Semiparametric Maximum Likelihood Estimation of
Di¤erentiated Product Demand Systems
Zsolt Sándor1
Sapientia University Miercurea Ciuc
November 2013
Abstract
We propose a semiparametric maximum likelihood (SPML) estimation procedure for di¤erentiated product demand systems. This method, developed by Ai (1997), replaces the unknown
density of the unobserved characteristics in the likelihood by the nonparametric kernel estimator of the residuals for given parameter value. For a special case of the model we show
that the proposed estimation method identi…es the parameters under simple regularity conditions. We show via Monte Carlo simulations that, with a reduced form supply, for the demand
parameters the proposed estimator has a …nite sample performance substantially better than
the commonly used generalized method of moments estimator regarding bias, variance and
mean squared error. We also …nd that the SPML estimator performs remarkably well when
the supply side is misspeci…ed such that the prices are determined as the Nash equilibrium of
a pricing game. The paper concludes that, in situations where the GMM estimator is poor,
the SPML estimator can be a slower but reliable and much more precise alternative.
Keywords: nonparametric, random coe¢ cient logit, discrete choice.
1
I thank József Molnár and Lars Nesheim for useful comments.
1
1
Introduction
Empirical models of market demand have become popular in various applications after
the in‡uential work by Berry et al. (1995, BLP hereafter). These authors constructed
a market equilibrium model with di¤erentiated products and price competition. An
important feature of the demand speci…cation is the presence of random coe¢ cients that
serve for modelling heterogeneity of consumers’ tastes for the various characteristics.
It has been shown (e.g., BLP, Nevo 2001) that these imply patterns of substitution
between products that are more realistic. Another important feature is the presence of
unobserved characteristics di¤erent from the previously used idiosyncratic errors, which
facilitates the modelling of price endogeneity.
These features are plausible from various perspectives, but they imply that the model
is di¢ cult to estimate with the originally proposed method, which is the GMM (i.e.,
generalized method of moments). The GMM estimator has the advantage that it does
not require speci…cation of the distribution of the unobserved characteristics, and for
the estimation of the demand parameters only, it does not require speci…cation of the
supply side. With GMM estimation, since prices are endogenous regressors, they need
to be instrumented. However, it is di¢ cult to …nd good instruments for this model. The
presence of random coe¢ cients in the demand system implies that in addition to the
price instrument other instruments are needed. This appears to have the consequence
that, in order to obtain GMM estimates of reasonable bias and precision, one needs a
very large number of observations. For example, BLP note that they needed to extend
their data set in order to obtain statistically meaningful estimates. In many situations
such large datasets are costly/di¢ cult to get; the parameters of the model may also
change over a long time period.
An alternative class of estimation methods is based on the likelihood constructed
as the density of the endogenous variables. In order to compute this density one needs
to specify the supply side of the model and the distribution of the unobserved characteristics. Then the likelihood of the endogenous variables is computed as the density
of the random vector of unobserved characteristics transformed by the mapping provided by the structural form of the model. Implementation of these likelihood-based
methods proceeds by either maximum likelihood or Bayesian estimation, where for both
methods the distribution of the unobserved characteristics is typically assumed to be
jointly normal, and the transformation of the unobserved characteristics is assumed to
be one-to-one. In order to get rid of the latter assumption, some authors (e.g., Yang
et al. 2003) assume a reduced form supply side structure according to which prices are
determined by an explicit expression of supply side variables.
Jiang et al. (2009) propose a Bayesian method for the estimation of the model with
reduced form supply, and construct an MCMC algorithm that samples from the posterior
2
distribution of the unknown parameters. They …nd in Monte Carlo simulations that their
estimator performs well relative to GMM in various cases that include misspeci…cation
of the distribution of unobserved characteristics like heteroskedasticity, autocorrelation,
asymmetric and U-shaped beta densities. It is unknown, however, how their estimator
performs if the unobserved characteristics have other types of distributions or if the
supply side di¤ers from the assumed reduced form pricing. This latter misspeci…cation
can occur in relevant cases; for example, when …rms compete in prices like in the original
BLP model.
In this paper we propose a semiparametric maximum likelihood (SPML) estimator
for which the distribution of unobserved characteristics need not be speci…ed. This
method, developed by Ai (1997), builds on the maximum likelihood estimator described
above, and it replaces the unknown density of the unobserved characteristics by the
nonparametric kernel estimator of the residuals for given parameter value. The estimator
proposed by Ai solves the trimmed …rst order conditions from maximizing the loglikelihood. Unlike this estimator, our estimator maximizes the semiparametric loglikelihood directly.
We contribute to the literature in two ways. First, under the assumption that the
observed and unobserved product characteristics are independent, we provide an equivalent condition for the identi…cation of the model through the SPML estimator. The
argument is similar in spirit to ideas explored by, for example, Brown (1983), Benkard
and Berry (2006), and Matzkin (2008). By specializing this result, we provide simple
regularity conditions under which the SPML estimator identi…es the model with simple
logit demand and reduced form supply. Second, we show via Monte Carlo simulations
that regarding bias, variance and mean squared error the SPML estimator has a …nite
sample performance substantially better than the commonly used GMM estimator and
a GMM estimator with approximately optimal instruments. We also …nd that our estimator performs well in cases when the supply side is misspeci…ed or the unobserved
characteristics are heteroskedastic. Speci…cally, in the case considered in our Monte
Carlo simulations, the SPML estimator performs remarkably well when the prices are
determined as the Nash equilibrium of a pricing game.
The SPML estimator is related to other semiparametric maximum likelihood estimators like the estimator developed by Gallant and Nychka (1987) or sieve maximum
likelihood (e.g., Chen 2007), where the sieve is used for the unknown distribution of the
unobserved characteristics. A practical disadvantage of these estimators compared to
SPML is that, due to the estimator of the unobserved characteristics, the number of
nonlinear parameters in the objective function increases, and it may reach a rather large
value, while for SPML the nonparametric kernel density estimator of the residuals has
closed form.
The next section presents the model. Section 3 presents the details of the SPML esti3
mation method. The proposed identi…cation condition is discussed in Section 4. Section
5 presents the Monte Carlo study on the …nite sample performance of the estimators.
Section 6 presents the Monte Carlo study on the performance under misspeci…cation.
Finally, Section 7 concludes.
2
The Model
The model determines the demand functions for the di¤erent products based on latent
utilities. The latent utility of consumer i from purchasing product j in market t is
assumed to be
uijt = i pjt + xjt i + jt + "ijt ;
where
i
N
;
2
;
i
N ( ; ) with
0
B
=@
2
1
0
..
