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  • Weilin Xiao (a1) and Jun Yu (a2)

This article develops an asymptotic theory for estimators of two parameters in the drift function in the fractional Vasicek model when a continuous record of observations is available. The fractional Vasicek model with long-range dependence is assumed to be driven by a fractional Brownian motion with the Hurst parameter greater than or equal to one half. It is shown that, when the Hurst parameter is known, the asymptotic theory for the persistence parameter depends critically on its sign, corresponding asymptotically to the stationary case, the explosive case, and the null recurrent case. In all three cases, the least squares method is considered, and strong consistency and the asymptotic distribution are obtained. When the persistence parameter is positive, the estimation method of Hu and Nualart (2010) is also considered.

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*Address correspondence to Jun Yu, School of Economics and Lee Kong Chian School of Business, Singapore Management University, 90 Stamford Road, Singapore 178903, Singapore; e-mail:
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We gratefully thank the editor, a co-editor, and two anonymous referees for constructive comments. All errors are our own. Xiao’s research is supported by the Humanities and Social Sciences of Ministry of Education Planning Fund of China (No. 17YJA630114). Yu’s research was supported by the Singapore Ministry of Education (MOE) Academic Research Fund Tier 3 grant MOE2013-T3-1-009.

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Anderson, T.W. (1959) On asymptotic distributions of estimates of parameters of stochastic difference equations. Annals of Mathematical Statistics 30, 676687.
Baillie, R.T. (1996) Long memory processes and fractional integration in econometrics. Journal of Econometrics 73, 559.
Baillie, R.T., Bollerslev, T., & Mikkelsen, H.O. (1996) Fractionally integrated generalized autoregressive conditional heteroskedasticity. Journal of Econometrics 74, 330.
Bayer, C., Friz, P., & Gatheral, J. (2016) Pricing under rough volatility. Quantitative Finance 16, 887904.
Belfadli, R., Es-Sebaiy, K., & Ouknine, Y. (2011) Parameter estimation for fractional Ornstein–Uhlenbeck processes: Non-ergodic case. Frontiers in Science and Engineering (An International Journal Edited by Hassan II Academy of Science and Technology) 1, 116.
Biagini, F., Hu, Y., Øksendal, B., & Zhang, T. (2008) Stochastic Calculus for Fractional Brownian Motion and Applications. Springer.
Breton, J.C. & Nourdin, I. (2008) Error bounds on the non-normal approximation of Hermite power variations of fractional Brownian motion. Electronic Communications in Probability 13, 482493.
Cheung, Y.W. (1993) Long memory in foreign-exchange rates. Journal of Business and Economic Statistics 11, 93101.
Chronopoulou, A. & Viens, F.G. (2012a) Estimation and pricing under long-memory stochastic volatility. Annals of Finance 8, 379403.
Chronopoulou, A. & Viens, F.G. (2012b) Stochastic volatility and option pricing with long-memory in discrete and continuous time. Quantitative Finance 12, 635649.
Comte, F., Coutin, L., & Renault, E. (2012) Affine fractional stochastic volatility models. Annals of Finance 8, 337378.
Comte, F. & Renault, E. (1996) Long memory continuous time models. Journal of Econometrics 73, 101149.
Comte, F. & Renault, E. (1998) Long memory in continuous-time stochastic volatility models. Mathematical Finance 8, 291323.
Corlay, S., Lebovits, J., & Véhel, J.L. (2014) Multifractional stochastic volatility models. Mathematical Finance 24, 364402.
Davidson, J. & De Jong, R.M. (2000) The functional central limit theorem and weak convergence to stochastic integrals II: Fractionally integrated processes. Econometric Theory 16, 643666.
Davydov, Y.A. (1970) The invariance principle for stationary processes. Theory of Probability and its Applications 15, 487498.
Duncan, T., Hu, Y., & Pasik-Duncan, B. (2000) Stochastic calculus for fractional Brownian motion I: Theory. SIAM Journal on Control and Optimization 38, 582612.
El Machkouri, M., Es-Sebaiy, K., & Ouknine, Y. (2016) Least squares estimator for non-ergodic Ornstein–Uhlenbeck processes driven by Gaussian processes. Journal of the Korean Statistical Society 45, 329341.
