C11 | Macro Paper Warehouse

Financial Frictions: Micro versus Macro Volatility

Mon, 01 Jan 0001 00:00:00 +0000

Overview

Research Question. How do consumer credit spreads — the gap between household borrowing rates and deposit rates — affect aggregate business cycle dynamics and the distribution of consumption across the wealth distribution? And what is the welfare trade-off between macroeconomic stabilization and household-level consumption volatility when bank capital requirements are tightened?

Data and Empirical Approach. The empirical analysis draws on Danish administrative register data for 2003–2018, combining approximately 15.5 million household-year observations. Income tax return data, which capture housing wealth, portfolio wealth, bank deposits, and bank and mortgage debt, are merged with bank-level reporting of interest rates submitted to Danmarks Nationalbank (MFI data). Household-specific credit spreads are constructed as the difference between the loan rate at a household’s primary loan bank and the deposit rate at its primary deposit bank in a given year. Consumption is imputed from household balance sheets following the method of Crawley and Kuchler (2023). The empirical specifications include household and time fixed effects, and quantile regressions are run across bins of the net wealth distribution.

Model. The authors develop a Heterogeneous Agent New Keynesian (HANK) model with explicit banking intermediation. Banks, subject to an agency friction following Gertler and Karadi (2011) — in which bankers can divert a fraction λ = 0.381 of assets — combine household deposits with net worth to invest in corporate equity and consumer loans. This leverage constraint generates an endogenous, countercyclical spread between borrowing and saving rates. Households face idiosyncratic income risk and a kink in their budget constraint at zero net worth due to the spread. The supply side features New Keynesian sticky prices (Rotemberg quadratic adjustment costs) and a Taylor rule. Aggregate shocks include monetary policy surprises, total factor productivity (TFP), and capital quality shocks (affecting bank net worth). The model is solved by first-order perturbation using the method of Bayer and Luetticke (2020) and calibrated to Danish macro and micro moments for 2003–2018.

Main Empirical Findings.

The average consumer credit spread in Denmark is strongly countercyclical, with a cross-correlation with HP-filtered output of −0.44 in the data (−0.31 in the model).
Higher credit spreads increase the transition rate into the zero net wealth state for households with moderately positive wealth at the beginning of the year, and reduce the outflow rate for households already at zero net wealth.
Pooled OLS (with household and time fixed effects) finds that a higher spread is negatively associated with consumption (coefficient −0.266), and the interaction between spread and log income is positive (coefficient 1.366), indicating that higher spreads raise income sensitivity of consumption. For below-median wealth households, the income–consumption link is stronger and the negative spread effect on consumption is larger.
The consumption-income elasticity derived from quantile regression estimates has a standard deviation of 2.4 percent and a cross-correlation with output of −0.53 when spread variation is incorporated; holding spreads constant roughly halves the volatility (to 1.3 percent) and reduces the countercyclicality (cross-correlation −0.31).

Model Aggregate Findings.

Consumer credit is procyclical (cross-correlation with output 0.56 in data, 0.67 in model) and more than twice as volatile as output (standard deviation ratio 2.11 in data, 1.51 in model).
Capital quality shocks and monetary policy shocks are amplified at the aggregate level through a financial accelerator working through endogenous spread movements. TFP shocks generate little spread amplification because households’ labor supply responses partially insulate banks’ net worth.
A 1 percentage point contractionary monetary policy shock leads to a sharp, persistent decline in aggregate output and investment, and is amplified relative to a constant-spread HANK benchmark.

Distributional Findings.

In response to a contractionary monetary policy shock, consumption of households at the 10th percentile of the consumption distribution (who are indebted) falls sharply in the short run, while consumption of the 90th percentile (wealthy households) rises in the short run due to higher returns on savings. The responses converge across the distribution in the medium run as spreads normalize.
When the consumer credit spread is held constant, consumption paths move in parallel across the wealth distribution, demonstrating that endogenous spread movements are the key driver of distributional effects for monetary policy and capital quality shocks.
The MPC is countercyclical in the model, with a cross-correlation with output of −0.60 (unconditional), compared with −0.53 for the empirically-estimated consumption-income elasticity. The consumption-income elasticity and MPC are correlated at 90 percent in the model at the annual rate.

Macroprudential Regulation.

A tightening of bank capital requirements reducing leverage by 10 percent (diversion parameter λ rising from 0.381 to 0.445) reduces output volatility by 5.5 percent and investment volatility by 10.1 percent, and does so at apparently no long-run aggregate cost in the HANK setting (precautionary savings stimulate output and consumption in the stationary equilibrium).
However, the regulation increases the annual consumer credit spread by 40 basis points, raises household consumption volatility across the wealth distribution (from about 8 percent to 10 percent for the poorest households under idiosyncratic shocks alone), and generates welfare losses across all deciles equivalent to 0.24–4.28 percent of consumption (with aggregate welfare loss of 0.79 percent).
When aggregate shocks are included, the lower cyclical sensitivity of spreads partially mitigates welfare losses for the poorest 80 percent of the population, but the overall welfare effect remains negative with an aggregate loss equivalent to 0.58 percent of consumption. The paper thus documents a trade-off between macro volatility (stabilized) and micro volatility (increased).
Results are robust to the extension of the model to three assets (including illiquid assets), which provides a better fit to micro data without materially changing the welfare conclusions.

Q&A

Q1: What is the specific Danish dataset used, and how is consumption constructed? A: The dataset covers 2003–2018 from Statistics Denmark administrative registers, combining income tax return data (which report end-of-year balances on all bank accounts, housing wealth, portfolio wealth, bank deposits, bank loans, and mortgage debt) with bank-level MFI interest rate reporting submitted to Danmarks Nationalbank. The total sample is approximately 15.5 million household-year observations (about 1.76–1.97 million households per year). Consumption is imputed as after-tax labor income plus after-tax financial income minus the change in end-of-year net worth, following Crawley and Kuchler (2023). Households with self-employment, housing transactions in the current or prior year, negative imputed consumption, or in the bottom and top 1 percent of wealth or income distributions are excluded.

Q2: How are household-specific credit spreads constructed from the administrative data? A: Each household’s primary loan bank is defined as the bank where it holds the largest loan balance at end of calendar year, and the primary deposit bank as the one holding the largest deposit balance. The household-specific spread is the difference between the loan rate applied by the primary loan bank and the deposit rate applied by the primary deposit bank, both measured as averages over the calendar year. If a household has no loans, the loan rate of the primary deposit bank is used. This construction yields a household-level interest rate spread that moves countercyclically at the aggregate level (cross-correlation with HP-filtered output of −0.44).

Q3: What do the empirical results say about the relationship between spreads and the probability of a household reaching zero net wealth? A: Equation (2) is estimated as a linear probability model for the transition to zero net wealth (defined as net assets within plus or minus two weeks of 2007 median weekly income). Higher spreads significantly increase the transition rate into zero net wealth for households with moderately positive net wealth at the beginning of the year (those in the third to sixth net wealth bins), and reduce the outflow rate from zero net wealth for households already in that state. Higher spreads also appear to increase debt repayments for indebted households (third to fifth bins), making it more difficult for them to accumulate wealth. Households at the extremes of the wealth distribution (very poor or very wealthy) show essentially no sensitivity of transition rates to spread movements.

Q4: What do the consumption regressions in Table 1 find, and what is the key identification caveat? A: The pooled regression (column 1) finds a positive income–consumption coefficient of 0.372, a negative spread coefficient of −0.266, and a positive income–spread interaction of 1.366, all statistically significant with standard errors clustered at the household level (15,610,327 observations, R² = 0.591). When interacted with below-median wealth (column 2), the income coefficient is larger (0.397 versus 0.335 for above-median), the spread effect is more negative for below-median wealth (−0.362 versus −0.101 for above-median), and the income–spread interaction is stronger for below-median wealth (1.640 versus 0.875). The authors explicitly note that these results should not be given a causal interpretation, as income and consumption are likely jointly determined. Institutional features of the Danish mortgage market (covered bonds, competitive market, rates independent of borrower credit situation) minimize confounding from mortgage rate correlation with consumer credit spreads.

