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From |
"David M. Drukker, StataCorp" <ddrukker@stata.com> |

To |
statalist@hsphsun2.harvard.edu |

Subject |
RE: st: Cumulative impulse response |

Date |
Wed, 03 Nov 2004 09:24:31 -0600 |

Alexander Cavallo <acavallo@lexecon.com> asked about how to compute the infinite-horizon cumulative impulse response functions and how to determine whether the estimated cumulatives have converged. As Alexander noted, the simple answer is to choose a -step(#)- that is large enough so that the estimated cumulatives settle down. It turns out that this is also the best method. The mathematical structure of the problem guarantees that if the estimates settle down to what looks like a convergence point, it is the convergence point. Lutkepohl (1993, page 97) provides the formulas for the infinite-horizon cumulative impulse response functions. Respectively, the simple and structural formulas are Psi_inf = (I_k-A1-A2-..-Ap)^(-1) Xi_inf = Psi_inf*P where I_k is the k x k identity matrix; Ai i=1, .., p are the p matrices of lag coefficients; P is the matrix which orthogonalizes the innovations. (For example, P is the Cholesky decomposition or P=inv(A)*B where A and B are the structural decompositions defined in SVAR.) The eigenvalue-stability condition guarantees that the inverse on the right-hand side exists and that simply increasing the step horizon provides an ever better approximation to Psi_inf and Xi_inf. The eigenvalue-stability condition also guarantees that after some finite horizon, the absolute value of each additional term decreases with the step horizon. Although it is possible to compare the asymptotic formulas with the cumulatives computed for a given -step(#), the best criteria is simply to choose a large enough -step(#)- so that the estimates settle down. Alexander also asked > Is anyone aware of guidance in the econometric literature on the > ratio of the number of lags in the VAR to the number of iterations > to compute in the IRF? The issue is not the number of lags but rather the eigenvalues of the companion matrix. The closer the modulus of the largest eigenvalue is to one, the longer it will take for the estimates to converge. However, assuming stability, the estimates will eventually settle down. --David ddukker@stata.com * * For searches and help try: * http://www.stata.com/support/faqs/res/findit.html * http://www.stata.com/support/statalist/faq * http://www.ats.ucla.edu/stat/stata/

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