Some limit properties for information based model selection criteria are given in the context of unit root evaluation and various assumptions about initial conditions. Allowing for a nonparametric short memory component, standard information criteria are shown to be weakly consistent for a unit root provided the penalty coeﬀicient C n → ∞ and C n /n → 0 as n → ∞. Strong consistency holds when C n /(log log n ) 3 → ∞ under conventional assumptions on initial conditions and under a slightly stronger condition when initial conditions are inﬁnitely distant in the unit root model. The limit distribution of the AIC criterion is obtained.
Phillips, Peter C.B., "Unit Root Model Selection" (2008). Cowles Foundation Discussion Papers. 1957.