A NONPARAMETRIC TEST OF EXOGENEITY

Richard Blundell Department of Economics University College London

London WC1E 6BT UNITED KINGDOM

and

Joel L. Horowitz

Department of Economics Northwestern University

Evanston, IL 60208 USA

January 2007

ABSTRACT This paper presents a test for exogeneity of explanatory variables that minimizes the need for

auxiliary assumptions that are not required by the definition of exogeneity. It concerns inference

about a nonparametric function g that is identified by a conditional moment restriction involving

instrumental variables. A test of the hypothesis that g is the mean of a random variable Y

conditional on a covariate X is developed that is not subject to the ill-posed inverse problem of

nonparametric instrumental variables estimation. The test is consistent whenever g differs from

( | )E Y X on a set of non-zero probability. The usefulness of this new exogeneity test is displayed

through Monte Carlo experiments and an application to estimation of nonparametric consumer

expansion paths.

KEYWORDS: Hypothesis test, instrumental variables, specification testing, consistent testing JEL Listing: C12, C14 ________________________________________________________________________ Acknowledgments: We thank Xiaohong Chen, James J. Heckman, the editor and two anonymous referees for comments. Part of this research was carried out while Joel Horowitz was a visitor at the Centre for Microdata Methods and Practice, University College London and Institute for Fiscal Studies. Horowitz’s research was also supported in part by NSF Grant SES 0352675. Richard Blundell thanks the ESRC Centre for the Microeconomic Analysis of Public Policy at the Institute for Fiscal Studies for financial support.

1. INTRODUCTION

The problem of endogeneity arises frequently in economics. In empirical

microeconomics, endogeneity usually occurs as a result of the joint determination of observed

variables by individual agents. For example, firms choose inputs and production levels, and

households choose consumption levels and labour supply. It has long been understood that the

econometric estimation methods needed when a model contains endogenous explanatory

variables are different from those that suffice when all variables are exogenous. For example,

ordinary least squares does not provide consistent estimates of the coefficients of a linear model

when one or more explanatory variables are endogenous. Therefore, it is important to have ways

of testing for exogeneity of a model’s explanatory variables.

This paper describes an exogeneity test that is applicable in a broad range of

circumstances and minimizes the need for auxiliary assumptions that are not required by the

definition of exogeneity. The specific case that motivates our research and provides the

application in this paper concerns the nonparametric analysis of consumer behaviour and, in

particular, the possible endogeneity of total outlay in the nonparametric estimation of Engel

curves (expansion paths). Suppose we are interested in estimating the structural relationship

between the quantity of leisure services bought by a consumer in a particular month and his total

consumption (or wealth) in that month. Knowledge of the shape of this kind of Engel curve is an

integral part of any analysis of consumer welfare (Deaton 1998) and is also a key input into

microdata-based revealed-preference bounds (see Blundell, Browning, and Crawford 2003). It is

important to allow this relationship to vary flexibly, which can be done by using nonparametric

regression methods. However, it is likely that the unobservables in the relationship, which

include individual tastes for leisure, are related to preferences for overall consumption or wealth.

If this is the case, then the total consumption (expenditure) variable will be endogenous for the

Engel curve, and standard nonparametric regression estimators will not recover the structural

relationship of interest for welfare or revealed-preference analysis.1

The structural function can be estimated in the presence of endogenous explanatory

variables if we have instruments for those variables that are mean-independent of the

unobservable error term in the structural relationship. Indeed, instrumental variables estimators

for linear models are well known and widely used in empirical economics. When the structural

(or regression) function is nonparametric, as in the case considered in this paper, and there is an

endogenous explanatory variable, the precision of any estimator is typically much lower than it is 1 See Blundell and Powell (2003) for a discussion of structural functions of interest in nonparametric regression.

1

when all explanatory variables are exogenous (Hall and Horowitz 2005). Consequently, there is a

large loss of estimation efficiency from unnecessarily treating one or more explanatory variables

as exogenous. On the other hand, erroneously assuming exogeneity produces a specification

error that may cause the estimation results to be highly misleading. Therefore, it is important to

have ways to test for exogeneity in nonparametric regression analysis. This paper presents the

first such test.

The approach taken in this paper is to test the orthogonality condition that defines the null

hypothesis of exogeneity. In a linear regression model, there are several asymptotically

equivalent tests of this condition (Smith 1994). In nonparametric regression, one possible

approach is to compare a nonparametric estimate of the regression function under exogeneity

with an estimate obtained by using nonparametric instrumental variables methods.

Nonparametric instrumental variables estimators of the structural function have been developed

by Newey and Powell (2003); Darolles, Florens, and Renault (2002); Blundell, Chen, and

Kristensen (2003); and Hall and Horowitz (2005).2 However, the moment condition that

identifies the structural function in the presence of endogeneity is a Fredholm equation of the first

kind, which leads to an ill-posed inverse problem (O’Sullivan 1986, Kress 1999). A consequence

of this is that in the presence of one or more endogenous explanatory variables, the rate of

convergence of a nonparametric estimator of the structural function is typically very slow.

Therefore, a test based on a direct comparison of nonparametric estimates obtained with and

without assuming exogeneity is likely to have very low power. Accordingly, it is desirable to

have a test of exogeneity that avoids nonparametric instrumental variables estimation of the

structural relationship. This paper presents such a test.

If the structural regression function is known up to a finite-dimensional parameter, then

exogeneity can be tested by using methods developed by Hausman (1978), Bierens (1990), and

Bierens and Ploberger (1997). However, these tests can give misleading results if the structural

function is misspecified. The nonparametric test we present avoids this problem. Another

possibility is to test for the exclusion of the instruments from the structural regression. However,

we will show that an omitted variables test imposes stronger restrictions than are implied by the

hypothesis of exogeneity. It is desirable to avoid these restrictions if exogeneity is the hypothesis

of interest. Our test accomplishes this and is no more difficult to implement than an omitted

variables test.

2 Newey, Powell, and Vella (1999) developed a nonparametric instrumental variables estimator based on “control functions.” This estimator avoids the problems described in the remainder of this paragraph, but its assumptions are considerably stronger than ours or those of the authors just cited.

2

Computation of the test statistic and its critical value require only finite-dimensional

matrix manipulations, kernel nonparametric regression, and kernel nonparametric density

estimation. A GAUSS program for computing the statistic is available at the Review’s web site.

Section 2 of this paper presents the test. This section also explains the difference

between testing for exogeneity and testing for omitted instrumental variables in a mean

regression. Section 3 describes the asymptotic properties of the test. In Section 4, we present the

results of a Monte Carlo investigation of the finite-sample performance of the test. Section 5

presents an application that consists of testing the hypothesis that the income variable in an Engel

curve is exogenous. Section 6 concludes. The proofs of theorems are in the appendix.

2. THE MODELING FRAMEWORK AND THE TEST STATISTIC

This section begins by presenting a detailed description of the model setting that we deal

with and the test statistic. Section 2.3 explains why our test is not a test for omitted variables.

2.1 The Model Setting

To be more precise about the setting for our analysis, let be a scalar random variable,

and W be continuously distributed random scalars or vectors, and

Y

X g be a structural function

that is identified by the relation

(2.1) . [ ( ) | ]E Y g X W− = 0

In (2.1), Y is the dependent variable, is the explanatory variable, and W is an instrument for

. The function

X

X g is nonparametric; it is assumed to satisfy mild regularity conditions but is

otherwise unknown.

Define the conditional mean function ( ) ( | )G x E Y X x= = . We say that is exogenous

if except, possibly, if

X

( ) ( )g x G x= x is contained in a set of zero probability. Otherwise, we say

that is endogenous. This paper presents a test of the null hypothesis, , that is

exogenous against the alternative hypothesis, , that is endogenous. It follows from (2.1)

that this is equivalent to testing the hypothesis

X 0H X

1H X

[ ( ) | ]E Y G X W 0− = . Under mild conditions, the

test rejects with probability approaching 1 as the sample size increases whenever

on a set of non-zero probability.

0H

( ) ( )g x G x≠

To understand the issues involved in estimating when is endogenous, write

(2.1) in the form

( )g x X

3

(2.2) , |( | ) ( ) X WE Y W g x dF= ∫where |X WF is the cumulative distribution function of conditional on the instrument W .

Equation (2.2) is an integral equation for the structural function

X

g . Identifiability of g is

equivalent to uniqueness of the solution of this integral equation. Assuming that that g is

identified, estimating it amounts to solving (2.2) after replacing ( | )E Y W and |X WF with

consistent estimators. Doing this is complicated, however, because (2.2) is a version of a

Fredholm integral equation of the first kind (O’Sullivan 1986, Kress 1999), and it produces a so-

called ill-posed inverse problem. Specifically, the solution to (2.2) is not a continuous functional

of ( | )E Y W , even if the solution is unique and ( | )E Y W and |X WF are smooth functions.

Therefore, very different structural functions g can yield very similar reduced forms ( | )E Y W .

