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Empirical Economics

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02.01.2016

The Beveridge–Nelson decomposition of mixed-frequency series

An application to simultaneous measurement of classical and deviation cycles

verfasst von: Yasutomo Murasawa

Erschienen in: Empirical Economics | Ausgabe 4/2016

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Abstract

Gibbs sampling for Bayesian VAR with mixed-frequency series draws latent high-frequency series and model parameters sequentially. Applying the multivariate Beveridge–Nelson (B–N) decomposition in each Gibbs step, one can simulate the joint posterior distribution of the B–N permanent and transitory components in latent and observable high-frequency series. This paper applies the method to mixed-frequency series of macroeconomic variables including quarterly real GDP to estimate the monthly natural rates and gaps of output, inflation, interest, and unemployment jointly. The resulting monthly real GDP and GDP gap are complementary coincident indices, measuring classical and deviation cycles, respectively.

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Classical analysis is also possible, but has the following difficulties:

With mixed-frequency series, ML estimation of a Gaussian VAR model requires direct maximization of the likelihood function instead of OLS. This is difficult with many VAR coefficients. Moreover, the likelihood function may be multimodal.

Since the B–N decomposition involves highly nonlinear transformation of the VAR coefficients, the delta method gives poor approximations to the exact distributions of the B–N components. Moreover, since we cannot use OLS, bootstrap is extremely time-consuming.

Note that the posterior medians of \(c_{t,i}\) and \(c_{t',i'}\) come from different draws of \(\varvec{\phi }\) and \(\varvec{Y}^*_T\); i.e., there is no single \(\varvec{\phi }\) that generates the “point estimate” of \(\{\varvec{c}_t\}\). This may look odd to frequentists but not to Bayesians, for whom \(\varvec{\phi }\) is random and the posterior median is just a summary statistic of the posterior pdf. I thank a referee for warning against this confusion.

The results of unit root tests are similar to the quarterly results in Murasawa (2014) and support this assumption for our data.

I thank a referee for pointing this out.

Perron and Wada (2009) assume a structural break in the drift term of US log real GDP in 1973Q1. As a robustness check of our result, we assume structural breaks in the drift terms of all variables in 1973M3, subtract the regime-specific means from the differenced series, and repeat the same analysis. We find that introducing a break in 1973M3 does not change our result drastically, which is in contrast to Perron and Wada (2009). Possible reasons for this difference are the decomposition method (univariate vs multivariate), the model (quarterly vs monthly), and the sample period. Though interesting and important, we leave further study on this issue for future work.

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Titel: The Beveridge–Nelson decomposition of mixed-frequency series
An application to simultaneous measurement of classical and deviation cycles
verfasst von: Yasutomo Murasawa
Publikationsdatum: 02.01.2016
Verlag: Springer Berlin Heidelberg
Erschienen in: Empirical Economics / Ausgabe 4/2016
Print ISSN: 0377-7332
Elektronische ISSN: 1435-8921
DOI: https://doi.org/10.1007/s00181-015-1061-5

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