Empirical Applications Based on Survey Data
Appendix A Determinacy of a forward-looking Taylor rule
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.
You are not authenticated to view the full text of this chapter or article.
This site requires a subscription or purchase to access the full text of books or journals.
Do you have any questions? Contact us.Or login to access all content.