Abstract
We develop a new consistent conditional moment test of functional form based on nuisance parameter indexed sample moments first presented in Bierens. We reduce the nuisance parameter space to known countable sets, which leads to a weighted average conditional moment test in the spirit of Bierens and Ploberger’s integrated conditional moment test (ICM). The weights are possibly stochastic in an arbitrary way, integer-indexed and flexible enough to cover a range of tests from average to higher quantile to maximum tests. The limit distribution under the null and local alternative belong to the same class as the ICM statistic, hence our test is admissible if the errors are Gaussian, and a flat weight leads to the greatest weighted average local power.
6 Appendix A: Assumption C
Assumption C1: ∈ ℝ × ℝkis an iid process with non-degenerate absolutely continuous marginal distributions and finite variances. The parameter space Φ is a compact subset of ℝk+1. ϕ0= ∈ interior {Φ} f(.,ϕ) is for each ∈ Borel measurable, and twice continuously differentiable on Φ.
Assumption C2: Let then An(ϕ) → A(ϕ) uniformly on Φ, where A(ϕ) is a non-stochastic positive definite matrix. The NLLS estimator satisfies
Assumption C3: Write where Then uniformly on ℕk ×Φ where b(m,ϕ) is a non-stochastic function satisfying
Assumption C4:
a finite non-stochastic matrix.
ut is governed by a non-generate distribution, andexists and is constant and finite. There exists a mapping η: ℕk→ ℝ such that uniformly on ℕk. iffor some q : ℕk→ ℝ then η(m) = O(q(m)).
There exists a matrix functional Γ(m1,m2) on ℕk×k such that and pointwise on ℕk×k. If for some q : ℕk→ ℝ then
Let. For each ζ ∈ ℝk, is uniformly integrable and for some t > 0. Moreover, for each u∈[0, ∞).
7 Appendix B: proofs of main results
Proof of Lemma 3.1. Recall from (6) and define for arbitrary Π ∈ ℝl, π′π=1, and l ≥ 1. Assumption C, Cramér’s Theorem, and the Lindeberg central limit theorem guarantee
The claim follows by invoking the Cramér-Wold Theorem. The bounds and follow from Assumptions A and C. .
Proof of Lemma 3.2. It suffices to show is uniformly tight on for each by straightforward extensions of results in Bickel and Wichura (1971) and Neuhaus (1971). We will apply Lemma A.1 of Bierens and Ploberger (1997). The functional g(xt,ξ) must satisfy a Lipschitz continuity condition on for arbitrary :
for every and for some Kt measurable with respect to that satisfies Finally, we need
for one arbitrary All requirements are met under Assumption C by choosing Since is bounded by construction of Gm(‧) under Assumption A and it follows
Proof of theorem 3.3 Using the non-stochastic limiting sequence {ωm}, apply expansion (6), Lemmas 3.1 and 3.2, and the continuous mapping theorem to verify under Assumption C
By (6) it therefore suffices to prove By the triangle inequality and ωm≥0
Assumptions A-C and Nn→∞ imply
hence by Chebyshev’s inequality
For A1,n observe
Assumption B states By Lemmas 3.1 and 3.2 where Therefore which completes the proof.3.
Proof of theorem 3.4 Denote by the inner product space of sequences of continuous functions y(ξ)∈C[0,∞)k with bound y(ξ)=O(ρξ), metrized with Let be the supporting inner product. Then H is separable and complete3, hence a separable Hilbert space. Separable inner product spaces have countably infinite orthonormal basis, say , (e.g. Giles, 2000: Theorem 3.27).
Now define Fourier coefficients
then z∈H admits a coordinate-wise expansion
Because is a symmetric positive-semi-definite bounded function under Lemma 3.1, it follows that is a linear compact self-adjoint operator (Giles, 2000: section 15). By the spectral theorem for compact self-adjoint operators on a Hilbert space there exists an orthonormal basis of H consisting of eigenfunctions of Γ, where each eigenvalue λi is real and non-negative (Giles, 2000: Theorem 20.4.1). It is immediate that denotes the eigenfunctions of Γ,
and Γ(m1,m2) obtains the series representation Use Parseval’s identity, (11) and (12) and orthonormality to get
Each z(m) under is mean zero Gaussian by Lemma 3.1, therefore each Fourier coefficient is Gaussian, completely characterized by means and pair-wise covariances
Thus which completes the proof.
Proof of Corollary 3.6 Since the argument simply mimics the proof of Theorem 3.4, we only present a sketch. Write
Exploit to deduce the following. First, if i = j and 0 otherwise; hence and hence the Fourier coefficients are w1:=z(m*) and wi = 0 a.s. ∀i≥2.
Second, hence λ1=Γ(m*, m*) and λi=03∀i≥2.
Third, use Parseval’s identity and orthonormality to obtain the trivial identity
Fourth, the Fourier coefficients are Gaussian, completely characterized by means if i = 1 and 0 ∀i≥2, and pair-wise covariances that reduce to if i≠j, i≠1 and λ1 otherwise.
