A Comparison of Johansen's,Bierens’ and the Subspace Algorithm Method for Cointegration Analysis*

The methods listed in the title are compared by means of a simulation study and a real world application. The aspects compared via simulations are the performance of the tests for the cointegrating rank and the quality of the estimated cointegrating space. The subspace algorithm method, formulated in the state space framework and thus applicable for vector autoregressive moving average (VARMA) processes, performs at least comparably to the Johansen method. Both the Johansen procedure and the subspace algorithm cointegration analysis perform significantly better than Bierens’ method. The real‐world application is an investigation of the long‐run properties of the one‐sector neoclassical growth model for Austria. The results do not fully support the implications of the model with respect to cointegration. Furthermore, the results differ greatly between the different methods. The amount of variability depends strongly upon the number of variables considered and huge differences occur for the full system with six variables. Therefore we conclude that the results of such applications with about five or six variables and 100 observations, which are typical in the applied literature, should possibly be interpreted with more caution than is commonly done.     

Environmental Kuznets Curves for Air Pollutant Emissions in Italy: Evidence from Environmental Accounts (NAMEA) Panel Data

This paper provides new empirical evidence on delinking and Environmental Kuznets Curves (EKC) for greenhouse gases and other air pollutant emissions in Italy. A panel dataset based on the Italian NAMEA (National Accounts Matrix including Environmental Accounts) for 1990–2001 is analysed. The highly disaggregated dataset (29 production branches, 12 years and nine air emissions) provides a large heterogeneity and can help to overcome the shortcomings of the usual approach to EKC based on cross-country data. Both value added and capital stock per employee are used as alternative drivers for analysing sectoral NAMEA emissions. Trade openness at the same sectoral level is also introduced among the covariates. We find mixed evidence supporting the EKC hypothesis. The analysis of NAMEA-based data shows that some of the pollutants such as two greenhouse gases (CO₂ and CH₄) and CO, produce inverted U-shaped curves with coherent within-range turning points. Other pollutants (SO_X, NO_X, PM₁₀) show a monotonic or even N-shaped relationship. Macro sectoral disaggregated analysis highlights that the aggregated outcome should hide some heterogeneity across different groups of production branches (industry, manufacturing only and services). Services tend to present an inverted N-shape in most cases. Manufacturing industry shows a mix of inverted U and N-shapes, depending on the emission considered. The same is true for industry (all industries, not only manufacturing): although a turning point has been experienced, N-shapes may lead to increased emissions with respect to very high levels of the economic driver. In general, EKC evidence is more pronounced for greenhouse gases. The results suggest that analysis at macro sector (whole industry, manufacturing only and services) can be the most promising approach to future research on EKC.     

A model of fractional cointegration, and tests for cointegration using the bootstrap

The paper proposes a framework for modelling cointegration in fractionally integrated processes, and considers methods for testing the existence of cointegrating relationships using the parametric bootstrap. In these procedures, ARFIMA models are fitted to the data, and the estimates used to simulate the null hypothesis of non-cointegration in a vector autoregressive modelling framework. The simulations are used to estimate p-values for alternative regression-based test statistics, including the F goodness-of-fit statistic, the Durbin–Watson statistic and estimates of the residual d. The bootstrap distributions are economical to compute, being conditioned on the actual sample values of all but the dependent variable in the regression. The procedures are easily adapted to test stronger null hypotheses, such as statistical independence. The tests are not in general asymptotically pivotal, but implemented by the bootstrap, are shown to be consistent against alternatives with both stationary and nonstationary cointegrating residuals. As an example, the tests are applied to the series for UK consumption and disposable income. The power properties of the tests are studied by simulations of artificial cointegrating relationships based on the sample data. The F test performs better in these experiments than the residual-based tests, although the Durbin–Watson in turn dominates the test based on the residual d.     

Determining the cointegrating rank in nonstationary fractional systems by the exact local Whittle approach

We propose to extend the cointegration rank determination procedure of Robinson and Yajima [2002. Determination of cointegrating rank in fractional systems. Journal of Econometrics 106, 217–242] to accommodate both (asymptotically) stationary and nonstationary fractionally integrated processes as the common stochastic trends and cointegrating errors by applying the exact local Whittle analysis of Shimotsu and Phillips [2005. Exact local Whittle estimation of fractional integration. Annals of Statistics 33, 1890–1933]. The proposed method estimates the cointegrating rank by examining the rank of the spectral density matrix of the d$d$th differenced process around the origin, where the fractional integration order, d$d$, is estimated by the exact local Whittle estimator. Similar to other semiparametric methods, the approach advocated here only requires information about the behavior of the spectral density matrix around the origin, but it relies on a choice of (multiple) bandwidth(s) and threshold parameters. It does not require estimating the cointegrating vector(s) and is easier to implement than regression-based approaches, but it only provides a consistent estimate of the cointegration rank, and formal tests of the cointegration rank or levels of confidence are not available except for the special case of no cointegration. We apply the proposed methodology to the analysis of exchange rate dynamics among a system of seven exchange rates. Contrary to both fractional and integer-based parametric approaches, which indicate at most one cointegrating relation, our results suggest three or possibly four cointegrating relations in the data.     

