Download Admissible, Similar Tests: A characterization

ADMISSIBLE, SIMILAR TESTS: A CHARACTERIZATION
José-Luis Montiel-Olea1
This paper studies two-sided testing problems with a nuisance parameter.
The main contribution is a characterization of two finite-sample properties:
admissibility and similarity. The characterization result has two parts. The
first part—established by Chernozhukov, Hansen, and Jansson (2009)—states
that maximizing Weighted Average Power (WAP) subject to a similarity constraint suffices to generate admissible, similar tests. The second part—hereby
established—states that constrained WAP maximization is (essentially) a necessary condition for a test to be admissible and similar. The characterization
result shows that if a researcher finds admissibility and similarity attractive,
any additional property of interest, in finite or large samples, could be searched
for inside the WAP-similar class.
In light of the relevance of the WAP criterion in the characterization result, this paper studies the question of how to select a weight function in an
Instrumental Variables model. Since there is no uniformly most powerful test
in this environment, a weight function facilitates comparisons among different
similar tests in empirical applications. The priors proposed by Chamberlain
(2007) are used to motivate weights for the model’s structural parameters.
Analytical results and simulation evidence suggest that the Conditional Likelihood Ratio test of Moreira (2003) almost maximizes WAP with respect to
the weights proposed in this paper.
Keywords: Admissibility, Instrumental Variables, Finite-Sample, Similarity, Statistical Decision Theory, Test, Weak Instruments..
1. INTRODUCTION
I study two-sided hypothesis testing problems with a nuisance parameter. The
main assumption is the existence of a parametric statistical model separately continuous in the parameters and the data. A common example of this environment is
a testing problem concerning the coefficient of a right-hand endogenous regressor in
an Instrumental Variables (IV) model.
1
A previous version of this paper was circulated under the title “Efficient Conditionally Similaron-the-Boundary Tests”. I am deeply indebted to my main advisers, Matías Cattaneo, Gary Chamberlain, Tomasz Strzalecki, and James Stock, for their continuous guidance, support, patience, and
encouragement. I would like to thank seminar audiences at Banco de México, Brown, Chicago Booth
(Econometrics), Cowles Summer Conference, Davis ARE, Duke, Imperial College London-Business
School, ITAM, Northwestern, University of Chicago, University of Michigan, NYU Economics, Penn
State, and TSE. I owe special thanks to Isaiah Andrews, Anna Mikusheva, Elie Tamer, Quang Vuong
and three anonymous referees for extremely helpful comments and suggestions. All errors are my
own. First draft: September 18th, 2013. This version: May 24th, 2016.
1
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JOSÉ LUIS MONTIEL OLEA
I focus on tests that satisfy two classical finite-sample properties: admissibility
and similarity. Admissibility (Wald (1950)) is a weak optimality requirement for a
test φ: there is no other test φ′ with better rates of Type I and Type II error.
Similarity (Neyman (1935)) is an stringent size-control condition for a test φ: the
rate of Type I error of an α-similar test equals α regardless of the the value of
nuisance parameters.
The main result herein shows that admissible, similar tests are—essentially—
Weighted Average Power (WAP) maximizers inside the class of α-similar tests.1
Thus, if a researcher finds admissibility and similarity attractive, the problem of
selecting an admissible similar test is tantamount to selecting an element of the
WAP-similar class.
More formally, the characterization result has two parts. The first part states
that maximizing WAP subject to a similarity constraint suffices to generate tests
that are both admissible and similar (a property that was established in the work
of Chernozhukov et al. (2009) in the context of IV models). The second part—the
main contribution of this paper—states that every admissible, similar procedure is
an extended WAP-similar test (a concept I make precise). As a note for the theorists,
the only assumption needed for the main result is a mild continuity requirement
for the statistical model under consideration. The proof is based on an essentially
Complete Class Theorem (see Theorem 2.9.2 and 2.10.3 in Ferguson (1967) and
also Le Cam (1986), Chapter 2, Theorem 1). A key step is to show that the set of
α-similar tests is compact in the weak∗ topology.
Application—Instrumental Variables: In light of the relevance of the WAP
criterion in the characterization result, I study the question of how to select a weight
function in an overidentified Instrumental Variables model. Since there is no uniformly most powerful test in this environment, a weight function has practical relevance: it facilitates power comparisons among different similar tests via the WAP
criterion. Thus, a weight function can help practitioners in empirical applications—
even if there is no final interest in implementing a WAP-similar test.
To illustrate my point, I propose weights w∗ based on the work of Chamberlain
(2007) and I use them to compare the WAP of different similar tests in Yogo (2004)’s
IV analysis of the Elasticity of Intertemporal Substitution (EIS).
1
WAP-similar tests have been used before in the context of the Instrumental Variables model to
construct a power envelope for the Conditional Likelihood Ratio test [Andrews, Moreira, and Stock
(2006)] and to show that there is no conventional sense in which the Anderson and Rubin (1949)
test wastes power in a Linear homoskedastic Instrumental Variables model.
ADMISSIBLE, SIMILAR TESTS: A CHARACTERIZATION
3
I approximate the WAP of the different tests under consideration using U.S. quarterly data. In find that both the robust version of the Anderson and Rubin (1949)
(AR) test and the Quasi-Conditional Likelihood Ratio (Q-CLR) test of Kleibergen
(2007) are less than three percentage points below the largest WAP that a similar
test can achieve (relative to w∗ ). This observation holds true for a wide-range of
hypothesized values of the EIS parameter. Interestingly, even though the difference
in WAP between the AR and the Q-CLR is negligible, these tests lead to very different confidence sets. For example, if the stock return is used as endogenous regressor
the AR rejects every value of the EIS parameter in the [0, 2] interval. The Q-CLR,
however, rejects none.
Finally, I analyze the properties of the WAP-similar test associated to the weights
w∗
using an i.i.d. homoskedastic design. This environment—which is typically used
to analyze the statistical power of tests in the IV model—is only of theoretical interest, as most data sets exhibit heteroskedasticity and/or autocorrelation of unknown
form. The main finding is that the WAP-similar test is ‘close’ (both analytically
and numerically) to the Conditional Likelihood Ratio of Moreira (2003). I view this
result as evidence of the reasonableness of the weights proposed in this paper.2
Organization of the Paper: The remainder of this paper is organized as
follows. Section 2 presents the finite-sample characterization of admissibility and
similarity. Section 3 focuses on the Instrumental Variables model. The proof of the
main theorem of this paper is presented in Appendix A. All the results related to
the IV model are collected in Appendix B.
