Variance-covariance Matrix Research Articles

Minimum-variance portfolio optimization: an asset pricing model approach

ABSTRACT This paper employs novel asset pricing model methods to construct the global minimum-variance portfolio G. Out-of-sample analyses of U.S. stock returns show that optimized G portfolios have relatively higher expected returns, lower variance, and higher Sharpe ratios than those based on traditional variance-covariance matrix estimation methods. Robustness checks confirm these findings and show that our G portfolio has lower variance than currently available minimum-variance exchange traded funds (ETFs). We conclude that asset pricing models can be used to build high performing G portfolios. Implications to portfolio management are discussed.

Modeling uncertainty with the truncated zeta distribution in mixture models for ordinal responses

In recent three decades, there has been a rapid increasing interest in the mixture models for ordinal responses, and the classical CUB model as a fundamental one has been extended to different preference and uncertainty models. In this article, based on a response style supported by Zipf’s law, we propose a novel mixture model for ordinal responses via replacing the uncertainty component of the CUB model with a truncated Zeta distribution. Parameters estimation with EM algorithm, inferential issues with respect to the approximation of a truncated Riemann Zeta function and estimators’ variance-covariance information matrix are investigated. The advantages of the proposed model over the CUB model have been illustrated with simulations of two sets of Monte Carlo experiments and practical applications of a health survey and a bicycle use. The intention of the article is to distinguish the respondents’ true preference from the response style of “the higher, the less”, so as to understand more reasonably the formation causes of ordinal responses.

Optimal Subsampling for Functional Quasi-Mode Regression with Big Data

We propose investigating optimal subsampling for functional regression with massive datasets based on the mode value, which is referred to as functional quasi-mode regression, to reduce data volume and alleviate computational burden. Using data-adaptive weights derived from regression residuals, the suggested regression offers enhanced robustness against nonnormal errors compared to traditional least squares or maximum likelihood estimation methods. To estimate the model, we employ B-spline basis functions to approximate the functional coefficient and include a penalty term in the objective function for enforcing smoothness in the resulting estimator. We adopt a computationally efficient mode-expectation-maximization algorithm, augmented by a Gaussian kernel, for numerical estimation. Under mild regularity conditions, we derive the asymptotic distributions of both full data and subsample quasi-mode estimators. The optimal subsampling probabilities by minimizing the asymptotic variance-covariance matrix under A- and L-optimality criteria are identified. These optimal probabilities rely on the full data estimate, prompting the development of a two-step algorithm to approximate the optimal subsampling procedure. The resultant algorithm is processing-efficient and can significantly reduce computational time compared to the full data approach. We also establish the asymptotic normality of the quasi-mode estimator obtained through this two-step algorithm. To assess finite sample performance, we conduct Monte Carlo simulations and analyze air quality data, showcasing the effectiveness of the developed estimator. Supplemental materials for this article are available online.

A Nonparametric Global Win Probability Approach to the Analysis and Sizing of Randomized Controlled Trials With Multiple Endpoints of Different Scales and Missing Data: Beyond O'Brien-Wei-Lachin.

Multiple primary endpoints are commonly used in randomized controlled trials to assess treatment effects. When the endpoints are measured on different scales, the O'Brien rank-sum test or the Wei-Lachin test for stochastic ordering may be used for hypothesis testing. However, the O'Brien-Wei-Lachin (OWL) approach is unable to handle missing data and adjust for baseline measurements. We present a nonparametric approach for data analysis that encompasses the OWL approach as a special case. Our approach is based on quantifying an endpoint-specific treatment effect using the probability that a participant in the treatment group has a better score than (or a win over) a participant in the control group. The average of the endpoint-specific win probabilities (WinPs), termed the global win probability (gWinP), is used to quantify the global treatment effect, with the null hypothesis gWinP = 0.50. Our approach involves converting the data for each endpoint to endpoint-specific win fractions, and modeling the win fractions using multivariate linear mixed models to obtain estimates of the endpoint-specific WinPs and the associated variance-covariance matrix. Focusing on confidence interval estimation for the gWinP, we derive sample size formulas for clinical trial design. Simulation results demonstrate that our approach performed well in terms of bias, interval coverage percentage, and assurance of achieving a pre-specified precision for the gWinP. Illustrative code for implementing the methods using SAS PROC RANK and PROC MIXED is provided.

