#### Publication Type

PhD Dissertation

#### Publication Date

2012

#### Abstract

Multivariate continuous time models are now widely used in economics and finance. Empirical applications typically rely on some process of discretization so that the system may be estimated with discrete data. The Chapter 2 introduces a framework for discretizing linear multivariate continuous time systems that includes the commonly used Euler and trapezoidal approximations as special cases and leads to a general class of estimators for the mean reversion matrix. Asymptotic distributions and bias formulae are obtained for estimates of the mean reversion parameter. Explicit expressions are given for the discretization bias and its relationship to estimation bias in both multivariate and in univariate settings. In the univariate context, we compare the performance of the two approximation methods relative to exact maximum likelihood (ML) in terms of bias and variance for the Vasicek process. The bias and the variance of the Euler method are found to be smaller than the trapezoidal method, which are in turn smaller than those of exact ML. Simulations suggest that for plausible parameter settings the approximation methods work better than ML, the bias formulae are accurate, and for scalar models the estimates obtained from the two approximate methods have smaller bias and variance than exact ML. For the square root process, the Euler method outperforms the Nowman method in terms of both bias and variance. Simulation evidence indicates that the Euler method has smaller bias and variance than exact ML, Nowman’s method and the Milstein method. The Chapter 3 examines the asymptotic properties of the maximum likelihood (ML) estimate of the mean reversion matrix that is obtained from the corresponding exact discrete model. Both the consistency and the asymptotic distribution are derived in the cases of stationarity and non-stationarity. Special attention is paid to the explicit expressions for the asymptotic covariance matrix, especially in low dimensional cases. This limit theory is facilitated by a new formula for the mapping from the discrete to the continuous system coefficients and its derivatives. An empirical application is conducted on daily realized volatility data on Pound, Euro and Yen exchange rates, illustrating the implementation of the theory. Recently, with the coming of the financial crisis, the interest of using explosive process to model asset bubbles has been growing tremendously. This underlies the importance of statistic properties of the explosive process. The Chapter 4 develops a double asymptotic limit theory for the persistent parameter (k) in explosive continuous time models driven by L´evy processes with a large number of time span (*N*) and a small number of sampling interval (*h*). The simultaneous double asymptotic theory is derived using a technique in the same spirit as in Phillips and Magdalinos (2007) for the mildly explosive discrete time model. Both the intercept term and the initial condition appear in the limiting distribution. In the special case of explosive continuous time models driven by the Brownian motion, we develop the limit theory that allows for the joint limits where *N* →∞ and *h*→ 0 simultaneously, the sequential limits where *N* →∞ is followed by *h*→ 0, and the sequential limits where *h*→ 0 is followed by *N* →∞ . All three asymptotic distributions are the same.

#### Keywords

continuous time, diffusion, bias reduction, limit theory, explosive process, levy process

#### Degree Awarded

PhD in Economics

#### Discipline

Econometrics

#### Supervisor(s)

PHILLIPS, Peter C.B.; YU, Jun

#### Citation

WANG, Xiaohu.
Three Econometric Essays on Continuous Time Models. (2012). Dissertations and Theses Collection (Open Access).
**Available at:** http://ink.library.smu.edu.sg/etd_coll/84

#### Creative Commons License

This work is licensed under a Creative Commons Attribution-Noncommercial-No Derivative Works 4.0 License.

## Comments

Jin, Sainan; Lim, Kian Guan