TY - UNPB

T1 - Likelihood Inference for a Fractionally Cointegrated Vector Autoregressive Model

AU - Johansen, Søren

AU - Nielsen, Morten Ørregaard

N1 - JEL classification: C32

PY - 2010

Y1 - 2010

N2 - We consider model based inference in a fractionally cointegrated (or cofractional) vector autoregressive model based on the conditional Gaussian likelihood. The model allows the process X(t) to be fractional of order d and cofractional of order d-b; that is, there exist vectors ß for which ß'X(t) is fractional of order d-b. The parameters d and b satisfy either d=b=1/2, d=b=1/2, or d=d0=b=1/2. Our main technical contribution is the proof of consistency of the maximum likelihood estimators on the set 1/2=b=d=d1 for any d1=d0. To this end, we consider the conditional likelihood as a stochastic process in the parameters, and prove that it converges in distribution when errors are i.i.d. with suitable moment conditions and initial values are bounded. We then prove that the estimator of ß is asymptotically mixed Gaussian and estimators of the remaining parameters are asymptotically Gaussian. We also find the asymptotic distribution of the likelihood ratio test for cointegration rank, which is a functional of fractional Brownian motion of type II.

AB - We consider model based inference in a fractionally cointegrated (or cofractional) vector autoregressive model based on the conditional Gaussian likelihood. The model allows the process X(t) to be fractional of order d and cofractional of order d-b; that is, there exist vectors ß for which ß'X(t) is fractional of order d-b. The parameters d and b satisfy either d=b=1/2, d=b=1/2, or d=d0=b=1/2. Our main technical contribution is the proof of consistency of the maximum likelihood estimators on the set 1/2=b=d=d1 for any d1=d0. To this end, we consider the conditional likelihood as a stochastic process in the parameters, and prove that it converges in distribution when errors are i.i.d. with suitable moment conditions and initial values are bounded. We then prove that the estimator of ß is asymptotically mixed Gaussian and estimators of the remaining parameters are asymptotically Gaussian. We also find the asymptotic distribution of the likelihood ratio test for cointegration rank, which is a functional of fractional Brownian motion of type II.

KW - Faculty of Social Sciences

KW - cofractional processes

KW - cointegration rank

KW - fractional cointegration

KW - likelihood inference

KW - vector autoregressive model

M3 - Working paper

BT - Likelihood Inference for a Fractionally Cointegrated Vector Autoregressive Model

PB - Department of Economics, University of Copenhagen

ER -