Hypothesis testing of multiple inequalities: the method of constraint chaining

Transcription

1 Hyothesis testing of multile inequalities: the method of constraint chaining Le-Yu Chen Jerzy Szroeter The Institute for Fiscal Studies Deartment of Economics, UCL cemma working aer CWP13/09

2 Hyothesis Testing of Multile Inequalities : The Method of Constraint Chaining 1 Le-Yu Chen a2 and Jerzy Szroeter b a Institute of Economics, Academia Sinica b Deartment of Economics, University College London Revised: May 2009 Abstract Econometric inequality hyotheses arise in diverse ways. Examles include concavity restrictions on technological and behavioural functions, monotonicity and dominance relations, one-sided constraints on conditional moments in GMM estimation, bounds on arameters which are only artially identi ed, and orderings of redictive erformance measures for cometing models. In this aer we set forth four key roerties which tests of multile inequality constraints should ideally satisfy. These are (1) (asymtotic) exactness, (2) (asymtotic) similarity on the boundary, (3) absence of nuisance arameters from the asymtotic null distribution of the test statistic, (4) low comutational comlexity and boostraing cost. We observe that the redominant tests currently used in econometrics do not aear to enjoy all these roerties simultaneously. We therefore ask the question : Does there exist any nontrivial test which, as a mathematical fact, satis es the rst three roerties and, by any reasonable measure, satis es the fourth? Remarkably the answer is a rmative. The aer demonstrates this constructively. We introduce a method of test construction called chaining which begins by writing multile inequalities as a single equality using zero-one indicator functions. We then smooth the indicator functions. The aroximate equality thus obtained is the basis of a well-behaved test. This test may be considered as the baseline of a wider class of tests. A full asymtotic theory is rovided for the baseline. Simulation results show that the nite-samle erformance of the test matches the theory quite well. KEYWORDS : Tests, multile inequality constraints, exactness, similarity on the boundary, smoothing, chaining methods JEL SUBJECT AREA : C1, C4 1 This aer is a revision of Chater 3 of the rst author s doctoral dissertation (Chen 2009). We thank Oliver Linton for helful comments on this research work. 2 Corresonding author : Le-Yu Chen, Institute of Economics, Academia Sinica, Taiei, Taiwan, ROC. lychen@econ.sinica.edu.tw (L. Chen), j.szroeter@ucl.ac.uk (J. Szroeter) 1

3 1 Introduction and related literature The growing imortance of research on testing econometric multile inequality hyotheses cannot be disuted. Such hyotheses now arise in diverse contexts, both classical and at the frontier. This is illustrated by the following selection of six non-exclusive grouings in which we give literature examles where formal inference on multile inequalities is called for. Additional references will be found within the aers listed. Grou 1 : Concavity and other restrictions characterizing consumer-roducer otimality and technology. See Hazilla and Ko (1986), Kodde and Palm (1987), Koebel, Falk and Laisney (2003), Reis (2006) and Wolak (2007). Grou 2 : Multieriod inequalities due to liquidityrisk remiums and random walk e ects in nancial markets. See Richardson, Richardson and Smith (1992), Boudoukh, Richardson, Smith and Whitelaw (1999), Fisher, Willson and Xu (1998) and Fleming, Kirby and Ostdiek (2006). Grou 3 : Stochastic dominance and monotonicity restrictions. See Anderson (1996), Davidson and Duclos (2000), Barrett and Donald (2003), Post (2003), Linton, Maassoumi and Whang (2005), Linton, Song and Whang (2008) and Lee, Linton and Whang (2009). Grou 4 : One-sided constraints on (conditional) moments in GMM estimation. See Rosen (2008) and Andrews and Soares (2009). Grou 5 : Bounds on arameters which are only artially identi ed. See Guggenberger, Hahn and Kim (2008) and McAdams (2008). Grou 6 : Orderings of redictive erformance measures for cometing models. See White (2000), Hansen (2005), Hansen and Lunde (2005) and Martin, Reidy and Wright (2009). It will be noticed that the dates of the aers above fall within the last 20 or so years. This is a re ection of the fact that econometric theory for handling multile inequalities has largely develoed in that eriod of time and is still very much incomlete. The urose of the resent aer is to address four outstanding concerns in test design and to show constructively that there does exist a theoretically sound test which can simultaneously satisfy four key roerties. That test is based on the novel concet of constraint chaining. The articles cited above consider null hyotheses of the tye where, for some nite or in nite subset S of the real line, the true values of arameters r ; r 2 S, in a given econometric model are all nonnegative whilst the alternative is that at least one is negative. The subscrit r may be simly an indexing argument but it could also be the value of an observable economic variable, in which case S may be a entire interval. This is the case in some aers of Grou 3 above. In other grous, S is generally the set of integers f1; 2; : : : ; g for some nite. The baseline statistical roerties we set forth shortly for tests of comosite null hyotheses are relevant regardless of S. The aer conducts most of its seci c analysis for nite but that extends to the case where S is a continuum. We allow the vector ( 1 ; 2 ; : : : ; ) 0 2

