Misspecification-Robust Standard Errors for the
Continuously-Updated GMM Estimator
Nikolay Gospodinov∗
Raymond Kan†
Cesare Robotti‡
August 2013
Abstract
This paper derives an explicit expression for the asymptotic variance of the continuously-updated
GMM estimator under potentially misspecified models. The proposed misspecification-robust
variance estimator allows the researcher to conduct valid inference on the model parameters
even when the model specification is rejected by the data. We illustrate the practical relevance
of our results using simulated data from linear asset pricing models and an empirical application.
Keywords: Model misspecification; Continuously-updated GMM; Generalized empirical likelihood; Misspecification-robust variance estimator; Asset pricing.
JEL classification numbers: C12; C13; G12.
∗
Nikolay Gospodinov, Research Department, Federal Reserve Bank of Atlanta, 1000 Peachtree Street N.E., Atlanta, GA 30309, USA; E-mail: [email protected]
†
Raymond Kan, Joseph L. Rotman School of Management, University of Toronto, 105 St. George Street, Toronto,
ON M5S 3E6, Canada; E-mail: [email protected]
‡
Cesare Robotti, Finance Department, Imperial College Business School, 53 Prince’s Gate, London SW7 2PG,
United Kingdom; E-mail: [email protected]
1
Introduction
Given the complexity of the economic and financial system, it seems natural to view all economic models only as approximations to the true data generating process (Watson, 1993; Canova,
1994; among others). Models for which the likelihood function is available are now routinely estimated in a quasi-maximum likelihood framework and the statistical inference is performed using
misspecification-robust standard errors (White, 1982). By contrast, misspecification-robust inference for moment condition models, estimated by the generalized method of moments (GMM), is
much less wide-spread among applied researchers and the common practice is to continue to use the
asymptotic standard errors of Hansen (1982), derived under a correctly specified model, even when
the model is rejected by the data. This is unfortunate since most economic models are defined by a
set of conditional or unconditional moment restrictions and not allowing for possible (global) misspecification of these moment restrictions would render the GMM inference asymptotically invalid.
Hall and Inoue (2003) studied the limiting behavior of the one- and multiple-step GMM estimators in globally misspecified models. They derived the asymptotic variance of these estimators in
the presence of model misspecification and showed that the misspecification correction depends on
the weighting matrix used in the estimation. The consequences of model misspecification for GMM
estimation and inference are summarized in Hall (2005). Despite these recent advances in the literature, the use of misspecification-robust standard errors in empirical work with GMM estimators
is largely absent.
In this paper, we derive an explicit expression for the asymptotic variance of the continuouslyupdated GMM (CU-GMM) estimator in globally misspecified models. We focus on the CU-GMM
estimator for several reasons. First, the CU-GMM estimator is an optimal one-step GMM estimator
(Hansen, 1982; Hansen, Heaton and Yaron, 1996) which is gaining increasing popularity in applied
work, especially when the invariance of the estimator to normalizations and transformations is
particularly desirable (Peñaranda and Sentana, 2013). Second, the CU-GMM estimator is a member
of the class of generalized empirical likelihood (GEL) estimators (Newey and Smith, 2004) which
provides an alternative look into the first- and higher-order asymptotic properties of the CU-GMM
estimator. In fact, we use the GEL framework to parameterize the degree of model misspecification
as the distance of the pseudo-true value of the vector of Lagrange multipliers, associated with the
moment conditions, from zero and cast the CU-GMM estimator as a solution to a quasi-likelihood
problem. This allows us to work directly with the score function and to sidestep some explicit
joint normality assumptions in the approach by Hall and Inoue (2003). Due to the quasi-likelihood
interpretation of the estimated augmented parameter vector (the parameters of interest and the
Lagrange multipliers), its asymptotic variance takes the usual sandwich form as in White (1982). In
1
this respect, we complement the results in Kitamura (1998) and Schennach (2007) and provide an
explicit expression for the asymptotic variance of the CU-GMM estimator in misspecified models.
The rest of the paper is structured as follows. Section 2 derives the limiting distribution of the
CU-GMM estimator in misspecified models. Section 3 specializes the expression of the asymptotic
variance to linear asset pricing models and provides simulation results on the empirical size of
t-tests computed with standard errors under correct model specification and misspecification robust standard errors. We also illustrate the economic significance of the proposed misspecification
adjustment using actual data for two popular asset pricing models. Section 4 concludes.
