Fusion of Data Sets in Multivariate Linear
Regression with Errors-in-Variables
Albert Satorra
1
Departament d'Economia i Empresa
Universitat Pompeu Fabra
Ramon Trias Fargas 23-25, 08005 BARCELONA, Spain
Summary: We consider the application of normal theory methods to the es-
timation and testing of a general type of multivariate regression models with
errors-in-variables, in the case where various data sets are merged into a single
analysis and the observable variables deviate possibly from normality. The vari-
ous samples to be merged can dier on the set of observable variables available.
We show that there is a convenient way to parametrize the model so that, despite
the possible non-normality of the data, normal-theory methods yield correct in-
ferences for the parameters of interest and for the goodness-of-t test. The theory
described encompasses both the functional and structural model cases, and can
be implemented using standard software for structural equations models, such as
LISREL, EQS, LISCOMP, among others. An illustration with Monte Carlo data
is presented.
among others.
1 Introduction
Consider the classical regression equation
Y
i
=  + x
i
+ v
i
; i = 1; : : : ; n; (1)
where, for individual i, Y
i
and x
i
are the values of the response and explanatory
variables respectively, v
i
is the value of the disturbance term, and  and  are
respectively the intercept and slope parameters. Suppose we are unable to observe
x
i
. Instead we observe X
i
which satisfy
X
i
= x
i
+ u
i
; (2)
where u
i
is an error term. We assume the v
i
and u
i
are iid with mean 0 and
variances 
2
v
and 
2
u
respectively. Equations (1) and (2) with the associated as-
sumptions dene the classical simple regression model with errors-in-variables.
When the x
i
are a set of xed values (across hypothetical sample replications)
the model is called a functional model. When the x
i
are random (i.e. varying
across sample replications) we have the so-called structural model. See Fuller
(1987) for a comprehensive overview of measurement error models in regression
analysis. For recent work on the importance of assessing measurement reliability
in multivariate linear regression, see Gleser (1992).
1
Work supported by the Spanish DGICYT grant PB93-0403
1
As it is well known, when 
2
u
> 0 and X
i
is used instead of x
i
in (1), the usual
least-squares estimate is not consistent for ; in fact, under normality, equations
(1) and (2) with the associated assumptions fail to identify the parameters of the
model. To obtain a consistent estimate of , additional information related with
the measurement error variance 
2
u
is required. The parameter  is identied, for
example, when we specify the value of 
2
u
, or the ratio of variances 
2
u
=
2
v
, amid
of other possibilities. In practice, however, it may be dicult to have such exact
information on the value of error variances.
An alternative to specifying the size of measurement error variances is the mul-
tiple indicator approach. Consider for example two indicators X
i1
and X
i2
(i =
1; : : : ; n) which satisfy
X
i1
= x
i
+ u
i1
; X
i2
= x
i
+ u
i2
; (3)
where the u
i1
and u
i2
are uncorrelated error terms. Equations (1) and (3) with
the associated assumptions yield an identied model. Inferences for this type of
models is usually carried out under normality and a single sample with complete
data (see, e.g., Fuller, 1987).
In practice, however, the data may be composed of several subsamples with ob-
servable variables missing in the dierent subsamples. In the described regression
model for example, we may have one sample with information only on Y
i
and X
i1
(X
i2
missing), and a second sample with information only on X
i1
and X
i2
(Y
i
missing). In the present paper we focus on the joint analysis of various samples of
this type using Normal Theory (NT) methods while allowing the data to deviate
from the normality assumption. NT methods are available in standard software
for structural equation modeling, such as LISREL (Joreskog & Sorbom, 1989),
EQS ( Bentler, 1989), LISCOMP (Muthen, 1987), the SAS Procedure CALIS
(e.g., Hatcher, 1994). A clear advantage of the NT methods is that they require
rst- and second-order moments only.
The present paper relates with work of Arminger and Sobel (1990), where Pseudo-
Maximum Likelihood (PML) analysis is applied to dierent data sets arising from
missing data; Arminger and Sobel, however, do not address the issue of analyzing
a functional model and the issue of correct NT inferences when data deviates
from the normality assumption. The present work can also be seen as generaliz-
ing to multi-sample analysis and non-normality the results of Dham and Fuller
(1986) that show the validity of NT-inferences in the case of a functional model.
Our paper relates also with the asymptotic robustness results of Anderson and
Amemiya (1988) and Amemiya and Anderson (1990); we extend such asymptotic
robustness results to the case of multiple samples. Finally, we should mention
that the following results are a specialization to the multivariate regression model
set-up of more general results developed in Satorra (1995).
The structure of the paper is as follows. In Section 2 we present the general
model set-up under consideration. Section 3 describes the NT generalized-least-
squares analysis and provides a theorem which summarizes the major results of
the paper. Finally, Section 4 presents a limited Monte Carlo study illustrating in
nite samples and a specic model context the practical implications of the paper.
2
2 Multivariate Regression andMoment Struc-
tures
Let

