Christian K. Wolf | Macro Paper Warehouse

Can Deficits Finance Themselves?

The paper asks whether a government can run a deficit today — issuing “stimulus checks” — and allow debt to return to its initial level without any future tax hike or spending cut. In environments combining (i) nominal rigidity and (ii) a violation of Ricardian equivalence (due to finite lives or liquidity constraints), this is possible through two complementary self-financing channels: (a) a Keynesian boom in real activity that expands the tax base and automatically raises revenue at existing tax rates; and (b) a surge in inflation that erodes the real value of outstanding nominal government debt. The paper’s headline result is that self-financing increases monotonically as fiscal adjustment is delayed, converging to full self-financing in the limit: if monetary policy does not lean too heavily against the fiscal stimulus, the initial deficit eventually returns debt to trend with no required future adjustment. Calibrated to empirical evidence on intertemporal MPCs, the speed of fiscal adjustment, the Phillips curve slope, and the monetary reaction, the model finds self-financing up to ν ≈ 0.95 — with the tax base channel dominant and inflation contributing negligibly.

Instrumental Variable Identification of Dynamic Variance Decompositions

This 2022 Journal of Political Economy paper by Mikkel Plagborg-Møller and Christian K. Wolf asks what an external instrument (a proxy correlated with one structural shock and uncorrelated with the others) can tell us about that shock’s dynamic variance decomposition – the share of a variable’s forecast-error variance it accounts for at each horizon – when the instrument may contain classical measurement error and the shock need not be “invertible” (recoverable from current and past values of the observed macro variables alone). Working in a structural moving-average (SVMA) model where the number of shocks need not equal the number of observables, they show that the forecast variance ratio (FVR) is generically only interval-identified: because the instrument’s relevance is governed by an unknown scale parameter (its strength net of measurement-error noise), the data pin down a sharp, generically nondegenerate identified set for the FVR rather than a point, with a lower bound coming from treating the instrument itself as if it were the shock (attenuated by measurement error) and an upper bound coming from projecting the instrument onto all observed macro leads and lags. The identified set collapses to a point only under one of two testable-or-assumable conditions: a perfect instrument (no measurement error) or “recoverability” (the shock spans all leads and lags of the observables, which is weaker than full invertibility). The paper also derives a Granger-causality pretest that can certify a shock is noninvertible but cannot certify invertibility, and shows results extend to multiple instruments, whose consistency with a rank-one cross-spectral restriction is itself testable. Applying the method to US data (January 1990-June 2012, monthly) with the Gertler-Karadi high-frequency federal-funds-futures surprise as the instrument, they find the data consistent with substantial noninvertibility of the monetary shock, and their identification-robust 90% confidence intervals rule out the shock explaining more than 31% of output growth’s forecast variance or more than 8% of inflation’s forecast variance at any horizon studied (up to 24 months); they read this as evidence that “monetary shocks are almost irrelevant for aggregate inflation” in the post-1990 sample, so that if inflation is a monetary phenomenon it is because of the systematic, rule-like part of policy rather than its unpredictable component. A companion application to an oil-supply-news instrument (Känzig 2021) finds that shock, too, is highly noninvertible, which the authors show causes conventional SVAR-IV analysis to reach spurious conclusions; a Monte Carlo study confirms the new confidence intervals achieve close-to-nominal coverage even when the true shock is noninvertible, unlike conventional SVAR-IV intervals, which undercover badly in that case.

Local Projections and VARs Estimate the Same Impulse Responses

This 2021 Econometrica paper by Mikkel Plagborg-Møller and Christian K. Wolf proves that local-projection (LP) and vector-autoregression (VAR) based impulse-response estimators are, at the population level, the same object up to a constant of proportionality, so that differences researchers observe between LP and VAR estimates in applied work reflect finite-sample and lag-truncation choices rather than the two methods identifying fundamentally different things. Working nonparametrically under covariance-stationary data with an everywhere-nonsingular spectral density and absolutely summable Wold decomposition (Assumptions 1-2), they show (Proposition 1) that the recursive-VAR impulse response theta_h at any horizon h equals the square root of E(x-tilde_t squared) times the LP coefficient beta_h, where x-tilde_t is the population residual of the impulse variable after controlling for contemporaneous and lagged covariates; the scale factor depends on neither the horizon nor the response variable, and the proof runs through the Frisch-Waugh theorem applied to the VAR(infinity) Wold representation. The equivalence extends beyond the simple recursive (Cholesky) case to nonrecursive structural rotations, to long-run restrictions (Blanchard-Quah), to sign restrictions (Uhlig; Rubio-Ramirez-Waggoner-Zha), and to instrumental-variables identification: “LP-IV” using an internal instrument recovers exactly the same relative impulse responses as an internal-instrument recursive SVAR (Corollary 1), in contrast to the popular external-instrument “SVAR-IV” approach (Stock-Watson 2012; Mertens-Ravn 2013), which is only consistent when the structural shock is invertible from current and past data and identifies absolute rather than relative responses. When lag length p is fixed rather than infinite, Proposition 2 shows the LP(p) and VAR(p) estimands agree only approximately at horizons h less than or equal to p and generally diverge at h greater than p – a divergence that vanishes when the impulse variable is a direct, serially unpredictable shock; the sample-based estimators are shown to converge to one another as p grows with the sample size T and both are asymptotically efficient at any fixed horizon under weak regularity conditions, though at finite p and T researchers still face a bias-variance trade-off at long horizons, a gap the authors explicitly flag for future research. An empirical illustration using Gertler and Karadi’s (2015) monthly monetary-policy data (January 1990-June 2012: industrial production growth, inflation, the one-year government bond rate, and the excess bond premium, with the Gertler-Karadi high-frequency futures surprise as instrument) compares LP against an internal-instrument recursive VAR at lag lengths p = 4 and p = 12: the two estimators track each other closely through horizon h = p and diverge noticeably beyond it, consistent with Proposition 2, and both recover the qualitative Gertler-Karadi finding that the excess bond premium rises initially after a contractionary monetary shock. On this basis the authors argue that four widely held claims are mistaken: that VAR estimators are generally more efficient than LP, that LP is generally more robust to misspecification than VARs, that nonrecursive non-IV identification schemes require a VAR, and that noninvertible shocks rule out simple SVAR methods. The result is explicitly a population-level equivalence for linear estimators; it does not itself resolve finite-sample estimator choice, and the paper deliberately leaves questions of inference and of multiple-instrument identification to other work.

