Discussion Paper: 2011/13
The performance of tests on endogeneity of subsets
of explanatory variables scanned by simulation
Jan F. Kiviet and Milan Pleus
www.ase.uva.nl/uva-econometrics
Amsterdam School of Economics
Department of Economics & Econometrics
Valckenierstraat 65-67
1018 XE AMSTERDAM
The Netherlands
The performance of tests on endogeneity of subsets
of explanatory variables scanned by simulation
Jan F. Kiviet
and
Milan Pleusy
30 December 2014
JEL-code: C01, C12, C15, C30
Keywords:
bootstrapping, classi…cation of explanatories, DWH orthogonality tests,
test implementation, test performance, simulation design
Abstract
Tests for classi…cation as endogenous or predetermined of arbitrary subsets of
regressors are formulated as signi…cance tests in auxiliary IV regressions and their
relationships with various more classic test procedures are examined and critically
compared with statements in the literature. Then simulation experiments are designed by solving the data generating process parameters from salient econometric
features, namely: degree of simultaneity and multicollinearity of regressors, and
individual and joint strength of external instrumental variables. Next, for various
test implementations, a wide class of relevant cases is scanned for ‡aws in performance regarding type I and II errors. Substantial size distortions occur, but these
can be cured remarkably well through bootstrapping, except when instruments
are weak. The power of the subset tests is such that they establish an essential
addition to the well-known classic full-set DWH tests in a data based classi…cation
of individual explanatory variables.
1. Introduction
In this study various test procedures are derived and examined for the classi…cation of
arbitrary subsets of explanatory variables as either endogenous or predetermined with
respect to a single adequately speci…ed structural equation. Correct classi…cation is
Emeritus Professor of Econometrics, Amsterdam School of Economics, University of Amsterdam,
Valckenierstraat 65, 1018 XE Amsterdam, The Netherlands (j.f.kiviet@uva.nl) and Visting Professor at
the Division of Economics, School of Humanities and Social Sciences, Nanyang Technological University,
14 Nanyang Drive, Singapore 637332 (jfkiviet@ntu.edu.sg).
y
Amsterdam School of Economics, University of Amsterdam, Valckenierstraat 65, 1018 XE Amsterdam, The Netherlands (m.pleus@uva.nl). Financial support from the Netherlands Organisation for
Scienti…c Research (NWO) grant "Statistical inference methods regarding e¤ectivity of endogous policy
measures" is gratefully acknowledged.
highly important because misclassi…cation leads to either ine¢ cient or inconsistent estimation. The derivations, which in essence are based on employing Hausman’s principle
of examining the discrepancy between two alternative estimators, formulate the various
tests as joint signi…cance tests of additional regressors in auxiliary IV regressions. Their
relationships are demonstrated with particular forms of classic tests such as DurbinWu-Hausman orthogonality tests, Revankar-Hartley covariance tests and Sargan-Hansen
overidenti…cation restriction tests. Various di¤erent and some under the null hypothesis
asymptotically equivalent implementations follow. The latter vary only regarding degrees of freedom adjustments and the type of disturbance variance estimator employed.
We run simulations over a wide class of relevant cases, to …nd out which versions have
best control over type I error probabilities and to get an idea of the power of these tests.
This should help to use these tests e¤ectively in practice when trying to avoid both evils
of inconsistency and ine¢ ciency. To that end a simulation approach is developed by
which relevant data generating processes (DGPs) are designed by deriving the values for
their parameters from chosen salient features of the system, namely: degree of simultaneity of individual explanatory variables, degree of multicollinearity between explanatory
variables, and individual and joint strength of employed external instrumental variables.
This allows scanning the relevant parameter space of wide model classes for ‡aws in
performance regarding type I and II errors of all implementations of the tests and their
bootstrapped versions. We …nd that testing orthogonality by standard methods is impeded for weakly identi…ed regressors. Like bootstrapped tests require resampling under
the null, we …nd here that testing for orthogonality by auxiliary regressions bene…ts from
estimating variances under the null, as in Lagrange multiplier tests, rather than under
the alternative, as in Wald-type tests. However, after proper size correction we …nd that
the Wald-type tests exhibit the best power properties.
Procedures for testing the orthogonality of all possibly endogenous regressors regarding the error term have been developed by Durbin (1954), Wu (1973), Revankar
and Hartley (1973), Revankar (1978) and Hausman (1978). Mutual relationships between these are discussed in Nakamura and Nakamura (1981) and Hausman and Taylor
(1981). This test problem has been put into a likelihood framework under normality by
Holly (1982) and Smith (1983). Most of the papers just mentioned, and in particular
Davidson and MacKinnon (1989, 1990), provide a range of implementations for these
tests that can easily be obtained from auxiliary regressions. Although this type of inference problem does address one of the basic fundaments of the econometric analysis
of observational data, relatively little evidence on the performance of the available tests
in …nite samples is available. Monte Carlo studies on the performance of some of the
implementations in static linear models can be found in Wu (1974), Meepagala (1992),
Chmelarova and Carter Hill (2010), Jeong and Yoon (2010), Hahn et al.(2011) and Doko
Tchatoka (2014), whereas such results for linear dynamic models are presented in Kiviet
(1985).
The more subtle problem of deriving a test for the orthogonality of subsets of the
regressors not involving all of the possibly endogenous regressors has also received substantial attention over the last three decades. Nevertheless, generally accepted rules for
best practice on how to approach this problem do not seem available yet, or are confusing as we shall see, and not yet supported by any simulation evidence. Self-evidently,
though, the situation where one is convinced of the endogeneity of a few of the regressors,
but wants to test some other regressors for orthogonality, is of high practical relevance.
2
If orthogonality is established, this permits to use these regressors as instrumental variables, which (if correct) improves the e¢ ciency and the identi…cation situation, because
it makes the analysis less dependent on the availability of external instruments. This
is important in particular when available external instruments are weak or of doubtful exogeneity status. Testing the orthogonality of subsets of the possibly endogenous
regressors was addressed …rst by Hwang (1980) and next by Spencer and Berk (1981,
1982), Wu (1983), Smith (1984, 1985), Hwang (1985) and Newey (1985), who all suggest
various test procedures, some of them asymptotically or even algebraically equivalent.
So do Pesaran and Smith (1990), who also provide theoretical arguments regarding an
ordering of the power of the various tests, although they are asymptotically equivalent
under the null and under local alternatives. Various of the possible sub-set test implementations are paraphrased in Ruud (1984, 2000), Davidson and MacKinnon (1993)
and in Baum et al. (2003), and occasionally their relationships with particular forms
of Sargan-Hansen (partial-)overidenti…cation test statistics are examined. As we shall
show, a few particular situations still call for further analysis and formal proofs, and
sometimes results from the studies mentioned above have to be corrected. As far as we
know, there are no published simulation results yet on the actual qualities of tests for
the exogeneity for arbitrary subsets of the regressors in …nite samples.
In this paper we shall try to elucidate the various forms of available test statistics
for the endogeneity of subsets of the regressors, demonstrate their origins and their
relationships, and also produce solid Monte Carlo results on their performance in single
static linear simultaneous models with IID disturbances. That yet no simulation results
are available on sub-set tests may be due to the fact that it is not straight-forward
how one should design a range of appealing and representative experiments. We believe
that in this respect the present study, which closely follows the rules set out in Kiviet
(2012), may claim originality. Besides exploiting some invariance properties, we choose
the remaining parameter values for the DGP indirectly from the inverse relationships
between the DGP parameter values and fundamental orthogonal econometric notions.
The latter constitute an insightful base for the relevant nuisance parameter space. The
present design can easily be extended to cover cases with a more realistic degree of
overidenti…cation and number of jointly dependent regressors. Other obvious extensions
would be: to include recently developed tests which are specially built to cope with weak
instruments, to consider non Gaussian and non IID disturbances, to examine dynamic
models, to include tests for the validity (orthogonality) of instruments which are not
included in the regression, etc. Regarding all these aspects the present study just o¤ers
an initial reference point.
The structure of the paper is as follows. In Section 2, we …rst de…ne the model’s
maintained properties and the hypothesis to be tested. Next, in a series of subsections,
various routes to develop test procedures are followed and their resulting test statistics
are discussed and compared analytically. Section 3 reviews earlier Monte Carlo designs
and results regarding orthogonality tests. In Section 4 we set out our approach to obtain
DGP parameter values from chosen basic econometric characteristics. A simulation design is obtained to parametrize a synthetic single linear static regression model including
two possibly endogenous regressors with an intercept and involving two external instruments. For this design Section 5 presents simulation results for a selection of practically
relevant parametrizations. Section 6 produces similar results for bootstrapped versions
of the tests, Section 7 provides an empirical case study and Section 8 concludes.
3
2. Testing the orthogonality of subsets of explanatory variables
2.1. The model and setting
We consider the single linear simultaneous equation model
(2.1)
y = X + u;
with IID unobserved disturbances u (0; 2 In ); K-element unknown coe¢ cient vector
, an n K regressor matrix X and n 1 regressand y: We also have an n L matrix
Z containing sample observations on identifying instrumental variables, so
E(Z 0 u) = 0; rank(Z) = L; rank(X) = K and rank(Z 0 X) = K:
(2.2)
In addition, we make asymptotic regularity assumptions to guarantee asymptotic identi…cation of all elements of too and consistency of its IV (or 2SLS) estimator
^ = (X 0 PZ X) 1 X 0 PZ y;
(2.3)
where PZ = Z(Z 0 Z) 1 Z 0 : Hence, we assume that
plim n 1 Z 0 Z =
Z0Z
and plim n 1 Z 0 X =
Z0X
(2.4)
are …nite and have full column rank, whereas ^ has limiting normal distribution
n1=2 ( ^
d
) ! N 0;
2
[
0
Z0X
1
Z0Z
Z0X ]
1
:
(2.5)
The matrices X and Z may have some (but not all) columns in common and can
therefore be partitioned as
X = (Y Z1 ) and Z = (Z1 Z2 );
(2.6)
where Zj has Lj columns for j = 1; 2: Because the number of columns in Y is K L1 > 0
we …nd from L = L1 + L2 K that L2 > 0; but we allow L1 0; so Z1 may be void.
Throughout this paper the model just de…ned establishes the maintained unrestrained
hypothesis, which allows Y to contain endogenous variables. Below we will examine
particular further curbed versions of the maintained hypothesis and develop tests to
verify these further limitations. These are not parametric restraints regarding but
involve orthogonality conditions in addition to the L maintained orthogonality conditions
embedded in E(Z 0 u) = 0. All these extra orthogonality conditions concern regressors
and not further external instrumental variables. Therefore, we consider a partitioning
of Y in Ke and Ko columns
Y = (Ye Yo );
(2.7)
where the variables Ye are maintained as possibly endogenous, whereas for the Ko vari¯
ables Yo their possible orthogonality will be examined, i.e. whether E(Yo0 u) = 0 seems
¯
to hold. We de…ne the n (L + Ko ) matrix
Zr = (Z Yo );
4
(2.8)
which relates to all the orthogonality conditions in the restrained model. Note that (2.2)
¯
implies that Zr has full column rank, provided n L + Ko : Now the null and alternative
hypotheses that we will examine can be expressed as
H 0 : y = X + u;
H 1 : y = X + u;
u
u
(0;
(0;
2
I);
2
I);
E(Zr0 u) = 0; and
E(Z 0 u) = 0; E(Yo0 u) 6= 0:
(2.9)
Hence, H 0 assumes E(Yo0 u) = 0.
Under the extended set of orthogonality conditions E(Zr0 u) = 0; i.e. under H 0 ; the
restrained IV estimator is
^ = (X 0 PZr X) 1 X 0 PZr y:
(2.10)
r
If H 0 is valid this estimator is consistent and, provided plim n 1 Zr0 Zr = Zr0 Zr exists
and is invertible, its limiting normal distribution has variance 2 [ 0Zr0 X Zr01Zr Zr0 X ] 1 ;
which involves an asymptotic e¢ ciency gain over (2.5). However, under the alternative
hypothesis H 1 estimator ^ r is inconsistent.
A test for (2.9) should (as always) have good control over its type I error probability1
and preferably also have high power, in order to prevent the acceptance of an inconsistent
estimator. In practice inference on (2.9) usually establishes just one link in a chain
of tests to decide on the adequacy of model speci…cation (2.1) and the maintained
instruments Z; see for instance Godfrey and Hutton (1994) and Guggenberger (2010).
Many of the …rm results obtained below require to make the very strong assumptions
embedded in (2.1) and (2.2) and leave it to the practitioner to make a balanced use of
them within an actual modelling context.
In the derivations to follow we make use of the following three properties of projection
matrices, which for any full column rank matrix A are denoted as PA = A(A0 A) 1 A0 : For
a full column rank matrix C = (A B) one has (i) PA = PC PA = PA PC ; (ii) PC = PA +
PMA B = P(A MA B) ; where MA = I PA ; (iii) for C = (A B); where A = A BD and D
an arbitrary matrix of appropriate dimensions, PC = PB + PMB A = PB + PMB A = PC :
2.2. The source of any estimator discrepancy
A test based on the Hausman principle focusses on the discrepancy vector
^
^r =
=
=
=
(X 0 PZ X)
(X 0 PZ X)
(X 0 PZ X)
(X 0 PZ X)
X 0 PZ y (X 0 PZr X) 1 X 0 PZr y
1 0
X PZ [I X(X 0 PZr X) 1 X 0 PZr ]y
1
(PZ X)0 u^r
1
(PZ Ye PZ Yo Z1 )0 u^r ;
1
(2.11)
where u^r = y X ^ r denotes the IV residuals obtained under H 0 : Although testing
whether the discrepancy between these two coe¢ cient estimators is signi…cantly di¤erent
from zero is not equivalent to testing H 0 ; we will show that in fact all existing test
procedures employ the outcome of this discrepancy to infer on the (in)validity of H 0 .
1
An actual type I error probability much larger than the chosen nominal value would more often
than intended lead to using an ine¢ cient estimator. A much lower actual type I error than the nominal
level would deprive the test from its power hampering the detection of estimator inconsistency.
5
Because (X 0 PZ X) 1 is non-singular ^ ^ r is close to zero if and only if the K
(PZ Ye PZ Yo Z1 )0 u^r is. So, we will examine now when its three sub-vectors
Ye0 PZ u^r ; Yo0 PZ u^r and Z10 u^r
1 vector
(2.12)
will jointly be close to zero. Note that due to the identi…cation assumptions both PZ Ye
and PZ Yo will have full column rank so cannot be O.
For the IV residuals u^r we have X 0 PZr u^r = 0; and since PZr X = (PZr Ye Yo Z1 ); this
yields
Ye0 PZr u^r = 0; Yo0 u^r = 0 and Z10 u^r = 0:
(2.13)
Note that the third vector of (2.12) is always zero according to the third equality from
(2.13). Using projection matrix property (ii) and the …rst equality of (2.13), we …nd for
the …rst vector of (2.12) that
Ye0 PZ u^r = Ye0 (PZr
PMZ Yo )^
ur =
Ye0 PMZ Yo u^r ;
so
Ye0 PZ u^r =
Ye0 MZ Yo (Yo0 MZ Yo ) 1 Yo0 MZ u^r :
(2.14)
This Ke element vector will be close to zero when the Ko element vector Yo0 MZ u^r is.
Due to the occurrence of the Ke Ko matrix Ye0 MZ Yo as a …rst factor in the right-hand
side of (2.14), it seems possible that Ye0 PZ u^r may be close to zero too in cases where
Yo0 MZ u^r 6= 0; we will return to that possibility below. For the second vector of (2.12)
we …nd, upon using the second equality of (2.13), that
Yo0 PZ u^r =
Yo0 MZ u^r :
(2.15)
Hence, the second vector of (2.12) will be close to zero if and only if the vector Yo0 MZ u^r
is close to zero.
From the above it follows that Yo0 MZ u^r being close to zero is both necessary and
su¢ cient for the full discrepancy vector (2.11) to be small. Checking whether Yo0 MZ u^r is
close to zero corresponds to examining to what degree the variables MZ Yo do obey the
orthogonality conditions, while using u^r as a proxy for u; which is asymptotically valid
under the extended set of orthogonality conditions. Note that by focussing on MZ Yo the
tested variables Yo have been purged from their components spanned by the columns of
Z: Since these are maintained to be orthogonal with respect to u; they should better be
excluded from the test indeed.
Since the inverse matrix in the right-hand side of (2.11) is positive de…nite, the probability limits of ^ and ^ r will be similar if and only if plim n 1 Yo0 MZ u^r = 0: Regarding
the power of any discrepancy based test of (2.9) it is now of great interest to examine
whether it could happen under H 1 to have plim n 1 Yo0 MZ u^r = 0: For that purpose we
specify the reduced form equations
Yj = Z
j
+ (u
0
j
+ Vj ); for j 2 fe; og;
(2.16)
where j is an L Kj matrix of reduced form parameters, j is a Kj 1 vector that
parametrizes the simultaneity and Vj establishes the components of the zero mean reduced form disturbances which are uncorrelated with u and of course with Z: After this
6
further parametrization the hypotheses (2.9) can now be expressed as H 0 : o = 0 and
0
H 1 : o 6= 0: Let (L + Ko ) (L + Ko ) matrix be such that
= (Zr0 Zr ) 1 : From
Yo0 PZ [In X(X 0 PZr X) 1 X 0 PZr ]u
Yo0 PZ [PZr PZr X(X 0 PZr X) 1 X 0 PZr ]u
Yo0 PZ Zr [IL+Ko P 0 Zr0 X ] 0 Zr0 u
Yo0 MZ u^r =
=
=
(2.17)
it follows that plim n 1 Yo0 MZ u^r = 0 if (L + Ko ) 1 vector plim n 1 Zr0 u = 2 (00 0o )0 is
in the column space spanned by plim n 1 Zr0 X = Zr0 X : This is obviously the case when
1 vector Zr0 X c; with
o = 0: However, it cannot occur for o 6= 0; because (L + Ko )
c a K 1 vector, has its …rst L
K elements equal to zero only for c = 0; due to
the identi…cation assumptions. This excludes the existence of a vector c 6= 0 yielding
2 0 0 0
(0 o ) when o 6= 0; so under asymptotic identi…cation the discrepancy will
Zr0 X c =
be nonzero asymptotically when Yo contains an endogenous variable.
