Determinacy and Large-Scale Solutions in the Sequence Space

The authors identify three problems with solving macro models in the sequence space: “there does not exist a standard criterion for determinacy and existence of solutions”; sequence-space Jacobians are in principle infinite-dimensional and must be truncated to some large horizon T to avoid error, which can require T to be impractically large; and models with many simultaneous unknown sequences (such as network or trade models) can require solving an nT-by-nT system that is “prohibitively expensive” (Introduction, p. 2). Their proposed remedy for all three is a single structural fact about sequence-space Jacobians: that they are quasi-Toeplitz operators.

A quasi-Toeplitz operator J = T(j) + E is a Toeplitz operator T(j) – whose matrix entries Jt,s = j(t-s) depend only on the distance between dates – plus a compact correction E whose entries “go to zero as t, s approach infinity,” so that “a quasi-Toeplitz Jacobian looks Toeplitz for shocks that occur far from zero – in other words, shocks that are sufficiently well-anticipated” (Section 2.3, pp. 6-7). The authors illustrate this with the Calvo pricing model’s Jacobian of aggregate prices with respect to marginal cost: for shocks anticipated far in advance, columns of the Jacobian are simple shifted copies of each other (Toeplitz), but the response to a completely unanticipated shock at date 0 is smaller, “since sticky price-setters do not react prior to date 0” – precisely the compact correction E (Section 2.3, Figure 1, p. 3).

The paper proves two structure theorems: that “the Jacobian J of a stationary heterogeneous-agent block is quasi-Toeplitz” (Proposition 4), building on a bound showing the underlying fake-news matrix entries decay geometrically in t+s (Lemma 2), and, more generally, that the solution mapping of any expectational linear difference equation satisfying the usual conditions for a stable rational-expectations solution is also quasi-Toeplitz (Proposition 5) (Sections 2.5-2.6, pp. 15-19). Because the class of quasi-Toeplitz operators is closed under both addition and multiplication (Propositions 1-2), and virtually any macro model is built by composing heterogeneous-agent blocks with difference equations describing production, market clearing, and policy rules, the authors conclude that “the vast majority of sequence-space Jacobians in economic applications are quasi-Toeplitz” (Section 2, p. 4).

For a Toeplitz operator J = T(j), Proposition 6 shows that the winding number of its symbol j(z) – the number of times the graph of j(z) circles the origin counterclockwise as z traces the unit circle – exactly characterizes the solution to Jx = y: a winding number of zero means J is invertible (a unique solution exists), a negative winding number means indeterminacy of that dimension “but a solution always exists,” and a positive winding number means nonexistence of that dimension “but all solutions are unique” (Section 3.1, pp. 20-21). The authors show algebraically that, for finite-order expectational difference equations written in the canonical Blanchard and Kahn (1980) form, the winding number test is exactly equivalent to counting stable versus unstable roots of the equation’s characteristic polynomial, reproducing the Blanchard-Kahn criterion as a special case (previously shown by Onatski 2006) – but the winding number test is more general, since it “applies even when [the equation] has infinitely many leads and lags” and can be evaluated directly in the sequence space (Section 3.1, pp. 22-23).

Proposition 7 shows that for quasi-Toeplitz operators, nonexistence and indeterminacy can occur simultaneously, but they always satisfy wind(j) = nonex(J) - indet(J), so that a winding number of zero can be a “false positive” for uniqueness but never a “false negative” – if a unique solution truly exists, the winding number must be zero (Section 3.2, pp. 24-25). Proposition 8 further shows the simple Toeplitz-style criterion (winding number alone determines existence and uniqueness) holds “generically” – on an open and dense subset of quasi-Toeplitz operators – addressing the critique raised by Sims (2007) against the genericity claims in Onatski (2006): the authors show that “non-generic economic models do exist, [but] these models suffer from both nonexistence and indeterminacy,” a condition they say is straightforward to detect by separately testing for nonexistence (Section 3.2, pp. 24-25; Introduction, p. 5).

Applying the winding number test to a heterogeneous-agent model with a New Keynesian Phillips curve and a Taylor rule (interest rate responding to inflation with coefficient phi), the authors trace out the exact boundary between determinacy and indeterminacy in (phi, Phillips-curve-slope kappa) space, finding that the determinacy threshold approaches the traditional Taylor principle (phi = 1) as the Phillips curve slope grows large, but “the Taylor rule threshold phi is strictly lower than 1” for finite slopes – for instance, “at kappa = 0.0062… the determinacy threshold… is phi = 0.96” (Section 3.6, pp. 30-32, Figure 10). A practical computational advantage they highlight is that recalculating the winding number for different values of phi or kappa requires only trivial computation, since only one factor of the operator’s symbol changes with these parameters (Section 3.6, p. 31).

The paper offers two complementary strategies (Section 4): using the cheaply computed Toeplitz part of the inverse as a “preconditioner” for iterative methods – either simple iteration or the Generalized Minimal Residual Method (GMRES) – whose cost per iteration “grows quadratically in T” rather than the cubic cost of direct solution, allowing a much higher truncation horizon T to be used without becoming impractical; and representing the compact correction term itself as a low-rank factorization, which the authors show “can often be good with surprisingly low rank” – for instance, “when the correction is truncated to T = 2000, a rank-8 approximation delivers error indistinguishable from using the full T-by-T matrix” (Section 4.2-4.3, pp. 37-39, Table 1, Figure 12). In a fiscal-shock application, GMRES with a Toeplitz-only preconditioner converges to a residual below 10^-8 in a maximum of 4 iterations, falling to 3 with an augmented preconditioner, versus up to 10 iterations for simple iteration with the same preconditioners (Section 4.2, p. 38, Table 1).

The authors build an “international intertemporal Keynesian cross” model in which 190 countries trade according to a realistic, asymmetric bilateral trade matrix calibrated from the BACI trade database, implying a sequence-space system of size 190,000-by-190,000 at a truncation horizon of 1,000 quarters – a system the paper states is “far too large for a direct solution” (Section 5, pp. 40-42). Applying GMRES with the Toeplitz part of the inverse as preconditioner, the model is solved “in 12 iterations and fewer than 3 seconds on a laptop”; the paper shows this iterative approach scales far better than direct truncated-matrix solution (which “takes almost a minute” even for calibrations with fewer than 30 countries) and, by an even larger margin, than an extrapolated state-space (Reiter-style) solution, for which “models that take years to solve in the state space take seconds in the sequence space” (Section 5, pp. 41-42, Figure 13). Applied to a deficit-financed US tax cut, the calibrated model shows all countries initially booming, with the size of each country’s boom scaling with how tightly it trades with the US, and close US trade partners such as Canada and Mexico also experiencing a subsequent “hangover” as US taxes rise to pay down the debt (Section 5, p. 42, Figure 14).

The analysis is explicitly restricted to “stationary models” whose Jacobians map into the space of square-summable sequences (ell-2), which the authors note “excludes… the representative-agent model, which has a unit root in consumption, so that [the Jacobian] maps to non-square-summable sequences” (Section 2.1, p. 7). In the conclusion, the authors list better handling of such unit-root models, extending the computational methods to other large-scale settings such as spatial trade or migration models, and finding a more parsimonious (e.g., rational-function-based) representation of Jacobians to solve even larger systems, as their main priorities for future work (Section 6, p. 48).