1. Introduction
This paper considers the instrumental variable (IV) estimation of the spatial autoregressive (SAR) model with SAR disturbances (SARAR model) in the presence of endogenous regressors and many instruments. We study the case where the number of instruments increases with the sample size and derive asymptotic distributions of the generalized spatial two stage least squares (GS2SLS) estimator and a bias-corrected GS2SLS (CGS2SLS) estimator based on the leading-order many-instrument bias. Using many moments may improve the asymptotic efficiency but can make inference inaccurate in finite samples. [
1] propose to minimize an approximate mean square error (MSE) as that of [
2] for choosing the number of instruments in a cross section data model with endogenous regressors. The MSE takes into account an important bias term, so the method can avoid cases where asymptotic inferences are poor due to the bias being large relative to the standard deviation.
Ref [
3] have derived the approximate MSEs of the two stage least squares (2SLS) and bias-corrected 2SLS (C2SLS) estimators for the SAR model with endogenous regressors and many instruments, but that SAR model has not included a SAR process in the disturbances. We extend the analysis to the SARAR model with endogenous regressors. The SARAR model combines spatial lag with spatial error dependence. The latter reflects spatial autocorrelation in measurement errors or in variables that are otherwise not crucial to the model [
4,
5]. It has a broader application than the simpler SAR model. It has been applied to empirical studies, e.g., Case’s work [
6,
7,
8,
9,
10]. Due to the presence of the spatial error dependence in addition to the spatial lag dependence, we consider the GS2SLS estimation of the model as in [
11]. (Ref [
12] have extended the estimation method in [
11] to the SARAR model with endogenous regressors. Our focus here is on choosing the number of instruments by minimizing the approximated MSEs.) The estimation has taken into account the spatial error structure, based on a transformed equation. Because the transformation uses an initial consistent estimator of the spatial error dependence parameter, the impact from this initial estimator creates extra complexity that should be investigated. The analytical difficulty lies in determining the leading order terms depending on the number of instruments due to the presence of the spatial error process, whose orders cannot be expressed using terms appeared only in a SAR model without SAR disturbances. The approximated MSEs of the GS2SLS and CGS2SLS estimators turn out to be more complicated than those of the corresponding 2SLS and C2SLS estimators for the SAR model but are still tractable for empirical use. For the GS2SLS, the expression for the approximate MSE is similar to that for the 2SLS in [
3], except for the presence of the filter for spatial error dependence in various matrices. If the formula for the approximate MSE in [
3] is used for the SARAR models, then the derived number of instruments will not be asymptotically optimal. For the CGS2SLS estimator, however, except for the filter, the approximate MSE has additional terms compared with that for the C2SLS in [
3], which are generated from the asymptotic distributions of the first two stage estimators.
We consider the following SARAR model:
where
n is the number of spatial units,
is an
n-dimensional vector of observations on the dependent variable, the
n-dimensional vector of disturbances
has i.i.d. elements with mean zero and variance
, and
is an
matrix of variables that are possibly correlated with
,
and
are
spatial weights matrices that can be equal or different from each other, scalars
λ and
ρ are spatial autoregressive parameters, and
γ is a parameter vector for
. Let
, where
. The
is assumed to be an unknown function of
, which is an
matrix of exogenous variables, and spatial lags of
:
,
, and so on. Model (
1) can be an equation of the spatial simultaneous system as in [
13]. In this case,
is a vector of observations on one of, say,
endogenous variables, and the equation for
, similar to those for other endogenous variables, is
, where
is the included exogenous variable matrix,
is the endogenous variable matrix including all observations on the other
endogenous variables and
and
are parameter vectors, then
, where
’s are matrices of parameters. Alternatively,
or some elements of
may be generated by an unknown nonlinear model [
14], and thus we have an unknown nonlinear functional form for the conditional mean
[
1]. For
, we assume that
’s are i.i.d. with mean zero and
,
is independent of
for
, but
. That is,
and
are correlated except with the exogenous explanatory variables. The
ith variable in
is exogenous if the
ith element of
is zero. Let
and
, then
.
We are interested in the parameter
δ. As in [
11], the final generalized estimator for
δ is based on the Cochrane–Orcutt transformed equation:
where
with
being a consistent estimator of
ρ. We consider the problem of choosing the number of instruments for
, which can be many due to the unknown functional form of
for its endogenous components. To derive
, we may first estimate the equation
by the 2SLS with a fixed number of instruments to obtain an initial estimator
of
δ, and then estimate
ρ with a fixed number of quadratic moment equations that have the form
, where the
matrix
has a zero trace, and
. (The equation
is a valid moment equation since
and
under regularity conditions.) The estimation thus involves three stages and the derivation of approximated MSEs is more complicated due to the presence of many terms with different orders. In [
11], the asymptotic distribution of the third stage estimator
is not affected by the estimators in the first two stages as long as
is a consistent estimator of
ρ. For the approximate MSE of our GS2SLS estimator in the third stage, one may expect that it involves the asymptotic distributions of the first two stage estimators, since we use higher-order asymptotic theory for IV. However, it turns out that the variance of the dominant component related to the first two stage estimators in the expression for the GS2SLS estimator has a smaller order compared with other terms because of the i.i.d. property of
’s. As a result, the leading order component of the MSE does not depend on the asymptotic distributions of the first two stage estimators and the expression for the approximate MSE is similar to that in [
3] except for the filter for spatial error dependence. However, for the CGS2SLS estimator, the expression for the approximate MSE is more complicated than that in [
3], because the term resulting from the estimation error of the leading order bias involves the asymptotic distributions of the first two stage estimators and an additional term appears due to the estimation of the spatial autoregressive parameter in the error process.
As
is an unknown function of
,
,
,
etc., we may assume an infinite series approximation for
and, in practice, use a known
matrix
to approximate
, where
depends on
,
and so on. To closely approximate
with a linear combination of
, we may need a large column number
q as well as appropriate form of
. The instruments for
can be based on
. Denote the true parameters for
δ and
ρ by
and
respectively. As model (
1) represents an equilibrium model,
can be assumed to be invertible, where
is the
identity matrix. (The SAR model is known as a simultaneous equation model in the spatial literature because the outcomes are determined by the interactions of spatial units. By assuming
to be invertible, we have the equilibrium vector
.) Then, if
for some matrix norm
, the equilibrium vector
can have an expansion
. Therefore, the instruments for
can be
,
and so on, and the instruments for
can be taken as the
matrix
where
. As an extension, we use the instrument matrix
for
. (Due to technical difficulties in the presence of many IVs that involve estimated parameters in the literature, we do not use
as the instrument matrix for
(see [
15]). If
, then
generates some identical IVs as those in
. In this case, we can simply take
.) The asymptotic variance of the 2SLS estimator decreases when a linear combination of IVs approximates the conditional mean of the endogenous variables more closely. The efficiency (lower bound) of IV estimators is achieved when a linear combination of IVs equals the conditional mean [
16]. Under regularity conditions, a linear combination of
can approximate
arbitrarily well as
. Thus, if a linear combination of
can approximate
well as
, a linear combination of
can approximate
arbitrarily well in probability as
. On the other hand, if the number of instruments increases too fast relative to the sample size, they will lead to a bias of certain order for the corresponding IV estimators. The tradeoff between variance and bias can be summarized by the MSE of the estimator. So, minimizing the (approximated) MSE can reduce inaccurate inference due to the presence of many instruments. Following [
1], we consider the case that the number of instruments
K increases with, but at a rate slower than, the sample size
n, which facilitates the investigation of the high order asymptotics of the MSEs.
The rest of the paper is organized as follows.
Section 2 establishes asymptotic properties of the GS2SLS and CGS2SLS estimators.
Section 3 derives the approximated MSEs for the estimators and gives a criterion function to choose the optimal number of IVs using the approximated MSEs.
Section 4 presents some Monte Carlo results on the performance of the instrumental variable selection procedure in finite samples.
Section 5 concludes. A list of notations, lemmas and proofs are collected in the appendices.
2. Properties of the GS2SLS and CGS2SLS Estimators
We establish the properties of the GS2SLS and CGS2SLS estimators in this section. Let , , with , and be the Frobenius matrix norm for a matrix A. UB stands for boundedness of the sequences of both row and column sum matrix norms for a sequence of matrices. For simplicity, denote , , , , , and . As , and . The following are some basic regularity conditions.
Assumption 1. ’s, , are i.i.d. with mean zero, , and . The moments , and are finite, where τ is some positive constant.
Assumption 2. (i) The sequences of matrices , , and are UB;
- (ii)
and have zero diagonals.
Since we use quadratic moments to estimate
ρ in model (
1), the existence of a moment of
higher than the fourth order is required to properly apply the central limit theorem for linear-quadratic forms of disturbances in [
17]. Some moment conditions are also imposed on
and
in Assumption 1. Assumption 2 (i), originated in [
11,
18], is a condition that bounds the degree of spatial dependence; Assumption 2 (ii) implies that no spatial unit is viewed as its own neighbor.
