A Pure-Jump Transaction-Level Price Model Yielding Cointegration, Leverage, and Nonsynchronous Trading Effects

Hurvich, Clifford and Wang, Yi (2009): A Pure-Jump Transaction-Level Price Model Yielding Cointegration, Leverage, and Nonsynchronous Trading Effects. Forthcoming in: Journal of Business and Economic Statistics

Abstract

We propose a new transaction-level bivariate log-price model, which yields fractional or standard cointegration. The model provides a link between market microstructure and lower-frequency observations. The two ingredients of our model are a Long Memory Stochastic Duration process for the waiting times between trades, and a pair of stationary noise processes which determine the jump sizes in the pure-jump log-price process. Our model includes feedback between the disturbances of the two log-price series at the transaction level, which induces standard or fractional cointegration for any fixed sampling interval. We prove that the cointegrating parameter can be consistently estimated by the ordinary least-squares estimator, and obtain a lower bound on the rate of convergence. We propose transaction-level method-of-moments estimators of the other parameters in our model and discuss the consistency of these estimators. We then use simulations to argue that suitably-modified versions of our model are able to capture a variety of additional properties and stylized facts, including leverage, and portfolio return autocorrelation due to nonsynchronous trading. The ability of the model to capture these effects stems in most cases from the fact that the model treats the (stochastic) intertrade durations in a fully endogenous way.

• Item Type: MPRA Paper
• Language: English
• Keywords: Tick Time; Long Memory Stochastic Duration; Information Share
• Subjects: C - Mathematical and Quantitative Methods > C3 - Multiple or Simultaneous Equation Models ; Multiple Variables > C32 - Time-Series Models ; Dynamic Quantile Regressions ; Dynamic Treatment Effect Models ; Diffusion Processes ; State Space Models

References

[1] Alexander, C. (1999), Optimal Hedging Using Cointegration, Philosophical Transactions of the Royal Society A 357, 2039{2058.39

[2] Alexander, C. and Dimitriu, A. (2005), Indexing, Cointegration and Equity Market Regimes, Inter- natiional Journal of Finance and Economics 10, 213{231.

[3] Amihud, Y. and H. Mendelson (1980), Dealership Markets: Market-Making with Inventory, Journal of Financial Economics 8, 31{53.

[4] Amihud, Y. and H. Mendelson (1982), Asset Price Behavior in a Dealershipt Market, Financial Analysts Journal, May/June.

[5] Amihud, Y. and H. Mendelson (1987), Trading Mechanisms and Stock Returns: An Empirical Inves- tigation, Journal of Finance XLII, 533{553.

[6] Andersen, T., Bollerslev, T., Frederiksen, P. and Nielsen M. (2006), Continuous-time Models, Realized Volatilities, and Testable Distributional Implications for Daily Stock Returns. Working Paper.

[7] Atchison, M., K. Butler and R. Simonds (1987), Nonsynchronous Security Trading and Market Index Autocorrelation, Journal of Finance 42, 111{118.

[8] Bertelli, S. and M. Caporin (2002), A Note on Calculating Autocovariances of Long-memory Processes, Journal of Time Series Analysis 23, 503{508.

[9] Bollerslev. T. (1986), Generalized Autoregressive Conditional Heteroskedasticity, Journal of Econo- metrics 31, 307{327.

[10] Boudoukh, J., M. Richardson and R. Whitelaw (1994), A Tale of Three Schools: Insights on Aut- correlations of Short-Horizon Returns, Review of Financial Studies 7, 539{573.

[11] Bowsher, C.G. (2007), Modelling Security Market Events in Continuous Time: Intensity Based, Multivariate Point Process Models, Journal of Econometrics 141, 876{912.

[12] Breidt, F., Crato, N. and de Lima, P. (1998), The Detection and Estimation of Long Memory in Stochastic Volatility, Journal of Econometrics 83, 325{348.

[13] Chen, W. and Hurvich, C. (2003a), Estimating Fractional Cointegration in the Presence of Polyno- mial Trends, Journal of Econometrics 117, 95{121. 40

[14] Chen, W. and Hurvich, C. (2003b), Semiparametric Estimation of Multivariate Fractional Cointe- gration, Journal of the American Statistical Association 98, 629{642.

