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Published online by Cambridge University Press:  07 June 2011


The Markov property is a fundamental property in time series analysis and is often assumed in economic and financial modeling. We develop a new test for the Markov property using the conditional characteristic function embedded in a frequency domain approach, which checks the implication of the Markov property in every conditional moment (if it exists) and over many lags. The proposed test is applicable to both univariate and multivariate time series with discrete or continuous distributions. Simulation studies show that with the use of a smoothed nonparametric transition density-based bootstrap procedure, the proposed test has reasonable sizes and all-around power against several popular non-Markov alternatives in finite samples. We apply the test to a number of financial time series and find some evidence against the Markov property.

Research Article
Copyright © Cambridge University Press 2011

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We thank Pentti Saikkonen (the co-editor), three referees, Frank Diebold, Oliver Linton, James MacKinnon, Katsumi Shimotsu, Kyungchul Song, Liangjun Su, George Tauchen, and seminar participants at Peking University, Queen’s University, University of Pennsylvania, the 2008 Xiamen University-Humboldt University Joint Workshop, the 2008 International Symposium on Recent Developments of Time Series Econometrics in Xiamen, the 2008 Symposium on Econometric Theory and Applications (SETA) in Seoul, and the 2008 Far Eastern Econometric Society Meeting in Singapore for their constructive comments on the previous versions of this paper. Any remaining errors are solely ours. Bin Chen thanks the Department of Economics, University of Rochester, for financial support. Yongmiao Hong thanks the outstanding overseas youth fund of the National Science Foundation of China for its support.


Aaronson, J., Gilat, D., & Keane, M. (1992) On the structure of 1-dependent Markov chains. Journal of Theoretical Probability 5, 545–561.CrossRefGoogle Scholar
Ahn, D., Dittmar, R., & Gallant, A.R. (2002) Quadratic term structure models: Theory and evidence. Review of Financial Studies 15, 243–288.CrossRefGoogle Scholar
Ahn, D. & Gao, B. (1999) A parametric nonlinear model of term structure dynamics. Review of Financial Studies 12, 721–762.CrossRefGoogle Scholar
Ait-Sahalia, Y. (1996) Testing continuous-time models of the spot interest rate. Review of Financial Studies 9, 385–426.CrossRefGoogle Scholar
Ait-Sahalia, Y. (1997) Do Interest Rates Really Follow Continuous-Time Markov Diffusions? Working paper, Princeton University.Google Scholar
Ait-Sahalia, Y., Fan, J., and Peng, H. (2009) Nonparametric transition-based tests for diffusions. Journal the of American Statistical Association 104, 1102–1116.CrossRefGoogle Scholar
Amaro de Matos, J. & Fernandes, M. (2007) Testing the Markov property with high frequency data. Journal of Econometrics 141, 44–64.CrossRefGoogle Scholar
Amaro de Matos, J. & Rosario, J. (2000) The Equilibrium Dynamics for an Endogenous Bid-Ask Spread in Competitive Financial Markets. Working paper, European University Institute and Universidade Nova de Lisboa.CrossRefGoogle Scholar
Anderson, T. & Lund, J. (1997) Estimating continuous time stochastic volatility models of the short term interest rate. Journal of Econometrics 77, 343–377.CrossRefGoogle Scholar
Aviv, Y. & Pazgal, A. (2005) A partially observed Markov decision process for dynamic pricing. Management Science 51, 1400–1416.CrossRefGoogle Scholar
