## wpiea2023179-print-pdf

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---

### Objective and scope
- Objective: help central banks model the demand for banks’ reserves (reserves = balance on banks’ demand deposits at the central bank), which the central bank targets to implement monetary policy.
- Motivation: reserves are the most liquid asset (“liquidity”); correct calibration of short-term interest rates is crucial but limited work exists to aid modeling reserve demand and thereby inform setting short-term interest rates.
- Prior literature: logistic curve identified as a stylised representation (Valimaki, 2001, 2008; Bindseil, 2014). Notable variants and approaches cited: Veyrune et al. (2018), Lopez-Salido and Vissing-Jorgensen (2022), El Gemayel et al. (2022), Afonso et al. (2022).

### Key contributions
- Proposes several alternative parametric sigmoid (and benchmark) models for the reserve demand that are able to account for external factors; provides non-parametric alternatives to demonstrate advantages and weaknesses.
- Prescribes a methodology to select between alternative reserve demand representations and regressors, automatically specifying the appropriate model.
- Provides asymmetric quantile estimates for the curves, communicating modeling confidence and indicating the risk of deviating from the estimated interest rate for a given reserve level; first work modeling uncertainty of reserve demand.
- Facilitates scenario analysis for changes in external factors, providing conditional predictive distributions that show shifts in the estimated reserve demand curve and its uncertainty.
- Embeds previous approximations (logistic, double exponential, arctangent) within a complete modeling framework.

### Importance for policy implementation
- Short-term interest rates are the principal tool for pursuing inflation objectives; reserves are central-bank-issued and subject to autonomous stock movements requiring forecasting and operational compensation.
- Banks’ demand for reserves depends chiefly on:
  - (i) the risk (distribution of payment shocks);
  - (ii) predictability of the payment shocks;
  - (iii) interbank market functioning;
  - (iv) the cost of liquidity management errors;
  - (iv) regulatory demand such as reserve requirements.
- Standing facilities imply short-term rates decline as reserves increase (converging to deposit facility rate); when scarce, banks borrow at the lending facility.
- Stable points on the reserve demand curve are liquidity levels where the reserve buffer absorbs shocks without impacting short-term rates; identifying stable points is central to effective monetary policy.
- Three policy implementation frameworks and their stable points:
  - Mid-corridor system: stable point equidistant from deposit and lending facility rates.
  - Floor system: sufficient recourse to deposit facility makes rates stable and close to deposit facility rate.
  - Ceiling system: standing lending facility serves as the stable point.
- Excess reserves from unconventional policies do not affect the methodology; reserve demand usually reflects factors under banks’ control.

### Modeling approach overview
- General form:
  - r_i = f(g(C_i)), where r_i is the short term interest rate for observation i, and C_i = [R_i X_i] contains excess reserve R_i and explanatory variables X_i.
  - g(C) = c + C w_g, with w_g a column vector of p+1 coefficients, and c a constant.
  - Additive error ε_i with no imposed distributional or homoscedasticity assumptions.
  - Vertical shifts modeled via η and indicator I_i: r_i = f(g(C_i)) + η I_i.
- Proposed functional forms for f(·):
  - Logistic: r_i = α + κ/(1 − β e^{g(C_i)}) + ε_i
  - Reduced Logistic: r_i = α + 1/(1 − β e^{g(C_i)}) + ε_i
  - Double exponential: r_i = α + β e^{ρ e^{g(C_i)}} + ε_i
  - Exponential: r_i = α + β e^{g(C_i)} + ε_i
  - Arctangent: r_i = α + β arctan(g(C_i)) + ε_i
  - Linear benchmark: r_i = g(C_i) + ε_i
- Parameter roles referenced to prior studies; linear serves as benchmark.

### Selection of explanatory variables and functional form
- p ≥ 0 explanatory variables may be included in g(·); selection must choose curve shape and variables simultaneously.
- Model selection procedure:
  - Use k-fold cross-validation (CV) with CV Mean Squared Error (MSE) as criterion to assess predictive performance.
  - Search strategy: common CV folds across curves; for each curve perform backwards variable selection up to p steps; compare best CV errors across curves.
  - Backwards general-to-specific approach for computational efficiency and parallelizability; greedy backward selection does not re-include rejected variables.
- Trials using an initial linear g(·) with AIC variable selection and plug-in predictions to f(·) introduced biases and scaling issues.

### Estimation of prediction intervals
- No parametric assumption on ε_i; intervals estimated from data allowing asymmetry.
- ε_i expected heteroscedastic with unknown scedastic function of C_i.
- Quantile estimation via the tick (pinball) loss S:
  - S = α |y_i − ȳ_i|, if ȳ_i ≤ y_t; (1 − α) |y_i − ȳ_i|, if ȳ_i > y_t.
  - Tick loss is proper for predicting quantiles at α ∈ (0,1). Example: to obtain a 95% interval, optimize S for α = 2.75% and α = 97.25%.

### Parameter estimation and handling nonconvexity
- Nonlinear curves produce difficult error surfaces with local minima; multiple parameter sets can yield visually similar curves.
- Estimation strategy:
  - Initialize g(·) coefficients by pre-optimizing with a standard numerical optimizer (e.g., Nelder-Mead).
  - Use Particle Swarm Optimization (PSO) metaheuristic for final estimates (PSO is stochastic and solution-sensitive to random initialization).
- Practical challenges:
  - Multiple runs of logistic estimation produced clustered parameter vectors with nearly identical fitted curves; averaging parameters produced poor fit.
- Remedies:
  - (i) multiple random initializations and pick initialization with lowest objective function value; or
  - (ii) combine initializations using an aggregation heuristic (modified “islands” pooling by Kourentzes et al. (2019)).
- Implementation: run j = 1,...,u random initializations, collect objective function values v_j; apply pooling/selection strategies.

