C32 | Macro Paper Warehouse

Committed to flexible fiscal rules

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

A central debate in fiscal policy is whether fiscal rules—numerical constraints on budget deficits or debt levels—impair a government’s ability to respond to adverse economic shocks, creating a fundamental trade-off between debt stabilization and macroeconomic stabilization. This paper uses data on large, random natural disasters as exogenous shocks to address the endogeneity of rule adoption and provides new empirical and theoretical evidence on this trade-off. Contrary to the trade-off hypothesis, countries with fiscal rules perform significantly better following such disasters than countries without rules: GDP and private consumption are persistently higher, and fiscal policy is significantly more expansionary. The superior performance is shown to depend on the existence of prior fiscal space and the presence of escape clauses in the rules. A model of sovereign default with endogenous fiscal space and tax plans rationalizes these findings: tight rules prevent myopic governments from accumulating excessive debt in good times, which creates fiscal space for deficit spending when disasters strike, keeping sovereign spreads lower and enabling more expansionary fiscal responses.

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 identification strategy and why do natural disasters solve the endogeneity problem?

Large natural disasters serve as a source of exogenous, random adverse economic shocks; by interacting disaster exposure with the presence or absence of fiscal rules, the paper identifies the effect of rules on macroeconomic performance without confounding from the non-random adoption of rules. Endogeneity is a central concern in the fiscal rules literature because countries that adopt rules may differ in politically or economically relevant ways from those that do not (e.g., more disciplined political environments, stronger institutions). Using large disasters as quasi-experimental variation removes this concern: the timing and magnitude of natural disasters are uncorrelated with which countries happened to adopt fiscal rules, isolating the effect of rules on crisis response.

Q2. What are the main empirical findings?

Countries with fiscal rules show significantly higher output and private consumption following large natural disasters, and implement significantly more expansionary fiscal policy, compared to countries without rules—holding over a 1970Q1–2018Q4 quarterly panel—with confidence bands at the 68% and 90% levels based on 500 Monte Carlo draws. The result directly contradicts the commonly held view that fiscal rules restrict governments’ ability to respond to shocks. Moreover, the paper finds that the superior performance of rule-constrained countries is conditional on two features: the existence of fiscal space prior to the shock (low debt or deficit positions), and the presence of escape clauses that allow rules to be suspended during severe adverse events.

Q3. What is the model mechanism?

In the sovereign default model, a fiscal rule prevents a myopic government from over-borrowing in good times out of political economy considerations (e.g., electoral incentives to spend); this forced restraint creates fiscal space—lower debt, lower sovereign spreads—which allows the government to run deficits when a shock hits without triggering a default episode or a sharp rise in borrowing costs. The model predicts that, relative to a no-rule economy, when a disaster strikes in a rule-constrained economy: sovereign spreads spike by less, the fiscal policy response is more expansionary, and output and consumption are higher. Escape clauses in the rules are important: they allow the government to depart from the rule explicitly in crisis situations without destroying the credibility of the rule in normal times.

Q4. What is the policy implication for the COVID-19 fiscal response?

The paper’s findings directly address the suspension of fiscal rules during COVID-19: the theoretical and empirical results suggest that rules with escape clauses do not impair crisis response and may actually improve it, by ensuring fiscal space is available when needed. The paper’s evidence implies that the COVID-era suspension of rules in many countries (including the EU’s Stability and Growth Pact) was not necessarily required to enable expansionary fiscal responses—countries with well-designed rules including escape clauses could have responded expansively while maintaining rule credibility.

Key concepts

escape clause : a provision in a fiscal rule that explicitly permits departure from the rule’s numerical target under defined circumstances (severe recessions, natural disasters, etc.); the paper finds that the presence of escape clauses is one of the two conditions for rule-constrained countries to outperform non-rule countries after adverse shocks.

fiscal space : the buffer of low debt and deficit levels that allows a government to increase spending or cut taxes during a shock without triggering unsustainable debt dynamics or elevated sovereign spreads; the paper shows fiscal space is created by rules in good times and consumed in bad times.

Double Robustness of Local Projections and Some Unpleasant VARithmetic

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

This paper provides formal theoretical results on the relative robustness of local projection (LP) and vector autoregression (VAR) confidence intervals for impulse response inference when the data generating process (DGP) is locally misspecified. The research question is whether the widely held belief that LP estimators are more robust to misspecification than VARs is theoretically justified, and if so, precisely under what conditions and with what consequences for VAR inference.

The analytical framework models the DGP as a stationary structural VARMA(1, ∞) that is local to an SVAR(1), of the form y_t = Ay_{t-1} + H[I + T^{-ζ}α(L)]ε_t, where the MA component T^{-ζ}α(L)ε_t represents misspecification that vanishes at rate T^{-ζ} as sample size T grows. The key rate parameter is ζ ∈ (1/4, 1/2), which corresponds to misspecification large enough to be detected with probability approaching 1 by conventional Hausman-type specification tests, yet small enough that the bias-variance trade-off between LP and VAR remains non-trivial asymptotically. The framework encompasses under-specification of lag length, omitted variables, temporal aggregation, measurement error, and failure of shock invertibility — essentially all sources of dynamic misspecification relevant to linearized DSGE models.

The main finding on LP is a “double robustness” result: the conventional LP confidence interval achieves correct asymptotic coverage for all ζ > 1/4, even when misspecification is large enough to be detected with certainty. The mechanism is that the omitted-variable bias in the LP regression is of order T^{-2ζ} = o(T^{-1/2}) when ζ > 1/4, because both the direct effect of omitted lags on the outcome and the covariance of the residualized regressor with omitted lags are each of order T^{-ζ}, so their product is negligible relative to the T^{-1/2} standard deviation. This is formally analogous to double robustness in partially linear regression and debiased machine learning: LP is consistent if either the outcome-equation controls or the first-stage controls are correctly specified.

In stark contrast, the VAR estimator carries asymptotic bias of order T^{-ζ}, which is non-negligible relative to its T^{-1/2} standard deviation for ζ ≤ 1/2. This causes the conventional VAR confidence interval to severely undercover: for ζ ∈ (1/4, 1/2) the coverage converges to zero, and for ζ = 1/2 it converges to a level strictly below the nominal level.

The “no free lunch” result formalizes the trade-off. Setting ζ = 1/2 and bounding the noise-to-signal ratio at M²/T, the worst-case scaled VAR bias equals M√(aVar(β̂_h)/aVar(δ̂_h) − 1). This worst-case bias is small if and only if the VAR asymptotic variance is close to that of LP. When the VAR standard error is less than half that of LP — which is typical in applied practice — worst-case coverage falls below 48% even for M = 1. Moreover, the least favorable misspecification takes the form of exponentially decaying MA coefficients peaking at horizon h, a pattern consistent with standard economic theories of adjustment costs, learning, or overshooting, and is difficult to rule out on prior grounds. The Hausman test also provides weak protection: when M = 1, the odds of the test failing to reject are nearly 3-to-1 at the 10% significance level.