Y
i
= Bx
i
+ v
i
X
i
= x
i
+ u
i
;
i = 1; : : : ; n; (4)
where for individual i, Y
i
(p  1), X
i
(q  1) and x
i
(m  1) are respectively the
values of the response, indicator and true (latent) regressor variables; v
i
(p  1)
and u
i
(q  1) are respectively the values of the disturbance and measurement
error terms; B (p  m) is a matrix of (unknown) parameters, and  (q  m) is
a matrix of known coecients. Assume that the v
i
and u
i
are independent and
both iid with mean 0 and variance matrices 
v
and 	
u
respectively; furthermore,
assume
n
 1
n
X
i=1
x
i
x
i
0
! 
x
; (5)
as n!1, where 
x
is a positive-denite matrix and the convergence is in prob-
ability in the case of a structural model. We further assume that for each i:
(structural model) x
i
, v
i
and u
i
are mutually independent; (functional model) v
i
and u
i
are independent with 
v
and 	
u
unrelated with the x
i
. We consider that
 is a 0-1 matrix of full column rank,
2
and 	
u
= H 
u
, where H is a 0-1 matrix
and  
u
is a vector of unrestricted parameters. Note that by allowing a compo-
nent of X
i
and x
i
to be constant to 1, we can accommodate as parameters of the
model the intercepts of the regression equations and the mean vector of x
i
. In the
model set-up of the present paper, we impose that the u
i
are normally distributed.
Clearly, (4) can be written as

Y
i
X
i

=

0
I

u
i
+

B


x
i
+

I
0

v
i
; (6)
or, alternatively, as

z
i
= 
?
z
?
i
+ u
?
i
z
?
i
= B
?
z
?
i
+ v
?
i
;
(7)
where
z
i
=

Y
i
X
i

; z
?
i
=

Y
i
x
i

;

?
= diag (I;); B
?
=

0 B
0 0

and
u
?
i
=

0
u
i

; v
?
i
=

v
i
x
i

:
(Here 0 and I denote zero and identity matrices respectively of dimensions clearly
determined by the context.). We denote by `  (p + q) the dimension of z
i
. We
should note that expression (7) enables the direct use of standard software for
2
In the context of a multiple indicator model,  is of the form 1
K

 I
m
where 1
K
denotes a vector of ones and K is the number of "replicates".
3
structural equation models for the purpose of model estimation and testing.
Now suppose that we have G dierent selection matrices T
g
; g = 1; : : : ; G; where
T
g
is of dimension `
g
 ` with `
g
 `. Suppose that instead of z
i
we observe only
~z
ig
= T
g
z
i
;
where each i (i = 1; : : : ; n) is associated to one of the matrices T
g
. Let the cases i
associated with T
g
form the group (or subsample) g, and denote by n
g
the size of
the gth group (g = 1; : : : ; G). For the validity of the asymptotic theory to be de-
veloped below, we require that G is "small" compared to n, so that n
g
=n! c
g
> 0
as n!1.
Dene now the (`
g
 `
g
) uncentered sample cross-product moment matrix
S
g
 n
 1
g
X
i
~z
ig
~z
0
ig
;
where here
P
i
denotes sum over the n
g
cases in group g. Clearly, under the
current model set-up,
S
g
P
! 
g
;
where
P
! denotes convergence in probability and 
g
is a nite `
g
 `
g
matrix.
From (6), and the denition of ~z
ig
, we have
~z
ig
= 
g0
u
i
(g)
+ 
g1
x
i
(g)
+ 
g2
v
i
(g)
; (8)
where

g0
 T
g

0
I

; 
g1
 T
g

B


; 
g2
 T
g

I
0

; (9)
and where u
i
(g)
and v
i
(g)
are subvectors of u
i
and v
i
respectively so that 
g0
and

