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ESTIMATION AND INFERENCE FOR VARYING-COEFFICIENT MODELS WITH NONSTATIONARY REGRESSORS USING PENALIZED SPLINES

Published online by Cambridge University Press:  14 October 2014

Haiqiang Chen ,
Ying Fang  and
Yingxing Li
Haiqiang Chen
Affiliation:
Xiamen University
Ying Fang*
Affiliation:
Xiamen University
Yingxing Li*
Affiliation:
Xiamen University
*
*Address correspondence to Ying Fang, Wang Yanan Institute for Studies in Economics, Xiamen University, Xiamen, China; e-mail: yifst1@gmail.com; or to: Yingxing Li, Wang Yanan Institute for Studies in Economics, Xiamen University, Xiamen, China; e-mail: yxli@xmu.edu.cn.
*Address correspondence to Ying Fang, Wang Yanan Institute for Studies in Economics, Xiamen University, Xiamen, China; e-mail: yifst1@gmail.com; or to: Yingxing Li, Wang Yanan Institute for Studies in Economics, Xiamen University, Xiamen, China; e-mail: yxli@xmu.edu.cn.
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Abstract

This paper considers estimation and inference for varying-coefficient models with nonstationary regressors. We propose a nonparametric estimation method using penalized splines, which achieves the same optimal convergence rate as kernel-based methods, but enjoys computation advantages. Utilizing the mixed model representation of penalized splines, we develop a likelihood ratio test statistic for checking the stability of the regression coefficients. We derive both the exact and the asymptotic null distributions of this test statistic. We also demonstrate its optimality by examining its local power performance. These theoretical findings are well supported by simulation studies.

Type
ARTICLES
Information
Econometric Theory , Volume 31 , Issue 4 , August 2015 , pp. 753 - 777
Copyright
Copyright © Cambridge University Press 2014 

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Footnotes

The authors are grateful to Peter Phillips, Wolfgang Härdle, and anonymous referees for their helpful comments. We also thank Zongwu Cai, Jiti Gao, Yongmiao Hong, and all participants of the 3rd WISE-Humboldt Workshop in Nonparametric Nonstationary High-dimensional Econometrics in May 2012. We acknowledge the financial support from the National Science Foundation of China with grant numbers 71201137, 71271179, 71131008, and 11201390, and from the Natural Science Foundation of Fujian Province with grant number 2013J01024, as well as the Deutsche Forschungsgemeinschaft through the SFB 649 “Economic Risk”.

