Alternative Errors-in-Variable Models and Their
Applications in Finance Research*
Cheng-Few Lee
Rutgers University , USA
Hong-Yi Chen
National Central University , Taiwan
Outline
1. Introduction
2. Effects of Errors-in-Variables in Different Cases
2.1 Bivariate Normal Case
2.2 Multivariate Case
2.2.1 The Classical Case
2.2.2 The Constrained Classical Case
3. Six Different Estimation Methods When Variables are Subject to Error
3.1 Classical Estimation Method
3.1.1 The Classical Method to a Simple Regression Analysis
3.1.2 The Classical Method to a Multiple Regression Analysis
3.1.3 The Constrained Classical Method
3.2 Grouping Method
3.3 Instrumental Variable Method
2
Outline
3.4 Mathematical Method
3.4.1 Bivariate Case
3.4.2 Multivariate Case
3.5 Maximum Likelihood Method
3.6 LISREL and MIMIC Methods
3.6.1 Structural model (LISEREL model)
3.6.2 MIMIC Model
4. Applications of Errors-in-Variables Models in Finance Research
4.1 Cost of Capital
4.2 Capital Asset Pricing Model
4.3 Capital Structure
4.4 Investment and Capital Asset Allocation
5. Conclusion
3
Abstract
The main purposes of this paper are (i) to review and extend
existing errors-in-variables (EIV) estimation methods; and (ii) to
discuss how these estimation methods have been used in finance
research. We first show how EIV problems affect estimators in
the regression model. We further extend six alternative
estimation methods dealing with EIV problem. These six
methods are (i) classical method, (ii) grouping method, (iii)
instrumental variable method, (iv) mathematical programming
method, (v) maximum likelihood method, and (vi) LISREL
method will be discussed and extended from the original simple
regression case to multiple regression case. Finally, we
investigate how EIV estimation methods have been used to
determine cost of capital and capital structure, and test capital
4
asset pricing model.
1. Introduction
• This paper studies and extents exiting EIV estimation methods
and to discuss how these estimation methods have been used
in finance research. We first show how EIV problems affect
estimators in the regression model. We further provide six
alternative estimation methods dealing with EIV problem.
Classical method, grouping method, instrumental variable
method, mathematical programming method, maxima
likelihood method, and LISREL method will be discussed and
extended from the original simple regression case to multiple
regression case. Finally, we investigate how EIV problem
associate to estimated beta can affect the second-pass
regression in testing capital asset pricing model.
5
2. Effects of Errors-in-Variables in Different Cases
• Bivariate Normal Case
• Multivariate Case
6
2.1 Bivariate Normal Case
• Two variate structural relationship
Vi    U i
Both Ui and Vi are unobserved, but we can observe X i  Ui   i and Yi  Vi  i .
where
( a)  i


N 0,12 and i

N 0, 22

(b) E  iUi   0, E  iVi   0, E  ii   0, E iUi   0, and E iVi   0.
(c) Ui


N E  X  ,U2 and Vi


N    E  X  ,  2 U2 .
Then
2


1
plimˆ    2
 U   12
=> ˆ is downward biased.
 12
plimˆ    2
EX 
 U   12
=>ˆ is upward biased.
7
2.2 Multivariate Case
• Trivariate structural relationship
Wi    U i   Vi
Wi , U i and Vi are unobserved,
but we can observe Zi  Wi   i , X i  U i   i , and Yi  Vi  i .
where
i


N 0,12 , i


N 0, 22 , and i

N 0, 32

Then
2
2
2
2
2
2
2
2 2
2 2










((




)








2 )
VW
2
VW
V
VW
UV
U
V
VW
U
2
V
1
1
plim ˆ   
2
(U2V2  VW
)  U2 22  V212  12 22
2
WV 12  WV V2  WV UV   ((U2V2  UV
)  U2 22  V212  12 22 )
plim ˆ   
2
(U2V2  UV
)  U2 22  V212  12 22
(12)
(13)
8
3. Six Different Estimation Methods When
Variables are Subject to Error
• Classical Method
• Grouping Method
• Instrumental Variable Method
• Mathematical Programming Method
• Maxima Likelihood Method
• LISREL and MIMIC Methods
9
3.1 Classical Method
  12
ˆ
plim    2
 U   12
(5)
 12
plimˆ    2
EX 
 U   12
2
2
2
2
2
2
2
2 2
2 2










((




)








