Hypothesis testing of multiple inequalities: the method of constraint chaining

Econometric inequality hypotheses arise in diverse ways. Examples include concavity restrictions on technological and behavioural functions, monotonicity and dominance relations, one-sided constraints on conditional moments in GMM estimation, bounds on parameters which are only partially identified, and orderings of predictive performance measures for competing models. In this paper we set forth four key properties which tests of multiple inequality constraints should ideally satisfy. These are (1) (asymptotic) exactness, (2) (asymptotic)similarity on the boundary, (3) absence of nuisance parameters from the asymptotic null distribution of the test statistic, (4) low computational complexity and boostrapping cost. We observe that the predominant tests currently used in econometrics do not appear to enjoy all these properties simultaneously. We therefore ask the question : Does there exist any nontrivial test which, as a mathematical fact, satisfies the first three properties and, by any reasonable measure, satisfies the fourth ? Remarkably the answer is affirmative. The paper demonstrates this constructively. We introduce a method of test construction called chaining which begins by writing multiple inequalities as a single equality using zero-one indicator functions. We then smooth the indicator functions. The approximate 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 asymptotic theory is provided for the baseline. Simulation results show that the finite-sample performance of the test matches the theory quite well.

[1]  M. Richardson,et al.  The Monotonicity of the Term Premium: Another Look , 1992 .

[2]  Thierry Post,et al.  Empirical Tests for Stochastic Dominance Efficiency , 2003 .

[3]  A. Kudô,et al.  A multivariate analogue of the one-sided test , 1963 .

[4]  F. Wolak.  Duality in testing multivariate hypotheses , 1988 .

[5]  O. Linton,et al.  Testing for Stochastic Monotonicity , 2006 .

[6]  O. Linton,et al.  Bootstrap Tests of Stochastic Dominance with Asymptotic Similarity on the Boundary , 2008 .

[7]  H. White,et al.  A Reality Check for Data Snooping , 2000 .

[8]  G. Fisher,et al.  An empirical analysis of term premiums using significance tests for stochastic dominance , 1998 .

[9]  Christian Gourieroux,et al.  Statistics and econometric models , 1995 .

[10]  Jean-Marie Dufour,et al.  Nonlinear hypotheses, inequality restrictions, and non-nested hypotheses: exact simultaneous tests in linear regressions , 1989 .

[11]  J. Horowitz.  A Smoothed Maximum Score Estimator for the Binary Response Model , 1992 .

[12]  Generalized Method of Moments Estimation When a Parameter Is on a Boundary , 2002 .

[13]  Jinde Wang,et al.  Likelihood ratio test against simple stochastic ordering among several multinomial populations , 2007 .

[14]  D. Andrews.  Inconsistency of the Bootstrap when a Parameter is on the Boundary of the Parameter Space , 2000 .

[15]  D. Andrews,et al.  Inference for Parameters Defined by Moment Inequalities Using Generalized Moment Selection , 2007 .

[16]  P. Hansen.  A Test for Superior Predictive Ability , 2005 .

[17]  R. Kopp,et al.  Testing for separable functional structure using temporary equilibrium models , 1986 .

[18]  Frank A. Wolak,et al.  An Exact Test for Multiple Inequality and Equality Constraints in the Linear Regression Model , 1987 .

[19]  A. Tamhane,et al.  Multiple Comparison Procedures , 2009 .

[20]  Peter E. Nuesch,et al.  ON THE PROBLEM OF TESTING LOCATION IN MULTIVARIATE POPULATIONS FOR RESTRICTED ALTERNATIVES , 1966 .

[21]  J. Hahn,et al.  Specification testing under moment inequalities , 2008 .

[22]  N. E. Savin,et al.  The Bonferroni and the Scheffé multiple comparison procedures , 1980 .

[23]  Chris Kirby,et al.  Bootstrap Tests of Multiple Inequality Restrictions on Variance Ratios , 2005 .

[24]  D. Andrews.  Estimation When a Parameter is on a Boundary , 1999 .

[25]  Frank A. Wolak,et al.  Local and Global Testing of Linear and Nonlinear Inequality Constraints in Nonlinear Econometric Models , 1989, Econometric Theory.