Simulations using the Smets and Wouters (2007) model with T = 240 observations confirm these results. With lag length selected by AIC (median selected p = 2), VAR confidence intervals materially undercover at all but very short horizons while LP achieves close to nominal coverage throughout. Increasing lag length to p = 4 or p = 8 ameliorates VAR undercoverage at short horizons but at the cost of making VAR confidence intervals essentially as wide as LP intervals, with substantial undercoverage persisting at longer horizons. For p = 4 the total misspecification measure is M ≈ 3.23; for p = 8, M ≈ 1.89.

Scope conditions: results are pointwise asymptotic in fixed model parameters and horizon; they abstract from order-T^{-1} small-sample biases from persistence or the nonlinearity of the impulse response transformation. The LP robustness result requires controlling for lags that are strong predictors of the outcome or impulse variables; omitting lags with small-to-moderate predictive power does not threaten coverage.

Q: What is the precise sense in which LP confidence intervals are “doubly robust”?

A: LP is doubly robust in the sense of partially linear regression: its bias from misspecified MA dynamics is the product of two errors, the estimation error in the outcome-equation lag controls γ̂ − γ_0 and the estimation error in the first-stage lag controls ν̂ − ν_0. In the local-to-SVAR model each error is of order T^{-ζ}, so their product is of order T^{-2ζ} = o(T^{-1/2}) whenever ζ > 1/4, making the omitted-variable bias negligible relative to the T^{-1/2} standard deviation. This means the asymptotic distribution of the LP estimator is completely invariant to the misspecification parameters α(L) and ζ.

Q: How large does misspecification need to be before LP coverage is threatened?

A: The LP double robustness result holds for all ζ > 1/4 regardless of the magnitude parameter M of the MA misspecification. Misspecification with ζ ∈ (1/4, 1/2) can be detected with probability approaching 1 asymptotically by standard specification tests — in particular, the Hausman test is consistent for this range — yet LP coverage remains exactly correct. There is no threshold M below which LP fails; robustness is structural, not contingent on misspecification being small.

Q: Under what conditions does the VAR estimator have zero asymptotic bias?

A: The VAR asymptotic bias is zero if and only if the lagged shocks ε_{j*,t-ℓ} for ℓ = 1, …, h lie in the span of the lagged data used for estimation. Two sufficient conditions from Corollary 3.2 are: (i) the true model is SVAR(p_0) and the estimation lag length p satisfies h ≤ p − p_0, so the extra lags absorb the residual MA structure; or (ii) the shock of interest is directly observed and ordered first, and h ≤ p. In these cases the VAR estimator is asymptotically equivalent to LP, with equal variance.

Q: What is the “no free lunch” result for VARs?

A: For ζ = 1/2 and noise-to-signal ratio bounded by M²/T, the worst-case scaled VAR bias equals M√(aVar(β̂_h)/aVar(δ̂_h) − 1) (Proposition 4.1). This quantity is small if and only if aVar(δ̂_h) ≈ aVar(β̂_h), meaning the VAR has little efficiency advantage over LP. Put differently, the only way to guarantee robust VAR coverage is to include enough lags that the VAR confidence interval becomes as wide as the LP interval. There is no procedure that simultaneously offers narrower intervals than LP and reliable coverage.

Q: How severe is the worst-case undercoverage of conventional VAR confidence intervals?

A: From Corollary 4.3, even for M = 1 (a noise-to-signal ratio of just 1/T), worst-case VAR coverage falls below 48% whenever the VAR asymptotic standard deviation is less than half that of LP — a configuration typical in applied practice. For larger M the undercoverage is worse: the formula 1 − r(M√(aVar(β̂_h)/aVar(δ̂_h) − 1); z_{1-α/2}) can approach zero. Furthermore, the worst-case probability that VAR fails to cover AND the Hausman test fails to reject misspecification simultaneously exceeds 46% when the VAR standard deviation is less than half that of LP (Corollary 4.4).

Q: Can the researcher detect the problematic misspecification using a Hausman test before it causes undercoverage?

A: Only weakly. When M = 1, the Hausman test fails to reject misspecification with probability approximately 74% (odds of nearly 3-to-1) at the 10% significance level, since r(1; z_{0.95}) = 26%. At the 5% level the odds of non-rejection are nearly 5-to-1, since r(1; z_{0.975}) = 17%. The least favorable misspecification also cannot be ruled out on economic-theory grounds: the least favorable MA polynomial has exponentially decaying coefficients peaking at horizon h, consistent with adjustment costs, learning, or overshooting.

Q: Does using a bias-aware critical value (Armstrong-Kolesár approach) resolve the VAR undercoverage problem?

A: The bias-aware VAR confidence interval CI_B(δ̂_h; M) achieves correct asymptotic coverage by inflating the critical value based on the known bound M on misspecification. However, the bias-aware VAR interval tends to be wider than the LP interval. Specifically, M must be quite small — apparently below 1 — for the bias-aware VAR to dominate LP in width regardless of DGP and horizon. For M ≥ 2 (noise-to-signal ratio above 4/T), bias-aware VAR is dominated by LP in interval width. The practical conclusion is that the simpler LP interval is preferable in most empirically relevant settings.

Q: What does the minimax model-averaging result say about optimal weighting of LP and VAR?

A: From Corollary 4.2, the minimax optimal weight on LP when estimating a convex combination of LP and VAR estimators is M²/(1 + M²). For M = 1 (equal noise-to-signal threshold), the optimal weight is 50% on each. For M = 2, the LP estimator receives 80% weight. In the Smets and Wouters simulations, M ≈ 3.23 for p = 4 lags, corresponding to an optimal LP weight of approximately 91%, and M ≈ 1.89 for p = 8 lags, giving an optimal LP weight of approximately 78%.

Q: What do the Smets and Wouters simulations show about AIC-selected VARs?

A: In 5,000 simulated samples of T = 240 observations from the Smets and Wouters (2007) model, the AIC selects a median lag length of p = 2. At all but very short horizons, VAR confidence intervals materially undercover while LP confidence intervals throughout achieve close to nominal coverage. A bootstrap correction for VARs somewhat improves coverage but leaves large distortions. Increasing lag length to p = 4 or p = 8 moves coverage closer to nominal at short horizons (h ≤ p) but makes VAR confidence intervals essentially as wide as LP, and substantial VAR undercoverage persists at longer horizons.

Q: Is the no-free-lunch result specific to univariate impulse responses?