2
In order to provide further support for the weights under consideration, I show that in the i.i.d.
homoskedastic model the WAP-similar test corresponding to w∗ satisfies two large-sample properties
that are typically required from testing procedures: it controls the rate of Type I error uniformly
over a large class of data generating processes, and it is locally as powerful as a Generalized Method
of Moments (GMM) Wald-test.
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JOSÉ LUIS MONTIEL OLEA
2. FINITE-SAMPLE THEORY
The main definitions in this section follow Chamberlain (2007); Chapters 2 and 5
in Ferguson (1967); and Chapter 4 in Linnik (1968). The main result of this section
is Theorem 1, which presents the characterization of admissibility and similarity.
2.1. Basic Elements of a Testing Problem and Main Assumption
The finite-sample parametric testing problem studied in this paper has the following components. There is a random vector X that takes values in the sample
space X ⊆ Rs . There is a parameter space Θ = B × Π ⊆ Rdβ ×dπ = Rp whose ele-
ments θ = (β, π) ∈ Θ are used to index a set of probability density functions (w.r.t.
to Lebesgue measure in Rs ) over the sample space, X ∼ f (x; θ). The collection
{ f (x; β, π) }(β,π)∈Θ is called a statistical model.
The null hypothesis H0 states X ∼ f (x; β0 , π) for some π ∈ Π: π is a nuisance
parameter. The null set Θ0 is the set of parameters (β, π) that satisfy H0 . The
alternative hypothesis H1 states X ∼ f (x; β, π) for β 6= β0 , π ∈ Π. The alternative
set Θ1 is defined as Θ\Θ0 .3 The testing problem studied in this paper is abbreviated
as:
H0 : β = β0 vs. H1 : β 6= β0 .
A test is a measurable mapping φ : X → [0, 1]. The scalar φ(x) is interpreted
as the probability of rejecting H0 after a realization x of X. The collection C will
denote the class of all tests.
In a finite-sample set-up, tests are usually compared on the basis of the Type I
and Type II error functions. The rate of Type I error of test φ at θ ∈ Θ0 is defined
as Eθ [φ(X)]. The rate of Type II error of φ at θ ∈ Θ1 is defined as 1 − Eθ [φ(X)].
The rates of Type I and Type II error are typically summarized by the risk func-
tion:
R(φ, θ) ≡









Eθ [φ(X)]
if
θ ∈ Θ0
1 − Eθ [φ(X)] if θ ∈ Θ1 .
Two tests φ, φ′ are said to be risk equivalent if R(φ, θ) = R(φ′ , θ) for all θ ∈ Θ.4
3
The null set is assumed to be nonempty, closed relative to the subspace topology in (Θ, T ), and
with an empty interior. This guarantees that Θ0 coincides with its topological boundary BdΘ0 .
4
This risk function implicitly assumes a “0-1” loss function, which is standard for testing problems; see Ferguson (1967), equation 5.4, p. 199.
ADMISSIBLE, SIMILAR TESTS: A CHARACTERIZATION
5
Assumption F0 below restricts the class of statistical models under consideration.
Assumption F0 (Separate Continuity): The statistical model f (x; θ) is:
i) continuous in θ for almost every x ∈ X,
ii) continuous in x for almost every θ ∈ Θ.
2.2. Admissibility, Similarity, and WAP-similar Tests
This section presents the formal definitions of admissibility and similarity. I then
characterize these properties in two-sided hypothesis problems in which the statistical model satisfies Assumption F0.
Admissibility: (Ferguson (1967), p. 54) The test φ is admissible within the class
C ∗ ⊆ C if there is no φ′ ∈ C ∗ such that R(φ′ , θ) ≤ R(φ, θ) for all θ ∈ Θ, with strict
inequality for at least one θ ∈ Θ.5
Admissibility was first introduced by Wald (1950) and it is a well-known concept
in mathematical statistics. Tests that violate admissibility within a class C ∗ can be
improved (that is, smaller rates of Type I and Type II error can be achieved) all over
the parameter space. Thus, admissibility seems a reasonable minimal requirement
that a test must satisfy. This paper focuses on admissibility with respect to the class
of all tests.
Similarity on Θ0 : A test φ is α-similar on Θ0 (α-s) if:
Eθ [φ(X)] = α,
∀ θ ∈ Θ0 .
Similarity was first introduced by Neyman (1935) and it has been extensively
studied by Linnik (1968). Note that in the two-sided problem studied in this paper
α-s is equivalent to invariance of the rate of Type I error with respect to the nuisance
5
Define an ‘ordering’ over tests as a binary relation ≻ in the space of all tests that verifies two
properties. The first one is asymmetry: φ ≻ φ′ =⇒ φ′ ⊁ φ. The second one is transitivity: φ ≻ φ′
and φ′ ≻ φ′′ implies φ ≻ φ′′ . Admissibility induces an ordering through the “weakly dominated”
binary relation: a test φ′ weakly dominates φ if R(φ′ , θ) ≤ R(φ, θ) with strict inequality for at least
one θ ∈ Θ.
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parameter π:
E(β0 ,π) [φ(X)] = α,
∀ π ∈ Π.
Therefore, in two-sided problems with a nuisance parameter similarity provides a
stringent size-control condition (‘= α’ as opposed to ‘≤ α’).
Characterization Result: The main contribution of this paper is a characterization of admissibility and similarity. As mentioned in the introduction, part i) of
the characterization result shows that WAP-similar tests are admissible in the class
of all tests. Part ii) shows that every admissible test that is α-similar is essentially
a WAP-similar test of level α. In order to state the main result and to formalize the
notion of ‘essentially’ this paper introduces the following definitions. Let w(β, π) be
a probability measure over B × Π.
WAP-similar tests: The α-similar test φw,α
WAP is (w)-WAP-similar if:
WAP(φw,α
WAP , w) ≡
Z
Θ1
Z
X
φw,α
WAP (x)f (x; θ)dx dw(θ) ≥ WAP(φ, w).
for any other α-similar test φ.
WAP-similar tests are indexed by a user-specified weight function w(β, π).6 The
weight function w(β, π) represents the part of the parameter space for which a WAPsimilar test directs its power. This is particularly relevant in problems that do not
admit a Uniformly Most Powerful (UMP) test.
I introduce the notion of an extended WAP-similar test based on the classical
notion of an extended Bayes test, as defined in Ferguson (1967) p. 50, Definition 3:
Extended WAP-similar Tests: The α-similar test φαE-WAP is extended WAPsimilar of level α if ∀ ǫ > 0 there exists a Borel probability measure wǫ (β, π) supported on a non-empty subset of Θ1 ≡ Θ\Θ0 such that:
ǫ ,α
WAP(φαE-WAP , wǫ ) ≥ WAP(φw
wap , wǫ ) − ǫ.
