Testing near or at the Boundary of the Parameter Space (Job Market Paper)

Transcription

1 Testing near or at the Boundary of the Parameter Space (Jo Market Paper) Philipp Ketz Brown University Novemer 7, 24 Statistical inference aout a scalar parameter is often performed using the two-sided t- test. In extremum prolems, where the estimator satisfies the restrictions on the parameter space - such as the nonnegativity of a variance parameter -, the test suffers from size distortions when the true parameter vector is near or at the oundary of the parameter space. Nevertheless, the two-sided t-test continues to e used when estimates are found to e close to the oundary. This can e attriuted to a lack of inference procedures that appropriately account for oundary effects on the asymptotic distriution of the estimator. To address this issue, we propose an estimator that is asymptotically normally distriuted, even when the true parameter vector is near or at the oundary, and the ojective function is not defined outside the parameter space. The novel estimator allows the implementation of several existing testing procedures and a new test ased on the Conditional Likelihood Ratio statistic (CLR). Compared to the existing procedures, the new test is easy to implement and has good power properties. Moreover, it offers power advantages over the two-sided t-test, when the latter controls size. We also show the test to e admissile when inference is performed with respect to a scalar parameter. We apply the test to the random coefficients logit model using data on the European car market and find more evidence of heterogeneity in consumer preferences than suggested y the two-sided t-test. Keywords: Boundary, asymptotic normality, testing, admissiility, random coefficients. I am very grateful to my advisor Frank Kleiergen as well as Adam McCloskey, Blaise Melly, and Eric Renault for their guidance, their constant support and encouragement, and many helpful comments and discussions. I also thank Andrew Elzinga, Joachim Freyerger, Bruno Gasperini, Hyojin Han, Pepe Montiel Olea, Daniela Scida, and seminar participants at Brown University for helpful comments and suggestions. Department of Economics, Brown University, 64 Waterman Street, Providence, RI philipp ketz@rown.edu.

2 Introduction Statistical inference aout a scalar parameter is often performed using the two-sided t-test, which relies on the asymptotic normality of the underlying estimator. In extremum prolems, however, the commonly employed estimator is not asymptotically normally distriuted when the true parameter vector is near or at the oundary of the parameter space (Andrews, 999). In that case, the two-sided t-test suffers from size distortions. While the test can suffer from overrejection (under some conditions), underrejection constitutes the more prevalent prolem. Due to the associated loss in power, a researcher is, for example, more likely to falsely conclude that a parameter is equal to zero, which has significant consequences in the context of model selection exercises. The random coefficients logit model (Berry, Levinsohn, and Pakes, 995), where variance parameters are restricted to e nonnegative, constitutes a prominent example of models in which estimates are frequently found to e close to the oundary, indicating that the assumption that the true parameter vector lies in the interior is violated. Nevertheless, the two-sided t-test continues to e used in practice (e.g., Nevo, 2; Goeree, 28). This can e attriuted to a lack of inference procedures that appropriately account for oundary effects on the asymptotic distriution of the estimator, which result from the restrictions on the parameter space to which the estimator is confined. 2 In this paper, we address this gap in the literature y introducing a modified extremum estimator that is asymptotically normally distriuted even when the true parameter vector is near or at the oundary of the parameter space. The novel estimator does not require the original extremum ojective function to e defined outside the parameter space and is, therefore, availale in a wide range of nonlinear models. The estimator is given y the unconstrained minimizer of a quadratic approximation to the original ojective function. It is easy to implement and otained y a single step of the Newton-Raphson algorithm starting at the constrained extremum estimator. Although the two-sided t-test ased on the novel estimator constitutes a valid and sizecorrect test, the nonstandard nature of the testing prolem can e utilized to construct tests that are more powerful. The testing prolem is nonstandard in that it is characterized y the presence of nuisance parameters that satisfy certain restrictions. We introduce two tests that take these restrictions into account. The tests are ased on the generalized Likelihood Ratio statistic (glr) and the Conditional Likelihood Ratio statistic (CLR), which previously have Other models with similar restrictions on the parameter space are given y random coefficients regression models (Andrews, 999), censored panel data models with slope heterogeneity (Arevaya and Shen, 24), multinomial discrete response models with random coefficients (Hausman and Wise, 978) or random effects (McFadden, 989). 2 Andrews and Guggenerger (2) show that ootstrap procedures are also inconsistent.

3 not een considered for the testing prolem at hand. The tests are easy to implement and, in many cases, lead to tighter confidence intervals than the enchmark, i.e., the two-sided t-test ased on the constrained extremum estimator. The null distriution of the glr depends on the true value of the nuisance parameter. In order to construct a valid test ased on the glr we consider the least favorale configuration approach. When the dimension of the nuisance parameter is small, the resulting test displays good power properties. In particular, it offers power advantages over the enchmark for a wide range of alternatives. However, the conservativeness of the test, resulting from the use of the least favorale configuration, leads to low power in large parts of the parameter space when the numer of nuisance parameters is large. 3 Conditional on a sufficient statistic for the nuisance parameter, the (conditional) null distriution of the CLR is independent of the true value of that parameter. As a result, the test ased on the CLR displays good power properties regardless of the dimension of the nuisance parameter. Furthermore, we show the test to e admissile when inference is performed with respect a scalar parameter. Another appealing feature is that the test reduces to the two-sided t-test when the true parameter vector lies in the interior of the parameter space. Consequently, confidence intervals otained y inverting the test can e interpreted as a natural extension of standard confidence intervals to the nonstandard setting where the true parameter vector might e near or at the oundary. Based on our novel estimator, other tests recently proposed in the literature ecome availale. Elliott, Müller, and Watson (23) propose tests that maximize weighted average power (WAP), while Montiel Olea (23) proposes tests that maximize WAP suject to a similarity constraint. WAP maximizing tests are attractive when the researcher has a particular weight function in mind, where the weight function specifies the alternatives towards which the test directs power. The tests proposed in this paper are ad hoc in that they are not designed to maximize WAP (or minimize a general loss function). However, from the results in Montiel Olea (23) it follows that there exist weights with respect to which the test ased on the CLR maximizes WAP suject to a similarity constraint. For the testing prolem with a scalar nuisance parameter, we conduct a power comparison of the tests ased on the glr, the CLR, and the two tests proposed y Elliott, Müller, and Watson (23) and Montiel Olea (23) for certain choices of the weight function. 4 All tests are found to have comparale power properties, displaying advantages over some parts of the parameters 3 Alternative methods that do not rely on the use of the least favorale configuration can also e implemented, see e.g., McCloskey (22). 4 The weight functions are taken from the two papers. Elliott, Müller, and Watson (23) use uniform weights that assign equal weights to all alternatives. Montiel Olea (23) uses weights that yield a simple closed form solution for his test statistic. 2

