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A UNIFIED THEORY FOR ARMA MODELS WITH VARYING COEFFICIENTS: ONE SOLUTION FITS ALL

Published online by Cambridge University Press:  27 February 2025

Menelaos Karanasos ,
Alexandros G. Paraskevopoulos ,
Tassos Magdalinos  and
Alessandra Canepa
Menelaos Karanasos
Affiliation:
Department of Economics and Finance, Brunel University London
Alexandros G. Paraskevopoulos
Affiliation:
University of Piraeus
Tassos Magdalinos*
Affiliation:
University of Southampton
Alessandra Canepa
Affiliation:
University of Turin
*
Address correspondence to Tassos Magdalinos, University of Southampton, Southampton, UK; e-mail: a.magdalinos@soton.ac.uk
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Abstract

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A new explicit solution representation is provided for ARMA recursions with drift and either deterministically or stochastically varying coefficients. It is expressed in terms of the determinants of banded Hessenberg matrices and, as such, is an explicit function of the coefficients. In addition to computational efficiency, the proposed solution provides a more explicit analysis of the fundamental properties of such processes, including their Wold–Cramér decomposition, their covariance structure, and their asymptotic stability and efficiency. Explicit formulae for optimal linear forecasts based either on finite or infinite sequences of past observations are provided. The practical significance of the theoretical results in this work is illustrated with an application to U.S. inflation data. The main finding is that inflation persistence increased after 1976, whereas from 1986 onward, the persistence declines and stabilizes to even lower levels than the pre-1976 period.

Type
ARTICLES
Information
Econometric Theory , First View , pp. 1 - 54
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 (https://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2025. Published by Cambridge University Press

Footnotes

The first two author names are in alphabetical order. We thank the Editor, the Co-Editor, and two anonymous referees for their thoughtful comments and constructive suggestions. We gratefully acknowledge the helpful conversations we had with L. Giraitis, and G. Kapetanios in the preparation of the paper. We have also benefited from the comments given by participants (R. Baillie, M. Brennan, L. Bauwens, A. Demos, W. Distaso, D. van Dijk, C. Francq, P. Fryzlewicz, E. Guerre, C. Gourieroux, M. Guidolin, A. Harvey, C. Hommes, E. Kyriazidou, S. Leybourne, P. Marsh, P. Minford, C. Robotti, W. Semmler, R. Smith, T. Teräsvirta, E. Tzavalis, P. Zaffaroni, and J.-M. Zakoian) at various seminars and conferences. We would like to thank S. Dafnos, who contributed to the interpretation and synthesis in the initial version of the paper, and P. Koutroumpis, who provided the empirical section in an earlier version. Magdalinos gratefully acknowledges financial support by the British Academy: grant SRG2324\241667. Alessandra Canepa acknowledges financial support under the National Recovery and Resilience Plan (NRRP), Mission 4, Component 2, Investment 1.1, Call for tender No. 104 published on February 2, 2022 by the Italian Ministry of University and Research (MUR), funded by the European Union—NextGenerationEU—Project Title 20223725WE—Methodological and computational issues in large-scale time series models for economics and finance – CUP J53D23003960006—Grant Assignment Decree No. 967 adopted on June 30, 2023 by the Italian Ministry of Ministry of University and Research (MUR).