2 )
VW V
VW UV
U V
VW
U 2
V 1
1
plim ˆ    VW 2
2
(U2V2  VW
)  U2 22  V212  12 22
(12)
2
WV 12  WV V2  WV UV   ((U2V2  UV
)  U2 22  V212  12 22 )
plim ˆ   
2
(U2V2  UV
)  U2 22  V212  12 22
(13)
10
3.2 Grouping Method
• Wald’s (1940) grouping method (2 groups) in deal with errors-in-variable
problem
- using groups instead of individual securities can minimize measurement error.
• CAPM test: (k-groups)
Black et al. (1972), Blume and Friend (1973), Fame and MacBath (1973),
Lizenberger and Ramaswamy (1979), etc.
  ˆ
k
ˆ1,t 
p 1
 ˆt
p ,t

n
i 1
pn
k

ˆ p ,t 
j 1 ( p 1)
n
k
p ,t
ˆ p ,t  ˆt

 Rt 
, where
2
pn
k

ˆ j ,t
n
k
 R
, R ip ,t
j 1 ( p 1)
n
k
n
R p ,t
n
k
, ˆt 
 ˆ j ,t
j 1
n
n
, and Rt 
R
j 1
n
j ,t
11
3.2 Grouping Method - continued
• Advantages
-
is intuitive and easy to implement with real data.
is convenient to incorporate other estimation methods or examine model
misspecification (multi-factor models).
• Limitations:
-
-
-
shrinks number or observations and the range of beta risk in the secondpass => the variance of the estimator becomes larger and reduce statistic
power.
The formation of portfolios for the second-pass estimation might cause a
loss of valuable information about cross-sectional behavior among
individual securities, since the cross-sectional variations would be
smoothed out.
may get different results by using different portfolio grouping methods.
(e.g., Ahn et al., 2009)
12
3.3 Instrumental Variable Method
Durbin (1953) proposes an instrumental variable method to deal with the
errors-in-variables problem in a regression model.
ˆ0,t 
ˆ
 ˆ   Zβ
 1,t 
 
1
ZR i , where
1 1 1
Z  
1 2 3
and Ri   RiN ,t  R f ,t
1 1 1

Z

 1 1 1
If

1
 1
, βi  
N 
 i1,t
RiN ,t  R f ,t
1
1
i 2,t
i 3,t
RiN ,t  R f ,t
1 
,
iN ,t 
RiN ,t  R f ,t 
1 1
1 1 , then instrumental variable method will reduce
to Wald’s 2-group method.
 Grouping method is a special case of instrumental method.
 Instrumental variable method is more generalized.
13
3.4 Mathematical Programming Method
Min S  Min  i

Yˆi  Yi

2
14
3.4 Mathematical Programming Method
• Deming (1943), York (1966) and Clutton-Brock (1967)

Min S  Min  i w  X i 

Uˆ i  X i

2

 w Yi  Vˆi  Yi

2
(53)
Vˆi    Uˆi ,
i  1,
, n
15
3.4 Mathematical Programming Method



where Vˆi    Uˆ i ,
 i  1,
Min S   i w  X i 
Uˆ i  X i
2

 w Yi  Vˆi  Yi
, n

2
We can obtain a “least-square cubic”

ki2 xi2
ki2 xi yi
ki2 yi2 
2
2
 i
 2  i
   i ki xi   i
   i ki xi yi  0
w Xi 
w Xi 
w  X i  

3
where ki 
w  X i  w Yi 
 w Yi   w  X i 
2
(66)
.
16
3.4 Mathematical Programming Method

ki2 xi2
ki2 xi yi
ki2 yi2 
2
2
 i
 2  i
   i ki xi   i
   i ki xi yi  0
w Xi 
w Xi 
w
X



i 

3
where ki 
w  X i  w Yi 
 2 w Yi   w  X i 
(66)
.
• If no error in X i ; Yi subject to errors=> w Xi   
w Y   X  X Y  Y 


=> weighted regression of Y on X.
 w Y   X  X 
i
i
i
i
2
i
i
i
• If no error in Yi ; X i subject to errors=> wYi   

i w  X i  Yi  Y 
 w X   X
i
i
i
2
 X Yi  Y 
=> weighted regression of X on Y.
17
3.4 Mathematical Programming Method
• If both X i and Yi subject to errors
=> numerical method to find  .
Two independent variable case:

Min i w  X i  xi  X i   w Yi  yi  Yi   w  Z i  zi  Z i 
 ,
2
2
s.t. zi     xi   yi
 3C3  2 2C2   C1  CO  0
2

(70)
(71)
(86)
where
Ki2
Ri  Ki Qi Pi 2
Ki2
 Ki Qi Ri
 Ki Ri Pi
C3  i
(Qi 
)
;
C


(
Q

R
)(
P

R
)
2
i
i
i
i
i
2
2
2
w( X i )
 Ki Ri
w( X i )
 Ki Ri
 Ki Ri
( Ki Qi Ri )2
Ki2
Ri  Ki Ri Pi
 Ki Qi Ri  Ki PQ
2
i i
C1   Ki Pi 


