1 Introduction

The local projection (LP) methodology of Jordà [2005] has become a workhorse for estimating dynamic causal effects—impulse response functions (IRFs)—in macroeconomics and applied microeconomics. The core LP idea is to estimate the effect of a shock at horizon h by directly regressing the h-period-ahead outcome on the shock at time t, without the auxiliary restrictions that structural vector autoregressions (SVARS) impose.

LPs enjoy several virtues: they are easy to implement (horizon-by-horizon OLS), flexible (covariates enter linearly), and robust to misspecification at long horizons where SVAR restrictions may fail. The theoretical equivalence between LP and VAR impulse responses under correct SVAR specification, established by Plagborg-Møller and Wolf [2021], provides a formal bridge between the two approaches.

A natural extension, increasingly prominent in the 2023-2025 literature, is to allow LP responses to depend on the state of the economy—to capture threshold effects, asymmetries, and regime-switching dynamics that are invisible in linear models. This article reviews the nonlinear LP framework, introduces state-dependent IRFs, discusses identification, and surveys recent developments.

2 Linear Local Projections: A Brief Review

In the linear LP framework [Jordà, 2005], the impulse response of variable Y at horizon h to shock εₜ is estimated by:

where Wₜ₋ⱼ are control variables (lags of Y and other variables in the system), α⁽ʰ⁾ is a horizon-specific intercept, and β⁽ʰ⁾ is the coefficient of interest—the IRF at horizon h. The sequence {β̂⁽ʰ⁾} for h = 0 to H traces the impulse response function.

Standard inference uses heteroskedasticity- and autocorrelation-consistent (HAC) standard errors to account for the moving-average error structure induced by overlapping horizon windows.

3 The Case for Nonlinearity

Linear LPs impose that the impulse response is the same regardless of the state of the economy at the time of the shock. This is often implausible:

• Fiscal multipliers may be larger in recessions (when monetary policy is at the zero lower bound and resources are underemployed) than in expansions [Auerbach and Gorodnichenko, 2012].
• Monetary policy may have asymmetric effects: rate increases (contractionary shocks) may contract output more than rate decreases (expansionary shocks) expand it.• Credit supply shocks may propagate differently when leverage is high versus low, reflecting balance-sheet amplification mechanisms.
• Productivity shocks may generate different employment responses depending on whether the labour market is tight or slack.

A model that forces symmetry across states averages these regime-specific effects and may miss important policy-relevant heterogeneity.

4 State-Dependent Local Projections

4.1 The Smooth Transition Framework

Auerbach and Gorodnichenko [2012] introduce a smooth-transition nonlinear LP:

where zₜ is a state variable (e.g., an output gap or unemployment rate), F(zₜ) is a smooth transition function (commonly the logistic function F(z) = (1 + e⁻ᵞᶻ)⁻¹), and (αₑ, βₑ) and (αᵣ, βᵣ) are expansion- and recession-state parameters.

The state-dependent impulse responses are:

The difference βᵣ⁽ʰ⁾ – βₑ⁽ʰ⁾ is the state-dependent heterogeneity in the IRF at horizon h.

4.2 Threshold LP

A simpler, discrete-state alternative uses a hard threshold:

where z̄ is a threshold value (possibly estimated). This is a regime-switching LP, where 1[zₜ > z̄] defines the high-state indicator. Plagborg-Møller and Wolf [2021] show that under correct specification, the linear LP and the threshold LP are both consistent estimators of regime-specific average IRFs in their respective populations. Neither is uniformly preferred; the choice depends on prior beliefs about the smoothness of the transition.

5 Identification in Nonlinear LPs

5.1 Exogenous Shocks and LP-IV

LP methods require the shock εₜ to be exogenous—uncorrelated with the residual uₜ₊ₕ⁽ʰ⁾ after controlling for Wₜ₋ⱼ. In macroeconomic applications, shocks are rarely directly observed. The LP-IV approach [Stock and Watson, 2018] uses an external instrument Zₜ—a variable correlated with the shock but orthogonal to the error—to identify the causal IRF:

The LP-IV estimator instruments εₜ with Zₜ at each horizon h separately:

In the nonlinear LP, LP-IV can be extended to allow state-dependent responses by interacting the instrument with the state indicator:

instrumenting (εₜ, εₜ · 1[zₜ > z̄]) with (Zₜ, Zₜ · 1[zₜ > z̄]).

