Understanding the relationship between Local Projections (LPs) and Vector Autoregressions (VARs) is crucial for applied macroeconomists and econometricians. While often portrayed as competing approaches, recent theoretical work shows they estimate the same population impulse responses under broad conditions. This equivalence has important implications for empirical practice: the choice between LPs and VARs is less about correctness and more about trade-offs in efficiency, robustness, and flexibility. The following expanded takeaways summarize the key results and practical guidance from Møller and Wolf (2021).
1. LPs and VARs estimate the same impulse responses in population.
The core theoretical result of the paper is that linear Local Projections (LPs) and Vector Autoregressions (VARs) identify the same impulse response functions (IRFs), as long as the lag structures are unrestricted. This equivalence holds nonparametrically under weak assumptions like stationarity and purely nondeterministic processes, even if the true data-generating process (DGP) is nonlinear or infinite-dimensional.
2. LPs and VARs are simply two methods of dimension reduction
Rather than fundamentally different models, LPs and VARs should be understood as alternative techniques for reducing a high-dimensional forecasting problem into something tractable. Both aim to estimate the same underlying response of a variable to a structural shock, but they differ in how they use and weight the available information.
3. LPs and VARs behave differently in finite samples.
In finite samples, LPs and VARs have different small-sample properties. VARs are more efficient under correct model specification because they use information iteratively and enforce internal consistency across horizons. LPs, by contrast, estimate each horizon independently and are thus more robust to misspecification but may suffer from higher variance, especially at longer horizons.
4. At short horizons, LP and VAR estimates often agree in practice.
If both LPs and VARs include the same number of lags (say, p), their impulse response estimates will approximately coincide for horizons h ≤ p. This provides practical reassurance that the two methods yield similar results in many empirical applications focused on short-term dynamics.
5. Structural identification is separate from estimation method.
The paper emphasizes that identification (e.g., sign restrictions, long-run restrictions) is a population-level conceptual issue, whereas the choice of estimation method (LP or VAR) is about how one estimates impulse responses. Therefore, identification strategies traditionally associated with VARs can also be implemented in LPs.
6. All standard SVAR identification schemes can be translated into LPs. Recursive (Cholesky), long-run, and sign restrictions commonly used in Structural VARs can all be expressed using LP frameworks. For example, recursive identification can be achieved by projecting on appropriately ordered variables, and sign restrictions can be implemented via linear programming on LP coefficients.
7. LP-IV and recursive VARs with the instrument ordered first are equivalent.
In cases where an external instrument (proxy variable) is used to identify shocks (as in LP-IV), the paper shows that the same relative impulse responses can be obtained using a recursive VAR where the instrument is placed first. This “internal instrument” VAR approach works even under noninvertibility of the structural shock.
8. LP-IV is more robust than SVAR-IV under noninvertibility.
When structural shocks are not invertible, SVAR-IV fails to identify the correct impulse response. However, LP-IV and recursive VARs with the instrument ordered first remain valid, making them preferable in settings with measurement error or weak shocks.
9. LPs and VARs both estimate best linear approximations in nonlinear models.
Even when the true DGP is nonlinear, both LPs and VARs estimate a best linear approximation of the true impulse response. This supports their use in practice, where fully nonlinear or regime-switching models are difficult to estimate reliably.
10. VARs extrapolate long-horizon dynamics from short-term patterns.
Because VARs use iterated forecasting, they impose structure across horizons. This makes them more efficient under correct specification but potentially biased at long horizons if the lag length is too short. LPs, which estimate each horizon separately, avoid this extrapolation and can be more accurate at longer horizons.
11. The difference between LP and VAR estimates vanishes asymptotically.
In large samples with sufficiently many lags, the difference between LP and VAR impulse response estimators converges to zero. The paper formally shows that the estimators are asymptotically equivalent under mild conditions.
12. Inference procedures should be chosen carefully for each method.
LPs allow for straightforward construction of confidence intervals using standard regression tools, while inference for VARs often relies on bootstrapping or asymptotic approximations.
13. LPs offer more flexibility in practice.
Because LPs estimate each horizon separately, they are easier to extend with time-varying coefficients, nonlinear terms, or state-dependent effects. This flexibility is useful in empirical work where structural relationships may vary over time or across regimes.
14. Empirical implication: both methods are valid, so choice depends on goals.
Researchers can choose either method depending on the empirical question, data characteristics, and preferences over bias versus variance. The paper’s results justify the use of either LPs or VARs as long as proper attention is paid to lag selection, sample size, and identification.
15. Misconceptions in the literature.
The paper directly addresses and refutes several widely held but incorrect claims in the literature:
• LPs and VARs estimate different objects — False.
• VARs are always more efficient — Only under correct specification.
• LPs are always more robust — Only in small samples or misspecified models.
• Structural restrictions require VARs — False; LPs can accommodate them too.
• SVAR-IV is superior to LP-IV — False if the shock is noninvertible.
In a nutshell, the paper demonstrates that, under certain conditions, local projections and vector autoregressions yield numerically equivalent impulse responses when subject to the same identifying restrictions. This finding challenges the common perception that LPs and VARs are fundamentally different approaches. Specifically, when a fixed number `p` of lags is used in both models, their impulse response estimands approximately coincide up to horizon p, without requiring parametric assumptions about the data-generating process. LPs offer greater modeling flexibility, making them especially useful in settings with nonlinearities, time variation, or potential model misspecification, whereas VARs are more rigid due to their imposed structure. However, this flexibility comes at a cost: VARs tend to be more efficient when correctly specified, highlighting a tradeoff between flexibility and efficiency. Importantly, identification strategies commonly associated with VARs—such as Cholesky decompositions, sign restrictions, and Blanchard-Quah long-run restrictions—can also be applied in the LP framework, validating LPs as a credible tool for structural analysis. Additionally, LPs offer simpler inference procedures, including the use of robust standard errors and separate regressions at each forecast horizon, which are complex to implement in a VAR.
Acknowledgements: I would express my sincere gratitude to Elyes for sharing this wonderful note.
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