## 4.1  The Loss Function

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### Momentary and life-time loss definitions
- Momentary loss function:
  - L_t(π_0) = (π_t − π^*)^2 + χ(u_t − u^*)^2
  - where π_t is the inflation rate and u_t is the unemployment rate (measure of economic activity).
- Life-time welfare loss:
  - L_0(π_0) = E_0 Σ_{t=0}^T β^t L_t(π_0)
  - where π_0 = {i_t}_{t=0}^T describes the chosen interest rate path and T is the policy horizon.
- Expectation operator bE accounts for both shock and parameter uncertainty perceived by the monetary authority.

### Two-state Markov chain for normal vs crisis states
- Time-t transition matrix:
  - M_t(π_0) = [[1 − p_t(π_0), p_t(π_0)], [ε, 1 − ε]]
  - p_t is probability of going from normal state to crisis state and depends on the interest rate path π_0 through credit variables.
  - ε represents probability of reverting from crisis to normal.
- m-period-ahead transition probabilities:
  - M_{t;t+m} = M_{t+m} M_{t;t+m−1}
- Welfare loss conditional on starting in normal times:
  - L_0(π_0) = [1;0] bE_0 Σ_{t=0}^T β^t M_{0;t}(π_0) L_t(π_0)

### Normal times vs crisis times behavior
- During normal times, evolution of target variables comes from BVAR conditional forecasts.
- During crisis times, the BVAR is considered unreliable; for simplicity, target variables (inflation and output) are assumed constant above their target levels (see Calibration).
- Linear-quadratic (benign-neglect) policy:
  - π^{LQ}_0 = arg max_{π_0} L_0(π_0) with p_t ≡ 0 at all times.
- Leaning-against-the-wind (LAW) policy welfare:
  - L_0(π^{LAW}_0) = [1;0] bE_0 Σ_{t=0}^T β^t M_{0;t}(π^{LAW}_0) L_t(π^{LAW}_0)
- Welfare comparison expressed as difference in welfare losses:
  - L_0(π^{LQ}_0) − L_0(π^{LAW}_0)

### Calibration (baseline)
- LAW baseline specification:
  - π^{LAW}_0 = π^{LQ}_0 + 0.25 for 8 quarters (25 basis points tighter).
- Policy horizon and targets:
  - T = 3 years
  - π^* = 2%
  - χ = 1 (same weight between inflation and output goals)
  - u^* = 6%
- Discount factor:
  - β = 0.996 (monetary authority discounts real future losses at an annualized rate of 1.5 percent)
- Transition parameter:
  - ε = 0.06 (implies average crisis duration of 4.2 years)
- Crisis cost calibration:
  - Moderate crisis: inflation is 2 percentage points below target and unemployment is 5 percentage points above target for entire crisis duration.
  - Severe crisis: unemployment is 7 percentage points above target (in line with IMF 2015).
- Notes on alternative conceptualizations of crisis costs:
  - Alternative (1): deviations measured relative to last pre-crisis observation rather than natural target (Svensson 2016) — implies LAW can make crisis costs more severe.
  - Alternative (2): deviations proportional to pre-crisis credit growth (BIS 2014 and IMF 2015) — implies LAW reduces both probability and severity of crises.
- Institutional note:
  - Canadian mortgage sector has extended government guarantee for mortgages insured through CMHC and a 90 percent Federal guarantee on mortgage insurance provided by two private insurers; may mitigate crisis costs but does not necessarily isolate real sector from financial turmoil.

### Result 1 — Baseline case
- Main finding:
  - Under baseline calibration it is very unlikely that a LAW policy would be beneficial in Canada under current circumstances.
- Quantified impacts:
  - Increase in average welfare loss due to LAW: about 0.39 percent (relative to benign-neglect policy).
  - Specification without debt: relative loss due to LAW increases to about 0.56 percent.
- Uncertainty:
  - Parameter uncertainty is substantial; some parameter draws yield L_0(π^{LQ}_0) > L_0(π^{LAW}_0) reflecting combinations where unemployment and inflation are little affected by monetary policy shocks while crisis probability is at highest percentiles.
- Timing of effects:
  - Benefits of LAW accrue slowly with peak effect after almost 8 years.
  - Costs are front loaded in the first 2 years of tighter policy.
- Additional projection detail:
  - Starting unemployment u_t = 6.8, projected to rise to 7.0 in the next quarter; CPI inflation projected to fall below 2 percent target in the following 2 quarters under the LQ policy.

