The Theory of Matrices. By F R. Gantmacher. Two volumes, pp. 374 and 276. 1959. (Translated from the Russian by K. A. Hirsch; Chelsea Publishing Company, New York)

Matrix Differential Calculus with Applications in Statistics and Econometrics

The commutation matrix $K$ is defined as a square matrix containing only zeroes and ones. Its main properties are that it transforms vecA into vecA', and that it reverses the order of a Kronecker product. An analytic expression for $K$ is given and many further properties are derived. Subsequently, these properties are applied to some problems connected with the normal distribution. The expectation is derived of $\varepsilon' A\varepsilon\cdot\varepsilon' B\varepsilon\cdot\varepsilon'C\varepsilon$, where $\varepsilon \sim N(0, V)$, and $A, B, C$ are symmetric. Further, the expectation and covariance matrix of $x \otimes y$ are found, where $x$ and $y$ are normally distributed dependent variables. Finally, the variance matrix of the (noncentral) Wishart distribution is derived.

The commutation matrix: Some properties and applications

This paper studies the pruned state-space system for higher-order perturbation ap- proximations to DSGE models. We show the stability of the pruned approximation up to third order and provide closed-form expressions for Orst and second unconditional moments and impulse response functions. Our results introduce GMM estimation and impulse-response matching for DSGE models approximated up to third order and pro- vide a foundation for indirect inference and SMM. As an application, we consider a New Keynesian model with Epstein-Zin-Weil preferences and two novel feedback e§ects from long-term bonds to the real economy, allowing us to match the level and variability of the 10-year term premium in the U.S. with a low relative risk aversion of 5.

/pdf/the-pruned-state-space-system-for-non-linear-dsge-models-3ix1kh9xho.pdf

The pruned state-space system for non-linear dsge models: theory and empirical applications

https://www.econstor.eu/bitstream/10419/76718/1/751117021.pdf

Uncertainty shocks, banking frictions and economic activity

https://www.econstor.eu/bitstream/10419/56731/1/68022257X.pdf

Solving DSGE models with a nonlinear moving average

We prove the existence of unique solutions for all undetermined coefficients of nonlinear perturbations of arbitrary order in a wide class of discrete tim e DSGE models under standard regularity and saddle stability assumptions for linear app roximations. Our result follows from the straightforward application of matrix analysis to our p erturbation derived with Kronecker tensor calculus. Additionally, we relax the assumptions needed for the local existence theorem of perturbation solutions and prove that the local solution is independent of terms first order in the perturbation parameter.

/pdf/existence-and-uniqueness-of-perturbation-solutions-to-dsge-rbld4i7bko.pdf

Existence and Uniqueness of Perturbation Solutions to DSGE Models

We derive recursive representations of nonlinear moving average (NLMA) perturbations of DSGE models. As the stability of higher order NLMA representations follows directly from stability at first order, these recursive representations provide rigorous support for the practice of pruning that is becoming widespread. Our recursive representation differs from pruned perturbations in that it centers the approximation and its coefficients at the approximation of the stochastic steady state consistent with the order of approximation. We compare our algorithm with six different pruning algorithms at second and third order, documenting the differences between these six algorithms and standard (non pruned) state space perturbations at first, second, and third order in a unified notation compatible with the popular software package Dynare. While our third order algorithm is the most accurate, the gains over two alternate algorithms are modest, suggesting that this choice is unlikely to be a potential source of error.

https://www.econstor.eu/bitstream/10419/79579/1/745850618.pdf

Pruning in perturbation DSGE models: Guidance from nonlinear moving average approximations

Solvability of perturbation solutions in DSGE models

This file contains the code from "Solving DSGE Models with a Nonlinear Moving Average." This add-on extends Dynare's (version 4) functionality to include nonlinear moving average policy functions for calculating impulse responses and simulations.

Hong Lan

Papers

Solving DSGE models with a nonlinear moving average

Existence and Uniqueness of Perturbation Solutions to DSGE Models

Pruning in perturbation DSGE models: Guidance from nonlinear moving average approximations

Solvability of perturbation solutions in DSGE models

Dynare add-on for "Solving DSGE Models with a Nonlinear Moving Average"