Semiparametric time series models with log-concave innovations: maximum likelihood estimation and its consistency. (English) Zbl 1378.62061
Summary: We study semiparametric time series models with innovations following a log-concave distribution. We propose a general maximum likelihood framework that allows us to estimate simultaneously the parameters of the model and the density of the innovations. This framework can be easily adapted to many well-known models, including autoregressive moving average (ARMA), generalized autoregressive conditionally heteroscedastic (GARCH), and ARMA-GARCH models. Furthermore, we show that the estimator under our new framework is consistent in both ARMA and ARMA-GARCH settings. We demonstrate its finite sample performance via a thorough simulation study and apply it to model the daily log-return of the FTSE100 index.
MSC:
| 62M10 | Time series, auto-correlation, regression, etc. in statistics (GARCH) |
| 62G32 | Statistics of extreme values; tail inference |
| 62G20 | Asymptotic properties of nonparametric inference |
Keywords:
ARMA; ARMA-GARCH; consistency; GARCH; log-concavity; maximum likelihood; semiparametric; shape constraint; time seriesReferences:
| [1] | Bagnoli, M. & Bergstrom, T. (2005). Log‐concave probability and its applications. Economet. Theor.26, 445-469. · Zbl 1077.60012 |
| [2] | Balabdaoui, F., Rufibach, K. & Wellner, J. A. (2009). Limit distribution theory for maximum likelihood estimation of a log‐concave density. Ann. Statist.37, 1299-1331. · Zbl 1160.62008 |
| [3] | Billingsley, P. (1995). Probability and measure, (3rd edn.)., John Wiley & Sons, New York. · Zbl 0822.60002 |
| [4] | Bollerslev, T. (1986). Generalized autoregressive conditional heteroscedasticity. J. Econometrics31, 307-327. · Zbl 0616.62119 |
| [5] | Bougerol, P. & Picard, N. (1992). Stationarity of GARCH processes and of some nonnegative time series. J. Econometrics52, 115-127. · Zbl 0746.62087 |
| [6] | Brockwell, P. J. & Davis, R. A. (1991). Time series: theory and methods, (2nd edn.)., Springer‐Verlag, New York. · Zbl 0709.62080 |
| [7] | Brockwell, P. J. & Lindner, A. (2010). Strict stationary solutions of autoregressive moving average equations. Biometrika97, 765-772. · Zbl 1195.62137 |
| [8] | Chen, Y. (2011). Semiparametric time series models with log‐concave innovations: maximum likelihood estimation and its consistency. arXiv:1111.6291. |
| [9] | Chen, X., Liao, Z. & Sun, Y. (2012). Sieve inference on semi‐nonparametric time series models, Cowles foundation discussion paper, No. 1849. |
| [10] | Chen, Y. & Samworth, R. J. (2013). Smoothed log‐concave maximum likelihood estimation with applications. Statist. Sinica23, 1373-1398. · Zbl 1534.62045 |
| [11] | Cule, M. & Samworth, R. (2010). Theoretical properties of the log‐concave maximum likelihood estimator of a multidimensional density. Electron. J. Stat.4, 254-270. · Zbl 1329.62183 |
| [12] | Cule, M., Samworth, R. & Stewart, M. (2010). Maximum likelihood estimation of a multi‐dimensional log‐concave density (with discussion). J. R. Stat. Soc. Ser. B Stat. Methodol.72, 545-607. · Zbl 1411.62055 |
| [13] | Damsleth, E. & El‐Shaarawi, A. H. (1989). ARMA models with double‐exponentially distributed noise. J. R. Stat. Soc. Ser. B Stat. Methodol.51, 61-69. |
| [14] | Diggle, P., Liang, K‐Y. & Zeger, S. (2002). Analysis of longitudinal data, (2nd edn.)., Oxford University Press, Oxford. |
| [15] | Drost, F. C. & Klaassen, C. A. J. (1997). Efficient estimation in semiparametric GARCH models. J. Econometrics81, 193-221. · Zbl 0944.62120 |
