Conditions for invariance of the multivariate versions of grubbs’ test and bartlett’s test under a general dependency structure

Summary This paper generalizes a result by Stadje (1984) by deriving conditions for which a general dependency structure for multivariate observations, given in Pavur (1987), yields a positive definite covariance structure. This general dependency structure allows the sample covariance matrix to be distributed as a constant times a Wishart random matrix. It is then demonstrated that the maximum squared-radii test and a test for equal population covariance matrices have null distributions which remain unchanged when the new general dependency structure, rather than the usual independence structure, for the vector observations, is assumed. Moreover, under a general dependency structure for which the population covariance matrices are unequal, it is shown that the distribution of the test statistic for testing equal covariance matrices is identical to the distribution of the same test statistic when the population covariance matrices are equal and the observations are independent.     

Some further remarks on a superiority problem in misspecified restricted singular linear models

Razzaghi (1987) examined the difference of covariance matrices of competing estimators in misspecified restricted linear models. Further, Gross, Trenkler and Liski (1998) extended Razzaghi's result by asserting his sufficient condition for nonnegative definiteness of the covariance matrix difference to be also necessary. In this paper, when the covariance matrix of the disturbance vector is nonnegative definite, the necessary and sufficient conditions for nonnegative definiteness of the covariance matrix difference are derived. So above results are strengthened. Received: February 1999     

On Some Characteristics of Gaussian Covariance Functions

The concepts of isotropy/anisotropy and separability/non‐separability of a covariance function are strictly related. If a covariance function is separable, it cannot be isotropic or geometrically anisotropic, except for the Gaussian covariance function, which is the only model both separable and isotropic. In this paper, some interesting results concerning the Gaussian covariance model and its properties related to isotropy and separability are given, and moreover, some examples are provided. Finally, a discussion on asymmetric models, with Gaussian marginals, is furnished and the strictly positive definiteness condition is discussed.     

Covariance measurement in the presence of non-synchronous trading and market microstructure noise

This paper studies the problem of covariance estimation when prices are observed non-synchronously and contaminated by i.i.d. microstructure noise. We derive closed form expressions for the bias and variance of three popular covariance estimators, namely realised covariance, realised covariance plus lead and lag adjustments, and the Hayashi and Yoshida estimator, and present a comprehensive investigation into their properties and relative efficiency. Our main finding is that the ordering of the covariance estimators in terms of efficiency crucially depends on the level of microstructure noise, as well as the level of correlation. In fact, for sufficiently high levels of noise, the standard realised covariance estimator (without any corrections for non-synchronous trading) can be most efficient. We also propose a sparse sampling implementation of the Hayashi and Yoshida estimator, study the robustness of our findings using simulations with stochastic volatility and correlation, and highlight some important practical considerations.     

Graphical Diagnostics for Modeling Unstructured Covariance Matrices

Prudent statistical analysis of correlated data requires accounting for the correlation among the measurements. Specifying a form for the covariance matrix of the data could reduce the high number of parameters of the covariance and increase efficiency of the inferences about the regression parameters. Motivated by the success of ordinary, partial and inverse correlograms in identifying parsimonious models for stationary time series, we introduce generalizations of these plots for nonstationary data. Their roles in detecting heterogeneity and correlation of the data and identifying parsimonious models for the covariance matrix are illuminated using a longitudinal dataset. Decomposition of a covariance matrix into "variance" and "dependence" components provides the necessary ingredients for the proposed graphs. This amounts to replacing a 3-D correlation plot by a pair of 2-D plots, providing complementary information about dependence and heterogeneity. Models identified and fitted using the variance-correlation decomposition of a covariance matrix are not guaranteed to be positive definite, but those using the modified Cholesky decomposition are. Limitations of our graphical diagnostics for general multivariate data where the measurements are not (time-) ordered are discussed.     

Forecasting multivariate realized stock market volatility

We present a new matrix-logarithm model of the realized covariance matrix of stock returns. The model uses latent factors which are functions of lagged volatility, lagged returns and other forecasting variables. The model has several advantages: it is parsimonious; it does not require imposing parameter restrictions; and, it results in a positive-definite estimated covariance matrix. We apply the model to the covariance matrix of size-sorted stock returns and find that two factors are sufficient to capture most of the dynamics.     