[26]  Joel L. Horowitz,et al.  Bootstrap Critical Values for Tests Based on the Smoothed Maximum Score Estimator , 2002 .

[27]  F. Palm,et al.  A parametric test of the negativity of the substitution matrix , 1987 .

[28]  F. Wolak.  Testing inequality constraints in linear econometric models , 1989 .

[29]  A. V. D. Vaart,et al.  Asymptotic Statistics: U -Statistics , 1998 .

[30]  C. Manski.  Semiparametric analysis of discrete response: Asymptotic properties of the maximum score estimator , 1985 .

[31]  Frank A. Wolak,et al.  The Local Nature of Hypothesis Tests Involving Inequality Constraints in Nonlinear Models , 1991 .

[32]  Frank A. Wolak,et al.  Quantifying the supply-side benefits from forward contracting in wholesale electricity markets , 2007 .

[33]  P. Hansen,et al.  A Forecast Comparison of Volatility Models: Does Anything Beat a Garch(1,1)? , 2004 .

[34]  Michael D. Perlman,et al.  One-Sided Testing Problems in Multivariate Analysis , 1969 .

[35]  C. Manski.  MAXIMUM SCORE ESTIMATION OF THE STOCHASTIC UTILITY MODEL OF CHOICE , 1975 .

[36]  Douglas A. McManus.  Who Invented Local Power Analysis? , 1991, Econometric Theory.

[37]  C. Gouriéroux,et al.  Likelihood Ratio Test, Wald Test, and Kuhn-Tucker Test in Linear Models with Inequality Constraints on the Regression Parameters , 1982 .

[38]  工藤 昭夫,et al.  A Multivariate Analogue of the One-Sided Testについての一注意 (多次元統計解析の数理的研究) , 1979 .

[39]  Franz C. Palm,et al.  Wald Criteria for Jointly Testing Equality and Inequality , 1986 .

[40]  A. Goldberger.  One-sided and Inequality Tests for a Pair of Means , 1992 .

[41]  D. Andrews,et al.  Hypothesis testing with a restricted parameter space , 1998 .

[42]  Jacob Boudoukh,et al.  Ex Ante Bond Returns and the Liquidity Preference Hypothesis , 1999 .

[43]  Adam M. Rosen,et al.  Confidence Sets for Partially Identified Parameters that Satisfy a Finite Number of Moment Inequalities , 2006 .

[44]  A. Shapiro.  Towards a unified theory of inequality constrained testing in multivariate analysis , 1988 .

[45]  N. Savin.  MULTIPLE HYPOTHESIS TESTING , 1984 .

[46]  Maxwell L. King,et al.  Joint one-sided tests of linear regression coefficients , 1986 .

[47]  Peter Reinhard Hansen,et al.  Asymptotic Tests of Composite Hypotheses , 2003 .

[48]  F. T. Wright,et al.  Order restricted statistical inference , 1988 .

[49]  Gordon Anderson,et al.  Nonparametric Tests of Stochastic Dominance in Income Distributions , 1996 .

[50]  M. Falk,et al.  Imposing and Testing Curvature Conditions on a Box-Cox Cost Function , 2003 .

[51]  O. Linton,et al.  Consistent Testing for Stochastic Dominance Under General Sampling Schemes , 2003 .

[52]  L.-Y. Chen.  Econometric inference involving discrete indicator functions , 2009 .

[53]  D. Andrews.  Testing When a Parameter Is on the Boundary of the Maintained Hypothesis , 2001 .

[54]  R. Davidson,et al.  Statistical Inference for Stochastic Dominance and for the Measurement of Poverty and Inequality , 1998 .

[55]  Stephen G. Donald,et al.  Consistent Tests for Stochastic Dominance , 2003 .

[56]  Jean-Marie Dufour,et al.  Invariance, Nonlinear Models and Asymptotic Tests , 1991 .

[57]  David McAdams.  Partial identification and testable restrictions in multi-unit auctions , 2008 .

[58]  G. Martin,et al.  Does the option market produce superior forecasts of noise‐corrected volatility measures? , 2009 .