Nonlinear Monetary Policy Tradeoffs

Layer 1: Overview

This paper measures how the inflation-unemployment tradeoff associated with monetary policy varies with both the sign of the monetary intervention (easing versus tightening) and the state of the business cycle (booms versus recessions) for the US economy over 1973:M1 to 2019:M6. The motivation is that standard linear Phillips-curve estimates implicitly impose a constant tradeoff, yet a flat Phillips curve would simultaneously predict that (i) stimulating activity during a recession costs nothing in terms of inflation and (ii) reducing inflation costs very large amounts of unemployment — both empirically extreme predictions that have very different policy implications. The paper challenges both extremes.

The empirical strategy extends the Proxy-SVAR approach of Mertens-Ravn (2013) and Stock-Watson (2018) to a nonlinear setting. The economy is described by a Vector Moving Average augmented with nonlinear functions of the monetary policy shock — specifically its absolute value (capturing sign dependence) and its interaction with a recession indicator (capturing state dependence). Under a finite-order VARX representation assumption and a linear monetary policy rule assumption, the paper proves (Proposition 1) that even though the underlying VARX is nonlinear, the monetary shock can be recovered as the projection of an external instrument onto residuals of a misspecified linear VAR. Once the shock is recovered, it and its nonlinear functions are used as regressors in a VARX to estimate nonlinear impulse responses. The instrument is the Degasperi-Ricco (2022) extension of Miranda-Agrippino and Ricco (2021), with a baseline span of 1991:M1-2015:M12 extrapolated to the full sample. The VAR contains five variables: the 1-year Treasury bond rate, industrial production growth, the Gilchrist-Zakrajsek excess bond premium, the unemployment rate, and CPI inflation, estimated with 7 lags. The recession indicator equals 1 when average GDP growth over the previous 12 months is negative.

The monetary policy tradeoff is defined analogously to the fiscal multiplier: the ratio of the cumulative average impulse response of inflation (unemployment) to the cumulative average impulse response of unemployment (inflation) over horizons H. In a nonlinear setting the easing tradeoff and tightening tradeoff are no longer inverses of one another and must be treated separately.

The main quantitative findings are as follows. For monetary easing during recessions, the inflation cost of reducing unemployment is small and statistically insignificant: point estimates of T+ range from -0.03 to -0.17 (in absolute value) across horizons H = 12 to H = 48 months, with 68% confidence intervals spanning from approximately -5.3 to +2.8 at H = 12 and -3.4 to +2.7 at H = 48. For monetary tightening during booms, the unemployment cost of reducing inflation is moderate and statistically significant: T- estimates range from -0.51 to -0.61 across H = 12 to H = 48, with 68% confidence intervals entirely below zero (e.g., -1.10 to -0.26 at H = 12 and -1.23 to -0.24 at H = 48). In other words, reducing inflation by 1 percentage point during a boom requires raising unemployment by roughly 0.5 to 0.6 percentage points. These results are qualitatively robust to excluding the post-2008 zero-lower-bound period (pre-2009 subsample) and to alternative specifications. By contrast, monetary tightening during recessions implies a very large and unfavorable tradeoff. Easing during booms is extremely inflationary with virtually no real effect.

A Likelihood Ratio test for the null hypothesis that all nonlinear terms are zero is rejected at the 1% level, confirming the statistical importance of nonlinearities. The null hypothesis of shock invertibility (Assumption A4) is not rejected at the 5% level across all combinations of VAR lags and residual leads tested.

A simple model with downward nominal wage rigidities — in which the wage floor introduces a kink in the aggregate supply curve — provides a theoretical rationale for the sign- and state-dependent tradeoff: an expansionary shock in a full-employment economy raises inflation with no output effect (the economy sits on the vertical AS segment), while a contractionary shock makes the wage rigidity binding and reduces output with no price effect (the horizontal AS segment). Monte Carlo validation using artificial data generated by the calibrated DSGE model shows that the proposed empirical procedure recovers the theoretical nonlinear impulse responses very accurately.

Layer 2: Deep Dive

What is the identification strategy and what are the main assumptions required?