References

REFERENCES

Abdrabbo, N. A., & Priestley, M. B. (1967). On the prediction of non-stationary processes. Journal of the Royal Statistical Society , 29, 570–585.CrossRefGoogle Scholar
Alj, A., Azrak, R., Ley, C., & Mélard, G. (2017). Asymptotic properties of QML estimators for VARMA models with time-dependent coefficients. Scandinavian Journal of Statistics , 44, 617–635.CrossRefGoogle Scholar
Andĕl, J. (1991). On stationarity of a multiple doubly stochastic model. Kybernetica , 2, 114–119.Google Scholar
Angeloni, I., Aucremanne, L., Ehrmann, M., Galí, J., Levin, A., & Smets, F. (2006). New evidence on inflation persistence and price stickiness in the euro area: Implications for macro modelling. Journal of European Economic Association , 4, 562–574.CrossRefGoogle Scholar
Azrak, R., & Mélard, G. (2006). Asymptotic properties of quasi-maximum likelihood estimators for ARMA models with time-dependent coefficients. Statistical Inference for Stochastic Processes , 9, 279–330.CrossRefGoogle Scholar
Bai, J., & Perron, P. (2003). Computation and analysis of multiple structural change models. Journal of Applied Econometrics , 18, 1–22.CrossRefGoogle Scholar
Baillie, R. T., Han, Y. W., & Kwon, T. G. (2002). Further long memory properties of inflationary shocks. Southern Economic Journal , 68, 496–510.Google Scholar
Brainard, W. C., & Perry, G. L. (2000). Making policy in a changing world. In Perry, G. L., & Tobin, J. (Eds.), Economic events, ideas, and policies: The 1960s and after (pp. 43–82). Brookings Institution Press.Google Scholar
Brockwell, P. J., & Davis, R. A. (2016). An introduction to time series and forecasting . Springer.CrossRefGoogle Scholar
Canarella, G., & Miller, S. F. (2017). Inflation targeting: New evidence from fractional integration and cointegration. Journal of Economics and Business , 92, 45–62.CrossRefGoogle Scholar
Canepa, A., Karanasos, M., Paraskevopoulos, A. G., & Zanetti Chini, E. (2022). Forecasting inflation: A GARCH-in mean-level model with time-varying predictability (Publication No. 11) [Working paper]. University of Bergamo.Google Scholar
Cavaliere, G., & Georgiev, I. (2008). Regime switching autoregressive coefficients and the asymptotics for unit root tests. Econometric Theory , 24, 1137–1148.CrossRefGoogle Scholar
Cavaliere, G., & Taylor, A. M. R. (2005). Stationarity tests under time-varying variances. Econometric Theory , 21, 1112–1129.CrossRefGoogle Scholar
Chambers, M. J., & Taylor, A. M. R. (2020). Deterministic parameter change models in continuous and discrete time. Journal of Time Series Analysis , 41, 134–145.CrossRefGoogle Scholar
Cogley, T., & Sargent, T. J. (2002). Evolving post-World War II U.S. inflation dynamics. In Getler, M., & Rogoff, K. (Eds.), NBER Macroeconomics Annual (pp. 331–373). MIT Press.Google Scholar
Conrad, C., & Karanasos, M. (2010). Negative volatility spillovers in the unrestricted ECCC-GARCH model. Econometric Theory , 26, 838–862.CrossRefGoogle Scholar
Cramér, H. (1961). On some classes of nonstationary stochastic processes. Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability , 2, 57–78.Google Scholar
Dahlhaus, R. (1997). Fitting time series models to nonstationary processes. Annals of Statistics , 25, 1–37.CrossRefGoogle Scholar
Devpura, N., Sharma, S. S., Harischandra, P. K. G., & Pathberiya, L. R. C. (2021). Is inflation persistent? Evidence from a time-varying unit root model. Pacific-Basin Finance Journal , 68(C), Article 101577.CrossRefGoogle Scholar
Diebold, F. X., & Inoue, A. (2001). Long memory and regime switching. Journal of Econometrics , 105, 131–159.CrossRefGoogle Scholar
Elaydi, S. (2005). An introduction to difference equations . Springer.Google Scholar
