Root-\(T\) consistent density estimation in GARCH models. (English) Zbl 1419.62225
Summary: We consider a new nonparametric estimator of the stationary density of the logarithm of the volatility of the \(\mathrm{GARCH}(1, 1)\) model. This problem is particularly challenging since this density is still unknown, even in cases where the model parameters are given. Although the volatility variables are only observed with multiplicative independent innovation errors with unknown density, we manage to construct a nonparametric procedure which estimates the log volatility density consistently. By carefully exploiting the specific GARCH dependence structure of the data, our iterative procedure even attains the striking parametric root-\(T\) convergence rate. As a by-product of our main results, we also derive new smoothness properties of the stationary density. Using numerical simulations, we illustrate the performance of our estimator, and we provide an application to financial data.
MSC:
| 62M10 | Time series, auto-correlation, regression, etc. in statistics (GARCH) |
| 62G07 | Density estimation |
| 62G20 | Asymptotic properties of nonparametric inference |
| 62P05 | Applications of statistics to actuarial sciences and financial mathematics |
Keywords:
autoregression; consistency; convergence rates; financial econometrics; nonparametric statistics; time seriesSoftware:
KernSmoothReferences:
| [1] | Aït-Sahalia, Y., Nonparametric pricing of interest rate derivative securities, Econometrica, 64, 527-560 (1996) · Zbl 0844.62094 |
| [2] | Berkes, I.; Horváth, L.; Kokoszka, P. S., Garch processes: structure and estimation, Bernoulli, 9, 201-227 (2003) · Zbl 1064.62094 |
| [3] | Bollerslev, T., Generalized autoregressive conditional heteroscedasticity, J. Econometrics, 31, 307-327 (1986) · Zbl 0616.62119 |
| [4] | Bollerslev, T.; Wooldridge, J., Quasi-maximum likelihood estimation and inference in dynamic models with time-varying covariances, Econometric Rev., 11, 143-172 (1992) · Zbl 0850.62884 |
| [5] | Bougerol, P.; Picard, N., S tationarity of GARCH processes and of some non-negative time series, J. Econometrics, 52, 115-127 (1992) · Zbl 0746.62087 |
| [6] | Carroll, R. J.; Hall, P., Optimal rates of convergence for deconvolving a density, J. Amer. Statist. Assoc., 83, 1184-1186 (1988) · Zbl 0673.62033 |
| [7] | Chang, C. C.; Politis, D. N., Bootstrap with larger resample size for root-n consistent density estimation with time series data, Statist. Prob. Lett., 81, 652-661 (2011) · Zbl 1213.62138 |
| [8] | Chen, X.; Christensen, T. M., Optimal uniform convergence rates and asymptotic normality for series estimators under weak dependence and weak conditions, J. Econometrics, 188, 447-465 (2015) · Zbl 1337.62101 |
| [9] | Comte, F.; Dedecker, J.; Taupin, M.-L., Adaptive density estimation for general ARCH models, Econometric Theory, 24, 1628-1662 (2008) · Zbl 1277.62103 |
| [10] | Delaigle, A., Nonparametric density estimation from data with a mixture of berkson and classical errors, Canad. J. Statist., 35, 89-104 (2007) · Zbl 1124.62018 |
| [11] | Delaigle, A.; Hall, P., Kernel methods and minimum contrast estimators for empirical deconvolution, (Bingham, N. H.; Goldie, C. M., Probability and Mathematical Genetics, Papers in Honour of Sir John Kingman. In: London Mathematical Society Lecture Note Series (2010), Cambridge University Press), (Chapter 8) · Zbl 1513.62075 |
| [12] | Diggle, P.; Hall, P., A Fourier approach to nonparametric deconvolution of a density estimate, J. Roy. Statist. Soc. Se . B, 55, 523-531 (1993) · Zbl 0783.62030 |
| [13] | Engle, R., Autoregressive conditional heteroskedasticity with estimates of the variance of the United Kingdom inflation, Econometrica, 50, 987-1007 (1982) · Zbl 0491.62099 |