Bangia, A., Diebold, F., Kronimus, A., Schagen, C., & Schuermann, T. (2002) Ratings migration and the business cycle, with application to credit portfolio stress testing. Journal of Banking and Finance 26, 445–474.CrossRefGoogle Scholar
Bierens, H. (1982) Consistent model specification tests. Journal of Econometrics 20, 105–134.CrossRefGoogle Scholar
Blume, L., Easley, D., & O’Hara, M. (1994) Market statistics and technical analysis: The role of volume. Journal of Finance 49, 153–181.CrossRefGoogle Scholar
Brown, B.M. (1971) Martingale limit theorems. Annals of Mathematical Statistics 42, 59–66.CrossRefGoogle Scholar
Chacko, G., & Viceira, L. (2003) Spectral GMM estimation of continuous-time processes. Journal of Econometrics 116, 259–292.CrossRefGoogle Scholar
Chan, K.C., Karolyi, G.A., Longstaff, F.A., & Sanders, A.B. (1992) An empirical comparison of alternative models of the short-term interest rate. Journal of Finance 47, 1209–1227.CrossRefGoogle Scholar
Chen, B. & Hong, Y. (2009) Diagnosing Multivariate Continuous-Time Models with Application to Affine Term Structure Models. Working paper, Cornell University and University of Rochester.Google Scholar
Cleveland, W.S. (1979) Robust locally weighted regression and smoothing scatterplots. Journal of the American Statistical Association 74, 829–836.CrossRefGoogle Scholar
Cox, J.C., Ingersoll, J.E., & Ross, S.A. (1985) A theory of the term structure of interest rates. Econometrica 53, 385–407.CrossRefGoogle Scholar
Dai, Q., & Singleton, K. (2000) Specification analysis of affine term structure models. Journal of Finance 55, 1943–1978.CrossRefGoogle Scholar
Darsow, W.F., Nguyen, B., & Olsen, E.T. (1992) Copulas and Markov processes. Illinois Journal of Mathematics 36, 600–642.CrossRefGoogle Scholar
Davies, R.B. (1977) Hypothesis testing when a nuisance parameter is present only under the alternative. Biometrika 64, 247–254.CrossRefGoogle Scholar
Davies, R.B. (1987) Hypothesis testing when a nuisance parameter is present only under the alternative. Biometrika 74, 33–43.Google Scholar
Duan, J.C. & Jacobs, K. (2008) Is long memory necessary? An empirical investigation of nonnegative interest rate processes. Journal of Empirical Finance 15, 567–581.CrossRefGoogle Scholar
Duffie, D. & Kan, R. (1996) A yield-factor model of interest rates. Mathematical Finance 6, 379–406.CrossRefGoogle Scholar
Duffie, D., Pan, J., & Singleton, K. (2000) Transform analysis and asset pricing for affine jump-diffusions. Econometrica 68, 1343–1376.CrossRefGoogle Scholar
Easley, D. & O’Hara, M. (1987) Price, trade size, and information in securities markets. Journal of Financial Economics 19, 69–90.CrossRefGoogle Scholar
Easley, D. & O’Hara, M. (1992) Time and the process of security price adjustment. Journal of Finance 47, 577–605.CrossRefGoogle Scholar
Edwards, R. & Magee, J. (1966) Technical Analysis of Stock Trends. John Magee.Google Scholar
Epps, T.W. & Pulley, L.B. (1983) A test for normality based on the empirical characteristic function. Biometrika 70, 723–726.CrossRefGoogle Scholar
Ericson, R. & Pakes, A. (1995) Markov-perfect industry dynamics: A framework for empirical work. Review of Economic Studies 62(1), 53–82.CrossRefGoogle Scholar
Fan, J. (1992) Design-adaptive nonparametric regression. Journal of the American Statistical Association 87, 998–1004.CrossRefGoogle Scholar
Fan, J. (1993) Local linear regression smoothers and their minimax efficiency. Annals of Statistics 21, 196–216.CrossRefGoogle Scholar
Fan, J. & Yao, Q. (2003) Nonlinear Time Series: Nonparametric and Parametric Methods. Springer Verlag.CrossRefGoogle Scholar
