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NEARLY OPTIMAL TEST FOR LONG-RUN PREDICTABILITY WITH NEARLY INTEGRATED REGRESSORS

Published online by Cambridge University Press:  27 April 2020

Natalia Sizova
Natalia Sizova*
Affiliation:
Microsoft
*
Address correspondence to Natalia Sizova, Microsoft, Bellevue, WA 98005, USA; e-mail: Natalia.Sizova@microsoft.com.
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Abstract

We develop a method for long-run predictability testing in series Y by a persistent series X. We consider a class of tests based on the long-run behavior of these series that are robust to short-run dynamics and attempt to attain the highest possible power. The test is based on the Whittle approximation to the likelihood ratio that is adjusted to remain accurate across a range of persistence in X. We verify the properties of this test in small simulations and compare this test against a group of recently proposed methods.

Type
ARTICLES
Information
Econometric Theory , Volume 37 , Issue 1 , February 2021 , pp. 82 - 137
Copyright
© Cambridge University Press 2020

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Footnotes

*

I am thankful to Michael Jansson for providing the code for the optimal inference method, as well as to Michalis P. Stamatogiannis and Tassos Magdalinos for providing the code for the IVX estimator.

References

REFERENCES

Andrews, D.W.K. & Guggenberger, P. (2012) Asymptotics for LS, GLS, and feasible GLS statistics in an AR(1) model with conditional heteroskedasticity. Journal of Econometrics 169(2), 196–210.CrossRefGoogle Scholar
Andrews, D.W.K., Moreira, M., & Stock, J.H. (2008) Efficient two-sided nonsimilar invariant tests in IV regression with weak instruments. Journal of Econometrics 146, 241–254.CrossRefGoogle Scholar
Bandi, F.M. & Perron, B. (2008) Long-run risk-return trade-offs. Journal of Econometrics 143, 349–374.CrossRefGoogle Scholar
Boudoukh, J., Richardson, M., & Whitelaw, R. (2016) New evidence on the forward premium puzzle. Journal of Financial and Quantitative Analysis 51(3), 875–897.CrossRefGoogle Scholar
Brillinger, D.R. (1975) Time Series: Data Analysis and Theory. Holt, Rinehart and Winston.Google Scholar
Campbell, J.Y. & Yogo, M. (2006) Efficient tests of stock return predictability. Journal of Financial Economics 81(1), 27–60.CrossRefGoogle Scholar
Cavanagh, C.L., Elliott, G., & Stock, J.H. (1995) Inference in models with nearly integrated regressors. Econometric Theory 11(05), 1131–1147.CrossRefGoogle Scholar
Chiburis, R.C. (2009) Approximately most powerful tests for moment inequalities. Chapter of Ph.D. dissertation, Department of Economics, Princeton University.Google Scholar
Christensen, B.J. & Nielsen, M.O. (2006) Asymptotic normality of narrow-band least squares in the stationary fractional cointegration model and volatility forecasting. Journal of Econometrics 133(1), 343–371.CrossRefGoogle Scholar
Davidson, J. (2002) Establishing conditions for the functional central limit theorem in nonlinear and semiparametric time series. Journal of Econometrics 106(2), 243–269.CrossRefGoogle Scholar
Corbae, D., Ouliaris, S., & Phillips, P.C.B. (2002) Band spectral regression with trending data. Econometrica 70(3), 1067–1110.CrossRefGoogle Scholar
Dahlhaus, R. (2000) A likelihood approximation for locally stationary processes. Annals of Statistics 28(6), 1762–1794.CrossRefGoogle Scholar
Davis, P.J. (1979) Circulant Matrices. Wiley.Google Scholar
Dzhaparidze, K. (1986) Parameter Estimation and Hypothesis Testing in Spectral Analysis of Stationary Time Series. Springer.CrossRefGoogle Scholar