Trend stationarity in the I(2) cointegration model

A vector autoregressive model for I(2) processes which allows for trend-stationary components and restricts the deterministic part of the process to be at most linear is defined. A two-step statistical analysis of the model is derived. The joint test of I(1) and I(2) cointegrating ranks is shown to be asymptotically similar with respect to the drift terms and the asymptotic distribution is tabulated. The cointegrating parameters are shown to be mixed Gaussian and an application for UK monetary data illustrates the proposed analysis.     

Stochastic cointegration: estimation and inference

This paper considers the estimation of a stochastically cointegrating regression within the stochastic cointegration modelling framework introduced in McCabe et al. (Stochastic cointegration: testing, 2001). A stochastic cointegrating regression allows some or all of the variables to be conventionally or heteroscedastically integrated. This generalizes Hansen's (J. Econom. 54 (1992) 139) heteroscedastic cointegrating regression model, where the dependent variable is heteroscedastically integrated, but all the regressor variables are restricted to being conventionally integrated. In contrast to conventional and heteroscedastic cointegrating regression, ordinary least-squares (OLS) estimation is shown to be inconsistent, in general, in a stochastically cointegrating regression. As a solution, a new instrumental variables (IVs) estimator is proposed and is shown to be consistent. Under a suitable exogeneity assumption, standard asymptotic inference on the stochastic cointegrating vector can be carried out based on the IV estimator. The finite sample properties of the test statistics, including their robustness to the exogeneity assumption, are examined by simulation.     

Testing for r versus r−1 cointegrating vectors

A parametric test for r versus r−1 cointegrating vectors is developed. The test exploits the fact that in a system of n I(1) variates the rth principal component is I(0) under the null but I(1) under the alternative. The statistic is parametric, is constructed using simple regression methods applied to principal components, follows a standard χ² distribution and does not require normalisation restrictions on the cointegrating vectors. A Monte Carlo investigation indicates that providing the lag length in the pre-whitening procedure is chosen by means of nested significance tests, the test has good size and power properties in small samples.     

Does financial development stimulate environmental sustainability? Evidence from a panel study of 115 countries

This paper investigates the moderation effect of financial development (FD) on the environmental Kuznets curve (EKC). Using comprehensive panel data of 115 countries spanning the 1990–2016 period and dividing the countries into different income groups, the researcher found systematic differences in the relationship between significant indicators of environmental degradation and economic growth. More specifically, the interaction effect of FD on the relationship between gross domestic product (GDP) growth and CO₂ emissions are positive; nevertheless, the interaction effect of FD on the relationship between GDP squared and CO₂ emissions become negative for all income groups and therefore supports the EKC. Moreover, the interaction effect of FD is negative on the relationship of GDP with both CH₄ and PM_2.5 emissions in the middle‐income groups, whereas the interaction effect of FD on the relationship between GDP squared and PM_2.5 emissions are negative for the high‐income group. Our results suggest that FD can reduce environmental degradation without adversely impacting on growth.     

Bootstrapping cointegrating regressions

In this paper, we consider bootstrapping cointegrating regressions. It is shown that the method of bootstrap, if properly implemented, generally yields consistent estimators and test statistics for cointegrating regressions. For the cointegrating regression models driven by general linear processes, we employ the sieve bootstrap based on the approximated finite-order vector autoregressions for the regression errors and the first differences of the regressors. In particular, we establish the bootstrap consistency for OLS method. The bootstrap method can thus be used to correct for the finite sample bias of the OLS estimator and to approximate the asymptotic critical values of the OLS-based test statistics in general cointegrating regressions. The bootstrap OLS procedure, however, is not efficient. For the efficient estimation and hypothesis testing, we consider the procedure proposed by Saikkonen [1991. Asymptotically efficient estimation of cointegration regressions. Econometric Theory 7, 1–21] and Stock and Watson [1993. A simple estimator of cointegrating vectors in higher order integrating systems. Econometrica 61, 783–820] relying on the regression augmented with the leads and lags of differenced regressors. The bootstrap versions of their procedures are shown to be consistent, and can be used to do asymptotically valid inferences. A Monte Carlo study is conducted to investigate the finite sample performances of the proposed bootstrap methods.     