(
)(

P
)
;
C

  Ki PQ
i
O
i i
2
2
2
 Ki Ri
w( X i )
 Ki
 Ki Ri
18
2
3.5 Maxima likelihood Method
• Kim (1995, 1997, 2010)
The second-pass regression and the measurement error can be jointly presented
as
 ε t   R t   0t   1t βt   2t Vt 1 
ηt  

  
ˆ
ξ
β

β
 t 1  
t 1
t

where
 
Ω
 
N  O, Ω 
(87)
 

 
ξ t 1 is a linear function of past idiosyncratic error terms ε s prior to the crosssectional regression at time  s  t 1 .
Additional information
it 1 

where ais  his Rms  Rm

t 1
a
s t T

 st T his Rms  Rm
t 1
his  1 Var   is  ; Rm   s t T his Rms
t 1
(90)
is is

2
 st T his
t 1
19
3.5 Maxima likelihood Method - continued
Conditional on the market return, the ratio of the idiosyncratic error variance
matrix to the measurement error variance matrix is  1  diag 1t , ,  Nt  ,
and it 
2
t 1
 R
ms
s t T
 Rm
 Var   Var  
it
(91)
is
The closed form solution
 

M   M 2  4 t m R2ˆ 1  ˆ RV ˆ ˆV ˆ Rˆ

ˆ1t 
2mRˆ 1  ˆ RV ˆ ˆV ˆ Rˆ
 

ˆ2t  mRV  ˆ1t mˆV



2 1/2


mVV
(94)
ˆ0t  Rt  ˆ1t ˆt 1  ˆ2tVt 1

2
where M  mRR 1  ˆ RV
   t mˆ ˆ 1  ˆ 2ˆV

N
N
mxy  1 N  x  1x  ˆ 1  y  1 y   1 N   i 1  j 1 wij  xi  x  yi  y 
x   i 1  j 1 wij xi
N
ˆ xy2  mxy
 