g2
are of full column rank. Note that the vector x
i
(g)
is just the whole vector x
i
.
The following moment structure for 
g
can be derived from (8) and the current
assumptions:

g
= 
g0

g0

g0
0
+
g1

g1

g1
0
+ 
g2

g2

g2
0
; (10)
where 
g0
 
u
(g)
, 
g1
 
x
(g)
and 
g2
 
v
(g)
are respectively the "population"
moment matrices associated to u
i
(g)
, x
i
(g)
and v
i
(g)
. Consider now the following
(unconstrained) vector of parameters
#
1
 (v
0
(
v
(1)
); : : : ; v
0
(
v
(G)
); v
0
(
x
(1)
); : : : ; v
0
(
x
(G)
))
0
; (11)
where the v(
v
(g)
) may be of varying dimensions (determined by T
g
) while the
v
0
(
x
(g)
) are all of dimension m(m + 1)=2 (For a symmetric matrix A, v(A) de-
notes the vectorized matrix A with the redundant elements due to symmetry
suppressed.). Furthermore, dene
 

vec(B)
 
u

and # 


#
1

: (12)
4
Let denote by t
?
and t the dimensions of  and # respectively.
In relation with (9), (10), (11) and (12), consider the (multiple-group) moment
structure

g
= 
g
(#); g = 1; : : : ; G; (13)
where 
g
(:) is the matrix-valued function of # associated with (10), with 
g`
=

g`
() and 
g0
= H 
u
( ` = 1; 2; 3; g = 1; : : : ; G) as specied above. Note that
in this model specication the non-redundant elements of 
x
and of 
v
are un-
constrained parameters (even across-groups) of the model (this is in fact a key
assumption for the theorem to be presented in Section 3).
3
We now consider the estimation and testing of the multiple-group moment struc-
ture (13). We will do so by tting simultaneously the S
g
to the 
g
(#). We will
use a tting function that is optimal under the following NT assumption: the z
i
are iid normally distributed. In fact, the major import of the present paper is a
theorem that identies conditions under which the NT-inferences are correct even
though the assumption NT does not hold. In structural equation modeling, the
validity of NT inferences when the NT does not hold has been called asymptotic
robustness (Anderson, 1987).
To facilitate notation, we dene
s  vecfs
g
; g = 1; : : : ; Gg and   vecf
g
; g = 1; : : : ; Gg;
where s
g
 v (S
g
) and 
g
 v (
g
), and we write (13) as
 = (#); (14)
where (:) is a continuously dierentiable vector-valued function of #. Now we
will t s to (#).
3 NT Generalized-Least-Squares
The NT Generalized-Least-Squares (NT-GLS) estimate of # is dened as
^
#  arg min[s  (#)]
0
^
V
?
[s  (#)]; (15)
where
^
V
?
converges in probability to the following block-diagonal matrix:
V
?
 diag
n
c
g
V
?
g
; g = 1; : : : ; G
o
;
where
n
g
n
! c
g
when n!1,
V
?
g

1
2
D
0
(
g
 1

 
g
 1
)D; (16)
3
We should note that the results to be described below apply also when B and  are
continuously dierentiable functions of the subvector of parameters  .
5
and D is the duplication matrix for symmetry of Magnus and Neudecker (1988).
It is well known that under the NT assumption this NT-GLS estimate is asymp-
totically optimal. For seminal work on GLS estimation of covariance structures
in single-sample analysis, see Browne (1974); NT-GLS estimation in multi-sample
analysis is treated in Lee and Tsui (1982).
We need to impose the following identication assumption: (#
?
) = (#) implies
(locally) that #
?
= #, and the Jacobian matrix  =
@(#)
@#
0
is of full column rank
in a neighborhood of #. This assumption is needed to ensure uniqueness and
consistency of the estimate
^
#.
An alternative to the NT-GLS estimate is the PML approach. Under the NT
assumption,
F
ML