Fan, Y. & Li, Q. (1999) Root-N-consistent estimation of partially linear time series models. Journal of Nonparametric Statistics 11, 251–269.CrossRefGoogle Scholar
Fan, Y., Li, Q., & Min, I. (2006) A nonparametric bootstrap test of conditional distributions. Econometric Theory 22, 587–613.CrossRefGoogle Scholar
Feller, W. (1959) Non-Markovian processes with the semi-group property. Annals of Mathematical Statistics 30, 1252–1253.CrossRefGoogle Scholar
Feuerverger, A. & McDunnough, P. (1981) On the efficiency of empirical characteristic function procedures. Journal of the Royal Statistical Society, Series B 43, 20–27.Google Scholar
Gallant, A.R., Hsieh, D., & Tauchen, G. (1997) Estimation of stochastic volatility models with diagnostics. Journal of Econometrics 81, 159–192.CrossRefGoogle Scholar
Gao, J. & Hong, Y. (2008) Central limit theorems for generalized U-statistics with applications in nonparametric specification. Journal of Nonparametric Statistics 20, 61–76.CrossRefGoogle Scholar
Hall, R. (1978) Stochastic implications of the life cycle permanent income hypothesis: Theory and practice. Journal of Political Economy 86, 971–987.CrossRefGoogle Scholar
Hamilton, J.D. (1989) A new approach to the economic analysis of nonstationary time series and the business cycle. Econometrica 57, 357–384.CrossRefGoogle Scholar
Hamilton, J.D. & Susmel, R. (1994) Autoregressive conditional heteroskedasticity and changes in regime. Journal of Econometrics 64, 307–333.CrossRefGoogle Scholar
Hansen, B.E. (1996) Inference when a nuisance parameter is not identified under the null hypothesis. Econometrica 64, 413–430.CrossRefGoogle Scholar
Hansen, B.E. (2008) Uniform convergence rates for kernel estimation with dependent data. Econometric Theory 24, 726–748.CrossRefGoogle Scholar
Heath, D., Jarrow, R., & Morton, A. (1992) Bond pricing and the term structure of interest rates: A new methodology for contingent claims valuation. Econometrica 60, 77–105.CrossRefGoogle Scholar
Hong, Y. (1999) Hypothesis testing in time series via the empirical characteristic function: A generalized spectral density approach. Journal of the American Statistical Association 94, 1201–1220.CrossRefGoogle Scholar
Hong, Y. & Li, H. (2005) Nonparametric specification testing for continuous-time models with applications to term structure of interest rates. Review of Financial Studies 18, 37–84.CrossRefGoogle Scholar
Hong, Y. & White, H. (2005) Asymptotic distribution theory for an entropy-based measure of serial dependence. Econometrica 73, 837–902.CrossRefGoogle Scholar
Horowitz, J.L. (2003) Bootstrap methods for Markov processes. Econometrica 71, 1049–1082.CrossRefGoogle Scholar
Ibragimov, R. (2007) Copula-Based Characterizations for Higher-Order Markov Processes. Working paper, Harvard University.Google Scholar
Inoue, A. (2001) Testing for distributional change in time series. Econometric Theory 17, 156–187.CrossRefGoogle Scholar
Jarrow, R., Lando, D., & Turnbull, S. (1997) A Markov model for the term structure of credit risk spreads. Review of Financial Studies 10, 481–523.CrossRefGoogle Scholar
Jarrow, R. & Turnbull, S. (1995) Pricing derivatives on financial securities subject to credit risk. Journal of Finance 50, 53–86.CrossRefGoogle Scholar
Jiang, G. & Knight, J. (1997) A nonparametric approach to the estimation of diffusion processes with an application to a short-term interest rate model. Econometric Theory 13, 615–645.CrossRefGoogle Scholar
Kavvathas, D. (2001) Estimating Credit Rating Transition Probabilities for Corporate Bonds. Working paper, University of Chicago.CrossRefGoogle Scholar
