Normalized least-squares estimation in time-varying ARCH models

Abstract

We investigate the time-varying ARCH (tvARCH) process. It is shown that it can be used to describe the slow decay of the sample autocorrelations of the squared returns often observed in financial time series, which warrants the further study of parameter estimation methods for the model. Since the parameters are changing over time, a successful estimator needs to perform well for small samples. We propose a kernel normalized-least-squares (kernel-NLS) estimator which has a closed form, and thus outperforms the previously proposed kernel quasi-maximum likelihood (kernel-QML) estimator for small samples. The kernel-NLS estimator is simple, works under mild moment assumptions and avoids some of the parameter space restrictions imposed by the kernel-QML estimator. Theoretical evidence shows that the kernel-NLS estimator has the same rate of convergence as the kernel-QML estimator. Due to the kernel-NLS estimator's ease of computation, computationally intensive procedures can be used. A prediction-based cross-validation method is proposed for selecting the bandwidth of the kernel-NLS estimator. Also, we use a residual-based bootstrap scheme to bootstrap the tvARCH process. The bootstrap sample is used to obtain pointwise confidence intervals for the kernel-NLS estimator. It is shown that distributions of the estimator using the bootstrap and the ``true'' tvARCH estimator asymptotically coincide. We illustrate our estimation method on a variety of currency exchange and stock index data for which we obtain both good fits to the data and accurate forecasts.Comment: Published in at http://dx.doi.org/10.1214/07-AOS510 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Locally stationary long memory estimation

“…In order to come up with a rigorous asymptotic theory of consistency and inference, the time-dependence of the spectral density f (u, λ) of such locally stationary processes is modelled to be in rescaled time u ∈ [0, 1], leading to a problem of non-parametric curve estimation: increasing the sample size T of the observed time series does no more mean to look into the future but to dispose of more and more observations to identify f (t/T, λ) locally around the "reference" rescaled time point u ≈ t/T . This approach was recently used for volatility estimation using time varying (short memory) non-linear processes, see [10,15]. We consider in this paper a locally stationary long-range dependent process with a singularity in the spectral density at zero frequency which is parameterized by a time-varying long-memory parameter d = d(u), u ∈ [0, 1], i.e.…”

Analisis Volatilitas Harga Eceran Beberapa Komoditas Pangan Utama dengan Model ARCH/GARCH

“…Karakteristik menggerombol dicirikan oleh adanya kecenderungan bahwa perubahan yang besar diikuti perubahan yang besar dan sebaliknya perubahan yang kecil cenderung diikuti pula oleh perubahan yang kecil (Diebold, 2004). Untuk kondisi seperti ini diperlukan pendekatan yang berbeda karena asumsi homoskedastisitas tak terpenuhi (Engle, 2003 Sampai saat ini berbagai modifikasi dan pengembangan model ARCH/GARCH telah banyak dilakukan sehingga bentuknya sangat banyak (Fryzlewicz et al, 2008;Bollerslev, 2008). Sebagai ilustrasi, ada yang nilai tengahnya (mean) mengandung komponen ARMA, mempunyai peubah penjelas (termasuk peubah boneka), ataupun kombinasi keduanya; namun ada pula yang semata-mata hanya mengandung suatu konstanta.…”

Estimation of semiparametric locally stationary diffusion models

“…We adjust their proofs to our setting. Recall (See also Lemma A.5 in Dahlhaus and Subba Rao (2006) and Lemma 2 of A.1 in Fryzlewicz et al (2008, for …”