1 Introduction

Many of the most important questions in economics are about dynamics: not just whether a policy has an effect, but how that effect evolves over time. Does a monetary policy shock affect output immediately, or with a delay? How long does the employment effect of a labour market programme persist? Do the benefits of early childhood intervention compound over time or fade out?

The traditional tool for estimating dynamic causal effects in macroeconomics and policy analysis is the vector autoregression (VAR), where impulse response functions trace out the dynamic effect of a structural shock. But VARs impose tight parametric restrictions on the dynamics and can deliver misleading impulse responses when the true model is misspecified.

Jordà [2005] proposed a simpler, more robust alternative: the local projection (LP). LPs estimate dynamic effects horizon by horizon using a sequence of direct regressions, without imposing the recursive structure of a VAR. This article explains LPs, their connection to the event-study estimator in microeconometrics, the LP-IV extension for causal identification, and recent developments including the LP-DML method for high-dimensional settings.

2 Local Projections: The Basic Idea

2.1 Setup

Let Yₜ be a scalar outcome and Dₜ a treatment variable (e.g. a policy change, a monetary shock). We want to estimate the impulse response function (IRF): the causal effect of a unit shock to D at time t on Y at time t + h for horizons h = 0, 1, 2, ..., H

In the LP framework, the IRF at horizon h is estimated by regressing the future outcome Yₜ₊ₕ directly on the current treatment Dₜ and appropriate controls:

where ψₕ(L) is a lag polynomial in Y (typically including p lags). The coefficient β̂ₕ is the estimated effect at horizon h. This is "local" in the sense that a separate regression is estimated at each horizon. The IRF is then the sequence {β̂₀, β̂₁, ..., β̂_H}.

2.2 Comparison to VARs

A key result of Plagborg-Møller and Wolf [2021] is that, in large samples, LP and VAR impulse responses are numerically equivalent when the correct lag length is used. The difference is in small samples and under misspecification: LPs are more robust because they do not impose the VAR's constraint that the impulse response must be consistent with a low-order VAR. The cost is efficiency: LPs typically have larger standard errors than correctly specified VARs.

Standard errors for LPs must account for the serial correlation in the residuals εₜ₊ₕ, which are correlated across horizons (since the same shock Dₜ appears in regressions with different dependent variables). Newey-West standard errors with bandwidth proportional to H are standard; panel settings typically use clustered standard errors.

3 LP-IV: Local Projections with an Instrumental Variable

3.1 Motivation

Equation (1) assumes that Dₜ is exogenous conditional on the lags. In most applications, treatment is endogenous. The LP-IV approach, developed by Stock and Watson [2018] and clarified by Plagborg-Møller and Wolf [2021], combines local projections with IV to identify dynamic causal effects:

where Zₜ is an instrument that is uncorrelated with εₜ₊ₕ and relevant for Dₜ.

Importantly, Plagborg-Møller and Wolf show that the LP-IV impulse response is numerically equivalent to the SVAR-IV (structural VAR with external instruments) impulse response under mild conditions. LP-IV is therefore a flexible, transparent implementation of the "Romer and Romer" approach to identifying monetary policy shocks and similar macro interventions.

3.2 Applications

LP-IV has been applied to:

• Fiscal multipliers: Using defense spending news as an instrument for government spending [Ramey and Zubairy, 2018].

• Monetary policy: Using high-frequency interest rate surprises around FOMC announcements as instruments for policy rate changes.

• Credit supply shocks: Using cross-sectional variation in bank exposure as a Bartik- style instrument.

