Estimation of fractional co-integration with unknown integration orders
This thesis presents different methods of estimating the co-integrating parameter in a bivariate fractionally co-integrated model. The proposed estimates enjoy optimal convergence rates and standard asymptotic distributions, yielding Wald test statistics with x2 null limit distribution. In the last few years increasing interest has developed in the issue of fractional co-integration, where both the observable series and the co-integrating error can be fractional processes, nesting the familiar situation where their respective orders are 1 and 0. These values have typically been assumed known. Chapter 1 is mainly devoted to reviewing this traditional prescription and motivate the relevance of fractional co-integration. In Chapter 2, we analyse a fully parametric model where the co-integrating gap, that is the difference between the integration order of the observables and that of the co-integrating error, is larger than 0.5. There, we show that our estimates share with the Gaussian maximum likelihood estimate the same limiting distribution, irrespective of whether the orders of integration are known or unknown, subject in the latter case to their estimation with adequate rates of convergence. Chapter 3, still in a parametric framework, proposes estimates of the parameter of co-integration in case the co-integrating gap is less than 0.5. Again, we cover both situations where the orders of integration are known and unknown. Our estimates are inefficient relative to the Gaussian maximum likelihood, but share with this estimate optimal rate of convergence and asymptotic normality, being computationally much more convenient. Chapter 4 concentrates on both situations described in the previous two chapters from a semiparametric perspective, that is without assuming knowledge of the parametric structure of the input series generating the fractional processes in the model. Finally, Chapter 5 describes a simple procedure of testing for the equality of orders of integration of different series. This is as essential step in any empirical work in order to asses for the presence of co-integration in a certain estimated model.