i

Empirical Tests of Asset Pricing Models with Individual Stocks

Narasimhan Jegadeesh and Joonki Noh ☆

Goizueta Business School

Emory University

May 12, 2013

First version: March 15, 2013

☆Narasimhan Jegadeesh is the Dean’s Distinguished Professor at the Goizueta Business School,

Emory University, Atlanta, GA 30322, and NBER, ph: 404-727-4821, email:

Jegadeesh@emory.edu. Joonki Noh is a Doctoral student at Emory University, ph: 734-276-

3162, email: joonki.noh@emory.edu.

ii

Empirical Tests of Asset Pricing Models with Individual Stocks

Abstract

We develop an instrumental variables methodology to test asset pricing models using

individual stocks. Simulation evidence indicates that this methodology yields unbiased estimates

and that the tests are well specified in finite samples. We use this method to test a number of

recently proposed asset pricing models. We find that the market risk under the CAPM, cash flow

risk and discount rate risk, and systematic illiquidity risk are not priced. In Fama-French three-

factor model, we find significantly positive market risk premium but significantly negative HML

risk premium.

3

One of the fundamental concepts of financial economics is that capital market investors

are compensated for higher systemic risks through higher returns. While this basic concept is

well accepted, there is very little agreement on the specific risk factors that indeed command risk

premiums. A number of recent papers propose a variety of risk factors that in theory should

explain the cross-sectional differences in expected returns. Some of these factors are human

capital risk (Jagannathan and Wang, 1996), productivity and capital investment risk (Cochrane,

1996, Chen, Novy-Marx and Zhang, 2011), different components of consumption risk (Lettau

and Ludvigson, 2001, Ait-Sahalia, Parker, and Yogo,2004 and Li, Vassalou, and Xing, 2006),

cash flow risk and discount rate risk (Campbell and Vuolteenaho, 2004) and systematic

illiquidity risk (Pastor and Stambaugh, 2003 and Acharya and Pedersen, 2005).

These studies also typically report empirical evidence supporting the hypothesis that the

risk factors that they propose are related to the cross-sectional differences in expected returns.

Their empirical tests use selected portfolios as test assets, and in many instances these portfolios

are sorted by size and book-to-market ratios to obtain the cross-sectional variation in average

returns. The ability of so many different factors to explain the cross-sectional differences in

expected returns have generated some skepticism in the literature. Lewellen, Nagel and Shanken

(2010) show that the strong factor structure inherent in the test portfolios enables any factors that

are even weakly correlated with the characteristics used to sort the test portfolios to explain the

differences in average returns across them regardless of the economic merits of the underlying

theory. Daniel and Titman (2012) find that the factors proposed in various papers exhibit low

time-series correlation with one another and argue that it is unlikely that all these factors can

simultaneously explain the cross-section of expected returns.

We develop a procedure to test asset pricing models using individual stocks as tests

assets, and thus our procedure is not susceptible to the low dimensionality problems inherent in

tests with portfolios. The main reason that the asset pricing tests in the literature typically use

portfolios is that the errors-in-variables (EIV) problem is less severe with portfolios than with

individual stocks. Our approach uses the instrumental variable technique to address the EIV

problem and obtain consistent estimates of risk premiums.

The EIV problem arises because factor sensitivities are estimated and employed as

independent variables in the subsequent cross-sectional regressions. The estimation error in

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factor sensitivities bias estimates of factor risk premiums. We also estimate factor sensitivities,

but in our approach we use factor sensitivities estimated in even months as independent variables

and the corresponding sensitivities estimated in odd months as their instruments or vice versa.

Since the errors in even month estimates are uncorrelated with those in the odd month estimates,

the IV estimator of factor risk premiums are consistent. Moreover, simulation evidence indicates

that this estimator yields unbiased estimates even in small samples.

Some papers propose a different approach to correct for the EIV problem. This

alternative approach assumes a factor structure for stock returns and attempts to adjust the OLS

estimator to undo the EIV bias. For instance, Litzenberger and Ramaswamy (1979) assume a

single-factor model and show that the EIV bias can be corrected with a modified regression.

More recently, Jagannathan and Wang (1998) propose an EIV correction assuming that

stock returns follow a multifactor model, and Chordia, Goyal and Shanken (2012) extend this

method to unbalanced panel data. Their correction method assumes that the residual returns from

a multifactor model are uncorrelated across stocks asymptotically as the number of assets

increases without bound. The empirical implementation of the correction method requires the

empiricist to correctly specify the number of common factors. For example, when Chorida,

Goyal and Shanken (2012) implement this correction procedure to estimate the Fama-French

three-factor model, they assume that after accounting for the three Fama-French factors, the

residuals do not contain any common factors.1 The procedure does not yield consistent

regression estimates if the number of common factors is understated and therefore residual

returns are cross-sectionally correlated.

Empirical studies generally find that there are more than three common factors in stock

returns. For example, Trzcinka (1986) finds that as he increases the number of stocks n in the

sample, his tests fail to reject the hypothesis that there are “at least 1/2 n (factors) in most cases.”

Therefore, the hypothesis that residual returns from a three-factor model do not have a common

factor is likely not tenable. Practitioners also seem to find that many more than three common

1 The factor structure that these papers assume pertains to the statistical description of common components in stock

returns. In general, the number of factors in the return generating process can be larger than the number of factors

that are priced in the market. In fact, it is possible that a large number of common factors are required to capture the

comovement in stock returns, but empirically none of them are priced.

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factors are required to capture the covariance structure for US stocks. For example, the widely

used Barra model contains 68 common factors.

The IV approach that we propose does not require the empiricist to specify a particular

factor structure and it is consistent for any finite number of common factors in the return

instrumental. To evaluate the small sample properties of the IV estimator we conduct a number

of simulation experiments. We choose the simulation parameters to be similar to that in the data.

The simulation experiment with a single factor model finds that the well-known bias due

to EIV when we use ordinary least squares regressions to estimate the market risk premium. We

find that the estimate is significantly biased even when we use the CGS approach to correct for

EIV bias. In contrast to the OLS bias, the CGS correction biases the estimate away from zero.

We find similar biases in risk premium estimates under the three factor model as well.

The IV approach yields unbiased small sample estimates both in the cases of the single

factor model as well as the three-factor model. We also examine the small sample properties of

tests based on the IV approach. We find that the conventional tests are well specified in small

samples.

We apply the IV approach to empirically test whether several of the risk factors proposed in

the literature command risk premiums with individual stocks as test assets. The specific risks that

we consider are the market risk in the CAPM, market, SMB and HML risks in the Fama-French

three-factor model, cash flow and discount rate risk proposed by Campbell and Vuolteenaho

(2004), and the systematic illiquidity risk proposed by Acharya and Pedersen (2005). We find

that the premiums for the market risk in the CAPM, cash flow and systematic illiquidity risks are

not priced. However, when we test the three-factor model, we find that the market risk

commands a significantly positive risk premium after controlling for SMB and HML risks. We

also find that SMB risk is not priced but HML risk earns a negative risk premium.

The rest of the paper is organized as follows: Section I presents our methodology. Section II

examines the finite sample properties using simulation experiments. Section III presents the

empirical tests and Section IV concludes.

