On central and non-central limit theorems in density estimation for sequences of long-range dependence

AbstractThis paper studies the asymptotic properties of the kernel probability density estimate of stationary sequences which are observed through some non-linear instantaneous filter applied to long-range dependent Gaussian sequences. It is shown that the limiting distribution of the kernel estimator can be, in quite contrast to the case of short-range dependence, Gaussian or non-Gaussian depending on the choice of the bandwidth sequences. In particular, if the bandwidth h(N) for sample of size N is selected to converge to zero fast enough, the usual √Nh(N) rate asymptotic normality still holds

On Berry--Esseen bounds for non-instantaneous filters of linear processes

Let $X_n=\sum_{i=1}^{\infty}a_i\epsilon_{n-i}$, where the $\epsilon_i$ are i.i.d. with mean 0 and at least finite second moment, and the $a_i$ are assumed to satisfy $|a_i|=O(i^{-\beta})$ with $\beta >1/2$. When $1/2<\beta<1$, $X_n$ is usually called a long-range dependent or long-memory process. For a certain class of Borel functions $K(x_1,...,x_{d+1})$, $d\ge0$, from ${\mathcal{R}}^{d+1}$ to $\mathcal{R}$, which includes indicator functions and polynomials, the stationary sequence $K(X_n,X_{n+1},...,X_{n+d})$ is considered. By developing a finite orthogonal expansion of $K(X_n,...,X_{n+d})$, the Berry--Esseen type bounds for the normalized sum $Q_N/\sqrt{N},Q_N=\sum_{n=1}^N(K(X_ n,...,X_{n+d})-\mathrm{E}K(X_n,...,X_{n+d}))$ are obtained when $Q_N/\sqrt{N}$ obeys the central limit theorem with positive limiting variance.Comment: Published in at http://dx.doi.org/10.3150/07-BEJ112 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

On functionals of linear processes with estimated parameters

Abstract: For a stationary linear process with independent and identically distributed innovations, the paper addresses asymptotic properties of partial sums of nonlinear functional applied to the process when an unknown parameter is estimated. General representations are established under the condition that the innovation coefficients are either summable or regularly varying with index in (−1, −1/2). The usefulness of the representations is demonstrated through the derivation of limiting distributions for several common examples such as kurtosis, the sign test, and the absolute deviation from the mean

How do Heterogeneous Beliefs Influence Asset Volatility?

[[abstract]]We investigate the influence of heterogeneous beliefs on asset volatility when agents' degrees of confidence differ. With a continuous-time model subsuming agent's heterogeneous beliefs in the expected increase in dividends, a stock price formula is derived. Based on this formula, the stock volatility is computed via Monte Carlo simulation. The results show that the influence of belief heterogeneity in expectation on volatility depends on the confident agents' level of optimism. Empirical results are also provided.[[abstract]]在本篇文章中，我們探討投資人自信程度不同時，異質信念對於資產波動率的影響。我們架構連續時間模型，在該模型中投資人對於股利的成長有不同的預期。藉由此模型，我們推導出股價公式。利用股價公式，我們以蒙地卡羅模擬計算出股價的波動率。我們的模擬結果發現異質信念對於波動率的影響會因為投資人的自信程度不同而有所差異。最後我們也提供相關實證結果。[[journaltype]]國外[[incitationindex]]SSCI[[ispeerreviewed]]Y[[booktype]]紙本[[booktype]]電子版[[countrycodes]]AU

Non-parametric Estimation of Conditional Tail Expectation for Long-Horizon Returns

When evaluating the tail risk of stock portfolio returns, providing statistically sound solutions for long return horizons is important, but difficult. Furthermore, there are drawbacks to using traditional parametric methods that rely on strong model assumptions or simulations. This study investigates the problem by focusing on an important risk measure, the conditional tail expectation (CTE), under a general multivariate stochastic volatility model. To overcome the estimation difficulties caused by the long period, we derive an asymptotic formula to approximate the CTE. Based on this formula, we propose a simple nonparametric estimate of the unconditional CTE, and show that it is both consistent and asymptotically normal. Next, we forecast the CTE using a modified form of the nonparametric estimator. With the help of the asymptotic formula, we evaluate the accuracy of the CTE predictor by treating it as an interval forecast for furure returns. Simulation studies demonstrate the applicability of our approach. Lastly, we apply the proposed estimation and predictor to daily S&P 500 index returns.補正完畢TW

Value at Risk for Integrated Returns and Its Applications to Equity-linked Insurance

補正完畢國際Melbourne, AustraliaTW