| [15] | Fan, J.; Yao, Q., Nonlinear Time Series: Nonparametric and Parametric Methods (2003), Springer · Zbl 1014.62103 |
| [16] | Francq, C.; Lepage, G.; Zakoian, J. M., Two-stage non Gaussian QML estimation of GARCH models and testing the efficiency of the Gaussian QMLE, J. Econometrics, 165, 246-257 (2011) · Zbl 1441.62692 |
| [17] | Francq, C.; Zakoian, J. M., Maximum likelihood estimation of pure GARCH and ARMA-GARCH processes, Bernoulli, 10, 605-637 (2004) · Zbl 1067.62094 |
| [18] | Francq, C.; Zakoian, J. M., GARCH Models: Structure, Statistical Inference and Financial Applications (2010), Wiley |
| [19] | Francq, C.; Zakoian, J. M., Strict stationarity testing and estimation of explosive and stationary generalized autoregressive conditional heteroskedasticity models, Econometrica, 80, 821-861 (2012) · Zbl 1274.62590 |
| [20] | Francq, C.; Zakoian, J. M., Estimating the marginal law of a time series with applications to heavy tailed distributions, J. Bus. Econom. Statist., 31, 412-425 (2013) |
| [21] | Hall, P.; Yao, Q., Inference in ARCH and GARCH models with heavy-tailed errors, Econometrica, 71, 285-317 (2003) · Zbl 1136.62368 |
| [22] | He, C.; Teräsvirta, T., Properties of moments of a family of GARCH processes, J. Econometrics, 92, 173-192 (1999) · Zbl 0929.62093 |
| [23] | Karanasos, M., The second moment and the autocovariance function of the squared errors of the garch model, J. Econometrics, 90, 63-67 (1999) · Zbl 1070.62528 |
| [24] | Kim, K. H.; Zhang, T.; Wu, W. B., Parametric specification test for nonlinear autoregressive models, Econometric Theory, 31, 1078-1101 (2015) · Zbl 1441.62774 |
| [25] | Lee, S.-W; Hansen, B. E., Asymptotic theory for the GARCH \((1, 1)\) quasi-maximum likelihood estimator, Econometric Theory, 10, 29-52 (1994) |
| [26] | Lindner, A. M., Stationarity, mixing, distributional properties and moments of GARCH \((p, q)\)-processes, (Andersen, T. G.; Davis, R. A.; Kreiß, J.-P.; Mikosch, T., Handbook of Financial Times Series (2009), Springer: Springer Berlin/Heidelberg), 43-69 · Zbl 1178.62101 |
| [27] | Lumsdaine, R. L., Consistency and asymptotic normality of the quasi-maximum likelihood estimator in IGARCH \((1, 1)\) and covariance stationary GARCH \((1, 1)\) models, Econometrica, 64, 575-596 (1996) · Zbl 0844.62080 |
| [28] | Meitz, M., A necessary and sufficient condition for the strict stationarity of a family of GARCH processes, Econometric Theory, 22, 985-988 (2006) · Zbl 1125.62106 |
| [29] | Mikosch, T.; Starica, C., Limit theory for the sample autocorrelations and extremes of a GARCH \((1, 1)\) process, Ann. Statist., 28, 1427-1451 (2000) · Zbl 1105.62374 |
| [30] | Mimoto, N., Convergence in distribution for the sup-norm of a kernel density estimator for GARCH innovations, Statist. Probab. Lett., 78, 915-923 (2008) · Zbl 1136.62369 |
| [31] | Nelson, D. B., Stationarity and persistence in the GARCH \((1, 1)\) model, Econometric Theory, 6, 318-334 (1990) |
| [32] | Robinson, P.; Zaffaroni, P., Pseudo-Maximum Likelihood Estimation of ARCH \(( \infty )\) Models, Ann. Statist., 34, 1049-1074 (2006) · Zbl 1113.62107 |
| [33] | Schick, A.; Wefelmeyer, W., Uniformly root-n consistent density estimators for weakly dependent invertible linear processes, Ann. Statist., 35, 815-843 (2007) · Zbl 1117.62036 |
| [34] | Silevrman, B., Density Estimation (1986), Chapman & Hall: Chapman & Hall London |
| [35] | Stefanski, L. A.; Carroll, R. J., Deconvoluting kernel density estimators, Statistics, 21, 169-184 (1990) · Zbl 0697.62035 |
| [36] | van Es, B.; Spreij, P.; van Zanten, H., Nonparametric volatility density estimation for discrete time models, J. Nonparametr Stat., 17, 237-251 (2005) · Zbl 1055.62038 |
| [37] | Wand, M. P.; Jones, M. C., Kernel Smoothing (1995), Chapman & Hall: Chapman & Hall London · Zbl 0854.62043 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