4 space, while lacking in power over others. Regardless of whether the researcher has a particular weight function in mind, the choice of a test is also guided y computational feasiility. The tests proposed y Elliott, Müller, and Watson (23) and Montiel Olea (23) ecome computationally expensive as the numer of nuisance parameters increases. 5 The computational costs associated with the tests introduced in this paper, on the other hand, are invariant to the numer of nuisance parameters, making them more attractive in practice. In order to illustrate their usefulness, we apply the proposed testing procedures to the random coefficients logit model (Berry, Levinsohn, and Pakes, 995), which is widely used in the industrial organization and marketing literatures to model demand for differentiated products. The random coefficients in this model are typically parameterized y a vector of means and variances and allow for heterogeneity in consumer preferences with respect to different product characteristics. In many applications, it is a priori unknown which of the product characteristics interact with a random coefficient, i.e., which variance parameters are non-zero. As a result, the empirical analysis often starts with a aseline model that allows for a random coefficient on all product characteristics. Then, a powerful test such as the one ased on the CLR can prove useful in determining a good model specification. In an application of the random coefficients model to the European car market using data from Reynaert and Veroven (24), we find evidence of consumer heterogeneity with respect to price (divided y income), horse power (divided y weight), and height of the car when using the CLR. 6 The two-sided t-test, on the other hand, which represents common practice, only suggests the presence of consumer heterogeneity with respect to horse power. The plan of this paper is as follows. Section 2 introduces the testing prolem, the two test statistics, the glr and the CLR, and the tests ased on them. This section also shows that the test ased on the CLR is admissile for testing hypotheses aout a scalar parameter. Section 3 provides asymptotic theory for general extremum estimators and introduces a novel estimator which is shown to e asymptotically normal. This section also contains details on how to implement the proposed testing procedures when using an asymptotically normal estimator. Section 4 contains a power comparison of our testing procedures, along with other tests recently proposed in the literature, in the context of several leading examples. Section 5 contains an application to the random coefficients logit model. In this section, we perform a small Monte Carlo study to illustrate the finite sample ehavior of our testing procedures 5 For certain choices of the weight function, the test statistic proposed y Montiel Olea (23) can e otained in closed form. Then the computational cost is equivalent to that of the test ased on the CLR and the glr. However, for a general weight function, the test statistic needs to e evaluated numerically, leading to an increase in computational cost. 6 The glr is not the preferred choice in this setting due to the large numer of nuisance parameters. 3

5 and present the empirical findings. Section 6 concludes. Throughout this paper, (a, ) denotes the vector (a, ), where a and are column vectors. R, R +, and R ++ denote (, ), [, ), and (, ), respectively. Furthermore, if A denotes an interval in R, then A N denotes A A with N N copies. 2 Testing In this section, we first descrie the testing prolem and motivate its relevance. Then, we introduce two testing procedures that have previously not een considered for the testing prolem at hand. They are ased on the generalized Likelihood Ratio statistic (glr) and the Conditional Likelihood Ratio statistic (CLR). The section concludes y showing that the test ased on the CLR is admissile for the testing prolem at hand. 2. Testing prolem This paper studies the prolem of testing hypotheses aout a scalar parameter, β. We consider the case where β is an element of a (K +) dimensional unknown parameter vector, θ = (β, δ), that enters a general extremum ojective function, Q n (θ), whose dependence on the data {W i : i n} is suppressed for notational convenience. Section 3 contains detailed information on what kind of ojective functions are permitted in our framework. The K dimensional vector δ is not specified under the null hypothesis and, therefore, constitutes a nuisance parameter in the testing prolem. The parameter space for θ is assumed to e restricted, and we are interested in testing hypotheses aout β when the true parameter vector is near or at the oundary of the parameter space. This is modeled y means of drifting sequences of true parameters, θ n. For the purpose of illustration, assume θ R K +. Then, the drifting sequences of interest would e such that nθ n µ, where µ denotes a localization parameter with µ <. Such drifting sequences are essential in order to derive asymptotic theory that provides good approximations to the finite sample ehavior of an estimator or a test statistic, when the true parameter vector is close to the oundary relative to the sample size. Throughout this paper, we use near and close interchangealy. Furthermore, the case where the true parameter vector is at the oundary, µ k = for some k {,..., K + }, constitutes a special case of the true parameter vector eing near the oundary and, hereafter, is sometimes referred to implicitly. Motivated y the recent literature, we introduce the hypothesis testing prolem in the limit experiment, or limiting prolem (see e.g., Van der Vaart, 2; Elliott, Müller, and Watson, 23; Müller, 2). In particular, we assume that we oserve a draw from a random 4

