Hostname: page-component-5d59c44645-dknvm Total loading time: 0 Render date: 2024-03-02T23:02:15.232Z Has data issue: false hasContentIssue false


Published online by Cambridge University Press:  17 February 2010

George Kapetanios*
Queen Mary, University of London
Andrew P. Blake
Bank of England
*Address correspondence to George Kapetanios, Department of Economics, Queen Mary, University of London, Mile End Road, London E1 4NS, UK; e-mail:


The martingale difference restriction is an outcome of many theoretical analyses in economics and finance. A large body of econometric literature deals with tests of that restriction. We provide new tests based on radial basis function (RBF) neural networks. Our work is based on the test design of Blake and Kapetanios (2000, 2003a, 2003b). However, unlike that work we provide a formal theoretical justification for the validity of these tests and present some new general theoretical results. These results take advantage of the link between the algorithms of Blake and Kapetanios (2000, 2003a, 2003b) and boosting. We carry out a Monte Carlo study of the properties of the new tests and find that they have very good power performance. A simplified implementation of boosting is found to have desirable properties and small computational cost. An empirical application to the S&P 500 constituents illustrates the usefulness of our new test.

Copyright © Cambridge University Press 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)



Beran, J. (1994) Statistics for Long-Memory Processes. Chapman and Hall.Google Scholar
Bierens, H.J. (1984) Model specification testing of time series regression. Journal of Econometrics 26, 323353.CrossRefGoogle Scholar
Bierens, H.J. (1990) A consistent conditional moment test of functional form. Econometrica 58, 14431458.CrossRefGoogle Scholar
Bierens, H.J. & Ploberger, W. (1991) Spectral-based test for the martingale hypothesis. Journal of Econometrics 50, 119.Google Scholar
Bierens, H.J. & Ploberger, W. (1997) Asymptotic theory of integrated conditional moment tests. Econometrica 65, 11291151.CrossRefGoogle Scholar
Blake, A.P. & Kapetanios, G. (2000) A radial basis function artificial neural network test for ARCH. Economics Letters 69, 1523.CrossRefGoogle Scholar
Blake, A.P. & Kapetanios, G. (2003a) Pure significance tests of the unit root hypothesis against nonlinear alternatives. Journal of Time Series Analysis 24(3), 253267.CrossRefGoogle Scholar
Blake, A.P. & Kapetanios, G. (2003b) A radial basis function artificial neural network test for neglected nonlinearity. The Econometrics Journal 6(2), 357373.CrossRefGoogle Scholar
Blake, A.P. & Kapetanios, G. (2007) Testing for ARCH in the presence of nonlinearity of unknown form in the conditional mean. Journal of Econometrics 137(2), 472488.CrossRefGoogle Scholar
Buhlmann, P. (2006) Boosting for high-dimensional linear models. Annals of Statistics 34, 559583.CrossRefGoogle Scholar
Cochrane, J.H. (2005) Asset Pricing, revised ed.Princeton University Press.Google Scholar
Davidson, J. (1994) Stochastic Limit Theory. Oxford University Press.CrossRefGoogle Scholar
de Jong, R.M. (1996) The Bierens tests under data dependence. Journal of Econometrics 72, 132.CrossRefGoogle Scholar
Deo, R.S. (2000) Spectral tests of the martingale hypothesis under conditional heteroskedasticity. Journal of Econometrics 99, 291315.CrossRefGoogle Scholar
Dominguez, M. & Lobato, I.N. (2003) A consistent test for the martingale difference hypothesis. Econometric Reviews 22, 351377.CrossRefGoogle Scholar
Escanciano, J.C. (2006) Consistent diagnostic test for regression models using projections. Econometric Theory 22(1), 10301051.CrossRefGoogle Scholar
Escanciano, J.C. & Velasco, C. (2006a) Generalised spectral tests for the martingale difference hypothesis. Journal of Econometrics 134(1), 151185.CrossRefGoogle Scholar
Escanciano, J.C. & Velasco, C. (2006b) Testing the martingale difference hypothesis using integrated regression functions. Computational Statistics and Data Analysis 51(1), 22782294.CrossRefGoogle Scholar