A similar problem arises in linear regression with multicollinearity, where large differences in

regression coefficients can correspond to small differences in the fitted values of the regression

function. As a consequence of the ill-posed inverse problem, the rate of convergence in

probability of a nonparametric instrumental variables estimator of g is typically very slow.

Depending on the details of the distribution of ( , , the rate may be slower than , )Y X W ( )pO n ε−

for any 0ε > (Hall and Horowitz 2005).

The test developed here does not require nonparametric estimation of g and is not

affected by the ill-posed inverse problem of nonparametric instrumental variables estimation.

Consequently, the “precision” of the test is greater than that of any nonparametric estimator of g .

Let denote the sample size used for testing. Under mild conditions, the test rejects with

probability approaching 1 as whenever

n 0H

n →∞ ( ) ( )g x G x≠ on a set of non-zero probability.

Moreover, the test can detect a large class of structural functions g whose distance from the

conditional mean function G in a suitable metric is . In contrast, the rate of

convergence in probability of a nonparametric estimator of

1/ 2(O n− )

g is always slower than .1/ 2( )pO n− 3

Throughout the remaining discussion, we will use an extended version of (2.1) that

allows g to be a function of a vector of endogenous explanatory variables, , and a set of

exogenous explanatory variables,

X

Z . We write this model as

3 Nonparametric estimation and testing of conditional mean and median functions is another setting in which the rate of testing is faster than the rate of estimation. See, for example, Guerre and Lavergne (2002) and Horowitz and Spokoiny (2001, 2002).

4

(2.3) , ( , ) ; ( | , ) 0Y g X Z U E U Z W= + =

where and U are random scalars, and W are random variables whose supports are

contained in a compact set that we take to be (

Y X

[0,1]p 1p ≥ ), and Z is a random variable whose

support is contained in a compact set that we take to be [0 ( ). The compactness

assumption is not restrictive because it can be satisfied by carrying out monotone increasing

transformations of any components of , W , and

,1]r 0r ≥

X Z whose supports are not compact. If 0r = ,

then Z is not included in (2.3). W is an instrument for . The inferential problem is to test the

null hypothesis, , that

X

0H

(2.4) ( | , ) 0U X x Z z= = =E

except, possibly, if ( , )x z belongs to a set of probability 0. The alternative hypothesis, , is that

(2.4) does not hold on some set

1H

[0,1]p rB +⊂ that has non-zero probability. The data,

are a simple random sample of . { , , , : 1,..., }i i i iY X Z W i n= ( , , , )Y X Z W

2.2 The Test Statistic

To form the test statistic, let denote the probability density function of ( ,XZWf , )X Z W .

Define . In what follows, we use operator notation that is taken

from functional analysis and is widely used in the literature on nonparametric instrumental

variables estimation. See, for example, Darolles, Florens, and Renault (2002); Carrasco, Florens,

and Renault (2005); Hall and Horowitz (2005), and Horowitz (2006). For each , define

the operator on by

( , ) ( | , )G x z Y X x Z z= =E =

[0,1]rz∈

zT 2[0,1]pL

( , ) ( , ) ( , )z zT x z t x z dψ ξ ψ ξ ξ= ∫ ,

where for each , 21 2( , ) [0,1] px x ∈

1 2 1 2( , ) ( , , ) ( , , )z XZW XZWt x x f x z w f x z w dw= ∫ .

Assume that is nonsingular for each . Then is equivalent to zT [0,1]rz∈ 0H

(2.5) ( , ) ( )( , ) 0zS x z T g G x z≡ − =%

for almost every . is equivalent to the statement that (2.5) does not hold on a

set with non-zero Lebesgue measure. A test statistic can be based on a sample

analog of , but the resulting rate of testing is slower than if . The rate

( , ) [0,1]p rx z +∈ 1H

[0,1]p rB +⊂

2( , )S x z dxdz∫ % 1/ 2n− 0r >

5

1/ 2n− can be achieved by carrying out an additional smoothing step. To this end, let

denote the kernel of a nonsingular integral operator, , on . That is, is defined by

1 2( , )z zl

L 2[0,1]rL L

( ) ( , ) ( )L z z dψ ζ ψ ζ ζ= ∫ l

and is nonsingular. Define the operator on T 2[0,1]p rL + by ( )( , ) ( ) ( ,zT x z LT x z)ψ ψ= . Then

is equivalent to 0H

(2.6) ( , ) ( )( , ) 0S x z T g G x z≡ − =

for almost every . is equivalent to the statement that (2.4) does not hold on a

set with non-zero Lebesgue measure. The test statistic is based on a sample analog

of .

( , ) [0,1]p rx z +∈ 1H

[0,1]p rB +⊂

2( , )S x z dxdz∫ The motivation for basing a test of on can be understood by observing that 0H ( , )S x z

1( , ) ( , )zg x z T Q x z−= , where |( , ) ( ) [ ( | , ) ( , , ) | )]Z W Z XZWQ x z f z Y Z z W f x z W Z z= = =E E and Zf

is the probability density function of Z (Hall and Horowitz 2005). 1zT − is a discontinuous

operator, and this discontinuity is the source of the ill-posed inverse problem in estimating g .

Basing the test of on avoids this problem because 0H ( , )S x z ( , ) ( )( , )zS x z L Q T G x z= − , which

does not involve 1zT − .

To form a sample analog of , observe that ( , )S x z ( , ) {[ ( , )]S x z Y G X Z= −E

( , , ) ( , )}XWf x z W Z z× l . Therefore, the analog can be formed by replacing and G XWf with

estimates and E with the sample average in {[ ( , )] ( , , ) ( , )}XWY G X Z f x z W Z z−E l . To do this,

let and , respectively, denote leave-observation-i-out “boundary kernel” estimators of

and (Gasser and Müller 1979; Gasser, Müller, and Mammitzsch 1985). To describe

these estimators, let denote a boundary kernel function with the property that for all

( )ˆ iXZWf − ( )ˆ iG −

XZWf G

( , )hK ⋅ ⋅

[0,1]ξ ∈ and some integer 2s ≥

(2.7) 1( 1) 1 if 0

( , )0 if 1 1.

j jh

jh u K u du

j sξ

ξξ

+− + =⎧= ⎨ ≤ ≤ −⎩∫

Here, denotes a bandwidth, and the kernel is defined in generalized form to overcome edge

effects. In particular, if is small and

0h >

h ξ is not close to 0 or 1, then we can set

6

( , ) ( / )hK u K u hξ = , where is an “ordinary” order s kernel. If K ξ is close to 1, then we can set

( , ) ( / )hK u K u hξ = , where K is a bounded, compactly supported function satisfying

0

1 if 0( )

0 if 1 1.j j

u K u duj s

∞ =⎧= ⎨ ≤ ≤ −⎩∫

If ξ is close to 0, we can set ( , ) ( / )hK u K u hξ = − . There are, of course, other ways of

overcoming the edge-effect problem, but the boundary kernel approach used here works

satisfactorily and is simple analytically.

Now define

( )( ) ( ),

1( , ) ,

pk k

p h hk

K x K xξ ξ=

=∏ ,

where ( )kx denotes the k ’th component of the vector x . Define ,r hK similarly. Then

( ) ( ) (1 1 1

( )

, , ,211

ˆ ( , , )

1 , ,

iXZW

n

p h j p h j r h jp rjj i

f x z w

K x X x K w W w K z Z znh

−

+=≠

=

− − −∑ ),

and

( ) (2 2

( ), ,( )

12

1ˆ ( , ) , , ,ˆ ( , )

ni

i p h j r h jp r ijXZj i

G x z Y K x X x K z Z znh f x z

−+ −

=≠

= −∑ )−

where and are bandwidths, and 1h 2h

( ) (2 2

( ), ,

12

1ˆ ( , ) , ,n

ip h j r h jXZ p r

jj i

)f x z K x X x K z Z znh

−+

=≠

= −∑ −

i

.

The sample analog of is ( , )S x z

. ( )1/ 2 ( )

1

ˆˆ( , ) [ ( , )] ( , , ) ( , )n

iin i i i i iXZW

iS x z n Y G X Z f x Z W Z z−− −

=

= −∑ l

The test statistic is

2 ( , )n nS x z dxdzτ = ∫0H is rejected if nτ is large.

7

2.3 Relation to Testing for Omitted Variables

As was mentioned in the Section 1, an omitted variables test imposes stronger restrictions

than are implied by the hypothesis of exogeneity. We now show that nτ is not a test of whether

is an omitted variable in the mean regression of Y on W ( , )X Z . Specifically, the null

hypothesis of the nτ test (that X exogenous) can be true and the null hypothesis of the omitted

variable test false simultaneously. The converse cannot occur. Therefore, the null hypothesis of

the omitted variable test is more restrictive than the exogeneity hypothesis of the n

is

τ test.

