Characteristics of some classes of space–time covariance functions

Although a wide list of classes of space–time covariance functions is now available, selecting an appropriate class of models for a variable under study is still difficult and it represents a priority problem with respect to the choice of a particular model of a specified class. Then, knowing the characteristics of various classes of covariances, and their auxiliary functions, and matching those with the characteristics of the empirical space–time covariance surface might be helpful in the selection of a suitable class. In this paper some characteristics, such as behavior at the origin, asymptotic behavior, nonseparability and anisotropy aspects, are studied for some well known classes of covariance models of stationary space–time random fields. Moreover, some important issues related to modeling choices are described and a case study is presented.     

A Note on Sufficient Conditions for Valid Unmodified t Testing in Correlation Analysis with Autocorrelated and Heteroscedastic Sample Data

Traditionally, sphericity (i.e., independence and homoscedasticity for raw data) is put forward as the condition to be satisfied by the variance–covariance matrix of at least one of the two observation vectors analyzed for correlation, for the unmodified t test of significance to be valid under the Gaussian and constant population mean assumptions. In this article, the author proves that the sphericity condition is too strong and a weaker (i.e., more general) sufficient condition for valid unmodified t testing in correlation analysis is circularity (i.e., independence and homoscedasticity after linear transformation by orthonormal contrasts), to be satisfied by the variance–covariance matrix of one of the two observation vectors. Two other conditions (i.e., compound symmetry for one of the two observation vectors; absence of correlation between the components of one observation vector, combined with a particular pattern of joint heteroscedasticity in the two observation vectors) are also considered and discussed. When both observation vectors possess the same variance–covariance matrix up to a positive multiplicative constant, the circularity condition is shown to be necessary and sufficient. “Observation vectors” may designate partial realizations of temporal or spatial stochastic processes as well as profile vectors of repeated measures. From the proof, it follows that an effective sample size appropriately defined can measure the discrepancy from the more general sufficient condition for valid unmodified t testing in correlation analysis with autocorrelated and heteroscedastic sample data. The proof is complemented by a simulation study. Finally, the differences between the role of the circularity condition in the correlation analysis and its role in the repeated measures ANOVA (i.e., where it was first introduced) are scrutinized, and the link between the circular variance–covariance structure and the centering of observations with respect to the sample mean is emphasized.     

Estimating and Testing a Structured Covariance Matrix for Three-Level Multivariate Data

This article considers an approach to estimating and testing a new Kronecker product covariance structure for three-level (multiple time points (p), multiple sites (u), and multiple response variables (q)) multivariate data. Testing of such covariance structure is potentially important for high dimensional multi-level multivariate data. The hypothesis testing procedure developed in this article can not only test the hypothesis for three-level multivariate data, but also can test many different hypotheses, such as blocked compound symmetry, for two-level multivariate data as special cases. The tests are implemented with two real data sets.     

Convergence of Sample Eigenvectors of Spiked Population Model

In this paper, for the general non Gaussian spiked population model, where a few fixed eigenvalues of the population covariance matrix are separated from others, we investigate the convergence properties of the eigenvectors of sample covariance matrices corresponding to the spiked population eigenvalues and angle between the population eigenvectors and sample eigenvectors as both the sample size and population size are large.     

The covariance matrix of normal order statistics

An approximation is given to calculate V, the covariance matrix for normal order statistics. The approximation gives considerable improvement over previous approximations, and the computing algorithm is available from the authors.     

Semiparametric model average prediction in panel data analysis

Forecasting in economic data analysis is dominated by linear prediction methods where the predicted values are calculated from a fitted linear regression model. With multiple predictor variables, multivariate nonparametric models were proposed in the literature. However, empirical studies indicate the prediction performance of multi-dimensional nonparametric models may be unsatisfactory. We propose a new semiparametric model average prediction (SMAP) approach to analyse panel data and investigate its prediction performance with numerical examples. Estimation of individual covariate effect only requires univariate smoothing and thus may be more stable than previous multivariate smoothing approaches. The estimation of optimal weight parameters incorporates the longitudinal correlation and the asymptotic properties of the estimated results are carefully studied in this paper.     

The exchange rate risk of Chinese yuan: Using VaR and ES based on extreme value theory

This paper applies extreme value theory (EVT) to estimate the tails of return series of Chinese yuan (CNY) exchange rates. We find that the degree of fitting Pareto distribution to the data of the tail of return series is extremely high. The empirical results indicate that expected shortfall cannot improve the tail risk problem of value-at-risk (VaR). The evidence of back testing indicates that EVT-based VaR values underestimate the risks of exchange rates such as USD/CNY and HKD/CNY, which may be caused by the continuous appreciation of CNY against USD and HKD. However, compared with VaR values calculated by historical simulation and variance–covariance method, VaR values calculated by EVT can measure the risk more accurately for the exchange rates of JPY/CNY and EUR/CNY.     

Testing for spurious and cointegrated regressions: A wavelet approach

This paper proposes a wavelet-based approach to analyze spurious and cointegrated regressions in time series. The approach is based on the properties of the wavelet covariance and correlation in Monte Carlo studies of spurious and cointegrated regression. In the case of the spurious regression, the null hypotheses of zero wavelet covariance and correlation for these series across the scales fail to be rejected. Conversely, these null hypotheses across the scales are rejected for the cointegrated bivariate time series. These nonresidual-based tests are then applied to analyze if any relationship exists between the extraterrestrial phenomenon of sunspots and the earthly economic time series of oil prices. Conventional residual-based tests appear sensitive to the specification in both the cointegrating regression and the lag order in the augmented Dickey–Fuller tests on the residuals. In contrast, the wavelet tests, with their bootstrap t-statistics and confidence intervals, detect the spuriousness of this relationship.     

Reduced-rank estimation of the difference between two covariance matrices

We consider m×m$m\times m$ covariance matrices, _Σ1${\Sigma }_1$ and _Σ2${\Sigma }_2$, which satisfy _Σ2-_Σ1=Δ${\Sigma }_2-{\Sigma }_1=\Delta$, where Δ$\Delta$ has a specified rank. Maximum likelihood estimators of _Σ1${\Sigma }_1$ and _Σ2${\Sigma }_2$ are obtained when sample covariance matrices having Wishart distributions are available and rank(Δ)$\text{rank}(\Delta )$ is known. The likelihood ratio statistic for a test about the value of rank(Δ)$\text{rank}(\Delta )$ is also given and some properties of its null distribution are obtained. The methods developed in this paper are illustrated through an example.