Let be a full rank instrument matrix for in the first stage of the GS2SLS estimation. The number of IVs is at least as large as the number of columns of , but is fixed for all n. Denote , where is a generalized inverse for the matrix A. The first stage 2SLS estimator for δ is . The following assumption about is maintained.
Assumption 3. The instrument matrix has full column rank for all n, is finite and nonsingular, and is finite and has full column rank, where in has uniformly bounded elements.
Proposition 1. Under Assumptions 1–3, .
In the second stage of the GS2SLS estimation, we use a fixed number, say
, of quadratic moments to estimate
ρ in model (
1). Let
, where
and
matrices
’s have zero traces. The
’s can be, e.g.,
and
. We maintain the following regularity condition on
.
Assumption 4. The sequences of matrices , , have zero traces and are UB.
Consider a generalized moments estimator
of
ρ, which is
for some
so that
contains
. It can be shown that
converges to zero in probability uniformly over
. For the identification of
, it requires
to be zero uniquely at
. Let
for any square matrix
A. Note that
, where
Assumption 5. The smallest eigenvalue of is bounded away from zero.
Assumption 5 is satisfied if the limit of the matrix exists and is nonsingular. With Assumption 5, there exists some such that for any . Thus for any , with probability approaching 1 as .
Proposition 2. Under Assumptions 1–5, is a consistent estimator of , andis asymptotically normal with a finite variance, whereandwith . In the expression for above, the term with the order is due to the usage of the first stage estimator . That is to say that the asymptotic distribution of has implication on the asymptotic distribution of .
We now consider the GS2SLS estimator using the transformed Equation (
2). With the instrument matrix
in Equation (
4), the GS2SLS estimator of
δ is
where
.
Assumption 6. (i) , where , is a finite nonsingular matrix; (ii) for each in Equation (
4)
, there exists such that as . Assumption 6 (i) gives a sufficient condition for the identification of
in Equation (
2); Assumption 6 (ii) requires
to be approximated arbitrarily well by a linear combination of
for large enough
K and
n, which is implied by Lemma 1 in
Section B under some other basic assumptions. For analytical tractability, we maintain the following assumption.
Assumption 7. The elements of in Equation (
4)
are uniformly bounded constants, and exists and is nonsingular for each K. The GS2SLS estimator
is characterized by the first order condition
. By a Taylor expansion of this condition at
, the first term is
, which has the dominant component
by Lemma 8. The expectation of this dominant component is
, where
with
Thus when , the GS2SLS estimator is generally inconsistent. When , is consistent, but if the number of instruments K grows somehow fast relative to the sample size n, the asymptotic distribution may not center at the true . The following proposition provides more information on this issue.
Proposition 3. Under Assumptions 1–7,- (i)
if , then , wherewithmight converge to a nonzero constant; - (ii)
if , then , wherewith .
From the above proposition, when , is consistent of , but whether its asymptotic distribution is centered at or not depends on the ratio as . The following corollary shows various scenarios.
Corollary 1. Under Assumptions 1–7,- (i)
if , ;
- (ii)
if and , ;
- (iii)
if but for some , .
When
, the number of instruments
K increases slow relative to the sample size
n and the asymptotic variance matrix
achieves the efficiency lower bound for the class of IV estimators. When
goes to a non-zero limit as
n goes to infinity,
is centered at
, which might be a non-zero finite constant and is a many instrument bias. Due to the spatial error dependence, the matrices
,
and
in Equation (
10) of the bias component in Equation (
9) play important roles. Without spatial error dependence, these matrices reduce to
and
. Although the GS2SLS estimation is based on the spatial Cochrane–Orcutt transformed model (
2), the asymptotic distribution of the estimator
in the transformation does not affect the asymptotic distribution of
, as usual for the GS2SLS estimation.
To correct the many instrument bias, we consider a bias corrected estimator based on the estimation of the leading order bias
in Equation (
12). Let
be an instrument matrix with a fixed number of instruments and
.
Assumption 8. The instrument matrix has full column rank for all n, is finite and nonsingular, and is finite and has full column rank.
The GS2SLS estimator
and
together can be used to estimate
. Let
,
,
,
and
. A bias-corrected GS2SLS ( CGS2SLS) estimator is
where
with
.
Proposition 4. Under Assumptions 1–8, if , then .
Note that the asymptotic distribution of
in Equation (
14) when
is the same as that of
in Equation (
8) when
. So the bias correction procedure has effectively relaxed some requirement on
K in order for the corrected estimator to have a properly centered asymptotic distribution. The asymptotic distributions of the initial estimators
in Equation (
13) and
in Proposition 2 used for the bias correction do not enter into the asymptotic distribution of
, when only the first order asymptotic expansion is considered. But when we investigate the approximated MSE of
later, as high order asymptotic expansions are considered, the asymptotic distributions of the estimators
and
used for the bias correction will generate additional terms for the approximated MSE.
3. Approximated MSE and Optimal K
For an estimator
satisfying
, [
1] have derived a lemma that gives conditions on the decompositions of
and
such that the leading order term of the MSE depending on
K is
, in the sense that
where
, and
and
are remainder terms that diminish faster than
, such that
as
. A criterion function for the optimal
K can be
, the leading order MSE depending on
K for a linear combination
. In particular, one may use the unweighted version
as a practical criterion. Let
be an estimator of
, then
K can be chosen by minimizing the function
.
In this section, we first derive the expression for
for both the GS2SLS and CGS2SLS estimators and then show that the chosen
K by minimizing
is asymptotically optimal in a sense in Equation (
20) originated in [
1]. Intuitively, this indicates that the error in the use of the feasible
criterion in place of the actual ideal
is asymptotically negligible.
Assumption 9. (i) , where , as ;
- (ii)
for , as , where is the th element of ;
- (iii)
and .
Assumption 9 (i) is for analytical tractability; Assumption 9 (ii) simplifies the expression for
by imposing a restriction on the rate at which
K increases with
n; Assumption 9 (iii) is also a condition that simplifies
. These simplifications are adopted in [
1,
3]. (Without Assumption 9 (iii),
for the GS2SLS will have an additional term
, and
for the CGS2SLS has an additional term that is much more complicated due to the estimator of
ρ in the second stage of the GS2LS estimation and its use to correct the many instrument bias. Without Assumption 9 (ii),
for the GS2SLS is not affected, but
for the CGS2SLS has an additional term. Those additional terms can be estimated along with other terms, but they are not included here for simplicity.)
Proposition 5. Under Assumptions 1–9, if and , then Equation (
15)
for the GS2SLS estimator is satisfied withwhere . Note that
above has a similar form as that in [
3] except for the transformation
involved due to the spatial error dependence. The
has a similar interpretation as that in [
3]:
is a variance term, which becomes smaller as a linear combination of
approximates the mean of
better;
is the leading order term in the MSE of
with the dominant component being from its expectation, which stands for the many instrument bias and increases as
K increases. The minimization of a criterion function
thus takes into account the trade-off between the bias and variance.
Proposition 6. Under Assumptions 1–9, if and , then Equation (
15)
for the CGS2SLS estimator is satisfied withwhere , and are given in Equations (
21)
, (
22)
and (
25)
respectively. The first term in Equation (
17) is the same as that in Equation (
16). The second term
is the leading order term in the variance of
. The third term
is due to the estimation error of the lead order bias of the GS2SLS estimator. This term becomes much more complicated than that for the SAR model because of the spatial error dependence. The last term
is an additional term compared with
in [
3], which is due to the estimation of
ρ. (Thus, the
in
is from
used for the bias correction, and the
in
is from
in the spatial Cochrane–Orcutt transformation of the GS2SLS estimation.) The
is a sum of different variance terms, because the bias terms have smaller orders compared with the variance terms.
We now consider the estimation of
. Estimators for the parameters in
can be constructed using a GS2SLS estimator. For the GS2SLS estimator, let the first stage IV matrix be
with
instruments, the matrices for the quadratic moments in the second stage be
, and the last stage IV matrix be
. (The
needs to increase with
n so that the estimators for
,
and
defined below are consistent.) Then the first stage estimator for
δ is
with
, and the last stage estimator for
δ is
with
and
being the estimator for
ρ in the second stage. Let the estimators for
,
and
be, respectively,
,
and
, where
and
. An estimator for
,
, can be derived by replacing the parameters with their respective estimators. An estimator for
is
. For
, note that
where
is in Equation (
11), thus
can be estimated, up to an additive constant not depending on
K, by
, where
is an estimator for
, derived by replacing the parameters in
by their estimators. Hence, for the GS2SLS,
can be estimated, up to an additive constant not depending on
K, by
Similarly, for the CGS2SLS,
can be estimated, up to an additive constant not depending on
K, by
where
is an estimator of
derived by replacing the parameters in
by their estimators,
is given in Equation (
26) and
is given in Equation (
27).