4 to be functionally deendent on other arameters in the model. This framework is then su ciently general to cover most nite multile inequality contexts. The di culty of designing tests arises because simle one-sided testing of the single scalar arameter case ( = 1) does not carry over in an obvious way to the situation where 2. It is temting to try to solve the roblem by conducting a family of individual tests each at reduced signi cance level in an attemt to kee the family-wide error rate (overall signi cance level) within a nominal value. However, since the actual overall signi cance level generally falls well short of the nominal, the family of tests is inexact and, being conservative, is likely to have lower ower than an e cient test which is designed to attain the nominal level. The family of tests (or multile comarisons ) aroach has useful features (Savin (1980, 1984), Hochberg and Tamhane (2007)) but we do not consider it in this aer since it does not enjoy the most imortant of the four key roerties which set the standard in tests of comosite null hyotheses. These are as follows. Proerty 1 : (Asymtotic) exactness. This roerty requires that the nominal test size is (asymtotically) achieved at some oint of the null hyothesis (and not exceeded at any oint). Proerty 2 : (Asymtotic) similarity on the boundary of a comosite null hyothesis. Alied to the resent context, this roerty requires that a test should have constant rejection robability at all oints 0 where at least one element of is exactly zero. Theoretical work indicates that bias and inadmissibility of test, for examle, can result from failure of Proerty 2. See Hansen (2003, 2005) and Linton, Song and Whang (2008). Proerty 3 : Absence of nuisance arameters from the (asymtotic) null distribution of the test statistic. The nuisance arameter (any arameter other than that subject to the null hyothesis) is usually a covariance matrix. When Proerty 3 fails, true test critical values cannot be evaluated even if the analytical form of the null distribution is known comletely. Estimates of critical values can be obtained either by lugging consistent estimates of the nuisance arameters into the analytical formula (if available) for the critical values. Alternatively, one may directly estimate the critical values as quantiles of the bootstra distribution of the test statistic. But the bootstra samles must be adjusted to ensure that they mimic a oulation in which the null hyothesis holds (Politis, Romano and Wolf (1999, Section 1.8)). Using the bootstra to eliminate nuisance arameters is quite searate from its otential in securing higher order imrovement over rst-order asymtotic aroximation to the true nite-samle cdf of test statistics Indeed, the latter may be adversely a ected by the former (Horowitz (2001)). Proerty 4 : Inexcessive comutational comlexity and boostraing cost. 3

5 Attemts to rescue tests from violation of Proerties 1-3 may comromise Proerty 4. In relation to the above roerties, we roceed now to review the two redominant aroaches used in econometrics for testing the validity of a set of inequality hyotheses. We initially assume that the exists an estimator b based on samle size T such that T (b ) is (asymtotically) multivariate normal with mean 0 and covariance matrix V consistently estimated by b V, both nonsingular. V and may deend on common arameters but this is generally ket imlicit for notational simlicity. The rst test aroach measures emirical discreancy between unrestricted and inequality-restricted models using a quadratic form such as min :0 T (b )0 b V 1 (b ): (1) This aroach extends classical Wald, LM and LR equality tests to the context of multile inequalities. The literature on this extension is vast and we therefore direct the reader to the key articles by Perlman (1969), Kodde and Palm (1986), Shairo (1988), Wolak (1987, 1988, 1989a, 1989b, 1991), Dufour (1989) and the textbooks by Robertson, Wright and Dykstra (1988), Gourieroux and Monfort (1995, Chater 27) and Silvaulle and Sen (2005, Chaters 3 and 4). The (asymtotic) distribution of (1) deends on the value of the DGP arameter from which data are drawn. For the case where V is known hence b V = V, the work of Perlman (1969) and Wolak (1987, 1988) shows that the (asymtotic) cdf of (1) at the articular oint = 0 is of the so-called chi-bar squared distribution 3 X w r; (V )F (xjr) (2) r=0 due to Kudo (1963), where w r; (V ) are nonnegative weights summing to 1 and F (xjr) is the cdf of a central chi-square variate with r degrees of freedom. At boundary oints of the comosite null hyothesis 0, the cdf is also chi-bar but with relaced by d, the number of binding constraints, and V relaced by a submatrix corresonding to those constraints 4. Since the quantiles of this distribution vary with d, a test using quantiles as critical values calculated assuming = 0 hence d = necessarily fails to have the desirable Proerty 2 (asymtotic similarity on the boundary) listed earlier. Nonetheless, in the 3 The use of quadratic form statistic and chi-bar squared test is also related to the test roblem of a simle null hyothesis against the one-sided alternative. See e.g., Kudo (1963), Nuesch (1966), Gourieroux, Holly and Monfort (1982), King and Smith (1986), Andrews (1998) and Feng and Wang (2007). See also Goldberg (1992) for discussions of the relation between the two testing roblems in the context of the chi-bar squared aroach. 4 See equation (4.116) in Silvaulle and Sen (2004, 204). 4

6 case V b = V, Proerty 1 (asymtotic exactness) still holds because Perlman (1969) roved that the articular value = 0 is least favourable (LF) to the null hyothesis in the sense that the (asymtotic) rejection robability of the test based on the quadratic form statistic (1) is maximized over oints satisfying 0 by the choice = 0, rovided that V is not functionally related to. Since the cdf of the test statistic at = 0 is known, a critical value can in rincile be calculated such that any nominal test size is attained. However, the weight w r; (V ) cannot be given a general algebraic form as it reresents the robability that r elements out of a dimensional vector of correlated normal variables with covariance V take value in a cone determined by the constraints. Comutation of these weights is non-trivial when, the number of inequality constraints exceeds three. When is large, numerical simulation of the weights or the tail robability of (2) is required. Silvaulle and Sen (2005,.78-81) rovide a helful recent review of available simulation techniques for the chi-bar squared distribution. Note that to ensure a reasonable estimate of the chi-bar squared distribution, iterations of constrained quadratic rogramming of (1) also have to be erformed. This cycle of otimization and simulation could hardly be described as routine and has an adverse imact on Proerty 4. There is unfortunately a further roblem with the chi-bar squared test described above. When V is related to, which inevitably haens if is a nonlinear function of some underlying arameters, the choice = 0 is no longer LF. In such cases, it is generally imossible to locate an LF value. One way out of this roblem is to derive conservative bounds on the LF distribution. See, e.g., Kodde and Palm (1986). The bounds aroach is not only oular, but it has even been roosed as necessary since, in most instances, it is quite di cult to obtain an emirically imlementable asymtotically exact test, esecially when the test is a NOS (nonlinear one-sided) hyothesis test. On the other hand, it runs the risk of inconclusive test results. Wolak (1991) discusses this roblem in deth and oints out that for tests with high dimensional inequality constraints, the bounds become very slack, making inconclusive test results more likely. The consequence is that a test based on the quadratic form statistic (1) using bounds of critical values will almost certainly not satisfy Proerty 1 (asymtotic exactness). Guggenberger, Hahn and Kim (2008) give an equivalence result which imlies that the roblem of testing seci cation of models containing moment inequality constraints involving artially identi ed arameters is equivalent to testing an aroriately-de ned set of nonlinear multile inequalities. Indeed Rosen (2008), using Wolak-style chi-bar theory in a artially identi ed setting, notes that his model seci cation test may be conservative. In a more general distributional setting also allowing artially identi ed arameters in moment inequalities, Andrews and Soares (2009,.22-23) warn that their GMS model seci cation test is conservative to unknown extent (hence may lose Proerty 1) in some cases. Finally, even when V is not related to, it is a nuisance arameter in the sense of Proerty 3 when its value is unknown. This causes the roblems discussed in the aragrah following the statement of Proerty 3 and thus comromises Proerty 4. The second redominant aroach to testing the hyothesis 0, which 5

