated separately for large-cap, mid-

cap and small-cap firms. For these regressions we use scaled decile ranked RPS specific to the

particular size group being analyzed. We observe that large-cap firms have far fewer

unidimensionally and multidimensionally priced RPS than do mid-cap and small-cap firms, although

this difference is more pronounced for 1-month ahead returns than for 2-12 or 13-36 months ahead

returns. This said, panel D shows that large-firm RPS explain twice as much cross-sectional return

variance as the RPS of mid-cap firms, and three times the variance as the RPS of small-cap firms.

This suggests that the multidimensional pricing of RPS is unlikely to be caused by market

microstructure or size-related frictions.

4.4 Unidimensional-to-multidimensional attenuation in RPS hedge returns

In this section we measure the degree to which MALSRets and t-statistics on MALSRets

attenuate when they are estimated multidimensionally rather than unidimensionally, holding constant

the future return horizon over which the MALSRets are computed. We do so by estimating

regressions of the form dk = f + g.bk + v, where bk and dk [k = 1 to 93] are the unidimensional and

multidimensional MALSRets, or t-statistics on MALSRets, respectively, as reported in Table 6 for a

given return horizon. We include all RPS listed in Table 6, not just those with absolute t-statistics

3.0. The degree of attenuation in moving from unidimensional to multidimensional measurement is

given by 1 – g.

Panel A of Figure 2 plots unidimensional MALSRets (x-axis) against multidimensional

MALSRets (y-axis) for each future return horizon. Also plotted is a dashed line that is the average of

the three separate fitted regressions whose very similar parameter estimates are reported individually

in panel B (intercept of 0.50, slope of 0.36). The t-statistics reported in panel B on the estimated

slopes show that multidimensional and unidimensional MALSRets are strongly positively related.

However, the third line of panel B makes clear that multidimensional MALSRets are on average two

thirds smaller than unidimensional MALSRets. Panels C and D tell a similar but not quite as

damaging a story regarding the t-statistics on MALSRets. In particular, a comparison of panels B

and D shows that while t-statistics on MALSRets attenuate substantially when calculated on

multidimensioned RPS relative to unidimensioned RPS, the degree of attenuation is less that that

experienced by MALSRets. This likely occurs because multidimensional estimation partially ‗cleans

up‘ noise and thereby yields smaller standard errors on the typical MALSRet than does

unidimensional estimation.

We draw two inferences from the large degree of dimensional attenuation shown in Figure 2.

First, the magnitude of the hedge returns reported in prior RPS studies have on average been highly

overstated, regardless of the future return horizon involved. We interpret our results as indicating

that a past study that used a unidimensional (or very partially multidimensional) empirical method

based on 1-month ahead returns and concluded that the RPS it was focused on earned an annualized

hedge return of say 10% would most likely have observed a hedge return of just 4.5% had its RPS

been multidimensioned within our framework of 93 RPS.25

25

From panel B of Figure 2, 4.5% = 0.66% + (0.38 x 10%).

16

Second, the t-statistic on the hedge return to an RPS estimated using unidimensional methods

should be shrunk by between 40% and 60%, depending on the future return horizon used.

Combining such shrinkage with Harvey, Liu and Zhu‘s (2013) argument that the presence of RPS

snooping biases point to a t-statistic cutoff of 3.0 being more appropriate than a cutoff of 1.96, and

noting that HLZ arrive at their 3.0 t-statistic cutoff solely within a unidimensional RPS framework,

we believe means that for RPS that are identified using unidimensional methods, the appropriate t-

statistic cutoff lies between (3.00 – 0.20) 0.59 = 4.7 for 1-month ahead returns, and (3.00 – 0.25)

0.38 = 7.2 for 13-36 months ahead returns.

4.5 Behavior of multidimensional RPS pricing before versus after publication

In a recent paper, McLean and Pontiff (2013) document that the monthly hedge returns to a

typical unidimensioned RPS decline by approximately 35% after the underlying RPS is published.

We replicate and extend McLean and Pontiff‘s work by measuring the degree to which

multidimensioned RPS also decline after publication. We show that the monthly hedge returns to

multidimensioned RPS on average decay by approximately the same absolute and relative amounts

as those of unidimensionsed RPS.

Figure 3 visually shows this result by first plotting in panel A the cumulative natural log of

mean MALSRets for unidimensioned (solid blue) and multidimensioned (dashed red) RPS in event

time relative to when each RPS was published.26 We limit our attention to the MALSRets obtained

by projecting RPS onto 1-month ahead returns. Event time begins 120 months before publication

and extends 120 months after publication. Since we construct our RPS to be as close as possible to

how they are defined in their originating papers, some RPS are intrinsically associated with negative

mean pre-publication hedge returns, while others have positive mean pre-publication hedge returns.

We therefore adopt the convention of multiplying by –1 the full set of 241 monthly hedge returns for

those RPS that we find have a negative pre-publication mean MALSRet. For visual emphasis, we

overlay onto the cumulative hedge returns their pre- and post-publication OLS trendlines.

Inspection of panel A shows that multidimensioned RPS exhibit a similar degree of absolute

and relative decay in their associated hedge returns as unidimensioned RPS. This can be seen by

noting that the size of the vertical gap between pre- and post-publication trendlines at event month

+120 is similar across unidimensioned and multidimensioned RPS. This is formalized in panel B

where we report the means, standard deviations and Sharpe ratios of monthly MALSRets pre-

publication, post-publication, and the difference between post- and pre-publication, together with the

t-statistic on the mean post- versus pre-publication mean monthly MALSRet. Panel B confirms the

findings of McLean and Pontiff by showing that the mean hedge return to unidimensoned RPS

decays by an average of 16 bps per month after publication, which equates to a 28% decline relative

to its pre-publication value, and extends their analysis to multidimensioned RPS by showing that the

mean hedge return to multidimensoned RPS decays by an average of 14 bps per month after

publication, which translates into a higher 37% decline relative to pre-publication levels.

5. Implications of Multidimensionality for Academic Practice and Market Efficiency

In this section we explore what we put forward as some implications of our results for

academic practice, and for the long-running question of whether RPS reflect the rational pricing of

economic risks, or mispricing, or a mixture of the two.

26

For simplicity, for published papers we define event month 0 as December of the year in which the RPS was

published, and for working papers as December of the year in which the first version of the working paper came out.

17

5.1 Academic practice

With regard to academic practice, we wish to make clear that we do not suggest that prior

research has been fatally flawed because it did not control for all 26 RPS that we estimate were

multidimensionally priced over the past 30 years. Nor do we argue that future research will be

fatally flawed if it ignores the apparently very high multidimensionality in returns. Instead, we offer

three opinions based on our results.

First, the replicable but at times crude choices we have to make in order to combine RPS

across companies, time periods and databases, our using just CRSP, Compustat and I/B/E/S

databases, our approach to dealing with missing data, our measuring the average of pre- and post-

publication relations, our use of scaled decile ranked RPS, and our using a large number but still less

than 50% of the RPS that have been documented by academics all lead us to suspect that the true

dimensionality in returns may well be larger than we infer it to be.

Second, we believe that future research that chooses in returns-based research (especially

when using monthly stock returns) to control only for firm size, book-to-market and 12-month

momentum is likely to arrive at unnecessarily low-powered inferences, and at times those inferences

will be predictably biased. We base our view on our findings that the hedge returns earned by

multidimensioned RPS are on average just one third of those earned by unidimensioned RPS, and

that the associated the t-statistic on the hedge return to a unidimensioned RPS should, depending on

the future return horizon used, be shrunk by between 40% and 60% to arrive at its value when

estimated multidimensionally. We emphasize that the uncertainty that our perspective implies for the

robustness and validity of the inferences made by prior RPS papers about the true novelty of the RPS

they study is separate from, but can be viewed as additive to, the concern expressed by Harvey, Liu

and Zhu‘s (2013) that the t-statistics found in prior RPS research suffer from a variety of data

snooping biases and therefore warrant a substantial haircut by readers.

Third, we believe that the NPV of academic investment into understanding the causes and

consequences of multidimensionality is currently almost surely higher than the NPV of academic

investment into discovering additional RPS. In light of the fact that 330+ RPS that have been already

been documented, it is hard to suppose that the net benefits of discovering one (or even ten) more

new RPS are positive, either to the producers or consumers of academic RPS research. We argue

that the costs to a researcher of rigorously documenting the presence of a genuinely new RPS from

this point on are very high. If the researcher does not control for the RPS we find to have been

multidimensionally priced in returns during 1980-2012, then per our calculations in section 4.4, they

will need to be able to report a t-statistic on their new RPS of between 4.7 and 7.2. If the researcher

does choose to control for the RPS we find to have been multidimensionally priced, or some even

larger set, then substantial programming costs may be involved. Furthermore, the benefits to

academic practice from establishing a truly new RPS would seem to be small, given the numbers of

multidimensionally priced RPS we have documented. In 1992, adding one new multidimensionally

priced RPS to the known set was a significant advance; that is not the case now. As such, we make

so bold as to recommend that scholars—particularly junior faculty—direct their time and energy

away from discovering new RPS and instead direct it toward theoretically and empirically

understanding the causes and consequences of the remarkable multidimensionality that appears to

exist in returns, and toward understanding what multidimensionality means for asset pricing and

other areas of finance.

Lastly for academics, we propose that the very large degree to which empirically-based

inferences about the dimensionality of returns appear to have changed over two decades suggests that

future inferences about the dimensionality of returns may warrant more caution in their production

18

and consumption by academics and practitioners—including those we arrive at in our study. Put

another way, we argue that it is not unreasonable to suppose that if the empirically established

dimensionality of as important a financial item as monthly stock returns can change by an order of

magnitude or more in just two decades, then most empirical findings—including ours—are not only

unlikely to robustly stand the test of time, but two decades from now future researchers may well

conclude that they are exactly the opposite of the truth.

5.2 Market efficiency

A clear question to ask of our key finding that the cross-section of expected U.S. stock

returns is an order of magnitude more multidimensional than reported by FF92 is what, if anything, it

implies for the long-running debate about the degree and nature of stock market efficiency—do

multidimensionally priced RPS reflect the rational pricing of economic risks, or mispricing, or a

mixture of the two. As evidenced by the awarding in 2013 of the Nobel Prize in Economics to

Eugene F. Fama, Lars Peter Hansen, and Robert J. Shiller, the debate continues to be seen as both

important and far from settled.27

In this subsection we contribute a new viewpoint to the debate and calibrate our viewpoint

using existing and additional empirical results. We begin by emphasizing that we cannot nor do we

try to confidently identify that any particular RPS reflects the rational pricing of economic risks over

mispricing, or vice-versa. Nor can we say with high confidence whether a given RPS actually is the

economic risk being priced or the actual information being mispriced, or whether the RPS is instead

merely a proxy for the underlying item.

Instead, what we think we do is seek to identify which RPS may be more versus less likely to

reflect which kind of pricing. Specifically, we propose that RPS that are priced in longer-length

returns and returns that start accumulating at a point in time beyond immediately after the RPS are

measured, or are persistently priced in near-term returns, are more likely to capture the rational

pricing of economic risks, while RPS that are inconsistently priced in near-term returns are more

likely to reflect mispricing. We reason that limits to arbitrage diminish the further away in time from

the RPS construction date the future hedge return period begins, making it unlikely that observing

multidimensionally priced RPS based on projecting RPS onto non-immediate future returns will

reflect mispricing. Immediately after the RPS measurement date is when the greatest opportunity

and incentive exists for sophisticated investors to exploit the mispricing, not several months or years

later.28 And RPS that are persistently priced in returns of different lengths and in both near and non-

near term returns would seem to manifest in-equilibrium behavior, which we argue is more likely to

be indicative of rational pricing than mispricing.

