SPECIFICATION TESTING IN NONPARAMETRIC INSTRUMENTAL .
SPECIFICATION TESTING IN NONPARAMETRIC INSTRUMENTAL VARIABLESESTIMATIONbyJoel L. HorowitzDepartment of EconomicsNorthwestern UniversityEvanston, IL 60208USAOctober 2009ABSTRACTIn nonparametric instrumental variables estimation, the function being estimated is thesolution to an integral equation. A solution may not exist if, for example, the instrument is notvalid. This paper discusses the problem of testing the null hypothesis that a solution existsagainst the alternative that there is no solution. We give necessary and sufficient conditions forexistence of a solution and show that uniformly consistent testing of an unrestricted nullhypothesis is not possible. Uniformly consistent testing is possible, however, if the nullhypothesis is restricted by assuming that any solution to the integral equation is smooth. Manyfunctions of interest in applied econometrics, including demand functions and Engel curves, areexpected to be smooth. The paper presents a statistic for testing the null hypothesis that a smoothsolution exists. The test is consistent uniformly over a large class of probability distributions ofthe observable random variables for which the integral equation has no smooth solution. Thefinite-sample performance of the test is illustrated through Monte Carlo experiments.Key Words: Inverse problem, instrumental variable, series estimator, linear operatorJEL Listing: C12, C14I thank Whitney Newey for asking a question that led to this paper. Xiaohong Chen, HidehikoIchimura, Sokbae Lee, and Whitney Newey provided helpful comments. This research wassupported in part by NSF grants SES-0352675 and SES-0817552.
SPECIFICATION TESTING IN NONPARAMETRIC INSTRUMENTAL VARIABLESESTIMATION1. IntroductionNonparametric instrumental variables (IV) estimation consists of estimating the unknownfunction g that is identified by the relation(1.1)E[Y g ( X ) W w] 0for almost every w in the support of the random variable W . Equivalently, g satisfies(1.2)Y g ( X ) U ; E (U W w) 0for almost every w . In this model, Y is a scalar dependent variable, X is a continuouslydistributed explanatory variable that may be endogenous (that is, E (U X x) may not be zero),W is an instrument for X , and U is an unobserved random variable. The function g isassumed to satisfy mild regularity conditions but does not belong to a known, finite-dimensionalparametric family.The data are an independent random sample from the distribution of(Y , X ,W ) .Methods for estimating g in (1.1) have been developed by Newey and Powell (2003);Darolles, Florens, and Renault (2006); Hall and Horowitz (2005); and Blundell, Chen, andKristensen (2007). Newey, Powell, and Vella (1999) developed a nonparametric estimator for gin a different setting in which a control function is available. Horowitz and Lee (2007); Chen andPouzo (2008); and Chernozhukov, Gagliardini, and Scaillet (2008) have developed methods forestimating quantile-regression versions of model (1.1).All methods for estimating g in model (1.1) assume the existence of a function thatsatisfies (1.1). However, as is explained in Section 2 of this paper, a solution need not exist if, forexample, the instrument W is not valid, that is if E (U W w) 0 on a set of w values that hasnon-zero probability. This raises the question whether it is possible to test for existence of asolution to (1.1). This paper provides an answer to the question. The null hypothesis is that asolution to (1.1) exists.The set of alternative hypotheses consists of the distributions of(Y , X ,W ) for which there is no solution to (1.1). We consider tests that are consistent uniformlyover this set. Uniform consistency is important because it ensures that there are not alternativesagainst which a test has low power even with large samples. If a test is not uniformly consistentover a specified set, then that set contains alternatives against which the test has low power.Some such alternatives may depart from the null hypothesis in extreme ways, as is illustrated byexample in Section 3. We show that the null hypothesis cannot be tested consistently uniformly1