The optimal choice of
K is the minimizer
of
. The
is optimal in the sense that
is asymptotically as small as
,
i.e.,
Assumption 10.
(i) , , , and ;
- (ii)
For the GS2SLS, , and for the CGS2SLS, , for some constant , where .
Assumption 11. For both the GS2SLS and CGS2SLS, .
We assume the
-consistency of
and consistency of other preliminary estimators in Assumption 10 (i). Assumption 10 (ii) and Assumption 11 are similar to those in [
3]. For the GS2SLS, from the proof of Proposition 5, the trace of the positive semi-definite matrix
has exactly the same order as
, then
has the order
. Assumption 10 (ii) requires
for the GS2SLS to have exactly the same order as
. A similar condition on
for the CGS2SLS is imposed. Assumption 11 imposes a restriction on the set of possible
K.
Proposition 7. Under Assumptions 1–11, for , Equation (
20)
is satisfied for both the GS2SLS and CGS2SLS. 4. Monte Carlo Study
We demonstrate the finite sample performance of our instrument selection procedure with Monte Carlo experiments. Except for the additional spatial error dependence, most parts of the experimental design follow [
3]. The model considered is
where
,
and
is a vector. The
’s are i.i.d. normal with mean zero,
and
both have unit variance, and the correlation coefficient between
and
is
, which will be varied by design. In the experiments,
,
, and
or
. Elements of the
matrix
are random samples from the standard normal distribution. The specification implies a theoretical first stage coefficient of determination
(with the spatial dependence being ignored), according to [
19]. The
will be designed later on.
As in [
3], we consider two models with different specifications of
. In Model 1, the coefficients are decreasing,
i.e., the
jth element of
is
where
is chosen such that
is equal to some specified value in the experiments; in Model 2, the coefficients are all equal,
i.e.,
These two specifications stand for, respectively, the case that some instruments are more important than others and the other case that no instrument should be preferred over others [
1]. In the experiments,
is equal to
or
,
is equal to
,
or
, and
or 490. The
is a block diagonal matrix with each block in the diagonal being the row normalized matrix used for the study of crimes across 49 districts in Columbus, OH in [
20]. The spatial weights matrix
in the error process is set to be the same as the spatial weights matrix
. The number of Monte Carlo repetitions is 2000.
Let
be a matrix consisting of the first
q columns of
, and
, for
and
. For
, we set
and
; for
, we set
and
. The following estimators are considered:
- (i)
GS2SLS-min: the GS2SLS with (as the instrument matrix in the third stage);
- (ii)
GS2SLS-max: the GS2SLS with ;
- (iii)
GS2SLS-op: the GS2SLS with
, where
minimizes
in Equation (
18) with
;
- (iv)
CGS2SLS-max: the CGS2SLS with ;
- (v)
CGS2SLS-op: the CGS2SLS with
, where
minimizes
in Equation (
19) with
.
The leading order bias for the CGS2SLS and the approximated MSEs are estimated using the GS2SLS with
as the instrument matrix in the third stage. For all the GS2SLS and CGS2SLS estimators considered, the instrument matrix used in the first stage is
, and the matrices used for the quadratic moments in the second stage are
and
. (As
is relatively large compared with the sample size, for the first stage estimator of the GS2SLS estimation and the estimator for the bias correction, we use
as suggested by [
11].)
For each estimator, the following robust measures of central tendency and dispersion are reported: (There are some outliers in the GS2SLS and CGS2SLS estimates, thus the mean and variance of the estimators are not reported.) the median bias (MB), the median of the absolute deviations (MAD), the difference between the 0.1 and 0.9 quantiles (DQ) in the empirical distribution, and the coverage rate (CR) of a nominal 95% confidence interval.
The summary statistics of the estimators for Model 1 are reported in
Table 1,
Table 2,
Table 3 and
Table 4. We first compare GS2SLS-min, GS2SLS-max and GS2SLS-op. The GS2SLS-max has the largest median bias in most cases, and the GS2SLS-op has the smallest median bias for half of the cases when
but it has the intermediate medium bias when
. The GS2SLS-max has the smallest MAD and DQ in all cases, the GS2SLS-op of
has the intermediate MAD and DQ, and GS2SLS-op of
has the intermediate MAD and DQ when
but largest MAD and DQ when
. The CR of GS2SLS-op is closest to the nominal level in most cases, while the CR of GS2SLS-max is significantly lower than the nominal level in many cases. The CGS2SLS-max generally reduces the bias of GS2SLS-max significantly, has similar magnitudes of MAD and DQ to those of GS2SLS-max, and has a CR closer to the nominal level compared with GS2SLS-max but still significantly lower than the nominal level in many cases. Compared with the GS2SLS-op, in most cases, the CGS2SLS-op has much larger MAD and DQ, similar CR, and has smaller median bias for
but larger median bias for
.
Table 1.
Estimation of Model 1 with and .
Table 1.
Estimation of Model 1 with and .
| | | | |
---|
| | MB | MAD | DQ | CR | MB | MAD | DQ | CR |
---|
| |
| GS2SLS-min | 0.174 | 0.375 | 2.327 | 1.000 | | −0.011 | 0.612 | 3.618 | 1.000 |
| GS2SLS-max | 0.242 | 0.083 | 0.323 | 0.810 | | −0.065 | 0.175 | 0.654 | 0.992 |
| GS2SLS-op | 0.171 | 0.297 | 1.702 | 0.999 | | 0.015 | 0.468 | 2.307 | 1.000 |
| CGS2SLS-max | −0.046 | 0.125 | 0.667 | 0.870 | | 0.098 | 0.295 | 1.248 | 0.974 |
| CGS2SLS-op | −0.375 | 0.581 | 10.489 | 0.991 | | 0.332 | 0.719 | 8.277 | 1.000 |
| GS2SLS-min | 0.157 | 0.428 | 2.917 | 1.000 | | 0.188 | 0.582 | 3.415 | 1.000 |
| GS2SLS-max | 0.156 | 0.071 | 0.279 | 0.921 | | 0.347 | 0.154 | 0.609 | 0.932 |
| GS2SLS-op | 0.129 | 0.235 | 1.357 | 1.000 | | 0.333 | 0.382 | 1.883 | 0.999 |
| CGS2SLS-max | −0.008 | 0.081 | 0.374 | 0.958 | | 0.407 | 0.246 | 0.983 | 0.824 |
| CGS2SLS-op | −0.190 | 0.383 | 5.713 | 0.999 | | 0.501 | 0.531 | 4.274 | 1.000 |
| GS2SLS-min | 0.148 | 0.295 | 2.039 | 1.000 | | 0.293 | 0.456 | 3.633 | 0.982 |
| GS2SLS-max | 0.064 | 0.031 | 0.120 | 0.968 | | 0.791 | 0.081 | 0.306 | 0.033 |
| GS2SLS-op | 0.074 | 0.129 | 0.814 | 1.000 | | 0.700 | 0.291 | 1.492 | 0.782 |
| CGS2SLS-max | 0.032 | 0.034 | 0.136 | 0.997 | | 0.723 | 0.152 | 0.608 | 0.189 |
| CGS2SLS-op | 0.011 | 0.160 | 1.544 | 1.000 | | 0.628 | 0.355 | 2.091 | 0.820 |
| |
| GS2SLS-min | 0.349 | 0.342 | 2.413 | 0.992 | | 0.026 | 0.572 | 3.234 | 1.000 |
| GS2SLS-max | 0.344 | 0.059 | 0.241 | 0.414 | | −0.070 | 0.176 | 0.696 | 0.992 |
| GS2SLS-op | 0.310 | 0.272 | 1.755 | 0.984 | | 0.031 | 0.428 | 2.375 | 1.000 |
| CGS2SLS-max | 0.057 | 0.160 | 1.432 | 0.720 | | 0.049 | 0.330 | 1.544 | 0.967 |
| CGS2SLS-op | −0.137 | 0.730 | 10.221 | 0.972 | | 0.203 | 0.810 | 7.607 | 1.000 |
| GS2SLS-min | 0.262 | 0.347 | 2.225 | 1.000 | | 0.195 | 0.556 | 3.487 | 1.000 |
| GS2SLS-max | 0.261 | 0.053 | 0.208 | 0.503 | | 0.335 | 0.155 | 0.578 | 0.934 |
| GS2SLS-op | 0.227 | 0.208 | 1.342 | 0.991 | | 0.350 | 0.403 | 2.108 | 1.000 |
| CGS2SLS-max | 0.092 | 0.077 | 0.401 | 0.855 | | 0.400 | 0.254 | 1.022 | 0.848 |
| CGS2SLS-op | −0.085 | 0.368 | 6.524 | 0.996 | | 0.464 | 0.565 | 4.658 | 1.000 |
| GS2SLS-min | 0.228 | 0.219 | 1.672 | 0.992 | | 0.339 | 0.460 | 3.461 | 0.973 |
| GS2SLS-max | 0.181 | 0.027 | 0.103 | 0.224 | | 0.775 | 0.066 | 0.264 | 0.023 |
| GS2SLS-op | 0.191 | 0.114 | 0.659 | 0.952 | | 0.690 | 0.262 | 1.390 | 0.768 |
| CGS2SLS-max | 0.140 | 0.029 | 0.124 | 0.546 | | 0.705 | 0.127 | 0.542 | 0.165 |
| CGS2SLS-op | 0.115 | 0.133 | 1.600 | 0.966 | | 0.639 | 0.310 | 2.055 | 0.809 |
Table 2.