For notational convenience, this paper uses φWAP (x1 , x2 ) instead of φw,α
WAP (x1 , x2 ) whenever
convenient.
6
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ADMISSIBLE, SIMILAR TESTS: A CHARACTERIZATION
Extended WAP-similar tests are essentially WAP-similar tests: for any ǫ > 0 there
is a WAP-similar test whose weighted average power exceeds that of the extended
test by at most ǫ.
Theorem 1:
Suppose Assumption F0 holds. Then,
i) φw,α
wap is admissible and α-similar, provided w(β, π) has full-support on Θ1 .
ii) Every admissible, α-similar test is an extended WAP-similar test of level α.
In both i) and ii), admissibility is taken with respect to the class of all tests.
Proof: See Appendix A.3
Q.E.D.
Comment on Part i): The proof of part i) is straightforward and generalizes the
results in Chernozhukov et al. (2009). The admissibility of WAP tests inside the
class of similar procedures follows directly from the WAP maximization property,
provided the weight function has full-support. Assumption F0 is then used to show
that the continuity of the risk function implies that a constrained WAP maximizer
cannot be dominated by a nonsimilar procedure.
Comment on Part ii): To the best of my knowledge part ii) of Theorem 1 is new.
The proof of ii) is based on an essentially complete class theorem [see Theorem
2.9.2 and 2.10.3 in Ferguson (1967) and also Le Cam (1986), Chapter 2, Theorem
1]. Broadly speaking, the theorem states that if C is an essentially complete class
relative to D (with C ⊆ D) and C is compact (relative to some topology that
makes the risk function continuous), then Extended Bayes rules in C are essentially
complete relative to D. Essential completeness of a class C (relative to D) means
that for any rule in D, there is a rule in C with smaller than or equal risk; see
Ferguson (1967), p. 55, Definition 3. The definition of an extended Bayes rule is
presented in Ferguson (1967) p. 50, Definition 3.
A key lemma in the proof of the paper shows that the set of α-similar tests—
denoted C(α-s)—is compact in the weak∗ topology (see Lemma 1 in Appendix
A). Since the risk function of the testing problem (Type I and Type II error) is
continuous with respect to the same topology, the essentially complete class theorem
applies: this is, the set of extended Bayes tests in C(α-s) is an essentially complete
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JOSÉ LUIS MONTIEL OLEA
class relative to C(α-s) (which is essentially complete relative to itself). The essential
completeness of extended Bayes tests in C(α-s) implies that any admissible test in
C(α-s) must be an extended Bayes test. I then show that extended Bayes tests in
C(α-s) are just extended WAP-similar tests. The desired result follows.
Remark 1:
Finding a WAP test typically requires a researcher to solve an opti-
mization problem over an infinite dimensional choice set. It is well-known that the
α-level (w)-WAP-similar Test for a two-sided testing problem where the data can be
partitioned as X = (x1 , x2 ) and x2 is a boundedly-complete, null-sufficient statistic
(i.e., f (x1 |x2 ; β0 ) ≡ f (x1 |x2 ; β0 , π) = f (x1 |x2 ; β0 , π ′ ) for all π) is given by:
(2.1)
φw,α
wap (x1 , x2 )
(
=
1
if
0 i.o.c.
z(x1 , x2 ) ≡ fw∗ (x1 , x2 )/f (x1 |x2 ; β0 ) > c(x2 ; α)
,
where
(2.2)
fw∗ (x1 , x2 ) ≡
Z
B×Π
f (x1 , x2 ; β, π)dw(β, π) < ∞
∀ (x1 , x2 ),
and for each x2 , c(x2 ; α) corresponds to the conditional (1-α) quantile of the random variable z(X1 , x2 ), with X1 ∼ f (x1 |x2 ; β0 ).7 See Lemma 3 in Appendix A for
the definition of a bounded-complete, null-sufficient statistic and the derivation of
equation (2.1).
7
The acronym i.o.c. stands for in other case
ADMISSIBLE, SIMILAR TESTS: A CHARACTERIZATION
9
3. INSTRUMENTAL VARIABLES MODEL
In light of the relevance of the WAP criterion in the characterization result, this
section revisits the question of how to select a weight function in an over-identified
Linear Instrumental Variables (IV) model. Following Chamberlain (2007), I propose
weights w∗ for the model’s structural parameters. In Section 3.1, I argue that the
weights w∗ can be useful for comparing different testing procedures in models with
heteroskedasticity and/or autocorrelation. In Section 3.2, I study the power properties of the WAP-similar test associated to w∗ using an i.i.d. homoskedastic design.
Set-up and Notation: In a sample of size n, let y ∈ Rn denote a vector that
collects the outcome variable for each observation in the sample. Likewise, x ∈ Rn
denotes the vector that collects all the observations of the endogenous regressor and
Z ∈ Rn×k is the matrix that collects the k instrumental variables. The parameter
β denotes the coefficient of the endogenous regressor of interest and π denotes the
first-stage coefficient. The testing problem is:
H0 : β = β0
vs. H1 : β 6= β0 ,
where β0 is some prespecified value and π, the first-stage coefficient, is the nuisance
parameter.
Statistical Model for IV: Let γbn denote the OLS estimator of the reducedform parameters in the IV regression. The parametric statistical model specifies a
multivariate normal distribution for γbn :
(3.1)
γbn ∼ N2k
βπ
π
!
!
Σ
,
,
n
where γbn ≡
(Z ′ Z)−1 Z ′ y
!
(Z ′ Z)−1 Z ′ x
.
Thus, the application of the theory in the first part of the paper will require
the Gaussian model above to approximate the finite-sample distribution of γbn .8
Fortunately, a Central Limit Theorem (CLT) guarantees that (3.1) will be, indeed,
a reasonable approximation for a wide range of data generating processes (including
stochastic instruments, time varying heteroskedasticity, autocorrelation of unknown
form, etc). The analysis provided in this paper does not cover the case in which the
8
This parametric model has been considered before by Müller (2011), p. 423; Moreira and Moreira
(2015), p. 10; and I.Andrews (2015), p. 6.
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JOSÉ LUIS MONTIEL OLEA
Gaussian model does not provide a good approximation for the distribution of γbn .
Auxiliary Notation to define Weights (ΨΣ and ΦΣ ): In order to define the
weights for the IV model define ΨΣ and ΦΣ as the matrices that solve the following
optimization problem:
(3.2)
min
Ψ∈R2×2 ,Φ∈Rk×k
Σ − (Ψ ⊗ Φ) ,
F
where || · ||F denotes the Frobenius norm; see Van Loan and Pitsianis (1993) pp.
