Variance-covariance Matrix Research Articles

A note on centering in subsample selection for linear regression

Centring is a commonly used technique in linear regression analysis. With centred data on both the responses and covariates, the ordinary least squares estimator of the slope parameter can be calculated from a model without the intercept. If a subsample is selected from a centred full data, the subsample is typically uncentred. In this case, is it still appropriate to fit a model without the intercept? The answer is yes, and we show that the least squares estimator on the slope parameter obtained from a model without the intercept is unbiased and it has a smaller variance covariance matrix in the Loewner order than that obtained from a model with the intercept. We further show that for noninformative weighted subsampling when a weighted least squares estimator is used, using the full data weighted means to relocate the subsample improves the estimation efficiency.

PENERAPAN PETA KENDALI T^2 HOTELLING ALGORITMA FAST MINIMUM COVARIANCE DETERMINANT PADA PENGENDALIAN KUALITAS BAWANG MERAH VARIETAS LEMBAH PALU

The physical condition of the Palu Valley shallots variety greatly affects the quality of the fried onions that are obtained. The poor quality of shallots will affect the product that will sale by the farmers. Therefore, it is necessary to monitor by conducting the quality control analysis of the physical condition of shallots. In this study we use the quality control method of the Hotelling control map with the fast-MCD algorithm. This method is used because the outlier in the data to be analyzed. The purpose of this study is to produce average vector estimates and variance-covariance matrix estimates in the formation of the Hotelling control map. From the calculation by using the mean vector and the variant-covariant matrix with fast-MCD estimation, 93 data were obtained that experienced out of control on the Hotelling control map with the fast-MCD algorithm where the observations that experienced out of control were more than the usual of Hotelling control map. This shows that the Hotelling control map with the fast-MCD algorithm is more effective in detecting observations which contain outliers. The value of the multivariate process capability analysis is less than one showing the process is uncapable.

Higher order asymptotic refinements in Bell regressions

The discrete Bell distribution and its associated regression model were introduced recently in the statistical literature. The Bell distribution has proved to be a useful alternative to the traditional Poisson distribution, mainly to deal with overdispersion. Likelihood‐based inference on the Bell regression parameters relies on asymptotic assumptions like the sample size going to infinity. In this paper, we focus on the small‐sample case and consider higher order asymptotic refinements in this class of regression models. In particular, we derive the second‐order biases of the maximum likelihood estimators, which are used to define bias‐corrected estimators. The preventive method to bias reducing is also considered. We provide a simple matrix formula for the skewness of the distributions of the maximum likelihood estimators, and also derive a simple matrix formula for the second‐order variance–covariance matrix of the maximum likelihood estimators in this class of regression models. We use Monte Carlo simulations to verify the performance of the proposed methods. Our simulation results suggest that the analytical expressions we derive deliver more reliable results in small‐sized samples. An empirical application to a real data set is considered for illustrative purposes.

Oil shocks and investor attention

In this paper, we examine the existence of sentiment exposure in oil price returns. We augment the SVAR model of Kilian and Park (International Economic Review, 2009, 50, 1267–1287) by including the effects of (1) investors sentiment proxied by Google’s search volume index, (2) economic policy uncertainty (EPU) and (3) time variation in both coefficients and the variance-covariance matrix. Our empirical results show that changes in investor attention do exhibit a significant long-lasting impact on oil and stock market returns. Aggregate oil demand and supply shocks have a transitory effect on investor sentiment. We reveal that the impact of EPU is temporary and significant, while EPU responds strongly to shocks on oil prices and stock market returns. In all cases, the magnitude and sign of responses are affected by the timing of the shock. Our findings are robust to an alternative sentiment indicator and once the role of oil inventories is considered.

ℓ 2 -Relaxation: With Applications to Forecast Combination and Portfolio Analysis

Abstract We propose ℓ2-relaxation, which is a novel convex optimization problem, to tackle forecast combination with many forecasts or minimum variance portfolio with many assets. ℓ2-relaxation minimizes the squared Euclidean norm of the weight vector subject to a set of relaxed linear inequalities to balance the bias and variance. It delivers optimality with approximately equal within-group weights when latent block equicorrelation patterns dominate the high dimensional sample variance-covariance matrix of the individual forecast errors or the assets. Its wide applicability is highlighted in three real data examples in microeconomics, macroeconomics, and finance.