I. Methodology

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A number of asset pricing models predict that expected returns on risky assets are linearly

related to their covariances with certain risk factors. A general specification of a K-factor asset

pricing model can be written as:

( ) ∑ (1)

where ( ) is the expected excess return on stock i, is the sensitivity of stock i to factor k,

and is the risk premium on factor k. is the excess return on the zero-beta asset. If riskless

borrowing and lending are allowed, then the zero-beta asset earns the risk-free rate and its excess

return is zero, i.e.

The CAPM predicts that only the market risk is priced in the cross-section of average returns.

Multifactor models identify additional risk factors that capture variations in investment

opportunity sets. For example, the models by Pastor and Stambaugh (2003) and Acharya and

Pedersen (2005) predict that the systematic illiquidity risk would be priced. Fama-French

propose an empirically motivated multifactor model with book-to-market and size risk as

additional priced factors.

Empirical tests of various models typically use Fama-MacBeth two-stage regression

procedure to test whether various risk factors are priced. The first stage estimates the factor

sensitivities ( ) using time-series regressions and the second stage runs the following

cross-sectional regressions where realized excess return is the dependent variable:

∑ (2)

Since is estimated with error, the OLS estimates of the coefficients of regression (2) are

biased due to the EIV problem. To mitigate such bias, the literature typically uses selected

portfolios as test assets rather than individual stocks since portfolio betas are estimated more

precisely.

The use of test portfolios presents a different set of problems. The test portfolios are typically

sorted on a few characteristics such as size and book-to-market. Sorting on such characteristics is

important to generate the reasonable variation in expected returns across test portfolios.

However, as Lewellen, Nagel and Shanken (2010) point out, the strong factor structure inherent

in such test portfolios enables any factors that are even weakly correlated with the sorting

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characteristics to explain the differences in average returns across them regardless of the

economic merits of the underlying theory.

Moreover, it is possible that the statistical significance and economic magnitudes of risk

premiums estimated in regression (2) could critically depend on the procedure used to form the

tests portfolios. For example, the Fama and French size and book-to-market factors are

significantly priced when test portfolios are sorted based on these characteristics, but they do not

command significant risk premiums if test portfolios are sorted only based on momentum.

This paper proposes a methodology that uses individual stocks as test assets. The use of

individual assets preserves the dimensionality of the variation in expected returns that we

observe in the stock market. Also, since the tests use all listed stocks individually, the results will

not be dependent on subjective choices made in the construction of test portfolios.

To address the EIV problem, we propose an instrumental variables approach. To describe out

estimator, we rewrite regression (2) as:

(3)

where is the vector of realized excess returns, is the ( +1) matrix containing the

intercept and K factor loadings, and is the vector of factor risk premiums if the N stocks is

used. The OLS estimator of is not consistent because of the well-known EIV problem. The

following instrumental variables estimator that we propose yields consistent estimates:

( )

(4)

where:

is the matrix of factor sensitivities estimated over odd months,

is the matrix of factor sensitivities estimated over even months and

otherwise.

We use odd month estimates as instruments when month t is even and the even month

estimates otherwise so that the measurement error in factor sensitivities is not correlated with

firm-specific return in month t. The appendix shows that this estimator is consistent.

Several papers in the literature propose an alternate approach to address the EIV problem.

Since the measurement error in factor loadings is the source of the EIV bias in the second stage

regression (3), one could in principle undo the EIV bias if we can correctly estimate the

variance-covariance matrix of the estimation errors. Intuitively, in the case of the CAPM or any

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single factor model, the measurement error in betas bias the slope coefficient estimate in

regression (2) by a factor equal to the variance of true betas divided the variance of estimated

betas (which equals the variance of the true betas plus the variance of measurement errors). So if

one can obtain a consistent estimate the variance of measurement errors in betas then one could

correct for the EIV bias and obtain the consistent estimates of risk premiums in regression (2).

Litzenberger and Ramaswamy (1979) apply an EIV correction to estimate the CAPM risk

premium with individual stocks. Their correction assumes that the market model residuals do not

contain any common factors, which is counterfactual. Also, their correction cannot be directly

extended to the cases of multifactor asset pricing models where factor loadings with respect to

more than one factor are estimated with errors.

More recently, Jagannathan and Wang (1998) propose an EIV correction assuming a

multifactor model and Chordia et al. (2012) extend this method to unbalanced panel data. For

example, Chordia et al. empirically apply their EIV correction assuming that the factor structure

in returns is fully captured by the Fama-French three-factor model, which implies that the

residual returns from the Fama-French model residual returns do not contain any common

factors.

A critical shortcoming with these methods is that these estimators are consistent only when

the number of common factors assumed to compute the EIV correction equals the number of

common factors in actual stock returns. For example, the EIV correction assuming the Fama-

French factor model would not be consistent if there are additional common factors (which are

not necessarily priced) besides the Fama-French factors. As we discussed earlier, Trzcinka

(1986) and others find that U.S. stock returns contain more than three common factors and hence

any three factor model will not fully capture all common factors. Also, later on we directly test

the hypothesis that the Fama-French factor residuals do not contain any common factors, and

reject it.

II. Small Sample Properties

Using simulations, this section examines the biases and small sample properties of different

variations of the two-stage methodology to estimate the factor risk premiums under the CAPM

and the Fama-French three-factor model. The three variations that we consider are:

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(i) FM: Fama-MacBeth methodology;

(ii) CGS: Fama-MacBeth methodology with EIV correction as in Chordia et al. (2012), and

(iii) IV: The instrumental variable methodology that we propose.

We use individual stocks as tests assets.

A. Return generating process

We consider a general setting where K common factors are priced. Under this model, we

can specify realized stock returns as:

∑ (5)

where :

Time t excess return of stock i;

Expected return of stock i

Sensitivity of stock i to the priced factor k;

Time t realization of the factor k; ( ) ; and

idiosyncratic return of stock i at time t.

Under the K-factor model, the expected excess returns are given by:

( ) ∑ (6)

where is the risk premium on the factor. The K-factor asset pricing model imposes no

restriction on the covariance structure of idiosyncratic returns across assets. In fact, idiosyncratic

returns would in general be correlated across assets. The EIV correction methods implemented

by Chordia et al. (2012) impose the assumption that idiosyncratic returns are uncorrelated across

assets. However, our IV approach does not impose any restrictions on the covariance structure of

idiosyncratic returns. To evaluate the small sample performance of the IV approach and the CGS

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approach, we assume in our simulations that idiosyncratic returns are generated by the following

model with J common factors:2

∑ (7)

where :

Sensitivity of stock i to the unpriced factor j;

Time t realization of the factor.

B. Simulation Experiments: Parameters and Design

We choose the parameters for the simulations based on the covariance structure in the actual

data over the sample period that spans January 1956 through December 2012. First, we

determine the parameters that are appropriate for simulation experiments for the single factor

model. Specifically, our simulation experiments match the average market risk premium, the risk

free rate, the distribution of betas, volatility of firm-specific returns, and the factor structure of

the residuals across stocks from the data.