Cases
in which the asymptotic
identi…cation assumptions are violated are e =
p
p
Ce = n and/or o = Co = n; where Ce and Co are matrices of appropriate dimensions
with full column rank and all elements …xed and …nite.2 Examining Zr0 X c closer yields
Zr0 X c
=
0
o
Z0Z
Z0Z
e
+
e
Vo0 Ve
+
2
0
o e
c1 +
Z0Z
o
Yo0 Yo
c2 +
0
o
Z 0 Z1
Z 0 Z1
c3 ;
(2.18)
p
where c = (c01 c02 c03 )0 and Vo0 Ve = plim n 1 Vo0 Ve : If only o = Co = n; so when all the
instruments Z are weak and asymptotically irrelevant for the set of regressors Yo whose
orthogonality is tested, we can set c1 = 0 and c3 = 0 and then for c2 = 2 Yo10 Yo o =
2
( 2 o 0o + Vo0 Vo ) 1 o 6= 0 we have Zr0 X c = 2 (00 0op
)0 6= 0; demonstrating that the
test will have no asymptotic power. If only e = Ce = n; thus all the instruments Z
are weak for Ye ; a solution c 6= 0 can be found upon taking c2 = 0; c3 = 0 and c1 6= 0,
provided Vo0 Ve + 2 o 0e 6= O or Ye and Yo are asymptotically not uncorrelated. Only c3
has to be set at zero to …nd a solution when Z is weak for both Yo and Ye : From (2.18)
it can also be established that when from Z2 at least Ke + Ko instruments are not weak
for Y the discrepancy will always be di¤erent from zero asymptotically when o 6= 0:
Using (2.16) we also …nd plim n 1 Ye0 MZ Yo = 0Vo0 Ve + 2 e 0o , which demonstrates
that the …rst vector of (2.12) would for o 6= 0 tend to zero also when e = 0 while the
reduced form disturbances of Ye and Yo are uncorrelated. This indicates the plausible
result that a discrepancy based test may loose power when Ye is unnecessarily treated
as endogenous and Yo is establishing a weak instrument for Ye after partialing out Z.
2.3. Testing based on the source of any discrepancy
Next we examine the implementation of testing closeness to zero of Yo0 MZ u^r in an auxiliary regression. Consider
y = X + PZ Y o + u ;
(2.19)
where u = u PZ Yo : Its estimation by IV employing the instruments Zr yields coe¢ cients that can be obtained by applying OLS to the second-stage regression of y on
PZr X and PZr PZ Yo = PZ Yo : For partitioned regression yields
^ = (Y 0 PZ MP
o
2
Zr X
PZ Yo ) 1 Yo0 PZ MPZr X y;
(2.20)
Doko Tchatoka (2014) considers a similar situation for the special case Ke = 0 and Ko = 1:
7
where, using rule (i), Yo0 PZ MPZr X y = Yo0 PZ [I X(X 0 PZr X) 1 X 0 PZr ]y = Yo0 PZ u^r : Thus,
by testing = 0 in (2.19) we in fact examine whether Yo0 PZ u^r = Yo0 MZ u^r di¤ers
signi…cantly from a zero vector, which is indeed what we aim for.3
Alternatively, consider the auxiliary regression
(2.21)
y = X + MZ Yo + v ;
where v = u MZ Yo : Using the instruments Zr involves here applying OLS to the
second-stage regression of y on PZr X and PZr MZ Yo = PZr Yo PZr PZ Yo = Yo PZ Yo =
MZ Yo : This yields
^ = (Y 0 MZ MP X MZ Yo ) 1 Y 0 MZ MP X y;
(2.22)
o
o
Zr
Zr
where
Yo0 MZ MPZr X y = Yo0 MZ [I PZr X(X 0 PZr X) 1 X 0 PZr ]y
= Yo0 [I X(X 0 PZr X) 1 X 0 PZr ]y Yo0 PZ [I
= Yo0 MZ u^r :
X(X 0 PZr X) 1 X 0 PZr ]y
(2.23)
Thus, like testing = 0 in (2.19), testing = 0 in auxiliary regression (2.21) examines
the magnitude of Yo0 MZ u^r : The estimator for resulting from (2.21) is
^ = (X 0 PZr MM Yo PZr X) 1 X 0 PZr MM Yo y:
r
Z
Z
Because PZr MMZ Yo = PZr PZr PMZ Yo = PZr (PZ + PMZ Yo )PMZ Yo = PZr PMZ Yo = PZ ;
we …nd ^ r = ^ : Hence, the IV estimator of exploiting the extended set of instruments
in the auxiliary model (2.21) equals the unrestrained IV estimator ^ : Many text books
mention this result for the special case Ke = 0:
From the above we …nd that testing whether included possibly endogenous variables
Yo can actually be used e¤ectively as valid extra instruments, can be done as follows:
Add them to Z; so use Zr as instruments, and add at the same time the regressors MZ Yo
(the reduced form residuals of the alleged endogenous variables Yo in the maintained
model) to the model, and then test their joint signi…cance. When testing = 0 in
(2.21) by a Wald-type statistic, and assuming for the moment that 2 is known, the test
statistic is
2 0
y PMPZ X MZ Yo y = 2 y 0 (MA MC )y;
(2.24)
r
where A = PZr X; B = MZ Yo and C = (A B): Hence, y 0 PMPZ X MZ Yo y is equal to the
r
di¤erence between the OLS residual sums of squares of the restricted (by = 0) and
the unrestricted second stage regressions (2.21). One easily …nds that testing = 0 in
(2.19) by a Wald-type test yields in the numerator
y 0 PMPZ
r
X P Z Yo
y = y 0 (MA
MC )y;
with again A = PZr X = (PZr Ye Yo Z1 ); but C = (A B ) with B = PZ Yo : Although
C 6= C, both span the same sub-space, so MC = MC and thus the two auxiliary
regressions lead to numerically equivalent Wald-type test statistics.
3
This procedure provides the explicit solution to the exercise posed in Davidson and MacKinnon
(1993, p.242).
8
Of course, 2 is in fact unknown and should be replaced by an estimator that is consistent under the null. There are various options for this. Two rather obvious choices
would be ^ 2 = u^0 u^=n or ^ 2r = u^0r u^r =n; giving rise to two under the null (and also under
local alternatives) asymptotically equivalent test statistics, both with 2 (Ko ) asymptotic null distribution. Further asymptotically equivalent variants can be obtained by
employing a degrees of freedom correction in the estimation of 2 and/or by dividing the
test statistic by Ko and then confronting it with critical values from an F distribution
with Ko and n l degrees of freedom with l some …nite number, possibly K + Ko :
Testing the orthogonality of Yo and u; while maintaining the endogeneity of Ye ; by a
simple 2 -form statistic and using as in a Wald-type test the estimate ^ 2 (without any
degrees of freedom correction) from the unrestrained model, will be indicated by Wo :
When using the uncorrected restrained estimator ^ 2r ; the statistic will be denoted here
as Do : So we have the two archetype test statistics
Wo = y 0 PMPZ
r
X M Z Yo
y=^ 2 and Do = y 0 PMPZ
r
X M Z Yo
y=^ 2r :
(2.25)
Using the restrained 2 estimator, as in a Lagrange-multiplier-type test under normality,
was already suggested in Durbin (1954, p.27), where Ke = L1 = 0 and Ko = L2 = 1:
Before we discuss further options for estimating 2 in general sub-set tests, we shall
…rst focus on the special case Ke = 0; where the full set of endogenous regressors is
tested. Then ^ 2r = y 0 MX y=n = n nK s2 stems from OLS. Wu (1973) suggested for this
case four test statistics, indicated as T1 ; :::; T4 ; where
T4 =
n
2Ko
n
n
L1 1
Do and T3 =
Ko
2Ko
n
L1 1
Wo :
Ko
(2.26)
On the basis of his simulation results Wu recommended to use the monotonic transformation of T4 (or Do )
T2 =
T4
1
Ko
T
n 2Ko L1 4
=
n
2Ko
n
L1 1
Do
:
Ko 1 Do =n
(2.27)
He showed that under normality of both structural and reduced form disturbances the
null distribution of T2 is F (Ko ; n 2Ko L1 ) in …nite samples.4 Because Ke = 0 implies
MPZr X = MX we …nd from (2.24) that in this case
Do
y 0 PMX MZ Yo y
y 0 P M X M Z Yo y
y 0 P M X M Z Yo y
=n 0
=n 0
=
:
1 Do =n
y (MX PMX MZ Yo )y
y M(X MZ Yo ) y
•2
Hence, from the …nal expression we see that T2 estimates 2 by • 2 = y 0 M(X MZ Yo ) y=n;
which is the OLS residual variance of auxiliary regression (2.21). Like ^ 2 and ^ 2r ; • 2 is
consistent under the null, because plim n 1 Yo0 MZ u^r = 0 implies, after substituting (2.23)
in (2.22), that plim ^ = 0:
Pesaran and Smith (1990) show that under the alternative
plim ^ 2
plim ^ 2r
4
plim • 2
Wu’s T1 test for case Ke = 0, which under normality has a F (Ko ; L2
reputation in terms of power and therefore we leave it aside.
9
Ko ) distribution, has a poor
and then invoke arguments due to Bahadur to expect that T2 (which uses • 2 ) has better
power than T4 (which uses ^ 2r ), whereas both T2 and T4 are expected to outperform T3
(which uses ^ 2 ). However, they did not verify this experimentally. Moreover, because
T2 is a simple monotonic transformation of T4 when Ke = 0; we also know that after a
fully successful size correction both should have equivalent power.
Following the same lines of thought for cases where Ke > 0; we expect (after proper
size correction) Do to do better than Wo ; but Pesaran and Smith (1990) suggest that an
even better result may be expected from formally testing = 0 in the auxiliary regression
(2.21) while exploiting instruments Zr : This amounts to the 2 (Ko ) test statistic To ,
which (omitting its degrees of freedom correction) generalizes Wu’s T2 for cases where
Ke 0; and is given by
To = y 0 PMPZ
r
X M Z Yo
y=• 2 = y 0 (MA
MC )y=• 2 ;
(2.28)
MZ Yo ^)=n:
(2.29)
with
• 2 = (y
X^
MZ Yo ^)0 (y
X^
Actually, it seems that Pesaran and Smith (1990, p.49) employ a slightly di¤erent estimator for 2 ; namely
(y
X^
MZ Yo ^ )0 (y
X^
MZ Yo ^ )=n
(2.30)
with
^ = (Y 0 MZ Yo ) 1 Y 0 MZ (y
o
o
X ^ ):
(2.31)
However, because OLS residuals are orthogonal to the regressors we have Yo0 MZ (y
X ^ MZ Yo ^) = 0; from which it follows that ^ = ^ ; so their test is equivalent with To :
When Ke > 0 the three tests Wo ; Do and To are not simple monotonic transformations
of each other, so they may have genuinely di¤erent size and power properties in …nite
samples. In particular, we …nd that for
y 0 PC y y 0 PA y
Do
= 0
;
1 Do =n
(^
ur u^r y 0 PC y + y 0 PA y)=n
the denominator in the right-hand expression di¤ers from • 2 (unless Ke = 0):5 Using
that ^ is given by (2.31) we …nd from (2.29) that • 2 = u^0 MMZ Yo u^=n ^ 2 ; so
Wo
To ;
(2.32)
whereas Do can be at either side of Wo and To :
5
Therefore, the test statistic (54) suggested in Baum et al. (2003, p.26), although asymptotically
equivalent to the tests suggested here, is built on an inappropriate analogy with the Ke = 0 case. Moreover, in their formulas (53) and (54) Q should be the di¤erence between the residual sums of squares
of second-stage regressions, precisely as in (2.25). The line below (54) suggests that Q is a di¤erence between squared IV residuals (which would mean that Q could be negative) of the (un)restricted
auxiliary regressions, although their footnote 25 seems to suggest otherwise.
10
2.4. Testing based on the discrepancy as such
Direct application of the Hausman (1978) principle yields the test statistic
Ho = ( ^
^ )0 [^ 2 (X 0 PZ X)
r
1
^ 2r (X 0 PZr X) 1 ] ( ^
^ );
r
(2.33)
which uses a generalized inverse for the matrix in square brackets. When 2 were known
the matrix in square brackets would certainly be singular though semi-positive de…nite.
Using two di¤erent 2 estimates might lead to nonsingularity but could yield negative
test statistics. As is obvious from the above, (2.33) will not converge to a 2K distribution
under H 0 ; but in our framework to one with Ko degrees of freedom, cf. Hausman and
Taylor (1981). Some further analysis leads to the following.
Let have separate components as follows from the decompositions
X = Ye
e
+ Yo
o
+ Z1
1
=Y
eo
+ Z1
whereas (X 0 PZ X) 1 has blocks Ajk ; j; k = 1; 2; where A11 is a Keo
Keo = Ke + Ko : Then we …nd from (2.11) and (2.13) that
^
^
^ = (X 0 PZ X)
r
eo
^
Y 0 PZ u^r
0
1
A11
A21
=
(2.34)
1;
Keo matrix with
Y 0 PZ u^r ;
(2.35)
= A11 Y 0 PZ u^r :
eo;r
Hence, the discrepancy vector of the two coe¢ cient estimates of just the regressors in
Y; but also those of the full regressor matrix X; are both linear transformations of rank
Keo of the vector Y 0 PZ u^r : Therefore it is obvious that the Hausman-type test statistic
(2.33) can also be obtained from
Ho = ( ^ eo
^
0 2
0
eo;r ) [^ (Y PMZ1 Z2 Y
)
1
^ 2r (Y 0 PMZ1 (Z2 Yo ) Y ) 1 ] ( ^ eo
^
eo;r ):
(2.36)
Both test statistics are algebraically equivalent, because of the unique linear relationship
^
^ =
r
IKeo
A21 A111
( ^ eo
^
eo;r ):
(2.37)
Calculating (2.36) instead of (2.33) just mitigates the numerical problems.
One now wonders whether an equivalent Hausman-type test can be calculated on
the basis of the discrepancy between the estimated coe¢ cients for just the regressors
Yo : This is not the case, because a relationship of the form ( ^ eo ^ eo;r ) = G( ^ o ^ o;r );
where G is a Keo Ko matrix, cannot be found6 . However, a matrix G can be found
such that ( ^ eo ^ eo;r ) = G^; indicating that test Ho can be made equivalent to the three
distinct tests of the foregoing subsection, provided similar 2 estimates are being used.
Using (2.14) and (2.15) in (2.35) we obtain
^ eo
^ eo;r = A11 Y 0 PZ u^r
=
A11
Ye0 MZ Yo (Yo0 MZ Yo )
IKo
1
(Yo0 MZ MPZr X MZ Yo )^;
(2.38)
6
Note that Wu (1983) and Hwang (1985) start o¤ by analyzing a test based on the descripancy
^
^ : Both Wu (1983) and Ruud (1984, p.236) wrongly suggest equivalence of such a test with
o
o;r
(2.33) and (2.36).
11
because (2.22) and (2.23) yield Yo0 MZ u^r = (Yo0 MZ MPZr X MZ Yo )^: So, under the null
hypothesis particular implementations of Wo ; Do ; To and Ho are equivalent.7 When Ho
is used with two di¤erent 2 estimates it may come close to a hybrid implementation
of Wo and Do where the two residual sums of squares in the numerator are scaled by
di¤erent 2 estimates as in
W Do =
y 0 M(PZr X MZ Yo ) y
:
^2
y 0 MPZr X y
^ 2r
(2.39)
2.5. Testing based on covariance of structural and reduced form disturbances
Auxiliary regression (2.21) is used to detect correlation of u and Vo (the reduced form
disturbances of Yo ) by examing the covariance of the residuals u^r and MZ Yo : This might
perhaps be done in a more direct way by augmenting regression (2.1) by the actual
reduced form disturbances, giving
y = X + (Yo
where w = u (Yo Z
(2.40) can be written as
y = Ye
= X
o)
e
with
a Ko
Z
o)
(2.40)
+w ;
1 vector. Let Z
+ Yo ( o + ) + Z1 (
+ Z2 + w
1
o1
)
Z2
o
= Z1
o2
o1
+ Z2
o2 ;
then
+w
(2.41)
in which we may assume that E(Z 0 w ) = 0; though E(Ye0 w ) 6= 0: However, testing
= 0; which corresponds to = 0 in (2.40), through estimating (2.41) consistently is
not an option, unless Ke = 0. For Ke > 0; which is the case of our primary interest
here, (2.41) contains all available instruments as regressors, so we cannot instrument Ye :
For the case Ke = 0 the test of
= 0 yields the test of Revankar and Hartley
(1973), which is an exact test under normality. When Ko = L2 (just identi…cation) it
specializes to Wu’s T2 .8 When L2 > Ko (overidenti…cation) Revankar (1978) argues that
testing the Ko restrictions = 0 by testing the L2 restrictions
= 0 is ine¢ cient. He
then suggests to test = 0 by a quadratic form in the di¤erence of the least-squares
estimator of o + in (2.41) and the IV estimator of o :9
From the above we see that the tests on the covariance of disturbances do not have
a straight-forward generalization for the case Ke > 0: However, a test that comes close
to it replaces the L L1 columns of Z2 in (2.41) by a set of L K regressors Z2 which
span a subspace of Z2 ; such that (PZ Ye Z1 Z2 ) spans the same space as Z: Testing these
7
This generalizes the equivalence result mentioned below (22.27) in Ruud (2000, p.581), which just
treats the case Ke = 0: Note, however, that because Ruud starts o¤ from the full discrepancy vector, the
transformation he presents is in fact singular and therefore the inverse function mentioned in his footnote
24 is non-unique (the zero matrix may be replaced with any other matrix of the same dimensions). To
obtain a unique inverse transformation, one should start o¤ from the coe¢ cient discrepancy for just the
regressors Y; and this is found to be nonsingular for Ke = 0 only.
