Local Projections

Definition

Local projections (Jordà 2005) estimate an impulse response function directly, horizon by horizon, by regressing the outcome $h$ periods ahead on the shock (or a shock proxy) plus controls — one separate regression per horizon $h$ — rather than iterating a fitted VAR forward. In the forecasting taxonomy, a local projection is the impulse-response counterpart of a direct multistep forecast, whereas the VAR-based impulse response is an iterated multistep forecast.

Key Ideas

Direct, not iterated. Each horizon- $h$ response comes from its own regression $Y_{t+h} = \theta_h X_t + (\text{controls}) + u_{t+h}$ ; the sequence $\{\theta_h\}$ traces the impulse response. No cross-horizon parametric restriction is imposed by an autoregressive structure.
Claimed robustness. LP is often argued to be robust to VAR misspecification (lag length, non-linearity, state dependence) because it does not extrapolate a single fitted model across horizons — though this robustness is not automatic and is set aside in linear treatments.
Efficiency cost. Relative to a correctly specified VAR, LP is less efficient (larger standard errors, especially at long horizons) and its estimates are serially correlated across horizons.
Inference. Long-horizon LP responses generally require HAR (heteroscedasticity- and autocorrelation-robust) standard errors.

How It Works

LP-IV (instrumented local projections)

When the regressor of interest is an endogenous shock, an external instrument $Z_t$ can be used, giving LP-IV (Jordà–Schularick–Taylor 2015; Ramey 2016). Stock and Watson (2018) show LP-IV identifies the structural impulse response without assuming invertibility, provided the instrument satisfies lead–lag exogeneity — uncorrelation with past and future shocks (after controls), plus contemporaneous relevance and exogeneity. This is the sense in which LP-IV can succeed where a VAR cannot: it never has to recover the shocks from current and lagged data.

The catch

The invertibility-freedom is conditional. Stock–Watson's "no free lunch" theorem shows that if the instrument depends on past shocks and lagged controls are added to compensate, the condition for those controls to validate LP-IV is generically equivalent to invertibility — in which case the VAR-based SVAR-IV estimator is more efficient.

Extensions

Cumulative effects / multipliers: ratios of cumulative responses (e.g. fiscal multipliers) are naturally estimated in the LP-IV framework.
Factor-augmented LP (factor-augmented local projections): augment the controls with factors from a dynamic factor model — the local-projection counterpart of a FAVAR.
Smooth local projections (Barnichon–Brownlees 2016; Plagborg-Møller 2016) shrink the horizon-by-horizon estimates toward smoothness.

Why It Matters

Local projections have become a standard alternative to VARs for estimating impulse responses in applied macroeconomics, valued for flexibility and transparency. Combined with external instruments, LP-IV is a leading tool for credibly estimating dynamic causal effects of monetary and fiscal shocks — and its comparison with SVAR-IV underpins a Hausman-type test of invertibility.

Open Questions

Whether the informal robustness of LP to non-linearity and misspecification holds up under scrutiny — a non-linear analogue of the no-free-lunch result is conjectured.
Optimal choice of controls and horizon-dependent lag structure.
Weak-instrument-robust inference for LP-IV at long horizons.