Equivalence of Cholesky identified shocks in VARs and Local Projections

Local projections have become a standard tool in applied macroeconometrics. They are widely used to estimate dynamic responses to shocks, especially when researchers do not want to impose the full dynamic structure of a vector autoregression. However, it is important to be precise about what local projections do and what they do not do.

Main point. Local projections do not solve the identification problem by themselves. The identifying assumption must come from the economic structure imposed on the shock. What local projections add is a flexible way to estimate the dynamic response to that shock.

This distinction is especially important when studying the macroeconomic effects of geopolitical risk. In the application considered here, the variables are annual country-specific geopolitical risk, inflation, and GDP growth. The key question is whether a geopolitical-risk innovation can be interpreted as predetermined with respect to contemporaneous macroeconomic innovations.

1. The baseline VAR system

Let the annual country-level vector be

Z_{i,t} = \begin{bmatrix} GPR_{i,t} \\ \pi_{i,t} \\ g_{i,t} \end{bmatrix}.

Here, GPR_i,t denotes country-specific geopolitical risk, π_i,t denotes inflation, and g_i,t denotes GDP growth.

The vector has size

Z_{i,t} \in \mathbb{R}^{3 \times 1}.

The baseline panel VAR with 1 lag is

Z_{i,t} = \alpha_i + A_1 Z_{i,t-1} + u_{i,t}.

The dimensions are

Z_{i,t} \in \mathbb{R}^{3 \times 1}, \qquad \alpha_i \in \mathbb{R}^{3 \times 1}, \qquad A_1 \in \mathbb{R}^{3 \times 3}, \qquad Z_{i,t-1} \in \mathbb{R}^{3 \times 1}, \qquad u_{i,t} \in \mathbb{R}^{3 \times 1}.

Therefore, the matrix multiplication is dimensionally consistent:

A_1 Z_{i,t-1} : (3 \times 3)(3 \times 1) = (3 \times 1).

The coefficient matrix is

A_1 = \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{bmatrix}.

Each row corresponds to one equation of the VAR. Each column corresponds to one lagged variable.

Thus, the system is equivalent to three equations:

GPR_{i,t} = \alpha_i^{GPR} + a_{11}GPR_{i,t-1} + a_{12}\pi_{i,t-1} + a_{13}g_{i,t-1} + u_{i,t}^{GPR},

\pi_{i,t} = \alpha_i^{\pi} + a_{21}GPR_{i,t-1} + a_{22}\pi_{i,t-1} + a_{23}g_{i,t-1} + u_{i,t}^{\pi},

g_{i,t} = \alpha_i^{g} + a_{31}GPR_{i,t-1} + a_{32}\pi_{i,t-1} + a_{33}g_{i,t-1} + u_{i,t}^{g}.

2. The residual covariance matrix

The reduced-form residual vector is

u_{i,t} = \begin{bmatrix} u_{i,t}^{GPR} \\ u_{i,t}^{\pi} \\ u_{i,t}^{g} \end{bmatrix}.

Its covariance matrix is

\Sigma_u = E(u_{i,t}u_{i,t}’).

Since

u_{i,t} \in \mathbb{R}^{3 \times 1}, \qquad u_{i,t}’ \in \mathbb{R}^{1 \times 3},

we have

u_{i,t}u_{i,t}’ : (3 \times 1)(1 \times 3) = (3 \times 3).

Therefore,

\Sigma_u \in \mathbb{R}^{3 \times 3}.

Explicitly,

\Sigma_u = \begin{bmatrix} \sigma_{11} & \sigma_{12} & \sigma_{13} \\ \sigma_{12} & \sigma_{22} & \sigma_{23} \\ \sigma_{13} & \sigma_{23} & \sigma_{33} \end{bmatrix}.

The reduced-form residuals are generally correlated. For example, the reduced-form innovation in inflation may be correlated with the reduced-form innovation in geopolitical risk. The identification problem is therefore to transform these correlated residuals into structural shocks that can be interpreted economically.

3. The Cholesky decomposition

The recursive identification uses the decomposition

\Sigma_u = BB’,

where B is lower triangular:

B = \begin{bmatrix} b_{11} & 0 & 0 \\ b_{21} & b_{22} & 0 \\ b_{31} & b_{32} & b_{33} \end{bmatrix}.