In a test that is an omitted variable, the null hypothesis is

)Y ZP E

W 0H :%

[ ( | , , ) ( | , ] 1X Z W Y X= =E . The alternative hypothesis is [ ( | , ,Y X Z W ) =P E

e been developed by Aït-Sahalia, Bickel,

96), Gozalo (1993), and Lavergne and Vuong (2000). The difference between

these tests and n

( | , )] 1Y X Z <E . Tests of hav and Stoker (2001),

Fan and Li (19

0H%

τ is that nτ assumes that [ ( | , ) 0] 1U Z W = =P E always and

[ ( | , ) 0]U X Z = =P E if 0H is true b not that [ ( | ,U X1 ut , ) 0] 1Z W = =P E asy to show that

, ) 0Z with probability ( | , , ) 0U X Z W =E with

null allows the conditional mean of U giv with

W. For example, let X ,

. It is e

U Z W U X= =E E 1 does not imply that

probability 1. The exog y

( | , ) ( |

eneit en X and Z to vary

Z , W , and ν be independent random variables with means of 0, and

set U XW ν= + . T n ( | , ) ( | , ) 0Z W U X Zhe U = =E E but ( | , , ) .U X Z W XW=E The null

hyp nothesis of the τ test is f the o false. Thus,

n

true, but the null hypothesis o mitted variable test is

τ is not a test of the hypothesis that W is an omitted variable. The hypothesis of exogeneity

tested by nτ is less restrictive than the hypothesis that W is an omitted variable.

3. ASYMPTOTIC PROPERTIES

3.1 Regularity Conditions

the assumptions that are used to obtain the asymptotic properties of This section states

nτ . Let 1 1 1 2 2 2( , , ) ( , , )x z w x z w− denote the Euclidean distance between the points 1 1 1( , , )x z w

and 2 2 2( , , )x z w in . Let 2[0,1] p r+j XZWD f denote any j ’th partial or mixed partial derivative o

f )

f

XZW . Set XZW XZW0 ( , , ) ( , ,D f f= . Let 2s ≥ e an integer. Define ( , )V Y G X Z= − ,

t XZ

x z w x z w

and le

b

f denote the density of ( , )X Z . The assumptions are as follows.

8

1. (i) The support of ( , , )X Z W is contained in 2[0,1] p r+ . (ii) ( , , )X Z W has a

probability density function with respect to Lebesgue measure. (ii) There is a constant

such that

XZWf

0XC > ( , )XZ Xf x z C≥ for all ( , ) supp( , )x z X Z∈ . (iv) There is a constant fC < ∞

such that | ( , , ) |j XZW fD f x z w C≤ for all 2( , , ) [0,1] p rx z w +∈ and 0,1,...,j s= , where derivatives

at the boundary of supp( , , )X Z W are defined as one-sided. (iv)

1 1 1 2 2 2| ( , , ) ( , , ) |s XZW s XZWD f x z w D f x z w− 1 1 1 2 2 2( , , ) ( , , )fC x z w x z w≤ − for any s ’th derivative

and any 21 1 1 2 2 2( , , ), ( , , ) [0,1] p rx z w x z w +∈ . (v) is nonsingular for almost every . zT [0,1]rz∈

2. (i) and for each ( | , ) 0U Z z W w= = =E 2( | , ) UVU Z z W w C= = ≤E ( , ) [0,1]p rz w +∈

and some constant . (ii) | (UVC < ∞ , ) | gg x z C≤ for some constant and all

.

gC < ∞

( , ) [0,1]p rx z +∈

3. The conditional mean function G satisfies | ( , ) |j fD G x z C≤ for all ( , ) [0,1]p rx z +∈

and (ii) | sD G0,1,...,j = .s ( , ) ( ,sx z D G x z− 1 1 2 2 ) | 1 1 2 2( , ) ( , )C x z x z≤ − for af ny s ’th deriva ive

and any 1 1 2 2( , , , )x z x z ∈ ,X x Z= = 0,1]p r

t

[0,1] p r+ . (iii) ) UVV z C≤E for each [x z2( ) 2( | ( , ) +∈

4. (i) satisfies (2.7) and

.

hK 2 1 2 1| ( , ) ( , ) | | | /h h KK u K u C u u hξ ξ− ≤ − for all , all 2 1,u u

[0,1]ξ ∈ , and some constant KC < ∞ . For each [0,1]ξ ∈ , ( , )hK h ξ ortedis supp on

, / ]h h[( 1) /ξ ξ ∩K , where K is a act interval not depend− comp ing on ξ . Mo

,h uξ

reover,

| ( ) |hK hu0, [0,1],

sup ξ> ∈ ∈K

(ii) The bandwidth satisfies

< ∞ .

1h 1/(2 2 )1 1

s p rhh c n− + += 1hc < ∞, where is a constant. (iii) The

sbandwidth, 2h , satisfie 2 2hh c n α−= , where 2hc < ∞ is a constant and 1/(2 ) 1/( )s p rα< < + .

Assu ption 1(ii) avoid im estimation of G in re XZfm is used to precise gions where is

close to 0. The assumption can be relaxed by replacing the fixed distribution of ( , , )X Z W y a

sequence of distributions with densities { }nXZWf and { }nXZf ( 1,2,...n

b

= ) that satisfy

( , )nXZ nf x z C≥ for all ( , ) [0,1]p qx z +∈ and a sequence { }nC of strictly constants that

s complicates the proofs but does not change the results

positive

converges to 0 sufficiently slowly

mption 1(v) comb o implies

. Thi

reported here. Assu ined with the m ment condition ( | , ) 0E U X Z =

9

that g is identified and the instruments W are valid in the sense of being suitably related to .X 4

Assumption 4(iii) implies that the estimator of is undersmoothed. Undersmoothing prevents

the asymptotic bias of from dominating the asymptotic distribution of

G( )ˆ iG −

nτ . Assumption 4

requires the use of a higher-order kernel if 4p r+ ≥ . The remaining assumptions are standard in

nonparametric estimation.

3.2 Asymptotic Properties of the Test Statistic

To obtain the asymptotic distribution of nτ under , define 0H ( , )i i i iV Y G X Z= −

1/ 2

1( , ) [ ( , ,i ) ( , ) / ( , )] ( , )

i

n

n i XZW i i Z i XZ i i ii

x z n U f x V t X x f X Z Z z−

=

= −∑ lZ WB ,

and

1 1 2 2 1 1 2 2( , ; , ) [ ( , ) ( , )]n nR x z x z B x z B x z= E .

Under , . The distinction between and in the definition of 0H iU V= i iU iV nB will be used later

to investigate the distribution of nτ when is false. Define the operator 0H Ω on by 2[0,1]p rL +

1

0( )( , ) ( , ; , ) ( , )x z R x z d dψ ξ ζ ψ ξ ζ ξ ζΩ = ∫ .

Let { : 1,2,...}j jω = denote the eigenvalues of Ω sorted so that 1 2 ... 0ω ω≥ ≥ ≥ .5 Let

denote independent random variables that are distributed as chi-square with one

degree of freedom. The following theorem gives the asymptotic distribution of

21{ : 1,2,...}j jχ =

nτ under .0H 6

4 is a self-adjoint, positive-semidefinite operator, so its eigenvalues are non-negative. Under 1(v), is positive-definite, and all its eigenvalues are strictly positive. If 1(v) does not hold, then some eigenvalues are 0. Let denote the linear space spanned by the eigenvectors of

zT zT

A zT corresponding to non-zero eigenvalues. If 1(v) does not hold, then one can test for deviations from 0H such that

( , ) ( , )g z G z⋅ − ⋅ has a non-zero projection into . It is not possible to test for deviations from A 0H for which ( , ) ( , )g z G z⋅ − ⋅ lies entirely in the complement of . In nonparametric IV estimation, validity of the instruments is equivalent to non-singularity of

A

zT . If zT is non-singular, then whether the instruments are “weak” or “strong” depends on the rate at which the eigenvalues of converge to 0. The instruments are weak if the eigenvalues converge rapidly and strong otherwise. There appears to be no simple, intuitive characterization of strength or weakness of instruments in this setting. In particular, in nonparametric IV estimation, the strength of correlation of

zT

X and W does not characterize the strength or weakness of W as an instrument. 5 R is a bounded function under the assumptions of Section 3.1. Therefore, Ω is a compact, completely continuous operator with discrete eigenvalues. 6 A referee asked whether nτ satisfies the Liapounov condition (Serfling 1980, p. 30), which would imply that nτ is asymptotically normal. The answer is that nτ does not satisfy the Liapounov condition. In our

10

Theorem 1: Let be true. Then under assumptions 1-4, 0H

21

1

dn j

jjτ ω χ

∞

=

→ ∑ .

3.3 Obtaining the Critical Value

The statistic nτ is not asymptotically pivotal, so its asymptotic distribution cannot be

tabulated. This section presents a method for obtaining an approximate asymptotic critical value.

The method is based on replacing the asymptotic distribution of nτ with an approximate

distribution. The difference between the true and approximate distributions can be made

arbitrarily small under both the null hypothesis and alternatives. Moreover, the quantiles of the

approximate distribution can be estimated consistently as . The approximate 1n →∞ α− critical

value of the nτ test is a consistent estimator of the 1 α− quantile of the approximate distribution.

