An Extension of Cointegration to Fractional Autoregressive Processes

TY - UNPB

T1 - An Extension of Cointegration to Fractional Autoregressive Processes

AU - Johansen, Søren

N1 - JEL classification: C32

PY - 2010

Y1 - 2010

N2 - This paper contains an overview of some recent results on the statistical analysis of cofractional processes, see Johansen and Nielsen (2010b). We first give an brief summary of the analysis of cointegration in the vector autoregressive model and then show how this can be extended to fractional processes. The model allows the process X_{t} to be fractional of order d and cofractional of order d-b=0; that is, there exist vectors ß for which ß'X_{t} is fractional of order d-b. We analyse the Gaussian likelihood function to derive estimators and test statistics. The asymptotic properties are derived without the Gaussian assumption, under suitable moment conditions. We assume that the initial values are bounded and show that they do not influence the asymptotic analysis. The estimator of ß is asymptotically mixed Gaussian and estimators of the remaining parameters are asymptotically Gaussian. The asymptotic distribution of the likelihood ratio test for cointegration rank is a functional of fractional Brownian motion.

AB - This paper contains an overview of some recent results on the statistical analysis of cofractional processes, see Johansen and Nielsen (2010b). We first give an brief summary of the analysis of cointegration in the vector autoregressive model and then show how this can be extended to fractional processes. The model allows the process X_{t} to be fractional of order d and cofractional of order d-b=0; that is, there exist vectors ß for which ß'X_{t} is fractional of order d-b. We analyse the Gaussian likelihood function to derive estimators and test statistics. The asymptotic properties are derived without the Gaussian assumption, under suitable moment conditions. We assume that the initial values are bounded and show that they do not influence the asymptotic analysis. The estimator of ß is asymptotically mixed Gaussian and estimators of the remaining parameters are asymptotically Gaussian. The asymptotic distribution of the likelihood ratio test for cointegration rank is a functional of fractional Brownian motion.

M3 - Working paper

BT - An Extension of Cointegration to Fractional Autoregressive Processes

PB - Department of Economics, University of Copenhagen

ER -