Òscar Jordà | Macro Paper Warehouse

Estimation and Inference by the Method of Projection Minimum Distance: An Application to the New Keynesian Hybrid Phillips Curve

This 2011 International Economic Review paper by Óscar Jordà and Sharon Kozicki proposes “projection minimum distance” (PMD), a two-step semiparametric estimator for macroeconomic models whose likelihood score function is nonlinear in the structural parameters but whose mapping from reduced-form Wold impulse-response coefficients to those parameters is linear – a class that includes forward-looking Euler equations and ARMA-type representations. In the first step, PMD estimates the Wold coefficients b_h semi-parametrically using Jordà’s (2005, 2009) local-projection regressions of y_{t+h} on lagged variables, run separately for each horizon; in the second step it exploits the model-implied linear restrictions on those coefficients via minimum distance (following Ferguson 1958) to estimate the structural parameters, with an information criterion following Hall, Inoue, Nason, and Rossi (2007) selecting the truncation horizon H by trading off identification gain from extra horizons against a many-weak-instruments cost. The paper establishes formally (Propositions 1-2 for the first-stage local-projection estimator; Lemmas 3-4 for the second-stage minimum-distance estimator) that both stages are consistent and asymptotically normal under stated rate conditions, and shows that in the simple case of i.i.d. shocks PMD coincides exactly with a GMM estimator that uses the lagged dependent variable as instrument – but the two estimators diverge once shocks are serially correlated, because the lagged dependent variable is then no longer a valid GMM instrument while the different (longer-lagged) instrument set implicitly used by local projections remains valid. Monte Carlo experiments (Section 4) show that in a univariate ARMA(1,1) design PMD converges to the true parameters even at T=50 and matches MLE’s standard errors closely by T=100-400, while MLE suffers numerical convergence difficulties in the pure-AR(1) and pure-MA(1) special cases that leave PMD unaffected; in a three-equation New Keynesian (Phillips-IS-Taylor) design calibrated from Lindé (2005) across 36 DGP variants, PMD has smaller bias than GMM in the majority of cases – both perform well when shocks are i.i.d. (though GMM’s output-gap estimates are somewhat downward biased), and both deteriorate once serial correlation and additional lagged terms are introduced, with GMM deteriorating more. Applying PMD to re-estimate Fuhrer and Olivei’s (2005) hybrid New Keynesian output and inflation Euler equations on 1966:Q1-2001:Q4 U.S. quarterly data, the paper finds PMD point estimates broadly similar to GMM, MLE, and OI-GMM but generally with similar or smaller standard errors, and uses the instability of the estimated coefficients across different choices of H together with frequent rejections of the overidentifying-restrictions test to argue that the hybrid Phillips curve is dynamically misspecified in ways that the other estimators’ point estimates alone do not reveal.

Estimation and Inference of Impulse Responses by Local Projections

This 2005 American Economic Review paper by Òscar Jordà introduces “local projections” (LP) — a way to estimate impulse responses directly from a sequence of single-horizon regressions of the future value of a variable on current shocks and lags, rather than by iterating forward a fitted vector autoregression (VAR) — and shows the method delivers valid inference even when the underlying data-generating process is not correctly captured by the fitted VAR. Formally, an impulse response at horizon s is the coefficient on the shock variable in a separate OLS regression of y_{t+s} on lagged variables, run once per horizon; when the true process actually is a VAR(p), Jordà shows the local-projection estimator coincides with the VAR-implied impulse response (a VAR is a restricted special case of LP), but LP remains valid even when the process lacks a standard Wold (linear, Gaussian) representation, and Newey-West standard errors correctly account for the resulting moving-average error structure at each horizon. Monte Carlo evidence based on a monetary VAR calibrated to Evans and Marshall (1998) shows that local-projection impulse responses stay within the true confidence bands at all horizons, while a misspecified low-order VAR generates a spurious, statistically significant “price puzzle” that the local projections do not exhibit; a second Monte Carlo exercise shows a cubic (nonlinear) local-projection specification tracks a genuinely nonlinear, time-varying-volatility data-generating process much more closely than a linear time-varying-parameter VAR. In an empirical application to U.S. inflation-output dynamics (1955-2003), the linear local-projection estimate of the output-gap response to a monetary tightening is roughly twice as large as the equivalent VAR(4) estimate, and a nonlinear, threshold-based local-projection specification finds statistically significant regime-dependence — with much larger output and inflation responses, and no price puzzle, in the low-inflation regime that has prevailed since the mid-1980s.