Estimation of Model 1 with and .
Table 2.
Estimation of Model 1 with and .
| | | | |
---|
| | MB | MAD | DQ | CR | MB | MAD | DQ | CR |
---|
| |
| GS2SLS-min | 0.081 | 0.283 | 1.528 | 1.000 | | −0.024 | 0.240 | 1.026 | 1.000 |
| GS2SLS-max | 0.225 | 0.076 | 0.302 | 0.835 | | −0.062 | 0.135 | 0.548 | 0.995 |
| GS2SLS-op | 0.116 | 0.254 | 1.437 | 1.000 | | 0.001 | 0.288 | 1.365 | 1.000 |
| CGS2SLS-max | −0.033 | 0.109 | 0.558 | 0.887 | | 0.039 | 0.197 | 0.829 | 0.979 |
| CGS2SLS-op | −0.212 | 0.379 | 5.158 | 0.997 | | 0.211 | 0.461 | 3.997 | 1.000 |
| GS2SLS-min | 0.082 | 0.231 | 1.368 | 1.000 | | −0.031 | 0.255 | 1.127 | 1.000 |
| GS2SLS-max | 0.149 | 0.066 | 0.263 | 0.903 | | 0.206 | 0.136 | 0.534 | 0.965 |
| GS2SLS-op | 0.078 | 0.213 | 1.203 | 0.998 | | 0.147 | 0.312 | 1.452 | 1.000 |
| CGS2SLS-max | −0.004 | 0.080 | 0.361 | 0.961 | | 0.174 | 0.174 | 0.693 | 0.949 |
| CGS2SLS-op | −0.155 | 0.283 | 3.963 | 0.999 | | 0.312 | 0.381 | 2.418 | 1.000 |
| GS2SLS-min | 0.103 | 0.270 | 1.683 | 1.000 | | 0.027 | 0.298 | 1.763 | 0.995 |
| GS2SLS-max | 0.075 | 0.044 | 0.171 | 0.914 | | 0.595 | 0.095 | 0.368 | 0.207 |
| GS2SLS-op | 0.071 | 0.182 | 1.185 | 0.998 | | 0.284 | 0.357 | 1.671 | 0.928 |
| CGS2SLS-max | 0.022 | 0.049 | 0.210 | 0.985 | | 0.407 | 0.153 | 0.605 | 0.598 |
| CGS2SLS-op | −0.034 | 0.190 | 2.795 | 1.000 | | 0.374 | 0.394 | 2.175 | 0.913 |
| |
| GS2SLS-min | 0.253 | 0.313 | 1.991 | 0.996 | | 0.021 | 0.273 | 1.243 | 1.000 |
| GS2SLS-max | 0.327 | 0.059 | 0.237 | 0.412 | | −0.058 | 0.146 | 0.568 | 0.995 |
| GS2SLS-op | 0.257 | 0.253 | 1.611 | 0.983 | | 0.019 | 0.290 | 1.374 | 1.000 |
| CGS2SLS-max | 0.055 | 0.127 | 1.086 | 0.766 | | 0.020 | 0.229 | 1.014 | 0.981 |
| CGS2SLS-op | −0.159 | 0.472 | 6.950 | 0.972 | | 0.144 | 0.557 | 5.132 | 1.000 |
| GS2SLS-min | 0.197 | 0.280 | 1.646 | 0.997 | | 0.002 | 0.278 | 1.253 | 1.000 |
| GS2SLS-max | 0.268 | 0.055 | 0.213 | 0.444 | | 0.214 | 0.138 | 0.527 | 0.959 |
| GS2SLS-op | 0.217 | 0.232 | 1.415 | 0.991 | | 0.166 | 0.316 | 1.515 | 1.000 |
| CGS2SLS-max | 0.087 | 0.083 | 0.421 | 0.826 | | 0.197 | 0.192 | 0.802 | 0.941 |
| CGS2SLS-op | −0.047 | 0.309 | 4.120 | 0.987 | | 0.282 | 0.400 | 2.706 | 1.000 |
| GS2SLS-min | 0.222 | 0.262 | 1.671 | 0.994 | | 0.013 | 0.239 | 1.165 | 0.995 |
| GS2SLS-max | 0.217 | 0.030 | 0.118 | 0.156 | | 0.488 | 0.080 | 0.322 | 0.334 |
| GS2SLS-op | 0.216 | 0.190 | 1.288 | 0.958 | | 0.148 | 0.246 | 1.216 | 0.968 |
| CGS2SLS-max | 0.140 | 0.043 | 0.185 | 0.669 | | 0.310 | 0.129 | 0.527 | 0.753 |
| CGS2SLS-op | 0.077 | 0.185 | 2.591 | 0.966 | | 0.249 | 0.327 | 1.756 | 0.947 |
Table 3.
Estimation of Model 1 with and .
Table 3.
Estimation of Model 1 with and .
| | | | |
---|
| | MB | MAD | DQ | CR | MB | MAD | DQ | CR |
---|
| |
| GS2SLS-min | 0.124 | 0.348 | 2.079 | 1.000 | | 0.000 | 0.383 | 1.736 | 1.000 |
| GS2SLS-max | 0.245 | 0.040 | 0.147 | 0.168 | | −0.116 | 0.094 | 0.360 | 0.958 |
| GS2SLS-op | 0.158 | 0.210 | 1.203 | 0.995 | | −0.009 | 0.307 | 1.431 | 1.000 |
| CGS2SLS-max | −0.023 | 0.056 | 0.310 | 0.866 | | 0.046 | 0.149 | 0.603 | 0.904 |
| CGS2SLS-op | −0.326 | 0.433 | 7.033 | 0.993 | | 0.265 | 0.425 | 4.096 | 1.000 |
| GS2SLS-min | 0.145 | 0.322 | 1.972 | 1.000 | | 0.024 | 0.364 | 1.766 | 1.000 |
| GS2SLS-max | 0.156 | 0.031 | 0.117 | 0.367 | | 0.306 | 0.080 | 0.302 | 0.587 |
| GS2SLS-op | 0.117 | 0.166 | 0.984 | 1.000 | | 0.301 | 0.276 | 1.301 | 0.998 |
| CGS2SLS-max | 0.014 | 0.035 | 0.138 | 0.978 | | 0.299 | 0.135 | 0.502 | 0.587 |
| CGS2SLS-op | −0.128 | 0.269 | 4.360 | 1.000 | | 0.364 | 0.360 | 1.961 | 0.999 |
| GS2SLS-min | 0.143 | 0.271 | 1.772 | 0.999 | | 0.016 | 0.295 | 1.569 | 0.995 |
| GS2SLS-max | 0.067 | 0.016 | 0.061 | 0.514 | | 0.757 | 0.041 | 0.155 | 0.000 |
| GS2SLS-op | 0.089 | 0.183 | 1.014 | 0.998 | | 0.348 | 0.274 | 1.361 | 0.898 |
| CGS2SLS-max | 0.038 | 0.019 | 0.076 | 0.934 | | 0.558 | 0.088 | 0.342 | 0.043 |
| CGS2SLS-op | −0.011 | 0.163 | 1.762 | 1.000 | | 0.423 | 0.284 | 1.513 | 0.850 |
| |
| GS2SLS-min | 0.241 | 0.333 | 2.121 | 0.996 | | 0.009 | 0.382 | 1.682 | 1.000 |
| GS2SLS-max | 0.338 | 0.029 | 0.111 | 0.001 | | −0.111 | 0.098 | 0.370 | 0.948 |
| GS2SLS-op | 0.248 | 0.220 | 1.452 | 0.978 | | 0.015 | 0.331 | 1.472 | 1.000 |
| CGS2SLS-max | 0.057 | 0.079 | 0.723 | 0.634 | | 0.015 | 0.188 | 0.860 | 0.855 |
| CGS2SLS-op | −0.241 | 0.530 | 9.160 | 0.936 | | 0.218 | 0.572 | 5.418 | 1.000 |
| GS2SLS-min | 0.241 | 0.266 | 1.641 | 0.996 | | 0.030 | 0.315 | 1.491 | 1.000 |
| GS2SLS-max | 0.265 | 0.025 | 0.094 | 0.002 | | 0.308 | 0.079 | 0.311 | 0.552 |
| GS2SLS-op | 0.230 | 0.163 | 0.956 | 0.971 | | 0.274 | 0.284 | 1.231 | 1.000 |
| CGS2SLS-max | 0.106 | 0.038 | 0.179 | 0.575 | | 0.302 | 0.140 | 0.551 | 0.572 |
| CGS2SLS-op | −0.077 | 0.292 | 4.467 | 0.984 | | 0.344 | 0.358 | 2.332 | 0.999 |
| GS2SLS-min | 0.218 | 0.263 | 1.765 | 0.994 | | 0.075 | 0.294 | 1.820 | 0.995 |
| GS2SLS-max | 0.184 | 0.012 | 0.046 | 0.000 | | 0.754 | 0.037 | 0.138 | 0.000 |
| GS2SLS-op | 0.204 | 0.161 | 0.961 | 0.963 | | 0.377 | 0.256 | 1.220 | 0.887 |
| CGS2SLS-max | 0.142 | 0.015 | 0.058 | 0.032 | | 0.580 | 0.084 | 0.319 | 0.031 |
| CGS2SLS-op | 0.111 | 0.151 | 2.019 | 0.950 | | 0.421 | 0.287 | 1.530 | 0.836 |
Table 4.