8
This is proved as follows: Both tests have regressors X under the null, and under the alternative
the full column rank matrices (X PZ Yo ) and (X Z2 ) respectively. These matrices span the same space
when X = (Yo Z1 ) and Z = (Z1 Z2 ) have the same number of columns.
9
Meepagala (1992) produces numerical results indicating that the descripancy based tests have lower
power than the Revankar and Hartley (1973) test when instruments are weak and than the Revankar
(1978) test when the instruments are strong.
12
L K exclusion restrictions yields the familiar Sargan-Hansen test for testing all the
so-called overidenti…cation restrictions of model (2.1). It is obvious that this test will
have power for alternatives in which Z2 and u are correlated, possibly because some of
the variables in Z2 are actually omitted regressors. In practical situations this type of
test, and also Hausman type tests for the orthogonality of particular instruments not
included as regressors in the speci…cation10 , are very useful. However, we do not consider
such implementations here, because right from the beginning we have chosen a context
in which all instruments Z are assumed to be uncorrelated with u: This allows focus
on tests serving only the second part of the two-part testing procedure as exposed by
Godfrey and Hutton (1994), who also highlight the asymptotic independence of these
two parts.
2.6. Testing by an incremental Sargan test
The original test of overidentifying restrictions initiated by Sargan (1958) does not enable
to infer directly on the orthogonality of individual instrumental variables, but a so-called
incremental Sargan test does. It builds on the maintained hypothesis E(Z 0 u) = 0 and
can test the orthogonality of additional potential instrumental variables. Choosing for
these the included regressors Yo yields a test statistic for the hypotheses (2.9) which is
given by
u^0 PZ u^r u^0 PZ u^
:
(2.42)
So = r 2r
^r
^2
When using for both separate Sargan statistics the same
PZ u^ = (PZ PPZ X )y; the numerator would be
u^0r PZr u^r
2
estimate, and employing
u^0 PZ u^ = y 0 (PZr PPZr X PZ + PPZ X )y
= y 0 (PMZ Yo + PPZ X PPZr X )y
= y 0 (P(PZ X MZ Yo ) PPZr X )y;
whereas that of Wo and Do in (2.24) is given by y 0 (PC
PA )y; where C = (A B)
11
with A = PZr X and B = MZ Yo : Equivalence is proved by using general result (iii)
on projection matrices, upon taking A = PZ X: Using PZr = PZ + PMZ Yo ; we have
A = A PB X = A B(B 0 B) 1 B 0 X; so D = (B 0 B) 1 B 0 X: Thus P(A B) = P(A B) =
P(PZ X MZ Yo ) giving
u^0r PZr u^r
u^0 PZ u^ = y 0 (P(PZr X MZ Yo )
PPZr X )y:
(2.43)
Hence, in addition to the Ho statistic, So establishes yet another hybrid form combining
elements of both Wo and Do ; but di¤erent from (2.39).
2.7. Concerns for practitioners
The foregoing subsections demonstrate that all available archetypical statistics Wo ; Do ;
To ; Ho and So for testing the orthogonality of a subset of the potentially endogenous
10
See Hahn et al. (2011) for a study on its behaviour under weak instruments.
Ruud (2000, p.582) proves this just for the special case Ke = 0: Newey (1985, p.238), Baum et al.
(2003, p.23 and formula 55) and Hayashi (2002) mention equivalence for Ke 0, but do not provide a
proof.
11
13
regressors basically just di¤er regarding the way in which the expresssions they are based
on are scaled with respect to 2 : Both So and Ho (and of course W Do ) show a hybrid
nature in this respect, because their most natural implementations require two di¤erent
2
estimates, which may lead to negative test outcomes. In addition to that, Ho has the
drawback that it involves a generalized inverse, whereas calculation of the other four is
rather straight-forward.12
Similar di¤erences and correspondences carry over to more general models, which
would require GMM estimation, see Newey (1985) and Ahn (1997). Although of no
concern asymptotically, these di¤erences may have major consequences in …nite samples,
thus practitioners are in need of clues which implementations should be preferred.13
Therefore, in the remainder of this study, we will examine the performance in …nite
samples of all these …ve archetypical tests. First, we will examine whether any simple
degrees of freedom corrections seem to lead to acceptable size control. Next, only for
those variants that pass this test we will perform some power calculations.
3. Earlier Monte Carlo designs and results
In the literature the actual rejection frequencies of tests on the independence between
regressors and disturbances have been examined by simulation only for situations where
all possibly endogenous regressors are tested jointly, hence Ke = 0. To our knowledge,
sub-set orthogonalty tests have not been examined yet.
Wu (1974) was the …rst to design a simulation study in which he examined the four
tests suggested in Wu (1973). He made substantial e¤orts, both analytically and experimentally, to assess the parameters and model characteristics which actually determine
the distribution of the test statistics and their power curves. His focus is on the case
where there is one possibly endogenous regressor (Ko = 1), an intercept and one other
included exogenous regressor (L1 = 2) and two external instruments (L2 = 2), giving
a degree of overidenti…cation of 1. All disturbances are assumed normal, all exogenous
regressors are mutually orthogonal and all consist of elements equal to either 1, 0, or
-1, whereas all instruments have coe¢ cient 1 in the reduced form. Wu demonstrates
that all considered test statistics are functions of statistics that follow Wishart distributions which are invariant with respect to the values of the structural coe¢ cients of the
equation of interest. The e¤ects of changing the degree of simultaneity and of changing the joint strength of the external instruments are examined. Because the design is
rather in‡exible regarding varying the explanatory part of the reduced form, no separate attention is paid to the e¤ects of multicollinearity of the regressors on the rejection
12
It is not obvious why Pesaran and Smith (1990, p.49,55) mention that they …nd To a computationally
more attractive statistic than Wo : All three test statistics are very easy to compute. However, To is the
only one that strictly applies a standard procedure (Wald) to testing zero restrictions in an auxiliary
regression, which eases its use by standard software packages. On the other hand Baum et al. (2003,
p.26) characterize tests like To as "computationally expensive and practically cumbersome", which we
…nd far fetched too.
13
Under the heading of "Regressor Endogeneity test" EViews 8.1 presents statistic So where for
both 2 estimates n K degrees of freedom are used, like it does for the J statistic. In Stata 13 the
"hausman" command calculates Ho by default and o¤ers the possibility to calculate Wo and Do . The
degrees of freedom reported is the rank of the estimated variance of the discrepancy vector. In case of
Ho this is not correct. It is possible to overwrite the degrees of freedom by an additional command.
The popular package "ivreg2" only reports Do with the correct degrees of freedom.
14
probabilities, nor to the e¤ects of weakness of individual instruments. Although none
of the tests examined is found to be superior under all circumstances, test T2 ; which is
exact under normality and generalized as To in (2.28), is found to be the preferred one.
Its power increases with the absolute value of the degree of simultaneity, with the joint
strength of the instruments and with the sample size.
Nakamura and Nakamura (1985) examine a design where Ke = 0; Ko = 1; L1 = 2;
L2 = 3 and all instruments are mutually independent standard normal. The structural equation disturbances u and the reduced form disturbances v are IID normal with
variances 2u and 2v respectively and correlation . They focus on the case where all
coe¢ cients in the structural equation and in the reduced form equation for the possibly
endogenous regressor are unity. Given the …xed parameters the distribution of the test
statistic T2 now depends only on the values of 2 ; 2u and 2v : Attention is drawn to the
fact that the power of an endogeneity test and its interpretation di¤ers depending on
whether the test is used to signal: (a) the degree of simultaneity expressed as , (b) the
simultaneity expressed as the covariance = u v , or (c) the extent of the asymptotic
bias of OLS (which in their design is also determined just by ; 2u and 2v ). When
testing (a) a natural choice of the nuisance parameters (which are kept …xed when is
varied to obtain a power curve) are u and v : However, when testing (b) or (c) ; u
and v cannot all be chosen independently. The study shows that, although the power
of test T2 does increase for increasing values of 2 while keeping u and v constant, it
may decrease for increasing asymptotic OLS bias. Therefore, test T2 is not very suitable
for signaling the magnitude of OLS bias. In this design 2v = 5(1 R2 )=R2 ; where R2 is
the population coe¢ cient of determination of the reduced form equation for the possibly
endogenous regressor. The joint strength of the instruments is a simple function of R2
and hence of v : Again, due to the …xed values of the reduced form coe¢ cients the e¤ects
of weakness of individual instruments or of multicollinearity cannot be examined from
this design.
The study by Kiviet (1985) demonstrates that in models with a lagged dependent
explanatory variable the actual type I error probability of test T2 may deviate substantially from the chosen nominal level. Then high rejection frequencies under the
alternative have little or no meaning.14 In the present study we will stick to static
cross-section type models.
Thurman (1986) performs a small scale Monte Carlo simulation of just 100 replications on a speci…c two equation simultaneous model using empirical data for the exogenous variables from which he concludes that Wu-Hausman tests may have substantial
power under particular parametrizations and none under others.
Chmelarova and Hill (2010) focus on pre-test estimation based on test T2 (for Ko = 1;
L1 = 2; L2 = 1) and two other forms of contrast based tests which use an improper
number of degrees of freedom15 . Their Monte Carlo design is very restricted, because
the possibly endogenous regressor and the exogenous regressor (next to the constant)
are uncorrelated, so multicollinearity does not occur, which makes the DGP unrealistic.
Moreover, all coe¢ cients in the equation of interest are kept …xed and are such that the
14
Because we could not replicate some of the presented …gures for the case of strong instruments, we
plan to re-address the analysis of DWH type tests in dynamic models in future work.
15
This may occur when standard software is employed based on a naive implementation of the Hausman test. Practitioners should be adviced never to use these standard options but always perform tests
based on estimator contrasts by running the relevant auxiliary regression.
15
signal to noise ratio is always 1. Therefore, the inconsistency of OLS is relatively large
(and in fact equal to the simultaneity correlation coe¢ cient ). Because the sample size
is not varied and neither is the instrument strength parameter16 the results do not allow
to form an opinion on how e¤ective the T2 test is to diagnose simultaneity.
Jeong and Yoon (2010) present a study in which they examine by simulation what
the rejection probability of the Hausman test is when an instrument is employed which
is actually correlated with the disturbances. Also for the sub-set tests to be examined
here the situation seems of great practical relevance that they might be implemented
while using some variable(s) as instruments which are in fact endogenous. In our Monte
Carlo experiments we will cover such situations, but we do not …nd the design as used
by Jeong and Yoon, in which the endogeneity/exogeneity status of variables depends on
the sample size very useful.
4. A more comprehensive Monte Carlo design
To examine the di¤erences between the various sub-set tests regarding their type I and II
error probabilities in …nite samples we want to lay out a Monte Carlo design which is less
restrictive than those just reviewed. It should allow to represent the major characteristics
of data series and their relationships as faced in empirical work, while avoiding the
imposition of awkward restrictions on the nuisance parameter space. Instead of picking
particular values for the coe¢ cients and further parameters in a simple DGP, and check
whether or not this leads to covering empirically relevant cases, we choose to approach
this design problem from the opposite direction.
4.1. The simulated data generating process
Model (2.1) is specialized in our simulations to
y=
y
(2)
y
(3)
=
=
1
21
31
+
+
+
2y
(2)
+
22 z
(2)
32 z
(2)
3y
+
+
(3)
(4.1)
+ u,
23 z
(3)
33 z
(3)
(2)
(4.2)
(3)
(4.3)
+v ,
+v ,
where is an n 1 vector consisting of ones. So, K = 3; L1 = 1 and L2 = 2; with Ko +
Ke = 2; Y = (y (2) y (3) ); Z1 = and Z = ( z (2) z (3) ): Since K = L; at this stage we only
investigate the case in which under the unrestrained alternative hypothesis the single
simultaneous equation (4.1) is just identi…ed according to the order condition. Because
the statistics to be analyzed will be invariant regarding the values of the intercepts, these
are all set equal to zero, thus 1 = 21 = 31 = 0. Ful…llment of the rank condition for
identi…cation then implies that the inequality
22 33
6=
23 32
(4.4)
has to be satis…ed.
The vectors z (2) and z (3) will be generated as mutually independent IID(0; 1) series.
They have been drawn only once and then were kept …xed over all replications. In fact
16
If the e¤ects of a weaker instrument had been checked the simulation estimates of the moments of
IV (which do not exist, because the model is just identi…ed) would have gone astray.
16
we drew two arbitrary series and next rescaled them such that their sample mean and
variance, and also their sample covariance correspond to the population values 0, 1 and
0 respectively.
To allow for simultaneity of both y (2) and y (3) ; as well as for any value of the correlation between the reduced form disturbances v (2) and v (3) ; these disturbances have
components
v (2) = (2) + 2 u and v (3) = (3) + (2) + 3 u,
(4.5)
(2)
(3)
where the series ui , i and i will be generated as mutually independent zero mean
IID series (for i = 1; :::; n). Without loss of generality, we may choose 2u = 1: Scaling
the variances of the potentially endogenous regressors simpli…es the model even further,
again without loss of generality. This scaling is innocuous, because it can be compensated
by the chosen values for 2 and 3 : We will realize 2y(2) = 2y(3) = 1 by choosing
appropriate values for 2(2) > 0 and 2(3) > 0 as follows. For the variance of the IID
series for the reduced form disturbances and for the possibly endogenous explanatory
variables we …nd
2
v (2)
=
2
=
2
(2)
2
y (2)
+
2
2;
+
2 2
=
2
22
=
2
32
+
2
23
+
2
33
+
2
v (2)
+
2
v (3)
= 1;
(4.6)
2
v (3)
(3)
(2)
2
y (3)
2
3;
+
= 1:
This requires
2
(2)
=1
2
22
2
23
2
2
> 0 and
2
(3)
=1
2
32
2
33
2 2
(2)
2
3
> 0:
(4.7)
In addition to (4.4), (4.7) implies two further inequality restrictions on the nine parameters of the data generating process, which are
f 2;
3;
;
22 ;
23 ;
32 ;
33 ;
2;
3 g:
(4.8)
However, more restrictions should be respected as we will see, when we consider further
consequences of a choice of particular values for these DGP parameters.
4.2. Simulation design parameter space
Assigning a range of reasonable values to the nine DGP parameters is cumbersome as
it is not immediately obvious what model characteristics they imply. Therefore, we now
…rst de…ne econometrically meaningful design parameters. These are functions of the
DGP parameters, and we will invert these functions in order to …nd solutions for the
parameters of the DGP in terms of the chosen design parameter values. Since the DGP is
characterized by nine parameters, we should de…ne nine variation free design parameters
as well. However, their relationships will be such, that this will not automatically imply
the existence nor the uniqueness of solutions.
Two obvious design parameters are the degree of simultaneity in y (2) and y (3) ; given
by
(j)
(4.9)
j = Cov(yi ; ui )=( y (j) u ) = j ; j = 2; 3:
Hence, by choosing 2y(2) = 2y(3) = 1, the degree of simultaneity in y (j) is directly
controlled by j for j = 2; 3; and it implies two more inequality restrictions, namely
j
< 1; j = 2; 3:
17
(4.10)
Another design parameter is a measure of multicollinearity between y (2) and y (3)
given by the correlation
23
=
22 32
+
23 33
2
22
+ (1
2
2)
2
23
+
2 3;
(4.11)
2 3
< 1:
(4.12)
implying yet another restriction
22 32
+
2
22
+ (1
23 33
2
23
2
2)
+
Further characterizations relevant from an econometric perspective are the marginal
strength of instrument z (2) for y (j) and the joint strength of z (2) and z (3) for y (j) ; which
are established by the (partial) population coe¢ cients of determination
2
j2
2
=
and Rj;z23
2
j2
2
=
Rj;z2
+
2
j3 ;
(4.13)
j = 2; 3:
In the same vain, and completing the set of nine design parameters, are two similar
characterizations of the …t of the equation of interest. Because the usual R2 gives
complications under simultaneity, we focus on its reduced form equation
y = ( 2
+(
+
2+
3 32 ) z
(2)
22
3
)
(2)
+(
+
3
+ 3 33 ) z (3)
+ (1 + 2 2 + 3 3 ) u:
2 23
(3)
(4.14)
This yields
2
y
= (
2 22
+(
2
2
3 32 )
2 2
+
+
3
)
+(
(2)
+
2
2 23 + 3 33 )
2 2
(3) + (1 +
2 2
3
+
2
3 3)
,
(4.15)
and in line with (4.13) we then have
2
R1;z2
= ( 2 22 + 3 32 )2 = 2y and
2
R1;z23
= [( 2 22 + 3 32 )2 + ( 2 23 +
2
2
3 33 ) ]= y :
(4.16)
The 9-dimensional design parameter space is given now by
f 2;
3;
2
2
2
2
2
2
23 ; R2;z2 ; R2;z23 ; R3;z2 ; R3;z23 ; R1;z2 ; R1;z23 g:
(4.17)
The …rst three of these parameters have domain ( 1; +1) and the six R2 values have to
obey the restrictions
2
2
< 1; j = 1; 2; 3:
(4.18)
0 Rj;z2
Rj;z23
However, without loss of generality we can further restrict the domain of the nine design
parameters, due to symmetry of the DGP with respect to: (a) the two regressors y (2)
and y (3) in (4.1), (b) the two instrumental variables z (2) and z (3) ; and (c) implications
which follow when all random variables are drawn from distributions with a symmetric
density function.