A: No. Proposition 4.2 extends the result to simultaneous inference on multiple impulse responses. For any k × 1 linear combination R of the impulse response vector, the worst-case squared bias is M² λ_max(R[aVar(β̂) − aVar(δ̂)]R’), where λ_max denotes the largest eigenvalue. Because VAR impulse response estimates are often highly correlated across horizons, undercoverage can be particularly severe in the multivariate (joint confidence ellipsoid) case. The no-free-lunch principle holds: the VAR ellipsoid offers non-negligible worst-case bias as long as it offers any efficiency gain relative to LP for any linear combination of horizon-specific impulse responses.

Q: What is the practical recommendation for lag selection in LP and VAR?

A: The paper offers three practical guidelines. First, LP researchers should control for those lags of the data that are strong predictors of the outcome or impulse variables, using conventional information criteria (such as AIC) applied to a VAR in all variables to select the number of lags for LP control — omitting lags with small-to-moderate predictive power does not threaten coverage. Second, VAR researchers should increase the lag length until the VAR confidence interval is no longer substantially narrower than the corresponding LP interval. Third, conventional specification tests do not suffice to guard against VAR coverage distortions.

Local Projection (LP) Estimator: The LP estimator for the impulse response at horizon h is the OLS coefficient on the shock variable y_{j*,t} in a direct regression of y_{i*,t+h} on y_{j*,t}, the variables ordered before it, and lagged data. It is a “direct” estimator in that it does not iterate a one-step VAR forward.

Double Robustness: A property of LP whereby its asymptotic bias from MA misspecification equals the product of two estimation errors — in the outcome-equation lag controls and in the first-stage residualization controls — each of order T^{-ζ}, making their product of order T^{-2ζ} = o(T^{-1/2}) for ζ > 1/4. This is the LP analogue of the double robustness of partially linear regression estimators in debiased machine learning.

Local-to-SVAR Misspecification: A DGP of the form y_t = Ay_{t-1} + H[I + T^{-ζ}α(L)]ε_t in which the MA term T^{-ζ}α(L)ε_t represents misspecification that vanishes at rate T^{-ζ}. The rate parameter ζ governs the magnitude; ζ ∈ (1/4, 1/2) is the empirically relevant range where bias is detectable by specification tests yet the bias-variance trade-off between LP and VAR remains non-trivial.

No Free Lunch (for VARs): The result that the worst-case scaled VAR bias equals M√(aVar(β̂_h)/aVar(δ̂_h) − 1), implying that the VAR confidence interval has reliable (robust) coverage if and only if the VAR asymptotic variance is close to that of LP — i.e., there is no way to simultaneously have shorter confidence intervals than LP and guaranteed coverage robustness.

Noise-to-Signal Ratio: The quantity T^{-1}||α(L)||² = trace{Var(T^{-1/2}α(L)ε_t) Var(ε_t)^{-1}}, which measures the total magnitude of the MA misspecification relative to the variance of the shocks. The paper bounds this at M²/T and uses M as the sufficient statistic for worst-case bias and coverage.

Bias-Aware Critical Value: An inflated critical value cv_{1-α}(b) solving r(b; cv_{1-α}(b)) = α, used to construct a VAR confidence interval CI_B(δ̂_h; M) that achieves correct asymptotic coverage by accounting for the worst-case bias M√(aVar(β̂_h)/aVar(δ̂_h) − 1). The paper shows this approach typically produces intervals at least as wide as LP for M ≥ 2.

Asymptotic Bias of VAR (aBias): The scaled bias term T^{ζ}E[δ̂_h − θ_{h,T}] converging to aBias(δ̂_h) = trace{S^{-1}Ψ_h H Σ_{ℓ=1}^∞ α_ℓ D H’(A’)^{ℓ-1}} − e’{i*,n} Σ{ℓ=1}^h A^{h-ℓ} H α_ℓ e_{j*,m}. This term is structurally absent from the LP asymptotics due to the double robustness mechanism.

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.

Heterogeneity and the Macro-Economic Effects of Changes in Loan-to-Value Limits

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

Layer 1 — Overview

Research Question

De Veirman and de Jong develop a new approach to estimating the macroeconomic effects of changes in regulatory loan-to-value (LTV) limits on mortgage loans. The central questions are: (1) how do changes in an LTV cap translate into changes in the average LTV and, through that channel, into house prices and real output; and (2) how do heterogeneity in the cross-sectional LTV distribution, non-linearity, and asymmetry shape those effects?

Motivation and Gap

Prior empirical literature on macroprudential LTV policy typically pools across countries using coded indicator variables, which imposes the restriction that all LTV policy actions have the same effect regardless of the size of the change or the position of the limit relative to the distribution. Standard TANK models with homogeneous borrowers imply either full symmetry or threshold asymmetry precisely at the point where the constraint ceases to bind. The authors are the first to relate borrower heterogeneity to non-linearity and asymmetry in LTV policy effects.

Data and Setting

The empirical application focuses on the Netherlands, which introduced an LTV cap of 106 percent on August 1, 2011, subsequently reduced in annual one-percentage-point steps to 100 percent by January 2018. Cross-sectional LTV distributions are constructed from the De Nederlandsche Bank Loan Level Data (LLD), covering 77-81 percent of outstanding Dutch mortgage debt in 2012Q4-2014Q4, restricted to borrowers aged 35 or younger as a proxy for first-time buyers. A survey-based average LTV series spanning 1979-2015 was fielded in January 2016 across the CentERpanel and LISS panel (7,943 respondents combined; 2,238 usable observations after cleaning), measuring LTV at the time of first home purchase. This survey-based annual LTV series, together with the log relative house price, log real GDP, and the real mortgage rate, forms a four-variable Vector Error Correction Model (VECM) estimated over 1981-2015, with a single cointegrating vector identified by Johansen maximum likelihood.

Methodology

The authors’ core innovation is to translate changes in the LTV cap into changes in the cross-sectional average LTV by applying each successive cap level to the underlying distribution: observations above the cap are moved to the cap value (with adjustments for exceptions in the ex post variant). These implied annual changes in the average LTV serve as a succession of impulses fed into the VECM. Two variants are implemented: an ex ante approach using only the pre-cap 2010M8-2011M7 distribution, and an ex post approach that uses the most recent empirical distribution prior to each cap change. The Cholesky identification ordering is [LTV, house prices, GDP, mortgage rate].

