Draft:Local Projections

Local projections (LPs) are an econometric method for estimating the dynamic effect of a shock, policy change, or other intervention on an outcome variable over time. The estimated sequence of horizon-specific coefficients is typically summarized and plotted as an impulse response function (IRF). Unlike methods based on a fully parameterized dynamic system such as a vector autoregression (VAR)—from which impulse responses are derived by simulation or algebra—local projections estimate a separate regression for each forecast horizon directly from the data. The method was introduced by Òscar Jordà in a 2005 paper in the American Economic Review.^[1] It has since become a standard tool in empirical macroeconomics and applied econometrics.^[2][3]

Jordà (2005) proposed local projections as a flexible alternative to VAR-based impulse response estimation, noting that LPs do not require correct specification of the full dynamic system.^[1] The method gained broad adoption in applied macroeconomics during the 2010s, particularly in fiscal and monetary policy research, where it was used alongside or in place of structural VARs.^[2] A theoretical comparison by Plagborg-Møller and Wolf (2021) established that, in linear settings with sufficiently rich controls, LPs and VARs identify the same population impulse responses, with differences arising from finite-sample efficiency and specification choices rather than from fundamentally different targets.^[4] Jordà and Taylor (2025) provide a comprehensive survey of the method and its extensions.^[5]

Let ${\displaystyle y_{t}}$ be an outcome of interest and let ${\displaystyle s_{t}}$ be a shock or treatment variable at time ${\displaystyle t}$. For each horizon ${\displaystyle h=0,1,\dots ,H}$, a local projection estimates:

$${\displaystyle y_{t+h}=\alpha _{h}+\beta _{h}s_{t}+\gamma _{h}'X_{t}+u_{t,h}}$$

where ${\displaystyle X_{t}}$ is a vector of predetermined controls (typically lags of ${\displaystyle y}$ and ${\displaystyle s}$), and ${\displaystyle u_{t,h}}$ is a horizon-specific error term. The coefficient ${\displaystyle {\hat {\beta }}_{h}}$ estimates the response of ${\displaystyle y}$ at horizon ${\displaystyle h}$ to a unit change in ${\displaystyle s_{t}}$, conditional on controls. Repeating this across horizons yields the estimated IRF ${\displaystyle \{{\hat {\beta }}_{h}\}_{h=0}^{H}}$, which can be plotted against ${\displaystyle h}$.^[1]

Under the assumption ${\displaystyle \mathbb {E} [u_{t,h}\mid s_{t},X_{t}]=0}$, the coefficient ${\displaystyle \beta _{h}}$ identifies the conditional expectation:

$${\displaystyle \mathrm {IRF} (h)={\frac {\partial }{\partial \delta }}\,\mathbb {E} \!\left[y_{t+h}\mid s_{t}=\delta ,\,{\mathcal {F}}_{t-1}\right]{\Big |}_{\delta =0},}$$

where ${\displaystyle {\mathcal {F}}_{t-1}}$ denotes the relevant information set spanned by the controls, provided these controls adequately approximate that information set.^[4]

Some applications report cumulative impulse responses:

$${\displaystyle \mathrm {CIRF} (h)=\sum _{j=0}^{h}\beta _{j},}$$

or ratios of cumulative responses across variables. Fiscal multipliers, for example, are often estimated as the ratio of the cumulative output response to the cumulative government spending response.^[2]

Local projections are an estimation strategy; causal interpretation requires identification assumptions on ${\displaystyle s_{t}}$.

