Likelihood Inference for a Fractionally Cointegrated Vector Autoregressive Model

Abstract

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 for which 0 X t is fractional of order d b:Our main technical contribution is the proof of consistency of the maximum likelihood estimators on the set 1=2 b d d 1 for any d 1 d 0 . 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 is asymptotically mixed Gaussian and estimators of the remaining parameters are asymptotically Gaussian. We also …nd the asymptotic distribution of the likelihood ratio test for cointegration rank, which is a functional of fractional Brownian motion of type II.

Crude oil and stock markets: Stability, instability, and bubbles

“…The reader is referred to Johansen's (1995) text for details on reduced rank regression estimation of a VECM.…”

“…This procedure is similar to one described by Johansen (1995) for implementing multiple restrictions on a cointegrating matrix.…”

“…We sort the equations with stock market prices first and the crude oil price last and then restrict the first r columns of A (corresponding to the N stock market prices) to be an identity matrix, an identification scheme suggested by Johansen (1995). This identifying restriction is natural, since the information criteria discussed above suggest that the last series (crude oil price) is cointegrated with each of the first r series (stock market prices), which are jointly cointegrated.…”

A fractionally cointegrated VAR model with deterministic trends and application to commodity futures markets

“…In our empirical analysis we use the FCVAR model of Johansen (2008) and Johansen and Nielsen (2010, 2012. This model is a generalization of Johansen's (1995) CVAR model to allow for fractional processes of order d that cointegrate to order d − b.…”

“…This model is a generalization of Johansen's (1995) CVAR model to allow for fractional processes of order d that cointegrate to order d − b. However, the development of the model in Johansen (2008) and Johansen and Nielsen (2010, 2012 does not accommodate drift in prices, i.e. deterministic linear trends in the observed variables.…”