Local Projections and VARs Estimate the Same Impulse Responses

The paper asks whether local projections (LP) and vector autoregressions (VARs) estimate the same impulse responses, and it answers yes: it proves a nonparametric population equivalence result showing the two methods recover the same impulse response function, up to a constant of proportionality, for reduced-form responses and for a wide class of structural identification schemes. This generalizes an informal observation by Jordà (2005), who noted the equivalence only under the assumption that the data are actually generated by a finite-order VAR(p); Plagborg-Møller and Wolf instead assume nothing about the true data-generating process beyond general regularity conditions, so the equivalence is not conditional on the VAR being “correctly specified” in any parametric sense.

The researcher observes a vector w_t = (r_t’, x_t, y_t, q_t)’ – contemporaneous controls r_t, a scalar impulse variable x_t, a scalar response variable y_t, and additional controls q_t ordered after y_t – and the paper’s two maintained assumptions are that {w_t} is covariance stationary and purely nondeterministic with an everywhere-nonsingular spectral density matrix and absolutely summable Wold decomposition coefficients (Assumption 1), and, for notational convenience only, that {w_t} is jointly Gaussian (Assumption 2). The authors note that all results go through with linear projections replacing conditional expectations if Gaussianity is dropped, so Assumption 2 is not doing substantive work.

Proposition 1 states that the recursive-VAR impulse response theta_h and the LP impulse response beta_h are equal up to a constant of proportionality at every horizon: theta_h = sqrt(E(x-tilde_t squared)) times beta_h for all h = 0, 1, 2, …, where x-tilde_t is the population residual of x_t after projecting out r_t and the full infinite history of lagged w’s, and the scale factor depends on neither the horizon nor which response variable y_t is examined. The proof uses the Frisch-Waugh theorem to write beta_h as Cov(y_{t+h}, x-tilde_t) / E(x-tilde_t squared), and shows separately, via the VAR(infinity) Wold decomposition, that the recursive-VAR object theta_h equals Cov(y_{t+h}, eta_{x,t}) where the orthogonalized shock eta_{x,t} is a rescaled version of the same population residual x-tilde_t; comparing the two expressions delivers the result. The authors describe the underlying intuition simply: a VAR(infinity) with enough lags perfectly captures the covariance structure of the data, iterated VAR forecasts coincide with direct LP forecasts, and since both impulse responses are linear functions of the same reduced-form forecasts, they must coincide in population.

It extends well beyond simple recursive identification: the paper shows that any structural rotation of the VAR’s reduced-form shocks – recursive or not – corresponds to an LP on a particular linear combination of the variables in the system, and works through this explicitly for three leading identification strategies. The Christiano-Eichenbaum-Evans (2005) recursive ordering can be implemented via an LP of the response variable on the funds rate ordered after real/price variables, giving the same population impulse response as the recursive VAR (Example 1). The Blanchard-Quah (1989) long-run restriction – that a demand shock has zero long-run effect on the level of output – can be imposed via a “long-difference” LP taken to a large horizon H, which as H approaches infinity correctly identifies the relative impulse responses with respect to the supply shock, up to a constant scale factor (Example 2). And sign restrictions in the style of Uhlig (2005) and Rubio-Ramírez, Waggoner, and Zha (2010) can be imposed on reduced-form LP coefficients via a linear program, recovering exactly the same identified set as the analogous SVAR sign-restriction exercise (Example 3).

No, not exactly: Proposition 2 shows that with a fixed lag length p, the LP(p) and VAR(p) impulse-response estimands agree only approximately at horizons h less than or equal to p, and generally disagree at horizons h greater than p, with the size of the discrepancy captured by a remainder term that is exactly zero only when the impulse variable is a “direct” shock uncorrelated with the controls and all past data. The paper illustrates this with a Smets-Wouters (2007) DSGE model used as the data-generating process: LP(p) and VAR(p) impulse responses agree exactly up to horizon h = p and then visibly diverge beyond it (Figure 1). The practical takeaway the authors draw is that choosing too short a lag length p is the main way researchers will see LP and VAR estimates disagree in practice, and that disagreement should show up specifically at horizons beyond p.

The paper shows the gap between the LP and VAR sample estimators vanishes asymptotically as the lag length p is allowed to grow with the sample size T at an appropriate rate, and that both estimators are asymptotically efficient at any fixed horizon under weak regularity conditions – but it stops short of saying which estimator performs better in any actual finite sample. At finite p and T, the authors state plainly that researchers must still navigate a bias-variance trade-off at long horizons, and they explicitly flag this as an open question for future research (their footnotes note the gap directly). The library’s wiki entry records that this specific gap was subsequently taken up by Li, Plagborg-Møller, and Wolf (2024), who study the finite-sample bias-variance trade-off left open here.

The paper shows that “LP-IV” – a two-stage local projection using an external instrument z_t – is numerically identical to an “internal instrument” recursive SVAR in which z_t is simply added to the VAR and ordered first (Corollary 1), and that this equivalence holds even when the structural shock is not invertible with respect to the original variables alone, because adding the instrument to the system resolves that noninvertibility. Both approaches identify only relative impulse responses (the response of y to the shock, scaled by the impact response of x to the shock). This is explicitly different from the popular “SVAR-IV” external-instrument procedure (Stock and Watson 2012; Mertens and Ravn 2013), which runs a VAR on the original variables alone and then projects the reduced-form residuals onto the instrument after the fact: that approach is only consistent when the underlying shock is invertible from current and past data, and under noninvertibility it misidentifies a different shock entirely, though when it is consistent it identifies absolute (not merely relative) impulse responses.

Using Gertler and Karadi’s (2015) monthly data from January 1990 to June 2012 (industrial-production growth, inflation, the one-year government bond rate, and the excess bond premium, with the Gertler-Karadi high-frequency futures surprise as the monetary-policy instrument), the paper compares LP against the internal-instrument recursive VAR at lag lengths p = 4 and p = 12 and finds the two estimators agree closely up to horizon h = p in each case and diverge substantially beyond it, directly illustrating Proposition 2 out of sample. Across specifications, the excess bond premium rises initially following a contractionary monetary shock, consistent with the original Gertler-Karadi (2015) finding. The wiki source flags that the specific numerical magnitudes plotted in the paper’s Figure 2 have not been independently re-verified against the figure itself, so only the qualitative pattern – agreement up to h = p, divergence beyond it, and the sign of the excess-bond-premium response – is reported here rather than point estimates.

The authors argue that four widely held beliefs are mistaken: that VAR impulse-response estimators are generally more efficient than LP estimators; that LP is generally more robust to misspecification than VARs; that SVAR analysis is required to implement nonrecursive, non-IV identification schemes such as long-run or sign restrictions; and that simple SVAR methods cannot be used when the structural shock of interest is noninvertible. At the same time, the paper is explicit about what it does not do: the equivalence is a population result that does not by itself determine which estimator has better finite-sample properties; the analysis is restricted to linear estimators, so the common LP/VAR estimand is only the best linear approximation to the true (possibly nonlinear) structural impulse response; the paper deliberately sets aside questions of inference, referring readers instead to Jordà (2005), Kilian and Lütkepohl (2017), and Stock and Watson (2018); and it restricts attention to a single instrument, leaving the case of multiple instruments for the same shock to future work. The results also require that LP and VAR use lag structures that are either both unrestricted or truncated to the same p – differing truncation choices are themselves a source of the divergence documented in Proposition 2.