External Instruments

Definition

An external instrument (or proxy variable) is a variable $Z_t$ that is correlated with the structural shock of interest $\varepsilon_{1,t}$ but uncorrelated with all other structural shocks, so it captures as-if-random variation in that one shock while remaining external to the macroeconomic system. External instruments identify the dynamic causal effects (structural impulse responses) of a shock by importing the microeconometric quasi-experimental instrumental-variables (IV) strategy into macroeconometrics — in contrast to conventional structural VAR identification, which uses restrictions internal to the system.

Key Ideas

External vs. internal instruments. Point-identified SVARs since Sims (1980) achieve identification through internal instruments — exclusion and timing restrictions internal to the system, in the tradition of three-stage least squares (Sargan 1964). External instruments instead use variation from outside the system (a partial, noisy measure of the shock): e.g. oil kept off-market by a political disruption, tax changes orthogonal to the business cycle, or the surprise in an FOMC announcement window (high-frequency monetary-policy identification).
Measurement error is fine. A constructed shock series is best thought of as an instrument, not as the shock itself. Its measurement error would bias it as a direct shock measure but need not compromise its validity as an instrument.
Two estimators. The instrument can be used (i) in a one-step direct regression — an IV version of a local projection, LP-IV; or (ii) together with a VAR to identify structural impulse responses — the IV version of an iterated multistep forecast, SVAR-IV (a.k.a. proxy-SVAR).
The central trade-off. LP-IV is valid without invertibility but requires strong lead–lag exogeneity; SVAR-IV is asymptotically more efficient and needs no lead–lag exogeneity but requires invertibility.

How It Works

Let $\varepsilon_{1,t}$ be the shock of interest and $\varepsilon_{2:n,t}$ the other structural shocks. An external instrument $Z_t$ must satisfy:

Relevance: $\mathbb{E}(Z_t\,\varepsilon_{1,t}) \neq 0$ .
Contemporaneous exogeneity: $\mathbb{E}(Z_t\,\varepsilon_{2:n,t}) = 0$ (uncorrelated with the other current shocks).

LP-IV (one-step, direct)

Regressing $Y_{t+h}$ on the (instrumented) current shock, controlling for $W_t$ , and instrumenting with $Z_t$ recovers the horizon- $h$ structural impulse response directly. Validity requires lead–lag exogeneity: $Z_t$ must be uncorrelated with past and future shocks (after conditioning on controls), not just contemporaneous ones. This condition is what lets LP-IV avoid invertibility — but it is demanding and disciplines both instrument construction and control choice.

SVAR-IV (two-step, VAR-based)

Estimate a reduced-form VAR to get innovations $e_t$ ; the instrument then identifies the column of the structural impact matrix corresponding to $\varepsilon_{1,t}$ (the impact response), which the VAR iterates forward into the full impulse response. This is more efficient under strong-instrument asymptotics and needs no lead–lag exogeneity, but is consistent only under invertibility.

The "no free lunch" theorem

If $Z_t$ violates lead–lag exogeneity by depending on past shocks, the natural remedy is to add lagged macro variables as controls. But the condition for those controls to deliver valid LP-IV inference is in general equivalent to invertibility of the corresponding VAR — in which case SVAR-IV is more efficient. LP-IV's freedom from invertibility is not free once lagged controls are needed.

Weak instruments

External instruments are often weak. Robust inference uses HAR (heteroscedasticity- and autocorrelation-robust) versions of Moreira's (2003) conditional likelihood ratio statistic, or the Montiel Olea–Pflueger (2013) / Andrews (2018) alternatives to first-stage $F$ screening.

Why It Matters

External instruments open a route to credible identification of macroeconomic dynamic causal effects using plausibly-exogenous variation, rather than the often-controversial timing/exclusion restrictions of internal-instrument SVARs. The programme (Stock 2008; Stock-Watson 2012; Mertens-Ravn 2013; Gertler-Karadi 2015) has become a workhorse for estimating the effects of monetary and fiscal shocks. Comparing LP-IV and SVAR-IV also delivers a practical Hausman-type test of invertibility.

Open Questions

Robustifying the invertibility test (and weak-instrument inference generally) to weak external instruments is open.
The framework assumes linear, homogeneous treatment effects; the frequently-claimed robustness of local projections to non-linearity is not established, and a non-linear analogue of the no-free-lunch result is conjectured.
Finding genuinely valid external instruments — satisfying lead–lag exogeneity — remains hard and application-specific.