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NEW ROBUST INFERENCE FOR PREDICTIVE REGRESSIONS

Published online by Cambridge University Press:  03 May 2023

Rustam Ibragimov ,
Jihyun Kim  and
Anton Skrobotov
Rustam Ibragimov
Affiliation:
Imperial College Business School and Center for Econometrics and Business Analytics, St. Petersburg University
Jihyun Kim*
Affiliation:
Sungkyunkwan University and Toulouse School of Economics
Anton Skrobotov
Affiliation:
Russian Presidential Academy of National Economy and Public Administration and St. Petersburg University
*
Address correspondence to Jihyun Kim, School of Economics, Sungkyunkwan University, Seoul 03063, South Korea; e-mail: kim.jihyun@skku.edu.
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Abstract

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We propose a robust inference method for predictive regression models under heterogeneously persistent volatility as well as endogeneity, persistence, or heavy-tailedness of regressors. This approach relies on two methodologies, nonlinear instrumental variable estimation and volatility correction, which are used to deal with the aforementioned characteristics of regressors and volatility, respectively. Our method is simple to implement and is applicable both in the case of continuous and discrete time models. According to our simulation study, the proposed method performs well compared with widely used alternative inference procedures in terms of its finite sample properties in various dependence and persistence settings observed in real-world financial and economic markets.

Information

Type
ARTICLES
Information
Econometric Theory , Volume 40 , Issue 6 , December 2024 , pp. 1364 - 1390
Copyright
© The Author(s), 2023. Published by Cambridge University Press

Footnotes

We thank the Editor (Peter C.B. Phillips), the Co-Editor (Anna Mikusheva), and three anonymous referees for many helpful comments and suggestions. We are also grateful to Walter Distaso, Jean-Marie Dufour, Siyun He, Nour Meddahi, Mikkel Plagborg-Møller, Aleksey Min, Ulrich K. Müller, Rasmus S. Pedersen, Artem Prokhorov, Rogier Quaedvlieg, Robert Taylor, Alex Maynard, Neil Shephard, Yang Zu, and the participants at the Center for Econometrics and Business Analytics (CEBA, St. Petersburg University) and University of Nottingham seminar series, the session on Econometrics of Time Series at the 12th World Congress of the Econometric Society, and iCEBA-2021, 2022 conferences for helpful discussions and comments. Financial support from a British Academy/Leverhulme Small Research Grant (R. Ibragimov, reference SG2122$\backslash$211040), the French Government and the ANR’ Investissements d’Avenir program (J. Kim, Grant ANR-17-EURE-0010) and the Russian Science Foundation (A. Skrobotov, Project No. 20-78-10113) for various and non-overlapping parts of this research is gratefully acknowledged.