N
m
xx
myy 
1/2
N
N
i 1
j 1
wij ; wij is the (i, j)-th element of ˆ 1
 
N
N
i 1
j 1
wij
20
3.6 LISREL and MIMIC Methods
•
The linear simultaneous equation system is widely used in finance and accounting
related research. However, a serious limitation of the simultaneous equation
approach is EIV problem. For example, the theoretical determinants of capital
structure in corporate finance can be attributed to unobservable constructs that are
usually measured in empirical studies by a variety of observable indicators or
proxies. These observable indicators or proxies can then be viewed as measures of
latent variables with measurement errors. Maddala and Nimalendran (1996) show
that the use of of these indicators as theoretical explanatory variables may cause
EIV problems. Bentler (1983) also emphasizes the estimated results of traditional
simultaneous equation model has no meaning when variables have measurement
errors. Therefore, the latent variable covariance structure model is provided and
applied in corporate finance. Titman and Wessels (1988), Chang et al. (2009), and
Yang et al. (2009), to mitigates the measurement problems of proxy variables,
apply structure equation models (e.g. LISREL model and MIMIC model) to
determining capital structure decision. Maddala and Nimalendran (1996) use
structure equation model to examine the effect of earnings surprises on stock prices,
trading volumes, and bid-ask spreads.
21
3.6 LISREL and MIMIC Methods - continued
•
Jöreskog and Goldberger (1975) develop a structure equation model with multiple
indicators and multiple causes of a single latent variable, MIMIC mode, and obtain
maximum likelihood estimates of parameters. Figure 1 shows the path diagram that
depicts a simplified MIMIC model in which variables in a rectangular box denote
observable variables, while variables in an oval box are latent constructs. In this
diagram, observable variables X1, X2, and X3 are causes of the latent variable η,
while Y1, Y2, and Y3 are indicators of η. In our study, X’s are determinants of
capital structure (η), which are then measured by Y’s.
•
Jöreskog and Sörbom (1989) show that the full structural equation (LIEREL model)
can be restricted to be a MIMIC model. We first discuss with specifying the
structural model, then we show how structural model can be restricted to a MIMIC
22
model.
3.6.1 Structural model (LISEREL model)
• A structural equation model is composed of two sub-models-structural sub-model and measurement sub-model. The structural
model can be defined as
 =  X + ,
(93)
Y = y   ,
(94)
where Y is a vector of indicators of the latent variable , and X is a
vector of causes of .
• The latent variable  is linearly determined by a set of observable
exogenous causes, X = (x1, x2, …, xq)’, and a disturbance . The
latent variable , in turn, linearly determines a set of observable
endogenous indicators, Y = (y1, y2, …, yp)’ and a corresponding set
of disturbance,  = (1, 2, …, p)’. Stapleton (1978) further develops
MIMIC with more latent variables.
23
3.6.2 MIMIC Model
• Substituting Eq. (93) into Eq. (94), we obtain a reduced form:
(95)
• In structural equation modeling, the total effect of a cause variable on an
indicator can be measured as the sum of the direct effect and the indirect
effect. Since a MIMIC model is a reduced form of a structural equation
model, the total effect of MIMIC model, denoted as ’ in Eq. (95), comes
merely from the indirect effect.
• Since the scale of the latent variable is unknown, the factor indeterminacy
is a common problem in the MIMIC model, as in other structure equation
models. We can obtain infinite parameter estimates from the reduced form
by arbitrarily changing the scale of the latent variables. However, by fixing
the scales of latent variables, one can solve the indeterminacy problem.
Two methods are usually adopted to fix the scale of latent variables. One
method is normalization in which a unit variance is assigned to each latent
variable, while another method is to fix a nonzero coefficient at unity for
24
each latent variable.
3.6.2 MIMIC Model - continued
• In terms of estimation of the parameters, Jöreskog and Goldberger
(1975) adopt the normalization method to deal with the factor
indeterminacy problem and use maximum likelihood estimation
method in structural equation modeling to estimate parameters. The
maximum likelihood estimates for the parameters of the model are
obtained at the minimization of the fit function as follows:
F = log |||| + tr(S-1) – log||S|| - (p + q)
(96)
where  is the population covariance matrix; S is the model-implied
covariance matrix; p is the number of exogenous observable
variables; and q is the number of endogenous observable variables.
Minimization of the fit function can be done by LISREL program
provided by Jöreskog and Goldberger (1981).
25
4. Applications of Errors-in-Variables Models in
Finance Research
• Cost of Capital
• Capital Asset Pricing Model
• Capital Structure
• Investment and Capital Asset Allocation
26
Study
Miller and Modigliani (1966)
Issue
Determinants
of cost of
capital
Method
Instrumental variable method
Results
- The extrapolation of historical population trends is
superior to the conventional use of change of capital,
and share prices are not a positive function of
dividends as often suggested.
Black et al. (1972)
Blume and Friend (1973)
CAPM test
CAPM test
Grouping (10 groups)
Grouping (12 groups)
- Reject both the CAPM and the zero-beta CAPM.