If ${\displaystyle s_{t}}$ is constructed to be plausibly exogenous—for example, a monetary policy surprise measured from high-frequency financial data, or a narrative shock series—then conditioning on appropriate controls may justify treating ${\displaystyle s_{t}}$ as conditionally exogenous, supporting a causal interpretation of ${\displaystyle {\hat {\beta }}_{h}}$.^[2][3]

When ${\displaystyle s_{t}}$ is endogenous, local projections can be combined with instrumental variables. A common implementation applies two-stage least squares at each horizon separately, instrumenting ${\displaystyle s_{t}}$ with an external instrument ${\displaystyle z_{t}}$ (sometimes called a proxy or external instrument). This approach is used to estimate dynamic causal effects in settings where the shock of interest cannot be directly observed or isolated.^[6][5]

Because LP regressions use overlapping windows of data—for example, ${\displaystyle y_{t+2}}$ appears in both the ${\displaystyle h=2}$ and the ${\displaystyle h=1}$ regressions run from different starting points—the residuals ${\displaystyle u_{t,h}}$ are typically serially correlated. Standard practice uses heteroskedasticity-robust and autocorrelation-robust standard errors, such as the Newey–West estimator, or cluster-robust standard errors in panel settings.^[1]

Montiel Olea and Plagborg-Møller (2021) show that including additional lags of the controls ("lag augmentation") enables asymptotically valid inference without requiring knowledge of the lag order of the data-generating process, simplifying applied practice.^[7]

In linear settings, LP and VAR impulse responses target the same population objects under appropriate conditions. Plagborg-Møller and Wolf (2021) establish this equivalence formally, showing that differences between LP and VAR estimates in practice reflect finite-sample efficiency and specification choices rather than different identification targets.^[4] Applied researchers have debated the relative merits of the two approaches, with LPs often preferred when robustness to misspecification is a priority and VARs preferred when efficiency at short samples is important.^[2]

Each horizon-specific LP regression is a "direct" multi-step forecasting regression, projecting the outcome ${\displaystyle h}$ periods ahead directly onto current and lagged variables, rather than iterating a one-step-ahead model forward. This connects LPs to the broader literature on distributed-lag models and multi-step forecasting.

LPs can be generalized to allow responses to vary across economic regimes by interacting the shock with a state indicator ${\displaystyle q_{t}}$:

$${\displaystyle y_{t+h}=\alpha _{h}+\beta _{h,0}s_{t}+\beta _{h,1}(s_{t}\cdot q_{t})+\gamma _{h}'X_{t}+u_{t,h}.}$$

This framework is used to study asymmetric dynamics, such as whether fiscal or monetary policy effects differ between recessions and expansions. Gonçalves et al. (2024) analyze inference in this setting.^[8]

With panel data indexed by unit ${\displaystyle i}$ and time ${\displaystyle t}$, local projections commonly include unit and time fixed effects:

$${\displaystyle y_{i,t+h}=\alpha _{h}+\beta _{h}s_{i,t}+\gamma _{h}'X_{i,t}+\mu _{i}+\tau _{t}+u_{i,t,h}.}$$

Standard errors are typically clustered by unit, or two-way clustered by unit and time period.

LP-style regressions are used in difference-in-differences and event study designs to estimate dynamic treatment effects, particularly under staggered treatment timing. Dube et al. (2025) develop a formal LP approach to difference-in-differences that accommodates heterogeneous and time-varying treatment effects.^[9]

To reduce sampling variability at longer horizons, Barnichon and Brownlees (2019) propose imposing smoothness on ${\displaystyle \{\beta _{h}\}}$ via basis expansions and penalization, trading some bias for lower variance relative to unconstrained LPs.^[10]

Local projections are implemented in several econometric software environments:

• Stata: the built-in command lpirf computes local-projection impulse-response functions.^[11]
• R: the package lpirfs estimates linear and nonlinear local-projection impulse responses.^[12]
• Python: the statsmodels library includes local projection functionality as part of its time series module.

• Impulse response
• Vector autoregression
• Structural vector autoregression
• Distributed lag model
• Instrumental variables estimation
• Newey–West estimator
• Panel data
• Fixed effects model
• Difference in differences
• Event study

Category:Econometrics Category:Time series analysis Category:Regression analysis Category:Causal inference