.
2
K
0
1
C
A;
pjt is the price of product j in market t, xjt is a 1 K-vector of characteristics of
product j in market t containing 1 for intercept, jt is a characteristic of product j in
market t that is not observed by the econometrician, and "ijt is an iid type I extreme
value distributed error term. In market t consumers can choose from Jt products and
the outside alternative, which represents the option of not purchasing any of the Jt
products, and is assumed to yield utility
ui0t = "i0t :
We adopt the simpli…cation of notation jt = pjt + xjt + jt for the part of utility
that is common to all consumers.
These utility speci…cations yield the predicted market share expression
R
sjt ( t ; pt ; xt ; ) = qjt ( t ; pt ; xt ; ; v) (v) dv
(1)
of product j in market t, where
qjt (
t ; pt ; xt ;
; v) =
1+
exp
PJt
r=1
jt
+
exp (
v pjt + xjt
rt
+
1=2
v
v prt + xrt
1=2 v
)
(2)
is the consumer-speci…c probability of purchasing product j. Here t = pt + xt + t ;
1 vectors of unobserved characteristics and prices in market t, xt
t ; pt are the Jt
0
is the Jt K matrix of observed characteristics in market t, = ( ; 0 ; 0 ) , where
0
= ( ; 1 ; :::; K )0 , v = v ; v 0
N (0; IK+1 ), and is the density function of the
K + 1-dimensional standard normal distribution. The predicted share of the outside
alternative in market t is
Z
1
s0t ( t ; pt ; xt ; ) =
(v) dv:
PJt
1 + r=1 exp ( rt + v prt + xrt 1=2 v )
4
We assume a reduced form supply side of the market so that the prices in market t are
determined as
pjt = wjt + ! jt ;
(3)
where wjt is a row vector of price-shifting variables corresponding to product j and ! jt
is a supply side unobserved characteristic.
3
Semiparametric Maximum Likelihood Estimation
The estimation is based on computing the demand and supply unobserved characteristics
in terms of the model ingredients. On the demand side this consists of solving the
nonlinear system of equations
sjt (
t ; pt ; xt ;
(4)
) = sjt ; j = 1; :::; Jt
in t in each market t, where sjt is the observed share of product j in market t. According
to Berry (1994) and BLP, the solution is unique and can be computed as the …xed point
of the contraction
G t : RJ t ! R Jt ; G t (
t)
=
t
+ ln st
ln st (
t ; pt ; xt ;
);
where st and st ( t ; pt ; xt ; ) are the column-vectors of the observed and predicted market
shares in market t. We note that the solution t (st ; pt ; xt ; ) does not depend on and
. Therefore, we can express the unobserved characteristics vectors in market t in terms
of the model ingredients as
t
(st ; pt ; xt ; ) =
t
! t (pt ; wt ; ) = pt
(st ; pt ; xt ; )
( pt + xt ) ;
(5)
wt ;
where wt is the column-vector of the price-shifting variables in market t. This de…nes a
simultaneous system of equations that is linear in ; ; and nonlinear in .
3.1
The log-likelihood
In order to derive the log-likelihood we use the assumption that the unobserved characteristics jt ; ! jt are independent for all products j and markets t with common density
denoted 0 , and they are independent of (xt ; wt ) for all markets t. The joint density
function of the endogenous variables (st ; pt ) in market t conditional on the observed
characteristics (xt ; wt ) can be computed by using the formula of the density of (st ; pt )
obtained as the one-to-one transformation ( t ; ! t ) 7 ! (st ; pt ) speci…ed by equations (1)
5
and (3).2 For the true value
density is equal to
f (st ; pt jxt ; wt ;
0
= ( 00 ;
0 ) = dt (st ; pt ; xt ;
0 0
0)
of the vector containing all parameters this
Jt
Y
0)
0
jt
(st ; pt ; xt ;
0 ) ; ! jt
(pt ; wt ;
0)
(6)
;
j=1
where
@ t
@s0t
@! t
@s0t
dt (st ; pt ; xt ; ) = det
@ t
@p0t
@! t
@p0t
!
(st ;pt ;xt ; )
is the Jacobian of the inverse of the transformation ( t ; ! t ) 7 ! (st ; pt ). By noting that
@! t
t
= 0 and @!
= I, we get
@s0t
@p0t
!
@ t
@ t
@ t
@s0t
@p0t
dt (st ; pt ; xt ; ) = det
= det
@s0t (st ;pt ;xt ; )
0
I
(st ;pt ;xt ; )
1
@st
(
@ 0t
= det
t (st ; pt ; xt ; ) ; pt ; xt ; )
;
where the latter equality follows by the implicit function theorem. Since
Z
@st
@qt
( t (st ; pt ; xt ; ) ; pt ; xt ; ) =
( (st ; pt ; xt ; ) ; pt ; xt ; ; v) (v) dv;
0
@ t
@ 0t t
where qt = (q1 ; :::; qJt )0 and
@qt
@ 0t
qt qt0 , we obtain that
R
(Qt qt q0t ) (v) dv ;
dt (st ; pt ; xt ; ) = det 1
= Qt
(7)
where Qt is the diagonal matrix with diagonal equal to qt , and qt ; Qt are evaluated at
( t (st ; pt ; xt ; ) ; pt ; xt ; ; v). Note that by equation (2) the Jacobian does not depend
on and , so we can write dt (st ; pt ; xt ; ). Further, dt (st ; pt ; xt ; ) > 0 for any market
t and any (st ; pt ; xt ; ) because the matrix (Qt qt q0t ) is positive de…nite. Writing the
Jacobian in the matrix form (7) is useful for coding it.
Based on the density (6), for arbitrary parameter values we de…ne the semiparametric
log-likelihood as
L( ) =
T
X
ln dt (st ; pt ; xt ; ) +
t=1
T X
Jt
X
t=1 j=1
where
t
XX
1
K H
N det (H) t=1 j=1
T
b ( ; !; ) =
ln b
jt
(st ; pt ; xt ; ) ; ! jt (pt ; wt ; ) ;
(8)
;
J
1
( ; !)
2
jt
(st ; pt ; xt ; ) ; ! jt (pt ; wt ; )
0
The argument that the transformation ( t ; ! t ) 7 ! (st ; pt ) is one-to-one is the following. Since
the pricing equation (3) is linear, there is a one-to-one correspondence between ! t and pt , so we can
regard pt as given in the predicted market share function (1). Since there is a unique unobserved
characteristic vector t for which the predicted market share vector is equal to the observed market
share vector (Berry 1994), the transformation ( t ; ! t ) 7 ! (st ; pt ) will be one-to-one.