7 we call the extreme value (EV) method, is based on the statistic min f T b 1 ; T b 2 ; : : : ; T b g (3) where b r denotes the r-th element of b. Recent heightened interest in this aroach is due to White (2000). If recisely d 1 elements of the true value of are zero (the rest being strictly ositive), the asymtotic null distribution of (3) is that of the minimum element of a normal vector with mean 0 and covariance equal to an aroriate dd submatrix of V. An analytic exression for the cdf of this distribution is not generally available, hence the cdf has to be obtained by simulation for each given V and d. That can be done to arbitrary accuracy but may imact on Proerty 4. Since the asymtotic null distribution of (3) deends on d, the test cannot satisfy Proerty 2. However, as long as V does not deend on, the cdf for the case d = is dominated by the cdfs for d <. So the oint = 0 is least favourable hence Proerty 1 will hold. When V is related to, this conclusion is not sustained and we are faced with the roblems discussed in connection with the chi-bar squared test. When V is not related to but its value is unknown, it is a nuisance arameter in the sense of Proerty 3 and the issues discussed in connection with that roerty come into lay. But note that Proerty 4 is less a ected than in the case of the chi-bar squared test whose statistic requires an inequality-constrained quadratic minimization couled with V inversion. The EV method of White has been modi ed in two ways by Hansen (2005) and Hansen and Lunde (2005). First, Hansen works with t-ratios of the b r rather than their raw values. Second, and more relevant to the four baseline roerties we have set forth in this aer, White comutes his critical values from an estimated null-hyothesis comliant cdf which corresonds to a oulation having mean = 0, whereas Hansen s mean is data-driven having a articular element equal to zero only if the corresonding element of b is negative or close to zero (the measure of closeness diminishing as samle size increases). Asymtotically, Hansen is consistently estimating the correct value of d, the true number of zeros to use in the asymtotic null distribution of the test statistic. In consequence, unlike White s test, Hansen s test will achieve the key Proerty 2. Nuisance arameter issues concerning V are the same as in White s test. The imortance of estimating the number of binding inequality constraints (or contact oints in the notion of Linton, Song, Whang (2008)) and the bene ts of automatic data-driven comonents in test rocedures has been further addressed in a number of more recent aers which include notably Linton, Song, Whang (2008), Andrews and Jia (2008), and Andrews and Soares (2009). In the light of the above discussion, it is comelling to ask if there exists any aroach which is theoretically guaranteed to satisfy Proerties 1 through 3 as a mathematical fact and, by any reasonable measure, satis es Proerty 4. For the last roerty, it is desired to develo a simle and reasonably e ective test of multile inequality constraints whenever the cost or comlexity of alternative testing rocedures is rohibitive. The test should be alicable under wide 6

8 assumtions and be easy to imlement without need for bootstraing or numerical simulation. Remarkably, the answer is a rmative. The rest of the resent aer demonstrates this. We set out an aroach to inequality test construction which is hilosohically and oerationally very di erent from the chi-bar squared and EV tests. The new idea is to chain multile inequalities into a single equality by exloiting zero-one indicator functions. Thus testing multile inequalities is reduced to testing a single equality. Then we smooth these functions to overcome the distribution roblems of discrete functions of continuous variates. The smoothing embodies a data-driven imortance weighting feature which at the outset automatically concentrates weight onto those arameter estimates having most inferential value. In resect of the role of smoothing, there is some a nity between our aer and the work of Horowitz (1992). The zero-one indicator in Horowitz s context reresents a binary choice robability deending on model arameters. By working with a smoothed version of the indicator which aroaches the original zero-one with increasing samle size, Horowitz nds that his smoothed maximum score estimation method results in arameter estimates which achieve root of samle size convergence rate and are asymtotically normal, quite unlike the original estimates obtained by Manski (1975, 1985) without smoothing which converge at slower rate (cubic root of samle size) and have very comlicated nonstandard asymtotic distribution. Our aer is therefore a counterart in testing of Horowitz s idea in estimation. Smoothed indicator functions invariably involve a function of samle size, called a bandwidth (Horowitz (1992, 2002)) or tuning arameter (see the works cited at the end of the revious aragrah), which controls the rate at which the smoothed indicator aroaches the original. Since smoothed indicator functions which simly shift the discontinuity away from the origin, the size of that shift is the bandwidth. The work of Horowitz (2002) suggests that, with some additional restrictions on bandwidth, Edgeworth-tye higher-order exansions are derivable for test statistics constructed using smoothed indicators, hence bootstraing may imrove test erformance. This is a otential future bonus for the test of this aer but our resent objective is restricted to setting out a comlete rst-order theory for an asymtotic test that enjoys Proerties 1-4. In fact, we also resent results of simulations which show that the test works quite well without bootstraing. At this junction, the related work by Andrews (1999, 2000, 2001, 2002) on hyothesis testing involving inequality restrictions should be noted since he raises issues which may at rst sight aear to restrict the methods of the resent aer. However, his work highlights reasons for nonstandard behaviour of test statistics other than those we have ointed out in our review above. For examle, Andrews considers estimation and testing when the arameter estimates lie in a set de ning a maintained hyothesis some of whose boundary oints are included in the set de ning the null hyothesis. In this case, the asymtotic distribution of the estimator cannot be symmetric under all oints of the null hyothesis and thus conventional quadratic-form test statistics based on the estimator cannot be standard chi-square (see Andrews (1999, 2001)). Indeed, even bootstra tests may not work (Andrews (2000)). This scenario 7

