A Stein-type shrinkage estimator of the covariance matrix for portfolio selections

The covariance matrix plays a crucial role in portfolio optimization problems as the risk and correlation measure of asset returns. An improved estimation of the covariance matrix can enhance the performance of the portfolio. In this paper, based on the Cholesky decomposition of the covariance matrix, a Stein-type shrinkage strategy for portfolio weights is constructed under the mean-variance framework. Furthermore, according to the agent’s maximum expected utility value, a portfolio selection strategy is proposed. Finally, simulation experiments and an empirical study are used to test the feasibility of the proposed strategy. The numerical results show our portfolio strategy performs satisfactorily.     

Model specification testing of time series regressions

Given the specification of the lag length and functional form of a (non)linear time series regression we shall propose a test of the null hypothesis that the expectation of the error conditional on the exogenous variables, all lagged exogenous variables and all lagged dependent variables equals zero with probability 1. In the case that the data-generating process is strictly stationary this test is consistent with respect to the alternative hypothesis that the null is false. The test is also applicable for a particular class of non-stationary time series regressions, although in that case consistency with respect to all possible alternatives is no longer guaranteed. The test involved is a generalization of a test proposed in Bierens (1982b). Moreover, we also present a similar but simpler test of the hypothesis that the errors are martingale differences.     

Segmenting mean-nonstationary time series via trending regressions

In this paper, we provide a segmentation procedure for mean-nonstationary time series. The segmentation is obtained by casting the problem into the framework of detecting structural breaks in trending regression models in which the regressors are generated by suitably smooth functions. As test statistics we propose to use the maximally selected likelihood ratio statistics and a related statistics based on partial sums of weighted residuals. The main theoretical contribution of the paper establishes the extreme value distribution of these statistics and their consistency. To circumvent the slow convergence to the extreme value limit, we propose to employ a version of the circular bootstrap. This procedure is completely data-driven and does not require knowledge of the time series structure. In an empirical part, we show in a simulation study and applications to air carrier traffic and S&P 500 data that the finite sample performance is very satisfactory.     

Higher order approximations for Wald statistics in time series regressions with integrated processes

Asymptotic expansions are developed for Wald test statistics in time series regressions with integrated processes. These expansions provide an opportunity to reduce size distortion in testing by suitable bandwidth selection, and automated rules for doing so are calculated. A band spectral regression and the associated Wald test are also considered. Both the first order and second order properties of the estimator are studied.     

A note on the covariance matrix of the maximum likelihood estimator in constrained multivariate linear regression

A useful result concerning variances and covariances of a linear function of a random matrix is applied to find the variance–covariance matrix of the maximum likelihood estimator in multivariate linear regression subject to zero constraints.     

Confidence sets for the date of a single break in linear time series regressions

This paper considers the problem of constructing confidence sets for the date of a single break in a linear time series regression. We establish analytically and by small sample simulation that the current standard method in econometrics for constructing such confidence intervals has a coverage rate far below nominal levels when breaks are of moderate magnitude. Given that breaks of moderate magnitude are a theoretically and empirically relevant phenomenon, we proceed to develop an appropriate alternative. We suggest constructing confidence sets by inverting a sequence of tests. Each of the tests maintains a specific break date under the null hypothesis, and rejects when a break occurs elsewhere. By inverting a certain variant of a locally best invariant test, we ensure that the asymptotic critical value does not depend on the maintained break date. A valid confidence set can hence be obtained by assessing which of the sequence of test statistics exceeds a single number.     

The minimum weighted covariance determinant estimator

In this paper, we introduce weighted estimators of the location and dispersion of a multivariate data set with weights based on the ranks of the Mahalanobis distances. We discuss some properties of the estimators like the breakdown point, influence function and asymptotic variance. The outlier detection capacities of different weight functions are compared. A simulation study is given to investigate the finite-sample behavior of the estimators. The research of Stefan Van Aelst was supported by a grant of the Fund for Scientific Research-Flanders (FWO-Vlaanderen) and by IAP research network grant nr. P6/03 of the Belgian government (Belgian Science Policy).     

On the strong uniform consistency of the mode estimator for censored time series

Let (T _n ) _n≥1 be a sequence random variables (rv) of interest distributed as T. In censorship models the rv T is subject to random censoring by another rv C. Let θ be the mode of T. In this paper we define a new smooth kernel estimator [^(q)]_n{\hat{\theta}_n} of θ and establish its almost sure convergence under an α-mixing condition.     

Estimation of fixed effect models for time series of cross-sections with arbitrary intertemporal covariance

Models for time series of cross-sections with fixed effects and with intertemporal correlation are considered. The regression coefficients, and their standard errors, can be estimated by generalized least squares applied to a transformed model. The procedure is given a conditional likelihood interpretation.     