Estimation of Model 1 with and .
Table 4.
Estimation of Model 1 with and .
| | | | |
---|
| | MB | MAD | DQ | CR | MB | MAD | DQ | CR |
---|
| |
| GS2SLS-min | 0.032 | 0.154 | 0.801 | 0.999 | | −0.016 | 0.131 | 0.526 | 1.000 |
| GS2SLS-max | 0.214 | 0.037 | 0.144 | 0.257 | | −0.068 | 0.076 | 0.274 | 0.984 |
| GS2SLS-op | 0.126 | 0.211 | 1.258 | 0.999 | | −0.004 | 0.296 | 1.475 | 1.000 |
| CGS2SLS-max | 0.007 | 0.044 | 0.176 | 0.970 | | 0.009 | 0.093 | 0.362 | 0.979 |
| CGS2SLS-op | −0.172 | 0.301 | 3.807 | 0.999 | | 0.209 | 0.388 | 2.928 | 1.000 |
| GS2SLS-min | 0.045 | 0.150 | 0.834 | 0.999 | | −0.015 | 0.136 | 0.553 | 1.000 |
| GS2SLS-max | 0.165 | 0.031 | 0.121 | 0.290 | | 0.199 | 0.067 | 0.260 | 0.792 |
| GS2SLS-op | 0.097 | 0.221 | 1.402 | 1.000 | | 0.110 | 0.301 | 1.433 | 1.000 |
| CGS2SLS-max | 0.029 | 0.035 | 0.139 | 0.967 | | 0.112 | 0.086 | 0.328 | 0.922 |
| CGS2SLS-op | −0.113 | 0.258 | 3.874 | 1.000 | | 0.248 | 0.338 | 2.203 | 1.000 |
| GS2SLS-min | 0.053 | 0.147 | 0.975 | 1.000 | | −0.003 | 0.136 | 0.574 | 0.998 |
| GS2SLS-max | 0.114 | 0.019 | 0.073 | 0.144 | | 0.503 | 0.044 | 0.167 | 0.003 |
| GS2SLS-op | 0.107 | 0.182 | 1.080 | 0.996 | | 0.106 | 0.220 | 0.980 | 0.986 |
| CGS2SLS-max | 0.060 | 0.026 | 0.103 | 0.861 | | 0.217 | 0.075 | 0.273 | 0.643 |
| CGS2SLS-op | −0.046 | 0.216 | 2.924 | 1.000 | | 0.280 | 0.364 | 2.083 | 0.957 |
| |
| GS2SLS-min | 0.072 | 0.189 | 1.255 | 0.996 | | 0.003 | 0.131 | 0.525 | 1.000 |
| GS2SLS-max | 0.316 | 0.030 | 0.115 | 0.006 | | −0.054 | 0.073 | 0.287 | 0.983 |
| GS2SLS-op | 0.211 | 0.238 | 1.563 | 0.986 | | 0.020 | 0.276 | 1.241 | 1.000 |
| CGS2SLS-max | 0.079 | 0.054 | 0.277 | 0.718 | | 0.014 | 0.108 | 0.431 | 0.957 |
| CGS2SLS-op | −0.137 | 0.382 | 5.654 | 0.967 | | 0.205 | 0.453 | 3.517 | 1.000 |
| GS2SLS-min | 0.097 | 0.173 | 1.275 | 0.993 | | −0.006 | 0.140 | 0.595 | 1.000 |
| GS2SLS-max | 0.264 | 0.025 | 0.101 | 0.006 | | 0.200 | 0.068 | 0.263 | 0.776 |
| GS2SLS-op | 0.191 | 0.219 | 1.377 | 0.991 | | 0.116 | 0.258 | 1.184 | 0.999 |
| CGS2SLS-max | 0.110 | 0.034 | 0.150 | 0.570 | | 0.108 | 0.092 | 0.361 | 0.910 |
| CGS2SLS-op | −0.028 | 0.291 | 4.901 | 0.985 | | 0.206 | 0.330 | 2.157 | 1.000 |
| GS2SLS-min | 0.098 | 0.156 | 1.341 | 0.989 | | −0.005 | 0.148 | 0.638 | 0.999 |
| GS2SLS-max | 0.210 | 0.017 | 0.064 | 0.000 | | 0.482 | 0.044 | 0.167 | 0.004 |
| GS2SLS-op | 0.150 | 0.180 | 1.114 | 0.977 | | 0.120 | 0.191 | 0.833 | 0.996 |
| CGS2SLS-max | 0.138 | 0.022 | 0.088 | 0.183 | | 0.195 | 0.078 | 0.300 | 0.702 |
| CGS2SLS-op | 0.039 | 0.213 | 3.970 | 0.974 | | 0.205 | 0.307 | 1.865 | 0.969 |
Table 5,
Table 6,
Table 7 and
Table 8 report the summary statistics of the estimators for Model 2. Among GS2SLS-min, GS2SLS-max and GS2SLS-op, in most cases, the GS2SLS-max has the largest median bias, the GS2SLS-op of
has the smallest median bias, and the GS2SLS-op of
has the intermediate median bias. The GS2SLS-max has the smallest MAD and DQ, and the GS2SLS-op has the intermediate MAD and DQ. The CR of GS2SLS-op is closest to the nominal level, while the CR of GS2SLS-max is significantly lower than the nominal level in many cases. The performance of CGS2SLS-max for Model 2 is similar to that for Model 1. Compared with the GS2SLS-op, the CGS2SLS-op has much larger MAD and DQ in most cases, similar CR, and has smaller median bias in more than half of the cases when
but larger median bias in most cases when
.
Table 5.
Estimation of Model 2 with and .
Table 5.
Estimation of Model 2 with and .