2
Asymptotic Variance of CU-GMM Estimator Under Model Misspecification
Let et (yt , λ) be an N × 1 vector of moment conditions that are known functions of the data yt and
a K × 1 (K < N ) parameter vector λ ∈ Λ. For notational convenience, we suppress the dependence
of the moment conditions on the data and let et (λ) ≡ et (yt , λ). The population moment conditions
in (globally) misspecified models are given by
E[et (λ)] = µ(λ),
(1)
where µ : Λ → RN such that kµ(λ)k > 0 for all λ ∈ Λ. Let λ∗ denote the pseudo-true value of λ
which is defined as λ∗ = argminλ kµ(λ)k2M , where kµ(λ)k2M = µ(λ)0 M µ(λ) and M is a symmetric,
positive definite weighting matrix. Note that different weighting matrices M give rise to different
pseudo-true values and λ∗ should be indexed by M but, for simplicity, we suppress the dependence
of λ∗ on M . In what follows, we let µ(λ∗ ) = µ∗ . In the case of correctly specified models, µ∗ = 0N
and λ∗ is the true value of λ.
Assumption A. Assume that (a) {yt }Tt=1 is a realization of a stationary and strongly mixing vector
P∞ 2
1−1/r for some r > 1; (b) e (λ) forms a martingale
process with mixing coefficients
t
j=1 j α(j)
difference sequence.
Assumption A imposes some restrictions on the dynamic behavior of the data and the moment
conditions. Assumption A(a) ensures that the process is stationary and ergodic. The martingale
difference sequence assumption in A(b) can be relaxed by modifying the structure of the estimation
problem along the lines suggested by Smith (2011). Given Assumption A, the continuously updated
GMM (CU-GMM) estimator of the K × 1 parameter vector λ is defined as
λ̂ = argminλ eT (λ)0 WT (λ)−1 eT (λ),
2
(2)
where eT (λ) =
1
T
PT
t=1 et (λ)
is an N × 1 vector of sample moment conditions and
T
1X
et (λ)et (λ)0 .
WT (λ) =
T
(3)
t=1
The first-order conditions for the CU-GMM estimator λ̂ are given by (Newey and Smith, 2004)1
DT (λ̂)0 WT (λ̂)−1 eT (λ̂) = 0K ,
(4)
where
DT (λ̂) =
T
1 X ∂et (λ̂)
wt
T
∂λ0
(5)
t=1
and wt = 1 − eT (λ̂)0 WT (λ̂)−1 et (λ̂).
Let GT (λ̂) =
1
T
PT
t=1 Gt (λ̂),
where Gt (λ) =
∂et (λ)
, and take a first-order Taylor series expan∂λ0
sion of eT (λ̂) about λ∗
eT (λ̂) = eT (λ∗ ) + GT (λ̄)(λ̂ − λ∗ ),
(6)
where λ̄ is an intermediate point on the line segment joining λ̂ and λ∗ . By substituting this
expression into the first-order conditions (4) and rearranging, we obtain
√
h
i−1
√
T (λ̂ − λ∗ ) = − DT (λ̂)0 WT (λ̂)−1 GT (λ̄)
DT (λ̂)0 WT (λ̂)−1 T eT (λ∗ ).
(7)
The approach of Hall and Inoue (2003) is to make the dependence on the degree of model misspecification µ∗ explicit by rewriting this expression as
√
h
i−1
T (λ̂ − λ∗ ) = − DT (λ̂)0 WT (λ̂)−1 GT (λ̄)
√
1 XT
[et (λ∗ ) − µ∗ ] + T DT (λ̂)0 WT (λ̂)−1 µ∗ ].
