Hostname: page-component-745bb68f8f-v2bm5 Total loading time: 0 Render date: 2025-02-10T01:17:13.281Z Has data issue: false hasContentIssue false
  • English
  • Français

CHRONOLOGICALLY TRIMMED LS FOR NONLINEAR PREDICTIVE REGRESSIONS WITH PERSISTENCE OF UNKNOWN FORM

Published online by Cambridge University Press:  07 February 2025

Zhishui Hu Open the ORCID record for Zhishui Hu [Opens in a new window] ,
Ioannis Kasparis Open the ORCID record for Ioannis Kasparis [Opens in a new window]  and
Qiying Wang Open the ORCID record for Qiying Wang [Opens in a new window]
Zhishui Hu
Affiliation:
University of Science and Technology of China
Ioannis Kasparis*
Affiliation:
University of Cyprus
Qiying Wang
Affiliation:
The University of Sydney
*
Address correspondence to Ioannis Kasparis, University of Cyprus, Nicosia, Cyprus, e-mail: [email protected].
Get access Rights & Permissions [Opens in a new window]

Abstract

Relatively, recent work by Jeganathan (2008, Cowles Foundation Discussion Paper 1649) and Wang (2014, Econometric Theory, 30(3), 509–535) on generalized martingale central limit theorems (MCLTs) implicitly introduces a new class of instrument arrays that yield (mixed) Gaussian limit theory irrespective of the persistence level in the data. Motivated by these developments, we propose a new semiparametric method for estimation and inference in nonlinear predictive regressions with persistent predictors. The proposed method that we term chronologically trimmed least squares (CTLS) is comparable to the IVX method of Phillips and Magdalinos (2009, Econometric inference in the vicinity of unity. Mimeo, Singapore Management University) and yields conventional inference in regressions where the nature and extent of persistence in the data are uncertain. In terms of model generality, our contribution to the existing literature is twofold. First, our covariate model space allows for both nearly integrated (NI) and fractional processes (stationary and nonstationary) as a special case, while the vast majority of articles in this area only consider NI arrays. Second, we allow for nonlinear regression functions. The CTLS estimator is obtained by applying certain chronological trimming to the OLS instruments using appropriate kernel functions of time trend variables. In particular, the instruments under consideration are a generalized (averaged) version of those widely used for time-varying parameter (TVP) models. For the purposes of our analysis, we develop a novel asymptotic theory for sample averages of various processes weighted by such kernel functionals which is of independent interest and highly relevant to the TVP literature. Leveraging our nonlinear framework, we also provide an investigation on the effects of misbalancing on the predictability hypothesis. A new methodology is proposed to mitigate misbalancing effects. These methods are used for exploring the predictability of SP500 returns.

Type
ARTICLES
Information
Econometric Theory , First View , pp. 1 - 39
Copyright
© The Author(s), 2025. 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

The authors thank a coeditor and three referees for very helpful comments on the original version of this article. Hu acknowledges research support from NSFC (No. 11671373); Wang acknowledges research support from the Australian Research Council.