Durlauf, S.N. & Phillips, P.C.B. (1986) Multiple time series regression with integrated processes. Review of Economic Studies 53(4), 473–495.Google Scholar
Nikolay, G. (2009) A new look at the forward premium puzzle. Journal of Financial Econometrics 7(3), 312–338.Google Scholar
Elliott, G. & Müller, U.K. (2014) Pre and post break parameter inference. Journal of Econometrics 180, 141–157.CrossRefGoogle Scholar
Elliott, G., Müller, U., & Watson, M. (2015) Nearly optimal tests when a nuisance parameter is present under the null hypothesis. Econometrica 83(2), 771–811.CrossRefGoogle Scholar
Elliott, G., Rothenberg, T.J., & Stock, J.H. (1996) Efficient tests for an autoregressive unit root. Econometrica 64, 813–836.CrossRefGoogle 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(3/4), 672–700.CrossRefGoogle Scholar
Fisher, M.E. & Seater, J.J. (1993) Long-run neutrality and superneutrality in an ARIMA framework. The American Economic Review 83(3), 402–415.Google Scholar
Goyal, A. & Welch, I. (2008) A comprehensive look at the empirical performance of equity premium prediction. The Review of Financial Studies 21(4), 1455–1508.Google Scholar
Granger, C.W.J. (1969) Investigating causal relations by econometric models and cross-spectral methods. Econometrica 37(3), 424–438.CrossRefGoogle Scholar
Granger, C.W.J. & Hatanaka, M. (1964) Spectral Analysis of Economic Time Series. Princeton University Press.CrossRefGoogle Scholar
Hamilton, J.D. (1994) Time Series Analysis. Princeton University Press.CrossRefGoogle Scholar
Hansen, B.E. (1999) The grid bootstrap and the autoregressive model. The Review of Economics and Statistics 81(4), 594–607.CrossRefGoogle Scholar
Hannan, E.J. (1973a) The asymptotic theory of linear time-series models. Journal of Applied Probability 10(1), 130–145.CrossRefGoogle Scholar
Hannan, E.J. (1973b) Central limit theorems for time series regression. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 26, 157–170.CrossRefGoogle Scholar
Hansen, B.E. (1992) Convergence to stochastic integrals for dependent heterogenous processes. Econometric Theory 8, 489–500.CrossRefGoogle Scholar
Hjalmarsson, E. (2011) New methods for inference in long-horizon regressions, Journal of Financial and Quantitative Analysis 46(3), 815–839.CrossRefGoogle Scholar
Hjalmarsson, E. (2007) Fully modified estimation with nearly integrated regressors. Finance Research Letters 4(2), 92–94.CrossRefGoogle Scholar
Hwang, J. & Sun, Y. (2018) Simple, robust, and accurate f and t tests in cointegrated systems. Econometric Theory 34(5), 949–984.CrossRefGoogle Scholar
Hurvich, C.M., Moulines, E., & Soulier, P. (2005) Estimating long memory in volatility. Econometrica 73(4), 1283–1328.CrossRefGoogle Scholar
Ibragimov, I.A. & Linnik, Y.V. (1971) Independent and Stationary Sequences of Random. Londo Mathematical Society.Google Scholar
Ibragimov, I.A. & Rozanov, Y.A. (1978) Gaussian Random Processes. Springer-Verlag.CrossRefGoogle Scholar
Ing, C.-K. (2003) Multistep prediction in autoregressive processes. Econometric Theory 19(2), 254–279.CrossRefGoogle Scholar
Jansson, M. & Moreira, M. (2006) Optimal inference in regression models with nearly integrated regressors. Econometrica 74, 681–714.CrossRefGoogle Scholar
Jeganathan, P. (1995) Some aspects of asymptotic theory with applications to time series models. Econometric Theory 11, 818–887.CrossRefGoogle Scholar
Hamilton, J.H. (1994) The Time Series Analysis. Princeton University Press.CrossRefGoogle Scholar
