Journal of Economic Dynamics and Control | Macro Paper Warehouse

A reconsideration of money growth rules

This 2022 Journal of Economic Dynamics & Control paper by Michael Belongia and Peter Ireland asks whether a policy rule that steers the growth rate of money, rather than the short-term nominal interest rate, could have delivered macroeconomic stabilization comparable to the Federal Reserve’s actual interest-rate policy — including through the 2009:1-2015:4 zero-lower-bound (ZLB) episode. They build a New Keynesian DSGE model with real Divisia M2 balances entered directly in household utility (with quadratic adjustment costs), habit formation in consumption, Rotemberg price-adjustment costs with backward-looking indexation, and five structural shocks (preference, productivity growth, money demand, cost-push, and monetary policy), and estimate it by Bayesian methods on quarterly U.S. data from 1983:1-2019:4, using the Kulish et al. (2017) piecewise-linear algorithm to handle the ZLB period (during which the funds rate is dropped from the observables and time-varying expected ZLB durations, informed by survey forecasts, are estimated as parameters). Comparing log marginal likelihoods, they find that augmenting the estimated Taylor rule with a contemporaneous money growth term raises the fit only slightly (2412.7 to 2413.3, with the money-growth response coefficient’s posterior mode a modest 0.0605), while adding lagged money growth actually lowers it — money growth adds a little information but not much on top of the interest-rate rule. Variance decompositions show monetary-policy shocks account for only 0.7% of output-growth variance and 5.6% of output-gap variance, with productivity, preference, and cost-push shocks dominating instead, and the model attributes the Great Recession’s declines in inflation and interest rates mainly to adverse preference and productivity shocks. The paper’s central counterfactual result is that a flexible money growth rule of the form mu-hat_t = mu-hat_{t-1} - 0.125 x-hat_{t-1} (found by grid search to minimize macroeconomic volatility and ZLB-equivalent duration) generates standard deviations of output growth (2.5511), inflation (1.1226), and the output gap (0.7095) that closely approximate those implied by the estimated Taylor rule (2.3762, 0.9774, and 0.5481 respectively), while producing a much shorter and milder episode of negative implied interest rates than the actual seven-year ZLB period. By contrast, a strictly constant money growth rule (all feedback coefficients set to zero) sharply amplifies volatility, raising the output-growth standard deviation to 3.6423 (more than 50% larger than under the Taylor rule) and the inflation standard deviation to 1.8871, confirming earlier findings by Ireland (2000), Collard and Dellas (2005), and Galí (2015) that fixed money growth performs poorly. The authors conclude that a flexible, output-gap-responsive money growth rule — not a rigid quantity-theoretic constant-growth rule — belongs on the list of policy alternatives capable of avoiding the ZLB while matching Taylor-rule-level stabilization performance.

DSGE pileups

This 2017 Journal of Economic Dynamics & Control paper by Stephen D. Morris is a methodological contribution explaining why maximum-likelihood or Bayesian estimates of DSGE structural parameters often “pile up” – concentrate at or near a boundary of the theoretically admissible parameter space, or otherwise depart from the asymptotic normal distribution – even when the model is correctly specified. Morris argues the underlying cause is weak identification of structural parameters by the reduced-form data, operating through three specific channels in DSGE models: observational equivalence (the map from structural parameters theta to reduced-form parameters pi sends multiple theta values, including economically implausible ones, to the same pi), functional boundaries created by the theta-to-pi mapping itself (rather than by the stated theoretical restrictions), and multiplicity of stable rational-expectations solutions. To study these mechanisms formally, the paper builds on the author’s own prior result (Morris 2016b) that, under stated regularity, stationarity, and left-invertibility assumptions, the ABCD state-space representation of a DSGE model has an exact finite-order VARMA(p, p-1) representation, and proposes a minimum chi-square estimator (MCSE) – asymptotically equivalent to MLE but numerically simpler and bootstrappable – together with an F-test and an overidentification chi-squared test to diagnose pileups. The paper contains no empirical application to real data; all results come from Monte Carlo experiments (T=225 quarters, N=1000 replications, mimicking a typical postwar quarterly sample) on three small illustrative models plus the Smets-Wouters (2007) medium-scale model. In the Brock-Mirman stochastic growth calibration, the discount-factor estimate beta-hat piles up at its upper boundary of 1 even when the true value is beta_0 = 0.99, with Kolmogorov-Smirnov tests rejecting Gaussianity at the 1%, 5%, and 10% levels; in the Krause-Lubik search-and-matching calibration, the match-elasticity parameter xi piles up near its lower boundary of 0.1 with a long left tail, driven by a “natural” functional boundary rather than the stated theoretical restriction; in the An-Schorfheide New Keynesian model, the CRRA coefficient tau and shock-persistence parameter rho_z show multimodal sampling distributions traceable to an observationally equivalent solution point with an economically infeasible tau of -45.4, a problem that Jeffreys priors only partly resolve and that informative conjugate priors can worsen for other parameters (Morris calls informative priors “a double-edged sword”); and in the Smets-Wouters model (41 structural parameters, VARMA(3,2) representation), all three pileup types recur simultaneously – a boundary pileup at 1 for TFP-shock persistence rho_z, a skewed distribution for the investment adjustment-cost parameter, and bimodality in the trend-growth parameter gamma – and persist under an alternative observable set that replaces hours worked with the labor income share. Morris stresses throughout that these phenomena reflect identification weakness intrinsic to DSGE models, not misspecification, and that while the MCSE framework helps diagnose pileups and enables valid bootstrap inference, it does not offer a general method to prevent them.