9 would aly, for examle, to testing that a variance arameter is zero using a variance estimate which is naturally nonnegative. By contrast, the boundary of the null hyothesis set considered in this aer lies within the interior of the maintained hyothesis set. So the scenario envisaged by Andrews is not alicable within the setu of this aer. The rest of the aer is organized as follows. Section 2 sets u the constraint chaining framework and the smoothing indicator functions. We note that, whilst the hilosohy of our aroach leads uniquely to a statistic which is of quasilinear form, this can be thought of as a secial case of a wider class of forms all using only roducts of arameter estimates and smoothed indicators. Section 3 states distributional assumtions and comletes the construction of the chaining test statistic. Section 4 investigates the asymtotic roerties of the chaining test. Section 5 discusses issues in choosing the smoothing indicator functions. Section 6 rooses a comutable interolating ower function which re ects the intentional di erential imact of imortance weighting on ower at local to origin oints comared with medium to distant oint. Section 7 studies the test s nite samle ower using Monte Carlo simulations. Section 8 resents roofs of the stated theorems. Section 9 sketches, without roof, how the constraint chaining method extends to a test of (x) 0; a x b, where (x) is a function of an observable variate x. Section 10 concludes the aer. 2 The chaining framework and the smoothing indicators Let = ( 1 ; 2 ; :::; ) 0 be a column vector of (functions of) arameters aearing in an econometric model. The roblem is to test : H 0 : j 0 for all j 2 f1; 2; :::; g versus H 1 : j < 0 for at least one j: (4) Note that in this roblem neither the number nor the seci c values of j are known for which j = 0 under H 0 or j < 0 under H 1. Let D(x); x 2 R, be the indicator function where D(x) = 1 if x 0 and D(x) = 0 if x > 0. Then we can chain the multile inequalities with the indicators to rewrite the hyothesis under test as follows : H 0 : X D( j ) j = 0 versus H 1 : j=1 X D( j ) j < 0 (5) Evidently, testing multile inequalities is equivalent to one-sided single equality testing. With arameter estimates b in hand, one can consider the constraint j=1 8

10 chaining test statistic X D(b j )b j : (6) j=1 However, a comlication arises from the discontinuity of the indicator function D(x) at the origin. Such discontinuity will result in the above chaining statistic having an asymtotic distribution which is the same as that of a sum of correlated and truncated random variables. The oerational roblems this incurs will hence be of similar di culty to those arising with the chi-bar squared test methods. To overcome this roblem, instead of directly working with D(x), we construct a sequence of functions f T (x)g each of which is continuous at the origin and converges to the indicator D(x) ointwise (excet ossibly at the origin) as the samle size T goes to in nity. In articular, we require that this sequence have a functional structure satisfying the following assumtions: T (x) = (K(T )x) [A1] (x) is non-negative, bounded, and non-increasing in x 2 R [A2] (x) is continuous at x = 0 and (0) > 0 [A3] K(T ) is ositive and increasing in T [A4] K(T )! 1 and K(T )= T! 0 as T! 1 [A5] (x)! 1 as x! 1 [A6] T (K(T )x)! 0 as T! 1 for x > 0 We shall refer to any T (x) = (K(T )x) that satis es the above assumtions as an (origin-) smoothing indicator. The technique of imroving asymtotic behaviour by relacing a discrete indicator function with a smoothed version was rst notably used by Horowitz (1992) in arameter estimation. Our aer shows the idea is also e ective in hyothesis testing. Note that our smoothness assumtions are not comletely the same as those in his (1992) aer or the follow-u (2002) and his bandwidth arameter is actually the inverse counterart of our tuning arameter K(T ). In our framework, the set of assumtions on T (x) is not very restrictive and the range of choice for a valid (x) is wide. In articular, one can use (x) = 1 F (x) where F (x) is the cdf of an arbitrary random variable with F (0) < 1. We shall discuss issues in choosing (x), K(T ) and articular examles in Section 5. All the assumtions on T (x) enable the smoothing indicator to aroximate X D(x) ointwise at just such a rate that the smoothed version T (b j )b j of X D(b j )b j has the roerty of asymtotic normality even though the latter j=1 itself is not asymtotically normal. Moreover, the idea of imosing [A6] is to create data-driven imortance weighting in the sense that each individual b j j=1 9