References

REFERENCES

Amihud, Y., & Hurvich, C. M. (2004). Predictive regressions: A reduced-bias estimation method. Journal of Financial and Quantitative Analysis , 39(4), 813–841.CrossRefGoogle Scholar
Andersen, T. G., & Varneskov, R. T. (2021). Consistent inference for predictive regressions in persistent economic systems. Journal of Econometrics , 224(1), 215–244.CrossRefGoogle Scholar
Ang, A., & Bekaert, G. (2007). Stock return predictability: Is it there? Review of Financial Studies , 20(3), 651–707.CrossRefGoogle Scholar
Bandi, F. M., & Perron, B. (2008). Long-run risk-return trade-offs. Journal of Econometrics , 143, 349–374.CrossRefGoogle Scholar
Bandi, F.M., Perron, B., Tamoni, A., & Tebaldi, C. (2019). The scale of predictability. Journal of Econometrics , 208(1), 120–140.CrossRefGoogle Scholar
Bollerslev, T., Osterrieder, D., Sizova, N., & Tauchen, G. (2013). Risk and return: Long-run relations, fractional cointegration, and return predictability. Journal of Financial Economics , 108, 409–424.CrossRefGoogle Scholar
Buchmann, B., & Chan, N. H. (2007). Asymptotic theory of least squares estimators for nearly unstable processes under strong dependence. Annals of Statistics , 35(5), 2001–2017.CrossRefGoogle Scholar
Campbell, J. Y., & Yogo, M. (2006). Efficient tests of stock return predictability. Journal of Financial Economics , 81, 27–60.CrossRefGoogle Scholar
Cavanagh, C. L., Elliot, G., & Stock, J. (1995). Inference in models with nearly integrated regressors. Econometric Theory , 11, 1131–1147.CrossRefGoogle Scholar
Chan, N., & Wang, Q. (2015). Nonlinear regressions with nonstationary time series. Journal of Econometrics , 185, 182–195.CrossRefGoogle Scholar
Chang, Y., Park, J. Y., & Phillips, P. C. B. (2001). Nonlinear econometric models with cointegrated and deterministically trending regressors. Econometrics Journal , 4, 1–36.CrossRefGoogle Scholar
Chen, W. W., & Deo, R. (2009). Bias reduction and likelihood-based almost exactly sized hypothesis testing in predictive regressions using the restricted likelihood. Econometric Theory , 25(5), 1143–1179.CrossRefGoogle Scholar
Christensen, B. J., & Nielsen, M. Ø. (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., & Nielsen, M. Ø. (2007). The effect of long memory in volatility on stock market fluctuations. Review of Economics and Statistics , 89, 684–700.CrossRefGoogle Scholar
Christopeit, N. (2009). Weak convergence of nonlinear transformations of integrated processes: The multivariate case. Econometric Theory , 25, 1180–1207.CrossRefGoogle Scholar
Demetrescu, M., Georgiev, I., Rodrigues, P., & Taylor, R. (2022). Testing for episodic predictability in stock returns. Journal of Econometrics , 227(1), 85–113.CrossRefGoogle Scholar
Duffy, J. A., & Kasparis, I. (2018). Regressions with fractional $d=1/2$ and weakly nonstationary processes. arXiv:1812.07944v1.Google Scholar
Duffy, J. A., & Kasparis, I. (2021). Estimation and inference in the presence of fractional d = 1/2 and weakly nonstationary processes. Annals of Statistics , 49(2), 1195–1217.CrossRefGoogle Scholar
Elliott, G., Müller, U. K., & Watson, M. W. (2015). Nearly optimal tests when a nuisance parameter is present under the null hypothesis. Econometrica , 83, 771–811.CrossRefGoogle Scholar
Francq, C., & Zakoian, J.-M. (2010). GARCH models: Structure, statistical inference and financial applications . John Wiley & Sons.CrossRefGoogle Scholar
Giraitis, L., Kapetanios, G., & Marcellino, M. (2021). Time-varying instrumental variable estimation. Journal of Econometrics , 224(2), 394–415.CrossRefGoogle Scholar
Giraitis, L., Kapetanios, G., & Yates, T. (2014). Inference on stochastic time-varying coefficient models. Journal of Econometrics , 179(1), 46–65.CrossRefGoogle Scholar
Gonzalo, J., & Pitarakis, J.-Y. (2012). Regime-specific predictability in predictive regressions. Journal of Business and Economic Statistics , 30(2), 229–241.CrossRefGoogle Scholar
Hansen, B. E. (1996). Inference when a nuisance parameter is not identified under the null hypothesis. Econometrica , 64, 413–430.CrossRefGoogle Scholar
Hu, Z., Kasparis, I., & Wang, Q. (2021a). Chronologically trimmed LS for nonlinear predictive regressions with persistence of unknown form. mimeo.Google Scholar
Hu, Z., Kasparis, I., & Wang, Q. (2024). Time varying parameter models with stationary persistent data. Econometric Theory, first published online 1 April 2024. https://doi.org/10.1017/S0266466624000082.CrossRefGoogle Scholar
Hu, Z., Phillips, P. C. B., & Wang, Q. (2021b). Nonlinear cointegrating power regression with endogeneity. Econometric Theory , 37(1), 138–168.CrossRefGoogle Scholar
Hualde, J., & Robinson, P. M. (2010). Semiparametric inference in multivariate fractionally cointegrated systems. Journal of Econometrics , 157(2), 492–511.CrossRefGoogle Scholar
Janson, M., & Moreira, J. M. (2006). Optimal inference in regression models with nearly integrated regressors. Econometrica , 74(3), 681–714.CrossRefGoogle Scholar
Jeganathan, P. (2008). Limit theorems for functionals of sums that converge to fractional Brownian and stable motions. Cowles Foundation Discussion Paper 1649.Google Scholar
Jin, C, & Wang, Q. (2024). Weighted nonlinear regression with nonstationary time series. Statistica Sinica , 34. https://doi.org/10.5705/ss.202021.0426 Google Scholar