| | | | |
---|
| | MB | MAD | DQ | CR | MB | MAD | DQ | CR |
---|
| |
| GS2SLS-min | 0.246 | 0.611 | 3.434 | 1.000 | | 0.046 | 0.774 | 4.234 | 1.000 |
| GS2SLS-max | 0.250 | 0.082 | 0.317 | 0.797 | | −0.079 | 0.177 | 0.661 | 0.994 |
| GS2SLS-op | 0.198 | 0.326 | 2.034 | 0.999 | | 0.059 | 0.472 | 2.364 | 1.000 |
| CGS2SLS-max | −0.055 | 0.132 | 0.730 | 0.849 | | 0.107 | 0.299 | 1.340 | 0.967 |
| CGS2SLS-op | −0.391 | 0.602 | 10.313 | 0.992 | | 0.360 | 0.766 | 7.293 | 1.000 |
| GS2SLS-min | 0.160 | 0.354 | 2.317 | 1.000 | | 0.408 | 0.796 | 4.989 | 1.000 |
| GS2SLS-max | 0.155 | 0.065 | 0.250 | 0.910 | | 0.338 | 0.146 | 0.576 | 0.943 |
| GS2SLS-op | 0.128 | 0.216 | 1.228 | 1.000 | | 0.399 | 0.393 | 1.964 | 1.000 |
| CGS2SLS-max | −0.008 | 0.078 | 0.354 | 0.960 | | 0.418 | 0.233 | 0.949 | 0.845 |
| CGS2SLS-op | −0.204 | 0.361 | 6.722 | 0.998 | | 0.572 | 0.531 | 4.621 | 1.000 |
| GS2SLS-min | 0.050 | 0.210 | 1.433 | 1.000 | | 0.741 | 0.523 | 3.243 | 0.963 |
| GS2SLS-max | 0.063 | 0.032 | 0.133 | 0.968 | | 0.793 | 0.080 | 0.316 | 0.038 |
| GS2SLS-op | 0.051 | 0.121 | 0.699 | 1.000 | | 0.763 | 0.238 | 1.278 | 0.775 |
| CGS2SLS-max | 0.030 | 0.036 | 0.155 | 0.993 | | 0.721 | 0.147 | 0.609 | 0.193 |
| CGS2SLS-op | −0.006 | 0.148 | 1.445 | 1.000 | | 0.714 | 0.286 | 2.544 | 0.800 |
| |
| GS2SLS-min | 0.289 | 0.363 | 2.264 | 0.994 | | 0.059 | 0.829 | 5.260 | 1.000 |
| GS2SLS-max | 0.342 | 0.061 | 0.238 | 0.367 | | −0.091 | 0.180 | 0.712 | 0.991 |
| GS2SLS-op | 0.267 | 0.274 | 1.645 | 0.985 | | 0.071 | 0.523 | 3.235 | 1.000 |
| CGS2SLS-max | 0.063 | 0.160 | 1.484 | 0.698 | | 0.023 | 0.356 | 1.665 | 0.958 |
| CGS2SLS-op | −0.167 | 0.675 | 9.228 | 0.966 | | 0.254 | 0.807 | 7.433 | 1.000 |
| GS2SLS-min | 0.277 | 0.342 | 2.408 | 0.997 | | 0.303 | 0.694 | 4.551 | 1.000 |
| GS2SLS-max | 0.264 | 0.052 | 0.203 | 0.449 | | 0.330 | 0.151 | 0.585 | 0.934 |
| GS2SLS-op | 0.226 | 0.196 | 1.324 | 0.986 | | 0.356 | 0.394 | 2.001 | 0.999 |
| CGS2SLS-max | 0.100 | 0.073 | 0.372 | 0.844 | | 0.356 | 0.242 | 1.027 | 0.853 |
| CGS2SLS-op | −0.098 | 0.362 | 7.297 | 0.989 | | 0.475 | 0.584 | 5.336 | 1.000 |
| GS2SLS-min | 0.182 | 0.181 | 1.172 | 0.992 | | 0.689 | 0.470 | 2.978 | 0.962 |
| GS2SLS-max | 0.184 | 0.027 | 0.105 | 0.240 | | 0.777 | 0.073 | 0.285 | 0.024 |
| GS2SLS-op | 0.179 | 0.109 | 0.682 | 0.969 | | 0.762 | 0.220 | 1.184 | 0.779 |
| CGS2SLS-max | 0.144 | 0.030 | 0.130 | 0.568 | | 0.710 | 0.137 | 0.559 | 0.183 |
| CGS2SLS-op | 0.099 | 0.146 | 1.737 | 0.972 | | 0.700 | 0.299 | 2.223 | 0.812 |
Table 6.
Estimation of Model 2 with and .
Table 6.
Estimation of Model 2 with and .
| | | | |
---|
| | MB | MAD | DQ | CR | MB | MAD | DQ | CR |
---|
| |
| GS2SLS-min | 0.199 | 0.439 | 2.673 | 1.000 | | −0.001 | 0.482 | 2.470 | 1.000 |
| GS2SLS-max | 0.230 | 0.076 | 0.295 | 0.804 | | −0.064 | 0.151 | 0.573 | 0.996 |
| GS2SLS-op | 0.190 | 0.290 | 1.703 | 0.999 | | 0.017 | 0.364 | 1.702 | 1.000 |
| CGS2SLS-max | −0.039 | 0.115 | 0.562 | 0.892 | | 0.069 | 0.209 | 0.839 | 0.983 |
| CGS2SLS-op | −0.285 | 0.461 | 6.720 | 0.994 | | 0.206 | 0.479 | 3.688 | 1.000 |
| GS2SLS-min | 0.002 | 0.385 | 2.408 | 1.000 | | 0.198 | 0.669 | 4.025 | 1.000 |
| GS2SLS-max | 0.137 | 0.068 | 0.266 | 0.907 | | 0.217 | 0.135 | 0.531 | 0.963 |
| GS2SLS-op | 0.058 | 0.217 | 1.323 | 1.000 | | 0.198 | 0.337 | 1.710 | 1.000 |
| CGS2SLS-max | −0.003 | 0.076 | 0.335 | 0.964 | | 0.177 | 0.178 | 0.709 | 0.942 |
| CGS2SLS-op | −0.173 | 0.302 | 4.685 | 0.999 | | 0.263 | 0.364 | 2.475 | 0.999 |
| GS2SLS-min | 0.058 | 0.364 | 2.209 | 0.999 | | 0.260 | 0.504 | 3.843 | 0.992 |
| GS2SLS-max | 0.102 | 0.042 | 0.170 | 0.887 | | 0.522 | 0.085 | 0.333 | 0.311 |
| GS2SLS-op | 0.103 | 0.231 | 1.521 | 0.999 | | 0.369 | 0.282 | 2.034 | 0.958 |
| CGS2SLS-max | 0.039 | 0.053 | 0.220 | 0.982 | | 0.331 | 0.134 | 0.528 | 0.728 |
| CGS2SLS-op | −0.023 | 0.197 | 2.793 | 1.000 | | 0.339 | 0.268 | 1.738 | 0.955 |
| |
| GS2SLS-min | 0.290 | 0.364 | 2.446 | 0.998 | | 0.053 | 0.601 | 3.690 | 1.000 |
| GS2SLS-max | 0.319 | 0.068 | 0.265 | 0.454 | | −0.064 | 0.162 | 0.632 | 0.989 |
| GS2SLS-op | 0.252 | 0.278 | 1.901 | 0.991 | | 0.056 | 0.443 | 2.357 | 1.000 |
| CGS2SLS-max | 0.063 | 0.120 | 1.088 | 0.777 | | 0.016 | 0.251 | 1.169 | 0.966 |
| CGS2SLS-op | −0.149 | 0.495 | 7.195 | 0.970 | | 0.220 | 0.642 | 5.969 | 1.000 |
| GS2SLS-min | 0.244 | 0.309 | 1.949 | 0.997 | | 0.329 | 0.728 | 4.353 | 1.000 |
| GS2SLS-max | 0.268 | 0.051 | 0.203 | 0.440 | | 0.233 | 0.129 | 0.507 | 0.961 |
| GS2SLS-op | 0.243 | 0.214 | 1.317 | 0.986 | | 0.222 | 0.366 | 1.924 | 1.000 |
| CGS2SLS-max | 0.091 | 0.082 | 0.445 | 0.825 | | 0.213 | 0.182 | 0.780 | 0.944 |
| CGS2SLS-op | −0.052 | 0.321 | 5.613 | 0.988 | | 0.259 | 0.395 | 2.986 | 1.000 |
| GS2SLS-min | 0.163 | 0.261 | 1.781 | 0.984 | | 0.088 | 0.387 | 2.616 | 0.991 |
| GS2SLS-max | 0.196 | 0.038 | 0.150 | 0.307 | | 0.487 | 0.086 | 0.330 | 0.371 |
| GS2SLS-op | 0.149 | 0.184 | 1.207 | 0.970 | | 0.290 | 0.247 | 1.503 | 0.965 |
| CGS2SLS-max | 0.117 | 0.050 | 0.220 | 0.774 | | 0.291 | 0.141 | 0.556 | 0.787 |
| CGS2SLS-op | 0.049 | 0.195 | 3.284 | 0.978 | | 0.279 | 0.270 | 1.742 | 0.970 |
Table 7.
Estimation of Model 2 with and .
Table 7.
Estimation of Model 2 with and .
