## Consistent Specification Testing With Nuisance Parameters Present Only Under The Alternative (1995)

Citations: | 55 - 10 self |

### BibTeX

@MISC{Stinchcombe95consistentspecification,

author = {Maxwell B. Stinchcombe and Halbert White},

title = {Consistent Specification Testing With Nuisance Parameters Present Only Under The Alternative},

year = {1995}

}

### Years of Citing Articles

### OpenURL

### Abstract

. The nonparametric and the nuisance parameter approaches to consistently testing statistical models are both attempts to estimate topological measures of distance between a parametric and a nonparametric fit, and neither dominates in experiments. This topological unification allows us to greatly extend the nuisance parameter approach. How and why the nuisance parameter approach works and how it can be extended bears closely on recent developments in artificial neural networks. Statistical content is provided by viewing specification tests with nuisance parameters as tests of hypotheses about Banach-valued random elements and applying the Banach Central Limit Theorem and Law of Iterated Logarithm, leading to simple procedures that can be used as a guide to when computationally more elaborate procedures may be warranted. 1. Introduction In testing whether or not a parametric statistical model is correctly specified, there are a number of apparently distinct approaches one might take. T...

### Citations

3012 |
Probability and Measure
- Billingsley
- 1986
(Show Context)
Citation Context ...nt with ~ oe 2 n replacingsoe 2 n is identical. (b) Because fZ(��) : �� 2 Tg is a continuous process and g is continuous with respect to k \Delta k 1 , it follows from the continuous mapping t=-=heorem (Billingsley 1968-=-, p.30) that g(n \Gamma 1 2 n X i=1 c M i =oe n ) ) R g(Z) and g(n \Gamma 1 2 n X i=1 c M i =~oe n ) ) R g(Z): \Xi (51) PROOF OF COROLLARY 5.5: Apply the continuous mapping theorem. \Xi PROOF OF THEOR... |

2716 | Learning internal representations by error propagation - Rumelhart, Hinton, et al. - 1986 |

1219 | Multilayer Multilayer feedforward networks are universal approximators,” Neural Networks 2 - Hornik, Stinchcombe, et al. - 1989 |

492 | Specification Tests in Econometrics - Hausman - 1978 |

453 |
The behavior of maximum likelihood estimation under nonstandard conditions
- Huber
- 1967
(Show Context)
Citation Context ... sequence of i.i.d. random variables on the probability space (\Omega ; F ; P ) taking values in R l and having the distribution of the random l \Theta 1 vector Z. To permit ` n to be an m-estimator (=-=Huber 1967-=-), we impose Assumption 2. For each n 2 N,s` n :\Omega ! \Theta (with \Theta a compact subset of R p ) is measurable. Further, there exists a function s : R l \Theta \Theta ! R p and a finite, nonsing... |

249 |
Inference when a Nuisance Parameter is not identified under the Null Hypothesis, Econometrica 64(2
- Hansen
- 1996
(Show Context)
Citation Context ...(37) for all 1sp ! 1,sfinite, where Z is the Gaussian process of Theorem 5.4. Obtaining asymptotic critical values for our statistics is computationally challenging but feasible, e.g. by Monte Carlo (=-=Hansen 1991-=-). We can also apply results of Gin'e and Zinn (1990) and Theorem 5.4 to guarantee applicability of the bootstrap. Nevertheless, a simple and sharp asymptotic bound follows from the Banach LIL (Ledoux... |

213 |
Hypothesis Testing when a Nuisance Parameter is Present Only under the Alternative
- Davies
- 1987
(Show Context)
Citation Context ...e parameter approach tests whether a statistic depending on a "nuisance parameter present only under the alternative" is zero for all values of the nuisance parameter, as is true under the n=-=ull (e.g. Davies 1977-=-, 1987; Bierens 1990). There are a variety of other possibilities and variants as well (e.g. Eubank and Hart 1992; Eubank and LaRiccia 1992, Blum, Kiefer and Rosenblatt 1961, Ghorai 1980, Holst and Ra... |

200 | Approximation capabilities of multilayer feedforward networks - Hornik - 1991 |

157 |
Real Analysis and Probability
- Ash
- 1972
(Show Context)
Citation Context ...e with (c) follows directly from the discussion of polar topologies (Robertson and Robertson 1973, III.2). \Xi PROOF OF THEOREM 3.3: (a) If sp H is uniformly dense in C(B), then Lusin's Theorem (e.g. =-=Ash 1972-=-, Corollary 4.3.17(b)) implies comprehensiveness. Suppose sp H is not uniformly dense in C(B). The uniform closure of sp H is a closed linear subspace of C(B). As the set of finite signed measures is ... |