Portfolio Optimization Using Machine Learning Method and Monte Carlo Simulation

Investment portfolio optimization is a crucial aspect of quantitative finance, aiming to maximize returns and minimize risks. This study focuses on optimizing investment allocation among five selected stocks (BAIC Blue Valley, BYD, Chang’ an Automobile, Kweichow Moutai, Sunwoda) over 100 days using Monte Carlo simulation and machine learning methods. This study uses the sliding window approach with the linear regression model to predict future returns and calculate the variance-covariance matrix to determine the optimal portfolio weights, leading to two key strategies: maximizing the Sharpe ratio and minimizing the risk. The results reveal that the Max Sharpe Ratio portfolio achieved a cumulative return of 0.0909, significantly outperforming the CSI 300 index’s return of -0.0348. Additionally, the Min Risk portfolio exceeded the market index with a cumulative return of 0.0062. These findings demonstrate that both the Max Sharpe Ratio and Min Risk strategies are effective in achieving a balance between risk and return and surpassing the market, offering valuable insights for investors seeking optimized portfolio allocations.

Form Deviation Uncertainty and Conformity Assessment on a Coordinate Measuring Machine

Coordinate measuring machines are widely used in the industrial field due to their ease of automation. However, estimating the measurement uncertainty is a delicate task, especially when controlling for deviation, given the large number of factors that influence the measurement. A precise estimate of the uncertainty is crucial to avoid incorrect conformity assessments. The purpose of this study is to control geometrical-form tolerance specifications, taking into consideration their associated uncertainty. A surface fitting model based on the least squares criterion is proposed, allowing one to obtain the variance–covariance matrix by iterative calculation according to the Levenberg–Marquard optimization method. The form deviation is then evaluated following the Geometrical Product Specifications (GPS) Standard, and its associated uncertainty is estimated using the guide to the expression of uncertainty in measurement (GUM) propagation of the uncertainty law. Finally, the conformity assessment is performed based on the measured deviation and its associated uncertainty. Different results for the measurement of straightness, flatness, circularity, roundness, and cylindricity are presented and detailed. This model is thereafter validated by a Monte Carlo simulation, and interlaboratory comparisons of the obtained results were performed, which showed satisfactory outcome. This contribution is of great use to manufacturing companies and metrology laboratories, allowing them to meet the normative guidelines, which stipulates that each measurement result must be accompanied by its associated uncertainty.

Effect of Co-exposure to Additional Substances on the Bioconcentration of Per(poly)fluoroalkyl Substances: A Meta-Analysis Based on Hydroponic Experimental Evidence.

A consensus has yet to emerge regarding the bioconcentration responses of per(poly)fluoroalkyl substances under co-exposure with other additional substances in aqueous environments. This study employed a meta-analysis to systematically investigate the aforementioned issues on the basis of 1,085 published datasets of indoor hydroponic simulation experiments. A hierarchical meta-analysis model with an embedded variance covariance matrix was constructed to eliminate the non-independence and shared controls of the data. Overall, the co-exposure resulted in a notable reduction in PFAS bioaccumulation (cumulative effect size, CES = - 0.4287, p < 0.05) and bioconcentration factor (R2 = 0.9507, k < 1, b < 0) in hydroponics. In particular, the inhibition of PFAS bioconcentration induced by dissolved organic matter (percentage form of the effect size, ESP = - 48.98%) was more pronounced than that induced by metal ions (ESP = - 35.54%), particulate matter (ESP = - 24.70%) and persistent organic pollutants (ESP = - 18.66%). A lower AS concentration and a lower concentration ratio of ASs to PFASs significantly promote PFAS bioaccumulation (p < 0.05). The bioaccumulation of PFASs with long chains or high fluoride contents tended to be exacerbated in the presence of ASs. Furthermore, the effect on PFAS bioaccumulation was also significantly dependent on the duration of co-exposure (p < 0.05). The findings of this study provide novel insights into the fate and bioconcentration of PFAS in aquatic environments under co-exposure conditions.

