Baoline Chen

Analytic derivatives of the matrix exponential for estimation of linear continuous-time models1

Abstract 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 loans (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.

Higher-Moments in Perturbation Solution of the Linear-Quadratic Exponential Gaussian Optimal Control Problem

The paper obtains two principal results. First, using a new definition ofhigher-order (>2) matrix derivatives, the paper derives a recursion forcomputing any Gaussian multivariate moment. Second, the paper uses this resultin a perturbation method to derive equations for computing the 4th-orderTaylor-series approximation of the objective function of the linear-quadraticexponential Gaussian (LQEG) optimal control problem. Previously, Karp (1985)formulated the 4th multivariate Gaussian moment in terms of MacRaesdefinition of a matrix derivative. His approach extends with difficulty to anyhigher (>4) multivariate Gaussian moment. The present recursionstraightforwardly computes any multivariate Gaussian moment. Karp used hisformulation of the Gaussian 4th moment to compute a 2nd-order approximationof the finite-horizon LQEG objective function. Using the simpler formulation,the present paper applies the perturbation method to derive equations forcomputing a 4th-order approximation of the infinite-horizon LQEG objectivefunction. By illustrating a convenient definition of matrix derivatives in thenumerical solution of the LQEG problem with the perturbation method, the papercontributes to the computational economists toolbox for solving stochasticnonlinear dynamic optimization problems.

An anticipative feedback solution for the infinite-horizon, linear-quadratic, dynamic, Stackelberg game

Abstract This paper derives and illustrates a new suboptimal-consistent feedback solution for an infinite-horizon, linear-quadratic, dynamic, Stackelberg game. This solution lies in the same solution space as the infinite-horizon, dynamic-programming, feedback solution but puts the leader in a preferred equilibrium position. The idea comes from Kydland (J. Econ. Theory 15 (1977)) who suggested deriving a consistent feedback solution for an infinite-horizon, linear-quadratic, dynamic, Stackelberg game by varying the coefficients in the players linear constant-coefficient decision rules. Here feedback is understood in the sense of setting a current control vector as a function of a predetermined state vector. The proposed solution is derived for discrete- and continuous-time games and is called the anticipative feedback solution. The solution is illustrated with a numerical example of a duopoly model.

Numerical Solution of an Endogenous Growth Model with Threshold Learning

This paper describes an application of numerical methods to solve a continuous time non-linear optimal growth model with technology adoption. In the model, a non-convex production function arises from a threshold level of knowledge required to operate new technology. The study explains and illustrates how to compute the complete transition path of the growth model by applying in concert three broad numerical techniques in particular specialized ways, in order to maintain certain regularity conditions and restrictions of the model. The three broad techniques are: (i) Gauss-Laguerre quadrature for computing discounted utility over an infinite horizon; (ii) Fourth-Order Runge-Kutta method for solving differential equations; and (iii) the Penalty Functions method for solving the constrained optimization problem. The particular specializations involve linear interpolation for solving the optimal adoption time in the model and quasi-Newton iterations for maximizing the penalty-weighted objective function, the latter aided by grid search for determining initial values and Richardson extrapolation for approximating the gradient vector.