Main Findings with Quantitative Magnitudes

Non-trivial macroeconomic effects of Dutch LTV policy: Under the ex post approach (the preferred estimate), the imposition of the cap at 106 percent in 2011 and its gradual reduction to 100 percent by 2018 imply, twenty years after the first shock, that relative house prices are 4.84 percent lower and real GDP is 1.15 percent lower than they would have been in the absence of the cap sequence. The bulk of these responses materializes within ten years, at 4.18 percent and 1.05 percent respectively.
Non-linearity: For a given underlying distribution, changes in the cap have progressively larger effects as the cap tightens. In the ex ante approach, the fraction of households constrained by the cap rises from approximately 20 percent at a limit of 105 percent to approximately 40 percent at a limit of 100 percent. A 10 percentage point tightening from 110 to 100 percent implies a long-run relative house price response of 6.12 percent, while a tightening from 100 to 90 percent implies a response of 14.27 percent — a pronounced non-linearity traceable to the substantial mass of observations in the 90-110 range of the Dutch distribution.
Heterogeneity matters substantially: In mean-preserving comparisons using Pearson-family approximations to the pre-cap Dutch distribution, the macroeconomic effects of the actual Dutch LTV policy sequence are 2.58 times larger in the high standard deviation case (standard deviation 25 percent above the Dutch baseline of 17.09) than in the low standard deviation case (standard deviation 25 percent below). Specifically, twenty-year house price responses are 12.34 percent (high SD) versus 4.79 percent (low SD), and GDP responses are 2.93 percent versus 1.14 percent.
Asymmetry is conditional on the position of the cap relative to the distribution: For the Dutch distribution, symmetry is a good approximation for LTV limits at around 80 percent or lower, where the cap is binding for the bulk of households. Asymmetry is pronounced for higher levels. At an initial cap of 100 percent, the absolute effect of a ten-percentage-point tightening is 2.33 times that of a ten-percentage-point loosening. At 80 percent, the asymmetry ratio is only 1.17. Tightenings have smaller effects when they start from a point where few households are constrained; conversely, loosenings can have larger effects when starting from a point where many are constrained.
Homogeneity assumption understates effects above the mean LTV: Under the homogeneous-borrower benchmark (all borrowers at the Dutch mean of 93.72 percent), asymmetry is infinite at cap levels of 100 and 95 percent but zero at other levels — a feature that causes effects to be entirely absent for caps above the mean. In the heterogeneous Dutch setting, an increase in the LTV limit from 95 to 105 percent raises house prices by 10.72 percent in the long run; the homogeneous case implies no effect at all.

Scope Conditions and Caveats

The paper does not address welfare or financial stability effects. The VECM impulse responses do not establish economic causality. Anticipation effects — if households front-loaded high-LTV purchases before the cap — would cause the procedure to overstate the effect. The LTI robustness check (which smooths the loan-to-income ratio due to noisy survey responses) yields twenty-year responses of 3.32 percent (house prices) and 0.74 percent (GDP), somewhat lower than the baseline, indicating that not controlling for LTI tends to overstate the LTV-macroeconomy connection. The approach requires a usable pre-cap or recent-prior LTV distribution; it is not directly portable to settings where a loosening is studied and no recent pre-cap distribution is available.

Layer 2 — Q&A

Q1: What is the fundamental identification challenge this paper faces, and how does the proposed approach address it?

A: The standard challenge is that LTV caps are changed infrequently and have no long time series suitable for regression, so panel studies typically pool countries and use coded dummy variables that impose size-independence of effects. The authors bypass this by using the cross-sectional LTV distribution itself: they measure how each cap level would truncate the underlying distribution and track the implied change in the cross-sectional mean LTV, which is then fed as a shock into a time-series VECM. This approach does not require the cap to have been in place previously, imposes no cross-country coefficient restrictions, and explicitly accounts for the size of the policy change.

Q2: What are the ex ante and ex post approaches to translating cap changes into average LTV changes, and how do their cumulative estimates differ?

A: The ex ante approach applies all successive cap levels to the single pre-cap distribution of 2010M8-2011M7 (after correcting for the June 2011 sales-tax reduction from 6 to 2 percent), without allowing for exceptions. The ex post approach uses the most recent empirical distribution prior to each cap change and accounts for the observed share of borrowers above the cap as exceptions. The ex ante approach yields a cumulative decline in the average LTV of 3.08 percentage points over 2011-2018; the ex post approach yields 1.96 percentage points, roughly one percentage point less. The difference is largely concentrated in 2011-2012 and stems from the ex ante approach not accounting for exceptions to the cap.

Q3: How does the paper correct for the coincident 2011 sales-tax reduction, and why does this matter?

A: In June 2011, the Dutch sales tax on housing purchases fell from 6 to 2 percent, approximately coinciding with the August 2011 imposition of the LTV cap. Without correction, the observed drop in high LTVs in the 106-cap period would conflate the two policy changes. The authors apply a tiered correction: LTVs at or below 100 percent are left unchanged (the data show no notable change in that range); LTVs between 100 and 110 percent are reduced proportionally to the share of total closing costs attributable to the tax; LTVs at or above 110 percent are reduced by the full magnitude of the tax decline. This yields the “tax-adjusted pre-cap distribution” with a mean of 93.72 percent, down from 94.46 percent in the unadjusted data.

Q4: Why does the fraction of constrained households matter so much, and how does it drive non-linearity?

A: The key mechanism is that the average LTV changes when and only when the cap binds for a given borrower. The larger the share of borrowers whose LTV (in the counterfactual uncapped distribution) would exceed the cap, the larger the share of individual LTVs that move in lockstep with any change in the cap, and therefore the larger the aggregate average LTV response and, through the VECM, the house price and GDP response. As the Dutch cap tightened from 105 to 100 percent, the constrained fraction rose from roughly 20 percent to roughly 40 percent, and the annual implied decline in the average LTV grew from 22 basis points to 42 basis points — illustrating monotonically increasing non-linearity within the ex ante approach.

Q5: How does the survey design address the risk of selection bias relative to alternative data sources such as the American Housing Survey?

A: The survey, fielded in January 2016 across both the CentERpanel and LISS panel, asks retrospectively about respondents’ first home purchase, irrespective of whether they still reside there. This avoids the selection bias in the American Housing Survey, where the first-time-buyer flag captures only those still living in the first home — disproportionately selecting homes that are traded less frequently. A single-wave design also avoids the methodological discontinuities that arise from combining multiple survey waves. The resulting series covers 2,238 observations over 1979-2015 (average 60.49 per year).

Q6: What does the VECM cointegration evidence suggest about the long-run relationship between LTV, house prices, GDP, and the real mortgage rate?

A: Augmented Dickey-Fuller tests do not reject a unit root in any of the four series in levels, while all four are stationary in first differences (with the borderline case of log relative house price inflation when an intercept is included). Both the Johansen L-Max and Trace tests reject no cointegration at the 1 percent level, and neither test indicates more than one cointegrating vector. The authors therefore estimate a single-cointegrating-vector VECM with one lag (selected by the Schwarz Information Criterion) over 1981-2015. The long-run relation is normalized so that the coefficient on the log relative house price is one.