G
X
g=1
n
g
n
[log j 
g
(#) j +tracefS
g

g
(#)
 1
g   log j S
g
j  `
g
] (17)
is an ane transformation of the log-likelihood function. Thus, under general
distribution assumption for the z
i
, maximization of F
ML
= F
ML
(#) yields the
so-called PML estimate of #. This PML estimate is in fact asymptotically equiv-
alent to the NT-GLS estimate described above. For a comprehensive overview of
PML estimation in structural equation modeling, see Arminger and Sobel (1990).
Multi-sample analysis of structural equation models using the maximum likelihood
method was rst proposed by Joreskog (1971) in the context of factor analysis.
Now we will review the asymptotic variance matrix of the NT-GLS and PML
estimates.
Let   denote the asymptotic variance matrix of the scaled vector of sample mo-
ments
p
ns. Under the NT assumption, the asymptotic variance matrix of
p
ns
is (e.g., Satorra, 1993)
 
?
= diag
n
c
 1
g
 
?
g
; g = 1; : : : ; G
o
;
where
 
?
g
 

g
 
g
;


g
 2D
+
(
g

 
g
)D
+0
; (18)

g
 2D
+
(
g

g
0

 
g

g
0
)D
+0
and

g
 E(z
g
);
where E denotes the expectation operator under the NT assumption. Further-
more, let

  diag
n
c
 1
g


g
; g = 1; : : : ; G
o
; (19)
and let
^

 denote the estimate of 
 obtained by substituting S
g
for 
g
and c
 1
g
for
n
n
g
in (18) and (19) respectively. Note that V
?
= 

 1
(and that
^
V
?
=
^


 1
).
6
By using the standard delta-method, under general distribution assumption for
the z
i
, the asymptotic variance matrix of the NT-GLS and PML estimates is
known to be (e.g., Satorra and Neudecker, 1994)
avar(
^
#) = (
0
V
?
)
 1

0
V
?
 V
?
(
0
V
?
)
 1
: (20)
Under the NT assumption we have
avar
NT
(
^
#) = (
0
V
?
)
 1
: (21)
In addition to parameter estimation, we are interested in testing the goodness-of-
t of the model. The following goodness-of-t test statistic will be used:
T
?
= n(s  ^)
0
(
^
V
?
 