Identification proceeds in two steps. First, the monetary shock is recovered by projecting an external instrument (Degasperi-Ricco 2022) onto the residuals of a standard linear VAR — this is justified by Proposition 1, which shows that even though the VAR is misspecified (it omits the nonlinear terms), the shock can still be recovered as a linear combination of VAR residuals under four assumptions: (A0) a structural VMA representation in which the shock is orthogonal to past observables and to the remaining structural shocks at all leads and lags; (A1) a finite-order VARX representation; (A2) invertibility of the Wold representation; (A3) a valid instrument (relevance and exogeneity); and (A4) informational sufficiency, meaning the monetary shock can be expressed as a linear combination of current and past observables — a condition implied by a linear monetary policy rule. Second, once the estimated shock and its nonlinear functions (absolute value and interaction with the state dummy) are in hand, they are used as exogenous regressors in a VARX to estimate nonlinear impulse response functions.

What are the main threats to identification?

Three main threats are acknowledged. (1) Instrument validity: if the instrument (Degasperi-Ricco 2022) is weak or contaminated by information shocks, the first-stage projection may recover a mislabeled shock. The authors note the first-stage F-statistic is adequate per Miranda-Agrippino and Ricco (2021) but acknowledge that the weak-instrument problem in the nonlinear context is non-trivial and left for future research. (2) Assumption A4 (informational sufficiency): if the central bank follows a nonlinear rule or the VAR variables are not sufficient to recover the shock, identification fails. The authors test this using the Forni-Gambetti-Ricco (2023) invertibility test — regressing the instrument on current and future VAR residuals and checking whether future residuals matter — and fail to reject invertibility at 5% across all lag/lead combinations. (3) Model misspecification in the nonlinear VARX: the VARX approximation may not capture all relevant nonlinearities generated by the true DSGE. The Monte Carlo validation on artificial DSGE data provides reassurance that the approach recovers the true nonlinear responses accurately.

How does the paper distinguish sign dependence from state dependence?

The paper includes two nonlinear terms as regressors in the VARX: the absolute value of the shock |u_t^r|, which captures sign-dependent effects (i.e., whether a tightening and an easing of equal magnitude have asymmetric effects), and the product s_{t-1} * u_t^r, which captures state-dependent effects (i.e., whether the same-sign shock has different effects depending on whether the economy was in a recession before the shock arrived). The two components are estimated simultaneously, allowing their separate contributions to be read off impulse responses in Figure 3. Robustness checks in the Online Appendix report models estimated with only sign dependence and only state dependence in isolation, with results described as qualitatively similar to Barnichon-Matthes (2018) and Tenreyro-Thwaites (2016), respectively.

What are the key quantitative results on impulse responses?

In the full nonlinear model, monetary tightening generates large and significant effects on real variables (unemployment, industrial production) regardless of the state, while monetary easing has more muted real effects. For prices, sign and state components operate in opposite directions: the largest inflation responses are associated with tightening during expansions. Numerically, the tradeoff estimates from Table 2 show: (a) easing during recessions — T+ point estimates of -0.03 at H=12, -0.12 at H=24, -0.17 at H=36, -0.17 at H=48 months (all statistically insignificant at 68%); (b) tightening during booms — T- point estimates of -0.51 at H=12, -0.61 at H=24, -0.59 at H=36, -0.53 at H=48 months (all statistically significant at 68%). For the pre-2009 subsample (excluding the ZLB period), tightening-in-booms estimates are somewhat larger in absolute value (-0.63 to -0.70) but confidence intervals widen to include zero at longer horizons.

What is the key implication for ‘pushing on a string’ results in the prior literature?

Tenreyro-Thwaites (2016) and Barnichon-Matthes (2018) document that monetary easing is less effective at stimulating real activity, especially during recessions — an apparent ‘pushing on a string’ result. The current paper accepts that the real effect of easing in recessions is muted, but adds a crucial dimension: price responses are also muted in the same circumstances, so the inflation-unemployment tradeoff is actually favorable even when the absolute size of real effects is small. The policy implication is that central banks can still usefully deploy monetary easing during recessions as long as interventions are sufficiently aggressive to achieve the desired stimulus, since the inflationary cost of doing so is low.

How does this paper measure the tradeoff differently from Phillips-curve regressions?

The tradeoff is defined as the ratio of the cumulative average impulse response of inflation to the cumulative average impulse response of unemployment (or vice versa) in response to an identified monetary shock, analogous to a fiscal multiplier. This approach avoids three problems that plague standard Phillips-curve estimates: (i) it does not require specifying a structural Phillips-curve equation, reducing misspecification risk; (ii) it does not require data on inflation expectations or the natural rate of unemployment, which are unobserved and introduce measurement error; (iii) identification comes from exogenous monetary shocks rather than OLS variation in unemployment, so the endogeneity problem is avoided.

