Christiano Notes - Solving Dynamic Equilibrium Models by a...

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Solving Dynamic Equilibrium Models by a Method of Undetermined Coe cients Macroeconomics II - Spring 2006 Department of Economics, University of Texas Instructor Dean Corbae Composed by Je f Thurk 1 1. INTRODUCTION You saw in class that one can solve the stochastic RBC model by linearizing the Euler (FOC) conditions around the steady-state values. You also saw that in even a relatively simple example, such linearization took a fair amount of algebra, paper/ chalkboard, and time. These factors increase exponentially when we introduce complications into the model. The objective of these notes is to introduce you to a simple ”recipe” for solving these problems relatively quickly. Christiano (2002) describes a method for solving a system of linear ex- pectational di f erence equations using relatively simple matrix algebra. His method follows the linearization procedure you saw in class, but reduces it to a series of mechanical steps which can be programmed into a computer. Below I will outline the matrix version of the undetermined coe cients method described in the paper as well as demonstrate how to use his method in a simple example. 2 2. GENERAL METHODOLOGY This section follows Christiano (2002) closely, attempting to remain consistent with his nomenclature and presentation. It is intended to provide some general theoretical foundation to the more speci f c model presented later. The solution to this model will be a linear feedback rule (decision rule) relating current period endogenous variables to a set of state variables. z t = Az t 1 + Bs t (1) where A (the “feedback” part) an ( n × n )matr ixand B (the “feedforward” part) an ( n × m ) matrix are the “coe cients to be determined”. The f rst vector z t 1 is a set of endogenous state variables (e.g. capital predetermined at the beginning of the current period) but determined in the previous period of the model. The second vector s t is a set of exogenous state variables generate outside the model by some stochastic process (e.g., the technology TFP shock). Let, z t = · z 1 t z 2 t ¸ (2) 1 These notes were prepared for a review session at The University of Texas at Austin, February 2006. The notes draw upon 1) Christiano, L. (2002), “Solving Dynamic Equilibrium Models by a Method of Undetermined Coe cients”, Computational Economics, Vol. 20, p. 21-55. 2) Previous lecture notes compiled by Michelle Petersen All errors are mine. 2 You can think of Christiano (2002) as a user’s manual for computing these problems. As such, there are a number of good examples in the back that may help you in the future. 1
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and be an ( n × 1) vector, which contains all endogenous state variables. z 1 t is an ( n 1 × 1) vector of all endogenous variables determined at time t and z 2 t is an ( qn 1 × 1) vector of q lagged z 1 t ’s. Since we’ll be modeling deviations from steady-state, we can think of q =0and therefore, z t = z 1 t . Furthermore let, s t be an ( m × 1) vector of exogenous shocks.
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This note was uploaded on 08/06/2008 for the course ECON 387 taught by Professor Corbae during the Spring '07 term at University of Texas at Austin.

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Christiano Notes - Solving Dynamic Equilibrium Models by a...

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