Instrumental Variable Identification of Dynamic Variance Decompositions

The paper asks what second moments of macro data and an external instrument reveal about a structural shock’s variance contribution – its dynamic variance decomposition and historical decomposition – when the instrument is imperfect (classical measurement error) and the shock is not assumed invertible. Stock and Watson (2018) had already shown that relative impulse responses (the shape of the response, normalized) are point-identified by such an external-instrument SVMA model without invertibility. This paper’s authors show that the scale needed for variance decompositions is a fundamentally different, harder problem: because the instrument’s underlying relevance parameter is not identified a priori, variance decompositions are in general only interval-identified, even when relative IRFs are exactly pinned down.

The FVR at horizon \u2113 for variable i is the fraction of that variable’s forecast-error variance (conditional on data through today) that would be eliminated if all future realizations of the structural shock of interest were revealed – formally, the sum of squared impulse responses up to \u2113-1 divided by the total forecast variance; it is always between 0 and 1 by construction. The paper treats this (and the closely related forecast variance decomposition and frequency-specific variance decomposition) as the primary target because policymakers and researchers usually want to know how important a shock is for business-cycle fluctuations, not merely the shape of its dynamic effect – and because, unlike relative IRFs, the FVR requires knowing the shock’s absolute scale, which is exactly what an imperfect, noisy instrument cannot pin down without further assumptions.

Because the instrument’s covariance with the data is proportional to both the true impulse response and an unknown “IV strength” scale parameter \u03b1, and only the ratio of impulse responses across variables or horizons – not their absolute magnitude – survives when \u03b1 is unknown; the FVR needs the absolute magnitude. In the static single-instrument case this is transparent: the covariance between the instrument and a variable equals \u03b1 times the true impact response, so the FVR is identified only up to the unknown factor 1/\u03b1\u00b2. The paper shows this indeterminacy is generic (not an artifact of the static case) and persists in the fully dynamic model once the instrument’s own serial correlation and measurement-error variance are accounted for.

The identified set for \u03b1\u00b2 is an interval [\u03b1\u00b2_LB, \u03b1\u00b2_UB]: the upper bound equals the total variance of the (serially uncorrelated) instrument-projection residual, obtained by assuming zero measurement error, while the sharp lower bound equals 2\u03c0 times the supremum over frequencies of that residual’s spectral density, obtained from the logic that macro aggregates cannot be more than fully informative about the shock at any frequency. “Sharp” means that for any value of \u03b1 strictly between these bounds, the authors can construct an SVMA-IV model consistent with the observed second moments – so the interval cannot be tightened without further assumptions. Because the sharp lower bound requires estimating a supremum of an unknown spectral density (difficult in finite samples), the paper’s actual implementation substitutes a weaker but asymptotically normal “practical” lower bound: the integral of the spectral density (i.e., the residual’s total variance) rather than its peak, which is easier to estimate even though it does not achieve the sharp bound.

Point identification holds under either of two conditions: a perfect instrument (zero measurement error, so \u03b1 hits its upper bound and the FVR lower bound becomes exact), which is not testable, or “recoverability” of the shock (it is spanned by all leads and lags, not just the past, of the observed variables), which is testable. The authors show recoverability’s identified-set condition holds if and only if the instrument’s projection residual is serially uncorrelated, and separately show that the degree of invertibility’s identified set contains 1 if and only if that residual does not Granger-cause the observed variables. This yields a practical pretest: reject invertibility if the instrument’s lags are jointly significant in a VAR for the observables (Wald test) – a rejection is decisive evidence of noninvertibility, but a failure to reject is not evidence of invertibility, since the test has no power against some noninvertible alternatives.

The empirical application uses monthly US data from January 1990 to June 2012 – industrial production growth, CPI inflation, the federal funds rate (or, as a robustness check, the 1-year Treasury rate), and the Gilchrist-Zakraj\u0161ek excess bond premium – with the Gertler-Karadi (2015) high-frequency federal-funds-futures surprise around FOMC announcements as the external instrument, a VAR(6) selected by AIC. The Granger-causality pretest strongly rejects invertibility in the federal-funds-rate specification (p = 0.0001) and is consistent with a very low degree of invertibility in the 1-year-rate specification (p = 0.390, i.e., not rejected, but the bound estimates for the degree of invertibility R\u00b2\u2080 are themselves low, ranging as low as [.118, .922] with a 90% confidence interval as wide as [.029, 1.000]). The authors read this as evidence that the conventional invertibility assumption underlying standard SVAR-IV analysis of monetary shocks is doubtful in this sample.

At every forecast horizon examined (1 to 24 months), the paper’s 90% identification-robust confidence intervals rule out the monetary shock explaining more than 31% of the forecast-error variance of output growth or more than 8% of the forecast-error variance of inflation; at horizons up to six months, they also rule out the shock explaining more than 19% of the forecast variance of the excess bond premium. For the excess bond premium at medium and long horizons, the intervals cannot rule out that the shock is completely unimportant. The authors’ summary judgment is that “monetary shocks are almost irrelevant for aggregate inflation” in this post-1990 sample, and that to the extent inflation is a monetary phenomenon, it reflects the systematic, rule-based component of policy rather than the erratic, shock-like component that external-instrument analysis isolates.

A companion application uses an external instrument built from OPEC-announcement surprises (K\u00e4nzig 2021) to bound the importance of international oil-supply-news shocks for US and global business cycles, finding that shock too is highly noninvertible and that conventional SVAR-IV analysis reaches spurious conclusions as a result; a Monte Carlo study (structural VARMA data-generating processes, 5,000 replications) shows the paper’s confidence sets achieve coverage close to or above the nominal 90% level even under noninvertibility (worst-case 82.9% for the identified set, never below 86.8% for the true parameter), while conventional SVAR-IV intervals undercover badly in the same noninvertible designs. Three further analytical examples (Section VI) illustrate the mechanics: with multiple observables, the FVR upper bound for a variable falls toward zero if the instrument induces comovement patterns inconsistent with the variables’ observed correlation structure (explaining the near-zero inflation bound in the empirical application); with a single observable, the sharp lower bound on \u03b1\u00b2 becomes exact in the limit where one shock dominates fluctuations at some frequency, even without invertibility; and with a stylized forward-guidance-type noninvertible shock, conventional SVAR-IV is shown analytically to overstate the shock’s true FVR by a factor of 1/R\u00b2\u2080, the inverse of the degree of invertibility.

The authors note the method is not robust to arbitrarily weak instruments (inference can fail as the measurement-error variance grows without bound), that the practical (integral-based) lower bound on \u03b1\u00b2 used in implementation is strictly weaker than the sharp (supremum-based) bound because the latter is difficult to estimate in finite samples, that the Granger-causality invertibility pretest has no power against some noninvertible alternatives, and that the analysis is confined to a stationary setting, with the Gaussianity assumption adopted for notational convenience only (the authors state results extend to conditionally heteroskedastic data). Extending the framework to nonstationary data is flagged as a topic for future research.