6 variale, Y, whose distriution is fully determined y a (K + ) dimensional unknown parameter vector µ. This µ corresponds to the localization parameter of the original testing prolem and is partitioned as µ = (, d), where is scalar and d is a K dimensional vector. We are interested in testing hypotheses aout, while treating d as a nuisance parameter. Similar testing prolems are also considered in Elliott, Müller, and Watson (23) and Montiel Olea (23). Our hypothesis testing prolem of interest is H : = B, d D vs. H :, B, d D, () where B equals (, ], [, ), or (, ), and D is a Cartesian product of intervals that equal (, ] or [, ). Let M = B D denote the parameter space for µ. Although the shape of the parameter space is restrictive, it arises in many models of interest. Random coefficient models, where variance parameters are restricted to e nonnegative, constitute a leading example. Another example is given y (semi-) parametric regression models, where some coefficients are known to satisfy sign restrictions. The motivation for why D does not contain intervals of the form (, ) is given y an invariance argument. In Section 3, we allow for an additional nuisance parameter in θ, whose true value is assumed to lie in the interior of the parameter space and which can e thought of as eing partialled out. The end points of the intervals in M are equal to zero, when they are not infinite, without loss of generality. When the original parameter space for (β, δ) is given y a Cartesian product of intervals, M can always e written as aove y appropriately recentering (β, δ). Similarly, the focus on two-sided testing prolems is without loss of generality. In this paper, we introduce tests for the hypothesis testing prolem given in () ased on the Gaussian shift experiment, i.e., ( Y = Y Y d ) (( ) ) N, Σ, (2) d where Σ is known and positive definite. Alternatively, we could derive tests ased on the limiting prolem that is otained under the sequence of (constrained) extremum estimators commonly employed in practice. The corresponding Y, say Y c, is distriuted as the projection of (2) onto a linear suspace of R K+, see Section 3.. However, tests ased on (2) are not only easier to derive, ut also susume tests ased on Y c, i.e., any test ased on Y c can also e implemented given (2), while the reverse is not true. Intuitively, a draw from a normal random variale is more informative (aout µ) than a draw from a truncated normal. Other tests recently proposed in the literature are also ased on the Gaussian shift experiment (see e.g., Elliott, Müller, and Watson, 23; Montiel Olea, 23). A main con- 5

7 triution of this paper is to propose an asymptotically normal estimator, which is availale under no additional assumptions eyond those made to derive the asymptotic distriution of a constrained extremum estimator, such that tests ased on the Gaussian shift experiment ecome availale in a wide range of nonlinear models. In standard settings, the support of µ = (, d) is R K+, and there are good reasons to ignore Y d when making inference aout. For example, for the two-sided testing prolem, H : = vs. H :, the test that rejects when Y > cv, where cv denotes the Σββ critical value of a standard normal distriution, is the uniformly most powerful uniased test. In the nonstandard setting considered in this paper, Y d contains information aout that can e exploited when testing (). Before introducing our testing procedures, we illustrate how tests developed for () ased on (2) can e used under drifting sequences of true parameters when an asymptotically normal estimator is availale. To that end, we consider a simple example taken from Andrews and Guggenerger (2). Example : We consider the testing prolem where a scalar nuisance parameter is near the oundary. Suppose {X i R 2 : i n} forms a triangular array of i.i.d. random vectors with X i = ( X i, X i,2 ), θ n E(X i ) = ( β n δ n ), and Σ Var(X i ), where Σ is positive definite. θ n is a drifting sequence of true parameters. The parameter space is given y Θ = B D, where B = (, ) and D = [, ), i.e., we know a priori that the mean of X i,2 is nonnegative. We assume that the drifting sequence of true parameters satisfies nβ n, where <, and nδ n d <. We are interested in testing H : = while leaving d unspecified under H. 7 The sample averages, X and X 2, are estimators of β n and δ n, respectively, and y a central limit theorem (CLT) for triangular arrays satisfy n ( X β n X 2 δ n ) d N(, Σ) or, equivalently, n ( X X 2 ) d N (( d ) ), Σ. Therefore, under the aove drifting sequences the scaled estimator, n X, is asymptotically distriuted as Y given in equation (2). Furthermore, the parameter spaces for and d are 7 Testing H : = can e understood as testing H : β n = / n at a given sample size, n. 6

8 given y B = (, ) and D = [, ), respectively. 8 As a result, any test derived for the testing prolem given in () and (2) can e applied here y replacing Y y its sample analogue, n X, and y replacing Σ y a consistent estimator, such as the sample variance of X i. 2.2 Testing procedures In the prolem without a nuisance parameter, i.e., without Y d in (2), and B = [, ), Feldman and Cousins (998) propose the use of the generalized Likelihood Ratio statistic (glr) in order to make inference. To the est of our knowledge, there has een no attempt yet to use the glr to make inference in the general testing prolem given in () and (2). Let φ(y, µ, Σ) denote the pdf of a (K + ) dimensional normal random vector with mean, µ, and positive definite variance, Σ. Then, the glr is defined as follows ( sup B glr(, y, y d ) = 2 log,d D φ((y ), y d ), (, d), Σ) sup d D φ((y, y d ), (, d), Σ) ( ) ( ) ( ) ( ) y = inf Σ y y inf Σ y. d D y d d y d d B,d D y d d y d d The distriution of glr(, Y, Y d ) depends on the true value of d, which is not specified under the null. One possiility to construct a level α test ased on the glr is y means of the least favorale configuration. The least favorale configuration, say d LFC, is such that inf d D P (glr(, Y, Y d ) < cv dlfc ) = α, where cv dlfc denotes the ( α) quantile of the glr(, Y, Y dlfc ) when the true parameter equals (, d LFC ). The corresponding test rejects whenever glr(, y, y d ) > cv dlfc. Another possile testing procedure is ased on the Conditional (generalized) Likelihood Ratio statistic (CLR), which was originally suggested y Moreira (23) for the linear Instrumental Variales regression model with weak instruments. Variates thereof have also een used in the context of weak identification in the Generalized Method of Moments (GMM) (see e.g., Kleiergen, 25). The test utilizes the glr without relying on the least favorale configuration. We consider the following transformation of Y ( Y Y d Σ δβ Σ ββ Y ) N 8 This relies on < and d <. (( d Σ δβ Σ ββ ), [ Σ ββ Σ δδ Σ δβ Σ ββ Σ βδ ]), 7