University Mathematics Major Students’ Ratings of Their Lecturers’ Preferred Teaching Methods: A Repeated-Measures Design

This study investigates the differences in lecturers’ preferred teaching methods based on the ratings of their students. It employed the repeated measures design, where four separate ratings for each participant were taken. The participants comprise eighty-two (82) university mathematics major students, made up of fifty-five (55) males and twenty-seven (27) females, who were selected from a university in the central region of Ghana, using a non-proportionate stratified sampling technique in two third-year cohort mathematics classes. The results indicated that Mauchly’s test of sphericity was not significant, χ^2 (5)=10.33,p>.05, (i.e., the assumption about the characteristics of the variance-covariance matrix was not violated). Thus, the within-subjects variable of the teaching method was highly significant, F (3, 243) = 468.17, p < .05, indicating that the mean students’ ratings differed significantly as a function of the four teaching methods. This was supported by the decrease in the mean students’ ratings from guided discovery to direct instruction methods. The pairwise comparisons (with Bonferroni adjustment) among the four teaching methods, showed a significant difference between any pair of teaching methods (p < .05). Thus, the students’ ratings for guided discovery were higher than the ratings for cooperative learning (p < .05), ratings for cooperative learning were higher than the ratings for inquiry-based learning (p < .05), and the ratings for inquiry-based learning were higher than the ratings for direct instruction (p < .05). The estimated marginal means for the ratings of guided discovery (M = 8.12; C. I = [7.94, 8.32]) were the highest, followed by the mean ratings for cooperative learning method (M = 6.68; C.I = [6.51, 6.86]), followed by the mean ratings for inquiry-based learning (M = 4 73; C. I = [4.60, 4.87]), and then followed by the mean ratings for direct instruction (M = 3.95; C.I = [3.80,4.11]). The study's implications are that, although many researchers recommend multiple teaching methods for mathematics instruction, lecturers should endeavour to use teaching methods that are popular and acceptable among students. This would enable them to understand the content their lecturers teach. The study concludes that lecturers should focus more on active teaching methods such as guided discovery and cooperative learning and focus less on the direct instruction teaching method.

Risk transmission from the oil market to Islamic and conventional banks in oil-exporting and oil-importing countries

This paper examines the volatility transmission from the oil market to Islamic banks' (IBs) share prices in two sets of data from oil exporters and importers. Our datasets comprise indices developed from banks' stocks of eight oil-exporting countries, including 41 IBs and 90 conventional banks (CBs), and five oil-importing countries, with 23 IBs and 63 CBs. We employ a trivariate version of the non-diagonal GARCH model, which allows for asymmetry in the variance-covariance matrix. Rather than relying on a single window, we perform the estimations through many recursive windows. The results reveal that oil volatility has higher predictive power (in majority recursive significant subsamples) in the exporter dataset compared to the importer dataset. We also find higher significant recursive subsamples for conventional counterparts in the same country. Holistically, IBs show greater bank stability during high oil price volatility horizons, a finding beneficial for policymakers, regulators, financial markets, IBs, and other concerned stakeholders.

Refreg: An R Package for Estimating Conditional Reference Regions

Multivariate reference regions (MVR) represent the extension of the reference interval concept to the multivariate setting. A reference interval is defined by two threshold points between which a high percentage of healthy subjects' results, usually 95%, are contained. Analogously, an MVR characterizes the values of several diagnostic tests most frequently found among non-diseased subjects by defining a convex hull containing 95% of the results. MVRs have great applicability when working with diseases that are diagnosed via more than one continuous test, e.g., diabetes or hypothyroidism. The present work introduces refreg, an R package for estimating conditional MVRs. The reference region is non-parametrically estimated using a multivariate kernel density estimator, and its shape allowed to change under the influence of covariates. The effects of covariates on the multivariate variable means, and on their variance-covariance matrix, are estimated by flexible additive predictors. Continuous covariate non-linear effects can be estimated by penalized spline smoothers. The package allows the user to propose, for instance, an age-specific diagnostic rule based on the joint distribution of two non-Gaussian, continuous test results. The usefulness of the refreg package in clinical practice is illustrated with a real case in diabetes research, with an age-specific reference region proposed for the joint interpretation of two glycemia markers (fasting plasma glucose and glycated hemoglobin). To show that the refreg package can also be used in other, and indeed very different fields, an example is provided for the joint prediction of two atmospheric pollutants (SO$_2$, and NO$_x$). Additionally, the text discusses how, conceptually, this method could be extended to more than two dimensions.

