Estimation Of Variance-covariance Matrix Research Articles

Evaluation of a flexible piecewise linear mixed-effects model in the analysis of randomized cross-over trials.

Cross-over designs are commonly used in randomized clinical trials to estimate efficacy of a new treatment. They have received a lot of attention, particularly in connection with regulatory requirements for new drugs. The main advantage of using cross-over designs over conventional parallel designs is increased precision, thanks to within-subject comparisons. In the statistical literature, more recent developments are discussed in the analysis of cross-over trials, in particular regarding repeated measures. A piecewise linear model within the framework of mixed effects has been proposed in the analysis of cross-over trials. In this article, we report on a simulation study comparing performance of a piecewise linear mixed-effects (PLME) model against two commonly cited models-Grizzle's mixed-effects (GME) and Jones & Kenward's mixed-effects (JKME) models-used in the analysis of cross-over trials. Our simulation study tried to mirror real-life situation by deriving true underlying parameters from empirical data. The findings from real-life data confirmed the original hypothesis that high-dose iodine salt have significantly lowering effect on diastolic blood pressure (DBP). We further sought to evaluate the performance of PLME model against GME and JKME models, within univariate modeling framework through a simulation study mimicking a 2 × 2 cross-over design. The fixed-effects, random-effects and residual error parameters used in the simulation process were estimated from DBP data, using a PLME model. The initial results with full specification of random intercept and slope(s), showed that the univariate PLME model performed better than the GME and JKME models in estimation of variance-covariance matrix (G) governing the random effects, allowing satisfactory model convergence during estimation. When a hierarchical view-point is adopted, in the sense that outcomes are specified conditionally upon random effects, the variance-covariance matrix of the random effects must be positive-definite. The PLME model is preferred especially in modeling an increased number of random effects, compared to the GME and JKME models that work equally well with random intercepts only. In some cases, additional random effects could explain much variability in the data, thus improving precision in estimation of the estimands (effect size) parameters.

On the consistency of the logistic quasi-MLE under conditional symmetry

For estimating the parameters of a linear conditional mean, I show that the quasi-maximum likelihood estimator (QMLE) obtained under the nominal assumption that the error term is independent of the explanatory variables with a logistic distribution is consistent provided the conditional distribution of the error term is symmetric. No other restrictions are required for Fisher consistency; in particular, the error and covariates need not be independent, and so general heteroskedasticity of unknown form is allowed. Importantly, the influence function of the logistic quasi-log likelihood is bounded, making it more resilient to outliers than ordinary least squares. Inference using the logistic QMLE is straightforward using a robust asymptotic variance–covariance matrix estimator.

Efficient estimation of high-dimensional dynamic covariance by risk factor mapping: Applications for financial risk management

This paper aims to explore a modified method of high-dimensional dynamic variance–covariance matrix estimation via risk factor mapping, which can yield a dependence estimation of asset returns within a large portfolio with high computational efficiency. The essence of our methodology is to express the time-varying dependence of high-dimensional return variables using the co-movement concept of returns with respect to risk factors. A novelty of the proposed methodology is to allow mapping matrices, which govern the co-movement of returns, to be time-varying. We also consider the flexible modeling of risk factors by a copula multivariate generalized autoregressive conditional heteroscedasticity (MGARCH) model. Through the proposed risk factor mapping model, the number of parameters and the time complexity are functions of a small number of risk factors instead of the number of stocks in the portfolio, making our proposed methodology highly scalable. We adopt Bayesian methods to estimate unknown parameters and various risk measures in the proposed model. The proposed risk mapping method and financial applications are demonstrated by an empirical study of the Hong Kong stock market. The assessment of the effectiveness of the mapping via risk measure estimation is also discussed.

Estimation of Matrix Variance-Covariance on Nonparametric Regression Spline Truncated for Longitudinal Data

Regression analysis is a statistical analysis used to determine the pattern of relationships between predictor variables and response variables. There are two models estimation approaches in regression analysis, namely parametric regression and nonparametric regression. The parametric regression approach is used if the shape of the regression curve is known. In cases with unknown relationship patterns, the development is done using nonparametric regression. Nonparametric regression is a model estimation method which is based on an approach that is not bound by certain assumptions of the regression curve shape. One of the most commonly used nonparametric regression curve estimates is the spline approach. Spline is one of the nonparametric approaches that is often used and has a very good ability to handle data characters that are smooth as well as data whose behavior changes at certain sub-intervals. Theoretically, the spline approach is not only able to estimate nonparametric regression curves for cross section data, but can also be developed for longitudinal data. Longitudinal data is a combination of cross section data and time series data where the longitudinal data between subjects are mutually independent, but between observations are dependent. This study aims to find the estimation curves of spline truncated multivariable nonparametric regression for longitudinal data using the Weighted Least Square (WLS) method and to find the estimation of variance-covariance matrix using the Maximum Likelihood Estimate (MLE) method.