To determine these parameters from the data, we use the CRSP value-weighted index as the

market index and the short-term T-bill rate as the risk free rate. We run the market model

regression for each stock to compute betas and firm-specific returns. We extract the first two

principal components from firm-specific returns using the approach proposed by Connor and

Korajczyk (1986).3 We then compute the sensitivities of the residual returns of all stocks to these

common residual factors by running time-series regressions corresponding to regression (7). We

use the regression estimates to determine the mean and standard deviations of and , and

.

2 Both the K priced common factors in equation (6) and the J unpriced common factors in equation (7) contribute to

return covariances across stocks. However, the sensitivities to the unpriced common factors do not command higher

expected returns. 3 We determine the number of residual factors in our simulations based on the three information criteria developed

by Bai and Ng (2002). Table A.1 in appendix presents these results. We can also reject the null hypothesis that there

are less than two common factors in the residual using the testing procedure proposed by Connor and Korajczyk

(1993) at any conventional level of significance.

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We then conduct the simulation with a sample of N stocks and a sample period of T months

as follows:

1) Randomly generate beta, and , and for each stock from normal

distributions with means and standard deviations equal to the corresponding sample

means and standard deviations from the data. For , we take the absolute values of the

random draws to make it positive.

2) Generate market returns for each month as a random draw from a normal

distribution with mean and standard deviation equal to the sample mean and standard

deviation from the data.

3) Generate from independent normal distributions with mean equal to

corresponding sample mean and standard deviation equal to one.4

4) Generate for each stock from independent normal distributions with mean

zero and standard deviation corresponding to the value generated in step (1).

For each stock i we compute returns in month t as:

(8)

We repeat simulation 3,000 times each for N =20, 50, 100, 200, 500 and 2000, and T=120 and

240.

For the first stage regression in the simulations, we fit the following market model

regression for each stock to estimate beta:

( ) (9)

For the Fama-MacBeth and for CGS approaches, we fit the regression over all T months.

For the IV approach, we fit the regression separately for odd and even months.

In the second stage, we fit the following cross-sectional regression for each month t:

( ) (10)

4 The principal components are scaled to have standard deviation equal to 1.

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where is the estimate from regression (9). For the Fama-MacBeth approach, we estimate the

OLS parameters each month. For the CGS approach, we apply their EIV correction each month.

For the IV approach, we use equation (4) to estimate the parameters. We compute Fama-

MacBeth standard errors (FMSEs) for all three approaches.5

We carry out the three-factor model simulation experiments analogously, but in addition

to market returns and market beta, we also generate factors and factor sensitivities corresponding

to Fama-French SMB and HML factors. We match means and standard deviations for these

parameters in the simulations to what we observe in the data. We then carry out the two-stage

procedure to estimate The Appendix presents more details on the

simulation design.

C. EIV-induced Bias

We examine the EIV-induced biases with number of stocks in the sample ranging from 20 to

2000 stocks. We use sample period lengths of 120 and 240 months. The median length of time

that a stock in our sample is publicly traded is 148 months and the mean length is 186 months

and the sample periods in our simulations are representatives of these lengths. The average

difference between the estimated price of risk using the two-stage procedure and the “true” price

of risk in the simulations is the EIV-induced bias.

Panel A1 of Table 1 presents the EIV-induced biases for the single factor model. With

T=120, the slope coefficient estimate using the FM procedure is biased towards zero for all

sample sizes, reflecting the well-known EIV problem. The average bias ranges from -.168 to -

.123 as we vary the number of stocks. The magnitude of the bias is not correlated with the

number of stocks in the sample, which is consistent with theoretical results.

5 Chordia et al. (2012) find that, in addition to the EIV correction, the one-month bias correction is also necessary

for the OLS estimate of the market risk premium when the panel is unbalanced or when the idiosyncratic

disturbances are conditionally heteroskedastic. In our simulations, we use the balanced panel with homoskedastic

disturbances, thus the one-month bias correction is not needed for the OLS estimate. However, we find that the one-

month bias plays a non-negligible role in computing the FMSEs even for the balanced panel with homoskedastic

idiosyncratic disturbances. Thus we also perform the one-month bias corrections for the FM and CGS approaches. It

is worthwhile to emphasize that the IV approach does not require the one-month bias correction since it uses the

even-month betas for odd-month cross-sectional regressions as instruments and vice versa.

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Panel A2 presents the biases with T=240. The bias ranges from -.103 to -.074 for different

cross-sectional sizes. The bias here is smaller than that in Panel A1 because the betas are now

estimated more precisely in the first stage regression because of the longer time-series.

In contrast, the biases are all positive when the CGS EIV correction is applied to the FM

estimates. For example, the biases range from .141 to .186 with T=120 and from .053 to .121

with T=240. Therefore, the CGS correction induces an upward bias to the univariate regression

estimates.

Why does the CGS correction induce an upward bias? It does so because the correction

assumes away the effect of the common factors in the residuals. Intuitively, the EIV problem

biases the slope coefficient towards zero by a factor related to the cross-sectional variance of the

measurement errors in betas. Empirically, the average cross-sectional correlation of residuals

due to common factors is positive. Therefore, the cross-sectional variance of measurement errors

in betas is smaller in the presence of common factors in residuals than that computed by

assuming away the residual correlation. As a result, the CGS procedure overcorrects for the EIV

bias and it leads to an upward biased slope coefficient estimate.

Table 1 also presents the biases under the IV approach. The results here indicate that the IV

approach yields unbiased estimates of the risk premium even when the sample period is only

120 months. Since the IV estimator makes no assumption about the number of common factors

in the residuals, the result is not dependent on the particular factor structure in residuals that the

simulation assumes.

The simulation results show that the biases decrease with the number of time-series

observations under both FM and CGS procedures. How long should the time-series extend

before the bias becomes negligible under these procedures? Figure 1 examines this issue. It

presents the percentage bias under the three procedures against the number of months in the

sample period, with the cross-sectional size of N=500. The vertical axis reports the bias as a

percentage of the true risk premium used in the simulations. The downward bias under the FM

procedure is about 10% even for T=300 months and the upward bias after CGS correction is

about the same magnitude. The IV procedure yields unbiased estimates for T=120 and the bias is

less that 5% for a sample period of five years (T=60).

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Panel B of Table 1 presents the results for the three-factor model. In univariate regressions,

the EIV problem always biases the slope coefficient estimates towards zero, but the directions of

biases are in general indeterminate in multivariate regressions. The simulation results indicate

that the FM approach biases all slope coefficients towards zero. The CGS correction leads to an

upward bias in all but one case. When we increase T=240, the direction of biases with the FM

and CGS procedures is the same as with T=120, but the magnitudes of the biases become

smaller. In contrast, the IV estimator is unbiased even with T=120.

As Figure 1 illustrates, with a sufficiently long sample period, the measurement errors in

beta estimates would converge to zeros and hence the FM procedure would yield consistent

estimates of the risk premiums. However, when we use individual stocks as test assets, there are

limitations on the length of the sample period that we can use to estimate betas. Over half the

firms in our sample are traded on the exchanges for less than 15 years after imposing the 60-

month survivorship (see Table 3). We also cannot restrict the sample to firms that survived for

say 240 months because such a restriction would lead to severe survivorship bias. Therefore, we

need a procedure such as the IV approach that yields unbiased estimates in small time-series

samples.

