Do Local Projections Help with Causal Identification?

1. Main point

Local projections (LPs) do not, by themselves, solve the identification problem. The identifying assumption must come from the economic structure imposed on the shock. In the application with annual country-level geopolitical risk (GPR), inflation, and GDP growth, the identifying restriction is the same as in the Cholesky panel VAR: the annual innovation to country-specific GPR is treated as predetermined with respect to contemporaneous macroeconomic innovations. LPs are useful because they estimate the dynamic effects of that same identified shock horizon by horizon, without imposing the VAR’s recursive propagation structure on all future responses.

The clean interpretation is therefore the following. The panel VAR and the LP can be designed to use the same short-run restriction and the same GPR shock. The difference between the two approaches is not identification. The difference is estimation of the impulse response. The VAR estimates a full dynamic system and iterates it forward. The LP estimates each horizon directly. This distinction matters especially with annual data, where the number of usable observations is limited and where misspecification of dynamic propagation can affect medium- and long-horizon responses.

2. Baseline recursive panel VAR

Let the annual country-level vector be

Z_i,t = [

GPR_i,t

π_i,t

g_i,t

]

where GPR_i,t is the country-specific geopolitical risk index, π_i,t is inflation, and g_i,t is GDP growth. A simple panel VAR with 1 lag can be written as

Z_i,t = α_i + A₁Z_i,t−1 + u_i,t,

with reduced-form residuals

u_i,t = [

u_i,t^GPR

u_i,t^π

u_i,t^g

The recursive identification uses a Cholesky decomposition of the residual covariance matrix:

Σ_u = BB′,

where B is lower triangular. Structural shocks ε_i,t are defined by

u_i,t = Bε_i,t.

With the ordering

[ GPR_i,t, π_i,t, g_i,t ],

the first structural shock is the GPR shock. Because GPR is ordered first, the GPR shock is simply proportional to the reduced-form innovation in GPR:

ε_i,t^GPR ∝ u_i,t^GPR.

The economic restriction is that inflation and GDP growth may respond within the same year to a GPR shock, but GPR does not respond within the same year to inflation or GDP-growth innovations. In words, annual macroeconomic conditions are not allowed to contemporaneously cause the identified GPR innovation. This is the short-run restriction embedded in the Cholesky ordering.

3. Equivalent LP implementation

The same short-run restriction can be implemented in local projections. The first step is to define the GPR innovation using the same information set as in the VAR. With 1 lag, this can be written as

GPR_i,t = a_i + ρ₁GPR_i,t−1 + ρ₂π_i,t−1 + ρ₃g_i,t−1 + η_i,t^GPR.

The residual

η̂_i,t^GPR

is the LP analogue of the Cholesky-identified GPR shock. The horizon-specific LP is then

y_i,t+h = α_i^h + β_hη̂_i,t^GPR + Γ_h′Z_i,t−1 + e_i,t+h^h, h = 0, 1, 2, …, H,

where y_i,t+h is either inflation or GDP growth at horizon h. The coefficient β_h gives the response of the outcome at horizon h to the same GPR innovation used in the recursive VAR.

Equivalently, by the Frisch-Waugh-Lovell theorem, the LP can be estimated directly as

y_i,t+h = α_i^h + β_hGPR_i,t + Γ_h′Z_i,t−1 + e_i,t+h^h.

In this direct regression, the coefficient on GPR_i,t is the coefficient on the part of GPR_i,t that remains after residualizing it with respect to the included lagged controls. That residualized component is precisely the GPR innovation under the recursive timing assumption.

4. Why the short-run restriction is equivalent in LP and VAR

The equivalence is easiest to see in a more general notation. Let x_i,t denote the impulse variable, here country-specific GPR. Let r_i,t denote variables ordered before x_i,t in a recursive Cholesky system. Let W_i,t−1 denote the vector of lagged controls. A local projection at horizon h can be written as

y_i,t+h = μ_i^h + θ_hx_i,t + γ_h′r_i,t + δ_h′W_i,t−1 + ξ_i,t+h^h.

By Frisch-Waugh-Lovell, θ_h is the coefficient from projecting y_i,t+h on the residualized impulse variable

x̃_i,t = x_i,t − proj( x_i,t | r_i,t, W_i,t−1 ).

In a recursive VAR with the same ordering and the same lag controls, the Cholesky shock to x_i,t is also the innovation in x_i,t after removing the information contained in variables ordered before it and in lagged variables:

ε_i,t^x ∝ x_i,t − proj( x_i,t | r_i,t, W_i,t−1 ).