| | | | |
---|
| | MB | MAD | DQ | CR | MB | MAD | DQ | CR |
---|
| |
| GS2SLS-min | 0.193 | 0.416 | 2.823 | 1.000 | | 0.070 | 0.704 | 4.351 | 1.000 |
| GS2SLS-max | 0.242 | 0.037 | 0.145 | 0.160 | | −0.100 | 0.093 | 0.351 | 0.967 |
| GS2SLS-op | 0.169 | 0.218 | 1.220 | 0.998 | | 0.014 | 0.347 | 1.673 | 1.000 |
| CGS2SLS-max | −0.017 | 0.056 | 0.273 | 0.893 | | 0.040 | 0.144 | 0.580 | 0.919 |
| CGS2SLS-op | −0.348 | 0.468 | 11.234 | 0.992 | | 0.321 | 0.517 | 5.647 | 1.000 |
| GS2SLS-min | 0.130 | 0.354 | 2.272 | 1.000 | | 0.259 | 0.652 | 3.957 | 1.000 |
| GS2SLS-max | 0.154 | 0.031 | 0.120 | 0.389 | | 0.316 | 0.078 | 0.297 | 0.557 |
| GS2SLS-op | 0.104 | 0.171 | 0.980 | 1.000 | | 0.349 | 0.288 | 1.408 | 0.999 |
| CGS2SLS-max | 0.015 | 0.033 | 0.136 | 0.977 | | 0.303 | 0.132 | 0.502 | 0.581 |
| CGS2SLS-op | −0.153 | 0.293 | 3.525 | 1.000 | | 0.405 | 0.384 | 2.560 | 0.999 |
| GS2SLS-min | 0.100 | 0.263 | 1.769 | 1.000 | | 0.412 | 0.541 | 4.162 | 0.986 |
| GS2SLS-max | 0.070 | 0.015 | 0.059 | 0.472 | | 0.748 | 0.041 | 0.159 | 0.000 |
| GS2SLS-op | 0.086 | 0.155 | 0.995 | 1.000 | | 0.546 | 0.263 | 1.447 | 0.860 |
| CGS2SLS-max | 0.041 | 0.020 | 0.074 | 0.925 | | 0.538 | 0.083 | 0.338 | 0.051 |
| CGS2SLS-op | −0.008 | 0.169 | 2.335 | 1.000 | | 0.490 | 0.247 | 1.964 | 0.863 |
| |
| GS2SLS-min | 0.322 | 0.398 | 2.574 | 0.997 | | −0.005 | 0.723 | 3.984 | 1.000 |
| GS2SLS-max | 0.338 | 0.029 | 0.110 | 0.002 | | −0.115 | 0.099 | 0.382 | 0.940 |
| GS2SLS-op | 0.271 | 0.243 | 1.508 | 0.976 | | −0.008 | 0.407 | 2.041 | 1.000 |
| CGS2SLS-max | 0.060 | 0.082 | 0.657 | 0.634 | | 0.014 | 0.189 | 0.862 | 0.855 |
| CGS2SLS-op | −0.300 | 0.587 | 11.233 | 0.939 | | 0.252 | 0.675 | 6.906 | 1.000 |
| GS2SLS-min | 0.251 | 0.281 | 1.692 | 0.997 | | 0.291 | 0.661 | 3.651 | 1.000 |
| GS2SLS-max | 0.263 | 0.025 | 0.096 | 0.004 | | 0.306 | 0.082 | 0.307 | 0.553 |
| GS2SLS-op | 0.239 | 0.172 | 1.055 | 0.971 | | 0.337 | 0.295 | 1.385 | 0.998 |
| CGS2SLS-max | 0.104 | 0.038 | 0.181 | 0.576 | | 0.302 | 0.140 | 0.554 | 0.580 |
| CGS2SLS-op | −0.086 | 0.316 | 6.578 | 0.984 | | 0.375 | 0.373 | 2.944 | 0.999 |
| GS2SLS-min | 0.252 | 0.236 | 1.595 | 0.991 | | 0.240 | 0.400 | 3.186 | 0.988 |
| GS2SLS-max | 0.184 | 0.012 | 0.046 | 0.000 | | 0.754 | 0.037 | 0.142 | 0.000 |
| GS2SLS-op | 0.212 | 0.134 | 0.943 | 0.961 | | 0.534 | 0.256 | 1.511 | 0.831 |
| CGS2SLS-max | 0.142 | 0.015 | 0.059 | 0.035 | | 0.584 | 0.082 | 0.320 | 0.026 |
| CGS2SLS-op | 0.098 | 0.156 | 2.048 | 0.958 | | 0.503 | 0.263 | 1.808 | 0.823 |
Table 8.
Estimation of Model 2 with and .
Table 8.
Estimation of Model 2 with and .
| | | | |
---|
| | MB | MAD | DQ | CR | MB | MAD | DQ | CR |
---|
| |
| GS2SLS-min | 0.138 | 0.318 | 1.886 | 1.000 | | 0.019 | 0.342 | 1.584 | 1.000 |
| GS2SLS-max | 0.215 | 0.038 | 0.147 | 0.246 | | −0.073 | 0.075 | 0.282 | 0.983 |
| GS2SLS-op | 0.123 | 0.241 | 1.339 | 1.000 | | 0.000 | 0.295 | 1.242 | 1.000 |
| CGS2SLS-max | 0.008 | 0.044 | 0.182 | 0.956 | | 0.002 | 0.099 | 0.368 | 0.982 |
| CGS2SLS-op | −0.261 | 0.416 | 7.521 | 0.998 | | 0.128 | 0.393 | 2.719 | 1.000 |
| GS2SLS-min | 0.143 | 0.279 | 1.654 | 1.000 | | 0.037 | 0.322 | 1.661 | 1.000 |
| GS2SLS-max | 0.164 | 0.032 | 0.121 | 0.286 | | 0.201 | 0.072 | 0.269 | 0.784 |
| GS2SLS-op | 0.094 | 0.223 | 1.218 | 0.999 | | 0.120 | 0.273 | 1.162 | 1.000 |
| CGS2SLS-max | 0.028 | 0.035 | 0.138 | 0.970 | | 0.108 | 0.091 | 0.343 | 0.912 |
| CGS2SLS-op | −0.181 | 0.357 | 7.336 | 1.000 | | 0.129 | 0.384 | 2.341 | 1.000 |
| GS2SLS-min | 0.193 | 0.206 | 1.298 | 1.000 | | 0.056 | 0.316 | 1.821 | 0.997 |
| GS2SLS-max | 0.118 | 0.019 | 0.075 | 0.117 | | 0.476 | 0.045 | 0.182 | 0.005 |
| GS2SLS-op | 0.117 | 0.185 | 1.127 | 0.997 | | 0.236 | 0.196 | 1.018 | 0.981 |
| CGS2SLS-max | 0.059 | 0.027 | 0.106 | 0.854 | | 0.200 | 0.071 | 0.285 | 0.703 |
| CGS2SLS-op | −0.073 | 0.284 | 5.465 | 0.998 | | 0.069 | 0.275 | 1.452 | 0.994 |
| |
| GS2SLS-min | 0.236 | 0.269 | 1.706 | 0.992 | | 0.026 | 0.275 | 1.236 | 1.000 |
| GS2SLS-max | 0.319 | 0.030 | 0.113 | 0.008 | | −0.057 | 0.071 | 0.275 | 0.982 |
| GS2SLS-op | 0.224 | 0.220 | 1.365 | 0.988 | | 0.030 | 0.237 | 1.036 | 1.000 |
| CGS2SLS-max | 0.078 | 0.054 | 0.306 | 0.724 | | 0.009 | 0.102 | 0.402 | 0.967 |
| CGS2SLS-op | −0.160 | 0.389 | 6.310 | 0.962 | | 0.065 | 0.298 | 1.734 | 1.000 |
| GS2SLS-min | 0.255 | 0.245 | 1.559 | 0.993 | | 0.082 | 0.374 | 1.865 | 1.000 |
| GS2SLS-max | 0.267 | 0.026 | 0.098 | 0.005 | | 0.215 | 0.073 | 0.267 | 0.740 |
| GS2SLS-op | 0.202 | 0.216 | 1.233 | 0.987 | | 0.139 | 0.253 | 1.200 | 1.000 |
| CGS2SLS-max | 0.109 | 0.037 | 0.162 | 0.566 | | 0.136 | 0.097 | 0.385 | 0.882 |
| CGS2SLS-op | −0.112 | 0.372 | 7.572 | 0.988 | | 0.091 | 0.406 | 2.593 | 1.000 |
| GS2SLS-min | 0.250 | 0.200 | 1.235 | 0.986 | | 0.060 | 0.271 | 1.456 | 0.996 |
| GS2SLS-max | 0.211 | 0.015 | 0.059 | 0.000 | | 0.492 | 0.042 | 0.158 | 0.001 |
| GS2SLS-op | 0.186 | 0.160 | 0.987 | 0.973 | | 0.247 | 0.172 | 0.857 | 0.978 |
| CGS2SLS-max | 0.142 | 0.022 | 0.089 | 0.164 | | 0.211 | 0.076 | 0.299 | 0.667 |
| CGS2SLS-op | 0.022 | 0.249 | 4.416 | 0.985 | | 0.078 | 0.266 | 1.447 | 0.993 |
From the Monte Carlo results of both models, we can see that the proposed CGS2SLS estimator can effectively reduce the many instrument bias, and the estimators derived by choosing the number of instruments to minimize their respective approximated MSEs, GS2SLS-op and CGS2SLS-op, have coverage rates closer to the nominal level than the estimators using very few or many instruments, i.e., GS2SLS-op and CGS2SLS-op can make inference more reliable. Between GS2SLS-op and CGS2SLS-op, no one is always better than the other in terms of central tendency or coverage rate, but the GS2SLS-op has much smaller dispersion in most cases.
The summary statistics of the estimated
p and
q are presented in
Table 9 and
Table 10. Consistent with [
3], in most cases for both models, only the first spatial lag (
) is used. For Model 1, in most cases,
is 1 or 2 with
, and is larger with
but is smaller than the maximum number of instruments
. For Model 2,
tends to be larger, which might be due to the fact that the variables in
of Model 2 are equally important but the importance of the variables in
of Model 1 is in decreasing order. For both models,
tends to be larger with a larger
.
Table 9.
The Distributions of and in Model 1.
Table 9.
