9

Structural Time Series Models for

Business Cycle Analysis

Tommaso Proietti

Abstract

The chapter deals with parametric models for the measurement of the business cycle in economic
time series. It presents univariate methods based on parametric trend-cycle decompositions and
multivariate models featuring a Phillips-type relationship between the output gap and inflation and
the estimation of the gap using mixed frequency data. We finally address the issue of assessing
the accuracy of the output gap estimates.

9.1 Introduction 386
9.2 Univariate methods 388
9.2.1 The random walk plus noise model 388
9.2.2 The local linear model and the Leser–HP filter 390
9.2.3 Higher-order trends and lowpass filters 392
9.2.4 The cyclical component 394
9.2.5 Models with correlated components 395
9.2.6 Model-based bandpass filters 400
9.2.7 Applications of model-based filtering: bandpass cycles and the estimation
of recession probabilities 403
9.2.8 Ad hocfiltering and the Slutsky–Yule effect 406
9.3 Multivariate models 407
9.3.1 Bivariate models of real output and inflation 407
9.3.2 A bivariate quarterly model of output and inflation for the US 409
9.3.2.1 ML estimation 410
9.3.2.2 Bayesian estimation 410
9.3.3 Multivariate extensions 416
9.3.4 A multivariate model with mixed frequency data 419
9.4 The reliability of the output gap measurement 421
9.4.1 Validity 422
9.4.2 Precision 423
9.5 Appendix A: Linear filters 424
9.6 Appendix B: The Wiener–Kolmogorov filter 425

385