Herrndorf, N. (1984) A functional central limit theorem for weakly dependent sequences of random variables. The Annals of Probability 12(1), 141–153.CrossRefGoogle Scholar
Hong, H. & Stein, J.C. (1999) A unified theory of underreaction, momentum trading, and overreaction in asset markets. The Journal of Finance 54(6), 2143–2184.CrossRefGoogle Scholar
Kapetanios, G. & Papailias, F. (2011) Block Bootstrap and Long Memory. Working papers 679, School of Economics and Finance, Queen Mary University of London.Google Scholar
Kim, Y.M. & Nordman, D.J. (2012) Properties of a block bootstrap under long-range dependence. Sankhya A 73(1), 79–109.CrossRefGoogle Scholar
Kostakis, A., Magdalinos, T., & Stamatogiannis, M.P. (2015) Robust econometric inference for stock return predictability. Review of Financial Studies 28(5), 1506–1553.CrossRefGoogle Scholar
Krafty, R.T. & Collinge, W.O. (2013) Penalized multivariate whittle likelihood for power spectrum estimation. Biometrika 100(2), 447–458.CrossRefGoogle Scholar
Lehmann, E.L. & Romano, J.P. (2005) Testing Statistical Hypotheses. Springer.Google Scholar
Lobato, I.N. (1997) Consistency of the averaged cross-periodogram in long memory series. Journal of Time Series Analysis 18, 137–155.CrossRefGoogle Scholar
Magdalinos, T. & Phillips, P.C.B. (2009) Econometric Inference in the Vicinity of Unity. CoFie Working Paper, Singapore Management University.Google Scholar
Marinucci, D. & Robinson, P.M. (2003) Semiparametric frequency domain analysis of fractional cointegration. In Robinson, P.M. (ed.), Time Series with Long Memory, pp. 334–373. Oxford University Press.Google Scholar
Marmol, F. & Velasco, C. (2004) Consistent testing of cointegrating relationships. Econometrica 72, 1809–1844.CrossRefGoogle Scholar
McLeish, D.L. (1974) Dependent central limit theorems and invariance principles. The Annals of Probability 2(4), 620–628.CrossRefGoogle Scholar
Mikusheva, A. (2007) Uniform inference in autoregressive models. Econometrica 75(5), 1411–1452.CrossRefGoogle Scholar
Mikusheva, A. (2012) One-dimensional inference in autoregressive models with the potential presence of a unit root. Econometrica 80(1), 173–212.Google Scholar
Müller, U. & Watson, M.W. (2013) Low-frequency robust cointegration testing. Journal of Econometrics 174, 66–81.CrossRefGoogle Scholar
Mishkin, F.S. (1990) What does the term structure of interest rates tell us about future inflation? Journal of Monetary Economics 25, 77–95.CrossRefGoogle Scholar
Müller, U.K. & Watson, M.W. (2008) Testing models of low-frequency variability. Econometrica 76, 979–1016.Google Scholar
Moreira, M.J., Mourão, R., & Moreira, H. (2014) A Critical Value Function Approach, with an Application to Persistent Time-Series. Centre for Microdata Methods and Practice Working Paper, The Institute for Fiscal Studies, UCL.Google Scholar
Nielsen, M.O. & Frederiksen, P. (2011) Fully modified narrow-band least squares estimation of weak fractional cointegration. Econometrics Journal 14, 77–120.CrossRefGoogle Scholar
Park, J.Y. (2006) A bootstrap theory for weakly integrated processes. Journal of Econometrics 133(2), 639–672.CrossRefGoogle Scholar
Phillips, P.C.B. & Solo, V. (1992) Asymptotics for linear processes. Annals of Statistics 20(2), 971–1001.CrossRefGoogle Scholar
Phillips, P.C.B. (1987) Towards a unified asymptotic theory for autoregression. Biometrika 74(3), 535–547.CrossRefGoogle Scholar
Phillips, P.C.B. (1987) Asymptotic expansions in nonstationary vector autoregressions. Econometric Theory 3(1), 45–68.CrossRefGoogle Scholar
Phillips, P.C.B. (1988) Regression theory for near-integrated time series. Econometrica 56(5), 1021–1043.CrossRefGoogle Scholar
