Statistics Seminar（2015-15）

Topic:Explosiveness and Initial Conditions: Theory and Application

Speaker:Jun Yu, Singapore Management University

Time:Thursday, 15th October, 14:00-15:30

Location:K01 of Guanghua Hotel

Abstract:This talk is based on the following two papers:

(1) Double Asymptotics for Explosive Continuous Time Models

This paper establishes a double asymptotic theory for explosive continuous time Lévy-driven processes and the corresponding exact discrete time models. The double asymptotic theory assumes the sample size diverges because the sampling interval (h) shrinks to zero and the time span (N) diverges. Both the simultaneous and sequential double asymptotic distributions are derived. In contrast to the long-time-span asymptotics (N diverging to infinity with fixed h) where no invariance principle applies, the double asymptotic distribution is derived without assuming Gaussian errors, so an invariance principle applies. Like the in-fill asymptotics (h converging to 0 with fixed N), the double asymptotic distribution explicitly depends on the initial condition. The convergence rate of the double asymptotics partially bridges that of the long-time-span asymptotics and that of the in-fill asymptotics. Monte Carlo evidence shows that the double asymptotic distribution works well in practically realistic situations and better approximates the finite sample distribution than the asymptotic distribution that is independent of the initial condition. Empirical applications to real Nasdaq prices highlight the difference between the new theory and the theory without taking the initial condition into account.

(2) Limit Theory for Continuous Time Systems with Mildly Explosive Regressors

Limit theory is developed for continuous co-moving systems with mildly explosive regressors. The theory uses double asymptotics with in-fill (as the sampling interval tends to zero) and large time span asymptotics. The limit theory explicitly involves initial conditions, allows for drift in the system, is provided for single and multiple explosive regressors, and is feasible to implement in practice. Simulations show that double asymptotics deliver a good approximation to the finite sample distribution, with both finite sample and asymptotic distributions showing sensitivity to initial conditions. The methods are implemented in the US real estate market for an empirical application, illustrating the usefulness of double asymptotics in practical work.