The dimensions are

\Sigma_u \in \mathbb{R}^{3 \times 3}, \qquad B \in \mathbb{R}^{3 \times 3}, \qquad B’ \in \mathbb{R}^{3 \times 3}.

The multiplication is

BB’ : (3 \times 3)(3 \times 3) = (3 \times 3).

The structural shocks are defined by

u_{i,t} = B\varepsilon_{i,t},

where

\varepsilon_{i,t} = \begin{bmatrix} \varepsilon_{i,t}^{GPR} \\ \varepsilon_{i,t}^{\pi} \\ \varepsilon_{i,t}^{g} \end{bmatrix}, \qquad E(\varepsilon_{i,t}\varepsilon_{i,t}’) = I_3.

Thus, the structural shocks are orthogonal, while the reduced-form residuals are allowed to be correlated.

The key calculation is

E(u_{i,t}u_{i,t}’) = E(B\varepsilon_{i,t}\varepsilon_{i,t}’B’) = B E(\varepsilon_{i,t}\varepsilon_{i,t}’) B’ = B I_3 B’ = BB’ = \Sigma_u.

Interpretation. The matrix B maps orthogonal structural shocks into reduced-form residuals with exactly the covariance structure observed in the VAR residuals.

4. What the recursive ordering means

With the ordering

\left[ GPR_{i,t}, \pi_{i,t}, g_{i,t} \right],

we have

u_{i,t} = \begin{bmatrix} u_{i,t}^{GPR} \\ u_{i,t}^{\pi} \\ u_{i,t}^{g} \end{bmatrix} = \begin{bmatrix} b_{11} & 0 & 0 \\ b_{21} & b_{22} & 0 \\ b_{31} & b_{32} & b_{33} \end{bmatrix} \begin{bmatrix} \varepsilon_{i,t}^{GPR} \\ \varepsilon_{i,t}^{\pi} \\ \varepsilon_{i,t}^{g} \end{bmatrix}.

Multiplying out gives

u_{i,t}^{GPR} = b_{11}\varepsilon_{i,t}^{GPR},

u_{i,t}^{\pi} = b_{21}\varepsilon_{i,t}^{GPR} + b_{22}\varepsilon_{i,t}^{\pi},

u_{i,t}^{g} = b_{31}\varepsilon_{i,t}^{GPR} + b_{32}\varepsilon_{i,t}^{\pi} + b_{33}\varepsilon_{i,t}^{g}.

This is the recursive contemporaneous structure.

The first equation says that the reduced-form GPR innovation depends only on the structural GPR shock:

u_{i,t}^{GPR} = b_{11}\varepsilon_{i,t}^{GPR}.

Therefore,

\varepsilon_{i,t}^{GPR} = \frac{u_{i,t}^{GPR}}{b_{11}}.

Central implication. When GPR is ordered first, the structural GPR shock is proportional to the reduced-form innovation in GPR.

5. The economic interpretation

The Cholesky ordering

\left[ GPR_{i,t}, \pi_{i,t}, g_{i,t} \right]

implies that geopolitical risk is predetermined within the year.

Inflation and GDP growth may respond contemporaneously to a GPR shock:

GPR \rightarrow \pi, \qquad GPR \rightarrow g.

Inflation may also affect GDP growth contemporaneously:

\pi \rightarrow g.

But the system rules out contemporaneous feedback from inflation and GDP growth to the identified GPR shock:

\pi \nrightarrow GPR, \qquad g \nrightarrow GPR.

In words, the annual innovation to country-specific geopolitical risk is treated as not contemporaneously caused by inflation or GDP-growth innovations.

This is not a statistical property of local projections. It is an economic identifying assumption.

6. The equivalent LP shock

The local projection should use the same identified shock as the recursive VAR.

The GPR innovation can be obtained from

GPR_{i,t} = a_i + \rho_1 GPR_{i,t-1} + \rho_2 \pi_{i,t-1} + \rho_3 g_{i,t-1} + \eta_{i,t}^{GPR}.

The residual

\widehat{\eta}_{i,t}^{GPR}

is the LP analogue of the Cholesky-identified GPR shock.

This residual is scalar:

\widehat{\eta}_{i,t}^{GPR} \in \mathbb{R}.

The key point is that this residual is the innovation in GPR after conditioning on lagged macroeconomic information. It corresponds to the same recursive timing assumption used in the VAR.