293-297 and p. 5 in Moreira and Moreira (2015) (MM15).9 Informally, the matrices
ΨΣ and ΦΣ decompose Σ into the variance of the instruments and the variance of
the endogenous regressors.
Weights for (β, π) in the IV-Model: Let ρ denote a random variable supported on R+ . Let φ denote a random vector supported on S 1 , the unit sphere. Let
ω denote a random vector supported on S k−1 , the k − 1 sphere.10 Let a0 = (β0 , 1)′ ,
b0 = (1, −β0 )′ . Consider weights on (β, π) of the following form:
(3.4)
βπ
π
!
=n
−1/2
ΨΣ C0′
⊗
1/2
ΦΣ
ρ(φ ⊗ ω), C0 ≡
(b′0 ΨΣ b0 )−1/2 b′0
−1/2 a′ Ψ−1
(a′0 Ψ−1
0 Σ
Σ a0 )
!
,
where ΨΣ and ΦΣ are defined as in (3.2); φ and ω are two independent random
variables, uniformly distributed in their domain:
(3.5)
φ ∼ U(S 1 ),
ω ∼ U(S k−1 ),
9
Partition Σ in blocks of size k × k and define the rearrangement of Σ, R(Σ), as the following
4 × k2 matrix:
Σ=
Σ11
Σ21
Σ12
Σ22
,
R(Σ) ≡ vec(Σ11 ),
vec(Σ21 ),
vec(Σ12 ),
′
vec(Σ22 ) .
Let s∗ denote the largest singular value of R(Σ). Let U denote the eigenvector corresponding to
the largest eigenvalue of R(Σ)R(Σ)′ and let V denote the eigenvector corresponding to the largest
eigenvalue of R(Σ)′ R(Σ). Thus, U and V are the left and right singular vectors corresponding to
the largest singular value of R(Σ), respectively. Define ΨΣ ∈ R2×2 and ΦΣ ∈ Rk×k as the auxiliary
matrices that satisfy:
(3.3)
vec(ΨΣ ) = s∗ U ∈ R4 ,
2
vec(ΦΣ ) = V ∈ Rk .
These matrices solve the minimum distance problem in (3.2).
10
For any m ∈ N, S m is the m unit sphere; that is S m = {x ∈ Rm+1 | ||x|| = 1}.
11
ADMISSIBLE, SIMILAR TESTS: A CHARACTERIZATION
and
(3.6)
ρ|φ, ω ∼ χ2k / (φ′ ⊗ ω ′ ) C0 Ψ′Σ ⊗ Φ′Σ
1/2
1/2
Σ−1 ΨΣ C0′ ⊗ ΦΣ
1/2
(φ ⊗ ω)
,
with a χ2k random variable independent of φ and ω.
These weights are based on a convenient reparameterization of the IV model
analyzed in Chamberlain (2007).11 In such reparameterization ω and φ represent—
respectively—the ‘direction’ of the first-stage coefficient, π, and also the direction
of the alternative hypothesis, (β, 1)′ . The parameter ρ represents the product of the
norms of each of these vectors and it can be thought of as a measure of instrument
strength. Chamberlain (2007) considered uniform priors for φ, ω (uninformative priors for the directions) but left the distribution of ρ unspecified. The priors for φ and
ω were motivated by studying a minimax problem (they are least-favorable). These
priors aimed to be context-free, although Chamberlain (2007) was explicit on the
fact that other prior specifications could be available depending on the application.
Remark 2 (A priori reasonableness of w∗ ):
Let γ(β, π) = (βπ, π)′ ∈ R2k denote
the vector of reduced-form parameters of the IV model. Define the Wald ellipsoid
for γ(β, π) (with radius r centered at γ ∗ ) as:
(3.7)
n
o
Wn (γ ∗ , r) ≡ γ ∈ R2k | n(γ − γ ∗ )′ Σ−1 (γ − γ ∗ ) ≤ r .
The choice of weights for (β, π) implies that:
P∗ (γ(β, π) ∈ Wn (0, r)) = P∗ ((βπ, π) ∈ Wn (0, r))

= P∗ n
βπ
π
!′
Σ−1
βπ
π
!
= P χ2k ≤ r ,
11

≤ r ,
The original parameters (β, π) induce the following “canonical” parameters (ρ, φ, ω):
ρ = (a′ Ω−1 a)1/2 (β ′ Z ′ Zβ)1/2 , φ = C0 a/(a′ Ω−1 a)1/2 , ω = (Z ′ Z)1/2 β/(β ′ Z ′ Zβ)1/2 .
This yields the following parameterization of the IV model:
Sn
Tn
∼ N2k
ρ(φ ⊗ ω),
I2k
,
ρ ∈ R+ ,
φ ∈ S 1,
ω ∈ S k−1 .
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JOSÉ LUIS MONTIEL OLEA
where P∗ denotes the probability induced by (3.5) and (3.6). Thus the a priori
probability that the reduced-form parameters belong to a Wald elliposid centered
around the origin (under the weights for ρ, φ and ω) is given by the c.d.f. of a
chi-square random variable with k degrees of freedom.
Now, fix π = 0 and note that:
Pπ=0 (γbn ∈ Wn (0, r)) = P χ2k ≤ r .
That is, the prior probability that reduced-form parameters γ belong to Wn (0, r) is
exactly the same as the probability that the estimated reduced-form parameters—in
repeated samples—fall in the same set.12 One can conclude that if the instrumental
variables have no correlation with the endogenous regressor, the weights w∗ and the
statistical model for the data will assign the same probability to any Wald ellipsoid
centered at the origin. I take this result as suggestive that the weights w∗ are a
priori reasonable when instrument weakness is a concern.
Note that I have chosen to motivate the a priori reasonableness of the weights
w∗
by relying solely on the behavior of the reduced-form parameters. Consequently,
researchers should not expect any hidden economics buried in this weight function.
The weighting scheme w∗ provides an automatic, context-free tool to direct power
in IV models. If researchers want to consider other weight functions based on their
knowledge of the structural parameters of the model, they can still conduct conduct
WAP comparisons (using the Gaussian statistical model for γbn .)
12
Assuming that the reduced-form coefficients are distributed according to the statistical model
(3.1) and that π = 0.
ADMISSIBLE, SIMILAR TESTS: A CHARACTERIZATION
13
3.1. WAP comparison in an empirical application
This section argues that the weights w∗ can be useful to choose among different
tests by relying on a WAP comparison using the model (3.1). Moreover, the WAPsimilar test for w∗ (which can be computed numerically) provides an upper bound
for the WAP of similar tests. I illustrate these points in the context of a concrete
empirical application.