- Linear model is better than quadratic model in
explaining expected return.
- Reject both the CAPM and the zero-beta CAPM.
Fama and MaBeth (1973)
CAPM test
Grouping (20 groups), period by
period
Lee (1977)
CAPM test
Wald’s Grouping / Instrumental
Variable
- Find a linear relationship between the expected
return and beta risk, beta is the only risk measure in
explaining expected return, and risk premium is
greater than zero.
- CAMP and efficient capital market hold.
- Adjust for measurement error of market return
(first-step).
- Estimated risk premium is larger than realized risk
premium.
- Reject CAPM.
- Before-tax expected rates of return are linearly
related to systematic risk and dividend yield.
- MLE can obtain consistent estimators without
losing efficiency.
- CAPM is rejected because of nonzero .
Litzenberger and Ramaswamy CAPM test
(1979)
MLE, OLS, GLS (individual stock)
Cheng and Grauer (1980 )
CAPM test
Grouping (20groups), Price-level
testing (Invariance Law)
- Neither framework of Invariance Law or security
market line can accommodate the possibility that the
CAPM may hold for each period.
- Reject CAPM.
Gibbons (1982 )
CAPM test
One-step Guass-Norman Procedure
(40 groups)
- Guass-Norman procedure can increase the
precision of estimated risk premium.
- Reject CAPM.
Titman and Wessels (1988)
Determinants
of capital
structure
LISREL model
- Do not support for four of eight propositions on the
determinants of capital structure.
- A firm’s capital structure is not significantly related
to its non-debt tax shields, volatility of earnings,
collateral value of assets, and future growth. 27
Study
MacKinlay and Richardson
(1991)
Shanken (1992)
Issue
CAPM test
GMM
Method
CAPM test
MLE (individual stock)
Fama and French (1992)
CAPM test
2-way grouping (10x10 groups)
Jagannathan and Wang(1993)
CAPM test
Multifactor Asset Pricing Model
Kim (1995, 2010)
CAPM test
MLE (individual stock – 20x20
groups)
Kim (1997)
CAPM test
Multifactor, MLE
Chang, Lee, and Lee (2009)
Determinants of MIMIC model
capital structure
Yang, Lee, Gu, and Lee (2009) Determinants of LISREL model
capital structure
Erickson and Whited (2000 )
Test q theory
GMM
Almeida, Campello, and
Galvao Jr. (2010)
Test q theory
GMM and instrumental variable
method
Results
- Conclusions of mean-variance efficiency vary by
settings.
- To deal with small-sample bias in the second-step
cross-sectional regression estimates due to
measurement error in the betas.
- The adjustment does not have much effect on Fama
and MacBeth’s (1973) conclusion.
- Support CAPM.
- The market capitalization and the book-to-market
ratio can replace beta altogether.
- Reject CAPM.
- Including human capital and business cycle can
increase explanatory power of expected return.
- Support CAPM
- MLE method can effectively adjust the errors-invariables bias and CAPM holds.
- Support CAPM.
- Linear relationship between beta and expected
return.
- Book-to-market ratio has significant explanatory
power for expected return, but size has not.
- Seven constructs, growth, profitability, collateral
value, volatility, non-debt tax shields, uniqueness, and
industry, as determinants of capital structure have
significant effects on capital structure decision.
- Stock returns, expected growth, uniqueness, asset
structure, profitability, and industry classification are
main determinants of capital structure.
- Leverage, expected growth, profitability, firm value,
and liquidity can explain stock returns.
- The capital structure and stock return, in addition,
are mutually determined by each other.
- Cash flow does not affect firms’ financial decision,
even for financially constrained firms.
- Support the q theory if measurement error is taken
into account.
- Estimators from GMM are unstable across different
specifications and not economically meaningful.
28
- Estimators from a simple instrumental method are
robust and conform to q theory.
4.1 Cost of Capital
The true relation between value and anticipated earnings, when
replaced by the observable estimates, implies a simultaneous system
of relationships:
*
X i  X i  vi
Vi*   X i*    j Zij  ui
j
Where
Vi* 
Vi   c Di
;
Ai
X i* 
X (1   c )
Ai
X   j Zij  wi
*
i
j
= (the true anticipated earnings);
ui and wi are regression residuals;
vi = Measurement errors associated with current earnings;
Xi = Observable estimate of earnings derived from the accounting
statements;
Zij = Other relevant variables determining earnings.
29
4.2 Capital Asset Pricing Model
•
Theoretical Model
- Sharpe (1964), Lintner (1965), and Mossian (1966)
the expected returns on securities are a positive linear function of their
market betas, and market betas have sufficient power to explain the crosssectional expected returns.
•
Empirical Tests (Two-passes regression)
 fails to explain the cross-sectional average returns
30
4.2 Capital Asset Pricing Model - continued
Why beta fails?
Model misspecification
Fama & French (1992), Carhart (1996), Chordia & Shivakumar (2006)
suggest new risk factors can explain cross-sectional average returns. (size,
B/M, price momentum factor, and earnings momentum factor.)
Errors-in-variables problem
Roll (1969 and 1977), Lee (1984)
The errors-in-variables problem underestimates the market beta which is
suffered measurement error and overestimates other risk factors with no
measurement error.
Correction:
Lee (1973), Gibbons (1985), Shanken (1992), Kim (1995, 1997, 2011), etc.
31