Due to (a) we may just consider cases where
2
2
2
3:
(4.19)
So, if one of the two regressors will have a more severe simultaneity coe¢ cient, it will
2
always be y (2) : Due to (b) we will limit ourselves to cases where 222
23 : Hence, if one
18
of the instruments for y (2) is stronger than the other, it will always be z (2) : On top of
(4.18) this implies
2
2
R2;z2
0:5R2;z23
:
(4.20)
If (c) applies, we may restrict ourselves to cases where next to particular values for
( 2 ; 3 ); we do not also have to examine ( 2 ;
3 ). This is achieved by imposing
+
0:
In
combination
with
(4.19)
this
leads
to
2
3
1>
j 3j
2
(4.21)
0:
Solving the DGP parameters in terms of the design parameters can now be achieved
as follows. In a …rst stage we can easily solve 7 of the 9 parameters, namely
9
>
j = j
>
>
>
=
1=2
2
; dj2 = 1; +1
j = 2; 3:
(4.22)
j2 = dj2 (Rj;z2 )
>
>
>
>
;
2
2
Rj;z2
)1=2 ; dj3 = 1; +1
j3 = dj3 (Rj;z23
With (4.11) these give
=(
23
22 32
2 3 )=(1
23 33
2
22
2
23
2
2 ):
(4.23)
Thus, for a particular case of chosen design parameter values, obeying the inequalities
(4.18) through (4.21), we may obtain 24 solutions from (4.22) and (4.23) for the DGP
parameters. However, some of these may be inadmissible, if they do not ful…ll the
requirements (4.4) and (4.7). Moreover, we will show that not all of these 24 solutions
necessarily lead to unique results on the distribution of the test statistics Wo ; Do and
To .
Finally, the remaining two parameters 2 and 3 can be solved from the pair of
nonlinear equations
9
2
2
>
(1 R1;z2
) ( 2 22 + 3 32 )2 = R1;z2
[( 2 23 + 3 33 )2
>
>
2
2 2
2 2
>
>
+ (1 + 2 2 + 3 3 ) + 3 (3) + ( 2 + 3 )
];
(2)
=
(1
2
)[(
R1;z23
2 22
+
2
3 32 )
+(
2 23
+
3 33 )
2
2
[(1 +
] = R1;z23
2 2
+ 3 (3) + (
2 2
+
+
3
2
3 3)
2 2
]:
(4.24)
Both these equations represent particular conic sections, specializing into either ellipses,
parabolas or hyperbolas, implying that there may be zero up to eight solutions. However,
it is easy to see that the three sub-set test statistics are all invariant with respect to
: Note that u^ = [I X(X 0 PZ X) 1 X 0 PZ ](X + u) = [I X(X 0 PZ X) 1 X 0 PZ ]u and
u^r = [I X(X 0 PZr X) 1 X 0 PZr ]u are invariant with respect to ; thus so are ^ 2 and
^ 2r : And • 2 is too, because y X ^ MZ Yo ^ = u^ MZ Yo ^ is, as follows from (2.22)
and (2.23). Moreover, because ^ is invariant with respect to also is the numerator of
2
2
the three test statistics.17 Therefore, R1;z2
and R1;z23
do not really establish nuisance
17
2
)
>
>
>
>
>
;
(2)
Wu (1974) …nds this invariance result too, but his proof suggests that it is a consequence of normality
of all the disturbances, whereas it holds more generally.
19
parameters, reducing the dimensionality of the nuisance parameter space to 7. Without
loss of generality we may always set 2 = 3 = 0 in the simulated DGP’s.
When (c) applies, not all 16 permutations of the signs of the four reduced form
coe¢ cients lead to unique results for the test statistics, because of the following. If
the sign of all elements of y (2) and (or) y (3) is changed, this means that in the general
formulas the matrix X is replaced by XJ; where J is a K K diagonal matrix with
diagonal elements +1 or 1; for which J = J 0 = J 1 : It is easily veri…ed that such
a transformation has no e¤ect on the quadratic forms in y which constitute the test
statistics Wo ; Do and To ; because it does not alter the space spanned by the matrices A
and C of (2.24) nor that of the projection matrices used in the three di¤erent estimators
of 2 : So, when changing the sign of all reduced form coe¢ cients, and at the same time
the sign of all the elements of the vectors u; (2) and (3) ; the same test statistics are
found, whereas the simultaneity and multicollinearity are still the same. This reduces
the 16 possible permutations to 8, which we achieve by choosing d22 = 1. From the
remaining 8 permutations four di¤erent couples yield similar 23 and values. We keep
the four permutations which genuinely di¤er by choosing d23 = 1, and will give explicit
attention to the four distinct cases
8
(1; 1; 1; 1)
>
>
<
(1; 1; 1; 1)
(d22 ; d23 ; d32 ; d33 ) =
(4.25)
(1; 1; 1; 1)
>
>
:
(1; 1; 1; 1);
when we generate the disturbances from a symmetric distribution, which at this stage
we will.
For the design parameters we shall choose various interesting combinations from
9
>
2 2 f0; :2; :5g
>
>
>
>
3 2 f :5; :2; 0; :2; :5g >
=
2
f
:5;
:2;
0;
:2;
:5g
(4.26)
23
>
>
2
>
Rj;z2 2 f:01; :1; :2; :3g
>
>
>
2
R
2 f:02; :1; :2; :4; :5; :6g;
j;z23
in as far as they satisfy the restrictions (4.18) through (4.21), provided they obey also
the admissibility restrictions given by (4.4), (4.7) and (4.12).
5. Simulation …ndings on rejection probabilities
In each of the R replications in the simulation study, new independent realizations are
drawn on u; (2) and (3) . The three test statistics Wo ; Do and To will be calculated for
both y (2) (then denoted as W 2 ; D2 ; T 2 ) and for y (3) (denoted W 3 ; D3 ; T 3 ) assuming the
other regressor to be endogenous. These genuine sub-set tests will be compared with
tests on the endogeneity of the full set. The latter are denoted W 23 ; D23 ; T 23 (these are
tests involving 2 degrees of freedom), W32 ; D32 ; T32 (when y (3) is treated as exogenous)
and W23 ; D23 ; T23 (when y (2) is treated as exogenous). The behavior under both the null
and the alternative hypothesis will be investigated. These full-set tests are included to
20
better appreciate the special nature of the more subtle sub-set tests under investigation
here.
Every replication it is checked whether or not the null hypothesis is rejected by test
statistic ; where is any of the tests indicated above. From this we obtain the Monte
Carlo estimate
1 PR
!
I (r) > c ( ) :
(5.1)
p =
R r=1
Here I (:) is the indicator function that takes value one when its argument is true and
zero when it is not. We take the standard form of the test statistics in which c ( ) is
the -level critical value of the 2 distribution (with either 1 or 2 degrees of freedom)
and in which 2 estimates have no degrees of freedom correction.
The Monte Carlo estimator !
p estimates the actual rejection probability of asymptotic test procedure . When H 0 is true it estimates the actual type I error probability
(at nominal level ) and when H 0 is false 1 !
p estimates the type II error probability,
whereas !
p is then a (naive, when there are size distortions) estimator of the power
function of the test in one particular argument (de…ned by the speci…c case of values of
the design and matching DGP parameters). Estimator !
p follows the binomial distribution and has standard errors given by SE(!
p ) = [!
p (1 !
p )=R]1=2 : For R large,
a 99:75% con…dence interval for the true rejection probability is
CI99:75% = [!
p
3 SE(!
p ), !
p + 3 SE(!
p )]:
(5.2)
We choose R = 10000; examine all endogeneity tests at the nominal signi…cance
level of 5%, and take the sample size equal to n = 40 (mostly). Note that the boundary
values for determining whether the actual type I error probability of these asymptotic
tests di¤ers at this particular small sample size signi…cantly (at the very small level of
:25%) from the nominal value 5% are :043 and :057 respectively.
5.1. At least one exogenous regressor
In this subsection we examine cases where either both regressors y (2) and y (3) are actually exogenous or just y (3) is exogenous. Hence, for particular implementations of the
sub-set and full-set tests on endogeneity the null hypothesis is true, but for some it is
false. In fact, it is always true for the sub-set tests on y (3) in the cases of this subsection. We present a series of tables containing estimated rejection probabilities and each
separate table focusses on a particular setting regarding the strength of the instruments.
Every case consists of potentially four subcases; "a" stands for (d32 ; d33 ) = (1; 1) , "b"
for (d32 ; d33 ) = ( 1; 1), "c" for (d32 ; d33 ) = (1; 1) and "d" for (d32 ; d33 ) = ( 1; 1).
When both instruments have similar strength for y (2) and also (but probably stronger
or weaker) for y (3) the identi…cation condition requires d32 6= d33 : Then only two of the
four combinations (4.25) are feasible so that every case just consists of the two subcases
"b" and "c".
In Table 1 we consider cases with mildly strong instruments. In the …rst …ve cases
both y (2) and y (3) are exogenous whereas the degree of multicollinearity changes. So in
the …rst ten rows of the table, for all …ve distinct implementations of the three di¤erent
test statistics examined, the null hypothesis is true. Because y (2) and y (3) are parametrized similarly here, the two sub-set test implementations are actually equivalent.
The minor di¤erences in rejection probabilities stem from random variation, both in the
21
disturbances and in the single realizations of the instruments. The same holds for the
two full-set implementations with one degree of freedom. For all implementations over
the …rst …ve cases (both "b" and "c") Do shows acceptable size control, whereas Wo
tends to underreject, whilst To overrejects. The sub-set version of Wo gets worse under
multicollinearity (irrespective of the sign of 23 ), whereas multicollinearity increases the
type I error probability of the full-set Wo tests. Both Do and To seem una¤ected by
multicollinearity for these cases.
When y (2) is made mildly endogenous, as in cases 6-10, the null hypothesis is still
true for the sub-set tests W 3 ; D3 and T 3 : Their type I error probability seems virtually
una¤ected by the actual values of 2 and 23 : For the sub-set tests W 2 ; D2 and T 2
the null hypothesis is false. Due to their di¤erences in type I error probability we
cannot conclude much about power yet, but that they have some and that it is virtually
una¤ected by 23 is clear. The next three columns demonstrate that it is essential that a
full-set test exploits genuinely exogenous regressors, because if it does not it may falsely
diagnose endogeneity of an exogenous regressor (but by a reasonably low probability
when the regressors are uncorrelated). However, the next tests reported, which exploit
the genuine exogeneity of y (3) ; demonstrate that in this case they do a much better
job in detecting the endogenous nature of y (2) than the sub-set tests, provided there is
(serious) multicollinearity. Here the full-set tests have the advantage of using an extra
valid instrument. The e¤ects of multicollinearity can be explained as follows. Using the
notation of the more general setup and auxiliary regression (2.19), the sub-set (full-set)
tests test here the signi…cance of the regressors PZ Yo (PZ Yo ) in a regression already
containing PZr X (PZr X = X), where Z = (Z Ye ) and Zr = (Zr Ye ): Regarding the
sub-set test it is obvious that, because the space spanned by PZr X = (PZr Ye Yo Z1 ) does
not change when Ye and Yo are more or less correlated, the signi…cance test of PZ Yo
is not a¤ected by 23 : However, PZ Yo is a¤ected (positively in a matrix sense) when
Yo and Ye are more (positively or negatively) correlated, which explains the increasing
probability of detecting endogeneity by the present full-set tests. Finally the two degrees
of freedom full-set tests demonstrate power, also when the null hypothesis tested is only
partly false. One would expect lower rejection probability here than for the full-set test
which correctly exploits orthogonality of y (3) ; but comparison is hampered again due to
the di¤erences between type I error probabilities. Note though that the …rst …ve cases
show larger type I error probabilities for T 23 than for T32 ; whereas cases 6-10 show fewer
correct rejections, which fully conforms to our expectations.
For a higher degree of simultaneity in y (2) (cases 11-13) we …nd for the sub-set
tests that W 3 still underrejects substantially but an e¤ect of multicollinearity is no
longer established, which is probably because DGP’s with a similar 2 and 3 but higher
3
3
23 are not feasible. Here D does no longer outperform T : For the other tests the
rejection probabilities that should increase with j 2 j do indeed, and we …nd that the
probability of misguidance by the full-set tests exploiting an invalid instrument is even
more troublesome now.
These results already indicate that sub-set tests are indispensable in a comprehensive
sequential strategy to classify regressors as either endogenous or exogenous. Because,
after a two degrees of freedom full-set test may have indicated that at least one of the two
regressors is endogenous, neither of the one degree of freedom full-set tests will be capable
of indicating which one is endogenous if there is one endogenous and one exogenous
regressor, unless these two regressors are mutually orthogonal. However, the two sub22
set tests demonstrate that they can be used to diagnose the endogeneity/exogeneity of
the regressors, especially when the endogeneity is serious, irrespective of their degree
of multicollinearity. We shall now examine how these capabilities are a¤ected by the
strength of the instruments.
The results in Table 2 stem from similar DGP’s which di¤er from the previous ones
only in the increased strength of both the instruments, which forces further limitations
on multicollinearity, due to (4.7). Note that the size properties have not really improved. Due to the limitations on varying multicollinearity its e¤ects can hardly be
assessed from this table. The rejection probabilities of false null hypotheses are larger
when the maintained hypothesis is valid, whereas the tests which impose an invalid orthogonality condition become even more confusing when the genuine instruments are
stronger. Multicollinearity still has an increasing e¤ect on the rejection probability of
all the full-set tests, which is very uncomfortable for the implementations which impose
a false exogeneity assumption.
Staiger and Stock (1997) found that full-set tests have correct asymptotic size, although being inconsistent under weak instrument asymptotics. The following three
tables illustrate cases in which the instruments are weak for one of the two potentially
endogenous variables or for both.
In the DGP’s used to generate Table 3, the instruments are weak for y (2) but strong for
y (3) . So now the two sub-set tests examine di¤erent situations (even when 2 = 3 = 0)
and so do the two one degree of freedom full-set tests. Especially the sub-set Wo tests and
the two degrees of freedom W 23 test are seriously undersized. When the endogeneity
of the weakly instrumented regressor is tested by W32 the type I error probability is
seriously a¤ected by (lack of) multicollinearity. All full-set To tests are oversized. Only
the Do tests would require just a (mostly) moderate size correction. The probability
that sub-set test D2 will detect the endogeneity is small, which was already predicted
immediately below (2.18). D23 will again provide confusing evidence, unless the regressors
are orthogonal. Full set tests D32 and D23 have power only under multicollinearity.
The latter result can be understood upon specializing (2.18) for this case, where the
contributions with c1 and c3 disappear because Ke = 0 and L1 = 0: Using Z 0 Z = I and
2
Yo0 Yo = I we have to …nd a solution c2 satisfying c2 =
o c2 = 0: Since
o 6= 0 and
0
=
(
0)
and
the
…rst
column
of
vanishes
asymptotically
there
is
such
a solution
o
o
2
indeed, but not if Yo0 Yo were nondiagonal.
The situation is reversed in Table 4, where the instruments are weak for y (3) and
strong for the possibly endogenous y (2) . Cases 23 and 24 are mirrored in cases 29 and
30. The Wo tests are seriously undersized, except W32 (building on exogeneity of y (3) ; it
is not a¤ected by its weak external instruments) and W23 (provided the multicollinearity
is substantial). The full-set To tests are again oversized. All Do implementations show
mild size distortions. Because of the …ndings below (2.18) it is no surprise that the subset tests on y (2) exhibit deminishing power for more severe multicollinearity. After size
correction it seems likely that W 2 or especially T 2 would do better than D2 . Also the
tests W32 ; D32 and T32 show power for detecting endogeneity of y (2) when the instruments
are weak for exogenous regressor y (3) ; and their power increases with multicollinearity
and of course with 2 :
Finally we construct DGP’s in which the instruments are weak for both regressors.
Given our predictions below (2.18) and because we found mixed results when the instruments are weak for one of the two regressors, not much should be expected when both
23
are a¤ected. The results in Table 5 do indeed illustrate this. The Wo tests underreject
severely, To gives a mixed picture, but Do would require only a minor size correction,
although it will yield very modest power.
In addition to cases in which the two instruments have similar strength for y (2) and
y (3) , we present a couple of cases in which this di¤ers. Note that the inequality (4.4) is
now satis…ed by all four combinations in (4.25). The reason that not every case in Table
6 consists of four subcases is that not every subcase satis…es the second part of (4.7).
The results for the sub-set tests di¤er greatly between the four subcases. Subcases "a"
and "d" show lower rejection probabilities for Wo and To ; whereas Do seems una¤ected
under the null hypothesis. This suggests that the estimate ^ r (and hence ^ 2r ) is probably
less a¤ected by (d23 ; d33 ) in these subcases than ^ 2 and • 2 :
The sub-set tests on y (2) and y (3) behave similar although the (joint) instrument
strength is a little higher for the former. Whereas the results between the subcases are
quite di¤erent for the sub-set tests and the two degrees of freedom full-set tests, the one
degree of freedom full-set test seem less dependent on the choice of (d23 ; d33 ).
