Explore the vast range of titles available.

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.

This article discusses identification problems in the fractionally cointegrated system of Johansen and Johansen and Nielsen. It is shown that several equivalent reparametrizations of the model associated with different fractional integration and cointegration parameters may exist for any choice of the lag-length when the true cointegration rank is known. The properties of these multiple nonidentified models are studied and a necessary and sufficient condition for the identification of the fractional parameters of the system is provided. The condition is named F(d). This is a generalization of the well-known I(1) condition to the fractional case. Imposing a proper restriction on the fractional integration parameter, d, is sufficient to guarantee identification of all model parameters and the validity of the F(d) condition. The article also illustrates the indeterminacy between the cointegration rank and the lag-length. It is also proved that the model with rank zero and lags may be an equivalent reparameterization of the model with full rank and k — 1 lags. This precludes the possibility to test for the cointegration rank unless a proper restriction on the fractional integration parameter is imposed.

This paper analyses comovement in housing prices across the Euro area. We use techniques based on the concepts of fractional integration and cointegration. Our results indicate that all the individual log-real price indices display orders of integration which are above one, implying long memory in their corresponding growth rates. Further, looking at the cointegration relationships, we observe that the series for the Euro area is cointegrated with those of Belgium, Germany and France. Focusing on the individual countries, we find cointegration relationships between Belgium and Spain, Belgium and the Netherlands, Germany and Spain, Germany and Ireland, France and Spain, and Ireland and the Netherlands. Other bilateral cointegration relationships can either clearly be rejected or the results are ambiguous. Finally, prices in Germany seem to move in the opposite direction from other countries, which may be related to capital flows associated with current account imbalances.

This paper examines the relationship between the logarithms of carbon dioxide (CO2) emissions and real Gross Domestic Product (GDP) in China by applying fractional integration and cointegration methods. These are more general than the standard methods based on the dichotomy between stationary and non-stationary series, allow for a much wider variety of dynamic processes, and provide information about the persistence and long-memory properties of the series and thus on whether or not the effects of shocks are long-lived. The univariate results indicate that the two series are highly persistent, their orders of integration being around 2, whilst the cointegration tests (using both standard and fractional techniques) imply that there exists a long-run equilibrium relationship between the two variables in first differences, i.e. their growth rates are linked together in the long run. This suggests the need for environmental policies aimed at reducing emissions during periods of economic growth.

Fractional cointegration in time series data has been explored by several authors, but panel data applications have been largely neglected. A previous study of ours discovered that the Chen and Hurvich fractional cointegration test for time series was fairly robust to a moderate degree of heterogeneity across sections of the six tests considered. Therefore, this paper advances a customized version of the Chen and Hurvich methodology to detect cointegrating connections in panels with unobserved fixed effects. Specifically, we develop a test statistic that accommodates variation in the long-term cointegrating vectors and fractional cointegration parameters across observational units. The behavior of our proposed test is examined through extensive Monte Carlo experiments under various data-generating processes and circumstances. The findings reveal that our modified test performs quite well comparatively and can successfully identify fractional cointegrating relationships in panels, even in the presence of idiosyncratic disturbances unique to each cross-sectional unit. Furthermore, the proposed modified test procedure established the presence of long-run equilibrium between the exchange rate and labor wage of 36 countries’ agricultural markets.

Purpose This paper aims to analyse the persistence of the S&P500 and DAX 30 stock indices as well as of the Fed’s Effective Federal Funds rate and of the European Central Bank’s Marginal Lending Facility rate, and the long-run linkages between stock prices and interest rates in the USA and Europe, respectively. Design/methodology/approach The methodology is based on the concepts of fractional integration and cointegration. Findings Using monthly data from January 1999 to December 2022, the results can be summarised as follows. All series examined are non-stationary: stock prices are found to be I(1) while interest rates display orders of integration substantially above 1, which implies a rejection of the hypothesis of mean reversion in all cases examined. Originality/value This paper uses an appropriate econometric framework to obtain new, reliable empirical evidence. All four series are highly persistent, and mean reversion does not occur in any single case. Moreover, the fractional cointegration analysis suggests that stock prices and interest rates are not linked in the long run.