Conversely, however, we propose that RPS that are multidimensionally priced in short

window returns that begin immediately after the RPS measurement date present more ambiguity, in

that they may reflect either mispricing or the rational pricing of short-term economic risks or market

frictions such as transactions costs or market microstructure. Although it seems more likely to us

27

For example, in its writing expressing the scientific background on the 2013 Nobel Prize in Economics, the

Economic Sciences Prize Committee of the Royal Swedish Academy of Sciences writes ―But which systematic risks

drive stock returns, and to what extent are investors compensated for them in terms of higher expected returns?

Alternatively, to the extent that fads and investor irrationality affect stock prices, as Shiller (1984) suggested, how

would this affect differences in expected returns across stocks?‖ (p.35). 28

We note that such reasoning is not without dispute. For example, in their study into whether the stock market

overreacts to winners and losers, De Bondt and Thaler (1985) highlight Graham‘s (1959, p.37) contention that ―the

interval required for a substantial undervaluation to correct itself averages approximately 1½ to 2½ years.‖

19

that short-lived return predictability is more likely to reflect the effects of mispricing than rational

compensation for short-lived risks, we acknowledge that confidently distinguishing between the

number and type of rationally priced economic risks and irrational mispricing based on RPS that are

multidimensionally priced in short-length future returns that begin cumulating immediately after

when an RPS is measured requires deeper economic and behavioral structure that we leave for future

work. This caveat stated, we nevertheless propose distinguishing between rational pricing and

mispricing in multidimensioned near-term short horizon future returns through the consistency of

pricing—thus, if multidimensional pricing is seen persistently over calendar time, then we propose

that such pricing likely captures rational risk pricing, leaving non-persistent multidimensional pricing

as likely reflecting mispricing.

We implement our approach of seeking to identify which of the 93 RPS we project onto

future returns more likely reflect rational pricing by returning to Table 6 and the results it contains

regarding the multidimensional pricing of RPS using for 2-12 months ahead and 13-36 months ahead

returns. In addition, we use new results we report in Table 8 on estimating the multidimensional

pricing of RPS separately by each of three decades within our full 1980-2012 window. For visual

highlighting, as in Table 6 we color in light blue those RPS that have a multidimensioned RPS with

an absolute t-statistic 3.0, and then color in light green the cells in each of the three by-decade

regressions that contain multidimensioned RPS with an absolute t-statistic 3.0. Given the large

amount of information contained in Table 6 and 8, we summarize the location of and overlap

between multidimensioned RPS in Figure 4.

In Figure 4, the solid red, dashed black, and dotted blue circles contain the number of RPS

that are multidimensionally priced in 1-month ahead, 2-12 months ahead, and 13-36 months ahead

returns, respectively. The RPS tabulated in the yellow box inside the 1-month ahead red circle are

RPS that are multidimensionally priced in 1-month ahead returns measured over the full 1980-2012

period but are not multidimensionally priced in all three decades (1980s, 1990s and 2000s). The RPS

tabulated in the grey box outside the 1-month ahead red circle are RPS that are not

multidimensionally priced in 1-month ahead returns measured over the full 1980-2012 period but are

multidimensionally priced in one or two of the 1980s, 1990s and 2000s but not all three. Inspection

of Figure 4 combined with our proposed reasoning on which RPS are most versus least likely to be

reflect rational pricing versus mispricing leads us to estimate that up to 19 RPS likely reflect the

rational pricing of economic risks, while a different group of up to 19 RPS likely reflect mispricing.

We summarize the details of what we proceed through below in Table 9.

In terms of up to 19 RPS that we propose likely reflect the rational pricing of economic risks,

we begin with the five RPS that are consistently multidimensionally priced across all three future

return horizons (1-month ahead, 2-12 months ahead, and 13-36 months ahead). We estimate that of

our 93 RPS, these RPS are the most likely to reflect the rational pricing of risk. The specific RPS

involved are unexpected quarterly earnings (sue #6), book-to-market (bm #8), leverage (lev #10),

analysts‘ mean annual forecasted earnings (sfe #49), and return volatility (retvol #68). Next most

likely are pchcapx_ia (#37), ill (#62) and mve (#4) since they are priced in 13-36 months ahead

returns, followed by dy (#5) because it is priced in 2-12 month ahead returns and ep (#3), saleinv

(#17), agr (#76), ear (#80), rsup (#81), roaq (#87), and convind (#100) because they are consistently

priced in both 2-12 month ahead returns and 1-month ahead returns. The final set of RPS that we

suggest based on our proposed reasoning may likely reflect the rational pricing of economic risks are

trading turnover (turn #34), trading volume (dolvol #46) and abnormal trading volume at the most

recent earnings announcement date (aeavol #74). Although these RPS are multidimensionally priced

only in 1-month ahead returns, they are consistently multidimensionally priced in 1-month ahead

20

returns in each of the 1980s, 1990s and 2000s, and plausibly characterize economic risks associated

with transactions costs or microstructure frictions.

Insofar as mispricing is concerned, we estimate that there are likely up to 19 different RPS in

this category. We arrive at this estimate by first identifying the RPS that are not multidimensionally

priced in 2-12 months ahead or 13-36 months ahead returns, but are multidimensionally priced in one

or two but not all three of the 1980s, 1990s and 2000s. These are tabulated in the grey box outside

the 1-month ahead red circle, these RPS are chfeps (#7), pchsaleinv (#18), mve_ia (#43), nanalyst

(#50), tb (#57), lgr (#60), ms (#64), stdcf (#82), and absacc (#90). We view these RPS as those most

likely to reflect mispricing. Next most likely in our view to reflect mispricing are the RPS that are

not multidimensionally priced in 2-12 months ahead or 13-36 months ahead returns, but are

multidimensionally priced based on the totality of the 1980-2012 window without reference toothier

subperiod pricing. As tabulated in the yellow box inside the 1-month ahead red circle, these RPS are

fgr5yr (#9), mom1m (#24), nincr (#40), indmom (#41), std_dolvol (#47), std_turn (#48), idiovol

(#53), zerotrade (#71), chtx (#92), and cash (#95).

6. Limitations of our Study

In this section we draw the reader‘s attention to several limitations of our paper. One of the

most important is that even though our paper is the first to study the multidimensional pricing of a

large number of RPS in the U.S. stock market, the set of 100 RPS that we use is only a portion of the

full population of the RPS that have been documented by business scholars (Green, Hand and Zhang,

2013; Harvey, Liu and Zhu, 2013). Our cost-benefit tradeoff led us not to include RPS that require

proprietary or specialized data to calculate, or RPS that employ infrequent events. As such, we

acknowledge that the degree of dimensionality we document, and the particular RPS that we estimate

to be multidimensionally priced in particular future return horizons, are likely quite approximate and

subject to change by more complete or advanced analysis by other scholars.

Our results are also dependent on the particular data and methodologies we use. To address

concerns that our findings may be sensitive to our methods and data, we assessed the robustness of

our results to certain alternative variable measurements and research design choices. For example, in

unreported analyses we found that our major results and the inferences about the multidimensionality

of stock market returns that we draw from them are robust to using raw or standardized RPS rather

than scaled decile rank RPS, winsorizing RPS values, not infilling as many missing observations,

using scaled decile ranks that depend on firm size, and calculating WLS rather than OLS Fama-

MacBeth regressions. Our results are also robust in their major aspects to including the seven highly

cross-correlated RPS that we excluded in the main bulk of our empirical tests.

7. Conclusions and Suggestions for Future Research

20+ years after Fama & French‘s seminal and highly influential study, we have taken up the

calls made by Cochrane (2011) and Goyal (2012) to re-measure the dimensionality of the cross-

section of expected U.S. stock returns. In light of the 300+ firm-level return predictive signals (RPS)

identified by business academics since 1990 we propose that this is a worthwhile endeavor. We

execute our empirical design using 93 academically documented RPS defined over CRSP, Compustat

and I/B/E/S data. We found that remarkably large 26 RPS are reliably multidimensionally priced as

defined by their mean coefficient estimates having an absolute t-statistic of 3.0 in Fama-MacBeth

regressions where the monthly recalculated scaled decile ranks of 93 documented RPS are

simultaneously projected onto 1-month ahead returns during the period 1980-2012, in part because

the mean absolute cross-correlation amongst RPS is a very small 0.03.

21

A second goal of our study was to contrast key economic aspects of unidimensionally and

multidimensionally priced RPS. We observed that large firms have far fewer multidimensionally

priced RPS than do small firms but their RPS explain three times as much cross-sectional return

variance, suggesting that the multidimensional pricing of RPS is unlikely to be caused by market

microstructure or size-related frictions. We also found that the multidimensional pricing of several

heavily cited RPS is quite different to their unidimensional pricing, and that none of firm size, book-

to-market, and 12-month momentum have t-statistics large enough to put them in the top 10 largest t-

statistics in 1-month ahead returns multidimensional regressions. Moreover, we observed that the

hedge returns earned by multidimensioned RPS are two thirds smaller than those earned by

unidimensioned RPS while the t-statistics on the hedge returns of multidimensioned RPS are only

between one and two thirds the size of the t-statistics on the hedge returns of unidimensioned RPS.

These findings suggest that many of the inferences arrived at in prior RPS research may be either

incorrect or much more fragile and low-powered than previously thought.

The final goal of our paper was to explore implications of our results for academic practice

and for the long-running question of whether RPS reflect the rational pricing of economic risks, or

mispricing, or a mixture of the two. With regard to academic practice, we offered four opinions.

First, we suspect that the true dimensionality in returns is larger than we infer it to be because of the

unavoidable distancing of many of our RPS from the constructs used by the papers that originally

documented the RPS. Second, we suggested that future returns-based research that controls only for

firm size, book-to-market and 12-month momentum is likely to arrive at unnecessarily low-powered

inferences that at times will be predictably biased. Third, based on the relative magnitudes of the

NPVs involved, we encouraged researchers to invest their time and talent into understanding the

causes and consequences of multidimensionality rather than uncovering yet more RPS. And fourth,

we conjectured that the very large degree to which empirically-based inferences about the

dimensionality of returns appear to have changed over two decades suggests that future inferences

about the dimensionality of returns warrant more caution in their production and consumption by

academics and practitioners—including those in our study.

On the long-running question of market efficiency, we proposed that our results may allow us

to say which RPS are more versus less likely to reflect which kind of pricing. We put forward the

view that RPS that are either priced in non-near-term returns or persistently priced in near-term

returns are more likely to capture the rational pricing of economic risks, while RPS that are

inconsistently priced in near-term returns are more likely to reflect mispricing. Conversely, RPS that

are multidimensionally priced in short window returns that begin immediately after the RPS

measurement date may reflect either mispricing or the rational pricing of short-term economic risks

or market frictions such as transactions costs or market microstructure. To discriminate between the

two, we proposed that if multidimensional pricing is seen persistently over calendar time, then such

pricing likely captures rational risk pricing, leaving non-persistent multidimensional pricing as likely

reflecting mispricing.

Adopting this view, we identified which of our 93 RPS more likely reflect rational pricing

versus mispricing by estimating an expanded set of multidimensional regressions in which we

projected the 93 RPS onto 2-12 months ahead and 13-36 months ahead returns. This led us to

estimate that up to 19 of our set of 93 RPS likely reflect the rational economic pricing, while an equal

number of RPS likely reflect mispricing. We suggest that it is not unreasonable to suppose that the

stock market is both efficient and inefficient, depending on the horizon involved and the information

being processed by the market, and this seems to be what our results are consistent with.

In conclusion, we are the first to propose that our paper may raise more questions than it

answers. Among such questions are: Why are returns so multidimensional? If we are right in which

22

RPS we identify as reflecting rational economic pricing versus mispricing, what are the economic

risks or mispricings that are involved? For mispricing RPS, where do the mispricings come from,

and are the RPS always indicative of mispricings being corrected rather than being created? Would

factors extracted from the RPS that we find to be multidimensionally priced in longer-horizon returns

dominate the RPS, and/or have explanatory power for non-equity assets and in countries outside the

U.S.? Do the RPS that matter for explaining the cross-section of expected returns also matter for

explaining the cross-section of higher moments of returns, and why? Do our multidimensionality

findings mean that portfolio managers have been misevaluated over the past 30 years? These and

other related questions we leave for our and others‘ future research.