123 |
Comparing nonparametric versus parametric regression fits
- HÄRDLE, E
(Show Context)
Citation Context ...ach compares a nonparametric estimator of the object of interest (say a conditional mean or density) to a parametric estimator (e.g. Eubank and Spiegelman 1990; Eubank and LaRiccia 1992; Gozalo 1993, =-=Hardle and Mammen 1993, Hong and-=- White 1995, Leung and Yu 1995, or Zheng 1994a). The nuisance parameter approach tests whether a statistic depending on a "nuisance parameter present only under the alternative" is zero for ... |

113 |
Estimation, Inference and Specification Analysis
- White
- 1996
(Show Context)
Citation Context ...fication of S for E(Y jX). It is this fact that ensures the power against various misspecifications of the Hausman (1978) test, because the Hausman test implicitly uses a particular choice for h (see =-=White 1994-=-, Chapter 10). Nevertheless, as Holly (1982) and Bierens (1982) showed, the Hausman test can fail to have power against potentially important departures from H 0 , the null hypothesis of correct speci... |

88 | On some global measures of the deviation of density function estimates - Bickel, Rosenblatt - 1973 |

87 |
A conditional moment test of functional form
- Bierens
- 1990
(Show Context)
Citation Context ...h tests whether a statistic depending on a "nuisance parameter present only under the alternative" is zero for all values of the nuisance parameter, as is true under the null (e.g. Davies 19=-=77, 1987; Bierens 1990-=-). There are a variety of other possibilities and variants as well (e.g. Eubank and Hart 1992; Eubank and LaRiccia 1992, Blum, Kiefer and Rosenblatt 1961, Ghorai 1980, Holst and Rao Date: January 1995... |

61 |
Some Asymptotic Results for Learning in Single HiddenLayer Feedforward Network Models
- WHITE
- 1989
(Show Context)
Citation Context ...nonparametric approach. In this context, we also will observe interesting connections between Bierens' approach and the approximation capabilities of artificial neural networks (Hornik, et. al. 1989, =-=White 1989-=-a). In Section 4 we show how the nuisance parameter approach to specification testing applies to testing probability models generally. Section 5 lays out the Banach random variable testing approach an... |

51 | Maximum Likelihood Specification Testing and Conditional Moment Tests", Econometrica 53 - Newey - 1985 |

51 | Diagnostic testing and evaluation of maximum likelihood models - Tauchen - 1985 |

49 | Consistent model specification tests - Bierens - 1982 |

42 | Bootstrapping general empirical measures - Giné, Zinn - 1990 |

41 | Distribution Free Tests of Independence Based on the Sample Distribution Functions - BLUM, KIEFER, et al. - 1961 |

40 | On nonparametric measures of dependence for random variables - Schweizer, Wolff - 1981 |

34 | Consequences and Detection of Misspecified Nonlinear Regression Model
- White
- 1981
(Show Context)
Citation Context ... \Theta regardless of the correctness of S, such that ` = ` 0 when S is correctly specified. For example, ` n can be a nonlinear least squares estimator from a random sample of size n on (Y; X) (e.g. =-=White 1981-=-). With ffl := Y \Gamma f(X; ` ), the correct specification of S for E(Y jX) is equivalent to e := E(ffljX) = 0 a.s.. Now, under suitable conditions on Y and f , e is an integrable function of X, that... |

33 | Consistent Specification Testing Via Nonparametric Series Regression,” Econometrica 63
- Hong, White
- 1995
(Show Context)
Citation Context ...tric estimator of the object of interest (say a conditional mean or density) to a parametric estimator (e.g. Eubank and Spiegelman 1990; Eubank and LaRiccia 1992; Gozalo 1993, Hardle and Mammen 1993, =-=Hong and White 1995, Leung an-=-d Yu 1995, or Zheng 1994a). The nuisance parameter approach tests whether a statistic depending on a "nuisance parameter present only under the alternative" is zero for all values of the nui... |

32 |
Consistent nonparametric entropy-based testing
- Robinson
- 1991
(Show Context)
Citation Context ...etworks. White's participation was supported by NSF Grant #SES-9209023. We are grateful to Rob Engle, Clive Granger and the participants of the UCSD econometrics workshops for helpful comments. 1980, =-=Robinson 1991-=-, Rosenblatt 1973, Schweizer and Wolff 1976, 1981, Wolff 1981, or Zheng 1994b.) Our purpose here is to unify the apparently disparate nonprametric and nuisance parameter approaches to testing models c... |