When y (2) is endogenous D2 has substantially less power in subcases "a" and "d"
even though under the null hypothesis it rejects less often in subcases "c" and "d".
For the full-set test things are di¤erent. These reject far more often in subcases "a"
and "d" when there is little or no multicollinearity. However, when multicollinearity is
more pronounced the tests reject less often in subcases "a" and "d" than in "b" and
"c". From these results we conclude that the relevant nuisance parameters for these
asymptotic tests are not just simultaneity, multicollinearity and instrument strength,
but also the actual signs of the reduced form coe¢ cients.
5.2. Both regressors endogenous
The rejection probabilities of the sub-set tests estimated under the alternative hypothesis
in the previous subsection are only of secondary interest, because the sub-set that was
treated as endogenous was actually exogenous. In such cases application of the onedegree of freedom full-set test is more appropriate. Now the not tested sub-set which
is treated as endogenous will actually be endogenous, so we will get crucial information
on the practical usefulness of the sub-set tests, and further evidence on the possible
misguidance by the here inappropriate one degree of freedom full-set tests. Similar
cases in terms of instrument strength have been chosen to keep comparability with the
previous subsection.
The DGP’s used for Table 7 mimic those of Table 1 in terms of instrument strength.
In most cases the sub-set tests behave roughly the same as when the maintained regressor
was actually exogenous, although multicollinearity is now found to have a small though
clear asymmetric impact on the rejection probabilities. When multicollinearity is of the
same sign as the simultaneity in y (3) , test statistics Wo and To reject less often than when
these signs di¤er. This is not caused by the …xed nature of the instruments, because
simulations (not reported) in which the instruments are random show the same e¤ect.
On the other hand, the di¤erences between subcases diminish when the instruments
are random. Multicollinearity decreases the rejection probabilities, but less so when
the endogeneity of the maintained regressor is more severe. The full-set tests with one
degree of freedom are a¤ected more by multicollinearity than the sub-set tests. As is to
be expected, the two degrees of freedom full-set tests reject more often now that both
24
regressors are endogenous. The rejection probabilities of these full-set tests, Do included,
decrease dramatically if 23 and 3 are of the same sign, and they do that much more
than for the sub-set tests. Note that the cases in which 3 takes on a negative value are
very similar to cases in which 3 is positive and the sign of 23 is changed, or those of
(d32 ; d33 ). More speci…cally, case 63b corresponds with case 59c and case 63c with case
59b. Therefore, we will exclude cases with negative values for 3 from future tables and
stick to their positive counterparts.
In Table 8 we examine stronger instruments. Comparing with Table 2 we …nd that the
rejection probabilities seem virtually una¤ected by choosing 3 6= 0. As we found before
the rejection probabilities are a¤ected in a positive manner by the increased strength of
the instruments. The sub-set tests reject almost every time if the corresponding degree
of simultaneity is .5. The e¤ect of having 23 and 3 both positive seems less severe. As
long as this is not the case, the two one degree of freedom full-set tests reject more often
than the sub-set tests. If 23 and 3 do not di¤er in sign Wo and Do reject more often
when applied to a sub-set than for their one degree and two degrees of freedom full-set
versions.
Because Table 3 and 4 are very similar and now both regressors are endogenous we
only need to consider the equivalent table of the latter. In Table 9 the instruments are
weak for y (3) but strong for y (2) . Obviously the sub-set tests for y (3) lack power now, as
was already concluded from Table 3. However, sub-set tests for y (2) show power also in
the presence of a maintained endogenous though weakly instrumented regressor. Note
that when 3 is increased all sub-set tests for y (2) reject more often. This dependence
was not apparent under non-weak instruments.
As we found in Table 5 the sub-set tests perform badly when the instruments are
weak for both regressors. From the results on the sub-set test for y (3) we expect the
same for the case in which 3 6= 0. This we found to be true in further simulations,
though we do not present a table on these as it is not very informative.
These simulations demonstrate that the sub-set tests are indispensable when there
is more than one regressor in a model that might be endogenous. Using only fullset tests will not enable to classify the individual variables as either endogenous or
exogenous. However, all tests examined here show substantial size distortions in …nite
samples. Moreover, these size distortions are found to be determined in a complex way
by the model characteristics. In fact the various tables illustrate that it are not just
the design parameters simultaneity, multicollinearity and instrument strength which
determine the size of these tests. The di¤erences between the subcases illustrate that
the size also depends on the actual reduced form coe¢ cients and therefore in fact on
the degree by which the multicollinearity stems from correlation between the reduced
form disturbances ( ): Trying to mitigate the size problems by simple degrees of freedom
adjustments or by transformations to F statistics seems therefore a dead-end.
6. Results for bootstrapped tests
Because all the test statistics that are under investigation here are based on appropriate
…rst order asymptotics, it should be feasible to mitigate the size problems by bootstrapping.
25
6.1. A bootstrap routine for sub-set DWH test statistics
Bootstrap routines for testing the orthogonality of all possibly endogenous regressors
have previously been discussed by Wong (1996). Implementation of these bootstrap
routines is relatively easy due to the fact that no regressors are assumed to be endogenous
under the null hypothesis. This in contrast to the test of sub-sets where some regressors
are endogenous also under the null hypothesis. Their presence complicates matters as
bootstrap realizations have to be generated on both the dependent variable and the
maintained set of endogenous regressors. We discuss two routines; …rst a parametric
and next a semiparametric bootstrap. For the former routine we have to assume a
distribution for the disturbances, which we choose to be the normal.
Consider the n (1 + Ke ) matrix U = (u Ve ): Its elements can be estimated by:
u^r = y X ^ r and V^er = Ye Zr ^ er , where ^ er = (Zr0 Zr ) 1 Zr0 Ye . Under the null
hypothesis ^ r and ^ er are consistent estimators and it follows that U^r = (^
ur V^er ) is
1
0
consistent for U , and hence ^ = n U^r U^r is a consistent estimator of the variance of its
rows. The following illustrates the steps that are required for the bootstrap procedure.
1. Draw pseudo disturbances of sample size n from the N (0; ^ ) distribution and
(b)
collect them in U (b) = (u(b) Ve ) . Obtain bootstrap realizations on the endogenous
(b)
(b)
explanatory variables and the dependent variable through: Ye = Zr ^ er + Ve
(b)
and y (b) = X (b) ^ r + u(b) , where X (b) = (Ye Yo Z1 ). Calculate the test statistic of
choice and store its value ^ (b) .
2. Repeat step (1) B times resulting in the B 1 vector ^ B = ( ^ (1) ::: ^ (B) )0 of which
the elements should be sorted in increasing order.
3. The null hypothesis should be rejected if for the empirical value ^ ; calculated on
the basis of y; X and Z; one …nds ^ > ^ bc , the (1
)(B + 1)-th value of the
sorted vector.
Applying the semiparametric bootstrap is very similar as it only di¤ers from the
parametric one in step (1). Instead of assuming a distribution for the disturbances we
resample by drawing rows with replacement from U^r .
6.2. Simulation results for bootstrapped test statistics
Wong (1996) concludes that bootstrapping the full-set test statistics yields an improvement over using …rst order asymptotics, especially in the case where the (in his case
external) instrument is weak. In this subsection we will discuss simulation results for
the bootstrapped counterparts of the various test statistics. Again all results are obtained with R = 10000 and n = 40, additionally we choose the number of bootstrap
replications to be B = 199. To mimic as closely as possible the way the bootstrap would
be employed in practice, for each case and each test statistic we calculated the bootstrap
critical value ^ bc again in each separate replication.
Table 10 is the bootstrapped equivalent of Table 1. Whereas we found that the
crude asymptotic version of Wo underrejects while To overrejects, bootstrapping these
test statistics results in a substantial improvement18 of their size properties. In fact,
18
Although the current implementation of the bootstrap already performs quite well, even better
results may be obtained by rescaling the reduced form residuals by a loss of degrees of freedom correction.
26
in this respect all three tests perform now equally well with mildly strong instruments,
because the estimated actual signi…cance level lies always inside the 99.75% con…dence
interval for the nominal level. Not only the sub-set tests pro…t from being bootstrapped,
the one degree and two degrees of freedom full-set tests do as well. In terms of power we
…nd that the bootstrapped versions of Wo ; To and Do perform almost equally well. We do
…nd minor di¤erences in rejection frequencies under the alternative, but often these seem
still to be the results of minor di¤erences in size. Nevertheless, on a few occasions test Do
seems to fall behind. Now we establish more convincingly that exploiting correctly the
exogeneity of y (2) in a full-set test provides more power, especially when multicollinearity
is present, than not exploiting it in a sub-set test. Of course, the unfavorable substantial
rejection probability of the exogeneity of the truly exogenous y (3) ; caused by wrongly
treating y (2) as exogenous in a full-set test, cannot be healed by bootstrapping. Similar
conclusions can be drawn from Table 11 which contains results for stronger instruments.
On the other hand, we …nd in Table 12 that bootstrapping does not achieve satisfactory size control for most of the sub-set tests, when the instruments are weak for one
regressor. Only D2 shows reasonable type I error probabilities, but when testing the
endogeneity of y (2) ; the regressor for which the instruments are weak, there is hardly
any power. The full-set tests do not show substantial size distortions and the one degree
of freedom full-set test on y (2) and the two degrees of freedom test demonstrate power
provided the regressors show multicollinearity. The results in Table 13 indicate that the
sub-set test is of more use when weakness of instruments does not concern the variable
under test. We can conclude that Wo and To have more power than Do ; since they reject
less often under the null hypothesis but more often under the alternative. Because we
were unable yet to properly size correct the sub-set test on the strongly instrumented
regressor in Tables 12 and 13, we know that we will be unable to do so too when all
regressors are weakly instrumented. This is supported by the results summarized in
Table 14. Again the results are slightly better for Wo and To but there is almost no
power.
For DGP’s in which both regressors are endogenous we again construct three tables.
From subsection 5.2 we learned that under the alternative hypothesis the tests behave
similar to cases in which only y (2) is endogenous. This is found here too as can be seen
from Table 15. We …nd further evidence that the sub-set version of Do performs less
than Wo and To . New in comparison with Table 7 is that the two degrees of freedom
full-set tests generally exhibit more power than the one degree of freedom full-set tests
when the instruments are mildly strong. However, this was already found for cases with
stronger instruments without bootstrapping. Increasing the instrument strength raises
the rejection probabilities as before as can be seen from Table 16. That our current
implementation of the bootstrap does not o¤er satisfactory size control for most sub-set
tests when y (3) is weakly instrumented was already demonstrated in Table 12 and we
conclude the same for the case when both regressors are endogenous as is obvious from
the results in Table 17.
7. Empirical case study
A classic application involving more than one possibly endogenous regressor is Griliches
(1976), which studies the e¤ect of education on wage. It is often used to demonstrate
27
instrumental variable techniques. Both education and IQ are presumably endogenous
due to omitted regressors. However, testing this assumption is often overlooked. Here
we shall examine the exogeneity status of both regressors jointly and individually by
means of the full-set tests and the sub-set tests. The same data are used as in Hayashi
(2000, p.236). We have the wage equation and reduced form equations
log Wi = 1 Si + 2 IQi + Z1i 1 + ui
Yi = Z1i 1 + Z2i 2 + Vi ;
(7.1)
(7.2)
where W is the hourly wage rate, S is schooling in years and IQ is a test score. All regressors that are assumed to be predetermined or exogenous are included in Z1 ; these are
an intercept (CON S), years of experience (EXP R), tenure in years (T EN ), a dummy
for southern states (RN S) and a dummy for metropolitan areas (SM SA). Additionally Z2 includes instruments age, age squared, mother education, KWW test score and
a marital status dummy. In accordance with our previous notation both potentially
endogenous regressors are included in Y .
Table 18 presents the results of four regressions. OLS treats both schooling and IQ
as exogenous, whereas they are assumed to be endogenous in the IV regression. In IV1
regressor Si is treated as predetermined and IQ as endogenous whereas in IV2 regressor
IQ is treated as predetermined and schooling as endogenous.
Next, in Table 19, we test various hypotheses regarding the exogeneity of one or both
potentially endogenous regressors. Joint exogeneity of schooling and IQ is rejected.
Hence, at least one of these regressors is endogenous and we should use the sub-set
tests to …nd out whether it is just one or both. However, …rst we examine the e¤ect
of using the full-set test on the individual regressors. In both cases the null hypothesis
is rejected. From the Monte Carlo simulation results we learned that the full-set tests
are inappropriate for correctly classifying individual regressors in the presence of other
endogenous regressors. Therefore, we better employ the sub-set tests. Again we reject
the null hypothesis that schooling is exogenous, but the null hypothesis that IQ is
exogenous is not rejected at usual signi…cance levels. Bootstrapping these two test
statistics does not lead to di¤erent conclusions. Based on these results one could greet
regression IV2 instead of IV , resulting in reduced standard errors and a less controversial
result on the e¤ect of IQ, as can be seen from Table 18.
8. Conclusions
In this study various tests on the orthogonality of arbitrary subsets of explanatory variables are motivated and their performance is compared in a series of Monte Carlo experiments. We …nd that genuine sub-set tests play an indispensable part in a comprehensive
sequential strategy to classify regressors as either endogenous or exogenous. Full-set
tests have a high probability to classify an exogenous regressor wrongly as endogenous
if it is merely correlated with an endogenous regressor.
Regarding type I error performance we …nd that sub-set tests bene…t from estimating
variances under the null hypothesis (Do ), as in Lagrange multiplier tests. Estimating
the variances under the alternative (Wo ), as in Wald-type tests, leads to underrejection
when the instruments are not very strong. However, bootstrapping results in good
size control for all test statistics as long as the instruments are not weak for one of
28
the endogenous regressors. When the various tests are compared in terms of power
the bootstrapped Wald-type tests behave often more favorable. This falsi…es earlier
theoretical presumptions on the better power of the To type of test. The outcome is
such that we do not expect that a better performance could have been obtained by the
computationally more involved implementations that result from strictly employing the
Hausman or the Hansen-Sargan principles.
Even when the instruments are weak for the maintained endogenous regressor but
strong for the regressor under inspection we …nd that the auxiliary regression tests
exhibit power, but there is insu¢ cient size control, also when bootstrapped. This is in
contrast to situations in which the instruments are not weak. Then, when bootstrapped,
the sub-set and full-set tests can jointly be used fruitfully to classify individual explanatory variables and groups of them as either exogenous or endogenous.
It must be noted though that the conclusions obtained from the experiments in this
study are limited, as they only deal with static linear models with Gaussian disturbances,
which are just identi…ed by genuinely exogenous external instruments. Apart from relaxing some of these limitations in future work, we plan to look further into e¤ects due to
weakness of instruments. Furthermore, tests on the orthogonality of sub-sets of external
instruments and joint tests on the orthogonality of included and excluded instruments
deserve further examination.