The Distributions of and in Model 1.
| GS2SLS | | CGS2SLS |
---|
| | | | | | |
---|
MO | LQ | ME | UQ | | MO | LQ | ME | UQ | | MO | LQ | ME | UQ | | MO | LQ | ME | UQ |
---|
| |
, , | 1 | 1 | 1 | 4 | | 1 | 1 | 1 | 5 | | 1 | 1 | 2 | 4 | | 1 | 1 | 2 | 5 |
| 1 | 1 | 1 | 4 | | 1 | 1 | 2 | 5 | | 1 | 1 | 1 | 4 | | 1 | 1 | 2 | 5 |
| 1 | 1 | 1 | 4 | | 1 | 1 | 1 | 5 | | 1 | 1 | 1 | 4 | | 1 | 1 | 1 | 5 |
, , | 1 | 1 | 1 | 4 | | 1 | 1 | 1 | 4 | | 4 | 1 | 3 | 4 | | 5 | 1 | 3 | 4 |
| 1 | 1 | 1 | 4 | | 1 | 1 | 1 | 5 | | 1 | 1 | 1 | 4 | | 1 | 1 | 2 | 5 |
| 1 | 1 | 1 | 4 | | 1 | 1 | 1 | 5 | | 1 | 1 | 1 | 4 | | 1 | 1 | 1 | 5 |
, , | 1 | 1 | 1 | 4 | | 1 | 1 | 1 | 5 | | 1 | 1 | 2 | 4 | | 1 | 1 | 3 | 5 |
| 1 | 1 | 1 | 4 | | 1 | 1 | 2 | 4 | | 1 | 1 | 2 | 4 | | 2 | 1 | 2 | 4 |
| 1 | 1 | 1 | 3 | | 1 | 1 | 1 | 3 | | 1 | 1 | 1 | 3 | | 1 | 1 | 2 | 3 |
, , | 1 | 1 | 1 | 4 | | 1 | 1 | 1 | 4 | | 1 | 1 | 2 | 4 | | 5 | 1 | 4 | 4 |
| 1 | 1 | 1 | 3 | | 1 | 1 | 2 | 4 | | 1 | 1 | 2 | 3 | | 1 | 1 | 2 | 4 |
| 1 | 1 | 1 | 2 | | 1 | 1 | 1 | 2 | | 1 | 1 | 1 | 2 | | 1 | 1 | 2 | 2 |
| |
, , | 1 | 1 | 2 | 9 | | 1 | 1 | 3 | 9 | | 1 | 1 | 2 | 9 | | 1 | 1 | 3 | 9 |
| 1 | 1 | 2 | 9 | | 2 | 1 | 3 | 9 | | 1 | 1 | 1 | 9 | | 2 | 1 | 3 | 9 |
| 1 | 1 | 1 | 3 | | 1 | 1 | 2 | 3 | | 1 | 1 | 1 | 3 | | 1 | 1 | 2 | 3 |
, , | 1 | 1 | 1 | 6 | | 1 | 1 | 2 | 7 | | 1 | 1 | 2 | 6 | | 1 | 1 | 3 | 7 |
| 1 | 1 | 1 | 7 | | 1 | 1 | 2 | 8 | | 1 | 1 | 1 | 7 | | 1 | 1 | 2 | 8 |
| 1 | 1 | 1 | 3 | | 1 | 1 | 2 | 3 | | 1 | 1 | 1 | 3 | | 1 | 1 | 2 | 3 |
, , | 1 | 1 | 1 | 7 | | 3 | 2 | 4 | 8 | | 1 | 1 | 2 | 7 | | 3 | 2 | 4 | 8 |
| 1 | 1 | 1 | 2 | | 3 | 2 | 3 | 5 | | 1 | 1 | 1 | 2 | | 3 | 2 | 4 | 5 |
| 1 | 1 | 1 | 1 | | 2 | 2 | 2 | 3 | | 1 | 1 | 1 | 1 | | 3 | 2 | 4 | 3 |
, , | 1 | 1 | 1 | 3 | | 3 | 1 | 3 | 5 | | 1 | 1 | 1 | 3 | | 3 | 2 | 4 | 5 |
| 1 | 1 | 1 | 2 | | 3 | 2 | 3 | 4 | | 1 | 1 | 1 | 2 | | 3 | 2 | 4 | 4 |
| 1 | 1 | 1 | 1 | | 2 | 2 | 2 | 3 | | 1 | 1 | 1 | 1 | | 3 | 2 | 3 | 3 |
Table 10.
The Distributions of and in Model 2.
Table 10.
The Distributions of and in Model 2.
| GS2SLS | | CGS2SLS |
---|
| | | | | | |
---|
MO | LQ | ME | UQ | | MO | LQ | ME | UQ | | MO | LQ | ME | UQ | | MO | LQ | ME | UQ |
---|
| |
, , | 1 | 1 | 1 | 4 | | 1 | 1 | 1 | 5 | | 1 | 1 | 2 | 4 | | 5 | 1 | 3 | 5 |
| 1 | 1 | 1 | 4 | | 1 | 1 | 2 | 5 | | 1 | 1 | 1 | 4 | | 1 | 1 | 2 | 5 |
| 1 | 1 | 1 | 4 | | 1 | 1 | 2 | 5 | | 1 | 1 | 1 | 4 | | 1 | 1 | 1 | 5 |
, , | 1 | 1 | 1 | 4 | | 1 | 1 | 1 | 5 | | 4 | 1 | 2 | 4 | | 5 | 1 | 3 | 5 |
| 1 | 1 | 1 | 4 | | 1 | 1 | 2 | 5 | | 1 | 1 | 1 | 4 | | 1 | 1 | 2 | 5 |
| 1 | 1 | 1 | 4 | | 1 | 1 | 1 | 5 | | 1 | 1 | 1 | 4 | | 1 | 1 | 1 | 5 |
, , | 1 | 1 | 1 | 4 | | 1 | 1 | 2 | 5 | | 1 | 1 | 2 | 4 | | 5 | 1 | 4 | 5 |
| 1 | 1 | 1 | 3 | | 1 | 1 | 3 | 5 | | 1 | 1 | 1 | 3 | | 5 | 1 | 4 | 5 |
| 1 | 1 | 1 | 2 | | 1 | 1 | 1 | 3 | | 1 | 1 | 1 | 2 | | 5 | 1 | 4 | 3 |
, , | 1 | 1 | 1 | 3 | | 1 | 1 | 2 | 5 | | 1 | 1 | 2 | 3 | | 5 | 2 | 5 | 5 |
| 1 | 1 | 1 | 3 | | 1 | 1 | 2 | 5 | | 1 | 1 | 1 | 3 | | 5 | 1 | 4 | 5 |
| 1 | 1 | 1 | 2 | | 1 | 1 | 1 | 3 | | 1 | 1 | 1 | 2 | | 5 | 1 | 4 | 3 |
| |
, , | 1 | 1 | 2 | 9 | | 1 | 1 | 4 | 10 | | 1 | 1 | 2 | 9 | | 1 | 1 | 4 | 10 |
| 1 | 1 | 2 | 9 | | 1 | 1 | 4 | 10 | | 1 | 1 | 1 | 9 | | 1 | 1 | 4 | 10 |
| 1 | 1 | 1 | 3 | | 1 | 1 | 2 | 4 | | 1 | 1 | 1 | 3 | | 1 | 1 | 4 | 4 |
, , | 1 | 1 | 1 | 6 | | 1 | 1 | 2 | 9 | | 1 | 1 | 2 | 6 | | 10 | 1 | 5 | 9 |
| 1 | 1 | 1 | 8 | | 1 | 1 | 3 | 10 | | 1 | 1 | 1 | 8 | | 1 | 1 | 4 | 10 |
| 1 | 1 | 1 | 3.5 | | 1 | 1 | 1 | 4 | | 1 | 1 | 1 | 3.5 | | 1 | 1 | 2 | 4 |
, , | 1 | 1 | 1 | 5 | | 10 | 3 | 9 | 10 | | 1 | 1 | 1 | 5 | | 10 | 8 | 10 | 10 |
| 1 | 1 | 1 | 1 | | 10 | 2 | 6 | 10 | | 1 | 1 | 1 | 1 | | 10 | 8 | 10 | 10 |
| 1 | 1 | 1 | 1 | | 3 | 1 | 2 | 5 | | 1 | 1 | 1 | 1 | | 10 | 8 | 10 | 5 |
, , | 1 | 1 | 1 | 1 | | 10 | 1 | 7 | 10 | | 1 | 1 | 1 | 1 | | 10 | 8 | 10 | 10 |
| 1 | 1 | 1 | 1 | | 4 | 2 | 4 | 10 | | 1 | 1 | 1 | 1 | | 10 | 8 | 10 | 10 |
| 1 | 1 | 1 | 1 | | 3 | 1 | 2 | 4 | | 1 | 1 | 1 | 1 | | 10 | 8 | 10 | 4 |