What theoretical mechanism rationalizes the nonlinear tradeoffs?

A simple New-Keynesian-style model with downward nominal wage rigidities (Wt >= theta * W_{t-1}) generates a kink in the aggregate supply curve. When the economy operates at full employment and inflation is non-negative, an expansionary monetary shock stimulates demand but the wage rigidity is non-binding, so the economy sits on the vertical segment of the AS curve: output cannot exceed its natural level, and the only effect is higher inflation. By contrast, a contractionary shock makes the wage rigidity binding, pushing the economy onto the flat segment of the AS curve: firms cut employment rather than nominal wages, so output falls but prices are unaffected. More generally, averaging over periods of full employment and periods of involuntary unemployment, tightening has larger real effects and weaker price effects than easing — matching the empirical pattern — because a contractionary shock keeps the economy below full employment for a longer time.

What robustness checks are conducted?

Three main robustness checks are reported in the main text, each presented with impulse-response figures (Figures 6, 7, 8): (1) replacing the authors’ state dummy (based on 12-month average GDP growth) with NBER recession dates; (2) replacing the 1-year Treasury bond rate with the Federal Funds rate and with the 6-month Treasury Bill rate; (3) replacing the baseline Degasperi-Ricco instrument with the Jarocinski-Karadi (2020) instrument both raw and cleaned (regressed on six lags of VAR variables). In all cases, the qualitative result — tightening in booms produces larger real effects than easing in recessions, while price responses are more muted in recessions — is preserved, and the tradeoff pattern remains favourable for easing in recessions and tightening in booms. The Online Appendix additionally reports results using: the unemployment rate as the state variable (instead of industrial production); the VAR extended with the 10-year Treasury Bill rate and M2 monetary aggregate; models with only sign dependence; models with only state dependence; and an alternative estimation using the instrument directly in place of the estimated shock (which yields implausible results, validating the two-stage procedure).

What does the Monte Carlo validation using the DSGE model establish?

The paper generates 1000 artificial realizations from a calibrated downward-nominal-wage-rigidity DSGE model (beta=0.99, sigma=1, theta=1, phi_pi=1.5, rho_m=0.5, sigma_r=0.25%, sigma_a=0.45%, solved by nonlinear global projection using Chebyshev polynomials). It then applies the nonlinear Proxy-SVAR procedure to each artificial dataset and compares average estimated impulse responses with average true (model-generated) generalized impulse responses. The two are described as ‘very similar’ (Figure 10), demonstrating that the empirical nonlinear VARX representation accurately approximates the nonlinearities of the DSGE even though the VARX is in principle misspecified relative to the true model. This validates both the econometric procedure and the interpretive link between the empirical findings and the theoretical mechanism.

Why does the paper estimate the shock from a misspecified linear VAR rather than the VARX directly?

The monetary shock is latent. Proposition 1 shows that, under the stated assumptions, the monetary shock equals (up to a scaling constant) the projection of the external instrument onto the VAR residuals of the linear VAR, even though the VAR omits the nonlinear terms. This is because the linear monetary policy rule implies the shock is a linear combination of current observables, and the VAR residuals span the same space. Using the instrument directly in the VARX instead of going through steps I and II introduces a non-proportional bias in the nonlinear case (unlike the linear case where the attenuation bias from measurement error in the instrument is proportional across units and corrects under normalization). The Online Appendix shows that bypassing the two-stage shock-estimation procedure yields implausible impulse response estimates.

What is the scope of the empirical findings and what caveats apply?

Three scope conditions are explicitly stated. (1) State uncertainty: the tradeoff varies significantly with the state of the economy, so if the central bank is uncertain about current economic conditions, interventions carry considerable risk — a disinflation during what turns out to be a weaker-than-anticipated economy could incur very large unemployment costs. (2) Historical average: estimates reflect the effects of average monetary interventions over 1973-2019 and may not generalize to unusually large, persistent, or unconventional policy actions. (3) Accompanying fiscal policy: the tradeoff could be influenced by fiscal policy measures that accompanied monetary interventions during the sample period. The sample also excludes the post-2019 inflation surge, so inference about that episode is not direct. The identification requires a valid external instrument, whose strength in the nonlinear context is an open question.

