An alternative explanation of the price puzzle

Giordani asks whether the price puzzle — inflation rising after a contractionary monetary policy shock in a standard VAR — can be caused by omitting the output gap from the VAR, rather than by omitting a forward-looking indicator of inflation such as a commodity price index (the Sims 1992 explanation). He shows analytically, in a class of commonly used backward-looking models, that omitting an output-gap measure and including only output instead spuriously produces a price puzzle, and — crucially — that this can happen even when the misspecified VAR still produces optimal forecasts of inflation, which means the informational-deficiency story (that the VAR simply doesn’t know something the Fed knows) is not required for the puzzle to appear.

The data-generating process is Svensson’s (1997) backward-looking model, consisting of an IS relation linking the output gap next period to its own lag and the ex ante real interest rate, an AR(1) process for potential output, a Phillips curve in which inflation depends on its own lag and the lagged output gap, and a (possibly deterministic) Taylor rule in which the policy rate responds to inflation and the output gap. A VAR that correctly includes the output gap, output, inflation, and the rate — or equivalently the gap, inflation, and rate — admits an exact VAR(1) representation and recovers monetary policy shocks correctly under a Cholesky decomposition; the “misspecified” VAR replaces the gap with output and drops it, keeping only output, inflation, and the rate.

Giordani shows that even without the output gap, the misspecified three-variable VAR produces optimal forecasts of inflation at all horizons — there is no loss of forecasting fit in the inflation equation. He states explicitly that “the misspecified VAR gives optimal forecasts of inflation at all time horizons, implying that the standard explanation for the price puzzle is not relevant in this setting.” This matters because it rules out an information-deficiency story: the VAR isn’t puzzling because it fails to predict inflation well, but because the interest rate variable is forced to double as a proxy for an omitted state variable, which distorts the shock identification even though forecasts are unaffected.

Because the Taylor rule responds positively to the output gap, movements in the interest rate carry information about the omitted output gap, and the misspecified VAR’s inflation equation picks up this correlation as a positive coefficient on the policy rate. Formally, the misspecified system does not admit a finite VAR but is instead a VARMA(2,1); the variance of the labeled monetary-policy shock is strictly positive even when the true policy rule is deterministic (σ*²_MP = γ²_y·σ²_AD·σ²_N/(σ²_AD+σ²_N) > 0), the recovered shock is positively correlated with true aggregate-demand shocks and negatively correlated with technology shocks, and the resulting price-puzzle magnitude at the first lag grows with the variance of technology (potential-output) shocks. Giordani describes the reason directly: “movements in the interest rate help retrieve movements in the output gap, which is omitted.”

The paper derives four sufficient conditions: (i) inflation responds to the output gap with a lag; (ii) monetary policy affects output with a lag and inflation with a longer lag; (iii) the output gap enters the policy rule with a positive coefficient; and (iv) the output gap cannot be written as a linear combination of a constant, current output, current inflation, and lagged variables. Forward-looking models such as Svensson (2000) and Christiano-Eichenbaum-Evans (2001) satisfy these conditions; Fuhrer-Moore (1995) and Clarida-Galí-Gertler (1999) do not satisfy them strictly, but numerical simulations still generate similar price puzzles in those models as long as the initial inflation response to a monetary shock is sufficiently small. The paper is explicit that these conditions are sufficient but not necessary, so the theory should be read as offering insight rather than a universal theorem.

Using quarterly U.S. data from 1966Q1 to 2001Q3, a recursively (Cholesky) identified VAR(3) with the federal funds rate ordered last, and Federal Reserve capacity utilization (scaled by roughly 0.5) as the output-gap proxy, the “VARgap” system’s inflation response to a contractionary shock is never significantly positive and is significantly negative for several quarters, while the otherwise identical system replacing the gap with log real GDP produces “a huge price puzzle,” with a significantly positive inflation response for several quarters. Both systems use the same Cholesky/recursive identification (the standard Christiano-Eichenbaum-Evans 1999 ordering) so the comparison isolates the effect of the omitted variable rather than a difference in identification strategy.

In VARgap, the 8-quarter-ahead inflation forecast-error variance is split 50.0% aggregate-demand shocks, 47.8% cost-push shocks, and 2.1% monetary-policy shocks, while the federal funds rate’s own variance at horizon 0 is 71.0% monetary-policy shocks; in the misspecified VAR, 2-quarter-ahead inflation variance is 4.3% aggregate demand, 92.1% cost-push, and 3.5% monetary policy, while funds-rate variance at horizon 0 is 82.9% monetary policy. Giordani reads this as showing monetary policy “looks much more endogenous” in VARgap, since the share of the funds-rate forecast-error variance attributed to policy shocks falls substantially once the output gap is included. Separately, the estimated standard deviation of the monetary-policy shock is about 12% lower in VARgap than in the misspecified VAR, and residual standard deviations in the inflation and funds-rate equations are 7% and 11% lower, respectively — consistent with the theoretical prediction that omitting the gap inflates the estimated policy-shock variance.

Giordani argues that a commodity price index reduces the price puzzle not because it improves inflation forecasts, but because it is correlated with the output gap and therefore proxies for it. The commodity index PcomCEE (from Christiano-Eichenbaum-Evans 2000/2001) correlates with capacity utilization at r = 0.58 over 1970Q1–1998Q4 and Granger-causes both capacity utilization and real GDP; in an F-test of the Fed’s reaction function, PcomCEE is redundant once capacity utilization is included (p = 0.41), capacity utilization is not redundant given PcomCEE (p ≈ 0.0000), and PcomCEE enters prominently only when capacity utilization is excluded (p = 0.0055). The paper concludes: “PcomCEE reduces the price puzzle because it is a cyclical variable, not for its inflation forecasting ability” — consistent with related findings (Hanson 2000; Barth and Ramey 2001) that a variable’s ability to forecast inflation is not what predicts its ability to resolve the puzzle.

The core VARgap-versus-misspecified-VAR comparison is reported as robust to lag lengths 1 through 8, using CPI in log levels or the GDP deflator instead of CPI inflation, switching the ordering of capacity utilization and inflation, adding M2, adding real GDP to VARgap (which barely changes the gap variable’s own responses), using industrial production instead of GDP, and substituting several alternative output-gap measures (linear- or quadratic-trend deviations, the CBO output gap, unemployment, HP-filtered output), all of which “reduce the puzzle substantially,” with capacity utilization giving the best fit. That said, the paper flags real limitations: the fix is not as clean in a 1966–1979 subsample, where some specifications still show a sizable (if generally insignificant) price puzzle; monthly-frequency results are less sharp than quarterly ones, possibly reflecting measurement error in core inflation and the gap; adding i.i.d. measurement error to either inflation or the output gap generates sizable price puzzles in simulation, underscoring that gap-measurement quality matters; and because the Fed’s capacity-utilization construction interpolates using future annual data (raising a Sims-1992-style concern that it proxies for future output), the authors checked that adding four leads of real GDP to the misspecified VAR does not mitigate the puzzle, though this does not fully rule out the concern.