Erhardsson, T. (2014). Conditions for convergence of random coefficient AR(1) processes and perpetuities in higher dimensions. Bernoulli , 20, 990–1005.CrossRefGoogle Scholar
Fuhrer, J., & Moore, G. (1995). Inflation persistence. The Quarterly Journal of Economics , 110, 127–159.CrossRefGoogle Scholar
Fuller, W. A. (1996). Introduction to statistical time series . John Wiley & Sons.Google Scholar
Giraitis, L., Kapetanios, G., & Yates, T. (2014). Inference on stochastic time-varying coefficient models. Journal of Econometrics , 179, 46–65.CrossRefGoogle Scholar
Glasserman, P., & Yao, D. D. (1995). Stochastic vector difference equations with stationary coefficients. Journal of Applied Probability , 32, 851–866.CrossRefGoogle Scholar
Goerg, G. M. (2010). Time series analysis of long memory versus structural breaks: A time-varying memory approach . VDM Verlag Dr. Müller.Google Scholar
Granger, C. W. J., & Hyung, N. (2004). Occasional structural breaks and long memory with an application to the S&P 500 absolute stock returns. Journal of Empirical Finance , 11, 399–421.CrossRefGoogle Scholar
Gray, R. M. (2006). Toeplitz and circulant matrices: A review. Foundation and Trends in Communications and Information Theory , 2, 155–239.CrossRefGoogle Scholar
Grillenzoni, C. (1990). Modelling time-varying dynamical systems. Journal of the American Statistical Association , 85, 499–507.CrossRefGoogle Scholar
Grillenzoni, C. (1993). ARIMA processes with ARIMA parameters. Journal of Business and Economic Statistics , 11, 235–250.CrossRefGoogle Scholar
Grillenzoni, C. (2000). Time-varying parameters prediction. Annals of the Institute of Statistical Mathematics , 52, 108–122.CrossRefGoogle Scholar
Hallin, M. (1979). Mixed autoregressive-moving average multivariate processes with time-dependent coefficients. Journal of Multivariate Analysis , 8, 567–572.CrossRefGoogle Scholar
Hallin, M. (1984). Spectral factorization of nonstationary moving average processes. Annals of Statistics , 12, 172–192.CrossRefGoogle Scholar
Hallin, M. (1986). Non-stationary q-dependent processes and time-varying moving-average models: Invertibility properties and the forecasting problem. Advances in Applied Probability , 18, 170–210.CrossRefGoogle Scholar
Hamilton, J. D. (1994). Time series analysis . Princeton University Press.CrossRefGoogle Scholar
Han, C. (2007). Determinants of covariance matrices of differenced AR (1) processes. Econometric Theory , 23, 1248–1253.CrossRefGoogle Scholar
Harvey, D., Leybourne, S., Georgiev, I., & Taylor, A. M. R. (2018). Testing for parameter instability in predictive regression models. Journal of Econometrics , 204, 101–118.Google Scholar
He, C., & Teräsvirta, T. (2004). An extended constant conditional correlation GARCH model and its fourth-moment structure. Econometric Theory , 20, 904–926.CrossRefGoogle Scholar
Hsu, C. C. (2005). Long memory or structural changes: An empirical examination on inflation rates. Economics Letters , 88, 289–294.CrossRefGoogle Scholar
Hwang, S. Y., & Basawa, I. V. (1998). Parameter estimation for generalized random coefficient autoregressive processes. Journal of Statistical Planning and Inference , 68, 323–337.CrossRefGoogle Scholar
Iacone, F., Leybourne, S. J., & Taylor, A. M. R. (2019). Testing the order of fractional integration of a time series in the possible presence of a trend break at an unknown point. Econometric Theory , 35, 1201–1233.CrossRefGoogle Scholar
Jeerawat, U., & Daowsud, K. (2022). Determinants of some Hessenberg matrices with generating functions. Special Matrices , 11, 1–8.Google Scholar