How does this paper relate to Barnichon-Mesters (2020, 2021) and Gali-Gambetti (2020)?

Barnichon-Mesters (2020, 2021) and Gali-Gambetti (2020) also exploit identified monetary shocks to estimate the conditional inflation-unemployment relationship (the ‘Phillips multiplier’) and to investigate whether the Phillips curve slope has changed over time. The main additional contribution of the present paper is to show that the relationship is not only time-varying but specifically sign- and state-dependent, driven by the direction of monetary intervention and the current phase of the business cycle. The sign- and state-dependent tradeoff framework provides a richer characterization that can explain why a flat aggregate Phillips curve is compatible with moderate costs of disinflation and low inflationary costs of stimulus — something a time-varying-slope model alone does not deliver.

What does the paper say about the implications for disinflation episodes like 2022-23?

The paper does not directly analyze the 2022-23 episode (the sample ends at 2019:M6 and the paper was written with November 2025 dating for the online appendix). However, the results imply that if the economy is in a boom when disinflation begins — as was broadly the case in 2022 — the unemployment cost of reducing inflation is moderate (roughly 0.5-0.6 percentage points of unemployment per percentage point of inflation at a 24-36 month horizon), substantially less than would be implied by a flat Phillips curve. The authors explicitly note that their results suggest central banks can pursue disinflation without necessarily incurring very large unemployment costs, subject to the caveats about state uncertainty and scale of the intervention.

Key Concepts

Monetary policy tradeoff: In this paper’s usage: the ratio of the cumulative average impulse response of inflation to the cumulative average impulse response of unemployment (for easing) or vice versa (for tightening), in response to an identified monetary shock, averaged over a horizon H. In a linear model easing and tightening tradeoffs are inverses; in the nonlinear model they must be estimated separately. The concept is deliberately defined without assuming a Phillips curve and without requiring inflation expectations or the natural rate.

Sign dependence: The property that a monetary easing and a monetary tightening of equal magnitude have asymmetric effects on inflation and unemployment, not just opposite-signed effects of the same absolute magnitude. Captured in the VARX by including the absolute value of the monetary shock as an exogenous regressor.

State dependence: The property that the effects of a monetary shock of given sign and magnitude differ depending on whether the economy was in a recession or a boom in the period before the shock arrived. Captured in the VARX by including the product of the recession indicator (s_{t-1}) and the monetary shock as an exogenous regressor.

Nonlinear Proxy-SVAR: The paper’s proposed econometric framework: a Vector Moving Average augmented with nonlinear functions of the monetary shock, which admits a VARX representation. Identification extends the standard Proxy-SVAR by showing — via Proposition 1 — that the latent monetary shock can be recovered from the residuals of a misspecified linear VAR, using an external instrument, under a linear monetary policy rule. The estimated shock and its nonlinear functions are then used as exogenous regressors to recover nonlinear impulse response functions.

Downward nominal wage rigidity: A labor market friction, modeled as the constraint W_t >= theta * W_{t-1}, that creates a kink in the aggregate supply curve. When the constraint binds (during downturns), firms respond to contractionary shocks by cutting employment rather than nominal wages, generating unemployment without deflation. When the constraint is non-binding (during expansions), expansionary shocks raise nominal wages and prices without affecting employment beyond full-employment output. In this paper the rigidity is the key mechanism generating a sign- and state-dependent monetary tradeoff.

Informational sufficiency (Assumption A4): The identifying assumption that the monetary policy shock can be expressed as a linear combination of current and past observable variables — equivalently, that the central bank follows a linear monetary policy rule. This allows the shock to be recovered from the residuals of a standard linear VAR even when the true model is nonlinear. Tested empirically via the Forni-Gambetti-Ricco (2023) invertibility test (checking whether the instrument Granger-causes future VAR residuals); not rejected at the 5% level in the authors’ data.

Generalized Impulse Response Function (GIRF): In this nonlinear context, defined as E(x_{t+h} | u_t^r = u-bar) - E(x_{t+h} | u_t^r = 0) for h = 0, 1, …, where u-bar is a given shock size. Unlike linear IRFs, GIRFs depend on the sign and magnitude of the shock and on the state of the economy, and are computed by summing the linear response alpha(L)*u-bar and the nonlinear response Phi(L)*g(u_t^r, …).