Q5: How do the quantile regression results and the derived consumption-income elasticity demonstrate countercyclical MPC? A: Quantile regressions across five-percent bins of the net wealth distribution show that income coefficients decline with wealth (from nearly 0.5 for the poorest to about 0.35 for the wealthiest households), spread coefficients are negative for households with negative, zero, and moderately positive wealth and positive for significantly wealthy households, and the income–spread interaction term is positive for all but the richest households (largest near zero net wealth). The consumption-income elasticity is computed as β₀,ⱼ + β₂,ⱼ × spread at the household level, then averaged cross-sectionally. When only wealth distribution shifts are allowed, the elasticity’s standard deviation is 1.3 percent and its cross-correlation with HP-filtered output is −0.31. When spread variation is also incorporated, standard deviation rises to 2.4 percent and the cross-correlation becomes −0.53. This measure is highly correlated (90 percent) with the model MPC, supporting the inference that the MPC is countercyclical.

Q6: What is the structure of the banking sector in the HANK model, and how does the agency friction generate a countercyclical spread? A: A continuum of banks combines household deposits with net worth to invest in corporate equity and consumer loans. Bankers can divert a fraction λ = 0.381 of assets, and if they do so, depositors can recover only the remaining fraction (1 − λ). This threat of diversion constrains the supply of deposits, resulting in banks needing to earn excess returns — Et(RK,t+1 − RS,t+1) > 0 — on their assets relative to the deposit rate. The leverage ratio is bounded above by ϱt/λ, where ϱt is a value multiplier that depends on current and expected future excess returns. When an adverse shock (capital quality shock or monetary tightening) reduces banking sector net worth, the leverage constraint tightens, banks reduce asset supply, and the spread between the return on capital (and hence the consumer loan rate, which is proportional to RK at markup ωB = 0.0075) and the deposit rate rises. This generates the observed countercyclical credit spread.

Q7: In the model, how do aggregate shocks affect the distribution of consumption, and why is the monetary policy shock particularly distributional? A: A one-percent capital quality shock reduces both wages and bank net worth, causing spreads to rise. In the baseline economy, rising borrowing rates lead to a large reduction in consumption for indebted households (10th percentile) while the constant spread model shows near-parallel movements across the distribution. A one-percentage-point monetary policy shock reduces equity returns, depressing bank net worth and (with a lag) raising spreads. Indebted households face both lower labor income and higher borrowing costs, producing a sharp consumption decline at the 10th percentile; wealthy households gain from higher returns on savings, so their consumption rises in the short run. Responses converge as spreads return to normal over the medium run. This matches empirical evidence from Holm, Paul, and Tischbirek (2021) for Norway. For TFP shocks, banks’ net worth is less affected because households’ higher labor supply partially offsets the productivity decline, so spreads move little and distributional effects are smaller (driven mainly by wage effects across the distribution).

Q8: How does the financial accelerator in the HANK model compare to the RANK version? A: In response to capital quality shocks and monetary policy shocks, the HANK model with banking frictions generates amplification relative to a constant-spread HANK benchmark, confirming the presence of a financial accelerator. However, relative to the RANK model, the incomplete markets model implies slightly less amplification of aggregate investment and consumption. This is because, in the HANK model, households facing higher credit spreads increase their labor supply (precautionary motive), which partially stabilizes aggregate income and moderates the financial accelerator. The finding that heterogeneous agent aspects are less important at the aggregate level is consistent with Berger, Bocola, and Dovis (2020). For TFP shocks, the financial accelerator through spreads is largely absent in both HANK and RANK, as spread changes are minor.

Q9: What are the long-run aggregate effects of tightening bank capital requirements (reducing leverage by 10 percent) in the HANK versus RANK model? A: In the RANK model, higher capital requirements increase the annual spread between the return on capital and the deposit rate by 25 basis points, reduce the aggregate capital stock by 2.4 percent, output by 0.5 percent, and aggregate consumption by 0.8 percent. In the HANK model, the spread increases by 40 basis points annually, but the mechanism differs: much of the spread change is absorbed by a reduction in the deposit rate (from 3.81 percent to 3.54 percent annually) rather than an increase in the capital return. Households respond to the lower deposit rate and higher credit costs by increasing precautionary savings and labor supply, so aggregate output and consumption actually rise slightly in the HANK stationary equilibrium. The capital requirements thus appear costless at the aggregate level in the HANK model — but this masks welfare costs that operate through the idiosyncratic risk channel.

Q10: What are the quantitative welfare costs of macroprudential regulation, and how do they vary across the wealth distribution and between idiosyncratic and aggregate shocks? A: Welfare is measured as the fraction of lifetime consumption households are willing to give up to stay in the unregulated baseline. In the face of idiosyncratic shocks only, welfare losses range from 0.24 to 0.43 percent of consumption for the first seven wealth deciles, and reach 4.28 percent for the richest decile (primarily because of the reduction in the return on their savings), with an average welfare loss of 0.79 percent. When aggregate shocks are added, the losses are substantially reduced for the poorest 80 percent (due to lower cyclical sensitivity of spreads), but remain large for the wealthiest decile (4.23 percent) and in aggregate (0.58 percent). These results are robust to the three-asset model extension, where the poorest households are approximately welfare-neutral under the regulation when aggregate shocks are included (0.00 percent), but aggregate welfare losses remain at 0.75 percent.

Q11: How does the three-asset model extension (with illiquid assets) affect the key results? A: In the three-asset extension, households can hold illiquid capital (calibrated with an adjustment probability of φk = 0.0025 per quarter, targeting the Danish ratio of bank deposits to output of 34 percent), creating wealthy hand-to-mouth households who have illiquid assets but no liquid assets. The consumption impulse responses across the wealth distribution remain very similar to the two-asset baseline: endogenous spread movements generate heterogeneous consumption dynamics in response to capital quality and monetary shocks, while constant-spread models produce near-parallel responses. The three-asset model provides a better fit to the micro data (consumption-spread-income relationship across the wealth distribution), but the welfare conclusions from macroprudential regulation are essentially unchanged: welfare losses across the distribution in the stationary equilibrium, partially mitigated when aggregate shocks are added, with losses concentrated in the richest decile.

Q12: What robustness checks are reported for the empirical consumption regressions? A: Three robustness exercises are reported. First, capitalizing car purchases using their official tax value (rather than treating car purchases as current expenditure) yields coefficients similar to the baseline (Table 10). Second, excluding households who purchase a car in the current or prior year (reducing the sample to 13.24 million observations) also leaves results unchanged. Third, first-differenced specifications (equation 42, with and without household fixed effects) produce results similar to the levels specification; the main exception is the spread effect for above-median wealth households when household fixed effects are omitted from the differenced specification (Table 11). The income–spread interaction is consistently positive and significant across all robustness checks.

Q13: What evidence does the paper provide that the model’s MPC is countercyclical and that credit spreads are the primary driver? A: Figure 7 shows impulse response functions of the average MPC to each of the three aggregate shocks. In all three cases, the MPC rises in recessions (countercyclical). The key mechanism is that adverse shocks cause spreads to rise, increasing the mass of households at the kink in the budget constraint (zero liquid assets), where MPCs are highest. When the consumer credit spread is held constant, the MPC remains countercyclical but close to constant, indicating that spread movements account for most of the cyclical variation in MPC. Eliminating the spread altogether implies an acyclical MPC (Table 12, Appendix D). The unconditional cross-correlation of the model MPC with output is −0.60, compared with −0.53 for the empirically estimated consumption-income elasticity in the Danish data.