×[DT (λ̂)0 WT (λ̂)−1 T − 2
t=1
(8)
While this approach is feasible, it is somewhat tedious due to the dependence of the weighting
matrix on the parameter vector λ. In what follows, we will pursue an alternative approach which
allows us to write the estimator of an augmented parameter vector as a solution to the score
function of a just-identified problem. The point of departure is the observation that the CU-GMM
estimator can be defined equivalently (see Newey and Smith, 2004) as the solution to the following
saddle point problem:
T
1X
ρ γ 0 et (λ) ,
λ∈Λ γ∈Γ(λ) T
λ̂ = arg min max
(9)
t=1
1
Newey and Smith (2004, footnote
2) establish the equivalence of this CU-GMM estimator and the CU-GMM
P
estimator based on WT (λ) = T1 Tt=1 [et (λ) − eT (λ)][et (λ) − eT (λ)]0 .
3
where ρ (v) = − 21 v 2 − v and γ is an N × 1 vector of Lagrange multipliers associated with the
moment conditions E [et (λ)] = 0N . The pseudo-true value for γ is defined accordingly as γ ∗ ≡
γ ∗ (λ) = arg maxγ∈Γ(λ) E[ρ (γ 0 et (λ))]. Note that this new setup allows us to change the parameters
that characterize model misspecification from µ∗ to γ ∗ . For correctly specified models, we have
γ ∗ = 0N while for misspecified models, kγ(λ)k > 0 for all λ ∈ Λ.
Let θ = (γ 0 , λ0 )0 ∈ Θ and θ∗ = (γ 0∗ , λ0∗ )0 . The first-order conditions are given by
T
1X
st (θ)
= 0N +K ,
sT (θ̂) ≡
T
t=1
where
∂ρ (γ 0 et (λ))
st (θ) ≡
=−
∂θ
(10)
θ=θ̂
[1 + γ 0 et (λ)] et (λ)
[1 + γ 0 et (λ)] Gt (λ)0 γ
.
(11)
The (N + K) vector st (θ) can be interpreted as the score function of a quasi-likelihood problem.
As argued above, we augment the first-order conditions for the parameter vector of interest λ with
the parameter vector of Lagrange multipliers γ in order to make the model misspecification, which
is reflected in γ, explicit in deriving the limiting distribution. Note also that from the first N
equations in (10), we have γ̂ = −WT (λ̂)−1 eT (λ̂).
A mean value expansion of sT (θ̂) about θ∗ yields
0N +K = sT (θ∗ ) + HT (θ̄)(θ̂ − θ∗ )
or
√
where HT (θ) =
1
T
PT
t=1 Ht (θ)
−1 √
T (θ̂ − θ∗ ) = − HT θ̄
T sT (θ∗ ) ,
(12)
(13)
with Ht (θ) = (∂/∂θ0 )st (θ) and θ̄ is an intermediate point on the line
segment joining θ̂ and θ∗ . More specifically,
#
"
et (λ)et (λ)0
ct Gt (λ) + et (λ)γ 0 Gt (λ)
,
Ht (θ) = −
(2)
ct Gt (λ)0 + Gt (λ)0 γe0t (λ) ct (IK ⊗ γ 0 )Gt (λ) + Gt (λ)0 γγ 0 Gt (λ)
(14)
(2)
where Gt (λ) = (∂/∂λ0 )vec(Gt (λ)) and ct = 1 + γ 0 et (λ). To derive the limiting distribution of θ̂,
we make the following assumption.
Assumption B. Assume that (a) the pseudo-true values λ∗ and γ ∗ are unique for all λ ∈ Λ
and λ ∈ Γ(λ); (b) et (λ) is continuous in λ, E [sup |ρ (γ 0 et (λ)) |] < ∞ for all λ ∈ Λ and λ ∈
Γ(λ), and the parameter space Λ is a compact subset of RK ; (c) E[et (λ)et (λ)0 ] is non-singular
for all λ ∈ Λ; (d) λ∗ is in the interior of Λ and et (λ) is twice continuously differentiable in
h
i
λ; (e) E supθ∈N (θ∗ ) ∂θ∂ 0 st (θ) < ∞ for some neighborhood N of θ∗ ; (f ) E [st (θ∗ )] = 0; (g)
E kst (θ∗ )st (θ∗ )0 k exists and is finite; (h) E ∂θ∂ 0 st (θ) is of full rank.
4
p
The regularity conditions in Assumption B are standard and ensure that θ̂ → θ∗ ,
p
sup kHT (θ) − E[HT (θ)]k → 0
(15)
θ∈Θ
and
√
d
T sT (θ∗ ) → N (0N +K , S),
(16)
where S = E[st (θ∗ )st (θ∗ )0 ]. Next, we state our main result.