^
V
?
^
(
^

0
^
V
?
^
)
 1
^

0
^
V
?
)(s  ^); (22)
under NT, and when the model holds, T
?
is asymptotically chi-square distributed
with
r = rank(P 
?
P
0
) (23)
degrees of freedom (df), where P = I  (
0
V
?
)
 1
V
?
(Satorra, 1993).
Direct specialization to the present model set-up of results of Satorra (1995) yield
the following theorem.
Theorem 1 (cf., Satorra, 1995, Theorem 1) Under (6) and (13) and the current
assumptions (NT is not included), the NT-GLS and PML estimates verify
1.
^
# is a consistent estimate of #.
2. The t
?
 t
?
leading principal submatrices of avar(
^
#) and avar
NT
(
^
#) (recall
(20) and (21)) coincide (that is, the asymptotic variance matrix avar(^) is
the same as under NT).
3. ^ is an ecient estimate within the class of GLS estimates of (15).
4. The asymptotic distribution of the goodness-of-t test statistic T
?
V
of (22)
is chi-square with degrees of freedom r given by (23).
Note that this theorem guarantees correct NT-inferences for the parameter vector
 , including the asymptotic eciency, and also asymptotic chi-squaredness for
the goodness-of-t test, without the requirement for the z
i
to be iid normally dis-
tributed. It is required, however, that for each i: (structural model) the x
i
; v
i
; u
i
are mutually independent, not only uncorrelated; (functional model) the v
i
; u
i
are
independent with the limit of the second-order moments of the x
i
to be nite.
With regard to the model specication, it is also required that the variances of
the possibly non-normal constituents of the model, such as the x
i
and v
i
, are not
constrained even across groups. See Satorra (1995) for full details of the proof of
this theorem (in this reference, the theorem is formulated in a more general model
context).
In the next section we present a limited Monte Carlo study to illustrate the
performance in nite samples of the asymptotic results of the paper.
7
4 Monte Carlo illustration
Consider the regression with errors-in-variables model set-up of equations (1) and
(3) of Section 1, from which we simulate two-sample data of sizes n
1
= 800 and
n
2
= 400, with X
i2
missing in the rst subsample and Y
i
missing in the second
subsample. The values of x
i
and v
i
are simulated as independent chi-square dis-
tributed of 1 degree of freedom (i.e., a highly skewed non-normal distribution)
conveniently scaled; the u
i1
and u
i2
are simulated as uncorrelated iid normally
distributed. Each Monte Carlo run consisted on replicating 1000 times the steps
of (a) generating two-sample data, and (b) estimating (for each two-sample data)
a specied model using the NT-GLS approach of Section 2.
The three models dier with respect to the restrictions imposed on equality across
groups of means and variances of x
i
. Two of the models considered are structural,
one is a functional model.
Unrestricted structural model (USM): The mean of x
i
is a parameter re-
stricted to be equal across groups; the variance of x
i
is a parameter dierent
for each group.
Restricted structural model (RSM): The variance of x
i
is a parameter re-
stricted to be equal across groups; the mean of x
i
is a parameter dierent
for each group.
Unrestricted functional model (UFM): the (pseudo) mean and variance of
x
i
are (pseudo) parameters dierent for each group.
In all the three models, the variance of v
i
in the rst group and the variance of
u
i2
in the second group are unrestricted parameters of the model; the parameters
 and  and the variance of u
i1
are restricted to be equal across groups. With re-
gard to the degrees of freedom of T
?
, the dierence between the number of distinct
moments (10 moments) and the number of free parameters in the model, yields
1 df in the case of models RSM and USM, and 2 df in the case of UFM. Tables
1-3 present the Monte Carlo results for models USM, UFM and RSM respectively.
Note that USM, UFM and RSM are in fact three alternative correctly specied
models, since the restrictions of equality of parameters across groups imposed by
the models are in fact true in the population. Only USM and UFM, however, sat-
isfy the conditions for invoking Theorem 1; RSM does not satisfy such conditions,
since it restricts across groups the variance of x
i
, a non-normal constituent of the
model.
In all the three models, we expect consistency of the parameter estimates. This
is in fact corroborated by comparing the second and third columns of Tables 1-3.
The second column corresponds to the population values of the parameters and
the third column corresponds to the mean (across 1000 replications) of NT-GLS
parameter estimates.
In the case of models USM and UFM, i.e. Tables 1 and 2 respectively, Theorem
1 guarantees asymptotic correctness of NT-inferences for the subvector of param-
eters  : the parameters , , 
2
u
1
, 
2
u
2
(x
(1)
i
) and (x
(2)
i
) in USM, and , , 
2
u
1
8
and 
2
u
2
in UFM. Thus, the rows of Tables 1-2 corresponding to the components
of  should show: (a) the Monte Carlo standard deviation of parameter estimates
(column sd(
^
#)) close to the Monte Carlo mean of NT standard errors (column
E(se)); (b) the Monte Carlo variance of d 
^
# #
se(
^
#)
(column var(d)) close to unity;
and, (c) the Monte Carlo two-sided tails of d close to nominal values (columns
5%, 10% and 20%). Theorem 1 ensures also the asymptotic chi-squaredness of the
goodness-of-t test T
?
; thus, in Tables 1-2, the Monte Carlo mean (and variance)
of T
?
should be close to the degrees of freedom (twice the degrees of freedom),
and the tails of the Monte Carlo distribution of T
?
should be close to the nominal
values. These expectations are clearly corroborated by the results shown in Ta-
bles 1 and 2. This correct performance of the NT-inferences is remarkable despite
clear non-normality of the z
i
.
In the case of model RSM, however, Table 3 shows that NT-inferences for param-
eters deviate dramatically from the correct performance (only NT-inferences for
the means (x
1
i
) and (x
2
i
) seem to be appropriate), and the empirical distribu-
tion of T
?
also deviates highly from the desired chi-square distribution. Note that
this lack of robustness of NT-inferences resulted from the restriction of equality
across groups of the variance of the x
i
, a restriction which is in fact true in the
population generating the data.
References:
AMEMIYA, Y. and ANDERSON, T.W. (1990): Asymptotic chi-square tests for
a large class of factor analysis models. The Annals of Statistics, 3, 1453-1463
ANDERSON, T. W. (1987): Multivariate linear relations. In T. Pukkila & S.
Puntanen (edits.) Proceedings of the Second International Conference in Statis-
tics, Pp. 9-36. Tampere, Finland: University of Tampere.
ANDERSON, T.W. and AMEMIYA, Y. (1988): The asymptotic normal distri-
bution of estimates in factor analysis under general conditions." The Annals of
Statistics, 16, 759-771.
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10
Table 1: Monte Carlo results for USM model, where 
2
(x
(1)
i
) and 
2
(x
(2)
i
)
are unrestricted parameters and (x
(1)
i
) = (x
(2)
i
).
distribution of estimates
para. true val. E(
^
#)
a
sd(
^
#)
b
E(se)
c
var(d)
d
5%
e
10% 20%