over the set of alternative hypotheses without placing restrictions on g beyond those needed toestimate it when (1.1) has a solution. Specifically, there is always a distribution of (Y , X ,W )such that no solution to (1.1) exists but any α -level test accepts the null hypothesis with aprobability that is arbitrarily close to 1 α .We also show that it is possible to test the hypothesis that a “smooth” solution to (1.1)exists. The test is consistent uniformly over a large class of non-smooth alternatives. The paperpresents such a test. Non-existence of a solution to (1.1) is an extreme form of non-smoothness,so in sufficiently large samples, the test presented here rejects the null hypothesis that g issmooth if no solution to (1.1) exists.We define g to be smooth if it has sufficiently many square-integrable derivatives. Witha sufficiently large sample, the test presented here rejects the null hypothesis that (1.1) has asmooth solution if no solution exists or if one exists but is not smooth. The possibility ofrejecting a non-smooth solution is desirable in many applications. For example, a demandfunction or Engel curve is unlikely to be discontinuous or wiggly.Thus, rejection of thehypothesis that a demand function or Engel curve is smooth implies misspecification of the modelthat identifies the curve or function (e.g., that W is not a valid instrument for X ). Accordingly,the test described here is likely to be useful in many applications.The test presented here is related to Horowitz’s (2006) test of the hypothesis that gbelongs to a specified, finite-dimensional parametric family.A smooth function can beapproximated accurately by a finite-dimensional parametric model consisting of a truncated seriesexpansion with suitable basis functions. See Section 4 for details. The approximation to a nonsmooth function is less accurate. Therefore, one can test for existence of a smooth solution to(1.1) by testing the adequacy of a truncated series approximation to g . The test statistic issimilar to Horowitz’s (2006) statistic for testing a finite-dimensional parametric model, but itsasymptotic behavior is different. In Horowitz (2006), the dimension of the parametric model isfixed. In the present setting, the dimension (or length) of the series approximation increases asthe sample size increases. This is necessary to ensure that the truncation error remains smallerthan the smallest deviation from the null hypothesis that the test can detect. The increasingdimension of the null hypothesis model changes the asymptotic behavior of the test statistic inways that are explained in Section 4.Section 2 of this paper gives a necessary and sufficient condition for existence of afunction g that solves (1.1). It also explains why a solution may not exist. Section 3 presents anexample showing that it is not possible to construct a uniformly consistent test of the hypothesis2
that (1.1) has a solution. No matter how large the sample is, there are always alternatives againstwhich any test has low power. Section 4 describes the statistic for testing the hypothesis that(1.1) has a smooth solution. This section also explains the test’s asymptotic behavior under thenull and alternative hypotheses. Section 5 presents the results of a Monte Carlo investigation ofthe test’s finite-sample behavior, and Section 6 presents conclusions. The proofs of theorems arein the appendix, which is Section 7.2. Necessary and Sufficient Conditions for a Solution to (1.1)Necessary and sufficient conditions for existence of a solution to (1.1) are given byPicard’s theorem (e.g., Kress 1999, Theorem 15.18). Before stating the theorem, we definenotation that will be used throughout the paper.Assume that X and W are real-valued random variables. Assume, also, that the supportof ( X ,W ) is contained in [0,1]2 . This assumption entails no loss of generality as it can alwaysbe satisfied by, if necessary, carrying out monotone transformations of X and W . Let f XW andfW , respectively, denote the probability density functions (with respect to Lebesgue measure) of( X ,W ) and W . For x, z [0,1] , definet ( x, z ) 1 0 f XW ( x, w) f XW ( z, w)dw .Define the operator T : L2 [0,1] L2 [0,1] by(Tν )( z ) 1 0 t ( x, z)ν ( x)dx ,where ν is any function in L2 [0,1] . Assume that T is non-singular. Denote its eigenvalues andeigenvectors by {λ j ,φ j : j 1, 2,.} . Sort these so that λ1 λ2 . 0 . Under the assumptionsstated in Section 4, T is a compact operator. Therefore, its eigenvectors form a complete,orthonormal basis for L2 [0,1] . Moreover, the eigenvalues are strictly positive and have 0 as theironly limit point.Now, for z [0,1] defineq( z ) EYW [Yf XW ( z ,W )]where EYW denotes the expectation with respect to (Y ,W ) . Hall and Horowitz (2005) show that(1.1) is equivalent to the operator equation(2.1)q Tg .3
