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WEAK-IDENTIFICATION ROBUST WILD BOOTSTRAP APPLIED TO A CONSISTENT MODEL SPECIFICATION TEST

Published online by Cambridge University Press:  02 June 2020

Jonathan B. Hill
Jonathan B. Hill*
Affiliation:
University of North Carolina at Chapel Hill
*
Address correspondence to Jonathan B. Hill, Department of Economics, University of North Carolina at Chapel Hill, Chapel Hill, NC, USA; e-mail: jbhill@email.unc.edu.
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Abstract

We present a new robust bootstrap method for a test when there is a nuisance parameter under the alternative, and some parameters are possibly weakly or nonidentified. We focus on a Bierens (1990, Econometrica 58, 1443–1458)-type conditional moment test of omitted nonlinearity for convenience. Existing methods include the supremum p-value which promotes a conservative test that is generally not consistent, and test statistic transforms like the supremum and average for which bootstrap methods are not valid under weak identification. We propose a new wild bootstrap method for p-value computation by targeting specific identification cases. We then combine bootstrapped p-values across polar identification cases to form an asymptotically valid p-value approximation that is robust to any identification case. Our wild bootstrap procedure does not require knowledge of the covariance structure of the bootstrapped processes, whereas Andrews and Cheng’s (2012a, Econometrica 80, 2153–2211; 2013, Journal of Econometrics 173, 36–56; 2014, Econometric Theory 30, 287–333) simulation approach generally does. Our method allows for robust bootstrap critical value computation as well. Our bootstrap method (like conventional ones) does not lead to a consistent p-value approximation for test statistic functions like the supremum and average. Therefore, we smooth over the robust bootstrapped p-value as the basis for several tests which achieve the correct asymptotic level, and are consistent, for any degree of identification. They also achieve uniform size control. A simulation study reveals possibly large empirical size distortions in nonrobust tests when weak or nonidentification arises. One of our smoothed p-value tests, however, dominates all other tests by delivering accurate empirical size and comparatively high power.

Type
ARTICLES
Information
Econometric Theory , Volume 37 , Issue 3 , June 2021 , pp. 409 - 463
Copyright
© The Author(s) 2020. Published by Cambridge University Press

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Footnotes

*

This paper was previously circulated under the title “Inference When There is a Nuisance Parameter under the Alternative and Some Parameters are Possibly Weakly Identified.” We thank two referees and Co-Editor Michael Jansson for helpful comments and suggestions.

