SEM Modeling with Singular Moment Matrices Part II: ML-Estimation of Sampled Stochastic Differential Equations

Abstract

Linear stochastic differential equations are expressed as an exact discrete model (EDM) and estimated with structural equation models (SEMs) and the Kalman filter (KF) algorithm. The oversampling approach is introduced in order to formulate the EDM on a time grid which is finer than the sampling intervals. This leads to a simple computation of the nonlinear parameter functionals of the EDM. For small discretization intervals, the functionals can be linearized, and standard software permitting only linear parameter restrictions can be used. However, in this case the SEM approach must handle large matrices leading to degraded performance and possible numerical problems. The methods are compared using coupled linear random oscillators with time-varying parameters and irregular sampling times.

Continuous-discrete state-space modeling of panel data with nonlinear filter algorithms

“…It is interesting that (33) is well defined also for N = 1 or N < K (Singer 2007), such that time series or small panels can be estimated with SEM. Unfortunately, most programs use a modified likelihood function which contains the log determinant of the sample covariance or moment matrix with rank min(N, K) and the analysis stops for N < K (Mardia and Kent 1979, p. 263;Singer 2011). It can be shown explicitly that the SEM likelihood (33) and the Kalman filter likelihood (29) coincide (for details, see Singer 2010).…”

“…The latter is a batch method, in the sense that all time points are processed simultaneously. A detailed comparison is given in Oud and Singer (2008), Singer (2010Singer ( , 2011.…”

“…It is seen that strength and even sign of the causal actions depend on the sampling interval Δt, even if the structural matrix A is constant. For small discretization intervals, the matrix exponentials may be linearized to obtain a time ordered approximate computation in programs with only linear restrictions (for details, see Singer 2011). In other words, the SDE is solved on a fine time scale δt such that all measurement times t i = τ j i lie on this grid (Singer 1995(Singer , 1998, i.e.…”

“…Filling all data in one response vector the SEM must be estimated on N = 1 (cf. Singer 2010Singer , 2011.…”

“…Modern programs admit appropriate nonlinear restrictions (cf. Oud and Jansen 2000;Singer 2007Singer , 2011Oud and Singer 2008). Panels with correlated units (e.g.…”