Therefore, the conditions for existence of a solution to (1.1) are the same as the conditions forexistence of a function g that satisfies (2.1).Let , denote the inner product in L2 [0,1] . The following theorem gives necessary andsufficient conditions for existence of a function g that solves (1.1) and (2.1).Theorem 2.1 (Picard): Let T be a compact, non-singular operator, and assume thatq 0 . Then (2.1) has a solution if and only if q, φ jj 12λ 2j .If a solution exists, it isg ( x) b jφ j ( x ) ,j 1wherebj q, φ jλj. Testing the hypothesis that (1.1) has a solution is equivalent to testing the hypothesis that j 1b2j against the alternative j 1b2j .The quantities q,φ jare the generalizedFourier coefficients of q using the basis functions {φ j } . That is,q( z ) q, φ j φ j ( z ) .j 1Therefore a solution to (1.1) exists if and only if the Fourier coefficients of q convergesufficiently rapidly relative to the eigenvalues of T . It is easy to construct examples in which thegeneralized Fourier coefficients converge more slowly than the eigenvalues so that j 1b2j .In applied econometric research g may be an Engel curve, demand function, or someother economically meaningful function whose existence is not in question. In this case, (1.1)may not have a solution if Wis not a valid instrument.Specifically, suppose thatE (U W w) 0 on a set of w values with positive probability. Then arguments like those usedto obtain (2.1) show that g solves not (1.1) or (2.1) but(2.2)(Tg )( z ) q ( z ) EUW [Uf XW ( z ,W )] .4
The misspecified models (1.1) and (2.1) need not have solutions when W is an invalidinstrument and (2.2) is the correct specification.3 The Impossibility of Uniformly Consistent Testing with Unrestricted Null and AlternativeHypothesesThis section presents an example in which uniformly consistent testing of the hypothesisthat (1.1) has a solution is not possible. The distributions used in the example are nested in anyreasonable class of probability distributions for (Y , X ,W ) in (1.1), so the impossibility resultobtained with the example holds generally.The example consists of a simple null-hypothesis and a simple alternative-hypothesis.“Simple” in this context means that there are no unknown population parameters in either the nullor alternative hypotheses. The null hypothesis is that a specific function g solves (1.1). Underthe alternative hypothesis, (1.1) has no solution. It follows from the Neyman-Pearson lemma thatthe likelihood ratio test is the most powerful test of the null hypothesis against the alternative.We show that regardless of the sample size, it is always possible to choose an alternativehypothesis against which the power of the likelihood ratio test is arbitrarily close to its size.Therefore, the likelihood ratio test is not uniformly consistent. It follows that no other test isuniformly consistent because no other test is more powerful than the likelihood ratio test.To construct the example, write (1.1) in the equivalent form(3.1)Y E[ g ( X ) W ] V ; E (V W ) 0 ,where V Y E (Y W ) . Assume that f XW is known and isf XW ( x, w) 1 2 λ1/j 2 cos( jπ x) cos( jπ w) ,j 1where the λ j ’s are constants satisfying λ1 λ2 . 0 andfunction, the eigenvalues of Tare {1, λ1 , λ2 ,.} . j 1λ1/j 2 .With this densityThe eigenvectors are φ1 ( x) 1 andφ j ( x) 21/ 2 cos[( j 1)π x] for j 2 .Assume that V is known to be distributed as N (0,1) and is independent of X and W .Let {b j : j 0,1, 2,.} denote the Fourier coefficients of g with the cosine basis. That is,(3.2)g ( x) b0 21/ 2 b j cos( jπ x) .j 1Then5
E[ g ( X ) W ] b0 21/ 2 b j λ1/j 2 cos( jπ W ) ,j 1and model (1.1) becomes(3.3)Y b0 21/ 2 b j λ1/j 2 cos( jπ W ) V ;V N (0,1) .j 1Now let J 0 be an integer. Consider testing the simple null hypothesis b0 1 H 0 : b j j 2 if 1 j J b j 0 if j Jagainst the simple alternative hypothesis b0 1 H1 : b j j 2 if 1 j J . b j 1if j JUnder H 0 , g in (3.2) is an ordinary function on [0,1] , and g solves (1.1). Under H1 , g is alinear combination of an ordinary function and a delta function, so g is not a function on [0,1] inthe usual sense and (1.1) has no solution.Let the data be the independent random sample {Yi , X i ,Wi : i 1,., n} . We show that forany fixed n , no matter how large, it is possible to choose J so that the power of the likelihoodratio test of H 0 against H1 is arbitrarily close to its size.The likelihood ratio statistic for testing H 0 against H1 is LR (1/ 2) Yi 1 21/ 2i 1 n(3.4) Yi 1 21/ 2 J 22j 2 λ1/j cos( jπ Wi ) j 1 21/ 2j 2 λ1/λ1/j 2 cos( jπ Wi ) j cos( jπ Wi ) 2 j 1j J 1 J 2 Substituting (3.3) into (3.4) shows that under H 0 , the likelihood ratio statistic isLR0 21/ 2n λ1/j 2 cos( jπ Wi )Vii 1 j J 12 λ1/j 2 cos( jπ Wi ) . i 1 j J 1n Under H1 , the likelihood ratio statistic is6 .
