On the Relation Between Linearity-Generating Processes and Linear-Rational Models

TL;DR

This paper explores the relationship between linearity-generating processes and linear-rational models, showing their equivalence and highlighting the advantages of LR models in financial modeling.

Contribution

It establishes a bidirectional representation between LG processes and LR models, emphasizing the practical benefits of LR models in finance.

Findings

LR models can be easily specified and ensure nonnegative interest rates.

LR models align with long-term risk factorization.

Every LG process under the long forward measure can be represented as a lower dimensional LR model.

Abstract

We review the notion of a linearity-generating (LG) process introduced by Gabaix (2007) and relate LG processes to linear-rational (LR) models studied by Filipovic, Larsson, and Trolle (2017). We show that every LR model can be represented as an LG process and vice versa. We find that LR models have two basic properties which make them an important representation of LG processes. First, LR models can be easily specified and made consistent with nonnegative interest rates. Second, LR models go naturally with the long-term risk factorization due to Alvarez and Jermann (2005), Hansen and Scheinkman (2009), and Qin and Linetsky (2017). Every LG process under the long forward measure can be represented as a lower dimensional LR model.