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THE LINEAR SYSTEMS APPROACH TO LINEAR RATIONAL EXPECTATIONS MODELS

Published online by Cambridge University Press:  17 April 2017

Majid M. Al-Sadoon
Majid M. Al-Sadoon*
Affiliation:
Universitat Pompeu Fabra & Barcelona GSE
*
*Address correspondence to Majid M. Al-Sadoon, Department of Economics and Business, Universitat Pompeu Fabra, Barcelona, Spain; e-mail: majid.alsadoon@gmail.com.
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Abstract

This paper considers linear rational expectations models from the linear systems point of view. Using a generalization of the Wiener-Hopf factorization, the linear systems approach is able to furnish very simple conditions for existence and uniqueness of both particular and generic linear rational expectations models. To illustrate the applicability of this approach, the paper characterizes the structure of stationary and cointegrated solutions, including a generalization of Granger’s representation theorem.

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ARTICLES
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Econometric Theory , Volume 34 , Issue 3 , June 2018 , pp. 628 - 658
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Copyright © Cambridge University Press 2017 

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Footnotes

I am grateful to Peter Phillips, Robert Taylor, and to three anonymous referees for their insightful comments and helpful suggestions. Thanks are also due to Manfred Deistler, Brian D. O. Anderson, Hashem Pesaran, Fabio Canova, Anastasia Kisil, Barbara Rossi, Alexei Onatski, Geert Mesters, Christian Brownlees, Davide Debortoli, Gábor Lugosi, Omiros Papaspiliopoulos, Todd Walker, and Fei Tan as well as seminar participants at Universitat Pompeu Fabra for helpful comments and suggestions. Any remaining errors are the author’s sole responsibility. This paper is dedicated to the memory of Marcelo Reyes (1962–2015). Research for this paper was supported by Spanish Ministry of Economy and Competitiveness projects ECO2012-33247 and ECO2015-68136-P (MINECO/FEDER, UE) and Fundación BBVA Scientific Research Grant PR16-DAT-0043.