Kiefer, N.M. & Larson, C.E. (2004) Testing Simple Markov Structures for Credit Rating Transitions. Working paper, Cornell University.Google Scholar
Kim, W. & Linton, O. (2003) A Local Instrumental Variable Estimation Method for Generalized Additive Volatility Models. Working paper, Humboldt University of Berlin, London School of Economics.Google Scholar
Kydland, F.E. & Prescott, E. (1982) Time to build and aggregate fluctuations. Econometrica 50, 1345–70.CrossRefGoogle Scholar
Lando, D. &Skødeberg, T. (2002) Analyzing rating transitions and rating drift with continuous observations. Journal of Banking & Finance 26, 423–444.CrossRefGoogle Scholar
LeBaron, B. (1999) Technical trading rule profitability and foreign exchange intervention. Journal of International Economics 49, 125–143.CrossRefGoogle Scholar
Lee, A.J. (1990) U-Statistics: Theory and Practice. Marcel Dekker.Google Scholar
Lévy, P. (1949) Exemple de processus pseudo-markoviens. Comptes Rendus de l’Académie des Sciences 228, 2004–2006.Google Scholar
Li, Q. & Racine, J.S. (2007) Nonparametric Econometrics: Theory and Practice. Princeton University Press.Google Scholar
Linton, O. & Gozalo, P. (1997) Conditional Independence Restrictions: Testing and Estimation. Working paper, Cowles Foundation for Research in Economics, Yale University.Google Scholar
Ljungqvist, L. & Sargent, T.J. (2000) Recursive Macroeconomic Theory. MIT Press.Google Scholar
Lobato, I.N. & Robinson, P.M. (1998) A nonparametric test for I(0). Review of Economic Studies 65, 475–495.CrossRefGoogle Scholar
Loretan, M. & Phillips, P.C.B. (1994) Testing the covariance stationarity of heavy-tailed time series: An overview of the theory with applications to several financial datasets. Journal of Empirical Finance 1, 211–248.CrossRefGoogle Scholar
Lucas, R. (1978) Asset prices in an exchange economy. Econometrica 46, 1429–45.CrossRefGoogle Scholar
Lucas, R. (1988) On the mechanics of economic development. Journal of Monetary Economics 22, 3–42.CrossRefGoogle Scholar
Lucas, R. & Prescott, E. (1971) Investment under uncertainty. Econometrica 39, 659–81.CrossRefGoogle Scholar
Lucas, R. & Stokey, N.L. (1983) Optimal fiscal and monetary policy in an economy without capital. Journal of Monetary Economics 12, 55–94.CrossRefGoogle Scholar
Masry, E. (1996a) Multivariate local polynomial regression for time series: Uniform strong consistency and rates. Journal of Time Series Analysis 6, 571–599.CrossRefGoogle Scholar
Masry, E. (1996b) Multivariate regression estimation local polynomial fitting for time series. Stochastic Processes and Their Applications 65, 81–101.CrossRefGoogle Scholar
Masry, E. & Fan, J. (1997) Local polynomial estimation of regression functions for mixing processes. Scandinavian Journal of Statistics 24, 165–179.CrossRefGoogle Scholar
Masry, E. & Tjøstheim, D. (1997) Additive nonlinear ARX time series and projection estimates. Econometric Theory 13, 214–252.CrossRefGoogle Scholar
Matús, F. (1996) On two-block-factor sequences and one-dependence. Proceedings of the American Mathematical Society 124, 1237–1242.CrossRefGoogle Scholar
Matús, F. (1998) Combining m-dependence with Markovness. Annales de l’Institut Henri Poincaré. Probabilités et Statistiques 34, 407–423.CrossRefGoogle Scholar
Mehra, R. & Prescott, E. (1985) The equity premium: A puzzle. Journal of Monetary Economics 15, 145–61.CrossRefGoogle Scholar
Mizutani, E. & Dreyfus, S. (2004) Two stochastic dynamic programming problems by model-free actor-critic recurrent network learning in non-Markovian settings. Proceedings of the IEEE-INNS International Joint Conference on Neural Networks.Google Scholar