Key Concepts

Consumer credit spread (borrowing-saving spread): In the paper, this is the difference between the gross real interest rate on consumer loans (RL,t) charged by banks and the gross real return on deposits (RS,t) received by savers. It is not an abstract measure of credit conditions but a household-specific, bank-derived rate gap that moves countercyclically due to banking agency frictions and creates a kink in households’ budget constraints at zero net worth. Distinct from mortgage spreads (which in Denmark are market-determined and independent of borrower credit conditions).

Kink in the budget constraint: The household budget constraint has a kink at zero net assets because borrowers face RL,t > RS,t; households at exactly zero liquid assets (type IV in the paper’s taxonomy) face a discrete jump in the cost of additional borrowing. This kink creates a mass point in the wealth distribution at zero net wealth, and households at this kink have higher MPCs than unconstrained savers or borrowers. The size of the mass point increases when the spread rises.

Financial accelerator (in the HANK-with-banking context): The amplification mechanism in which shocks that reduce banking sector net worth tighten banks’ leverage constraints, raise credit spreads, reduce asset supply to both the corporate sector and households, and further depress investment and consumption — which in turn reduces bank net worth further. In this paper, the accelerator operates through the consumer credit spread channel in addition to the standard corporate lending channel, and is present for capital quality and monetary policy shocks but not materially for TFP shocks.

Countercyclical MPC: The MPC — defined as the response of consumption to a small transitory income shock — rises during recessions and falls during expansions in this model. The mechanism is that recessions are associated with higher consumer credit spreads, which expand the mass of households at or near the zero net wealth kink (high MPC), and contract the mass of unconstrained savers (low MPC). This is a distinct source of MPC cyclicality from the wealth distribution channel alone.

Agency friction (diversion problem): Banks can divert a fraction λ of their assets; if they do so, depositors can recover only the fraction (1 − λ) and the bank is liquidated. This threat limits depositors’ willingness to supply funds, resulting in an incentive-compatibility constraint on bank leverage: assets cannot exceed ϱt/λ (where ϱt is the bank’s franchise value multiplier). When ϱt declines (because expected excess returns fall), the constraint binds more tightly and the spread between the return on assets and the deposit rate must be positive to sustain bank participation.

Macro versus micro volatility trade-off: The paper uses this phrase to describe the finding that tighter bank capital requirements (restricting leverage) reduce the cyclical volatility of aggregate output and investment (macro volatility falls) while simultaneously increasing the volatility of individual household consumption streams due to higher credit spreads and lower deposit returns (micro volatility rises). Welfare costs from increased micro volatility outweigh the aggregate stabilization benefits.

Consumption-income elasticity (d log c / d log y): A time-varying cross-sectional average measure derived from quantile regression parameter estimates, equal to β₀,ⱼ + β₂,ⱼ × RSi,t for household i in wealth bin j. It is used in the paper as an empirical proxy for the MPC (not a direct estimate), and is shown to be highly correlated with the model MPC (cross-correlation of 90 percent at the annual rate). Its cyclicality is stronger when spread variation is incorporated (standard deviation 2.4 percent, cross-correlation with output −0.53) than when spreads are held fixed (standard deviation 1.3 percent, cross-correlation −0.31).

Financial shocks and leverage of financial institutions: When do they matter?

Mon, 01 Jan 0001 00:00:00 +0000

This paper investigates the role of leverage of financial institutions in amplifying the transmission of financial shocks to the macroeconomy, with particular attention to whether that amplification differs across economic regimes. The authors develop a new endogenous regime-switching structural vector autoregression (RS-SVAR) model with time-varying transition probabilities, in which the probability of switching regime depends on the contemporaneous state of the economy (endogenous switching). The model extends the Sims and Zha (2006) and Sims, Waggoner, and Zha (2008) Markov-switching SVAR framework by: (1) incorporating a time-varying transition matrix in which the probability of staying in a regime is a logistic function of lagged endogenous variables; and (2) introducing new identification techniques for RS-SVARs, including non-recursive zero restrictions, sign restrictions, and narrative sign restrictions, which can in some cases uniquely identify structural shocks rather than merely set-identify them.

The leverage measure is market-based — book assets divided by market equity — constructed from CRSP/Compustat institution-level data covering publicly listed depository institutions, bank holding companies, and nonbank financial institutions. The sample runs monthly from December 1988 to December 2019. The five-variable VAR includes industrial production growth, core CPI inflation, the 2-year Treasury rate, market leverage of financial institutions, and the Chicago Fed’s National Financial Conditions Index (NFCI). The authors estimate three model variants that substitute in turn the leverage of: (i) all depository institutions, (ii) Global Systemically Important Banks (GSIBs), and (iii) securities brokers and dealers.

The model identifies two coefficient regimes — a “financial constraint” regime and “normal times” — using the criterion that the first regime has higher smoothed probability during September 2008 to August 2009. The financial constraint regime covers the end of the Savings and Loan crisis, the 1990/91 recession, the Russian debt default, the Global Financial Crisis (GFC), and the European sovereign debt crisis.

The core finding is that real effects of financial shocks are amplified in the financial constraint regime but not in normal times. In the financial constraint regime, the output response to a financial shock is significantly negative, large, and protracted; GSIB leverage initially rises sharply (as falling asset prices erode equity) and then declines as institutions deleverage. In normal times, the output growth response is negative but non-persistent, and market leverage remains insignificant over the entire horizon.

The counterfactual experiment holding GSIB market leverage constant as of October 2008 is the sharpest quantitative result: if GSIB leverage had not risen further at the onset of the GFC, the decline in industrial production growth would have been approximately 20 percentage points smaller, with a faster subsequent recovery in output growth and inflation and higher short-term interest rates. The counterfactual probability of staying in the financial constraint regime would have fallen as low as 0.1 for some draws, compared to the actual probability remaining elevated. By contrast, for a system using depository institution leverage, the lower-bound counterfactual probability of staying in the constraint regime does not fall below 0.90, indicating substantially weaker heterogeneity effects for the broader depository sector.

Securities brokers and dealers show leverage that rises more on impact than other institutions and then declines immediately, consistent with their willingness to expand balance sheets going into the crisis amplifying losses and forcing a sharp post-crisis contraction.

A separate counterfactual holding the NFCI constant (rather than leverage) shows that the probability of staying in the constraint regime does not decline, confirming that market leverage and the financial conditions index provide distinct characterizations of the financial system and have different implications for shock propagation and regime persistence. Results are robust to substituting the GZ corporate spread for the NFCI and to imposing narrative restrictions for shock identification.

Q: What is the central research question? A: The paper asks whether and how the leverage of financial institutions amplifies the transmission of financial shocks to the real economy, and whether this amplification differs between a financial constraint regime and normal times. A secondary question concerns heterogeneity: do GSIBs, depository institutions broadly, and nonbank securities dealers transmit shocks differently?

Q: What is novel about the econometric framework? A: The RS-SVAR model allows the probability of remaining in a given coefficient regime to vary over time as a logistic function of lagged endogenous variables, so regime switching is endogenous to the state of the economy rather than governed by a fixed transition matrix. The paper also introduces sign restrictions, zero restrictions, and narrative sign restrictions into the RS-SVAR class, enabling identification of both structural shocks and regimes within a single framework; in roughly 20 percent of posterior draws these sign restrictions uniquely identify the financial shock.

