A Test for Multimodality of Regression Derivatives with an Application to Nonparametric Growth Regressions

Henderson, Daniel J. (2008): A Test for Multimodality of Regression Derivatives with an Application to Nonparametric Growth Regressions.

Abstract

This paper presents a method to test for multimodality of an estimated kernel density of parameter estimates from a local-linear least-squares regression derivative. The procedure is laid out in seven simple steps and a suggestion for implementation is proposed. A Monte Carlo exercise is used to examine the finite sample properties of the test along with those from a calibrated version of it which corrects for the conservative nature of Silverman-type tests. The test is included in a study on nonparametric growth regressions. The results show that in the estimation of unconditional β-convergence, the distribution of the parameter estimates is multimodal with one mode in the negative region (primarily OECD economies) and possibly two modes in the positive region (primarily non-OECD economies) of the parameter estimates. The results for conditional β-convergence show that the density is predominantly negative and unimodal. Finally, the application attempts to determine why particular observations posess positive marginal effects on initial income in both the unconditional and conditional frameworks.

• Item Type: MPRA Paper
• Language: English
• Keywords: Nonparametric Kernel; Convergence; Modality Tests
• Subjects: O - Economic Development, Innovation, Technological Change, and Growth > O1 - Economic Development > O10 - General; C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General > C14 - Semiparametric and Nonparametric Methods: General; C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General > C15 - Statistical Simulation Methods: General; O - Economic Development, Innovation, Technological Change, and Growth > O4 - Economic Growth and Aggregate Productivity > O40 - General

References

[1] Abramovitz, M. (1986), “Catching Up, Forging Ahead and Falling Behind,” Journal of Economic History, 46, 386-406.

[2] Aitchison, J. and C. G. G. Aitken (1976), “Multivariate Binary Discrimination by Kernel Method,” Biometrika, 63, 413-420.

[3] Barro, R. (1991), “Economic Growth in a Cross Section of Countries,” Quarterly Journal of Economics, 106, 407-443.

[4] Barro R. J. and J.-W. Lee (2001), “International Data on Educational Attainment: Updates and Implications,” Oxford Economic Papers, 3, 541-563.

[5] Barro, R., and X. Sala-i-Martin (1992), “Convergence,” Journal of Political Economy, 100, 223-251.

[6] Baumol, W. J. (1986), “Productivity Growth, Convergence, and Welfare: What the Long-Run Data Show,” American Economic Review, 76, 1072-1085.

[7] Brock, W. A., and S. N. Durlauf (2001), “What Have we Learned from a Decade of Empirical Research on Growth? Growth Empirics and Reality ,” World Bank Economic Review, 15, 229-272.

[8] Bianchi, M. (1997), “Testing for Convergence: Evidence from Non-Paramteric Multimodality Tests,” Journal of Applied Economics, 12, 393-409.

[9] Bowman, A. W., M. C. Jones, and I. Gijbels (1998), “Testing Monotonicity of Regression,” Journal of Computational and Graphical Statistics, 7, 489-500.

[10] Cheng, M.-Y. and P. Hall (1998), “Calibrating the Excess Mass and Dip Tests of Modality,” Journal of the Royal Statistical Society, Series B, 60, 579-590.

[11] De Long, J. B. (1988), “Productivity Growth, Convergence, and Welfare: Comment,” American Economic Review, 78, 1138-1154.

[12] Durlauf, S. N. and P. A. Johnson (1995), “Multiple Regimes and Cross-Country Growth Behaviour,” Journal of Applied Econometrics, 10, 365-384.

[13] Durlauf, S. N., A. Kourtellos, and A. Minkin (2001), “The Local Solow Growth Model,” European Economic Review, 45, 928-940.

[14] Durlauf, S. N. and D. T. Quah (1999), “The New Empirics of Economic Growth,” in the Handbook of Macroeconomics, J. Taylor andM.Woodford (Eds.), North- Holland.