| [16] | Drost, F. C., Klaassen, C. A. J. & Werker, B. J. M. (1997). Adaptive estimation in time‐series models. Ann. Statist.25, 786-817. · Zbl 0941.62093 |
| [17] | Dümbgen, L. & Rufibach, K. (2009). Maximum likelihood estimation of a log‐concave density and its distribution function: basic properties and uniform consistency. Bernoulli15, 40-68. · Zbl 1200.62030 |
| [18] | Dümbgen, L. & Rufibach, K. (2011). logcondens: computations related to univariate log‐concave density estimation. J. Stat. Softw.39, 1-28. |
| [19] | Dümbgen, L., Hüsler, A. & Rufibach, K. (2011). Active set and EM algorithms for log‐concave densities based on complete and censored data. Technical report 61, IMSV, Univ. Bern. arXiv:0707.4643v4. |
| [20] | Dümbgen, L., Samworth, R. & Schuhmacher, D. (2011). Approximation by log‐concave distributions with applications to regression. Ann. Statist.39, 702-730. · Zbl 1216.62023 |
| [21] | Dümbgen, L., Samworth, R. J. & Schuhmacher, D. (2013). Stochastic search for semiparametric linear regression models. In From probability to statistics and back: high‐dimensional models and processes – a Festschrift in honor of Jon A. Wellner (eds M.Banerjee (ed.), F.Bunea (ed.), J.Huang (ed.), V.Koltchinskii (ed.) & M. H.Maathuis (ed.)), Institute of Mathematical Statistics, Ohio, USA; 78-90. · Zbl 1327.62204 |
| [22] | Engle, R. F. (1982). Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation. Econometrica50, 987-1007. · Zbl 0491.62099 |
| [23] | Engle, R. F. & Gonzalez‐Rivera, G. (1991). Semiparametric ARCH models. J. Bus. Econom. Statist.9, 345-359. |
| [24] | Fan, J., Qi, L. & Xiu, D. (2014). Quasi maximum likelihood estimation of GARCH models with heavy‐tailed likelihoods (with discussion). J. Bus. Econom. Statist. to appear. |
| [25] | Francq, C. & Zakoïan, J. ‐M. (2004). Maximum likelihood estimation of pure GARCH and ARMA‐GARCH processes. Bernoulli10, 605-637. · Zbl 1067.62094 |
| [26] | Francq, C. & Zakoïan, J‐M. (2010). GARCH models: structure, statistical inference and financial applications, John Wiley & Sons, New York. |
| [27] | Francq, C., Lepage, G. & Zakoïan, J. ‐M. (2011). Two‐stage non Gaussian QML estimation of GARCH models and testing the efficiency of the Gaussian QMLE. J. Econometrics165, 246-257. · Zbl 1441.62692 |
| [28] | Granger, C. W. J. & Ding, Z. (1995). Some properties of absolute return: an alternative measure of risk. Ann. Econ. Stat.40, 67-91. |
| [29] | Groeneboom, P., Jongbloed, G. & Wellner, J. A. (2001). Estimation of a convex function: characterizations and asymptotic theory. Ann. Statist.29, 1653-1698. · Zbl 1043.62027 |
| [30] | Haas, M., Mittnik, S. & Paolella, M. S. (2006). Modelling and predicting market risk with Laplace‐Gaussian mixture distributions. Applied Financial Economics16, 1145-1162. |
| [31] | Hannan, E. J. (1970). Multiple time series, John Wiley & Sons, New York. · Zbl 0211.49804 |
| [32] | Koenker, R. & Hallock, K. F. (2001). Quantile regression. J. Econ. Perspect.15, 143-156. |
| [33] | Koenker, R. & Mizera, I. (2010). Quasi‐concave density estimation. Ann. Statist.38, 2998-3027. · Zbl 1200.62031 |
| [34] | Kreiss, J. ‐P. (1987). On adaptive estimation in stationary ARMA processes. Ann. Statist.15, 112-133. · Zbl 0616.62042 |
| [35] | Li, W. ‐K. & Mcleod, A. (1988). ARMA modelling with non‐Gaussian innovations. J. Time Series Anal.9, 155-168. · Zbl 0637.62079 |
| [36] | Ling, S. & McAleer, M. (2003). On adaptive estimation in nonstationary ARMA models with GARCH errors. Ann. Statist.31, 642-674. · Zbl 1039.62084 |