References

REFERENCES

Baillie, R.T. (1996) Long memory processes and fractional integration in econometrics. Journal of Econometrics 73, 5–59.CrossRefGoogle Scholar
Bercu, B. & Touati, A. (2008) Exponential inequalities for self-normalized martingales with applications. Annals of Applied Probability 18(5), 1848–1869.CrossRefGoogle Scholar
Billingsley, P. (1986) Convergence of Probability Measures . Wiley.Google Scholar
Brown, D. & Ibragimov, R. (2019) Sign tests for dependent observations. Econometrics and Statistics 10, 1–8.CrossRefGoogle Scholar
Campbell, B. & Dufour, J.-M. (1995) Exact nonparametric orthogonality and random walk tests. Review of Economics and Statistics 77, 1–16.CrossRefGoogle Scholar
Campbell, J. & Yogo, M. (2006) Efficient tests of stock return predictability. Journal of Financial Econometrics 81, 27–60.CrossRefGoogle Scholar
Carrasco, M. & Chen, X. (2002) Mixing and moment properties of various GARCH and stochastic volatility models. Econometric Theory 18, 17–39.CrossRefGoogle Scholar
Cauchy, A. (1836) On a new formula for solving the problem of interpolation in a manner applicable to physical investigations. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 8(49), 459–468.CrossRefGoogle Scholar
Cavaliere, G. (2004) Testing stationarity under a permanent variance shift. Economics Letters 82, 403–408.CrossRefGoogle Scholar
Cavaliere, G. & Taylor, A.R. (2007) Testing for unit roots in time series models with non-stationary volatility. Journal of Econometrics 140, 919–947.CrossRefGoogle Scholar
Cavaliere, G. & Taylor, A.R. (2009) Heteroskedastic time series with a unit root. Econometric Theory 25, 1228–1276.CrossRefGoogle Scholar
Chan, N. & Wang, Q. (2015) Nonlinear regressions with nonstationary time series. Journal of Econometrics 185, 182–195.CrossRefGoogle Scholar
Chen, W. & Deo, R. (2009) Bias reduction and likelihood-based almost exactly sized hypothesis testing in predictive regressions using the restricted likelihood. Econometric Theory 25, 1143–1179.CrossRefGoogle Scholar
Choi, Y., Jacewitz, S., & Park, J.Y. (2016) A reexamination of stock return predictability. Journal of Econometrics 192, 168–189.CrossRefGoogle Scholar
Chung, H. & Park, J.Y. (2007) Nonstationary nonlinear heteroskedasticity in regression. Journal of Econometrics 137, 230–259.CrossRefGoogle Scholar
Cont, R. (2001) Empirical properties of asset returns: Stylized facts and statistical issues. Quantitative Finance 1, 223–236.CrossRefGoogle Scholar
Davis, R.A. & Mikosch, T. (1998) The sample autocorrelations of heavy-tailed processes with applications to ARCH. Annals of Statistics 26, 2049–2080.CrossRefGoogle Scholar
de la Peña, V.H. (1999) A general class of exponential inequalities for martingales and ratios. Annals of Probability 27(1), 537–564.CrossRefGoogle Scholar
de la Peña, V.H. & Ibragimov, R. (2017) Sharp probability inequalities for random polynomials, generalized sample cross-moments, and Studentized processes. In Pinelis, I. (ed.), Inequalities and Extremal Problems in Probability and Statistics: Selected Topics , pp. 159–187. Elsevier.CrossRefGoogle Scholar
de la Peña, V.H., Ibragimov, R., & Sharakhmetov, S. (2003) On extremal distributions and sharp ${L}_p$ -bounds for sums of multilinear forms, Annals of Probability 31, 630–675.Google Scholar
Dufour, J.-M. & Hallin, M. (1993) Improved Eaton bounds for linear combinations of bounded random variables, with statistical applications. Journal of the American Statistical Association 88, 1026–1033.CrossRefGoogle Scholar
Edelman, D. (1990) An inequality of optimal order for the tail probabilities of the $t$ statistic under symmetry. Journal of the American Statistical Association 85, 120–122.Google Scholar
Efron, B. (1969) Student’s $t$ test under symmetry conditions. Journal of the American Statistical Association 64, 1278–1302.Google Scholar
Elliott, G. & Stock, J.H. (1994) Inference in time series regression when the order of integration of a regressor is unknown. Econometric Theory 10, 672–700.CrossRefGoogle Scholar
Embrechts, P., Klüppelberg, C., & Mikosch, T. (1997) Modelling Extremal Events for Insurance and Finance . Springer.CrossRefGoogle Scholar
Francq, C. & Zakoïan, J.-M. (2006) Mixing properties of a general class of GARCH (1, 1) models without moment assumptions on the observed process. Econometric Theory 22, 815–834.CrossRefGoogle Scholar
Ghysels, E., Hill, J.B., & Motegi, K. (2020) Testing a large set of zero restrictions in regression models, with an application to mixed frequency Granger causality. Journal of Econometrics 218, 633–654.CrossRefGoogle Scholar
Granger, C.W.J. & Orr, D. (1972) Infinite variance and research strategy in time series analysis. Journal of the American Statistical Association 67, 275–285.Google Scholar
Hansen, B.E. (1992) Convergence to stochastic integrals for dependent heterogeneous processes. Econometric Theory 8, 489–500.CrossRefGoogle Scholar
Hansen, B.E. (1995) Regression with nonstationary volatility. Econometrica 63, 1113–1132.CrossRefGoogle Scholar
Hansen, P.R. & Lunde, A. (2014) Estimating the persistence and the autocorrelation function of a time series that is measured with error. Econometric Theory 30, 60–93.CrossRefGoogle Scholar
Harvey, D.I., Leybourne, S.J., & Zu, Y. (2019) Testing explosive bubbles with time-varying volatility. Econometric Reviews 38, 1131–1151.CrossRefGoogle Scholar
Ibragimov, M., Ibragimov, R., & Walden, J. (2015) Heavy-Tailed Distributions and Robustness in Economics and Finance . Lecture Notes in Statistics, vol. 214. Springer.CrossRefGoogle Scholar
