Robust estimation of a high-dimensional integrated covariance matrix

In this article, we consider a robust method of estimating a realized covariance matrix calculated as the sum of cross products of intraday high-frequency returns. According to recent articles in financial econometrics, the realized covariance matrix is essentially contaminated with market microstructure noise. Although techniques for removing noise from the matrix have been studied since the early 2000s, they have primarily investigated a low-dimensional covariance matrix with statistically significant sample sizes. We focus on noise-robust covariance estimation under converse circumstances, that is, a high-dimensional covariance matrix possibly with a small sample size. For the estimation, we utilize a statistical hypothesis test based on the characteristic that the largest eigenvalue of the covariance matrix asymptotically follows a Tracy–Widom distribution. The null hypothesis assumes that log returns are not pure noises. If a sample eigenvalue is larger than the relevant critical value, then we fail to reject the null hypothesis. The simulation results show that the estimator studied here performs better than others as measured by mean squared error. The empirical analysis shows that our proposed estimator can be adopted to forecast future covariance matrices using real data.     

数据随机缺失下分位数回归模型的诱导光滑估计法

在数据随机缺失的分位数回归模型中,运用诱导光滑思想构造光滑的估计方程,得到了回归参数的诱导光滑估计及渐近协方差估计。接着证明了诱导光滑估计的渐近正态性质,并给出诱导光滑估计及其渐近协方差估计的算法。模拟研究表明新方法在有限样本中表现出色。     

Artificial neural networks for some macroeconomic series: A first report

There is a tendency for the true variability of feasible GLS estimators to be understated by asymptotic standard errors. For estimation of SUR models, this tendency becomes more severe in large equation systems when estimation of the error covariance matrix, C, becomes problematic. We explore a number of potential solutions involving the use of improved estimators for the disturbance covariance matrix and bootstrapping. In particular, Ullah and Racine (1992) have recently introduced a new class of estimators for SUR models that use nonparametric kernel density estimation techniques. The proposed estimators have the same structure as the feasible GLS estimator of Zellner (1962) differing only in the choice of estimator for C. Ullah and Racine (1992) prove that their nonparametric density estimator of C can be expressed as Zellner's original estimator plus a positive definite matrix that depends on the smoothing parameter chosen for the density estimation. It is this structure of the estimator that most interests us as it has the potential to be especially useful in large equation systems.

Atkinson and Wilson (1992) investigated the bias in the conventional and bootstrap estimators of coefficient standard errors in SUR models. They demonstrated that under certain conditions the former were superior, but they caution that neither estimator uniformly dominated and hence bootstrapping provides little improvement in the estimation of standard errors for the regression coefficients. Rilstone and Veal1 (1996) argue that an important qualification needs to be made to this somewhat negative conclusion. They demonstrated that bootstrapping can result in improvements in inferences if the procedures are applied to the t-ratios rather than to the standard errors. These issues are explored for the case of large equation systems and when bootstrapping is combined with improved covariance estimation.     

Testing sphericity and intraclass covariance structures under a growth curve model in high dimension

In this article, we consider the problem of testing (a) sphericity and (b) intraclass covariance structure under a growth curve model. The maximum likelihood estimator (MLE) for the mean in a growth curve model is a weighted estimator with the inverse of the sample covariance matrix which is unstable for large p close to N and singular for p larger than N. The MLE for the covariance matrix is based on the MLE for the mean, which can be very poor for p close to N. For both structures (a) and (b), we modify the MLE for the mean to an unweighted estimator and based on this estimator we propose a new estimator for the covariance matrix. This new estimator leads to new tests for (a) and (b). We also propose two other tests for each structure, which are just based on the sample covariance matrix.

To compare the performance of all four tests we compute for each structure (a) and (b) the attained significance level and the empirical power. We show that one of the tests based on the sample covariance matrix is better than the likelihood ratio test based on the MLE.     