[15] Eren, O. and D. J. Henderson (2006), “The Impact of Homework on Student Achievement,” Econometrics Journal, forthcoming.

[16] Fan, J. and I. Gijbels (1992), “Variable Bandwidth and Local Linear Regression Smoothers,” Annals of Statistics, 20, 2008-2036.

[17] Fiaschi, D. and A. M. Lavezzi (2003), “Distribution Dynamics and Nonlinear Growth,” Journal of Economic Growth, 8, 379-401.

[18] Galor, O. (1996), “Convergence? Inferences from TheoreticalModels,” Economic Journal, 106, 1056-1069.

[19] Guerre, E., I. Perrigne, and Q. Vuong (2000), “Optimal Nonparametric Estimation of First-Price Auctions,” Econometrica, 68, 525-574.

[20] Hall, P., Q. Li and J. Racine (2007), “Nonparametric Estimation of Regression Functions in the Presence of Irrelevant Regressors,” Review of Economics and Statistics, 784-789.

[21] Hall, P. and M. York (2001), “On the Calibration of Silverman’s Test for Multimodality,” Statistica Sinica, 11, 515-536.

[22] Härdle W. (1993), Applied Nonparametric Regression, Cambridge University Press, Cambridge.

[23] Henderson, D. J., C. F. Parmeter, and R. R. Russell (2006), “Modes, Weighted Modes and Calibrated Modes: Evidence of Clustering Using Modality Tests,” Journal of Applied Econometrics, forthcoming.

[24] Henderson, D. J. and R. R. Russell (2005), “Human Capital and Convergence: A Production-Frontier Approach,” International Economic Review, 46, 1167-1205.

[25] Hsiao, C., Q. Li, and J. Racine (2007), “A Consistent Model Specification Test with Mixed Discrete and Continuous Data,” Journal of Econometrics, 140, 802- 826.

[26] Hurvich, C. M., J. S. Simonoff, and C.-L. Tsai (1998), “Smoothing Parameter Selection in Nonparametric Regression Using an Improved Akaike Information Criterion,” Journal of the Royal Statistical Society, Series B, 60, 271-293.

[27] Islam, N. (1995), “Growth Empirics: A Panel Data Approach,” Quarterly Journal of Economics, 110, 1127-1170.

[28] Johnson, P. (2005), “A Continuous State Space Appraoch to “Convergence by Parts”,” Economics Letters, 86, 317-321.

[29] Jones, C. I. (1997), “On the Evolution of the World Income Distribution,” Journal of Economic Perspectives, 11, 19-36.

[30] Kalaitzidakis, P., T. P. Mamuneas, A. Savvides, and T. Stengos (2001), “Measures of Human Capital and Nonlinearities in Economic Growth,” Journal of Economic Growth, 6, 229-254.

[31] Kelly, A. C. (1988), “Economic Consequences of Population Change in the Third World,” Journal of Economic Literature, 26, 1685-1728.

[32] Ketteni, E., T. P.Mamuneas, and T. Stengos (2007), “Nonlinearities in Economic Growth: A Semiparametric Approach Applied to Information Technology Data,” Journal of Macroeconomics, 29, 555-568.

[33] Li, Q. (1996), “Nonparametric Testing of Closeness between Two Unknown Distribution Functions,” Econometric Reviews, 15, 261-274.

[34] Li, Q. and J. Racine (2004), “Cross-Validated Local Linear Nonparametric Regression,” Statistica Sinica, 14, 485-512.

[35] Li, Q. and J. Racine (2007), Nonparametric Econometrics: Theory and Practice, Princeton University Press, Princeton.

[36] Liu, Z. and T. Stengos (1999), “Non-linearities in Cross-country Growth Regressions: A Semiparametric Approach,” Journal of Applied Econometrics, 14, 527-538.

[37] Maasoumi, E., J. Racine, and T. Stengos (2007), “Growth and Convergence: A Profile of Distribution Dynamics and Mobility,” Journal of Econometrics, 136, 483-508.

