Mikkel Plagborg-Møller | Macro Paper Warehouse

Dominant Currency Paradigm

Standard open-economy macro models assume that export prices are sticky either in the producer’s currency, in which case a depreciation improves the terms of trade and competitiveness, or in the destination’s currency, in which case it worsens them. Neither matches the invoicing evidence: the vast majority of world trade is priced in a small number of dominant currencies, with the dollar playing an outsized role. This paper builds an alternative “dominant currency paradigm” from three joint ingredients – infrequently adjusted prices set in a dominant currency, strategic complementarities in pricing that make desired markups variable, and roundabout production using imported inputs – and derives four sharp testable implications: the bilateral terms of trade should be insensitive to bilateral exchange rates; for non-US countries import price pass-through should be high but driven by the dollar rather than the bilateral exchange rate, and more so the higher the country’s dollar invoicing share; import quantities should likewise be driven by the dollar rate, with US import quantities much less responsive; and a uniform appreciation of the dollar should reduce trade among countries other than the United States. The tests use two new datasets: bilateral non-commodity price and volume indices built from UN Comtrade for more than 2,500 country pairs covering 91 percent of world trade, 1992-2015, and firm-10-digit-product-country-quarter customs records for Colombia, an economy that invoices 98 percent of its exports in dollars. All four implications hold. Regressing bilateral terms of trade growth on bilateral exchange rate growth gives a contemporaneous coefficient of 0.037 with a 95 percent confidence interval of [0.02, 0.05], against a predicted 1 under producer currency pricing and −1 under local currency pricing, and the coefficient shrinks further toward zero once relative producer prices are controlled for. A standard bilateral pass-through regression implies near-complete pass-through – a 10 percent depreciation of the importer’s currency against the exporter’s raises import prices about 8 percent within the year – but adding the dollar exchange rate and time fixed effects knocks the bilateral coefficient from 0.76 to 0.16, with the dollar coefficient at 0.78 absorbing almost all of it, and raising a country’s dollar invoicing share by 10 percentage points raises contemporaneous dollar pass-through by 3.5 to 7.6 percentage points. On volumes the contemporaneous dollar elasticity is roughly −0.19 to −0.13 while the bilateral elasticity is an order of magnitude smaller; the euro is far less important than the dollar in both sets of regressions. Consistent with 97 percent of US exports and 93 percent of US imports being dollar-invoiced, bilateral pass-through into US export prices is complete on impact and close to zero for US import prices, and US import volumes are essentially unresponsive to the bilateral exchange rate (an implied 0.003 percent contemporaneous response to a 1 percent dollar depreciation, against −0.12 percent for non-US importers), so US trade balance adjustment runs through exports rather than imports. Aggregating the bilateral panel, a 1 percent ceteris paribus dollar appreciation against all other currencies predicts a 0.6 percent contraction in rest-of-world trade volume within the year, persisting for at least two years, controlling for proxies for the global business and financial cycles; dollar pass-through into foreign CPI and PPI averages 11 and 28 percent within the year and rises with the dollar invoicing share. The Colombian microdata reproduce all of this and additionally let the authors estimate the model: the estimated invoicing shares are essentially DCP, the estimated model tracks the observed dynamics of pass-through while PCP and LCP counterfactuals do not, and removing strategic complementarities and imported inputs halves four-quarter export pass-through from 65 to 30 percent. The authors are explicit about interpretation: the volume regressions “do not capture structural demand elasticity parameters” and “conflate expenditure switching and shifts in aggregate import demand,” so they are predictive relationships rather than structural estimates; and the invoicing currency is taken as given, with the argument that the model’s own ingredients are the ones that would generate dominant-currency pricing endogenously.

Double Robustness of Local Projections and Some Unpleasant VARithmetic

This paper provides formal theoretical results on the relative robustness of local projection (LP) and vector autoregression (VAR) confidence intervals for impulse response inference when the data generating process (DGP) is locally misspecified. The research question is whether the widely held belief that LP estimators are more robust to misspecification than VARs is theoretically justified, and if so, precisely under what conditions and with what consequences for VAR inference.

The analytical framework models the DGP as a stationary structural VARMA(1, ∞) that is local to an SVAR(1), of the form y_t = Ay_{t-1} + H[I + T^{-ζ}α(L)]ε_t, where the MA component T^{-ζ}α(L)ε_t represents misspecification that vanishes at rate T^{-ζ} as sample size T grows. The key rate parameter is ζ ∈ (1/4, 1/2), which corresponds to misspecification large enough to be detected with probability approaching 1 by conventional Hausman-type specification tests, yet small enough that the bias-variance trade-off between LP and VAR remains non-trivial asymptotically. The framework encompasses under-specification of lag length, omitted variables, temporal aggregation, measurement error, and failure of shock invertibility — essentially all sources of dynamic misspecification relevant to linearized DSGE models.

