This paper is concerned with frequency domain analysis of Markov mean-switching autoregressive (MMSAR) models, linear Markov switching autoregressive (LMSAR) model and transitional Markov switching autoregressive (TMSAR) model. We derive the general expressions of autocovariance functions and spectra for these three models. Simulation studies of theoretical spectral density functions of these three models are presented. The results show that Markov chain seems to be the most important determinants of the frequency distribution of the volatility. A time series is analysed and both smoothed periodogram and theoretical spectra (of LMSAR and TMSAR models) show similar pattern and give clear ideas of business cycle.
AndersonT.W., Goodness-of-fit tests for autoregressive processes, Journal of Time Series Analysis18(4) (1997), 321-339.
2.
CavicchioliM., Spectral density of Markov-switching VARMA models, Economics Letters121 (2013), 218-220.
3.
CavicchioliM., Autoregressive and linear transformations of Markov switching VARMA arocesses, Central European Journal of Economic Modelling and Econometrics6 (2014), 275-289.
4.
ChengJ., A transitional Markov switching autoregressive model, Communications in Statistics Theory and Methods45(10) (2016), 2785-2800.
5.
CooleyJ.W. and TukeyJ.W., An algorithm for the machine calculation of complex fourier series, Math Comp19 (1965), 297-301.
6.
CostaO.L.V., FragosoM.D. and MarquesR.P., Discrete-Time Markov Jump Linear Systems, Springer, London, 2005.
7.
FrancqC. and Zako"ianJ.M., Stationarity of multivariate Markov-switching ARMA models, Journal of Econometrics102 (2001), 339-364.
8.
FrancqC. and Zako"ianJ.M., Comments on the paper by minxian yang ``Some properties of vector autoregressive processes with Markov switching coefficients'', Econometric Theory18 (2002), 815-818.
9.
DonohoD.L. and JohnstoneI.M., Ideal spatial adaptation by wavelet shrinkage, Biometrika81 (1994), 425-455.
10.
GaoH.Y., Wavelet estimation of spectral densities in time series analysis, Ph.D. Thesis, University of California, Berkeley, CA, 1993.
11.
GaoH.Y., Choice of thresholds for wavelet shrinkage estimate of the spectrum, Journal of Time Series Analysis18(3) (1997), 231-251.
12.
GarciaR. and PerronP., An analysis of the real interest rate under regime shifts, The Review of Economics and Statistics78(1) (1996), 111-125.
13.
HamiltonJ., A new approach to the economic analysis of nonstationary time series and the business cycle, Econometrica57 (1989), 357-384.
14.
HamiltonJ., Analysis of time series subject to changes in regime, Journal of Econometrics45 (1990), 39-70.
15.
HamiltonJ., Time Series Analysis, Princeton University Press, 1994.
16.
HamiltonJ., Specification testing in Markov-switching time-series models, Journal of Econometrics70 (1996), 127-157.
17.
IoannidisE.E., Spectra of bivariate VAR(p) models, Journal of Statistical Planning and Inference137(2) (2007), 554-566.
18.
KimC., Dynamic linear models with Markov-switching, Journal of Econometrics60(1-2) (1994), 1-22.
19.
PataracchiaB., The spectral representation of Markov switching ARMA models, Economics Letters112 (2011), 11-15.
20.
PriestleyM.B., Spectral analysis and time series, Volume 1, Univariate series, Academic Press, Inc., 1981.
21.
YangM., Some properties of vector autoregressive processes with Markov-switching coefficients, Econometric Theory16 (2000), 23-43.
