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MULTIVARIATE AR SYSTEMS AND MIXED FREQUENCY DATA: G-IDENTIFIABILITY AND ESTIMATION

Published online by Cambridge University Press:  02 April 2015

Brian D.O. Anderson ,
Manfred Deistler ,
Elisabeth Felsenstein ,
Bernd Funovits ,
Lukas Koelbl  and
Mohsen Zamani
Brian D.O. Anderson
Affiliation:
Australian National University and National ICT Australia Ltd.
Manfred Deistler*
Affiliation:
Vienna University of Technology and Institute for Advanced Studies
Elisabeth Felsenstein
Affiliation:
Vienna University of Technology
Bernd Funovits
Affiliation:
University of Vienna and Vienna University of Technology
Lukas Koelbl
Affiliation:
Vienna University of Technology
Mohsen Zamani
Affiliation:
Australian National University
*
*Address correspondence to Manfred Deistler, Institute of Statistics and Mathematical Methods in Economics, Unit: Econometrics and System Theory, Vienna University of Technology, Vienna, Austria; e-mail: [email protected].
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Abstract

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This paper is concerned with the problem of identifiability of the parameters of a high frequency multivariate autoregressive model from mixed frequency time series data. We demonstrate identifiability for generic parameter values using the population second moments of the observations. In addition we display a constructive algorithm for the parameter values and establish the continuity of the mapping attaching the high frequency parameters to these population second moments. These structural results are obtained using two alternative tools viz. extended Yule Walker equations and blocking of the output process. The cases of stock and flow variables, as well as of general linear transformations of high frequency data, are treated. Finally, we briefly discuss how our constructive identifiability results can be used for parameter estimation based on the sample second moments.

Type
ET LECTURE
Information
Econometric Theory , Volume 32 , Issue 4 , August 2016 , pp. 793 - 826
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/3.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
Copyright © Cambridge University Press 2015

Footnotes

The authors want to thank Hans Havlicek, Vienna University of Technology, and Benedikt Pötscher, University of Vienna, for valuable suggestions concerning the proof of Theorem 2. In addition we thank Peter Phillips, several anonymous referees and handling co-editors for valuable comments.

Support by the FWF (Austrian Science Fund under contracts P20833/N18 and P24198/N18), the ARC (Australian Research Council under Discovery Project Grant DP1092571) and NICTA is gratefully acknowledged. NICTA is funded by the Australian Government as represented by the Department of Broadband, Communications and the Digital Economy and the ARC through the ICT Centre of Excellence program.