Q7: What do the impulse responses in the baseline VECM specification imply for the long-run macro effects of Dutch LTV policy?

A: Under the preferred ex post approach, twenty years after the first shock in 2011 the VECM implies that relative house prices are 4.84 percent lower and real GDP is 1.15 percent lower than the no-cap counterfactual. The bulk of the response materializes within ten years, with house prices 4.18 percent lower and GDP 1.05 percent lower at the ten-year horizon. The twenty-year real mortgage rate response is positive but negligibly small. When the ex ante approach is used instead, responses are larger owing to the larger cumulative LTV impulse.

Q8: How does the paper conduct the mean-preserving heterogeneity exercise, and what are the key quantitative results?

A: The authors generate Pearson-family distributions that match the first four moments of the Dutch pre-cap distribution (mean 93.72, standard deviation 17.09, skewness -1.16, kurtosis 5.97 under the convention that a normal has kurtosis 3), truncated to support (0, 200]. Two alternative distributions are constructed with standard deviations 25 percent below (12.97) and 25 percent above (21.61) the Pearson proxy, holding mean, skewness, and kurtosis constant. The same VECM and Cholesky ordering are applied. Twenty-year house price responses are 12.34 percent (high SD), 8.46 percent (Pearson proxy), and 4.79 percent (low SD). Twenty-year GDP responses are 2.93, 2.01, and 1.14 percent respectively. The ratio of high-to-low-SD responses is 2.58 for both variables.

Q9: How does asymmetry vary across different initial levels of the LTV cap for the Dutch distribution, and what is the intuition?

A: At a starting cap of 100 percent, a ten-percentage-point tightening produces a long-run house price response 2.33 times larger (in absolute value) than a ten-percentage-point easing from the same starting point. At 80 percent the asymmetry ratio falls to 1.17, meaning the effects of tightening and easing are nearly symmetric. The intuition is that at 80 percent the cap is binding for the bulk of the distribution, so both tightenings and easings move a similarly large fraction of borrowers and have large, roughly comparable effects. At 100 percent, far fewer borrowers are currently constrained, so an easing from 100 to 110 moves almost no one whereas a tightening from 100 to 90 moves substantially more.

Q10: What does the comparison of the heterogeneous-borrower and homogeneous-borrower cases reveal about the implications for TANK and HANK models?

A: Under the homogeneous benchmark — all borrowers at the mean Dutch LTV of 93.72 percent — changes in the cap produce infinite asymmetry at cap levels of 100 and 95 percent (tightening has a full effect, easing has zero effect) but zero asymmetry and zero effect for any cap level above 95 percent. For example, an increase in the cap from 95 to 105 percent has no effect in the homogeneous case but raises house prices by 10.72 percent in the heterogeneous case. In sum, homogeneous-borrower models — including TANK frameworks and linearized models with always-binding constraints such as Iacoviello (2005) — overstate asymmetry in a narrow range around the mean LTV and simultaneously understate the effects of cap changes above the mean LTV. The results are more consistent with heterogeneous-agent frameworks, though the authors note they are not aware of any existing HANK paper that investigates asymmetry and non-linearity specifically in response to changes in the borrowing limit.

Q11: What do the robustness checks show about sensitivity of results to LTV measurement choices?

A: The results are robust to all alternative Cholesky orderings, to using the real mortgage rate computed as the nominal rate minus current (rather than two-year moving average) inflation, to using the computed LTV without cross-checking, and to using the directly reported LTV after cross-checking. The most notable alternative is the directly reported LTV without cross-checking, which yields a twenty-year house price response of 3.81 percent and a GDP response of 0.72 percent (ex post approach), somewhat lower than the baseline of 4.84 and 1.15 percent but in the same direction. A further robustness check using an LTV series that extrapolates 2011-2015 values from the Loan Level Data yields larger estimates (cumulative twenty-year house price response of 6.65 percent and GDP response of 1.40 percent), reflecting the LLD series’ more moderate drop in 2014.

Q12: What is the policy implication regarding the importance of distributional information for gauging LTV policy effects?

A: The results imply that knowing the mean of the LTV distribution is not sufficient for estimating the effects of cap changes: the variance — and specifically the fraction of borrowers constrained by the cap — is critical. This is analogous in spirit to the finding of Krueger, Mitman, and Perri (2016) that matching the tails of the wealth distribution, and not just the mean, is essential for determining the aggregate consumption effects of shocks. Existing empirical literature that focuses on the first moment of the LTV distribution will therefore systematically mismeasure the macro effects of LTV limits, and the direction of the bias depends on where the cap stands relative to the distribution.

Key Concepts

Loan-to-value (LTV) cap / limit: The regulatory maximum on the ratio of total mortgage loan amount to the purchase price of the property (excluding buyer-incurred closing costs such as sales taxes and notary fees). In the Netherlands, this was set at 106 percent from August 2011 and reduced annually by one percentage point to 100 percent by January 2018. The paper explicitly distinguishes the cap (the regulatory threshold) from the average LTV (the cross-sectional mean of the distribution, which the cap may or may not bind for all borrowers).

Underlying (or pre-cap) LTV distribution: The cross-sectional distribution of LTV ratios that would prevail in the absence of any LTV cap — approximated in the paper by the empirical distribution in the twelve months before the cap was introduced (2010M8-2011M7, adjusted for the June 2011 sales-tax cut). The shape, mean, and variance of this distribution determine the fraction of borrowers who are constrained by any given cap level and therefore govern the magnitude and symmetry of policy effects.

Mean-preserving change in heterogeneity: A change in the standard deviation of the LTV distribution that holds the mean (and, in the paper’s stylized scenarios, also the skewness and kurtosis) constant. The paper uses this construct to isolate the effect of dispersion per se on the macroeconomic consequences of cap changes, showing that a 25 percent increase in the standard deviation relative to the Dutch baseline more than doubles the macro effects relative to a 25 percent decrease.

Ex ante approach: The method of translating cap changes into average LTV changes that uses only the pre-cap distribution, applying successive cap levels to that single distribution. It does not require an LTV cap to have been in place and is therefore applicable for prospective analysis. It does not account for exceptions to the cap.

Ex post approach: The method that uses the most recent empirical LTV distribution preceding each cap change as the proxy for the counterfactual uncapped distribution, and that explicitly accounts for the observed share of borrowers above the cap (treated as exceptions). Preferred by the authors when feasible because it incorporates information about how the underlying distribution has evolved for reasons unrelated to the current cap change.

Asymmetry ratio: The ratio of the absolute value of the long-run house price (or GDP) response to a ten-percentage-point tightening in the cap to the absolute value of the response to a ten-percentage-point easing from the same initial cap level. A ratio exceeding one indicates that tightenings have larger effects than easings of equal magnitude from the same starting point. In the paper, this ratio is shown to depend critically on where the initial cap sits relative to the underlying distribution.

