On the relation between linearity‐generating processes and linear‐rational models

We review the notion of a linearity-generating (LG) process introduced by Gabaix (2009) and relate LG processes to linear-rational (LR) models studied in Filipovic, Larsson, and Trolle (2014). We show that every LG process can be represented as an LR model of the same dimension. More importantly, we identify those (m 1)-dimensional LG processes that can be represented as m-dimensional LR models. We show that these are the only LG processes that are stationary and mean-reverting after exponential scaling. We highlight the ease with which LR models can be specified and be made consistent with nonnegative interest rates. We also show that LR models fit naturally into the state price density factorization due to Alvarez and Jermann (2005) and Hansen and Scheinkman (2009).