We now describe the approximation to the asymptotic distribution of nτ . Under , 0H nτ

is asymptotically distributed as

21

1j j

jτ ω χ

∞

=

≡∑% .

Given any 0ε > , there is an integer Kε < ∞ such that

21

10 (

K

j jj

t tε

)ω χ τ=

⎛ ⎞⎜ ⎟< ≤ − ≤⎜ ⎟⎝ ⎠∑P P % ε< .

uniformly over t . Define

21

1

K

j jj

ε

ετ ω χ=

=∑% .

Let zεα denote the 1 α− quantile of the distribution of ετ% . Then 0 ( )zεατ α ε< > − <P % . Thus,

using zεα to approximate the asymptotic 1 α− critical value of nτ creates an arbitrarily small

error in the probability that a correct null hypothesis is rejected. Similarly, use of the

approximation creates an arbitrarily small change in the power of the nτ test when the null

/ 0n n → →∞ ( )1/

1n

n jja

νsetting, the condition is a b as n , where νω

== ∑ ( )1/ 2

21

nn jj

b ω=

= ∑2

and for

some ν > . Boundedness of XZWf1

cjj

ω∞

= implies that 0 < < ∞∑ 2c ≥ for any , so the convergence to

0 required by the Liapounov condition does not happen.

11

hypothesis is false. The approximate 1 α− critical value for the nτ test is a consistent estimator

of the 1 α− quantile of the distribution of ετ% . Specifically, let ˆ jω ( 1, ..., )2,j Kε= be

consistent estimator of

a

jω under . Then the approximate critical v lue of 0H a nτ is the 1 α−

quantile of the distribution of

ˆ ˆKε

21

1n j j

jτ ω χ=∑

This qu

A e cost

=

.

antile can be estimated with arbitrary accuracy by simulation.

t th of additional analytic complexity, it may be possible to let 0ε → and

→∞ as n →∞ , thereby obtaining a consistent estimator of the asymptotic critical value of

n

Kε

τ . However, this would likely require stronger assumptions than a made here while providing

little insight into the accuracy of the estimator or t choic of K

re

he e ε in applications. This is

because the difference between the dis butions of ˆntri τ and τ% is a complicated function of the

spacings and multiplicities of the jω ’s (Hall and Horowitz 2006). The sp and

multipli n

acings

cities are unknow in applications and appear difficult to estimate reliably.

In applications, Kε can be chosen informally by sorting the ˆ jω ’s in decreasing order and

plotting them as a function of j . Th typically plot as random noise near ˆ 0jey ω = when j is

sufficiently large. One can choose Kε to be a value of j that is near the lower end of the

“random noise” range. The rejection probability of the nτ test is not highly sensitive to Kε , so it

is not necessary to attempt precision in making the choice.

The remainder of this section explains how estimated eigenvalues ˆ{ } to obtain the jω .

Because V U= under 0H , a consistent estimator of 1 1( , ; 2 2, )R x z x z can be obtained by

unknown quantities with estimators on the right-hand side of

replacing

1 1 2 2

21 21 2( , , ) ( , , ) (

( , ) ( , )XZ XZ1 2

( , ; , )

( , ) ( , ) , ) ( , )Z ZXZW

R x z x z

t X x t X xf XZWx Z W x Z W Zf X Z f X Z

f z Z z V

=

⎧ ⎫⎡ ⎤ ⎡ ⎤⎪ ⎪− − ⎬⎢⎨ ⎥ ⎢ ⎥⎪ ⎪⎣ ⎦ ⎣ ⎦⎩ ⎭

l

To do this, let

E l

ˆXZWf be a kernel estimator of with bandwidth . Define

1 2ˆ ˆ( , , ) ( , , )XZW XZWx f x z w f x z w dw .

Estimate the ’s by

XZWf h

11 2 0

ˆ ( , )zt x = ∫iV

12

( )ˆ ( , )ii iG X Z−− . i iV Y=

1 1 2 2( , ; , )R x z x z is estimated consistently by

1 1 2 2

1 21 21

1

ˆ ˆ( , , ), )XZW i i

iin f x W

Z=⎢⎢⎣

∑ 2 1 2

ˆ( , , , )

ˆ ˆ( , ) ( , ) ˆ( , , ) ( , ) ( , ) .ˆ ˆ( ( , )i i

nZ i Z i

XZW i i i i iXZ i XZ i i

R x z x z

t X x t X xZ f x Z W Z z Z z V

f X f X Z−

=

⎡ ⎤ ⎡ ⎤− −⎥ ⎢ ⎥

⎥ ⎢ ⎥⎦ ⎣ ⎦l l

efine the operator on by

, ( , )

Ω 2[0,1]p rL +D

0

ˆ ˆ( )( , ) ( ; , )1

x z R x z d dψ ξ ζΩ = ∫Denote the eigenvalues o Ω by : 1,2,...}j j

ψ ξ ζ ξ ζ .

f ˆ{ω = and order them so that 1 2ˆ ˆ ... 0M Mω ω≥ ≥ ≥ .

The relation between the ˆ jω ’s and jω ’s is given by the following theorem.

Theorem 2: Let assumptions 1-4 hold. Then as

rical approximation to the

2 1/ 2ˆ [(log ) /( ) ]p rj j po n nhω ω +− =

n →∞ ,for each 1,2,...j =

To obtain an accurate nume ˆ jω ’s, let denˆ ( , )F x z ote the n 1×

vector who ’th component is ˆ ˆˆ[ ( , ) ( , ) /iXZW i i Z XWse ( , )] ( , )i i i i i ,f x X X Z Z zl , and let Z W t x f− ϒ

d ment is enote the diagonal matrix whose ele . Then n n× ( , )i i 2iV

11 1 2 2 1 1 2 2

ˆ ˆ ˆ( , ; , ) ( , ) ( , )R x z x z n F x z F x z− ′= ϒ .

The computation of the eigenvalues can d to finding the eigenvalues of a finite-

l matrix

now be reduce

dimensiona . To this end, let { : 1,2,...}j jφ = be a complete, orthonormal basis for

. Then 2[0,1]p rL +

1 1

ˆ ˆ( , , ) ( , ) ( , ) ( , )XZW jk j kj k

f x z W Z z d x z Z Wφ φ∞ ∞

= =

=∑ ∑l ,

where

w

and

k

where

1 1 1 11 2 1 2 1 1 20 0 0 0

ˆ ˆ ( , , ) ( , ) ( , ) ( , )jk XZW j kd dx dz dz dwf x z w z z x z zφ φ= ∫ ∫ ∫ ∫ l ,

1 1

ˆ ˆ( , ) ( , ) ( , ) ( , )Z jk jj k

t X x Z z a x z X Zφ φ∞ ∞

= =

=∑ ∑l ,

13

1

1 1 1 12 2 2 2 1 1ˆˆ ) ( , ) ( ,jk z j ka dz t x z x zφ φ= 1 2 1 1 2 10 0 0 0

( , ) ( , )dx dx dz x x z z∫ ∫ ∫ ∫ l .

Approximate ˆ ( , , ) ( , )XZWf x z W Z zl and by the finite sums

k

ˆ ( , ) ( , )zt X x Z zl

1 1

ˆ( , , , ) ( , ) ( , )L L

f jk jj k

x z W Z d x z Z Wφ φ= =

Π =∑ ∑

and

)) ( , ,L L

k jj k1 1

ˆ( , , , ) ( ).t j kx z X a z XΠ ∑ ∑

for some integer . Since ˆ

Z x Zφ φ= =

=

L < ∞ XZWf l and Zt l are known functio can be chosen to ns,

approximate

L

ˆXZWf l and Zt l be the n L× with any desired accuracy. Let Φ matrix whose

component is

(L

ik

X=

( , )i j

1/ 2 ˆ ˆˆ[ ( , ) ( , ) / , )]ij jk k i i jk k i i XZ in d Z W a X Z f Zφ φ−Φ = −∑ . 1

The eigenvalues of Ω are approximated by those of the L L× matrix ′Φ ϒΦ .

nsistency of the Test against a Fi odel

In this section, it is assumed that is false. That is, .

Define . Let

3.4 Co xed Alternative M

0H [ , : ( , ) ( , )] 1X Z g X Z G X Z= <P

( , ) ( , ) ( , )q x z g x z G x z= − zα% denote the 1 α− quantile of the distribution of nτ

under sampling from the null-hypothesis model ( , ) , ( | , ) 0Y G X Z V V X Z= + =E . The following

theorem establishes consistency of the nτ test against a fixed alternative hypothesis.