### Combination of solutions and inclusion criteria
- Compute correlations ρ = −ρ_j of all parameter vectors with the solution with minimum v (negative sign ranks more correlated solutions lower).
- Sort ρ from lowest to highest.
- Construct ρ′ = {0, ∆ρ}, where ∆ρ is the first difference of the sorted ρ.
- Include initializations where ρ′ ≤ τ, with τ calculated iteratively as Q3(ρ′_j) + IQR(ρ′_j); eliminates poorly performing estimates.
- Additional inclusion criterion: ρ_j ≥ 0.8 to avoid combining very different parameters.
- Final combination: linearly combine remaining parameter sets using weights z_l / P_z, where z_l is the l-th value of v included in the pool.
- Purpose: retain high performing solutions (“islands”), reducing number of combination weights and variability.

### Parameter restrictions, optimization, and variable selection
- Analysts can impose sign restrictions (non-negativity/non-positivity) on coefficients in g(·) when domain knowledge implies a sign.
- Restrictions can speed optimization and aid variable selection by prohibiting spuriously signed variables.
- Restrictions improve acceptability by demonstrating capture of accepted domain knowledge and reduce PSO stochasticity.
- In prediction-interval estimation: starting from obtained curve parameters, retain sign restrictions and add restriction ensuring intervals do not cross the fitted curve.

### Use of the method and scenario analysis
- Overall methodology:
  - From multiple functional forms and explanatory variables use cross-validation to obtain best curve f*(·).
  - Optimize f*(·) across complete sample using MSE, and optimize twice with tick loss for upper and lower prediction intervals.
  - Can estimate parameters for any number of quantiles to sample predictive distribution in detail.
- Applications:
  - Obtain predictions for r_i or evaluate effects of scenarios via conditions on X_i (e.g., changing market segmentation).
  - Scenario analysis supports planning under uncertainty.
- Caveats:
  - Scenario narratives and the statistical model are not simultaneously valid; scenario point predictions often omit uncertainty.
  - Example: bivariate regression y_t = c_0 + c_1 x_t + ε_t, ε ∼ N(0, σ^2): when x_{T+1} is scenario-driven, assuming variance σ^2 for y_{T+1} is restrictive.
  - Advantages: proposed approach can construct mean point predictions for scenarios and sample predictive distributions at desired quantiles, including changes in shape and location of quantiles due to explanatory variables.

---

### Application to Case Studies — Experimental design and methods
- Two experiments using ECB and US FED data with identical setup.
- Cross-validation:
  - Outer 30-fold CV for test observations.
  - Inner 10-fold CV for variable and curve selection.
- Test metrics tracked:
  - RMSE = sqrt(1/o Σo (y_o − ˆy_o)^2)
  - MAE = 1/o Σo |y_o − ˆy_o|
  - MIS_α = 1/o Σo [ (U_α − L_α) + (2/α)(L_α − y_o)1(y_o < L_α) + (2/α)(y_o − U_α)1(y_o > U_α) ]
- Methods compared:
  - Parametric curves (Table 1) with linear benchmark.
  - Random Forest (quantregForest), GAMs/spline regression (qgam).
- Implementation:
  - PSO with 10 random initializations (psopackage for R).
  - Variable and curve selection via 10-fold CV in training set.
- Experiments per dataset:
  - (1) model using only excess reserve as predictor;
  - (2) model using additional explanatory variables.

### ECB case — using only excess reserves (selected metrics from Table 2; mean over 30-fold CV test errors)
- Logistic: RMSE 0.0631; MAE 0.0495; MIS 0.1750; Valid.⋆ 3.77; Test† 4.43; St. Dev. 59.03; Combination gains: RMSE -2.82; MAE -0.86; MIS 2.65
- Red. Logistic: RMSE 0.0637; MAE 0.0497; MIS 0.1740; Valid.⋆ 2.10; Test† 4.33; St. Dev. -11.94; Combination gains: RMSE 0.04; MAE 1.10; MIS 0.21
- Double Exp.: RMSE 0.0624; MAE 0.0489; MIS 0.1750; Valid.⋆ 4.73; Test† 3.93; St. Dev. 14.68; Combination gains: RMSE -6.93; MAE -4.73; MIS -26.80
- Exponential: RMSE 0.0639; MAE 0.0498; MIS 0.1770; Valid.⋆ 3.20; Test† 4.47; St. Dev. 2.24; Combination gains: RMSE -1.31; MAE -1.13; MIS -1.78
- Arctangent: RMSE 0.0616; MAE 0.0484; MIS 0.1730; Valid.⋆ 4.50; Test† 3.70; St. Dev. -36.40; Combination gains: RMSE -7.89; MAE -5.01; MIS -7.04
- Linear: RMSE 0.0811; MAE 0.0694; MIS 0.2430; Valid.⋆ 8.00; Test† 6.10; St. Dev. 0.00; Combination gains: RMSE 0.00; MAE 0.00; MIS 0.00
- Random Forest: RMSE 0.0676; MAE 0.0510; MIS 0.1320; Valid.⋆ 6.70; Test† 5.10
- Spline Regr.: RMSE 0.0635; MAE 0.0487; MIS 0.3140; Valid.⋆ 3.00; Test† 3.93
- Key findings:
  - Accuracy (RMSE and MAE): Arctangent performs best; linear substantially less accurate.
  - MIS (interval quality): Random Forest ranks first; spline regression produces wide intervals reflecting uncertainty.
  - Combination of curve parameters reduces parameter standard deviation for most curves (except reduced logistic and arctangent), often with small accuracy loss.

