What are the effects of monetary policy on output? Results from an agnostic identification procedure

Uhlig asks what a contractionary monetary policy shock does to real GDP when the identifying restrictions imposed on the VAR do not themselves force an answer. The paper’s stated purpose is “to at least provide a systematic procedure and a framework to assess the conventional wisdom” that tight money lowers output, rather than assuming that conclusion through the identification scheme (Abstract; Introduction, pp. 382-385). Standard recursive (Cholesky) identification typically orders GDP so that it cannot respond within the period of the shock – baking in a zero-impact restriction on the very variable whose response is the object of interest, which risks answering the question by construction rather than letting the data speak.

The VAR is estimated on six monthly U.S. series – real GDP, the GDP deflator, a commodity price index, total reserves, nonborrowed reserves, and the federal funds rate (all but the funds rate in log levels) – over January 1965 to December 2003, with 12 lags, following the variable list and lag choice of Bernanke and Mihov (1998a,b) (Section 3, p. 389; Appendix B, p. 409). A Section 3.3 extension adds the 1-year Treasury bill rate as a seventh variable so that inflation and real interest rate responses can be constructed.

Assumption A.1 defines a contractionary shock as any impulse vector for which, in each of the six months following the shock (K=5, the benchmark horizon), the GDP deflator response is not positive, the commodity price index response is not positive, the nonborrowed reserves response is not positive, and the federal funds rate response is not negative – with the real GDP response left completely unrestricted (Section 3, p. 389). Formally, any admissible impulse vector a can be written as a linear combination of the reduced-form covariance matrix’s Cholesky factor (or, equivalently, of its eigenvectors weighted by the square roots of the eigenvalues), and the impulse response at horizon k is a weighted sum of the responses to each underlying orthogonal shock (Appendix A, Eq. 4, pp. 406-409).

Uhlig uses a Bayesian procedure with a weak Normal-Wishart prior whose posterior collapses to the OLS/MLE estimates of the VAR coefficients and residual covariance (Eq. 9, p. 410), then draws unit-length impulse-vector candidates uniformly from the unit sphere for each posterior draw of the VAR, keeping only draws that satisfy Assumption A.1 and discarding the rest (“pure-sign-restriction,” Appendix B.1, using 200 draws). A second “penalty-function” approach (Appendix B.2) instead always returns a best impulse vector for every posterior draw by minimizing a penalty function that penalizes sign violations steeply (slope 100) while rewarding negative responses only mildly (slope 1) (Eq. 10); this always yields an answer even when the pure-sign-restriction constraint set is empty for a given draw, but Uhlig notes the reward feature is “in effect, imposing somewhat more than just the sign restrictions” (Appendix B.2, p. 414) and that this approach is more sensitive to the choice of K (Section B.3, p. 417).

Leaving GDP unrestricted, the response is ambiguous: it moves up or down by up to about ±0.2% with two-thirds probability, and monetary policy shocks account for probably less than 25% – and may easily be less than 3% – of the forecast-error variance of real output (Section 4, pp. 405-406; Fig. 6, p. 396; Fig. 8, p. 399, giving an approximate 5-10% variance share across horizons under the pure-sign-restriction benchmark). This contrasts with a Cholesky-ordered VAR, in which “nearly half of all the variance in the 5-year ahead forecast revision for real GDP is explained as due to monetary policy shocks,” which Uhlig calls “unplausibly large” (Section 3.2, p. 402; Fig. 9, p. 401).

By construction, the GDP deflator and commodity prices do not rise on impact: the deflator falls slowly (about -0.1% within a year, about -0.4% within five years) and the commodity price index falls faster (about -1.5% after a year); nonborrowed reserves drop about 1% and total reserves about 0.6%, while the federal funds rate rises roughly 20 basis points on impact before reversing to about -10 basis points within the restriction horizon (Fig. 6, p. 396; Section 4, p. 406). Monetary shocks account for roughly a quarter of long-horizon price variance and about 25% of federal-funds-rate variance at horizons under six months (Section 3.2, p. 399). The standard Cholesky decomposition, by contrast, generates a price puzzle in which prices rise following the contractionary shock (Fig. 5, p. 394).

Uhlig argues the conventional contractionary-output finding rests on one additional restriction – forcing real GDP’s response to be zero on impact – which he calls “a rather spurious identification restriction.” Adding that restriction back into the agnostic framework (Fig. 12) does produce “considerable evidence that real GDP does indeed fall,” but Uhlig characterizes the shocks eliminated by this restriction as ones that “move up interest rates and real GDP, while moving down prices and nonborrowed reserves,” which are “hard to view … as the endogenous response of all other variables to, say, technology shocks or demand shocks” (Section 3.4, p. 403). He summarizes: “The life of the conventional wisdom hangs on the thin thread of a rather spurious identification restriction” (Section 3.4, p. 403). Fixing instead the initial deflator response to zero (Fig. 11) simply widens the already-ambiguous output bands further.

Lengthening the restriction horizon K shifts the real GDP band modestly upward (toward positive values), though the median response stays near zero throughout (Section 3.1, p. 396; Fig. 7, p. 397). The penalty-function approach gives qualitatively similar results with somewhat sharper (more precisely estimated) and slightly larger-magnitude error bands: real GDP “stays above zero for most of the first year at 0.1% above the no-shock scenario,” and the identified funds-rate response is about 30 basis points on impact, rising a further 10 basis points (Section B.3, pp. 415-416). With 64% posterior probability, the penalty-function GDP response never leaves the ±0.2% band of the pure-sign-restriction approach; Uhlig describes the penalty-function result as either “a sharpening” or “a distortion” of the pure-sign-restriction finding, and notes it is more sensitive to K (Section B.3, p. 417).

Adding the 1-year Treasury bill rate and constructing inflation and real-rate responses, Uhlig finds the real interest rate stays positive for up to two years after the shock before returning to zero, with some overshooting to the negative side visible thereafter (Section 3.3, pp. 402-403; Fig. 10, p. 403). The author is explicit that the agnostic method’s key feature is also its limitation: it cannot confirm the conventional wisdom, only test whether that wisdom is an artifact of the identifying restrictions (Abstract; Section 4); the Bayesian framing also means every posterior draw is treated as a “candidate truth,” so a Cholesky-implied price puzzle counts as “a violation by candidate truths, and worrisome” rather than mere sampling noise (Appendix B, pp. 409-410). The paper concludes: “Good monetary policy should be predictable policy, and should not rock the boat. From that perspective, monetary policy in the U.S. during this time span has been successful indeed” (Section 4, p. 406).