23

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28

TABLE 1

Return predictive signals (RPS) used in the study, listed by publication or working paper year

# RPS Acronym Author(s) Date, Journal

1 Beta beta Fama & MacBeth 1973, JPE

2 Beta squared betasq Fama & MacBeth 1973, JPE

3 Earnings-to-price ep Basu 1977, JF

4 Firm size (market cap) mve Banz 1981, JFE

5 Dividends-to-price dy Litzenberger & Ramaswamy 1982, JF

6 Unexpected quarterly earnings sue Rendelman, Jones & Latane 1982, JFE

7 Change in forecasted annual EPS chfeps Hawkins, Chamberlin & Daniel 1984, FAJ

8 Book-to-market bm Rosenberg, Reid & Lanstein 1985, JPM

9 36-month momentum mom36m De Bondt & Thaler 1985, JF

10 Forecasted growth in 5-year EPS fgr5yr Bauman & Dowen 1988, FAJ

11 Leverage lev Bhandari 1988, JF

12 Current ratio currat Ou & Penman 1989, JAE

13 % change in current ratio pchcurrat Ou & Penman 1989, JAE

14 Quick ratio quick Ou & Penman 1989, JAE

15 % change in quick ratio pchquick Ou & Penman 1989, JAE

16 Sales-to-cash salecash Ou & Penman 1989, JAE

17 Sales-to-receivables salerec Ou & Penman 1989, JAE

18 Sales-to-inventory saleinv Ou & Penman 1989, JAE

19 % change in sales-to-inventory pchsaleinv Ou & Penman 1989, JAE

20 Cash flow-to-debt cashdebt Ou & Penman 1989, JAE

21 Illiquidity (bid-ask spread) baspread Amihud & Mendelson 1989, JF

22 1-month momentum mom1m Jegadeesh 1990, JF

23 6-month momentum mom6m Jegadeesh & Titman 1990, JF

24 12-month momentum mom12m Jegadeesh 1990, JF

25 Depreciation-to-gross PP&E depr Holthausen & Larcker 1992, JAE

26 % change in depreciation-to-gross PP&E pchdepr Holthausen & Larcker 1992, JAE

27 Industry-adjusted firm size mve_ia Asness, Porter & Stevens 1994, WP

28 Industry-adjusted cash flow-to-price ratio cfp_ia Asness, Porter & Stevens 1994, WP

29 Industry-adjusted book-to-market bm_ia Asness, Porter & Stevens 1994, WP

30 Annual sales growth sgr Lakonishok, Shleifer & Vishny 1994, JF

31 Industry-adjusted change in employees chempia Asness, Porter & Stevens 1994, WP

32 New equity issue IPO Loughran, Ritter & Ritter 1995, JF

33 Dividend initiation divi Michaely, Thaler & Womack 1995, JF

34 Dividend omission divo Michaely, Thaler & Womack 1995, JF

35 Sales-to-price sp Barbee, Mukherji & Raines 1996, FAJ

36 Working capital accruals acc Sloan 1996, TAR

37 Share turnover turn Datar, Naik & Radcliffe 1998, JFM

38 % change in sales - % change in inventory pchsale_pchinvt Abarbanell & Bushee 1998, TAR

39

% change in sales - % change in accounts

receivable pchsale_pchrect Abarbanell & Bushee 1998, TAR

40

% change in CAPEX - industry % change in

CAPEX pchcapx_ia Abarbanell & Bushee 1998, TAR

41 % change in gross margin - % change in sales pchgm_pchsale Abarbanell & Bushee 1998, TAR

42 % change in sales - % change in SG&A pchsale_pchxsga Abarbanell & Bushee 1998, TAR

43 # of consecutive earnings increases nincr Barth, Elliott & Finn 1999, JAR

44 Industry momentum indmom Moskowitz & Grinblatt 1999, JF

45 Financial statements score ps Piotroski 2000, JAR

46 Dollar trading volume in month t-2 dolvol Chordia, Subrahmanyam & Anshuman 2001, JFE

47 Volatility of dollar trading volume std_dolvol Chordia, Subrahmanyam & Anshuman 2001, JFE

48 Volatility of share turnover std_turn Chordia, Subrahmanyam & Anshuman 2001, JFE

49

Scaled analyst forecast of one year ahead

earnings sfe Elgers, Lo & Pfeiffer 2001, TAR

50 # of analysts covering stock nanalyst Elgers, Lo & Pfeiffer 2001, TAR

29

TABLE 1 (continued)

Return predictive signals (RPS) used in the study, listed by publication or working paper year

# RPS Acronym Author(s) Date, Journal

51 Dispersion in forecasted eps disp Diether, Malloy & Scherbina 2002, JF

52 Changes in inventory chinv Thomas & Zhang 2002, RAS

53 Idiosyncratic return volatility idiovol Ali, Hwang & Trombley 2003, JFE

54 Growth in long term net operating assets grltnoa Fairfield, Whisenant & Yohn 2003, TAR

55 RD_increase rd Eberhart, Maxwell & Siddique 2004, JF

56 Corporate investment cinvest Titman, Wei & Xie 2004, JFQA

57 Taxable income to book income tb Lev & Nissim 2004, TAR

58 Cash flow-to-price cfp Desai, Rajgopal & Venkatachalam 2004, TAR

59 Earnings volatility roavol Francis, LaFond, Olsson & Schipper 2004, TAR

60 Change in long-term debt lgr Richardson, Sloan, Soliman & Tuna 2005, JAE

61 Change in common shareholder equity egr Richardson, Sloan, Soliman & Tuna 2005, JAE

62 Illiquidity ill Acharya & Pedersen 2005, JF

63 # of years since first Compustat coverage age Jiang, Lee & Zhang 2005, RAS

64 Financial statements score ms Mohanram 2005, RAS

65 Price delay pricedelay Hou & Moskowitz 2005, RFS

66 R&D-to-sales rd_sale Guo, Lev & Shi 2006, JBFA

67 R&D-to-market cap rd_mve Guo, Lev & Shi 2006, JBFA

68 Return volatility retvol Ang, Hodrick, Xing & Zhang 2006, JF

69 Industry sales concentration herf Hou & Robinson 2006, JF

70 % change over two years in CAPEX grcapex Anderson & Garcia-Feijoo 2006, JF

71 Zero-trading days zerotrade Liu 2006, JFE

72 Change in 6-month momentum chmom Gettleman & Marks 2006, WP

73 Return on invested capital roic Brown & Rowe 2007, WP

74 Abnormal volume in earnings announcement

month

aeavol Lerman, Livnat & Mendenhall 2007, WP

75 Change in # analysts chnanalyst Scherbina 2007, WP

76 Asset growth agr Cooper, Gulen & Schill 2008, JF

77 Change in shares outstanding chcsho Pontiff & Woodgate 2008, JF

78 Industry-adjusted change in profit margin chpmia Soliman 2008, TAR

79 Industry-adjusted change in asset turnover chatoia Soliman 2008, TAR

80 3-day return around earnings announcement ear Kishore, Brandt, Santa-Clara & Venkatachalam 2008, WP

81 Revenue surprise rsup Kama 2009, JBFA

82 Cash flow volatility stdcf Huang 2009, JEF

83 Debt capacity-to-firm tangibility tang Hahn & Lee 2009, JF

84 Sin stock sin Hong & Kacperczyk 2009, JFE

85 Employee growth rate hire Bazdresch, Belo & Lin 2009, WP

86 Cash productivity cashpr Chandrashekar & Rao 2009, WP

87 ROA roaq Balakrishnan, Bartov & Faurel 2010, JAE

88 CAPEX and inventory invest Chen & Zhang 2010, JF

89 Real estate holdings realestate Tuzel 2010, RFS

90 Absolute accruals absacc Bandyopadhyay, Huang & Wirjanto 2010, WP

91 Accrual volatility stdacc Bandyopadhyay, Huang & Wirjanto 2010, WP

92 Change in tax expense chtx Thomas & Zhang 2010, WP

93 Maximum daily return in prior month maxret Bali, Cakici & Whitelaw 2011, JFE

94 Percent accruals pctacc Hafzalla, Lundholm & Van Winkle 2011, TAR

95 Cash holdings cash Palazzo 2012, JFE

96 Gross profitability gma Novy-Marx 2012, WP

97 Organizational capital orgcap Eisfeldt & Papanikolaou 2013, JF

98 Secured debt-to-total debt secured Valta 2013, WP

99 Secured debt indicator securedind Valta 2013, WP

100 Convertible debt indicator convind Valta 2013, WP

30

TABLE 2

Definitions of and data availability statistics for the return predictive signals (RPS) used in the study

# Acronym RPS definition (annual figures are for most recent fiscal year prior to signal date)

1 beta Beta estimated from 3 years of weekly firm and EW market returns ending month t-1 (with at least 52 weeks of returns

2 betasq Beta squared.

3 ep Annual income before extraordinary items (ib) divided by market cap at end of prior fiscal year.

4 mve Natural log of market cap at month-end immediately prior to signal date (prc*shrout).

5 dy Total dividends (dvt) divided by market cap at fiscal year end.

6 sue Unexpected quarterly earnings divided by market cap at end of most recent fiscal quarter. Unexpected earnings is IBES actual

earnings minus median forecasted earnings if available; else unexpected earnings is the seasonally differenced quarterly earnings

before extraordinary items from Compustat quarterly file.

7 chfeps Mean analyst forecast of annual EPS in month prior to fiscal period end date from IBES summary file minus same mean forecast

for prior fiscal period.

8 bm Book value of equity (ceq) divided by end of fiscal year end market cap.

9 mom36m 24-month cumulative return ending month t-13.

10 fgr5yr Most recently available analyst forecasted 5-year growth.

11 lev Total liabilities (lt) divided by fiscal year end market cap.

12 currat Current assets (act) divided by current liabilities (lct).

13 pchcurrat % change in currat.

14 quick Current assets (act) minus inventory (invt), divided by current liabilities (lct).

15 pchquick % change in quick.

16 salecash Annual sales (sale) divided by cash and cash equivalents (che).

17 salerec Annual sales (sale) divided by accounts receivable (rect).

18 saleinv Annual sales (sale) divided by total inventory (invt).

19 pchsaleinv % change in saleinv.

20 cashdebt Earnings before depreciation and extraordinary items (ib+dp) divided by avg total liabilities (lt).

21 baspread Monthly avg of daily bid-ask spread divided by avg of daily bid-ask spread.

22 mom1m Return in month t-1.

23 mom6m 5-month cumulative return ending month t-2.

24 mom12m 11-month cumulative returns ending month t-2.

25 depr Depreciation expense (dp) divided by gross PPE (ppegt).

26 pchdepr % change in depr.

27 mve_ia Industry-adjusted fiscal year-end market cap.

28 cfp_ia Industry-adjusted cfp.

29 bm_ia Industry-adjusted book-to-market.

30 sgr Annual % change in sales (sale).

31 chempia Industry-adjusted change in number of employees (emp).

32 IPO Indicator = 1 if first year available on Compustat monthly file.

33 divi Indicator = 1 if company pays dividends but did not in prior year.

34 divo Indicator = 1 if company does not pay dividends but did in prior year.

35 sp Annual revenue (sale) divided by fiscal year end market cap.

36 acc Annual income before extraordinary items (ib) minus operating cash flows (oancf) divided by avg total assets (at). If oancf is

missing then oancf is set to ib minus change in act minus change in che minus change in lct plus change in dlc plus change in {txp

less dp}, where each item is set to zero if missing.

37 turn Avg monthly trading volume for the three months t-3 to t-1 scaled by number of shares outstanding at end of month t-1.

