VAR, Structural Identification, and Local Projections

Separate multivariate forecasting from structural shocks using companion matrices, impact restrictions, and impulse responses.

VAR, Structural Identification, and Local Projections

Reduced form first

A $K$ -variable VAR( $p$ ) is

\mathbf y_t =c+A_1\mathbf y_{t-1}+\cdots+A_p\mathbf y_{t-p} +\mathbf u_t, \qquad E(\mathbf u_t\mathbf u_t^\top)=\Sigma_u.

It provides:

conditional forecasts;
forecast-error covariance;
Granger-predictive relations;
a compact representation of dynamic propagation.

It does not label $\mathbf u_t$ as an economic shock. Reduced-form innovations can be contemporaneously correlated and can combine several underlying disturbances.

Companion matrix

Stack the state

\mathbf s_t= (\mathbf y_t^\top,\mathbf y_{t-1}^\top,\ldots, \mathbf y_{t-p+1}^\top)^\top.

Then

\mathbf s_t=F\mathbf s_{t-1}+G\mathbf u_t,

where the first block row of $F$ contains $A_1,\ldots,A_p$ and lower blocks shift lags. Stability requires every eigenvalue of $F$ to lie inside the unit circle. The $h$ -step propagation is $F^h$ .

This is the direct bridge from the classical companion matrix to multivariate finance and macroeconomics.

Structural impact matrix

Suppose

\mathbf u_t=B\boldsymbol\varepsilon_t,\qquad E(\boldsymbol\varepsilon_t\boldsymbol\varepsilon_t^\top)=I.

Then

\Sigma_u=BB^\top.

The covariance matrix supplies $K(K+1)/2$ distinct equations, while unrestricted $B$ has $K^2$ entries. Additional restrictions are required. Common approaches include:

Restriction	Meaning	Main risk
recursive/Cholesky	some contemporaneous effects are zero under an ordering	conclusions inherit the ordering
long-run	specified shocks have zero long-run effects on selected variables	sensitive to long-horizon estimation
sign	responses obey theory-based signs over selected horizons	set identification and researcher choices
external instrument	a proxy correlates with one shock but not others	instrument relevance and exclusion
narrative	historically documented episodes reveal a shock	episode classification and representativeness

Identification is an economic argument represented by matrix restrictions.

R laboratory: ordering changes the named shock

Consider a reduced-form VAR(1) with the same $A$ and $\Sigma_u$ under two recursive orderings.

Compare recursive impulse responses under two orderings

Both factorizations reproduce the same reduced-form covariance. Yet the response of variable 2 on impact differs because “a shock to variable 1” has been defined differently. Statistical fit cannot select the ordering.

Local projections

At each horizon $h$ , a local projection estimates

y_{i,t+h} =\alpha_h+\theta_h s_t+\Gamma_h^\top w_t+e_{t+h},

where $s_t$ is the identified shock or treatment-like innovation and $w_t$ contains controls and lags. The sequence $\{\theta_h\}$ forms the response.

Local projections make horizon-specific controls and nonlinearities convenient, but:

overlapping horizons require serial-correlation-robust inference;
the shock still requires identification;
flexible horizon-by-horizon estimates can be noisy;
post-selection of controls or horizons changes inference.

Jordà (2005) introduced this approach for impulse responses. Plagborg-Møller and Wolf (2021) establish the population equivalence between unrestricted LP and VAR impulse responses under common identification; regularisation and finite-sample choices drive practical differences.

Finance and economics use VARs differently

Use	Finance	Economics
system	returns, volatility, liquidity, flows, yields	output, inflation, employment, rates, money
horizon	often short and high-frequency	monthly to multi-year
innovation label	market, liquidity, order-flow, volatility shock	monetary, fiscal, supply, demand shock
key threat	asynchronous trading and rapidly changing covariance	policy anticipation, simultaneity, regime change
evidence beyond fit	executable timing and market mechanism	institutional timing and identification restrictions

In finance, connectedness or variance decomposition is descriptive unless shocks are identified. In economics, a policy response is structural only to the extent that the restrictions isolate an exogenous policy disturbance.

Identification checklist

Before reporting an impulse response, state:

the reduced-form variables, transformations, frequency, and lags;
the matrix restriction or instrument defining the shock;
why that restriction matches institutional timing;
shock scale and sign;
whether the response is cumulative, in levels, or in growth;
uncertainty method and horizon multiplicity;
sensitivity to ordering, lag length, sample, trends, and alternative identification.

Practice

Does a diagonal $\Sigma_u$ solve all structural identification problems?
If reversing a Cholesky order changes the conclusion, what has been learned?
Can an LP turn a poorly identified innovation into a causal shock?

Answers

No. It removes contemporaneous correlation under the reduced form but does not attach economic labels without assumptions.
The substantive result depends on contemporaneous zero restrictions and must be justified or reported as fragile.
No. LP changes response estimation, not the credibility of the shock.

Next: State Space, Mixed Frequency, and Nowcasting.

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