Quantile Regression Random Effects

This paper develops a random effects model for quantile regression (QR). We establish identification of the QR coefficients, and develop practical estimation and inference procedures. We employ a simple pooled QR estimator to estimate the coefficients of interest, and derive its statistical properties. The random effects induce cluster dependence hence we use a cluster-robust variance-covariance matrix estimator for inference, and establish its uniform consistency over the set of quantiles. We also develop a new test procedure for uniform testing of linear hypotheses in QR models. This procedure is a modified Wald test applied on a growing number of quantiles such that, asymptotically, the test is uniform over the quantiles. We show this procedure can be applied to test the random effects hypothesis in QR panel data models. Two significant differences between our model and fixed-effects QR models are that effects of time-invariant regressors can be estimated, and that the time-series dimension can be small and finite. We provide Monte Carlo simulations to evaluate the finite sample performance of the estimation and inference procedures. Finally, we apply the proposed methods to study the roles of education and ability in wage determination. We document strong heterogeneity in returns to education along the conditional distribution of earnings.

Higher order expansions for error variance matrix estimates in the Gaussian AR(1) linear regression model

We derive a stochastic expansion of the error variance–covariance matrix estimator for the linear regression model under Gaussian AR(1) errors. The higher order accuracy terms of the refined formula are not directly derived from formal Edgeworth-type expansions but instead, the paper adopts Magadalinos’ (1992) stochastic order of ω which is a convenient device to obtain the equivalent relation between the stochastic expansion and the asymptotic approximation of corresponding distribution functions. A Monte Carlo experiment compares tests based on the new estimator with others in the literature and shows that the new tests perform well.

Early intervention in cauda equina syndrome associated with better outcomes: a myth or reality? Insights from the Nationwide Inpatient Sample database (2005–2011)

Background ContextEvidence-based consensus on timing to surgical decompression following symptom onset in patients with cauda equina syndrome (CES) is limited or widely debated. PurposeThis study aimed to investigate whether timing to intervention in the management of patients with CES has an impact on outcomes. Study Design/SettingThis is a retrospective cohort study. Patient SampleThe patient sample included 4,066 adult patients with CES registered in the Nationwide Inpatient Sample database (2005–2011) and undergoing elective decompression surgery. Outcome MeasuresThe outcome measures are inpatient mortality, unfavorable discharge (discharge to rehabilitation), prolonged length of stay (LOS>75th percentile), and high hospital charges in patients undergoing decompression for CES. MethodsThe patients were stratified into three categories based on timing to surgical intervention: (1) within 24 hours (n=1,846, 45.6%); (2) between 24 and 48 hours (n=1,080, 26.6%), and (3) beyond 48 hours (n=1,130, 27.8%). Multivariable logistic regression fitted with generalized estimating equations using the sandwich variance-covariance matrix estimator to account for the clustering of similar outcomes within hospitals was used to examine the association of timing to surgical intervention categories with binary primary end points. For metric end points (charges), we used the ordinary least squares model to test the effect of timing to intervention. ResultsThe mean age of the cohort was 50.19±17.55 years and 41% were female. In comparison to patients operated within 24 hours, increased likelihood of inpatient mortality (odds ratio [OR]: 3.61, 95% confidence interval [CI]: 1.32–9.85, p=.012), unfavorable discharge (OR: 2.23, 95% CI: 1.87–2.66, p<.001), prolonged postsurgical LOS (OR: 1.76, 95% CI: 1.44–2.14, p<.001), and high hospital charges (OR:1.92, 95% CI: 1.81–2.05, p<.001) were observed in patients operated on over 48 hours since admission. Likewise, patients with incomplete CES with intervention beyond 48 hours had higher odds for unfavorable discharge (OR: 2.51, 95% CI: 1.99–3.17, p<.001), prolonged postsurgical LOS (OR: 1.73, 95% CI: 1.35–2.20, p<.001), and high hospital charges (OR: 1.94, 95% CI: 1.79–2.10, p<.001). Likewise, patients with complete CES with interventions beyond 48 hours had higher odds for unfavorable discharge (OR: 1.86, 95% CI: 1.41–2.45, p<.001), prolonged postsurgical LOS (OR: 2.06, 95% CI: 1.53–2.77, p<.001), and high hospital charges (OR: 1.39, 95% CI: 1.15–1.68, p<.001). ConclusionsEarly intervention in CES, regardless of the subtype (complete or incomplete), has higher likelihood of improved inpatient outcomes. The odds of getting better were higher, however, with incomplete CES. The timing of intervention did not seem to matter in traumatic CES as compared with degenerative etiology. Prospective randomized controlled trials may further help elucidate the impact of early intervention on outcomes in patients with CES.

