^{1} Department of Economics, Faculty of Social Sciences, Københavns Universitet^{2} Økonomisk Institut, Department of Economics, Faculty of Social Sciences, Københavns Universitet^{3} Queens University^{4} Økonomisk Institut, Department of Economics, Faculty of Social Sciences, Københavns Universitet

DOI:

10.3982/ECTA9299

Abstract:

We consider model based inference in a fractionally cointegrated (or cofractional) vector autoregressive model with a restricted constant term, ¿, based on the Gaussian likelihood conditional on initial values. The model nests the I(d) VAR model. We give conditions on the parameters such that the process X_{t} is 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, and no other fractionality order is possible. We define the statistical model by 0<b=d, but conduct inference when the true values satisfy b0¿1/2 and d0-b0<1/2 for which ß0'X_{t}+¿0' is (asymptotically) a mean zero stationary process and ¿0 can be estimated consistently. Our main technical contribution is the proof of consistency of the maximum likelihood estimators. To this end we prove weak convergence of the conditional likelihood as a continuous stochastic process in the parameters when errors are i.i.d. with suitable moment conditions and initial values are bounded. When the limit is deterministic this implies uniform convergence in probability of the conditional likelihood function. If the true value b0>1/2, we prove that the limit distribution of (ß',¿')' is mixed Gaussian and for the remaining parameters it is Gaussian. The limit distribution of the likelihood ratio test for cointegration rank is a functional of fractional Brownian motion of type II extended by u^{-(d0-b0)}. If b0<1/2 all limit distributions are Gaussian or chi-squared.

Type:

Journal article

Language:

English

Published in:

Econometrica, 2012, Vol 80, Issue 6, p. 2667-2732

Keywords:

Faculty of Social Sciences; Cofractional processes, cointegration rank, fractional cointegration, likelihood inference, vector autoregressive model