4 The Connection to Event Studies in Microeconometrics

The LP estimator has a direct analogue in microeconometric panel data: the event study specification. Suppose units i are treated at different times gᵢ (the staggered adoption set- ting). The event study regression estimates:

The coefficient βₕ is the average effect at h periods after treatment exactly an LP estimated on a panel. Pre-treatment coefficients β₋ₖ for k > 0 serve as pre-trend tests. Montiel Olea et al. [2025] formalise this connection and extend it by showing that LP event-study estimators remain valid even under heterogeneous treatment effects and staggered adoption, provided they are combined with appropriate DML-style controls. Their LP-DML approach projects Yᵢₜ – Ē[Yᵢₜ] and Dᵢₜ – Ê[Dᵢₜ] onto the event-time indicators, using machine learning to residualise the outcome and treatment on high-dimensional co- variates.

5 LP-DML: High-Dimensional Controls

The LP-DML framework of Montiel Olea et al. [2025] extends DML [Chernozhukov et al., 2018] to dynamic settings. The algorithm is:

1. For each horizon h, split the sample into K folds.

1. In each fold, train a machine learning model to predict Yₜ₊ₕ from controls Wₜ (lags, covariates) using the other K - 1 folds. Compute residuals Ỹₜ₊ₕ

1. Train a separate model to predict Dₜ from Wₜ. Compute residuals D̃ₜ

1. Regress Ỹₜ₊ₕ on D̃ₜ to obtain β̂ₕ. The cross-fitting in step 2 and 4 ensures that the Neyman orthogonality condition holds, making β̂ₕ robust to overfitting in the nuisance models. The result is a valid asymptotic t-test for βₕ even when the controls are high-dimensional.

6 Practical Guidance

6.1 Choosing the Lag Length

The LP regression should include enough lags of Y to ensure that the residuals εₜ₊ₕ are approximately uncorrelated with Zₜ or Dₜ. AIC/BIC on the reduced-form VAR can guide lag selection.

6.2 Horizon-Robust Inference

Because the same data are used for all H + 1 horizon regressions, hypothesis tests across hori- zons are correlated. Plagborg-Møller and Wolf [2021] recommend using the full covariance matrix of {β̂₀, ..., β̂_H} for joint inference. In practice, sup-t bands simultaneous confidence bands that control size across all horizons - are preferred over pointwise confidence intervals.

6.3 Available Software

• R: The lpirfs package implements local projections with standard and instrumental- variable specifications, including panel LPs with fixed effects. The fixest package's feols can estimate LP panels with fixed effects and clustered SEs.

• Stata: The irf command in the VAR framework can be adapted; the user-written lpoly and local projections packages provide direct LP estimation.

• Python: The statsmodels library supports LP with adequate custom coding.

7 Conclusion

Local projections have become the standard tool for estimating dynamic causal effects in macroeconomics and increasingly in applied microeconomics. Their main advantages are transparency (one regression per horizon), robustness to VAR misspecification, and com- patibility with IV identification via LP-IV. The LP-DML extension brings the method to high-dimensional settings. The connection between LP and panel event studies reinforces the conceptual unity of dynamic causal inference across macro and micro applications.

References

1. Chernozhukov, V., Chetverikov, D., Demirer, M., Duflo, E., Hansen, C., Newey, W., and Robins, J. Double/debiased machine learning for treatment and structural parameters. The Econometrics Journal, 21(1):C1-C68, 2018.
2. Jordà, Ò. Estimation and inference of impulse responses by local projections. American Economic Review, 95(1):161-182, 2005.
3. Montiel Olea, J. L., Qian, E., and Wolf, C. K. Double robustness of local projections inference. Working paper, 2025.
4. Plagborg-Møller, M. and Wolf, C. K. Local projections and VARs estimate the same impulse responses. Econometrica, 89(2):955-980, 2021.
5. Ramey, V. A. and Zubairy, S. Government spending multipliers in good times and in bad: Evidence from US historical data. Journal of Political Economy, 126(2):850-901, 2018.
6. Romer, C. D. and Romer, D. H. A new measure of monetary shocks: Derivation and implications. American Economic Review, 94(4):1055-1084, 2004.
7. Stock, J. H. and Watson, M. W. Identification and estimation of dynamic causal effects in macroeconomics using external instruments. Economic Journal, 128(610):917-948, 2018.

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