5.2 Plagborg-Møller and Kolesár (2025)

A recent and important contribution is Plagborg-Møller and Kolesár [2025], published in the Journal of Business & Economic Statistics. Their paper provides a formal causal-inference foundation for nonlinear LPs, addressing several gaps in the earlier literature:

1. Causal interpretation. They show that the nonlinear LP identifies the average effect of the shock conditional on the state variable, under a potential outcomes framework. The state-dependent IRF is interpreted as an average treatment effect for units (time periods) in that state.‍
2. Weak identification. When the shock has little variation in one state (e.g., there are few recession observations in a short time series), the state-specific IRF is weakly identified. They derive robust inference procedures analogous to Lee et al. [2022] for binary IV that are valid regardless of the strength of within-state shock variation.‍
3. Misspecification. They analyse the behaviour of the nonlinear LP estimator when the true model is neither a smooth transition nor a hard threshold. The estimator recovers a projection coefficient that minimises a weighted mean-squared error, analogous to the OLS projection interpretation of IV under heterogeneous effects.

Their result on weak identification is particularly important in practice: with quarterly post-war US data, there are only 15-20 NBER recession quarters, making recession-specific IRFs estimated with high uncertainty. Conventional heteroskedasticity-robust standard errors may undercover in this regime.

6 Practical Implementation

6.1 Step-by-Step Estimation

1. Choose the state variable zₜ. Common choices: NBER recession indicator, output gap (HP-filtered), unemployment rate, financial conditions index, term spread.‍
2. Specify the transition. Hard threshold vs. smooth logistic transition. Estimate z̄ by grid search or use economic priors (e.g., z̄ = mean unemployment rate).‍
3. Estimate equation (2) by OLS or 2SLS at each horizon h. Include enough lags of Wₜ₋ⱼ to ensure residual uₜ₊ₕ⁽ʰ⁾ is approximately serially uncorrelated.‍
4. Compute HAC standard errors. Use Newey-West with bandwidth h + 1 to account for the MA(h) error structure.‍
5. Plot state-dependent IRFs β̂ₑ⁽ʰ⁾ and β̂ᵣ⁽ʰ⁾ with 90% confidence bands. Test H₀: βᵣ⁽ʰ⁾ = βₑ⁽ʰ⁾ jointly across horizons using a Wald test with clustered covariance matrix.

6.2 Weak-Instrument Robust Inference in State-Specific LP-IV

When the instrument Zₜ has low within-state variation, use the Anderson-Rubin-style test of Plagborg-Møller and Kolesár [2025]: construct confidence sets for βₛ⁽ʰ⁾ by inverting the test of the null βₛ⁽ʰ⁾ = b for a grid of b values. This is robust to weak identification and avoids the distorted size of conventional 2SLS Wald tests.

7 An Application: Fiscal Multipliers Across the Cycle

Auerbach and Gorodnichenko [2012] apply nonlinear LP to quarterly US data (1947-2008) to test whether government spending multipliers differ in recessions versus expansions. Their state variable is the seven-quarter moving average of GDP growth, with recession defined as periods of below-average growth.

They find:

• ‍Recession multiplier: approximately 2.0 at a 1-year horizon—government spending of $1 raises GDP by $2.
• ‍Expansion multiplier: approximately 0.7 at a 1-year horizon—far below the recession multiplier.

This finding has informed debates about the appropriate size of fiscal stimulus packages during recessions and is consistent with New Keynesian models featuring zero lower bounds on interest rates. It also illustrates the substantive importance of allowing for state dependence: the linear LP estimate (averaging over states) yields a multiplier of about 1.1, masking the strong asymmetry.

Figure 1: Stylised state-dependent fiscal multipliers from a nonlinear LP `a la Auerbach andGorodnichenko [2012]. Recession (red) and expansion (blue) multipliers diverge substantially, while the linear LP (dashed) averages over both states.

8 Comparison to SVAR Methods

Nonlinear SVARS (Markov-switching VARs, threshold VARs) can also estimate state-dependent IRFs but require full model specification and are computationally intensive. Nonlinear LPs are semiparametric—only the projection equation is specified—and therefore more robust to misspecification of system dynamics. The trade-off is efficiency: under a correctly specified SVAR, structural methods can be more precise. The LP approach is preferred when researchers are primarily interested in a small number of horizon-specific parameters and want robustness to dynamic misspecification.

9 Conclusion

The nonlinear extension of local projections provides a flexible, robust, and increasingly theoretically grounded framework for estimating state-dependent dynamic causal effects. From the smooth-transition LP of Auerbach and Gorodnichenko [2012] to the formal causal-inference treatment of Plagborg-Møller and Kolesár [2025], the methodology has matured rapidly. The primary practical challenges—weak within-state identification and the choice of state variable—have principled solutions. For macroeconomists and applied researchers interested in whether causal effects depend on economic conditions, nonlinear LPs are now an essential part of the toolkit.

References

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3. Lee, D. S., McCrary, J., Moreira, M. J., and Porter, J. (2022). Valid t-ratio inference for IV. American Economic Review, 112(10):3260-3290.
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