Hostname: page-component-848d4c4894-8bljj Total loading time: 0 Render date: 2024-07-06T12:28:17.210Z Has data issue: false hasContentIssue false
  • English
  • Français

CONSISTENT LOCAL SPECTRUM INFERENCE FOR PREDICTIVE RETURN REGRESSIONS

Published online by Cambridge University Press:  03 August 2022

Torben G. Andersen  and
Rasmus T. Varneskov
Torben G. Andersen*
Affiliation:
Northwestern University
Rasmus T. Varneskov
Affiliation:
Copenhagen Business School
*
Address correspondence to Torben G. Andersen, Department of Finance, Kellogg School of Management, Northwestern University, Evanston, IL 60208, USA; e-mail: t-andersen@northwestern.edu.
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper studies the properties of predictive regressions for asset returns in economic systems governed by persistent vector autoregressive dynamics. In particular, we allow for the state variables to be fractionally integrated, potentially of different orders, and for the returns to have a latent persistent conditional mean, whose memory is difficult to estimate consistently by standard techniques in finite samples. Moreover, the predictors may be endogenous and “imperfect.” In this setting, we develop a consistent local spectrum (LCM) estimation procedure, that delivers asymptotic Gaussian inference. Furthermore, we provide a new LCM-based estimator of the conditional mean persistence, that leverages biased regression slopes as well as new LCM-based tests for significance of (a subset of) the predictors, which are valid even without estimating the return persistence. Simulations illustrate the theoretical arguments. Finally, an empirical application to monthly S&P 500 return predictions provides evidence for a fractionally integrated conditional mean component. Our new LCM procedure and tools indicate significant predictive power for future returns stemming from key state variables such as the default spread and treasury interest rates.

Type
ARTICLES
Information
Econometric Theory , Volume 38 , Issue 6: YALE 2018 CONFERENCE IN HONOR OF PETER C. B. PHILLIPS: PART II , December 2022 , pp. 1253 - 1307
Copyright
© The Author(s), 2022. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

We wish to thank the Co-Editor (Guido Kuersteiner), two anonymous referees, and seminar participants at the 2018 conference in honor of Peter C.B. Phillips at Yale University, the 13th annual SoFiE conference, Durham University Business School, and Singapore Management University for helpful comments. Financial support from the Center for Research in Econometric Analysis of Time Series (CREATES), funded by the Danish National Research Foundation, is gratefully acknowledged. Varneskov further acknowledges support from the Danish Finance Institute (DFI).