Tests about the Elasticity of Intertemporal Substitution: I.Andrews
(2015) uses Yogo (2004)’s Elasticity of Intertemporal Substitution (EIS) exercise
to analyze the finite-sample performance of different tests in an IV model. I use
the same application and report WAP for different tests that are common in the
literature.
The outcome variable in the EIS exercise is the first difference of log consumption growth. The right-hand endogenous regressor is either the risk-free interest
rate or the stock market return. The coefficient of the endogenous regressor, β0 , is
interpreted as the Elasticity of Intertemporal Substitution. There are four instrumental variables: the nominal interest rate, inflation, consumption growth, and the
log dividend-price ratio, all lagged twice.
The data is available at a quarterly frequency for each of 11 countries: Australia,
Canada, France, Germany, Italy, Japan, Netherlands, Sweden, Switzerland, United
Kingdom and the United States. Montiel Olea and Pflueger (2013) have argued that
except for 5 countries (Australia, Canada, France, Italy, and Sweden), it is not possible to reject the null hypothesis of weak instruments using the effective F statistic.
Test Inversion for the EIS parameter: I ‘invert’ three well-known 5%-similar
tests to construct a 95% confidence interval for the EIS parameter: the robust
Anderson and Rubin (1949) test, the K test, and the Quasi-Conditional Likelihood
Ratio (both proposed by Kleibergen (2007)). I consider values for β0 in a uniform
grid over the interval [0, 2] with steps of size .01. I estimate the matrix Σ (required by the three tests) using the Newey and West (1987) estimator with three
lags (Script_1_ReducedForm_Yogo.m). The results are reported on Table I.1 and
Table I.2. These tables were generated using the script Script_2_CIntervals.m.
The sample periods are the same in both tables.
14
JOSÉ LUIS MONTIEL OLEA
Table I.1: Confidence Intervals for the EIS
(Endogenous regressor: Risk-Free Interest Rate)
Country
Sample
AR
K
QCLR
Australia
Canada
1970.3-1998.4
1970.3-1999.1
[0 , .17]
[.89 , 2]
France
1970.3-1998.3
∅
∅
[0 , .26]
[0 , .12]
[0 , .17]
∅
Germany
1979.1-1998.3
[0 , .15]
Italy
1971.4-1998.1
[0 , .29]
∅
[0 , .16]
[0 , .16]
Japan
1970.3-1998.4
[0 , .10]
[0 , .39]∪[1.59 , 2]
[0 , .36]
Netherlands
1977.3-1998.4
[0 , .03]∪[1.61 , 2]
[0 , .01]
Sweden
1970.3-1999.2
∅
[0 , .17]
[0 , .13]
[0 , .13]
Switzerland
1976.2-1998.4
[0 , .03]
[1.62 , 2]
UK
1970.3-1999.1
[0 , .41]∪[1.75 , 2]
[0 , .4]
US
1970.1-1998.4
∅
∅
[0 , .39]
[0 , .37]
∅
[0 , .13]
∅
Table I.2: Confidence Intervals for the EIS
(Endogenous regressor: Stock Return)
Country
AR
K
QCLR
Australia
∅
[0 , 2]
[0 , 2]
Canada
[.04 , 2]
[.07 , .35]
[.06 , .49]
France
[0 , .04]
[0 , .02]∪[.12 , .16]
[0 , 0.02]
[0 , .29]∪[1.8, 2]
[0, .94]
[0 , .27]∪[.36 , .77]
Italy
[0 , 2]
[0 , 2]
[0 , 2]
Japan
[0 , .12]
[0 , .11]∪[.46 , 2]
[0 , .11]
Netherlands
[.06 , 2]
[0 , .02]∪[0.09 , 2]
[0 , .01]∪[0.09 , 2]
Sweden
[0 , 2]
[0 , 2]
[0 , 2]
Switzerland
[0 , 2]
[0 , .04]∪[.12 , 2]
[0 , 2]
UK
[.15 , 2]
[0.03 , .05]∪[.11 , 2]
[.14 , 2]
US
∅
[0 , 2]
[0 , 2]
Germany
ADMISSIBLE, SIMILAR TESTS: A CHARACTERIZATION
15
WAP comparison of Different tests: The asymptotic validity and the local
power properties of the AR, K, and Q-CLR rely on the assumption that:
γbn
approx
∼
N2k
βπ
π
!
!
Σ
,
.
n
I use this statistical model and the weights w∗ to approximate the Weighted
b I also include the 5%-WAP-similar
Average Power of each test, replacing Σ by Σ.
test corresponding to w∗ , which provides an upper bound on the WAP that any
5%-similar test can achieve.
Figure 1 presents the WAP comparison for the United States. The figure suggests
that the Q-CLR and the AR are close to maximizing WAP relative to w∗ . Although
the difference in WAP between the AR and the Q-CLR is negligible, they lead to
very different confidence sets. For example, if the stock return is chosen as the endogenous regressor the AR rejects every null hypothesis in the [0, 2] interval, while
the Q-CLR rejects none. The WAP in each figure is computed using 500 realizations
from the weights w∗ for (β, π). In addition, 1,000 MC draws of the Gaussian model
are used to compute the power of each test at each alternative (β, π). The figure is
generated using the script file Script_3_WAPComparisonUS.m.
Is the EIS above or below 1?: Bansal and Yaron (2004) have proposed a ‘longrun risks’ model to explain the importance of the expectations of long-run consumption growth and its volatility in the movements of stock prices. A key assumption
in their framework is that the representative investor in the economy has an EIS
parameter above 1. Schorfheide, Song, and Yaron (2014) have developed a novel
non-linear state-space model to identify long-run risks, and have estimated an EIS
parameter above 1 using U.S. data.
The tests under consideration in this paper are only valid for two-sided problems
(β = β0 vs. β 6= β0 ). However, one can still look at the values of the EIS parameter
that are rejected in Yogo (2004)’s empirical framework. Table I.1 shows that all the
values of the EIS above 1 are rejected by the AR, the K, and the Q-CLR in 7 out of
the 11 countries under consideration. The results are different if the stock return is
used as an endogenous regressor: in Table I.2 there is only one country (France) for
which the three tests reject all the values of the EIS above 1. In the particular case
of the United States, the AR test (which has the highest WAP) rejects every value
of the EIS in the [0, 2] interval (regardless of the choice of endogenous regressor).