Freund, Y. & Schapire, R. (1996) Experiments with a new boosting algorithm. Machine Learning: Proceedings of the 13th International Conference 148156.Google Scholar
Friedman, J. (2001) Greedy function approximation: A gradient boosting machine. Annals of Statistics 29, 11891232.CrossRefGoogle Scholar
Friedman, J., Hastie, T., & Tibshirani, R. (2000) Additive logistic regression: A statistical view of boosting. Annals of Statistics 28, 337374.CrossRefGoogle Scholar
Girosi, F. & Anzelloti, G. (1993) Rates of convergence for radial basis functions and neural networks. In Mammone, R.J. (ed.), Artificial Neural Networks for Speech and Vision. Chapman and Hall.Google Scholar
Guay, A. & Guerre, E. (2006) A data-driven nonparametric specification test for dynamic regression models. Econometric Theory 22, 543586.CrossRefGoogle Scholar
Hong, Y. (1999a) Hypothesis testing in time series via the empirical characteristic function: A generalized spectral density approach. Journal of the American Statistical Association 84, 12011220.CrossRefGoogle Scholar
Hong, Y. (1999b) Testing serial independence via the empirical characteristic function. Working paper.Google Scholar
Hornik, K., Stinchcombe, M., & White, H. (1989) Multi-layer feedforward networks and universal approximators. Neural Network 2, 359366.CrossRefGoogle Scholar
Koul, H.L. & Stute, W. (1999) Nonparametric model checks for time series. Annals of Statistics 27, 204236.CrossRefGoogle Scholar
Lee, T.H., White, H., & Granger, C.W.J. (1993) Testing for neglected nonlinearity in time series models: A comparison of neural network methods and alternative tests. Journal of Econometrics 56, 269290.CrossRefGoogle Scholar
Lettau, M. & Ludvigson, S. (2001) Consumption, aggregate wealth and expected stock returns. Journal of Finance 56, 815849.CrossRefGoogle Scholar
Moody, J. & Darken, C. (1989) Fast learning in networks of locally-tuned processing units. Neural Computation 1(2), 289303.CrossRefGoogle Scholar
Orr, M.J. (1995) Regularisation in the selection of radial basis function centers. Neural Computation 7(3), 606623.CrossRefGoogle Scholar
Park, J. & Sandberg, I.W. (1991) Universal approximation using radial-basis-function networks. Neural Computation 3(4), 246257.CrossRefGoogle ScholarPubMed
Park, J.Y. & Whang, Y.J. (1999) Testing for the martingale hypothesis. Working paper.Google Scholar
Poskitt, D.S. (2005) Autoregressive Approximation in Nonstandard Situations: The Non-Invertible and Fractionally Integrated Cases. Working paper 16/05, Monash University.Google Scholar
Schapire, R. (2002) The boosting approach to machine learning: An overview. In Denison, D., Hansen, M., Holmes, C., Mallick, B., & Yu. Springer, B. (eds.), MSRI Workshop on Nonlinear Estimation and Classification. Mathematical Sciences Research Institute.Google Scholar
Stinchcombe, M.B. & White, H. (1998) Consistent specification testing with nuisance parameters present only under the alternative. Econometric Theory 14, 295325.CrossRefGoogle Scholar
Stute, W. (1997) Nonparametric model checks for regression. Annals of Statistics 25, 613641.CrossRefGoogle Scholar
Temlyakov, V.N. (2000) Weak greedy algorithms. Advances in Computational Mathematics 12, 213227.CrossRefGoogle Scholar
Teräsvirta, T., Lin, C.F., & Granger, C.W.J. (1993) Power of the neural network linearity test. Journal of Time Series Analysis 14, 209220.CrossRefGoogle Scholar
Whang, Y.J. (2000) Consistent bootstrap tests of parametric regression functions. Journal of Econometrics 98, 2746.CrossRefGoogle Scholar
White, H. (1999) Asymptotic Theory for Econometricians. Academic Press.Google Scholar
White, H. (2006) Approximate nonlinear forecasting methods. In Elliott, G., Granger, C.W.J., & Timmermann, A. (eds.), Handbook of Economics Forecasting. Elsevier.Google Scholar
White, H. & Wooldridge, J. (1991) Some results on sieve estimation with dependent observations. In Barnett, W., Powell, J., & Tauchen, G. (eds.), Nonparametric and Semiparametric Methods in Econometrics and Statistics. Cambridge University Press.Google Scholar