[15] Chen, W. and Hurvich, C. (2006), Semiparametric Estimation of Fractional Cointegrating Subspaces, Annals of Statistics 34, 2939{2979.

[16] Clark, P.K. (1973), A subordinated stochastic process model with ¯nite variance for speculative prices, Econometrica 41, 135{155.

[17] Comte, F. (1998), Discrete and continuous time cointegration - Le cas multidimensionnel, Journal of Econometrics 88, 207{226.

[18] Comte, F. and Renault, E. (1996), Long Memory Continuous Time Models, Journal of Economet- rics 73, 101{149.

[19] Comte, F. and Renault, E. (1998), Long Memory in Continuous-time Stochastic Volatility Models, Mathematical Finance 8, 291{323.

[20] Daley, D.J. and Vere-Jones, D. (2003), An Introduction to the Theory of Point Processes, 2'nd Ed. New York: Springer.

[21] Deo, R., Hsieh, M. and Hurvich, C. (2006), The Persistence of Memory: From Durations to Realized Volatility. Working Paper, Stern School of Business, New York University.

[22] Deo, R., Hurvich, C., Soulier, P. and Wang, Y. (2007), Conditions for the Propagation of Memory Parameter from Durations to Counts and Realized Volatility, to appear in Econometric Theory.

[23] Duan, J.C. and Pliska, S. (2004), Option Valuation with Cointegrated Asset Prices, Journal of Economic Dynamics and Control 28, 727{754.

[24] Dueker, M. and Startz, R. (1998), Maximum-likelihood estimation of fractional cointegration with an application to U.S. and Canadian bond rates, The Review of Economics and Statistics 80, 420{426.

[25] Dunis, C.L. and Ho, R. (2005), Cointegration Portfolios of European Equities for Index Tracking and Market Neutral Strateries, Journal of Asset Management 6, 33-52. 41

[26] Engle, R. and Granger C.W.J. (1987), Co-integration and Error Correction: Representation, Esti- mation and Testing, Econometrica 66, 1127{1162.

[27] Engle, R. and Russell J. (1998), Autoregressive Conditional Duration: a New Model for Irregularly Spaced Transaction Data, Econometrica 66, 1127{1162.

[28] Fisher, L. (1966), Some New Stock Market Indices, Journal of Business 39, 191{225.

[29] Frijns, B. and Schotman, P. (2006), Price Discovery in Tick Time, Working Paper.

[30] Geweke, J. and Porter-Hudak, S. (1983), The Estimation and Application of Long Memory Time Series Models, Journal of Time Series Analysis 4, 221{237.

[31] Glosten, L.R. (1987), Components of the Bid-Ask Spread and the Statistical Properties of Transac- tion Prices, Journal of Finance XLII, 1293{1307.

[32] Harvey, A.C. (1998), Long Memory in Stochastic Volatility, in: J. Knight, S. Satchell (Eds.), Fore- casting Volatility in Financial Markets, Butterworth-Heineman, Oxford, 307{320.

[33] Hasbrouck, J. (1995), One Security, Many Markets: Determining the Contributions to Price Discov- ery, Journal of Finance 50, 1175{1199.

[34] Hausman, J. (1978), Speci¯cation tests in econometrics, Econometrica 46, 1251{1271.

[35] Iglehart, D.L. and Whitt, W. (1971) The Equivalence of Functional Central Limit Theorems for Counting Processes and Associated Partial Sums. Annals of Mathematical Statistics 42, 1372{1378.

[36] Isserlis, L. (1918), On a Formula for the Product-Moment Coe±cient of any Order of a Normal Frequency Distribution in any Number of Variables, Biometrika 12, 134{139.

[37] Johansen, S. (1988), Statistical Analysis of Cointegration Vectors, Journal of Economic Dynamics and Control 12, 231{254.

