Computing Sequence‐Space Jacobians along a Transition: Path-SSJ and the Accuracy of Linearizing at a Steady State

Sequence‐space Jacobians are computed at a steady state and increasingly used on long transitions. A steady‐state linearization makes two choices, where to take the prediction's level from (its anchor) and where to evaluate its Jacobian, and imposes two boundary conditions, the starting distribution and what households expect after the window. Path‐SSJ is the sequence‐space Jacobian evaluated on the nonlinear transition path, computed by re‐solving the household problem along it. At present the literature offers little guidance on how accurately a steady‐state linearization describes a transition, either the path or the responses to shocks along it. I vary all four on a smooth life‐cycle household block and a lumpy durables block, against nonlinear perfect‐foresight transitions with a small shock partway through. On the smooth block a steady‐state linearization given the transition's boundary conditions misses assets by under half a percent (5 percent if truncated at the window), and only marginal responses need the path Jacobian: the steady‐state one misses 5 to 20 percent of them. On the durables block the same boundary‐corrected linearization misses the adjustment share by 69 percent, because a stationary Jacobian has the timing of lumpy adjustment wrong. Anchoring on the nonlinear path removes seven tenths of that error and the path Jacobian beats the steady‐state one at some shock dates and loses at others; what remains, a fifth to a half of the marginal response, no anchor or Jacobian removes. This yields clear rules for practice: anchor on the nonlinear base path or, on a smooth block, give a steady‐state linearization the boundary conditions of the transition it describes, and verify against a nonlinear solve wherever a shock moves a discrete choice.