6
is a nonparametric kernel estimator of the joint density of the residuals jt (st ; pt ; xt ; )
P
and ! jt (pt ; wt ; ), N = Tt=1 Jt , H is a 2 2 matrix of bandwidths and K is a bivariate
kernel function. The SPML estimator maximizes the log-likelihood function given in
(8).
A closely related estimator was introduced by Ai (1997), who used a trimmed version
of the score function of the log-likelihood in a more general framework de…ned by an
index restriction similar to that in equation (6). The trimming is needed for deriving the
asymptotic distribution of Ai’s estimator in order to deal with values of the nonparametric kernel estimator b that are close to zero. Under mild regularity conditions Ai
shows that his estimator has asymptotic variance equal to the infeasible ML estimator
that uses the unknown density 0 .3
In the Monte Carlo simulations reported below we use the bivariate two-sided exponential density K (u1 ; u2 ) = 41 exp ( ju1 j ju2 j) for the bivariate kernel function.4
This is motivated by the fact that, using the univariate two-sided exponential density
kernel, Eggermont and LaRiccia (1999) show that under weak regularity conditions the
univariate SPML log-likelihood divided by the number of observations, say N , conp
verges almost surely at rate N to the expected value of the log-likelihood. This result
can be used in the single equation case to show the consistency of the SPML estimator without the trimming proposed by Ai (1997). For the choice of bandwidth we
follow the practical suggestion from Härdle and Linton (1994), that is, we use the matrix bandwidth H = hS 1=2 , where S is the sample covariance matrix of the residuals
jt (st ; pt ; xt ; ) ; ! jt (pt ; wt ; ) , j = 1; :::; Jt , t = 1; :::; T and h is a scalar bandwidth.
4
Identi…cation
The fact that the SPML estimator that maximizes the semiparametric log-likelihood (8)
involves the nonparametric estimator b ( ; ; ) that is computed from the current parameter values makes the issue of identi…cation nontrivial. In this section we provide
conditions under which the SPML estimator identi…es the parameters of a general nonlinear simultaneous system, which has the demand system from Section 2 as a special
case. Let y and " denote the vectors of endogenous variables and errors, respectively,
about which we assume to have the same dimension.5 Let x be the matrix of observed exogenous variables (covariates) and the vector of parameters. We assume that equation
3
Without trimming it appears to be di¢ cult to show asymptotic normality of the SPML estimator
when the true density 0 is not assumed to be bounded away from zero.
4
In fact we truncate this so that K (u1 ; u2 ) = 0 for ju1 j + ju2 j > 300; the integral of the truncated
density is equal to 1 2 exp ( 300) ' 1 10 130 .
5
For the demand system this assumption means that the number of products Jt in di¤erent markets
t is the same. This assumption is not necessary, and is only adopted for simplicity. Our identi…cation
results can easily be extended to the case when Jt are di¤erent by treating Jt as random and including
it in the vector of covariates.
7
r (y; x; ) = u has a unique solution in y for every x; ; u, which we denote by h (u; x; );
we assume that r is continuous and continuously di¤erentiable in y. We maintain the
assumption of independence of x and ".
For the demand system from Section 2, indexing the variables by t we have that xt
is the vector of demand characteristics and price-shifting variables,
0
0
0
= ( 0 ; 0 ) ; r (yt ; xt ; ) = (
yt = (s0t ; p0t ) ; "t = ( 0t ; ! 0t ) ;
0
t
0
(st ; pt ; xt ; ) ; ! 0t (pt ; wt ; )) :
Our identi…cation argument relies on the independence of the covariates x and the
errors ", which translates into the fact that the density of the residuals does not depend
functionally on x. Similar ideas are explored by, for example, Brown (1983), Benkard
and Berry (2006), Matzkin (2008).
The conditional density of y given x is
f (yjx;
where
0
0)
= d (y; x;
is the density of " = r (y; x;
0)
0)
0
(r (y; x;
0 )) ;
and
d (y; x; ) = det
@r (y; x; )
@y 0
:
We assume that 0 is continuous and d (y; x; 0 ) > 0. For an arbitrary in the parameter
space let u = r (y; x; ); denote the density of u by ( j ). Clearly, it holds that ( j 0 ) =
0.
The likelihood of y at conditional on x is based on the conditional density
f (yjx; ) = d (y; x; )
The density
(r (y; x; ) j ) :
(uj ) of u can be computed as
Z
(uj ) = f (ujx; ) f (x) dx;
where
f (ujx; ) =
d (h (u; x; ) ; x; 0 )
d (h (u; x; ) ; x; )
0
(r (h (u; x; ) ; x;
(9)
0 ))
(10)
is the density of u conditional on x and f (x) is the density of x. This can be derived
by noting that the density of y conditional on x can be written in two ways, that is,
f (yjx;
0)
= d (y; x; ) f (ujx; ) ;
(11)
which is the density of y given x based on the transformation u 7 ! y, and putting
y = h (u; x; ). This means that ( j ) is completely determined by and 0 .
8
The identi…cation condition for the SPML estimator corresponding to the identi…cation condition typically used for ML estimators is6
E [ln f (yjx; )] < E [ln f (yjx;
0 )]
for any
6=
0:
The usual su¢ cient condition invoked for this is that f (yjx; ) = f (yjx; 0 ) for all y; x
implies = 0 .7 This means that the density of y conditional on x uniquely determines
0 . The following result establishes a further equivalent condition, proved in Appendix
A.1.
Proposition 1 The condition
f (yjx; ) = f (yjx;
implies
=
0
for all y; x
(12)
is satis…ed if and only if the fact that the expression
d (h (u; x; ) ; x; 0 )
d (h (u; x; ) ; x; )
does not depend on x for all u implies
4.1
0)
0
(r (h (u; x; ) ; x;
(13)
0 ))
0.
=
Illustration: Simple logit
The model presented in Section 2 implies market share formulas whose integrals have
no closed form, which causes that the expressions involved are highly nonlinear. As a
consequence, the expression (13) is di¢ cult to evaluate analytically in order to verify
the identi…cation condition from Proposition 1. The simple logit model is a special case
of the model discussed in Section 2 in that the utility does not account for consumer
heterogeneity. In this case i = ; i = . Although, due to this the simple logit model
generates restrictive substitution patterns, the fact that all formulas are in closed form
makes it useful to illustrate some phenomena that are di¢ cult to demonstrate in the
general cases. For example, Berry et al. (2004) also use the simple logit model to verify
the conditions for consistency and asymptotic normality of their IV estimator.