References

REFERENCES

Ahlfors, L.V. (1979) Complex Analysis, 3rd ed. Mathematics Series. McGraw Hill International Edition.Google Scholar
Al-Sadoon, M.M. (2014) Geometric and long run aspects of granger causality. Journal of Econometrics 178, Part 3(0), 558–568.CrossRefGoogle Scholar
Anderson, B.D.O. & Moore, J.B. (1979) Optimal Filtering. Prentice-Hall, Inc.Google Scholar
Binder, M. & Pesaran, M.H. (1995) Multivariate rational expectations models and macroeconomic modelling: A review and some new results. In Pesaran, M.H. & Wickens, M.R. (eds.), Handbook of Applied Econometrics. Volume 1: Macroeconomics, chapter 3, pp. 139–187. Blackwell Publishers Ltd.Google Scholar
Binder, M. & Pesaran, M.H. (1997) Multivariate linear rational expectations models. Econometric Theory 13(06), 877–888.CrossRefGoogle Scholar
Blanchard, O.J. & Fischer, S. (1989) Lectures on Macroeconomics. MIT Press.Google Scholar
Blanchard, O.J. & Kahn, C.M. (1980) The solution of linear difference models under rational expectations. Econometrica 48(5), 1305–1311.Google Scholar
Broze, L., Gouriérou, C., & Szafarz, A. (1990) Reduced Forms of Rational Expectations Models. Fundamentals of Pure and Applied Economics, vol. 42. Harwood Academic Publishers.Google Scholar
Broze, L., Gourieroux, C., & Szafarz, A. (1985) Solutions of linear rational expectations models. Econometric Theory 1(3), 341–368.CrossRefGoogle Scholar
Broze, L., Gouriroux, C., & Szafarz, A. (1995) Solutions of multivariate rational expectations models. Econometric Theory 11, 229–257.CrossRefGoogle Scholar
Cagan, P.D. (1956) The monetary dynamics of hyperinflations. In Friedman, M. (ed.), Studies in the Quantity Theory of Money. Chicago University Press.Google Scholar
Clancey, K.F. & Gohberg, I. (1981) Factorization of matrix functions and singular integral operators. In Gohberg, I. (ed.), Operator Theory: Advances and Applications, vol. 3. Birkhäuser Verlag Basel.Google Scholar
Desoer, C.A. & Vidyasagar, M. (2009) Feedback Systems: Input-Output Properties. Classics in Applied Mathematics. SIAM.CrossRefGoogle Scholar
Engle, R. & Yoo, S. (1991) Cointegrated economic time series: An overview with new results. In Engle, R.F. & Granger, C.W.J. (eds.), Long-Run Economic Relationships. Advanced Texts in Econometrics, pp. 237–266. Oxford University Press.Google Scholar
Engle, R.F. & Granger, C.W.J. (1987) Co-integration and error correction: Representation, estimation, and testing. Econometrica 55(2), 251–276.CrossRefGoogle Scholar
Farmer, R.E.A. (1999) Macroeconomics of Self-fulfilling Prophecies, 2nd ed. MIT University Press.Google Scholar
Funovits, B. (2014) Implications of Stochastic Singularity in Linear Multivariate Rational Expectations Models. Technical report, University of Vienna, Department of Economics.Google Scholar
Gohberg, I., Kaashoek, M.A., & Spitkovsky, I.M. (2003) An overview of matrix factorization theory and operator applications. In Gohberg, I., Manojlovic, N., & dos Santos, A.F. (eds.), Factorization and Integrable Systems: Summer School in Faro, Portugal, September 2000. Operator Theory: Advances and Applications, vol. 141, chapter 1, pp. 1–102. Springer Basel AG.CrossRefGoogle Scholar
Gohberg, I. & Krein, M.G. (1960) Systems of integral equations on a half-line with kernel depending upon the difference of the arguments. American Mathematical Society Translations 14(2), 217–287.Google Scholar
Gohberg, I.C. & Fel’dman, I.A. (1974) Convolution Equations and Projection Methods for Their Solution. Translations of Mathematical Monographs, vol. 41. American Mathematical Society.Google Scholar
Haldrup, N. & Salmon, M. (1998) Representations of I(2) cointegrated systems using the smith-mcmillan form. Journal of Econometrics 84(2), 303–325.CrossRefGoogle Scholar
Hall, R.E. (1978) Stochastic implications of the life cycle-permanent income hypothesis: Theory and evidence. Journal of Political Economy 86(6), 971–987.CrossRefGoogle Scholar
Hannan, E.J. & Deistler, M. (2012) The Statistical Theory of Linear Systems. Classics in Applied Mathematics. SIAM.CrossRefGoogle Scholar
Hansen, L.P. & Sargent, T.J. (1980) Formulating and estimating dynamic linear rational expectations models. Journal of Economic Dynamics and Control 2(1), 7–46.CrossRefGoogle Scholar
Hansen, L.P. & Sargent, T.J. (1981) Linear rational expectations models for dynamically interrelated variables. In Lucas, R.E. Jr. & Sargent, T.J. (eds.), Rational Expectations and Econometric Practice, pp. 127–156. University of Minnesota Press.Google Scholar
Hylleberg, S., Engle, R.F., Granger, C.W.J., & Yoo, B.S. (1990) Seasonal integration and cointegration. Journal of Econometrics 44(1–2), 215–238.CrossRefGoogle Scholar
Johansen, S. (1995) Likelihood-Based Inference in Cointegrated Vector Autoregressive Models. Oxford University Press.CrossRefGoogle Scholar
Juselius, M. (2008) Cointegration Implications of Linear Rational Expectation Models. Research Discussion papers 6/2008, Bank of Finland.Google Scholar
Kailath, T. (1980) Linear Systems. Prentice Hall.Google Scholar
Klein, P. (2000) Using the generalized schur form to solve a multivariate linear rational expectations model. Journal of Economic Dynamics and Control 24(10), 1405–1423.CrossRefGoogle Scholar
Lubik, T.A. & Schorfheide, F. (2003) Computing sunspot equilibria in linear rational expectations models. Journal of Economic Dynamics and Control 28(2), 273–285.CrossRefGoogle Scholar
Onatski, A. (2006) Winding number criterion for existence and uniqueness of equilibrium in linear rational expectations models. Journal of Economic Dynamics and Control 30(2), 323–345.CrossRefGoogle Scholar
Pesaran, M.H. (1987) The Limits to Rational Expectations. Basil Blackwell Inc.Google Scholar
Reinsel, G.C. (2003) Elements of Multivariate Time Series Analysis, 2nd ed. Springer Series in Statistics. Springer.Google Scholar
Schumacher, J. (1991) System-theoretic trends in econometrics. In Antoulas, A. (ed.), Mathematical System Theory: The Influence of R.E. Kalman, pp. 559–578. Springer.CrossRefGoogle Scholar
Shiller, R. (1978) Rational expectations and the dynamic structure of macroeconomic models: A critical review. Journal of Monetary Economics 4(1), 1–44.CrossRefGoogle Scholar
Sims, C.A. (2002) Solving linear rational expectations models. Computational Economics 20(1–2), 1–20.CrossRefGoogle Scholar
Sims, C.A. (2007) On the Genericity of the Winding Number Criterion for Linear Rational Expectations Models. Mimeo.Google Scholar
Tan, F. & Walker, T.B. (2015) Solving generalized multivariate linear rational expectations models. Journal of Economic Dynamics and Control 60, 95–111.CrossRefGoogle Scholar
White, H., Al-Sadoon, M.M., & Chalak, K. (2015) Rational Expectations and Causality: A Settable Systems View. Mimeo.Google Scholar
White, H., Chalak, K., & Lu, X. (2011) Linking granger causality and the pearl causal model with settable systems. Journal of Machine Learning Research Workshop and Conference Proceedings 12, 1–29.Google Scholar
White, H. & Lu, X. (2010) Granger causality and dynamic structural systems. Journal of Financial Econometrics 8(2), 193–243.CrossRefGoogle Scholar
White, H. & Pettenuzzo, D. (2014) Granger causality, exogeneity, cointegration, and economic policy analysis. Journal of Econometrics 178, 316–330.CrossRefGoogle Scholar
Whiteman, C.H. (1983) Linear Rational Expectations Models: A User’s Guide. University of Minnesota Press.Google Scholar
Whiteman, C.H. (1985) Spectral utility, wiener-hopf techniques, and rational expectations. Journal of Economic Dynamics and Control 9(2), 225–240.CrossRefGoogle Scholar
Williams, D. (1991) Probability with Martingales. Cambridge University Press.CrossRefGoogle Scholar
Youla, D., Bongiorno, J.J., & Jabr, H. (1976) Modern wiener–hopf design of optimal controllers part I: The single-input-output case. IEEE Transactions on Automatic Control 21(1), 3–13.CrossRefGoogle Scholar
Youla, D., Jabr, H., & Bongiorno, J.J. (1976) Modern wiener-hopf design of optimal controllers–part II: The multivariable case. IEEE Transactions on Automatic Control 21(3), 319–338.CrossRefGoogle Scholar
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