Phillips, P.C.B. (1991) Spectral regression for cointegrated time series. In Barnett, W.A., Powell, J., Tauchen, G. (eds.), Nonparametric and Semiparametric Methods in Economics and Statistics, pp. 413–435. Cambridge University Press.Google Scholar
Phillips, P.C.B. (2007) Unit root log periodogram regression. Journal of Econometrics 138, 104–124.CrossRefGoogle Scholar
Phillips, P.C.B. (2014a) On confidence intervals for autoregressive roots and predictive regression. Econometrica 82(3), 1177–1195.Google Scholar
Phillips, P.C.B. (2014b) Optimal estimation of cointegrated systems with irrelevant instruments. Journal of Econometrics 178, 210–224.CrossRefGoogle Scholar
Phillips, P.C.B. & Hansen, B.E. (1990) Statistical inference in instrumental variables regression with I(1). The Review of Economic Studies 57(1), 99–125.CrossRefGoogle Scholar
Politis, D.N., Romano, J.P., & Wolf, M. (1999) Subsampling. Springer.CrossRefGoogle Scholar
Poskit, D.S. (2007) Properties of the sieve boostrap for fractionally integrated and non-invertible processes. Journal of Time Series Analysis 29(2), 224–250.CrossRefGoogle Scholar
Richardson, M. & Stock, J. (1989) Drawing inferences from statistics based on multi-year asset. Journal of Financial Economics 25(2), 323–348.CrossRefGoogle Scholar
Ramey, V.A. & Zubairy, S. (2018) Government spending multipliers in good times and in bad: Evidence from U.S. historical data. Journal of Political Economy 126(2), 850–901.CrossRefGoogle Scholar
Robinson, P.M. (1994) Semiparametric analysis of long-memory time series. Annals of Statistics 22, 515–539.CrossRefGoogle Scholar
Robinson, P.M. (2003) Time Series with Long Memory. Oxford University Press.Google Scholar
Sizova, N. (2013) Long-horizon return regressions with historical volatility. Journal of Business & Economic Statistics 31(4), 546–559.CrossRefGoogle Scholar
Rozanov, Y.A. (1967) Stationary random processes. In Jenkins, G.M., Parzen, E. (eds.), Holden-Day Series in Time Series Analysis. Holden-Day.Google Scholar
Sizova, N. (2014) A frequency-domain alternative to long-horizon regressions with application to return predictability. Journal of Empirical Finance 28, 261–272.CrossRefGoogle Scholar
Stambaugh, R.F. (1999) Predictive regressions. Journal of Financial Economics 54, 375–421.CrossRefGoogle Scholar
Stock, J.H. (1994) Unit Roots Structural Breaks and Trends. Elsevier Science B.V.Google Scholar
Stock, J. & Watson, M. (1996) Confidence Sets in Regressions with Highly Serially Correlated Regressors. Working Paper, Harvard University.Google Scholar
Valkanov, R. (2003) Long-horizon regressions: Theoretical results and applications. Journal of Financial Economics 68(2), 201–232.CrossRefGoogle Scholar
Ventosa-Santaulària, D. (2009) Spurious regression. Journal of Probability and Statistics.CrossRefGoogle Scholar
Wolf, M. (2000) Stock returns and dividend yields revisited: A new way to look at an old problem. Journal of Business & Economic Statistics 18(1), 18–30.Google Scholar
Wooldridge, J.M. & White, H. (1988) Some invariance principles and central limit theorems for dependent heterogeneous processes. Econometric Theory 4(2), 210–230.CrossRefGoogle Scholar
Velasco, C. (1999) Gaussian semiparametric estimation of nonstationary time series. Journal of Time Series Analysis 20, 87–127.CrossRefGoogle Scholar
Whittle, P. (1953) The analysis of multiple stationary time series. Journal of the Royal Statistical Society Series B (Methodological) 15(1), 125–139.CrossRefGoogle Scholar
Whittle, P. (1951) Hypothesis Testing in Times Series Analysis. Hafner.Google Scholar