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We consider model based inference in a fractionally cointegrated (or cofractional) vector autoregressive model, based on the Gaussian likelihood conditional on initial values. We give conditions on the parameters such that the process X t is fractional of order d and cofractional of order d — b; that is, there exist vectors β for which βʹX t is fractional of order d — b and no other fractionality order is possible. For b = 1, the model nests the I(d — 1) vector autoregressive model. We define the statistical model by 0 < b ≤ d, but conduct inference when the true values satisfy 0 ≤ d₀ — b₀ < 1/2 and b₀ ≠ 1/2, for which ${{\\mathrm{\\beta }}^{\\prime }}_{0}{\\mathrm{X}}_{\\mathrm{t}}$ is (asymptotically) a stationary process. Our main technical contribution is the proof of consistency of the maximum likelihood estimators. To this end, we prove weak convergence of the conditional likelihood as a continuous stochastic process in the parameters when errors are independent and identically distributed with suitable moment conditions and initial values are bounded. Because the limit is deterministic, this implies uniform convergence in probability of the conditional likelihood function. If the true value b₀ > 1/2, we prove that the limit distribution of ${\\mathrm{T}}^{{\\mathrm{b}}_{0}}(\\hat{\\mathrm{\\beta }}-{\\mathrm{\\beta }}_{0})$ is mixed Gaussian, while for the remaining parameters it is Gaussian. The limit distribution of the likelihood ratio test for cointegration rank is a functional of fractional Brownian motion of type II. If b₀ < 1/2, all limit distributions are Gaussian or chi-squared. We derive similar results for the model with d = b, allowing for a constant term.

Johansen's (1988, 1991) likelihood ratio test for cointegration rank of a vector autoregression (VAR) depends only on the squared sample canonical correlations between current changes and past levels of a simple transformation of the data. We study the asymptotic behavior of the empirical distribution of those squared canonical correlations when the number of observations and the dimensionality of the VAR diverge to infinity simultaneously and proportionally. We find that the distribution weakly converges to the so-called Wachter distribution. This finding provides a theoretical explanation for the observed tendency of Johansen's test to find \"spurious cointegration.\"

This paper evaluates the link between economic growth, renewable energy, tourism arrivals, trade openness, and carbon dioxide emissions in the European Union (EU-28). As an econometric strategy, the research uses panel data. In the first step, we apply the unit root test, and the results demonstrated that the variables used in this study are integrated I (1) in the first difference. In the second step, we apply the Pedroni cointegration test, and Kao Residual cointegration test, and we observe that the variables are cointegrated in the long run. The panel fully modified least squares (FMOLS), panel dynamic least squares (DOLS), and generalized moments system (GMM-System) estimator are considered in this research. The econometric results proved that trade openness and renewable energy decreased climate change and environmental degradation. The empirical study also found a positive effect of economic growth on carbon dioxide emissions. Moreover, tourism arrivals are negatively correlated with carbon dioxide emissions, showing sustainability practices of the tourism sector on the environment. Furthermore, carbon dioxide emissions in the long run present a positive impact, indicating that climate change increases. In this study, we also consider the recent methodology of Dumitrescu–Hurlin to observe the causality and the relationship between renewable energy, trade openness, economic growth, tourism arrivals, and carbon dioxide emissions.

This study aims to investigate the effects of information and communication technology (ICT), energy consumption, economic growth, and financial development on carbon dioxide emissions using 1993–2013 panel data from 12 Asian countries. The study employs a panel unit root test accounting for the presence of cross-sectional dependence and found that Internet usage is stationary and carbon dioxide emissions, energy consumption, gross domestic production (GDP), and financial development are first-difference stationary. The results form Pedroni panel cointegration test confirms that the variables are cointegrated. The results of the cointegration test indicate that the ICT-energy-GDP-carbon dioxide emissions nexus has long-run equilibrium. Both energy consumption and GDP have significant, positive impacts on carbon dioxide emissions; energy consumption and GDP have an effect on carbon dioxide emissions growth. ICT has a significantly negative effect on carbon dioxide emissions; the promotion of ICT becomes one of the important strategies introduced to mitigate carbon dioxide emissions for various countries. Causality results show that energy consumption, GDP, and financial development cause more carbon dioxide emissions. Energy consumption, GDP, and carbon dioxide emissions cause ICT. GDP causes financial development, whereas energy consumption and GDP are interdetermined. The feedback hypothesis exists in the region; those countries need to develop alternative energy to replace fossil fuels. ICT does not threaten the environment and ICT policy can be seen as a part of carbon dioxide emissions reduction policy.

This paper analyzes the legitimacy of the Environmental Kuznets Curve (EKC) hypothesis for a group of seven (G7) countries over the period 1995–2015. In addition to testing the EKC speculation, the authors also would like to understand the ways in which increases in renewable energy consumption and the international tourism receipt affect the CO2 emissions in G7 countries, because the energy and tourism sectors may have considerable direct impacts on CO2 emissions. In this investigation, a panel bootstrap cointegration test and an augmented mean group (AMG) estimator were applied. The empirical findings indicate that the tourism-induced EKC hypothesis is valid only for France. Additionally, it was detected that a rise in renewable energy consumption has a negative (reduction) impact on CO2 emissions in France, Italy, the UK, and the US. However, an increase in the receipt of international touristm has a positive (additional) impact on Italy’s CO2 emissions. Hence, this country’s decision-makers should re-review their tourism policy to adopt a renewable-inclusive one for sustainable tourism and the environment.