38 pchsale_pchinvt Annual % change in sales (sale) minus annual % change in inventory (invt).

39 pchsale_pchrect Annual % change in sales (sale) minus annual % change in receivables (rect).

40 pchcapx_ia 2 digit sic fiscal year mean adjusted % change in capital expenditures (capx).

41 pchgm_pchsale Annual % change in gross margin (sale minus cogs) minus % change in sales (sale).

42 pchsale_pchxsga Annual % change in sales (sale) minus annual % change in SG&A (xsga).

43 nincr Number of consecutive quarters (up to eight) with an increase in earnings (ibq) over same quarter in the prior year.

44 indmom EW avg industry 12-month returns.

45 ps Sum of 9 indicator variables that form fundamental financial health F-score.

46 dolvol Natural log of trading volume times price per share from month t-2.

47 std_dolvol Monthly standard deviation of daily dollar trading volume.

48 std_turn Monthly standard deviation of daily share turnover.

49 sfe Analysts' mean earnings forecast of current year annual earnings from IBES summary files scaled by price per share at end of

most recent fiscal quarter.

50 nanalyst Number of analyst forecasts from most recently available IBES summary files in month prior to month of portfolio formation,

number of analysts set to zero of not covered in IBES summary file.

31

TABLE 2 (continued)

Definitions of and data availability statistics for the return predictive signals (RPS) used in the study

# Acronym RPS definition (annual figures are for most recent fiscal year prior to signal date)

51 disp Standard deviation of analyst forecasts in month prior to fiscal period end date divided by the absolute value of the mean

forecast. If meanest=0 then scalar set to 1. Forecast data from IBES summary files.

52 chinv Change in inventory (invt) divided by avg total assets (at).

53 idiovol Standard deviation of residuals of weekly returns on weekly EW market returns for 3 years prior to month end.

54 grltnoa Growth in long term net operating assets.

55 rd Indicator = 1 if R&D expense as a percentage of total assets has year-to-year increase of more than 5%.

56 cinvest Change over one quarter in net PPE (ppentq) divided by sales (saleq) scaled by avg of this variable for prior 3 quarters. If saleq

= 0 then scale by 0.01.

57 tb Taxable income, set as current tax expense divided by maximum federal tax rate, divided by income before extraordinary items

58 cfp Operating cash flows (oancf) divided by FYE market cap.

59 roavol Standard deviation for 16 quarters of income before extraordinary items (ib) divided by avg total assets (at).

60 lgr Annual % change in total liabilities (lt).

61 egr Annual % change in book value of equity (ceq).

62 ill Avg of daily values of absolute return divided by dollar volume.

63 age Number of years since first year of Compustat coverage.

64 ms Sum of 9 indicator variables that form fundamental performance measure M-score.

65 pricedelay Proportion of variation in weekly returns for 36 months ending in month t explained by 4 lags of weekly market returns

incremental to contemporaneous market return.

66 rd_sale R&D expense divided by sales (xrd/sale).

67 rd_mve R&D expense (xrd) divided by end of fiscal year market cap.

68 retvol Standard deviation of daily returns in month t-1.

69 herf 2 digit sic-fiscal year sales concentration [sum of squared % of sales in industry for each company].

70 grcapex % change in capital expenditures (capx) from year t-2 to year t.

71 zerotrade Turnover-weighted number of zero trading days for most recent 1 month.

72 chmom Cumulative return for months t-6 to t-1 minus cumulative return for months t-12 to t-7.

73 roic Annual earnings before interest and taxes (ebit) minus non-operating income (nopi), divided by non-cash enterprise value (ceq+lt-

74 aeavol Avg daily trading volume (vol) for 3 days centered on earnings announcement date minus avg daily volume in month ending 2

weeks before earnings announcement divided by 1 month avg daily volume; earnings announcement day is from Compustat

75 chnanalyst Change in nanalyst from month t-3 to month t.

76 agr Annual % change in total assets (at).

77 chcsho Annual % change in shares outstanding (csho).

78 chpmia 2-digit sic fiscal year mean-adjusted change in income before extraordinary items (ib) divided by sales (sale).

79 chatoia 2-digit sic fiscal year mean-adjusted change in sales (sale) divided by avg total assets (at) .

80 ear Sum of daily returns in three days around earnings announcement; earnings announcement from Compustat quarterly file (rdq).

81 rsup Sales from quarter t minus sales from quarter t-4 (saleq) divided by fiscal quarter end market cap (cshoq * prccq).

82 stdcf Standard deviation for prior 16 quarters of cash flows divided by sales (saleq); if saleq equal to zero then scale by 0.01; cash

flows defined as ibq minus accruals as defined in stdacc.

83 tang [Cash holdings + (0.715 × receivables) + (0.547 × inventory (invt)) + (0.535 × PPE (ppegt))] divided by total assets (at).

84 sin Indicator = 1 if a company's primary industry classification is in smoke, tobacco, beer, alcohol or gaming.

85 hire % change in number of employees (emp).

86 cashpr Fiscal year end market cap plus long term debt (dltt) minus total assets (at) divided by cash and equivalents (che).

87 roaq Income before extraordinary items (ibq) divided by one quarter lagged total assets (atq).

88 invest Annual change in gross property, plant, and equipment (ppegt) + annual change in inventories (invt) all divided by lagged total

assets (at)

89 realestate buildings and capitalized leases divided by gross PP&E

90 absacc Absolute value of acc

91 stdacc Standard deviation for 16 quarters of accruals scaled by sales. If saleq=0 then scale by 0.01, accruals is defined as change in

non-cash current assets minus change in current liabilities minus change in debt in current liabilities (change in actq minus change

in cheq minus change in lctq plus change in dlcq). If item is missing it is set to zero. Change is for 1 quarter change.

92 chtx % change in total taxes (txtq) from quarter t-4 to t.

93 maxret Max daily return in calendar month t-1.

94 pctacc Annual income before extraordinary items (ib) minus operating cash flows (oancf) divided by the abs{ib}; unless ib=0 then

ib=0.01. If oancf is missing then oancf is set to ib minus change in act minus change in che minus change in lct plus change in

dlc plus change in {txp less dp}, where each item is set to zero if missing.

95 cash Cash + cash equivalents (che) divided by avg total assets (at).

96 gma Sales (sale) minus cost of goods sold (cogs) divided by lagged total assets (at).

97 orgcap Capitalized SG&A expenses.

98 secured Total liability scaled secured debt.

99 securedind Indicator = 1 if company has secured debt obligations.

100 convind Indicator = 1 if the firm has convertible debt obligations.

32

TABLE 3

Description of CRSP and Compustat data items that were either deleted or reset to zero,

and RPS missing values that were rest to the real-time monthly mean value for that RPS

Panel A: Firm-month observations deleted due to no valid market cap or book value of equity

Panel B: Firm-month observations deleted due to extreme or impossible CRSP stock returns

Panel C: Frequency of missing Compustat or I/B/E/S data items reset to zero

Missing market cap or book equity

# obs./month

46

Return outlier condition # obs.

Monthly stock return > 10,000% 1

Monthly stock return < -100% 52

Compustat data item

Compustat

annual file

identifier

% of obs.

reset to zero

R&D expense xrd 54.6%

Capitalized leases--PP&E fatl 52.3%

Buildings--PP&E fatb 47.4%

Debt mortgages and other secured dm 18.9%

Number of employees emp 12.2%

Intangible assets intan 9.7%

Convertible debt dcvt 8.5%

Other current liabilities lco 7.7%

Other current assets aco 7.7%

Depreciation dp 3.9%

Total receivables rect 2.4%

Total inventory invt 2.1%

Accounts payable ap 1.0%

Total dividends dvt 0.6%

Other assets ao 0.3%

Cash and cash equivalents che 0.3%

Non-operating income nopi 0.2%

Other liabilities lo 0.1%

Total assets at 0.0%

I/B/E/S data item (only for years starting 1989)

I/B/E/S

identifier

% of obs.

reset to zero

Number of analysts issuing earnings forecast nanalyst 35.6%

33

TABLE 3 (continued)

Panel D: Number of firm-month observations after Compustat and I/B/E/S resetting to zero, and

percentage of firm-month observations then set to real-time monthly mean for that RPS

RPS #

RPS

Acronym

# firm-month obs.

post-Compustat and

I/B/E/S resets

= NOBS

% of firm-

month obs.

reset to RPS

month mean RPS #

RPS

Acronym

# firm-month obs.

post-Compustat and

I/B/E/S resets

= NOBS

% of firm-

month obs.

reset to RPS

month mean

10 fgr5yr 750,195 62% 45 ps 1,849,990 7%

51 disp 802,803 60% 52 chinv 1,849,990 7%

49 sfe 959,759 52% 55 rd 1,849,990 7%

7 chfeps 977,712 51% 60 lgr 1,843,840 7%

19 pchsaleinv 1,437,861 28% 61 egr 1,849,793 7%

59 roavol 1,425,620 28% 76 agr 1,849,957 7%

82 stdcf 1,425,620 28% 77 chcsho 1,849,007 7%

91 stdacc 1,425,620 28% 85 hire 1,844,528 7%

97 orgcap 1,453,762 27% 96 gma 1,845,300 7%

38 pchsale_pchinvt 1,461,189 26% 46 dolvol 1,897,150 5%

75 chnanalyst 1,493,809 25% 73 roic 1,896,790 5%

9 mom36m 1,512,004 24% 17 salerec 1,915,569 4%

50 nanalyst 1,515,638 24% 23 mom6m 1,910,139 4%

42 pchsale_pchxsga 1,530,539 23% 47 std_dolvol 1,918,606 3%

18 saleinv 1,561,123 21% 48 std_turn 1,924,420 3%

70 grcapex 1,650,117 17% 62 ill 1,922,298 3%

6 sue 1,694,392 15% 71 zerotrade 1,922,328 3%

12 currat 1,693,059 15% 1 beta 1,949,992 2%

14 quick 1,693,059 15% 2 betasq 1,949,992 2%

79 chatoia 1,695,660 15% 25 depr 1,955,524 2%

81 rsup 1,694,412 15% 37 turn 1,946,489 2%

92 chtx 1,694,224 15% 53 idiovol 1,949,992 2%

43 nincr 1,718,825 14% 65 pricedelay 1,949,961 2%

56 cinvest 1,702,206 14% 66 rd_sale 1,955,550 2%

64 ms 1,718,825 14% 16 salecash 1,967,987 1%

74 aeavol 1,706,462 14% 83 tang 1,969,631 1%

80 ear 1,717,417 14% 86 cashpr 1,964,487 1%

87 roaq 1,717,053 14% 3 ep 1,987,340 0%

95 cash 1,718,746 14% 4 mve 1,987,340 0%

36 acc 1,724,752 13% 5 dy 1,987,340 0%

90 absacc 1,724,752 13% 8 bm 1,987,340 0%

94 pctacc 1,724,740 13% 11 lev 1,981,906 0%

28 cfp_ia 1,755,669 12% 20 cashdebt 1,980,259 0%

57 tb 1,755,372 12% 21 baspread 1,987,294 0%

58 cfp 1,755,669 12% 22 mom1m 1,987,340 0%

26 pchdepr 1,760,216 11% 32 IPO 1,987,340 0%

39 pchsale_pchrect 1,769,186 11% 35 sp 1,981,691 0%

40 pchcapx_ia 1,795,780 10% 44 indmom 1,987,214 0%

78 chpmia 1,815,857 9% 27 mve_ia 1,987,340 0%

24 mom12m 1,820,468 8% 29 bm_ia 1,987,340 0%

30 sgr 1,820,506 8% 63 age 1,987,340 0%

41 pchgm_pchsale 1,820,295 8% 67 rd_mve 1,987,340 0%

54 grltnoa 1,831,854 8% 68 retvol 1,987,278 0%

72 chmom 1,820,468 8% 69 herf 1,987,328 0%

88 invest 1,829,166 8% 84 sin 1,987,340 0%

13 pchcurrat 1,849,990 7% 89 realestate 1,987,340 0%

15 pchquick 1,849,990 7% 93 maxret 1,987,308 0%

31 chempia 1,844,528 7% 98 secured 1,987,340 0%

33 divi 1,849,990 7% 99 securedind 1,987,340 0%

34 divo 1,849,990 7% 100 convind 1,987,340 0%

34

TABLE 4

Descriptive statistics on the degree of cross-correlation among the 100 RPS used in this study. Each

firm’s scaled decile ranked RPS are recalculated monthly by ranking every RPS into deciles (0-9)

and dividing by 9. Before ranking, certain missing data items are set to zero (per Table 3, panel A)

and all RPS with missing values are reset to that month’s mean RPS value.