References

REFERENCES

Akaike, H. (1974) Stochastic theory of minimal realization. IEEE Transactions on Automatic Control 19(6), 667–674.CrossRefGoogle Scholar
Amemiya, T. & Wu, R.Y. (1972) The effect of aggregation on prediction in the autoregressive model. Journal of the American Statistical Association 67(339), 628–632.CrossRefGoogle Scholar
Anderson, B.D.O., Deistler, M., Felsenstein, E., Funovits, B., Zadrozny, P.A., Eichler, M., Chen, W., & Zamani, M. (2012) Identifiability of regular and singular multivariate autoregressive models. doi:10.1109/CDC.2012.6426713.CrossRefGoogle Scholar
Aruoba, S.B., Diebold, F.X., & Scotti, C. (2007) Real-time measurement of business conditions. Journal of Business and Economic Statistics 27(4), 417–427.CrossRefGoogle Scholar
Bai, J., Ghysels, E., & Wright, J.B. (2013) State space models and MIDAS regressions. Econometric Reviews 32(7), 779–813.CrossRefGoogle Scholar
Bergstrom, A. (1988) The history of continuous-time econometric models. Econometric Theory 4, 365–383.CrossRefGoogle Scholar
Bernanke, B.S., Gertler, M., Watson, M.W., Sims, C.A., & Friedman, B.M. (1997) Systematic monetary policy and the effects of oil price shocks. In Brainard, W.C. and Perry, G. L. (eds.), Brookings Papers on Economic Activity. Brookings Institution Press, 91–157.Google Scholar
Bittanti, S., Colaneri, P., & De Nicolao, G. (1988) The difference periodic Riccati equation for the periodic prediction problem. IEEE Transactions on Automatic Control 33(8), 706–712.CrossRefGoogle Scholar
Bochnak, J., Coste, M., & Roy, M.-F. (1998) Real Algebraic Geometry. Springer Verlag.CrossRefGoogle Scholar
Brockwell, P.J. (1995) A note on the embedding of discrete time ARMA processes. Journal of Time Series Analysis 16(5), 451–460.CrossRefGoogle Scholar
Chambers, M.J. & Thornton, M.A. (2012) Discrete time representation of continuous time ARMA processes. Econometric Theory 28, 219–238.CrossRefGoogle Scholar
Chen, W., Anderson, B.D.C., Deistler, M., & Filler, A. (2012) Properties of blocked linear systems. Automatica 48, 2520–2525.CrossRefGoogle ScholarPubMed
Chen, B. & Zadrozny, P.A. (1998) An extended Yule-Walker method for estimating a vector autoregressive model with mixed-frequency. Advances in Econometrics 13, 47–73.CrossRefGoogle Scholar
Deistler, M., Anderson, B.D.C., Filler, A., Zinner, C., & Chen, W. (2010) Generalized linear dynamic factor models - An approach via singular autoregressions. European Journal of Control 16(3), 211–224.CrossRefGoogle Scholar
Deistler, M., Filler, A., & Funovits, B. (2011) AR systems and AR processes: The singular case. Communications in Information and Systems 11(3), 225–236.CrossRefGoogle Scholar
Deistler, M., Peternell, K., & Scherrer, W. (1995) Consistency and relative efficiency of subspace methods. Automatica 31(12), 1865–1875.CrossRefGoogle Scholar
Deistler, M. & Seifert, H.-G. (1978) Identifiability and consistent estimability in econometric models. Econometrica 46(6), 969–980.CrossRefGoogle Scholar
Doz, C., Giannone, D., & Reichlin, L. (2011) A two-step estimator for large approximate dynamic factor models based on Kalman filtering. Journal of Econometrics 164(1), 188–205.CrossRefGoogle Scholar
Felsenstein, E. (2014) Regular and singular AR and ARMA models: The single and the mixed frequency case. PhD thesis, Vienna University of Technology.Google Scholar
Filler, A. (2010) Generalized dynamic factor models - structure theory and estimation for single frequency and mixed frequency data. PhD thesis, Vienna University of Technology.Google Scholar
Forni, M., Hallin, M., Lippi, M., & Reichlin, L. (2000) The generalized dynamic factor model: Identification and estimation. Review of Economics and Statistics 82(4), 540–554.CrossRefGoogle Scholar
Ghysels, E. (2012) Macroeconomics and the Reality of Mixed Frequency Data. Working paper, Department of Economics, University of North Carolina.CrossRefGoogle Scholar
Ghysels, E., Hill, J.B., & Motegi, K. (2014) Testing for Granger Causality in Mixed Frequency. Working paper.CrossRefGoogle Scholar
Ghysels, E., Santa-Clara, P., & Valkanov, R. (2006) Predicting volatility: Getting the most out of return data sampled at different frequencies. Journal of Econometrics 131(1), 59–95.CrossRefGoogle Scholar