Therefore,

x̃_i,t ∝ ε_i,t^x.

This is the core equivalence. LP and VAR use the same residualized shock when they impose the same recursive ordering and the same control set. The short-run restriction is not a property of the VAR estimator alone. It is a restriction on the construction of the shock. Once the shock is defined as the residual of x_i,t after conditioning on the variables ordered before it and on the lagged information set, the same shock can be used either in a VAR or in an LP.

For the specific GPR-first ordering, there are no contemporaneous variables ordered before GPR. Hence r_i,t is empty and the residualized shock becomes

GPR̃_i,t = GPR_i,t − proj( GPR_i,t | Z_i,t−1 ).

This is exactly the annual GPR innovation used by both approaches. The implied short-run restriction is that contemporaneous inflation and GDP growth do not enter the GPR equation. They are variables ordered after GPR, so they may respond contemporaneously to a GPR shock, but they do not help define that shock.

This point is important for implementation. If the aim is to reproduce the Cholesky ordering [GPR, π, g] in LPs, one should not include contemporaneous inflation or contemporaneous GDP growth as controls in the GPR-shock LP. Including them would partial out part of the within-year response that the Cholesky ordering intentionally allows. The LP should include lags of GPR and macroeconomic variables, and possibly fixed effects and other predetermined controls, but not contemporaneous variables ordered after GPR.

5. What LP adds relative to the Cholesky VAR

The VAR estimates the reduced-form law of motion

Z_i,t = α_i + A₁Z_i,t−1 + u_i,t

and then computes impulse responses by iterating the estimated dynamics forward:

IRF_h^VAR = A₁^hBe_GPR.

This gives smooth impulse responses and usually narrower uncertainty bands. The cost is that medium- and long-horizon responses are strongly shaped by the finite-lag VAR approximation. If the true propagation mechanism is richer than the estimated VAR, the VAR can be precise but biased.

The LP instead estimates

y_i,t+h = α_i^h + β_hη̂_i,t^GPR + Γ_h′Z_i,t−1 + e_i,t+h^h

separately for each horizon h. The coefficient β_h is not obtained by iterating a VAR coefficient matrix. It is estimated directly from the covariance between the GPR shock at time t and the outcome at time t+h. This makes LP responses less smooth and typically less precise, but less dependent on the assumed VAR propagation structure.

This is why LPs help in a causal-response exercise. They do not make the shock more exogenous. Rather, conditional on the same identifying assumption, they make the estimated dynamic causal effect less dependent on dynamic specification. For annual country data, this is valuable because the sample is long historically but still limited for dynamic macroeconomic estimation, and because the effects of geopolitical shocks may propagate through heterogeneous and slow-moving channels such as military spending, public debt, money growth, trade disruptions, shortages, and output losses.

6. Interpretation with annual GPR, inflation, and GDP growth

In the annual GPR application, the baseline causal statement should be formulated carefully. Under the recursive timing restriction, a country-specific GPR innovation is interpreted as an annual geopolitical shock that is not contemporaneously caused by inflation or GDP growth. The response of inflation or GDP growth in the same year and subsequent years can then be interpreted as the dynamic effect of this GPR innovation.

This assumption is plausible for many geopolitical episodes, such as wars, invasions, terrorist attacks, foreign military threats, and sudden diplomatic crises. It is stronger for annual data than for monthly or quarterly data because a year is a long period. Some domestic political crises may themselves be partly triggered by inflation, debt distress, or social unrest. For this reason, the recursive design is more convincing when complemented by robustness checks: excluding economically driven episodes, adding global GPR controls, including year fixed effects, using narrative GPR shocks, or comparing VAR results with LP responses.

References

Caldara, D., Conlisk, S., Iacoviello, M., and Penn, M. (2026). “Do geopolitical risks raise or lower inflation?” Journal of International Economics, 159, 104188.

Jordà, Ò. (2005). “Estimation and inference of impulse responses by local projections.” American Economic Review, 95(1), 161–182.

Montiel Olea, J. L., Plagborg-Møller, M., Qian, E., and Wolf, C. K. (2025). “Local projections or VARs? A primer for macroeconomists.” NBER Macroeconomics Annual draft.

Plagborg-Møller, M., and Wolf, C. K. (2021). “Local projections and VARs estimate the same impulse responses.” Econometrica, 89(2), 955–980.