Exploiting MIT shocks in heterogeneous-agent economies: the impulse response as a numerical derivative

This paper proposes a new way to compute the equilibrium of heterogeneous-agent models with aggregate uncertainty: rather than building a recursive, linearized law of motion for a high-dimensional state such as the cross-sectional wealth distribution – the strategy behind existing linearization methods including Reiter (2009, 2010), Childers (2017), and Ahn, Kaplan, Moll, Winberry, and Wolf (2017) – the authors solve, using entirely standard nonlinear methods, a single deterministic perfect-foresight transition path that follows a one-time, unexpected (“MIT”) shock away from the model’s non-stochastic steady state. Under the working assumption that the true stochastic equilibrium is well approximated by a linear system, that one nonlinear transition path is literally the model’s numerical derivative at each future time horizon, so the response to any sequence of aggregate shocks – and to any number of independent shocks, with computation time rising only linearly in the number of shocks – can be recovered by scaling and summing copies of this single impulse response, with no analytical differentiation and no explicit treatment of the distribution as a linearized state. The only nontrivial numerical tool the method requires is value-function iteration: once to solve the model’s non-stochastic steady state, and again, backward over a finite horizon, to solve for the transition path, with the passage of calendar time serving as the sole state variable added along the way. Applied to a standard Aiyagari-style economy with valued leisure and two aggregate technology shocks, the method reproduces conditional-moment results close to Dynare’s linearization- and second-order-perturbation-based solutions in a representative-agent benchmark, delivers accurate impulse responses (including for distributional statistics like the Gini coefficient and the hand-to-mouth share) in the heterogeneous-agent case, and, in an extension with a consumption externality and deficit-financed transfers, shows that Ricardian equivalence breaks down and fiscal transfers have real stabilizing effects – all without added numerical difficulty. The paper reports that its own scalability and additivity checks are satisfied “with flying colors” in this application, but is explicit that the whole approach rests on the assumption that linearization is in fact a good approximation for the model at hand, offers no formal stability or determinacy characterization of the transition path it computes, and, unlike recursive methods, provides no separate goodness-of-fit metric for that assumption.

Long-term inflation expectations and the transmission of monetary policy shocks: Evidence from a SVAR analysis