D. Small sample distribution

We are primarily interested in using the IV estimator to test whether risk premiums

associated with various factors are reliably different from zero. We propose to use the

conventional t-statistic to test this hypothesis. We compute the t-statistic as follows:

(11)

where is the IV estimate of the risk premium and is the corresponding Fama-MacBeth

standard error. This subsection examines the small sample distribution of the t-statistics.

To examine the small sample distribution, we simulate the data under the null hypothesis.

Specifically, we follow the same steps as in the last section to generate simulated data, but we set

all risk premiums equal to zero. We then examine the percentage of simulations when the t-

statistics are significant at the various levels (two-sided) using critical values based on the

asymptotic normal distribution of the test statistic.

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Table 2 presents the test sizes for the CAPM and the Fama-French three factor model. For

example, in both single factor model and multifactor model, the test sizes at the 5% level of

significance are between 4.2% and 5.9%. The tests sizes are not statistically different from the

theoretical sizes for any value of N for both T=120 and T=240. Therefore, we conclude that the

conventional t-test statistics based on the IV approach can be used to draw reliable inferences

about the significance of risk premiums.

III. Empirical Tests

A. Data

We obtain stock return and market capitalization data from CRSP files and book value data

from COMPUSTAT during the sample period that spans January 1956 through December 2012.

We exclude American depository receipts (ADRs), shares of beneficial interest, Americus Trust

components, close-end funds, preferred stocks, and real estate investment trusts (REITs). We

include only common stocks in our sample (CRSP share codes of 10 or 11). We also exclude

stocks with prices below $1 and market capitalization less than $500,000 at the end of a month

from the sample for the following month. We include all stocks that meet these criteria for which

returns and book values are available.

Table 3 presents the summary statistics of the stocks in the sample universe. There were a

total of 9057 different stocks which entered and left the sample at different points in time. On

average, there are 2473 stocks per month in the sample.

Figure 2 presents the frequency distribution of the length of time that a stock is in the sample

before imposing the 60-month survivorship. The average life of a firm in our sample universe is

186 months after imposing the 60-month survivorship (see Table 3).6 Twenty five percent of the

firms have less than 94 monthly observations.

B. CAPM and Fama-French three-factor models

We first estimate the premium for bearing market risk in a single factor model. We

include only stocks with at least 60 months of data in our sample. For the Fama-MacBeth

approach, we fit the market model regression with monthly returns to estimate betas for each

6 Before imposing the 60-month survivorship of firms, we find that the average life of a firm is 114 months.

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stock. For the IV approach, we estimate odd and even month betas using all available data for

odd and even months. To compute the t-statistics, we employ thet Newey-West HAC

covariance matrix estimators.

Table 4 reports the estimates of the market risk premium over the entire sample period

and over two roughly equal subperiods. The point estimate of the market risk premium is

positive but not reliably different from zero. Therefore, our results using individual stocks as

test assets are consistent with the findings in the literature using portfolios.

Table 5 reports the regression estimates of risk premiums for the Fama-French three-

factor model. The risk premium on the SMB factor is not different from zero in any sample

period. Interestingly, the only risk premium that is positive and significant is the market risk

premium. Over the full sample period, the market risk premium is .557%, which is

significant at conventional levels. The point estimates of the market risk premiums in the two

subperiods are about the same magnitudes although the estimate is not significant in the first

subperiod but significant at the 10% level in the second subperiod.

Another striking result is that the risk premium on the HML factor is significantly

negative.7 The negative risk premium on HML is inconsistent with the interpretation in the

literature that HML represents a distress risk factor. We leave further analysis of this result

for future research.

C. Cash flow (CF) beta and discount rate (DR) beta

Market prices change due to the changes in discount rates and changes in expectations

about future cash flows. Therefore, we can decompose the market return ( ) at time t as:

= (12)

where are the cash flow and discount rate components. Campbell and Vuolteenaho

(2004) decompose a stock’s beta into the cash flow beta, or sensitivity to the cash flow

component, and discount rate beta, or sensitivity to the discount rate component. So stock i’s

beta can be decomposed as:

7 Chordia et al. (2012) report a similar result when they assume constant betas over the sample period.

17

= , (13)

where

= ( )

[ ( )] and (14)

= ( )

[ ( )],

and ( ) is the expected market return at the end of the previous period. In this

decomposition, is the cash flow beta and is the discount rate beta. Campbell and

Vuolteenaho (2004) present a model where the cash flow betas command a bigger risk premium

than discount rate betas.

Campbell and Vuolteenaho (2004) estimate the cash flow and discount rate components

of the market returns using a VAR model with the following four state variables: stock market

excess returns, yield spread between long- and short-term bonds, 10-years smoothed price

earnings ratio, and small-stock value spread. Campbell and Vuolteenaho find empirical support

for their model using 25 Fama-French portfolios and 20 risk-sorted portfolios as test assets. They

conclude that their model could explain the size and book-to-market anomalies.

We test the Campbell and Vuolteenaho model using individual stocks as test assets. We

first fit the VAR model using the same model as Campbell and Vuolteenaho (2004) and

decompose the market returns into cash flow and discount rate components. We use the sample

analogues of variances and covariances to estimate cash flow betas and discount rate betas. We

then fit the following cross-sectional regression each month:

(15)

( )

(16)

18

where:

N×3 matrix with unit vector as the first column and month t estimates of cash

flow and discount betas for the N stocks as the second and third columns. We

estimate the even betas using all even month observations.

Analogous to , estimated with observations in odd months.

if month t is odd and otherwise.

We use the Fama-MacBeth approach to compute the estimates of risk premiums and their

standard errors. We use these estimates to test the null hypotheses that

. The alternate hypotheses implied by the Campbell and Vuolteenaho model

are .

Table 6 presents the regression estimates to test these hypotheses. To facilitate

comparison, the sample period in this subsection is from January 1929 through December 2001,

which is the same as that in Campbell and Vuolteenaho (2004). With individual stocks, the risk

premiums on both discount rate beta and cash flow beta are not significantly different from zero

over the entire sample period.

Campbell and Vuolteenaho split the sample period into two subperiods, one from

January1929 to June 1963 and the other from July 1963 to December 2001.They report that the

risk premium on the cash flow beta is particularly large in the recent sample period. With

individual stocks, the point estimates of the cash flow risk premium is larger in the second

sample than in the first sample, but these risk premiums are insignificant in both subperiods.

For comparison, Table 6 also presents the risk premium estimates with portfolios as test

assets. We use the same 45 test portfolios as Campbell and Vuolteenaho in this test, say 45-CV

portfolios. Using the FM approach we find that cash flow risk premium is significant in the full

sample period. This risk premium is larger in the second subperiod, which is consistent with the

findings in Campbell and Vuolteenaho (2004).

Even with the 45-CV portfolios as test assets, the betas that we use in the second stage

regression contain measurement errors. Therefore, we estimate the risk premiums using the IV

approach. For the full sample, the IV estimates lead to similar inference as the FM estimates.

However, when we estimate the regression within subperiods, our results are different. The cash

flow risk premiums are about equal in the two subperiods but both are significant only at the

19

10% level. In particular, we do not find that the cash flow risk premium is significantly larger in

the second subperiod.