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 Projection Inference Is Simpler and More Robust Than You Think

This 2021 Econometrica paper by José Luis Montiel Olea and Mikkel Plagborg-Møller studies what inference procedure applied researchers should use for local-projection (LP) estimates of impulse response functions, and argues that a specific variant, lag-augmented LP (LA-LP) paired with ordinary Eicker-Huber-White (EHW) heteroskedasticity-robust standard errors, is both simpler to implement than commonly assumed and uniformly valid across a much wider range of data persistence than previously understood. The key theoretical mechanism, worked out first in an AR(1) setting, is that adding a single extra lag as a control (regressing y_{t+h} on both y_t and y_{t-1} rather than on y_t alone) makes the resulting regression scores serially uncorrelated even though the LP residual itself follows an MA(h-1) process, so heteroskedasticity-robust standard errors alone suffice for correct coverage: no HAC long-run-variance estimator or bandwidth choice is needed. Extending this to a general VAR(p) system, Proposition 1 shows the LA-LP estimator’s studentized statistic converges uniformly to a standard normal distribution across the entire parameter space of stationary, near-unit-root, and unit-root processes (each variable’s persistence parameter ρ_i ranging over [-1,1]) and across every horizon h that grows no faster than the sample size (h̄_T/T → 0), so the resulting confidence intervals have correct asymptotic coverage without the researcher pretesting for unit roots or tailoring the procedure to the persistence regime. Monte Carlo evidence (T = 240, 5,000 replications, nominal 90% coverage) shows LA-LP combined with a wild recursive-VAR bootstrap achieves approximately nominal coverage across all persistence/horizon combinations examined, including the unit-root case and long horizons, whereas a delta-method confidence interval built from an estimated AR(1) badly under-covers at longer horizons when the process is persistent, and a bootstrap confidence interval built from an augmented AR model achieves coverage only by becoming extremely wide near the unit root (e.g., a reported median CI length of 23.050 for the augmented-AR bootstrap versus 0.942 for LA-LP at ρ = 0.95, h = 60 under homoscedastic innovations, with the gap growing far more extreme under ARCH innovations). The efficiency ranking between LA-LP and its competitors is not uniform, however: LA-LP is more efficient than the augmented-AR estimator for persistent processes at long horizons, and more efficient than non-augmented LP (which is biased near unit roots) for large |ρ|, but for stationary processes at short horizons the ranking is ambiguous and AR-based estimators can be tighter. On this basis the authors recommend LA-LP with EHW standard errors, or the wild bootstrap, as a default inference procedure for LP-based impulse responses, recommend choosing the lag length conservatively (more lags rather than fewer, with the theory accommodating lag order growing at rate T^{1/3} or slower), and note that the results concern pointwise, single-horizon confidence intervals rather than confidence bands that are simultaneously valid across a whole range of horizons.

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.

Simultaneous Confidence Bands: Theory, Implementation, and an Application to SVARs

This 2019 Journal of Applied Econometrics paper by José Luis Montiel Olea and Mikkel Plagborg-Møller addresses a practical problem in applied time-series econometrics: which “simultaneous” (joint, across-parameter) confidence band should researchers use by default when reporting an entire vector of estimated quantities – such as impulse responses across horizons from a structural VAR – rather than one interval per parameter reported separately. The authors set up a general framework in which a possibly nonlinear transformation theta = h(mu) of an asymptotically normal estimator mu-hat is to be covered jointly, and show that a wide class of popular bands (pointwise, Bonferroni, Sidak, projection, and the “sup-t” band) can all be written as members of a single “one-parameter class” that scales every pointwise standard error by one common critical value c. Within this class, the sup-t band – whose critical value is the quantile of the maximum absolute studentized draw from the estimator’s joint asymptotic distribution – is the narrowest band that still achieves exact asymptotic simultaneous coverage, and the paper adds a decision-theoretic result (Proposition 1) showing it uniquely minimizes worst-case regret across all degree-one-homogeneous loss functions, which makes it a defensible default when the researcher does not know which feature of the band matters most to different readers. The paper gives three computationally convenient ways to construct it – a plug-in (delta-method) simulation algorithm, a bootstrap algorithm, and a Bayesian algorithm delivering exact finite-sample simultaneous credibility – and notes all three are first-order asymptotically equivalent. In an empirical application to a monthly U.S. SVAR (July 1979-June 2012, 12 lags; identified two ways, via a recursive/Cholesky scheme and via a Gertler-Karadi (2015)-style external instrument using federal-funds-futures surprises from January 1990) with industrial production, CPI, a one-year bond yield, and the excess bond premium, the sup-t band is substantially narrower than the Bonferroni or Sidak bands – around 35% narrower in the external-instrument specification at 68% confidence – and the narrowing is not merely cosmetic: at the 68% simultaneous level the plug-in sup-t band excludes zero for the industrial-production response at horizons of roughly 13-36 months, letting the authors reject the no-effect null at some horizon in that range, whereas the Bonferroni band does not permit that rejection; conversely, an output response that looks pointwise significant at the 2-month horizon is no longer simultaneously significant once the sup-t multiple-comparison adjustment is applied. A companion Monte Carlo study of bivariate VARs finds the sup-t band 20-25% narrower than Bonferroni/Sidak at 68% confidence and 10-20% narrower at 90% confidence, though for highly persistent data only the Bayesian sup-t implementation is reported to achieve satisfactory finite-sample coverage. The theory is developed for point-identified models with a continuously differentiable transformation h(.); partially identified (e.g., sign-restricted) SVARs require additional considerations, though the authors suggest the Bayesian sup-t band may still be usable there for subjective Bayesian analysis.