7. The horizon-specific local projection

For each horizon h, the LP is

y_{i,t+h} = \alpha_i^h + \beta_h \widehat{\eta}_{i,t}^{GPR} + \Gamma_h’Z_{i,t-1} + e_{i,t+h}^h.

Here, y_i,t+h is either inflation or GDP growth at horizon h.

The dimensions are

y_{i,t+h} \in \mathbb{R}, \qquad \alpha_i^h \in \mathbb{R}, \qquad \beta_h \in \mathbb{R}, \qquad \widehat{\eta}_{i,t}^{GPR} \in \mathbb{R},

Z_{i,t-1} \in \mathbb{R}^{3 \times 1}, \qquad \Gamma_h \in \mathbb{R}^{3 \times 1}, \qquad \Gamma_h’ \in \mathbb{R}^{1 \times 3}.

Thus,

\Gamma_h’Z_{i,t-1} : (1 \times 3)(3 \times 1) = (1 \times 1).

The coefficient β_h measures the response of the outcome at horizon h to the identified GPR innovation.

8. Why LP and VAR use the same short-run restriction

The equivalence is easiest to see in 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 (if the country GPR is order first, then this term can be dropped). Let W_i,t−1 denote lagged controls.

The local projection can be written as

y_{i,t+h} = \mu_i^h + \theta_h x_{i,t} + \gamma_h’ r_{i,t} + \delta_h’ W_{i,t-1} + \xi_{i,t+h}^h.

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

\widetilde{x}_{i,t} = x_{i,t} – \operatorname{proj} \left( x_{i,t} \mid r_{i,t}, W_{i,t-1} \right).

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:

\varepsilon_{i,t}^{x} \propto x_{i,t} – \operatorname{proj} \left( x_{i,t} \mid r_{i,t}, W_{i,t-1} \right).

Therefore,

\widetilde{x}_{i,t} \propto \varepsilon_{i,t}^{x}.

Core equivalence. LP and VAR use the same residualized shock when they impose the same recursive ordering and the same control set. The difference is not identification. The difference is how the dynamic response is estimated.

9. What local projections add

The VAR estimates the reduced-form law of motion

Z_{i,t} = \alpha_i + A_1Z_{i,t-1} + u_{i,t}

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

IRF_h^{VAR} = A_1^h B e_{GPR}.

The dimensions are

A_1^h \in \mathbb{R}^{3 \times 3}, \qquad B \in \mathbb{R}^{3 \times 3}, \qquad e_{GPR} \in \mathbb{R}^{3 \times 1}.

The selection vector is

e_{GPR} = \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}.

Hence,

A_1^h B e_{GPR} : (3 \times 3)(3 \times 3)(3 \times 1) = (3 \times 1).

The resulting impulse-response vector is

IRF_h^{VAR} = \begin{bmatrix} IRF_h^{GPR} \\ IRF_h^{\pi} \\ IRF_h^{g} \end{bmatrix}.

The VAR therefore gives a full dynamic response of all variables to the GPR shock.

The advantage is efficiency and smoothness. The cost is that medium- and long-horizon responses are shaped by the finite-lag VAR approximation.

By contrast, the LP estimates

y_{i,t+h} = \alpha_i^h + \beta_h \widehat{\eta}_{i,t}^{GPR} + \Gamma_h’Z_{i,t-1} + e_{i,t+h}^h

separately for each horizon h.

The coefficient β_h is not obtained by iterating A₁. It is estimated directly from the relationship between the GPR shock at time t and the outcome at time t+h.

This is why LPs are useful. They do not make the shock more exogenous. Instead, conditional on the same identifying assumption, they make the estimated dynamic causal effect less dependent on the VAR propagation structure.

10. Why this matters with annual geopolitical-risk data

Annual data create a specific challenge. A year is a long period, and many macroeconomic channels can operate within that horizon.

Geopolitical shocks may affect inflation and GDP growth through:

military spending,
public debt,
energy prices,
money growth,
trade disruptions,
shortages,
confidence effects,
exchange-rate movements,
and output losses.

A low-lag VAR may impose too restrictive a dynamic propagation mechanism. The LP approach is useful because it estimates the response horizon by horizon and is therefore less tied to a specific finite-lag approximation.