An Updated Literature Review of Distance Correlation and Its Applications to Time Series

The concept of distance covariance/correlation was introduced recently to characterise dependence among vectors of random variables. We review some statistical aspects of distance covariance/correlation function, and we demonstrate its applicability to time series analysis. We will see that the auto‐distance covariance/correlation function is able to identify non‐linear relationships and can be employed for testing the i.i.d. hypothesis. Comparisons with other measures of dependence are included.     

Covariance functions that are stationary or nonstationary in space and stationary in time

There is a great demand for statistical modelling of phenomena that evolve in both space and time, and thus, there is a growing literature on covariance function models for spatio-temporal processes. Although several nonseparable space–time covariance models are available in the literature, very few of them can be used for spatially anisotropic data. In this paper, we propose a new class of stationary nonseparable covariance functions that can be used for both geometrically and zonally anistropic data. In addition, we show some desirable mathematical features of this class. Another important aspect, only partially covered by the literature, is that of spatial nonstationarity. We show a very simple criteria allowing for the construction of space–time covariance functions that are nonseparable, nonstationary in space and stationary in time. Part of the theoretical results proposed in the paper will then be used for the analysis of Irish wind speed data as in HASLETT and RAFTERY ( Applied Statistics , 38 , 1989, 1).     

Multivariate rotated ARCH models

This paper introduces a new class of multivariate volatility models which is easy to estimate using covariance targeting, even with rich dynamics. We call them rotated ARCH (RARCH) models. The basic structure is to rotate the returns and then to fit them using a BEKK-type parameterization of the time-varying covariance whose long-run covariance is the identity matrix. This yields the rotated BEKK (RBEKK) model. The extension to DCC-type parameterizations is given, introducing the rotated DCC (RDCC) model. Inference for these models is computationally attractive, and the asymptotics are standard. The techniques are illustrated using data on the DJIA stocks.     

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.     

Are the profitability and investment factors valid ICAPM risk factors? Pre-1963 evidence

We empirically evaluate whether the profitability and investment factors from Novy-Marx (2013) and Fama and French (2015, 2018) are compatible with Merton’s (1973) intertemporal CAPM (ICAPM) framework in the pre-1963 period. We show that: (i) the covariance risk price estimates of the profitability factors are positive and statistically significant, which indicates that they have explanatory power with respect to the cross-section of stock returns; (ii) the investment factors carry insignificant covariance risk prices and are therefore not valid ICAPM risk factors; and (iii) the profitability factors forecast the first moment of the aggregate stock return and economic activity with the correct sign, which is consistent with their positive covariance risk price estimates and satisfies the sign restrictions associated with the ICAPM.     

Optimale Zweifachblockpläne bei Seriell Korrelierten Fehlern

Summary Williams-designs are proved to be optimal within the class of latin 4×4-squares in rowcolumn-models, where serial correlations are fitted with an autoregressive scheme of first order.Designs are presented that are not generalized Youden-designs and that are better than Williams-designs for strong negative correlations. However, it will be shown that optimal designs do not depend on the special structure of the error covariance in models with only two columns and in models with a completely symmetric covariance matrix.     

Explaining movements in UK stock prices

We examine movements in aggregate UK stock prices by decomposing the variance of unexpected real stock returns into components due to revisions in expectations of future dividends, discount rates, and the covariance between the two. The contribution of news about future discount rates is about four times that of news about future dividends, with no significant covariance between them. Our analysis of excess returns uncovers a positive covariance between news about dividends and news about real interest rates. Since these two elements have opposite effects on current stock prices, their combined effect is negligible. Persistence in expected returns, as well as predictability, are found to be important in explaining stock price movements.     

Two kinds of variance/covariance estimates in linear mixed models

For longitudinal data, the within-subject covariance matrix plays an important role in statistical inference and it is of great interest to investigate this. In the paper, two kinds of estimators are investigated for the random effect covariance matrix D ₁ and the error variance σ ² in linear mixed models. One is to estimate D ₁ first and then to estimate σ ²; the other kind is to estimate σ ² first and then for D ₁. Both kinds of estimators are consistent. The covariance matrices of these covariance estimators and the variances of these two error variance estimators are calculated. In particular, the mean square errors of these estimators are also derived for one dimensional random effects. Besides, a simulation study is conducted to investigate the performances of these estimators.     