References

Andrews, D.W.K. & Cheng, X. (2012a) Estimation and inference with weak, semi-strong and strong identification. Econometrica 80, 2153–2211.Google Scholar
Andrews, D.W.K. & Cheng, X. (2012b) Supplement to “Estimation and Inference with Weak, Semi-Strong, and Strong Indentification”. Department of Economics, Yale University.Google Scholar
Andrews, D.W.K. & Cheng, X. (2013) Maximum likelihood estimation and uniform inference with sporadic identification failure. Journal of Econometrics 173, 36–56.CrossRefGoogle Scholar
Andrews, D.W.K. & Cheng, X. (2014) GMM estimation and uniform subvector inference with possible identification failure. Econometric Theory 30, 287–333.CrossRefGoogle Scholar
Andrews, D.W.K. & Guggenberger, P. (2010) Asymptotic size and a problem with subsampling and the m out of n bootstrap. Econometric Theory 26, 426–468.CrossRefGoogle Scholar
Andrews, D.W.K., Moreira, M.J. & Stock, J. (2006) Optimal two-sided invariant similar tests for instrumental variables regression. Econometrica 74, 715–752.CrossRefGoogle Scholar
Andrews, D.W.K. & Ploberger, W. (1994) Optimal tests when a nuisance parameter is present only under the alternative. Econometrica 62, 1383–1414.CrossRefGoogle Scholar
Andrews, I. & Mikusheva, A. (2016) A geometric approach to nonlinear econometric models. Econometrica 84, 1249–1264.CrossRefGoogle Scholar
Arcones, M.A. & Yu, B. (1994) Central limit theorems for empirical U-processes of stationary mixing sequences. Journal of Theoretical Probability 7, 47–71.CrossRefGoogle Scholar
Bickel, P.J. & Freedman, D.A. (1981) Some asymptotic theory for the bootstrap. Annals of Statistics 9, 1196–1217.CrossRefGoogle Scholar
Bierens, H.J. (1982) Consistent model specification tests. Journal of Econometrics 20, 105–134.CrossRefGoogle Scholar
Bierens, H.J. (1990) A consistent conditional moment test of functional form. Econometrica 58, 1443–1458.CrossRefGoogle Scholar
Bierens, H.J. & Ploberger, W. (1997) Asymptotic theory for integrated conditional moment tests. Econometrica 65, 1129–1151.CrossRefGoogle Scholar
Caner, M. (2010) Testing, estimation in GMM and CUE with nearly weak identification. Econometric Reviews 29, 330–363.CrossRefGoogle Scholar
Chan, K.S. & Tong, H. (1986) On estimating thresholds in autoregressive models. Journal of Time Series Analysis 7, 179–190.CrossRefGoogle Scholar
Cheng, X. (2015) Robust inference in nonlinear models with mixed identification strength. Journal of Econometrics 189, 207–228.CrossRefGoogle Scholar
Chernoff, H. & Zacks, S. (1964) Estimating the current mean of a normal distribution which is subject to changes in time. Annals of Mathematical Statistics 35, 999–1028.CrossRefGoogle Scholar
Choi, I. & Phillips, P.C.B. (1992) Asymptotic and finite sample distribution theory for IV estimators and tests in partially identified structural equations. Journal of Econometrics 51, 113–150.CrossRefGoogle Scholar
Cox, G. (2016) Weak Identification in a Class of Generically Identified Models with an Application to Factor Models. Discussion Paper, Yale University.Google Scholar
Davidson, J. & Halunga, A.G. (2014) Consistent testing of functional form in time series models. In Haldrup, N., Meitz, M. & Saikkonen, P. (eds.), Essays in Nonlinear Time Series Econometrics. Oxford University Press.Google Scholar
Davies, R. (1977) Hypothesis testing when a nuisance parameter is present only under the alternative. Biometrika 64, 247–254.CrossRefGoogle Scholar
Davies, R. (1987) Hypothesis testing when a nuisance parameter is present only under the alternative. Biometrika 74, 33–43.Google Scholar
de Jong, R.M. (1996) On the Bierens test under data dependence. Journal of Econometrics 72, 1–32.CrossRefGoogle Scholar
de Jong, R.M. (1997) Central limit theorems for dependent heterogeneous random variables. Econometric Theory 13, 353–367.CrossRefGoogle Scholar
Delgado, M.A., Dominguez, M.A. & Lavergne, P. (2006) Consistent tests of conditional moment restrictions. Annales d’Economie et de Statistique 81, 33–67.CrossRefGoogle Scholar
Dette, H. (1999) A consistent test for the functional form of a regression based on a difference of variance estimators. Annals of Statistics 27, 1012–1040.CrossRefGoogle Scholar
Dudley, R.M. (1978) Central limit theorems for empirical measures. Annals of Probability 6, 899–929.CrossRefGoogle Scholar
Dudley, R.M. (1984) A course on empirical processes. Lecture Notes in Math, vol. 1097, chap. École d’Été de Probabilités de Saint-Flour XII—1982, pp. 2–142. Springer, Berliln.Google Scholar
Dufour, J.-M. (1997) Some impossibility theorems in econometrics with applications to structural and dynamic models. Econometrica 65, 1365–1387.CrossRefGoogle Scholar
Eberlein, E. (1984) Weak rates of convergence of partial sums of absolute regular sequences. Statistics and Probability Letters 2, 291–293.CrossRefGoogle Scholar
Elliott, G., Muller, U. & Watson, M. (2015) Nearly optimal tests when a nuisance parameter is present under the null hypothesis. Econometrica 83, 771–811.CrossRefGoogle Scholar
Gine, E. & Zinn, J. (1986) Lectures on the central limit theorem for empirical processes. Probability and Banach Spaces: Lecture Notes in Math, vol. 1221, pp. 50–113, Berliln. Springer.CrossRefGoogle Scholar
Gine, E. & Zinn, J. (1990) Bootstrapping general empirical measures. Annals of Statistics 18, 851–869.Google Scholar
Gonzalez-Rivera, G. (1998) Smooth-transition GARCH models. Studies in Nonlinear Dynamics and Econometrics 3, 61–78.Google Scholar
Granger, C.W.J. & Terasvirta, T. (1993) Modelling Non-Linear Economic Relationships. Oxford University Press.Google Scholar