Figure 1: WAP Comparsion for the United States
(β0 ∈ [0, 2])
20
20
WAP Bound
AR
K
Q-CLR
WAP Bound
AR
K
Q-CLR
18
16
16
14
14
Weighted Average Power (%)
Weighted Average Power (%)
JOSÉ LUIS MONTIEL OLEA
18
12
10
8
6
12
10
8
6
4
4
2
2
0
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
β0
(a) Risk-Free Interest Rate
1.6
1.8
2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
β0
(b) Stock Return
16
Description: Figure 1 presents a WAP comparison between the AR test, the K test, and the Q-CLR. The WAP is computed
using the weights w∗ . The testing problem is H0 : β = β0 vs. H1 : β 6= β0 and β0 is taken from a uniform grid over [0, 2] with
b ). The AR test has slightly more
steps of size 0.1. The statistical model used to compute WAP assumes that b
γn ∼ N2k ((βπ ′ , π ′ )′ , Σ
power than the Q-CLR for each of the hypothesis β0 ∈ [0, 2]. The WAP bound (black, cross) corresponds to the Weighted Average
Power of the w∗ -WAP similar test based on the Gaussian model for b
γn . Both the AR and the Q-CLR are less than 3 percentage
points below this bound for every β0 ∈ [0, 2].
1
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
F(β)
1
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
-3
-2
-1
0
1
β
(a) Risk-Free Interest Rate
2
3
0
-3
-2
-1
0
1
2
3
β
(b) Stock Return
Description: Figure 2 presents the weights over the EIS induced by the weights w∗ . The testing problem is H0 : β = 1 vs.
H1 : β 6= 1. Note that the weights for (β, π) depend on the estimated covariance matrix Σ, so the c.d.f. of β changes depending
on the endogenous regressor used in the IV specification. Even though negative values of the EIS parameter are not a priori
reasonable from an economic point of view, the weights w∗ (motivated by statistical behavior of the reduced-form parameters)
still direct power towards negative alternatives.
ADMISSIBLE, SIMILAR TESTS: A CHARACTERIZATION
F(β)
Figure 2: Weights over β induced by w∗ when β0 = 1
(Cumulative Distribution Function)
17
18
JOSÉ LUIS MONTIEL OLEA
3.2. WAP comparison for the i.i.d. homoskedastic design
In models with i.i.d. conditionally homoskedastic data Σ can be written as Ω ⊗
Q−1 ,
where Ω is the matrix of second moments of reduced-form residuals, Q is
the matrix of second moments of the instrumental variables, and ⊗ denotes the
Kronecker product. This ‘Kronecker’ case—which is commonly used to analyze the
power properties of tests in the IV model—is primarily of theoretical relevance.
In this section I study how the WAP similar test based on w∗ compares to the Con-
ditional Likelihood Ratio (CLR) of Moreira (2003), which is the literature’s current
recommendation for the Kronecker model. The main finding is that WAP-similar
test is ‘close’to the CLR. As mentioned in the introduction, this is taken as further
support of the reasonableness of the weights w∗ .
IV weights specialized to the Kronecker case: If Σ is of the form Ψ ⊗ Φ
then the matrices ΨΣ and ΦΣ defined in (3.2) are given by:13
ΨΣ = (vec(Φ)′ vec(Φ))1/2 Ψ,
ΦΣ = Φ/(vec(Φ)′ vec(Φ))1/2 .
Therefore,
ρ|φ, ω ∼
=
χ2k / (φ′ ⊗ ω ′ ) C0 Ψ′Σ ⊗ Φ′Σ
1/2
1/2
Σ−1 ΨΣ C0′ ⊗ ΦΣ
1/2
(φ ⊗ ω)
1/2
(vec(Φ)′ vec(Φ))−1/2 χ2k / (φ′ ⊗ ω ′ ) C0 ΨC0′ ⊗ Ik (φ ⊗ ω)
∼
,
q
χ2k .
This means that in the Kronecker case ρ is distributed as the square root of a χ2k
independently of φ and ω (which are uniform on S 1 and S k−1 , respectively).
Appendix B.1 shows that, in the Kronecker case, the weights for (β, π) proposed
in this section—which
are Chamberlain (2007)’s weights with the additional assumpq
tion that ρ ∼
χ2k —coincide with the MM2 weights proposed by Moreira and Moreira
(2015) (up to a scaling parameter). The equivalence does not hold if Σ does not have
a Kronecker form.
13
To see this, note that in the Kronecker case, the rearrangement R(Σ) is given by:
vec(Ψ) ⊗ vec(Φ)′ .
Therefore, R(Σ)R(Σ)′ = (vec(Ψ)vec(Ψ)′ )vec(Φ)′ vec(Φ) ∈ R2×2 and the eigenvector corresponding
to the largest eigenvalue of this matrix given by vec(Ψ)/(vec(Ψ)′ vec(Ψ))1/2 . Likewise the matrix
R(Σ)′ R(Σ) = (vec(Ψ)′ vec(Ψ))vec(Φ)vec(Φ)′ ∈ Rk×k and the eigenvector corresponding to the
largest eigenvalue is vec(Φ)/(vec(Φ)′ vec(Φ))1/2 .
19
ADMISSIBLE, SIMILAR TESTS: A CHARACTERIZATION
Distribution of
√
λ(β − β0 ) and λ: The i.i.d. homoskedastic model has been
analyzed in detail in the work of Andrews et al. (2006). The Monte-Carlo exercises
reported in that paper depend on the parameters:
λ ≡ nπΦ−1 π, and
√
λ(β − β0 ).
√
The probability density function of ( λ(β − β0 ), λ) is given in Figure 3 below.
The WAP-similar test directs its power according to this density:
√
Figure 3: Weights for ( λ(β−β0 ), λ) induced by the weights on (ρ, φ, ω)
0.1
0.08
0.06
0.04
0
0.02
5
10
0
6
4
2
15
0
λ
-2
1/2
-4
λ
-6
20
(β-β 0 )
Description: Figure 3 is based on weights (3.5), (3.6) for the parameters (φ, ω, ρ)
and the formulas in equations (B.7), (B.8) in the Appendix. The matrix Ψ is assumed
to have unit diagonal elements and correlation parameter r = .5. The matrix Φ is
assumed to be the identity. The null hypothesis is β0 = 0. There are 4 instruments,
k = 4. The bivariate density is generated by a Monte-Carlo exercise with 50,000
independent draws from (ρ2 , φ) and the Matlab bivariate density estimator gkde2.
The script used to generate this figure is Weights_WAP2016.m.
20
JOSÉ LUIS MONTIEL OLEA
I now show that, under the proposed weights, there is a closed-form solution for
the test statistic of the corresponding WAP-similar test.
Result 1 (WAP-similar test for Kronecker IV):
Ψ ⊗ Φ where Ψ ∈
R2×2
and Φ ∈
Rk×k
Suppose that Σ =
are positive definite, symmetric matrices.