Ibragimov, R. & Müller, U. (2010) $t$ -statistic based correlation and heterogeneity robust inference. Journal of Business & Economic Statistics 28, 453–468.CrossRefGoogle Scholar
Ibragimov, R., Pedersen, R.S., & Skrobotov, A. (2021) New Approaches to Robust Inference on Market (Non-)Efficiency, Volatility Clustering and Nonlinear Dependence. Working paper, Imperial College Business School and the University of Copenhagen. https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3580916.Google Scholar
Ibragimov, R. & Phillips, P.C.B. (2008) Regression asymptotics using martingale convergence methods. Econometric Theory 24, 888–947.CrossRefGoogle Scholar
Jacquier, E., Polson, N.G., & Rossi, P.E. (2004) Bayesian analysis of stochastic volatility models with fat-tails and correlated errors. Journal of Econometrics 122, 185–212.CrossRefGoogle Scholar
Kim, C.S. & Phillips, P.C.B. (2006) Log Periodogram Regression: The Nonstationary Case. Cowles Foundation Discussion Paper no. 1587. https://papers.ssrn.com/sol3/papers.cfm?abstract_id=937915.Google Scholar
Kim, J. & Meddahi, N. (2020) Volatility regressions with fat tails. Journal of Econometrics 218, 690–713.CrossRefGoogle Scholar
Kim, J. & Park, J.Y. (2017) Asymptotics for recurrent diffusions with application to high frequency regression. Journal of Econometrics 196, 37–54.CrossRefGoogle Scholar
Kristensen, D. (2009) Uniform convergence rates of kernel estimators with heterogeneous dependent data. Econometric Theory 25, 1433–1445.CrossRefGoogle Scholar
Kurtz, T.G. & Protter, P. (1991) Weak limit theorems for stochastic integrals and stochastic differential equations. Annals of Probability 19, 1035–1070.CrossRefGoogle Scholar
Leadbetter, M. & Rootzén, H. (1988) Extremal theory for stochastic processes. Annals of Probability 16, 431–478.CrossRefGoogle Scholar
Liebscher, E. (1996) Strong convergence of sums of $\alpha$ -mixing random variables with applications to density estimation. Stochastic Processes and their Applications 65, 69–80.CrossRefGoogle Scholar
Mikosch, T. & Stărică, C. (2000) Limit theory for the sample autocorrelations and extremes of a GARCH $\left(1,1\right)$ process. Annals of Statistics 28, 1427–1451.Google Scholar
Park, J.Y. (2003) Weak Unit Roots. Unpublished manuscript. http://www.ruf.rice.edu/econ/papers/2003papers/17park.pdf.Google Scholar
Phillips, P.C.B. (1987a) Time-series regression with a unit root. Econometrica 55, 277–301.CrossRefGoogle Scholar
Phillips, P.C.B. (1987b) Towards a unified asymptotic theory for autoregression. Biometrika 74, 535–547.CrossRefGoogle Scholar
Phillips, P.C.B. (1999) Discrete Fourier Transforms of Fractional Processes. Cowles Foundation Discussion paper no. 1243.Google Scholar
Phillips, P.C.B. (2015) Halbert White Jr. Memorial JFEC Lecture: Pitfalls and possibilities in predictive regression. Journal of Financial Econometrics 13, 521–555.CrossRefGoogle Scholar
Phillips, P.C.B. & Magdalinos, T. (2007) Limit theory for moderate deviations from a unit root. Journal of Econometrics 136, 115–130.CrossRefGoogle Scholar
Phillips, P.C.B. & Magdalinos, T. (2009) Econometric Inference in the Vicinity of Unity, Working paper, Yale University and University of Nottingham.Google Scholar
Phillips, P.C.B., Park, J.Y., & Chang, Y. (2004) Nonlinear instrumental variable estimation of an autoregression. Journal of Econometrics 118, 219–246.CrossRefGoogle Scholar
Phillips, P.C.B. & Solo, V. (1992) Asymptotics for linear processes. Annals of Statistics 20, 971–1001.CrossRefGoogle Scholar
Pinelis, I. (1994) Extremal probabilistic problems and Hotelling’s $t$ test under a symmetry condition. Annals of Statistics 22, 357–368.CrossRefGoogle Scholar
Pollard, D. (1984) Convergence of Stochastic Processes, Springer Series in Statistics . Springer.CrossRefGoogle Scholar
Samorodnitsky, G., Rachev, S.T., Kurz-Kim, J.-R., & Stoyanov, S.V. (2007) Asymptotic distributions of unbiased linear estimators in the presence of heavy-tailed stochastic regressors and residuals. Probability and Mathematical Statistics 27, 275–302.Google Scholar
Shephard, N. (2020) An Estimator for Predictive Regression: Reliable Inference for Financial Economics. Working paper, Harvard University.Google Scholar
So, B. & Shin, D. (1999) Cauchy estimators for autoregressive processes with applications to unit root tests and confidence intervals. Econometric Theory 15, 165–176.CrossRefGoogle Scholar
So, B. & Shin, D. (2001) An invariant sign tests for random walks based on recursive median adjustment. Journal of Econometrics 102, 197–229.CrossRefGoogle Scholar
Stambaugh, R.F. (1999) Predictive regressions. Journal of Financial Economics 54, 375–421.CrossRefGoogle Scholar
Vogt, M. (2012) Nonparametric regression for locally stationary time series. Annals of Statistics 40, 2601–2633.CrossRefGoogle Scholar
Wang, Q. & Chan, N. (2014) Uniform convergence rates for a class of martingales with application in non-linear cointegrating regression. Bernoulli 20, 207–230.CrossRefGoogle Scholar
Wang, Q., Lin, Y.-X., & Gulati, C.M. (2003) Asymptotics for general fractionally integrated processes with applications to unit root tests. Econometric Theory 19, 143–164.CrossRefGoogle Scholar
Xu, K.-L. & Guo, J. (2022) A new test for multiple predictive regression. Journal of Financial Econometrics , nbac030. https://doi.org/10.1093/jjfinec/nbac030.Google Scholar
Xu, K.-L. & Phillips, P.C.B. (2008) Adaptive estimation of autoregressive models with time-varying variances. Journal of Econometrics 142, 265–280.CrossRefGoogle Scholar
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