Transmuted Power Gumbel Distribution: Estimation and Applications

In recent years, generalized distributions have been widely studied in statistics as they possess flexibility in applications. This is justified because the traditional distributions often do not provide good fit in relation to the real data set studied. This paper develops a Power Gumbel distribution using the quadratic rank transmutation map (QRTM). The new generalization is called the transmuted Power-Gumbel distribution. Various mathematical properties of this distribution including moments, moment generating function, quantile function, mean deviation and order statistics were also studied. These features support the legitimacy and robustness of the proposed distribution. The maximum likelihood method is used for estimating the model parameters, and the finite sample performance of the estimators are assessed by simulation studies indicating that their precision improves with larger sample sizes. The asymptotic confidence intervals for the parameters are also obtained based on asymptotic variance-covariance matrix. Finally, the usefulness of the proposed model is illustrated in an application to two real data sets and conclude that the four-parameter transmuted Power Gumbel distribution provides better fit than the other five models.

Real-time LEO satellite clock estimation with predicted LEO satellite orbits constrained

Low Earth Orbit (LEO) satellites can augment the traditional GNSS-based positioning, navigation and timing services, which require real-time high-precision LEO satellite clock products. As the complicated systematic effects contained in the LEO satellite clock estimates limit their high-precision mid- to long-term prediction, high-frequency LEO satellite clocks need to be estimated within a Kalman filter, resulting in a short prediction time for real-time applications. Compared to the clock estimation using Batch Least-Squares (BLS) adjustment, filter-based clock estimation experiences a lower precision. Increasing the model strength by introducing external orbital information, thus, de-correlating the orbital and clock parameters, will benefit real-time clock precision. In this contribution, reduced-dynamic LEO satellite orbits are first estimated using BLS adjustment in near real-time and predicted in the short term. The predicted orbits are then constrained during the Kalman-filter-based clock estimation process. The variance–covariance matrix of the introduced orbital errors is tested for different sets of values in the radial, along-track and cross-track directions when constraining orbits of different prediction times. One week of GPS data from the Sentinel-3B satellite in 2018 was used for validation of the proposed method. When weakly constraining high-accuracy predicted orbits within a prediction time of 20 min, i.e., with a standard deviation of the constraint set to 2–3 dm in the radial and cross-track directions, and 4–6 dm in the along-track direction, the estimated clock accuracy can be improved from about 0.27 to 0.23 ns, with a 13.4% improvement. Depending on the prediction period of the introduced orbits, the Signal-In-Space Range Error (SISRE) of the LEO satellite to Earth can also be improved, from about 9.59 cm without constraints, to 7.38–8.07 cm after constraining the predicted orbits, with an improvement of 16–23%. The improvements in the SISRE also indicate a better consistency between the real-time clock and orbital estimates.

Geometric Morphometric Study on Distinguishing Metopic Craniosynostosis from Metopic Ridging.

Craniosynostosis, a common congenital anomaly, results from premature fusion of the cranial sutures. One of the forms of craniosynostosis is premature fusion of the metopic suture, referred to as trigonocephaly, but the diagnosis of metopic suture synostosis remains controversial. The purpose of this study was to clarify, using geometric morphometric analysis, if a metopic ridge alone observed in cases of mild trigonocephaly represents a pathological phenomenon. Three different cranial morphologies were compared among patients up to 2 years old who were categorized into the true group, the mild group, and the normal group, based on the presence or absence of specific symptoms, history of cranioplasty for trigonocephaly, or lack of any abnormality on computed tomography. Using the obtained computed tomography images, 235 anatomical landmarks and semi-landmarks were plotted on the entire cranial surface for analysis of neurocranial morphology, and the cranial shapes represented by landmarks were analyzed using geometric morphometrics. Principal components of shape variations among specimens were then computed, based on the variance-covariance matrix of the Procrustes residuals of all specimens, and statistically analyzed. The principal component analyses of the variations in endocranial shape, frontal bone shape, and occipital bone shape did not show any significant differences in cranial morphology between mild trigonocephaly and normal skulls; however, true trigonocephaly was found to differ significantly from mild trigonocephaly and normal skulls. These findings suggest that in assessments of cranial morphology, the presence of a ridge alone cannot be diagnosed as fundamentally pathological, and may represent normal morphology.

Penerepan Analisis Diskriminan Kuadratik Robust Pada Klasifikasi Desa

Discriminant analysis is a method used in separating objects into different groups and allocating objects into a predetermined group. Discriminant analysis is bound by the assumption that the mean vector for each group is different, the data is normally distributed multivariate and the covariance variance matrix between groups is the same. If there is a covariance variance matrix between different groups, then quadratic discriminant analysis is used for the classification process. However, sometimes it is found that data contains outliers, so a robust estimator is used, namely the Minimun Covariance Determinant with the fast-MCD algorithm. Therefore, robust quadratic discriminant analysis can be used to classify 128 villages and 48 sub-districts in Wajo district. It was found that 106 villages were correctly classified into village groups and 22 villages were misclassified into sub-district groups and 35 sub-districts were correctly classified as sub-district groups and 13 sub-districts were misclassified into village groups and produced an accuracy of classification results of 80.11%.