### Result 2 — Optimal deviation is 6 basis points
- Search for optimal deviation x (basis points) from LQ path over 8 quarters:
  - π^A_0 = π^{LQ}_0 + x/100
- Optimal welfare-minimizing deviation:
  - x = 6 basis points higher than LQ path.
- Welfare improvement:
  - Improves welfare by only 0.02 percent relative to benign-neglect policy.
- Intuition:
  - Momentary loss is quadratic in inflation and unemployment; small deviations around targets have negligible welfare cost.
  - Marginal benefits of higher policy rate on crisis probability are strictly positive (except when crisis probability is exactly zero).
  - There exists a sufficiently small increase in policy rate where marginal benefits exceed marginal costs; that increase is quantitatively just 6 basis points.

### Result 3 — High-credit-growth and severe-crisis scenario
- Motivation:
  - BVAR projects slowdown in credit growth overall, but credit growth is strong in buoyant housing markets (e.g., Vancouver and Toronto).
- Scenario assumptions:
  - Higher credit growth and potentially severe crisis costs.
- Findings:
  - Under this scenario, benefits of LAW are increased because: (1) potential severity of crisis is higher, (2) marginal effects of higher real rates on reducing crisis probability are stronger when crisis predictors are larger.
- Credit growth threshold for LAW (25 basis points for 8 quarters) to be beneficial:
  - When real credit growth exceeds 9.1 percent, the benefits of a LAW policy outweigh costs.

### Key cross-cutting insights and implications
- Initial conditions, non-linearities, long lags, and parameter uncertainty substantially affect policy comparison:
  - (1) The level of debt interacts with credit growth: higher debt amplifies the marginal effect of credit growth on crisis probability.
  - (2) Monetary policy shock affects crisis probability with long lags; peak effect after 8 years.
  - (3) Costs in terms of lower inflation and economic activity are front loaded.
  - (4) Substantial parameter uncertainty in both the credit–crisis probability relation and the effect of monetary policy on credit aggregates.
- Policy implementation challenges:
  - Credibility, implementation, and communication challenges would be substantial for a central bank considering LAW.
  - Systematic use of a non-systematic policy would confront the Lucas critique; more research needed on implications of a systematic reaction to financial risk-taking.

*Source: 4.1  The Loss Function (content unit)_wp1673 - 4.1  The Loss Function*

### 4.1  The Loss Function

### 4.1  The Loss Function

### Momentary and life-time loss definitions
- Momentary loss function:
  - L_t(π_0) = (π_t − π^*)^2 + χ(u_t − u^*)^2
  - where π_t is the inflation rate and u_t is the unemployment rate (measure of economic activity).
- Life-time welfare loss:
  - L_0(π_0) = E_0 Σ_{t=0}^T β^t L_t(π_0)
  - where π_0 = {i_t}_{t=0}^T describes the chosen interest rate path and T is the policy horizon.
- Expectation operator bE accounts for both shock and parameter uncertainty perceived by the monetary authority.

### Two-state Markov chain for normal vs crisis states
- Time-t transition matrix:
  - M_t(π_0) = [[1 − p_t(π_0), p_t(π_0)], [ε, 1 − ε]]
  - p_t is probability of going from normal state to crisis state and depends on the interest rate path π_0 through credit variables.
  - ε represents probability of reverting from crisis to normal.
- m-period-ahead transition probabilities:
  - M_{t;t+m} = M_{t+m} M_{t;t+m−1}
- Welfare loss conditional on starting in normal times:
  - L_0(π_0) = [1;0] bE_0 Σ_{t=0}^T β^t M_{0;t}(π_0) L_t(π_0)

### Normal times vs crisis times behavior
- During normal times, evolution of target variables comes from BVAR conditional forecasts.
- During crisis times, the BVAR is considered unreliable; for simplicity, target variables (inflation and output) are assumed constant above their target levels (see Calibration).
- Linear-quadratic (benign-neglect) policy:
  - π^{LQ}_0 = arg max_{π_0} L_0(π_0) with p_t ≡ 0 at all times.
- Leaning-against-the-wind (LAW) policy welfare:
  - L_0(π^{LAW}_0) = [1;0] bE_0 Σ_{t=0}^T β^t M_{0;t}(π^{LAW}_0) L_t(π^{LAW}_0)
- Welfare comparison expressed as difference in welfare losses:
  - L_0(π^{LQ}_0) − L_0(π^{LAW}_0)