### ECB case — using additional predictors (selected metrics from Table 3)
- Logistic: RMSE 0.0471; MAE 0.0372; MIS 0.1840; Valid.⋆ 2.97; Test† 4.00
- Red. Logistic: RMSE 0.0527; MAE 0.0417; MIS 0.3430; Valid.⋆ 4.40; Test† 4.50
- Double Exp.: RMSE 0.0483; MAE 0.0389; MIS 0.1800; Valid.⋆ 3.40; Test† 4.33
- Exponential: RMSE 0.0701; MAE 0.0591; MIS 47.5000; Valid.⋆ 6.73; Test† 6.00
- Arctangent: RMSE 0.0653; MAE 0.0529; MIS 0.1950; Valid.⋆ 6.60; Test† 6.13
- Linear: RMSE 0.0557; MAE 0.0459; MIS 0.2420; Valid.⋆ 6.57; Test† 4.87
- Random Forest: RMSE 0.0385; MAE 0.0295; MIS 0.1520; Valid.⋆ 2.10; Test† 2.83
- Spline Regr.: RMSE 0.0390; MAE 0.0320; MIS 0.1720; Valid.⋆ 3.23; Test† 3.33
- Key findings:
  - Random Forest most accurate across RMSE, MAE, and MIS, followed by spline regression.
  - Additional regressors improve most parametric curves (exceptions: exponential, arctangent).
  - Non-parametric curves can over-explain variability and may imply counter-theoretical behavior at extremes.
  - Frequently selected variables include monetary aggregates (M1, M2, M3), rgdp, bond_yield, CDS, VIX, and turnover measures.

### US Fed case — data and normalization
- Short-term rate: Effective Federal Funds Rate (EFFR); weekly data 2003–2022 with 1042 observations.
- Post-2008 features: IOER introduced; overnight reverse repurchase (repo) introduced at slightly lower rate than IOER.
- Normalization: primary credit rate used as ceiling and 0 as floor; after 2008 replace 0 by IOER.
- Additional regressors as in ECB case plus binary indicator for period before/after overnight reverse repo program.

### US Fed case — using only excess reserves (selected metrics from Table 4; mean over 30-fold CV test errors)
- Logistic: RMSE 0.11500; MAE 0.07850; MIS 0.3450; Valid.⋆ 2.97; Test† 3.60; St. Dev. 43.29; Combination gains: RMSE -0.91; MAE -4.25; MIS -2.41
- Red. Logistic: RMSE 0.11500; MAE 0.07900; MIS 0.3460; Valid.⋆ 3.30; Test† 4.00; St. Dev. -1.31; Combination gains: RMSE 0.02; MAE -0.50; MIS 0.01
- Double Exp.: RMSE 0.11500; MAE 0.07870; MIS 0.3450; Valid.⋆ 3.93; Test† 3.40; St. Dev. 17.28; Combination gains: RMSE -7.59; MAE -11.27; MIS -115.80
- Exponential: RMSE 0.11600; MAE 0.08070; MIS 0.3490; Valid.⋆ 5.57; Test† 5.00; St. Dev. 23.98; Combination gains: RMSE -1.11; MAE -2.75; MIS -1.09
- Arctangent: RMSE 0.11600; MAE 0.07860; MIS 0.3570; Valid.⋆ 4.60; Test† 3.97; St. Dev. -38.39; Combination gains: RMSE -3.03; MAE -5.42; MIS -1.39
- Linear: RMSE 0.20400; MAE 0.15500; MIS 0.7180; Valid.⋆ 8.00; Test† 8.00; St. Dev. 0.00
- Random Forest: RMSE 0.13600; MAE 0.11300; MIS 0.8210; Valid.⋆ 6.63; Test† 6.97
- Spline Regr.: RMSE 0.1080; MAE 0.07530; MIS 0.3940; Valid.⋆ 1.00; Test† 1.07
- Key findings:
  - Spline regression appears best in RMSE, closely followed by parametric curves.
  - Non-parametric methods sometimes violate expected monotonic behavior; random forest shows overfitting evidence.
  - Parametric curves capture period-indicator influence (the “leaky” floor).
  - Combination of parameters reduces volatility, often at cost of accuracy.

### US Fed case — using additional predictors (selected metrics from Table 5)
- Logistic: RMSE 0.10400; MAE 0.07080; MIS 0.5840; Valid.⋆ 5.30; Test† 5.20
- Red. Logistic: RMSE 0.10000; MAE 0.07020; MIS 0.7160; Valid.⋆ 4.60; Test† 4.57
- Double Exp.: RMSE 0.10700; MAE 0.07400; MIS 0.5910; Valid.⋆ 6.53; Test† 5.73
- Exponential: RMSE 0.10300; MAE 0.07200; MIS 0.7620; Valid.⋆ 5.60; Test† 4.97
- Arctangent: RMSE 0.09840; MAE 0.06680; MIS 0.5480; Valid.⋆ 3.00; Test† 4.43
- Linear: RMSE 0.13600; MAE 0.09211; MIS 1.0100; Valid.⋆ 7.97; Test† 7.87
- Random Forest: RMSE 0.0468; MAE 0.0252; MIS 0.1350; Valid.⋆ 1.00; Test† 1.03
- Spline Regr.: RMSE 0.06710; MAE 0.04020; MIS 0.2540; Valid.⋆ 2.00; Test† 2.20
- Key findings:
  - Additional regressors improve accuracy for all methods; Random Forest ranks first.
  - Among parametric curves, arctangent ranks first, followed by reduced logistic and logistic.
  - Improvements in RMSE/MAE do not always translate to narrower MIS for parametric curves; upper intervals often remain wide.
  - Binary indicator for overnight reverse repo program effectively models changes in excess reserve curve.

