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# Monetary Policy Rules

## Dirk Bleich

This work provides different studies of how econometric evaluation of monetary policy based on forward-looking Taylor rules is conducted. The first part discusses theoretical results regarding the Taylor principle and can be used as a guideline for the evaluation of the following three empirical applications based on survey data of Consensus Economics. The first application deals with the question whether the introduction of inflation targeting affects monetary policy. The second application investigates the consequences of oil price movements for monetary policy. The third application analyzes monetary policy conditions in Spain before and after the changeover to the Euro by estimating forward-looking Taylor rules.

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# Appendix A Determinacy of a forward-looking Taylor rule

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We have to transform (2.12) into the form given by (2.5). Thus we premultiply both sides of (2.12) with the inverse of A1: (A.1) [ xt πt ] = A [ Etxt+1 Etπt+1 ] + [A1] −1 vt, where A = [A1] −1A2 = [ 1− ϕφx ϕ(1− φπ) κ(1− ϕφx) β + κϕ(1− φπ) ] is the relevant matrix for determinacy. Because both the inﬂation rate, πt, and the output gap, xt, depend on their expectations formed at date t, we have two non- predetermined variables (see deﬁnition 2). Thus, we need both eigenvalues of A to be inside the unit circle (i.e. to have moduli strictly less than 1) for uniqueness. To obtain the eigenvalues, λi, of A we have to consider the characteristic polynomial which is for 2 x 2 matrices in general given by: (A.2) |λI − A| = λ2 − tr(A)λ+ det(A), where det(A) = β(1 − ϕφx) and −tr(A) = κϕ(φπ − 1) + ϕφx − 1 − β. According to Bullard and Mitra (2002), both eigenvalues of A are inside the unit circle if and only if the following conditions hold (LaSalle, 1986, p. 28): (A.3) |det(A)| < 1, 104 Appendix (A.4) |tr(A)| < 1 + det(A). From (A.3) follows inequality (2.13) and from (A.4) we obtain inequalities implies (2.14) and (2.15). Thus we arrive at proposition 1.

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