Karanasos, M. (1998). A new method for obtaining the autocovariance of an ARMA model: An exact form solution. Econometric Theory , 14, 622–640.CrossRefGoogle Scholar
Karanasos, M. (1999). The second moment and the autocovariance function of the squared errors of the GARCH model. Journal of Econometrics , 90, 63–76.CrossRefGoogle Scholar
Karanasos, M. (2001). Prediction in ARMA models with GARCH in mean effects. Journal of Time Series Analysis , 22, 555–576.CrossRefGoogle Scholar
Karanasos, M., Hu, Y., & Yfanti, S. (2022a). Constrained QML for vector MEM: The practicality of matrix inequality constraints. Unpublished paper.Google Scholar
Karanasos, M., Koutroumpis, P., Karavias, Y., Kartsaklas, A., & Arakelian, V. (2016). Inflation convergence in the EMU. Journal of Empirical Finance , 39, 241–253.CrossRefGoogle Scholar
Karanasos, M., Paraskevopoulos, A. G., & Canepa, A. (2020). A unified theory for the large family of time-varying models with ARMA representations: One solution fits all. Cognetti de Martiis Department of Economics and Statistics.Google Scholar
Karanasos, M., Paraskevopoulos, A. G., & Dafnos, S. (2014a). A univariate time-varying analysis of periodic ARMA processes. Preprint, arXiv:1403.4803.CrossRefGoogle Scholar
Karanasos, M., Paraskevopoulos, A. G., & Dafnos, S. (2022). The fundamental properties of AR models with deterministically varying coefficients. In Pempetzoglou, M., & Hristou-Varsakelis, D. (Eds.), Essays in economic theory and policy in honor of Professor Stella Karagianni . Springer.Google Scholar
Karanasos, M., Paraskevopoulos, A. G., Karoglou, M., Menla Ali, F., & Yfanti, S. (2014b). Modelling returns and volatilities during financial crises: A time-varying coefficient approach. Journal of Empirical Finance , 29, 113–128.CrossRefGoogle Scholar
Kittappa, R. K. (1993). A representation of the solution of the nth order linear difference equation with variable coefficients. Linear Algebra and Its Applications , 193, 211–222.CrossRefGoogle Scholar
Kowalski, A., & Szynal, D. (1991). On a characterization of optimal predictors for nonstationary ARMA processes. Stochastic Processes and Their Applications , 37, 71–80.CrossRefGoogle Scholar
Kozicki, S., & Tinsley, P. (2002). Alternative sources of the lag dynamics of inflation (Publication No. 02-12) [Working paper]. FRB Kansas City Research.Google Scholar
Levin, A., & Piger, J. (2004) Is inflation persistence intrinsic in industrial economics? (Publication No. 202-023A) [Working paper]. Federal Reserve Bank of St. Louis.Google Scholar
Lim, A., & Dai, J. (2011). On product of companion matrices. Linear Algebra and Its Applications , 435, 2921–2935.CrossRefGoogle Scholar
Mallik, R. K. (1998). Solutions of linear difference equations with variable coefficients. Journal of Mathematical Analysis and Applications , 222, 79–91.CrossRefGoogle Scholar
Marrero, A. J., & Tomeo, V. (2012). On the closed representation for the inverse of Hessenberg matrices. Journal of Computational and Applied Mathematics , 236, 2962–2970.CrossRefGoogle Scholar
Marrero, A. J., & Tomeo, V. (2017). On compact representations for the solutions of linear difference equations with variable coefficients. Journal of Computational and Applied Mathematics , 318, 411–421.CrossRefGoogle Scholar
Massacci, D., & Trapani, L. (2022). High dimensional threshold regression with common stochastic trends. SSRN. https://ssrn.com/abstract=4133488; https://doi.org/10.2139/ssrn.4133488 CrossRefGoogle Scholar
McConnell, M., & Perez-Quiros, G. (2000). Output fluctuations in the United States: What has changed since the early 1980’s? American Economic Review , 90, 1464–1476.CrossRefGoogle Scholar