Multivariate test applied to single Pu isotopic composition measurement

The determination of the plutonium isotopic composition by gamma spectrometry is one of the tasks performed by nuclear inspectors. It is typically measured using planar germanium detectors. The spectra are analysed with deconvolution codes, which calculate all isotopic abundances. The experimental results are then compared with the declared values and the acceptance criterion is often based on one unique isotope, 240Pu, to distinguish between reactor and weapon grade Pu.If the measurement is a part of a measurement series, it is possible to calculate covariance terms and carry out a more comprehensive comparison using Hotelling's test as recently presented in Berndt and Mortreau (2023) [1]. In this study, it was shown that the T2 test often rejects simulated false cases which are accepted if only the 240Pu abundance is considered. Hence, the T2 test should be preferred to the 240Pu criterion.In case of a single measurement, which is typical for inspections, the co-variance cannot be determined and the comprehensive calculation of the T2 test as performed in Berndt and Mortreau (2023) [1] cannot be applied. The purpose of the present paper is to consider approximations of the variance-covariance matrix to allow the application of the T2 test to isolated measurements. The results were then applied to more than 400 single measurements of Pu samples covering many different experimental conditions.

Covariate-adjusted generalized pairwise comparisons in small samples.

Semiparametric probabilistic index models allow for the comparison of two groups of observations, whilst adjusting for covariates, thereby fitting nicely within the framework of generalized pairwise comparisons (GPC). As with most regression approaches in this setting, the limited amount of data results in invalid inference as the asymptotic normality assumption is not met. In addition, separation issues might arise when considering small samples. In this article, we show that the parameters of the probabilistic index model can be estimated using generalized estimating equations, for which adjustments exist that lead to estimators of the sandwich variance-covariance matrix with improved finite sample properties and that can deal with bias due to separation. In this way, appropriate inference can be performed as is shown through extensive simulation studies. The known relationships between the probabilistic index and other GPC statistics allow to also provide valid inference for example, the net treatment benefit or the success odds.

O-315 Familial endocrine disease increases the risk of pregnancy loss and recurrent pregnancy loss– a nationwide register-based study of 366,548 Danish women