The α-WAP similar test for the problem H0 : β = β0 vs. H1 : β 6= β0 given the
statistical model (3.1) and the weights over (β, π) induced by (3.5), (3.6) rejects the
null hypothesis if the statistic zwap (Sn , Tn ):
1h
(Sn′ Sn −Tn′ Tn )+8 ln I0
i1/2 (Sn′ Sn −Tn′ Tn )2 +4(Sn′ Tn )2 )
8
+4 ln(2pi)+4 ln((1/8)Tn′ Tn )
exceeds the critical value function cwap (Tn , α). The critical value function is defined
as the (1 − α) quantile of the distribution of the statistic above with S ∼ Nk (0, Ik )
and Tn fixed. The function I0 (·) is the modified Bessel function of the first kind
of order zero defined in Section 9.6, p. 374 of Abramowitz and Stegun (1964). The
value 3.1415 . . . is abbreviated as ‘pi’. The statistics Sn and Tn are given by:
Sn
Tn
([b′0 ⊗ Ik )Σ(b0 ⊗ Ik )]−1/2
0
!
≡
0
[(a′0
⊗
Ik )Σ−1 (a0
⊗ Ik
)]−1/2
!
(b′0 ⊗ Ik )
(a′0
⊗ Ik
with a = (β, 1)′ , a0 = (β0 , 1)′ , b0 = (1, −β0 )′
Proof: See Appendix B.2.
)Σ−1
!
√
nγbn ,
Q.E.D.
Comment on Result 1: If the expression:
8 ln I0
1h
8
i1/2 (Sn′ Sn − Tn′ Tn )2 + 4(Sn′ Tn )2 )
were to be replaced by:
1/2
(Sn′ Sn − Tn′ Tn )2 + 4(Sn′ Tn )2 )
,
the test in Result 1 would be equivalent to the Conditional Likelihood Ratio test.
Note that for large values of Tn′ Tn it is possible to motivate the desired substitution,
up to an error term. In fact, it is common practice to evaluate the modified Bessel
ADMISSIBLE, SIMILAR TESTS: A CHARACTERIZATION
21
function I0 (x) using the asymptotic approximation in Olver (1997), p. 435:
I0 (x) ≈
ex
, which implies zwap (s, t) ≈ 2CLR(s, t).
(2pix)1/2
To further analyze the similarities (and differences) between the test in Result 1 and
the CLR, I report a) the correlation between the two tests, b) conventional power
plots, and the c) weighted average power of the two tests.
Correlation Between the WAP-similar test and the CLR: Under the null
hypothesis, any nonrandomized test is a Bernoulli random variable with success
probability equal to its rate of Type I error. Thus, one way to understand whether
the CLR is ‘close’ to the test in Result 1 is by reporting the correlation between
the two tests under the null hypothesis. Figure 6 in Appendix B.3 suggests that the
test in Result 1 and the CLR have a high correlation under the null hypothesis.
Power Plots: The analysis of correlation has nothing to say about the similarities
(or differences) in power perfomance. Figure 4 presents a standard power comparison
between the WAP-similar test in Result 1 and the Anderson and Rubin (1949) test
(AR), the Lagrange Multiplier Test (LM), the CLR, and the POSI2 (Point-Optimal
Similar, Invariant Two-sided test). The figure reports the average power computed
over the grid of values of λ1/2 (β −β0 ) using uniform weights and fixing the value of λ
under consideration (λ = nπ ′ Φ−1 π). The figures also contain an (infeasible) power
curve of the test that rejects whenever |π ′ Φ−1/2 S|/(π ′ Φ−1 π)1/2 > 1.96. Figure 4
suggests that the power curves of the WAP-similar test in Result 1 and the CLR are
almost indistinguishable. Also, the figure shows that there are alternatives (β, λ) for
which the curve of the POSI2 test seems closer to the AR test than to the CLR.
Weighted Average Power Comparsion: The weights that define the WAPsimilar test Result 1 facilitate numerical comparison of testing procedures. For instance, in the context of Figure 4, the WAP comparison using weights (3.5) and
(3.6) is as follows: 24% for the WAP-similar test, 23.8% for the CLR, 22.2% for the
AR and 18.5% for the LM. Figure 5 compares the WAP of the WAP-similar test
against that of the CLR, LM, and AR for a wider range of reduced-form correlations
(r ∈ [−.9, .9]). One can claim that with 4 instruments the CLR is .15% away from
maximizing weighted average power among the class of α-similar tests, given the
22
JOSÉ LUIS MONTIEL OLEA
proposed weights in (3.5) in (3.6). I also report the WAP using a uniform distribution over the area displayed in Figure 3 in Appendix B.3. The WAP comparison is
as follows: 62.33% for the WAP-similar test; 62.24% for the CLR; 56.10% for the
LM; and 59.02% for the AR.
Conditional Rejection Region: The WAP-similar test in Result 1 (R1) is measurable with respect to the triplet (AR, LM, T ′ T ). It is natural to ask whether the
WAP-similar test rejects the null hypothesis when both the AR and LM do. Figure
7 in Appendix B.3 reports ‘conditional’ critical regions in the (AR, LM) space for
two different values of T ′ T . The conditional critical regions suggest that the WAPsimilar test in Result 1 can be well approximated by a Linear Combination between
the AR and the LM as the tests in I.Andrews (2015).
Large-Sample properties: The test in Result 1 was derived under the assumption
that the rotated reduced-form OLS estimators (Sn′ , Tn′ )′ have the exact distribution:
Qnβ,π,Σ
≡ N2k
√
([b′0 ⊗ Ik )Σ(b0 ⊗ Ik )]−1/2 (β − β0 ) nπ
!
√ , I2k ,
[(a′0 ⊗ Ik )Σ−1 (a0 ⊗ Ik )]−1/2 (a′0 ⊗ Ik )Σ−1 (a ⊗ Ik ) nπ
where Σ is of the form Ψ ⊗ Φ. In any finite sample, however, the law of (Sn′ , Tn′ )′ is
a function of (β, π), the sample size, and the joint distribution between the instru-
mental variables and reduced-form residuals, denoted F . In fact, one can write:
b 0 ⊗ I )]−1/2 (b′ ⊗ I )
([b′0 ⊗ Ik )Σ(b
k
k
0
[(a′0
b −1
b −1 (a0 ⊗ I )]−1/2 (a′ ⊗ I )Σ
⊗ Ik )Σ
k
k
0
!
√
n
nγbn ∼ Pβ,π,F
,
b is an estimator of the variance of √nγ
bn . This variance depends on F and
where Σ
b need not have the Kronecker
such dependence is denoted Σ(F ). The estimator Σ
form, even when Σ(F ) does.