---

### Discussion — scenario analysis, stability, endogeneity, and conclusion

Scenario analysis (5.1)
- Goal: provide tools for central banks to calibrate decisions; scenario analysis makes assumptions explicit.
- Illustration (ECB): baseline (stronger euro) vs alternative (parity USD/euro) scenarios, each with upper/lower quantiles showing implied uncertainty (Figure 13).
- Effects:
  - Scenario shifts can change stable points and alter shape and width of uncertainty — not just location shifts.
- Limitations:
  - Multiple parameter sets can produce similar curves; remedy is to impose sign restrictions on coefficients.
  - Scenario generation requires future values for inputs and may not ensure quantiles do not cross out-of-sample.
- Advantages:
  - Approach can produce scenario-based predictive distributions without relying on ad-hoc judgmental probabilities.

Stability of reserve demand curve (5.2)
- Empirical test: split sample into three periods and fit bivariate curves versus multivariate fit across all periods.
- Finding:
  - When additional regressors are available, a single curve estimated on the full sample is sufficient; regressors explain changes in reserve demand, implying no fundamental curve shift.

Endogeneity (5.3)
- Concern: excess reserves and short-term rates can be endogenous due to central bank reactions.
- Treatment:
  - Including lags of the rate can address endogeneity; methodology emphasizes predictive performance (CV) rather than inference.
- Empirical ECB assessment (percentage gains/losses in Table 6 when including lags of the rate):
  - Logistic: -10.75
  - Red. Logistic: 2.19
  - Double Exp.: -8.94
  - Exponential: -3.72
  - Arctangent: 9.14
  - Linear: 19.36
  - Random Forest: -0.98
  - Spline Regr.: -19.94
  - Average: -1.71
  - Average w/o Linear: -4.71
- Interpretation:
  - Majority of nonlinear curves lost performance with lag inclusion; linear model improved by 19.36 percent.
  - Little overall benefit from including lagged rates for nonlinear curves.

Conclusion and implications
- Contribution: complete modeling framework to identify reserve demand curves, select regressors, and produce quantile estimates reflecting uncertainty; contrasts parametric sigmoid and non-parametric approaches in Euro area and US studies.
- Scenario utilities: framework enables scenario analysis with predictive distributions that incorporate curve-modeling and regressor forecast uncertainty.
- Open issues:
  - Scenario-generation methodologies require further research.
  - Nonlinear quantile models cannot guarantee absence of quantile crossing out-of-sample.
  - Nonlinear parametric curves pose estimation challenges; remedies increase computational cost; Bayesian or shrinkage estimators merit exploration.
- Policy relevance: reserve demand modeling supports high-impact, frequent decisions in monetary policy implementation and warrants further research.

*Source: wpiea2023179-print-pdf - 1. Introduction; 4. Application to Case Studies; 5. Discussion*

### 1. Introduction

### 1. Introduction

### Objective and scope
- Objective: help central banks model the demand for banks’ reserves (reserves = balance on banks’ demand deposits at the central bank), which the central bank targets to implement monetary policy.
- Motivation: reserves are the most liquid asset (“liquidity”); correct calibration of short-term interest rates is crucial but limited work exists to aid modeling reserve demand and thereby inform setting short-term interest rates.
- Prior literature: logistic curve identified as a stylised representation (Valimaki, 2001, 2008; Bindseil, 2014) but calibration guidance limited. Veyrune et al. (2018) estimate a bivariate logistic curve; Lopez-Salido and Vissing-Jorgensen (2022) modify the logistic by fixing an asymptote; El Gemayel et al. (2022) use a double exponential; Afonso et al. (2022) estimate interest-rate elasticity via a locally bi-variate linear approximation and later impose an arctangent functional form without assessing goodness of fit.

### Key contributions (as stated)
- Proposes several alternative parametric sigmoid (and benchmark) models for the reserve demand that are able to account for external factors. Non-parametric alternatives are provided to demonstrate the advantages and weaknesses of the proposed parametric curves.
- Prescribes a methodology to select between the alternative reserve demand curve representations and regressors, automatically specifying the appropriate model.
- Provides asymmetric quantile estimates for the curves, both communicating the modeling confidence to the analyst and indicating the risk of deviating from the estimated interest rate for a given reserve level. This is the first work that models the uncertainty of the reserve demand.
- Facilitates scenario analysis for changes in the external factors, while providing conditional predictive distributions to the analyst, demonstrating both shifts in the estimated reserve demand curve and its uncertainty. This helps to forecast the change in the demand and calibrate monetary operations accordingly.
- Embeds previous approximations (logistic, double exponential, arctangent) within a complete modeling framework.

### Importance of reserve-demand modeling for policy implementation
- Short-term interest rates represent the marginal cost of funding financial intermediaries and are the principal tool central banks use to pursue inflation objectives.
- Reserves are issued by central banks; some stock movements are driven by autonomous factors (e.g., demand for bank notes, government account) not under direct central bank control and may need to be forecasted and compensated by operations.
- Banks’ demand for reserves depends chiefly on:
  - (i) the risk (distribution of payment shocks);
  - (ii) predictability of the payment shocks (the central bank liquidity forecast if publicly available);
  - (iii) interbank market functioning;
  - (iv) the cost of liquidity management errors (the cost of being short or long of reserves);
  - and (iv) regulatory demand for reserve such as the reserve requirement.
- Standing lending and deposit facilities imply short-term rates decline as reserves increase, converging to the deposit facility rate; when liquidity is scarce, banks borrow at the lending facility.
- The concept of “stable points” on the reserve demand curve: levels of liquidity where the reserve buffer can absorb shocks without impacting short-term rates; identifying these stable points is central to effective monetary policy.
- Three monetary policy implementation frameworks and their stable points:
  - Mid-corridor system: stable point equidistant from deposit facility rate and lending facility rate; small excess reserves usually suffice.
  - Floor system: sufficient recourse to the deposit facility makes rates stable and close to the deposit facility rate (the effective policy rate).
  - Ceiling system: banks finance shortfalls via the standing lending facility, which then serves as the stable point.
- Excess reserves (e.g., from unconventional policies providing ample reserves) do not affect the proposed methodology; reserve demand usually reflects factors under banks’ control.

