A note on the limit theory of a Dickey–Fuller unit root test with heavy tailed innovations

We are occupied with the limit theory of the OLSE and of a subsequent Dickey–Fuller test when the unit root process has heavy tailed and dependent innovations that do not possess moments of order α for some α∈0,2. The innovation process has the form of a “martingale-type” transform constructed as a pointwise product between an iid sequence in the domain of attraction of an α stable distribution with a non existing α moment, for some α∈0,2, and a positive scaling mixing sequence that has a slowly varying at infinity truncated α moment. We derive a functional limit theorem with complex rates and limits that depend on Levy α-stable processes. The OLSE remains superconsistent with rate n, and the limiting distribution is a functional of the previous process. When α=2 we recover the standard Dickey–Fuller distribution.