A new estimator of covariance matrix

The problem of estimating a covariance matrix is considered in this paper. Using the so-called partial Iwasawa coordinates of the covariance matrix, a new improved estimator dominating the James-Stein estimator is proposed. The results of a simulation study verifies that the new estimator provides a substantial improvement in risk under Stein's loss.     

Estimation and Test for Multi-Dimensional Regression Models

This work is concerned with the estimation of multi-dimensional regression and the asymptotic behavior of the test involved in selecting models. The main problem with such models is that we need to know the covariance matrix of the noise to get an optimal estimator. We show in this article that if we choose to minimize the logarithm of the determinant of the empirical error covariance matrix, then we get an asymptotically optimal estimator. Moreover, under suitable assumptions, we show that this cost function leads to a very simple asymptotic law for testing the number of parameters of an identifiable and regular regression model. Numerical experiments confirm the theoretical results.     

An alternative Wald type test for two linear restrictions with applications to non-linear models

This paper proposes a new test procedure called the rel test to resolve the problem of small-sample local biasedness and non-monotonic power behavior of the Wald test for two linear restrictions caused by inaccuracy of the estimated covariance matrix of the estimator. This new test procedure, which does not need the covariance matrix of the estimator, involves finding the critical region based on contour points of the percentile confidence limit of a rel utilizing the bootstrap in order to obtain a test with the desired size and good power properties. Simulation results indicate that this new test procedure, the rel test, performs rather well both with respect to controlling size and having monotonic increasing power.     

Estimating the Spot Covariation of Asset Prices—Statistical Theory and Empirical Evidence

ABSTRACT

We propose a new estimator for the spot covariance matrix of a multi-dimensional continuous semimartingale log asset price process, which is subject to noise and nonsynchronous observations. The estimator is constructed based on a local average of block-wise parametric spectral covariance estimates. The latter originate from a local method of moments (LMM), which recently has been introduced by Bibinger et al.. We prove consistency and a point-wise stable central limit theorem for the proposed spot covariance estimator in a very general setup with stochastic volatility, leverage effects, and general noise distributions. Moreover, we extend the LMM estimator to be robust against autocorrelated noise and propose a method to adaptively infer the autocorrelations from the data. Based on simulations we provide empirical guidance on the effective implementation of the estimator and apply it to high-frequency data of a cross-section of Nasdaq blue chip stocks. Employing the estimator to estimate spot covariances, correlations, and volatilities in normal but also unusual periods yields novel insights into intraday covariance and correlation dynamics. We show that intraday (co-)variations (i) follow underlying periodicity patterns, (ii) reveal substantial intraday variability associated with (co-)variation risk, and (iii) can increase strongly and nearly instantaneously if new information arrives. Supplementary materials for this article are available online.     

Estimation of the Length Distribution of Marine Populations in the Gaussian-multinomial Setting using the Method of Moments

In this paper, the problem of estimation of the length distribution of marine populations in the Gaussian-multinomial model is considered. For the purpose of the mean and covariance parameter estimation, the method of moments estimators are developed. That is, minimum variance linear unbiased estimator for the mean frequency vector is derived and a consistent estimator for the covariance matrix of the length observations is presented. The usefulness of the proposed estimators is illustrated with an analysis of real cod length measurement data.     

Heteroskedasticity-consistent covariance matrix estimation:white's estimator and the bootstrap ∗

This paper considers the issue of estimating the covariance matrix of ordinary least squares estimates in a linear regression model when heteroskedasticity is suspected. We perform Monte Carlo simulation on the White estimator, which is commonly used in.

empirical research, and also on some alternatives based on different bootstrapping schemes. Our results reveal that the White estimator can be considerably biased when the sample size is not very large, that bias correction via bootstrap does not work well, and that the weighted bootstrap estimators tend to display smaller biases than the White estimator and its variants, under both homoskedasticity and heteroskedasticity. Our results also reveal that the presence of (potentially) influential observations in the design matrix plays an important role in the finite-sample performance of the heteroskedasticity-consistent estimators.     