References

REFERENCES

Andersen, T.G. and Benzoni, L. (2009) Stochastic volatility. In Meyers, R. A. (ed.), Complex Systems in Finance and Econometrics , pp. 694–726. Springer.CrossRefGoogle Scholar
Andersen, T.G. and Bollerslev, T. (1998) Answering the skeptics: Yes, standard volatility models do provide accurate forecasts. International Economic Review 39, 885–905.CrossRefGoogle Scholar
Andersen, T.G., Bollerslev, T., Diebold, F.X., and Labys, P. (2003) Modeling and forecasting realized volatility. Econometrica 71, 579–625.CrossRefGoogle Scholar
Andersen, T.G. and Varneskov, R.T. (2021a) Consistent inference for predictive regressions in persistent economic systems. Journal of Econometrics 224, 215–244.CrossRefGoogle Scholar
Andersen, T.G. and Varneskov, R.T. (2021b) Testing for parameter instability and structural change in persistent predictive regressions. Journal of Econometrics, published online 10 November 2021. doi:10.1016/j.jeconom.2021.05.011 CrossRefGoogle Scholar
Bansal, R., Kiku, D., Shaliastovich, I., and Yaron, A. (2014) Volatility, the macroeconomy and asset prices. The Journal of Finance 69, 2471–2511.CrossRefGoogle Scholar
Bansal, R. and Yaron, A. (2004) Risks for the long run: A potential resolution of asset pricing puzzles. Journal of Finance 59, 1481–1509.CrossRefGoogle Scholar
Barndorff-Nielsen, O.E. and Shephard, N. (2002) Econometric analysis of realized volatility and its use in estimating stochastic volatility models. Journal of the Royal Statistical Society Series B 64, 253–280.CrossRefGoogle Scholar
Bauer, D. and Maynard, A. (2012) Persistence-robust surplus-lag Granger causality testing. Journal of Econometrics 169, 293–300.CrossRefGoogle Scholar
Boudoukh, J., Richardson, M., and Whitelaw, R.F. (2008) The myth of long-horizon predictability. Review of Financial Studies 21(4), 1577–1605.CrossRefGoogle Scholar
Breitung, J. and Demetrescu, M. (2015) Instrumental variable and variable addition based inference in predictive regressions. Journal of Econometrics 187, 358–375.CrossRefGoogle Scholar
Brillinger, D.R. (1981) Time Series. Data Analysis and Theory , Classics in Applied Mathematics. SIAM.Google Scholar
Campbell, J.Y. (2018) Financial Decisions and Markets: A Course in Asset Pricing . Princeton University Press.Google Scholar
Campbell, J.Y., Giglio, S., Polk, C., and Turley, R. (2018) An intertemporal CAPM with stochastic volatility. Journal of Financial Economics 128, 207–233.CrossRefGoogle Scholar
Campbell, J.Y. and Yogo, M. (2006) Efficient tests of stock return predictability. Journal of Financial Economics 81, 27–60.CrossRefGoogle Scholar
Cavanagh, C., Elliott, G., and Stock, J. (1995) Inference in models with nearly integrated regressors. Econometric Theory 11, 1131–1147.CrossRefGoogle Scholar
Choi, I. (1993) Asymptotic normality of the least-squares estimates for higher order autoregressive integrated processes with some applications. Econometric Theory 9, 263–282.CrossRefGoogle Scholar
Christensen, B.J. and Nielsen, M.O. (2006) Asymptotic normality of narrow-band least squares in the stationary fractional cointegration model and volatility forecasting. Journal of Econometrics 133, 343–371.CrossRefGoogle Scholar
Christensen, B.J. and Varneskov, R.T. (2017) Medium band least squares estimation of fractional cointegration in the presence of low-frequency contamination. Journal of Econometrics 197, 218–244.CrossRefGoogle Scholar
Dolado, J.J. and Lütkepohl, H. (1996) Making Wald tests work for cointegrated VAR systems. Econometric Reviews 15, 369–386.CrossRefGoogle Scholar
Duffy, J.A. and Kasparis, I. (2021) Estimation and inference in the presence of fractional $d=1 / 2$ and weakly nonstationary processes. Annals of Statistics 49, 1195–1217.CrossRefGoogle Scholar
Elliott, G., Müller, U., and Watson, M. (2015) Nearly optimal tests when a nuisance parameter is present under the null hypothesis. Econometrica 83, 771–811.CrossRefGoogle Scholar
Fama, E.F. (1970) Efficient capital markets: A review of theory and empirical work. Journal of Finance 25, 383–417.CrossRefGoogle Scholar
Ferson, W.E., Sarkissian, S., and Simin, T. (2003) Spurious regressions in financial economics. Journal of Finance 58, 1393–1414.CrossRefGoogle Scholar