### Calibration (baseline)
- LAW baseline specification:
  - π^{LAW}_0 = π^{LQ}_0 + 0.25 for 8 quarters (25 basis points tighter).
- Policy horizon and targets:
  - T = 3 years
  - π^* = 2%
  - χ = 1 (same weight between inflation and output goals)
  - u^* = 6%
- Discount factor:
  - β = 0.996 (monetary authority discounts real future losses at an annualized rate of 1.5 percent)
- Transition parameter:
  - ε = 0.06 (implies average crisis duration of 4.2 years)
- Crisis cost calibration:
  - Moderate crisis: inflation is 2 percentage points below target and unemployment is 5 percentage points above target for entire crisis duration.
  - Severe crisis: unemployment is 7 percentage points above target (in line with IMF 2015).
- Notes on alternative conceptualizations of crisis costs:
  - Alternative (1): deviations measured relative to last pre-crisis observation rather than natural target (Svensson 2016) — implies LAW can make crisis costs more severe.
  - Alternative (2): deviations proportional to pre-crisis credit growth (BIS 2014 and IMF 2015) — implies LAW reduces both probability and severity of crises.
- Institutional note:
  - Canadian mortgage sector has extended government guarantee for mortgages insured through CMHC and a 90 percent Federal guarantee on mortgage insurance provided by two private insurers; may mitigate crisis costs but does not necessarily isolate real sector from financial turmoil.

### Result 1 — Baseline case
- Main finding:
  - Under baseline calibration it is very unlikely that a LAW policy would be beneficial in Canada under current circumstances.
- Quantified impacts:
  - Increase in average welfare loss due to LAW: about 0.39 percent (relative to benign-neglect policy).
  - Specification without debt: relative loss due to LAW increases to about 0.56 percent.
- Uncertainty:
  - Parameter uncertainty is substantial; some parameter draws yield L_0(π^{LQ}_0) > L_0(π^{LAW}_0) reflecting combinations where unemployment and inflation are little affected by monetary policy shocks while crisis probability is at highest percentiles.
- Timing of effects:
  - Benefits of LAW accrue slowly with peak effect after almost 8 years.
  - Costs are front loaded in the first 2 years of tighter policy.
- Additional projection detail:
  - Starting unemployment u_t = 6.8, projected to rise to 7.0 in the next quarter; CPI inflation projected to fall below 2 percent target in the following 2 quarters under the LQ policy.

### Result 2 — Optimal deviation is 6 basis points
- Search for optimal deviation x (basis points) from LQ path over 8 quarters:
  - π^A_0 = π^{LQ}_0 + x/100
- Optimal welfare-minimizing deviation:
  - x = 6 basis points higher than LQ path.
- Welfare improvement:
  - Improves welfare by only 0.02 percent relative to benign-neglect policy.
- Intuition:
  - Momentary loss is quadratic in inflation and unemployment; small deviations around targets have negligible welfare cost.
  - Marginal benefits of higher policy rate on crisis probability are strictly positive (except when crisis probability is exactly zero).
  - There exists a sufficiently small increase in policy rate where marginal benefits exceed marginal costs; that increase is quantitatively just 6 basis points.

### Result 3 — High-credit-growth and severe-crisis scenario
- Motivation:
  - BVAR projects slowdown in credit growth overall, but credit growth is strong in buoyant housing markets (e.g., Vancouver and Toronto).
- Scenario assumptions:
  - Higher credit growth and potentially severe crisis costs.
- Findings:
  - Under this scenario, benefits of LAW are increased because: (1) potential severity of crisis is higher, (2) marginal effects of higher real rates on reducing crisis probability are stronger when crisis predictors are larger.
- Credit growth threshold for LAW (25 basis points for 8 quarters) to be beneficial:
  - When real credit growth exceeds 9.1 percent, the benefits of a LAW policy outweigh costs.

