Local Projections

• Comment: This appears to be written to seek to create a faux-notability for a neologism which is net yet notable, and may never be notable. The concatenation of the words 'local' and 'projections' does not appear likely to enter general usage as a closely defined entity. Rather, it is likely to mean different things in different topic areas. 🇵🇸‍🇺🇦 Fiddle^Timtrent Faddle^Talk to me 🇺🇦‍🇵🇸 11:39, 12 September 2026 (UTC)

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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 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. It has since become a standard tool in empirical macroeconomics and applied econometrics.

History and reception

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. 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. 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. Jordà and Taylor (2025) provide a comprehensive survey of the method and its extensions.

Method

Basic regression

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

$$y_{t+h}={\alpha }_h+{\beta }_h{s}_t+{{\gamma }_h}^{\prime }{X}_t+u_{t,h}$$

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

Under the assumption $𝔼[u_{t,h}\mid s_t,X_t]=0$, the coefficient ${\beta }_h$ identifies the conditional expectation:

$$\mathrm{I}\mathrm{R}\mathrm{F}(h)=\frac{\partial }{\partial \delta }𝔼[y_{t+h}\mid s_t=\delta ,{ℱ}_{t-1}]{|}_{\delta =0},$$

where ${ℱ}_{t-1}$ denotes the relevant information set spanned by the controls, provided these controls adequately approximate that information set.

Cumulative responses

Some applications report cumulative impulse responses:

$$\mathrm{C}\mathrm{I}\mathrm{R}\mathrm{F}(h)=\sum _{j=0}^h{\beta }_j,$$

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.

Identification

Local projections are an estimation strategy; causal interpretation requires identification assumptions on $s_t$.

Exogenous shocks

If $s_t$ 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 $s_t$ as conditionally exogenous, supporting a causal interpretation of ${\hat{\beta }}_h$.

Instrumental variables (LP-IV)

When $s_t$ is endogenous, local projections can be combined with instrumental variables. A common implementation applies two-stage least squares at each horizon separately, instrumenting $s_t$ with an external instrument $z_t$ (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.

Inference

Because LP regressions use overlapping windows of data—for example, $y_{t+2}$ appears in both the $h=2$ and the $h=1$ regressions run from different starting points—the residuals $u_{t,h}$ 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.

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.

Relationship to other methods

Local projections and VARs

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. 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.

Distributed-lag and multi-step forecasting

Each horizon-specific LP regression is a "direct" multi-step forecasting regression, projecting the outcome $h$ 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.

Extensions

State-dependent local projections

LPs can be generalized to allow responses to vary across economic regimes 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 asymmetric dynamics, such as whether fiscal or monetary policy effects differ between recessions and expansions. Gonçalves et al. (2024) analyze inference in this setting.

Panel local projections

With panel data indexed by unit $i$ and time $t$, local projections commonly 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 typically clustered by unit, or two-way clustered by unit and time period.

Difference-in-differences and event studies

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.

Smooth local projections

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

Software

Local projections are implemented in several econometric software environments:

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

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

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

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