Non-linearity in LTV effects: The property that changes in the cap from a lower starting point have larger macroeconomic effects than changes from a higher starting point, for a given underlying distribution. This arises because the fraction of constrained borrowers increases as the cap is tightened, so a further tightening moves a larger share of individual LTVs. In the paper, this is documented through the increasing year-on-year effects in Table 1 and the large difference between the house price response to a tightening from 110 to 100 percent (6.12 percent) versus from 100 to 90 percent (14.27 percent).

Pearson system (as used in this paper): A parametric family of distributions in which every combination of the first four moments (mean, variance, skewness, kurtosis) corresponds to a unique distribution. The authors use it to construct smooth approximations to the empirical Dutch distribution with the same mean, skewness, and kurtosis but varying standard deviations, enabling a controlled comparison of heterogeneity scenarios.

Local Projection-Based Inference under General Conditions

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

This paper develops a uniform asymptotic theory for local projection (LP) regression under general conditions, addressing a gap in the literature where existing results required restrictive assumptions about lag order, data persistence, and shock processes. The research question is: how can one conduct valid statistical inference on impulse responses from LP regressions when the true lag order is unknown (possibly infinite), data exhibit arbitrary persistence including unit roots and near-unit roots, horizons are allowed to grow with sample size, and shocks follow general conditionally heteroskedastic martingale difference sequences (MDS)?

The paper works within a VAR(infinity) data-generating process framework, where the vector autoregression may have an unknown and potentially infinite number of lags. The LP regression truncates this at a chosen model order p, with the truncation bias controlled by tail decay conditions on the VAR coefficients. The theoretical framework accommodates a class of VARMA models as a specific illustration, showing that Assumptions 1 and 2 hold for VARMA(q+1, r) processes when the model lag order p diverges at least as fast as log n.

The main theoretical result (Theorem 1) establishes uniform asymptotic normality of the LP estimator, simultaneously over: the coefficient parameter space A, model lag orders p in [p_low, p_high], horizons h in [1, h_bar], and configurations of the linear combination vector gamma (covering both individual and cumulated impulse responses). The convergence rate is pi_1(h; gamma)^{-1/2} n^{1/2}, which depends on persistence level and horizon. For an AR(1) process, the individual response rate is (sum_{i=0}^{h-1} a_1^{2i})^{-1/2} n^{1/2} and the cumulative response rate is h^{-3/2} n^{1/2}, which is slower.

The paper makes two principal contributions. First, LP is shown to be semiparametrically efficient when the controlled lag order diverges. Under classical assumptions (homoskedastic MDS shocks, stationarity, fixed horizon), the LP estimator achieves the same asymptotic distribution as the VAR-implied iterative estimator, and reaches the semiparametric efficiency bound of Chamberlain (1987) under the conditional moment restriction model. Under Gaussianity, LP is asymptotically Cramer-Rao efficient. This extends Plagborg-Moller and Wolf (2021) from distributional equivalence of estimands to equivalence of asymptotic distributions. The commonly held view that LP is inefficient relative to VAR-implied methods holds only under finite small-order VAR models; with a diverging lag order, the efficiency gain from the parsimonious VAR structure vanishes. The alternative LP estimator of Lusompa (2022), shown to be more efficient than standard LP under a known AR(1) model, is likewise shown (Proposition 2) to be asymptotically equivalent to standard LP when a sufficiently large lag order is used (p_u/sqrt(n) -> 0 and sqrt(n)(1-|rho|)^{p_u} -> 0).

Second, two new standard errors are proposed, neither involving HAR-type correction or bandwidth selection. SE_1 is a White-style heteroskedasticity-robust standard error applied after partialling out controls; it is uniformly consistent under a zero fourth cumulant condition on shocks (e.g., zero excess kurtosis with conditional homoskedasticity), but not for general MDS shocks. SE_2, the paper’s main methodological contribution, constructs the variance estimator using martingale-transformed scores: the LP residual Delta_t is projected onto forward residuals (Delta_{t+1}, …, Delta_{t+h-1}) to partial out serial dependence, recovering the true MDS error xi_{1t}(h; gamma) asymptotically. SE_2 is uniformly consistent for general MDS shocks (Proposition 4) and, under a finite-order VAR DGP, requires only p = p_true lags (rather than p >= p_true + 1 required by SE_1 and HAR-type methods).

Simulations using univariate ARMA(1,1) models with rho in {0, 0.5, 0.95, 1} and theta in {-0.5, 0, 0.5}, and bivariate VAR(1) models, confirm that SE_2-based 95% confidence intervals maintain coverage close to the nominal level across all cases including unit roots, while SE_1 shows degraded coverage under conditional heteroskedasticity (GARCH). Both outperform MOPM for cumulated responses at longer horizons.

Scope conditions: the framework accommodates data with unit roots and near-unit roots but not explosive roots or integration of order greater than one (for which differencing is prescribed before applying the LP). The growing-horizon rate condition p^2 h^2 / n -> 0 becomes binding as h grows, requiring h and p to grow at comparable rates or p more slowly. The results are for the VAR framework and do not directly apply to structural (SVAR) identification without additional assumptions.

Q: What is the central inferential problem that motivates this paper?

A: Applied macroeconomists estimating impulse responses via LP regressions face a trilemma: the true lag order is unknown and may be infinite, data may be highly persistent or integrated, and shocks may be conditionally heteroskedastic. Existing uniform validity results (chiefly Montiel Olea and Plagborg-Møller 2021) assume a finite and known model order and require mean-independent shocks, leaving inference potentially invalid when these conditions fail. The paper constructs a theory and inference procedures that remain valid simultaneously over all these dimensions.

Q: What is the VAR(infinity) data-generating process assumed, and what are the key restrictions on it?

A: The DGP is yt = sum_{j=1}^{infinity} a_j y_{t-j} + u_t, where u_t is serially uncorrelated. Assumption 1 bounds the impulse responses uniformly over the parameter space (ruling out explosive roots and integration of order greater than one). Assumption 2 imposes that the tail coefficients a_j decay fast enough that the truncation bias is asymptotically negligible: the rate condition requires sqrt(n) * p * sum_{j=1}^{infinity} j |a_{p+j}| -> 0, implying p must diverge for infinite-order processes. For VARMA models, p need only diverge as slowly as log n.

Q: What does Theorem 1 establish, and what is the convergence rate?

