×

Properties of higher order stochastic cycles. (English) Zbl 1113.62110

The author analyses higher order cycles and derives expressions for autocorrelation functions, spectra and initial covariance matrices of the state vector. An unobserved component \(\psi_{n,t}\) is an \(n\)-th order stochastic cycle, for positive integer \(n\), if \[ \begin{aligned} \left[\begin{matrix}\psi_{1,t}\\ \psi_{1,t}^{*} \end{matrix}\right]&= \rho \left[\begin{matrix}\cos\lambda_{c} & \sin\lambda_{c}\\ -\sin\lambda_{c} & \cos\lambda_{c} \end{matrix} \right] \left[\begin{matrix}\psi_{1,t-1}\\ \psi_{1,t-1}^{*} \end{matrix} \right]+ \left[\begin{matrix}\kappa_{t}\\ \kappa_{t}^{*} \end{matrix} \right], \quad t=1,2,\dots,T,\\ \left[\begin{matrix}\psi_{i,t}\\ \psi_{i,t}^{*} \end{matrix} \right]&= \rho\left[\begin{matrix}\cos\lambda_{c} & \sin\lambda_{c}\\ -\sin\lambda_{c} & \cos\lambda_{c} \end{matrix} \right]\left[\begin{matrix} \psi_{i,t-1}\\ \psi_{i,t-1}^{*} \end{matrix} \right]+\left[\begin{matrix}\psi_{i-1,t-1}\\ \psi_{i-1,t-1}^{*} \end{matrix} \right],\quad i=2,\dots,n,\;t=1,2,\dots,T, \end{aligned} \]
where \(\kappa_{t}\) and \(\kappa_{t}^{*}\) are uncorrelated white-noise processes with mean zero and variance \(\sigma_{\kappa}^2\); \(0<\rho<1\) is the damping factor; \(0\leq\lambda_{c}\leq\pi\). The \(n\)-th order cyclical state vector is given by \[ { \psi}_{t}=(\psi_{n,t},\psi_{n,t}^{*},\psi_{n-1,t},\psi_{n-1,t}^{*},\dots, \psi_{1,t},\psi_{1,t}^{*})'. \] The author presents the key properties of higher order cycles, with general analytical expressions derived for the spectrum and autocorrelation function. The analysis is extended to multivariate models.

MSC:

62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
62M15 Inference from stochastic processes and spectral analysis
62M20 Inference from stochastic processes and prediction

Software:

STAMP
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] DOI: 10.1162/003465399558454 · doi:10.1162/003465399558454
[2] DOI: 10.1198/073500101681019909 · Zbl 04566951 · doi:10.1198/073500101681019909
[3] Harvey A. C., Forecasting, Structural Time Series Models and the Kalman Filter (1989)
[4] Harvey A. C., Journal of Applied Econometrics 8 pp 231– (1993)
[5] Harvey A. C., System Dynamics in Economic and Financial Models (1997)
[6] DOI: 10.1111/1467-9892.00106 · Zbl 0913.62082 · doi:10.1111/1467-9892.00106
[7] DOI: 10.1162/003465303765299774 · doi:10.1162/003465303765299774
[8] A. C. Harvey, T. M. Trimbur, and H. K. Dijk(2003 ) Cyclical components in economic time series: a Bayesian approach . DAE Discussion Paper 0302, Cambridge.
[9] Kitagawa G., Smoothness Priors Analysis of Time Series (1996) · Zbl 0853.62069 · doi:10.1007/978-1-4612-0761-0
[10] Koopman S. J., STAMP 6.0 Structural Time Series Analysis Modeller and Predictor (2000)
[11] Murray C. J., Review of Economics and Statistics 85 pp 471– (2003)
[12] T. M. Trimbur(2003 ) Cycles and trends in time series. PhD Dissertation , University of Cambridge.
[13] Young P., Recursive Estimation and Time Series Analysis (1984) · Zbl 0544.62081 · doi:10.1007/978-3-642-82336-7
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.