Related Other Work on Gaussian Estimation and Continuous- Continuous-Time Models Continuous-Time Models

Over the last two decades a series of studies have expanded the range of alternative differential models, providing closed forms for such models, ready for estimation and forecasting analysis. This was driven by exploring more flexible and complex continuous-time models that would better fit continuous-time series data with particular dynamic features. The exact discrete model approach presented in Bergstrom (1983) represents one of the major methods applied in the estimation of linear stochastic differential equation systems besides approaches based on Kalman filtering of state space forms developed by Harvey and Stock (1985, 1988) and spectral representations considered by Robinson (1976a, b).

Inspired by a model developed by Bailey et al. (1987), Chambers (1991) extended Bergstrom’s econometric framework and derived the exact discrete model (EDM) equivalent to a more general continuous-time system, more specifically a second order

71 differential equations system that included the first and second derivatives of the exogeneous variables in addition to their levels. Another alternative that would accommodate for the new dynamics while still using the framework in Bergstrom (1986), was the adjustment algorithm applied to the exogeneous variables prior to the estimation suggested by Nowman (1991). In a theoretical paper Chambers (1998) presented a detailed estimation technique that involved the derivation of a frequency domain Gaussian estimator of the parameters of a joint differential-difference equation system⁷. It was shown that this estimator is strongly consistent and asymptotically normally distributed without requiring the Gaussianity of the data. A more flexible specification of continuous-time models incorporated unobservable stochastic trends instead of deterministic trends. Studies exploring this feature include Phillips (1991), Simos (1996) and Harvey and Stock (1988, 1989, 1993). Extending the estimation algorithm of Bergstrom (1986), a new exact Gaussian estimation procedure was developed in Bergstrom (1997) with unobservable stochastic trends in a continuous-time model combining first and second order differential equations with white noise innovations and mixed data. Chambers (1999) derived the formulae for an EDM corresponding to a continuous system of higher order stochastic differential equations that can be applied to stationary, non-stationary and even explosive systems. The differential-difference type equations were employed by Chambers and McGarry (2002) in modelling cyclical behaviour in an unobserved components framework⁸. Using the discrete form of the Whittle likelihood the authors have proposed a flexible estimation technique for the derivation of a frequency domain Gaussian estimator of the parameters of a more dynamic model than those models previously considered in the literature. On the same line of research, Ercolani and Chambers (2006) and Ercolani (2009, 2011) conducted rigorous econometric analyses of various continuous-time specifications with unknown lag-parameters or driven by fractional noise.

Overcoming the complications brought by the inclusion of exogeneous variables constituted the central objective of many studies. Following suggestions made by Robinson (1992) regarding a pragmatic approach to estimating the exogeneous component in the EDM, McCrorie (2001) provided an order-selection criterion for

7 Among relatively few previous attempts there are Robinson (1976a) and Robinson (1977b).

8 The authors considered a univariate first-order three-component (trend, seasonal and cyclical) continuous time model and provided conditions for the parameters of the differential-difference equation concerning the cyclical component (containing lags), so that the initial process becomes stationary and allows for a business cycle.

72 choosing the optimal interpolant⁹ that would close the model. He also showed in a Monte Carlo experiment that the choice of a wrong degree polynomial could lead to seriously biased estimates of the variables of interest. Some early studies, including Telser (1967), had mentioned the aliasing problem in a continuous-time differential equation system.

However, it was Phillips (1973) who looked first at possible ways of minimizing the identification problem of the structural coefficient matrix in a first order linear differential equation system. Assuming that a priori restrictions on the system are simple linear functions of the elements of the coefficient matrix, Phillips (1973) showed that structural parameters are in some cases identifiable. In McCrorie (2003) a sharper characterization of the identification problem is presented, allowing for the joint treatment of the coefficient and the covariance matrices.

McGarry (2003) derived the EDM equivalent to a novel continuous-time formulation that included seasonal dummies, avoiding in this way the widely practiced seasonal data adjustments. The SDE system was of forth order allowing for a mixture of stock and flow inside all the vector processes. When open systems were considered exogeneous variables assumed a higher degree of smoothness which, according to Phillips (1974), should reduce the asymptotic bias induced in the estimation procedure. Another EDM was obtained by Simos and Taylor (2009) from a third order differential underlying equation system with fixed initial condition driven by I(1) observable stochastic and white noise disturbances.

Cointegrated continuous-time models form another class of models studied in the continuous-time modelling literature. An early approach to estimating the parameters of cointegrated systems was proposed by Phillips (1991) with two different procedures: a frequency domain regression method for the cointegrating parameters and a non-parametric treatment for the dynamic parameters. Chambers (2009) derived the EDM analogue to a first order cointegrated continuous-time system in a triangular error correction format with mixed stock and flow variables and observable stochastic trends.

Following the recursive computation algorithms presented in Bergstrom (1985, 1990), he also provided a time domain full Gaussian estimation procedure applied to both sets of parameters. The statistical properties of the Gaussian estimators are revealed by Chambers and McCrorie (2007) where frequency domain Gaussian estimators had been derived in a more general continuous-time context.

9 In most of the empirical work a quadratic interpolation is used; Bergstrom et al. (1992) and Bergstrom and Nowman (1999) have used this type of interpolation with successful results.

73 More recently econometricians explored different ways of estimating continuous-time models driven by moving average innovations, a complex feature that is retained in the discrete time representation. Following the exact discrete time approach in Bergstrom (1983), Chambers and Thornton (2012) derived the exact discrete models for a general

( , )

ARMA p q specification of the continuous-time model with stock or flow variables. In another recent study, Park and Jeong (2010) developed an asymptotic theory for maximum likelihood estimators of the parameters of continuous dynamic processes that possess a zero root.