Theorem 3: Suppose that

Let assumptions 1-4 hold. Then for any

1 2[( )( , )] 0Tq x z dxdz >∫ . 0

α such that 0 1α< < ,

lim ( ) 1nn

zατ→∞

=P % . >

test is consistent whenever differs from ( , )g x z Because is nonsingular, the T nτ

( , )G x z on a set of ( , )x z values whose probability exceeds zero.

ution under Local Alternatives

This section obtains the asymptotic distribution of

3.5 Asymptotic Distrib

nτ under the sequence of local

alternative hypotheses

14

(3.1) 1/ 2( , ) ; ( | , ( | , ) ( , )g X Z U U Z W U X Z n X Z−= + = ΔE E ,

where on on [0

) 0;Y =

is a bounded functiΔ ,1]p r+ . Under (2.6), the distributions of U and V depend

on n , 1/ 2 ( ) ( , )n U V X Z− = Δ , and 1/ 2( , ) ( , ) ( , )G X Z g X Z n X Z−= + Δ . To provide a complete

characterization of the sequence of alt ativ hypotheses, it is necessern e a

d utions of and on . Here, it is assumed that

V

ry to specify the

depen ibence of the distr U V n

(3.2) 2n 1/ν ε−= + ,

where ε and ν are random variables whose distributions do not depend on

X Z

n ,

( ( | , ) 0Z W| , )ν ν =E E , ( )Var= ν < ∞ , ( | , ) 0X Zε =E , ( | , ) [ ( , ) | , ]Z W X Z Z Wε = − ΔE E ,

and Var( )ε < ∞ . It follows fro

(3.3)

m (3.1)-(3.2) that

1/ 2 1/ 2( , )U n X Z nν ε− −= +

The following additional notation is used. Define n

n i XZW i i i ii

Δ + ,

1/ 2

1( , ) [ ( , , ) ( , ) / ( , )] ( , )B

iZ i XZ ix z n f x Z W t X x f X Z Z zν−

=

= −∑% l

and 1 1 2 2 1 1 2 2( , ; , ) [ ( , ) ( , )]n nR x z x z B x z B x z= E% % % on by 2[0,1]p rL +. Define the operator Ω%

1

0( )( , )x ( , ; , ) ( , )z R x z d dψ ξ ζ ψ ξ ζ ξ ζΩ = ∫% %

Let j j j

.

,2,...}{( , ) : 1ω ψ =% denote the eigenvectors and orthonormal eigenvectors of . Define Ω%

( , ) ( )( , )x z T x zμ = Δ and

j j1

0( , ) ( ,x )z x zμ μ ψ= ∫ dxdz .

Let 2 21{ ( / ) : 1,2,...}j j j j

of freedom and non-central parameters

χ μ ω =% denote independent random variables that are distributed as non-

central chi-square with one degree 2{ / }j jμ ω% . The

Theorem 4

following theorem states the result.

: Let assumptions 1-4 hold. Under the sequence of local alternatives (3.1)-

(3.3),

d 2 21

1( / )n j j j j

jτ ω χ μ ω

−

→ ∑ % % . ∞

It follows from Theorems 2 and 4 that under (3.1)-(3.3),

nzlimsup | ( ) ( ) |n n zεα ατ τ ε− > ≤P P

→∞>

15

for any 0ε > , where zεα denotes the estimated approximate α -level critical value.

3.6 Uniform Consistency

This section shows that for any 0ε > , the nτ test rejects 0H with probability exceeding

1 ε− uniformly over a set of functions g whose distance from G is ( )O n . This set contains

deviations from 0H that cannot be represented as sequences of local alternatives. Thus, the set is

larger than the class of local alternatives against

1/ 2−

ch the fwhi power o nτ exceeds 1 ε− . The

ca ly large class of alternatives againstpracti l consequence of this result is to define a relative

which the nτ test has hi r in large samp

notation is used. Define (q

gh powe les.

The following additional x z g x z G x z, ) ( , ) ( , )= − .

, define s a set of distributi

n 1; (ii) | , ] 0Z W

Let XZWf

be fixed. For each 1,2,...n = and finite ncF a of

( , , , )Y X Z W such that: (i) XZWf satisfies assumptio , )g X Z

0C > ons

[ (Y − =E for some

function g that satisfies assu tion 2 withmp ( , )U Y g X Z= − ; (iii) 0[ ( , ) | , ]Y G X Z X Z− =E for

that satisfisome function G mption 3 with ( , )V Yes assu G X Z= − ; (iv) 1/ 2Tq n C−≥ , where ⋅

denotes the 2L norm; and (v) 1 (log ) / (1)h n q Tq o= as n →∞ of distributions of

( , , , )Y X Z W for which the distance of

s . is a set ncF

g fr shrinks to zero at the rate 1/ 2n− in the sen

that ncF includes distributions for which

om se G

1/ 2( )q O n−= . Conditi out d tributions for

which q dep

on (v) rules

on

is

ends ( , )x z only through sequences of eigenvectors of T whose eigenvalues

too rapidly. For example, let converge to 0 1p , 0r == , so Z is not in the model. Let

1,2,...}{ , :j j jλ φ enote the eig eigenvecto= d alues and rs of T ordered so that env

1 2 ... 0λ λ≥ ≥ > . Suppose that 1( ) ( )G x xφ= , 1( ) ( ) ( )ng x x xφ φ= + , and the instr

%

ument is

1( )Wφ= . Then W 2 21 1/ / nh q Tq h λ= . Because n∝ , condition (v) is violated if

1/ 3( )n o nλ −= . The practical significance of condition (v) is that the n

1/ 61h −

τ test has low power when

g differs from G only through eigenvectors of with very small lues. Such differences

unlikely to be important in most

the result of this section.

Theorem 5

T eigenva

tend to oscillate rapidly (that is, to be very wiggly) and are

applications.

The following theorem states

0δ > , any α such that 0 1α< < , : Let assumption 4 hold. Then given any

and any sufficiently large (but finite) , C

16

lim inf ( ) 1nc

nn

zατ δ→∞

> ≥ −PF

and

εα ˆlim inf ( n z ) 1 2τncn

δ≥ − .

→∞ F

3.7 Alternative Weights

This section compares n

>P

τ with a generalization of the test of Bierens (1990 Bierens

and Ploberger

) and

(1997). To m ize the complexity of the discussion, assume that inim 1p = and

, so 0r = Z is not in the model. Let ( , )H ⋅ ⋅ be a bounded, real-valued function on with the

property that

2[0,1]

21

0( , ) ( ) 0H z w s w dw =∫

( ) 0s w = for almost every [0,1]w∈only if . Then a test of can be based on the statistic

z

where

0H

1 20

( )nH nHS z dτ = ∫ ,

1/ 2 ( )

1

ˆ( ) [ ( )] ( , )n

inH i i i

iS z n Y G X H z W− −

=

= −∑ .

If ( , ) ( )H z w H zw= % for a suitably chosen function H% , then nHτ is a modification of the statistic

mean function belongs to a specified, finite-dimensional parametric family. In this section, it is

e power of the

of Bierens (1990) and Bierens and Ploberger (1997) pothesis that a conditional

shown that th

for testing the hy

nHτ test can be low relative to that of the nτ test. Specifically, there

are combinations of density functions XWf a atnd local altern ive models (2.6)-(2.8) such that an

α -level nHτ test based on a fixed H that does not depend on the sampled population has

asymptotic local power arbitrarily close to α , whereas the α -level nτ test has asymptotic local

power that is bounded away from α . The opposite situation cannot e assumptions occur under th

of this paper. That is, it is not possible for the asymptotic power of the α -level nτ test to

approach α while the power of the α -level nHτ t remains bounded away from tes α .

The conclusion that the power of nHτ can be low relative to that of nτ is reached by

constructing an example in which the α -level nτ test has asymptotic power that is bounded

away from α test has asy ptotic power that is arbitrarily close to m α . but the nHτ To minimize

17

the complexity of the example, assume that is known and does not have to be estimated.

Define

G

1/ 2

1( ) ( , )

n

n i XWi

iB z n U f z W−

=

= ∑ ,

1/ 2

1( ,

i( ) )

n

nH i iB z n U−= ∑ , H z W=

1 2 1 2( , ) [ ( ) ( )]n nR z z B z B z= E , and 1 2 1 2( , ) [ ( ) ( )]H nH nHR z z B z B z= E . Also, define the operators

Ω and HΩ on by

2[0,1]L

1

0( )( ) ( , ) ( )z R z x x dψ ψΩ = ∫ x

and 1

0

Let

( )( ) ( , ) ( )H Hz R z x x dxψ ψΩ = ∫ .

{ , : 1,2,...}j j jω ψ = and { , : 1,2,...}jH jH jω ψ = denote the eigenvalues and eigenvectors of

Ω and HΩ , respectively, with the eigenvalues sorted in decreasing order. For defined as in

(3.1), define

Δ

( ) ( )( )z T zμ = Δ ,

1 1

0 0( ) ( ) ( , ) ( , )H XWz x H x w f x w dxdwμ = Δ∫ ∫ ,

1

0( ) ( )j jz z dμ μ ψ= ∫ , z

and 1

0( ) ( )jH H jHz z dzμ μ ψ= ∫ .