Fractional cointegration has been extensively examined in time series analysis, but its extension to heterogeneous panel data with unobserved heterogeneity and cross-sectional dependence remains underdeveloped. This paper develops a robust framework for testing fractional cointegration in heterogeneous panel data, where unobserved heterogeneity, cross-sectional dependence, and persistent shocks complicate traditional approaches. We propose the Bayesian Tapered Narrowband Least Squares (BTNBLS) estimator, which addresses three critical challenges: (1) spectral leakage in long-memory processes, mitigated via tapered periodograms; (2) precision loss in fractional parameter estimation, resolved through narrowband least squares; and (3) unobserved heterogeneity in cointegrating vectors (θi) and memory parameters (ν,δ), modeled via hierarchical Bayesian priors. Monte Carlo simulations demonstrate that BTNBLS outperforms conventional estimators (OLS, NBLS, TNBLS), achieving minimal bias (0.041–0.256), near-nominal coverage probabilities (0.87–0.94), and robust control of Type 1 errors (0.01–0.07) under high cross-sectional dependence (ρ=0.8), while the Bayesian Chen–Hurvich test attains near-perfect power (up to 1.00) in finite samples. Applied to Purchasing Power Parity (PPP) in 18 fragile Sub-Saharan African economies, BTNBLS reveals statistically significant fractional cointegration between exchange rates and food price ratios in 15 countries (p<0.05), with a pooled estimate (θ^=0.33, p<0.001) indicating moderate but resilient long-run equilibrium adjustment. These results underscore the importance of Bayesian shrinkage and spectral tapering in panel cointegration analysis, offering policymakers a reliable tool to assess persistence of shocks in institutionally fragmented markets.

In this paper, we introduce test procedures for no fractional cointegration against possible breaks to a fractional cointegrating relationship in a segment of the data. We base the proposed tests on the supremum of the Hassler and Breitung (Econom Theor 22(6):1091–1111, 2006) test statistic for no cointegration over possible breakpoints in the long-run equilibrium. We show that the new tests correctly standardized converge to the supremum of a Chi-squared distribution and that this convergence is uniform. An in-depth Monte Carlo analysis provides results on the finite sample performance of our tests. We then use the new procedures to investigate whether there was a dissolution of fractional cointegrating relationships between the yields of government bonds of eleven EMU countries (Spain, Italy, Portugal, Ireland, Greece, Belgium, Austria, Finland, the Netherlands, Germany and France) as a consequence of the European debt crisis and to understand the degree of interdependence of lending rates to non-financial corporations across these eleven countries.

This paper estimates the seasonal and zonal CO2 marginal emissions factors (MEFs) from electricity production in the Italian electricity system. The inclusion of the zonal configuration of the Italian wholesale power market leads to a complete measurement of marginal emission factors which takes into account the heterogeneous distribution of RES power plants, their penetration rate and their variability within the zonal power generation mix. This article relies on a flexible econometric approach that includes the fractional cointegration methodology to incorporate the typical features of long-memory processes into the estimation of MEFs. We find high variability in annual MEFs estimated at the zonal level. Sardinia reports the highest MEF (0.7189 tCO2/MWh), followed by the Center South (0.7022 tCO2/MWh), the Center North (0.4236 tCO2/MWh), the North (0.2018 tCO2/MWh) and Sicily (0.146 tCO2/MWh). The seasonal analysis also shows a large variability of MEFs in each zone across time. The heterogeneity of results leads us to recommend that policymakers consider the zonal configuration of the power market and the large seasonal variability related to carbon emissions and electricity generation when designing incentives for renewable energy sources expansion and for achieving emission reduction targets.

This paper is concerned with a dependence analysis of returns, return volatility and trading volume for five companies listed on the Vienna Stock Exchange and five from the Warsaw Stock Exchange. Taking into account high frequency data for these companies, tests based on a comparison of Bernstein copula densities using the Hellinger distance were conducted. The paper presents some patterns of causal and other relationships between stock returns, realized volatility and expected and unexpected trading volume. There is a linear causality running from realized volatility to expected trading volume, and a lack of nonlinear dependence in the opposite direction. The authors detected strong linear and nonlinear causality from stock returns to expected trading volume. They did not find causality running in the opposite direction. In addition, the existence of fractional cointegration was examined. Despite the equality of the long memory parameters of realized volatility and trading volumes, they do not move together in the long term horizon.