2
(v
(1)
i
) 1:00 1.00 .25 .23 1.24 7.10 13.20 23.50

2
(x
(1)
i
) 1:00 1.00 .15 .08 3.43 29.40 37.10 49.50

2
(x
(2)
i
) 1:00 1.00 .20 .08 5.62 39.80 49.90 59.40

2
(u
(j)
2i
) :30 .30 .05 .05 1.01 5.40 10.40 19.90

2
(u
(2)
2i
) :40 .40 .05 .05 .89 3.70 8.20 16.60
 2:00 2.00 .12 .12 1.04 4.90 9.20 20.20
 1:00 1.00 .37 .37 1.03 4.40 9.00 20.50
(x
(j)
) 3:00 3.00 .03 .03 1.05 4.50 10.10 20.90
distribution of T
?
(df=2 )
mean var 5% 10% 20%
2.12 4.44 6.40 11.40 21.40
a
Monte Carlo mean of estimates
b
Monte Carlo standard deviation of estimates
c
Monte Carlo mean of NT standard errors
d
Monte Carlo variance of d 
^
# #
se(
^
#)
e
Monte Carlo nominal 5%, 10% and 20% two-sided tails for d
11
Table 2: Monte Carlo results for UFM model, where 
2
(x
(1)
i
), 
2
(x
(2)
i
),
(x
(1)
i
) and (x
(2)
i
) are unrestricted (pseudo) parameters.
distribution of estimates
para. true val. E(
^
#)
a
sd(
^
#)
b
E(se)
c
var(d)
d
5%
e
10% 20%

2
(v
(1)
i
) 1:00 1.00 .20 .16 1.41 8.70 16.80 26.60

2
(u
(j)
1i
) :30 .30 .03 .03 1.08 6.20 10.90 21.80

2
(u
(2)
2i
) :40 .40 .03 .04 .99 5.10 10.50 19.70
 2:00 2.00 .08 .08 1.04 6.10 10.40 21.60
 1:00 1.00 .17 .17 1.00 5.10 9.50 18.40
distribution of T
?
(df=1 )
mean var 5% 10% 20%
1.05 2.27 5.40 10.00 19.90
a
Monte Carlo mean of estimates
b
Monte Carlo standard deviation of estimates
c
Monte Carlo mean of NT standard errors
d
Monte Carlo variance of d 
^
# #
se(
^
#)
e
Monte Carlo nominal 5%, 10% and 20% two-sided tails for d
12
Table 3: Monte Carlo results for RSM model, where (x
(1)
i
) and (x
(2)
i
) are
unrestricted parameters and 
2
(x
(1)
i
) = 
2
(x
(2)
i
).
distribution of estimates
para. true val. E(
^
#)
a
sd(
^
#)
b
E(se)
c
var(d)
d
5%
e
10% 20%

2
(v
(1)
i
) 1:00 .99 .29 .22 1.72 13.50 20.80 32.80

2
(x
(j)
i
) 1:00 .98 .12 .06 4.37 34.90 43.60 55.30

2
(u
(j)
1i
) :30 .30 .06 .05 1.49 10.80 17.30 29.50

2
(u
(2)
2i
) :40 .40 .05 .05 1.12 5.50 10.90 22.70
 2:00 2.01 .15 .11 1.85 14.80 23.20 35.00
 1:00 .98 .45 .33 1.84 15.10 22.70 34.80
(x
(1)
) 3:00 3.00 .04 .04 1.06 5.50 12.20 20.90
(x
(2)
) 3:00 3.00 .05 .05 1.07 6.40 10.70 21.10
distribution of T
?
(df=2)
mean var 5% 10% 20%
5.68 40.81 33.20 41.30 53.80
a
Monte Carlo mean of estimates
b
Monte Carlo standard deviation of estimates
c
Monte Carlo mean of NT standard errors
d
Monte Carlo variance of d 
^
# #
se(
^
#)
e
Monte Carlo nominal 5%, 10% and 20% two-sided tails for d
13