11 corresonding to a strictly ositive j is likely to contribute ever less to the value of the test statistic as samle size T increases. As a consequence, the test statistic will be asymtotically dominated by those b j corresonding to zero or negative j, detection of which is indeed the urose of the test. By such imortance weighting, we are therefore able to design a simle consistent one-sided test of H 0 versus H 1. Note that the idea of chaining constraints with indicators is quite general and also exible in terms of formulation. In articular, for arbitrary strictly ositive scalars j, j 2 f1; 2; :::; g, it is clear that the original roblem could also be osed more generally as the following. H 0 : X D( j j ) j j = 0 versus H 1 : j=1 X D( j j ) j j < 0: (7) One key choice of j which can be motivated from standardization of arameter estimators is j = (V jj ) 1=2 where V jj denotes the asymtotic variance of b j 5. Furthermore, we note that the unsmoothed and smoothed quasi-linear chaining test quantities X D(b j )b j and j=1 X j=1 j=1 T (b j )b j may be thought of as secial cases of a class of lausible test quantities generated by taking the unsmoothed and the smoothed ( 1 ; 2 ; :::; ) where j D(b j )b j e (e 1 ; e 2 ; :::; e ) where e j T (b j )b j in a suitable class of functions g(); 2 R including at least the following g() = X j ; g() = 0 U; g() = minf 1 ; 2 ; :::; g j=1 where U is a ositive semi-de nite matrix. This aer develos a comlete test theory for the case of linear function. It is the benchmark. The quadratic case is considered in Chen (2009, Chater 4) 6. 5 See Hansen (2005) for motivating arguments of the use of standardized individual statistic for the test construction in the context of multile inequality testing. 6 Chen (2009, Chater 4) shows that the smoothed indicator test based on quadratic form of g(e) can also simultaneously satisfy the four roerties discussed in Section 1. In the contrast, the extreme value form of g(e) is not smooth in e and thus even used with smoothed indicator, such extreme value test still su ers from the roblem of nuisance arameter from unknown covariance V as in White (2000) and Hansen (2005). 10

12 3 Distributional assumtions and the chaining test statistic Let the true value and estimator of the arameter vector be denoted resectively by = ( 1; 2; :::; ) 0 and b = (b 1 ; b 2 ; :::; b ) 0. In the simlest distributional set-u, b is assumed to be T consistent and [D1] T (b ) d! N(0; V ) where V is some nite ositive de nite matrix. For notational ease, we kee imlicit the ossible deendence of V on the values of any underlying arameters. We assume there is available an estimator V b, almost surely ositive de nite, such that [D2] b V! V. Regarding the scalars introduced at the end of Section 2, we assume the availability of strictly ositive estimators b j ; j 2 f1; 2; :::; g, such that [D3] b j! j : To construct the chaining test statistic, we introduce the following objects. Let and b be the dimensional square matrices Let b be the dimensional column vector De ne Q 1 and Q 2 as diag( 1 ; 2 ; :::; ) (8) b diag( b 1 ; b 2 ; :::; b ) (9) b ( T ( b 1 b 1 ); T ( b 2 b 2 ); :::; T ( b b )) 0 : (10) Q 1 T b 0 b b (11) Q 2 b 0 b b V b b : (12) We de ne a test statistic Q as follows : (Q1 =Q Q = 2 ) if Q 2 > 0 1 if Q 2 = 0 (13) where (x) is the standard normal distribution function. For asymtotic signi cance level, we reject H 0 if Q <. The chaining statistic Q is therefore a form of tail robability or -value. We now sketch the reasoning which motivates our test rocedure. Formal theorems and rigorous roofs are given later. Intuitively, we should reject H 0 if Q 1 is too small. For those arameter oints 11

13 under H 0 for which the robability limit of Q 2 is nonzero, Q 2 will be strictly ositive with robability aroaching one as samle size grows. Then the ratio Q 1 =Q 2 will exist and be asymtotically normal. By contrast, for all oints under H 1, the value of Q 1 will go in robability to minus in nity as samle size grows. Therefore, in cases where Q 2 is ositive, we roose to reject H 0 if Q 1 =Q 2 is too small comared with the normal distribution. Note that our assumtions on the smoothing indicators do not exclude discrete functions. Only continuity at origin is required. Therefore, a discrete function that is continuous at origin can be a candidate of the smoothing indicator and such an examle is resented in Section 5. If all arameters are strictly ositive and is a discrete function, then b T is zero with robability aroaching one as samle size grows. In this case, Q 2 is also zero with robability aroaching one. Therefore, occurrence of the event that Q 2 = 0 signals that we should not reject H 0. Note that it is not an adhoc choice to set Q to be one for which the event Q 2 = 0 occurs because the robability limit of (Q 1 =Q 2 ) is also one when all arameters are strictly ositive and is an everywhere continuous function 7. 4 Asymtotic roerties of the chaining test The rst-order asymtotics of the chaining test can be justi ed by the following four theorems, whose roofs are given in Section 8. To facilitate the statement of those theorems, we introduce some convenient notation. Let S denote the set f1; 2; :::; g. Decomose S as S = A [ M [ W, where A fj 2 S : j > 0g; M fj 2 S : j = 0g; W fj 2 S : j < 0g: Let d MW be -dimensional selection vector whose jth element is unity or zero according as j 2 M [ W or j =2 M [ W. Let d S be the -dimensional vector in which all its elements are unity. De ne the scalars,! MW and! S as! MW d 0 W M V d W M (14)! S d 0 SV d S : (15) Note that these scalars are in fact the variances of the asymtotic distribution of the sum X T j (b j j j ) over j 2 M [ W and j 2 S resectively. Theorem 1 (Asymtotic Null Distribution) Given [D1], [D2] and [D3], the following are true under H 0 : j j 2 f1; 2; :::; g. (1) If M 6=?, then Q d! U(0; 1): 0 for all 7 The case of being everywhere continuous is more comlicated because Q 2 in this case is almost surely strictly ositive. If all arameters are strictly ositive, then both numerator and denominator in the "t ratio", Q 1 =Q 2 go to zero in robability. See Section 8 for derivation of asymtotic roerties of the test statistic for such non-standard scenario. 12