For the simple logit the predicted market share of product j is
sj ( ; p; x; ) =
6
1+
exp
PJ
r=1
pj + xj +
j
exp ( pr + xr +
r)
;
We note that this condition is implied by the identi…cation condition used by Ai (1997) under
standard regularity conditions that allow the interchange of di¤erentiation and the expectation operator.
To be precise we recall that Ai uses the condition that E[d ln fd(yjx; )]
= 0 implies = 0 .
7
=
If
where
0
0
is continuous and strictly positive then the two conditions are equivalent due to the inequalities
Z
Z
Z
2
2
1
j
j
j
j
ln
;
2
and
are two strictly positive densities (see, e.g., Eggermont and LaRiccia 2001, p.353).
9
while the predicted share of the outside alternative is
s0 ( ; p; x; ) =
1+
0
1
PJ
r=1 exp ( pr + xr +
r)
;
where = ( ; 0 ) , and the rest of the notation is the same as in Sections 2 and 3 but
we omit the subscript t for market. Making these equal to the observed market shares
sj and s0 , we can compute the unobserved characteristic of product j as
j
sj
s0
(s; p; x; ) = ln
( pj + xj ) :
We assume the same reduced form supply side as in equation (3); the Jacobian of the
inverse of the transformation ( ; !) 7 ! (s; p) evaluated at (s; p; x; ) is
d (s; p; x; ) = det
1
ss0 ) ;
(S
(14)
where S is the diagonal matrix with diagonal equal to s. Note that the Jacobian only
depends on the observed market shares.
In this section, although we use the same notation, we omit the 1 for the intercepts
from xj and wj and move the intercepts to be the expected values of the unobserved
characteristics. This will make it possible to identify all the parameters of the simple
logit. Speci…cally, by using Proposition 1 we show that the parameters apart from the
intercepts are identi…ed, and the intercepts can be identi…ed as the expected values of
(s; p; x; w;
0)
= ln
! j (s; p; x; w;
0)
= pj
j
sj
s0
wj
(
0 pj
+ xj
0) ;
0:
Next we show that, if 0 is continuous and its support is the whole Euclidean space,
then for the simple logit the second statement of Proposition 1 holds. Proposition 1
0
then will imply identi…cation through SPML. For the simple logit = ( 0 ; 0 ) and 0 is
the density of the true vectors of unobserved characteristics ( ; !) in a generic market.
In order to compute the expression (13) for the simple logit, …rst note that since the
Jacobian (14) does not depend on , this expression is equal to 0 (r (h (u; x; ) ; x; 0 )).
The argument in 0 is
r (h (u; x; ) ; x;
where
e
j
= (
!
e j = wj (
e1 ; :::; eJ ; !
e 1 ; :::; !
eJ
0) =
0 ) (wj
0)
+ ! j ) + xj (
+ !j ;
10
0
e0 ; !
e0
0)
+
j;
0
;
(for more details on this we refer to Appendix A.1). According to Lemma 3 in Appendix
A.1, if
0
e0 ; !
e0
0
0
e 0 does not
does not depend on (x; w) for all ( 0 ; ! 0 ), then e ; !
depend on (x; w) for all ( 0 ; ! 0 ). The latter statement means that for any j, ej ; !
e j do
not depend on xj ; wj for all j ; ! j . The derivatives of these are
dej
=
dxj
0;
dej
=(
dwj
0)
;
de
!j
=
dwj
0;
which must be 0; therefore, = 0 , (
= 0, = 0 . These imply that = 0 ,
0)
provided that 0 6= 0, that is, at least one component of the true value of the supply
side parameters is not zero. This establishes the second statement of Proposition 1 for
the simple logit.
The arguments presented above can be summarized in the following identi…cation
result.
Proposition 2 Suppose that at least one component of the true value of the supply side
parameters 0 is not zero, the density of the unobserved characteristics 0 is continuous, its support is the whole Euclidean space, and the expected value of the unobserved
characteristics is 0. Then the SPML estimator identi…es the parameters of the model
with simple logit demand.
5
Monte Carlo Study
In this Monte Carlo study we simulate the model presented in Section 2 and estimate it
by GMM and SPML. By GMM we estimate only the demand parameters . We implement GMM in two ways. In the …rst implementation we use the vector of instruments
2
2
zjt = xjt2 ; xjt3 ; wjt2 ; wjt3 ; x2jt2 ; x2jt3 ; wjt2
; wjt3
;p
j
;
(15)
corresponding to product j in market t, where p j is the average of the prices of the
products with the same observed characteristics that appear in the other markets; this
quantity was proposed by Nevo (2001) as an instrument. In the rest we follow standard 2-step GMM estimation procedure, whose details we describe in Appendix A.2.1.
The performance of this estimator may depend on the speci…c choice of instruments.
Therefore, in the second implementation of GMM we use instruments obtained by approximating the asymptotically optimal instruments derived by Chamberlain (1987);
for details we refer to Appendix A.2.2. We implement the SPML estimator for the
bandwidth value h = 0:09.
11
5.1
Simulation design
There are T = 50 markets with J = 5 products in each, so the number of observations
is N = 250. The products have K = 3 observed demand characteristics, where the
…rst is 1 and the other two are generated as uniform on [1; 2]. We generated 12 such
products in this way, and for each market we draw 5 out of these randomly. This way
the observed characteristics will be discrete random variables; we believe this pattern of
products is rather realistic. The price-shifting variables include 1 for the constant and
the logarithms of the two demand characteristics.
For the distribution of the unobserved characteristics jt ; ! jt we consider two di¤er0
0
ent distributions. For both distributions we use the structure jt ; ! jt =
jt ; $ jt ,
where is the Cholesky decomposition of the matrix
2:329 1:269
1:269 1:751
0
and
is a random vector with mean zero and variance equal to the identity
jt ; $ jt
matrix. This implies a relatively high correlation of 0:63 between jt and ! jt . We take
the …rst distribution to be the standard normal, that is, jt ; $jt
N (0; 1). For the
second distribution we take $jt
N (0; 1) and for the distribution of jt we assume
a mixture of two normals with density equal to 32 (3x + 2) + 21 (x 2=3), where
denotes the standard normal density; note that this mixture distribution has mean 0
and variance 1 and it is asymmetric bimodal.8
The true values of the demand parameters are presented in the tables below that
contain the results. The true values of the supply side parameters are 0 = (1:5; 2; 2:5)0 .9
The integral involved in the market share expression (1) is a 4-dimensional analytically intractable integral and thus needs to be evaluated numerically. A standard way
to do so is by Monte Carlo simulations by computing the average of the integrand function over several draws from the 4-dimensional standard normal distribution. In order to
reduce the computational time, instead of using the so-called pseudo-random draws generated by the computer, we evaluate the market share integrals by using quasi-random
draws. Speci…cally, we use a so-called (0; 3; 4)-net in base 4, which contains 64 draws
from the 4-dimensional standard uniform distribution, and we transform it componentwise by the inverse normal distribution function. Sándor and Train (2004) show that,
for the estimation of a random coe¢ cient logit model, this quasi-random sample is the
best among a number of other samples including the commonly used randomized Halton
sequence.