| [37] | Mammen, E. & Park, B. U. (1997). Optimal smoothing in adaptive location estimation. J. Statist. Plann. Inference58, 333-348. · Zbl 0877.62033 |
| [38] | Nelder, J. A. & Mead, R. (1965). A simplex algorithm for function minimization. Comput. J.7, 308-313. · Zbl 0229.65053 |
| [39] | Newey, W. K. & Steigerwald, D. G. (1997). Asymptotic bias for quasi‐maximum‐likelihood estimators in conditional heteroscedasticity models. Econometrica65, 587-599. · Zbl 0870.62091 |
| [40] | Pal, J. K., Woodroofe, M. & Meyer, M. (2007). Estimating a Polya frequency function. In Complex datasets and inverse problems: tomography, networks and beyond, vol. 54 (eds R. Y.Liu (ed.), W. E.Strawderman (ed.) & C.‐H.Zhang (ed.)), Lecture notes ‐ monograph seriesInstitute of Mathematical Statistics, Ohio; 239-249. |
| [41] | Pan, J., Wang, H. & Yao, Q. (2007). Weighted least absolute deviations estimation for ARMA models with infinite variance. Economet. Theor.23, 852-879. · Zbl 1237.62122 |
| [42] | Price, K., Storn, R. M. & Lampinen, J. A. (2005). Differential evolution: a practical approach to global optimization, Springer‐Verlag, Berlin. · Zbl 1186.90004 |
| [43] | Rockafellar, R. T. (1997). Convex analysis, Princeton University Press, Princeton, New Jersey. · Zbl 0932.90001 |
| [44] | Rufibach, K. (2012). A smooth ROC curve estimator based on log‐concave density estimates. Int. J. Biostat.8, 1-29. |
| [45] | Samworth, R. J. & Yuan, M. (2012). Independent component analysis via nonparametric maximum likelihood estimation. Ann. Statist.40, 2973-3002. · Zbl 1296.62084 |
| [46] | Schuhmacher, D., Hüsler, A. & Dümbgen, L. (2011). Multivariate log‐concave distributions as a nearly parametric model. Statistics and Risk Modeling28, 277-295. · Zbl 1245.62060 |
| [47] | Seregin, A. & Wellner, J. A. (2010). Nonparametric estimation of convex‐transformed densities. Ann. Statist.38, 3751-3781. · Zbl 1204.62058 |
| [48] | Silverman, B. W. (1982). On the estimation of a probability density function by the maximum penalized likelihood method. Ann. Statist.10, 795-810. · Zbl 0492.62034 |
| [49] | Sun, Y. & Stengos, T. (2006). Semiparametric efficient adaptive estimation of asymmetric GARCH models. J. Econometrics133, 373-386. · Zbl 1345.62122 |
| [50] | Tao, P. ‐C., Yevjevich, V. & Kottegoda, N. (1976). Distribution of hydrologic independent stochastic components, Hydrology Papers 82, Colorado State University, Fort Collins. |
| [51] | Trindade, A. A., Zhu, Y. & Andrews, B. (2010). Time series models with asymmetric Laplace innovations. J. Stat. Comput. Sim.80, 1317-1333. · Zbl 1205.62135 |
| [52] | Villani, C. (2009). Optimal transport: old and new, Springer‐Verlag, Berlin. · Zbl 1156.53003 |
| [53] | Walther, G. (2002). Detecting the presence of mixing with multiscale maximum likelihood. J. Amer. Statist. Assoc.97, 508-513. · Zbl 1073.62533 |
| [54] | Wuertz, D. & Chalabi, Y. (2013). fGarch: Rmetrics ‐ autoregressive conditional heteroskedastic modelling. Rpackage version 2150.82. Available from http://cran.r‐project.org/web/packages/fGarch/. |
| [55] | Xia, Y. & Tong, H. (2010). Discussion of the paper Maximum likelihood estimation of a multi‐dimensional log‐concave density by Cule, Samworth and Stewart. J. R. Stat. Soc. Ser. B Stat. Methodol.72, 585. |
| [56] | Yao, Q. (2010). Discussion of the paper Maximum likelihood estimation of a multi‐dimensional log‐concave density by Cule, Samworth and Stewart. J. R. Stat. Soc. Ser. B Stat. Methodol.72, 588. |
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