### Key cross-cutting insights and implications
- Initial conditions, non-linearities, long lags, and parameter uncertainty substantially affect policy comparison:
  - (1) The level of debt interacts with credit growth: higher debt amplifies the marginal effect of credit growth on crisis probability.
  - (2) Monetary policy shock affects crisis probability with long lags; peak effect after 8 years.
  - (3) Costs in terms of lower inflation and economic activity are front loaded.
  - (4) Substantial parameter uncertainty in both the credit–crisis probability relation and the effect of monetary policy on credit aggregates.
- Policy implementation challenges:
  - Credibility, implementation, and communication challenges would be substantial for a central bank considering LAW.
  - Systematic use of a non-systematic policy would confront the Lucas critique; more research needed on implications of a systematic reaction to financial risk-taking.

*Source: 4.1  The Loss Function (content unit)_wp1673 - 4.1  The Loss Function*

### References

### References

### Bibliographic references
- [1] Ajello, Andrea, Thomas Laubach, David Lopez-Salido and Taisuke Nakata (2015). "Financial Stability and Optimal Interest-Rate Policy", working paper, Federal Reserve Board.
- [2] Altig, David., Lawrence. Christiano, Martin. Eichenbaum, and Jesper. Linde. (2011). ìFirm-SpeciÖc Capital, Nominal Rigidities and the Business Cycle.î Review of Economic Dynamics 14 (2): 225ñ47.
- [3] Bank for International Settlements (2014), ì84th BIS Annual Report, 2013/14,î Bank for International Settlements.
- [4] Christiano, Lawrence J., Martin Eichenbaum, and Charles L. Evans. (1999). ìMonetary Policy Shocks: What Have We Learned And To What End?î In Handbook of Macroeconomics. Vol. 1, ed. John B. Taylor and Michael Woodford. Elsevier Science B.V.
- [5] Coibion, Olivier, (2012), "Are the E§ects of Monetary Policy Shocks Big or Small?," American Economic Journal: Macroeconomics, American Economic Association, vol. 4(2), pages 1-32, April.
- [6] Diaz Kalan, Federico, Stefan Laseen, Andrea Pescatori, David Vestin, and Aleksandra Zdzienicka, (2015), ìLeaning Against the Wind: A Dynamic Cost and BeneÖt Analysis.î IMF Working Paper (Washington: International Monetary Fund) (forthcoming).
- [7] Dynan, Karen, (2012). ìIs a Household Debt Overhang Holding Back Consumption?î Brookings Papers on Economic Activity, Spring 2012, 299-344.
- [8] Iacoviello, Matteo. (2002), ìHouse prices and business cycles in Europe: a VAR analysisî, Boston College Working Papers in Economics 540, Boston College Department of Economics (also ECB Working Paper No 18).
- [9] International Monetary Fund (2015), ìMonetary Policy and Financial Stability,î Policy Papers, August28, International Monetary Fund, www.imf.org.
- [10] Laseen, Stefan, Andrea Pescatori and Jarkko Turunen, (2015). "Systemic Risk: A New Trade-o§for Monetary Policy?," IMF Working Papers 15/142, International Monetary Fund.
- [11] Litterman, Robert B. (1986). ìForecasting with Bayesian Vector Autoregressionsó Five Years of Experience.î Journal of Business and Economic Statistics 4 (1): 25ñ38.
- [12] Mian Atif, Kamalesh Rao and Amir SuÖ (2013), "Household balance sheets, consumption, and the economic slump", Quarterly Journal of Economics 2013;128:1687-726.
- [13] Sveriges Riksbank. (2013). ìFinancial Imbalances in the Monetary Policy Assessment.î Box in Monetary Policy Report July 2013. Available at http://www.riksbank.se.
- [14] Schularick, Moritz, and Alan M. Taylor, (2012), "Credit Booms Gone Bust: Monetary Policy, Leverage Cycles, and Financial Crises, 1870-2008", American Economic Review, 102(2): 1029-61.
- [15] Stein, Jeremy C., (2013), ìOverheating in Credit Markets: Origins, Measurement, and Policy Responses,î speech given at the "Restoring Household Financial Stability after the Great Recession: Why Household Balance Sheets Matter" research symposium sponsored by the Federal Reserve Bank of St. Louis, St. Louis, Missouri.
- [16] Stein, Jeremy C., (2014), ìIncorporating Financial Stability Considerations into a Monetary Policy Framework,î speech given at the International Research Forum on Monetary Policy, Washington, D.C., March 21, 2014.
- [17] Svensson, Lars E.O. (2010), ìInáation Targeting,î in Friedman, Benjamin M., and Michael Woodford, eds., Handbook of Monetary Economics, Volume 3B, chapter 22, Elsevier 2010.
- [18] Svensson, Lars E. O. (2014), "Ináation Targeting and "Leaning against the Wind", International Journal of Central Banking, 10(2), 103-114.
- [19] Svensson, Lars E.O (2015), ìInáation targeting and leaning against the wind,î in South African Reserve Bank (2015), Fourteen Years of Ináation Targeting in South Africa and the Challenge of a Changing Mandate: South African Reserve Bank Conference Proceedings 2014. Pretoria: South African Reserve Bank, 19-36. Available at www.larseosvensson.se.
- [20] Svensson, Lars E.O (2016), ìCost-BeneÖt Analysis of Leaning Against the Wind : Are Costs Larger Also with Less E§ective Macroprudential Policy?,î IMF Working Papers 16/3, International Monetary Fund.
- [21] Zmitrowicz, K, and M. Khan, (2014), ìBeyond the Unemployment Rate: Assessing Canadian and US Labour Markets Since the Great Recession.î Bank of Canada Review, Spring 2014.