### Modeling approach overview
- General model form:
  - r_i = f(g(C_i)), where r_i is the short term interest rate for observation i, and C_i = [R_i X_i] contains excess reserve R_i and explanatory variables X_i.
  - g(C) = c + C w_g, where w_g is a column vector of p+1 coefficients, and c is a constant.
  - Additive error term ε_i with no imposed distributional or homoscedasticity assumptions.
  - Vertical shifts in the reserve demand curve are modeled via parameter η and binary indicator I_i: r_i = f(g(C_i)) + η I_i.
- Proposed functional forms for f(·) (Table 1) include:
  - Logistic: r_i = α + κ/(1 − β e^{g(C_i)}) + ε_i
  - Reduced Logistic: r_i = α + 1/(1 − β e^{g(C_i)}) + ε_i
  - Double exponential: r_i = α + β e^{ρ e^{g(C_i)}} + ε_i
  - Exponential: r_i = α + β e^{g(C_i)} + ε_i
  - Arctangent: r_i = α + β arctan(g(C_i)) + ε_i
  - Linear benchmark: r_i = g(C_i) + ε_i
- Parameter roles and references:
  - Logistic and Reduced Logistic: parameters α, β, κ control curve shape (Veyrune et al., 2018; Lopez-Salido and Vissing-Jorgensen, 2022).
  - Double exponential: parameters α, β, ρ control shape (El Gemayel et al., 2022).
  - Arctangent: parameter α changes location, β controls spread (Afonso et al., 2022).
  - Linear form serves as a benchmark.

### Selection of explanatory variables and functional form
- p ≥ 0 explanatory variables may be included in g(·) to explain variance in r_i, but the functional form f(·) also explains variance; thus selection must choose curve shape and variables simultaneously.
- Model selection procedure:
  - Use k-fold cross-validation (CV) with CV Mean Squared Error (MSE) as the criterion to assess predictive performance (MSE chosen to accurately model expected r_i).
  - Search strategy: generate common CV folds; for each curve, perform backwards variable selection (remove variables whose elimination improves CV error) up to p steps; then compare best CV errors across curves (common folds make errors comparable).
  - Backwards general-to-specific approach is used for computational efficiency and parallelizability; greedy backward selection does not re-include rejected variables.
  - Trials that used an initial linear g(·) with AIC variable selection and plug-in predictions to f(·) were found to introduce biases and scaling issues (necessitating a scaling factor φ which proved difficult to optimize).

### Estimation of prediction intervals
- No parametric assumption on ε_i; intervals estimated from data to avoid bias from incorrect distributional assumptions and to allow asymmetry.
- ε_i is expected to be heteroscedastic with unknown scedastic function of C_i (e.g., variance of r_i smaller for high R_i).
- Quantile estimation via re-estimating curves using the tick loss (pinball) S:
  - S = α |y_i − ȳ_i|, if ȳ_i ≤ y_t; (1 − α) |y_i − ȳ_i|, if ȳ_i > y_t.
  - The tick loss is a proper score for predicting quantiles at levels α ∈ (0,1). Example: to obtain a 95% interval, optimize S for α = 2.75% and α = 97.25%.

### Parameter estimation and handling nonconvexity
- Nonlinear curves have difficult error surfaces with local minima; multiple parameters interact and different parameter sets can yield visually similar curves.
- Estimation strategy:
  - Initialize coefficients in g(·) by pre-optimizing with a standard numerical optimizer (e.g., Nelder-Mead).
  - Use Particle Swarm Optimization (PSO) metaheuristic for final estimates to effectively search the error surface (PSO is stochastic and solution-sensitive to random initialization).
- Practical challenges illustrated:
  - Multiple runs of logistic estimation produced clustered parameter vectors with nearly identical fitted curves; averaging parameters produced a poor fit.
- Two strategies to resolve initialization sensitivity:
  - (i) perform multiple random initializations and pick the initialization with lowest objective function value; or
  - (ii) use a combination/aggregation heuristic (modified “islands” pooling by Kourentzes et al. (2019)) to combine multiple initializations more effectively than simple averaging.
- Implementation notes:
  - Multiple random initializations are run (j = 1,...,u), collecting objective function values v_j; subsequent pooling/selection strategies are applied.

*Source: wpiea2023179-print-pdf - 1. Introduction*

### 1.  Find  the  correlationsρ=−ρ

### wpiea2023179-print-pdf - 1.  Find  the  correlationsρ=−ρ

### Combination of solutions and inclusion criteria
- Compute the correlations ρ = −ρ_j of all vectors of parameters with the solution with the minimum v. The negative sign ensures that the more correlated solutions rank lower in ρ.
- Sort ρ from lowest to highest.
- Construct ρ′ = {0, ∆ρ}, where ∆ρ is the first difference of the sorted ρ.
- Include in the combination pool all initializations where ρ′ ≤ τ, with τ being a threshold calculated iteratively as Q3(ρ′_j) + IQR(ρ′_j), with Q3 and IQR being the third quartile and the interquartile range respectively, and ρ′_j being a vector including up to the j-th value of ρ′. This step eliminates from the combination poorly performing estimates.
- Ensure a second criterion of inclusion: ρ_j ≥ 0.8, so solutions with very different parameters are not included. This helps avoid combining counterbalancing estimated parameters that can perform poorly.