Abstract Study question Is endocrine disease and pregnancy loss associated, and can the association be due to inherited genetic causes? Summary answer Endocrine disease is associated with an increasing number of pregnancy losses. Endocrine disease in parents and siblings increases a woman’s risk of pregnancy loss. What is known already One in four pregnancies results in a pregnancy loss, but only 60% hereof are due to detectable chromosome abnormalities. The remaining 40% are suspected to be caused by other factors such as maternal comorbidity. Recurrent pregnancy loss (RPL) is defined as 2 or 3 consecutive losses and affects 1-3% of couples trying to conceive. Pregnancy loss has been associated with endocrine diseases, e.g., thyroid disease, diabetes, or PCOS. Furthermore, thyroid autoimmunity has been associated with RPL. It is unknown if other endocrine diseases are associated with RPL. To date, there is no evidence for a genetic basis of the risk. Study design, size, duration Danish nationwide cohort study including data from the Danish National Health Registers on ICD-8/ICD-10 diagnosis codes, all pregnancy outcomes and redeemed medicine prescriptions. Endocrine disease was defined as any endocrine disease (e.g., diabetes, thyroid disease, PCOS etc.). The study included in total 366,548 women, including 54,394 women with endocrine disease and 312,154 without endocrine diseases. The study included data from 1973 to 2022. Linkage to parent and siblings was available through the Multi-Generation Registry Lite. Participants/materials, setting, methods The cohort consisted of women born between 1977-1993 with ≥1 pregnancy. The exposure was endocrine disease. Logistic regression models adjusted for birth year provided odds ratios (ORs) for endocrine diseases according to number of pregnancy losses. Parent-offspring and sibling regression provided estimates for familial aggregation, assessing the degree of resemblance between relatives to infer the potential influence of shared familial factors. The variance-covariance matrix was adjusted using the Huber-White method to account for familial clustering. Main results and the role of chance We found a significant association between endocrine disease and pregnancy loss increasing with the number of pregnancy losses; OR and 95%CI for one loss 1.15 (1.12-1.17), two losses 1.31 (1.24-1.38) and ≥ three losses 1.81 (1.70-1.93). Further, endocrine disease was associated with RPL (OR 1.87, (1.74-2.00)). The association was strongest with primary RPL (OR 2.13 (1.94-2.35)), compared to secondary RPL (OR 1.58 (1.42-1.76)). In cases where either of a woman’s parents was diagnosed with an endocrine disease, she also had a higher risk of pregnancy loss (OR 1.06 (1.04–1.08)) and RPL (OR 1.07 (1.01–1.14)), compared to a woman with parents without endocrine disease. If a woman’s sibling had an endocrine disease, the OR for pregnancy loss was 1.07 (1.03–1.11) and RPL 1.17 (1.03–1.33). Further stratification into individual endocrine phenotypes revealed that type 2 diabetes conferred a comparable risk: for offspring of parents with this condition, the risk of pregnancy loss increased to an OR of 1.08 (1.06–1.11), and for siblings, the OR was 1.13 (1.02–1.25). This effect persisted even when adjusting for the disease in the individual experiencing the loss. Limitations, reasons for caution Very early pregnancy losses are often not handled at the hospital and therefore not part of the Danish registers. Thus, there is a risk of underdiagnosed pregnancy losses, although this would be the case for both the exposed and control cohort. Wider implications of the findings Women with RPL should be investigated for endocrine disease, and family history of endocrine disease is an important and novel risk factor that should guide the diagnostic work-up. Identifying the shared mechanism or pleiotropic effects between pregnancy loss and endocrine disease should be a focus area. Trial registration number Research related to this project was supported by the Novo Nordisk Foundation (ID: NNF22OC0077221, NNF23OC0087269) and the William Demant Foundation (salary for Pia Egerup).

Data reconciliation of indoor positioning data: Improve position data accuracy in warehouse environment

This article focuses on improving indoor positioning data through data reconciliation. Indoor positioning systems are increasingly used for resource tracking to monitor manufacturing and warehouse processes. However, measurement errors due to noise can negatively impact system performance. Redundant measurement involves the use of multiple sensor tags that provide position data on the same resource, to identify errors in the physical environment. If we have measurement data from the entire physical environment, a map-based average measurement error can be determined by specifying the points in the examined area where measurement data should be compensated and to what extent. This compensation is achieved through data reconciliation, which improves real-time position data by considering the measurement error in the actual position as an element of the variance-covariance matrix. A case study in a warehouse environment is presented to demonstrate how discrepancies in position data from two sensor tags on forklifts can be used to identify layout-based errors. The algorithm is generally capable of handling the multi-sensor problem in the case of indoor positioning systems. The key points are as follows:•The layout-based error detection is determined with the indoor positioning system measurement error.•This article shows how redundant measurements and data reconciliation can improve the accuracy of such systems.•Improving the accuracy of position data with the layout-based error map using a data reconciliation algorithm.

The Modified Ambiguity Function Approach with regularization for instantaneous precise GNSS positioning

Abstract The Modified Ambiguity Function Approach (MAFA) implicitly conducts the search procedure of carrier phase GNSS integer ambiguity resolution (IAR) in the coordinate domain using the integer least squares (ILS) principle, i.e. MAFA-ILS. One of the still open scientific problems is an accurate definition of the search region, especially in the context of instantaneous IAR. In doing so, the float solution results, which encompass float position (FP) and its variance-covariance (VC) matrix, must be improved as these are necessary for defining the search region. For this reason, the ambiguity parameters are separately regularized, and then the baseline parameters are conditioned on regularized float ambiguities. The conditional-regularized estimation is thus designed, obtaining the regularized FP (RFP) and its VC-matrix. This solution is promising because its accuracy is enhanced in the sense of mean squared error (MSE) thanks to the improved precision at the cost of regularized bias. The optimal regularization parameter (RP) values obtained for ambiguity parameters balance the contributions of improved precision and bias in the regularized float baseline solution’s MSE. Therefore, the regularized search region is defined accurately in the coordinate domain to contain such approximate coordinates that more frequently give the correct ILS solution. It also contains fewer MAFA-ILS candidates, improving the search procedure’s numerical efficiency. The regularized ILS estimator performs well with the presence of bias, increasing the probability of correct IAR in the coordinate domain.

