# Practical DSGE modelling - University of Oxford Advanced dynamic models Martin Ellison University of Warwick and CEPR Bank of England, December 2005 More complex models Frisch-Slutsky paradigm Impulses Propagation Fluctuations Impulses Can add extra shocks to the model xt Et xt 1 1 (it Et t 1 ) g t t Et t 1 xt ut

it t vt Shocks may be correlated vt 1 11 ut 1 21 g t 1 31 12 22 32 13 vt 11 12 13 tv1

u 23 ut 21 22 23 t 1 33 g t 31 32 33 tg1 Propagation Add lags to match dynamics of data (Del Negro-Schorfeide, Smets-Wouters) h 1 xt xt 1 Et xt 1 1 (it Et t 1 ) 1 h 1 h p

t t 1 Et t 1 xt 1 p 1 p Taylor rule it t x xt vt h 0.35 1 h p 0.29 1 p Solution of complex models

Blanchard-Kahn technique relies on invertibility of A0 in state-space form. A0 Et X t 1 A1 X t B0 vt 1 1 0 1 0 A B Et X t 1 A A1 X t A B0 vt 1

Et X t 1 AX t Bvt 1 QZ decomposition For models where A0 is not invertible A0 Et X t 1 A1 X t B0 vt 1 QZ decomposition: Q, , Z , s.t. Q' Z ' A0 Q' Z ' A1 upper triangular Recursive equations

11 0 ~ 12 w ~t 1 11 22 Et yt 1 0 ~ 12 w ~t (Q' ) 1 B0 vt 1 22 yt ~ E ~ ~ ~

11 w y w t 1 12 t t 1 11 t 12 yt R1vt 1 stable 22 Et ~yt 1 22 ~yt R2 vt 1 unstable

Recursive structure means unstable equation can be solved first Solution strategy Solve unstable transformed equation ~ yt Substitute into stable transformed equation ~ wt Translate back into

original problem wt y t Simulation possibilities Stylised facts Impulse response functions Forecast error variance decomposition Optimised Taylor rule What are best values for parameters in Taylor rule it t x xt vt ? Introduce an (ad hoc) objective function for policy

2 2 2 min ( t x xt i it ) i 0 i Brute force approach Try all possible combinations of Taylor rule parameters Check whether Blanchard-Kahn conditions are

satisfied for each combination For each combination satisfying B-K condition, simulate and calculate variances Brute force method Calculate simulated loss for each combination Best (optimal) coefficients are those satisfying B-K conditions and leading to smallest simulated loss Grid search For each point check B-K conditions

2 x 1 0 1 2 Find lowest loss amongst points

satisfying B-K condition Next steps Ex 14: Analysis of model with 3 shocks Ex 15: Analysis of model with lags Ex 16: Optimisation of Taylor rule coefficients