Panel A: Distribution of variance inflation factors (VIFs) from pooled time-series cross-sectional

regressions of 1-month ahead stock returns on 100 or 93 RPS

Panel B: Percentiles of absolute values of pooled time-series cross-sectional data cross-correlations

among the full set of 100 RPS and among the VIF-outlier-reduced set of 93 RPS

Panel C: Histogram of the 396 mean absolute cross-correlations among RPS calculated each month

Jan. 1980 - Dec. 2012

Before

removing

RPS with

VIF > 5

After

removing

RPS with

VIF > 5

# RPS 100 93 • betasq • chempia

Min. 1.0 1.0 • currat • stdacc

Median 1.2 1.1 • pchcurrat • maxret

Mean 3.4 1.3 • mom6m

max. 58.0 4.0

# VIFs > 5 14 0

RPS removed in reducing the

set of 100 RPS to 93 RPS

Min.

1st

pctile

25th

pctile Median Mean

75th

pctile

99th

pctile Max.

0.00 0.00 0.00 0.01 0.03 0.03 0.29 0.99

0.00 0.00 0.00 0.01 0.03 0.03 0.27 0.68

N = 100 RPS

N = 93 RPS

35

TABLE 5

Mean annualized long/short hedge returns (MALSRets) and associated t-statistics implied by slope

coefficients estimated in Fama-MacBeth regressions of 1 month ahead, 2-12 months ahead and 13-36

months ahead firm-specific returns RET = a + b1mve + b2.bm + b3.mom12m + e over 1980-2012. Each

of the three RPS mve, bm, mom12m is recalculated monthly by first ranking the RPS into deciles (0-9)

and then dividing by 9. Before ranking, certain missing data items are set to zero (see Table 3), and

missing RPS values are reset to that month’s mean RPS value. Estimated MALSRets are annualized

by multiplying by 12 (12/11, 0.5) for 1 month (2-12, 13-36) ahead horizons. t-statistics use Newey-

West adjustments of 12 (1) lags for 1 (2-12, 13-36) month ahead returns. Return windows are

defined in non-overlapping calendar time, so that 2-12 months ahead returns cumulate from Aug. of

year t thru June of year t+1, while 13-36 months ahead returns cumulate in non-overlapping two

year periods starting Jan. 1981 - Dec. 1982. Large-Cap are the largest 1,000 companies by market

cap; Mid-Cap are the next largest 2,000 companies; Small-Cap are all remaining firms.

Panel A: 1 month ahead returns (396 non-overlapping monthly regressions)

RPS Pred. sign MALSRet t-stat. MALSRet t-stat. MALSRet t-stat. MALSRet t-stat.

mve - -3.8% -0.9 -1.1% -0.7 2.5% 1.5 -13.3% -3.9

bm + 13.4% 3.5 4.2% 1.1 11.9% 2.8 14.1% 3.5

mom12m + 6.1% 5.6 6.3% 4.4 6.8% 5.9 4.5% 3.3

Panel B: 2-12 months ahead returns (33 regressions each with a non-overlapping 11 month span)

RPS Pred. sign MALSRet t-stat. MALSRet t-stat. MALSRet t-stat. MALSRet t-stat.

mve - 0.7% 0.2 -1.9% -1.0 3.5% 2.4 -2.2% -0.8

bm + 13.4% 3.7 3.9% 1.0 11.4% 2.5 14.9% 4.2

mom12m + 1.2% 1.6 0.5% 0.7 2.1% 3.7 0.7% 0.6

Panel C: 13-36 months ahead returns (11 regressions each with a non-overlapping 2-year span)

RPS Pred. sign MALSRet t-stat. MALSRet t-stat. MALSRet t-stat. MALSRet t-stat.

mve - 4.3% 1.6 -0.2% -0.2 3.7% 2.0 3.0% 1.0

bm + 11.9% 3.2 3.6% 1.4 10.2% 2.0 14.7% 4.4

mom12m + 0.6% 0.6 -0.2% -0.2 -0.5% -0.8 1.1% 0.6

Mean adjusted R2

2.2% 1.5% 2.8% 1.2%

Mean # obs. per regression

Mean adjusted R2

Mean # obs. per regression

Mean adjusted R2

Mean # obs. per regression

All firms Large-Cap Mid-Cap Small-Cap

All firms Large-Cap Mid-Cap Small-Cap

5,018

1.5%

996 1,997 2,019

0.7%2.3% 1.5%

5,026 996 1,997 2,027

All firms Large-Cap Mid-Cap Small-Cap

5,140 996 1,997 2,141

2.6% 3.1% 2.8% 1.5%

36

TABLE 6

Mean annualized long/short hedge returns (MALSRets) and associated t-statistics derived from the slope coefficients estimated in

unidimensional (N = 1) and multidimensional (N = 93) Fama-MacBeth regressions RET = a + b1RPS1 + ... bN.RPSN + e on 1 month ahead,

2-12 months ahead and 13-36 months ahead firm-specific returns for 1980-2012. Each firm’s scaled decile ranked RPS is recalculated

monthly by first ranking every RPS into deciles (0-9) and then dividing by 9. Before ranking, certain missing data items are set to zero

(see Table 3), and missing RPS values are reset to that month’s mean RPS value. Estimated MALSRets are annualized by multiplying by

12 (12/11, 0.5) for 1 month (2-12, 13-36) ahead horizons. t-statistics use Newey-West adjustments of 12 (1) lags for 1 (2-12, 13-36) month

ahead returns. Predicted coefficient signs are from the RPS literature. Return windows are defined in non-overlapping calendar time, so

that 2-12 months ahead returns cumulate from Aug. of year t thru June of year t+1, while 13-36 months ahead returns cumulate starting

the 2nd year of non-overlapping three year periods starting Jan. 1981. Multidimensional regressions use the subset of N = 93 RPS

described in Table 4 for which the maximum RPS VIF is 4.0. MALSRets with an absolute t-statistic 3.0 are highlighted.

Panel A: Results for all 93 RPS

RPS Pred. sign MALSRet t-stat. MALSRet t-stat. MALSRet t-stat. MALSRet t-stat. MALSRet t-stat. MALSRet t-stat.

1 beta + -3.9% -0.8 0.1% 0.1 -4.0% -0.9 0.2% 0.1 -6.8% -1.7 -1.2% -0.6

2 betasq + -4.1% -0.8 -4.4% -1.0 -7.0% -1.8

3 ep + 6.6% 1.2 6.4% 4.3 9.5% 2.1 6.6% 3.5 9.8% 2.8 2.6% 1.2

4 mve - -6.8% -1.8 12.1% 3.8 -2.5% -0.6 1.4% 0.4 1.5% 0.5 6.8% 7.3

5 dy + 0.9% 0.2 -2.6% -2.6 1.5% 0.4 -4.6% -4.3 5.3% 1.2 -1.5% -1.8

6 sue + 18.4% 12.1 10.2% 13.2 7.3% 4.1 3.3% 3.4 1.1% 0.6 2.4% 3.2

7 chfeps + 10.8% 3.6 2.4% 1.2 2.1% 1.2 0.4% 0.2 0.2% 0.1 2.4% 0.6

8 bm + 15.1% 4.6 10.8% 5.4 13.8% 3.9 9.5% 3.8 11.3% 2.9 9.9% 4.7

9 mom36m - 8.7% 5.2 0.8% 1.9 0.2% 0.3 0.2% 0.4 -2.3% -1.8 -0.6% -1.2

10 fgr5yr - -0.9% -0.2 -4.2% -4.1 -0.6% -0.2 0.3% 0.2 -1.0% -0.5 0.2% 0.9

11 lev + 6.6% 1.6 7.1% 3.3 6.1% 1.3 8.2% 3.7 7.1% 1.7 7.3% 4.9

12 currat + 1.3% 0.8 2.5% 1.0 1.8% 0.9

13 pchcurrat + -2.5% -2.0 -1.1% -0.7 1.2% 1.2

14 quick + 1.3% 0.6 1.1% 0.9 1.6% 0.6 3.3% 2.0 0.1% 0.1 1.6% 1.6

15 pchquick + -1.7% -1.4 -0.7% -1.1 -1.1% -0.7 -1.2% -1.7 1.1% 1.3 0.5% 0.6

16 salecash + 1.9% 0.6 -1.6% -1.2 3.6% 1.1 -1.2% -0.8 3.2% 1.1 -1.0% -0.7

17 salerec + 4.4% 2.1 2.6% 1.7 4.9% 2.2 2.8% 1.4 3.5% 1.1 1.4% 0.6

18 saleinv + 3.1% 2.2 5.4% 5.1 2.0% 1.3 3.6% 4.0 0.2% 0.1 1.1% 0.8

19 pchsaleinv + 1.3% 0.9 1.9% 1.3 -0.4% -0.3 0.9% 0.7 -2.5% -3.0 0.6% 0.9

20 cashdebt + 3.8% 1.0 -1.0% -0.6 7.0% 2.4 0.4% 0.2 6.8% 6.3 5.4% 2.2

9 8# abs{t-stat} 3.0

# abs{t-stat} 1.96 48 47 46 30

37 26 23 13

All firms, 2-12 month ahead returns All firms, 13-36 month ahead returns

Unidimensional MultidimensionalUnidimensional Multidimensional Unidimensional

All firms, 1 month ahead returns

27 21

Multidimensional

37

TABLE 6 (continued) – multidimensionally priced RPS

RPS Pred. sign MALSRet t-stat. MALSRet t-stat. MALSRet t-stat. MALSRet t-stat. MALSRet t-stat. MALSRet t-stat.

