## Online Annex 2.2. Theoretical Modeling Framework

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### Model setup and key features
- Nonlinear dynamic stochastic general equilibrium (DSGE) model with occasionally binding housing collateral constraint, as in Guerrieri and Iacoviello (2017).
- Focuses on the household sector; corporate sector interactions are ignored.
- Two household types distinguished by discount factors: borrowers and lenders.
- Households choose consumption, leisure, and housing subject to budget and collateral constraints.
- Housing is the only collateral for borrowing; house prices determined by a standard forward-looking asset pricing formula.
- Collateral constraint is occasionally binding depending on house prices, income, and debt level.
- Other model components: Euler equations for each household type, a new Keynesian Phillips curve, and a monetary policy rule.

### How the model replicates housing crises
- Model produces an ergodic distribution of output gaps showing huge declines in output during a housing crisis (Panel 1 of Annex Figure 2.2.1).
- Mechanism: when household debt is high, an income decline induces further borrowing until the collateral constraint binds; households then undertake "fire sales" (or foreclosures), house prices fall, the collateral constraint tightens further, consumption is reduced, aggregate demand and output decline, wages and household income fall, and a vicious cycle ensues.
- The collateral-constraint-driven tightening propagates declines in house prices into large real economy contractions.

### Debt-to-GDP as an early-warning indicator
- Panel 2 of Annex Figure 2.2.1: scatter plots of house price growth in period t (vertical axis) against debt-to-GDP ratios in period t-1 (horizontal axis).
- The three lines represent the estimated 5th, 50th, and 95th percentile.
- Findings:
  - Higher debt-to-GDP ratios have no effects (or slightly positive effects) on median growth of house prices (50th percentile).
  - Higher debt-to-GDP ratios significantly increase the probability of housing crises (lower tail, 5th percentile).
- Conclusion: the model provides theoretical foundations for using debt-to-GDP ratios as an early warning indicator, consistent with empirical quantile regression analysis.

### Policy experiments evaluated
- Three policy measures examined:
  1. Macroprudential measures (MPMs) modeled as a Pigouvian tax on debt; rule approximated by a linear function of household debt (high taxes on debt when debt level is high). Note: caps on loan-to-value ratios and surcharges on lending rates are mathematically equivalent to taxes on debt in the model.
  2. Monetary policy augmented by a response to household debt (nominal interest rates positively associated with household debt level): central bank raises rates during run-up and lowers rates in deleveraging.
  3. Monetary policy augmented by a response to credit spreads (gap between secured and unsecured lending), following Cúrdia and Woodford (2011): central bank lowers interest rates in crises in response to higher credit spreads.
- Crisis definition in policy comparisons: output growth less than −3 percent in the baseline model.

### Macroprudential measures: effects and channels
- Ergodic distribution (Panel 1): with MPMs (green line) volatility of the output gap is significantly decreased relative to baseline (blue line), particularly by shrinking negative tails.
- Crisis-period effects (Panel 3A, 3B, 3D):
  - MPMs mitigate declines in output growth during a crisis.
  - MPMs mitigate declines in house price growth during a crisis.
  - Average household debt level is significantly lower than baseline, implying lower debt not only in run-up periods but also in normal times when policymakers commit to activate MPMs in response to overborrowing.
- Mechanism: by reducing debt accumulation and preventing large house price declines, MPMs prevent tightening of the collateral constraint and thereby cushion the real economy.

### Monetary policy responding to household debt (MP1): effects and limitations
- Calibration: parameters set so steady-state debt equals the steady-state debt in the economy with macroprudential measures (Annex Figure 2.2.1, panel 3D).
- Findings:
  - Slight mitigation of adverse effects on output growth in a crisis (panel 3A).
  - No effect on house price growth in a crisis (panel 3B), unlike MPMs.
  - Larger declines in nominal interest rates in a crisis than baseline (panel 3C) as central bank responds to deleveraging.
- Overall assessment: monetary policy responding to household debt only slightly mitigates crisis output losses and does not influence house prices; monetary policy is described as potentially too blunt an instrument for crisis management (citing Bernanke 2011).

### Monetary policy responding to credit spreads (MP2): effects and feasibility concerns
- Findings:
  - Slight mitigation of adverse effects on output growth in a crisis (panel 3A).
  - No effect on house price growth in a crisis (panel 3B).
  - Very large declines in nominal interest rates in a crisis (panel 3C) as central bank responds to increases in credit spreads by lowering rates.
- Feasibility concern: such large declines in nominal interest rates may not be possible in some countries in a low interest rate environment, making the practical feasibility of this rule doubtful.
- Conclusion: like the debt-responsive rule, the credit-spread-responsive monetary rule does not prevent crises but can modestly mitigate output losses if significant policy room exists.

### Comparative policy conclusions
- Macroprudential measures:
  - Lower the probability of crises and mitigate negative effects substantially.
  - Reduce both output and house price declines during crises and lower average household debt.
- Monetary policy augmented by household debt response:
  - Slightly mitigates output declines but does not affect house prices.
  - Performs worse than MPMs because monetary policy affects a broader set of interest rates (including deposit rates), not just lending rates.
- Monetary policy augmented by credit spread response:
  - Slightly mitigates output declines but not house price declines.
  - Requires considerable room for policy reactions; may be infeasible in low-rate environments.
- Overall implication: macroprudential policy is more effective than the explored forms of monetary policy augmentation for preventing and mitigating housing crises, while monetary policy can provide some crisis-period relief but is limited and potentially constrained by the policy rate lower bound.

*Source: IMF staff calculations, Online Annex 2.2 (April 2019 — Global Financial Stability Report).*

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_Source: https://www.imf.org/-/media/files/publications/gfsr/2019/april/english/onlineannex22.pdf_