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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 ff 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 ff 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 fi 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 ) matrix and B (the “feedforward” part) an ( n × m ) matrix are the “coe cients to be determined”. The fi 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 = 0 and therefore, z t = z 1 t . Furthermore let, s t be an ( m × 1) vector of exogenous shocks.
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