21 baspread - -2.8% -0.5 6.0% 2.7 -5.8% -1.2 -0.2% -0.1 -4.2% -1.2 7.1% 2.2

22 mom1m - 4.2% 4.5 -1.4% -3.7 1.2% 1.7 0.1% 0.1 0.1% 0.6 0.1% 0.2

23 mom6m + 8.6% 5.4 2.4% 3.1 0.1% 0.2

24 mom12m + 9.8% 6.0 0.7% 1.5 1.8% 2.2 1.1% 1.4 -0.2% -0.2 0.5% 0.6

25 depr + 6.0% 1.7 3.1% 2.5 4.9% 1.4 4.1% 2.7 -1.6% -1.0 1.2% 1.6

26 pchdepr - -0.8% -0.5 -1.9% -2.4 -0.2% -0.2 -0.2% -0.4 -2.0% -1.8 -1.5% -1.9

27 mve_ia - -0.6% -0.3 2.8% 2.3 1.0% 0.6 -0.6% -0.4 0.1% 0.0 -2.0% -1.6

28 cfp_ia + -0.2% -0.1 0.0% 0.0 -1.2% -0.6 -0.8% -0.7 0.2% 0.1 -1.4% -1.0

29 bm_ia + 7.4% 3.2 -2.6% -2.2 5.8% 2.7 -2.2% -1.4 3.2% 1.7 -4.4% -2.3

30 sgr - -10.4% -5.5 -0.8% -0.6 -6.3% -3.7 3.1% 2.1 -3.6% -1.8 2.2% 0.9

31 chempia - -6.5% -4.7 -2.5% -1.9 -0.7% -0.9

32 IPO - -9.0% -4.3 -1.9% -1.3 -7.7% -4.5 -2.1% -1.2 -4.8% -2.0 -0.5% -0.4

33 divi + -2.5% -1.9 -1.4% -1.2 -3.2% -3.2 -1.5% -1.5 -1.6% -1.4 0.1% 0.1

34 divo - -0.3% -0.3 -0.4% -0.5 0.5% 0.3 0.1% 0.1 -0.7% -0.5 0.2% 0.1

35 sp + 14.0% 4.0 -1.6% -0.8 14.3% 3.7 2.0% 1.0 10.3% 2.5 1.4% 0.5

36 acc - -8.0% -4.1 -3.3% -2.8 -4.3% -2.8 -1.1% -0.8 -1.6% -1.1 -0.6% -0.4

37 turn + 21.9% 9.1 25.2% 11.1 0.3% 0.2 -2.3% -2.3 -0.8% -0.9 -2.5% -2.2

38 pchsale_pchinvt + 5.6% 6.8 -1.1% -0.8 3.8% 3.8 -2.0% -1.6 1.0% 0.7 -1.4% -0.9

39 pchsale_pchrect - 4.1% 3.4 0.4% 0.7 3.0% 2.4 -0.4% -0.5 1.0% 0.9 1.2% 2.5

40 pchcapx_ia - -5.6% -3.7 -2.2% -2.8 -2.3% -1.7 -1.0% -1.1 -3.5% -2.7 -3.0% -3.3

41 pchgm_pchsale + 5.1% 3.7 1.8% 2.3 5.5% 4.4 1.5% 1.9 2.9% 2.2 0.2% 0.2

42 pchsale_pchxsga + -2.2% -1.8 1.6% 2.1 -1.3% -1.1 1.5% 1.7 -1.5% -1.2 -1.0% -1.1

43 nincr + 12.8% 8.7 2.7% 4.1 3.4% 2.5 -0.4% -0.7 -1.3% -1.2 -0.2% -0.4

44 indmom + 25.8% 7.3 7.2% 6.0 7.6% 3.0 2.8% 1.8 -2.4% -0.8 1.1% 0.7

45 ps + 5.6% 2.5 -1.3% -1.6 7.3% 4.2 -0.9% -1.1 5.1% 6.5 0.4% 0.3

46 dolvol - 1.0% 1.5 -10.0% -10.1 1.0% 1.5 1.7% 2.6 2.1% 2.4 2.6% 2.8

47 std_dolvol - 4.5% 1.9 -6.5% -3.1 3.5% 1.3 1.6% 1.5 2.6% 1.2 1.1% 1.1

48 std_turn - 4.0% 1.2 10.5% 3.8 -5.1% -1.9 -1.3% -0.9 -5.2% -2.3 -1.7% -1.9

49 sfe + -6.0% -1.4 -16.1% -8.6 -3.9% -1.5 -20.2% -6.8 0.7% 0.5 -20.9% -4.5

50 nanalyst - -4.5% -1.4 3.4% 1.1 1.2% 0.5 10.7% 2.6 2.4% 1.0 12.0% 2.0

51 disp - -0.1% 0.0 -2.0% -1.6 -0.8% -0.3 -1.3% -1.0 -1.6% -0.6 -2.1% -1.2

52 chinv - -9.2% -5.8 -0.8% -0.8 -5.4% -3.7 -1.7% -1.5 -0.5% -0.4 0.6% 0.4

53 idiovol - -2.4% -0.4 7.6% 3.1 -5.6% -1.1 1.2% 0.5 -9.1% -2.1 -3.4% -1.4

54 grltnoa - -11.1% -5.6 -0.5% -0.5 -8.2% -3.3 -2.2% -2.0 -3.0% -1.6 -1.6% -1.7

All firms, 2-12 month ahead returns All firms, 13-36 month ahead returns

Unidimensional MultidimensionalUnidimensional Multidimensional Unidimensional

All firms, 1 month ahead returns

Multidimensional

38

TABLE 6 (continued) – multidimensionally priced RPS

RPS Pred. sign MALSRet t-stat. MALSRet t-stat. MALSRet t-stat. MALSRet t-stat. MALSRet t-stat. MALSRet t-stat.

55 rd + 6.2% 2.3 1.2% 2.1 4.1% 1.5 0.0% 0.0 -0.4% -0.3 -0.3% -0.3

56 cinvest - 0.4% 0.3 -1.0% -1.4 1.3% 0.9 -1.5% -2.5 1.0% 0.7 1.2% 1.1

57 tb + 4.3% 1.5 0.9% 1.1 5.4% 2.6 2.2% 2.2 5.7% 2.8 1.3% 1.0

58 cfp + 1.1% 0.4 -1.9% -2.1 2.8% 1.2 -0.7% -0.6 1.4% 1.3 -0.1% -0.1

59 roavol + -1.2% -0.3 0.7% 0.8 -3.9% -1.0 0.8% 0.8 -5.9% -3.1 0.6% 0.9

60 lgr - -12.9% -8.3 -2.9% -2.5 -10.0% -7.6 -3.3% -2.9 -3.7% -2.2 1.8% 1.3

61 egr - -8.2% -3.2 -0.5% -0.6 -5.2% -2.3 -1.7% -2.1 -1.5% -1.7 -0.1% -0.1

62 ill + 2.7% 0.8 10.7% 1.8 2.1% 0.6 5.6% 1.5 1.2% 0.4 10.6% 3.2

63 age + 6.6% 2.2 -0.1% -0.1 7.1% 2.8 0.6% 0.4 6.0% 1.7 0.9% 0.5

64 ms + 1.8% 0.8 -1.1% -1.5 4.7% 2.3 0.2% 0.3 2.2% 1.3 0.3% 0.3

65 pricedelay + 2.1% 1.7 1.3% 2.5 -1.4% -1.6 0.0% 0.1 -0.4% -0.5 -0.1% -0.2

66 rd_sale + 5.0% 0.9 4.1% 1.3 5.4% 0.8 0.8% 0.2 -1.7% -0.4 -4.3% -1.5

67 rd_mve + 9.0% 1.7 6.3% 1.9 8.9% 1.5 10.7% 2.7 1.5% 0.3 9.1% 1.6

68 retvol - -5.8% -1.1 -13.9% -6.7 -6.9% -1.6 -4.1% -3.4 -6.4% -1.8 -4.8% -3.0

69 herf - -2.4% -1.1 -3.0% -2.1 -1.8% -0.6 -1.3% -0.8 -4.2% -1.8 -2.5% -1.5

70 grcapex - -8.3% -4.4 -2.0% -2.8 -3.6% -2.0 -0.3% -0.5 0.7% 0.4 1.0% 1.2

71 zerotrade + 3.2% 1.0 12.3% 4.9 7.8% 2.2 3.4% 1.6 6.0% 2.3 -0.3% -0.3

72 chmom - 0.6% 1.2 0.3% 0.6 0.9% 2.1 1.2% 2.1 -0.1% -0.1 0.1% 0.1

73 roic + 4.6% 1.2 1.7% 1.3 7.0% 2.3 1.9% 1.4 6.1% 4.4 3.2% 2.2

74 aeavol + 7.7% 6.2 3.3% 6.1 3.1% 2.7 1.3% 2.6 0.6% 1.1 1.6% 2.0

75 chnanalyst - -3.1% -1.6 0.5% 0.7 -0.3% -0.2 -0.3% -0.3 1.0% 0.5 -0.5% -0.7

76 agr - -15.8% -5.9 -6.9% -4.2 -11.4% -4.9 -5.2% -3.6 -3.3% -2.2 -3.1% -2.4

77 chcsho - -12.7% -5.6 -2.0% -2.8 -10.5% -4.9 -1.0% -1.5 -7.2% -4.3 -2.6% -2.4

78 chpmia + -0.4% -0.3 0.7% 0.7 -1.0% -0.7 0.1% 0.1 -2.5% -1.7 -0.5% -0.6

79 chatoia + 4.0% 5.8 1.0% 1.7 4.7% 7.1 1.4% 1.7 1.1% 1.8 0.1% 0.3

80 ear + 16.5% 16.6 9.4% 13.8 4.2% 5.0 2.1% 4.2 0.5% 0.5 0.0% 0.0

81 rsup + 7.9% 3.5 7.4% 7.5 3.2% 1.4 2.9% 3.4 1.5% 0.5 -0.3% -0.3

82 stdcf - -7.8% -2.7 -2.2% -2.0 -9.9% -3.9 -3.0% -2.7 -8.1% -4.2 -2.4% -1.9

83 tang + 4.2% 1.4 2.3% 1.8 3.6% 1.0 2.8% 1.9 1.3% 0.5 2.5% 2.1

84 sin + 4.6% 1.9 5.4% 2.7 5.5% 1.8 6.6% 2.5 5.3% 1.8 5.3% 2.0

85 hire - -10.9% -5.9 -0.2% -0.2 -6.8% -4.0 1.2% 1.4 -3.1% -1.9 0.3% 0.4

86 cashpr - -9.5% -2.9 -0.2% -0.2 -8.8% -2.5 0.2% 0.2 -8.7% -2.5 -0.2% -0.2

87 roaq + 13.6% 3.3 10.8% 7.7 8.0% 2.5 7.3% 5.6 2.8% 1.6 -0.2% -0.2

88 invest - -12.9% -6.0 -0.7% -0.6 -8.4% -3.5 1.5% 0.9 -2.3% -1.0 1.2% 0.6

All firms, 2-12 month ahead returns All firms, 13-36 month ahead returns

Unidimensional MultidimensionalUnidimensional Multidimensional Unidimensional

All firms, 1 month ahead returns

Multidimensional

39

Table 6 (continued) – multidimensionally priced RPS

Note: A few t-statistics shown as 3.0 lie between 2.95 and 2.99, and thus only appear to be 3.0. Only those t-statistics that are 3.0 or more are highlighted in in Panel A.

Panel B: Listing of the largest 10 multidimensionally priced RPS at each future return horizon, subject to abs{t-stat} 3.0

RPS Pred. sign MALSRet t-stat. MALSRet t-stat. MALSRet t-stat. MALSRet t-stat. MALSRet t-stat. MALSRet t-stat.

89 realestate + 2.1% 1.4 -0.4% -0.4 2.4% 1.6 0.4% 0.5 2.3% 1.7 0.9% 0.6

90 absacc - -0.6% -0.2 -2.1% -2.5 -1.5% -0.7 -1.0% -0.8 -4.4% -2.8 -0.8% -0.5

91 stdacc - -8.1% -3.0 -10.3% -4.2 -7.8% -4.2

92 chtx + 12.5% 9.7 3.3% 4.0 2.5% 2.3 0.4% 0.5 1.4% 1.0 1.0% 2.1

93 maxret - -8.8% -1.8 -5.2% -1.4 -6.4% -2.2

94 pctacc - -5.8% -3.3 -2.2% -1.5 -3.3% -2.0 -3.0% -1.8 -5.3% -5.5 -1.4% -1.9

95 cash + 5.1% 1.3 3.8% 3.5 2.5% 0.6 3.5% 1.9 -0.8% -0.3 1.8% 1.1

96 gma + 4.3% 2.5 2.4% 1.2 7.0% 3.6 4.8% 2.7 3.4% 2.6 -0.7% -0.4

97 orgcap + 8.7% 3.0 3.4% 2.2 6.9% 2.3 1.7% 1.1 3.2% 1.3 3.0% 1.5

98 secured + 1.6% 0.5 0.1% 0.0 0.4% 0.2 0.8% 0.4 0.9% 0.5 3.2% 1.6

99 securedind + 0.6% 0.3 0.9% 0.8 0.3% 0.2 0.2% 0.1 0.8% 1.1 -0.1% -0.2

100 convind + -5.0% -4.1 -3.1% -5.0 -4.6% -4.8 -3.2% -4.9 -2.7% -2.8 -1.4% -1.6

All firms, 2-12 month ahead returns All firms, 13-36 month ahead returns

Unidimensional MultidimensionalUnidimensional Multidimensional Unidimensional

All firms, 1 month ahead returns

Multidimensional

Mean # obs. per regression

Mean adjusted R2

Sharpe ratio on portfolio of RPS with abs{t-stat} 3 2.98 1.92 2.03

6,362

7.4%

5,035

0.4%

4,930

6.5%

5,157

0.5%

5,309

9.0%

5,048

0.4%

# RPS Pred. sign MALSRet t-stat. MALSRet t-stat. RPS Pred.

sign

MALSRet t-stat. RPS Pred.

sign

MALSRet t-stat.