26 |
Testing the Goodness of Fit of a Linear Model Via Nonparametric Regression
- Eubank, Spiegelman
- 1990
(Show Context)
Citation Context ...arently distinct approaches one might take. The nonparametric approach compares a nonparametric estimator of the object of interest (say a conditional mean or density) to a parametric estimator (e.g. =-=Eubank and Spiegelman 1990; Eub-=-ank and LaRiccia 1992; Gozalo 1993, Hardle and Mammen 1993, Hong and White 1995, Leung and Yu 1995, or Zheng 1994a). The nuisance parameter approach tests whether a statistic depending on a "nuis... |

26 | A Central Limit Theorem Under Metric Entropy with L2 - Ossiander - 1987 |

22 |
Linear Operators, Part 1: General Theory
- Dunford, Schwartz
- 1988
(Show Context)
Citation Context ...ever e 6= 0, there is an h 2 H exp with hffl; hi 6= 0. This leads to a consistent test because sp H is weakly dense. For p; q 2 (1; 1), the weak closure and the norm closure of convex sets are equal (=-=Dunford and Schwartz 1958-=-, V.2.14, p. 418), so that weak denseness of sp H exp is the same as norm denseness in these spaces. The nonparametric approach works because it is based on a class of functions that comes arbitrarily... |

22 |
An additional hidden unit test for neglected nonlinearity in multilayer feedforward networks
- White
- 1989
(Show Context)
Citation Context ...nonparametric approach. In this context, we also will observe interesting connections between Bierens' approach and the approximation capabilities of artificial neural networks (Hornik, et. al. 1989, =-=White 1989-=-a). In Section 4 we show how the nuisance parameter approach to specification testing applies to testing probability models generally. Section 5 lays out the Banach random variable testing approach an... |

21 | Specification testing in dynamic models - White - 1987 |

20 |
Testing goodness-of-fit in regression via order selection criteria
- Eubank, Hart
- 1992
(Show Context)
Citation Context ... alternative" is zero for all values of the nuisance parameter, as is true under the null (e.g. Davies 1977, 1987; Bierens 1990). There are a variety of other possibilities and variants as well (=-=e.g. Eubank and Hart 1992-=-; Eubank and LaRiccia 1992, Blum, Kiefer and Rosenblatt 1961, Ghorai 1980, Holst and Rao Date: January 1995. Key words and phrases. Specification testing; Nuisance parameters; Goodness of fit; Model m... |

20 | Approximation and Learning Unknown Mappings Using Multilayer Feedforward Networks with Bounded Weights - Stinchcombe, White - 1990 |

19 | Testing the (parametric) null model hypothesis in (semiparametric) partial and generalized spline models - Cox, Koh, et al. - 1988 |

13 |
A Consistent Model Specification Test for Nonparametric Estimation of Regression Function Models, Econometric Theory
- Gozalo
- 1993
(Show Context)
Citation Context ...ametric approach compares a nonparametric estimator of the object of interest (say a conditional mean or density) to a parametric estimator (e.g. Eubank and Spiegelman 1990; Eubank and LaRiccia 1992; =-=Gozalo 1993, Har-=-dle and Mammen 1993, Hong and White 1995, Leung and Yu 1995, or Zheng 1994a). The nuisance parameter approach tests whether a statistic depending on a "nuisance parameter present only under the a... |

12 |
Asymptotic comparison of Cramér-von Mises and nonparametric techniques for testing goodness-of-fit
- Eubank, LaRicca
- 1992
(Show Context)
Citation Context ...one might take. The nonparametric approach compares a nonparametric estimator of the object of interest (say a conditional mean or density) to a parametric estimator (e.g. Eubank and Spiegelman 1990; =-=Eubank and LaRiccia 1992; Goz-=-alo 1993, Hardle and Mammen 1993, Hong and White 1995, Leung and Yu 1995, or Zheng 1994a). The nuisance parameter approach tests whether a statistic depending on a "nuisance parameter present onl... |

9 |
A smoothing spline based test of model adequacy in polynomial regression
- COX, E
(Show Context)
Citation Context ...urier series, or Eubank and Speckman's (1990) polynomial-trigonometric series, and regression splines, or Gallant's (1981) Flexible Fourier Form, as considered by Hong and White (1995), splines as in =-=Cox and Koh 1989-=-, Cox et. al. 1988. With an indirect approach, e need not be estimated. 3. Revealing Test Functions and Duality From the preceeding section, we see that totally revealing classes of test functions pla... |

7 | Curve Fitting by Polynomial-Trigonometric Regression - Eubank, Speckman - 1990 |

7 | A remark on Hausman’s specification test - Holly - 1982 |

5 | Using Feedforward Network to Distinguish Multivariate Populations," Discussion Paper - Stinchcombe, White - 1991 |

4 | Commonality of cusum, von Neumann and smoothingbased goodness-of-fit tests - Eubank, Hart - 1993 |

3 | Optimal Tests When a Nuisance Parameter is Identified Only Under the Alternative - Andrews, Ploberger - 1994 |