This 2021 Journal of Economic Dynamics & Control paper by Max Diegel and Dieter Nautz asks whether U.S. long-term inflation expectations respond to monetary policy shocks in a way consistent with a “re-anchoring channel,” and how quantitatively important that channel is for how monetary policy shocks pass through to inflation and unemployment. The authors estimate a four-variable Bayesian structural VAR (inflation, unemployment, a policy rate that splices the federal funds rate with the Krippner 2013 shadow rate through the zero lower bound, and the Survey of Professional Forecasters’ median 10-year-ahead CPI inflation expectation) on quarterly U.S. data from 1991Q4 to 2019Q4, with four lags. Rather than leaving the monetary policy shock’s effect on expectations unrestricted only in the impulse-response sign pattern, they identify it with sign and zero restrictions on the structural impact matrix and, crucially, an additional restriction on the systematic component of the policy rule requiring that the central bank raise the policy rate when long-term inflation expectations rise (a restriction the authors show is binding: without it, only 50% of posterior draws would satisfy that sign, so imposing it substantially shrinks the identified set). A separate expectations shock is identified via zero restrictions reflecting that Survey of Professional Forecasters respondents typically report before the current quarter’s CPI and unemployment releases. The main finding is that, in contrast to earlier studies that found essentially no response, a one-standard-deviation contractionary monetary policy shock causes long-term inflation expectations to fall significantly and persistently (though the effect is ultimately transitory), with monetary policy shocks accounting for roughly 16-28% of the forecast-error variance of long-term expectations across horizons out to ten years. A counterfactual analysis that shuts down this re-anchoring channel shows it matters a great deal for the transmission of monetary policy to inflation – the monetary-policy contribution to inflation’s forecast-error variance collapses from about 15% to under 1% on impact (and from roughly 34% to about 13% at long horizons) once the channel is switched off – but “virtually no effect” on the transmission to unemployment, which is governed instead by the conventional interest-rate channel. A further structural-scenario exercise finds that had the Federal Reserve not responded to expectations shocks since its 2012 inflation-target announcement, median inflation would have been about 57 basis points lower and median unemployment about 99 basis points higher on average, suggesting the Fed’s systematic response to below-target long-term expectations has itself helped stabilize both variables.

Modeling inflation expectations in forward-looking interest rate and money growth rules

This 2025 Journal of Economic Dynamics and Control paper by Zhengyang Chen and Victor J. Valcarcel proposes a “rational expectations structural VAR” (RE-SVAR) — a way to embed rational expectations directly into a low-dimensional structural VAR without mapping the system to a fully specified DSGE model — and uses it to compare two candidate monetary-policy indicators, the Wu and Xia (2016) shadow federal funds rate and Divisia M4 money growth, on which produces a larger share of theoretically sensible (“non-puzzling”) impulse responses. The model starts from a three-equation consensus New Keynesian system (a forward-looking monetary policy rule written in either an interest-rate or a money-growth form, an IS equation, and a Phillips-curve AS equation, all estimated simultaneously) and identifies the monetary policy shock using a rational-expectations forecast-revision restriction that recovers it as a linear combination of reduced-form VAR residuals — not a Cholesky/recursive scheme, and without imposing any delayed-reaction exclusion restriction on the policy indicator. Rather than estimating the policy rule’s forward-looking parameters, the paper runs a “pseudo-calibration” grid search over the inflation- and output-response coefficients (φ_π, φ_y, each cycling over 61 values from 0 to 4) and the horizons over which expectations are formed (h_π = 0,…,12 months for inflation, h_y = 0,…,4 months for output), yielding 241,865 structural VAR specifications (8 lags, monthly U.S. data) and a resulting “cloud” of impulse responses; a response counts as a “puzzle” if output or inflation turns negative at any point within the first year following an expansionary shock to the policy indicator. In the main October 1988-February 2020 sample (377 monthly observations), the shadow federal funds rate produces output puzzles in 98.68% of the 241,865 specifications and inflation puzzles in 99.13%, with only 0.87% (2,109 specifications) surviving a no-joint-puzzle criterion; replacing it with Divisia M4 growth as the policy indicator cuts output puzzles to 4.02% and inflation puzzles to 4.13%, with 95.85% (231,825 specifications) surviving. This Divisia advantage holds up under a coarser 25,137-specification grid, a post-Global-Financial-Crisis effective-lower-bound sample (December 2008-February 2020, where even the shadow rate’s best-case output and inflation puzzle rates remain high at 72% and 93%), a long historical sample back to January 1967, a narrower Divisia M2 aggregate, and a PCE-based inflation measure — with one partial exception: in the long historical sample under PCE, the output-puzzle gap between indicators narrows substantially (53.3% for the shadow rate vs. 56.0% for DM4 vs. 47.9% for DM2). Longer inflation-expectation horizons help Divisia far more than the shadow rate (at a 12-month horizon, 18,430 of 241,865 DM4 specifications are non-puzzling versus only 5 for the shadow rate), and extending the model to four variables by adding the Gilchrist-Zakrajsek excess bond premium (July 1979-February 2020 sample) still yields an 81.45% no-joint-puzzle survival rate with DM4 as the indicator, alongside IS- and AS-shock responses that are broadly consistent with textbook signs. The authors read the results as evidence that the federal funds rate — especially its shadow-rate extension through the effective lower bound — is an inadequate standalone policy indicator in low-dimensional VARs, and that Divisia M4’s richer information content lets it capture forward-looking central bank behavior without added information variables, while cautioning that the RE-SVAR’s validity is conditional on the sensibility of its underlying three-equation theoretical structure and that, unlike a standard recursive VAR, it is not modular: adding any variable (as they do, only loosely, for the excess bond premium) requires specifying a full structural equation for it.

