A new class of large-sample covariance and spectral density matrix estimators is proposed based on the notion of flat-top kernels. The new estimators are shown to be higher-order accurate when higher-order accuracy is possible. A discussion on kernel choice is presented as well as a supporting finite-sample simulation. The problem of spectral estimation under a potential lack of finite fourth moments is also addressed. The higher-order accuracy of flat-top kernel estimators typically comes at the sacrifice of the positive semi-definite property. Nevertheless, we show how a flat-top estimator can be modified to become positive semi-definite (even strictly positive definite) while maintaining its higher-order accuracy. In addition, an easy (and consistent) procedure for optimal bandwidth choice is given; this procedure estimates the optimal bandwidth associated with each individual element of the target matrix, automatically sensing (and adapting to) the underlying correlation structure.

The evolution of financial markets is a complicated real-world phenomenon that ranks at the top in terms o fdifficulty of modeling and/or prediction. One reason for this difficulty is the well-documented nonlinearity that is inherently at work. The state-of-the-art on the nonlinear modeling of financial returns is given by the popular ARCH (Auto-Regressive Conditional Heteroskedasticity) models and their generalization but they all have their short-comings. Foregoing the goal of finding the "best" model, we propose an exploratory, model-free approach in trying to understand this difficult type of data. In particular, we propose to transform the problem into a more manageable setting such as the setting of linearity. The form and properties of such a transformation are given, and the issue of one-step-ahead prediction using the new approach is explicitly addressed.

We address the problem of estimating the autocovariance matrix of a stationary process. Under short range dependence assumptions, convergence rates are established for a gradually tapered version of the sample autocovariance matrix and for its inverse. The proposed estimator is formed by leaving the main diagonals of the sample autocovariance matrix intact while gradually down-weighting o�-diagonal entries towards zero. In addition we show the same convergence rates hold for a positive de�nite version of the estimator, and we introduce a new approach for selecting the banding parameter. The new matrix estimator is shown to perform well theoretically and in simulation studies. As an application we introduce a new resampling scheme for stationary processes termed the linear process bootstrap (LPB). The LPB is shown to be asymptotically valid for the sample mean and related statistics. The e�ectiveness of the proposed methods are demonstrated in a simulation study.

We consider the problem of estimating the variance of the partial sums of a stationary time series that has either long memory, short memory, negative/intermediate memory, or is the ¯rst- di®erence of such a process. The rate of growth of this variance depends crucially on the type of memory, and we present results on the behavior of tapered sums of sample autocovariances in this context when the bandwidth vanishes asymptotically. We also present asymptotic results for the case that the bandwidth is a ¯xed proportion of sample size, extending known results to the case of °at-top tapers. We adopt the ¯xed proportion bandwidth perspective in our empirical section, presenting two methods for estimating the limiting critical values { both the subsampling method and a plug-in approach. Extensive simulation studies compare the size and power of both approaches as applied to hypothesis testing for the mean. Both methods perform well { although the subsampling method appears to be better sized { and provide a viable framework for conducting inference for the mean. In summary, we supply a uni¯ed asymptotic theory that covers all di®erent types of memory under a single umbrella.

We review the notion of linearity of time series, and show that ARCH or stochastic volatility (SV) processes are not only non-linear: they are not even weakly linear, i.e., they do not even possess a martingale representation. Consequently, the use of Bartlett’s formula is unwarranted in the context of data typically modelled as ARCH or SV processes such as financial returns. More surprisingly, we show that even the squares of an ARCH or SV process are not weakly linear. Finally, we present an alternative to Bartlett’s formula that is applicable (and consistent) in the context of financial returns data.

This paper considers the problem of distribution estimation for the studentized sample mean in the context of Long Memory and Negative Memory time series dynamics, adopting the fixed-bandwidth approach now popular in the econometrics literature. The distribution theory complements the Short Memory results of Kiefer and Vogelsang (2005). In particular, our results highlight the dependence on the employed kernel, whether or not the taper is nonzero at the boundary, and most importantly whether or not the process has short memory. We also demonstrate that small-bandwidth approaches fail when long memory or negative memory is present since the limiting distribution is either a point mass at zero or degenerate. Extensive numerical work provides approximations to the quantiles of the asymptotic distribution for a range of tapers and memory parameters; these quantiles can be used in practice for the construction of confidence intervals and hypothesis tests for the mean of the time series.

In this paper we propose a subsampling estimator for the distribution of statistics diverging at either known or unknown rates when the underlying time series is strictly stationary and strong mixing. Based on our results we provide a detailed discussion how to estimate extreme order statistics with dependent data and present two applications to assessing financial market risk. Our method performs well in estimating Value at Risk and provides a superior alternative to Hill's estimator in operationalizing Safety First portfolio selection.