More important than the portfolio results is the fact that the risk premiums are not

different from zero when we use individual stocks as test assets. It is likely that the significant

premiums that Campbell and Vuolteenaho (2004) found are attributable to the correlation

between cash flow betas and some omitted variables. As Lewellen et al. (2010) point out, the use

of portfolios as tests assets reduces the dimensionality of expected return variations, and as a

result any factors that is only weakly correlated with the true driver of expected returns may

seem significant.

D. Liquidity-adjusted CAPM

Acharya and Pedersen (2005) propose a liquidity-adjusted capital asset pricing model

(LCAPM). In this model, in addition to the CAPM beta, the sensitivities of stock returns to a

market-wide illiquidity and covariances of a firm’s illiquidity to market returns and market

illiquidity are also priced in the cross-section of expected returns. Specifically, the unconditional

expected excess return under the LCAPM is:

[ ] [ ] ( ) , (17)

where is risk-free rate, is the illiquidity cost, the net risk premium is defined as

with the market return and aggregate illiquidity cost, and the associated

betas take forms of

( ( ))

( ( ) ( ) ) (18)

( ( ) ( ))

( ( ) ( ) )

( ( ))

( ( ) ( ) )

( ( ) ( ))

( ( ) ( ) )

The term [ ] is the compensation for firm-specific illiquidity level, which is a compensation

for holding an illiquid asset as in Amihud and Mendelson (1986). The betas in this expression

20

are the compensation for different types of systematic risk. Since the risk premiums are the same

for all betas, we can define a net beta, say LMKT beta, as in AP as follows:

(19)

To be consistent with AP, we compute firm-specific illiquidity costs using the same

approach as theirs. Specifically, we use the monthly illiquidity measure ( ) proposed by

Amihud (2002) as a proxy for illiquidity cost, and adjust the costs for inflation, and cap it by

30%. Thus for stock i in month t, we have

∑

(20)

( )

where is the return and is the dollar volume (in millions) on the dth

trading day, and

is the number of non-zero trading days in the month. is the ratio of the capitalizations

of the market portfolio at the end of month t-1 and of the market portfolio at the end of July

1962.

For each month t, we compute liquidity costs using equation (20). For each stock, we

estimate and separately using the generalized method of moments (GMMs)

based on the moment conditions in equation (18), and then sum the estimates to obtain as

in equation (19). Finally, we estimate the premiums using the following cross-sectional

regression:

(21)

The IV estimator of the premiums is given by:

( )

( )

where:

21

N×3 matrix with unit vector as the first column and and estimated even-

month LMKT betas for the N individual stocks as the second and third columns,

respectively. We estimate the even-month LMKT betas using all even-month

observations.

Analogous to , estimated using all odd-month observations.

We use the Fama-MacBeth approach to compute the point estimates and standards errors of

coefficients from the time-series of monthly risk premium estimates.

Panel A of Table 7 presents the regression estimates of premiums for the full sample

period. When individual stocks are used as test assets, the premium on the Amihud Illiquidity

measure is .099% and .102% with the FM and IV approaches, respectively, and these estimates

are reliably different from zero. The slope coefficient on is .081%, which is not reliably

different from zero. This point estimate is also economically fairly small. These results indicate

that firm-specific illiquidity level, which is a firm characteristic, earns a premium but a stock’s

sensitivity to market-wide illiquidity does not earn a risk premium.

One possible explanation for the difference between our findings and AP’s findings that

the liquidity risk is priced is that the IV approach is not sufficiently powerful. To examine this

explanation, we conduct our tests using the two sets of test portfolios that are the same as those

used by AP. The first set of test portfolios is the 25 illiquidity portfolios and the second set is the

25 (illiquidity) portfolios. We follow the same procedure as AP to construct these portfolios.

Table 7 also presents the results with these tests portfolios. The slope coefficients on

are 1.465% and 1.490% with the 25 illiquidity portfolios and 25 (illiquidity) portfolios,

respectively. Both these slope coefficients are reliably different from zero. These results are

consistent with the AP’s results. Therefore, the difference between the findings with individual

stocks as test assets and with portfolios as test assets is not due to the lack of power of the IV

estimator.

IV. Conclusion

The empirical tests in the literature typically use portfolios rather than individual stocks

as test assets to mitigate the errors-in-variables problems. Specifically, since the factor loadings

of portfolios are estimated more precisely than those of individual stocks, the estimates of factor

risk premiums in the second stage of the Fama-MacBeth regressions will be more biased if one

were to use individual stocks rather than portfolios. The cost of using portfolios in the empirical

22

tests is that they limit the number of dimensions along which expected returns vary. Therefore,

even factors that are not important from an economic perspective can easily explain the cross-

section of expected returns in a statistical sense.

This paper develops an instrumental variables methodology to obtain unbiased estimates of

risk premiums using individual stocks as test assets. Our simulation evidence indicates that this

methodology yields unbiased estimates and that the tests are well specified in finite samples.

We apply the IV approach to empirically test whether several of the risk factors proposed

in the literature command risk premiums. The specific risks that we consider are the market risk

in the CAPM, market, SMB and HML risks in the Fama-French three-factor model, cash flow

and discount rate risk proposed by Campbell and Vuolteenaho (2004), and the systematic

illiquidity risk proposed by Acharya and Pedersen (2005). We find that the premiums for the

market risk in the CAPM, cash flow and systematic illiquidity risks are not priced. However,

when we test the three-factor model, we find that the market risk commands a significantly

positive risk premium after controlling for SMB and HML risks. We also find that SMB risk is

not priced but HML risk earns a negative risk premium.

The asset pricing models that we test in this paper were all proposed after several papers

in the literature found that the premium on the market risk in the context of the CAPM is not

statistically different from zero. Fama and French (1992) propose a three-factor model that

augments the market factor with SMB and HML factors to better explain the cross-sectional

differences in average returns. Several subsequent papers build theoretical models to identify risk

factors. The cash flow and discount rate risks proposed by Campbell and Vuolteenaho (2004)

and the systematic illiquidity risk proposed by Acharya and Pedersen (2005) are two such

examples. However, none of these factors are priced when we estimate their risk premiums using

individual stocks. Somewhat surprisingly, only market risk earns a positive risk premium in a

three factor model.

23

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26

Figure 1: Percentage Bias versus Firm Time-series Length T

The figure presents the percentage biases in the three premium estimates of the market risk against the

length of firm time series. The simulations are conducted with N=500 and varying T under the CAPM.

Figure 2: Frequency Distribution of Firm Time-series Lengths

This figure provides the frequency distribution of firm time-series lengths in the sample universe before

imposing the 60 months survivorship of firm time-series.