A drunk and her dog: a spurious relation? Cointegration tests as instruments to detect spurious correlations between integrated time series

A significant correlation between integrated time series does not necessarily imply a meaningful relation. The relation can also be meaningless, i.e. spurious. Cointegration is sometimes illustrated by the metaphor of ‘a drunk and her dog’. The relation between integrated processes is meaningful, if they are cointegrated. To prevent spurious correlations, integrated series are usually transformed. This implies a loss of information. In case of cointegration, these transformations are no longer necessary. Moreover, it can be shown that cointegration tests are instruments to detect spurious correlations between integrated time series. This paper compares the Dickey–Fuller and the Johansen cointegration test. By means of Monte Carlo simulations, we found that these cointegration tests are a much more accurate alternative for the identification of spurious relations compared to the rather imprecise method of utilizing the R ²-and DW-statistics recommended by some authors. Furthermore, we demonstrate that cointegration techniques are precise methods of distinguishing between spurious and meaningful relations even if the dependency between the processes is very low. Using these tests, the researcher is not in danger of either neglecting a small but meaningful relation or regarding a relation as meaningful which is actually spurious.     

On the structure of moving average processes

This paper uses Hilbert space methods to develop a rigorous proof that the sum of two uncorrelated moving average processes of order q₁ and q₂ is an MA process of order q ≦ max (q₁, q₂). The methods establish the existence of suitable random shocks for the summed process, they illuminate relationships between the coefficients of such processes and their random shocks, and they provide means for proving that the random shocks of the summed processes are normal when the shocks of the underlying processes are normal. The role of the Wold decomposition is examined in terms of multiple representations of an MA process.     

Low-frequency robust cointegration testing

Standard inference in cointegrating models is fragile because it relies on an assumption of an I(1)$I\left(1\right)$ model for the common stochastic trends, which may not accurately describe the data’s persistence. This paper considers low-frequency tests about cointegrating vectors under a range of restrictions on the common stochastic trends. We quantify how much power can potentially be gained by exploiting correct restrictions, as well as the magnitude of size distortions if such restrictions are imposed erroneously. A simple test motivated by the analysis in Wright (2000) is developed and shown to be approximately optimal for inference about a single cointegrating vector in the unrestricted stochastic trend model.     

Asymptotic properties of the efficient estimators for cointegrating regression models with serially dependent errors

In this paper, we analytically investigate three efficient estimators for cointegrating regression models: Phillips and Hansen’s [Phillips, P.C.B., Hansen, B.E., 1990. Statistical inference in instrumental variables regression with I(1) processes. Review of Economic Studies 57, 99–125] fully modified OLS estimator, Park’s [Park, J.Y., 1992. Canonical cointegrating regressions. Econometrica 60, 119–143] canonical cointegrating regression estimator, and Saikkonen’s [Saikkonen, P., 1991. Asymptotically efficient estimation of cointegration regressions. Econometric Theory 7, 1–21] dynamic OLS estimator. We consider the case where the regression errors are moderately serially correlated and the AR coefficient in the regression errors approaches 1 at a rate slower than 1/T$1/T$, where T$T$ represents the sample size. We derive the limiting distributions of the efficient estimators under this system and find that they depend on the approaching rate of the AR coefficient. If the rate is slow enough, efficiency is established for the three estimators; however, if the approaching rate is relatively faster, the estimators will have the same limiting distribution as the OLS estimator. For the intermediate case, the second-order bias of the OLS estimator is partially eliminated by the efficient methods. This result explains why, in finite samples, the effect of the efficient methods diminishes as the serial correlation in the regression errors becomes stronger. We also propose to modify the existing efficient estimators in order to eliminate the second-order bias, which possibly remains in the efficient estimators. Using Monte Carlo simulations, we demonstrate that our modification is effective when the regression errors are moderately serially correlated and the simultaneous correlation is relatively strong.     

Asymptotic distribution of the cointegrating vector estimator in error correction models with conditional heteroskedasticity

This paper explores the asymptotic distribution of the cointegrating vector estimator in error correction models with conditionally heteroskedastic errors. Asymptotic properties of the maximum likelihood estimator (MLE) of the cointegrating vector, which estimates the cointegrating vector and the multivariate GARCH process jointly, are provided. The MLE of the cointegrating vector follows mixture normal, and its asymptotic distribution depends on the conditional heteroskedasticity and the kurtosis of standardized innovations. The reduced rank regression (RRR) estimator and the regression-based cointegrating vector estimators do not consider conditional heteroskedasticity, and thus the efficiency gain of the MLE emerges as the magnitude of conditional heteroskedasticity increases. The simulation results indicate that the relative power of the t-statistics based on the MLE improves significantly as the GARCH effect increases.     