Theorem 1. Let W = E[et (λ∗ )et (λ∗ )0 ], G = E[Gt (λ∗ )], B = E[ct Gt (λ∗ )] + E[et (λ)γ 0 Gt (λ)] and
(2)
C = (IK ⊗ γ 0∗ )E[Gt (λ∗ )] + E[Gt (λ∗ )0 γ ∗ γ 0∗ Gt (λ∗ )]. Under Assumptions A and B, it follows that
√
d
T (θ̂ − θ∗ ) → N (0N +K , Σ),
(17)
0 , l0 ] and
where Σ ≡ E[lt lt0 ], lt ≡ [l1t
2t
l1t = W −1 [ct et (λ∗ ) − Bl2t ] ,
l2t = (C − B 0 W −1 B)−1 ct Gt (λ∗ )0 γ ∗ − B 0 W −1 et (λ∗ ) .
(18)
(19)
Proof. See Appendix A.
(2)
Note that for linear models, Gt (λ∗ ) is a null matrix and C = E[Gt (λ∗ )0 γ ∗ γ 0∗ Gt (λ∗ )]. Furthermore, for correctly specified models, the limiting distribution in Theorem 1 specializes to the result
in Theorem 3.2 of Newey and Smith (2004). More specifically, for correctly specified models, we
have γ ∗ = 0N , (C − B 0 W −1 B)−1 = −(G0 W −1 G)−1 and
l1t = W −1 [et (λ∗ ) − Gl2t ] ,
(20)
l2t = (G0 W −1 G)−1 G0 W −1 et (λ∗ ).
√
Since the asymptotic variance matrix of T γ̂, given by
(21)
W −1 − W −1 G(G0 W −1 G)−1 G0 W −1 ,
(22)
is singular, the following corollary provides an alternative representation of the limiting distribution
of θ̂ for correctly specified models.
Corollary 1. Let PW G denote an N ×(N −K) orthonormal matrix whose columns are orthogonal
1
to W − 2 G. Then, under Assumptions A and B and for γ ∗ = 0N ,
!
√
1
0
2 γ̂
IN −K
0(N −K)×K
T
P
W
d
W
G
√
→ N 0N +K ,
.
0K×(N −K) (G0 W −1 G)−1
T (λ̂ − λ∗ )
(23)
A consistent estimator, Σ̂, of the variance matrix of θ̂ in Theorem 1 can be obtained by replacing
the population quantities with their corresponding sample analogs. The sample variances of γ̂ and
λ̂ are then given by the upper left (N × N ) and the bottom right (K × K) blocks of Σ̂, respectively.
5
3
Numerical Assessment: Linear Asset Pricing Models
3.1
Monte Carlo Simulations
In this section, we evaluate the performance of the proposed variance estimator by reporting the
empirical size and power of t-tests that are constructed using standard errors under correct specification and misspecification-robust standard errors. We consider linear asset pricing models with a
constant term and one risk factor, ft = [1, f˜t ]0 , that are either (i) correctly specified or (ii) misspecified. The returns on the test assets and the risk factor f˜t are drawn from a multivariate normal
distribution. In all simulation designs, the covariance matrix of the simulated test asset returns Rt
is set equal to the estimated covariance matrix from the 1959:2–2012:12 sample of monthly gross
returns on the 25 Fama-French size and book-to-market ranked portfolios from Kenneth French’s
website (N = 25). For misspecified models, the means of the simulated returns are set equal to
the means of the actual returns. For correctly specified models, the means of the simulated returns
are set such that the asset pricing model restrictions are satisfied (i.e., the pricing errors are zero).
Similarly, the mean and variance of the simulated risk factor are calibrated to the mean and variance of the value-weighted market excess return. The covariances between the risk factor and the
returns are chosen based on the covariances estimated from the data. The time-series sample size
is taken to be T = 300, 600, 1200, and 2400. The number of Monte Carlo replications is 100,000.