Local Projections

This 2025 Journal of Economic Literature survey by Òscar Jordà (who introduced the method in a 2005 AER paper) and Alan M. Taylor takes stock of local projections (LP) as a general framework for estimating impulse responses, arguing that LPs “help bridge the divide between current best practices in applied microeconomics and standard time series methods in macroeconomics.” The baseline LP estimates the impulse response at horizon h directly as the coefficient beta_h in a single OLS regression of y_{t+h} on the intervention s_t and lagged controls x_t, requiring one separate regression per horizon rather than iterating forward a fitted VAR; the authors identify four practical advantages over VARs – no cross-equation system constraints, straightforward accommodation of nonlinearities and heterogeneity, direct estimation of cumulative multipliers, and a natural extension to panel and difference-in-differences settings – while showing LP remains asymptotically equivalent to VAR-implied impulse responses (and to VAR-based Cholesky and long-run identification) whenever the underlying process truly is a VAR of the assumed order (citing Plagborg-Møller and Wolf 2021), and retains identification via LP-IV even in “non-invertible” settings where VAR identification from the reduced-form covariance matrix breaks down. The survey then works through the full LP toolkit: levels versus long-difference specifications (long-differencing “considerably reduces” the O(T^-1) small-sample bias that is severe for persistent, near-unit-root outcomes in short samples); horizon-by-horizon smoothing via B-splines or a Gaussian basis function; inference via Newey-West HAC standard errors, lag-augmented LP (which restores ordinary White standard errors with, per Montiel Olea and Plagborg-Møller 2021, correct uniform coverage across stationary, near-unit-root, and unit-root processes), system GMM for joint hypothesis tests across horizons, sup-t simultaneous confidence bands, and Lagrange-multiplier significance bands constructed under the null; identification via selection on observables, inverse-propensity weighting, long-run restrictions, and LP-IV; projection minimum distance (PMD) for recovering structural parameters from LP-estimated auxiliary parameters; counterfactual policy-path analysis subject to a Mahalanobis-distance “modesty” test; state-dependent and nonlinear LPs; and panel LP / LP-DiD for staggered-adoption settings. Worked illustrations throughout convey magnitudes without claiming new empirical results of their own: a state-dependent local-projection estimate of the OECD fiscal-consolidation multiplier (1978-2019) finds the average four-year output response is more than twice as large in slumps (-1.78, chi-squared(5)=58.3, p=0) as in booms (-0.80, chi-squared(5)=17.2, p=0.004); a projection-minimum-distance estimate of a UK Phillips curve (1975m1-2007m12, using the Cloyne-Hürtgen 2016 monetary shock) yields coefficients of 0.838 (SE 0.593) on inflation and -1.990 (SE 0.180) on the output term; and an LP-IV estimate of the unemployment response to a Romer-Romer monetary shock (1985:1-1999:12) peaks at roughly 1.25 percentage points around 24 months before returning to zero by about 48 months.

Loose Monetary Policy and Financial Instability

This paper provides the first long-run causal evidence that a persistently loose stance of monetary policy — defined as extended periods of low interest rates relative to the neutral rate — significantly raises the probability of a financial crisis several years later. Using a long historical panel of 18 advanced economies (approximately 1870–2020, excluding world wars), the paper estimates local projection (LP) regressions in which the stance is measured as the 5-year backward moving average of (r – r*), with r* from the Del Negro–Giannoni–Gaballo–Tambalotti (DGGT) factor model. The OLS baseline finds that a 1 percentage-point (pp) looser average stance over a 5-year window raises the 3-year financial crisis probability by 2.2pp at a 5–7 year horizon and 3.3pp at a 7–9 year horizon, against an unconditional base of 10.5%. To address the endogeneity of monetary policy to pre-existing economic conditions, the authors construct an instrumental variable based on the international trilemma of open-economy finance: for countries pegging their exchange rate, changes in the base-country interest rate orthogonal to domestic economic conditions provide exogenous variation in domestic rates, weighted by a capital mobility index. IV estimates are substantially larger: 1pp looser average stance raises crisis probability by 5.5pp at 5–7 years and 15.5pp at 7–9 years, indicating that OLS understates the causal effect because accommodative policy is endogenously adopted during recessions when crisis risk is already low. The same loose-policy stance significantly raises the probability of entering R-zones — periods of credit market overheating identified by Greenwood, Hanson, Shleifer, and Sørensen (2022) as harbingers of financial crisis — and, with a lag of 6–9 years, raises the probability of historically low GDP growth (below the 20th percentile of the cross-country distribution). The evidence supports a growth-risk tradeoff: loose policy may deliver short-term stimulus, but at a meaningful cost in medium-term financial fragility and real tail risk.

The Response of Term Rates to Fed Announcements

Demiralp and Jorda argue that since the Fed began publicly announcing its Federal funds rate target in 1994, it has manipulated short-term rates through a Pavlovian “announcement effect” – moral suasion backed by a credible threat – requiring far smaller open-market operations than the conventional liquidity effect would predict, and they show, using a new hazard-model decomposition of target changes into anticipated and surprise components, that unanticipated target changes and surprise Fed inaction both move the term structure in ways consistent with the rational expectations hypothesis. The paper’s starting observation is that the February 3-4, 1994 FOMC meeting inaugurated the practice of publicly disclosing the new Federal funds rate target immediately after each meeting, and the authors argue this gave the Fed a genuinely new policy tool distinct from the textbook liquidity effect, in which the Fed moves rates only by manipulating the supply of nonborrowed reserves. Exploiting institutional developments in the reserves market (declining required reserves, retail sweep programs, minimal use of the discount window), the authors show with structural VARs that the estimated liquidity effect is artificially inflated when target-change dates are included in the sample, and that this inflation is much smaller after a 1989 “Thanksgiving effect” episode after which banks began reading Fed intentions with near-certainty – evidence that an announcement/credibility channel was already emerging before the formal 1994 disclosure policy. To measure the effect of target changes on the term structure, the authors build an autoregressive conditional hazard (ACH) model, paired with an ordered-probit model for the size of changes, that decomposes each Federal funds rate target change into its anticipated and unanticipated components without relying solely on the federal funds futures market (which only began in 1989). Regressing Treasury rates of various maturities on these components, they find that anticipated target changes have no significant effect on term rates (as the rational-expectations hypothesis predicts), while unanticipated target changes move rates significantly, an effect that becomes stronger and extends to longer maturities after 1989; strikingly, after 1989 markets also react significantly to surprising Fed inaction – an expected target change that fails to occur – at medium and long maturities, evidence that the market had come to understand and price the Fed’s own reaction function closely enough to be surprised when it did not act as expected.