Gabaix, X. (2012) Variable rare disasters: An exactly solved framework for ten puzzles in macro finance. Quarterly Journal of Economics 127, 645–700.CrossRefGoogle Scholar
Georgiev, I., Harvey, D.I., Leybourne, S.J., and Taylor, A.R. (2019) A bootstrap stationarity test for predictive regression invalidity. Journal of Business & Economic Statistics 37, 528–541.CrossRefGoogle Scholar
Granger, C.V.J. and Newbold, P. (1974) Spurious regression in econometrics. Journal of Econometrics 2, 111–120.CrossRefGoogle Scholar
Hamilton, J.D. (1994) Time Series Analysis . Princeton University Press.CrossRefGoogle Scholar
Hong, Y. (1996) Testing for independence between two covariance stationary time series. Biometrika 83, 615–625.CrossRefGoogle Scholar
Hualde, J. and Robinson, P.M. (2011) Gaussian pseudo-maximum likelihood estimation of fractional time series models. Annals of Statistics 39, 3152–3181.CrossRefGoogle Scholar
Johansen, S. and Nielsen, M.O. (2012) Likelihood inference for a fractionally cointegrated vector autoregressive model. Econometrica 80, 2667–2732.Google Scholar
Kendall, M. (1954) Note on bias in the estimation of autocorrelation. Biometrika 41, 403–404.CrossRefGoogle Scholar
Kostakis, A., Magdalinos, T., and Stamatogiannis, M.P. (2015) Robust econometric inference for stock return predictability. Review of Financial Studies 28, 1506–1553.CrossRefGoogle Scholar
Leeb, H. and Pötscher, B. (2005) Model selection and inference: Fact and fiction. Econometric Theory 21, 21–59.CrossRefGoogle Scholar
Lettau, M. and Ludvigson, S. (2001) Consumption, aggregate wealth, and expected stock returns. Journal of Finance 56, 815–849.CrossRefGoogle Scholar
Lettau, M. and Ludvigson, S. (2010) Measuring and modeling variation in risk-return trade-off. In Aït-Sahalia, Y. and Hansen, L.P. (eds.), Handbook of Financial Econometrics . Elsevier Science B.V.Google Scholar
Lettau, M., Ludvigson, S., and Wachter, J. (2008) The declining equity premium: What role does macroeconomic risk play? Review of Financial Studies 21, 1653–1687.CrossRefGoogle Scholar
Lin, Y. and Tu, Y. (2020) Robust inference for spurious regressions and cointegrations involving processes moderately deviated from a unit root. Journal of Econometrics 219, 52–65.CrossRefGoogle Scholar
Liu, X., Yang, B., Cai, Z., and Peng, L. (2019) A unified test for predictability of asset returns regardless of properties of predicting variables. Journal of Econometrics 208, 141–159.CrossRefGoogle Scholar
Lobato, I. (1999) A semiparametric two-step estimator in a multivariate long memory model. Journal of Econometrics 90, 129–155.CrossRefGoogle Scholar
Marriott, F. and Pope, J. (1954) Bias in the estimation of autocorrelations. Biometrika 41, 390–402.CrossRefGoogle Scholar
Mikusheva, A. (2007) Uniform inference in autoregressive models. Econometrica 75, 1411–1452.CrossRefGoogle Scholar
Neuhierl, A. and Varneskov, R.T. (2021) Frequency dependent risk. Journal of Financial Economics 140, 644–675.CrossRefGoogle Scholar
Newey, W.K. and West, K.D. (1987) A simple positive semi-definite, heteroskedasticity and autocorrelation consistent covariance matrix. Econometrica 55, 703–708.CrossRefGoogle Scholar
Nielsen, M.O. (2015) Asymptotics for the conditional-sum-of-squares estimator in mutivariate fractional time series models. Journal of Time Series Analysis 36, 154–188.CrossRefGoogle Scholar
Nielsen, M.O. and Shimotsu, K. (2007) Determining the cointegrating rank in nonstationary fractional systems by the exact local whittle approach. Journal of Econometrics 141, 574–596.CrossRefGoogle Scholar
Ortu, F., Tamoni, A., and Tebaldi, C. (2013) Long-run risk and the persistence of consumption shocks. Review of Financial Studies 26, 2876–2915.CrossRefGoogle Scholar
Park, J.Y. and Phillips, P.C.B. (1989) Statistical inference in regressions with integrated processes: Part 2. Econometric Theory 5, 95–131.CrossRefGoogle Scholar
Pastor, L. and Stambaugh, R.F. (2009) Predictive systems: Living with imperfect predictors. Journal of Finance 64, 1583–1628.CrossRefGoogle Scholar
Phillips, P.C.B. (1986) Understanding spurious regressions in econometrics. Journal of Econometrics 33, 311–340.CrossRefGoogle Scholar