The pricing errors (moment conditions) of the linear asset pricing model are given by
e(λ) = Gλ − 1N ,
(24)
P
P
where G = E[Rt ft0 ]. Let Gt = Rt ft0 , GT = T1 Tt=1 Gt = T1 Tt=1 Rt ft0 , et (λ) = Rt ft0 λ − 1N
P
and eT (λ) = T1 Tt=1 et (λ) = GT λ − 1N . Then, the CU-GMM estimators of λ and γ are obtained
as
λ̂ = argminλ eT (λ)0 WT (λ)−1 eT (λ)
(25)
γ̂ = −WT (λ̂)−1 eT (λ̂).
(26)
and
Also, let ĉt = 1 + γ̂ 0 et (λ̂), B̂ =
1
T
PT
1
0
t=1 ĉt Rt ft + T
PT
t=1 et (λ̂)γ̂
0
Gt and Ĉ =
1
T
PT
h
i
ˆl1t = WT (λ̂)−1 ĉt et (λ̂) − B̂ ˆl2t ,
h
i
ˆl2t = (Ĉ − B̂ 0 WT (λ̂)−1 B̂)−1 ĉt G0 γ̂ − B̂ 0 WT (λ̂)−1 et (λ̂) ,
t
0
0
t=1 Gt γ̂γ̂ Gt .
Then,
(27)
(28)
which are used to construct a consistent estimator of the asymptotic variance matrix of θ̂, Σ̂,
in Theorem 1. The square root of the last diagonal element of Σ̂ is then used to construct the
6
misspecification-robust t-test, denoted by tm . The variance estimator of θ̂ under correct specification is obtained from
h
i
ˆl1t = WT (λ̂)−1 et (λ̂) − GT ˆl2t ,
(29)
ˆl2t = (G0 WT (λ̂)−1 GT )−1 G0 WT (λ̂)−1 et (λ̂),
T
T
(30)
and the square root of the last diagonal element is used to construct the t-test under correct
specification, denoted by tc .
Tables I and II report the empirical size and (raw, size-unadjusted) power of H0 : λ2 = λ∗2
(size)2 and H0 : λ2 = 0 (power) for the t-tests tc (λ̂2 ) and tm (λ̂2 ) using standard normal critical
values. The null hypothesis H0 : λ2 = 0 is often of primary interest to financial economists since it
provides information on whether this risk factor is priced or not. Table I presents the results for
data simulated from a correctly specified model.
Table I about here
Although the true model is correctly specified, the t-test under correct specification tc tends
to overreject even for sample sizes as large as T = 1200. Interestingly, the misspecification-robust
t-test tm corrects these relatively large size distortions in small samples. The slight conservativeness
of the misspecification-robust t-test does not appear to adversely affect its power.
Table II reports the empirical rejection probabilities for data simulated from a misspecified asset
pricing model.
Table II about here
When the true model is misspecified, the t-tests tc are no longer valid which is reflected in the
substantial overrejections under the null H0 : λ2 = λ∗2 . This should serve as a warning signal
to applied researchers who routinely use standard errors constructed under the assumption of a
correctly specified model in evaluating the statistical significance of risk premium parameters. This
suggests that the researcher will conclude erroneously (with relatively high probability) that the
risk factor is important for the pricing of the test assets. In contrast, the empirical size of the
misspecification-robust t-tests is very close to the nominal level for sample sizes T ≥ 600. As in
the case of correctly specified models, the effective size correction that the misspecification-robust
t-tests perform does not reflect negatively on the power of the tests.
2
The pseudo-true values are λ∗2 = 2.931 for the correctly specified model and λ∗2 = 7.223 for the misspecified
model.
7
3.2
Empirical Illustration
In this empirical application, we estimate the parameters λ of two asset pricing models by CUGMM and report their corresponding t-statistics under correctly specified and misspecified models.
To quantify the degree of misspecification of these models, we also present the p-value of the test for
overidentifying restrictions (OIR). The test asset returns Rt are the monthly gross returns on the
value-weighted 25 Fama-French size and book-to-market ranked portfolios for the period February
1959 – December 2012. Since these portfolios are believed to be characterized by a strong factor
structure (see Lewellen, Nagel and Shanken, 2010), we also augment the 25 Fama-French portfolios
with the 17 industry portfolios from Kenneth French’s website. As argued in Lewellen, Nagel and
Shanken (2010), the inclusion of the industry portfolios presents a greater challenge to the various
asset pricing models.
