Local Projection Inference Is Simpler and More Robust Than You Think
📄 Summarized from the full manuscript · Human-reviewed for faithfulness before publication
In brief
How should researchers put error bands around estimates of how the economy responds over time to a shock? This 2021 paper shows that one simple change, adding an extra lagged variable as a control, makes the plainest error formula sufficient, with no need for the complicated corrections and tuning choices usually thought necessary. The resulting bands hold their stated coverage whether the data are mildly or extremely persistent, and at long horizons, where rival methods either badly understate uncertainty or become enormously wide. It matters because it gives applied researchers a default that works without first diagnosing persistence, though it is not always the most precise choice.
What this paper finds — and why it matters
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.
Summary of a classic paper, AI-assisted and human-reviewed. See the linked original for the authoritative claims and full conditions.
Questions & answers
Q1. What question does the paper address, and what is its central answer?
Should applied researchers use local projections (LP) or VARs for inference on impulse response functions, and the paper’s answer is that a particular LP variant, lag-augmented LP (LA-LP) with simple Eicker-Huber-White (EHW) heteroskedasticity-robust standard errors, achieves uniform asymptotic validity across stationary, near-unit-root, and unit-root processes and across a wide range of horizons. This makes standard LP inference simpler (no HAC estimator needed) and more robust to the underlying degree of persistence than had previously been appreciated (Abstract; Section 1, pp. 1789-1791).
Q2. What is “lag-augmented” LP, and why does adding one extra lag change the inference problem?
Lag-augmented LP (LA-LP) is the OLS coefficient on y_t in a regression of y_{t+h} on both y_t and its own lag y_{t-1}, rather than on y_t alone (the non-augmented LP of Eq. 2). In the baseline AR(1) model y_t = ρy_{t-1} + u_t, the non-augmented LP residual ξ_t(ρ,h) follows an MA(h-1) process, which is why researchers have conventionally reached for HAC (Newey-West-type) standard errors. The paper’s Lemma 1 shows that once y_t is regressed on y_{t-1} to obtain a residualized regressor û_t(h) (Eq. 6), the product of the LP residual and this residualized regressor, ξ_t(ρ,h)·û_t(h), is serially uncorrelated at all leads and lags even though ξ_t itself is not. The extra lag “absorbs” the serial correlation that would otherwise contaminate the regression scores, so the scores behave like a martingale difference sequence and ordinary EHW (heteroskedasticity-only) standard errors are sufficient for valid inference (Section 2.1, pp. 1795-1797).
Q3. What is the formal uniform-coverage result for the AR(1) case, and how far does “uniform” extend?
For the AR(1) model, the LA-LP 1-α confidence interval Ĉ(h,α) satisfies inf over ρ ∈ [-1,1] and over horizons 1 ≤ h ≤ h̄_T of P_ρ(β(ρ,h) ∈ Ĉ(h,α)) → 1-α as T → ∞, provided h̄_T/T → 0 (Eq. 8, p. 1797). The infimum is taken jointly over the entire persistence range, including the exact unit root ρ = 1, and jointly over every horizon up to h̄_T — so the same confidence interval formula delivers correct asymptotic coverage no matter where the true process sits on the stationary-to-unit-root spectrum, as long as the researcher does not push the horizon out faster than the sample grows (Section 2.2, p. 1797).
Q4. How does the result generalize beyond a single AR(1) variable to a full VAR system?
