Determinacy and Large-Scale Solutions in the Sequence Space

This paper studies the mathematical structure of sequence-space Jacobians – the derivative operators, mapping perfect-foresight paths of shocks to paths of aggregate outcomes, that underlie the increasingly popular “sequence-space” approach to solving macroeconomic models with rich heterogeneity. The authors prove that under general conditions these Jacobians are “quasi-Toeplitz operators”: a Toeplitz operator (whose matrix has constant diagonals, reflecting time-invariant responses to well-anticipated shocks) plus a compact correction that captures the extra effect of a shock’s not being anticipated before the initial date, and that vanishes for shocks announced sufficiently far in advance. They establish two structure theorems – that the Jacobian of any stationary heterogeneous-agent block is quasi-Toeplitz, and, more generally, that the solution operator of any expectational linear difference equation satisfying standard stability conditions is quasi-Toeplitz – implying that quasi-Toeplitz structure is close to universal in sequence-space macroeconomics. The authors exploit this structure in three ways. First, they derive a “winding number” test, building on Onatski (2006), that determines whether a sequence-space system has a unique solution, suffers from indeterminacy, or has no solution at all, by counting how many times a related complex-valued “symbol” function winds around the origin; they show this test agrees with the classic Blanchard-Kahn root-counting criterion when applicable, but extends to a much broader class of models, including heterogeneous-agent models with no finite-dimensional canonical form, and they show the test holds “generically” for quasi-Toeplitz operators, addressing a genericity critique previously raised by Sims (2007) against Onatski’s original test. Second, they show that quasi-Toeplitz structure can be exploited computationally to sharply reduce the cost of avoiding truncation error, either by using the (cheap-to-compute) Toeplitz part of a Jacobian’s inverse as a preconditioner for iterative solvers such as GMRES, or by representing the compact correction term with a low-rank approximation. Third, and most strikingly, they apply these methods to solve a heterogeneous-agent, multi-country fiscal policy model in which 190 countries trade according to a realistic, asymmetric bilateral trade network – a sequence-space system with roughly 190,000 unknowns at each of 1,000 time periods, far too large to solve by direct matrix inversion – in just 12 iterations and under three seconds on a laptop, versus an extrapolated multi-year cost for a comparable state-space solution method. Throughout, the paper’s applications center on stationary models (technically, Jacobians mapping into the space of square-summable sequences), explicitly excluding representative-agent models with a unit root in consumption, which the authors flag as a limitation and direction for future work.

Evan Majic | Macro Paper Warehouse

Determinacy and Large-Scale Solutions in the Sequence Space