Enriched standard conjugate priors and the right invariant prior for Wishart distributions

The prediction of the variance–covariance matrix of the multivariate normal distribution is important in the multivariate analysis. We investigated Bayesian predictive distributions for Wishart distributions under the Kullback–Leibler divergence. The conditional reducibility of the family of Wishart distributions enables us to decompose the risk of a Bayesian predictive distribution. We considered a recently introduced class of prior distributions, which is called the family of enriched standard conjugate prior distributions, and compared the Bayesian predictive distributions based on these prior distributions. Furthermore, we studied the performance of the Bayesian predictive distribution based on the reference prior distribution in the family and showed that there exists a prior distribution in the family that dominates the reference prior distribution. Our study provides new insight into the multivariate analysis when there exists an ordered inferential importance for the independent variables.

Time series (re)sampling using Generative Adversarial Networks

We propose a novel bootstrap procedure for time series data based on Generative Adversarial networks (GANs). We show that the dynamics of common stationary time series processes can be learned by GANs and demonstrate that GANs trained on a single sample path can be used to generate additional samples from the process. We find that temporal convolutional neural networks provide a suitable design for the generator and discriminator, and that convincing samples can be generated on the basis of a vector drawn from a normal distribution with zero mean and an identity variance–covariance matrix. We demonstrate the finite sample properties of GAN sampling and the suggested bootstrap using simulations where we compare the performance to circular block bootstrapping in the case of resampling an AR(1) time series processes. We find that resampling using the GAN can outperform circular block bootstrapping in terms of empirical coverage. Finally, we provide an empirical application to the Sharpe ratio.

Continuous wavelet estimation for multivariate fractional Brownian motion

In this paper, we propose a method using continuous wavelets to study the multivariate fractional Brownian motion through the deviations of the transformed random process to find an efficient estimate of Hurst exponent using eigenvalue regression of the covariance matrix. The results of simulations experiments shown that the performance of the proposed estimator was efficient in bias but the variance get increase as signal change from short to long memory the MASE increase relatively. The estimation process was made by calculating the eigenvalues for the variance-covariance matrix of Meyer’s continuous wavelet details coefficients.

Simultaneous Maximum Likelihood Estimation for Piecewise Linear Instrumental Variable Models.

Analysis of instrumental variables is an effective approach to dealing with endogenous variables and unmeasured confounding issue in causal inference. We propose using the piecewise linear model to fit the relationship between the continuous instrumental variable and the continuous explanatory variable, as well as the relationship between the continuous explanatory variable and the outcome variable, which generalizes the traditional linear instrumental variable models. The two-stage least square and limited information maximum likelihood methods are used for the simultaneous estimation of the regression coefficients and the threshold parameters. Furthermore, we study the limiting distribution of the estimators in the correctly specified and misspecified models and provide a robust estimation of the variance-covariance matrix. We illustrate the finite sample properties of the estimation in terms of the Monte Carlo biases, standard errors, and coverage probabilities via the simulated data. Our proposed model is applied to an education-salary data, which investigates the causal effect of children’s years of schooling on estimated hourly wage with father’s years of schooling as the instrumental variable.