Then arguments like those used to prove Theorem 4 show that under the sequence of local

alternatives (3.1)-(3.3) with a known function , G

2 21

1( /d

n j j jj

)jτ ω χ μ ω∞

−

→ ∑

and

2 21

1( / )d

jH j jH jHω χ μ ω∞

−

→ ∑ nHj

τ

18

as . Therefore, to the fi tion, ffices to w that for a

fixed function H

n →∞ establish rst conclusion of this sec it su sho

, XWf and can be chosen so that Δ 21

/ jjμ ω

∞

=∑ is bounded away from 0

and 21

/H jHjμ ω

∞

=∑ is arbitrarily close to 0.

To this end, let 1( ) 1xφ = and 1/ 21( ) 2 cos( )j x j xφ π−+ = for . Let be a finite

integer. Define

j j

j

eλ −

=⎧⎪= ⎨⎪⎩

l

Let

( , ) 1 ( ) ( )XW j j j

1j ≥ 1>l

2

1 if 1 or

otherwise.

1j

1/ 21 1 1x w x wλ φ φ+ + += +∑ .

1= fo

f∞

=

Let 2( |U W w=E r all [0,1]w∈ . Then ) 1 2( , , and 1 jjω

∞

=∑1 2) ( , )R z z t z z= , j jω λ= is

non-zero and finite. Set ( ) ( )x D xφΔ = for some finite . Then l 0D > 2 2 2 2D Dμ λ= =l . Since

2HμH is fixed, it suffices to show that l can be chosen so that is arbitrari

this, observe that has the Fourier representation

H z w h z wφ φ= ∑ ,

here

ly close to 0. To do

, 1j k

∞

=

( , )H z w

( , ) ( ) ( )jk j k

w : , 1,2,...}jkh j k = are constants. Moreover, { 2 2 21H jj

D hμ∞

== ∑ l . Sin H is bounded,

l can b sen so that 2 2/h Dε∞

<∑ for any 0

ce

e cho jj= l

1ε > . With this l , 2

Hμ ε< , which

establishes the first conclusion.

he op on (a sequence of local alternatives for which T posite situati 2μ ap 0

while is

proaches

2Hμ

remains bounded away from 0) cannot occur. To show this, assume without loss

of generality that the marginal distributions of and are 1 for all

, and

X W [0,1]U , 2( | )U W w= =E

[0,1]w∈1

1jHjω

∞

==∑ . Also, assume that 2 CΔΔ < for some constant . Then, CΔ < ∞

1 1 20 0

1( , ) jH

jH z w dzdw ω

∞

=

=∑∫ ∫ .

It follows from the Cauchy-Schwartz inequality that

19

21 1 12 20 0 0 0

21

0 0

2 2

2

( , ) ( , ) ( )

( , ) ( )

.

H XW

XW

H z w dzdw f x w x dx dw

f x w x dx dw

T

C

μ

μΔ

⎡ ⎤ ⎡≤ Δ⎢ ⎥ ⎢⎣ ⎦ ⎣

⎡ ⎤= Δ⎢ ⎥⎣ ⎦

≤ Δ Δ

≤

∫ ∫ ∫ ∫

∫ ∫

⎤⎥⎦

Therefore, 2μ can approach 0 only if 2Hμ also approaches 0.

4. MONTE CARLO EXPERIMENTS

This section reports the results of a Monte Carlo investigation of the finite-sample

performance of the nτ test. In the experiments, 1p = and 0r = , so Z does not enter the model.

Realizations of ( , )X W were generated by ( )X ξ= Φ and ( )W ζ= Φ , where is the cumulative

normal distribution function,

Φ

~ (0,1)Nζ , 2 1/ 2(1 )ξ ρζ ρ ε= + − , (0,1)Nε , and 0.35ρ = or

0.7ρ = , depending on the experiment. Realizations of Y were generated from

(4.1) 0 1 UY X Uθ θ σ= + + ,

where 0 0θ = , 1 0.5θ = , 2 1/ 2(1 )U ηε η ν= + − (0,1)N, ν , 0.2Uσ = , and η is a constant

parameter whose value varies among experiments. is true if 0H 0η = and false otherwise. To

provide a basis for judging whether the power of the nτ test is high or low, we also report the

results of a Hausman (1978) type test of the hypothesis that the ordinary least squares and

instrumental variables (IV) estimators of 1θ in (4.1) are equal. The instruments used for IV

estimation of (4.1) are . In addition, we report the results of simulations with (1, )W nHτ . The

weight function is ( , ) exp( )H x w xw= and is taken from Bierens (1990). The bandwidth used to

estimate XWf was selected by cross-validation. The bandwidth used to estimate Xf is

times the cross-validation bandwidth. The kernel is

1/ 5 7 / 24n −

2 2( ) (15/16)(1 ) (| | 1)K v v I v= − ≤

f

, where is

the indicator function. The asymptotic critical value was estimated by setting . The

results of the experiments are not sensitive to the choice o K

I

25Kε =

ε , and the estimated eigenvalu es

20

ˆ jω are very close to 0 wh 25> . The experiments use sample sizes n = 250, 500, and 750

and the nominal 0.05 level. There are 1000 Monte Carlo replications in each experime

en j of

nt.

The results of the experiments are shown in Table 1. The differences between the

nominal and empirical rejection probabilities of the nτ and Hausman-type tests are small when

is true. When is false, the power of the 0H 0H nτ test is, not surprisingly, somewhat smaller

than the power of the Hausman-type test, which is parametric, but the differences in power are

not great. The performance of nHτ is worse than that of nτ . When is true, difference

between the nominal and empirical rejection probabilities of the

0H

nHτ test is relatively large, and

the power of the nHτ test is usually lower than that of the nτ test.

5. EXOGENEITY AND CONSUMER EXPANSION PATHS

The empirical analysis of consumer expansion paths (or Engel curves) concerns the

relationship between expenditures on specific commodities and total consumption; see Deaton

(1998), for example. The shape of the expansion path defines whether a good is a necessity or a

luxury and knowledge of the expansion path allows the researcher to measure reactions to

policies that change the resources allocated to individuals across the wealth distribution. The

relationship we wish to recover for policy analysis is the structural function that describes

changes in commodity demands in response to exogenous changes in overall consumption. In

household expenditure survey data it is likely that the total consumption (expenditure) variable

will be endogenous for the expansion path and standard nonparametric regression estimators will

not recover the structural relationship of interest.

Here we explore the expansion path relationship for leisure services bought by consumers

in a particular month. This curve has been shown to be nonlinear in nonparametric regression

analysis (see Blundell, Duncan and Pendakur (1998)) and so poses a particularly acute problem in

estimation under endogeneity. We present an empirical application of our test statistic nτ to this

expansion path problem and assess whether we can reject the exogeneity hypothesis for total

consumption in the leisure services expansion path. The curve is given by (2.3) with 1p = and

, where denotes the expenditure share of services, denotes the logarithm of total

expenditures, and W denotes annual income from wages and salaries of the head of household.

Engel curves are important in the analysis of consumer behavior.

0r = Y X

21

The data consist of household-level observations from the British Family Expenditure

Survey, which is a popular data source for studying consumer behavior.7 This is a diary-based

household survey that is supplemented by recall information. We use a subsample of 1518

married couples with one or two children and an employed head of household.8 W should be a

good instrument for if income from wages and salaries is not influenced by household

budgeting decisions.

X

The bandwidths for estimating XWf were selected by the method described in the Monte

Carlo section. The kernel is the same as the one used in the Monte Carlo experiments. As in the

experiments, the critical value of nτ was estimated by setting 25Kε = . The nτ test of the

hypothesis that is exogenous gives X 0.162nτ = with a 0.05-level critical value of 0.151. Thus,

the test rejects the hypothesis that is exogenous. X

Parametric specifications are often linear or quadratic in (Muellbauer 1976; Banks,

Blundell, and Lewbel 1998). Consequently, the hypothesis was also tested by comparing the

OLS and IV estimates of

X

1θ and 2θ in the quadratic model

20 1 2Y X Xθ θ θ= + + +U .

The instruments are . The hypothesis that the OLS estimates of 2(1, , )W W 1θ and 2θ equal the

IV estimates is rejected at the 0.05 level. Thus, the nτ test and the parametric test both reject the

hypothesis that the logarithm of total expenditures is exogenous.

6. CONCLUSIONS

Endogeneity of explanatory variables is an important problem in applied econometrics.

Erroneously assuming that explanatory variables are exogenous can cause estimation results to be

highly misleading. Conversely, unnecessarily assuming that one or more variables are

endogenous can greatly reduce estimation precision, especially in the nonparametric setting

considered in this paper. This paper has described a test for exogeneity of explanatory variables

that minimizes the need for auxiliary assumptions that are not required by the definition of

exogeneity. Specifically, the test does not make parametric functional form assumptions, thereby

avoiding the possibility of obtaining a misleading result due to model misspecification. In

7 See Blundell, Pashardes and Weber (1993), for example. 8 The data is available on the Review’s website. This is also the sample selection used in the Blundell, Chen and Kristensen (2003) study.