Monetary transmission in money markets: The not-so-elusive missing piece of the puzzle

This 2021 Journal of Economic Dynamics and Control paper by Zhengyang Chen and Victor Valcarcel shows that the Wu-Xia (2016) shadow federal funds rate — a standard modern-sample proxy for the stance of monetary policy that extends below the zero lower bound — produces a persistent, statistically significant price puzzle in a time-varying-parameter VAR estimated on 1988-2020 U.S. data, and that this puzzle survives the standard fixes (adding commodity prices, federal funds futures, or forward rates) that resolved the price puzzle in earlier, pre-1988 samples. In its place, the authors propose Divisia monetary aggregates — weighted monetary aggregates that account for the different liquidity services of component assets, rather than simply summing dollar balances — as an alternative policy indicator: using Divisia M4 (and, with more muted magnitudes, the narrower Divisia M2) in place of the shadow rate produces no puzzling price response in the first three months and the theoretically correct price-level increase at 18-, 30-, and 60-month horizons following an expansionary shock, a correction the authors show is not an artifact of their time-varying estimation approach since it also holds in a constant-parameter VAR. Extending the analysis to the transmission of monetary shocks into 14 disaggregated money-market components (spanning currency, deposits, retail and institutional money-market funds, time deposits, repurchase agreements, commercial paper, and Treasury bills), the paper documents that transmission strengthened substantially after the 2007 financial crisis, with patterns in the responses of savings deposits and less-liquid institutional instruments that the authors interpret as evidence of a “flight-to-safety” effect among both households and firms during and after the crisis. The authors attribute the shadow rate’s breakdown as a policy indicator to the Federal Reserve’s increased forward-looking transparency (making it harder to generate a true interest-rate “surprise”) and to the post-crisis shift from reserve scarcity to reserve abundance, concluding that reintroducing monetary aggregates — measured correctly via the Divisia index rather than simple summation — may be “the missing piece of the puzzle” in a low-rate environment where a key short-term policy rate is highly persistent.

Solving heterogeneous-agent models by projection and perturbation

This paper proposes a numerical method for solving stochastic general-equilibrium models with incomplete markets and a continuum of heterogeneous agents – a class of problems where, as the paper puts it, “the state vector includes the whole cross-sectional distribution of wealth,” an infinite-dimensional object in principle. The dominant approach at the time, pioneered by Krusell and Smith (1998), represents that distribution with only a small number of statistics (typically the mean) and works very well for the models it was built for; but the paper argues this cannot serve as a general solution, since in models where the shape of the distribution itself drives the dynamics – such as (S,s) pricing or inventory models, or the redistributive-shock example this paper constructs – a low-dimensional summary can miss essential dynamics. The proposed method instead computes a solution that is fully nonlinear in the idiosyncratic (individual) shocks but only linear in the aggregate shocks: it first solves precisely for the steady-state cross-sectional distribution and consumption function (with no aggregate shocks but the full idiosyncratic shock process), using cubic splines for the consumption function and a fine histogram (up to 1000, and as a robustness check 5000, intervals) for the wealth distribution; it then computes a first-order perturbation of that whole high-dimensional representation with respect to small aggregate shocks, using Sims (2001)’s solver for linear rational-expectations systems. Applied to a test model of household saving with uninsurable income risk, liquidity constraints, an aggregate technology shock, and an i.i.d. redistributive capital-tax shock, the method reproduces the Krusell-Smith “approximate aggregation” finding when only the technology shock is active (a one-moment forecast of future aggregate capital is nearly exact), but shows that this breaks down once the tax shock is introduced, in which case even a four-moment forecast leaves sizable error while the paper’s high-dimensional, spline-based solution remains accurate to roughly 10^-6 in absolute forecast error. The method is explicitly a linear approximation in the aggregate dimension – suited to cases “where individual shocks are much bigger than aggregate shocks” – and the paper proposes it as a first step that can be combined with state-space reduction (via spline or principal-component bases) before, in future work, attempting higher-order perturbations in a reduced aggregate state space.