Phillips, P.C.B. (1987) Towards a unified asymptotic theory for autoregression. Biometrika 74, 535–547.CrossRefGoogle Scholar
Phillips, P.C.B. (2014) On confidence intervals for autoregressive roots and predictive regression. Econometrica 82, 1177–1195.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. and Lee, J.H. (2013) Predictive regression under various degrees of persistence and robust long-horizon regression. Journal of Econometrics 177, 250–264.CrossRefGoogle Scholar
Phillips, P.C.B. and Lee, J.H. (2016) Robust econometric inference with mixed integrated and mildly explosive regressors. Journal of Econometrics 192, 433–450.CrossRefGoogle Scholar
Phillips, P.C.B. and Magdalinos, T. (2007) Limit theory for moderate deviations from a unit root. Journal of Econometrics 136, 115–130.CrossRefGoogle Scholar
Phillips, P.C.B. and Magdalinos, T. (2009) Econometric inference in the vicinity of unity. CoFie Working Paper 7, Singapore Management University.Google Scholar
Phillips, P.C.B. and Shimotsu, K. (2004) Local whittle estimation in nonstationary and unit root cases. Annals of Statistics 32, 656–692.CrossRefGoogle Scholar
Ren, Y., Tu, Y., and Yi, Y. (2019) Balanced predictive regressions. Journal of Empirical Finance 54, 118–142.CrossRefGoogle Scholar
Robinson, P.M. (1995) Gaussian semiparametric estimation of long range dependence. Annals of Statistics 23, 1630–1661.CrossRefGoogle Scholar
Robinson, P.M. (2005) The distance between rival nonstationary fractional processes. Journal of Econometrics 128, 283–300.CrossRefGoogle Scholar
Robinson, P.M. and Hualde, J. (2003) Cointegration in fractional systems with unknown integration orders. Econometrica 71, 1727–1766.CrossRefGoogle Scholar
Robinson, P.M. and Marinucci, D. (2001) Narrow-band analysis of nonstationary processes. Annals of Statistics 29, 947–986.Google Scholar
Robinson, P.M. and Marinucci, D. (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
Robinson, P.M. and Yajima, Y. (2002) Determination of cointegrating rank in fractional systems. Journal of Econometrics 106, 217–241.CrossRefGoogle Scholar
Shao, X. (2009) A generalized portmanteau test for independence between two stationary time series. Econometric Theory 25, 195–210.CrossRefGoogle Scholar
Shiller, R. (2000) Irrational Exuberance . Princeton University Press.Google Scholar
Shimotsu, K. (2010) Exact local whittle estimation of fractional integration with unknown mean and time trend. Econometric Theory 26, 501–540.CrossRefGoogle Scholar
Shimotsu, K. and Phillips, P.C.B. (2005) Exact local whittle estimation of fractional integration. Annals of Statistics 32, 656–692.Google Scholar
Sims, C.A., Stock, J.H., and Watson, M.W. (1990) Inference in linear time series models with some unit roots. Econometrica 58, 113–144.CrossRefGoogle Scholar
Stambaugh, R. F. (1986), Bias in regressions with lagged stochastic regressors. Working Paper 156, CRSP, Graduate School of Business, University of Chicago.Google Scholar
Stambaugh, R.F. (1999) Predictive regressions. Journal of Financial Economics 54, 783–820.CrossRefGoogle Scholar
Toda, H. and Yamamoto, T. (1995) Statistical inference in vector autoregressions with possibly integrated processes. Journal of Econometrics 66, 225–250.CrossRefGoogle Scholar
Tsay, W.-J. and Chung, C.-F. (2000) The spurious regression of fractionally integrated processes. Journal of Econometrics 96, 155–182.CrossRefGoogle Scholar
Valkanov, R. (2003) Long-horizon regressions: Theoretical results and applications. Journal of Financial Economics 68, 201–232.CrossRefGoogle Scholar
Welch, I. and Goyal, A. (2008) A comprehensive look at the empirical performance of equity premium prediction. Review of Financial Studies 21, 1455–1508.CrossRefGoogle Scholar
Xu, K.-L. (2020), Testing for return predictability with co-moving predictors of unknown form. Department of Economics, Indiana University, unpublished manuscript.Google Scholar
Supplementary material: PDF

Andersen and Varneskov supplementary material

Andersen and Varneskov supplementary material

Download Andersen and Varneskov supplementary material(PDF)
PDF 344.4 KB