Q: Why does the paper use market leverage rather than book leverage? A: Market leverage (book assets divided by market equity) is argued to be more timely than book leverage because book equity incorporates losses with a delay, giving institutions time to adjust book leverage to avoid regulatory limits. Market capitalization reflects market participants’ assessment of an institution’s creditworthiness, and low market-to-book ratios signal that institutions are more leveraged than their books indicate. Market leverage is therefore a more informative early-warning indicator of financial fragility and the need for rapid deleveraging.

Q: How are the two regimes identified? A: For each estimated regime, the authors count the number of months between September 2008 and August 2009 (inclusive) for which the smoothed probability of being in that regime exceeds 0.70; the regime with the higher count is labeled “financial constraint” and ordered first. Shock identification uses sign restrictions: in the financial constraint regime, a positive financial shock must have a contemporaneously negative effect on output, inflation, and the short-term interest rate, but positive effects on the financial conditions index and leverage; in normal times, only the financial conditions index is required to respond positively on impact.

Q: What regimes does the model assign historically? A: The smoothed probability of the financial constraint regime is elevated during the end of the Savings and Loan crisis, the 1990/91 recession, the Russian debt default, the GFC and associated recession (where the probability reaches 1.0 at end-2008 and beginning-2009 before declining sharply to approximately 0.6 percent in 2009/2010), and the European sovereign debt crisis.

Q: What do the impulse responses show in the financial constraint regime? A: In the financial constraint regime, the output response to a positive financial shock (tightening) is significantly negative, large, and protracted. GSIB leverage initially rises due to a sharp decline in asset prices eroding market equity, then falls as GSIBs deleverage in response. The authors interpret this pattern as evidence that deleveraging produces procyclical financial amplification effects with adverse real consequences.

Q: What do the impulse responses show in normal times? A: In normal times, the output growth response is large and negative but non-persistent, in contrast to the financial constraint regime. Market leverage remains statistically insignificant across the entire horizon in normal times, indicating that the leverage amplification channel is inactive outside of financial constraint episodes.

Q: What does the GSIB leverage counterfactual show quantitatively? A: Holding GSIB market leverage constant as of October 2008 implies a decline in industrial production growth that is approximately 20 percentage points smaller than actually occurred, along with a faster recovery in output growth and inflation and higher short-term interest rates. The counterfactual probability of staying in the financial constraint regime declines to as low as 0.1 for some posterior draws, compared to remaining elevated in the actual data.

Q: How do depository institutions compare to GSIBs in the counterfactual? A: For the model using broad depository institution leverage, the lower-bound counterfactual probability of staying in the financial constraint regime does not fall below 0.90, compared to as low as 0.1 for the GSIB specification. This implies that GSIB deleveraging has substantially more detrimental macroeconomic effects and a much larger effect on regime persistence than the broader depository sector.

Q: What is distinctive about securities brokers and dealers? A: Broker-dealer market leverage rises more on impact than leverage of other financial institutions following a financial shock, and then immediately declines due to rapid deleveraging. The authors interpret this as reflecting that dealers’ willingness to expand balance sheets ahead of the crisis amplified growth and losses, followed by a sharp post-crisis contraction — a pattern consistent with the procyclical leverage mechanism described in Adrian and Shin (2014).

Q: How do the authors distinguish the role of market leverage from the financial conditions index? A: A counterfactual holding the NFCI constant (rather than leverage) as of October 2008 shows that the probability of staying in the financial constraint regime does not decline, unlike the leverage counterfactual. This demonstrates that market leverage and the NFCI provide distinct characterizations of financial conditions and have different implications for the propagation of shocks and the persistence of the constraint regime.

Q: How robust are the results? A: Substituting the GZ corporate bond spread for the NFCI yields very similar results, specifically that the probability of staying in the constraint regime declines much more in the counterfactual than in the actual data, suggesting the findings are not driven by the choice of financial conditions proxy. Imposing narrative restrictions for shock identification (exploiting the known high-stress period around Lehman’s failure in September 2008) yields results that are “rather robust” relative to the baseline sign-restriction identification.

Q: What are the policy implications? A: The results confirm the leverage ratio as a useful financial stability indicator, with particular emphasis on market leverage as providing timely information for monitoring. The heterogeneity findings suggest that regulatory attention to GSIB leverage is especially warranted, since GSIB deleveraging can have substantially more detrimental macroeconomic effects and a much larger influence on the persistence of financial constraint regimes than deleveraging by the broader depository sector. The leverage ratio is characterized as complementary to the risk-weighted capital ratio as a regulatory tool.

Market leverage: Measured as book assets divided by market equity (not book equity), constructed from CRSP/Compustat institution-level data at monthly frequency. The paper argues market leverage is more timely than book leverage because market equity immediately reflects losses, preventing institutions from masking fragility through delayed book adjustments.

Financial constraint regime: One of two identified coefficient regimes in the RS-SVAR, characterized by a significantly negative, large, and protracted output response to financial shocks and by active leverage amplification. Identified empirically as the regime with the highest smoothed probability during September 2008 to August 2009.

Endogenous regime switching: A modeling approach in which the probability of transitioning between regimes depends on lagged values of the endogenous variables themselves (via a logistic function), rather than being governed by a fixed constant transition matrix. This allows regime dynamics to respond to the state of the economy.

Time-varying transition probabilities: The diagonal elements of the coefficient-regime transition matrix follow a logistic transformation of a linear function of lagged endogenous variables, so the probability of remaining in any given regime changes each period as a function of current financial and macroeconomic conditions.

Procyclical financial amplification: The mechanism by which financial institution deleveraging in response to falling asset prices further tightens financial conditions and reduces real output, generating a feedback loop. The paper provides empirical evidence for this channel operating specifically in financial constraint regimes.

Heterogeneity of financial institutions: The finding that GSIBs, broad depository institutions, and securities brokers and dealers differ substantially in how their leverage affects the transmission of financial shocks. GSIB deleveraging is shown to have much more detrimental macroeconomic effects and a much larger influence on the probability of remaining in the financial constraint regime than depository institution deleveraging more broadly.

Narrative sign restrictions in RS-SVARs: An identification technique extended from Antolin-Diaz and Rubio-Ramirez (2018) to the regime-switching context, which uses known historical episodes (here, the Lehman failure in September 2008) to impose restrictions on which regime the economy was in or on the sign of structural shocks at particular dates, thereby aiding identification of both shocks and regimes.

Inference Based on Time-Varying SVARs Identified with Sign Restrictions

Mon, 01 Jan 0001 00:00:00 +0000

Layer 1 — Overview

Research Question. The paper asks how to conduct valid Bayesian inference in time-varying structural vector autoregressions (SVARs) identified with sign restrictions, a setting in which existing algorithms are shown to be theoretically flawed. As an empirical illustration, the authors use the new framework to examine three questions about the 2022–2023 Federal Reserve tightening cycle: (i) how did the Fed respond to the state of the economy; (ii) how would more dovish or hawkish stances have fared; and (iii) was the Fed behind the curve in 2021, and at what cost?

Methodology. The paper defines a class of rotation-invariant time-varying SVARs, building on Bognanni (2018). A model belongs to this class when its prior over sequences of structural parameters is invariant to orthogonal transformations of those sequences—i.e., it assigns equal prior density to all observationally equivalent structural parameter sequences (Proposition 1 establishes that observational equivalence corresponds exactly to orthogonal rotation of the sequence). The authors prove an if-and-only-if characterization (Proposition 2): a prior belongs to this class if and only if the induced prior over sequences of orthogonal matrices is uniform and independent of the time-varying reduced-form parameters.