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31
Table 1: One endogenous regressor and mildly strong instruments:
2
2
2
2
R2;z2
= :2, R2;z23
= :4, R3;z2
= :2, R3;z23
= :4
0
0
-.2
-.2
.2
.2
-.5
-.5
.5
.5
W3
.033
.036
.032
.034
.034
.035
.024
.024
.026
.025
D3
.054
.056
.055
.057
.056
.057
.058
.056
.060
.057
T3
.064
.069
.064
.067
.065
.069
.069
.067
.070
.066
W2
.030
.030
.029
.031
.028
.029
.024
.023
.023
.023
D2
.050
.053
.055
.053
.051
.053
.057
.056
.056
.056
T2
.061
.063
.064
.062
.061
.063
.068
.066
.067
.067
W23
.040
.042
.044
.044
.045
.047
.057
.059
.054
.057
D23
.050
.050
.050
.050
.049
.052
.052
.052
.048
.052
T23
.070
.073
.069
.074
.069
.073
.073
.076
.071
.075
W32
.040
.037
.043
.040
.041
.041
.058
.054
.053
.055
D32
.046
.047
.047
.048
.046
.046
.052
.050
.048
.049
T32
.068
.069
.067
.068
.068
.068
.073
.072
.071
.070
W 23
.021
.023
.023
.024
.023
.023
.027
.029
.027
.028
D23
.044
.046
.044
.047
.044
.044
.046
.047
.045
.047
T 23
.088
.086
.090
.086
.087
.086
.090
.090
.083
.086
0
0
0
0
0
0
0
0
0
0
0
0
-.2
-.2
.2
.2
-.5
-.5
.5
.5
.033
.036
.032
.032
.034
.034
.025
.026
.027
.025
.056
.058
.057
.059
.058
.059
.059
.058
.060
.058
.064
.067
.064
.067
.064
.067
.067
.066
.071
.066
.122
.125
.120
.122
.117
.122
.103
.092
.093
.102
.177
.180
.174
.174
.168
.173
.155
.143
.144
.154
.199
.203
.198
.198
.197
.198
.192
.183
.182
.192
.039
.043
.068
.075
.076
.072
.701
.705
.709
.707
.047
.051
.076
.085
.085
.082
.682
.687
.691
.690
.074
.074
.108
.117
.116
.112
.743
.744
.754
.748
.134
.135
.172
.177
.177
.172
.759
.763
.767
.763
.161
.157
.186
.192
.195
.190
.744
.745
.751
.746
.208
.205
.239
.243
.246
.243
.798
.800
.804
.800
.067
.067
.083
.088
.090
.090
.609
.609
.614
.612
.128
.128
.147
.148
.150
.150
.644
.645
.651
.646
.200
.202
.228
.233
.231
.230
.747
.748
.755
.753
0
0
0
0
0
0
0
0
-.2
-.2
.2
.2
.028
.031
.028
.030
.031
.030
.063
.065
.064
.065
.065
.066
.059
.064
.060
.064
.062
.063
.816
.810
.768
.763
.762
.771
.865
.865
.818
.809
.808
.816
.885
.888
.862
.850
.852
.859
.041
.042
.322
.333
.335
.321
.051
.052
.344
.355
.358
.343
.075
.075
.410
.421
.425
.406
.825
.822
.929
.930
.934
.932
.848
.846
.933
.934
.938
.936
.886
.884
.952
.954
.958
.956
.634
.638
.814
.820
.818
.815
.782
.783
.895
.898
.898
.900
.861
.858
.939
.941
.944
.943
Case
1b
1c
2b
2c
3b
3c
4b
4c
5b
5c
2
3
23
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
6b
6c
7b
7c
8b
8c
9b
9c
10b
10c
.2
.2
.2
.2
.2
.2
.2
.2
.2
.2
11b
11c
12b
12c
13b
13c
.5
.5
.5
.5
.5
.5
Table 2: One endogenous regressor and stronger instruments:
2
2
2
2
R2;z2
= :3, R2;z23
= :6, R3;z2
= :3, R3;z23
= :6
Case
14b
14c
15b
15c
16b
16c
2
3
23
0
0
0
0
0
0
0
0
0
0
0
0
0
0
-.2
-.2
.2
.2
W3
.048
.052
.045
.048
.048
.046
D3
.058
.061
.058
.059
.060
.058
T3
.068
.072
.068
.070
.070
.072
W2
.042
.045
.045
.045
.042
.044
D2
.052
.056
.057
.055
.054
.056
T2
.065
.068
.070
.068
.065
.068
W23
.052
.055
.054
.056
.053
.057
D23
.051
.053
.050
.052
.049
.053
T23
.070
.076
.071
.076
.070
.076
W32
.046
.048
.052
.051
.050
.051
D32
.046
.047
.048
.046
.047
.045
T32
.068
.070
.070
.070
.069
.066
W 23
.039
.042
.042
.042
.040
.042
D23
.044
.047
.045
.047
.044
.046
T 23
.089
.086
.092
.087
.087
.088
17b
17c
18b
18c
19b
19c
.2
.2
.2
.2
.2
.2
0
0
0
0
0
0
0
0
-.2
-.2
.2
.2
.049
.051
.045
.046
.047
.046
.058
.061
.058
.059
.061
.059
.068
.072
.069
.071
.070
.071
.328
.329
.306
.304
.303
.307
.358
.356
.331
.329
.326
.334
.392
.391
.372
.370
.370
.375
.047
.049
.213
.220
.225
.219
.046
.047
.202
.210
.212
.208
.066
.067
.258
.260
.270
.259
.329
.328
.482
.485
.478
.480
.325
.324
.468
.471
.464
.464
.387
.388
.535
.536
.538
.534
.224
.229
.363
.361
.361
.359
.241
.247
.372
.367
.368
.365
.348
.353
.491
.488
.488
.483
20b
20c
21b
21c
22b
22c
.5
.5
.5
.5
.5
.5
0
0
0
0
0
0
0
0
-.2
-.2
.2
.2
.045
.048
.043
.041
.043
.040
.063
.066
.065
.061
.061
.061
.066
.069
.067
.065
.065
.064
1
1
.994
.994
.994
.994
1
1
.992
.993
.993
.992
1
1
.996
.995
.996
.996
.023
.025
.978
.981
.980
.980
.023
.024
.975
.978
.977
.976
.037
.039
.987
.988
.987
.987
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
.999 .999
.999 .999
1
1
1
1
1
1
1
1
1
1
1
1
1
1
32
Table 3: One endogenous regressor and weak instruments for y (2) :
2
2
2
2
R2;z2
= :01, R2;z23
= :02, R3;z2
= :3, R3;z23
= :6
0 0
0 0
0 .5
0 .5
W3
.013
.014
.001
.001
D3
.031
.032
.052
.054
T3
.021
.023
.012
.016
W2
.001
0
.001
.001
D2
.054
.054
.054
.055
T2
.051
.050
.056
.057
W23
.051
.055
.056
.060
D23
.050
.055
.050
.053
T23
.072
.076
.071
.076
W32
.001
.002
.047
.051
D32
.048
.047
.049
.052
T32
.070
.070
.071
.075
W 23
.005
.005
.008
.007
D23
.044
.046
.045
.047
T 23
.086
.089
.084
.087
.2
.2
.2
.2
0 0
0 0
0 .5
0 .5
.013
.011
.003
.004
.032
.033
.053
.056
.021
.021
.026
.026
.001
0
.001
0
.057
.058
.059
.060
.053
.054
.061
.062
.050
.052
.345
.337
.048
.052
.321
.317
.071
.075
.389
.381
.001
.002
.317
.309
.047
.049
.324
.316
.072
.070
.390
.383
.005
.006
.083
.083
.047
.047
.248
.238
.089
.092
.353
.342
.5
.5
.5
.5
0 0
0 0
0 .5
0 .5
.012
.012
.023
.023
.039
.040
.060
.062
.019
.020
.126
.122
.002
.002
.002
.002
.089
.087
.091
.091
.080
.081
.103
.109
.049 .047 .069
.050 .048 .069
1
1
1
1
1
1
Case
23a
23b
24a
24b
2
3
0
0
0
0
25a
25b
26a
26b
27a
27b
28a
28b
23
.002 .062 .090
.003 .059 .086
1
1
1
1
1
1
.005 .061 .113
.005 .063 .114
.694
1
1
.696
1
1
Table 4: One endogenous regressor and weak instruments for y (3) :
2
2
2
2
R2;z2
= :3, R2;z23
= :6, R3;z2
= :01, R3;z23
= :02
0 0
0 0
0 .5
0 .5
W3
.001
.001
.001
.001
D3
.058
.058
.057
.060
T3
.054
.055
.060
.062
W2
.012
.011
.002
.001
D2
.032
.031
.056
.058
T2
.020
.021
.012
.013
W23
.002
.001
.047
.046
D23
.045
.049
.047
.048
T23
.067
.074
.070
.069
W32
.048
.049
.053
.054
D32
.046
.046
.047
.047
T32
.069
.070
.069
.068
W 23
.004
.004
.007
.006
D23
.044
.044
.044
.045
T 23
.088
.088
.087
.088
.2
.2
.2
.2
0 0
0 0
0 .5
0 .5
.001
.001
.001
.001
.059
.059
.058
.061
.056
.057
.069
.069
.100
.101
.017
.016
.137
.141
.072
.072
.141
.145
.086
.094
.006
.005
.867
.866
.156
.156
.869
.868
.201
.197
.904
.901
.328
.329
.891
.892
.324
.325
.880
.880
.386
.384
.912
.910
.068
.070
.428
.433
.242
.244
.810
.809
.347
.349
.881
.878
.5
.5
.5
.5
0 0
0 0
0 .2
0 .2
.001
.001
.001
.001
.063
.064
.063
.064
.074
.075
.074
.081
.572
.570
.414
.420
.460
.460
.298
.311
.643
.637
.536
.543
.068
.068
.429
.430
.600
.593
.870
.868
.630
.626
.883
.884
1
1
1
1
1
1
1
1
1
1
1
1
.667 .999
.658 .999
.699
1
.700
1
1
1
1
1
Case
29b
29c
30b
30c
2
3
0
0
0
0
31b
31c
32b
32c
33b
33c
34b
34c
23
Table 5: One endogenous regressor and weak instruments:
2
2
2
2
R2;z2
= R3;z2
= :01, R2;z23
= R3;z23
= :02
Case
35b
35c
36b
36c
2
3
0
0
0
0
0 0
0 0
0 .5
0 .5
W3
0
0
0
0
D3
.033
.034
.035
.033
T3
.016
.016
.020
.017
W2
0
0
.001
0
D2
.032
.031
.037
.034
T2
.017
.016
.018
.017
W23
.002
.002
.003
.002
D23
.046
.047
.046
.048
T23
.068
.070
.069
.068
W32
.002
.002
.002
.002
D32
.047
.047
.046
.048
T32
.068
.067
.065
.070
W 23
0
0
0
0
D23
.043
.044
.044
.044
T 23
.084
.083
.086
.083
37b
37c
38b
38c
.2
.2
.2
.2
0 0
0 0
0 .5
0 .5
0
0
0
0
.033
.033
.037
.038
.017
.016
.019
.018
0
0
0
0
.033
.032
.036
.036
.016
.017
.018
.017
.002
.002
.003
.002
.050
.050
.049
.052
.071
.071
.074
.076
.001
.001
.002
.002
.047
.049
.050
.054
.068
.070
.075
.079
0
0
0
0
.046
.047
.050
.049
.089
.090
.095
.092
39b
39c
40b
40c
.5
.5
.5
.5
0 0
0 0
0 .5
0 .5
0
0
0
0
.037
.038
.051
.050
.018
.017
.026
.024
0
.001
0
.001
.042
.041
.054
.057
.022
.021
.032
.031
.002
.002
.005
.005
.058
.057
.086
.082
.086
.084
.118
.115
.003
.003
.007
.005
.063
.061
.095
.091
.088
.089
.125
.125
0
0
0
0
.060
.059
.093
.092
.112
.112
.157
.152
23
33
Table 6: One endogenous regressor and asymmetric instrument strength:
2
2
2
2
R2;z2
= :3, R2;z23
= :5, R3;z2
= :1, R3;z23
= :4
Case
1a
1b
1c
1d
2b
2c
2d
3a
3b
3c
4c
4d
5a
5b
2
3
23
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
-.2
-.2
-.2
.2
.2
.2
-.5
-.5
.5
.5
W3
.001
.034
.033
.002
.030
.037
.002
.002
.037
.033
.032
.004
.005
.034
D3
.059
.056
.059
.060
.059
.057
.058
.058
.056
.059
.055
.054
.054
.060
T3
.036
.066
.071
.036
.069
.070
.024
.025
.068
.068
.069
.023
.026
.071
W2
.001
.037
.036
.001
.031
.038
.002
.003
.036
.030
.033
.006
.007
.030
D2
.057
.053
.052
.059
.058
.053
.057
.056
.049
.055
.054
.050
.047
.054
T2
.028
.063
.064
.028
.066
.064
.018
.018
.061
.063
.066
.022
.022
.066
W23
.057
.041
.043
.054
.049
.043
.043
.044
.041
.049
.056
.020
.022
.054
D23
.051
.050
.052
.048
.050
.051
.047
.049
.051
.050
.052
.047
.050
.049
T23
.072
.069
.073
.069
.073
.073
.069
.073
.071
.073
.075
.070
.071
.069
W32
.057
.045
.044
.054
.050
.045
.047
.048
.044
.047
.054
.035
.032
.054
D32
.050
.048
.047
.049
.048
.047
.047
.048
.046
.045
.048
.048
.046
.048
T32
.073
.070
.068
.070
.069
.071
.069
.071
.073
.067
.071
.071
.068
.069
W 23
.008
.026
.027
.008
.027
.028
.006
.008
.027
.025
.033
.006
.007
.031
D23
.048
.045
.043
.047
.045
.045
.044
.045
.044
.045
.046
.044
.044
.043
T 23
.090
.089
.086
.089
.093
.086
.089
.088
.088
.088
.089
.088
.089
.085
6a
6b
6c
6d
7b
7c
7d
8a
8b
8c
9c
9d
10a
10b
.2
.2
.2
.2
.2
.2
.2
.2
.2
.2
.2
.2
.2
.2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
-.2
-.2
-.2
.2
.2
.2
-.5
-.5
.5
.5
.002
.033
.034
.003
.030
.035
.004
.003
.036
.031
.031
.005
.008
.033
.059
.057
.059
.059
.060
.059
.058
.059
.058
.060
.056
.056
.056
.061
.044
.066
.069
.046
.068
.069
.033
.032
.067
.068
.068
.032
.031
.072
.005
.190
.192
.005
.161
.203
.010
.009
.202
.162
.166
.030
.035
.164
.073
.233
.234
.072
.206
.240
.079
.080
.239
.206
.200
.104
.107
.198
.073
.264
.265
.074
.245
.272
.056
.057
.272
.245
.250
.076
.077
.248
.703
.050
.053
.710
.186
.053
.249
.246
.055
.188
.615
.056
.055
.622
.684
.059
.062
.692
.189
.062
.265
.264
.066
.192
.599
.115
.114
.608
.742
.087
.087
.751
.246
.090
.322
.322
.093
.246
.665
.151
.147
.670
.714
.224
.224
.721
.349
.231
.285
.284
.232
.343
.724
.134
.130
.729
.693
.231
.230
.700
.343
.239
.287
.286
.239
.337
.708
.172
.167
.715
.753
.289
.288
.758
.410
.301
.348
.349
.298
.406
.763
.218
.215
.771
.304
.123
.128
.310
.207
.134
.080
.082
.134
.217
.586
.037
.038
.589
.590
.177
.179
.596
.258
.185
.223
.220
.182
.266
.604
.139
.135
.607
.701
.265
.267
.706
.370
.275
.325
.317
.275
.369
.710
.214
.211
.720
11b
11c
12b
12c
12d
13a
13b
13c
14d
15a
.5
.5
.5
.5
.5
.5
.5
.5
.5
.5
0
0
0
0
0
0
0
0
0
0
0
0
-.2
-.2
-.2
.2
.2
.2
-.5
.5
.032
.032
.030
.032
.014
.013
.032
.030
.025
.022
.062
.064
.065
.064
.063
.064
.063
.063
.066
.067
.064
.066
.064
.066
.089
.089
.063
.063
.074
.074
.954
.955
.853
.960
.100
.102
.961
.850
.302
.300
.958
.959
.848
.963
.201
.197
.963
.843
.432
.427
.972
.973
.911
.973
.330
.328
.975
.912
.498
.497
.145
.147
.930
.156
.989
.991
.159
.930
.438
.443
.165
.165
.930
.176
.991
.992
.179
.930
.619
.627
.208
.213
.951
.219
.995
.995
.228
.951
.674
.681
.979
.980
1
.986
.997
.997
.986
1
.867
.871
.979
.979
1
.986
.997
.996
.986
1
.909
.910
.987
.987
1
.990
.998
.998
.992
1
.934
.934
.937
.942
.996
.960
.701
.699
.963
.996
.495
.493
.960
.963
1
.971
.995
.994
.973
1
.872
.873
.981
.983
1
.987
.998
.998
.987
1
.926
.926
34
Table 7: Two endogenous regressors and mildly strong instruments:
2
2
2
2
R2;z2
= :2, R2;z23
= :4, R3;z2
= :2, R3;z23
= :4
Case
57b
57c
58b
58c
59b
59c
60b
60c
61b
61c
2
3
23
.2
.2
.2
.2
.2
.2
.2
.2
.2
.2
-.2
-.2
-.2
-.2
-.2
-.2
-.2
-.2
-.2
-.2
0
0
-.2
-.2
.2
.2
-.5
-.5
.5
.5
W3
.135
.125
.113
.106
.139
.130
.078
.072
.118
.111
D3
.192
.179
.182
.172
.182
.172
.156
.149
.150
.142
T3
.213
.202
.197
.184
.217
.204
.172
.165
.208
.201
W2
.120
.126
.109
.109
.130
.135
.079
.071
.116
.125
D2
.181
.183
.176
.177
.171
.176
.158
.146
.145
.156
T2
.203
.203
.186
.189
.207
.208
.173
.163
.202
.212
W23
.154
.142
.076
.073
.390
.382
.057
.056
1
1
D23
.177
.163
.085
.081
.414
.405
.051
.050
1
1
T23
.225
.212
.121
.113
.483
.474
.071
.072
1
1
W32
.142
.143
.073
.076
.383
.380
.057
.055
1
1
D32
.167
.168
.083
.086
.408
.405
.052
.050
1
1
T32
.215
.211
.113
.116
.478
.475
.075
.074
1
1
W 23
.138
.132
.076
.072
.338
.335
.044
.045
.999
.999
D23
.231
.229
.154
.148
.433
.426
.108
.101
1
1
T 23
.336
.327
.238
.231
.553
.546
.177
.170
1
1
62b
62c
63b
63c
64b
64c
65b
65c
66b
66c
.2
.2
.2
.2
.2
.2
.2
.2
.2
.2
.2
.2
.2
.2
.2
.2
.2
.2
.2
.2
0
0
-.2
-.2
.2
.2
-.5
-.5
.5
.5
.120
.131
.124
.134
.102
.112
.111
.120
.069
.075
.174
.185
.164
.177
.171
.176
.141
.148
.145
.159
.195
.207
.196
.214
.183
.192
.194
.212
.162
.173
.126
.126
.135
.131
.104
.110
.127
.114
.070
.078
.181
.182
.178
.175
.173
.176
.155
.145
.146
.158
.199
.201
.208
.208
.187
.187
.210
.203
.162
.173
.139
.149
.376
.391
.072
.075
1
1
.054
.059
.163
.173
.400
.417
.080
.086
1
1
.048
.052
.208
.221
.471
.485
.111
.117
1
1
.070
.077
.143
.143
.384
.378
.073
.074
1
1
.053
.056
.168
.166
.409
.402
.081
.084
1
1
.046
.049
.215
.214
.477
.473
.115
.114
1
1
.070
.073
.124
.132
.330
.342
.065
.068
.999
.999
.041
.045
.218
.234
.426
.431
.145
.154
1
1
.096
.103
.329
.332
.544
.547
.227
.239
1
1
.164
.177
67b
67c
68b
68c
69b
69c
70b
70c
.5
.5
.5
.5
.5
.5
.5
.5
-.2
-.2
-.2
-.2
-.2
-.2
-.2
-.2
0
0
-.2
-.2
.2
.2
-.5
-.5
.137
.127
.094
.091
.159
.147
.047
.045
.210
.196
.202
.191
.195
.184
.172
.163
.216
.205
.186
.174
.229
.223
.141
.131
.811
.808
.781
.775
.746
.756
.598
.592
.857
.860
.853
.847
.767