Johansen, S. (1995). Likelihood-based inference in cointegrated vector auto-regressive models . Oxford University Press.CrossRefGoogle Scholar
Kasparis, I. (2010). The Bierens test for certain nonstationary models. Journal of Econometrics , 158, 221–230.CrossRefGoogle Scholar
Kasparis, I. (2011). Functional form misspecification in regressions with a unit root. Econometric Theory , 27, 285–311.CrossRefGoogle Scholar
Kasparis, I., Andreou, E., & Phillips, P. C. B. (2015). Nonparametric predictive regression. Journal of Econometrics , 185(2), 468–494.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
Lewellen, J. (2004). Predicting returns with financial ratios. Journal of Financial Economics , 74(2), 209–235.CrossRefGoogle Scholar
Magdalinos, T. (2022). Least squares and IVX limit theory in systems of predictive regressions with GARCH innovations. Econometric Theory , 38(5), 875–912.CrossRefGoogle Scholar
Magdalinos, T., & Petrova, K. (2022). Uniform and distribution-free inference with general autoregressive processes, Working Paper 1344, Barcelona School of Economics.Google Scholar
Marmer, V. (2008). Nonlinearity, nonstationarity and spurious forecasts. Journal of Econometrics , 142(1), 1–27.CrossRefGoogle Scholar
McCracken, M. W. (2007). Asymptotics for out of sample tests of Granger causality. Journal of Econometrics , 140(2), 719–752.CrossRefGoogle Scholar
Mikusheva, A. (2007). Uniform inference in autoregressive models. Econometrica , 75(5), 1411–1452.CrossRefGoogle Scholar
Park, J. Y., & Phillips, P. C. B. (1999). Asymptotics for nonlinear transformations of integrated time series. Econometric Theory , 15(3), 269–298.CrossRefGoogle Scholar
Park, J. Y., & Phillips, P. C. B. (2001). Nonlinear regressions with integrated time series. Econometrica , 69(1), 117–161.CrossRefGoogle Scholar
Phillips, P. C. B. (1995). Fully modified least squares and vector autoregression. Econometrica , 63(5), 1023–1078.CrossRefGoogle Scholar
Phillips, P. C. B. (2014). On confidence intervals for autoregressive roots and predictive regressions. Econometrica , 82(3), 1177–1195.Google Scholar
Phillips, P. C. B. (2015). Pitfalls and possibilities in predictive regression. Journal of Financial Econometrics , 13(3), 521–555.CrossRefGoogle Scholar
Phillips, P. C. B., & Hansen, B. (1990). Statistical inference in instrumental variables regression with I(1) processes. Review of Economic Studies , 57(1), 99–125 CrossRefGoogle Scholar
Phillips, P. C. B., Li, D., & Gao, J. (2017). Estimating smooth structural change in cointegration models. Journal of Econometrics , 196, 180–195.CrossRefGoogle Scholar
Phillips, P. C. B., & Magdalinos, T. (2009). Econometric inference in the vicinity of unity. Mimeo , Singapore Management University.Google Scholar
Robinson, P. M. (1991). Time varying nonlinear regression. In: Hackl, P. (Ed.), Economic structural change: Analysis and forecasting (pp. 179–190). Springer.CrossRefGoogle Scholar
Robinson, P. M., & Hualde, J. (2003). Cointegration in fractional systems with unknown integration orders. Econometrica, 71(6), 1727–1766.CrossRefGoogle Scholar
Shimotsu, K., & Phillips, P. C. B. (2005). Exact local Whittle estimation of fractional integration. Annals of Statistics , 33(4), 1890–1933.CrossRefGoogle Scholar
Stambaugh, R. F. (1999). Predictive regressions. Journal of Financial Economics , 54, 783–820.CrossRefGoogle Scholar
Sun, Y., Phillips, P. C. B., & Jin, S. (2008). Optimal bandwidth selection in heteroskedasticity-autocorrelation robust testing. Econometrica , 76(1), 175–194.CrossRefGoogle Scholar
Valkanov, R. (2003). Long-horizon regressions: Theoretical results and applications. Journal of Financial Economics , 68, 201–232.CrossRefGoogle Scholar
Wang, Q. (2014). Martingale limit theorem revisited and nonlinear cointegrating regression. Econometric Theory , 30(3), 509–535.CrossRefGoogle Scholar
Wang, Q. (2015). Limit theorems for nonlinear cointegrating regression . Singapore: World Scientific.CrossRefGoogle Scholar
Wang, Q. (2021). Least squares estimation for nonlinear regression models with heteroscedasticity. Econometric Theory , 37(6), 1173–1213.CrossRefGoogle Scholar
Wang, Q., & Phillips, P. C. B. (2009a). Asymptotic theory for local time density estimation and nonparametric cointegrating regression. Econometric Theory , 25(3), 710–738.CrossRefGoogle Scholar
Wang, Q., & Phillips, P. C. B. (2009b). Structural nonparametric cointegrating regression. Econometrica , 77(6), 1901–1948.Google Scholar
Wang, Q., & Phillips, P. C. B. (2011). Asymptotic theory for zero energy functionals with nonparametric regression applications. Econometric Theory , 27(2), 235–259.CrossRefGoogle Scholar
Wang, Q., & Phillips, P. C. B. (2012). A specification test for nonlinear nonstationary models. Annals of Statistics , 40(2), 727–758.CrossRefGoogle Scholar
Welch, I., & Goyal, A. (2008). A comprehensive look at the empirical performance of equity premium prediction. Review of Financial Studies , 21, 1455–1508.CrossRefGoogle Scholar
Supplementary material: File

Hu et al. supplementary material

Hu et al. supplementary material
Download Hu et al. supplementary material(File)
File 1.1 MB