### Appendix — Data series used in the BVAR analysis
- U.S. GDP ñ Bureau of Economic Analysis real chained gross domestic product (2005 dollars, SAAR).
- U.S. unemployment rate ñ Bureau of Labor Statistics.
- U.S. CPI ñ Bureau of Labor Statistics all-item consumer price index.
- Oil price ñ Spot West Texas Intermediate.
- Canada GDP ñ Statistics Canada real chained gross domestic product (2007 dollars, SAAR).
- Canada unemployment rate ñ Statistics Canada.
- Canada CPI ñ The non-seasonally adjusted Statistics Canada all-item consumer price index is used as the seasonally adjusted series only dates back to 1992.
- Canada Core: Bank of Canada core CPI excluding indirect taxes.
- Canada Household Credit: Bank of Canada household credit series.
- Canada real e§ective exchange rate ñ Bank of Canada, Canadian dollar e§ective exchange rate index (CERI, 1992 = 100).
- Canada HPI: Canadian Real Estate Association national house price index.

- Note on credit series mapping: Schularick and Taylor's variable "loansgdp" (bank loans to household and private non Önancial corporations as ratio of GDP) is about 70 percent in 2008 while the StatCan equivalent series is about 130 percent in the same year. Instead of using total private credit the authors use Bank of Canada household credit series which is about 76 percent in 2008.

### Tables — key reported numeric results
- Table 1: Financial Crisis Prediction – Logit Estimates (table content includes coefficient lines and significance markers; see source for full matrix).
- Table 2: Marginal Effects on Crisis Probability by debt-to-GDP
  - Marginal effects on crisis probability
    - Initial debt-to-GDP
      - Low debt  Low Debt
        - +1% Credit: 0.02  0.04
          - [0.01 , 0.07]  [0.01 , 0.12]
        - +10 p.p. Debt: 0.06  0.20
          - [0.01 , 0.21]  [0.04 , 0.88]
  - Marginal effects are median estimates. Square brackets represent the 5th and 90th percent confidence bands. Low (High) debt is the Canadian household debt-to-GDP in 1985 (2014); credit growth is at its sample mean.
- Selected panel regression / logit table summary values:
  - L2. ∆log (loans/P): 7.138***  (2.631); 7.543***  (2.795); 8.097***  (2.867)
  - L. ∆log (loans/P): -0.398(2.110); -0.204(2.261); -0.649(2.255)
  - L2. Exp (Loan/GDP): 0.615***  (0.233); 0.454*(0.262)
  - L. Short Term Real Rate: -0.954(8.360)
  - Interaction: L.Real rate & L2.Loan/GDP: 3.513(4.039)
  - Constant: -3.968, -4.953, -4.812
  - Observations: 1,272; 1,272; 1,031
  - Groups: 14; 14; 14
  - R2 † : 0.06590; 0.08240; 0.0818
  - Pseudolikelihood: -205.8; -202.2; -189.1
  - Test for country effects = 0 †† : 7.674; 12.551; 0.72
  - Robust standard errors in parentheses. *** p<0.01, ** p<0.05, * p<0.1.
  - † Ps eudo R2. †† Chi-squared statistic.