The first model that we consider is the CAPM with ft = [1, vwt ]0 , where vw is the excess
return (in excess of the one-month T-bill rate) on the value-weighted stock market index (NYSEAMEX-NASDAQ). The second model is the three-factor model of Fama and French (FF3, 1993)
with ft = [1, vwt , smbt , hmlt ]0 , where smb is the return difference between portfolios of stocks with
small and large market capitalizations, and hml is the return difference between portfolios of stocks
with high and low book-to-market ratios (“value” and “growth” stocks, respectively). The choice of
these models is driven by their popularity in empirical work as well as the fact that these models do
not appear to violate the rank condition in Assumption B which may render the standard statistical
inference invalid (see Gospodinov, Kan and Robotti, 2013, for theoretical and empirical analyses of
unidentified asset pricing models). All the data for the test assets and the factors are obtained from
Kenneth French’s website. The results for the two models and the two sets of test asset returns
(Panel A and Panel B, respectively) are presented in Table III.
Table III about here
The first result that emerges from Table III is that both models are strongly rejected by the
data. Hence, to ensure valid statistical inference, the standard errors need to be adjusted to
take into account the additional uncertainty arising from model misspecification. However, if the
applied researcher continued to use the usual standard errors (derived under the assumption of
correct model specification), she would have concluded that all of the risk factors in CAPM and
FF3 are priced as their t-statistics exceed the standard normal critical value at the 5% significance
level (all factors, except for hml in Panel A are also significant at the 1% significance level). In
contrast, accounting for model misspecification produces standard errors that are almost twice as
large as those constructed under correct model specification. As a result, the misspecification-robust
8
inference suggests that only smb appears to be priced at the 1% significance level. This suggests
that the evidence of pricing in arguably one of the most successful empirical asset pricing models,
such as the three-factor model of Fama and French (1993), is relatively weak once the uncertainty
associated with potential model misspecification is incorporated into the inference procedure.
4
Conclusions and Implications for Empirical Practice
This paper derives the asymptotic variance of the CU-GMM estimator in potentially misspecified
models. This fills an important gap in the literature given the increasing popularity of the CUGMM estimator and the wide-spread belief that economic models are inherently misspecified. The
expression for the asymptotic variance that we derive is explicit and easy-to-use in practice.
We illustrate the importance of the model misspecification adjustment of the standard errors
in the context of linear asset pricing models. While, as expected, the misspecification-robust
tests deliver impressive improvements when the true model is misspecified, these tests also tend
to provide substantial small-sample corrections when the model is correctly specified. All these
size corrections are achieved at no apparent cost associated with loss of power. As a result, the
main recommendation that emerges from our analysis is that standard errors which are robust
to potential model misspecification should also be used in applied work regardless of whether the
model is believed (based, for example, on the outcome of a pre-test for overidentifying restrictions)
to be correctly specified or misspecified.
9
A
Appendix: Proof of Theorem 1
From Assumptions A and B, it follows that
√
d
T (θ̂ − θ∗ ) → N (0N +K , H −1 SH 0−1 ),
where
H ≡ E[HT (θ∗ )] =
W B
B0 C
(A1)
,
(A2)
where W , B and C are defined in the text.
To derive the explicit expression for the asymptotic covariance matrix of θ̂ in Theorem 1, we
write
H −1 SH 0−1 = E[lt lt0 ],
where
"
lt ≡
l1t
l2t
(A3)
#
= H −1 st (θ∗ ).
(A4)
From the definition of H in (A2), we can use the partitioned matrix inverse formula to obtain
"
#
−1 (I + B H̃D 0 W −1 ) −W −1 B H̃
W
N
H −1 =
,
(A5)
−H̃ 0 B 0 W −1
H̃
where H̃ = (C − B 0 W −1 B)−1 . The invertibility of C − B 0 W −1 B follows from Assumption B(h).
Using (A5) and (11), we can express l1t and l2t as
l1t = W −1 [ct et (λ∗ ) − Bl2t ] ,
l2t = H̃ct Gt (λ∗ )0 γ ∗ − B 0 W −1 et (λ∗ ) .
This delivers the desired result.
