It has been a challenging problem to determine the co-integrating rank in the vector error correction (VEC) model when its noise is a heavy-tailed random vector. This paper proposes an automated approach via adaptive shrinkage techniques to determine the co-integrating rank and estimate parameters simultaneously in the VEC model with unknown order p when its noises are i.i.d. heavy-tailed random vectors with tail index α ∈ (0, 2). It is showed that the estimated co-integrating rank and order p equal to the true rank and the true order p0, respectively, with probability trending to 1 as the sample size n → ∞, while other estimated parameters achieve the oracle property, that is, they have the same rate of convergence and the same limiting distribution as those of estimated parameters when the co-integrating rank and the true order p0 are known. This paper also proposes a data-driven procedure of selecting the tuning parameters. Simulation studies are carried to evaluate the performance of this procedure in finite samples. Our techniques are applied to explore the long-run and short-run behavior of prices of wheat, corn and wheat in USA. Our results may provide a new insight to the Lasso approach for both stationary and non-stationary heavy-tailed time series.