22

addition, the test described here does not make the auxiliary assumptions that are implied by a

test for omitted instrumental variables. We have shown that the hypothesis of exogeneity can be

true and the hypothesis of no omitted variables false simultaneously. The opposite situation

cannot occur. Thus, an omitted variables test requires assumptions that are stronger than implied

by exogeneity and can erroneously reject the hypothesis of exogeneity due to failure of the

auxiliary conditions to hold. We have illustrated the usefulness of the new exogeneity test

through Monte Carlo experiments and an application to estimation of Engel curves.

APPENDIX: PROOFS OF THEOREMS

To minimize the complexity of the presentation, we assume that 1p = , , and 0r = 2s = .

The proofs for 1p > , , and/or are identical after replacing quantities for ,

and with the analogous quantities for the more general case. Let

0r > 2s > 1, 0p r= =

2s = XWf denote the density

function of ( , )X W .

Define

1/ 21

1( ) ( , )

n

n i XWi

S z n U f z W−

=

= ∑ i

XW i

XW i

i

W i

W i

j

,

1/ 22

1( ) [ ( ) ( )] ( , )

n

n i ii

S z n g X G X f z W−

=

= −∑ ,

1/ 2 ( )3

1

ˆ( ) [ ( ) ( )] ( , )n

in i i

iS z n G X G X f z W− −

=

= −∑ ,

( )1/ 24

1

ˆ( ) [ ( , ) ( , )]n

in i i XWXW

iS z n U f z W f z W−−

=

= −∑ ,

( )1/ 25

1

ˆ( ) [ ( ) ( )][ ( , ) ( , )]n

in i i i XXW

iS z n g X G X f z W f z W−−

=

= − −∑ ,

and

( )1/ 2 ( )6

1

ˆˆ( ) [ ( ) ( )][ ( , ) ( , )]n

iin i i i XXW

iS z n G X G X f z W f z W−− −

=

= − −∑ .

Then 6

1( ) ( )n n

jS z S z

=

=∑ .

Define . ( )i i iV Y G X= −

Lemma 1: As , n →∞

23

, 1/ 23

1( ) ( , ) / ( ) ( )

n

n i i X ii

S z n V t X z f X r z−

=

= − +∑ n

where . 1 20

( ) (1)n pr z dz o=∫ Proof: Define

2

( )1

2 1

1( ) ( , )( )

ni

j h jnX j

j i

R x V K xnh f x

−

=≠

= −∑ X x ,

2

( )2

2 1

1( ) [ ( ) ( )] ( , )( )

ni

j hnX j

j i

jR x G X G x K xnh f x

−

=≠

= −∑ X x− ,

( )1/ 23 1

1( ) [ ( ) ( , )]

ni

n a i i XW ini

S z n R X f z W−−

=

= ∑E

where iE denotes the expected value over i-subscripted random variables,

( ) ( )1/ 23 1 1

1( ) { ( ) ( , ) [ ( ) ( , )]}

ni i

n b i XW i i i XW in ni

S z n R X f z W R X f z W− −−

=

= −∑ E ,

and

( )1/ 23 2

1( ) ( ) ( , )

ni

n c i XW ini

S z n R X f z W−−

=

= ∑

Standard calculations for kernel estimators show that

2

2( ) 4

22 21

1 (ˆ ( ) ( ) [ ( )] ( , )( )

ij h j

X jj i

nG x G x Y G x K x X x O hnh f x nh

−

=≠

log )⎡ ⎤− = − − + +⎢ ⎥

⎢ ⎥⎣ ⎦∑

uniformly over . Therefore, [0,1]x∈

3 3 3 3( ) [ ( ) ( ) ( )] (1)n n a n b n cS z S z S z S z o= − + + + p

uniformly over . Lengthy but straightforward calculations show that [0,1]z∈

1 12 23 30 0

( ) (1), ( ) (1)n b n cS z dz o S z dz o= =∫ ∫E E

as . Therefore, n →∞

(A.1) 1 2

30( ) (1)n b pS z dz o=∫

and

(A.2) 1 2

30( ) (1)n c pS z dz o=∫

24

by Markov’s inequality. Moreover, we can write

2

( )1

1

02 1

11

[ ( ) ( , )]

1 [ ( , ) ( , ) / ( )] ( , )

1 [ ( , ) / ( ) ( , )],

ii i XW in

n

j XW XW X h jjj i

n

j j X j n jjj i

R X f z W

V f x w f z w f x K x X x dxdnh

V t X z f X X zn

ρ

−

=≠

=≠

= −

= +

∑ ∫

∑

E

w

2

where 21( , ) ( )n x z O hρ = uniformly over . Therefore, 2( , ) [0,1]x z ∈

(A.3) 1/ 23 2

1( ) ( , ) / ( ) ( )

n

n a i i X i ni

S z n V t X z f X zρ−

=

= +∑

where as . The lemma follows by combining (A.1)-(A.3). Q.E.D. 1 2

20( ) (1)n z dz oρ =∫E n →∞

Lemma 2: As , . n →∞1 2

40( ) (1)n pS z dz o=∫

Proof: Define

1 ( ) ( )10

1 1

ˆ ˆ[ ( , ) ( , )][ ( , ) ( , )]n n

i jn i j i XW i j XW jXW XW

i jj i

D n U U f z W f z W f z W f z W dz− −−

= =≠

= −∑ ∑ ∫E −

2

.

Then

1 1 ( )2 1 240 0

1

ˆ( ) [ ( , ) ( , )]

(A.4) (1).

ni

n n i i X iXi

n

S z dz D n U f z W f z W dz

D o

−−

=

= + −

= +

∑∫ ∫E E

Now define

( ) (1 1

( , )21 1

,

1ˆ ( , ) , ,n

i jh k h kXW

kk i j

)f z w K z X z K w W wnh

− −

=≠

= − −∑

and

( ) (1 121

1( , ) , ,j h j hz w K z X z K w W wnh

δ = − − )j .

25

Then 1 22n n n n3D D D D= + + , where

1

1 ( , ) ( , )10

1 1

ˆ ˆ[ ( , ) ( , )][ ( , ) ( , )]

n

n ni j j i

i j i XW i i XW jXW XWi j

j i

D

n U U f z W f z W f z W f z W dz− − − −−

= =≠

=

− −∑ ∑ ∫E

1 ( , )12 0

1 1

ˆ[ ( , ) ( , )] ( ,, )n n

i jn i j i XW i jXW

i jj i

iD n U U f z W f z W z Wδ− −−

= =≠

= −∑ ∑ ∫E dz

i z

,

and

113 0

1 1( , ) ( ,, )

n n

n i j i j ji j

j i

D n U U z W z W dδ δ−

= =≠

= ∑ ∑ ∫E .

But . Therefore, ( | ) 0U W =E 1 2 0n nD D= = , and 2 13 1[( ) ]nD O nh −= . The lemma now follows

from Markov’s inequality. Q.E.D.

Lemma 3: As , n →∞ 6 ( ) (1)n pS z o= uniformly over [0,1]z∈ .

Proof: This follows from ( ) 2 1/ 2 21

ˆ ( , ) ( , ) [(log ) /( ) ]iXWXW 1f x w f x w O n nh h− − = + almost

surely uniformly over and almost surely

uniformly over . Q.E.D.

2( , ) [0,1]x w ∈ ( ) 1/ 2 22

ˆ ( ) ( ) [(log ) /( ) ]iG x G x O n nh h− − = + 2

[0,1]x∈

Proof of Theorem 1: Under , 0H 2 5( ) ( ) 0n nS z S z= = for all [0,1]z∈ . Therefore, it

follows from Lemmas 1-3 that 1 20

( ) (1)n n pB z dz oτ = +∫ .

The result follows by writing 1 20[ ( ) ( ) ]n n

2B z B z d−∫ E z as a degenerate U statistic of order two.

See, for example Serfling (1980, pp. 193-194). Q.E.D.

Proof of Theorem 2: ( )ˆˆ| |j j Oω ω− = Ω−Ω%% by Theorem 5.1a of Bhatia, Davis, and

McIntosh (1983). Moreover, standard calculations for kernel density estimators show that 2 1/ 21

ˆ [(log ) /( ) ]O n nhΩ −Ω =% . Part (i) of the theorem follows by combining these two results.

Part (ii) is an immediate consequence of part (i). Q.E.D.

26

Proof of Theorem 3: Let zα% denote the 1 α− quantile of the distribution of

211 j jj

ω χ∞

=∑ % . Because of Theorem 2, it suffices to show that if holds, then under sampling

from ,

1H

( )Y g X U= +

lim ( ) 1nn

zατ→∞

> =P % .

This will be done by proving that 11 20

plim [( )( )] 0nn

n Tq z dzτ−

→∞= >∫ .

