Distribution of test statistics under parameter uncertainty for time series data: an application to testing skewness, kurtosis and normality

Year 2022, Volume: 51 Issue: 1, 253 - 272, 14.02.2022

Anil Bera Osman Doğan , Suleyman Taspinar

https://doi.org/10.15672/hujms.802601

Cited By: 1

Abstract

In this paper, we provide a general result under some high level assumptions that shows how to account for the parameter uncertainty problem in test statistics formulated with the quasi maximum likelihood (QML) estimator. We use our general result to develop various test statistics for testing skewness, kurtosis and normality for time series data. We show that the asymptotic distributions of our test statistics coincide with the asymptotic distributions of some tests suggested in the literature. Thus, our general result provides a unified approach for test statistics formulated with the QML estimator for time series data.

Keywords

Normality tests , Skewness , Kurtosis , Asymptotic Variance , MLE , QMLE , Inference , Parameter Uncertainty , test statistics

References

• [1] D.W.K. Andrews, Heteroskedasticity and autocorrelation consistent covariance matrix estimation, Econometrica 59 (3), 817-858, 1991.
• [2] J. Bai and S. Ng, Tests for skewness, kurtosis, and normality for time series data, J. Bus. Econom. Statist. 23 (1), 49-60, 2005.
• [3] A.K. Bera, O. Doğan and S. Taşpınar, Asymptotic variance of test statistics in the ML and QML frameworks, J. Stat. Theory Pract. 15 (2), 1-26, 2021.
• [4] C. Bontemps, Moment-based tests under parameter uncertainty, Rev. Econ. Stat. 101 (1), 146-159, 2019.
• [5] C. Bontemps and N. Meddahi, Testing normality: a GMM approach, J. Econometrics 124 (1), 149-186, 2005.
• [6] C. Bontemps and N. Meddahi, Testing distributional assumptions: A GMM approach, J. Appl. Econometrics 27 (6), 978-1012, 2012.
• [7] J. Davidson, Stochastic Limit Theory: An Introduction for Econometricians, Oxford University Press, 1994.
• [8] R. Davidson and J.G. MacKinnon, Model specification tests based on artificial linear regressions, Internat. Econom. Rev. 25 (2), 485-502, 1984.
• [9] A.R. Gallant, Explicit estimators of parametric functions in nonlinear regression, J. Amer. Statist. Assoc. 75 (369), 182-193, 1980.
• [10] A.R. Gallant and H. White, A Unified Theory of Estimation and Inference for Nonlinear Dynamic Models, B. Blackwell, 1988.
• [11] P. Glewwe, A test of the normality assumption in ordered probit model, Econometric Rev. 16 (1), 1-19, 1997.
• [12] S. Goncalves and H. White, Maximum likelihood and the bootstrap for nonlinear dynamic models, J. Econometrics 119 (1), 199-219, 2004.
• [13] J.A. Hausman, Specification tests in econometrics, Econometrica 46 (6), 1251-1271, 1978.
• [14] C.M. Jarque and A.K. Bera, A test for normality of observations and regression residuals, Int. Stat. Rev. 55 (2), 163-172, 1987.
• [15] N.M. Kiefer and M. Salmon, Testing normality in econometric models, Econom. Lett. 11 (1), 123-127, 1983.
• [16] I.N. Lobato and C. Velasco, A simple test of normality for time series, Econometric Theory 20 (4), 671-689, 2004.
• [17] Z.A. Lomnicki, Tests for departure from normality in the case of linear stochastic processes, Metrika 4 (1), 37-62, 1961.
• [18] J.G. MacKinnon, Model specification tests and artificial regressions, J. Econ. Lit. 30 (1), 102-146, 1992.
• [19] M.W. McCracken, Robust out-of-sample inference, J. Econometrics 99 (2), 2000.
• [20] M.W. McCracken and K.S. West, Inference about predictive ability, A companion to economic forecasting, 14, 299-321, Blackwell Publishing, 2002.
• [21] J. Neyman, Optimal Asymptotic Tests of Composite Statistical Hypotheses, Probability and Statistics, the Harald Cramer Volume, Wiley, 1959.
• [22] W.K. Newey, Maximum likelihood specification testing and conditional moment tests, Econometrica 53 (5), 1047-1070, 1985.
• [23] W.K. Newey, Generalized method of moments specification testing, J. Econometrics 29 (3), 229-256, 1985.
• [24] W.K. Newey and K.D. West, A simple, positive semi-definite, heteroskedasticity and autocorrelation consistent covariance matrix, Econometrica 55 (3), 703-708, 1987.
• [25] A. Pagan and F. Vella, Diagnostic tests for models based on individual data: a survey, J. Appl. Econometrics 4 (1), 29-59, 1989.
• [26] D.A. Pierce, The asymptotic effect of substituting estimators for parameters in certain types of statistics, Ann. Statist. 10 (2), 475-478, 1982.
• [27] B.M. Pötscher and I.R. Prucha, Basic structure of the asymptotic theory in dynamic nonlinear econometric models, part I: consistency and approximation concepts, Econometric Rev. 10 (2), 125-216, 1991.
• [28] B.M. Pötscher and I.R. Prucha, Basic structure of the asymptotic theory in dynamic nonlinear econometric models, part II: consistency and approximation concepts, Econometric Rev. 10 (3), 253-325, 1991.
• [29] R.H. Randles, On the asymptotic normality of statistics with estimated parameters, Ann. Statist. 10 (2), 462-474, 1982.
• [30] P.A. Ruud, Tests of specification in econometrics, Econometric Rev. 3 (2), 211-242, 1984.
• [31] G. Tauchen, Diagnostic testing and evaluation of maximum likelihood models, J. Econometrics 30 (1), 415-443, 1985.
• [32] K.D. West, Asymptotic inference about predictive ability, Econometrica 64 (5), 1067- 1084, 1996.
• [33] K.D. West, Forecast Evaluation, Handbook of Economic Forecasting, North-Holland, 2006.
• [34] K.D.West and M.W. McCracken, Regression-based tests of predictive ability, Internat. Econom. Rev. 39 (4), 817-840, 1998.
• [35] H. White, Maximum likelihood estimation of misspecified models, Econometrica 50 (1), 1-25, 1982.
• [36] H. White, Specification Testing in Dynamic Models, Advances in Econometrics - Fifth World Congress, Cambridge University Press, 1987.
• [37] H. White, Estimation, Inference and Specification Analysis, Cambridge University Press, 1994.
• [38] J.M. Wooldridge, A unified approach to robust, regression-based specification tests, Econometric Theory 6 (1), 17-43, 1990.
• [39] Z. Yang, Y.K. Tse and Z. Bai, Statistics with estimated parameters, Statist. Sinica 17 (2), 817-837, 2007.