8
We also run some of the simulations for jt and $jt having a t (3) distribution standardized so
that their variance be equal to 1. Since the results are qualitatively rather similar to those obtained
for the standard normal, we do not present them here.
9
With these values the price shifting equation (3) implies correlation coe¢ cients between price and
the two observed price shifters of 0:34 and 0:42. This means that these price shifters are relatively
strong instruments.
12
For both the GMM and SPML estimation procedures, we optimize the objective
function by the nonderivative simplex search algorithm. After each run of the algorithm,
if the improvement of the objective function is higher than a given tolerance, we re-run
the algorithm starting with a simplex containing the previously obtained optimal value
together with new starting values of the parameters. We allow for a maximum of …ve
runs of the algorithm. In the simulations we have found that in the case of the GMM the
algorithm stops after one re-run most of the times, and rarely after two re-runs. In the
case of the SPML the algorithm needs all the …ve re-runs most of the times, and rarely
stops after three/four re-runs. This –together with the fact that the SPML objective
function is optimized with respect to all parameters, while the GMM objective function
is optimized only with respect to the four standard deviation parameters– results in
computing time for the SPML estimator about twelve times as much as for the GMM
estimator.
5.2
Results
The results of the Monte Carlo study are presented in Table 1, where by GMM we denote
the version with instruments (15) and by GMM-OI the version with (approximate)
optimal instruments. The table has an upper and a lower block for the two distributions
of the unobserved characteristics, which are the normal and the mixture of normals. In
the table the leftmost column contains the true values of the demand parameters. Each
block for each estimator contains the bias, the standard deviation (StD) and the rootmean squared error (RMSE) of the estimates from 50 replications.
[Table 1 about here]
The general impression from the performance of both GMM estimators in both cases
is that they can have large bias and are rather imprecise. As a consequence, the RMSEs
are rather high, in fact most of them are above 1:5. Similar conclusions regarding
the performance of GMM for this model have also been found by other authors. For
example, BLP mention that they needed to extend their data set on cars to …fteen years
in order to obtain meaningful estimates. Jiang et al. (2009) also obtain large RMSE for
the GMM estimator in their Monte Carlo study, which for some parameters are even
higher than ours. Based on the RMSEs of the two GMM estimators we …nd that in
the normal case GMM-OI has a performance poorer than the other GMM estimator,10
while in the mixture case their relative performance cannot be ordered unambiguously.
In contrast, the SPML estimates are nearly unbiased and have very low standard
deviations in both cases. As a consequence, SPML substantially outperforms GMM also
10
Goeree (2008, Table D.III) reports a …nding similar in that the standard errors obtained by using
BLP-type instruments are lower than those obtained by using approximately optimal instruments.
13
regarding the RMSE criterion. In fact most of the RMSEs are below 0:4, which is a
remarkable performance compared to the GMM estimators.
6
Performance under misspeci…cation
The SPML estimator uses more information than the GMM estimator because it also
uses the supply side structure of the model. We would like to know if this additional
information is responsible for the fact that SPML outperforms GMM so substantially.
Therefore, we simulate the model by misspecifying the supply side in two essential
ways, and estimating the model described in Section 2. In the …rst misspeci…cation the
prices are determined as the Nash equilibrium of a pricing game, while in the second
misspeci…cation the prices are determined by a functionally misspeci…ed equation. The
details of this are presented in the …rst subsection. In the second subsection we present
the performance of the GMM and SPML estimators under heteroskedasticity.
6.1
Misspeci…cation of the pricing equation
This section presents a Monte Carlo study on the performance of GMM and SPML
when the pricing equation is misspeci…ed. In the …rst case, in each market the prices
are determined as the Nash equilibrium of a pricing game between two …rms, of which
the …rst has 2 and the second has 3 out of the 5 products in the market. The marginal
cost of product j in market t is given by cjt = wjt 0 + ! jt , where 0 is the same
as in Section 5. For the demand parameters we use true values for ,
and 1
di¤erent from those in Section 5 (see Table 2) because these values allow the numerical
computation of the Nash equilibrium of prices.11 In the second case the price of product
j in market t depends exponentially on the price-shifters, that is, pjt = exp (wjt 0 + ! jt ),
where 0 = (0:2; 0:5; 0:7)0 . In both cases the unobserved characteristics have the normal
distribution given in Section 5.1.
[Table 2 about here]
Table 2 presents the results of the GMM and SPML estimators. Both GMM estimators can have large bias and in general have large standard deviation; consequently,
most of their RMSEs are again above 1:5.12 The performance of the SPML estimators,
is di¤erent in the two cases considered. In the …rst case, when the price equilibrium is
11
We compute the price Nash equilibrium in each market as the solution of the nonlinear system of
equations that are determined as the …rst order conditions for pro…t maximization (see equation (3.3)
in BLP). We use the Eqsolve function in Gauss to solve this system. We veri…ed equilibrium uniqueness
numerically.
12
In the case when the price equilibrium is misspeci…ed we also do the Monte Carlo simulations for
GMM by using the BLP-instruments (for details see BLP, p.861). The performance of this estimator
is poorer than the performance of GMM and GMM-OI.
14
misspeci…ed, the SPML estimator is virtually unbiased with low standard deviations.13
In the second case, when the pricing function is misspeci…ed, the SPML estimator has a
small bias, and most of its RMSEs are below 0:5. We can conclude that SPML performs
well under misspeci…cation of the pricing equation. Regarding the question whether the
additional supply side information is responsible for the substantial di¤erence in the
performance of GMM and SPML, we can claim that it does not appear to be so in the
case when the price equilibrium is misspeci…ed. In the case when the pricing function
is misspeci…ed the performance di¤erence between the two estimators is still large, but
slightly smaller than in the other cases.
