The Tail Behavior of Maximum Likelihood Estimates of Cointegrating Coefficients in Error Correction Models
This paper derives exact ﬁnite sample distributions of maximum likelihood estimators of the cointegrating coeﬀicients in error correction models. The distributions are derived for the leading case where the variables in the system are independent random walks. But important aspects of the theory, in particular the tail behavior of the distributions, continue to apply when the system is cointegrated. The reduced rank regression estimator is shown to have a distribution with Cauchy-like tails and no ﬁnite moments of integer order. The maximum likelihood estimator of the coeﬀicients in the triangular system representation has matrix t -distribution tails with ﬁnite integer moments in order T - n + r where T is the sample size, n is the total number of variables in the system and r is the dimension of the cointegration space. These results help to explain simulation studies where extreme outliers are found to occur more frequently for the reduced rank regression estimator than for alternative asymptotically eﬀicient procedures that are based on the triangular representation.
Phillips, Peter C.B., "The Tail Behavior of Maximum Likelihood Estimates of Cointegrating Coefficients in Error Correction Models" (1991). Cowles Foundation Discussion Papers. 1242.