Proposition 1 extends the uniform-coverage result to an n-variable VAR(p): under Assumptions 1-3 and h̄_T/T → 0, √T(β̂_1(h) - β_1(h))/ŝ_1(h,ν) converges in distribution to a standard normal uniformly over the parameter space of Definition 1 and over all horizons 1 ≤ h ≤ h̄_T (pp. 1805-1806). The parameter space is defined via a companion-matrix factorization A(L) = B(L)(I_n - diag(ρ_1,…,ρ_n)L) with each ρ_i ∈ [-1,1], which spans stationary, near-unit-root, and unit-root multivariate processes in one unified class. The assumptions require strictly stationary innovations with conditional mean zero (permitting ARCH/GARCH heteroskedasticity, Assumption 1), finite eighth moments and summable cumulants up to order four (Assumption 2), and non-singularity of the LA-LP estimator’s denominator matrix (Assumption 3). The practical upshot: a researcher using LA-LP with EHW standard errors gets valid confidence intervals without pretesting for unit roots, choosing a HAC bandwidth, or otherwise tailoring the procedure to the persistence regime of the data (pp. 1805-1806, 1809-1811).
Q5. What do the Monte Carlo experiments (Tables I and II) show about actual finite-sample coverage?
In a design with T = 240 observations and 5,000 replications targeting 90% nominal coverage, LA-LP combined with a wild bootstrap (LP-LA_b) achieves approximately 90% coverage across all (ρ, h) combinations examined, including the unit-root case (ρ = 1) and long horizons, under both homoscedastic (Table I) and ARCH (Table II) innovations. By contrast, an AR(1) delta-method confidence interval severely under-covers at moderate-to-long horizons when the process is persistent (e.g., ρ = 0.95 at h = 20), and a bootstrap CI built from an augmented AR model (AR-LA_b) does achieve nominal coverage but only by becoming impractically wide near the unit root (Section 3.1-3.2, pp. 1797-1801). The wiki source flags that the exact coverage percentage and CI-length figures for every individual (ρ, h) cell in Tables I and II have not been independently re-verified against the published tables — only the specific summary figures quoted below are drawn directly from the source text.
Q6. How much wider are the competing confidence intervals near the unit root, and why does that matter?
At ρ = 0.95 and h = 60 under homoscedastic innovations, the reported median CI length is 23.050 for the augmented-AR bootstrap (AR-LA_b) versus 0.942 for LA-LP (LP-LA_b) — roughly a 24-fold difference — and at the exact unit root (ρ = 1) the AR-LA_b interval widens further; under ARCH innovations (Table II) the gap becomes far more extreme, with a reported median AR-LA_b CI length of 5,593.663 at ρ = 1, h = 60. The point is that formally valid coverage is not by itself sufficient: an interval that achieves nominal coverage only by being enormously wide near persistent, heteroskedastic processes is of little practical use, whereas LA-LP achieves comparable coverage with intervals that stay tight (Section 3.1-3.2, pp. 1799-1801).
Q7. When is LA-LP not the best choice, and how does its relative efficiency vary?
The efficiency ranking between LA-LP and alternative estimators depends on where the true process sits (Figure 1, Section 3.3, p. 1802): LA-LP is more efficient than the augmented-AR estimator for large |ρ| and long horizons, and more efficient than non-augmented LP for large |ρ| (non-augmented LP is biased near unit roots), but for stationary processes with small |ρ| at short horizons the ranking is ambiguous and AR-based estimators can be tighter. The paper derives closed-form efficiency-boundary formulas for the AR(1) case (AsyVar(β̂(h)) = Σ_{ℓ=0}^{h-1} ρ^{2ℓ}, Eq. 22): LA-LP beats non-augmented LP when |ρ| is below a threshold ρ̄(h), and beats augmented-AR when |ρ| is above a (generally different) threshold ρ(h) (Appendix B.2.1, pp. 1817-1818). The wiki source flags that the exact boundary curves in Figure 1 were not independently re-verified against the figure itself. Correspondingly, the applied recommendations (Section 6) note two specific cases where LA-LP may be inferior: (i) a stationary process at a short horizon, where a researcher confident in stationarity may prefer an AR-based estimator, and (ii) a near-unit-root process at a very long horizon (close to the h̄_T/T → 0 boundary), where both LA-LP and the augmented-AR bootstrap can degrade (Section 6, pp. 1812-1815).