References

REFERENCES

Bierens, H.J. & Martins, L. (2010) Time-varying cointegration. Econometric Theory 26, 1453–1490.CrossRefGoogle Scholar
Cai, Z., Li, Q., & Park, J.Y. (2009) Functional-coefficient models for nonstationary time series data. Journal of Econometrics 148, 101–113.CrossRefGoogle Scholar
Campbell, J. & Yogo, M. (2006) Efficient tests of stock return predictability. Journal of Financial Econometrics 81, 27–60.CrossRefGoogle Scholar
Chen, H., Fang, Y., & Li, Y. (2013) Estimation and inference for varying-coefficient models with nonstationary regressors using penalized splines. Available at http://ssrn.com/abstract=2287449.CrossRefGoogle Scholar
Claeskens, G., Krivobokova, T., & Opsomer, J. (2009) Asymptotical properties of penalized spline estimators. Biometrika 96, 529–544.CrossRefGoogle Scholar
Crainiceanu, C.M. & Ruppert, D. (2004) Likelihood ratio tests in linear mixed models with one variance component. Journal of the Royal Statistical Society, Series B 66, 165–185.CrossRefGoogle Scholar
Crainiceanu, C.M., Ruppert, D., Claeskens, G., & Wand, M.P. (2005) Exact likelihood ratio tests for penalized splines. Biometrika 92, 91–103.CrossRefGoogle Scholar
de Boor, C. (1978) A Practical Guide to Splines. Springer-Verlag.Google Scholar
Eilers, P.H.C. & Marx, B.D. (1996) Flexible smoothing with B-splines and penalties. Statistical Science 11, 89–121.CrossRefGoogle Scholar
Engle, R.F. & Granger, C.W.J. (1987) Cointegration and error correction: Representation, estimation, and testing. Econometrica 55, 251–276.CrossRefGoogle Scholar
Fan, J., Zhang, C., & Zhang, J. (2001) Generalized likelihood ratio statistics and Wilks phenomenon. Annals of Statistics 29, 153–193.CrossRefGoogle Scholar
Gao, J., King, M., Lu, Z., & Tjøstheim, D. (2009) Nonparametric specification testing for nonlinear time series with nonstationarity. Econometric Theory 25, 1869–1892.CrossRefGoogle Scholar
Goldfajn, I. & Baig, T. (1998) Monetary Policy in the Aftermath of Currency Crises: The Case of Asia. Working paper, IMF.Google Scholar
Goyal, A. & Welch, I. (2003) Predicting the equity premium with dividend ratios. Management Science 49, 639–654.CrossRefGoogle Scholar
Hall, P. & Opsomer, J.D. (2005) Theory for penalised spline regression. Biometrika 92, 105–118.CrossRefGoogle Scholar
Hansen, B.E. (1992) Tests for parameter instability in regressions with I(1) processes. Journal of Business and Economic Statistics 10, 321–335.Google Scholar
Hansen, H. & Johansen, S. (1999) Some tests for parameter constancy in cointegrated VAR-models. Econometrics Journal 2, 306–333.CrossRefGoogle Scholar
Hao, K. (1996) Testing for structural change in cointegrated regression models: Some comparisons and generalizations. Econometric Reviews 15, 401–429.CrossRefGoogle Scholar
Härdle, W. & Mammen, E. (1993) Comparison nonparametric versus parametric regression fits. Annals of Statistics 21, 1926–1947.CrossRefGoogle Scholar
Harris, D., McCabe, B., & Leybourne, S. (2002) Stochastic cointegration: Estimation and inference. Journal of Econometrics 111, 363–384.Google Scholar
Hong, S. & Phillips, P.C.B. (2010) Testing linearity in cointegrating relations with an application to purchasing power parity. Journal of Business and Economic Statistics 28, 96–113.CrossRefGoogle Scholar
Hong, Y. & White, H. (1995) Consistent specification testing via nonparametric series regression. Econometrica 63, 1133–1159.CrossRefGoogle Scholar
Johansen, S., Mosconi, R., & Nielsen, B. (2000) Cointegration analysis in the presence of structural breaks in the deterministic trend. Econometrics Journal 3, 216–249.CrossRefGoogle Scholar
Juhl, T. (2005) Functional-coefficient models under unit root behavior. Econometrics Journal 8, 197–213.CrossRefGoogle Scholar
Kasparis, I. (2008) Detection of functional form misspecification in cointegrating relations. Econometric Theory 24, 1373–1404.CrossRefGoogle Scholar
Kuo, B.S. (1998) Test for partial parameter instability in regressions with I(1) processes. Journal of Econometrics 86, 337–368.CrossRefGoogle Scholar
Lettau, M. & Ludvigsson, S. (2001) Consumption, aggregate wealth, and expected stock returns. Journal of Finance 56, 815–849.Google Scholar
Li, Y. & Ruppert, D. (2008) On the asymptotics of penalized splines. Biometrika 95, 415–436.CrossRefGoogle Scholar
Nelson, R. & Plosser, C.I. (1982) Trends and random walks in macroeconomic time series: Some evidence and implications. Journal of Monetary Economics 10, 139–162.CrossRefGoogle Scholar
Park, J. & Hahn, S. (1999) Cointegrating regressions with time varying coefficients. Econometric Theory 15, 664–703.CrossRefGoogle Scholar
Park, J. & Phillips, P.C.B. (2001) Nonlinear regressions with integrated time series. Econometrica 69, 117–161.CrossRefGoogle Scholar
Patterson, H.D. & Thompson, R. (1971) Recovery of inter-block information when block sizes are unequal. Biometrika 58, 545–554.CrossRefGoogle Scholar
Paye, B.S. & Timmermann, A. (2006) Instability of return prediction models. Journal of Empirical Finance 13, 274–315.CrossRefGoogle Scholar
Quintos, C.E. (1997) Stability tests in error correction models. Journal of Econometrics 82, 289–315.CrossRefGoogle Scholar
Ruppert, D. (2002) Selecting the number of knots for penalized splines. Journal of the Computational and Graphical Statistics 11, 735–757.CrossRefGoogle Scholar
Ruppert, D., Wand, M.P., & Carroll, R.J. (2003) Semiparametric Regression. Cambridge University Press.CrossRefGoogle Scholar
Saikkonen, P. (1991) Asymptotically efficient estimation of cointegration regressions. Econometric Theory 7, 1–20.CrossRefGoogle Scholar
Saikkonen, P. & Choi, I. (2004) Cointegrating smooth transition regressions. Econometric Theory 20, 201–340.CrossRefGoogle Scholar
Shao, Q. & Lu, C. (1987) Strong approximation for partial sums of weakly dependent random variables. Scientia Sinica 15, 576–587.Google Scholar
Shi, X. & Phillips, P.C.B. (2012) Nonlinear cointegrating regression under weak identification. Econometric Theory 28, 509–547.CrossRefGoogle Scholar
Wang, Q. & Phillips, P.C.B. (2009a) Structural nonparametric cointegrating regression. Econometrica 77, 1901–1948.Google Scholar
Wang, Q. & Phillips, P.C.B. (2009b) Asymptotic theory for local time density estimation and nonparametric cointegrating regression. Econometric Theory 25, 710–738.CrossRefGoogle Scholar
Xiao, Z. (2009) Functional-coefficient cointegration models. Journal of Econometrics 152, 81–92.CrossRefGoogle Scholar