4.2 Capital Asset Pricing Model - continued
This study:
1. We investigate the effect of errors-in-variables problem on asset
pricing test.
-
The errors-in-variables problem underestimates the market beta which is suffered
measurement error and overestimates other risk factors with no measurement error.
2. We provide three alternative estimation methods for errors-invariable problem.
-
(1) Grouping method; (2) Instrumental variable method; (3) Maxima likelihood
method
3. We will reexamine asset pricing models after correcting errors-invariable problem.
-
Market beta does not fail
32
Figure1. Flow Chart
The Static CAPM
(single-period)
Dividend and Taxation Effect
Models
Miller and Modigliani(1961,Journal
of Business)
Brennan (1970, National Tax
Journal)
Black and Scholes (1974,Journal of
Financial Economics)
Sasson and Kolodny(1976, The
Review of Economics and Statistics)
Miller and Scholes (1978,Journal of
Financial Economics)
Litzenberger and Ramaswamy
(1979, Journal of Financial
Economics)
Morgan (1982,The Journal of
Finance)
Litzenberger and Ramaswamy
(1982,The Journal of Finance)
Hagiwara and Herce (1997, The
American Economic Review)
Skewness Effect Models
Borch (1969, Review of
Economics Studies)
Feldstein (1969, Review of
Economics Studies)
Jean (1971, Journal of Financial
and Quantitative Analysis)
Tsiang (1972, American Economic
Review)
Ingersoll (1975, Journal of
Financial and Quantitative
Analysis)
Schweser (1978, Journal of
Financial and Quantitative
Analysis)
The Original CAPM
Sharpe (1964), Lintner (1965),and Mossin (1966)
Existence of Equilibrium
Hart (1974, Journal of Economic
Theory)
Nielsen (1989, Review of Economic
Studies)
Equilibrium Models with
Heterogeneity Investment Horizon
Lee (1976, The Review of Economics
and Statistics)
Levhari and Levy (1977, The Review
of Economics and Statistics)
Lee, Wu, and Wei (1990, Journal of
Financial and Quantitative Analysis)
Supply-Side Effect Models
Black (1976, American
Economic Review)
Grinols (1984, Journal of
Finance)
Lee, Tsai, and Lee (2009,
Quarterly Review of Economics
and Finance)
The Dynamic CAPM
(multi-period)
Intertemporal CAPM－Merton
Model
Merton (1973, Econometrica)
Behavioral Finance
Kahneman and Tversky
(1979, Econometrica)
Tversky and Kahneman
(1992, Journal of Risk and
Uncertainty)
Levy (2010, European
Financial Management)
International CAPM
Stulz (1981a, Journal of Finance)
Stulz (1981b, Journal of Financial
Economics)
Stulz (1982, Journal of
International Economics)
Stulz (1984, Journal of
International Business Studies)
Equilibrium Models with
Heterogeneity Beliefs and Investors
Constantinides (1982, Journal of
Business)
Constantinides and Duffie (1996,
Journal of Political Economy)
Brav, Constantinides, and Geczy
(2002, Journal of Political Economy)
Basak (2005, Journal of Banking and
Finance)
Levy, Levy, and Benita (2006, Journal
of Business)
Intertemporal CAPM－Consumption-based Models
Breeden (1979, Journal of Financial Economics)
Campbell (1993, American Economic Review)
Campbell and Cochrane (1999, Journal of Political Economy)
Jagannathan and Wang (1996, Journal of Finance)
Lettau and Ludvigson (2001a, Journal of Finance)
Lettau and Ludvigson (2001b, Journal of Political Economy)
Lewellen and Nagel (2006, Journal of Financial Economics)
Balvers and Huang (2009, Journal of Financial and Quantitative
Analysis)
Liquidity-based Models
Pastor and Stambaugh (2003,
Journal of Political Economy)
Acharya and Pedersen (2005,
Journal of Financial Economics)
Yoel(2009,working paper)
Intertemporal CAPM－Production-based Models
Balvers, Cosimano, and McDonald (1990, Journal of Finance)
Cochrane (1991, Journal of Finance) (1996, Journal of Political
Economy)
Balvers and Huang (2007, Journal of Financial Economics)
Figure2. The Dynamic CAPM
The Dynamic CAPM
(multi-period)
Supply-Side Effect Models
Black (1976, American Economic Review)
Grinols (1984, Journal of Finance)
Lee, Tsai, and Lee (2009, Quarterly Review of
Economics and Finance)
Intertemporal CAPM－
Merton Model
International CAPM
Merton (1973, Econometrica)
Stulz (1981b, Journal of Financial Economics)
Stulz (1981a, Journal of Finance)
Stulz (1982, Journal of International Economics)
Stulz (1984, Journal of International Business
Studies)
Chang and Hung (2000, Review of Quantitative
Finance and Accounting)
Intertemporal CAPM－Consumption-based Models
Breeden (1979, Journal of Financial Economics)
Campbell (1993, American Economic Review)
Campbell and Cochrane (1999, Journal of Political Economy)
Jagannathan and Wang (1996, Journal of Finance)
Lettau and Ludvigson (2001a, Journal of Finance)
Lettau and Ludvigson (2001b, Journal of Political Economy)
Lewellen and Nagel (2006, Journal of Financial Economics)
Balvers and Huang (2009, Journal of Financial and Quantitative Analysis)
Intertemporal CAPM－Production-based Models
Balvers, Cosimano, and McDonald (1990, Journal of Finance)
Cochrane (1991, Journal of Finance) (1996, Journal of Political Economy)
Balvers and Huang (2007, Journal of Financial Economics)
Figure 3. The Static CAPM
The Static CAPM
(single-period)
Dividend and Taxation Effect
Models
Miller and Modigliani(1961,Journal of
Business)
Brennan (1970, National Tax Journal)
Black and Scholes (1974,Journal of Financial
Economics)
Sasson and Kolodny(1976, The Review of
Economics and Statistics)
Miller and Scholes (1978,Journal of Financial
Economics)
Litzenberger and Ramaswamy (1979, Journal
of Financial Economics)
