Impulse Responses by Local Projections

One equation per horizon

Local projections estimate the response at horizon $h$ directly (Jorda, 2005).

The sequence $\{\beta_h\}$ is the impulse response. Each horizon can use the same controls and a separate covariance estimate.

A four-horizon response

The cumulative response through horizon three is $1.00+0.70+0.30-0.10=1.90$.

Assumptions and implementation

Shock identification and control selection determine the causal interpretation. Serial correlation and overlapping outcomes create horizon-dependent error dependence.

Newey-West inference requires a kernel, a lag length, and a finite-sample convention. Software can also shift lead and lag indexing. The horizon-flow figure tracks one shock through separate outcome equations.

Reading the response path

Each local-projection coefficient answers a separate horizon question. If $x_t$ is a one-unit innovation, $\beta_h$ is the expected change in $y_{t+h}$ associated with that innovation after the stated controls. Plot the estimates against $h$ with confidence bands and a zero line. State whether the response is in levels, growth rates, or cumulative units.

A cumulative response through horizon $H$ is a sum of horizon-specific effects only when the equations and outcome transformation support that definition. Its standard error must include covariance across estimates. Pointwise bands describe each horizon separately. Simultaneous bands address coverage of the full plotted path.

Specification and uncertainty

Use a common sample across horizons when direct curve comparisons require it. Otherwise, report the changing observation count. Select lags before viewing the preferred response and keep the control set consistent with the shock identification. Forecast overlap induces serial correlation in horizon residuals, especially for $h>0$. Use a covariance estimator or bootstrap that reflects this dependence and any panel clustering.

Local projections are flexible, but long horizons can be noisy because fewer effective observations remain. Inspect sensitivity to lag length, trend terms, horizon range, and extreme shocks. If $x_t$ is not externally identified, the curve describes a conditional association. Recursive, proxy, or narrative identification adds assumptions that must be stated. Label the shock normalization because a one-standard-deviation and one-unit response have different scales. (Jorda, 2005)

Further reading

Jordà introduces local projections (Jorda, 2005). Plagborg-Møller and Wolf compare their population impulse responses with VAR responses (Plagborg-Moller and Wolf, 2021).