[38] Mamuneas, T. P., A. Savvides, and T. Stengos (2006), “Economic Development and the Return to Human Capital: A Smooth Coefficient Semiparametric Approach,” Journal of Applied Econometrics, 21, 111-132.

[39] Masanjala, W. and C. Papageorgiou (2004), “The Solow Model with CES Technology: Nonlinearities and Parameter Heterogeneity,” Journal of Applied Econometrics, 19, 171-201.

[40] Mammen, E., J. S. Marron, and N. I. Fisher, (1992), “Some Asymptotics for Multimodality Tests Based on Kernel Density Estimates,” Probability Theory and Related Fields, 91, 115-132.

[41] Mankiw, N. G., D. Romer, and D. Weil (1992), “A Contribution to the Empirics of Economic Growth,” Quarterly Journal of Economics, 107, 407-37.

[42] Paap, R. and H. K. van Dijk (1998), “Distribution and Mobility of Wealth of Nations,” European Economic Review, 42, 1269-1293.

[43] Pagan, A. and A. Ullah (1999), Nonparametric Econometrics, Cambridge University Press, Cambridge.

[44] Pittau, M. G. (2005), “Fitting Regional Income Distributions in the European Union,” Oxford Bulletin of Economics and Statistics, 67, 135-161.

[45] Pittau, M. G. and R. Zelli (2006), “Empirical Evidence of Income Dynamics Across EU Regions,” Journal of Applied Econometrics, 21, 605-628.

[46] Quah, D. T. (1993a), “Empirical Cross-Section Dynamics in Economic Growth,” European Economic Review, 37, 426-434.

[47] Quah, D. T. (1993b), “Galton’s Fallacy and Tests of the Convergence Hypothesis,” Scandinavian Journal of Economics, 95, 427-443.

[48] Quah, D. T. (1996a), “Twin Peaks: Growth Convergence in Models of Distribution Dynamics,” Economic Journal, 106, 1045-1055.

[49] Quah, D. T. (1996b), “Empirics for Economic Growth and Convergence,” European Economic Review, 40, 1353-1375.

[50] Quah, D. T. (1997), “Empirics for Growth and Distribution: Stratification, Polarization, and Convergence Clubs,” Journal of Economic Growth, 2, 27-59.

[51] Racine, J. and Q. Li (2004), “Nonparametric Estimation of Regression Functions with Both Categorical and Continuous Data,” Journal of Econometrics, 119, 99- 130.

[52] Sala-i-Martin, X. (1996), “The Classical Approach to Convergence Analysis,” Economic Journal, 106, 1019-1036.

[53] Sala-i-Martin, X. (2006), “The World Distribution of Income: Falling Poverty and ... Convergence, Period,” Quarterly Journal of Economics, 121, 351-397.

[54] Silverman, B. W. (1981), “Using Kernel Density Estimates to Investigate Multimodality,” Journal of the Royal Statistical Society, Series B, 43, 97-99.

[55] Silverman, B. W. (1986), Density Estimation for Statistics and Data Analysis, Chapman and Hall, London.

[56] Summers, R. and A. Heston (1988), “A New Set of International Comparisons of Real Product and Prices: Estimates for 130 Countries,” Review of Income and Wealth, 34, 1-26.

[57] Temple, J. (1999), “The New Growth Evidence,” Journal of Economic Literature, 37, 112-156.

[58] Vavona, A. and S. Schiavo (2007), “Nonparametric and Semiparametric Evidence on the Long-Run Effects of Inflation on Growth,” Economics Letters, 94, 452- 458.

[59] Wang, M. C. and J. Van Ryzin (1981), “A Class of Smooth Estimators for Discrete Estimation,” Biometrika, 68, 301-309.

[60] York, M. G. (1998), “Some Problems in Testing for Modality,” PhD Thesis, Australian National University.