Structured low-rank matrix completion for forecasting in time series analysis

This paper considers the low-rank matrix completion problem, with a specific application to forecasting in time series analysis. Briefly, the low-rank matrix completion problem is the problem of imputing missing values of a matrix under a rank constraint. We consider a matrix completion problem for Hankel matrices and a convex relaxation based on the nuclear norm. Based on new theoretical results and a number of numerical and real examples, we investigate the cases in which the proposed approach can work. Our results highlight the importance of choosing a proper weighting scheme for the known observations.     

A minimal characterization of the covariance matrix

Summary LetX be ak-dimensional random vector with mean vectorμ and non-singular covariance matrix Σ. We show that among all pairs (a, Δ),a ∈ IR ^k , Δ ∈ IR ^k×k positive definite and symmetric andE(X−a)′ Δ⁻¹(X−a)=k, (μ, Σ) is the unique pair which minimizes det Δ. This motivates certain robust estimators of location and scale. Research supported by the Nuffield Foundation.     

Heteroskedasticity-consistent covariance matrix estimators for spatial autoregressive models

In the presence of heteroskedasticity, conventional test statistics based on the ordinary least squares (OLS) estimator lead to incorrect inference results for the linear regression model. Given that heteroskedasticity is common in cross-sectional data, the test statistics based on various forms of heteroskedasticity-consistent covariance matrices (HCCMs) have been developed in the literature. In contrast to the standard linear regression model, heteroskedasticity is a more serious problem for spatial econometric models, generally causing inconsistent extremum estimators of model coefficients. This paper investigates the finite sample properties of the heteroskedasticity-robust generalized method of moments estimator (RGMME) for a spatial econometric model with an unknown form of heteroskedasticity. In particular, it develops various HCCM-type corrections to improve the finite sample properties of the RGMME and the conventional Wald test. The Monte Carlo results indicate that the HCCM-type corrections can produce more accurate results for inference on model parameters and the impact effects estimates in small samples.     

A copula-based time series model for global horizontal irradiation

The increasing importance of solar power for electricity generation leads to increasing demand for probabilistic forecasting of local and aggregated photovoltaic (PV) yields. Based on publicly available irradiation data, this paper uses an indirect modeling approach for hourly medium to long-term local PV yields. We suggest a time series model for global horizontal irradiation that allows for multivariate probabilistic forecasts for arbitrary time horizons. It features several important stylized facts. Sharp time-dependent lower and upper bounds of global horizontal irradiations are estimated. The parameters of the beta distributed marginals of the transformed data are allowed to be time-dependent. A copula-based time series model is introduced for the hourly and daily dependence structure based on simple vine copulas with so-called tail dependence. Evaluation methods based on scoring rules are used to compare the model’s power for multivariate probabilistic forecasting with other models used in the literature showing that our model outperforms other models in many respects.     

Extreme-quantile tracking for financial time series

Time series of financial asset values exhibit well-known statistical features such as heavy tails and volatility clustering. We propose a nonparametric extension of the classical Peaks-Over-Threshold method from extreme value theory to fit the time varying volatility in situations where the stationarity assumption may be violated by erratic changes of regime, say. As a result, we provide a method for estimating conditional risk measures applicable to both stationary and nonstationary series. A backtesting study for the UBS share price over the subprime crisis exemplifies our approach.     

Prediction regions for interval-valued time series

We approximate probabilistic forecasts for interval-valued time series by offering alternative approaches. After fitting a possibly non-Gaussian bivariate vector autoregression (VAR) model to the center/log-range system, we transform prediction regions (analytical and bootstrap) for this system into regions for center/range and upper/lower bounds systems. Monte Carlo simulations show that bootstrap methods are preferred according to several new metrics. For daily S&P 500 low/high returns, we build joint conditional prediction regions of the return level and volatility. We illustrate the usefulness of obtaining bootstrap forecasts regions for low/high returns by developing a trading strategy and showing its profitability when compared to using point forecasts.     

High dimensional covariance matrix estimation using a factor model

High dimensionality comparable to sample size is common in many statistical problems. We examine covariance matrix estimation in the asymptotic framework that the dimensionality p$p$ tends to ∞$\infty$ as the sample size n$n$ increases. Motivated by the Arbitrage Pricing Theory in finance, a multi-factor model is employed to reduce dimensionality and to estimate the covariance matrix. The factors are observable and the number of factors K$K$ is allowed to grow with p$p$. We investigate the impact of p$p$ and K$K$ on the performance of the model-based covariance matrix estimator. Under mild assumptions, we have established convergence rates and asymptotic normality of the model-based estimator. Its performance is compared with that of the sample covariance matrix. We identify situations under which the factor approach increases performance substantially or marginally. The impacts of covariance matrix estimation on optimal portfolio allocation and portfolio risk assessment are studied. The asymptotic results are supported by a thorough simulation study.