Feigin, P.D. (1976) Maximum likelihood estimation for continuous-time stochastic processes. Advances in Applied Probability 8, 712736.
Geweke, J. & Porter-Hudak, S. (1983) The estimation and application of long memory time series models. Journal of Time Series Analysis 4, 221238.
Gradinaru, M. & Nourdin, I. (2006) Approximation at first and second order of m-order integrals of the fractional Brownian motion and of certain semimartingales. Electronic Journal of Probability 8, 126.
Granger, C.W. & Hyung, N. (2004) Occasional structural breaks and long memory with an application to the S&P 500 absolute stock returns. Journal of Empirical Finance 11, 399421.
Hu, Y. & Nualart, D. (2010) Parameter estimation for fractional Ornstein–Uhlenbeck processes (with the online supplement for Section 5). Statistics and Probability Letters 80, 10301038.
Hu, Y., Nualart, D., & Zhou, H. (2018) Parameter estimation for fractional Ornstein–Uhlenbeck processes of general Hurst parameter. Statistical Inference for Stochastic Processes, forthcoming.
Isserlis, L. (1918) On a formula for the product-moment coefficient of any order of a normal frequency distribution in any number of variables. Biometrika 12, 134139.
Kleptsyna, M. & Le Breton, A. (2002) Statistical analysis of the fractional Ornstein–Uhlenbeck type process. Statistical Inference for Stochastic Processes 5, 229248.
Kloeden, P. & Neuenkirch, A. (2007) The pathwise convergence of approximation schemes for stochastic differential equations. LMS Journal of Computation and Mathematics 10, 235253.
Lo, A.W. (1991) Long-term memory in stock market prices. Econometrica 59, 12791313.
Mandelbrot, B.B. & Van Ness, J.W. (1968) Fractional Brownian motions, fractional noises and applications. Society for Industrial and Applied Mathematics Review 10, 422437.
Mishura, Y. (2008) Stochastic Calculus for Fractional Brownian Motion and Related Processes. Springer.
Nualart, D. (2006) The Malliavin Calculus and Related Topics, 2nd ed. Springer.
Phillips, P.C.B. (1987) Time series regression with a unit root. Econometrica 55, 277301.
Phillips, P.C.B. & Magdalinos, T. (2007) Limit theory for moderate deviations from a unit root. Journal of Econometrics 136, 115130.
Phillips, P.C.B. & Perron, P. (1988) Testing for a unit root in time series regression. Biometrika 75, 335346.
Prakasa Rao, B.L.S. (2010) Statistical Inference for Fractional Diffusion Processes. Wiley, Chichester.
Robinson, P. (1995a) Log-periodogram regression of time series with long-range dependence. Annals of Statistics 23, 10481072.
Robinson, P. (1995b) Gaussian semiparametric estimation of long-range dependence. Annals of Statistics 23, 16301661.
Shimotsu, K. & Phillips, P.C.B. (2005) Exact local Whittle estimation for fractional integration. Annals of Statistics 33, 18901933.
Sowell, F. (1990) The fractional unit root distribution. Econometrica 58, 495505.
Tanaka, K. (2013) Distributions of the maximum likelihood and minimum contrast estimators associated with the fractional Ornstein–Uhlenbeck process. Statistical Inference for Stochastic Processes 16, 173192.
Tanaka, K. (2014) Distributions of quadratic functionals of the fractional Brownian Motion based on a martingale approximation. Econometric Theory 30, 10781109.
Tanaka, K. (2015) Maximum likelihood estimation for the non-ergodic fractional Ornstein–Uhlenbeck process. Statistical Inference for Stochastic Processes 18, 315332.
Taqqu, M.S. (1977) Law of the iterated logarithm for sums of non-linear functions of Gaussian variables that exhibit a long range dependence. Probability Theory and Related Fields 40, 203238.
Tudor, C. & Viens, F. (2007) Statistical aspects of the fractional stochastic calculus. Annals of Statistics 35, 11831212.
Vasicek, O. (1977) An equilibrium characterization of the term structure. Journal of Financial Economics 5, 177188.
Wang, X. & Yu, J. (2015) Limit theory for an explosive autoregressive process. Economics Letters 126, 176180.
Wang, X. & Yu, J. (2016) Double asymptotics for explosive continuous time models. Journal of Econometrics 193, 3553.
White, J.S. (1958) The limiting distribution of the serial correlation coefficient in the explosive case. Annals of Mathematical Statistics 29, 11881197.
Young, L.C. (1936) An inequality of the Hölder type, connected with Stieltjes integration. Acta Mathematica 67, 251282.
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Econometric Theory
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