6.2
Heteroskedasticity
The homoskedastic unobserved characteristics are independent of the covariates. Estimation in the heteroskedastic case serves the purpose of verifying how the SPML estimator performs under deviations from this independence assumption. In this section,
instead of assuming that in market t the vectors ( t ; ! t ) and (xt ; wt ) are independent,
we assume that for product j in market t
jt ; ! jt
0
=
(xjt ; wjt )
jt ; $ jt
0
;
(16)
where (xjt ; wjt ) is a 2 2 invertible matrix function of (xjt ; wjt ), and jt ; $jt is a
random vector independent of (xt ; wt ), with mean zero and variance equal to the identity
matrix. This implies the rather standard heteroskedastic structure for jt ; ! jt given
by
h
i
0
E
;
!
jx
;
w
=
0;
var
;
!
jx
;
w
(xjt ; wjt ) ;
jt
t
t
jt
t
t =
jt
jt
where (xjt ; wjt ) = (xjt ; wjt ) (xjt ; wjt )0 . The expected correlation implied is similar
to the correlation in the homoskedastic case.
Table 3 presents the Monte Carlo simulation results for GMM and SPML when the
data generating process follows the type of heteroskedasticity described above. The
main conclusion regarding the comparison of the GMM and SPML estimators is the
same as that from the homoskedastic case. The RMSEs for this estimator are much
lower than those for the GMM estimators; the latter ones can also have very high bias.
We can conclude that SPML performs well under deviations from the assumption of
independence of the observed and unobserved characteristics.
[Table 3 about here]
13
The RMSEs are in fact lower than those in Table 1 when there is no misspeci…cation. However,
this is possible because the two data generating processes are di¤erent.
15
7
Conclusions
In this paper we study SPML (i.e., semiparametric maximum likelihood) estimation of
a class of random coe¢ cient demand systems for di¤erentiated products. The SPML
estimator is based on the likelihood function, where the unknown density of the errors
is replaced by the kernel nonparametric estimator of the residuals for given parameter
value. We provide an equivalent condition for the identi…cation of the model by SPML,
and by specializing this result we provide simple regularity conditions for the identi…cation of the model with simple logit demand. In a number of Monte Carlo studies we
compare the …nite sample performance of the SPML estimator to the performance of the
commonly used standard GMM estimator and a GMM estimator with approximately
optimal instruments.
We …nd that the SPML estimator outperforms the GMM estimators substantially in
terms of bias, standard deviation and mean squared error, irrespective of the assumed
distribution of the underlying errors. We also study the performance of the SPML
estimator in situations when the pricing equation is misspeci…ed. This estimator turns
out to work rather well under such misspeci…cations. Although the standard GMM
estimator by construction does not make use of the structure of the supply side, this
positive feature remains unexplored due to the poor …nite sample properties of this
estimator. The good performance of SPML under misspeci…cation reported in this
paper call for more work on the theoretical properties of SPML regarding its robustness
to misspeci…cation.
One weakness of the SPML estimator is its computing time. As we note in Section
5.1, this can be twelve times as much as the computing time of the GMM estimator.
In order to improve the computational performance of SPML one may try to adapt the
MPEC procedure proposed by Dubé, Fox and Su (2011). However, since this does not
appear to be a trivial extension, we will study it in future work. The current paper
concludes that, in situations where the GMM estimator is poor, the SPML estimator
can be a slower but reliable and much more precise alternative.
A
A.1
Appendix
Identi…cation in the simple logit
Proof of Proposition 1. "(=". Suppose that if (13) does not depend on x for all u
then = 0 . Take an arbitrary and suppose that (12) holds. We are going to prove
that = 0 . From (12)
d (y; x; )
(r (y; x; ) j ) = d (y; x;
16
0)
0
(r (y; x;
0 ))
for all y; x:
Now we put y = h (u; x; ) and obtain
d (h (u; x; ) ; x; )
(uj ) = d (h (u; x; ) ; x;
0)
0
(r (h (u; x; ) ; x;
0 ))
for all u; x;
that is,
(uj ) =
d (h (u; x; ) ; x; 0 )
d (h (u; x; ) ; x; )
0
(r (h (u; x; ) ; x;
0 ))
for all u; x:
Since the left hand side does not depend on x, the equality can only hold if the right
hand side does not depend on x for all u. By hypothesis this implies = 0 .
"=)". Suppose now that (12) implies = 0 ; we are going to prove that if (13) does
not depend on x for all u then = 0 . Take an arbitrary for which (13) does not depend
on x for all u. Then by equation (10) the density f (ujx; ) of u = r (y; x; ) conditional
on x does not depend on x for all u, which by (9) implies that f (ujx; ) = (uj ) for
all u. By (11) this implies further that (12) holds, so = 0 .
Lemma 3 Suppose that the density of the unobserved characteristics 0 is continuous
0
e0 ; !
e0
and its support is the whole Euclidean space. Then 0
does not depend on
0
e 0 does not depend on (x; w) for all ( 0 ; ! 0 ).
(x; w) for all ( 0 ; ! 0 ) implies that e ; !
Proof. By contradiction assume that for some
there is ( 0 ; ! 0 ) such that
e0 ; !
e0
0
e 0 depends on at
depends on (x; w). This means that at least one component of e ; !
least one component of (x; w). In particular, this means that there is ` such that either
e` = (
0 ) (w` + ! ` )+x` (
0 )+ ` depends on at least one component of (x` ; w` )
or !
e ` = w` (
0 ) + ! ` depends on at least one component of w` . We treat the former
case here; the latter can be treated similarly. Due to the additive structure we obtain
that e` = (
0 ) (w` + ! ` ) + x` (
0 ) + ` depends on at least one component
of (x` ; w` ) for any ` . Since the support of 0 is the whole Euclidean space, it is not
constant in any component of its argument. Consequently 0 is not constant in ` for
any values of the other components of the argument. Since e` and !
e ` take values in the
whole real line,
0
is a contradiction.
e0 ; !
e0
0
will depend on at least one component of (x` ; w` ), which
In the rest of this section we derive the expression r (h (u; x; ) ; x; 0 ) for the simple
logit that is needed to verify identi…cation by Proposition 1. First recall that
0
r (y; x; ) = ( 0 (s; p; x; ) ; ! 0 (p; w; ))
17
0
( 0 (s; p; x; w; ) ; ! 0 (s; p; x; w; )) ;
where
j
sj
( p j + xj ) ;
s0
wj :
(s; p; x; w; ) = ln
! j (s; p; x; w; ) = pj
(17)
(18)
Next recall that h (u; x; ) solves r (y; x; ) = u in y, so analogously
(s ( ; !; x; w; ) ; p ( ; !; x; w; )) solves
(s; p; x; w; ) = ; ! (s; p; x; w; ) = !
