Exact Discrete Time Representation Research Articles

Random coefficient continuous systems: Testing for extreme sample path behavior

This paper studies a continuous time dynamic system with a random persistence parameter. The exact discrete time representation is obtained and related to several discrete time random coefficient models currently in the literature. The model distinguishes various forms of unstable and explosive behavior according to specific regions of the parameter space that open up the potential for testing these forms of extreme behavior. A two-stage approach that employs realized volatility is proposed for the continuous system estimation, asymptotic theory is developed, and test statistics to identify the different forms of extreme sample path behavior are proposed. Simulations show that the proposed estimators work well in empirically realistic settings and that the tests have good size and power properties in discriminating characteristics in the data that differ from typical unit root behavior. The theory is extended to cover models where the random persistence parameter is endogenously determined. An empirical application based on daily real S&P 500 index data over 1928–2018 reveals strong evidence against parameter constancy over the whole sample period leading to a long duration of what the model characterizes as extreme behavior in real stock prices.

Continuous time ARMA processes: Discrete time representation and likelihood evaluation

This paper explores the representation and estimation of mixed continuous time ARMA (autoregressive moving average) systems of orders p, q. Taking the general case of mixed stock and flow variables, we discuss new state space and exact discrete time representations and demonstrate that the discrete time ARMA representations widely used in empirical work, based on differencing stock variables, are members of a class of observationally equivalent discrete time ARMA(p+1, p) representations, which includes a more natural ARMA(p, p) representation. We compare and contrast two approaches to likelihood evaluation and computation, namely one based on an exact discrete time representation and another utilising a state space representation and the Kalman–Bucy filter. We demonstrate the value of our approach in two applications: a univariate study of the yield curve at different frequencies; and, a multivariate study of the relationship between US GDP and oil prices, taking account of the mixed frequencies with which these data are available.

Random Coefficient Continuous Systems: Testing for Extreme Sample Path Behaviour

This paper studies a continuous time dynamic system with a random persistence parameter. The exact discrete time representation is obtained and related to several discrete time random coefficient models currently in the literature. The model distinguishes various forms of unstable and explosive behaviour according to specific regions of the parameter space that open up the potential for testing these forms of extreme behaviour. A two-stage approach that employs realized volatility is proposed for the continuous system estimation, the asymptotic theory is developed, and test statistics to identify the different forms of extreme sample path behaviour are proposed. Simulations show that the proposed estimators work well in empirically realistic settings and that the tests have good size and power properties in discriminating characteristics in the data that differ from typical unit root behaviour. The theory is extended to cover models where the random persistence parameter is endogenously determined. An empirical application based on daily real S&P500 index data over 1928-2018 reveals strong evidence against parameter constancy over the whole sample period leading to a long duration of what the model characterizes as extreme behaviour in real stock prices.

DISCRETE TIME REPRESENTATIONS OF COINTEGRATED CONTINUOUS TIME MODELS WITH MIXED SAMPLE DATA

This paper derives an exact discrete time representation corresponding to a triangular cointegrated continuous time system with mixed stock and flow variables and observable stochastic trends. The discrete time model inherits the triangular structure of the underlying continuous time system and does not suffer from the apparent excess differencing that has been found in some related work. It can therefore serve as a basis for the study of the asymptotic sampling properties of estimators of the model's parameters. Some further analytical and computational results that enable Gaussian estimation to be implemented are also provided.

The exact discrete time representation of a system of fourth-order differential equations

The exact discrete time representations of both a closed and an open system of fourth-order differential equations are considered, in which all vector processes may contain both stock and flow variables. For extra generality, the closed system of equations contains seasonal dummies. The open system here contains levels and derivatives (up to fourth. order) of the vector of exogenous variables whose continuous time paths are approximated using a quartic interpolant.

Discrete time representation of stationary and non-stationary continuous time systems

This paper derives the formulae for an exact discrete time representation corresponding to a system of higher-order stochastic differential equations. The formulae are applicable in stationary, non-stationary and explosive systems and for data observed as a mixture of both stock and flow variables. Expressions are also provided for an explicit moving average representation of the disturbance vector in the discrete time model, which can be used, under the assumption of white noise continuous time disturbances, to derive formulae for the computation of the exact Gaussian likelihood function.

Exact Discrete Time Representation of Continuous Time Systems

Abstract The solutions proposed in the literature for the representation of a continuous time system by a difference equation all have the disadvantage to be accurate at very large sampling frequencies only. In this paper a new exact discrete time representation, based on the instantaneous sampling method, is presented. It requires almost no oversampling and uses broadband digital differentiators. The proposed technique results in less complex discrete time models than the integrated sampling method.