If one assumes that for n large enough the distributions P n and Qn are close to
each other (under the null), then one would expect the rate of Type I error computed
under P n to be close to that obtained under Qn . I formalize this statement in
Appendix B.4. I also show that the test in Result 1 is as powerful (locally) as the
GMM-Wald test for β0 (Appendix B.5) based on the sample moment condition:
1
√ Z ′ (y − β0 x) = 0.
n
Figure 4: WAP-similar test (R1) vs. CLR, LM, AR, POSI2 (Power Plots)
(k = 4, r = .5)
1
1
0.9
0.9
χ 21(λ(β -β 0)2)--67.60%
R1--60.96%
CLR--60.71%
LM--50.86%
AR--58.94%
POSI2--59.70%
0.8
0.7
Power
Power
0.6
0.5
0.3
0.3
0.2
0.2
0.1
0.1
-4
-2
0
2
4
0
-6
6
-4
-2
0
λ 1/2 (β-β 0 )
λ 1/2 (β-β 0 )
(a) λ = 1
(b) λ = 5
2
4
6
2
4
6
1
0.9
0.9
χ 21(λ(β -β 0)2)--67.60%
R1--63.94%
CLR--63.98%
LM--54.84%
AR--58.99%
POSI2--59.83%
0.8
0.7
χ 21(λ(β -β 0)2)--67.60%
R1--65.40%
CLR--65.45%
LM--63.23%
AR--58.95%
POSI2--59.85%
0.8
0.7
0.6
Power
0.6
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
-4
-2
0
λ 1/2 (β-β 0 )
4
6
0
-6
-4
-2
0
λ 1/2 (β-β 0 )
(d) λ = 20
Description: Figure 4 reports power curves for WAP-similar test in Result 1 (denoted R1), the CLR, LM, AR, the POSI2 in
Andrews et al. (2006), and an infeasible test that rejects whenever |π ′ Φ−1/2 S|/(π ′ Φ−1 π)1/2 > 1.96. The POSI2 test is evaluated
at β = 2.1 and λ = 1. The figure suggests that the power curves of the test R1 and that of the CLR are almost indistinguishable.
The numbers in the box are weighted average power for a each fixed λ and a uniform grid of 121 points for λ1/2 (β − β0 ) ∈ [−6, 6].
The script used to generate this figure is PowerPlots.m.
23
(c) λ = 10
2
ADMISSIBLE, SIMILAR TESTS: A CHARACTERIZATION
0.4
1
Power
0.5
0.4
0
-6
R1--62.30%
CLR--62.15%
LM--49.86%
AR--58.95%
POSI2--59.73%
0.7
0.6
0
-6
χ 21(λ(β -β 0)2)--67.60%
0.8
Figure 5: WAP-similar test (R1) vs. CLR, LM, AR (WAP)
(k = 4, Φ = I4 , Ψ(1, 1) = Ψ(2, 2) = 1)
0.25
0.24
Weighted Average Power
JOSÉ LUIS MONTIEL OLEA
0.23
0.22
Test in Result 1
CLR
LM
AR
POSI2
0.21
0.2
0.19
0.18
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
24
Reduced-Form Correlation of Ψ
Description: Figure 5 presents a comparison of Weighted Average Power between the test in Result 1 (denoted R1), the CLR,
LM, AR, and the POSI2 in Andrews et al. (2006). The weights in (3.4) for the parameters (β, π) are evaluated using Φ = I4 and
Ψ = [1, r; r, 1], with r ∈ [−.9 : .1 : .9]. The null hypothesis is β0 = 0. The sample size is n = 100. The figures use 500 draws from
(β, π) and 1,000 Monte-Carlo draws to compute the power at each point. The POSI2 test is evaluated at β = 2.1 and λ = 1.
The figure suggests that the WAP of R1 and that of the CLR are almost the same. The script used to generate this figure is
WAPComparison_KronIV.m.
ADMISSIBLE, SIMILAR TESTS: A CHARACTERIZATION
25
4. CONCLUSION
This paper studied two-sided testing problems with a nuisance parameter. The
leading example was a testing problem concerning the coefficient of a single righthand endogenous regressor in an Instrumental Variables (IV) model.
The main result in this paper showed that WAP-similar tests characterize two important finite-sample properties: admissibility and similarity. The characterization
result stated that WAP-similar tests are admissible and similar; but more important,
that every admissible, similar procedure is essentially a WAP-similar test. Thus, if
a researcher finds admissibility attractive and at the same time desires to make the
rate of Type I error invariant to nuisance parameters the WAP-similar class should
be of interest.
In light of the relevance of the WAP criterion in the characterization result, this
paper proposed a weight function w∗ for the IV model. I argued that the weights
herein proposed are useful to compare different α-similar tests via the WAP criterion.
WAP comparisons are thus relevant in the IV model: there is no uniformly most
powerful test and different α-similar tests (e.g., AR, K, Q-CLR) can lead to different
interpretations of the same data. Therefore, this paper suggested researchers to
report results based on the α-similar test with the largest WAP (relative to w∗ or
to any other reasonable weight function).
To illustrate this point, the paper inverted three common tests (AR, K, Q-CLR) to
construct a confidence interval for the Elasticity of Intertemporal Substitution (EIS)
using U.S. data. Numerical approximations to the WAP showed that both the AR
and the Q-CLR were close to maximizing WAP relative to w∗ (Figure 1). Although
the difference in WAP between the AR and the Q-CLR was less than one percent,
both tests led to quite different confidence sets. For instance, when the stock return
was chosen as endogenous regressor, the AR—which had the largest WAP—rejected
all EIS values in the interval [0, 2], whereas the Q-CLR rejected none of them. The
EIS application also showed that the WAP criterion can be useful even if there is
no interest in computing confidence intervals based on the w∗ -WAP-similar test.
The paper also studied the WAP-similar test associated to w∗ using an i.i.d.,
homoskesdastic design. Result 1 showed that there is a closed-form expression for the
test statistic of the WAP-similar test corresponding to w∗ . Analytical and numerical
evidence suggested that the w∗ -WAP-similar test is comparable to the Conditional
Likelihood Ratio of Moreira (2003). This observation was taken as further evidence
of the reasonableness of the weights w∗ .
26
JOSÉ LUIS MONTIEL OLEA
REFERENCES
Abramowitz, M. and I. Stegun (1964): Handbook of Mathematical Functions with Formulas,
Graphs, and Mathematical Tables, vol. 55, Dover publications, Inc. New York.
Aliprantis, C. and K. Border (2006): Infinite Dimensional Analysis: a Hitchhiker’s Guide,
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