Study of Spatial Autoregressive Regression With Heteroskedasticity Using the Generalized Method of Moments and Bayesian Approach

Spatial dependence and spatial heteroskedasticity are problems in spatial regression. Spatial autoregressive regression (SAR) concerns only to the dependence on lag. The estimation of SAR parameters containing heteroskedasticity using the maximum likelihood estimation (MLE) method provides biased and inconsistent estimators. The alternative method that can be used are generalized method of moments (GMM) and Bayesian method. GMM uses a combination of linear and quadratic moment functions simultaneously so that the computation is easier than MLE. Bayesian method solves heteroskedasticity by modeling the structure of variance-covariance matrix. The bias are used to evaluate the GMM and Bayes in estimating parameters of SAR model with heteroskedasticity disturbances in simulation data. The results show that GMM and Bayes provides the bias of parameter estimates relatively consistent and smaller with larger number of observations. GMM and Bayes methods are applied to district/city GRDP data in Indonesia. The result show GMM method with Eksponential Distance Weights (EDW) matrix produces the minimum variance and the largest pseudo-R2

Higher-order asymptotic refinements in a multivariate regression model with general parameterization

This paper derives a general Bartlett correction formula to improve the inference based on the likelihood ratio test in a multivariate model under a quite general parameterization, where the mean vector and the variance-covariance matrix can share the same vector of parameters. This approach includes a number of models as special cases such as non-linear regression models, errors-in-variables models, mixed-effects models with non-linear fixed effects, and mixtures of the previous models. We also employ the Skovgaard adjustment to the likelihood ratio statistic in this class of multivariate models and derive a general expression of the correction factor based on Skovgaard approach. Monte Carlo simulation experiments are carried out to verify the performance of the improved tests, and the numerical results confirm that the modified tests are more reliable than the usual likelihood ratio test. Applications to real data are also presented for illustrative purposes.

Forecasting interval‐valued returns of crude oil: A novel kernel‐based approach

AbstractThis paper proposes a novel kernel‐based generalized random interval multilayer perceptron (KG‐iMLP) method for predicting high‐volatility interval‐valued returns of crude oil. The KG‐iMLP model is constructed by utilizing the distance based on a kernel function, which outperforms the conventional Euclidean distance. Additionally, the optimal kernel function is estimated using the variance–covariance matrix of the prediction error, contributing to a better understanding of the overall characteristics of interval‐valued data. The introduction of the kernel function renders the algorithms used for estimating machine learning parameters ineffective. Therefore, this paper further proposes a backward distance of accumulative error propagation algorithm to estimate both the kernel function and model parameters, which provides a feasible approach for utilizing kernel function in interval neural networks. In the empirical analysis of weekly and daily returns of WTI crude oil, the superior predictive performance of the proposed method is demonstrated, enabling stable and accurate predictions for both point values and interval values. The model exhibits consistent outstanding performance across different network structures, showcasing the potential of KG‐iMLP for crude oil price forecasting.

Carbon Emissions Pricing: Linkages Between EU ETS Spot and Future Prices and Completeness of EU ETS Market

The study examines the relationship and linkages between spot and future prices of European Union Emission Trading Systems (EU-ETS) during 2019–2021. Dynamic conditional correlation specification of multivariate generalized autoregressive conditional heteroskedasticity (MGARCH) model was employed to examine the time-varying correlation between spot and future prices. Also, vector autoregression mean model and MGARCH-Baba-Engle-Kroner-Kraft volatility model was jointly estimated to model the spillover between EU ETS spot and future prices in the first and second moments. Lastly, we utilize the variance-covariance matrix of joint mean-variance model estimation to derive the optimal conditional hedge ratio as well as the hedge effectiveness of EU ETS future contracts. Our findings reveal a high conditional correlation and significant spillover between carbon spot and future markets in EU. Further, our study uncovers a high degree of hedge effectiveness for EU ETS future contracts. This is possibly the first study that examines the linkages between EU ETS spot and future prices pertaining to the recent transition stage of phase III and the initial stage of ongoing phase IV of the ETS market. Our findings pinpoint to ETS markets becoming more complete and in turn offering optimal hedging avenues. JEL Codes: G15, C58, Q38