### Figures — captions, notes, and key numeric labels
- Figure 1
  - Source: StatCan, Bank of Canada, and IMF
- Figure 2
  - Note: house price index deflated by CPI.
  - Source: CREA, Bank of Canada, and IMF
- Figure 3: Crisis Probability in Canada with and w/o debt-to-GDP
  - Note: The black line is the point-estimate probability of observing a financial crisis in the next quarter. Grey shaded areas represent 66th and 90th confidence bands. The red-circled line is the point-estimate probability of observing a financial crisis, in the baseline specification that includes debt.
  - Y-axis label example: quarterly prob. %
  - X-axis range shown: 1993 1995 1998 2001 2004 2006 2009 2012 2015; Y-axis ticks: 0, 0.5, 1, 1.5, 2, 2.5, 3
  - Series: w/o debt, with debt
- Figure 4: Responses to a typical Monetary Policy Shock.
  - Note: BVAR impulse response function to a 1 standard deviation monetary policy shock under the baseline specification.
  - Panels with axis labels and scales include:
    - 3m yield x100 a.r. (horizontal scale 0 10 20 30 40)
    - real house price growth x100 a.r.
    - real credit growth x100 a.r.
    - debt-to-GDP x100
    - unemploymeny x100 a.r.
    - real GDP growth x100 a.r.
    - CPI Inflation x100 a.r.
    - C PIX Inflation x100 a.r.
  - Example plot value ranges shown: -1 to 1; -2 to 2; -5 to 5; -0.5 to 0.5; -0.2 to 0.2
- Figure 5: Historical Decomposition
  - Note: Black solid lines are actual data, red solid line is the contribution of BVAR internal dynamics, and dashed blue lines are the contributions of the set of shocks described on the Y-axis label.
  - X-axis: 1990 1995 2000 2005 2010 2015
  - Panels include series for: real house price growth, real credit growth, household debt to GDP, shocks: housing & credit, shocks: monetary policy
  - Example Y-axis ranges shown: -30 to 20; -10 to 15; -120 to 0; -30 to 20
- Figure 6: Evaluating the Policy Tradeoffs: Baseline case.
  - Note: Histograms represent the distribution of loss differentials based on 10000 draws from the BVAR and panel regression distributions. The two histograms compare the specification with and without debt.
- Figure 7: Evaluating the Policy Tradeoffs: Costs and Benefits
  - Note: Grey shaded areas represent confidence intervals based on 10000 draws from the BVAR and panel regression distributions.
  - Illustrative axes and labels:
    - Welfare Loss LQ vs. LAW (range -5 -4 -3 -2 -1 0 1)
    - Prob % (range 0 50 100 150)
    - Change in Crisis Prob due to LAW (quarters axis 5 10 15 20 25 30 35 40)
    - traditional loss in each quarter; Change in period traditional loss due to LAW (y-axis ranges -0.15 to 0.05; -0.2 to 0.2)
  - Series labels: without debt; with debt (baseline)
- Figure 8: Evaluating the Policy Tradeoffs: the Best Policy Reaction
  - Note: Histograms represent the distribution of loss differentials based on 10000 draws from the BVAR and panel regression distributions. The two histograms compare the specification with the best LAW (6 bps) and the baseline LAW (25 bps).
- Figure 9: The Best Policy Reaction: Costs and Benefits
  - Note: Grey shaded areas represent confidence intervals based on 10000 draws from the BVAR and panel regression distributions. The red-dashed line is the best LAW (6 bps) the black is baseline LAW (25 bps).
  - Axes and labels similar to Figure 7; policy lines: 25 bps and 6 bps.
- Figure 10: Evaluating the Policy Tradeoffs: A Critical Scenario
  - Note: Histograms represent the distribution of loss differentials based on 10000 draws from the BVAR and panel regression distributions. The two histograms compare the baseline specification (grey) with the severe scenario.
- Figure 11: A Critical Scenario: Costs and Benefits
  - Note: Grey shaded areas represent confidence intervals based on 10000 draws from the BVAR and panel regression distributions. The dashed line represents the severe scenario.
  - Axes and labels similar to previous figures; series: baseline case and critical scenario.

*Source: _wp1673 - References*

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_Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2016/_wp1673.pdf_