Estimation and Inference of Impulse Responses by Local Projections

Questions & answers

Q1. What problem does the local-projections method solve, and how is it defined?

Jordà seeks a way to estimate impulse responses — the difference between the conditional expectation of a future outcome under a given shock and under no shock — directly from the data, without committing to (and being potentially misspecified by) a particular VAR lag structure, while still producing standard errors that remain valid when the fitted VAR is wrong. The local-projection estimator at horizon s is obtained from a separate linear regression of y_{t+s} on lagged values of all variables in the system, run independently for each horizon s = 0, …, H; the impulse response at that horizon is simply the estimated coefficient on the shock (or its associated variable) in that horizon-specific regression, rather than being derived by iterating a single estimated VAR forward multiple periods.

Q2. How does local projections relate to VAR-based impulse responses, and why can LP remain valid when a VAR is misspecified?

When the true data-generating process actually is a VAR(p), the local-projection point estimates coincide exactly with the VAR-implied impulse responses — a VAR is, in this sense, a restricted special case of local projections — but local projections remain consistent even when the underlying process does not have a standard Wold (linear, Gaussian) representation, for instance when the process is nonlinear or non-Gaussian. The reason is structural: at horizon s, the local-projection residual has a moving-average structure of order s (it accumulates the shocks between period t and t+s), a feature that Newey-West heteroskedasticity-and-autocorrelation-consistent standard errors can correctly account for by setting the lag-correction parameter equal to the horizon s itself — in contrast, VAR-based standard errors computed by iterating a single fitted model forward can behave anomalously at long horizons (documented in Sims and Zha 1999), a problem local projections’ horizon-by-horizon regression approach avoids.

Q3. How does the paper extend local projections to allow for nonlinear impulse responses?

Standard linear VAR/LP models impose four implicit restrictions on impulse responses: symmetry between positive and negative shocks of the same size, shape invariance regardless of shock size, independence from the economy’s prior history, and (in multivariate systems) restrictions on cross-variable interactions; Jordà relaxes these using a Volterra series expansion, approximated in practice by a cubic local-projection specification that includes quadratic and cubic interaction terms in the lagged variables. The resulting nonlinear impulse response depends on both the size of the shock and the state of the economy (the lagged values) at the time it occurs — a feature linear VARs rule out by construction, and one the paper’s Monte Carlo and empirical exercises both exploit.

Q4. What does the Monte Carlo evidence show about local projections’ consistency and efficiency relative to a VAR?

Using a six-variable monthly VAR(12) calibrated to Evans and Marshall (1998) as the true data-generating process (494 observations, 1,000 replications), a two-lag local-projection estimator (LP(2)) stays within the true ±2-standard-error bands at every horizon, while a two-lag VAR (VAR(2)) — deliberately underspecified relative to the true 12-lag process — exhibits a statistically significant price puzzle, with the price level’s response to a contractionary funds-rate shock remaining significantly positive for the first 17 periods, a misspecification artifact absent from the local-projection estimates. When the VAR is correctly specified, LP’s Newey-West standard errors are approximately equal to the VAR’s Monte Carlo standard errors, but the VAR’s standard errors decline anomalously at long horizons (a known issue documented elsewhere), whereas the LP standard errors remain stable or grow, which the paper treats as evidence LP’s inference is more reliable at longer horizons.

Q5. What does the nonlinear Monte Carlo exercise show?

Using a three-variable structural VAR with GARCH (time-varying volatility) as the true nonlinear data-generating process (300 observations, 500 replications), the cubic local-projection specification closely tracks the true nonlinear impulse response at all horizons, while a time-varying-parameter VAR (following Cogley and Sargent) fails to capture the nonlinearity — its estimated impulse responses show essentially no variability over the first 6-7 periods, a direct consequence of the model’s underlying linearity assumption even though its coefficients are allowed to drift over time.

Q6. What does the empirical application find about the output-gap response to a monetary tightening, comparing VAR and local-projection estimates?

Using a quarterly three-variable system (the CBO output gap, GDP deflator inflation, and the federal funds rate, 1955:I-2003:I, Cholesky-identified with the funds rate ordered last), a 0.8-percentage-point contractionary funds-rate shock produces a peak output-gap decline of about −0.25% at a 12-quarter horizon in a linear VAR(4), while the equivalent linear local-projection estimate implies a peak decline of roughly twice that magnitude (about −0.5% at 12 quarters), a finding the cubic (nonlinear) local projection confirms. For inflation, the VAR(4) response is mostly positive but statistically insignificant, while the cubic local projection reveals a pronounced disinflationary decline beginning around the 7-quarter horizon — suggesting the linear VAR was masking a genuinely nonlinear response of inflation to monetary policy that the local-projection approach is able to recover.

Q7. What do the paper’s threshold tests reveal about regime-dependence in the inflation-output relationship?

Applying Hansen’s (2000) LM threshold test equation-by-equation, the paper finds statistically significant threshold effects: lagged inflation is a significant threshold variable in the inflation equation (with an identified threshold around 4.75% annual inflation), and the lagged federal funds rate is a significant threshold variable in the funds-rate equation (with an identified threshold around 6%). Estimating regime-dependent impulse responses around these thresholds, the paper finds the low-inflation regime — corresponding broadly to the post-mid-1980s period, when inflation has generally been below the 4.75% threshold — displays substantially larger output-gap and inflation responses to funds-rate shocks than the high-inflation regime of the 1970s and early 1980s, leading Jordà to conclude that “the price puzzle does not characterize the current economic environment.”

Q8. What costs or limitations does the paper itself identify for the local-projections approach?

Jordà notes that local projections use more degrees of freedom than a VAR for a given horizon, since each horizon’s regression estimates a fresh set of coefficients rather than reusing a single fitted model iterated forward — an efficiency cost the author frames as “the price of robustness.” He also notes that Newey-West standard errors must have their lag-correction parameter set equal to the specific forecast horizon to correctly account for the resulting moving-average error structure (understating this correction produces overly narrow confidence bands), that the cubic approximation to the general Volterra series captures only up to third-order nonlinearities, and that the paper’s own empirical application relies on a simple three-variable Cholesky-identified system subject to the standard critiques of that identification approach (e.g., simultaneity among financial variables and the Fed’s informational advantage).