A: Theorem 1 establishes uniform asymptotic normality of the LP estimator, with the supremum taken jointly over the coefficient space A, lag orders p in [p_low, p_high], horizons h in [1, h_bar], and the linear combination vector gamma. The convergence rate is pi_1(h; gamma)^{-1/2} n^{1/2}, where pi_1(h; gamma) = sum_{i=1}^{h} |phi_{1i}|^2 captures persistence and horizon effects. For an AR(1) process, the individual response rate is (sum_{i=0}^{h-1} a_1^{2i})^{-1/2} n^{1/2} and the cumulative response rate is the slower h^{-3/2} n^{1/2}.

Q: In what sense is LP semiparametrically efficient, and under what assumptions?

A: Under classical assumptions — homoskedastic MDS shocks, stationarity, and fixed horizon — when the controlled lag order p diverges at the appropriate rate, the LP estimator reaches the semiparametric efficiency bound of Chamberlain (1987) under the conditional moment restriction model E(yt - sum a_j y_{t-j} | ys, s <= t-1) = 0. It achieves the same asymptotic distribution as the VAR-implied estimator, which itself has the same distribution as the LP estimator under these conditions (established by extending Lutkepohl 1990). Under Gaussianity, LP is asymptotically Cramer-Rao efficient.

Q: Why does the efficiency advantage of VAR-implied methods over LP vanish with a large lag order?

A: Under a finite, small-order VAR model, imposing the functional relationship between all impulse responses and a small set of VAR slope parameters — analogous to dimension reduction in a factor model — yields an efficiency gain for the iterative VAR-implied estimator. However, as the model lag order grows, the number of parameters to estimate grows correspondingly, eroding the dimension-reduction benefit. With a diverging lag order, the extraction of common parameters through a parsimonious model no longer tightens the asymptotic variance of the VAR-implied estimator relative to the direct LP estimator.

Q: How does SE_2 avoid the need for HAR (heteroskedasticity and autocorrelation robust) bandwidth selection?

A: The LP regression error Delta_t(h; gamma) is serially correlated for h >= 2 (it contains MA terms of order h-1), which would normally require HAR correction. SE_2 avoids this by constructing the variance estimator from the martingale-transformed score: the LP residual Delta_t is regressed on the forward residuals (Delta_{t+1}, …, Delta_{t+h-1}) and the fitted residual hat{xi}{1t} is used in place of Delta_t. Asymptotically, hat{xi}{1t} recovers the true LP(infinity) error xi_{1t}(h; gamma) = sum_{i=1}^{h} phi’{1i} u{t+i}, which is a MDS with respect to {u_t, u_{t-1}, …}. Since MDS sums have a martingale structure, their variance can be estimated as a simple sum of squares without bandwidth selection.

Q: Under what condition is SE_1 uniformly consistent, and when does it fail?

A: SE_1 is the standard White heteroskedasticity-robust variance estimator applied to the partialled-out score. It is uniformly consistent under the zero fourth cumulant condition on shocks — that is, when u_t has zero excess kurtosis and is conditionally homoskedastic. This condition fails for general MDS shocks (e.g., GARCH-type shocks), because the cross-moment Cov((tau’w_0)^2, (tau’w_k)^2) does not vanish in general. Simulation results confirm that SE_1-based confidence intervals show degraded coverage under GARCH shocks, while SE_2 maintains coverage.

Q: What is the relationship between this paper and Montiel Olea and Plagborg-Møller (2021)?

A: Montiel Olea and Plagborg-Møller (2021) (MOPM) established uniform validity of LP inference under a finite-order, known VAR model and required mean-independent (not merely MDS) shocks. The current paper extends MOPM in five dimensions: it allows an unknown and potentially infinite true lag order; allows the controlled lag order to diverge; develops new asymptotic theory for general MDS shocks; proposes SE_2 whose consistency does not require mean-independent shocks; and unifies inference for both individual and cumulated impulse responses. The lag-augmented LP regression of MOPM (setting p = p_true + 1) is a special case of the framework here.

Q: What does the paper show about the alternative LP estimator of Lusompa (2022)?

A: Lusompa (2022) showed that, under a known AR(1) model with the true lag order, an alternative LP estimator that exploits the serial dependence structure of the LP error is asymptotically more efficient than standard LP across horizons. Proposition 2 of the current paper shows this efficiency gain does not survive when a sufficiently large lag order is used for the preliminary VAR used to compute the transformation. Specifically, when p_u/sqrt(n) -> 0 and sqrt(n)(1-|rho|)^{p_u} -> 0, the alternative and standard LP estimators are asymptotically equivalent: sqrt(n)[tilde{beta}_1(h) - beta_1(h)] - sqrt(n)[hat{beta}_1(h) - beta_1(h)] = o_p(1). The discrepancy arises from estimation errors in the preliminary residuals entering the asymptotic distribution.

Q: What are the rate conditions on the lag order p and horizon h, and how do they compare to VAR-implied methods?

A: Under a fixed horizon, the condition p^2/n -> 0 suffices for LP, which is weaker than the p^3/n -> 0 typically required for VAR-implied methods (the stricter condition arises because VAR-implied methods must estimate all p slope matrices jointly, while LP treats all but the first as nuisance). Under growing horizons (h -> infinity), the rate condition is p^2 h^2/n -> 0, and the analysis shows p = O(h) is sometimes optimal — p and h should grow at the same rate or p more slowly. By contrast, VAR-implied methods require p = o(n^{1/3}/h^{2/3}) under growing horizons.

Q: What is the lag order flexibility advantage of SE_2 under a finite-order VAR DGP?

A: When the true DGP is a finite-order VAR(p_true), SE_2 achieves consistent inference using exactly p = p_true lags — the exact order. In contrast, SE_1 and HAR-type standard errors require p >= p_true + 1 (at least one extra lag) because at p = p_true the LP residuals Delta_t(h; gamma) contain MA terms of order h-1 that create serial dependence. SE_2’s martingale transformation handles this serial dependence directly, without requiring the extra lag to purge it.

Q: What scope conditions limit the paper’s framework?

A: The framework rules out explosive roots (violating the uniform impulse response bound in Assumption 1) and integration of order two or higher (violating Assumption 1(iii)). For I(2) variables, the prescribed solution is to take differences before applying the LP, and then use the cumulated response (gamma = gamma_CIR) to recover original level responses. The growing-horizon results require the tension condition h_bar * p^2 / n -> 0 (for gamma with ||gamma||_1 = O(1)), implying a binding tradeoff between the range of allowed horizons and the range of allowed lag orders. Results do not directly extend to structural identification without additional assumptions.

Local Projection (LP) regression: A direct regression of the outcome h periods ahead on current and lagged endogenous variables, as in Jorda (2005). The LP estimator of the horizon-h impulse response is the OLS coefficient on the current endogenous variable in this regression, with p-1 lags included as controls. It estimates impulse responses directly for each horizon without imposing the recursive structure of a VAR model.

