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Signal extraction for nonstationary multivariate time series with illustrations for trend inflation

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  • Tucker S. McElroy
  • Thomas M. Trimbur

Abstract

This paper advances the theory and methodology of signal extraction by introducing asymptotic and finite sample formulas for optimal estimators of signals in nonstationary multivariate time series. Previous literature has considered only univariate or stationary models. However, in current practice and research, econometricians, macroeconomists, and policy-makers often combine related series - that may have stochastic trends--to attain more informed assessments of basic signals like underlying inflation and business cycle components. Here, we use a very general model structure, of widespread relevance for time series econometrics, including flexible kinds of nonstationarity and correlation patterns and specific relationships like cointegration and other common factor forms. First, we develop and prove the generalization of the well-known Wiener-Kolmogorov formula that maps signal-noise dynamics into optimal estimators for bi-infinite series. Second, this paper gives the first explicit treatment of finite-length multivariate time series, providing a new method for computing signal vectors at any time point, unrelated to Kalman filter techniques; this opens the door to systematic study of near end-point estimators/filters, by revealing how they jointly depend on a function of signal location and parameters. As an illustration we present econometric measures of the trend in total inflation that make optimal use of the signal content in core inflation.

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Bibliographic Info

Paper provided by Board of Governors of the Federal Reserve System (U.S.) in its series Finance and Economics Discussion Series with number 2012-45.

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Date of creation: 2012
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Handle: RePEc:fip:fedgfe:2012-45

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References

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  1. Arabinda Basistha & Richard Startz, 2008. "Measuring the NAIRU with Reduced Uncertainty: A Multiple-Indicator Common-Cycle Approach," The Review of Economics and Statistics, MIT Press, vol. 90(4), pages 805-811, November.
  2. Koopman, S.J.M. & Shephard, N. & Doornik, J.A., 1998. "Statistical Algorithms for Models in State Space Using SsfPack 2.2," Discussion Paper 1998-141, Tilburg University, Center for Economic Research.
  3. Nyblom, Jukka & Harvey, Andrew, 1999. "Tests of Common Stochastic Trends," Cambridge Working Papers in Economics 9902, Faculty of Economics, University of Cambridge.
  4. Durbin, James & Koopman, Siem Jan, 2001. "Time Series Analysis by State Space Methods," OUP Catalogue, Oxford University Press, number 9780198523543.
  5. Valle e Azevedo, Joao & Koopman, Siem Jan & Rua, Antonio, 2006. "Tracking the Business Cycle of the Euro Area: A Multivariate Model-Based Bandpass Filter," Journal of Business & Economic Statistics, American Statistical Association, vol. 24, pages 278-290, July.
  6. Johansen, Soren, 1988. "Statistical analysis of cointegration vectors," Journal of Economic Dynamics and Control, Elsevier, vol. 12(2-3), pages 231-254.
  7. McElroy, Tucker & Sutcliffe, Andrew, 2006. "An iterated parametric approach to nonstationary signal extraction," Computational Statistics & Data Analysis, Elsevier, vol. 50(9), pages 2206-2231, May.
  8. McElroy, Tucker, 2008. "Matrix Formulas For Nonstationary Arima Signal Extraction," Econometric Theory, Cambridge University Press, vol. 24(04), pages 988-1009, August.
  9. Michael T. Kiley, 2008. "Estimating the common trend rate of inflation for consumer prices and consumer prices excluding food and energy prices," Finance and Economics Discussion Series 2008-38, Board of Governors of the Federal Reserve System (U.S.).
  10. Trimbur, Thomas M., 2010. "Stochastic level shifts and outliers and the dynamics of oil price movements," International Journal of Forecasting, Elsevier, vol. 26(1), pages 162-179, January.
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Cited by:
  1. Tucker McElroy & Michael W. McCracken, 2012. "Multi-step ahead forecasting of vector time series," Working Papers 2012-060, Federal Reserve Bank of St. Louis.

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