10
(A6)
(A7)
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11
Table I
Empirical size and power of t-tests: correctly specified model
Panel A: tc
T
300
600
1200
2400
Size: H0 : λ2 = λ∗2
Power: H0 : λ2 = 0
Level of significance
Level of significance
10%
0.226
0.153
0.125
0.111
10%
0.434
0.535
0.744
0.938
5%
0.149
0.088
0.067
0.058
1%
0.057
0.024
0.016
0.012
5%
0.337
0.423
0.639
0.891
1%
0.182
0.229
0.410
0.736
Panel B: tm
T
300
600
1200
2400
Size: H0 : λ2 = λ∗2
Power: H0 : λ2 = 0
Level of significance
Level of significance
10%
0.078
0.089
0.095
0.097
10%
0.229
0.428
0.700
0.930
5%
0.035
0.042
0.046
0.048
1%
0.006
0.007
0.009
0.009
5%
0.136
0.303
0.579
0.875
1%
0.037
0.120
0.331
0.702
The table presents the actual probabilities of rejection for the t-tests of H0 : λ2 = λ∗2 (size of the
test) and H0 : λ2 = 0 (power of the test) for different levels of significance. The model includes a
constant term and one risk factor (CAPM specification) and is calibrated to monthly data for the
period February 1959 – December 2012. The risk factor and the returns on the test assets are generated
from a multivariate normal distribution. Panel A presents the empirical size and power for t-tests
that are constructed assuming that the model is correctly specified (tc ). Panel B reports the empirical
size and power for misspecification-robust t-tests (tm ). The results for different number of time series
observations (T ) are based on 100,000 simulations.
12
Table II
Empirical size and power of t-tests: misspecified model
Panel A: tc
T
300
600
1200
2400
Size: H0 : λ2 = λ∗2
Power: H0 : λ2 = 0
Level of significance
Level of significance
10%
0.555
0.458
0.408
0.375
10%
0.877
0.953
0.995
1.000
5%
0.479
0.377
0.323
0.291
1%
0.348
0.244
0.192
0.165
5%
0.843
0.936
0.993
1.000
1%
0.765
0.889
0.984
1.000
Panel B: tm
T
300
600
1200
2400
Size: H0 : λ2 = λ∗2
Power: H0 : λ2 = 0
Level of significance
Level of significance
10%
0.125
0.102
0.099
0.098
10%
0.493
0.743
0.952
0.999
5%
0.065
0.049
0.048
0.048
1%
0.014
0.009
0.009
0.009
5%
0.366
0.629
0.911
0.997
1%
0.165
0.379
0.762
0.985
The table presents the actual probabilities of rejection for the t-tests of H0 : λ2 = λ∗2 (size of the
test) and H0 : λ2 = 0 (power of the test) for different levels of significance. The model includes a
constant term and one risk factor (CAPM specification) and is calibrated to monthly data for the
period February 1959 – December 2012. The risk factor and the returns on the test assets are generated
from a multivariate normal distribution. Panel A presents the empirical size and power for t-tests
that are constructed assuming that the model is correctly specified (tc ). Panel B reports the empirical
size and power for misspecification-robust t-tests (tm ). The results for different number of time series
observations (T ) are based on 100,000 simulations.
13
Table III
Empirical results for CAPM and FF3
Panel A: 25 size and book-to-market portfolios
tc
models
CAPM
FF3
vw
4.29
3.92
smb
-4.23
tm
hml
vw
2.47
2.28
-2.01
smb
-2.91
OIR
hml
-1.41
0.0000
0.0017
Panel B: 25 size and book-to-market and 17 industry portfolios
tc
models
vw
CAPM
FF3
4.26
4.92
smb
-5.02
tm
hml
vw
-3.75
1.93
2.27
smb
-2.80
OIR
hml
-1.89
0.0000
0.0000
The table reports the t-statistics under correct model specification (tc ) and model misspecification (tm )
as well as the p-value of the test for overidentifying restrictions (OIR). The models are CAPM and the
three-factor model (FF3) of Fama and French (1993). The sample period is February 1959 – December
2012. Panel A presents the results for the 25 Fama-French size and book-to-market portfolios as test
assets and Panel B presents the results for the 25 Fama-French size and book-to-market and 17 industry
portfolios as test assets.
14