A specific member of this class, the Random Correlations SVAR (RC-SVAR), is constructed by combining a prior over time-varying reduced-form parameters based on Archakov and Hansen’s (2021) parametrization of correlation matrices with a uniform prior over sequences of orthogonal matrices. The RC-SVAR is preferred over alternatives (Primiceri 2005’s decomposition, which is order-dependent; Bognanni’s 2018 discounted Wishart model, whose marginal likelihood significantly underperforms) because, for the type of empirical applications considered, it generally implies a higher log-predictive score than most orderings of the Primiceri (2005) model.

The authors introduce three algorithms. Algorithm 1 (simple acceptance sampling) is theoretically correct but computationally infeasible when sign restrictions span many periods because the probability of satisfying all restrictions simultaneously converges to zero as sample length T grows. Algorithm 2, the current approach in the literature (Baumeister and Peersman 2013; Bognanni 2018; Debortoli, Galí and Gambetti 2020), draws orthogonal matrices period-by-period from the sign-restriction-truncated uniform distribution; the authors show this does not draw from the correct target posterior because the resulting prior over orthogonal matrices is not independent of the reduced-form parameters and therefore the prior does not satisfy the rotation-invariance condition. Algorithm 3, the paper’s contribution, uses a Gibbs sampler that incorporates the Particle Gibbs with Ancestor Sampling (PGAS) method of Lindsten, Jordan and Schon (2014) to draw sequentially from the correct target posterior conditional on sign restrictions over an arbitrary number of periods.

An important additional contribution is the allowance for time-varying sign restrictions—restrictions that are imposed only in selected periods—enabling researchers to tailor identification to institutional knowledge about when particular restrictions are economically appropriate.

Data and Empirical Application. The RC-SVAR is estimated at a quarterly frequency with five variables: output growth (log difference of real GDP), core inflation (log difference of core PCE price index), the federal funds rate, money growth (log difference of M2), and the Moody’s Baa corporate bond yield relative to the 10-year Treasury yield (credit spread). The sample runs from 1959:Q1 to 2023:Q2, with a constant and two lags (n=5, p=2, m=11). Four independent MCMC chains of 20,000 draws are used, keeping every tenth draw after discarding the first 2,500; 1,800 particles approximate the reduced-form posterior and 3,600 particles approximate the posterior of the orthogonal matrices.

Main Findings. Decomposing the unexpected change in the federal funds rate from 2022:Q2 to 2023:Q2 into contributions from the predictable component, the systematic monetary policy response to non-monetary-policy shocks, and pure monetary policy shocks, the authors find that the lion’s share of the unpredictable rate increase was a systematic response to non-monetary policy shocks. Monetary policy shocks contributed about 100 basis points of the unexpected change in the federal funds rate by 2023:Q2 (out of roughly 4.99 percentage points of cumulative actual funds rate).

In the Dovish Fed counterfactual—where the response of the federal funds rate to contemporaneous inflation is halved for the first quarter of 2022—the economy would have marginally overheated, with inflation running persistently above 5 percent. In the Hawkish Fed counterfactual—where the response to inflation is doubled—inflation would have quickly declined at a small output cost: focusing on posterior medians, real GDP in 2023:Q2 would have been about 0.7 percent lower than in the data, though the lower envelope of the 68 percent probability bands indicates the output cost could have been as large as 3.1 percent.

Regarding the “behind the curve” question, the model finds evidence that the Fed was accommodative in 2021 (expansionary monetary policy shocks in that period), consistent with Summers (2021b). However, monetary policy shocks contributed only about 0.6 percentage points to annualized core inflation during 2021:Q2–2021:Q4 on a cumulative basis; the larger and dominant source of the unexpected inflation surge was non-monetary policy shocks. A comparison of the RC-SVAR with a constant-parameter SVAR identified only by Restriction 1 (Uhlig 2005) shows substantively different conclusions: the constant-parameter model attributes the unexpected increase in the federal funds rate to shocks that affect money growth and credit spreads, without a clear connection to the real economy, whereas the RC-SVAR links the rate increases to shocks that made the economy run hotter.

Layer 2 — Q&A

Q1: What is the fundamental theoretical flaw in existing algorithms for time-varying SVARs identified with sign restrictions, and why does it matter?

Existing algorithms (e.g., Baumeister and Peersman 2013; Bognanni 2018; Debortoli, Galí and Gambetti 2020) draw orthogonal matrices period-by-period from the uniform distribution restricted to those matrices satisfying the sign restrictions at each t. This construction implicitly defines a marginal density for the orthogonal matrices conditional on the reduced-form parameters that is not uniform: it is proportional to the reciprocal of the volume of the sign-restriction-satisfying subset of the orthogonal group, which depends on the reduced-form parameters. Consequently, the prior over structural parameters implied by these algorithms does not assign equal density to observationally equivalent sequences of structural parameters, violating Proposition 2’s necessary and sufficient condition. The resulting posteriors are therefore not correctly targeted to the desired posterior, meaning inference is distorted in a way that cannot be corrected by importance reweighting without prohibitive computation.

Q2: What does Proposition 1 establish, and how does it generalize the constant-parameter case?

Proposition 1 proves that two sequences of time-varying structural parameters are observationally equivalent if and only if there exists a sequence of orthogonal matrices such that one sequence is obtained from the other by post-multiplying each period’s structural parameters by the corresponding orthogonal matrix. This directly mirrors the constant-parameter result in Rubio-Ramírez, Waggoner and Zha (2010) and Uhlig (2005), where a single orthogonal matrix produces observational equivalence. The extension to sequences is non-trivial because the law of motion couples parameter draws across time, but the likelihood’s separability across periods preserves the period-by-period orthogonal rotation structure.

Q3: What is Proposition 2, and what is its practical implication for constructing valid priors?

Proposition 2 states that the prior over time-varying structural parameters satisfies the rotation-invariance condition (Equation 3) if and only if the induced prior over the time-varying orthogonal reduced-form parameters does not depend on the sequence of orthogonal matrices—equivalently, the prior over (Qt) is uniform over the product of orthogonal groups and is independent of the reduced-form parameters (Bt, Σt). The practical implication is constructive: any prior over time-varying reduced-form parameters (Bt, Σt), combined with an independent uniform prior over sequences of orthogonal matrices, automatically produces a rotation-invariant SVAR. This means that widely-used priors for reduced-form time-varying VARs (Primiceri 2005, Bognanni 2018, the new RC prior) can all be adapted for structural analysis without modification, as long as the orthogonal matrices are drawn uniformly and independently of the reduced-form parameters.

Q4: Why do models with heteroskedastic structural shocks (identification via heteroskedasticity) not belong to the class of rotation-invariant SVARs?

In models identified through heteroskedasticity, the time-varying structural parameters take the form (A Ψt^{-1/2}, F Ψt^{-1/2}), where Ψt is a time-varying diagonal matrix. For any permissible sequence, post-multiplying by a non-diagonal orthogonal matrix at one period produces a sequence where the ratio of structural parameters across consecutive periods is not diagonal, which violates the permissibility constraint of those models. Thus, the class of rotation-invariant SVARs and models identified through heteroskedasticity are mutually exclusive when the heteroskedastic specification has constant impulse responses up to scale—a restriction that the authors note has been criticized as a potential weakness of the heteroskedasticity-based approach.

Q5: Why is the Random Correlations SVAR (RC-SVAR) chosen as the baseline, and how does it compare to alternatives?

The RC-SVAR uses the Archakov and Hansen (2021) parametrization of correlation matrices to define a prior over time-varying reduced-form parameters that is order-invariant (unlike Primiceri 2005, which produces n! different elements depending on variable ordering) and avoids the highly restrictive structure of Bognanni’s (2018) discounted Wishart model, which significantly underperforms in marginal likelihood. For the empirical applications considered, Arias, Rubio-Ramírez and Shin (2023) show the RC-SVAR generally achieves a higher log-predictive score than most orderings of the Primiceri (2005) model, motivating its use as the baseline. The theoretical results apply to any member of the rotation-invariant class, so the algorithm is not specific to the RC-SVAR.