Ghysels, E., Sinko, A., & Valkanov, R. (2007) MIDAS regressions: Further results and new directions. Econometric Reviews 26(1), 53–90.CrossRefGoogle Scholar
Ghysels, E. & Wright, J.H. (2009) Forecasting professional forecasters. Journal of Business and Economic Statistics 27(4), 504–516.CrossRefGoogle Scholar
Hannan, E.J. (1970) Multiple Time Series. Wiley.CrossRefGoogle Scholar
Hannan, E.J. & Deistler, M. (2012) The Statistical Theory of Linear Systems. SIAM Classics in Applied Mathematics. Society for Industrial and Applied Mathematics.CrossRefGoogle Scholar
Hansen, L.P. & Sargent, T.J. (1983) The dimensionality of the aliasing problem in models with rational spectral densities. Econometrica 51(2), 377–387.CrossRefGoogle Scholar
Harvey, A.C. & Pierse, R.G. (1984) Estimating missing observations in economic time series. Journal of the American Statistical Association 79(385), 125–131.CrossRefGoogle Scholar
Ho, B.L. & Kalman, R.E. (1966) Effective construction of linear state-variable models from input/output functions. Regelungstechnik 14, 545–548.Google Scholar
Kailath, T. (1980) Linear Systems. Prentice Hall.Google Scholar
Kalman, R.E. (1965) Irreducible realizations and the degree of a rational matrix. Journal of the Society for Industrial & Applied Mathematics 13(2), 520–544.CrossRefGoogle Scholar
Kohn, R. & Ansley, C.F. (1986) Estimation, prediction and interpolation for ARIMA models with missing data. Journal of the American Statistical Association 81(395), 751–761.CrossRefGoogle Scholar
Lee, E.B. & Markus, L. (1967) Foundations of Optimal Control Theory. Wiley.Google Scholar
Marcellino, M. (1998) Temporal disaggregation, missing observations, outliers, and forecasting: A unifying non-model based procedure. Advances in Econometrics 13, 181–202.CrossRefGoogle Scholar
Marcellino, M. & Schumacher, C. (2010) Factor MIDAS for nowcasting and forecasting with ragged-edge data: A model comparison for German GDP. Oxford Bulletin of Economics and Statictics 72(4), 518–550.CrossRefGoogle Scholar
Mariano, R.S. & Murasawa, Y. (2003) A new coincident index of business cycles based on monthly and quarterly series. Journal of Applied Econometrics 18(4), 427–443.CrossRefGoogle Scholar
Nijman, T.E. (1985) Missing observations in dynamic macroeconomic modeling. PhD thesis, VU University Amsterdam.Google Scholar
Phillips, P.C.B. (1973) The problem of identification in finite parameter continuous time models. Journal of Econometrics 1(4), 351–362.CrossRefGoogle Scholar
Phillips, P.C.B. (1974) The estimation of some continuous time models. Econometrica 42(5), 803–823.CrossRefGoogle Scholar
Phillips, P.C.B. (1976) Some computations based on observed data series of the exogenous variable component in continuous systems. In Bergstrom, A. R. (ed), Statistical Inference in Continuous Time Economic Models. North Holland, 175–214.Google Scholar
Rozanov, Y.A. (1967) Stationary Random Processes. Holden-Day.Google Scholar
Stock, J.H. & Watson, M.W. (2002) Forecasting using principal components from a large number of predictors. Journal of the American Statistical Association 97(460), 1167–1179.CrossRefGoogle Scholar
Tiao, G.C. (1972) Asymptotic behaviour of temporal aggregates of time series. Biometrika 59(3), 525–531.CrossRefGoogle Scholar
Wohlrabe, K. (2008) Forecasting with mixed-frequency time series models. PhD thesis, Ludwig-Maximilians-Universitt Mnchen.Google Scholar
Wonham, W.M. (1985) Linear Multivariable Control - A Geometric Approach, 3rd ed. Springer Verlag.CrossRefGoogle Scholar
Zadrozny, P.A. (1988) Gaussian likelihood of continuous-time ARMAX models when data are stocks and flows at different frequencies. Econometric Theory 4(1), 108–124.CrossRefGoogle Scholar
Zadrozny, P.A. (1990a) Estimating a multivariate ARMA model with mixed frequency data: An application to forecasting U.S. GNP at monthly intervals. Federal Reserve Bank of Atlanta, 90(6), 1–58.Google Scholar
Zadrozny, P.A. (1990b) Forecasting US GNP at monthly intervals with an estimated bivariate time series model. Economic Review 75, 2–15.Google Scholar
Zamani, M. (2014) Modeling multivariable time series using regular and singular autoregressions. PhD thesis, The Australian National University.Google Scholar