Inference from high-frequency data: A subsampling approach

In this paper, we show how to estimate the asymptotic (conditional) covariance matrix, which appears in central limit theorems in high-frequency estimation of asset return volatility. We provide a recipe for the estimation of this matrix by subsampling; an approach that computes rescaled copies of the original statistic based on local stretches of high-frequency data, and then it studies the sampling variation of these. We show that our estimator is consistent both in frictionless markets and models with additive microstructure noise. We derive a rate of convergence for it and are also able to determine an optimal rate for its tuning parameters (e.g., the number of subsamples). Subsampling does not require an extra set of estimators to do inference, which renders it trivial to implement. As a variance–covariance matrix estimator, it has the attractive feature that it is positive semi-definite by construction. Moreover, the subsampler is to some extent automatic, as it does not exploit explicit knowledge about the structure of the asymptotic covariance. It therefore tends to adapt to the problem at hand and be robust against misspecification of the noise process. As such, this paper facilitates assessment of the sampling errors inherent in high-frequency estimation of volatility. We highlight the finite sample properties of the subsampler in a Monte Carlo study, while some initial empirical work demonstrates its use to draw feasible inference about volatility in financial markets.

Quantile Regression Random Effects

This paper develops a random effects model for quantile regression (QR). We establish identification of the QR coefficients, and develop practical estimation and inference procedures. We employ a simple pooled QR estimator to estimate the coefficients of interest, and derive its statistical properties. The random effects induce cluster dependence hence we use a cluster-robust variance-covariance matrix estimator for inference, and establish its uniform consistency over the set of quantiles. We also develop a new test procedure for uniform testing of linear hypotheses in QR models. This procedure is a modified Wald test applied on a growing number of quantiles such that, asymptotically, the test is uniform over the quantiles. We show this procedure can be applied to test the random effects hypothesis in QR panel data models. Two significant differences between our model and fixed-effects QR models are that effects of time-invariant regressors can be estimated, and that the time-series dimension can be small and finite. We provide Monte Carlo simulations to evaluate the finite sample performance of the estimation and inference procedures. Finally, we apply the proposed methods to study the roles of education and ability in wage determination. We document strong heterogeneity in returns to education along the conditional distribution of earnings.

Impact of associated injuries in conjunction with fracture of the axis vertebra on inpatient outcomes and postoperative complications: a Nationwide Inpatient Sample analysis from 2002 to 2011

Background ContextThere are limited data available on the impact of associated spinal (other spinal injuries [OSIs]) and extra-spinal injuries (ESIs) occurring in conjunction with fractures of the axis vertebra (C2) on clinical outcomes. PurposeThis study aimed to compare outcomes in patients with isolated C2 fractures versus patients with associated injuries in conjunction with C2 fractures. Study Design/SettingA retrospective cohort study. Patient SampleA total of 30,472 adult patients with C2 fractures (International Classification of Diseases, Ninth Revision, Clinical Modification code 805.02) registered in the Nationwide Inpatient Sample (NIS) database (2002–2011) comprised the patient sample. Outcome MeasuresInpatient mortality, unfavorable discharge, prolonged length of stay (LOS) and high-end hospital charges in the non-operative and operative cohorts, and postoperative complications (deep venous thrombosis [DVT]; acute renal failure [ARF]; respiratory complications and wound infections) for the operative cohort were the outcome measures. MethodsPatients were stratified into four categories based on injury type: (1) isolated C2 fracture (n=10,135; 33.3%); (2) C2 fracture+OSI (8.7%); (3) C2 fracture+ESI (37.2%); and (4) C2 fracture+OSI+ESI (20.8%). Multivariable logistic regression fitted with generalized estimating equations using the sandwich variance-covariance matrix estimator to account for clustering of similar outcomes within hospitals was used to examine the association of primary endpoints for each of the associated injury categories with reference to isolated C2 fractures. ResultsMean age of the cohort was 66.27±21.67 years and 52% were female. Of the cohort, 52% underwent surgical intervention for C2 fracture. In a pooled regression analysis involving the operative cohort, the risks for inpatient mortality (odds ratio [OR]: 3.77; 95% confidence interval [CI]: 3.02–4.70; p<.001), unfavorable discharge (OR: 1.83; 95% CI: 1.66–2.01; p<.001), prolonged LOS (OR: 1.33; 95% CI: 1.18–1.50; p<.001), high hospital charges (OR: 1.49; 95% CI: 1.31–2.69; p<.001), DVT (OR: 2.08; 95% CI: 1.61–2.68; p<.001), and ARF (OR: 1.46; 95% CI: 1.16–1.83; p=.001) were significantly higher in patients with additional injuries when compared with patients with C2 fractures alone. Likewise, increased chances of inpatient mortality (OR: 1.40; 95% CI: 1.21–1.62; p<.001), unfavorable discharge (OR: 1.24; 95% CI: 1.15–1.34; p<.001) and high hospital charges (OR: 1.31; 95% CI: 1.21–1.43; p<.001) were observed in a pooled analysis of patients with concomitant associated injuries in the non-operative cohort. ConclusionsAssociated injuries occurring concomitantly with C2 fractures adversely influence postoperative outcomes. In comparison to isolated C2 fractures, patients with associated injuries tend to have a greater propensity for higher health-care resource use because of more complicated and longer hospital inpatient stay.