Therefore and wi = 0 a.s. ∀i≥2. Coupled with this completes the proof under . Under we have hence by (4)
8 Appendix C
- 1
Recall a real analytic function is infinitely differentiable and therefore has an infinite order Taylor expansion.
- 2
The data are iid Gaussian and the nuisance parameter space is countable, hence it is easy to show Hansen’s approximate p-value is consistent (cf. de Jong 1996, Hill 2008).
- 3
It is straightforward to show Davidson’s (1994: Theorem 5:15) argument carries over to H due to boundedness y(ξ) = O(ρ–ξ) of every y∈H.
References
Andrews, D. W. K., and W. Ploberger. 1994. “Optimal Tests when a Nuisance Parameter is Present Only under the Alternative.” Econometrica 82: 1383–1414.10.2307/2951753Search in Google Scholar
Bickel, P. J., and M. J. Wichura. 1971. “Convergence Criteria for Multiparameter Stochastic Processes and Some Applications.” Annals of Mathematical Statistics 42: 1656–1670.10.1214/aoms/1177693164Search in Google Scholar
Bierens, H. J. 1982. “Consistent Model Specification Tests.” Journal of Econometrics 20: 105–134.10.1016/0304-4076(82)90105-1Search in Google Scholar
Bierens, H. J. 1984. “Model Specification Testing of Time Series Regressions.” Journal of Econometrics 26: 323–353.10.1016/0304-4076(84)90025-3Search in Google Scholar
Bierens, H. J. 1990. “A Consistent Conditional Moment Test of Functional Form.” Econometrica 58: 1443–1458.10.2307/2938323Search in Google Scholar
Bierens, H. J. 1991. “Least Squares Estimation of Linear and Nonlinear ARMAX Models Under Data Heterogeneity.” Annales d’Economie et de Statistique 20/21: 143–169.Search in Google Scholar
Bierens, H. J. 1994. Comment on Artificial Neural Networks: An Econometric Perspective, by C-M. Kuan and H. White, Econometric Reviews 13: 93–97.Search in Google Scholar
Bierens, H. J., and W. Ploberger. 1997. “Asymptotic Theory of Integrated Conditional Moment Tests.” Econometrica 65: 1129–1151.10.2307/2171881Search in Google Scholar
Bierens, H. J., and L. Wang. 2012. “Integrated Conditional Moment Tests for Parametric Conditional Distributions.” Econometric Theory 28: 328–362.10.1017/S0266466611000168Search in Google Scholar
Billingsley, P. 1999. Convergence of Probability Measures. New York: John Wiley & Sons.10.1002/9780470316962Search in Google Scholar
Blake, A.P., and G. Kapetanios. 2007. “Testing for Neglected Nonlinearity in Cointegrating Relationships.” Journal of Time Series Analysis 28: 807–826.10.1111/j.1467-9892.2007.00532.xSearch in Google Scholar
Boning, B. W., and F. Sowell. 1999. “Optimality for the Integrated Conditional Moment Test.” Econometric Theory 15: 710–718.10.1017/S026646669915504XSearch in Google Scholar
Davies, R. B. 1977. “Hypothesis Testing When a Nuisance Parameter is Present Only under the Alternative.” Biometrika 64: 247–254.10.2307/2335690Search in Google Scholar
de Jong, R. 1996. “The Bierens Test under Data Dependence.” Journal of Econometrics 72: 1–32.10.1016/0304-4076(94)01712-3Search in Google Scholar
de Jong, R., and H. J. Bierens. 1994. “On the Limit Behavior of a Chi-Squared Type Test if the Number of Conditional Moments Tested Approaches Infinity.” Econometric Theory 9: 70–90.10.1017/S0266466600008239Search in Google 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: 1012–1040.10.1214/aos/1018031266Search in Google Scholar
Fan, Y., and Q. Li. 1996. “Consistent Model Specification Tests: Omitted Variables and Semiparametric Functional Forms.” Econometrica: 64: 865–890.10.2307/2171848Search in Google Scholar
Fan, Y. and Q. Li. 2000. “Consistent Model Specification Tests: Kernel-Based Tests versus Bierens’ ICM Tests.” Econometric Theory 16: 1016–1041.10.1017/S0266466600166083Search in Google Scholar
Giacomini, R., and H. White. 2006. “Tests of Conditional Predictive Ability.” Econometrica 74: 1545–1578.10.1111/j.1468-0262.2006.00718.xSearch in Google Scholar
Giles, J. R. 2000. Introduction to the Analysis of Normed Linear Spaces. U.K: Cambridge University Press10.1017/CBO9781139168465Search in Google Scholar
Hansen, B. 1996. “Inference When a Nuisance Parameter Is Not Identified Under the Null Hypothesis.” Econometrica 64: 413–430.10.2307/2171789Search in Google Scholar
Härdle, W., and E. Mammen. 1993. “Comparing Nonparametric Versus Parametric Regression Fits.” Annals of Statistics 21: 1926–1947.10.1214/aos/1176349403Search in Google Scholar