Q6: Why are time-varying sign restrictions important, and how are they implemented in the monetary policy application?

Time-varying sign restrictions allow researchers to impose identification restrictions only in periods where those restrictions are economically appropriate, adhering to the principle “If you know it, impose it; if you do not know it, do not impose it” (Uhlig 2017). In the monetary policy application, Restriction 2 (which constrains the contemporaneous elasticities in the policy rule to plausible ranges, following Arias, Caldara and Rubio-Ramírez 2019) is not imposed during three exceptional periods: 1979:Q4–1982:Q4 (non-borrowed reserves targeting under Volcker), 2009:Q1–2015:Q3 (quantitative easing following the Great Recession), and 2020:Q2–2021:Q4 (QE and effective zero lower bound during COVID-19). Restriction 1 (sign restrictions on impulse responses to a monetary policy shock, following Uhlig 2005) is imposed throughout the entire sample.

Q7: What do the estimated contemporaneous elasticities reveal about how monetary policy has changed over time?

The model estimates show substantial time variation. The contemporaneous elasticity of the federal funds rate to output growth exhibits three peaks: during Arthur Burns’s chairmanship in 1974 (capturing the sharp rate cut during the 1974–1975 recession), during Volcker’s chairmanship in 1983–1984 (when annualized real GDP growth averaged 6.8 percent), and during Greenspan’s tenure in 2001 (when the federal funds rate fell from 6.4 percent in December 2000 to 1.8 percent by end-2001). Outside these peaks, the elasticity averaged about 0.1, implying a 0.1 percentage point rise in the annualized federal funds rate per 1 percentage point increase in annualized GDP growth. The elasticity to inflation averaged about 0.3 percentage points per 1 percentage point rise in annualized core inflation, with a range from above 0.5 in the early 1970s and early Volcker years down to about 0.15 during Yellen’s tenure. The elasticity to the credit spread moved from about −1.4 at the beginning of Burns’s tenure to −2.2 at the end of Nixon’s presidency, then declined through the mid-1970s to the Great Recession, and stood at about −1 by mid-2023.

Q8: What is the exact decomposition of the 2022–2023 tightening cycle into predictable, systematic non-monetary, and monetary policy shock components?

Table 1 from the paper shows the federal funds rate decomposition. In 2022:Q2, the predictable component was 0.27 percentage points, the unpredictable component due to systematic response to non-monetary shocks was 0.24 pp, and the unpredictable component due to monetary policy shocks was 0.26 pp, summing to 0.77 pp. By 2023:Q2, these were 1.70 pp (predictable), 2.25 pp (systematic/non-monetary), and 1.04 pp (MP shocks), totaling 4.99 pp. Thus, at the tightening cycle’s end in 2023:Q2, the systematic response to non-monetary shocks accounted for about two-thirds of the unpredictable component (2.25 / (2.25 + 1.04) ≈ 68 percent), consistent with the broader literature finding that most variation in policy instruments is driven by the systematic component of policy.

Q9: How do the Hawkish and Dovish Fed counterfactuals work, and what do they imply?

The Hawkish (Dovish) counterfactual replaces the estimated contemporaneous response to inflation in the policy rule with one that is twice (half) as large as the estimated response for the first quarter of 2022, then simulates history forward from 2022:Q2 under the modified rule. Under the Dovish Fed, the economy would have marginally overheated with output rising above CBO potential GDP estimates, and inflation would have run persistently above 5 percent. Under the Hawkish Fed, posterior medians show inflation quickly declining at a cost of about 0.7 percent of real GDP in 2023:Q2 relative to the data; the lower envelope of the 68 percent probability bands shows the output cost could have been as large as 3.1 percent. A parallel set of counterfactuals, designed to be robust to the Lucas critique by working through one-time monetary policy shocks rather than changes to the reaction function, yields broadly similar results.

Q10: What does the comparison with Romer and Romer (2023a) reveal about the model’s monetary policy shock series?

Romer and Romer (2023a) identify a contractionary monetary policy shock in July 2022 (2022:Q3) using a narrative approach. The RC-SVAR’s estimated monetary policy shock series is broadly consistent with this finding: the model detects a contractionary shock in 2022:Q3 and, like Romer and Romer, also finds some evidence of a contractionary shock in 2022:Q2 (though they characterized it as “signs but not definitive evidence”). Beyond the Romer-Romer estimation window, the RC-SVAR additionally finds evidence of an expansionary monetary policy shock in 2023:Q1, when the Fed decelerated the pace of rate increases from 50 to 25 basis points.

Q11: How does the RC-SVAR’s inference on the 2022–2023 tightening cycle differ from that of a constant-parameter SVAR identified only with Restriction 1?

Two salient differences emerge. First, through the lens of the constant-parameter SVAR, monetary policy shocks contribute insignificantly to unexpected output growth between 2022:Q2 and 2023:Q2; in fact, the posterior median output response to a contractionary monetary policy shock is positive in that model (consistent with Uhlig 2005’s finding), implying that the positive monetary policy shocks needed to explain the rate increase would propel rather than reduce output. In the RC-SVAR, the posterior median output response to a contractionary shock is negative, so contractionary monetary policy shocks worked to decelerate output against a backdrop of non-monetary shocks that made the economy run hotter. Second, in the constant-parameter SVAR, non-monetary policy shocks that drive the unexpected increase in the federal funds rate do not propagate through output or inflation, whereas in the RC-SVAR they do—yielding a much more coherent macroeconomic narrative for the tightening cycle.

Q12: What does the model find about whether the Fed was behind the curve in 2021, and what were the consequences?

The model’s 2021:Q1 forecasts predicted the federal funds rate would reach about 0.6 percent by end-2021, consistent with a view that rate normalization was already warranted. The actual federal funds rate remained at its effective lower bound through 2021:Q4, and the shock decomposition shows that the cumulative unexpected change in the funds rate during 2021:Q2–2021:Q4 was driven by expansionary monetary policy shocks—supporting the view that monetary policy was accommodative and the FOMC fell behind the curve. However, monetary policy shocks contributed only about 0.6 percentage points (annualized) to the unexpected increase in core inflation during this period; the dominant and larger source of the inflation surge was non-monetary policy shocks. The model therefore finds that the delay in tightening was not the primary driver of the 2021 inflation surge.

Q13: Do time-varying sign restrictions materially affect inference, as demonstrated in Section 6.8?

Yes. Comparing the baseline identification scheme (Restrictions 1 and 2, with Restriction 2 not imposed during exceptional periods) against an alternative scheme that imposes both restrictions throughout the entire sample reveals differences in the estimated monetary policy shocks, particularly in 2021:Q4. Under the alternative scheme, there was an expansionary monetary policy shock in 2021:Q4, while the baseline finds the shock was nearly centered around zero. Additionally, for 2021:Q2, the alternative scheme implies the contemporaneous output response to an expansionary monetary policy shock is more likely to have been positive, whereas the baseline scheme yields a different posterior distribution for this response. These differences illustrate that imposing or omitting restrictions in specific periods affects inference about structural shocks and impulse responses at economically important junctures.

Key Concepts

Rotation-Invariant Time-Varying SVAR: A class of time-varying SVAR models whose prior over sequences of structural parameters satisfies: for every permissible sequence of structural parameters and every sequence of orthogonal matrices, the orthogonally-rotated sequence is also permissible and receives the same prior density. This ensures the prior does not break the observational equivalence among structural parameter sequences related by orthogonal rotation, so that identification comes solely from the imposed restrictions.