Pagan, A.R. & Schwert, G.W. (1990) Testing for covariance stationarity in stock market data. Economics Letters 33, 165–70.CrossRefGoogle Scholar
Paparoditis, E. & Politis, D.N. (2000) The local bootstrap for kernel estimators under general dependence conditions. Annals of the Institute of Statistical Mathematics 52, 139–159.CrossRefGoogle Scholar
Paparoditis, E. & Politis, D.N. (2002) The local bootstrap for Markov processes. Journal of Statistical Planning and Inference 108, 301–328.CrossRefGoogle Scholar
Platen, E. & Rebolledo, R. (1996) Principles for modelling financial markets. Journal of Applied Probability 31, 601–613.CrossRefGoogle Scholar
Romer, P. (1986) Increasing returns and long-run growth. Journal of Political Economy 5, 1002–1037.CrossRefGoogle Scholar
Romer, P. (1990) Endogenous technological change. Journal of Political Economy 5, 71–102.CrossRefGoogle Scholar
Rosenblatt, M. (1960) An aggregation problem for Markov chains. In Machol, R.E. (ed.), Information and Decision Processes, pp. 87–92. McGraw-Hill.Google Scholar
Rosenblatt, M. & Slepian, D. (1962) N th order Markov chains with every N variables independent. Journal of the Society for Industrial and Applied Mathematics 10, 537–549.CrossRefGoogle Scholar
Ruppert, D. & Wand, M.P. (1994) Multivariate weighted least squares regression. Annals of Statistics 22, 1346–1370.CrossRefGoogle Scholar
Rust, J. (1994) Structural estimation of Markov decision processes. Handbook of Econometrics 4, 3081–3143.CrossRefGoogle Scholar
Sargent, T. (1987) Dynamic Macroeconomic Theory. Harvard University Press.Google Scholar
Singleton, K. (2001) Estimation of affine asset pricing models using the empirical characteristic function. Journal of Econometrics 102, 111–141.CrossRefGoogle Scholar
Skaug, H.J. & Tjøstheim, D. (1993) Nonparametric tests of serial independence. In Subba Rao, T. (ed.), Developments in Time Series Analysis: The Priestley Birthday Volume, pp. 207–229. Chapman & Hall.CrossRefGoogle Scholar
Skaug, H.J. & Tjøstheim, D. (1996) Measures of distance between densities with application to testing for serial independence. In Robinson, P. & Rosenblatt, M. (eds.), Time Series Analysis in Memory of E. J. Hannan, pp. 363–377. Springer.Google Scholar
Stinchcombe, M.B. & White, H. (1998) Consistent specification testing with nuisance parameters present only under the alternative. Econometric Theory 14, 295–325.CrossRefGoogle Scholar
Stone, C.J. (1977) Consistent nonparametric regression. Annals of Statistics 5, 595–645.CrossRefGoogle Scholar
Su, L. & White, H. (2007) A consistent characteristic-function-based test for conditional independence. Journal of Econometrics 141, 807–834.CrossRefGoogle Scholar
Su, L. & White, H. (2008) Nonparametric Hellinger metric test for conditional independence. Econometric Theory 24, 829–864.CrossRefGoogle Scholar
Uzawa, H. (1965) Optimum technical change in an aggregative model of economic growth. International Economic Review 6, 18–31.CrossRefGoogle Scholar
Vasicek, O. (1977) An equilibrium characterization of the term structure. Journal of Financial Economics 5, 177–188.CrossRefGoogle Scholar
Weintraub, G.Y., Benkard, L.C., & Van Roy, B. (2008) Markov perfect industry dynamics with many firms. Econometrica 76, 1375–1411.Google Scholar
Yoshihara, K. (1976) Limiting behavior of U-statistics for stationary, absolutely regular processes. Z. Wahrsch. Verw. Gebiete 35, 237–252.CrossRefGoogle Scholar
Zhu, X. (1992) Optimal fiscal policy in a stochastic growth model. Journal of Economic Theory 2, 250–289.CrossRefGoogle Scholar
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