14 (2) If M =?, then Q! 1: Note that the asymtotic size of the test is the suremum of the asymtotic rejection robability with resect to those restricted to the set de ned under H 0. Therefore, Theorem 1 imlies that the asymtotic test size is equal to. Furthermore, we observe that the size is attained uniformly over all oints on M, the boundary of null-restricted arameter sace and thus the test achieves asymtotic similarity on the boundary. In contrast to the chi-bar squared and the extreme value test aroaches, the limiting null distribution of the chaining test, when non-degenerate, is uniform, which clearly ful lls ease of imlementation as required in Proerty 4. Theorem 2 (Consistency) Given [D1], [D2], and [D3], the following is true under H 1 : j < 0 for some j 2 f1; 2; :::; g : P (Q < )! 1 as T! 1: Theorem 2 shows that the chaining test is a consistent test. Besides consistency, we are also interested in the local behavior of the test. In order to derive a local ower function of the chaining test, we need to view the magnitudes of j for j 2 W as becoming smaller when T! 1 (See McManus 1991). Seci cally, we imagine that [L1] For j 2 W, j = cj T, where c j is a xed negative scalar. Then we have the following result. Theorem 3 (Neyman-Pitman Local Power) Assume [D1], [D2] and [D3] hold with the elements j of taking the T- deendent forms [L1] for j 2 W but keeing xed values for j 2 SnW where W is non-emty. De ne c W X j c j < 0: j2w Then as T! 1 P (Q < )! ( 1 ()! 1=2 MW c W ): (16) Theorem 3 shows that the asymtotic local ower of the test is increasing in the absolute magnitude of each negative c j. Further, this ower is not diluted 13

15 by any xed ositive j. On the other hand, suose we extend the imagined condition [L1] to [L2] For j 2 A [ W, j = cj T, where c j is a xed scalar taking negative value for j in W but ositive value for j in A. Then Theorem 3 would be relaced by the following. Theorem 4 (Two-Sided Local Analysis) Assume [D1], [D2] and [D3] hold with the elements j of taking the T-deendent forms [L2] for j 2 A [ W, where A and W are both non-emty. De ne c AW X j c j : Then as T! 1 j2a[w P (Q < )! ( 1 ()! 1=2 S c AW ): (17) In the case of two-sided local analysis, it is clear from Theorem 4 that the chaining test could be locally biased 8. However, any arameter region of local bias will be shrinking as samle size increases, and its ractical relevance will become negligible. An argument we resent later in Section 7.2 using a seci c origin-smoothed indicator function sheds further light on this matter. Moreover, the simulations we reort there are encouraging of the view that local bias is not an issue of ractical concern and rather a small rice to ay for an otherwise e ective test that is so easy to aly. 5 The choice of smoothing indicator For xed T, our roosed chaining aroach imlicitly relaces the testing roblem (5) with the technically easier roblem of testing H 0 : X j=1 T ( j ) j = 0 versus H 1 : X j=1 T ( j ) j < 0. Though the former testing roblem is equivalent to the original roblem of testing multile inequalities, the latter is only asymtotically equivalent to the 8 Though Theorem 1 shows that the chaining test is asymtotically similar on the boundary, such roerty is only necessary rather than su cient for the test to be asymtotically unbiased (See Theorem 5 in Hansen (2003)). However, as shown by Hansen (2003, Theorem 3 and Corollary 6), the LF-based test that is asymtotically non-similar on the boundary is asymtotically inadmissible when comared with another test which is asymtotically similar on the boundary. In other words, while both are asymtotically exact tests, the ower of the former test against local alternatives is dominated by the latter. 14

16 former and hence for any xed samle size T, the smoothed version of the test is indeed testing a deformed hyothesis. In rincile, otimization exercises are desired to nd choices of (x) and K(T ) that minimize the discreancy between the limiting distribution of the chaining statistic and its actual nite samle distribution. This is neither the same as nor incomatible with the objective of minimizing the discreancy the original (unsmoothed) and the deformed (smoothed) hyotheses subject to the restrictions on the smoothing indicators imosed under [A1] ~[A6]. The whole roblem is comlicated by the fact that any such otimization has to consider all relevant values of samle size T simultaneouly since (x);as distinct from T (x) = (K(T )x); has to be indeendent of T: Thus a formal objective function will need to be in the nature of a sum of terms over T: Because any choice of (x) and K(T ) will in theory yield the same rst order asymtotic roerties as long as the conditions [A.1] to [A.6] are met, such otimization is unlikely to roduce a tractable solution without some judicious aroximation. In Aendix A of this aer, we give a reliminary analysis where we suggest that the "interolating ower function" to be introduced in the next section could lay a useful role. Nonetheless, we leave a thorough analysis of this second-level roblem for further research. In this aer, we roose a set of valid smoothing indicators by aealing to simlicity of functional form (an Occam s razor tye of argument) and then imlementing simulation exercises to assess ractical size and ower otential. The simlest choice of (x) that satis es [A.1]-[A.2] is the following. Ste-at-unity : (x) = 1 if x 1; and (x) = 0 if x > 1. Another convenient choice of (x) that is everywhere continuous is the following. Logisitic 9 : (x) = (1 + ex(x)) 1. For the choice of K(T );we roose to set K(T ) T= log(t ); which is simle and comlies fully with the conditions [A.3] ~[A.6] on smoothing indicators. To imlement the chaining test in ractice, we also argue for an adjustment to account for the otentially overwhelming e ect of the total number of inferentially uninformative arameter estimates. To understand this oint, note that we should reject H 0 even if only one of j, j 2 f1; 2; :::; g is negative. Hence, though the asymtotic roerties of the test are not a ected, it is still desirable that the smoothing indicator be made deendent on, the total number of arameters in such a way that it is decreasing in for x > 0. Then, in nite samles, the inferential value of a single negative b j is less likely to be overwhelmed by additional numbers of ositive b j as increases. To achieve this, an easy way is to relace (x) with (h()x) where h() is a ositive function increasing in. For examle, we can use the quadratic seci cation for h() such that h() ( 1) for 2 and h(1) 1: 9 Note that this (x) is the comlement of the logisitc distribution: (x) = 1 F (x), where F (x) is the logistic cdf. 15