The paper analyzes cointegration in vector autoregressive processes (VARs) for the cases when both the number of coordinates, N, and the number of time periods, T, are large and of the same order. We propose a way to examine a VAR of order 1 for the presence of cointegration based on a modification of the Johansen likelihood ratio test. The advantage of our procedure over the original Johansen test and its finite sample corrections is that our test does not suffer from overrejection. This is achieved through novel asymptotic theorems for eigenvalues of matrices in the test statistic in the regime of proportionally growing N and T. Our theoretical findings are supported by Monte Carlo simulations and an empirical illustration. Moreover, we find a surprising connection with multivariate analysis of variance (MANOVA) and explain why it emerges.

The health and wellness of people through life expectancy, mortality rate improvement, and sustaining the productivity of labor contributes a lot to national income. Infrastructure development consumes energy and releases carbon dioxide at different stages of the construction process. The current study explores the nexus between CO2 emission, energy consumption, mortality, life expectancy, and GDP in the top five carbon-emitting countries by using time series data from 1975 to 2015. The study used a cointegration technique to find the long- and short-run relationships between study variables. The study also used a structural break test to identify the break time. The results of the correlation matrix show strong positive correlation between CO2 emissions and energy consumption. It also reflects a weak correlation with mortality and life expectancy in Japan and Russia. The results of the ADF test indicated that the series are stationary at first difference and provided evidence to use Johansen cointegration test for long- and short-run relationships between independent series. Vector error correction term and ECT method are used to find long-run relationships between cointegrated series and adjustment parameters. For the structural breaks of health indicators and energy consumption study, we used the Gregory Hanson structural break. Mortality rate and life expectancy rate of China, U.S., Russia, India, and Japan show relevant policy changes with economic policies of each country.

Recent economic and environmental literature suggests that the current state of energy use in South Africa amidst rapid growing population is unsustainable. Researchers in this area mostly focus on the effect of fossil energy use on carbon (CO 2 ) emission, which represents only an aspect of environmental quality. In contrast, the current study evaluates the influence of renewable energy use, human capital, and trade on ecological footprint––a more comprehensive measure of environmental quality. To this end, the study employs multiple structural breaks cointegration tests (Maki cointegration tests), dynamic unrestricted error correction model through Autoregressive Distributed Lag (ARDL) model, and VECM Granger causality tests. The results of the Maki cointegration tests reveal the existence of a cointegration between the variables in all the models with evidence of multiple structural breaks. Further, the ARDL results divulge that an increase in renewable energy use, human capital, and trade improves environmental quality through a decrease in ecological footprint, while an increase in income stimulates ecological footprint. Moreover, causal relationship is found, running from all the variables to renewable energy and trade flow in the long run, while in the short run, economic growth causes ecological footprint. Trade is found to Granger-cause human capital, while human capital causes renewable energy. Additionally, human capital, renewable energy, and economic growth are predictors of trade. The study therefore recommends South African policymakers to consider the importance of renewable energy, human capital development, and trade as a policy option to reduce ecological footprint and improve environmental quality.

Data-driven models have become powerful tools for structural and condition monitoring of engineering systems, particularly wind turbines. This paper presents a comparative analysis of common machine learning (ML) algorithms (artificial neural networks, linear regression, random forests, and gradient boosting) and a cointegration-based approach for fault detection using Supervisory Control and Data Acquisition (SCADA) data. While ML models offer early fault prediction, the cointegration method is simpler, requires less training data, and has lower computational costs. However, it is less effective for early detection. To balance these trade-offs, we propose a cascading monitoring framework, where the ML model provides long-term predictions (outer monitoring process) and the cointegration model offers short-term verification (inner monitoring process). The cointegration model serves to confirm anomalies flagged by the ML model. By combining both models in a cascade structure, the system reduces the risk of false alarms triggered by uncertainties in the ML model alone. Furthermore, the short-term cointegration-based prediction model helps pinpoint immediate risks and mitigate the issue of prolonged downtime. This combination enhances both accuracy and reliability, as demonstrated through testing on a five-year SCADA dataset from a commercial wind turbine with a known gearbox fault.

We propose a new and easy-to-use method for identifying cointegrated components of nonstationary time series, consisting of an eigenanalysis for a certain nonnegative definite matrix. Our setting is model-free, and we allow the integer-valued integration orders of the observable series to be unknown, and to possibly differ. Consistency of estimates of the cointegration space and cointegration rank is established both when the dimension of the observable time series is fixed as sample size increases, and when it diverges slowly. The proposed methodology is also extended and justified in a fractional setting. A Monte Carlo study of finite-sample performance, and a small empirical illustration, are reported. Supplementary materials for this article are available online.