Observational Equivalence in Time-Varying SVARs: Two sequences of time-varying structural parameters are observationally equivalent if and only if there exists a sequence of orthogonal matrices such that one sequence equals the other sequence post-multiplied period-by-period by the corresponding orthogonal matrix. This definition extends Rothenberg’s (1971) concept to the time-varying setting and directly implies the rotation-invariance restriction.

Random Correlations SVAR (RC-SVAR): A specific member of the rotation-invariant class constructed by using the Archakov and Hansen (2021) parametrization of correlation matrices to define the prior over time-varying reduced-form parameters, combined with a uniform prior over sequences of orthogonal matrices. The prior is order-invariant and, for the empirical applications considered, generally achieves higher log-predictive scores than the workhorse Primiceri (2005) model.

Time-Varying Sign Restrictions: Sign restrictions imposed only on selected time periods rather than uniformly across the sample, implemented by allowing the restriction function St() to differ across t (including the possibility that no restriction is imposed at some t). This allows researchers to tailor identification to periods in which the theoretical or institutional knowledge motivating the restriction is deemed applicable—e.g., imposing policy-rule contemporaneous restrictions only when the federal funds rate is the primary policy instrument.

Particle Gibbs with Ancestor Sampling (PGAS): The sequential Monte Carlo method (from Lindsten, Jordan and Schon 2014) used in the paper’s Algorithm 3 to draw the sequence of structural parameters At from its conditional posterior given the sign restrictions. PGAS conditions on the previous Gibbs draw of the structural parameter sequence to ensure an invariant distribution, which is the key property that makes the Gibbs sampler valid for drawing from the correct target posterior.

Systematic Component of Monetary Policy: In the paper’s structural monetary policy equation, the linear combination of contemporaneous endogenous variables (output growth, inflation, money growth, credit spread) that enters the federal funds rate equation, weighted by the contemporaneous elasticities ψ. It represents the portion of interest rate variation that is a predictable, rule-based response to economic conditions, as distinguished from the monetary policy shock (the residual).

Contemporaneous Elasticity: The coefficient ψi,t in the monetary policy equation measuring the response of the federal funds rate to a one-unit contemporaneous change in variable i at time t, defined directly in terms of the structural parameter matrix At. The paper’s time-varying framework allows these elasticities to evolve over the sample, revealing historically distinct episodes of how aggressively the Fed responded to output growth, inflation, money growth, and credit spreads.

Mixing It Up: Inflation at Risk

Mon, 01 Jan 0001 00:00:00 +0000

This paper introduces a Bayesian Gaussian mixture density regression framework that estimates the complete forecast distribution of inflation — not just selected quantiles — and decomposes the entire risk outlook into contributions from individual economic predictors. The methodology accommodates multimodality, skewness, and fat tails without parametric restrictions, and allows construction of risk measures calibrated to the central bank’s own loss function rather than generic percentile-based measures. Applied to the recent U.S. inflation surge, the framework finds that post-pandemic inflation risk was primarily driven by the recovery of the U.S. business cycle and surging commodity prices, while adjustments in monetary policy contributed negatively — partially mitigating the increase in right-tail inflation risk — and credit spreads also offset some risk. The Gaussian mixture structure enables fast MCMC estimation and produces well-calibrated density forecasts across a range of macroeconomic variables.

Summary of a forthcoming paper, AI-assisted and human-reviewed. See the linked original for the authoritative claims and full conditions.

In depth

Q1. What is the key methodological contribution relative to existing inflation-at-risk approaches?

Existing approaches to macroeconomic at-risk measures focus on specific quantiles of the forecast distribution — typically the 5th or 25th percentile — discarding information contained in the rest of the distribution; this paper redirects attention to the full forecast distribution while retaining the nonparametric flexibility of quantile regression. The Gaussian mixture density regression estimates a conditional distribution that is a weighted mixture of Gaussians, capturing multimodality, asymmetry, and fat tails simultaneously. The key innovation is decomposability: each predictor’s contribution to any region of the forecast distribution can be quantified, enabling a driver-level accounting of what generates tail risk in any given period.

Q2. What does the U.S. application reveal about the inflation surge?

The framework attributes the increase in right-tail U.S. inflation risk during 2021–2023 primarily to surging commodity prices and the recovery of the domestic business cycle, while monetary policy tightening contributed negatively — its effect partially offset the upward pressure from commodity and cycle drivers. Credit spreads also partially mitigated the risk. The decomposition implies that the dominant drivers of inflation risk were supply-side and aggregate-demand factors, and that monetary policy, when it tightened, reduced the right-tail risk as intended — providing quantitative support for the interpretation that policy was reactive but directionally correct.

Q3. How does the framework construct policy-relevant risk measures?

The framework allows weighting probability mass over the forecast distribution by any user-specified loss function, including asymmetric central bank preferences, yielding risk measures that integrate the full distributional information in proportion to the policymaker’s actual valuation of different inflation outcomes. A central bank that penalizes above-target inflation more heavily than below-target inflation (consistent with empirical evidence on CB loss functions) would weight the upper tail more, producing a risk statistic that is higher than a symmetric measure for the same distribution. This policy-preference-aligned risk measure could have provided a more accurate signal of the urgency of the 2021–2023 inflation risk than standard percentile measures.

Key concepts

inflation at risk : the quantile-based or distribution-based characterization of future inflation uncertainty; extended in this paper from a single quantile to the complete forecast distribution and its risk decomposition by driver.

density regression : a regression model in which the conditional distribution of the outcome — not just its mean or a specific quantile — is the object of estimation; the paper uses a Gaussian mixture density regression to capture non-standard distributional shapes.

risk decomposition : the attribution of shifts in the full forecast distribution to individual predictor variables; the paper’s key tool for identifying which economic factors drive right-tail inflation risk in any period.

CB-preference-aligned risk measure : a summary statistic constructed by weighting probability mass over the forecast distribution by the central bank’s loss function; captures asymmetric preferences and goes beyond standard percentile measures.

Optimal monetary policy with uncertain private sector foresight

Mon, 01 Jan 0001 00:00:00 +0000

Central banks must set policy under uncertainty about how private-sector expectations form, which changes how monetary policy transmits to output and inflation. This paper studies optimal time-consistent monetary policy using a New Keynesian finite-horizon planning (NK-FHP) model in which households and firms have limited foresight: they solve structural problems only up to a finite horizon, and update their beliefs about longer-run inflation by averaging over past data. In this setting—unlike in standard New Keynesian models—an “inflation scares” problem can arise: agents’ longer-run inflation expectations can deviate persistently from the central bank’s target, generating costly and prolonged disinflations. The authors formally characterize optimal policy when the planning horizons of private-sector agents are uncertain and a risk of inflation scares is present, showing that risk-management considerations modify the standard “leaning against the wind” principle with a novel preemptive motive: the optimal policy responds more aggressively to the risk of unanchoring to prevent inflation scares from materializing. An estimated version of the model is used to quantify how much this preemptive motive mattered during the post-pandemic inflation surge.

Summary of a forthcoming paper, AI-assisted and human-reviewed. See the linked original for the authoritative claims and full conditions.

In depth

Q1. What is the NK-FHP model and how does it differ from standard New Keynesian models?

In the NK-FHP model, households and firms are boundedly rational because they evaluate state-contingent paths only up to a finite horizon; beyond that horizon they extrapolate longer-run beliefs by averaging over past data, making those beliefs adaptive rather than anchored at the target. When agents have long planning horizons, the model approximates rational expectations: inflation expectations are well anchored, disinflations are relatively costless, and policy transmits quickly. When planning horizons are short, longer-run inflation expectations can become unanchored, disinflations are costly, and policy transmission lags lengthen. The NK-FHP model thus nests both extremes and provides micro-foundations for the “inflation scares” discussed by Goodfriend (1993).

