Local Projections

This article is about the econometrics method. For the cartographic use of “local projection”, see Map projection.

Not to be confused with Local regression.

Local projections (LPs) are an econometric method for estimating the dynamic effect of a shock, policy change, or other intervention on an outcome over time, typically summarized as an impulse response function (IRF). Rather than estimating a full dynamic system such as a vector autoregression (VAR) and then deriving impulse responses from the fitted system, local projections estimate a sequence of horizon-specific regressions—one regression for each forecast horizon. The method was introduced by Òscar Jordà (2005) and is widely used in macroeconomics and applied econometrics due to its flexibility, including straightforward extensions to instrumental variables (IV), panel data with fixed effects, and nonlinear or state-dependent responses.

Overview

The core idea of local projections is to estimate how a variable responds at horizons $h=0,1,\dots ,H$ after a shock at time $t$ by running a regression that directly relates $y_{t+h}$ to the shock at $t$ and suitable controls. Repeating this for many horizons yields an estimated response path $\{{\hat{\beta }}_h\}$ that can be plotted as an impulse response function.

Method

Basic local projection regression

Let $y_t$ be an outcome of interest (e.g., output, inflation, employment), and let $s_t$ be a shock, policy variable, or other treatment measure at time $t$. For each horizon $h$, a baseline local projection estimates: $$y_{t+h}={\alpha }_h+{\beta }_h{s}_t+{{\gamma }_h}^{\prime }{X}_t+u_{t,h},h=0,1,\dots ,H.$$

Here:

• ${\beta }_h$ is interpreted as the response of $y$ at horizon $h$ to a one-unit change in $s_t$, conditional on controls.
• $X_t$ is a vector of predetermined controls, often including lags of $y$ and lags of $s$ (and possibly other variables).
• $u_{t,h}$ is the horizon-specific error term.

Estimating the regression separately for each horizon produces the sequence $\{{\hat{\beta }}_h{\}}_{h=0}^H$. Plotting ${\hat{\beta }}_h$ against $h$ yields an estimated impulse response function.

Interpretation as a linear projection

Each ${\beta }_h$ is the coefficient of the best linear predictor (population linear projection): $$({\alpha }_h,{\beta }_h,{\gamma }_h)=\arg \underset{a,b,c}{\min }\mathbb{𝔼}\left[{(y_{t+h}-a-b{s}_t-c^{\prime }{X}_t)}^2\right].$$ Under conditions such as $\mathbb{𝔼}[u_{t,h}\mid s_t,X_t]=0$, ${\beta }_h$ is often interpreted as a conditional causal response at horizon $h$.

Impulse responses as conditional expectations

A common reduced-form definition of an impulse response at horizon $h$ is a change in a conditional expectation of $y_{t+h}$ following a change in the shock or intervention $s_t$, holding fixed an information set ${{ℱ}}_{t-1}$: $$\mathrm{I}\mathrm{R}\mathrm{F}(h)=\frac{\partial }{\partial \delta }\mathbb{𝔼}[y_{t+h}\mid s_t=\delta ,{{ℱ}}_{t-1}]{|}_{\delta =0}.$$ In linear settings, if the controls $X_t$ span (or adequately approximate) the relevant information set ${{ℱ}}_{t-1}$, this object coincides with the coefficient in the corresponding linear projection of $y_{t+h}$ on $s_t$ and $X_t$.

Cumulative responses and multipliers

Some applications report cumulative effects: $$\mathrm{C}\mathrm{I}\mathrm{R}\mathrm{F}(h)=\sum _{j=0}^h{\beta }_j,$$ or ratios of cumulative responses (e.g., fiscal multipliers), depending on the shock and outcomes studied.

Identification

Local projections are an estimation strategy; causal interpretation depends on identification.

Exogenous or externally measured shocks

If $s_t$ is constructed to be plausibly exogenous (e.g., a policy “surprise” measured from high-frequency data or a narrative shock), then conditioning on appropriate controls may justify treating $s_t$ as conditionally exogenous.

Instrumental variables (LP-IV)

When $s_t$ is endogenous, LPs can be combined with two-stage least squares using an instrument $z_t$. A common approach estimates horizon-specific IV regressions with $s_t$ instrumented by $z_t$. This strategy is widely used in macroeconomics to estimate dynamic causal effects using external instruments (“proxy” identification).

Inference

Because LP regressions use overlapping dependent variables (e.g., $y_{t+h}$ and $y_{t+h+1}$ share observations) and because multi-step forecast errors can be serially correlated, the regression errors $u_{t,h}$ often exhibit autocorrelation. Consequently, empirical work commonly uses heteroskedasticity-robust and autocorrelation-robust standard errors such as the Newey–West estimator, or resampling methods for confidence intervals and bands.

Montiel Olea and Plagborg-Møller (2021) analyze inference for LPs and show that “lag augmentation”—including additional lags in $X_t$—can simplify inference in settings with persistence and long horizons under certain conditions.

Relationship to other methods

Connection to VAR impulse responses

In linear settings, LP impulse responses are closely related to VAR-based impulse responses. Plagborg-Møller and Wolf (2021) show that, under appropriate conditions and with a sufficiently rich set of controls, linear LPs and VARs target the same population impulse responses; differences arise from finite-sample performance and specification choices.

Distributed-lag and direct forecasting interpretation

Each horizon-specific LP is related to a distributed lag model and to “direct” multi-step forecasting regressions, in contrast to iterating one-step-ahead forecasts from a fitted model.

Extensions

State dependence and nonlinear local projections

LPs are frequently generalized to allow responses to vary across regimes or states by interacting the shock with a state indicator $q_t$: $$y_{t+h}={\alpha }_h+{\beta }_{h,0}{s}_t+{\beta }_{h,1}(s_t\cdot q_t)+{{\gamma }_h}^{\prime }{X}_t+u_{t,h}.$$ This framework is used to study asymmetries and regime dependence in macroeconomic responses.

Panel local projections

With panel data (units $i$ over time $t$), LPs often include unit and time fixed effects: $$y_{i,t+h}={\alpha }_h+{\beta }_h{s}_{i,t}+{{\gamma }_h}^{\prime }{X}_{i,t}+{\mu }_i+{\tau }_t+u_{i,t,h}.$$ Standard errors are often clustered by unit (and sometimes two-way clustered).

Event studies and difference-in-differences

LP-style regressions are used to estimate dynamic treatment effects in difference-in-differences and event study designs, especially with staggered treatment timing.

Smooth and regularized local projections

To reduce sampling variability across horizons, some methods impose smoothness or regularization on $\{{\beta }_h\}$. “Smooth local projections” use basis expansions and penalties to trade bias for variance at longer horizons.

Software

Local projections are implemented in a range of econometrics software:

• Stata: lpirf for local-projection impulse responses.
• R: packages such as lpirfs for linear and nonlinear local-projection impulse responses.

See also

• 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

References

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