Supplement to “ Numerically stable and accurate stochastic simulation approaches for solving dynamic economic models ” : Appendices

Appendix A: Nonlinear regression model and nonlinear approximation methods In this section, we extend the approximation approaches that we developed in Sections 4.2 and 4.3 to the case of the nonlinear regression model y = Ψ (kk a; b) + εε (A.1) where b ∈ R n+1 , k ≡ (k 0 k T −1) ∈ R T , a ≡ (a 0 a T −1) ∈ R T , and Ψ (kk a; β) ≡ (Ψ (k 0 a 0 ; β) Ψ (k T −1 a T −1 ; β)) ∈ R T. 1 We first consider a nonlinear LS (NLLS) problem and then formulate the corresponding LAD problem. The NLLS problem is min b y − Ψ (kk a; b) 2 2 = min b [y − Ψ (kk a; b)] [y − Ψ (kk a; b)] (A.2) The typical NLLS estimation method linearizes (A.2) around a given initial guess b by using a first-order Taylor expansion of Ψ (kk a; b) and makes a step b toward a solution b b + bb (A.3) Using the linearity of the differential operator, we can derive an explicit expression for the step b. This step is given by a solution to the system of normal equations J J b = J y (A.4) 1 The regression model with the exponentiated polynomial Ψ (k t a t ; b) = exp(b 0 + b 1 ln k t + b 2 ln a t + · · ·), used in Marcet's (1988) simulation-based PEA, is a particular case of (A.1).