0 100 200 300 400 500 600 700 800 900 1000-40

-20

0

20

40

60

80

Length of Firm Time-series (T)

Bia

s /

Tru

e M

KT

Pre

miu

m [

%]

Percentage Bias vs Time-series Length T

FM

CGS

IVFM

0 100 200 300 400 500 600 7000

200

400

600

800

1000

1200

1400

1600

1800

Number of Months

Nu

mb

er o

f Fir

ms

Histogram for Firm Time-series Length (# of Bins = 66)

27

Table 1: Simulation Evidence

Using simulations, this table presents the biases and Fama-MacBeth standard errors (FMSEs) of three

different estimators of risk premiums for the CAPM (Panel As) and Fama-French three factor model

(Panel Bs) when different cross-sectional sizes (N=20, 50, 100, 200, 500, 2000) and time-series lengths

(T=120 and 240 months) are employed. The three different estimators include two standard Fama-

MacBeth OLS estimators without and with EIV-corrections proposed by Chordia et al. (FM and CGS),

and IV estimator (IV). For given N and T, each simulation is repeated 3000 times to compute the biases

and FMSEs. The true monthly risk-free rate and risk premiums are assumed as follows: 0.0833% for risk-

free rate, 0.6667% for MKT, 0.333% for SMB, 0.1667% for HML, respectively. In the beginning of each

simulation, we generate the “true” simulated loadings of priced and residual factors, and standard

deviations of idiosyncratic disturbances for each stock based on actual data that span January 1956 to

December 2012, and keep them the same across 3000 repetitions. However, we allow the priced and

residual factor realizations alter across 3000 repetitions, which are simulated by incorporating their actual

factor means and covariance structures. The idiosyncratic disturbances are generated based on the

simulated standard deviations. The detailed simulation procedure is provided in appendix.

Panel A1: CAPM (T=120 months)

Number of

Stocks (N)

Bias in MKT Fama-MacBeth SE

FM CGS IV FM CGS IV

20 -0.123 0.168 0.037 0.501 0.762 0.770

50 -0.121 0.141 0.007 0.411 0.602 0.581

100 -0.154 0.162 0.029 0.379 0.607 0.589

200 -0.168 0.186 0.013 0.341 0.578 0.558

500 -0.168 0.143 -0.004 0.312 0.504 0.490

2000 -0.145 0.151 0.011 0.303 0.473 0.461

Panel A2: CAPM (T=240 months)

Number of

Stocks (N)

Bias in MKT Fama-MacBeth SE

FM CGS IV FM CGS IV

20 -0.078 0.121 0.011 0.438 0.583 0.560

50 -0.103 0.074 -0.001 0.350 0.459 0.469

100 -0.075 0.059 0.000 0.286 0.350 0.349

200 -0.074 0.068 0.006 0.265 0.328 0.330

500 -0.085 0.053 -0.010 0.253 0.312 0.316

2000 -0.076 0.068 0.000 0.244 0.303 0.306

28

Panel B1: Fama-French Three-Factor Model (T=120 months)

Number of

Stocks (N)

Bias in MKT Fama-MacBeth SE

FM CGS IV FM CGS IV

20 -0.098 -0.155 -0.038 0.552 1.727 1.079

50 -0.117 0.528 0.102 0.447 1.658 1.073

100 -0.166 0.200 0.053 0.391 0.661 0.691

200 -0.142 0.151 0.006 0.351 0.516 0.516

500 -0.136 0.128 -0.001 0.340 0.499 0.499

2000 -0.151 0.143 -0.003 0.328 0.496 0.501

Number of

Stocks (N)

Bias in SMB Fama-MacBeth SE

FM CGS IV FM CGS IV

20 -0.049 0.150 -0.005 0.322 1.567 0.674

50 -0.032 0.076 0.109 0.284 0.467 0.487

100 -0.046 0.049 0.015 0.252 0.327 0.342

200 -0.045 0.044 0.001 0.245 0.307 0.309

500 -0.039 0.026 0.001 0.237 0.307 0.309

2000 -0.047 0.031 -0.002 0.233 0.300 0.304

Number of

Stocks (N)

Bias in HML Fama-MacBeth SE

FM CGS IV FM CGS IV

20 -0.086 -0.628 -0.366 0.491 5.656 1.716

50 -0.094 0.863 0.344 0.327 2.747 1.353

100 -0.089 0.113 0.018 0.258 0.497 0.518

200 -0.081 0.113 0.009 0.225 0.380 0.369

500 -0.059 0.070 0.008 0.215 0.313 0.313

2000 -0.078 0.071 -0.004 0.207 0.318 0.321

Panel B2: Fama-French Three-Factor Model (T=240 months)

Number of

Stocks (N)

Bias in MKT Fama-MacBeth SE

FM CGS IV FM CGS IV

20 -0.088 0.158 -0.015 0.474 0.729 0.687

50 -0.084 0.151 0.022 0.326 0.459 0.450

100 -0.066 0.075 0.012 0.296 0.367 0.372

200 -0.076 0.059 -0.003 0.273 0.328 0.336

500 -0.070 0.059 0.003 0.260 0.311 0.316

2000 -0.079 0.069 0.002 0.252 0.310 0.316

29

Number of

Stocks (N)

Bias in SMB Fama-MacBeth SE

FM CGS IV FM CGS IV

20 -0.025 0.017 -0.008 0.225 0.256 0.261

50 -0.022 0.023 0.004 0.230 0.300 0.297

100 -0.012 0.005 -0.001 0.202 0.226 0.228

200 -0.024 0.010 -0.005 0.186 0.213 0.217

500 -0.022 0.016 0.001 0.181 0.204 0.206

2000 -0.021 0.015 0.002 0.177 0.202 0.204

Number of

Stocks (N)

Bias in HML Fama-MacBeth SE

FM CGS IV FM CGS IV

20 -0.036 0.100 0.027 0.342 0.583 0.524

50 -0.065 0.095 0.002 0.233 0.360 0.340

100 -0.035 0.047 0.010 0.184 0.232 0.238

200 -0.037 0.042 0.004 0.176 0.219 0.224

500 -0.039 0.026 -0.002 0.165 0.198 0.202

2000 -0.037 0.035 -0.001 0.159 0.198 0.202

30

Table 2: Small Sample Distribution of the Test Statistic using the IV Estimator

This table reports the size of the test of the null hypothesis that the risk premiums equal zero using the t-

statistic of the slope coefficients. We estimate the slope coefficients using the IV estimator and the Fama-

Macbeth standard errors of the slope coefficients to compute the t-statistics. The table reports the sizes at

the five significance levels from 1% to 10%. Two lengths of time-series (T=120 and 240 months) are

examined in simulations. Panel As and Bs present the results from the CAPM and Fama-French three

factor model, respectively.

Panel A: CAPM (T=120 months)

Number of Stocks

MKT

Significance Level

10.0% 7.5% 5.0% 2.5% 1.0%

20 0.097 0.074 0.058 0.028 0.013

50 0.111 0.082 0.052 0.023 0.011

100 0.111 0.072 0.047 0.031 0.011

200 0.106 0.073 0.051 0.025 0.012

500 0.111 0.074 0.052 0.029 0.009

2000 0.099 0.079 0.059 0.027 0.007

Panel B: CAPM (T=240 months)

Number of Stocks

MKT

Significance Level

10.0% 7.5% 5.0% 2.5% 1.0%

20 0.099 0.074 0.051 0.026 0.009

50 0.102 0.076 0.048 0.026 0.008

100 0.092 0.077 0.054 0.025 0.010

200 0.093 0.074 0.049 0.028 0.011

500 0.104 0.080 0.055 0.024 0.010

2000 0.103 0.071 0.051 0.027 0.010

31

Panel C: Fama-French Three-Factor Model (T=120 months)