.777
.686
.680
.879
.884
.875
.870
.830
.842
.769
.758
.194
.181
.045
.050
.882
.878
.995
.997
.221
.209
.052
.057
.891
.888
.995
.995
.272
.255
.078
.082
.919
.918
.997
.997
.848
.853
.733
.735
.996
.997
1
1
.867
.872
.750
.753
.997
.997
1
1
.899
.902
.802
.806
.998
.999
1
1
.773
.773
.586
.590
.993
.992
.987
.988
.889
.884
.766
.761
.999
.999
1
1
.936
.934
.848
.844
.999
.999
1
1
71b
71c
72b
72c
73b
73c
74b
74c
.5
.5
.5
.5
.5
.5
.5
.5
.2
.2
.2
.2
.2
.2
.2
.2
0
0
-.2
-.2
.2
.2
.5
.5
.119
.131
.142
.158
.086
.094
.042
.046
.192
.202
.178
.193
.190
.196
.159
.170
.198
.210
.215
.227
.171
.177
.130
.142
.813
.806
.756
.747
.777
.779
.588
.599
.861
.855
.778
.767
.848
.853
.679
.690
.885
.880
.845
.830
.871
.873
.759
.768
.178
.189
.876
.882
.052
.049
.997
.996
.203
.216
.884
.892
.058
.056
.996
.995
.256
.270
.914
.920
.084
.081
.998
.997
.845
.846
.997
.997
.735
.734
1
1
.866
.866
.998
.997
.753
.753
1
1
.900
.897
.999
.998
.806
.802
1
1
.771
.775
.992
.993
.585
.586
.988
.987
.883
.885
.998
.998
.761
.762
1
1
.928
.935
.999
.999
.847
.846
1
1
75b
75c
76b
76c
77b
77c
.5
.5
.5
.5
.5
.5
-.5
-.5
-.5
-.5
-.5
-.5
0
0
-.2
-.2
-.5
-.5
.805
.800
.803
.799
.603
.598
.838
.831
.893
.889
.804
.795
.870
.862
.895
.889
.820
.809
.794
.802
.794
.795
.599
.595
.831
.836
.886
.888
.803
.793
.862
.873
.887
.892
.816
.807
.936
.937
.416
.408
.061
.057
.947
.947
.439
.431
.055
.052
.963
.962
.504
.496
.078
.078
.939
.940
.404
.401
.061
.061
.948
.949
.427
.425
.054
.054
.963
.966
.496
.492
.081
.079
.986
.987
.849
.849
.342
.340
1
1
.972
.970
.683
.677
1
1
.985
.984
.783
.774
78b
78c
79b
79c
80b
80c
.5
.5
.5
.5
.5
.5
.5
.5
.5
.5
.5
.5
0
0
.2
.2
.5
.5
.800
.809
.795
.804
.596
.603
.835
.843
.889
.892
.794
.806
.865
.872
.889
.892
.807
.816
.802
.793
.794
.793
.593
.601
.838
.828
.886
.887
.792
.799
.872
.861
.890
.887
.808
.813
.933
.935
.408
.416
.057
.060
.944
.944
.434
.440
.051
.052
.959
.961
.503
.507
.077
.078
.939
.936
.414
.407
.059
.060
.948
.946
.435
.431
.054
.054
.963
.961
.501
.496
.077
.077
.985
.987
.842
.844
.340
.346
1
1
.971
.975
.677
.684
1
1
.986
.987
.776
.783
35
Table 8: Two endogenous regressors and stronger instruments:
2
2
2
2
R2;z2
= :3, R2;z23
= :6, R3;z2
= :3, R3;z23
= :6
Case
81b
81c
82b
82c
83b
83c
2
3
23
.2
.2
.2
.2
.2
.2
.2
.2
.2
.2
.2
.2
0
0
-.2
-.2
.2
.2
W3
.323
.334
.305
.319
.290
.296
D3
.350
.361
.314
.328
.333
.343
T3
.384
.393
.368
.383
.358
.367
W2
.331
.330
.318
.317
.293
.296
D2
.362
.359
.326
.324
.339
.343
T2
.394
.391
.380
.378
.363
.368
W23
.332
.337
.934
.935
.100
.105
D23
.327
.332
.928
.930
.093
.098
T23
.391
.399
.948
.949
.129
.137
W32
.338
.336
.937
.936
.106
.105
84b
84c
85b
85c
.5
.5
.5
.5
.2 0
.2 0
.2 .2
.2 .2
.327
.342
.273
.283
.360
.375
.357
.366
.389
.404
.349
.360
.999
.999
.999
.999
.999 .999
.999
1
.999
1
.999 .999
.402
.408
.200
.192
.395
.403
.189
.179
.475
.483
.250
.240
1
1
1
1
1
1
.998 .998 .999
.997 .997 .998
86b
86c
.5 .5 .2
.5 .5 .2
1
1
.999 .999
1
1
1
1
.635 .613 .699
.637 .619 .700
.640 .622 .702
.631 .610 .692
1
1
1
1
D32
.333
.331
.932
.931
.098
.097
T32
.400
.397
.951
.952
.133
.133
W 23
.479
.481
.951
.954
.232
.244
D23
.497
.502
.948
.950
.272
.282
T 23
.618
.619
.970
.974
.384
.390
1
1
1
1
.999
1
.999 .999
1
1
1
1
1
1
1
1
1
1
W 23
.071
.072
.158
.157
.051
.050
.735
.731
.074
.074
D23
.256
.258
.435
.438
.194
.193
1
1
.224
.226
T 23
.361
.364
.556
.553
.285
.285
1
1
.326
.327
.676
1
.669
1
.731
1
.737
1
.668 .999
.662 .999
1
1
1
1
1
1
Table 9: Two endogenous regressors and weak instruments for y (3) :
2
2
2
2
R2;z2
= :3, R2;z23
= :6, R3;z2
= :01, R3;z23
= :02
Case
87b
87c
88b
88c
89b
89c
90b
90c
91b
91c
2
3
23
.2
.2
.2
.2
.2
.2
.2
.2
.2
.2
.2
.2
.2
.2
.2
.2
.2
.2
.2
.2
0
0
-.2
-.2
.2
.2
-.5
-.5
.5
.5
W3
0
.001
0
.001
.001
.001
.001
.001
.001
.001
D3
.060
.063
.060
.064
.061
.064
.062
.067
.060
.063
T3
.058
.060
.060
.065
.059
.061
.074
.084
.062
.066
W2
.105
.105
.107
.105
.045
.047
.042
.041
.006
.005
D2
.147
.146
.136
.131
.099
.102
.078
.075
.071
.069
T2
.147
.146
.157
.158
.077
.081
.195
.182
.038
.040
W23
.004
.005
.089
.092
.032
.030
1
1
.272
.270
D23
.162
.161
.380
.390
.172
.169
1
1
.280
.276
T23
.206
.203
.443
.450
.218
.213
1
1
.340
.336
W32
.338
.340
.547
.546
.259
.258
1
1
.311
.307
D32
.335
.334
.537
.535
.252
.250
1
1
.293
.290
T32
.398
.398
.602
.605
.311
.309
1
1
.353
.349
92b
92c
93b
93c
94b
94c
.5
.5
.5
.5
.5
.5
.2
.2
.2
.2
.2
.2
0
0
-.2
-.2
.2
.2
.001
.001
0
.001
.001
.001
.065
.069
.068
.070
.066
.068
.074
.078
.083
.092
.077
.076
.584
.576
.449
.451
.407
.404
.466
.463
.304
.295
.317
.316
.655
.643
.572
.570
.526
.524
.075
.074
.477
.483
.388
.384
.613
.612
.900
.906
.829
.828
.644
.640
.910
.916
.851
.850
1
1
1
1
1
1
1
1
1
1
.999 .999
1
1
1
1
1
1
95b
95c
96b
96c
97b
97c
.5
.5
.5
.5
.5
.5
.5
.5
.5
.5
.5
.5
0
0
.2
.2
.5
.5
.001
.002
.001
.002
.001
.001
.093
.101
.087
.098
.088
.092
.110
.119
.097
.106
.102
.104
.615
.605
.416
.413
.068
.071
.477
.473
.361
.359
.156
.168
.682
.671
.541
.534
.273
.285
.102
.100
.356
.348
.998
.997
.679
.675
.788
.790
.998
.997
.706
.699
.812
.815
.999
.998
1
1
.999
.999
1
.999
36
1
1
1
1
.999 .999
.999 .999
.999
1
.999
1
.702
.695
.634
.622
.676
.670
1
1
1
1
.997 .999
.997 .999
.999 .999
.998
1
Table 10: Bootstrapped: One endogenous regressor and mildly strong instruments:
2
2
2
2
R2;z2
= :2, R2;z23
= :4, R3;z2
= :2, R3;z23
= :4
0
0
-.2
-.2
.2
.2
-.5
-.5
.5
.5
W3
.053
.055
.053
.054
.052
.056
.053
.049
.052
.050
D3
.048
.052
.051
.051
.050
.052
.051
.049
.050
.049
T3
.051
.055
.052
.053
.052
.054
.052
.048
.052
.050
W2
.050
.051
.052
.051
.049
.050
.049
.047
.048
.049
D2
.046
.047
.049
.047
.046
.048
.049
.048
.048
.047
T2
.048
.050
.050
.049
.048
.049
.050
.047
.048
.048
W23
.051
.051
.052
.052
.052
.054
.053
.053
.050
.051
D23
.051
.051
.052
.052
.052
.054
.053
.053
.050
.051
T23
.051
.051
.052
.052
.052
.054
.053
.053
.050
.051
W32
.047
.049
.049
.046
.048
.046
.054
.051
.049
.047
D32
.047
.049
.049
.046
.048
.046
.054
.051
.049
.047
T32
.047
.048
.049
.046
.048
.046
.054
.051
.049
.047
W 23
.049
.049
.050
.050
.048
.049
.051
.054
.047
.051
D23
.050
.050
.050
.051
.048
.049
.049
.052
.050
.050
T 23
.050
.050
.050
.051
.048
.049
.049
.052
.050
.050
0
0
0
0
0
0
0
0
0
0
0
0
-.2
-.2
.2
.2
-.5
-.5
.5
.5
.054
.056
.052
.056
.053
.053
.051
.051
.051
.049
.049
.052
.051
.052
.051
.052
.051
.048
.051
.049
.051
.054
.051
.053
.052
.053
.052
.049
.052
.049
.176
.177
.173
.172
.169
.171
.159
.147
.150
.162
.164
.168
.162
.159
.155
.159
.138
.126
.128
.138
.172
.174
.169
.170
.166
.169
.159
.148
.150
.161
.051
.051
.079
.088
.088
.083
.680
.684
.692
.687
.051
.051
.079
.088
.088
.083
.680
.684
.692
.687
.051
.051
.079
.088
.088
.083
.680
.684
.692
.687
.162
.159
.191
.192
.195
.190
.740
.742
.750
.743
.162
.159
.191
.192
.195
.190
.740
.742
.750
.743
.162
.159
.191
.192
.195
.190
.740
.743
.750
.743
.125
.130
.150
.156
.158
.155
.697
.699
.710
.705
.132
.138
.153
.156
.160
.156
.652
.650
.657
.654
.132
.138
.153
.156
.160
.156
.652
.650
.657
.654
0
0
0
0
0
0
0
0
-.2
-.2
.2
.2
.050
.055
.050
.052
.051
.052
.051
.055
.053
.053
.054
.053
.049
.053
.048
.051
.050
.050
.864
.861
.833
.823
.827
.833
.847
.845
.788
.781
.777
.785
.861
.858
.833
.824
.825
.832
.053
.054
.342
.354
.356
.344
.053
.054
.342
.354
.356
.344
.053
.054
.342
.354
.356
.344
.844
.842
.932
.932
.938
.935
.844
.842
.932
.932
.938
.935
.844
.842
.932
.932
.938
.935
.775
.774
.896
.899
.902
.901
.788
.786
.896
.900
.901
.901
.788
.786
.896
.900
.901
.901
Case
1b
1c
2b
2c
3b
3c
4b
4c
5b
5c
2
3
23
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
6b
6c
7b
7c
8b
8c
9b
9c
10b
10c
.2
.2
.2
.2
.2
.2
.2
.2
.2
.2
11b
11c
12b
12c
13b
13c
.5
.5
.5
.5
.5
.5
Table 11: Bootstrapped: One endogenous regressor and stronger instruments:
2
2
2
2
R2;z2
= :3, R2;z23
= :6, R3;z2
= :3, R3;z23
= :6
Case
14b
14c
15b
15c
16b
16c
2
3
23
0
0
0
0
0
0
0
0
0
0
0
0
0
0
-.2
-.2
.2
.2
W3
.053
.057
.051
.052
.053
.053
D3
.051
.054
.051
.050
.053
.051
T3
.052
.056
.051
.052
.053
.053
W2
0.047
0.050
0.051
0.050
0.049
0.049
D2
.045
.049
.050
.049
.049
.049
T2
.046
.049
.050
.049
.048
.049
W23
.052
.053
.052
.054
.050
.055
D23
.052
.053
.052
.054
.050
.055
T23
.052
.053
.052
.054
.050
.055
W32
.048
.048
.051
.048
.048
.046
D32
.048
.048
.051
.048
.048
.046
T32
.048
.048
.051
.048
.048
.046
W 23
.049
.050
.050
.051
.048
.049
D23
.049
.051
.050
.051
.049
.049
T 23
.049
.051
.050
.051
.049
.049
17b
17c
18b
18c
19b
19c
.2
.2
.2
.2
.2
.2
0
0
0
0
0
0
0
0
.2
.2
-.2
-.2
.052
.057
.052
.051
.051
.053
.052
.056
.052
.050
.051
.052
.051
.056
.051
.050
.050
.053
0.332
0.333
0.313
0.323
0.320
0.315
.326
.328
.299
.308
.306
.298
.329
.331
.312
.320
.319
.312
.047
.047
.214
.208
.205
.208
.047
.047
.214
.208
.205
.208
.047
.047
.214
.208
.205
.208
.324
.323
.465
.464
.465
.468
.324
.323
.465
.464
.465
.468
.324
.323
.466
.464
.465
.468
.249
.255
.389
.386
.389
.387
.254
.258
.378
.376
.382
.379
.254
.258
.378
.376
.382
.379
20b
20c
21b
21c
22b
22c
.5
.5
.5
.5
.5
.5
0
0
0
0
0
0
0
0
.2
.2
-.2
-.2
.050
.055
.049
.049
.050
.049
.052
.055
.049
.049
.053
.051
.050
.054
.050
.048
.050
.049
1
1
.995
.995
.994
.994
1
1
.989
.989
.989
.989
1
1
.995
.994
.994
.994
.025
.025
.976
.976
.974
.975
.025
.025
.976
.976
.974
.975
.025
.025
.976
.976
.974
.975
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
37
.999 .999 .999
.999 .999 .999
1
1
1
1
1
1
1
1
1
1
1
1
Table 12: Bootstrapped: One endogenous regressor and weak instruments for y (2) :
2
2
2
2
R2;z2
= :01, R2;z23
= :02, R3;z2
= :3, R3;z23
= :6
0 0
0 0
0 .5
0 .5
W3
.034
.037
.014
.016
D3
.038
.038
.046
.049
T3
.033
.036
.024
.026
W2
.029
.027
.029
.028
D2
.046
.047
.047
.045
T2
.042
.042
.045
.042
W23
.051
.053
.051
.055
D23
.051
.054
.051
.055
T23
.051
.054
.051
.055
W32
.050
.048
.050
.054
D32
.050
.048
.050
.054
T32
.050
.048
.050
.054
W 23
.050
.053
.050
.054
D23
.049
.050
.049
.051
T 23
.049
.050
.049
.051
.2
.2
.2
.2
0 0
0 0
0 .5
0 .5
.033
.034
.016
.018
.036
.038
.046
.049
.032
.035
.032
.033
.029
.033
.030
.033
.052
.051
.052
.052
.049
.047
.048
.050
.049
.054
.323
.316
.049
.054
.322
.316
.049
.054
.323
.316
.049
.052
.325
.315
.049
.052
.325
.315
.049
.052
.325
.315
.049
.051
.261
.259
.052
.052
.257
.248
.052
.052
.257
.248
.5
.5
.5
.5
0 0
0 0
0 .5
0 .5
.034
.034
.031
.033
.044
.044
.048
.050
.034
.034
.076
.072
.047
.046
.046
.048
.079
.079
.077
.077
.069
.071
.077
.083
.049 .049 .049
.050 .050 .050
1
1
1
1
1
1
Case
23b
23c
24b
24c
2
3
0
0
0
0
25b
25c
26b
26c
27b
27c
28b
28c
23
.064 .064 .064
.060 .060 .060
1
1
1
1
1
1
.046 .068 .068
.050 .069 .069
.848
1
1
.847
1
1
Table 13: Bootstrapped: One endogenous regressor and weak instruments for y (3) :
2
2
2
2
R2;z2
= :3, R2;z23
= :6, R3;z2
= :01, R3;z23
= :02
0 0
0 0
0 .5
0 .5
W3
.029
.031
.033
.032
D3
.050
.050
.051
.052
T3
.045
.046
.048
.050
W2
.034
.033
.013
.013
D2
.039
.035
.051
.052
T2
.033
.032
.024
.024
W23
.047
.050
.047
.049
D23
.047
.050
.047
.049
T23
.047
.050
.047
.049
W32
.049
.049
.050
.048
D32
.049
.049
.050
.048
T32
.049
.049
.050
.048
W 23
.047
.048
.048
.047
D23
.048
.050
.049
.050
T 23
.048
.050
.049
.050
.2
.2
.2
.2
0 0
0 0
0 .5
0 .5
.031
.031
.030
.032
.051
.053
.050
.051
.047
.050
.051
.054
.177
.186
.038
.041
.141
.145
.062
.060
.178
.186
.074
.079
.157
.159
.867
.866
.157
.159
.867
.866
.157
.159
.867
.866
.320
.323
.876
.879
.320
.323
.876
.879
0.320
0.323
0.876
0.879
.252
.257
.688
.692
.253
.253
.813
.813
.253
.253
.813
.813
.5
.5
.5
.5
0 0
0 0
0 .2
0 .2
.030
.031
.030
.030
.050
.052
.052
.053
.054
.057
.052
.059
.592
.590
.388
.392
.399
.404
.230
.239
.616
.613
.451
.455
.599
.594
.871
.866
.599
.594
.871
.866
.599
.594
.871
.866
1
1
1
1
1
1
1
1
1
1
1
1
Case
29b
29c
30b
30c
2
3
0
0
0
0
31b
31c
32b
32c
33b
33c
34b
34c
23
.844 .999 .999
.842 .999 .999
.861
1
1
.861
1
1
Table 14: Bootstrapped: One endogenous regressor and weak instruments:
2
2
2
2
R2;z2
= R3;z2
= :01, R2;z23
= R3;z23
= :02
Case
35b
35c
36b
36c
2
3
0
0
0
0
0 0
0 0
0 .5
0 .5
W3
.030
.030
.031
.029
D3
.038
.040
.040
.036
T3
.029
.030
.032
.030
W2
.028
.029
.027
.026
D2
.038
.036
.043
.038
T2
.028
.029
.028
.027
W23
.047
.049
.048
.049
D23
.047
.049
.048
.049
T23
.047
.049
.048
.049
W32
.048
.048
.046
.049
D32
.048
.048
.046
.049
T32
.048
.048
.046
.049
W 23
.046
.051
.045
.047
D23
.047
.049
.049
.049
T 23
.047
.049
.049
.049
37b
37c
38b
38c
.2
.2
.2
.2
0 0
0 0
0 .5
0 .5
.029
.026
.027
.030
.038
.039
.042
.043
.028
.028
.031
.030
.026
.030
.028
.026
.037
.037
.040
.040
.027
.029
.030
.028
.051
.050
.050
.055
.051
.050
.050
.055
.051
.050
.050
.055
.049
.050
.053
.056
.049
.050
.053
.056
.049
.050
.053
.056
.048
.051
.048
.052
.051
.053
.056
.054
.051
.053
.056
.054
39b
39c
40b
40c
.5
.5
.5
.5
0 0
0 0
0 .5
0 .5
.029
.032
.035
.035
.042
.043
.057
.054
.031
.032
.040
.039
.033
.034
.045
.040
.048
.046
.058
.062
.036
.036
.047
.047
.059
.059
.087
.086
.059
.059
.087
.086
.059
.059
.087
.086
.065
.060
.094
.092
.065
.060
.094
.092
.065
.060
.094
.092
.053
.056
.072
.073
.065
.067
.100
.098
.065
.067
.100
.098
23
38
Table 15: Bootstrapped: Two endogenous regressors and mildly strong instruments:
2
2
2
2
R2;z2
= :2, R2;z23
= :4, R3;z2
= :2, R3;z23
= :4
Case
57b
57c
58b
58c
59b
59c
60b
60c
61b
61c
2
3
23
.2
.2
.2
.2
.2
.2
.2
.2
.2
.2
-.2
-.2
-.2
-.2
-.2
-.2
-.2
-.2
-.2
-.2
0
0.