3 | ARMAX Model Specification Testing, with an Application to Unemployment in the Netherlands - Bierens - 1987 |

3 |
Asymptotic Normality of a Quadratic Measure of Orthogonal Series Type Density Estimate
- Ghorai
- 1980
(Show Context)
Citation Context ...he null (e.g. Davies 1977, 1987; Bierens 1990). There are a variety of other possibilities and variants as well (e.g. Eubank and Hart 1992; Eubank and LaRiccia 1992, Blum, Kiefer and Rosenblatt 1961, =-=Ghorai 1980-=-, Holst and Rao Date: January 1995. Key words and phrases. Specification testing; Nuisance parameters; Goodness of fit; Model misspecification; Neural networks. White's participation was supported by ... |

2 | ARMA Memory Index Modeling of Economic Time Series (with discussion)," Econometric Theory 4 - Bierens - 1988 |

2 | Lagrange Multiplier Tests for Parametric Instability in Nonlinear Models - Hansen - 1990 |

2 | Asymptotic Theory for Some Families of Two-sample Nonparametric Statistics," Sankhy��a Series A 42 - Holst, Rao - 1980 |

2 |
Topological Vector Spaces, 2'nd ed
- Robertson, Robertson
- 1973
(Show Context)
Citation Context ...ae R k . If H ae L q (X) is any norm bounded set with weakly dense span, then kgkH := supfjhg; hij : h 2 Hg defines a norm on L p (X). This type of norm gives rise to what is called a polar topology (=-=Robertson and Robertson 1973, II-=-I.2), and this is the topological basis of Bierens' second type of test. Again, no direct estimation of e need be made because hffl; hi j he; hi. Denote the maximizing value ass�� n . The asymptot... |

2 |
Sur une mesure de d'ependance pour les variable al'eatoires, Comptes Rendus de l'Academie des Sciences de Paris 283A
- Schweizer, Wolff
- 1976
(Show Context)
Citation Context ...as supported by NSF Grant #SES-9209023. We are grateful to Rob Engle, Clive Granger and the participants of the UCSD econometrics workshops for helpful comments. 1980, Robinson 1991, Rosenblatt 1973, =-=Schweizer and Wolff 1976-=-, 1981, Wolff 1981, or Zheng 1994b.) Our purpose here is to unify the apparently disparate nonprametric and nuisance parameter approaches to testing models consistently for abitrary misspecification (... |

1 |
A New Regression Specification Error Test," photocopy
- Leung, Yu
- 1995
(Show Context)
Citation Context ... object of interest (say a conditional mean or density) to a parametric estimator (e.g. Eubank and Spiegelman 1990; Eubank and LaRiccia 1992; Gozalo 1993, Hardle and Mammen 1993, Hong and White 1995, =-=Leung and Yu 1995, or Zheng-=- 1994a). The nuisance parameter approach tests whether a statistic depending on a "nuisance parameter present only under the alternative" is zero for all values of the nuisance parameter, as... |

1 |
Approximation Theorems of Mathematial Statistics
- Serfling
- 1980
(Show Context)
Citation Context ...scalar and G is chosen to be the indicator function G(a) = 1[a ? 0], using G(~x 0 ��) = 1[x ? �� 0 ] with �� := (�� 0 ; 1) leads to tests closely related to the familiar Kolmogorov-Smi=-=rnov test (e.g. Serfling 1980-=-, pp. 57-58) when the uniform norm is applied, as described in the next section; choosing the L 2 -norm instead leads to tests related to the Cram'er-von Mises statistic (e.g. Serfling 1980, p. 58). S... |

1 |
N-Dimensional
- Wolff
- 1981
(Show Context)
Citation Context ...209023. We are grateful to Rob Engle, Clive Granger and the participants of the UCSD econometrics workshops for helpful comments. 1980, Robinson 1991, Rosenblatt 1973, Schweizer and Wolff 1976, 1981, =-=Wolff 1981-=-, or Zheng 1994b.) Our purpose here is to unify the apparently disparate nonprametric and nuisance parameter approaches to testing models consistently for abitrary misspecification (i.e. with power ap... |

1 |
A Residual-Based Consistent Test of Parametric Regression Models," Working Paper
- Zheng
- 1994
(Show Context)
Citation Context ...ay a conditional mean or density) to a parametric estimator (e.g. Eubank and Spiegelman 1990; Eubank and LaRiccia 1992; Gozalo 1993, Hardle and Mammen 1993, Hong and White 1995, Leung and Yu 1995, or =-=Zheng 1994a). The nu-=-isance parameter approach tests whether a statistic depending on a "nuisance parameter present only under the alternative" is zero for all values of the nuisance parameter, as is true under ... |