17 6 An interolating comutable theoretical ower function The results of Section 4 show that the ower function (rejection robability) of the chaining test deends on which of the scenarios [L1] or [L2] is aroriate. Scenario [L2] alies if all j are viewed as being small enough to be local. Scenario [L1] alies if only the negative (and zero) j are viewed as small enough to be local, but in that case the imortance weighting via smoothing indicators comes into lay to eliminate the nuisance ositive ones. If negative j cannot be considered small enough to be local, then local ower theory does not aly and instead consistency of the chaining test simly gives an asymtotic ower of unity. But how do we judge what is small enough to be local? In other words, how do we know which of the two local ower functions or the ower of unity is aroriate? A way to resolve this issue is to use smoothing indicator functions which automatically interolate between the two local owers and the unity. In other words, we let the numbers do the talking and avoid the adhoc suggestion that we should x an arbitrary oint value which exlicitly distinguish local from nonlocal. Accordingly, our roosal is to use an interolating theoretical ower function de ned by P ( 1 ()! 1=2 c), where c = T X T ( j j ) j j ; (18) j2auw! = d 0 V d; (19) d = ( T ( 1 1); T ( 2 2); :::; T ( )) 0 : (20) Note the limit of P as T tends to in nity is the ower function (16) under the Neyman-Pitman local sequence [L1] yet is (17) under the two-sided local sequence [L2]. Moreover, for xed ( nonlocal ) and negative j, (18) tends to minus in nity as T aroaches in nity, thus correctly yielding asymtotic ower of unity in such a nonlocal case. Note that the interolating ower P can also be comuted for those cases of imlied under H 0. If all j are non-local and ositive, then! 1=2 c tends to in nity as T goes to in nity and the limit of P in such case is zero 10. If all j are non-local and non-negative with at least one j being zero, then the limit of P in this case is equal to, the size of the test. Therefore, in between these extreme scenarios the formula of P acts as an interolation 11. The algebraic form of the function used in interolating ower need not be the same as used in the construction of the test statistic since it serves for a di erent urose. That urose is to automatically interolate 10 Note that in this case, both c and! go to zero. However, the ratio! 1=2 c goes to in nity as a consequence of the roerties of the smoothing indicators. Proof of this result can be based on the same arguments as that of art (2) of Theorem 1 given in Section Indeed the ower function P automatically interolates the rejection robabilities obtained in the four theorems of Section 4. 16

18 between the three di erent scenarios. The function that is used in comuting P should be continuous (such as the logistic seci cation) because interolation is a form of continuous smoothing 12. In the next section we make some illustrative comutations of the interolating ower function and comare its grahs with those of ower functions obtained by simulation. 7 Monte Carlo studies of nite samle test ower In this section we conduct a series of Monte Carlo simulations to study the nite samle ower of the test statistic and to comare the nite samle erformance of di erent choices of the smoothing indicators. 7.1 The simulation setu The smoothing indicators (h()k(t )x) are formed using the following seci- cation : (i) Ste-at-unity : (x) = 1fx 1g Logistic : (x) = (1 + ex(x)) 1 (ii) K(T ) = T= log(t ) (iii) No dimension adjustment : h() = 1 Quadratic adjustment : h() = ( 1) Microeconomic and nancial datasets are nowadays tyically large, but we decided nonetheless to check the validity of our asymtotic theory by going down to a samle as small as 50. In fact, we generated R relications for each of two samle sizes and according to the following scheme : (iv) Parameters : r and r (v) Two ( = 2) jointly normal variates br 1 and br 2 such that br1 r1 N( ; V 1 br 2 T ), where V = with known. 1 r 2 The variates in (v) are generated via br 1 = r 1 + " 1 T br 2 = r 2 + (br 1 r 1 ) T " 2 12 The transition from ower at scenarios of local oints such as [L1] or [L2] to ower at nonlocal large oints is continuous. A further investigation of this issue would ideally be covered by a higher-order asymtotic theory and (Edgeworth) exansions in T. In the absence of that, we roose the novel technique of interolation formula P based on everywhere continuous interolating weights. 17

19 where " 1 and " 2 are indeendently drawn from the N(0; 1) distribution. The number of draws made was R = Each draw led to a articular value of our test statistic and to a articular decision (reject or accet). The simulated rejection robabilities enabled grahical resentations of a quality which suggests that the number of draws used was more than adequate for the accuracy of our general and seci c conclusions. We focused on a nominal test size of = 0:05. We shall also refer to the simulated rejection robability as the (simulated) ower of the test. Hence under H 1, the ower of an unbiased test should be no smaller than. Furthermore, the test size is not distorted if the ower of the test under H 0 is not greater than. We considered several re- xed values of r 2 and for each such value we conducted simulations yielding an emirical ower function calculated on a grid of 1000 values of r 1. In articular, we chose r 2 2 f 0:5; 0; 0:5g because they illustrate the issues and deict an informative range of behaviour with visual clarity. Corresonding to each emirical ower function we calculated an interolating theoretical ower function for comarison. Based on Section 6, the corresonding interolating ower in current simulation setu is where P ( 1 ()! 1=2 c); (21) c = T ( T (r 1 )r 1 + T (r 2 )r 2 ); (22)! = T (r 1 ) 2 + T (r 2 ) T (r 1 ) T (r 2 ): (23) Since interolation is a form of continuous smoothing, we use the continuous logistic function for interolating ower even when the ste-at-unity function is used in constructing the test statistic. 7.2 Simulation results for large samles To study how well the asymtotic roerties of the test hold in nite samles, we rst consider a very simle simulation con guration : (x) is ste-at-unity and there is no adjustment for. Table shows the simulated ower curves in the left column and the corresonding interolating theoretical ower curves in the right column when r 2 is ket xed and the samle size T is These curves are functions of r 1 varying from 1 to 1 when r 2 is ket xed at the values 0:5 (dashed curve), 0 (continuous curve) and 0:5 (dotted curve). These choices allow r 1 to vary equally widely, small and large, in both ositive and negative con gurations. They are thus concisely informative about test rejection robability both under the null hyothesis and under the alternative. The to air of anels in Table 1 show rejection robability when br 1 and br 2 are uncorrelated ( = 0), the middle anels when negatively correlated ( = 0:8), and the bottom anels when ositively correlated ( = 0:8). The non-zero values 0:8 were chosen to reresent serious correlation. 13 All tables in this aer can be found at the end of Section 7. 18