Number of Stocks

MKT

Significance Level

10.0% 7.5% 5.0% 2.5% 1.0%

20 0.106 0.082 0.053 0.028 0.013

50 0.117 0.078 0.054 0.028 0.011

100 0.108 0.079 0.055 0.028 0.009

200 0.102 0.088 0.053 0.023 0.011

500 0.095 0.065 0.050 0.032 0.010

2000 0.111 0.074 0.043 0.031 0.008

Number of Stocks

SMB

Significance Level

10.0% 7.5% 5.0% 2.5% 1.0%

20 0.093 0.076 0.052 0.026 0.015

50 0.104 0.084 0.052 0.028 0.011

100 0.102 0.076 0.055 0.027 0.011

200 0.098 0.075 0.059 0.023 0.014

500 0.104 0.077 0.050 0.027 0.007

2000 0.098 0.072 0.050 0.024 0.011

Number of Stocks

HML

Significance Level

10.0% 7.5% 5.0% 2.5% 1.0%

20 0.099 0.087 0.055 0.027 0.015

50 0.109 0.079 0.057 0.024 0.011

100 0.100 0.076 0.042 0.024 0.016

200 0.110 0.081 0.055 0.034 0.012

500 0.099 0.069 0.055 0.024 0.013

2000 0.099 0.082 0.045 0.029 0.010

32

Panel D: Fama-French Three-Factor Model (T=240 months)

Number of Stocks

MKT

Significance Level

10.0% 7.5% 5.0% 2.5% 1.0%

20 0.106 0.073 0.049 0.023 0.009

50 0.095 0.082 0.053 0.028 0.008

100 0.105 0.074 0.049 0.032 0.009

200 0.100 0.071 0.053 0.029 0.011

500 0.097 0.079 0.053 0.023 0.008

2000 0.098 0.071 0.052 0.028 0.009

Number of Stocks

SMB

Significance Level

10.0% 7.5% 5.0% 2.5% 1.0%

20 0.101 0.076 0.051 0.034 0.010

50 0.101 0.069 0.054 0.027 0.008

100 0.104 0.081 0.056 0.024 0.007

200 0.108 0.069 0.052 0.030 0.011

500 0.096 0.079 0.051 0.027 0.011

2000 0.098 0.064 0.055 0.026 0.009

Number of Stocks

HML

Significance Level

10.0% 7.5% 5.0% 2.5% 1.0%

20 0.092 0.082 0.048 0.027 0.013

50 0.099 0.081 0.058 0.027 0.007

100 0.098 0.081 0.050 0.022 0.013

200 0.105 0.068 0.052 0.025 0.009

500 0.087 0.076 0.052 0.031 0.011

2000 0.097 0.076 0.051 0.024 0.008

33

Table 3: Summary Statistics

This table presents the various summary statistics (mean, median, standard deviation, first and third

quartiles) of individual stocks used in the subsequent real data analyses after imposing the 60-month

survivorship. For the first row, the time-series statistics of the number of stocks per month are shown. For

the second row, the cross-sectional statistics of the length of firm time-series are presented. For all other

rows, the time-series averages of cross-sectional statistics are provided. Market capitalization is defined

as price times number of shares outstanding. Book-to-market is defined following Davis et al. (2000).

Excess return is the return in the excess of risk-free rate. Return volatility is defined as the standard

deviation of daily returns for each month. Following Amihud (2002), Amihud illiquidity is defined as the

average of its daily counterparts each month. The volatility of Amihud illiquidity is the standard deviation

of daily Amihud measures for each month. The sample period is from January 1956 through December

2012.

Mean Median STD Q1 Q3

Number of Stocks (N) 2473 2067 1318 1571 3575

Length of Time-series (T) 186 148 125 94 234

Market Cap (in billion $) 1.385 0.169 6.063 0.046 0.680

Book-to-market 0.913 0.763 0.672 0.472 1.160

Excess Return (in %) 0.887 0.058 11.187 -5.268 5.891

Return Volatility (in %) 2.711 2.336 1.652 1.628 3.351

Amihud Illiquidity of NYSE/AMEX 1.833 0.196 6.712 0.041 0.964

Illiquidity Volatility of NYSE/AMEX 2.446 0.210 9.708 0.039 1.134

Amihud Illiquidity of NASDAQ 4.811 0.338 27.462 0.052 2.029

Illiquidity Volatility of NASDAQ 9.371 0.559 63.776 0.073 3.553

34

Table 4: CAPM Test

This table provides the results of asset pricing tests of the CAPM using individual stocks. The column

titled FM reports the estimates using the Fama-MacBeth approach and the column titled IV reports the

estimates using the IV estimator. The sample period is from January 1956 through December 2012. The

slope coefficients on MKT are expressed in percentage. To report the t-statistics in parentheses (bold if

significant in 5% level), Newey-West HAC covariance matrix estimators are employed.

Sample Period FM IV

1956 to 2012 Const 0.004 0.006

(2.79) (2.95)

MKT 0.434 0.280

(1.87) (1.05)

1956 to 1985 Const 0.004 0.006

(1.90) (2.08)

MKT 0.377 0.115

(1.13) (0.28)

1986 to 2012 Const 0.004 0.005

(1.77) (1.74)

MKT 0.544 0.485

(1.63) (1.35)

35

Table 5: Fama-French Three Factor Model Test

This table provides the results of asset pricing tests of the Fama-French three factor model using

individual stocks. This table provides the results of asset pricing tests of the Fama-French three factor

model using individual stocks. The column titled FM reports the estimates using the Fama-MacBeth

approach and the column titled IV reports the estimates using the IV estimator. The sample period is from

January 1956 through December 2012. The slope coefficients on MKT, SMB, and HML are expressed in

percentage. To report the t-statistics in parentheses (bold if significant in 5% level), Newey-West HAC

covariance matrix estimators are employed.

Sample Period FM IV

1956 to 2012 Const 0.005 0.006

(4.34) (3.61)

MKT 0.443 0.557

(2.23) (2.45)

SMB 0.108 -0.115

(0.87) (-0.84)

HML -0.273 -0.517

(-2.00) (-2.92)

1956 to 1985 Const 0.005 0.004

(2.82) (1.54)

MKT 0.338 0.455

(1.23) (1.24)

SMB 0.161 0.154

(0.94) (0.88)

HML -0.370 -0.742

(-2.35) (-3.19)

1986 to 2012 Const 0.005 0.008

(3.61) (4.20)

MKT 0.486 0.558

(1.63) (1.91)

SMB 0.111 -0.358

(0.59) (-1.56)

HML -0.252 -0.387

(-1.09) (-1.50)

36

Table 6: Bad-Beta and Good-Beta Model Test

This table provides the results of asset pricing tests of the bad-beta and good-beta (BBGB) model by

Campbell and Vuolteenaho (2004) using individual stocks and their original test portfolios (45 CV-

Portfolios). The BBGB model has the risk premiums of discount rate (DR) news and cash flow (CF)

news. The column titled FM reports the estimates using the Fama-MacBeth approach and the column

titled IV reports the estimates using the IV estimator. The sample period is from January 1929 through

December 2001. Following CV, the two subperiods are from January 1929 through June 1963 and July

1963 through December 2001. The slope coefficients on DR and CF are expressed in percentage. To

report the t-statistics in parentheses (bold if significant in 5% level), Newey-West HAC covariance matrix

estimators are employed.