### Forming the final combination
- Linearly combine the remaining sets of parameters using weights: z_l / P_z, where z_l is the l-th value of v that was included in the combination pool.
- Objective: retain only high performing solutions in the combination (combination “islands”), reducing the number of combination weights to be estimated and reducing variability of the final set of parameters.
- Contrast: the first option (not using islands) does not mitigate variability of the solutions but can still provide good quality estimates.

### Parameter restrictions, optimization, and variable selection
- The approach enables incorporation of restrictions on coefficients in g(·) that model the impact of explanatory variables:
  - Analysts can impose non-negativity or non-positivity constraints when domain knowledge implies a positive or negative effect.
  - Introducing these restrictions can achieve faster optimization and help selection of variables.
  - If a variable is included spuriously and its sign is inappropriate, the sign restriction can prohibit such solutions, pushing the variable out of the model.
  - If no expectation about sign exists for a variable, its coefficient remains unrestricted.
- Benefits beyond statistics:
  - Parameter restrictions help acceptability and adoption of the method by demonstrating capture of accepted domain knowledge and allowing analyst expertise.
  - Restrictions limit apparent stochasticity of PSO metaheuristic solutions, providing stable and consistent predictions.
- In estimating prediction intervals:
  - Obtained curve parameters are used as starting points.
  - Restrictions retain the same signs for estimated parameters.
  - An additional restriction ensures provided intervals do not cross the fitted curve.

### Use of the method and scenario analysis
- Overall modeling methodology (summarized in Figure 4):
  - From multiple potential functional forms and explanatory variables (see Table 1), use cross-validation to obtain best curve f*(·).
  - Optimize f*(·) across the complete sample using MSE, and optimize twice with tick loss for upper and lower prediction intervals.
  - Can estimate parameters for any number of quantiles to sample predictive distribution in detail, resembling quantile regression.
- Applications:
  - Obtain predictions for r_i or evaluate effects of scenarios via conditions on variables in X_i (e.g., increasing or decreasing market segmentation).
  - Scenario analysis supports planning under uncertainties.
- Caveats on scenarios vs probabilistic predictions:
  - Example: bivariate regression y_t = c_0 + c_1 x_t + ε_t, ε ∼ N(0, σ^2). Prediction for t = T+1: ˆy_{T+1} = c_0 + c_1 x_{T+1} with variance σ^2 assumes x_{T+1} known.
  - When x_{T+1} is outcome of scenarios, obtaining conditional variance for y_{T+1} is challenging; assuming variance remains σ^2 is overly restrictive because omitted variables driving scenarios may affect variance.
  - Scenario narratives and the statistical model are not simultaneously valid; scenario point predictions often omit uncertainty and may rely on ad-hoc probabilities subject to judgmental biases.
  - Multiple experts can partially mitigate bias, but ad-hoc probabilities are not rigorously data-based.
- Illustration (Figure 5):
  - Panel (i): density of y_{T+1} contrasted with three scenarios produced by three values of x_{T+1}; scenarios fail to provide a probabilistic view.
  - Panel (ii): predictive distributions for the three scenarios based solely on variance σ of ε_t, which does not account for scenario-generating conditions.
- Advantages of the proposed approach for scenarios:
  - Can construct mean point predictions for scenarios and sample predictive distribution at desired quantiles.
  - Intervals generated via curves include estimated effects of explanatory variables that change across scenarios, so changes in shape and location of quantiles are modeled.
  - Enables extraction of conventional predictive distributions, scenario-based point estimates, or full scenario-based predictive distributions without requiring judgmental estimates from participants, making the approach scalable for operational decision-making.

*Source: Authors’ Calculation (wpiea2023179-print-pdf)*

### 4. Application to Case Studies

### 4. Application to Case Studies

### 4.1 Experimental Design
- Setup:
  - Two experiments using data from the European Central Bank (ECB) and the US Federal Reserve (FED).
  - Identical setup for both experiments.
- Cross-validation:
  - Outer 30-fold cross-validation (CV) to iterate across test observations.
  - Inner 10-fold CV used to generate validation observations for variable and curve selection.
  - Double CV ensures selection/validation and test evaluation use different data.
- Test metrics tracked:
  - RMSE = sqrt(1/o Σo (yo − ˆyo)^2)
  - MAE = 1/o Σo |yo − ˆyo|
  - MISα = 1/o Σo [ (Uα − Lα) + (2/α)(Lα − yo)1(yo < Lα) + (2/α)(yo − Uα)1(yo > Uα) ]

### 4.1.1 Methods
- Parametric curves:
  - Parametric curves described in Table 1 tested; linear option acts as benchmark.
- Non-parametric alternatives:
  - Random Forest (Breiman, 2001); implemented via randomForest (Liaw and Wiener, 2002) in R; quantiles from quantregForest (Meinshausen, 2017).
  - Generalized Additive Models (GAMs) using thin-plate regression splines; implemented via mgcv (Wood, 2006) and quantiles via qgam (Fasiolo et al., 2021b).
- Implementation details:
  - Particle Swarm Optimization (PSO): 10 random initializations using psopackage (Bendtsen, 2022) for R.
  - Variable and curve selection: 10-fold CV in training set.
- Experiments:
  - For each dataset two experiments: (1) model using only excess reserve as predictor; (2) model using additional explanatory variables (listed separately per case study).

### 4.2 European Central Bank

#### 4.2.1 Data
- Short-term interest rate: volume-weighted Euro Overnight Index Average (EONIA), normalized between 0 and 1:
  - ̃ri = (ri − Deposit Ratei) / (Lending Ratei − Deposit Ratei). (Equation (3))
- EONIA averaged over the reserve maintenance period to smooth intra-period volatility.
- EONIA discontinued in October 2019 and replaced with ESTR; ESTR may drift below deposit facility rate.
- Variables considered:
  - Macroeconomic and financial: monetary aggregates, domestic production, consumer prices, volatility index, foreign exchange rate, sovereign bond yields, credit default swap, market turnover.
- Sample:
  - Data collected from 1999 to 2019, resulting in 239 maintenance periods with all data available.