20 In interreting Table 3.1, we note that our roosed interolating theoretical ower formula describes very well the nite samle behavior of the test. All interolating theoretical ower curves t remarkably well the corresonding curves of simulated ower. Regarding the erformance of the test, we rst look to check whether rejection robability under the null anywhere exceeds nominal size = 0:05. The null hyothesis is e ective on those fragments of the dotted (r 2 = +0:5) and continuous (r 2 = 0) curves lying to the right of r 1 = 0 (inclusive). The following observations aly to both simulated and interolating theoretical curves in all anels of Table 3.1. It is clear that no oint on those curve fragments is above 0.05 in height. The continuous curve soon climbs back at height 0.05 for values of r 1 exceeding 0. The dotted curve lies almost immediately at a height of zero and is barely discernible as it coincides with the horizontal axis. This haens because the dotted line reresents the case where both r 1 and r 2 are strictly ositive and so the imortance weighting e ect of the smoothing indicator wies out the imact of these values with high robability in large samles. A second check of Table 3.1 should look for high rejection robability at least equal to the nominal size = 0:05 under the alternative hyothesis. The alternative hyothesis is e ective on all curves at oints to the left of r 1 = 0 in all anels of Table 3.1. It is additionally e ective on the dashed curves (r 2 = 0:5) at oints to the right of r 1 = 0 (inclusive). Note that the dashed curve is always at height 1, thus indicating that the test is extremely successful even when only one arameter value is strictly negative. The dotted and continuous curves are at height no lower than the nominal size, and mostly at height 1, at all oints strictly to the left of r 1 = 0. It is evident that the test is successful. 7.3 Simulation results for small samles Table 3.2 shows the simulation results using the same con guration as Table 3.1 but a smaller samle size (T = 50). The ndings are similar to those in Tables 3.1. Rejection robability under the null hyothesis is bounded by the nominal size = 0:05. The rejection robability under that region of the alternative hyothesis reresented by the dashed curves is reduced comared with that of Table 3.1; however it remains at value 1 for the most art and everywhere still exceeds ten times the nominal size. The heights of the continuous curves stay above the nominal size in the alternative hyotheses region to the left of r 1 = 0 and increase towards 1 as becomes ever more negative. The height of the dotted curve in the lower left-hand anel falls below the nominal size in the tiny local alternative region 0:04 r 1 < 0. As r 1 becomes ever more negative (to -1), the height increases rogressively to 1. Overall the theoretical curves match the simulated well, though not as well as in the large samle case of Table 3.1. The above simulation results show that the chaining test seems to erform quite well in nite samles even under a very simle choice of the smoothing indicator. In the next simulations, we consider the e ect of making an adjustment for the number of arameters,. Hence, we use the same simle con gurations as above to study the e ect of introducing the quadratic version of the adjust- 19

21 ment function h(). 7.4 E ect of adjustment for number of arameters Tables 3.3 and 3.4 show the simulated di erence in ower between using a quadratic adjustment function h() = ( 1) and using no adjustment h() = 1. In comuting this ower, r 2 is ket xed and the samle size T is set to be 1000 and 50, resectively. We nd that the introduction of quadratic adjustment h() has indeed raised the nite samle ower. The ower gain occurs over a much wider range of r 1 values for small samles than for large. The maximal gains achievable are greatest in small samles when the correlation between arameter estimators is nonnegative. 7.5 E ect of di erent choices of the (x) function on ower Tables 3.5 and 3.6 shows the curves of di erence of rejection robabilities of the logistic and the ste-at-unity seci cations of (x) when r 2 is ket xed and the samle size T is 1000 and 50 resectively. These tables suggest that the logistic seci cation generally incurs no greater robability of Tye 1 error (rejecting the null hyothesis when both r 1 and r 2 are non-negative) than the ste-at-unity. Also, the logistic is generally more owerful than the ste-atunity against the alternative hyothesis when r 2 is negative (dashed curves). The non-negativity of the continuous curves in the region r 1 < 0 indicates that the logistic smoothing indicator yields ower at least as great as that of the ste-at-unity when r 2 = 0. The dotted curves concern the case r 2 > 0. They almost coincide with the horizontal axis where r 1 < 0 in Table 3.5, thus suggesting that the logistic and ste-at-unity seci cations have similar ower in large samles when r 2 > 0. But in Table 3.6 the dotted curves go below the horizontal axis where r 1 < 0. Thus, in small samles, the logistic seems to be is less owerful than ste-atunity when r 2 > 0. This nding can be attributed to the fact that the ste-atunity indicator is caable of eliminating ositive arameter estimates comletely from the test statistic. In the contrast, the logistic does not comletely eliminate these estimates and hence the ower-reducing e ect of such ositives may have more imact in small samles comared to large where asymtotic equivalences between all smoothing indicators satisfying the theory assumtions [A1]-[A6]. 20