To do this, observe that by a uniform law of large numbers of Pakes and Pollard (1989, Lemma

2.8), uniformly over 1/ 22 ( )nn S z− = ( )( ) (1)pTq z o+ [0,1]z∈ . Moreover,

uniformly over because

1/ 25 ( ) (1)n pn S z o− =

[0,1]z∈ ( ) 2 1/ 2 21

ˆ ( , ) ( , ) [(log ) /( ) ]iXWXW 1f z w f z w O n nh h− − = + a.s.

uniformly over . Combining these results with Lemmas 1-3 yields 2( , ) [0,1]z w ∈

1/ 2 1/ 2( ) ( ) ( )( ) ( )n nn S z n B z Tq z r z− −= + + n

dz

where as . It follows from Theorem 1 that .

Therefore, . Q.E.D.

1 20

( ) (1)n pr z dz o=∫ n →∞11 20

( ) (1)n pn B z dz o− =∫11 20[( )( )]p

nn Tq zτ− → ∫Proof of Theorem 4: The conclusions of lemmas 1-3 hold under (2.6)-(2.8). Therefore,

2 5( ) ( ) ( ) ( ) ( )n n n n nS z B z S z S z r z= + + + ,

where . Moreover, 1 20

( ) (1)nr z dz o=∫ p

i

i

( )15

1

ˆ( ) ( )[ ( , ) ( , )]

(1)

ni

n i i XWXWi

S z n X f z W f z W

o

−−

=

= Δ −

=

∑

almost surely uniformly over . In addition z

12

1( ) ( ) ( , )

( ) (1)

n

n i XWi

S z n X f z W

z oμ

−

=

= Δ

= +

∑

almost surely uniformly over . Therefore, z ( ) ( ) ( ) ( )n n nS z B z z r zμ= + + . But

( ) ( ) (1)n n pB z B z o= +%

27

uniformly over . Therefore, it suffices to find the asymptotic distribution of [0,1]z∈

1 2 20

1[ ( ) ( )] ( )n j

jB z z dz bμ μ

∞

=

+ = +∑∫ %%j

dz

,

where 1

0( ) ( )j n jb B z zψ= ∫% % .

The random variables j jb μ+% are asymptotically distributed as independent ( , )j jN μ ω% variates.

Now proceed as in , for example, Serfling’s (1980, pp. 195-199) derivation of the asymptotic

distribution of a degenerate, order-2 U statistic. Q.E.D.

The following definitions are used in the proof of Theorem 5. For each distribution

ncπ ∈F , let ( )A π be a random variable. Let { : 1,2,...}nc n = be a sequence of positive constants.

Write ( )p nA O c= uniformly over if for each ncF 0ε > there is a constant Mε such that

sup [| ( ) | / ]nc

nA c Mεπ

π ε∈

> <PF

.

For each ncπ ∈F , let { ( ) : 1,2,...}nA nπ = be a sequence of random variables. Write

uniformly over if for each

(1)n pA o=

ncF 0ε >

lim sup [| ( ) | ] 0nc

nn

Aπ

π ε→∞ ∈

> =PF

.

Proof of Theorem 5: Let zα denote the critical value of nτ . Observe that zα is bounded

uniformly over . The arguments used to prove lemmas 1-3 show that for

and uniformly over . In addition, an application of Markov’s

inequality shows that uniformly over . Define

ncF1 20

( ) (1)nj pS z dz o=∫

4,6j =1 2

30( ) (1)nS z dz O=∫ p

p

ncF

1 210( ) (1)nS z dz O=∫ ncF

1 3 4 6( ) ( ) ( ) ( ) ( )n n n n nS z S z S z S z S z= + + +%

and

2 5( ) ( ) ( )n n nD z S z S z= + .

Let ⋅ denote the norm. Use the inequality with and

to obtain

2[0,1]L 2 20.5 ( )a b b a≥ − − 2

5

na S=

2n nb S S= +

22( ) 0.5n g n nz D S zα ατ ⎛ ⎞> ≥ − >⎜ ⎟⎝ ⎠

P P % .

For any finite , 0M >

28

( )

2 22 2

2 22

22

0.5 0.5 ,

0.5 ,

0.5 .

n n n n n

n n n

n n

D S z D z S S M

D z S S M

D z M S M

α α

α

α

⎛ ⎞ ⎛− ≤ = ≤ + ≤⎜ ⎟ ⎜⎝ ⎠ ⎝

⎛ ⎞+ ≤ + >⎜ ⎟⎝ ⎠

⎛ ⎞≤ ≤ + + >⎜ ⎟⎝ ⎠

P P

P

P P

% %

% %

%

2 ⎞⎟⎠

%

(1)n pS O=% uniformly over . Therefore, for each ncF 0ε > there is Mε < ∞ such that for all

M Mε>

( )22 20.5 .5n n nD S z D z Mα α ε⎛ ⎞− ≤ ≤ ≤ + +⎜ ⎟⎝ ⎠

P P%

for all distributions in . Equivalently, ncF

( )22 20.5 .5n n nD S z D z Mα α ε⎛ ⎞− > ≥ > + −⎜ ⎟⎝ ⎠

P P%

and

(A8) ( )2( ) .5n nz D z Mα ατ ε> ≥ > + −P P .

Now

( )1/ 2

1

ˆ( ) [ ( ) ( )] ( , )n

in i i XW

iiD z n g X G X f z W−−

=

= −∑ .

Therefore,

1/ 2 21

1( ) [ ( ) ( )][ ( , ) ( )]

n

n i i XW ii

nD z n g X G X f z W h R z−

=

= − +∑E E ,

where ( )nR z is nonstochastic, does not depend on g or , and is bounded uniformly over

. It follows that

G

[0,1]z∈

1/ 2 1/ 2 21( ) ( )( )nD z n Tq z O n h q⎡ ⎤= + ⎣ ⎦E

and

1/ 2( ) 0.5 ( )( )nD z n Tq z≥E

for all distributions in and all sufficiently large . Moreover, ncF n

29

( ) ( )1/ 2 ( )

1

( ) ( )1/ 2 ( )

1

1 2

ˆ ˆ( ) ( ) [ ( ) ( , ) ( ) ( , )]

ˆ( )[ ( , ) ( , )]

( ) ( ),

ni ii

n n i iXW XWi

ni ii

i i iXW XWi

n n

iD z D z n q X f z W q X f z W

n q X f z W f z W

D z D z

− −− −

=

− −− −

=

− = −

+ −

≡ +

∑

∑

E E E

E

where ( )i−E denotes the expectation with respect to the distribution of

. It is clear that { , : 1,..., ; }j jX W j n j i= ≠ 21 (1)n pD O= uniformly over . Moreover, it

follows from the properties of kernel estimators that

ncF

21 1

1/ 2

1

log| ( ) | | ( ) |

log [ | ( ) | ( )],

nn

n in i

np

r nD z q Xnh

r n q X O nh

=

−

≤

= +

∑

E

uniformly over , where almost surely as and depends only on the

distribution of

ncF (1)nr O= n →∞

( , )X W . Therefore,

22 2

1/ 21

log ( | |) (1) (1)nn n p p

r nD D n q O On h

⎛ ⎞⎜ ⎟− ≤ +⎜ ⎟⎝ ⎠

E E .

A further application of with 2 20.5 ( )a b b a≥ − − 2na D= and nb D= E gives

2 4 22 2 1

1/ 2 3 21

22

1/ 2 31

log ( | |).125 (1) (1)

log.125 (1) (1)

nn p

np p

r n h qD n Tq O On h Tq

r nn Tq o On h

⎡ ⎤⎛ ⎞⎢ ⎥≥ − +⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

⎡ ⎤⎛ ⎞⎢ ⎥= − +⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

Ep

uniformly over . Therefore, if is sufficiently large, ncF C 20.5 nD z Mα> + with probability

approaching 1 as uniformly over . Q.E.D. n →∞ ncF

30

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33

Table 1: Results of Monte Carlo Experiments Empirical Probability that H0 Is Rejected Using

n η nτ Hausman test nHτ __ __________ __ __

0.35ρ =

250 0.0 0.042 0.050 0.012 0.10 0.062 0.072 0.022 0.15 0.077 0.126 0.039 0.20 0.076 0.164 0.068 0.25 0.119 0.265 0.116 500 0.0 0.048 0.055 0.025 0.10 0.256 0.304 0.187 0.15 0.539 0.590 0.429 0.20 0.814 0.876 0.724 0.25 0.945 0.971 0.922 750 0.0 0.048 0.053 0.035 0.10 0.137 0.172 0.131 0.15 0.274 0.313 0.232 0.20 0.422 0.468 0.379 0.25 0.596 0.675 0.601

0.70ρ = 250 0.0 0.047 0.051 0.028 0.10 0.156 0.188 0.079 0.15 0.293 0.366 0.192 0.20 0.464 0.568 0.360 0.25 0.705 0.802 0.563 500 0.0 0.048 0.055 0.025 0.10 0.256 0.304 0.187 0.15 0.539 0.590 0.429 0.20 0.814 0.876 0.724 0.25 0.945 0.971 0.922 750 0.0 0.050 0.049 0.025 0.10 0.383 0.479 0.298 0.15 0.728 0.806 0.646 0.20 0.929 0.958 0.896 0.25 0.994 0.997 0.983

34