Morgan (1982,The Journal of Finance)
Litzenberger and Ramaswamy (1982,The
Journal of Finance)
Hagiwara and Herce (1997, The American
Economic Review)
Skewness Effect Models
Borch (1969, Review of Economics Studies)
Feldstein (1969, Review of Economics
Studies)
Jean (1971, Journal of Financial and
Quantitative Analysis)
Tsiang (1972, American Economic Review)
Ingersoll (1975, Journal of Financial and
Quantitative Analysis)
Schweser (1978, Journal of Financial and
Quantitative Analysis)
Equilibrium Models with
Heterogeneity Investment
Horizon
Lee (1976, The Review of Economics and
Statistics)
Levhari and Levy (1977, The Review of
Economics and Statistics)
Lee, Wu, and Wei (1990, Journal of Financial and
Quantitative Analysis)
Equilibrium Models with
Heterogeneity Beliefs and
Investors
Constantinides (1982, Journal of Business)
Constantinides and Duffie (1996, Journal of
Political Economy)
Brav, Constantinides, and Geczy (2002, Journal
of Political Economy)
Basak (2005, Journal of Banking and Finance)
Levy, Levy, and Benita (2006, Journal of
Business)
Liquidity-based Models
Pastor and Stambaugh (2003, Journal of Political
Economy)
Acharya and Pedersen (2005, Journal of Financial
Economics)
Yoel(2009,working paper)
Errors-in-Variables Problem in testing CAPM
• In testing CAPM,
Rt   0,t   1,t βt  εt
However, beta is unknown.
 Two passes regression method to test CAPM
First-pass: estimate beta coefficient (market model).
Ri ,t  i ,t  ˆi ,t Rm,t  i ,t
Second-pass: using estimated beta coefficient to test CAPM.
Rt   0,t   1,t ˆt 1   t
If independent variable ˆ is measured with error, the estimated
coefficient  1,t (market premium) by OLS is underestimated.
=> Empirical CAPM test reject the linear relationship between
expected return and beta risk might be due to errors-in-variable
problem.
36
Study
Testing Period
Method
Results
Black et al. (1972)
1931-1965
Grouping (10 groups)
- Reject both the CAPM and the zero-beta CAPM.
Blume and Friend (1973)
1955-1968
Grouping (12 groups)
- Linear model is better than quadratic model in explaining expected return.
- Reject both the CAPM and the zero-beta CAPM.
Fama and MaBeth (1973)
1955-1968
Grouping (20 groups),
period by period
Lee (1977)
1967-1972
Wald’s Grouping /
Instrumental Variable
Litzenberger and
Ramaswamy (1979)
1936-1977
MLE, OLS, GLS
(individual stock)
Cheng and Grauer (1980)
1935-1977
Gibbons (1982)
1926-1975
MacKinlay and Richardson
(1991)
Shanken (1992)
1926-1988
Grouping (20groups),
Price-level testing
(Invariance Law)
One-step GuassNorman Procedure (40
groups)
GMM
- Find a linear relationship between the expected return and beta risk, beta is the only
risk measure in explaining expected return, and risk premium is greater than zero.
- CAMP and efficient capital market hold.
- Adjust for measurement error of market return (first-step).
- Estimated risk premium is larger than realized risk premium.
- Reject CAPM.
- Before-tax expected rates of return are linearly related to systematic risk and
dividend yield.
- MLE can obtain consistent estimators without losing efficiency.
- CAPM is rejected because of nonzero .
- Neither framework of Invariance Law or security market line can accommodate the
possibility that the CAPM may hold for eachˆ0period.
- Reject CAPM.
- Guass-Norman procedure can increase the precision of estimated risk premium.
- Reject CAPM.
Fama and French (1992)
1963-1990
1935-1968
- Conclusions of mean-variance efficiency vary by settings.
MLE (individual stock) - To deal with small-sample bias in the second-step cross-sectional regression
estimates due to measurement error in the betas.
- The adjustment doesn’t have much effect on Fama and MacBeth’s (1973)
conclusion.
- Support CAPM.
2-way grouping (10x10) - The market capitalization and the book-to-market ratio can replace beta altogether.
- Reject CAPM.
Jagannathan and Wang(1993) 1962-1990
Multifactor Asset
Pricing Model
Kim (1995, 2010)
1936-1991
MLE (individual
stock – 20x20 groups)
Kim (1997)
1963-1993
Multifactor, MLE
- Including human capital and business cycle can increase explanatory power of
expected return.
- Support CAPM
- MLE method can effectively adjust the errors-in-variables bias and CAPM holds.
- Support CAPM.
- Linear relationship between beta and expected return.
37
- Book-to-market ratio has significant explanatory power for expected return, but
size has not.
4.3 Capital Structure
• Titman and Wessels (1988), Chang et al. (2009), and Yang et al. (2009) use
structure equation models (e.g. LISREL model and MIMIC model) to
mitigates the measurement problems of proxy variables when working on
capital structure theory. Titman and Wessel (1998) use LISREL method to
investigate determinates of capital structure. In the structure equation
model, they use 15 indicators associated with eight latent variables and set
105 restrictions on the coefficient matrix. Empirical results, however, do
not support for four of eight propositions on the determinants of capital
structure.Specifically, their results show that a firm’s capital structure is not
significantly related to its non-debt tax shields, volatility of earnings,
collateral value of assets, and future growth. One possible reason for the
poor results is that the indicators used in the empirical study do not
adequately reflect the nature of the attributes suggested by financial theory.
38
4.3 Capital Structure - continued
•
•
Chang et al. (2009), therefore, apply a MIMIC model with refined indicators to