9 where [ Σ ββ Σ δβ Σ βδ Σ δδ ] denotes a conformale partition of Σ. Let X Y d Σ δβ Σ ββ Y. Note that the distriution of X depends on d, while that of Y does not. Since X and Y are independent, X is a sufficient statistic for d. In fact, X is a complete sufficient statistic for d, where completeness follows from Theorem 4.3. in Lehmann and Romano (25). Now define CLR(, y, x) = inf d D ( inf B,d D y x + Σ δβ Σ ββ y d ( y The CLR test, ϕ CLR (y, x), is given y x + Σ δβ Σ ββ y d ) ( ) Σ y x + Σ δβ Σ ββ y d ) ( Σ y x + Σ δβ Σ ββ y d ). (3) if CLR(, y, x) > cv(α, x) ϕ CLR (y, x) =, (4) otherwise where cv(α, x) denotes the conditional ( α) quantile of the distriution of CLR(, Y, x), where Y N(, Σ ββ ). Unlike the (unconditional) distriution of the glr statistic, the (conditional) distriution of the CLR statistic does not depend on the true value of d due to the conditioning on X = x, which is a sufficient statistic for d. The critical values, cv dlfc e otained y means of simulation. and cv(α, x), although not availale in closed form, can easily Another test statistic of interest is the two-sided t-statistic ased on the constrained maximum likelihood estimator for, t ML hereafter, where the constraint is given y µ M. The power function of the test that compares the t ML to the standard critical value of a normal random variale matches, under certain conditions, the local asymptotic power function of the two-sided t-test ased on a constrained extremum estimator when the true parameter vector is near or at the oundary. The latter corresponds to common practice where potential oundary effects on the distriutions of the underlying estimator are ignored. Therefore, the performance of the test ased on t ML is of particular interest and analyzed, in detail, elow. In what follows, glr, CLR, and t ML can refer to the statistic or the respective test ased on that statistic with the understanding that the glr uses the least favorale configuration approach, while the t ML uses the critical value of a standard normal random variale. Before 8

10 turning to the asymptotic theory, we show that the CLR is admissile in the aove testing prolem. 2.3 Admissiility of the CLR In order to show that the CLR is admissile in the class of all tests for the testing prolem at hand, we first introduce some additional notation. Define M and M such that the testing prolem given in equation () can e written as H : µ M vs. H : µ M, where y definition M = M \M. Let C denote the class of all tests. A test is defined as a measurale function ϕ : Y [, ]. 9 Here, Y = R K+ denotes the support of Y. Similarly, let X = R K denote the support of X. ϕ(y) is to e understood as the proaility of rejecting the null hypothesis given a realization of the data, y. The type I error of ϕ for µ M defined as E µ [ϕ(y )] = Y ϕ(y)f(y µ)dy, where f(y µ) denotes the pdf of Y, which in the model on hand is the pdf of a multivariate normal. Since Σ is known, we suppress the dependence of f(y µ) on Σ. The type I error is only defined for parameter values in the null set, M, and signifies the proaility with which the null hypothesis is falsely rejected. The type II error of ϕ for µ M E µ [ϕ(y )] = Y ϕ(y)f(y µ)dy. is is defined as The type II error is only defined for parameter values in the alternative set, M, and signifies the proaility of falsely failing to reject the null hypothesis. Define the risk function associated with ϕ as E µ [ϕ(y )] if µ M R ϕ (µ) = E µ [ϕ(y )] if µ M The risk function allows the comparison of tests. In particular, a test ϕ is said to dominate ϕ, if R ϕ (µ) R ϕ (µ) for all µ M with strict inequality, R ϕ (µ) < R ϕ (µ), for some µ M. A test ϕ is called admissile in a class of tests, C C, if there exists no test ϕ C such that ϕ dominates ϕ. Admissiility is a minimal optimality requirement for a test within a certain class of tests. If a test is admissile, it cannot uniformly e improved 9 The CLR aove is defined as a function of Y and X, ut since Y is a one-to-one function of Y and X, the CLR could equivalently e expressed as a function of Y.. 9

11 upon, i.e., the proaility of making an incorrect decision cannot e lowered somewhere in the parameter space without an increase in that proaility elsewhere. Before stating the admissiility result for the CLR, we introduce the concept of similarity, which is used in its derivation. A test, ϕ, is said to e conditionally similar if E µ [ϕ(y ) X = x] = α x X and µ M and similar if E µ [ϕ(y )] = α µ M. The CLR is y construction conditionally similar. It follows, y the law of iterated expectations, that it is also (unconditionally) similar. The following Theorem asserts that the CLR introduced in Section 2.2 is admissile in the class of all tests pertaining to the testing prolem given in () and (2). Theorem. The ϕ CLR defined in (4) is admissile in the class of all tests, C, pertaining to the testing prolem given in () and (2). The proof of Theorem is given in Appendix A.. It consists of two parts. First, it is shown that any similar test with convex acceptance sections is admissile. Second, it is shown that the CLR has convex acceptance sections. A test is said to have convex acceptance sections if for any x X the acceptance region of the test as a function of y is closed and convex. The acceptance region of a test is the part of the sample space, for which the test fails to reject the null hypothesis. The first part of the proof follows from Theorem 3. in Matthes and Truax (967). For the prolem at hand, the Theorem asserts that the class of similar tests with convex acceptance sections is complete. A class of tests, C C, is complete, if for any ϕ C, there exists a ϕ C such that ϕ dominates ϕ. Admissiility of a similar test with convex acceptance sections can then e derived when Y is scalar. The result relies on X eing a complete sufficient statistic for d. The second part of the proof follows from the definition of the CLR statistic given in equation (3). The testing prolem at hand satisfies assumption F-F2 in Montiel Olea (23). Therefore, his Theorem applies, which states that any admissile and similar test is an extended Efficient Conditionally Similar (ECS) test. The appeal of an extended ECS test is that there exist weights (with full support on M ) with respect to which the test is aritrarily close to the weighted average power (WAP) maximizer suject to a similarity constraint. Put differently, there exists weights such that the CLR is the essential WAP maximizing test (with respect to those weights) suject to a similarity constraint. A priori, it is not clear whether similarity implies good power properties. For example, in moment inequality models similar tests have een shown to have poor power (Andrews, 22). The power analysis in Section 4 shows that the CLR has good power properties for the testing prolem at hand. The definition of convex acceptance sections given in equation (3.) in Matthes and Truax (967) is slightly different, since it allows for randomized tests. Here, we can restrict ourselves to non-randomized tests, since Y is a continuous random variale.

12 As mentioned aove, the proof of Theorem depends crucially on Y eing scalar. For testing prolems, where Y is vector-valued, the CLR can e shown to e admissile in the conditional prolem, ut it appears to e an open question whether it is admissile in the unconditional prolem. 3 Asymptotic Theory In this section, we introduce a general class of extremum prolems. In Section 3., we show that constrained extremum estimators, which y construction satisfy the restrictions on the parameter space, are not asymptotically normally distriuted when the true parameter vector is near or at the oundary. Consequently, the glr and the CLR, as well as other tests defined in the Gaussian shift experiment cannot e implemented ased on such estimators. Unconstrained estimators are often unavailale, ecause the ojective function is not defined outside the parameter space, e.g., a likelihood function may not e defined for negative values of a variance parameter. In Section 3.2, we propose a modified extremum estimator that is asymptotically normally distriuted and that does not require the ojective function to e defined outside the parameter space. This novel estimator consideraly roadens the applicaility of tests defined in the Gaussian shift experiment. Details on how to implement such tests ased on an asymptotically normal estimator are provided in Section 3.3. Throughout this section, we orrow notation from Andrews and Cheng (22a) (AC hereafter) and Andrews and Cheng (22) (AC2 hereafter). The criterion function of the extremum prolem is denoted Q n (θ). The class of extremum prolems is large and includes (quasi) Maximum Likelihood (ML), Generalized Method of Moments (GMM) and Minimum Distance (MD) prolems among others. i.e., The constrained estimator ˆθ n is defined as the approximate minimizer of Q n (θ) over Θ, Q n (ˆθ n ) = inf θ Θ Q n(θ) + o p (/n), (5) where Θ denotes the true parameter space. In AC, Θ denotes the optimization parameter space, and it is assumed that the true parameter space is a strict suset of the optimization parameter space. This is done to assume away oundary effects on the asymptotic distriution of the estimator. Here, we make the assumption that the true parameter space and the optimization parameter space are identical, since we are interested in the ehavior of the estimator near the oundary. We assume that θ can e partitioned as follows: θ = (β, δ, ξ), where β denotes the scalar parameter of interest, δ denotes a K dimensional nuisance parameter, and ξ denotes

13 an additional L dimensional nuisance parameter. Our setup differs from that of AC as none of the parameters determine the identification strength of any other parameter. In fact, all parameters are assumed to e well identified. The difference etween δ and ξ is that δ is modeled as close to the oundary, whereas ξ is modeled as in the interior of the parameter space. The ojective function Q n (θ) depends on data {W i : i n}, which may e i.i.d., independent and nonidentically distriuted, or temporally dependent. In most applications the distriution of the data is not fully specified y the vector θ, ut it depends on an additional, commonly infinite-dimensional, parameter, φ. For example in conditional maximum likelihood prolems, φ denotes the distriution of the data on which we condition. The parameter γ = (θ, φ) is assumed to fully specify the distriution of the data. The true parameter space is assumed to e of the following form Γ = {γ = (θ, φ) : θ Θ, φ Φ(θ)}, (6) where the true parameter space for θ, Θ R K+L+, is compact. In particular, we assume that Θ equals a Cartesian product of intervals equal to [ c, ], [, c] or [ c, c] for c R +, i.e., some of the parameters are ounded elow or aove y, where the normalization to is without loss of generality. 2 The oundary we are interested in is at, and not at c. 3 The form of the parameter space is restrictive, ut it is otained for many models of interest, most notaly in the context of random coefficients models, see e.g., Berry, Levinsohn, and Pakes (995) or Arevaya and Shen (24). 4 As in AC, we assume that Φ(θ) Φ θ Θ for some compact metric space Φ with a metric that induces weak convergence of the ivariate distriution (W i, W i+m ) for all i, m. When the true parameter vector lies in the interior of the parameter space, standard asymptotic theory provides good approximations to the finite sample ehavior of the estimator for large enough sample sizes. But, at any given sample size, the true parameter vector might e too close to the oundary for standard asymptotic theory to provide good approximations. Intuitively, standard asymptotic theory provides poor approximations if the estimates are within a few standard errors from the oundary. In that case, modeling the true parameter vector as close to the oundary relative to the sample size provides etter finite sample approximations. This is achieved y means of drifting sequences of true param- 2 The use of c as common endpoint for all intervals is merely for notational convenience. The endpoints are free to vary etween all K + L + sets in Θ. 3 In fact, here we implicitly assume that the true parameter space is a strict suset of the optimization parameter space, since we do not allow the true parameter to e equal to c, ut crucially we do not assume that for the oundary at. 4 With the exception of random coefficients models which allow the random coefficients to e correlated, see e.g., Andrews (2). 2

14 eters, which approach the oundary at a rate inversely related to the sample size, n, where the rate is chosen such that the distance to the oundary shows up in the asymptotic distriution. A drifting sequence of true parameters is denoted γ n = (θ n, φ n ). The set of all such drifting sequences is given y Γ(γ ) = {{γ n Γ : n } : γ n γ Γ}. The drifting sequences of primary interest are given y Γ(γ,, d) = {{γ n Γ(γ ) : n } : nβ n B and nδ n d D }, (7) where B equals (, ], [, ), or (, ) when the first coordinate of Θ equals [ c, ], [, c], or [ c, c], and D equals the product space of sets equaling (, ] or [, ) in accordance with the coordinates of Θ, which equal [ c, ] or [, c]. Thus, the set of drifting sequences given in (7) implicitly imposes < and d <. Throughout this paper, we use the terminology under {γ n } Γ(γ ) to mean when the true parameters are γ n Γ(γ ) for any γ Γ and under {γ n } Γ(γ,, d) to mean when the true parameters are γ n Γ(γ,, d) for any γ Γ with β =, δ =, R, and d R K. In order to help fix ideas, we introduce a running example: Example 2 : Consider the following random coefficients model y i = x i η i + u i, where for expository purposes x i is assumed to e scalar. (x i, y i ) is assumed to e i.i.d.. We assume further that η i N(µ η, ση) 2 and u i N(, σu) 2 such that η i u i. Then, the model can e written as y i = x i µ η + u i + x i v i σ η = x i µ η + ε i, where v i N(, ) and ε i = u i + x i v i σ η. Note that ε i x i N(, σu 2 + x 2 i ση). 2 The conditional individual log likelihood function is given y l(µ η, σu, 2 ση y 2 i, x i ) = 2 log(2π) 2 log ( ) σu 2 + x 2 i ση 2 (y i x i µ η ) 2 2 ( ). σu 2 + x 2 i σ2 η The (scaled) conditional log likelihood function, which for notational consistency is denoted 3