Q2. What is the “inflation scares problem” and why does it matter for optimal policy?

An inflation scare occurs when agents’ longer-run inflation expectations deviate persistently from the central bank’s target because agents with short planning horizons update beliefs adaptively, and past high inflation feeds forward into current expectations. This creates a welfare-relevant asymmetry: once expectations become unanchored, a disinflation is costly in output because the central bank must build credibility against backward-looking expectations. The standard NK model with rational expectations does not generate this problem—rational agents’ inflation expectations are pinned to the target irrespective of history—so it cannot address the design of policy specifically to prevent scares from materializing.

Q3. What is the “preemptive motive” and how does it modify the leaning-against-the-wind principle?

Optimal time-consistent policy under uncertain private-sector foresight adds a preemptive motive to the standard leaning-against-the-wind (LATW) principle: the central bank contracts demand not just in response to current output-gap and inflation deviations, but also to prevent expectations from becoming unanchored. Under the standard NK LATW result (Clarida, Galí, and Gertler 1999), the policymaker responds to the means of output and inflation. Under uncertain and potentially short-horizon foresight, optimal policy also depends on the distribution of output and inflation, as well as agents’ beliefs about future inflation—specifically, whether those beliefs risk drifting away from target. The preemptive motive implies a more aggressive policy response to the risk of an inflation scare even before the scare has fully materialized.

Q4. How does the paper relate to the post-pandemic inflation experience?

Using parameter estimates from an estimated version of the NK-FHP model, the paper applies the optimal policy framework to quantify how much the preemptive motive matters during the recent post-pandemic inflation surge. The model—which has been shown in related work (Gust, Herbst, and López-Salido 2022, 2024) to fit macroeconomic time series substantially better than hybrid NK models and to account for initial underreaction and subsequent overreaction of inflation forecasts—is well suited to analyze an episode where longer-run inflation expectations initially remained anchored but later showed signs of drift. The paper’s quantification indicates that risk-management considerations, including the preemptive motive, significantly affect the optimal policy path.

Key concepts

finite-horizon planning (NK-FHP) : a bounded-rationality framework (Woodford 2018) in which agents evaluate only those state-contingent paths within a finite planning horizon, updating beliefs about events beyond the horizon adaptively from past data.

inflation scares : episodes (Goodfriend 1993) in which agents’ longer-run inflation expectations deviate persistently from the central bank’s target, making disinflation costly; the NK-FHP model provides micro-foundations for such scares.

preemptive motive : the additional incentive for a policymaker to tighten beyond what current output-gap and inflation deviations alone would prescribe, specifically to prevent longer-run inflation expectations from becoming unanchored.

time-consistent policy under uncertainty : optimal policy that does not rely on commitment and hence accounts for future re-optimization; in this model it must also account for the non-additive uncertainty arising from a distribution of planning horizons.

Uniform Priors for Impulse Responses

Mon, 01 Jan 0001 00:00:00 +0000

Structural vector autoregressions (SVARs) identified with sign restrictions are a widely used tool for estimating dynamic causal effects in macroeconomics. Critics—notably Baumeister and Hamilton (2015) and Watson (2020)—have called for caution because the standard practice of using a uniform prior over the set of orthogonal matrices (with respect to the Haar measure) induces non-uniform marginal prior distributions over the identified sets of individual impulse responses. This paper formally challenges that caution: through an if-and-only-if theorem the authors show that the uniform prior over orthogonal matrices is not only sufficient but also necessary to induce a uniform joint prior distribution over the identified set for the vector of impulse responses—a result that holds for any prior distribution over the reduced-form parameters. The paper additionally shows how to conduct posterior inference based on a uniform joint prior for the vector of impulse responses, which requires modifying the prior for the reduced-form parameters away from the standard Minnesota prior while retaining the uniform prior over orthogonal matrices. An application to Watson’s (2020) empirical example finds that joint credible sets under this new prior are similar to, but wider than, those obtained under the standard approach, and that imposing tighter identifying restrictions sharpens inference under both priors.

Summary of a forthcoming paper, AI-assisted and human-reviewed. See the linked original for the authoritative claims and full conditions.

In depth

Q1. What is the core result and what does it imply for applied work?

The central result is an if-and-only-if theorem: the uniform prior over the set of orthogonal matrices is both sufficient and necessary for the conventional Bayesian approach to induce a uniform joint prior distribution over the identified set for the vector of impulse responses, for any prior over the reduced-form parameters. The critics’ concern about non-uniform individual marginal priors does not extend to the joint object: when inference targets the full vector of impulse responses, the standard Haar prior is exactly appropriate. Practitioners interested in joint inference on the shape and comovement of the impulse response function need not heed the call for caution.

Q2. Why does non-uniformity of individual marginal priors not imply non-uniformity of the joint distribution?

The marginal distribution extracted from a uniform joint distribution over a compact manifold need not be uniform; marginal uniformity and joint uniformity are different properties, and only the latter is required for observationally equivalent vectors to be distinguished solely by the identifying restrictions. Baumeister and Hamilton (2015) and Watson (2020) correctly note that individual impulse responses have non-uniform marginal priors under the Haar measure, but this is not the relevant criterion when the object of interest is the entire impulse response vector. The paper’s theorem shows the joint distribution is uniform, which is the property that ensures the identification restrictions—not the prior—drive the posterior shape.

Q3. How does one implement a uniform joint prior for the vector of impulse responses?

The authors show that a uniform joint prior for the vector of impulse responses requires a modified prior for the reduced-form parameters: one that is independent between (B, Σ) and Q, takes a model-dependent non-standard form for (B, Σ), and retains a uniform prior over orthogonal matrices. The induced reduced-form prior resembles but differs from both the standard Minnesota prior and Uhlig’s (2005) “weak prior.” Because the induced prior for (B, Σ, Q) is still a uniform-normal-inverse-Wishart (UNIW) distribution, the conventional sampling algorithm applies without modification; analysts supply the modified reduced-form prior while continuing to draw Q uniformly from the Haar measure.

Q4. What does the empirical illustration show?

In Watson’s (2020) empirical example, joint credible sets under the uniform-joint-prior approach are similar to but wider than those under the standard Minnesota-prior approach. The widening is consistent with theory: the uniform joint prior spreads probability mass more evenly over the identified set rather than concentrating it toward regions favored by the Minnesota prior. The finding that tighter identifying restrictions sharpen inference under both approaches reinforces the conclusion of Inoue and Kilian (2022b) that many sign restrictions help when the focus is on joint distributions.

Q5. How is the analysis generalized?

The paper extends the results to a broader class of objects of interest—any smooth function of impulse responses, such as combinations of structural elasticities and standard deviations—with an importance-sampling correction when the induced prior over orthogonal matrices is not uniform in the extended case. The generalization exploits the diffeomorphism between IR parameters and orthogonal reduced-form parameters, which allows the change-of-variables formula to apply to any smooth object of interest.

Key concepts

vector of impulse responses : the collection of impulse responses across all variables, shocks, and horizons, treated as a single vector object for joint inference; contrasted with individual impulse responses (the response of one variable to one shock at one horizon).

uniform prior over orthogonal matrices (Haar measure) : the unique probability measure on the set of n×n orthogonal matrices invariant under left and right multiplication; the standard prior used in Bayesian sign-restricted SVARs.

identified set : the set of vectors of impulse responses that are observationally equivalent given the data and the sign restrictions; the conventional approach draws uniformly from this set under the Haar prior.

uniform-normal-inverse-Wishart (UNIW) prior : the joint prior over orthogonal reduced-form parameters consisting of the Haar prior over Q and a normal-inverse-Wishart prior over (B, Σ); conjugate and computationally tractable.