#### 4.2.2 Using Only the Excess Reserves
- Table 2 summary (mean over 30-fold CV test errors). Columns include RMSE, MAE, MIS, validation CV RMSE rank (Valid.⋆), test CV RMSE rank (Test†), St. Dev. (% reduction of average standard deviation of curve parameters), and combination gains (%). Best per column highlighted in source.
- Selected metrics in Table 2:
  - Logistic: RMSE 0.0631; MAE 0.0495; MIS 0.1750; Valid.⋆ 3.77; Test† 4.43; St. Dev. 59.03; Combination gains: RMSE -2.82; MAE -0.86; MIS 2.65
  - Red. Logistic: RMSE 0.0637; MAE 0.0497; MIS 0.1740; Valid.⋆ 2.10; Test† 4.33; St. Dev. -11.94; Combination gains: RMSE 0.04; MAE 1.10; MIS 0.21
  - Double Exp.: RMSE 0.0624; MAE 0.0489; MIS 0.1750; Valid.⋆ 4.73; Test† 3.93; St. Dev. 14.68; Combination gains: RMSE -6.93; MAE -4.73; MIS -26.80
  - Exponential: RMSE 0.0639; MAE 0.0498; MIS 0.1770; Valid.⋆ 3.20; Test† 4.47; St. Dev. 2.24; Combination gains: RMSE -1.31; MAE -1.13; MIS -1.78
  - Arctangent: RMSE 0.0616; MAE 0.0484; MIS 0.1730; Valid.⋆ 4.50; Test† 3.70; St. Dev. -36.40; Combination gains: RMSE -7.89; MAE -5.01; MIS -7.04
  - Linear: RMSE 0.0811; MAE 0.0694; MIS 0.2430; Valid.⋆ 8.00; Test† 6.10; St. Dev. 0.00; Combination gains: RMSE 0.00; MAE 0.00; MIS 0.00
  - Random Forest: RMSE 0.0676; MAE 0.0510; MIS 0.1320; Valid.⋆ 6.70; Test† 5.10
  - Spline Regr.: RMSE 0.0635; MAE 0.0487; MIS 0.3140; Valid.⋆ 3.00; Test† 3.93
- Key findings:
  - Accuracy (RMSE and MAE): Arctangent performs best, followed by other parametric curves; linear substantially less accurate than alternatives.
  - MIS (prediction interval quality): Random Forest ranks first; parametric curves follow; spline regression ranks last (wide intervals).
  - Non-parametric methods:
    - Random Forest: very non-smooth approximation but narrow intervals.
    - Spline Regression: good fit but wide intervals, reflecting modeling uncertainty when data is limited.
  - Combination of curve parameters:
    - For most curves (except reduced logistic and arctangent) combination reduces standard deviation of parameters, typically with a small loss of accuracy — combination stabilizes estimates.

#### 4.2.3 Using Additional Predictors
- Table 3 summary (best parameter estimates shown). Key metrics:
  - Logistic: RMSE 0.0471; MAE 0.0372; MIS 0.1840; Valid.⋆ 2.97; Test† 4.00
  - Red. Logistic: RMSE 0.0527; MAE 0.0417; MIS 0.3430; Valid.⋆ 4.40; Test† 4.50
  - Double Exp.: RMSE 0.0483; MAE 0.0389; MIS 0.1800; Valid.⋆ 3.40; Test† 4.33
  - Exponential: RMSE 0.0701; MAE 0.0591; MIS 47.5000; Valid.⋆ 6.73; Test† 6.00
  - Arctangent: RMSE 0.0653; MAE 0.0529; MIS 0.1950; Valid.⋆ 6.60; Test† 6.13
  - Linear: RMSE 0.0557; MAE 0.0459; MIS 0.2420; Valid.⋆ 6.57; Test† 4.87
  - Random Forest: RMSE 0.0385; MAE 0.0295; MIS 0.1520; Valid.⋆ 2.10; Test† 2.83
  - Spline Regr.: RMSE 0.0390; MAE 0.0320; MIS 0.1720; Valid.⋆ 3.23; Test† 3.33
- Key findings:
  - Random Forest provides most accurate results across RMSE, MAE, ME, and MIS, followed by spline regression.
  - Inclusion of additional regressors improves performance for most parametric curves (exceptions: exponential and arctangent).
  - Error distributions become narrower and shifted lower with additional regressors.
  - Non-parametric curves can over-explain variability and may imply counter-theoretical behavior (e.g., a rise in interest rates at high excess reserves).
- Variable inclusion:
  - Barplots (Figure 10) show percentage of times each variable was included across CV samples by curve.
  - Frequently selected variables include monetary aggregates (M1, M2, M3), real GDP growth (rgdp), sovereign bond yields (bond_yield), and risk measures (CDS, VIX, exchange rate volatility).
  - Interpretation: monetary aggregates capture reserve demand; risk perception variables capture hoarding behavior; market functioning variables (turnover) also included.

### 4.3 US Federal Reserve

#### 4.3.1 Data
- Short-term rate: Effective Federal Funds Rate (EFFR), volume-weighted median of overnight federal funds transactions.
- Post-2008 changes:
  - Interest on excess reserves (IOER) introduced; 0 replaced by IOER as floor after 2008.
  - Overnight reverse repurchase agreements (repo) introduced at slightly lower rate than IOER (currently 10 basis point) for counterparts without IOER access.
  - At end-April 2022, reverse repo balances close to IOER account balances.
- Normalization:
  - Primary credit rate used as ceiling and 0 as floor; after 2008 replace 0 by IOER.
- Additional regressors as in ECB case plus binary indicator for period before/after introduction of overnight reverse repo program.
- Sample:
  - Weekly data from 2003 to 2022, resulting in 1042 observations with all variables available.

