This paper analyses a class of nonlinear time series models exhibiting long memory. These processes exhibit short memory fluctuations around a local mean (regime) which switches randomly such that the durations of the regimes follow a power law. We show that if a large number of independent copies of such a process are aggregated, the resulting processes are Gaussian, have a linear representation, and converge after normalisation to fractional Brownian motion. Alternatively, an aggregation scheme with Gaussian common components can yield the same result. However, a non-aggregated regime process is shown to converge to a Levy motion with infinite variance, suitably normalised, emphasising the fact that time aggregation alone fails to yield a FCLT. Two cases arise, a stationary case in which the partial sums of the process converge, and a nonstationary case in which the process itself converges, the Hurst coefficient falling in the ranges (1/2, 1) and (0, 1/2) respectively. We comment on the relevance of our results to the interpretation of the long memory phenomenon, and also report some simulations aimed to throw light on the problem of discriminating between the models in practice.

modi…ed least squares. The weak limit is derived of the sample covariance of a pair of fractionally integrated 1 processes, one I(dY) for 2 < dY < 1 2, and the other I(1 + dX) 1 for 2 < dX < 1

This paper considers alternative methods of testing cointegration in fractionally integrated processes, using the bootstrap. The special feature of the fractional case is the dependence of the asymptotic null distributions of conventional statistics on the fractional integration parameter. Such tests are said to be asymptotically non-pivotal, and conventional asymptotic tests are therefore not available. Bootstrap tests can be constructed, although these may be less reliable in small samples than in the case of asymptotically pivotal statistics. Bias correction techniques, including the double bootstrap of Beran (#988) and the fast double bootstrap of Davidson and MacKinnon (2000) are considered. The investigation focuses on the issues of (a) choice of statistic, (b) bias correction, and also (c) designing the simulation of the null hypothesis. The latter consideration is crucial for ensuring tests are both correctly sized and powerful. Three types of test are considered, all based on residuals from a putative cointegrating regression. Two are of the null hypothesis of non-cointegration; a conventional residual-based test using the Durbin-Watson statistic, and a test based on the F -statistic, as proposed in Davidson (200#). The third is the Shin (#994) residual-based test of the null hypothesis that cointegration exists. The tests are compared in Monte Carlo experiments whose main object is to throw light on the relative roles of issues (a), (b) and (c) in test performance. 1

Abstract. Since its first inception in the debate on the relationship between environment and growth in 1992, the Environmental Kuznets Curve has been subject of continuous and intense scrutiny. The literature can be roughly divided in two historical phases. Initially, after the seminal contributions, additional work aimed to extend the investigation to new pollutants and to verify the existence of an inverted-U shape as well as assessing the value of the turning point. The following phase focused instead on the robustness of the empirical relationship, particularly with respect to the omission of relevant explanatory variables other than GDP, alternative datasets, functional forms, and grouping of the countries examined. The most recent line of investigation criticizes the Environmental Kuznets Curve on more fundamental grounds, in that it stresses the lack of sufficient statistical testing of the empirical relationship and questions the very existence of the notion of Environmental Kuznets Curve. Attention is in particular drawn on the stationarity properties of the series involved – per capita emissions or concentrations and per capita GDP – and, in case of presence of unit roots, on the cointegration property that must be present for the Environmental Kuznets Curve to be a well-defined concept. Only at that point can the researcher ask whether the long-run relationship exhibits an inverted-U pattern. On the basis of panel integration and cointegration tests for sulphur, Stern (2002, 2003) and Perman and Stern (1999, 2003) have presented evidence and forcefully stated that the Environmental Kuznets Curve does not exist. In this paper we ask

The weak limit is derived of the sample covariance of a pair of fractionally integrated The processes, one I(dy) for - dy , and the other I(1 q- dx) for - dx .

In this paper we consider polynomial cointegrating relationships among stationary processes with long range dependence. We express the regression functions in terms of Hermite polynomials and we consider a form of spectral regression around frequency zero. For these estimates, we establish consistency by means of a more general result on continuously averaged estimates of the spectral density matrix at frequency zero.