On the estimation of the mean vector of a multivariate normal distribution under symmetry

The problem of estimation of the mean vector of a multivariate normal distribution with unknown covariance matrix, under uncertain prior information (UPI) that the component mean vectors are equal, is considered. The shrinkage preliminary test maximum likelihood estimator (SPTMLE) for the parameter vector is proposed. The risk and covariance matrix of the proposed estimato are derived and parameter range in which SPTMLE dominates the usual preliminary test maximum likelihood estimator (PTMLE) is investigated. It is shown that the proposed estimator provides a wider range than the usual premilinary test estimator in which it dominates the classical estimator. Further, the SPTMLE has more appropriate size for the preliminary test than the PTMLE.     

The Asymptotic Covariance Matrix of the Least Squares Estimator in the Stochastic Linear Regression Model: The Case of Elliptically Symmetric Distribution

This article considers the unconditional asymptotic covariance matrix of the least squares estimator in the linear regression model with stochastic explanatory variables. The asymptotic covariance matrix of the least squares estimator of regression parameters is evaluated relative to the standard asymptotic covariance matrix when the joint distribution of the dependent and explanatory variables is in the class of elliptically symmetric distributions. An empirical example using financial data is presented. Numerical examples and simulation experiments are given to illustrate the difference of the two asymptotic covariance matrices.     

Statistical inference on transformation models: a self-induced smoothing approach

This paper deals with a general class of transformation models that contains many important semiparametric regression models as special cases. It develops a self-induced smoothing for the maximum rank correlation estimator, resulting in simultaneous point and variance estimation. The self-induced smoothing does not require bandwidth selection, yet provides the right amount of smoothness so that the estimator is asymptotically normal with mean zero (unbiased) and variance–covariance matrix consistently estimated by the usual sandwich-type estimator. An iterative algorithm is given for the variance estimation and shown to numerically converge to a consistent limiting variance estimator. The approach is applied to a data set involving survival times of primary biliary cirrhosis patients. Simulation results are reported, showing that the new method performs well under a variety of scenarios.     

Covariance Matrix Estimation via Network Structure

In this article, we employ a regression formulation to estimate the high-dimensional covariance matrix for a given network structure. Using prior information contained in the network relationships, we model the covariance as a polynomial function of the symmetric adjacency matrix. Accordingly, the problem of estimating a high-dimensional covariance matrix is converted to one of estimating low dimensional coefficients of the polynomial regression function, which we can accomplish using ordinary least squares or maximum likelihood. The resulting covariance matrix estimator based on the maximum likelihood approach is guaranteed to be positive definite even in finite samples. Under mild conditions, we obtain the theoretical properties of the resulting estimators. A Bayesian information criterion is also developed to select the order of the polynomial function. Simulation studies and empirical examples illustrate the usefulness of the proposed methods.     

Time series count data regression

The count data model studied in the paper extends the Poisson model by al-lowing for overdispersion and serial correlation. Alternative approaches to esti-mate nuisance parameters, required for the correction of the Poisson maximum likelihood covariance matrix estimator and for a quasi-likelihood estimator, are studied. The estimators are evaluated by finite sample Monte Carlo experi-mentation. It is found that the Poisson maximum likelihood estimator with corrected covariance matrix estimators provide reliable inferences for longer time series. Overdispersion test statistics are wellbehaved, while conventional portmanteau statistics for white noise have too large sizes. Two empirical illustrations are included.     