LR1 21/ 2n λ1/j 2 cos( jπ Wi )Vii 1 j J 12 λ1/j 2 cos( jπ Wi ) . i 1 j J 1n Therefore, LR1 LR0 2 λ1/j 2 cos( jπ Wi ) i 1 j J 1n2 2 λ1/j 2 . 2n j J 1 Because j 1 λ1/j 2 , LR1 LR0 can be made arbitrarily small by making J sufficientlylarge. Therefore, we obtain the following result, which is proved in the Appendix.Proposition 3.1: Let cnα denote the α -level critical value of LR when the sample sizeis n . That is, P ( LR cnα H 0 ) α . Let n be fixed. For each ε 0 there is a J 0 such that thepower of the α -level likelihood ratio test of H 0 against H1 is less than or equal to α εwhenever J J 0 . That is, P ( LR cnα H1 ) α ε whenever J J 0 . Now consider the class of alternative hypotheses consisting of distributions of (Y , X ,W )for which H1 is true for some J . Because no test is more powerful than the likelihood ratiotest, it follows from Proposition 3.1 that no test of H 0 is consistent uniformly over this class.Regardless of the sample size n , there are always distributions in H1 for some finite J againstwhich the power of any test is arbitrarily close to the test’s level. The intuitive reason is thatFourier components of g corresponding to eigenvectors of T with small eigenvalues have littleeffect on Y and, therefore, are hard to detect empirically. This is illustrated by (3.3), where Y isinsensitive to changes in Fourier coefficients b j that are associated with very small eigenvaluesλ j . This problem can be overcome by restricting the null and alternative hypotheses so as toavoid the need for estimating or testing Fourier coefficients associated with very smalleigenvalues of T . Section 4 presents a way of doing this.4 A Uniformly Consistent Test of the Hypothesis That (1.1) Has a Smooth SolutionIn this section, we restrict the null hypothesis by requiring g to be smooth in the sensethat it has s square-integrable derivatives, where s is a sufficiently large integer. Under thealternative hypothesis, (1.1) has no smooth solution. In sufficiently large samples, the resulting7
test rejects the null hypothesis if (1.1) has no solution or if (1.1) has a non-smooth solution. Aswas explained in the introduction, a non-smooth solution to (1.1) is an indicator ofmisspecification (possibly due to an invalid instrument) in many applications. Therefore, theability to reject non-smooth solutions to (1.1) can be a desirable property of a test.4.1 MotivationWe begin with an informal discussion that provides intuition for the test that is developedhere. Let {ψ j : j 1, 2,.} be a complete, orthonormal basis for L2 [0,1] . Suppose for themoment that under H 0 , the solution to (1.1) has the finite-dimensional representation(4.1)g ( x) J b jψ j ( x) ,j 1for some fixed J and (generalized) Fourier coefficients {b j : j 1,., J } . Equation (4.1)restricts g to a finite-dimensional parametric family. The null hypothesis that (4.1) is a solutionto (1.1) for some set of b j ’s can be tested against the alternative that it is not by using the test ofHorowitz (2006). The test statistic is(4.2)1τ Pn Sn ( z ) 2 dz ,0whereSn ( z ) n 1/ 2 Yi i 1 n J j 1 ( i )( z ,Wi ), bˆ jψ j ( X i ) fˆXW( i )is a leavebˆ j is an estimator of b j that is n 1/ 2 -consistent under the null hypothesis, and fˆXWobservation- i -out nonparametric kernel estimator of f XW . Specifically,(4.3)( i )fˆXW( x, w) n1(n 1)h2 x X j w Wj,hh K j 1j i , where h is a bandwidth and K is a kernel function of a 2-dimensional argument. Horowitzshows that if (4.1) is a solution to (1.1), then τ Pn is asymptotically distributed as a weighted sumof independent chi-square random variables with one degree of freedom. Horowitz (2006) alsoshows that τ Pn is consistent uniformly over a class of nonparametric alternative hypotheseswhose distance from (4.1) is proportional to n 1/ 2 .8
Now let g be nonparametric, but suppose that its derivatives through order s are squareintegrable on [0,1] . Then g has the infinite-dimensional Fourier representationg ( x) b jψ j ( x)j 1but can be approximated accurately by the finite-dimensional model that is obtained by truncatingthis series. Specifically, let {J n : n 1, 2,.} be a sequence of positive integers with J n asn . Define(4.4)g n ( x) Jn b jψ j ( x) .j 1Then for a wide variety of basis functions {ψ j } that includes trigonometric functions andorthogonal polynomials, the error of g n as an approximation to g satisfiesg g n O ( J n s ) ,where denotes the norm in L2 [0,1] . Thus, a smooth function g can be approximatedaccurately by a finite-dimensional parametric function. This suggests testing the null hypothesisthat (1.1) has a smooth solution by using Horowitz’s (2006) procedure to test the hypothesis that(4.4) is the solution. If J n is sufficiently large, the approximation error will be small comparedto the minimum deviation from (4.4) that the test can detect. On the other hand, if the solution to(1.1) is non-smooth or does not exist, then (4.4) will be a poor approximation to the solution to(1.1), and the test will reject the null hypothesis if the sample is sufficiently large.The main difference between this version of Horowitz’s (2006) test and the test based onτ Pn in (4.2) is that when g is nonparametric, J n must increase as n increases to ensure that theapproximation error remains too small to be detected by the test. This changes the asymptoticdistributional properties of the test statistic.Among other things, the test statistic isasymptotically degenerate (its asymptotic distribution is concentrated at a single point) under thenull hypothesis. We deal with this problem here by splitting the sample into halves. One half isused to estimate the b j ’s and the other half is used to construct the test statistic. The samplesplitting procedure is explained in more detail in Section 4.3.
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