in (s; p). We obtain
exp
PJ
sj ( ; !; x; w; ) =
(wj + ! j ) + xj +
j
1 + r=1 exp ( (wr + ! r ) + xr +
pj ( ; !; x; w; ) = wj + ! j ; j = 1; :::; J:
r)
(19)
;
(20)
So (17), (18), (19), (20) imply
ej
A.2
A.2.1
!
ej
j
= (
(s ( ; !; x; w; ) ; p ( ; !; x; w; ) ; x; w;
0 ) (wj
+ ! jt ) + xj (
0)
+
0)
jt ;
! j (s ( ; !; x; w; ) ; p ( ; !; x; w; ) ; x; w;
0)
= wj (
0)
+ !j :
Estimation by GMM
2-step GMM
GMM estimation is based on the conditional moment restriction E jt jxt ; wt = 0 for
P
t = 1; :::; T and j = 1; :::; Jt . Let N = Tt=1 Jt be the total number of products. Let
Z = (Z10 ; :::; ZT0 )0 be an (N L) matrix of instruments whose block Zt corresponds to
market t. The row of Z used in our Monte Carlo corresponding to product j in market t
is speci…ed in (15). As before, denotes the vector of all demand parameters. Let ( )
and ( ) be the column-vector of all unobserved characteristics and the column-vector
of t (st ; pt ; xt ; ) ; t = 1; :::; T , whose dependence on market shares, prices and observed
characteristics is suppressed. In this case ( ) = ( ) ( p + x ), where p and x are
the vector and matrix of all prices and observed characteristics. GMM estimation is
based on the sample moment expression
g( ) =
1 0
Z ( ):
T
The GMM estimator minimizes the function
g0 ( ) W
1
g( )
in , where the weighting matrix W will be speci…ed below.
18
This GMM objective function can be concentrated because g ( ) is linear in
0
; let X = (p; x) and = ( ; 0 ) . Then the objective function
g0 ( ) W
is quadratic in
1
g( ) =
1
T2
0
X
( )
ZW
1
Z0
( )
and
X
and , so it is minimized at
b ( ) = X 0 ZW
1
1
Z 0X
0
1
X ZW
Z0 ( )
for arbitrary , so the concentrated objective function depends only on , that is,
g0 b ( ) ;
W
1
g b( );
=
1
T2
0
X b ( ) ZW
( )
1
Z0
Xb ( ) :
( )
We minimize this with respect to by the non-derivative simplex search method.
Our GMM estimates reported in the tables are the result of a 2-step procedure, where
P
0
in the …rst step we use W = Z 0 Z=T . In the second step we use W = Tt=1 Zt0btbt Zt =T 2 ,
where bt is the (column-)vector of residuals in market t computed from the …rst-step
GMM estimator of the parameters. We note that Jiang et al. (2009) use the same 2-step
GMM procedure in their Monte Carlo simulations.
A.2.2
Approximation of optimal instruments
As Chamberlain (1987) showed, the optimal set of instruments for the conditional moment restrictions E [ t jxt ; wt ] = 0 is
H (xt ; wt ) = E
@
@
t
0
( 0 ) xt ; wt
E
t
( 0)
1=2
( 0 )0 jxt ; wt
t
:
Here the matrix E t ( 0 ) t ( 0 )0 jxt ; wt is often approximated by the identity matrix;
this approximation is exact up to a constant factor in the case of homoskedasticity. The
derivative involved in the other matrix is
@
@
t
0
( 0) =
@(
t
(st ; pt ; xt ; )
@ 0
( pt + xt ))
=
=
pt ; xt ;
0
@
t
(st ; pt ; xt ;
@ 0
0)
:
By the implicit function theorem
@
t
(st ; pt ; xt ;
@ 0
0)
=
@st (
t
(st ; pt ; xt ; 0 ) ; pt ; xt ;
@ 0t
0)
where the derivatives involved can be computed as
Z
@st
( (st ; pt ; xt ; 0 ) ; pt ; xt ; 0 ) =
(Qt
@ 0t t
Z
@st ( t (st ; pt ; xt ; 0 ) ; pt ; xt ; 0 )
=
(Qt
@ 0
19
1
@st (
t
(st ; pt ; xt ;
@ 0
qt qt0 ) (v) dv;
qt qt0 ) X t V (v) dv;
0 ) ; pt ; xt ; 0 )
;
where qt qt ( t (st ; pt ; xt ; 0 ) ; pt ; xt ; 0 ; v), X t = (pt ; xt ) and V is the diagonal matrix
with main diagonal elements v ; v 0 . It will be useful to work with the variables t ; ! t
instead of st ; pt . By doing so, we obtain
t
(st ; pt ; xt ;
0)
=
t
(st ( t ; ! t ; xt ; wt ;
0 ) ; pt
=
0
(wt
0
0
+ ! t ) + xt
+
( t ; ! t ; xt ; wt ;
0 ) ; xt ;
0)
t
so
qjt (
t
(st ; pt ; xt ;
q jt ( t ; ! t ; xt ; wt ;
0 ) ; pt ; xt ; 0 ; v)
exp ( 0 +
=
Pt
1 + Jr=1
exp (
0
0v
) (wjt
+
0v
0
+ ! jt ) + xjt
) (wrt
+
0
0 + ! rt ) + xrt
0 ; v)
1=2
0 v
0+
+
jt
1=2
0 v
:
+
rt
Therefore,
@
@
t
0
( 0 ) xt ; wt
"
Z
wt 0 ; xt ; E
E
=
Let
G ( t ; ! t ; xt ; wt ;
Then
E
=
Z
"
Z
0)
=
1
q t q 0t
Qt
Z
(v) dv
1
q t q 0t
Qt
(v) dv
1
Qt
q t q 0t
G ( t ; ! t ; xt ; wt ;
Z
(v) dv
0) f
Z
Qt
Qt
Z
Qt
!
xt ; wt
q t q 0t X t V (v) dv:
(21)
q t q 0t X t V (v) dv
q t q 0t X t V (v) dv
!
xt ; wt
#
( t ; ! t jxt ; wt ) d t d! t ;
which can be estimated by Monte Carlo integration as
1X
G
R i=1
R
i
i
t ; ! t ; xt ; wt ; 0
;
(22)
where it ; ! it ; i = 1; :::; R are draws from the distribution with density f ( t ; ! t jxt ; wt ).
However, this Monte Carlo estimator cannot be computed because the true value 0 of
the parameters and the conditional distribution ( t ; ! t jxt ; wt ) are not known. Therefore,
we replace the true value of the parameters 0 by the …rst-step GMM estimator b1 and
the true value of 0 by the OLS estimator b1 from (3). Then we approximate the Monte
Carlo estimator (22) by G 0; 0; xt ; wt ; b1 , that is, by taking R = 1, it = 0, ! it = 0,
0
0
and b1 = b1 ; b01 , as suggested by Berry et al. (1999), and we take the corresponding
Monte Carlo estimators of the integrals involved in G in equation (21). We replace the
matrix of instruments Zt from the previous section by the approximated H (xt ; wt ), take
W to be equal to the identity matrix, and do GMM estimation as described there.