reexamine Titman and Wessel’s (1998) work on determinants of capital structure.
Chang et al. (2009) examine seven indicator factors as follows: growth,
profitability, collateral value, volatility, non-debt tax shields, uniqueness, and
industry. Their empirical results show that the growth is the most influential
determinant on capital structure, followed by profitability, and then collateral value.
Under a simultaneous cause-effect framework, their seven constructs as
determinants of capital structure have significant effects on capital structure
decision.
Yang et al. (2009) apply a LISREL model to find determinants of capital structure
and stock returns, and estimate the impact of unobservable attributes on capital
structure decision and stock returns. Using leverage ratio and stock returns as two
endogenous variables and 11 latent factors as exogenous variables, Yang et al.
(2009) find that stock returns, expected growth, uniqueness, asset structure,
profitability, and industry classification are main determinants of capital structure,
while leverage, expected growth, profitability, firm value, and liquidity can explain
stock returns. The capital structure and stock return, in addition, are mutually
39
determined by each other.
4.4 Investment and Capital Asset Allocation
•
Modern q theory developed by Lucas and Prescott (1971) and Mussa (1977) shows
that the shadow value of capital, marginal q, is the firm manager’s expectation of
the marginal contribution of new capital goods to future profits. Marginal q,
therefore, should summarize the effects of all factors relevant to the investment
decision. However, empirical work in testing association between the investment
decision and cash flow is inconsistent to the q theory (e.g. Fazzari et al., 1988).
Lucas and Prescott (1971) and Hayashi (1982) show the equality of marginal q with
average q under the assumptions of constant returns to scale and perfect
competition. In addition, output, sales, and, measures of internal funds have
statistically significant explanation in determining investment decision (e.g. Fazzari
et al., 1988; Schaller (1990); Blundell et al., 1992; and Gilchrist and Himmelberg,
1995).
40
4.4 Investment and Capital Asset Allocation continued
•
•
If financial markets valuation of the capital will be equal to the manager’s valuation, average q
should equal an observable value, Tobin’s q, defined as the ratio of the market value to the
replacement value. Most empirical studies therefore use Tobin’s q as a proxy for marginal q to test
the q theory of investment. Erickson and Whited (2000) argue that the measurement error of
marginal q can result different implications in empirical q models. They therefore incorporate an
EIV model to reexamine the empirical work done by Fazzari et al. (1988). By using generalized
method of moments (GMM), Erickson and Whited (2000) obtain consistent estimators that the
information contained in the third- and higher-order moments of the joint distribution of the
observed regression variables. The estimator precision and consistency can be increased by exploit
the information afforded by an excess of moment equations over parameters. Results show that cash
flow does not affect firms’ financial decision, even for financially constrained firms, and the q theory
is held if measurement error is taken into account.
Almeida et al. (2010) use Monte Carlo simulations and real data to compare the performance of
alternative approaches dealing with measurement error problem. In Monte Carlo simulations, they
find estimators of GMM proposed by Erickson and Whited (2000) are biased for both mismeasured
and well-measured regressors when the data have individual-fixed effects, heteroscedasticity, or no
high degree of skewness.Erickson and Whited (2000). In contrast, the instrumental variable method
results fairly unbiased estimators under those same conditions. Almeida et al. (2010) further
examine empirical investment equation introduced by Fazzari et al. (1988) by using GMM and
instrumental variable method. They conclude that estimators from GMM are unstable across
different specifications and not economically meaningful, while estimators from a simple
41
instrumental method are robust and conform to q theory.
Conclusion
• In this paper, we study how EIV problem affect the estimation of
regression models, extent exiting EIV estimation methods, and discuss the
empirical research applying these estimation methods in deal with
measurement error problem. We first show how EIV problems affect
estimators in the regression model. More specifically, in a multivariate
regression, we show that the EIV leads an underestimation of the
independent variable with measurement error and an overestimation of the
independent variable without measurement error. We further provide six
alternative estimation methods dealing with EIV problem. Classical
method, grouping method, instrumental variable method, mathematical
programming method, maxima likelihood method, and LISREL method are
discussed in detailed. Finally, we investigate how EIV problem can affect
empirical results in issues of cost of capital, asset pricing models, capital
structure, and investment decision and how alternative EIV methods have
been used to correct estimation bias in those issues. We suggest future
studies should pay more efforts on dealing with EIV and obtain robust 42
empirical results from EIV models.