Uniform asymptotic validity: A distributional approximation (here, standard normal) that holds simultaneously over a parameter space A, a range of model lag orders [p_low, p_high], a range of horizons [1, h_bar], and specifications of the linear combination vector gamma — not merely pointwise for fixed parameter values. Uniformity is the operative concept ensuring finite-sample reliability across empirically relevant configurations.

Semiparametric efficiency: In the paper’s usage, the LP estimator achieves the efficiency bound of Chamberlain (1987) for the semiparametric conditional moment restriction model E(yt - sum a_j y_{t-j} | ys, s <= t-1) = 0 when the controlled lag order diverges. Under Gaussianity, this coincides with Cramer-Rao efficiency. The key result is that the efficiency loss of LP relative to VAR-implied methods — well-documented under finite small-order VAR — is asymptotically negligible once the lag order diverges.

Martingale difference sequence (MDS) shocks: The shock process u_t satisfying E(u_t | u_s, s <= t-1) = 0 almost surely — a condition weaker than mean independence (E(u_t | u_s, s <= t-1) = 0 for all functions of past shocks). MDS shocks include GARCH and stochastic volatility processes. The paper’s SE_2 is designed to be consistent for general MDS shocks, while SE_1 and MOPM require the stronger mean-independence condition.

SE_2 (martingale-transformed standard error): The paper’s proposed standard error, constructed by first regressing LP residuals Delta_t on their forward values (Delta_{t+1}, …, Delta_{t+h-1}) to partial out serial dependence, then using the residual hat{xi}{1t} in the variance estimator as a simple sum of squares. SE_2 is uniformly consistent for general MDS shocks and requires no bandwidth selection, because the residual hat{xi}{1t} asymptotically recovers the MDS LP(infinity) error xi_{1t}(h; gamma).

VAR(infinity) model: A vector autoregression yt = sum_{j=1}^{infinity} a_j y_{t-j} + u_t with potentially infinitely many lags. The paper’s framework treats the true lag order as unknown and possibly infinite, requiring the controlled lag order p in the LP regression to diverge (at a rate constrained by Assumption 2) so that truncation bias becomes asymptotically negligible. VARMA processes are a special case shown to satisfy the paper’s assumptions.

Cumulated impulse response: The linear combination beta_1(h; gamma_CIR) = sum_{j=1}^{h} beta_1(j), corresponding to gamma = (1, …, 1)’. Cumulated responses exhibit slower convergence rates than individual responses — h^{-3/2} n^{1/2} versus (sum_{i=0}^{h-1} a_1^{2i})^{-1/2} n^{1/2} for an AR(1) — and are especially relevant when the response variable is in differences and the researcher seeks level responses of the original variable.

Political Pressure on the Fed

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

This paper combines a hand-collected archival data set of over 800 personal interactions between U.S. Presidents and Federal Reserve officials from 1933 to 2016 with a narrative structural VAR to identify shocks to political pressure on the Fed and quantify their macroeconomic effects. The identification strategy exploits the well-documented Nixon-Burns episode of 1971—corroborated by Nixon Tapes recordings and Burns’s personal diary—as a narrative restriction that the spike in personal interactions that year was driven primarily by a political pressure shock rather than by economic conditions. Political pressure shocks are found to (i) increase inflation strongly and persistently, (ii) lead to statistically weak negative effects on activity, (iii) contribute to inflationary episodes outside the Nixon era, and (iv) transmit differently from standard expansionary monetary policy shocks because political pressure can be publicly observed, generating a stronger direct effect on inflation expectations. Quantitatively, increasing political pressure by half as much as Nixon, sustained for six months, is estimated to raise the price level by more than 8%.

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 narrative identification strategy and how is the Nixon-Burns episode exploited?

The identification strategy imposes that the spike in President-Fed personal interactions in 1971 is mainly driven by a political pressure shock, exploiting the well-documented fact that Nixon pressured Burns to ease monetary policy in the run-up to his 1972 re-election. Recordings from the “Nixon Tapes” and Burns’s personal diary corroborate this interpretation: Burns wrote that “the President will do anything to be reelected” and that Nixon urged him to “start expanding the money supply.” Romer and Romer (2004) estimated large easing shocks to monetary policy prior to Nixon’s re-election, contrasting with a large systematic tightening after it, further supporting that Burns eased in response to the pressure. Narrative evidence from Johnson’s pressure in the 1960s is additionally used to strengthen the identification.

Q2. What does the new data on President-Fed personal interactions show?

The paper hand-collects over 800 personal interactions between U.S. Presidents and Fed officials from the historical daily schedules made available by the Presidential Libraries from Franklin D. Roosevelt (1933) through Barack Obama (2016). The average interaction lasts 53 minutes; 36% are one-on-one; 11% occur on weekends; 16% are in social settings such as dinners; 92% involve the Fed Chair and 8% other Fed officials. There is large variation across administrations: President Nixon interacted with Fed officials 160 times, while only 6 interactions occurred under Clinton. These interactions arise endogenously in response to economic conditions, which is why narrative identification is needed to isolate the political pressure component.

Q3. What are the estimated macroeconomic effects of political pressure shocks?

Political pressure shocks are found to increase inflation strongly and persistently, to have statistically weak negative effects on activity, and a pressure shock half as large as Nixon’s sustained over six months is estimated to raise the price level by more than 8%. The weak activity effect distinguishes these shocks from standard demand expansions; the mechanism operates more through expectations channels than through aggregate demand, consistent with the public observability of political pressure on the central bank. The evidence also suggests political pressure shocks contributed to inflationary episodes in periods beyond the Nixon era.

Q4. Why do political pressure shocks transmit differently from conventional monetary policy easing shocks?

Political pressure shocks transmit differently from standard expansionary monetary policy shocks primarily because political pressure on the Fed can be publicly observed, which generates a stronger direct effect on inflation expectations than a private Fed decision to ease. The paper finds a stronger effect of political pressure shocks on inflation expectations relative to the activity effect, consistent with this channel: when the public observes that the President is pressuring the central bank, expected inflation rises even before the Fed acts on that pressure.

Key concepts

President-Fed personal interactions : face-to-face or telephone contacts between U.S. Presidents and Federal Reserve officials recorded in historical presidential daily schedules 1933–2016; used as a noisy observable proxy for political attention to the Fed, from which a political pressure shock series is extracted via narrative restrictions.

political pressure shock : an exogenous, structurally identified shock to the intensity of political influence on Fed policy, isolated using a narrative SVAR restriction that the 1971 Nixon-Burns spike in interactions was driven by political pressure rather than economic conditions.

narrative identification : an approach that imposes sign or zero restrictions on a structural VAR at specific historical episodes known from external archival evidence to be driven predominantly by a particular structural shock; here used to exploit the Nixon-Burns and Johnson-Fed pressure episodes.

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.