Hill, J. B. 2008. “Consistent and Non-Degenerate Model Specification Tests Against Smooth Transition and Neural Network Alternatives.” Annale’s d’Economie et de Statistique 90: 145–179.10.2307/27739822Search in Google Scholar
Hill, J. B. 2012. “Heavy-Tail and Plug-In Robust Consistent Conditional Moment Tests of Functional Form,” by X. Chen and N. Swanson, Springer, New York: Festschrift in Honor of Hal White.10.2139/ssrn.1914104Search in Google Scholar
Hill, J. B. 2013. “Consistent GMM Residuals-Based Tests of Functional Form.” Econometric Reviews 32: 361–383.10.1080/07474938.2012.690662Search in Google Scholar
Holley, A. 1982. “A Remark on Hausman’s Specification Test.” Econometrica 50: 749–760.10.2307/1912612Search in Google Scholar
Hong, Y., and H. White. 1995. “Consistent Specification Testing via Nonparametric Series Regression.” Econometrica 63: 1133–1159.10.2307/2171724Search in Google Scholar
Hong, Y., and Y. -J. Lee. 2005. “Generalized Spectral Tests for Conditional Mean Models in Time Series with Conditional Heteroscedasticity of Unknown Form.” Review of Economic Studies 72: 499–541.10.1111/j.1467-937X.2005.00341.xSearch in Google Scholar
Hornik, K., M. Stinchcombe, and H. White. 1989. “Multilayer Feedforward Networks are Universal Approximators.” Neural Networks 2: 359–366.10.1016/0893-6080(89)90020-8Search in Google Scholar
Huber, P. J. 1977. Robust Statistical Procedures. Philadelphia: Society for Industrial and Applied Mathematics.Search in Google Scholar
Koul, H. L., and W. Stute. 1999. “Nonparametric Model Checks for Time Series.” Annals of Statistics 27: 204–236.10.1214/aos/1018031108Search in Google Scholar
Leadbetter, M. R., G. Lindgren, and H. Rootzén. 1983. Extremes and Related Properties of Random Sequences and Processes. New York: Springer-Verlag.10.1007/978-1-4612-5449-2Search in Google Scholar
Lee T., H. White, and C. W. J. Granger. 1993. “Testing for Neglected Nonlinearity in Time-Series Models: A Comparison of Neural Network Methods and Alternative Tests.” Journal of Econometrics 56: 269–290.10.1016/0304-4076(93)90122-LSearch in Google Scholar
Li, Q., C. Hsiao, and J. Zinn. 2003. “Consistent Specification Tests for Semiparametric/Nonparametric Models Based on Series Estimation Methods.” Journal of Econometrics 112: 295–325.10.1016/S0304-4076(02)00198-7Search in Google Scholar
Neuhaus, G. 1971. “On Weak Convergence of Stochastic Processes with Multidimensional Time Parameter.” Annals of Mathematical Statistics 42: 1285–1295.10.1214/aoms/1177693241Search in Google Scholar
Newey, W. K. 1985. “Maximum Likelihood Specification Testing and Conditional Moment Tests.” Econometrica 53: 1047–1070.10.2307/1911011Search in Google Scholar
Ramsey, J. B. 1969. “Tests for Specification Errors in a Classical Linear Least-Squares Regression Analysis.” Journal of the Royal Statistical Society Series B 31: 350–371.10.1111/j.2517-6161.1969.tb00796.xSearch in Google Scholar
Stinchcombe, M. B., and H. White. 1998. “Consistent Specification Testing with Nuisance Parameters Present Only Under the Alternative.” Econometric Theory 14: 295–325.10.1017/S0266466698143013Search in Google Scholar
Stute, W. 1997. “Nonparametric Model Checks for Regression.” Annals of Statistics 35: 613–641.10.1214/aos/1031833666Search in Google Scholar
Stute, W., and L.-X. Zhu. 2005. “Nonparametric Checks for Single-Index Models.” Annals of Statistics 33: 1048–1083.10.1214/009053605000000020Search in Google Scholar
Teräsvirta, T. 1994. “Specification, Estimation, and Evaluation of Smooth Transition Autoregressive Models.” Journal of the American Statistical Association 89: 208–218.Search in Google Scholar
White, H. 1982. “Maximum Likelihood Estimation of Misspecified Models.” Econometrica 50: 1–2.10.2307/1912526Search in Google Scholar
White, H. 1989. “An Additional Hidden Unit Test of Neglected Nonlinearity in Multilayer Feedforward Networks.” In Proceedings of the International Joint Conference on Neural Networks. Vol. 2. New York, Washington D.C: IEEE Press.10.1109/IJCNN.1989.118281Search in Google Scholar
Yatchew, A. J. 1992. “Nonparametric Regression Tests Based on Least Squares.” Econometric Theory 8: 435–451.10.1017/S0266466600013153Search in Google Scholar
Zheng, J. 1996. “A Consistent Test of Functional Form via Nonparametric Estimation Techniques.” Journal of Econometrics 75: 263–289.10.1016/0304-4076(95)01760-7Search in Google Scholar
©2013 by Walter de Gruyter Berlin Boston