Analytic derivatives of the matrix exponential for estimation of linear continuous-time models

Linear-in-variables continuous-time processes are estimated nonlinearly, because the coefficients of the implied linear-in-variables discrete-time estimating equations are the exponential of a matrix formed with the continuous-time parameters. Even with sampling complications such as irregular intervals, mixed frequencies, and stock and flow variables, using Van loan's (1978) results, the mapping from continuous- to discrete-time parameters and its derivatives can be expressed as the submatrix of a matrix exponential. For quicker estimation and more accurate hypothesis testing or sensitivity analysis, it is often better to compute analytically the first-order derivatives of the mapping. This paper explains how to compute efficiently the continuous- to discrete-time parameter mapping and its derivatives, without computing an eigenvalue decomposition, the common way of doing this. By linking present results with previous ones, a complete chain rule is obtained for computing the Gaussian likelihood function and its derivatives with respect to the continuous-time parameters.