Local Projections

A local projection (LP) estimates the impulse response at each horizon h directly as the coefficient beta_h in a single OLS regression of the future outcome y_{t+h} on the current intervention s_t and controls x_t – one separate regression per horizon, rather than fitting one VAR and iterating it forward to trace out the whole impulse-response path. Formally the impulse response is defined as the difference in expected future outcomes under an intervention of size delta versus none, R_{s->y}(h,delta) = E[y_{t+h}|s_t=s_0+delta;x_t] - E[y_{t+h}|s_t=s_0;x_t] for h=0,1,…,H (Eq. 1); the baseline LP regression y_{t+h} = alpha_h + beta_h s_t + gamma’h x_t + v{t+h} (Eq. 2) delivers beta_h as a direct estimate of that object, with v_{t+h} a moving-average error of order h by construction and H+1 separate regressions needed to trace the full horizon profile. The authors identify four practical advantages LPs have over VARs: it is a single-equation method with no cross-equation system constraints to impose; nonlinearities and heterogeneity are straightforward to accommodate; cumulative responses and multipliers are directly obtained; and the method extends naturally to panel data and difference-in-differences settings.

LPs and VARs are asymptotically equivalent whenever the underlying data-generating process truly is a VAR of the assumed order – citing Plagborg-Møller and Wolf (2021) – and this equivalence extends to identification: Cholesky and long-run identification schemes implemented within an LP framework are asymptotically equivalent to their VAR counterparts. For Cholesky identification, adding contemporaneous values of variables ordered earlier in the Wold causal chain as additional controls in x_t makes LP-Cholesky asymptotically equivalent to VAR-Cholesky. Where LP has a genuine edge is in “non-invertible” settings – where the number of underlying shocks exceeds the number of observables (for example, news shocks about future technology) – because “VAR identification methods based on the covariance matrix of reduced-form residuals will not work directly” there, whereas LP-IV, using an external instrument, “achieves identification even in non-invertible settings.” LP-IV was first introduced in Jordà, Schularick, and Taylor (2015).

The baseline LP can be run in levels (Eq. 11) or in long differences of the outcome (Eq. 12), and the levels specification carries a small-sample bias of order O(T^-1) that becomes considerable for persistent outcomes (autoregressive root rho near 1) combined with a small sample size T. Monte Carlo experiments (T=100, rho in {0.95, 1.0}, Figure 1) illustrate that “small sample biases appear to be considerably reduced when using long differencing” instead of levels – a practical recommendation rather than a claim that long-differencing dominates in every setting.

The cumulative multiplier at horizon h is the ratio of the cumulative outcome response to the cumulative response of the intervention itself, m(h) = R^c_{sy}(h)/R^c_{ss}(h) (Eq. 16), and when estimation uses LP-IV with a single instrument z_t, this cumulative multiplier can be recovered from a single regression (Eq. 19) that instruments the cumulative treatment path with z_t, rather than requiring separate numerator and denominator regressions at each horizon.

Horizon-by-horizon LP estimates can be noisy, so the survey describes two ways to impose a parametric functional form on beta_h as a function of h: B-splines (piecewise polynomials with continuity constraints at the knots, following Eilers and Marx 1996) and a three-parameter Gaussian basis function (GBF), phi(h;Theta) = a*exp{-[(h-b)^2/c^2]}, where a is the amplitude, b the location of the peak response, and c governs its width. GBF parameters are estimated jointly within the same GMM system used for inference; a worked example fits a = 1.388 (SE 0.278), b = 26.189 (SE 0.922), c = 12.997 (SE 0.891) to an unemployment response to a Romer-Romer shock (1985:1-1999:12), implying a peak response around month 26.

Because the LP error v_{t+h} is a moving average of order h by construction, it is serially correlated in a way that standard OLS standard errors ignore, so the survey recommends either Newey-West HAC standard errors with J = h+p lags (Bartlett kernel, p the LP’s own lag length) or “lag-augmented” LP, which adds extra lags of the outcome variable to the regression (Eq. 28) so that ordinary White heteroskedasticity-robust standard errors apply. Following Montiel Olea and Plagborg-Møller (2021), lag-augmented LP is reported to give “correct uniform probability coverage under a wide range of scenarios (stationarity, near unit roots, nonstationarity) and even for long-distance horizons (as long as the sample size is large enough relative to the horizon).”