Multivariate selection and the making and breaking of mutational pleiotropy

The role of mutations have been subject to many controversies since the formation of the Modern Synthesis of evolution in the early 1940ties. Geneticists in the early half of the twentieth century tended to view mutations as a limiting factor in evolutionary change. In contrast, natural selection was largely viewed as a “sieve” whose main role was to sort out the unfit but which could not create anything novel alone. This view gradually changed with the development of mathematical population genetics theory, increased appreciation of standing genetic variation and the discovery of more complex forms of selection, including balancing selection. Short-term evolutionary responses to selection are mainly influenced by standing genetic variation, and are predictable to some degree using information about the genetic variance–covariance matrix (G) and the strength and form of selection (e. g. the vector of selection gradients, β). However, predicting long-term evolution is more challenging, and requires information about the nature and supply of novel mutations, summarized by the mutational variance–covariance matrix (M). Recently, there has been increased attention to the role of mutations in general and M in particular. Some evolutionary biologists argue that evolution is largely mutation-driven and claim that mutation bias frequently results in mutation-biased adaptation. Strong similarities between G and M have also raised questions about the non-randomness of mutations. Moreover, novel mutations are typically not isotropic in their phenotypic effects and mutational pleiotropy is common. Here I discuss the evolutionary origin and consequences of mutational pleiotropy and how multivariate selection directly shapes G and indirectly M through changed epistatic relationships. I illustrate these ideas by reviewing recent literature and models about correlational selection, evolution of G and M, sexual selection and the fitness consequences of sexual antagonism.

Multivariate Bayesian Arm-Based Network Meta-Analysis of Pharmacological Interventions for the Treatment of Acute Bipolar Mania in Adults.

In a network meta-analysis (NMA), multiple treatments can be compared simultaneously by aggregating pieces of evidence from direct as well as indirect treatment comparisons in different randomized controlled trials (RCTs). Conventional NMA are performed using a normal approximation approach and can be applied for arm-level binary outcome data as well. This study aimed to estimate the treatment effects within a Bayesian framework using a binomial likelihood for a multivariate NMA model. The dataset consists of 57 RCTs comparing the effect of ten pharmacological drugs and a placebo for acute bipolar mania in adults. The binary outcomes of interest were treatment response and all-cause dropouts measured three weeks from the baseline. Binomial distribution was adopted for the number of events and the probability of event occurrence modeled on the logit scale. Jeffrey's Beta prior was considered for the heterogeneity and inconsistency of standard deviation (SD) parameters. Cholesky and spherical decomposition strategies were adopted for the between-study variance-covariance matrix. Deviance information criterion (DIC) indices were computed to determine the model fit. All results pertaining to Markov chain Monte Carlo simulations and all analyses were carried out in WinBUGS software. The estimated common heterogeneity SDs were similar, and the DIC values did not provide any evidence for superiority between the two decomposition strategies. The correlation (95% credible interval) between the outcomes was estimated as -0.31 (-0.71, -0.02) and -0.37 (-0.73, -0.03) for the Cholesky and spherical decompositions, respectively. Gelman-Rubin convergence statistics were stable, and Monte Carlo errors for all the parameters were around 0.005. Overall, olanzapine, paliperidone, and quetiapine were both significantly more effective and acceptable than a placebo when both the study outcomes were considered simultaneously. The findings favoring olanzapine, paliperidone, and quetiapine possess an excellent concordance with the one adopted in clinical practice, and the Canadian Network for Mood and Anxiety Treatments and Royal Australian and New Zealand College of Psychiatrists guidelines recommend these as first-line drugs for treating bipolar disorder.

Improved deformation modelling of structures by least-squares variance component estimation based on multi-sensor data integration

In this contribution, to improve the deformation modelling based on data integration, the LS-VCE algorithm is proposed by obtaining a stochastic model of input multi-sensor data. So, one can achieve the accurate variance-covariance matrix of multi-sensor observations to participate in iterative least-squares. A practical application was made for the settlement observations from geotechnical settlement-meters and geodetic levelling (respectively known as internal and external sensors) to model the surface settlement variation of the Karkhe earth-dam. The determined variance component shows less contribution of the geotechnical settlements in the deformation modelling. An achievement of this paper is that the LS-VCE method improves the integration of the geotechnical with geodetic data by estimating an optimal stochastic model resulting in deformation model optimization. Validation results of estimated surface settlements on the check-points show an RMSE of about 3 cm and a relative-error of about 14%, which indicates the success of the modelling.