1 ear + 16.5% 16.6 9.4% 13.8 sfe + -20.2% -6.8 mve - 6.8% 7.3

2 sue + 18.4% 12.1 10.2% 13.2 roaq + 7.3% 5.6 lev + 7.3% 4.9

3 turn + 21.9% 9.1 25.2% 11.1 convind + -3.2% -4.9 bm + 9.9% 4.7

4 dolvol - 1.0% 1.5 -10.0% -10.1 dy + -4.6% -4.3 sfe + -20.9% -4.5

5 sfe + -6.0% -1.4 -16.1% -8.6 ear + 2.1% 4.2 pchcapx_ia - -3.0% -3.3

6 roaq + 13.6% 3.3 10.8% 7.7 saleinv + 3.6% 4.0 ill + 10.6% 3.2

7 rsup + 7.9% 3.5 7.4% 7.5 bm + 9.5% 3.8 sue + 2.4% 3.2

8 retvol - -5.8% -1.1 -13.9% -6.7 lev + 8.2% 3.7 retvol - -4.8% -3.0

9 aeavol + 7.7% 6.2 3.3% 6.1 agr - -5.2% -3.6

10 saleinv + 3.1% 2.2 5.4% 5.1 sue + 3.3% 3.4

Multidimensional Multidimensional

All firms, 1 month ahead returns

Unidimensional Multidimensional

All firms, 2-12 month ahead returns All firms, 13-36 month ahead returns10 largest multidimensional t-stats

40

TABLE 7

Analysis by decade and by firm size: Number of significant t-statistics on RPS MALSRets derived

from slope coefficients estimated in unidimensional (N = 1) and multidimensional (N = 93) Fama-

MacBeth regressions RET = a + b1RPS1 + ... bN.RPSN + e on 1 month ahead, 2-12 months ahead and

13-36 months ahead returns for 1980-2012. The regression analysis is identical to that in Table 6

except that panel A reports the results of regressions estimated separately by decade using all firms,

and panels B-D estimate regressions separately for Large-Cap, Mid-Cap and Small-Cap stocks.

Large-Cap are defined as the largest 1,000 companies by market cap, Mid-Cap are the next largest

2,000 companies, and Small-Cap are all remaining firms.

Panel A: By decade: Number of multidimensional MALSRet with abs{t-stat.} 1.96 shown in the first

part of the cell, and with abs{t-stat} 3.0 in the second part of the cell

Panel B: By firm size: Number of unidimensional MALSRet with abs{t-stat.} 1.96 shown in the first part

of the cell, and with abs{t-stat} 3.0 in the second part of the cell

Panel C: By firm size: Number of multidimensional MALSRet with abs{t-stat.} 1.96 shown in the first

part of the cell, and with abs{t-stat} 3.0 in the second part of the cell

Panel D: By firm size: Mean Fama-MacBeth multidimensional RPS regression adjusted R2

RET horizon 1980-2012 1980-1989 1990-1999 2000-2012

1 month 47, 26 24, 15 32, 19 36, 22

2-12 months 30, 13 15, 2 20, 6 21, 6

Estimation decade

RET horizon All firms Large-Cap Mid-Cap Small-Cap

1 month 48, 37 22, 6 61, 40 51, 34

2-12 months 46, 23 7, 1 51, 29 40, 22

13-36 months 27, 9 8, 3 20, 6 38, 20

RET horizon All firms Large-Cap Mid-Cap Small-Cap

1 month 47, 26 16, 5 35, 17 34, 21

2-12 months 30, 13 13, 5 23, 6 22, 4

13-36 months 21, 8 15, 4 23, 11 18, 5

RET horizon All firms Large-Cap Mid-Cap Small-Cap

1 month 6.5% 16.8% 8.7% 4.4%

2-12 months 9.0% 21.0% 12.6% 7.1%

13-36 months 7.4% 16.5% 10.4% 4.2%

41

TABLE 8

Mean annualized long/short hedge returns (MALSRets) and associated t-statistics derived from the

slope coefficients estimated in multidimensional Fama-MacBeth regressions of the 93 RPS described

in Table 4 for which the maximum RPS VIF is 4.0 onto 1-month ahead returns, both for all years

1980-2012 and separately by decade. T-statistics 3.0 and their associated MALSRets are shaded.

Predicted coefficient signs are taken from extant RPS literature.