Sample Period Individual Stocks 45 CV-Portfolios

FM IV FM IV

192901 to

200112

Const 0.008 0.007 0.005 0.003

(4.19) (3.71) (2.83) (1.49)

DR 0.318 0.419 -0.385 0.014

(1.78) (1.32) (-1.20) (0.04)

CF -0.428 -1.388 2.736 2.623

(-1.16) (-1.49) (2.86) (2.22)

192901 to

196306

Const 0.011 0.010 0.004 0.005

(3.09) (3.28) (1.77) (1.90)

DR 0.015 0.123 0.069 -0.165

(0.09) (0.37) (0.18) (-0.22)

CF 0.213 0.340 0.909 4.597

(0.41) (0.35) (0.88) (1.72)

196307 to

200112

Const 0.004 0.005 0.001 0.001

(2.7) (1.93) (0.26) (0.42)

DR 0.696 0.153 0.124 0.548

(2.11) (0.28) (0.27) (0.88)

CF -0.201 1.625 7.797 3.767

(-0.31) (0.86) (2.98) (1.68)

37

Table 7: Liquidity-adjusted CAPM Test

This table provides the results of asset pricing tests of the liquidity-adjusted CAPM (LCAPM) by

Acharya and Pedersen (2005) using individual stocks and the two sets of their original test portfolios (25

Illiquidity and 25 Sig(Illiquidity) Portfolios). Following AP, 25 illiquidity and 25 Sig(illiquidity)

portfolios are constructed based on the means and standard deviations of annualized Amihud illiquidity

measures from prior year, respectively. The column titled FM reports the estimates using the Fama-

MacBeth approach and the column titled IV reports the estimates using the IV estimator after estimating

the AP betas by GMMs. The sample period is from January 1956 through December 2012. The slope

coefficients on Amihud illiquidity and LMKT are expressed in percentage. To report the t-statistics in

parentheses (bold if significant in 5% level), Newey-West HAC covariance matrix estimators are

employed.

Panel A: Full-Sample Period

Individual Stocks 25 Illiquidity Portfolios 25 Sig(Illiquidity) Portfolios

FM IV FM IV FM IV

Const 0.005 0.007 -0.007 -0.006 -0.007 -0.006

(3.68) (4.06) (-2.17) (-1.95) (-2.35) (-2.08)

Amihud 0.099 0.102 0.131 0.139 0.088 0.098

Illiquidity (2.90) (2.94) (2.74) (2.92) (1.38) (1.54)

LMKT 0.264 0.081 1.537 1.465 1.584 1.490

(1.01) (0.29) (3.73) (3.48) (3.95) (3.71)

Panel B: Sub-Sample Periods

Sample Period Individual Stocks

FM IV

1956 to 1985 Const 0.005 0.007

(2.50) (2.95)

Amihud Liquidity 0.155 0.159

(2.55) (2.57)

LMKT 0.260 0.001

(0.70) (0.00)

1986 to 2012 Const 0.005 0.005

(2.60) (2.43)

Amihud Liquidity 0.037 0.037

(3.31) (3.41)

LMKT 0.303 0.238

(0.81) (0.59)

38

APPENDIX

A.I. Residual Factor Structure

This appendix describes the parameters that we use in the simulations. We first determine the

mean risk premiums and the covariance structure of the common factors based on the realization of the

three Fama-French factors over the 1956 to 2012 sample period. Panel A of Table A.1 presents the values

that we use in the simulations.

We determine the factor structure of the residuals separately for the market model and for the

Fama-French three-factor model. We first run the market model regression and three factor model

regression for each stock within five-year subperiods. We determine the realizations of common factors in

the residuals using the asymptotic principal components (APCs) proposed by Connor and Korajczyk

(1986). We use the -test described in Trzcinka (1986) and the information criteria developed in Bai and

Ng (2002) to examine the number of common residual factors. In untabulated results, we find that the -

test always rejects the hypothesis that there are less than five common factors in the residuals at the 10%

significance level.

The information criteria by Bai and Ng are more conservative. Table A.2 presents the number of

residual factors based on these criteria. When we estimate the residuals using the market model, we find

that there are between two and five common factors in the residuals. When we estimate the residuals

based on the Fama-French model, there are between one and five common residual factors. Based on

these results, we use two common factors in the residuals in the simulation experiments. We use the

common parameters from the 1971 through 1975 subperiod to simulate the residual factor sensitivities

and the idiosyncratic volatilities.8 Panel B of Table A.1 presents the parameters that we use in the

simulations.

8 The major reasons for choosing five-year subperiods for APCs are 1) the factors that we estimate are unique only

up to their sign and hence we cannot splice them across subperiods and 2) to avoid survivorship biases that would

arise with longer sample periods. When the PCs are applied to stock returns, the same choices of five-year

subperiods are also employed by Bai and Ng (2002) and Goyal et al. (2008).

39

TABLE A.1: Simulation Parameters

This table presents the parameters used in simulations to assess the performances of different risk

premium estimators (FM, CGS, and IV approaches) and the small sample properties of the IV

approach.

Panel A: Time-series Means and Standard Deviations of common factors

Single Factor Model Fama-French Three-Factor Model

Mean (%) StdDev (%) Mean (%) StdDev (%)

Factors MKT 0.6667 4.4299 0.6667 4.4299

SMB

0.3330 3.0146

HML

0.1667 2.8126

Panel B: Cross-sectional Means and Standard Deviations of factor sensitivities

Single Factor Model Fama-French Three-Factor Model

Mean StdDev Mean StdDev

Factor Loadings MKT 1.17 0.35 1.04 0.36

SMB

1.05 0.68

HML

0.59 0.55

Residual Factor Loadings 1st 0.0852 0.0348 0.0685 0.0308

2nd 0.0008 0.0161 0.0037 0.0256

Idiosyncratic Volatility 0.07 0.04 0.07 0.04

40

Table A.2: Number of Residual Factors

This table provides the number of residual factors determined by the three information criteria, i.e. PCp1 to PCp3, developed by Bai and Ng (2002).

The asymptotic principal components (APCs) by Connor and Korajczyk (1986) are used to extract the residual factors and the maximum number

of residual factors that we examine is set to 15 as in Connor and Korajczyk (1993). Panel A and B report the numbers of residual factors by PCps

for the CAPM and Fama-French three factor model, respectively.

Panel A: CAPM

1956-

1960

1961-

1965

1966-

1970

1971-

1975

1976-

1980

1981-

1985

1986-

1990

1991-

1995

1996-

2000

2001-

2005

2006-

2010

PCp1 4 3 2 3 2 2 2 2 4 3 5

PCp2 4 3 2 3 2 2 2 2 4 2 5

PCp3 5 5 2 4 2 2 2 2 4 3 5

Panel B: Fama-French Three-factor Model

1956-

1960

1961-

1965

1966-

1970

1971-

1975

1976-

1980

1981-

1985

1986-

1990

1991-

1995

1996-

2000

2001-

2005

2006-

2010

PCp1 3 2 1 2 2 1 1 1 4 1 5

PCp2 3 2 1 2 2 1 1 1 4 1 5

PCp3 4 5 2 2 2 1 1 1 4 1 5