Asymptotic Standard Errors of IRT Linking Coefficients for the Bifactor Extension of the 3PL Model

The bifactor extension of the three-parameter logistic (B3PL) model has been used in applications of multidimensional item response theory (IRT) such as test equating and vertical scaling. Developing a common multidimensional IRT scale (that is, multidimensional coordinate system) is critical in those applications. Three common-item scale linking methods, the direct least squares (DLS), mean/least squares (MLS), and item response function (IRF) methods, for the B3PL model have been found to be effective in developing a common multidimensional IRT ability scale between two test forms to be linked. In this paper, the asymptotic standard errors (SEs) of IRT linking coefficients estimated by the DLS, MLS, and IRF methods are derived assuming that the B3PL model holds and the asymptotic variance-covariance matrix of item parameter estimates from separate calibrations is available. The delta method is used for the derivations. Computer simulations which investigate the accuracy of the derivations under various conditions are given, showing that the derivations are reasonably accurate when sufficiently large samples are used and that in general the SEs of the IRF method are smaller than those of the DLS and MLS methods. The simulation results also suggest that the SEs of linking coefficient estimates are, approximately, inversely proportional to the square root of the sample size when two test forms are administered to the same number of examinees.

On Sensitivity for Portfolio Optimisation Based on a High-dimensional Jump-diffusion Merton Model

The problem of singularity of the variance-covariance matrix and its impact on the sensitivity of Markowitz portfolio optimization has been extensively studied in the literature when the underlying model does not include jump terms. In this paper, we first use a jump-diffusion multivariate Merton model to evaluate sensitivity of portfolio optimization and apply principal component analysis (PCA) for dimensionality reduction as a solution to singularity of the variance-covariance matrix. Finally, we provide a numerical study based on the adjusted daily closing price of $S\&{P}\, 500$ stocks to explore the impact of the dimension of the reduced space and jump terms on the sensitivity of the portfolio optimization. Empirical experiments confirm that for models without jump terms, the sensitivity analysis may not reflect the correct assessment of the impact of dimensionality reduction on the portfolio optimization.

TvReg: Time-varying Coefficients in Multi-Equation Regression in R

This article explains the usage of R package [tvReg](https://CRAN.R-project.org/package=tvReg), publicly available for download from the Comprehensive R Archive Network, via its application to economic and finance problems. The six basic functions in this package cover the kernel estimation of semiparametric panel data, seemingly unrelated equations, vector autoregressive, impulse response, and linear regression models whose coefficients may vary with time or any random variable. Moreover, this package provides methods for the graphical display of results, forecast, prediction, extraction of the residuals and fitted values, bandwidth selection and nonparametric estimation of the time-varying variance-covariance matrix of the error term. Applications to risk management, portfolio management, asset management and monetary policy are used as examples of these functions usage.

Joinpoint Regression Methods of Aggregate Outcomes for Complex Survey Data

Abstract Joinpoint regression can model trends in time-specific estimates from aggregated data. These methods have been developed mainly for nonsurvey data such as cancer registry data assuming that the time-specific estimates are uncorrelated from time point to time point. This independence assumption can be violated for trends in time-specific estimates from complex survey samples due to using the same primary sampling units across time and, therefore, the full variance–covariance matrix of the time-specific estimates should be incorporated into the regression model fitting. This article extends these joinpoint methods for analyzing complex survey data within the National Cancer Institute’s Joinpoint software and empirically compares the extended method to existing methods for analyses of time trends in three surveys.

Rényi entropy and divergence for VARFIMA processes based on characteristic and impulse response functions

Rényi entropy based on characteristic function has been used as an information measure contained in wide-sense and real stationary vector autoregressive and moving average (VARMA) processes. These classes of processes have been extended by fractionally integrated VARMA (VARFIMA) ones, composed of a VARMA process, a vector of fractional differencing parameters, and independent and identically distributed multivariate normal random errors. Such processes have often been used to explicitly account for persistence to incorporate long-term correlations into multivariate data. The purpose of this paper is to extend Rényi entropy from VARMA to VARFIMA processes, addressing long-memory behavior of time series by adding a fractional differencing parameter. The characteristic function of the process can be derived directly from the asymptotic form of the impulse response function using the Wold representation. Then, assuming multivariate Gaussian white noise with known fractional differencing, autoregressive and moving average matrix parameters, the differential and Rényi entropies and Kullback–Leibler and Rényi divergences were obtained by evaluating the variance-covariance matrix identified with VARFIMA process distribution. The influences of the fractional differencing parameters on the Rényi entropy increment were analyzed, as were comparisons between VARFIMA processes using the Kullback–Leibler and Rényi divergences. Finally, numerical examples and an application to U.S. daily temperature time series are presented.