[38] Johansen, S. (1991), Estimation and Hypothesis Testing of Cointegration Vectors in Gaussian Vector Autoregressive Models, Econometrica 59, 1551{1580. 42

[39] Kadlec, G. and Patterson, D. (1999), A Transactions Data Analysis of Nonsynchronous Trading, Review of Financial Studies 12, 609{630.

[40] Lieberman, O. and Phillips, P.C.B. (2006), A Complete Asymptotic Series for the Autocovariance Function of a Long Memory Process. Working Paper, Yale University.

[41] Liu, J., Longsta®, F. and Pan, J. (2003), Dynamic Asset Allocation with Event Risk, Journal of Finance 58, 231{259.

[42] Lee, C. and M. Ready (1991), Inferring Trade Direction from Intraday Data, Journal of Finance 46, 733{746.

[43] Lo, A. and A.C. MacKinlay (1990a), An Econometric Analysis of Nonsynchronous Trading, Journal of Econometrics 45, 181{211.

[44] Lo, A. and A.C. MacKinlay (1990b), When are Constrarian Pro¯ts Due to Stock Market Overreac- tion?, Review of Financial Studies 3, 175{205.

[45] Marinucci, D. and Robinson, P.M. (2001), Semiparametric Fractional Cointegration Analysis. Fore- casting and empirical methods in ¯nance and macroeconomics, Journal of Econometrics 105, 225{247.

[46] Nijman, T.E., Spierdijk, L. and Soest, Arthur H.O. (2004), Modeling Co-Movements in Trading Intensities to Distinguish Stock and Sector-Speci¯c News, Working Paper, Tilburg University.

[47] Oomen, R.C.A., (2006), Properties of Realized Variance under Alternative Sampling Schemes, Jour- nal of Business and Economic Statistics 24, 219{237.

[48] Perry, P. (1985), Portfolio Serial Correlation and Nonsynchronous Trading, Journal of Financial and Quantitative Analysis 20, 517{523.

[49] Phillips, P.C.B. (1991a), Optimal Inference in Cointegrated Systems, Econometrica 59, 283{306.

[50] Phillips, P.C.B. (1991b), Error Correction and Long-run Equilibrium in Continuous Time, Econo- metrica 59, 967{980. 43

[51] Phillips, P.C.B. and Durlauf, S. (1986), Multiple Time Series Regression with Integrated Processes, The Review of Economic Studies 53, 473{495.

[52] Press, S.J. (1967), A Compound Events Model for Security Prices, Journal of Business 40, 317{335.

[53] Prigent, J-L. (2005), Option Pricing with a General Marked Point Process, Mathematics of Opera- tions Research 26, 50{66.

[54] Robinson, P.M. (1995), Gaussian Semiparametric Estimation of Long Range Dependence, Annals of Statistics 23, 1630{1661.

[55] Robinson, P.M. and Haulde, J. (2003), Cointegration in Fractional Systems with Unknown Integra- tion Orders, Econometrica 71, 1727{1766.

[56] Robinson, P.M. and Marinucci, D. (2001), Narrow-band analysis of nonstationary processes, Annals of Statistics 29, 947{986.

[57] Robinson, P.M. and Yajima, Y. (2002), Determination of cointegrating rank in fractional systems, Journal of Econometrics 106, 217{241.

[58] Scholes M. and J. Williams (1977), Estimating Betas from Nonsynchronous Data, Journal of Finan- cial Economics 5, 309{327.

[59] Shanken J. (1987), Nonsynchronous Data and the Covariance-Factor Structure of Returns, Journal of Finance 42, 221{232.

[60] Stock, J. (1987), Asymptotic Properties of Least Squares Estimators of Cointegrating Vectors, Econo- metrica 55, 1035{1056.

[61] Stock, J. and Watson M. (1988), Testing for Common Trends, Journal of American Statistical Association 83, 1097{1107.

[62] Velasco, C. (2003), Gaussian Semiparametric Estimation of Fractional Cointegration, Journal of Time Series Analysis 24, 345{378. 44

[63] Velasco, C. and Marmol, F. (2004), Consistent Testing of Cointegration Relationships, Economet- rica 72, 1809{1844.