However, this flexibility does not remove the need for credible identification. The causal interpretation still depends on the recursive timing assumption that country-specific GPR innovations are not contemporaneously caused by inflation and GDP-growth innovations.

11. Recommended interpretation

A precise formulation is the following:

We implement the same recursive short-run restriction in both the panel VAR and the local projections. In the VAR, country-specific GPR is ordered first, so the GPR shock is the annual innovation in GPR after conditioning on lagged macroeconomic variables. In the LP, the impulse variable is the same residualized GPR innovation. Hence, the LP-VAR comparison does not reflect different identification assumptions. It evaluates whether the estimated dynamic effects of GPR on inflation and GDP growth are robust to the VAR’s finite-lag propagation restrictions.

This wording is important because it avoids overstating the role of local projections. The LP is not a new identification strategy. It is a more flexible estimator of the response to the same identified shock.

12. Case with global GPR and country-specific GPR

If global GPR is added, the vector becomes

Z_{i,t} = \begin{bmatrix} Global\ GPR_t \\ Country\ GPR_{i,t} \\ \pi_{i,t} \\ g_{i,t} \end{bmatrix}.

Now,

Z_{i,t} \in \mathbb{R}^{4 \times 1}.

The VAR is still

Z_{i,t} = \alpha_i + A_1Z_{i,t-1} + u_{i,t},

but now

A_1 \in \mathbb{R}^{4 \times 4}, \qquad u_{i,t} \in \mathbb{R}^{4 \times 1}, \qquad \Sigma_u \in \mathbb{R}^{4 \times 4}, \qquad B \in \mathbb{R}^{4 \times 4}.

If the ordering is

\left[ Global\ GPR_t, Country\ GPR_{i,t}, \pi_{i,t}, g_{i,t} \right],

then the country-specific GPR shock is the innovation in country GPR after conditioning on contemporaneous global GPR and lagged variables:

\widetilde{GPR}_{i,t}^{country} = GPR_{i,t}^{country} – \operatorname{proj} \left( GPR_{i,t}^{country} \mid Global\ GPR_t, Z_{i,t-1} \right).

This shock captures country-specific geopolitical risk orthogonal to same-year global geopolitical risk.

The corresponding selection vector is

e_{Country\ GPR} = \begin{bmatrix} 0 \\ 1 \\ 0 \\ 0 \end{bmatrix}.

The VAR impulse response is

IRF_h^{VAR} = A_1^h B e_{Country\ GPR}.

The dimensions are

A_1^h B e_{Country\ GPR} : (4 \times 4)(4 \times 4)(4 \times 1) = (4 \times 1).

Thus, the impulse-response vector contains the responses of global GPR, country GPR, inflation, and GDP growth to a country-specific GPR shock.

13. Practical implication for empirical work

For a clean LP counterpart to the annual Cholesky panel VAR, the implementation should follow four steps.

Define the GPR innovation using lagged GPR, lagged inflation, lagged GDP growth, and other predetermined controls.
Estimate horizon-by-horizon LPs of inflation and GDP growth on that GPR innovation.
Do not control for contemporaneous inflation or contemporaneous GDP growth if GPR is ordered first. These variables are ordered after GPR, and controlling for them would remove part of the contemporaneous response that the Cholesky ordering allows.
Compare LP impulse responses with VAR impulse responses as a robustness check on dynamic propagation, not as a separate identification strategy.

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.

14. Main takeaway

Local projections do not make shocks exogenous.

They make the estimation of dynamic responses more flexible once the shock has been credibly identified.

In the annual GPR-inflation-growth application, the identifying assumption is the same in the recursive VAR and in the LP. GPR is ordered first, so the GPR shock is the innovation in geopolitical risk after conditioning on lagged macroeconomic information.

The VAR and LP differ in the estimation of impulse responses. The VAR estimates a dynamic system and iterates it forward. The LP estimates each horizon directly. Therefore, comparing VAR and LP responses is a useful way to assess whether the estimated macroeconomic effects of geopolitical risk are robust to the VAR’s finite-lag propagation restrictions.

For annual country-level data, this is not a minor issue. Geopolitical shocks may propagate slowly and heterogeneously across countries. Local projections are therefore a valuable complement to recursive VARs, provided the identifying assumption is stated clearly and implemented consistently.

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.