#### 4.3.2 Using Only the Excess Reserves
- Table 4 summary (mean over 30-fold CV test errors). Selected metrics:
  - Logistic: RMSE 0.11500; MAE 0.07850; MIS 0.3450; Valid.⋆ 2.97; Test† 3.60; St. Dev. 43.29; Combination gains: RMSE -0.91; MAE -4.25; MIS -2.41
  - Red. Logistic: RMSE 0.11500; MAE 0.07900; MIS 0.3460; Valid.⋆ 3.30; Test† 4.00; St. Dev. -1.31; Combination gains: RMSE 0.02; MAE -0.50; MIS 0.01
  - Double Exp.: RMSE 0.11500; MAE 0.07870; MIS 0.3450; Valid.⋆ 3.93; Test† 3.40; St. Dev. 17.28; Combination gains: RMSE -7.59; MAE -11.27; MIS -115.80
  - Exponential: RMSE 0.11600; MAE 0.08070; MIS 0.3490; Valid.⋆ 5.57; Test† 5.00; St. Dev. 23.98; Combination gains: RMSE -1.11; MAE -2.75; MIS -1.09
  - Arctangent: RMSE 0.11600; MAE 0.07860; MIS 0.3570; Valid.⋆ 4.60; Test† 3.97; St. Dev. -38.39; Combination gains: RMSE -3.03; MAE -5.42; MIS -1.39
  - Linear: RMSE 0.20400; MAE 0.15500; MIS 0.7180; Valid.⋆ 8.00; Test† 8.00; St. Dev. 0.00; Combination gains: RMSE 0.00; MAE 0.00; MIS 0.00
  - Random Forest: RMSE 0.13600; MAE 0.11300; MIS 0.8210; Valid.⋆ 6.63; Test† 6.97
  - Spline Regr.: RMSE 0.1080; MAE 0.07530; MIS 0.3940; Valid.⋆ 1.00; Test† 1.07
- Key findings:
  - Accuracy: spline regression appears best in RMSE, closely followed by parametric curves (similar performance).
  - MIS (uncertainty): spline regression performs about 32% worse on average in MIS relative to best methods.
  - Non-parametric methods do not follow expected monotonic behavior; random forest shows evidence of overfitting.
  - Parametric curves reflect the period-indicator influence: mean predictions shift modeling the “leaky” floor.
  - Combination of curve parameters reduces parameter volatility on average, often at cost of accuracy.

#### 4.3.3 Using Additional Predictors
- Table 5 summary (all regressors). Selected metrics:
  - Logistic: RMSE 0.10400; MAE 0.07080; MIS 0.5840; Valid.⋆ 5.30; Test† 5.20
  - Red. Logistic: RMSE 0.10000; MAE 0.07020; MIS 0.7160; Valid.⋆ 4.60; Test† 4.57
  - Double Exp.: RMSE 0.10700; MAE 0.07400; MIS 0.5910; Valid.⋆ 6.53; Test† 5.73
  - Exponential: RMSE 0.10300; MAE 0.07200; MIS 0.7620; Valid.⋆ 5.60; Test† 4.97
  - Arctangent: RMSE 0.09840; MAE 0.06680; MIS 0.5480; Valid.⋆ 3.00; Test† 4.43
  - Linear: RMSE 0.13600; MAE 0.09211; MIS 1.0100; Valid.⋆ 7.97; Test† 7.87
  - Random Forest: RMSE 0.0468; MAE 0.0252; MIS 0.1350; Valid.⋆ 1.00; Test† 1.03
  - Spline Regr.: RMSE 0.06710; MAE 0.04020; MIS 0.2540; Valid.⋆ 2.00; Test† 2.20
- Key findings:
  - Inclusion of additional regressors improves accuracy for all methods.
  - Non-parametric methods (random forest, spline) remain best; random forest ranks first.
  - Parametric: arctangent ranks first among parametric curves, followed by reduced logistic and logistic.
  - Improvements in RMSE/MAE are not always reflected in MIS for parametric curves; upper intervals remain wide for many parametric specifications.
- Variable selection:
  - Figure 12 (not reproduced here) shows regressors consistently selected across curves (e.g., m1 and hicp) and others selected only by some curves (e.g., m2 and rgdp).
- Additional notes:
  - Inclusion of a binary indicator for the change due to the overnight reverse repo program is effective in modeling changes in the excess reserve curve in both bivariate and multivariate cases.
  - Section 5.2 (in source) investigates further the stability of estimated curves given changes over the sample.

*Source: Authors’ Calculation (content from "4. Application to Case Studies" in the provided PDF chapter).*

### 5. Discussion

### 5. Discussion

### 5.1. Scenario Analysis
- Modeling objective: provide tools for central banks to calibrate decisions and operations; scenario analysis makes implicit policymaker assumptions explicit.
- Example (ECB): Baseline scenario based on a stronger euro; alternative scenario assumes parity between USD and euro. Each scenario accompanied by upper and lower quantiles corresponding to implied uncertainty (Figure 13).
- Effects of scenario changes:
  - Transitioning from the baseline to the alternative scenario shifts the curve, implying different stable points, and alters the shape and width of the uncertainty.
  - Scenario analysis reveals both location shifts and changes in uncertainty — a more complete view than changing only the curve location.
- Limitations and cautions:
  - Multiple parameter sets can correspond to similar-looking curves (see Figure 3), potentially invalidating analyst expectations about variable impacts.
  - Remedy: impose restrictions on the signs of the coefficients for covariates to restrict solutions to those consistent with economic understanding.
  - