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AN ALTERNATIVE ASYMPTOTIC ANALYSIS of RESIDUAL-BASED STATISTICS Elena Andreou and Bas J

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AN ALTERNATIVE ASYMPTOTIC ANALYSIS of RESIDUAL-BASED STATISTICS Elena Andreou and Bas J AN ALTERNATIVE ASYMPTOTIC ANALYSIS OF RESIDUAL-BASED STATISTICS Elena Andreou and Bas J. M. Werker* Abstract—This paper presents an alternative method to derive the limiting estimated propensity score that can be a nonsmooth function distribution of residual-based statistics. Our method does not impose an explicit assumption of (asymptotic) smoothness of the statistic of interest of the estimated parameters and for which standard boot- with respect to the model’s parameters and thus is especially useful in cases strap inference is often not valid (Abadie & Imbens, 2008). where such smoothness is difficult to establish. Instead, we use a locally In applications where the statistic of interest is smooth, our uniform convergence in distribution condition, which is automatically sat- isfied by residual-based specification test statistics. To illustrate, we derive conditions can be checked along the traditional lines. In order the limiting distribution of a new functional form specification test for dis- to illustrate our approach, we derive the limiting distribution crete choice models, as well as a runs-based tests for conditional symmetry of a new test based on Kendall’s tau for omitted variables in in dynamic volatility models. binary choice models and a runs-based test for conditional symmetry in dynamic volatility models. I. Introduction Our proposed method applies to general model specifica- tions as long as they satisfy the uniform local asymptotic ESIDUAL-BASED tests are generally used for diag- normality (ULAN) condition. Most of the standard econo- R nostic checking of a proposed statistical model. Such metric models satisfy this condition (see section IIIA for a specification tests are covered in many textbooks and remain more detailed discussion). The ULAN condition is central in of interest in ongoing research. Similarly, residual-based esti- Hájek and Le Cam’s theory of asymptotic statistics (Bickel mators, often referred to as two-step estimators, are widely et al., 1993; Le Cam & Yang, 1990; Pollard, 2004; van der applied in econometric work. Traditionally the asymptotic Vaart, 1998). We use this theory to derive our results. Other distribution of residual-based statistics (be it tests or esti- advances in econometric theory using the LAN approach can mators) is derived using a particular model specification, be found in Abadir and Distaso (2007), Jeganathan (1995), some more or less stringent assumptions about the statistic, and Ploberger (2004). For ULAN models, our results offer and conditions on the first-step estimator employed. A key a simple yet general method to derive the asymptotic√ distri- assumption is some form of (asymptotic) smoothness of the bution of residual-based statistics using initial n-consistent statistic with respect to the parameter to be estimated as for- estimators. Under the conditions imposed, our main theo- malized first in Pierce (1982) and Randles (1982). Since then, rem, theorem 3.1, asserts that the residual-based statistic is this approach has been significantly extended (for example, asymptotically normally distributed with a variance that is Pollard, 1989; Newey & McFadden, 1994; Andrews, 1994). a simple function of the limiting variances and covariances We present a new and alternative approach that does of the innovation-based statistic, the central sequence (the not involve explicit smoothness conditions for the statis- ULAN equivalent of the derivative of the log likelihood), and tic of interest. Instead, we rely on a locally uniform weak the estimator.1 Using this approach, we can readily obtain the convergence assumption shown to be generally (automat- local power of such residual-based tests, which can also be ically) satisfied by residual-based statistics. Our approach interpreted in terms of specification tests with locally mis- offers a useful and unifying alternative, especially when specified alternatives such as in Bera and Yoon (1993). In smoothness conditions are nontrivial to establish or require particular, this allows US to assess in which situations the additional regularity. Some examples of such statistics are, local power of the residual-based test exceeds, falls below, or for instance, rank-based statistics (Hallin & Puri, 1991) equals that of the innovation-based test. and statistics based on nondifferentiable forecast error loss To illustrate our method, we consider two applications. functions (McCracken, 2000). Abadie and Imbens (2009) First, we derive the asymptotic distribution of a new non- present an application of our method to derive the asymp- parametric test for omitted variables in a binary choice totic distribution theory of matching estimators based on the model. Second, we discuss a runs-based test for conditional symmetry in dynamic volatility models. These applications Received for publication November 20, 2009. Revision accepted for publication August 6, 2010. purposely focus on nonparametric statistics as these are usu- * Andreou: University of Cyprus; Werker: Tilburg University. ally defined in terms of inherently nonsmooth statistics like Part of this research was completed when E.A. held a Marie Curie fel- ranks, signs, and runs. For these applications, an appropriate lowship at Tilburg University (MCFI-2000-01645) and while both authors were visiting the Statistical and Applied Mathematical Sciences Insti- form of asymptotic smoothness can probably be established, tute. E.A. acknowledges support of the European Research Council under but our technique offers a useful alternative for which this the European Community FP7/2008-2012 ERC grant 209116. B.W. also is not necessary. We introduce our applications in section II. acknowledges support from Mik. Comments by two anonymous referees, Anil Bera, Christel Bouquiaux, Rob Engle, Eric Ghysels, Lajos Horváth, A number of additional applications of our method can be Nour Meddahi, Bertrand Melenberg, Werner Ploberger, Eric Renault, found in Andreou and Werker (2009). Enrique Sentana, conference participants at the ESEM 2004, MEG 2004, and NBER time series 2004 conferences, and seminar participants at 1 Throughout the paper, we use the term innovation-based statistic for the the London School of Economics, Université de Montréal, and Tilburg statistic applied to the true innovations in the model—the statistic obtained University are kindly acknowledged. if the true value of the model parameters were used. The Review of Economics and Statistics, February 2012, 94(1): 88–99 © 2011 by the President and Fellows of Harvard College and the Massachusetts Institute of Technology Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/REST_a_00151 by guest on 25 September 2021 AN ALTERNATIVE ASYMPTOTIC ANALYSIS OF RESIDUAL-BASED STATISTICS 89 Although this paper deals mainly with residual-based test- In applications, the unknown parameter θ is replaced by ing, the results can be applied directly in the area of two-step an estimator θˆn—for instance, the maximum likelihood esti- θˆ(ML) θˆ estimation when assessing the estimation error in a second- mator n . This leads to the residual-based statistic Tn( n). step estimator calculated from the residuals of a model The traditional way of deriving the limiting distribution of estimated in a first step. This problem has received large Tn(θˆn) relies on linearizing the statistic Tn(θ) (see Pagan & attention in the econometrics literature (see Murphy & Topel, Vella, 1989). This approach leads to 1985, 2002; Pagan, 1986). In the notation below, this would L T T T −1 T merely mean that the statistic Tn should be taken as the Tn(θˆn) −→ N(0, EWZZ − EWXZ (EWXX ) EWXZ ), second-step estimation error. (5) The rest of this paper is organized as follows. The next section introduces the applications we use to illustrate the as n →∞, with scope of our technique. Section IIIA then presents the con- f (XT θ)2 ditions we need to derive the limiting distribution of a W = . residual-based statistic. Our main result is stated and dis- F(XT θ)(1 − F(XT θ)) cussed in section IIIB. Section IIIC uses our main theorem to derive the (local) power of residual-based tests and compares The test statistic, equation (4), checks for linear correlation this with the local power of the underlying innovation-based between the generalized residuals and the possibly omitted tests. We indicate that a technical issue arises when making variable Z. One could also be interested in a test with power our ideas rigorous. Section IV addresses this by discretiza- against nonlinear forms of dependence based on Kendall’s tau εG θ tion, and we provide a formal proof of our main result. Section applied to the pairs ( i ( ), Zi). For simplicity, we consider the case where the possibly omitted variables are univariate: V concludes, and the appendix contains the proofs and some ∈ R auxiliary results. Zi . Recall that the population version of Kendall’s tau is defined as II. Two Motivational Applications τ = P εG θ εG θ − = 4 i ( )< j ( ), Zi < Zj 1, i j. (6) A. Omitted Variable Test for the Binary Choice Model An appropriately scaled innovation-based version of Consider the binary choice model, Kendall’s tau is the U-statistic, T − P{Y = 1|X}=F(X θ), (1) 1 n i−1 √ n T τ(θ) = n 4I εG(θ)<εG(θ), Z < Z − 1 where Y denotes a binary response variable, X some exoge- n i j i j 2 = = nous explanatory variables, and F a given probability distri- i 1 j 1 L 4 bution function. We assume that the distribution function F −→ N 0, . admits a continuous density f and that the Fisher information 9 matrix, This limiting distribution, under the null hypothesis of inde- f (XT θ)2 pendent εG and Z , can be obtained using the projection I (θ) = E XXT ,(2)i i F F(XT θ)(1 − F(XT θ)) theorem for U-statistics, for example, theorem 12.3 in van der Vaart (1998). exists and is continuous in θ. For inference, an i.i.d. sample Deriving the limiting distribution of the residual-based of observations (Y , X ), i = 1, ..., n, is available.
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