-.2
-.2
.2
.2
-.5
-.5
.5
.5
W3
.186
.175
.172
.159
.191
.179
.141
.132
.173
.163
D3
.175
.166
.168
.160
.164
.154
.141
.132
.127
.120
T3
.182
.171
.169
.157
.189
.176
.140
.131
.176
.162
W2
.179
.179
.159
.162
.183
.183
.139
.129
.165
.176
D2
.169
.168
.160
.162
.155
.159
.142
.131
.123
.134
T2
.175
.175
.157
.160
.179
.179
.139
.131
.168
.177
W23
.183
.169
.087
.083
.414
.405
.052
.053
1
1
D23
.183
.169
.087
.083
.414
.405
.052
.053
1
1
T23
.183
.169
.087
.083
.414
.405
.052
.053
1
1
W32
.167
.170
.084
.085
.408
.408
.052
.051
1
1
D32
.167
.170
.084
.085
.408
.408
.052
.051
1
1
T32
.167
.170
.084
.085
.408
.408
.052
.052
1
1
W 23
.229
.227
.138
.135
.469
.453
.084
.082
1
1
D23
.242
.239
.164
.158
.447
.438
.117
.108
1
1
T 23
.242
.239
.164
.158
.447
.438
.117
.108
1
1
62b
62c
63b
63c
64b
64c
65b
65c
66b
66c
.2
.2
.2
.2
.2
.2
.2
.2
.2
.2
.2
.2
.2
.2
.2
.2
.2
.2
.2
.2
0
0
-.2
-.2
.2
.2
-.5
-.5
.5
.5
.171
.182
.171
.188
.158
.166
.161
.176
.128
.141
.160
.172
.149
.161
.158
.165
.119
.128
.130
.140
.167
.179
.169
.185
.155
.164
.163
.178
.128
.141
.176
.176
.184
.184
.161
.164
.175
.167
.132
.141
.167
.168
.160
.157
.160
.163
.135
.123
.133
.143
.173
.173
.180
.181
.158
.161
.179
.170
.132
.141
.165
.174
.397
.414
.082
.086
1
1
.047
.053
.165
.174
.397
.414
.082
.086
1
1
.047
.053
.165
.174
.397
.414
.082
.086
1
1
.047
.053
.169
.165
.409
.399
.083
.083
1
1
.048
.051
.169
.165
.409
.399
.083
.083
1
1
.048
.051
.169
.165
.409
.399
.083
.083
1
1
.048
.051
.220
.232
.456
.463
.128
.138
1
1
.076
.084
.233
.244
.435
.440
.152
.164
1
1
.102
.111
.233
.244
.435
.440
.152
.164
1
1
.102
.111
67b
67c
68b
68c
69b
69c
70b
70c
.5
.5
.5
.5
.5
.5
.5
.5
-.2
-.2
-.2
-.2
-.2
-.2
-.2
-.2
0
0
-.2
-.2
.2
.2
-.5
-.5
.198
.185
.158
.147
.212
.205
.104
.096
.183
.170
.179
.167
.165
.153
.150
.144
.193
.179
.156
.145
.208
.200
.107
.099
.861
.863
.847
.842
.806
.815
.714
.705
.835
.836
.833
.830
.722
.729
.654
.644
.859
.861
.845
.842
.806
.817
.722
.716
.219
.209
.054
.058
.890
.883
.994
.995
.219
.209
.054
.058
.890
.883
.994
.995
.219
.209
.054
.058
.890
.883
.994
.995
.861
.866
.750
.750
.997
.997
1
1
.861
.866
.750
.750
.997
.997
1
1
.861
.866
.750
.750
.997
.997
1
1
.883
.880
.732
.737
.998
.998
.997
.997
.891
.886
.768
.771
.999
.998
1
1
.891
.886
.768
.771
.999
.998
1
1
71b
71c
72b
72c
73b
73c
74b
74c
.5
.5
.5
.5
.5
.5
.5
.5
.2
.2
.2
.2
.2
.2
.2
.2
0
0
-.2
-.2
.2
.2
.5
.5
.181
.192
.199
.211
.147
.155
.097
.102
.167
.177
.148
.163
.166
.174
.141
.149
.176
.187
.195
.206
.145
.153
.098
.106
.870
.860
.820
.809
.842
.846
.708
.718
.842
.833
.734
.722
.828
.833
.644
.651
.866
.857
.822
.808
.840
.845
.716
.727
.205
.215
.882
.889
.060
.058
.995
.994
.205
.215
.882
.889
.060
.058
.995
.994
.205
.215
.882
.889
.060
.058
.995
.994
.864
.861
.997
.997
.749
.746
1
.999
.864
.861
.997
.997
.749
.746
1
.999
.864
.861
.997
.997
.749
.746
1
.999
.874
.878
.998
.998
.734
.736
.997
.997
.883
.889
.998
.998
.769
.767
1
1
.883
.889
.998
.998
.769
.767
1
1
75b
75c
76b
76c
77b
77c
.5
.5
.5
.5
.5
.5
-.5
-.5
-.5
-.5
-.5
-.5
0
0
-.2
-.2
-.5
-.5
.867
.856
.875
.864
.769
.759
.793
.785
.873
.864
.779
.769
.862
.854
.874
.863
.771
.762
.859
.864
.863
.868
.765
.757
.783
.791
.860
.862
.776
.766
.854
.861
.862
.867
.766
.757
.944
.945
.440
.430
.054
.052
.944
.945
.440
.430
.054
.052
.944
.945
.440
.430
.055
.052
.946
.949
.427
.422
.054
.055
.946
.949
.427
.422
.054
.055
.946
.949
.427
.422
.054
.055
.998
.998
.945
.940
.506
.501
1
1
.972
.969
.693
.685
1
1
.972
.969
.693
.685
78b
78c
79b
79c
80b
80c
.5
.5
.5
.5
.5
.5
.5
.5
.5
.5
.5
.5
0
0
.2
.2
.5
.5
.862
.867
.865
.872
.754
.765
.789
.797
.864
.871
.767
.775
.858
.865
.865
.871
.758
.767
.866
.856
.867
.865
.756
.761
.789
.781
.862
.861
.766
.771
.861
.852
.867
.864
.760
.763
.940
.944
.431
.440
.053
.054
.940
.944
.431
.440
.053
.054
.940
.944
.431
.440
.053
.054
.945
.944
.432
.431
.055
.055
.945
.944
.432
.431
.055
.055
.945
.944
.432
.431
.055
.055
.998
.998
.938
.944
.498
.510
1
1
.970
.974
.685
.694
1
1
.970
.974
.685
.694
39
Table 16: Bootstrapped: Two endogenous regressors and stronger instruments:
2
2
2
2
R2;z2
= :3, R2;z23
= :6, R3;z2
= :3, R3;z23
= :6
Case
81b
81c
82b
82c
83b
83c
2
3
23
.2
.2
.2
.2
.2
.2
.2
.2
.2
.2
.2
.2
0
0
-.2
-.2
.2
.2
W3
.331
.340
.318
.330
.304
.308
D3
.322
.333
.282
.294
.308
.312
T3
.329
.338
.316
.329
.303
.308
W2
.339
.338
.333
.328
.308
.312
D2
.332
.331
.297
.292
.311
.315
T2
.337
.337
.330
.326
.306
.309
W23
.324
.330
.924
.927
.097
.102
D23
.324
.330
.924
.927
.097
.102
T23
.324
.330
.924
.927
.097
.102
W32
.336
.332
.928
.928
.099
.099
D32
.336
.332
.928
.928
.099
.099
T32
.336
.332
.928
.928
.099
.099
84b
84c
85b
85c
.5
.5
.5
.5
.2 0
.2 0
.2 .2
.2 .2
.345
.357
.298
.307
.325
.333
.324
.331
.343
.354
.294
.305
.999
.999
.999
.999
.998
.999
.998
.998
.999
.999
.999
.999
.395
.403
.189
.181
.395
.403
.189
.181
.395
.403
.189
.181
1
1
1
1
1
1
.998 .998 .998
.996 .996 .996
86b
86c
.5 .5 .2
.5 .5 .2
.999 .999 .999
.999 .999 .999
1
1
1
1
1
1
.609 .609 .608
.618 .618 .618
.618 .618 .618
.603 .603 .603
W 23
.506
.508
.954
.958
.260
.272
D23
.510
.507
.948
.952
.283
.293
T 23
.510
.507
.948
.952
.283
.293
1
1
1
1
1
1
.999 .999 .999
.999 .999 .999
1
1
1
1
1
1
Table 17: Bootstrapped: Two endogenous regressors and weak instruments for y (3) :
2
2
2
2
R2;z2
= :3, R2;z23
= :6, R3;z2
= :01, R3;z23
= :02
Case
87b
87c
88b
88c
89b
89c
90b
90c
91b
91c
2
3
23
.2
.2
.2
.2
.2
.2
.2
.2
.2
.2
.2
.2
.2
.2
.2
.2
.2
.2
.2
.2
0
0
-.2
-.2
.2
.2
-.5
-.5
.5
.5
W3
.031
.031
.032
.032
.031
.031
.031
.035
.034
.035
D3
.053
.055
.052
.056
.053
.056
.051
.056
.053
.055
T3
.050
.050
.049
.053
.049
.050
.054
.059
.048
.051
W2
.187
.186
.159
.158
.089
.095
.056
.051
.028
.027
D2
.149
.148
.125
.119
.097
.099
.062
.060
.062
.063
T2
.187
.189
.171
.168
.095
.101
.127
.117
.048
.049
W23
.165
.162
.378
.389
.175
.168
1
1
.281
.279
D23
.165
.162
.378
.389
.175
.169
1
1
.281
.279
T23
.165
.162
.378
.389
.175
.169
1
1
.281
.279
W32
.334
.335
.535
.534
.254
.252
1
1
.292
.288
92b
92c
93b
93c
94b
94c
.5
.5
.5
.5
.5
.5
.2
.2
.2
.2
.2
.2
0
0
-.2
-.2
.2
.2
.032
.032
.032
.033
.031
.033
.055
.056
.051
.056
.056
.057
.057
.058
.060
.067
.058
.055
.599
.584
.404
.408
.392
.389
.399
.401
.223
.215
.250
.252
.626
.613
.487
.484
.451
.450
.611
.611
.898
.906
.829
.827
.611
.611
.898
.906
.829
.827
.611
.611
.898
.906
.829
.827
1
1
1
1
1
1
1
1
1
1
1
1
.999 .999 .999
.999 .999 .999
.846
1
1
.837
1
1
.877
1
1
.878
1
1
.843 .999 .999
.839 .999 .999
95b
95c
96b
96c
97b
97c
.5
.5
.5
.5
.5
.5
.5
.5
.5
.5
.5
.5
0
0
.2
.2
.5
.5
.041
.046
.042
.045
.044
.045
.076
.084
.072
.082
.075
.078
.082
.089
.073
.080
.076
.078
.613
.602
.411
.407
.097
.102
.399
.397
.296
.294
.132
.142
.648
.635
.468
.464
.198
.213
.678
.674
.788
.789
.998
.997
.678
.674
.788
.789
.998
.997
.678
.674
.788
.789
.998
.997
1
1
.998
.999
.999
.999
.856
.847
.811
.801
.837
.831
40
D32
.334
.335
.535
.534
.254
.251
1
1
.292
.288
1
1
.998
.999
.999
.999
T32
.334
.335
.535
.534
.254
.251
1
1
.293
.289
1
1
.998
.999
.999
.999
W 23
.260
.269
.415
.418
.204
.203
.876
.877
.237
.232
D23
.268
.266
.447
.446
.202
.203
1
1
.230
.236
1
1
.997
.997
.998
.998
T 23
.268
.266
.447
.446
.202
.203
1
1
.230
.236
1
1
.997
.997
.998
.998
Table 18: Regression results for Griliches data, n = 758
OLS
log W
S
IQ
EXPR
RNS
TEN
SMSA
CONS
IV
Coef. Std. Err.
.0928
.0067
.0033
.0011
.0393
.0063
-.0745
.0288
.0342
.0077
.1367
.0279
3.8952
.1091
IV1
Coef. Std. Err.
.1783
.0186
-.0099
.0052
.0461
.0076
-.1014
.0358
.0398
.0090
.1291
.0321
4.1049
.3552
IV2
Coef. Std. Err.
.1289
.0162
-.0088
.0050
.0348
.0070
-.1096
.0341
.0394
.0086
.1475
.0305
4.6600
.3285
Coef. Std. Err.
.1550
.0113
-.0017
.0013
.0495
.0068
-.0771
.0304
.0363
.0082
.1212
.0296
3.5641
.1244
Table 19: DWH tests for Griliches data
Variables
Test type
Full-set
Full-set
Full-set
Sub-set
Sub-set
Tested
S, IQ
S
IQ
S
IQ
Instruments
Z1 ,Z2
Z1 , Z2 , IQ
Z 1 , Z2 , S
Z 1 , Z2
Z 1 , Z2
Test Statistics
W
D
T
46.87 59.42 65.13
50.56 55.90 60.96
6.28 7.24 7.38
41.16 45.24 46.74
2.72 3.12 2.88
41
Critical values
2
:05
5.99
3.84
3.84
3.84
3.84
^ bc
W
:05
6.27
4.69
3.12
4.46
3.28
^ bc
D
:05
6.66
4.70
3.32
4.64
3.32
bc
T^:05
6.78
4.77
3.37
4.52
3.54