The financial market effects of unwinding the Federal Reserve's balance sheet

This 2023 Journal of Economic Dynamics and Control paper by A. Lee Smith and Victor J. Valcarcel asks whether the financial-market effects of shrinking (“unwinding”) the Federal Reserve’s balance sheet are simply quantitative easing (QE) in reverse, studying the Fed’s 2014-2019 balance-sheet normalization episode with U.S. data only and no DSGE model. The paper combines three empirical strategies: a reduced-form regression of the federal-funds/IOR spread on log reserves, where the liquidity-effect coefficient rises in magnitude from -0.086 (s.e. 0.009) during the 2009Q1-2014Q3 expansion to -0.299 (s.e. 0.015) during the 2014Q4-2019Q3 normalization – an effect roughly three-and-a-half times larger in magnitude, attributed to the zero lower bound having put a floor under the spread during QE while no symmetric ceiling constrained it during QT; a two-day event study around 11 quantitative-tightening announcements (January 2008-December 2018, 4,016 daily observations) that splits 2 “taper” announcements from 9 “unwind” announcements, finding that while 9 QE announcements produced large, statistically significant cumulative declines (-1.49 percentage points on the 10-year Treasury yield, -3.70pp on MBS, -1.00pp on 8-quarter Eurodollar futures), the 11 QT announcements produced no significant cumulative effect on any of these series, and within QT only the 2 taper announcements – not the 9 unwind announcements – had statistically significant effects across the yield curve and other assets; and a block-triangular structural VAR that orders reserves first, identifying “reserve supply shocks” tied to the Fed’s own balance-sheet decisions rather than to Treasury-bill supply (which affects reserves only with a one-week settlement lag), estimated both as a Bayesian time-varying-parameter VAR (October 2015-October 2019, following Cogley and Sargent 2005 and Primiceri 2005) and, over the Asset Runoff sample alone (October 2017-August 2019), as a constant-parameter Bayesian SVAR for each of several individual asset prices. The TVP-VAR shows that negative reserve-supply shocks produced no significant response in the Goldman Sachs Financial Conditions Index during the September 2014-September 2017 Full Reinvestment phase but a significant financial-tightening response emerging in early 2018 during the October 2017-August 2019 Asset Runoff phase, with the FF-IOR spread’s response to reserve shocks increasing roughly threefold in magnitude between the two phases; the granular asset-level SVAR finds that a negative one-standard-deviation reserve-supply shock raises the 10-year Treasury yield by roughly 4 basis points within 2-4 weeks, with the VAR-implied term premium accounting for nearly all of that increase (suggesting a term-premium channel rather than revised short-rate expectations), raises BBB corporate and MBS yields by comparable amounts (BBB remaining elevated near 2bp at 28 weeks), raises the broad dollar index by about 0.3% around week 8, and produces only a small, imprecisely estimated, statistically insignificant decline in the S&P 500; and a counterfactual simulation against continued full reinvestment implies the balance-sheet unwind raised the 10-year Treasury yield by a peak of approximately 40 basis points around mid-2018 and by approximately 8 basis points on average over the full Asset Runoff period. The paper’s central conclusion is that unwinding tightened financial conditions even in the absence of significant announcement effects, but transmitted through liquidity and implementation channels rather than through the signaling and duration-driven announcement channels that dominated QE – an asymmetry the authors attribute to the FOMC’s deliberate post-taper-tantrum strategy of communicating balance-sheet normalization as separate from the policy-rate path (Yellen’s 2017 “watching paint dry” characterization) and to the absence of a zero-lower-bound-style ceiling on rates during reserve drawdowns. The sample deliberately stops short of the September 2019 repo-rate spike because of endogeneity concerns about aggregate reserve dynamics, so the authors note the results may understate the ultimate tightening effect as reserves approached scarcity.