Local Projections vs. VARs: Lessons from Thousands of DGPs

This 2024 Journal of Econometrics paper by Dake Li, Mikkel Plagborg-Møller, and Christian K. Wolf asks a purely practical question rather than proposing a new estimator or identification scheme: when researchers estimate structural impulse responses, does the local-projection (LP) estimator or the vector-autoregression (VAR) estimator perform better in realistic macroeconomic settings, and under what conditions does the ranking flip? Because no single real-world dataset can answer this — the true DGP is never known — the authors build an “encompassing model,” a non-stationary dynamic factor model with six latent factors estimated on the 207-series Stock and Watson (2016) quarterly U.S. dataset (1959Q1-2014Q4), and use it to generate 6,000 simulated economies (3,000 built around a monetary policy shock with the federal funds rate as instrument, 3,000 around a fiscal policy shock with government spending as instrument), each simulated for T=200 quarters with 5,000 Monte Carlo draws. Across this population of realistically calibrated DGPs, comparing least-squares, bias-corrected, and penalized LP against least-squares, bias-corrected, Bayesian, and model-averaged VARs (plus SVAR-IV for the instrumented case), the paper documents a clear and pervasive bias-variance trade-off: at short horizons (h ≤ the p=4 lag length) LP and VAR have similar bias, but at longer horizons VAR bias grows substantially larger than LP bias, while LP’s standard deviation rises steeply with horizon — by h=20 roughly double the VAR’s. Bias-corrected LP removes only about a third of LP’s bias while adding variance, so it is preferred over uncorrected LP only when a researcher places very high weight (ω ≥ 0.9 in the paper’s bias-variance loss function) on bias at intermediate horizons; otherwise VAR-type methods, especially a Bayesian VAR with a Minnesota-type prior, dominate essentially throughout. A further headline finding concerns SVAR-IV: because roughly 90% of the simulated DGPs exhibit a degree of shock invertibility below 49%, the external-instrument SVAR-IV estimator carries substantially higher bias than internal-instrument alternatives at every horizon, though it also has notably lower dispersion. The authors are explicit that these are simulation-based lessons conditional on the choice of encompassing model and DGP class (quarterly, five-variable systems, no sign or long-run restrictions), not universal theorems.

Optimal Policy Rules in HANK

This paper characterizes optimal monetary and fiscal policy rules in a rich heterogeneous-agent New Keynesian (HANK) business-cycle model with nominal rigidities, in which the policymaker has two instruments – the short-term nominal interest rate and uniform lump-sum transfer (stimulus-check) payments – and asks whether, and how, household inequality should change how each instrument is set. For a policymaker with a conventional “dual mandate” that targets aggregate output and inflation, the paper proves the optimal interest-rate targeting rule is exactly the same as in the textbook representative-agent New Keynesian model, because in this economy household heterogeneity affects only the demand side, which is a slack constraint once the supply-side Phillips curve is left unchanged; empirically disciplined HANK and RANK models therefore prescribe essentially the same policy-rate paths. The paper then adds an explicit distributional objective – a planner who wants to insure households against business-cycle-driven swings in their consumption shares – and derives a linear-quadratic optimal rule with an additional term governed by the causal effect of each instrument on consumption inequality. Because the calibrated model, built to match evidence on monetary transmission, implies that interest-rate changes move household consumption by roughly similar percentages up and down the wealth and income distribution, this distributional term turns out to matter little in practice: optimal monetary policy stays close to the dual-mandate benchmark even when the planner cares about inequality, because using rates to fight inequality would require costly departures from aggregate stabilization for limited distributional gain. Stimulus checks, by contrast, have strongly progressive effects in the model – both from elevated marginal propensities to consume among low-income, low-wealth households and from a fixed dollar transfer mattering more as a share of low incomes – so they are shown to be an effective complementary tool for offsetting shocks with a strong distributional tilt, such as a simulated income-redistribution shock resembling the Covid-19 recession. These conclusions are explicitly conditional on the paper’s calibration of policy transmission channels; the authors show that alternative model specifications implying larger distributional effects of monetary policy (as in some other recent HANK papers) would restore a more significant role for distributional considerations in interest-rate policy.