Gary S. Anderson

A linear algebraic procedure for solving linear perfect foresight models

Abstract This paper presents a failsafe method for analyzing any linear perfect foresight model. It describes a procedure which either computes the reduced-form solution or indicates why the model has no reduced form.

Higher-order perturbation solutions to dynamic, discrete-time rational expectations models

We present an algorithm and software routines for computing nthorder Taylor series approximate solutions to dynamic, discrete-time rational expectations models around a nonstochastic steady state. The primary advantage of higher-order (as opposed to first- or secondorder) approximations is that they are valid not just locally, but often globally (i.e., over nonlocal, possibly very large compact sets) in a rigorous sense that we specify. We apply our routines to compute first- through seventh-order approximate solutions to two standard macroeconomic models, a stochastic growth model and a life-cycle consumption model, and discuss the quality and global properties of these solutions.

A reliable and computationally efficient algorithm for imposing the saddle point property in dynamic models

This paper describes a set of algorithms for quickly and reliably solving linear rational expectations models. The utility, reliability and speed of these algorithms are a consequence of 1) the algorithm for computing the minimal dimension state space transition matrix for models with arbitrary numbers of lags or leads, 2) the availability of a simple modeling language for characterizing a linear model and 3) the use of the QR Decomposition and Arnoldi type eigenspace calculations. The paper also presents new formulae for computing and manipulating solutions for arbitrary exogenous processes.

A Weekly Perfect Foresight Model of the Nonborrowed Reserve Operating Procedure

Of the many studies analyzing the Federal Reserves post-October 6, 1979 nonborrowed reserve (NBR) operating procedure, none has focused upon weekly money market dynamics under rational expectations. This paper employs the rational expectations assumption in an explicit institutional model of the NBR procedure. The paper is positive rather than normative, isolating the policy elements that comprise the procedure and investigating their dynamic interaction.

A procedure for differentiating perfect-foresight-model reduced-from coefficients

Anderson and Moore (1985) present a fail-safe method for analyzing any linear perfect-foresight model. They describe a procedure which either computes the reduced-form solution or indicates why the model has no reduced form. This paper presents formulae for differentiating singular vectors and vectors spanning an invariant space, and shows how to use these formulae to differentiate Anderson and Moores structural-model to reduced-form-model transformation.

Characteristics of discrete housing market model equilibria

Abstract A discrete microeconomic model of a market for heterogeneous housing units is presented. It is proven that whenever bid rents are lower semicontinuous monotone decreasing functions of utility and asked rents are continuous monotone increasing functions of profits an equilibrium solution exists whose graph corresponds to a bipartite tree. This graphical representation of the solutions provides a mechanism for analyzing important theoretical issues. Using graph theoretic techniques, it is demonstrated that bargaining power assumptions are sufficient to ensure unique equilibrium price and utility levels. Alternatively ruling out the possibility of isolated submarkets guarantees unique market clearing prices. Unique equilibrium prices imply almost unique location decisions: One may still be able to reassign households among housing units for which they are indifferent. The model reconciles the dichotomy between open and closed market models by providing a robust mechanism for characterizing conditions when agents leave the market. The reconciliation underscores the necessity for carefully choosing bargaining power, market exit, reservation rent, and reservation utility assumptions in closed as well as open models. Without at least implicitly making assumptions of these sorts, neither closed nor open models can determine anything more than relative prices. The model generalizes versions of The Urban Institute Housing Market Model, and the market clearing section of the National Bureau of Economic Research Urban Simulation Model. It presents an algorithm for computing equilibria and describes an application of this algorithm to The Urban Institute Housing Market Model. The algorithm computes equilibrium solutions at one-sixth of the cost of using the original U.I. Model algorithm.

A Reliable and Computationally Efficient Algorithm for Imposing the Saddle Point Property in Dynamic Models

Abstract (Anderson and Moore, 1983; Anderson and Moore, 1985) describe a powerful method for solving linear saddle point models. The algorithm has proved useful in a wide array of applications including analyzing linear perfect foresight models, providing initial solutions and asymptotic constraints for for nonlinear models. Although widely used at the Federal Reserve, few outside the central bank know about or have used the algorithm. This paper attempts to present the current algorithm in a more accessible fonnat in the hope that economists outside the Federal Reserve may also find it useful. In addition, over the years there have been many undocumented changes in approach that have improved the efficiency and reliability of algorithm. This paper describes the present state of development of this set of tools. This paper analyzes a general linear saddle point model with a unique steady state and a unique solution converging to that steady state for any set of temporally predetermined variables. We prove that any such model has a reduced form relating the solution sequence entirely to its history, and we present an efficient procedure for computing the reduced form coefficients. The procedure is a generalization of the familiar saddlepoint analysis, and it is straightforward to program. The procedure consists of efficient library routines for matrix rank determination and invariant space calculation embedded in a simple control structure. The algorithm solves linear probles with dozens of lags and leads and hundreds of equations in seconds. The technique works well for both symbolic algebra and numerical computation.