Optimum allocation in multivariate stratified random sampling: stochastic matrix mathematical programming

The allocation problem for multivariate stratified random sampling as a problem of stochastic matrix integer mathematical programming is considered, minimizing the estimated covariance matrix of estimated means subject to fixed cost or fixed total sample size. With these aims the asymptotic normality of sample covariance matrices for each strata is established. Some alternative approaches are suggested for its solution. An example is solved by applying the proposed techniques.     

Asymptotic expansion of S-estimators of location and covariance

By means of a straightforward application of empirical process theory, we show that S-estimators of multivariate location and covariance are asymptotically equivalent to a sum of independent vector and matrix valued random elements respectively. This provides an alternative proof of asymptotic normality of S-estimators and clearly explains the limiting covariance structure. It also leads to a relatively simple proof of asymptotic normality of the length of the shortest α-fraction.     

Modeling and Forecasting Large Realized Covariance Matrices and Portfolio Choice

We consider modeling and forecasting large realized covariance matrices by penalized vector autoregressive models. We consider Lasso‐type estimators to reduce the dimensionality and provide strong theoretical guarantees on the forecast capability of our procedure. We show that we can forecast realized covariance matrices almost as precisely as if we had known the true driving dynamics of these in advance. We next investigate the sources of these driving dynamics as well as the performance of the proposed models for forecasting the realized covariance matrices of the 30 Dow Jones stocks. We find that the dynamics are not stable as the data are aggregated from the daily to lower frequencies. Furthermore, we are able beat our benchmark by a wide margin. Finally, we investigate the economic value of our forecasts in a portfolio selection exercise and find that in certain cases an investor is willing to pay a considerable amount in order get access to our forecasts. Copyright © 2016 John Wiley & Sons, Ltd.     

Portfolio selection based on semivariance and distance correlation under minimum variance framework

In the minimum variance model, the covariance matrix plays an important role because it measures the risk and relationship of asset returns simultaneously under the normality assumption. However, in practice, the distribution of asset returns is nonnormal and has an obvious fat‐tail nature. In addition, the risk is one‐sided. In this paper, the main objective is to propose a better tool to replace the covariance matrix. The covariance matrix can be decomposed into two parts: a diagonal variance matrix and a square matrix with its elements being the Pearson correlation coefficient. A substitution of the covariance matrix is presented by replacing the variance and Pearson correlation coefficient in the decomposition of the covariance matrix with a semivariance and distance correlation coefficient, respectively. The proposed portfolio optimization strategy is applied to empirical data, and the numerical studies show the strategy performs well.     

Combining competing forecasts of inflation using a bivariate arch model

This paper sets out the basic structure of the bivariate generalization of Engle's ARCH model. Conditions which guarantee that the conditional covariance matrix is well defined are summarized, as are estimation and hypothesis testing.The process is used to combine forecasts where the weights are allowed to vary over time. Forecast errors from competing models are treated as a bivariate ARCH process so that the conditional covariance matrix adapts over time. At each point in time these conditional estimates of the variances and covariances are used to construct the optimal weights for combining the forecasts. Consequently, when one model is fitting well, its variance will be reduced and its weight will be increased.Two models of US inflation are constructed; one is a stylized monetarist model while the other is a mark-up model. The forecast errors are modeled as a simple bivariate ARCH process. Diagnostic tests reveal that this has overly restricted the parameterization of the covariance matrix. An approximation to the theoretically anticipated factor structure model is then estimated. The results in both cases show the weights varying over the sample period in moderately interpretable fashion.     

Minimax robust designs for field experiments

Experimental designs for field experiments are useful in planning agricultural experiments, environmental studies, etc. Optimal designs depend on the spatial correlation structures of field plots. Without knowing the correlation structures exactly in practice, we can study robust designs. Various neighborhoods of covariance matrices are introduced and discussed. Minimax robust design criteria are proposed, and useful results are derived. The generalized least squares estimator is often more efficient than the least squares estimator if the spatial correlation structure belongs to a small neighborhood of a covariance matrix. Examples are given to compare robust designs with optimal designs. The work was partially supported by research grants from the Natural Science and Engineering Research Council of Canada.