A fractional cointegration var analysis of exchange rate dynamics

In their seminal work, Baillie and Bollerslev (1994) carried out an analysis of deviations from the cointegrating relationship of seven important exchange rates. They suggested that the exchange rate series possess long memory and therefore such processes could be well described as fractionally integrated processes. Hence, the influence of shocks to the equilibrium exchange rates may only vanish at very long horizons. In this work we analyze the cointegrating structure of five exchange rates to the US dollar, namely the British pound, the Euro, the Swedish Krona, the Canadian Dollar and the Swiss Franc. The series possess long memory and we show that they can be modeled through fractional integration. In fact, standard cointegration is rejected with the more traditional Johansen CVAR methodology. By using the recently introduced Fractionally Cointegrated VAR by Johansen and Nielsen (2012) we provide a cointegrating relationship taking into account fractional integration.     

Does Financial Development Affect Environmental Degradation? Evidence from the OECD Countries

In this study, building a simple model that incorporates static and dynamic elements, the relationship of financial development and economic growth to environmental degradation is investigated together with the validation of the Environmental Kuznets Curve (EKC) hypothesis. Our analysis is based on an unbalanced panel data set covering the OECD countries over the period 1970–2014. Our approach thoroughly accounts for the presence of cross‐sectional dependence between the sample variables and utilizes second generation panel unit root tests in order to investigate possible cointegration relationships. The empirical findings do indicate that local (NO_x per capita emissions) and global (CO₂ per capita emissions) pollutants redefine the EKC hypothesis when we account for the presence of financial development indicators. Specifically, in the case of global pollution an N‐shape relationship is evident in both static and dynamic frameworks, with a very slow adjustment. Lastly, our study calls for a strengthening of the effectiveness of environmental degradation policies by ensuring sustainability of the OECD banking system in order to drastically reduce emissions. Copyright © 2017 John Wiley & Sons, Ltd and ERP Environment     

TEN THINGS WE SHOULD KNOW ABOUT TIME SERIES

Abstract Time series data affect many aspects of our lives. This paper highlights 10 things we should all know about time series, namely, a good working knowledge of econometrics and statistics, an awareness of measurement errors, testing for zero frequency, seasonal and periodic unit roots, analysing fractionally integrated and long memory processes, estimating VARFIMA models, using and interpreting cointegrating models carefully, choosing sensibly among univariate conditional, stochastic and realized volatility models, not confusing thresholds, asymmetry and leverage, not underestimating the complexity of multivariate volatility models, and thinking carefully about forecasting models and expertise.     

Singular value decomposition in cointegration analysis: a note regarding the difference stationary series

Many methods and techniques have been developed gradually to compute cointegration vectors. We present here a comparatively simple method for computing the matrix of cointegrating vectors, by applying singular value decomposition. With this method, one can easily accommodate in the cointegrating vectors any deterministic factors, such as a dummy, apart from the constant term and the trend. Besides the errors corresponding to the finally selected cointegrating vector have the property of minimum variance. It is noted that this procedure is not mentioned in the relevant literature. Additionally, a sequential technique is introduced, for determining the order of integration for a given series. With this procedure one can directly detect whether the differencing process produces a stationary series or not, since it seems to be a common belief that differencing a variable (one or more times) we will always get a stationary series [Harris, R.: Using Cointegration Analysis in Econometric Modelling. Prentice Hall, London (1995)]. It will be seen that this is not necessarily the case.     

A Wald test for the cointegration rank in nonstationary fractional systems

This paper develops new methods for determining the cointegration rank in a nonstationary fractionally integrated system, extending univariate optimal methods for testing the degree of integration. We propose a simple Wald test based on the singular value decomposition of the unrestricted estimate of the long run multiplier matrix. When the “strength” of the cointegrating relationship is less than 1/2, the test statistic has a standard asymptotic distribution, like Lagrange Multiplier tests exploiting local properties. We consider the behavior of our test under estimation of short run parameters and local alternatives. We compare our procedure with other cointegration tests based on different principles and find that the new method has better properties in a range of situations by using information on the alternative obtained through a preliminary estimate of the cointegration strength.     

EKC-type transitions and environmental policy under pollutant uncertainty and cost irreversibility

Previous studies have suggested that some pollutant levels first increases due to the economic growth and then start decreasing, the pattern being called the “environmental Kuznets curve” (EKC). We examine EKC-type transitions of pollutant levels not with respect to economic growth but more generally in time. Assuming that each policy maker optimally executes the two switching options of regulation and unregulation for pollution, the switching dynamics of environmental policy can be described by an alternating renewal process. It is shown that the double Laplace transform of transition density of a pollutant level can be obtained by a novel application of renewal theory. The expected level of overall pollutants is then calculated numerically and found to exhibit either a Λ‐shaped or an N-shaped pattern in time. Our results present a simple explanation for the EKC-type transitions of pollutant levels within a real options framework.