Han, S. & McCloskey, A. (2016) Estimation and Inference with a (Nearly) Singular Jacobian. Discussion Paper, Brown University.CrossRefGoogle Scholar
Hansen, B.E. (1996) Inference when a nuisance parameter is not identified under the null hypothesis. Econometrica 64, 413–430.CrossRefGoogle Scholar
Härdle, W. & Mammen, E. (1993) Comparing nonparametric vs parametric regression fits. Annals Statistics 21, 1926–1947.CrossRefGoogle Scholar
Hill, J.B. (2008) Consistent and non-degenerate model specification tests against smooth transition and neural network alternatives. Annales D’ Economie et de Statistique 90, 145–179.CrossRefGoogle Scholar
Hill, J.B. (2018) A Smoothed p-Value Test When There is a Nuisance Parameter under the Alternative. Discussion Paper, Department of Economics, University of North Carolina.Google Scholar
Hoffman-Jørgensen, J. (1984) Convergence of Stochastic Processes on Polish Spaces, mimeo.Google Scholar
Hoffman-Jørgensen, J. (1991) Convergence of stochastic processes on Polish spaces. Various Publications Series 39. Aarhus Universitet, mimeo.Google Scholar
Hong, Y. & White, H. (1995) Consistent specification testing via nonparametric series regression. Econometrica 63, 1133–1159.CrossRefGoogle Scholar
Kullback, S. & Leibler, R.A. (1951) On Information and sufficiency. Annals of Mathematical Statistics 22, 79–86.CrossRefGoogle Scholar
Lehmann, E.L. (1994) Testing Statistical Hypotheses, 2nd edition. Chapman and Hall New York.Google Scholar
Li, H., Li, Q. & Liu, R. (2016) Consistent model specification tests based on k-nearest-neighbor estimation method. Journal of Econometrics 194, 187–202.CrossRefGoogle Scholar
Li, Q. (1999) Consistent model specification tests for time series econometric models. Journal of Econometrics 92, 101–147.CrossRefGoogle Scholar
Liu, R.Y. (1988) Bootstrap procedures under some NON-I.I.D. models. Annals of Statistics 16, 1696–1708.CrossRefGoogle Scholar
Lundbergh, S. & Terasvirta, T. (2006) A time series model for an exchange rate in a target zone with applications. Journal of Econometrics 131, 579–609.CrossRefGoogle Scholar
Luukkonen, R., Saikkonen, P. & Terasvirta, T. (1988) Testing linearity against smooth transition autoregressive models. Biometrika 75, 491–499.CrossRefGoogle Scholar
McCloskey, A. (2017) Bonferroni-based size-correction for nonstandard testing problems. Journal of Econometrics 200, 17–35.CrossRefGoogle Scholar
Moreira, M.J. (2003) A conditional likelihood ratio test for structural models. Econometrica 71, 1027–1048.CrossRefGoogle Scholar
Newey, W.K. (1985) Maximum likelihood specification testing and conditional moment test. Econometrica 53, 1047–1070.CrossRefGoogle Scholar
Phillips, P.C.B. (1990) Partially identified econometric models. Econometric Theory 5, 181–240.CrossRefGoogle Scholar
Pollard, D. (1984) Convergence of Stochastic Processes. Springer, New York.CrossRefGoogle Scholar
Pollard, D. (1990) Empirical processes: Theory and applications. Regional Conference Series in Probability and Statistics, Volume 2.CrossRefGoogle Scholar
Romano, J.P. (1989) Do bootstrap confidence procedures behave well uniformly in P? Canadian Journal of Statistics 17, 75–80.CrossRefGoogle Scholar
Sargan, J.D. (1983) Identification and lack of identification. Econometrica 51, 1605–1633.CrossRefGoogle Scholar
Sawa, T. (1978) Information criteria for discriminating among alternative regression models. Econometrica 46, 1273–1291.CrossRefGoogle Scholar
Sheehy, A. & Wellner, J. (1992) Uniform Donsker classes of functions. Annals of Statistics 20, 1983–2030.Google Scholar
Stinchcombe, M. & White, H. (1998) Consistent specification testing with nuisance parameters present only under the alternative. Econometric Theory 14, 295–325.CrossRefGoogle Scholar
Stock, J.H., & Wright, J. (2000) GMM with weak identification. Econometrica 68, 1055–1096.CrossRefGoogle Scholar
Tauchen, G. (1985) Diagnostic testing and evaluation of maximum likelihood models. Journal of Econometrics 30, 415–443.CrossRefGoogle Scholar
Terasvirta, T. (1994) Specification, estimation, and evaluation of smooth transition autoregressive models. Journal of the American Statistical Association 89, 208 218.Google Scholar
Tong, H. & Lim, K.S. (1980) Threshold autoregression, limit cycles and cyclical data (with discussion). Journal of the Royal Statistical Society, Series B 42, 245–292.Google Scholar
van der Vaart, A. & Wellner, J. (1996) Weak Convergence and Empirical Processes. Springer, New York.CrossRefGoogle Scholar
Vapnik, V.K. & Červonenkis, A.Y. (1971) On the uniform convergence of relative frequencies of events to their probabilities. Theory of Probability and its Applications 16, 264–280.CrossRefGoogle Scholar
Whang, Y.-J. (2000) Consistent bootstrap tests of parametric regression functions. Journal of Econometrics 98, 27–46.CrossRefGoogle Scholar
White, H. (1982) Maximum likelihood estimation of misspecified models. Econometrica 50, 1–25.CrossRefGoogle Scholar
White, H. (1989) An additional hidden unit test for neglected nonlinearity in multilayer feedforward networks. Proceeding of the International Joint Conference on Neural Networks, II. pp. 451–455, Washington, D.C. New York, NY. IEEE Press.CrossRefGoogle Scholar
Wu, C.F.J. (1986) Jackknife, bootstrap and other subsampling methods in regression analysis. Annals of Statistics 14, 1261–1295.Google Scholar
Zheng, J.X. (1996) A consistent test of functional form via nonparametric estimation techniques. Journal of Econometrics 75, 263–289.CrossRefGoogle Scholar
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