Mélard, G. (2024) Time-dependent time series models: Comments on Marc Hallin’s early contributions and a pragmatic view on estimation. In: Barigozzi, M., Hörmann, S. and Paindaveine, D. (Eds.), Recent advances in econometrics and statistics: Festschrift in honour of Marc Hallin . Springer.Google Scholar
Miller, K. S. (1968). Linear difference equations . Benjamin.CrossRefGoogle Scholar
Nelson, D. (2008). The Penguin dictionary of mathematics (4th ed.). Penguin Books Limited.Google Scholar
Paraskevopoulos, A. G. (2012). The infinite Gauss–Jordan elimination on row-finite $\omega \times \omega$ matrices. Preprint, arXiv:1201.2950.Google Scholar
Paraskevopoulos, A. G. (2014). The solution of row-finite linear systems with the infinite Gauss–Jordan elimination. Preprint, arXiv:1403.2624.Google Scholar
Paraskevopoulos, A. G., & Karanasos, M. (2021). Explicit and compact representations for the Green function and the solution of linear difference equations with variable coefficients. Preprint, arXiv:1906.07130.Google Scholar
Peiris, S. (1986). On prediction with time dependent ARMA models. Communications in Statistics – Theory and Methods , 15, 3659–3668.CrossRefGoogle Scholar
Perron, P., & Qu, Z. (2006). Estimating restricted structural change models. Journal of Econometrics , 134, 373–399.CrossRefGoogle Scholar
Petrova, K. (2019). A quasi-Bayesian local likelihood approach to time-varying parameter VAR models. Journal of Econometrics , 212, 286–306.CrossRefGoogle Scholar
Phillips, P. C. B. (2005). Econometric analysis of Fisher’s equation. The American Journal of Economics and Sociology , 64, 125–168.CrossRefGoogle Scholar
Phillips, P. C. B. (2007). Unit root log periodogram regression. Journal of Econometrics , 138, 104–124.CrossRefGoogle Scholar
Pivetta, F., & Reis, R. (2007). The persistence of inflation in the United States. Journal of Economic Dynamics and Control , 31, 1326–1358.CrossRefGoogle Scholar
Rao, S. T. (1970). The fitting of non-stationary time-series models with time-dependent parameters. Journal of the Royal Statistical Society , 32, 312–322.CrossRefGoogle Scholar
Sibbertsen, P. (2004). Long memory versus structural breaks: An overview. Statistical Papers , 45, 465–515.CrossRefGoogle Scholar
Singh, N., & Peiris, M. S. (1987). A note on the properties of some nonstationary ARMA processes. Stochastic Processes and Their Applications , 24, 151–155.CrossRefGoogle Scholar
Stock, J. H., & Watson, M. W. (2002). Has the business cycle changed and why? In Gertler, M., & Rogoff, K. (Eds.), NBER Macroeconomics Annual (pp. 159–218). MIT Press.Google Scholar
Stock, J. H., & Watson, M. W. (2009). Phillips curve inflation forecasts. In Fuhrer, J. C., Kodrzycki, Y., Little, J., & Olivei, G. (Eds.), Understanding inflation and the implications for monetary policy (pp. 99–202). MIT Press.Google Scholar
Sun, Y., & Phillips, P. C. B. (2004). Understanding the Fisher equation. Journal of Applied Econometrics , 19, 869–886.CrossRefGoogle Scholar
Taylor, J. (2000). Low inflation, pass through, and the pricing power of firms. European Economic Review , 44, 1389–1408.CrossRefGoogle Scholar
Vein, R., & Dale, P. (1999). Determinants and their application in mathematical physics . Springer.Google Scholar
Whittle, P. (1965). Recursive relations for predictors of non-stationary processes. Journal of the Royal Statistical Society, Series B , 27, 523–532.CrossRefGoogle Scholar
Widom, H. (1958). On the eigenvalues of certain Hermitian operators. Transactions of the American Mathematical Society , 88, 491–522.CrossRefGoogle Scholar
Wright, M. (2006). Green function or Green’s function? Nature Physics , 2, Article 646.CrossRefGoogle Scholar
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