The Greek equity market in European equity portfolios

The present paper emphasizes on the importance of the Greek equity market to European equity portfolios. The portfolio performance is higher for portfolios for which the Greek equity market is included. This result is consistent across a variety of variance–covariance matrix estimators, portfolio types, and evaluation measures as well. Results are also robust to the 2008 financial crisis. In terms of the variance–covariance matrix estimation, the realized volatility estimators result in higher portfolio performance than the daily squared returns estimator does. The realized volatility estimator which is optimally sampled and bias corrected is the most accurate variance–covariance matrix estimator. The Capital Market Line portfolio type is the portfolio type with the best portfolio performance. Overall, the inclusion of the Greek equity market to the European equity portfolios results in higher European equity portfolio performance.

Inference from High-Frequency Data: A Subsampling Approach

In this paper, we show how to estimate the asymptotic (conditional) covariance matrix, which appears in central limit theorems in high-frequency estimation of asset return volatility. We provide a recipe for the estimation of this matrix by subsampling; an approach that computes rescaled copies of the original statistic based on local stretches of high-frequency data, and then it studies the sampling variation of these. We show that our estimator is consistent both in frictionless markets and models with additive microstructure noise. We derive a rate of convergence for it and are also able to determine an optimal rate for its tuning parameters (e.g., the number of subsamples). Subsampling does not require an extra set of estimators to do inference, which renders it trivial to implement. As a variance-covariance matrix estimator, it has the attractive feature that it is positive semi-definite by construction. Moreover, the subsampler is to some extent automatic, as it does not exploit explicit knowledge about the structure of the asymptotic covariance. It therefore tends to adapt to the problem at hand and be robust against misspecification of the noise process. As such, this paper facilitates assessment of the sampling errors inherent in high-frequency estimation of volatility. We highlight the finite sample properties of the subsampler in a Monte Carlo study, while some initial empirical work demonstrates its use to draw feasible inference about volatility in financial markets.

A Comparison of Item Parameter Standard Error Estimation Procedures for Unidimensional and Multidimensional Item Response Theory Modeling

The present study was motivated by the recognition that standard errors ( SEs) of item response theory (IRT) model parameters are often of immediate interest to practitioners and that there is currently a lack of comparative research on different SE (or error variance–covariance matrix) estimation procedures. The present study investigated item parameter SEs based on three error variance–covariance matrix estimation procedures for unidimensional and multidimensional IRT models: Fisher information, empirical cross-product, and supplemented expectation maximization. This study centers on the direct comparisons of SEs from different procedures and complements a recent study by Tian, Cai, Thissen, and Xin by providing insights and suggestions on the nature of the differences and similarities as well as on practical matters such as the computational cost. The simulation results show that all three procedures produced similar results with respect to bias in the SE estimates for most conditions. When the number of items is large and sample size is small, empirical cross-product, which was the most computationally efficient procedure, appeared to be affected most, producing slight upward bias.

Some remarks on linear least squares fits with errors in both coordinates

The exact least squares line fit with errors in both coordinates (Reed 1992) is investigated together with the approximate solution based on the formalism of the linear Gauss-Helmert model or the unified adjustment approach of the classical textbook by Mikhail (1976). The similarities and the differences are described in details. In spite of the small differences the exact solution is preferable and the calculations are simpler. This paper does not deal with the errors-in-variables (EIV) models solved by the total least squares (TLS) principle, since the exact line fit solution is used to validate this general approach, which is basically designed to solve more sophisticated nonlinear tasks. In the most general case the fit of Person’s data with York’s weights is iteratively solved starting with the arbitrary zero initial value of the slope. The test computation with different but systematically chosen weights proved that in special cases — e.g. the weighted least squares sum of the distances between the data points and the estimated line is minimised — there is no need for iterations at all. It is shown that methods described by Detrekői (1991) and Zavoti (2012) are special cases of the general exact solutions. Reed (1992) derived the variances of the slope and intercept parameters without their covariance. The simple linear estimation of variance-covariance matrix of the exact solution is also demonstrated. The importance of the stochastic models coupled with exact solution is also demonstrated.