15 Q n (θ), is given y Q n (θ) = n n l(µ η, σu, 2 ση y 2 i, x i ). i= We can change the order of the elements in θ to conform with the aove notation according to which parameter we are interested in. The parameter space for µ η, ση, 2 and σu 2 is given y [ c, c], [, c], and [a, c], respectively, where < a < c. We assume that σ u 2 > a, i.e., only the variance parameter σ η 2 may e at the oundary. 5 A reparameterization from σu 2 to σu 2 a results in a parameter space of the form [ c, c], which conforms with the aove definitions. There are three possile orderings of the elements in θ, depending on which of the three parameters is the parameter of interest: ) β = µ η, δ = ση, 2 and ξ = σu 2 a, 2) β = ση, 2 δ empty, and ξ = (µ η, σu 2 a), and 3) β = σu 2 a, δ = ση, 2 and ξ = µ η. The parameter space for φ is given y Φ = {φ : E φ x i 8+ɛ C}, (8) where ɛ R ++ and C R ++. Note that here, Φ does not depend on θ. Next, we introduce the assumptions underlying the asymptotic distriution theory derived in this paper. The assumptions are stronger than those in Andrews (999), A hereafter, as they to not allow for non-stationary time series or complicated shapes of the parameter space. However, the assumptions allow for drifting sequences of true parameters. We make the high level assumption that ˆθ n is consistent for θ. Assumption. ˆθ n = θ + o p () γ Γ. Note that Assumption implies that ˆθ n = θ n + o p (). This follows trivially as θ n θ. Thus, ˆθ n can e thought of as a consistent estimator for the drifting sequence of true parameters, θ n. A sufficient condition for Assumption is given y the following. Assumption. (a) Under {γ n } Γ(γ ), sup θ Θ Q n (θ) Q(θ; γ ) = o p () for some non-stochastic realvalued function Q(θ; γ ). () Q(θ; γ ) is continuous on Θ γ Γ. 5 As aove the superscript denotes the limit of the drifting sequence of true parameters. 4

16 (c) Q(θ; γ ) is uniquely minimized y θ γ Γ. (d) Θ is compact. As in A and AC, we consider a quadratic approximation to the ojective function. In particular, we consider an approximation around the true value as in A with the difference that here the true value is given y a drifting sequence and, thus, depends on the sample size, n. We do not have to consider the approximation around the point of discontinuity as in AC, since we assume that identification does not depend on the true parameter value. The quadratic approximation is given y Q n (θ) = Q n (θ n ) + θ Q n(θ n ) (θ θ n ) + 2 (θ θ n) 2 θ θ Q n(θ n )(θ θ n ) + R n (θ), (9) where Q θ n(θ n ) and 2 Q θ θ n (θ n ) are defined in the following assumption, which assures that the quadratic approximation exists and that the remainder term is asymptotically negligile. Assumption 2. (a) Q n (θ) has continuous left/right (l/r) partial derivatives of order two on Θ n with proaility. () Under {γ n } Γ(γ ), for all constants ɛ n, sup θ Θ: θ θ n ɛ n 2 θ θ Q n(θ) 2 θ θ Q n(θ n ) = o p(), where ( / θ)q n (θ) and ( 2 / θ θ )Q n (θ) denote the (K + L + ) vector and (K + L + ) (K + L + ) matrix of l/r partial derivatives of Q n (θ) of orders one and two, respectively. In many cases, Assumption 2 () can e verified using a uniform LLN, see e.g., Andrews (992). 6 The last two assumptions concern the asymptotic ehavior of the first and second order partial derivatives of the ojective function under drifting sequences of true parameters. Assumption 3. Under {γ n } Γ(γ ), J n = and symmetric. 2 Q θ θ n (θ n ) p J(γ ), where J(γ ) is nonsingular Assumption 4. (i) Under {γ n } Γ(γ ), n θ Q n(θ n ) d N(, V (γ )) for some symmetric V (γ ). (ii) V (γ ) is positive definite γ Γ. 6 Assumption 2 corresponds to Assumption 2 2 in A and Assumption Q in AC2. Assumption 2 2 in A is sufficient for Assumption 2 in A, while Assumption Q in AC2 is sufficient for D in AC. 5

17 Assumption 3 can often e verified using a uniform LLN for triangular arrays, while Assumption 4 typically follows from a CLT for triangular arrays. 7 A.4. Example 2 (continued): The verification of Assumptions -4 can e found in Appendix We introduce some additional notation. Let Z n = Jn n θ Q n(θ n ), such that Z n d Z(γ ), where Z(γ ) N(, J(γ ) V (γ )J(γ ) ). () With J n and Z n thus defined, the quadratic approximation to the ojective function given in equation (9) can e written as Q n (θ) = Q n (θ n ) 2n Z nj n Z n + 2n q n( n(θ θ n )) + R n (θ), () where q n (λ) = (λ Z n ) J n (λ Z n ). Under the assumptions made aove, the remainder, R n (θ), is small enough such that the asymptotic distriution of the centered and scaled minimizer of Q n (θ), n(ˆθ n θ n ), is asymptotically equivalent to the asymptotic distriution of the centered and scaled minimizer of Q n (θ) R n (θ). The latter function only depends on θ through the function q n ( ), which is quadratic in θ. The distriution of the minimizer of a quadratic function is much easier to characterize than the distriution of the minimizer of Q n (θ) explaining the use of rewriting (9) as (). The asymptotic distriution of the minimizer of q n (λ) is given y the distriution of where ˆλ = arg min q(λ), (2) λ Λ(,d) q(λ) = (λ Z(γ )) J(γ )(λ Z(γ )) and where Λ(, d) denotes the limit of the shifted and scaled parameter space, n(θ θ n ). 7 Assumptions 3 and 4 correspond to Assumptions D2 and D3 in AC, respectively, and to Assumption 3 in A. 6