For testing hypotheses across multiple horizons at once, the survey estimates the full set of LP coefficients as a system by GMM (Eqs. 30-33), which nests standard 2SLS/IV as the single-horizon special case; for simultaneous confidence bands that keep joint coverage at the nominal level (rather than the pointwise 1.96-SE band, which “understates actual joint coverage”), it uses the sup-t bands of Montiel Olea and Plagborg-Møller (2019), computed by simulating from the joint normal distribution of all H+1 coefficients – bands the authors describe as “relatively conservative, since they accommodate unspecified nulls.” A third tool, significance bands (Inoue, Jordà, and Kuersteiner, forthcoming), is constructed under the null hypothesis beta_h=0 via the Lagrange-multiplier principle – analogous to the standard +/-1.96/sqrt(n) bands used for autocorrelation functions – and, unlike error bands, straddles zero rather than the point estimate; the authors suggest “current practice could be extended to display significance bands alongside error bands.” An empirical illustration (price level and inflation responses to an extended Romer-Romer shock, 1969:I-2007:IV) reports joint-significance test p-values of 1.72e-23 for the price level and 1.12e-17 for inflation.

The survey organizes LP identification into four approaches: LP-OLS (“selection on observables”), which assumes variation in the intervention s_t is as good as random conditional on controls x_t; LP-IPW, which reweights by the estimated propensity score for a binary intervention (with a doubly robust LP-IPWRA variant adding regression adjustment); long-run restrictions, where Plagborg-Møller and Wolf (2021) show Blanchard-Quah-style long-run identification can be implemented as a two-step LP procedure; and LP-IV, which uses an external instrument and requires relevance and lead-lag exogeneity (Assumption 1). As noted in Q2, LP-IV’s distinguishing advantage is that it remains valid in non-invertible settings where covariance-matrix-based VAR identification does not.

Projection minimum distance (PMD, Jordà and Kozicki 2011) estimates structural parameters theta = g(pi) by minimum distance from LP-estimated auxiliary parameters pi-hat, minimizing (theta - g(pi-hat))‘W_T(theta - g(pi-hat)) to get an asymptotically normal theta-hat – a method the authors note “does not require the user to use LPs to estimate the impulse responses” in the first place, and which extends to rational-expectations and DSGE models. A worked illustration estimates a UK Phillips curve (using the Cloyne-Hürtgen 2016 monetary policy shock, residualized against UK oil-price and exchange-rate controls, sample 1975m1-2007m12) with PMD point estimates theta-hat_pi = 0.838 (SE 0.593) and theta-hat_x = -1.990 (SE 0.180). Building on the same PMD logic, Barnichon and Mesters (2023) use LP-estimated impulse responses to test whether a policymaker minimizing a quadratic loss over inflation and unemployment deviations is at its optimum (H0: delta=0), with the notable feature that “nowhere in the discussion did we have to explicitly write down the policy rule.”

For a predetermined binary state D_{t-r}, the survey recommends estimating separate state-specific long-difference LPs (Eq. 64) rather than pooling, since “the correct approach is to condition on the state as well, and to estimate a state-dependent LP”; a worked fiscal-consolidation illustration (OECD, 1978-2019, treatment = change in the cyclically adjusted primary balance, instrumented with the Guajardo et al./Adler et al. narrative measure) finds the average output response over years 0-4 is -1.78 (chi-squared(5)=58.3, p=0) in economic slumps versus -0.80 (chi-squared(5)=17.2, p=0.004) in booms, i.e., “much larger when fiscal consolidations are implemented during slumps.” The survey cautions this state-dependent approach breaks down “if the state includes current information,” since then the intervention and the state are mutually influenced and instruments may be needed for both. A related but distinct nonlinear-LP specification (Eq. 67, following Cloyne, Jordà, and Taylor 2023) decomposes the total response into a direct effect and an indirect “state-amplification” effect that varies continuously with x_t rather than switching between two discrete regimes. Separately, counterfactual policy-path analysis (following Leeper and Zha’s “modest policy interventions,” 2003) uses the joint normality of LP coefficient estimates to compute a counterfactual response to an alternative policy path (Eq. 61), valid only when a Mahalanobis-distance “modesty test” (Eq. 63) is not too large; a worked example (funds-rate GBF parameters for 1985:1-2000:12) shifting the peak-timing parameter by one standard deviation yields a borderline modesty p-value of 0.08, and the authors caution that results in such cases “should be interpreted with some caution.” Finally, panel LP (Eq. 73) adds individual and time fixed effects (absorbing the former by long-differencing) and uses Driscoll-Kraay standard errors as T->infinity or a wild cluster bootstrap as N->infinity for small T; LP-DiD (Dube et al. 2023) adapts this to staggered-adoption settings by restricting the estimation sample to newly treated or not-yet-treated (“clean control”) observations, which the authors note is “trivial” to impose in LP’s forward-looking regression framework compared with algorithmic pairwise comparisons across control units in other DiD estimators.