The relative efficiency of the restricted estimators in linear regression models

This paper deals with the problem of multicollinearity in a multiple linear regression model with linear equality restrictions. The restricted two parameter estimator which was proposed in case of multicollinearity satisfies the restrictions. The performance of the restricted two parameter estimator over the restricted least squares (RLS) estimator and the ordinary least squares (OLS) estimator is examined under the mean square error (MSE) matrix criterion when the restrictions are correct and not correct. The necessary and sufficient conditions for the restricted ridge regression, restricted Liu and restricted shrunken estimators, which are the special cases of the restricted two parameter estimator, to have a smaller MSE matrix than the RLS and the OLS estimators are derived when the restrictions hold true and do not hold true. Theoretical results are illustrated with numerical examples based on Webster, Gunst and Mason data and Gorman and Toman data. We conduct a final demonstration of the performance of the estimators by running a Monte Carlo simulation which shows that when the variance of the error term and the correlation between the explanatory variables are large, the restricted two parameter estimator performs better than the RLS estimator and the OLS estimator under the configurations examined.     

A new heteroskedasticity-consistent covariance matrix estimator and inference under heteroskedasticity

To solve the heteroscedastic problem in linear regression, many different heteroskedasticity-consistent covariance matrix estimators have been proposed, including HC0 estimator and its variants, such as HC1, HC2, HC3, HC4, HC5 and HC4m. Each variant of the HC0 estimator aims at correcting the tendency of underestimating the true variances. In this paper, a new variant of HC0 estimator, HC5m, which is a combination of HC5 and HC4m, is proposed. Both the numerical analysis and the empirical analysis show that the quasi-t inference based on HC5m is typically more reliable than inferences based on other covariance matrix estimators, regardless of the existence of high leverage points.     

Criterion for the Selection of a Working Correlation Structure in the Generalized Estimating Equation Approach for Longitudinal Balanced Data

The generalized estimating equation is a popular method for analyzing correlated response data. It is important to determine a proper working correlation matrix at the time of applying the generalized estimating equation since an improper selection sometimes results in inefficient parameter estimates. We propose a criterion for the selection of an appropriate working correlation structure. The proposed criterion is based on a statistic to test the hypothesis that the covariance matrix equals a given matrix, and also measures the discrepancy between the covariance matrix estimator and the specified working covariance matrix. We evaluated the performance of the proposed criterion through simulation studies assuming that for each subject, the number of observations remains the same. The results revealed that when the proposed criterion was adopted, the proportion of selecting a true correlation structure was generally higher than that when other competing approaches were adopted. The proposed criterion was applied to longitudinal wheeze data, and it was suggested that the resultant correlation structure was the most accurate.     

Lower Bounds for the Efficiency of Designs with Respect to the D-Criterion when the Observations are Correlated

In the general linear model consider the experimental design problem for the Gauß-Markov estimator or least squares estimator when the observations are correlated. We prove new formulas for the efficiency of an exact design with respect to the D-criterion. For models with intercept term, for example, these formulas are useful to derive better lower bounds for the efficiency than the bounds recently given for an arbitrary linear model. These bounds are applied in examples to symmetrical regular circulants as covariance matrices. A byproduct of the investigations is some insight as to what kinds of designs might retain their optimality or high efficiency (for the uncorrelated homoscedastic case) under correlated observations.     

Developing Ridge Parameters for SUR Model

This paper proposes a number of procedures for developing new biased estimators of the seemingly unrelated regression (SUR) parameters, when the explanatory variables are affected by multicollinearity. Several ridge parameters are proposed and then compared in terms of the trace mean squared error (TMSE) and (PR) criteria. The PR criterion is the proportion of replication (out of 1,000) for which the SUR version of the generalized least squares (SGLS) estimator has a smaller TMSE than others. The study was performed using Monte Carlo simulations where the number of equations in the system, the number of observations, the correlation among equations, and the correlation between explanatory variables have been varied. For each model, we performed 1,000 replications. Our results show that under certain conditions some of the proposed SUR ridge parameters, (R _Sgeom , R _Skmed , R _Sqarith , and R _Sqmax ), performed well when compared, in terms of TMSE and PR criteria, with other proposed and popular existing ridge parameters. In large samples and when the collinearity between the explanatory variables is not high, the unbiased SUR estimator (SGLS), performed better than the other ridge parameters.