Q8. How is the recommended bootstrap implemented, and what are the scope limits of the whole framework?
The recommended “wild recursive VAR bootstrap” (Section 5) estimates a VAR(p) on the data, generates bootstrap samples recursively as y_t = Â_1 y_{t-1} + … + Â_p y_{t-p}* + û_tw_t using i.i.d. wild-bootstrap weights w_t* with mean zero and unit variance, and — critically — centers the bootstrap t-statistic at the VAR-implied impulse response ν’β̂_{1,VAR}(h) rather than at the LP point estimate itself, which the paper identifies as the feature that makes the bootstrap uniformly valid; the resulting percentile-t interval is uniformly valid under the same conditions as Proposition 1 and accommodates ARCH/GARCH heteroskedasticity (Section 5, pp. 1810-1812).** The theory has explicit scope limits: results are for pointwise (single-horizon) confidence intervals, not confidence bands simultaneously valid across a range of horizons (for which the authors point to their companion paper, Montiel Olea and Plagborg-Møller 2019); uniformity requires h̄_T/T → 0, so horizons growing proportionally with the sample size fall outside the theory; the VAR(p) generalization requires lag order p fixed or growing slowly with T; Assumption 1’s conditional-mean-zero requirement permits ARCH/GARCH innovations but not innovations whose conditional mean itself depends on the past; and the paper does not cover non-linear or state-dependent local projections (Caveats and limitations; Section 1, p. 1791; Proposition 1 conditions, p. 1806).
Key terms in this paper
Definitions below follow the paper's own usage.
- Lag-augmented local projection (LA-LP)
- in this paper, the LP estimator obtained by regressing y_{1,t+h} on the full set of contemporaneous and lagged regressors (y_t, y_{t-1}, ..., y_{t-p}) rather than on y_t and its own lags up to y_{t-p+1} alone -- the "augmentation" is the inclusion of one additional lag beyond what a non-augmented LP of the same order would use, and it is this extra lag that removes the serial correlation problem in the regression scores (Eqs. 2-3, 10, 12).
- EHW (Eicker-Huber-White) standard errors
- the paper's term for ordinary heteroskedasticity-robust ("White") standard errors that correct only for conditional heteroskedasticity, not for serial correlation across time -- distinguished throughout from HAC (heteroskedasticity-and-autocorrelation-consistent, Newey-West-type) standard errors, which additionally require a bandwidth/kernel choice to estimate a long-run variance; the paper's central claim is that EHW suffices for LA-LP because lag augmentation already removes the serial-correlation problem (Eq. 5; Section 2.1).
- Uniform asymptotic validity
- coverage accuracy that holds simultaneously -- not just pointwise for one fixed data-generating process -- across the entire parameter space of persistence values rho_i in [-1,1] (spanning stationary, near-unit-root, and exact-unit-root processes) and across every horizon h up to h-bar_T, formalized as an infimum over that whole space converging to the nominal level (Eq. 8; Proposition 1).
- Parameter space (Definition 1)
- the class of VAR(p) data-generating processes the theory covers, defined by factoring the VAR's companion polynomial as A(L) = B(L)(I_n - diag(rho_1,...,rho_n)L) with each rho_i in [-1,1]; this single parameterization nests stationary, near-unit-root, and unit-root multivariate processes so that the same asymptotic theory applies across all of them without the researcher having to specify in advance which regime the data are in (Eq. 13, p. 1805).
- Wild recursive VAR bootstrap
- the paper's recommended resampling scheme, which simulates data recursively from the estimated VAR(p) using i.i.d. wild-bootstrap multipliers applied to the VAR residuals, and forms the bootstrap t-statistic by centering it at the VAR-implied impulse response rather than at the LP point estimate -- a centering choice the paper identifies as essential for the bootstrap interval to inherit the uniform validity of the analytic LA-LP interval (Section 5, pp. 1810-1812).