# RPS

# RPS Pred. sign MALSRet t-stat. MALSRet t-stat. MALSRet t-stat. MALSRet t-stat.

1 beta + 0.1% 0.1 2.6% 0.7 3.9% 1.0 -4.7% -1.3

3 ep + 6.4% 4.3 8.7% 3.7 4.5% 1.9 6.1% 3.0

4 mve - 12.1% 3.8 7.9% 1.3 12.8% 2.2 14.9% 2.9

5 dy + -2.6% -2.6 -0.4% -0.2 -0.6% -0.3 -5.9% -3.7

6 sue + 10.2% 13.2 9.5% 6.2 12.8% 8.4 8.9% 6.6

7 chfeps + 2.4% 1.2 9.5% 4.4 -2.6% -1.4

8 bm + 10.8% 5.4 8.7% 3.2 12.2% 4.5 11.4% 4.7

9 mom36m - 0.8% 1.9 1.5% 1.7 1.1% 1.3 0.0% 0.0

10 fgr5yr - -4.2% -4.1 -2.8% -1.5 -5.1% -3.2

11 lev + 7.1% 3.3 3.3% 0.9 9.1% 2.5 8.6% 2.7

14 quick + 1.1% 0.9 3.5% 1.6 -2.4% -1.1 1.9% 1.0

15 pchquick + -0.7% -1.1 -0.2% -0.2 -1.4% -1.2 -0.4% -0.4

16 salecash + -1.6% -1.2 -4.8% -1.8 -0.5% -0.2 -0.1% 0.0

17 salerec + 2.6% 1.7 5.3% 2.7 -1.6% -0.8 3.8% 2.2

18 saleinv + 5.4% 5.1 7.2% 4.4 6.8% 4.2 3.0% 2.1

19 pchsaleinv + 1.9% 1.3 6.4% 3.3 -1.2% -0.6 0.8% 0.5

20 cashdebt + -1.0% -0.6 -2.2% -0.9 -2.3% -0.9 1.0% 0.5

21 baspread - 6.0% 2.7 -2.8% -0.8 9.9% 2.8 9.9% 3.2

22 mom1m - -1.4% -3.7 -1.2% -1.3 -1.6% -1.7 -1.5% -1.9

24 mom12m + 0.7% 1.5 1.9% 2.1 -0.3% -0.3 0.4% 0.6

25 depr + 3.1% 2.5 3.8% 2.1 4.6% 2.6 1.5% 0.9

26 pchdepr - -1.9% -2.4 -2.4% -2.0 -2.6% -2.1 -0.9% -0.9

27 mve_ia - 2.8% 2.3 1.8% 1.0 1.0% 0.6 4.8% 3.1

28 cfp_ia + 0.0% 0.0 -0.9% -0.6 2.9% 1.8 -1.5% -1.1

29 bm_ia + -2.6% -2.2 -4.7% -2.6 -0.6% -0.3 -2.5% -1.6

30 sgr - -0.8% -0.6 -0.5% -0.2 -0.6% -0.3 -1.1% -0.6

32 IPO - -1.9% -1.3 -1.4% -0.6 0.6% 0.3 -4.2% -2.1

33 divi + -1.4% -1.2 -2.9% -1.5 -3.4% -1.7 1.3% 0.8

34 divo - -0.4% -0.5 1.3% 0.7 -1.1% -0.6 -1.3% -0.8

35 sp + -1.6% -0.8 1.9% 0.6 -3.4% -1.1 -2.9% -1.0

32

19

36

22

1980s 1990s 2000sALL YEARS

87 93 93

24

15

93

# abs{t-stat} 1.96

# abs{t-stat} 3.0

47

26

42

TABLE 8 (continued) – multidimensional 1-month ahead pricing of RPS by decade

# RPS

# RPS Pred. sign MALSRet t-stat. MALSRet t-stat. MALSRet t-stat. MALSRet t-stat.

36 acc - -3.3% -2.8 -4.4% -1.8 -3.2% -1.3 -2.5% -1.2

37 turn + 25.2% 11.1 29.2% 12.2 33.1% 13.8 16.1% 7.7

38 pchsale_pchinvt + -1.1% -0.8 -2.3% -1.2 -1.2% -0.6 -0.2% -0.1

39 pchsale_pchrect - 0.4% 0.7 -0.5% -0.4 1.4% 1.2 0.4% 0.4

40 pchcapx_ia - -2.2% -2.8 -1.5% -1.0 -1.9% -1.3 -3.1% -2.5

41 pchgm_pchsale + 1.8% 2.3 2.1% 1.6 1.4% 1.0 1.8% 1.5

42 pchsale_pchxsga + 1.6% 2.1 1.7% 1.2 -0.1% -0.1 2.9% 2.5

43 nincr + 2.7% 4.1 1.8% 1.6 4.7% 4.4 1.9% 2.0

44 indmom + 7.2% 6.0 5.5% 3.2 4.6% 2.7 10.3% 6.9

45 ps + -1.3% -1.6 -1.0% -0.7 -1.4% -1.0 -1.4% -1.1

46 dolvol - -10.0% -10.1 -10.6% -8.3 -13.0% -10.1 -7.3% -6.5

47 std_dolvol - -6.5% -3.1 -1.4% -0.3 -3.2% -0.7 -13.0% -3.4

48 std_turn - 10.5% 3.8 1.1% 0.3 8.1% 2.4 19.6% 6.6

49 sfe + -16.1% -8.6 -23.1% -10.0 -11.2% -5.9

50 nanalyst - 3.4% 1.1 14.4% 3.7 -4.2% -1.3

51 disp - -2.0% -1.6 -4.3% -2.4 -0.4% -0.3

52 chinv - -0.8% -0.8 1.7% 0.9 -2.5% -1.3 -1.3% -0.7

53 idiovol - 7.6% 3.1 0.6% 0.2 18.3% 4.7 4.7% 1.4

54 grltnoa - -0.5% -0.5 0.3% 0.2 -1.3% -0.8 -0.3% -0.2

55 rd + 1.2% 2.1 0.1% 0.1 1.5% 1.3 1.8% 1.8

56 cinvest - -1.0% -1.4 -1.4% -1.2 -2.2% -1.8 0.2% 0.2

57 tb + 0.9% 1.1 -1.4% -1.3 -0.8% -0.7 4.0% 4.1

58 cfp + -1.9% -2.1 -0.4% -0.2 -2.6% -1.6 -2.6% -1.7

59 roavol + 0.7% 0.8 -0.8% -0.5 0.2% 0.1 2.3% 1.6

60 lgr - -2.9% -2.5 -6.8% -3.6 0.7% 0.4 -2.7% -1.6

61 egr - -0.5% -0.6 -2.0% -1.1 -0.4% -0.2 0.5% 0.3

62 ill + 10.7% 1.8 -0.2% 0.0 8.0% 0.8 21.3% 2.3

63 age + -0.1% -0.1 3.3% 2.1 -1.4% -0.9 -1.7% -1.3

64 ms + -1.1% -1.5 -0.7% -0.6 1.7% 1.3 -3.4% -3.1

65 pricedelay + 1.3% 2.5 2.2% 2.1 1.4% 1.3 0.5% 0.5

66 rd_sale + 4.1% 1.3 2.4% 0.4 5.0% 0.9 4.7% 0.9

67 rd_mve + 6.3% 1.9 4.6% 0.9 8.5% 1.6 5.9% 1.3

68 retvol - -13.9% -6.7 -5.5% -1.8 -15.9% -5.2 -18.9% -7.0

69 herf - -3.0% -2.1 0.0% 0.0 -4.4% -2.2 -4.1% -2.3

70 grcapex - -2.0% -2.8 -2.6% -2.0 -3.4% -2.6 -0.5% -0.4

71 zerotrade + 12.3% 4.9 9.9% 2.2 10.8% 2.4 15.2% 3.9

72 chmom - 0.3% 0.6 0.3% 0.4 -0.4% -0.4 0.9% 1.1

73 roic + 1.7% 1.3 4.2% 2.0 3.0% 1.4 -1.2% -0.7

74 aeavol + 3.3% 6.1 4.0% 4.3 3.1% 3.2 2.9% 3.5

75 chnanalyst - 0.5% 0.7 1.5% 1.2 -0.2% -0.2

1980s 1990s 2000sALL YEARS

87 93 9393

43

TABLE 8 (continued) – multidimensional 1-month ahead pricing of RPS by decade

# RPS

# RPS Pred. sign MALSRet t-stat. MALSRet t-stat. MALSRet t-stat. MALSRet t-stat.

76 agr - -6.9% -4.2 -1.9% -0.8 -12.7% -5.2 -6.2% -2.9

77 chcsho - -2.0% -2.8 -2.0% -1.3 -1.5% -1.0 -2.3% -1.7

78 chpmia + 0.7% 0.7 -1.5% -1.0 2.8% 1.9 0.7% 0.6

79 chatoia + 1.0% 1.7 -0.2% -0.2 2.8% 2.3 0.5% 0.5

80 ear + 9.4% 13.8 7.3% 5.1 10.1% 7.2 10.4% 8.4

81 rsup + 7.4% 7.5 4.5% 3.2 6.1% 4.3 10.5% 8.5

82 stdcf - -2.2% -2.0 1.2% 0.7 -2.4% -1.4 -4.6% -3.0

83 tang + 2.3% 1.8 -0.6% -0.4 4.3% 2.7 3.0% 2.2

84 sin + 5.4% 2.7 4.4% 1.1 6.2% 1.6 5.4% 1.6

85 hire - -0.2% -0.2 1.2% 0.9 0.8% 0.6 -2.0% -1.7

86 cashpr - -0.2% -0.2 -3.0% -1.7 -0.5% -0.3 2.2% 1.4

87 roaq + 10.8% 7.7 10.1% 4.9 9.9% 4.8 11.9% 6.5

88 invest - -0.7% -0.6 -1.6% -0.8 -3.0% -1.4 1.8% 1.0

89 realestate + -0.4% -0.4 0.3% 0.2 -2.2% -1.4 0.5% 0.3

90 absacc - -2.1% -2.5 0.9% 0.6 -5.2% -3.3 -2.0% -1.4

92 chtx + 3.3% 4.0 7.7% 6.7 2.2% 1.9 0.8% 0.8

94 pctacc - -2.2% -1.5 2.2% 1.0 -4.2% -1.9 -4.1% -2.1

95 cash + 3.8% 3.5 2.7% 1.4 6.8% 3.5 2.3% 1.3

96 gma + 2.4% 1.2 2.9% 1.0 1.0% 0.4 3.1% 1.3

97 orgcap + 3.4% 2.2 3.8% 1.9 4.0% 2.0 2.7% 1.5

98 secured + 0.1% 0.0 0.2% 0.0 -1.1% -0.2 0.8% 0.2

99 securedind + 0.9% 0.8 4.6% 1.9 -1.3% -0.5 -0.3% -0.1

100 convind + -3.1% -5.0 -3.0% -3.1 -3.2% -3.3 -3.1% -3.6

1980s 1990s 2000sALL YEARS

87 93 9393

44

TABLE 9

Identification based on a distillation of the results in Table 6 and Table 8 of the RPS that we identify

as most versus least likely to reflect the rational pricing of economic risks versus mispricing.

Rational pricing of economic risks Mispricing

sue #6 pchcapx_ia #40 dy #5 turn #37 fgr5yr #10 chfeps #7

bm #8 ill #62 ep #3 dolvol #46 mom1m #22 pchsaleinv #19

lev #11 mve #4 saleinv #18 aeavol #74 nincr #43 mve_ia #27

sfe #49 agr #76 indmom #44 nanalyst #50

retvol #68 ear #80 std_dolvol #47 tb #57

rsup #81 std_turn #48 lgr #60

roaq #87 idiovol #53 ms #64

convind #100 zerotrade #71 stdcf #82

chtx #92 absacc #90

cash #95

MispricingRational pricing of economic risks

Estimated likelihood of multidimensionally priced RPS reflecting:

Next most likely Most likelyMost likely Next most likely Next most likely Next most likely

45

FIGURE 1

Cumulative number of firm-specific return predictive signals (RPS) publicly reported

by accounting, finance and other business academics, 1970-2010.

Panel A: Cumulative number of total RPS

Panel B: Cumulative RPS by type (accounting-based, finance-based, other-based)

An accounting-based RPS is one where the signal is in the firm‘s financial statements (e.g., accruals; cash flows; assets). A

finance-based RPS is one where the signal was directly or indirectly dependent on the firm‘s stock price (e.g., return

momentum; implied skewness in returns derived from option prices). The exceptions to these rules are that P/E is classified

as an accounting signal, book-to-market is classified as a finance signal, and any accounting signal interacted with a finance

signal is classified as a finance signal. Other-based signals are those that are neither accounting nor finance, such as labor

mobility, stock repurchases, or a stock‘s ticker symbol.

0

50

100

150

200

250

300

350

Cumulative # of ABTS vs. PBTS vs. OBTS, 1970 - 2010

Total RPS

0

20

40

60

80

100

120

140

160

Cumulative # of ABTS vs. PBTS vs. OBTS, 1970 - 2010

Accounting-based RPS Finance-based RPS Other-based RPS

46

FIGURE 2

Scatterplots and regressions describing the relations between unidimensional and multidimensional

MALSRets, and t-statistics on those MALSRets, of the form dk = f + g.bk + v. The variables bk and

dk are the unidimensional and multidimensional MASRets (or t-statistics on MALSRets) reported in

Table 6 for a given future return horizon. MALSRets (or t-statistics on MALSRets) estimated from

1-month ahead returns are denoted by blue diamonds; from 2-12 months ahead returns by red

squares; and from 13-36 months ahead returns by green triangles.

Panel A: Scatterplots of unidimensional MALSRets (x-axis) vs. multidimensional MALSRets (y-axis). The

dashed black line has a slope of 0.36, the average attenuation reported in panel B.

Panel B: Estimated slope coefficients g and the implied degree of unidimensional-to-multidimensional

MALSRet attenuation = 1 – g

-20%

-10%

0%

10%

20%

30%

-20% -10% 0% 10% 20% 30%

1 month ahead returns 2-12 months ahead returns 13-36 months ahead returns

1 month 2-12 months 13-36 months

f 0.66 0.38 0.47

t-stat.{f} (1.4) (1.1) (1.3)

g 0.38 0.36 0.35

t-stat.{g} (6.2) (5.8) (3.7)

Attenuation = 1 - g 0.62 0.64 0.65

t-stat.{1 - g} (10.1) (10.4) (7.0)

Future return horizon used in

estimating MALSRet

47

FIGURE 2 (continued) – dimensional attenuation

Panel C: Scatterplots of t-statistics on unidimensional MALSRets (x-axis) vs. t-statistics on

multidimensional MALSRets (y-axis).

Panel D: Estimated slope coefficients g and the implied degree of unidimensional-to-multidimensional

attenuation on the t-statistics on MALSRets = 1 – g

-15

-10

-5

0

5

10

15

-10 -5 0 5 10 15 20

1-month ahead returns 2-12 month ahead returns 13-36 months ahead returns

1 month 2-12 months 13-36 months

f 0.20 0.17 0.25

t-stat.{f} (0.6) (0.9) (1.5)

g 0.59 0.45 0.38

t-stat.{g} (7.9) (6.3) (4.5)

Attenuation = 1 - g 0.41 0.55 0.62

t-stat.{1 - g} (5.4) (7.8) (7.4)

Future return horizon used in

estimating t-stat.{MALSRet}

48

FIGURE 3

Mean unidimensional and multidimensional RPS MALSRets derived from 1-month ahead returns in

event time centered on the estimated month the RPS was published.

Panel A: Cumulative ln(1 + mean event month RPS MALSRets) starting 120 months prior to publication

and ending 120 months after publication. Beta and ep are not included because their

publication dates were before the start of our 1980-2012 window. In order to properly

aggregate across RPS with both positive and negative mean MALSRets, if a given RPS has a

negative mean monthly pre-publication MALSRet then we multiply all pre- and post-publication

monthly MALSRets for that RPS by –1

Panel B: Pre- vs. post-publication unidimensional MALSRets derived from 1-month ahead returns

Panel C: Pre- vs. post-publication multidimensional MALSRets derived from 1-month ahead returns

0%

20%

40%

60%

80%

100%

120%

140%

-12

0

-11

5

-11

0

-10

5

-10

0

-95

-90

-85

-80

-75

-70

-65

-60

-55

-50

-45

-40

-35

-30

-25

-20

-15

-10 -5 0 5

10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95

100

105

110

115

120

Mean monthly unidimensional MALSRets Mean monthly multidimensional MALSRets

t-stat.

Mean monthly MASLRet -2.8

Std.dev(monthly MALSRets)

Monthly Sharpe ratio 1.35 0.97 -0.38

0.58% 0.42% -0.16%

0.43% 0.43% 0.01%

Post-publication vs.

pre-publicationPre-publication Post-publication

t-stat.

Mean monthly MASLRet -3.1

Std.dev(monthly MALSRets)

Monthly Sharpe ratio

-0.03%

-0.33

Post-publication vs.

pre-publicationPre-publication Post-publication

0.38%

0.35%

1.08

0.24%

0.32%

0.75

-0.14%

49

FIGURE 4

Venn diagram of multidimensionally priced RPS compiled from Table 6 and Table 8. The solid red,

dashed black, and dotted blue circles contain the number of RPS that are multidimensionally priced

in 1-month ahead, 2-12 months ahead, and 13-36 months ahead returns, respectively. The RPS

tabulated in the yellow box inside the 1-month ahead red circle are RPS that are multidimensionally

priced in 1-month ahead returns measured over the full 1980-2012 period but are not

multidimensionally priced in all three decades (1980s, 1990s and 2000s). The RPS tabulated in the

grey box outside the 1-month ahead red circle are RPS that are not multidimensionally priced in 1-

month ahead returns measured over the full 1980-2012 period but are multidimensionally priced in

one or two of the 1980s, 1990s and 20002 (but not all three).

mve #4

ep #3

saleinv #18

agr #76

ear #80

rsup #81

roaq #87

convind #100

dy #5

pchcapx_ia #40

ill #62

sue #6

bm #8

lev #11

sfe #49

retvol #68

fgr5yr #9

mom1m #22

nincr #43

indmom #44

std_dolvol #47

std_turn #48

idiovol #53

zerotrade #71

chtx #92

cash #95

turn #37

dolvol #46

aeavol #74

1-month ahead

returns

2-12 months

ahead

returns

13-36 months

ahead returns

chfeps #7

pchsaleinv #19

mve_ia #27

nanalyst #50

tb #57

lgr #60

ms #64

stdcf #82

absacc #90