18 Below, we formally show that n(ˆθ n θ n ) is asymptotically distriuted as ˆλ. The distriution of ˆλ crucially depends on Λ(, d) and is given y the projection of a normal random variale, Z(γ ), onto Λ(, d) with respect to the norm q(λ) 2. In the standard case, where the limit of drifting sequence of true parameters, θ, lies in the interior of the parameter space, we have that n(θ θ n ) Λ = R K+L+ such that ˆλ = Z(γ ), recall equation (2). This illustrates that the approach of quadratically approximating the ojective function, which conceptually differs slightly from the typical linear approximation to the first order condition, constitutes another way of otaining the standard asymptotic normality result for n(ˆθ n θ n ) when θ lies in the interior of the parameter space. 3. Asymptotic distriution of constrained extremum estimator In this section, we present the asymptotic distriution result for n(ˆθ n θ n ) when the true parameter vector is near or at the oundary. Although the result is not new (Andrews, 999), it is helpful in estalishing uniformity results, as it is derived using drifting sequences of true parameters as in Andrews and Cheng (22a). We also discuss conditions under which the two-sided t-test ased on a constrained extremum estimator controls asymptotic size. For all sequences {γ n } Γ(γ,, d), we have that n(θ θ n ) Λ(, d), where Λ(, d) denotes a cone with nonzero vertex. In what follows, we also refer to Λ(, d) as the local parameter space. We illustrate the shape of Λ(, d) in the context of our running example. Example 2 (continued): We consider ordering numer ) β = µ η, δ = ση, 2 and ξ = σu 2 a such that B = [ c, c], D = [, c], and Ξ = [ c, c]. Furthermore, let nβ n, where <, nδ n d <, and ξ n ξ, where c < ξ < c. Then, n(θ θ n ) Λ(, d) = (, ) [ d, ) (, ), which is a cone with vertex (, d, ). 8 Here, Λ(, d) does not depend on. More generally, Λ(, d) is equal to a product set of intervals. In particular it takes the form (, ], [, ) or (, ) when B equals [ c, ], [, c] or [ c, c] times a K dimensional product set, where the k th set equals (, d k ] or [ d k, ) when D k equals [ c, ] or [, c] for k =,..., K, times a L dimensional product set, where each set equals (, ). 8 Note, that since Λ(, d) is not a proper cone the vertex is not uniquely defined with respect to the first and the third element. We choose without loss of generality. 7

19 Proposition. Under Assumptions -4 and under {γ n } Γ(γ,, d), n(ˆθ n θ n ) d ˆλ, where ˆλ is defined in (2) with Λ(, d) defined as in the preceding paragraph. The proof of Proposition follows from the arguments in A. Details can e found in Appendix A.2. The results in Section 6 of A concerning the asymptotic distriution of suvectors of n(ˆθ n θ n ) apply here as well with the slight modification that here Λ(, d) is a cone with non zero vertex. Since the asymptotic distriution of n(ˆθ n θ n ) is not the main interest of this paper, we refrain from reproducing the results here. As mentioned in Section 2.2, the asymptotic distriution of n( ˆβ n β n ) is given y the distriution of the maximum likelihood estimator for in (2) suject to µ M under some condition. This condition is given y V (γ ) = aj(γ ) for some constant a R +. Put differently, the correspondence is otained if the matrix defining the norm q(λ) 2 is proportional to the variance matrix of Z(γ ). To gain some intuition for this, we consider a simple example. Let K = and L = with B = [ c, c] and D = [, c]. Then, letting a denote asymptotically distriuted as, it can e shown that n ˆβn a Y + J J 2 min(, Y d ) (3) where Y and Y d are scalar and J = J(γ ) = [ J J 2 J 2 J 22 ]. Generally, the variance matrix of Y = (Y, Y d ), Σ, is given y J(γ ) V (γ )J(γ ). However, if V (γ ) = cj(γ ), Σ simplifies, and the expression in equation (3) reduces to n ˆβn a Y Σ βδ Σ δδ min(, Y d). It is easy to see that the distriution of Y Σ βδ Σ δδ min(, Y d) is also the distriution of the maximum likelihood estimator for in the corresponding Gaussian shift experiment. Since the t ML controls size, as illustrated in Section 4 elow, it follows that the two-sided t-test ased on ˆβ n controls asymptotic size. V (γ ) = aj(γ ) holds, for instance, when, in the context of GMM, the efficient weighting matrix is employed or when, in the context of ML, the likelihood function is correctly specified. If V (γ ) cj(γ ), the two-sided t-statistic ased on ˆβ n can suffer from overrejection. Note, however, that overrejection does not occur in the part of the sample space where the estimate is not restricted or, put differently, not at the oundary. One way to think of this is that, if the estimate is not found to e restricted, then it could have een otained using 8