A new noise-compensated estimation scheme for multichannel autoregressive signals from noisy observations

In many engineering applications concerning the recovery of signals from noisy observations, a common approach consists in adopting autoregressive (AR) models. This paper is concerned with not only the estimation of multichannel autoregressive (MAR) model parameters but also the recovery of signals. A new noise compensated parameter estimation scheme is introduced in this paper. It contains an advanced least square vector (ALSV) algorithm which not only keeps the advantage of blindly estimating the MAR parameters and the variance-covariance matrix of observation noises, but also aims at ensuring the variance-covariance matrix to be symmetric in each iterative procedure. Moreover, the estimation of variance-covariance matrix of input noise is proposed, and then we form an optimal filtering to recover the signals. In the numerical simulations, the estimation performance of the ALSV estimation algorithm significantly outperforms that of other existed methods. Moreover, the optimal filtering based on the ALSV algorithm leads to more accurate recovery of the true signals.

A Bootstrap Study of Variance Estimation under Heteroscedasticity Using Genetic Algorithm

The conventional ordinary least squares (OLS) variance-covariance matrix estimator for a linear regression model under heteroscedastic errors is biased and inconsistent. Accordingly, several estimators have so far been proposed by various researchers. However, none of these perform well under the finite-sample situation. In this paper, the powerful optimization technique of Genetic algorithm (GA) is used to modify these estimators. Properties of these newly developed estimators are thoroughly studied by Monte Carlo method for various sample sizes. It is shown that GA-versions of the estimators are superior to corresponding non-GA versions as there are significant reductions in the Total relative bias as well as Total root mean square error.

Spatial and Temporal Time Series Conversion

Spatial and Temporal Time Series Conversion - A Consistent Estimator of the Error Variance-Covariance Matrix. Abstract: We focus on the problem of time series conversion from low to high frequency satisfying the twofold temporal and spatial constraint. We offer a simple solution to variance-covariance matrix estimation of the error terms. Since the residuals of high frequency equations of the indicated indicator model are not observable, we inferred the characteristics of their stochastic process through both a specific hypothesis (VAR 1 process) and estimation of the related annual model. We derive a consistent estimator of the variance-covariance matrix and we prove that Di Fonzo's (1990) estimator based on this matrix is asymptotically equivalent to a GLS estimator.

Yes, Implied Volatilities Are Not Informationally Efficient. An Empirical Estimate Using Options on Interest Rate Futures Contracts.

The accuracy of volatility forecast estimators has been assessed using daily overlapping and non overlapping observations on two major short-term interest rate futures contracts traded in London. The use of a panelized data set has eliminated some of the drawbacks usually associated with non overlapping data estimation, such as the lack of accuracy due to an insufficient number of observations or the arbitrariness of the choice of tenor. In the same way non stationarity and long memory characteristics of daily overlapping time series are disposed of. Information content estimation in levels associated with the Hansen (1982) variance covariance matrix estimator provides reasonably accurate estimates, broadly similar to the corresponding benchmark panel data ones.

Heteroskedasticity and Neglected Parameter Heterogeneity

The paper studies the consequences of neglecting parameter heterogeneity for the linear regression model and cross‐sectional data. Monte‐Carlo experiments are used to illustrate that neglected parameter heterogeneity typically leads to (a) regression coefficients that are economically meaningless and (b)significant test statistics for heteroskedasticity and, possibly non‐normality. The paper concludes that evidence for heteroskedasticity should not routinely lead to the use of White's well‐known heteroskedasticity‐consistent variance covariance matrix estimator. If heteroskedasticity is caused by neglected parameter heterogeneity or other causes of heteroskedasticity, such as wrong functional form, White's estimator will not serve any useful purpose.

A comparison of some estimators of the variance covariance matrix when the population mean is known

In this article, we consider the problem of estimation of variance covariance matrix a normally distributed population, when the population mean is known. We, specifically consider the maximum likelihood estimators derived under the known as well as unknown mean assumptions and test their relative merits under various loss functions. Definite results are obtained under entropy as well as squared error loss functions. Finally some open problems in this area are suggested.