Empirical Applications Based on Survey Data
Appendix B E-Stability of a Forward-Looking Taylor Rule
In a ﬁrst step we derive the minimal state variable (MSV) solution (McCal- lum, 1983) for the system of diﬀerence equations (A.1) for the REE by using the method of undetermined coeﬃcients. With y = ( xt πt )′ and B = [A1]−1 we rewrite (A.1) as (B.1) yt = AEtyt+1 +Bvt. We guess a solution of the form (B.2) yt = c¯vt, where c¯ is a 2 x 2 matrix that has to be determined. Because of (B.1) and (2.3) and (2.4) we can set up the following conditions (B.3) Etyt = c¯vt, (B.4) Etyt+1 = c¯vt+1 = c¯Fvt = Fyt. Plugging (B.4) into (B.1) and rearranging gives (B.5) yt = (I − FA)−1Bvt, 106 Appendix where I denotes a 2 x 2 identity matrix. It is easy to see that with c¯ = (I − FA)−1B, (2.34) gives the MSV solution for the REE. Suppose the PLM is given by (B.6) yt = a+ cvt. Note that we allow for a structural deviation of (B.6) from the MSV solution for the REE (B.2) in form of an intercept a = 0. Moreover, it is pos- sible that c = c¯. Based on the PLM economic agents form their expectations according to (B.7) E∗t yt+1 = a+ cFvt. Plugging (B.7) into (B.1) gives the ALM (B.8) yt = Aa+ (AcF +B)vt. Now we can carry out the mapping from the PLM (B.6) to the ALM (B.8): (B.9) T ( a c ) = ( Aa AcF +B ) . To obtain the conditions for E-stability we set up the following diﬀerential equation:...
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