20 an unrestricted estimator, which, if used in the construction of the two-sided t-test, allows for size-correct inference. Next, we illustrate the asymptotic distriution result given in Proposition in the context of our running example. Example 2 (continued): Since our model is a well-specified likelihood model, we make use of the fact that the information equality holds. In particular, this implies that J(γ ) = V (γ ), such that Z(γ ) N(, V (γ ) ). We consider the three different orderings of elements in θ separately. For the last two orderings the asymptotic distriution is not normal. In order to investigate how well the asymptotic distriution approximates the finite sample distriution, we provide Monte Carlo results for these two orderings. The asymptotic distriution and the finite sample distriution are oth evaluated using. Monte Carlo draws. We choose n = 4. x i is drawn from a U(, 2) distriution. The parameter values are given y µ η =, σ 2 u =, and σ 2 η is varied as indicated elow. ) β = µ η, δ = ση, 2 and ξ = σu 2 a: Since β n and δ n, we have that θ = (,, ξ ) = (,, σu 2 a). The Fisher Information, V (γ ), is given y 9 E φ σu 2 x2 i x 4 2 (σu 2 ) 2 i x 2 2 (σu 2 ) 2 i x 2 2 (σu 2 ) 2 i 2 (σu 2 ) 2 where E φ [ ] denotes the expectation with respect to the distriution of x i under φ. Due to the information orthogonality the asymptotic distriution of n( ˆβ n β n ) does not depend on δ, and we otain n( ˆβn β n ) d N(, Σ ββ ),, where Σ ββ denotes σu 2 E φ [x 2]. i 2) β = σ 2 η, δ empty, and ξ = (µ η, σ 2 u a): Since β n, we have that θ = (, ξ ) = (, µ η, σ 2 u a). The Fisher Information, V (γ ), is given y E φ x 4 2 (σu 2 ) 2 i x 2 2 (σu 2 ) 2 i σu 2 x2 i x 2 2 (σu 2 ) 2 i 2 (σu 2 ) 2 9 The first and second order partial derivatives of the likelihood function can e found in Appendix A.4.. 9

21 Since ξ does not impact the asymptotic distriution of n( ˆβ n β n ) (see e.g., A), we have that n( ˆβ n β n ) d max(, N(, Σ ββ )) or, equivalently, n ˆβn d max(, N(, Σ ββ )), (4) where Σ ββ denotes 2(σ2 u ) 2 E φ [x 4]. i Asy, = Asy, = 3 Asy, = n = 4, =.5 n = 4, = 3.5 n = 4, = Figure : Asymptotic and finite sample densities of ˆβ n = ˆσ 2 η for different values of. Figure shows the asymptotic and finite sample densities of ˆβ n = ˆσ 2 η for different values of = nβ n. takes on the values, 3, and 6 from left to right, which corresponds to β n taking on the values,.5, and.3, respectively. 2 3) β = σ 2 u a, δ = σ 2 η, and ξ = µ η : Since β n and δ n, we have that θ = (,, ξ ) = (,, µ η). The Fisher Information, V (γ ), is given y E φ 2 (σu 2 ) 2 2 x 2 2 (σu 2 ) 2 i 2 x 2 (σu 2 ) 2 i x 4 (σu 2 ) 2 i σu 2 x2 i 2 In fact, the asymptotic approximation is otained y evaluating V (γ ) at β equal to,.5, and.3 rather than,, and. This provides a much etter finite sample approximation and reflects common practice, where plug-in estimates of V (γ ) are utilized. This also holds true for Figure 2 elow. 2.

22 The asymptotic distriution of n( ˆβ n β n ) can e deduced from that of n ˆβ n, which is given y the distriution of Y Σ βδ Σ δδ min(, Y d), where ( Y Y d Here, Σ denotes the inverse of ) (( ) ) [ N, Σ with Σ = d [ x 2 2 (σ E 2 φ u ) 2 2 (σu 2 ) 2 i x 2 2 (σu 2 ) 2 i x 4 2 (σu 2 ) 2 i ]. Σ ββ Σ βδ Σ βδ Σ δδ ]. Asy, d = Asy, d = 3 Asy, d = n = 4, d = 2 n = 4, d = 3 2 n = 4, d = Figure 2: Asymptotic and finite sample densities of ˆβ n = ˆσ 2 u for different values of d. Figure 2 shows the asymptotic and finite sample densities of ˆβ n = ˆσ u 2 for different values of d = nδ n. d takes on the values, 3, and 6 from left to right, which corresponds to δ n taking on the values,.5, and.3, respectively. The Monte Carlo results illustrate that the finite sample distriution is well approximated y the asymptotic distriution derived aove. 2

23 The reason why the asymptotic distriution of a constrained extremum estimator is not normal is that the estimator is restricted to the parameter space. If it were possile to otain an unrestricted estimator, then that estimator would e asymptotically normal under standard regularity conditions. But as seen in Example 2, an unrestricted estimator is not always availale. 3.2 Asymptotic distriution of modified extremum estimator In the preceding section, we showed that constrained extremum estimators are not asymptotically normally distriuted when the true parameter vector is near or at the oundary of the parameter space. In this section, we show that it is possile to otain an asymptotically normal estimator, even if the ojective function is not defined outside the parameter space. As mentioned aove, this is useful as it roadens the applicaility of testing procedures defined in the Gaussian shift experiment. In order to give some intuition for the estimator proposed elow, consider the asymptotic distriution of the estimator of ση 2 in Example 2 given in equation (4). It is given y a normal distriution truncated elow at. The truncation of the asymptotic distriution results from the restriction on the parameter space, which prevents the estimator from taking on negative values. If an unrestricted estimator were availale, it would e asymptotically distriuted as the underlying normal random variale, N(β, ω ββ ). Our ojective is, thus, to construct an estimator that ehaves like an unrestricted estimator. The distriutional result in the previous section is otained y showing that constrained extremum estimators ehave asymptotically like minimizers of quadratic functions over a strict suset of the Euclidean space, the local parameter space. A quadratic function, in contrast to the original ojective function, is always defined over the entire Euclidean space and can, therefore, also e minimized over the entire Euclidean space. The proposed estimator is given y the unrestricted minimizer of a quadratic function that approximates the original ojective function. Another way of motivating the estimator is as follows. When the estimate is at the oundary of the parameter space, we know that the estimate does not satisfy the first order condition of the optimization prolem. But intuitively, there is a point outside the parameter space that would satisfy the first order condition, if that point were allowed in the sense of the ojective function eing well defined at that point or, more precisely, in an open set around that point. This point is approximated y the minimum of the quadratic function that approximates the ojective function and that passes through the estimate at the oundary. Figure 3 illustrates the aove intuition graphically, where the quadratic approximation is 22