20
#!
:
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22
Table 1. Performance of the GMM and SPML estimators in the correctly speci…ed case
T = 50
True
Bias
GMM
StD RMSE
= -3.0
= 1.5
=
-5.0
1
2 = 2.0
3 = 2.5
1 = 2.5
2 = 2.0
3 = 2.0
0.801
-0.106
0.073
-0.941
-1.010
-0.028
-0.169
-0.490
1.154
0.900
1.854
0.959
1.180
2.666
1.498
1.551
1.404
0.906
1.856
1.344
1.554
2.666
1.507
1.627
= -3.0
= 1.5
1 = -5.0
2 = 2.0
3 = 2.5
1 = 2.5
2 = 2.0
3 = 2.0
-0.509
0.929
0.316
-0.521
-0.953
1.469
-0.013
0.162
1.637
1.175
1.745
1.058
1.523
3.206
2.140
2.012
1.714
1.498
1.773
1.179
1.796
3.526
2.140
2.019
GMM-OI
Bias
StD RMSE
Normal
0.425 1.604
1.660
0.040 1.180
1.181
0.433 2.219
2.261
-0.685 1.623
1.761
-1.592 2.308
2.804
0.156 2.690
2.694
0.209 2.292
2.302
0.729 2.437
2.544
Mixture
-0.027 1.537
1.537
0.456 0.837
0.953
0.028 3.810
3.810
-0.030 1.985
1.985
-1.568 3.546
3.877
1.368 3.073
3.363
-0.255 1.600
1.620
1.239 2.978
3.225
23
Bias
SPML
StD RMSE
-0.184
0.072
-0.009
-0.079
-0.043
-0.114
0.135
0.113
0.268
0.139
0.382
0.306
0.273
0.337
0.277
0.292
0.326
0.156
0.382
0.316
0.276
0.355
0.308
0.313
-0.218
0.167
-0.004
-0.054
-0.050
-0.024
0.130
0.161
0.222
0.149
0.367
0.266
0.229
0.302
0.281
0.284
0.312
0.223
0.367
0.271
0.234
0.303
0.309
0.326
Table 2. Performance of the GMM and SPML estimators when the pricing equation is
misspeci…ed
T = 50
True
Bias
= -5.0
= 1.0
1 = -2.0
2 = 2.0
3 = 2.5
1 = 2.5
2 = 2.0
3 = 2.0
0.752
-0.045
1.437
-1.392
-1.570
-0.113
0.032
0.251
= -3.0
= 1.5
=
-5.0
1
2 = 2.0
3 = 2.5
1 = 2.5
2 = 2.0
3 = 2.0
1.444
-0.601
-0.717
-0.658
-0.902
0.036
-0.346
-0.394
GMM
GMM-OI
StD RMSE
Bias
StD RMSE
Misspeci…ed price equilibrium
1.649
1.812
0.295 2.743
2.758
0.615
0.616
0.129 0.871
0.881
3.156
3.468
1.035 2.224
2.453
1.514
2.056
-1.057 1.375
1.735
2.230
2.727
-1.267 1.909
2.291
2.321
2.323
0.370 3.318
3.338
1.442
1.442
0.158 1.498
1.506
1.649
1.668
0.064 1.109
1.111
Misspeci…ed pricing function
1.202
1.879
0.914 1.863
2.075
0.864
1.052
-0.221 1.299
1.318
1.483
1.648
0.014 1.807
1.807
1.028
1.220
-0.737 1.010
1.250
1.329
1.606
-0.957 1.520
1.796
2.351
2.351
0.075 2.889
2.890
1.351
1.395
0.322 1.715
1.745
1.750
1.794
0.305 2.037
2.060
24
Bias
SPML
StD RMSE
-0.097
0.006
0.030
-0.066
-0.055
-0.046
0.017
0.081
0.244
0.068
0.348
0.244
0.296
0.262
0.242
0.151
0.262
0.069
0.349
0.252
0.301
0.266
0.243
0.172
0.292
-0.179
-0.009
-0.288
-0.344
-0.167
-0.061
0.020
0.435
0.187
0.466
0.368
0.406
0.395
0.276
0.367
0.524
0.259
0.466
0.467
0.532
0.429
0.283
0.367
Table 3. Performance of the GMM and SPML estimators under heteroskedasticity
T = 50
True
Bias
GMM
StD RMSE
= -3.0
= 1.5
=
-5.0
1
2 = 2.0
3 = 2.5
1 = 2.5
2 = 2.0
3 = 2.0
0.828
-0.103
0.001
-0.948
-1.074
0.518
-0.149
-0.448
1.149
0.923
2.106
0.861
1.198
2.751
1.249
1.512
1.416
0.929
2.106
1.281
1.609
2.800
1.258
1.577
= -3.0
= 1.5
1 = -5.0
2 = 2.0
3 = 2.5
1 = 2.5
2 = 2.0
3 = 2.0
-0.327
0.811
0.276
-0.675
-0.886
1.652
0.008
-0.101
1.835
1.369
2.136
1.099
1.170
3.122
1.906
1.958
1.863
1.591
2.154
1.289
1.468
3.532
1.906
1.961
GMM-OI
Bias
StD RMSE
Normal
0.438 1.375
1.443
0.080 0.931
0.934
0.651 2.328
2.418
-0.685 1.646
1.783
-1.829 2.590
3.171
-0.028 2.423
2.423
0.196 2.097
2.106
1.000 2.350
2.554
Mixture
0.241 2.097
2.111
0.409 1.232
1.299
0.589 3.495
3.544
-0.723 2.402
2.509
-1.781 3.524
3.949
1.096 3.309
3.486
0.193 1.846
1.856
0.801 2.478
2.605
25
Bias
SPML
StD RMSE
-0.222
0.124
-0.051
-0.074
-0.032
-0.043
0.118
0.144
0.294
0.170
0.433
0.290
0.278
0.351
0.278
0.340
0.368
0.211
0.436
0.299
0.280
0.353
0.302
0.369
-0.444
0.237
-0.202
-0.054
-0.003
0.114
0.111
0.204
0.264
0.164
0.380
0.272
0.291
0.328
0.338
0.348
0.517
0.288
0.431
0.277
0.291
0.347
0.356
0.403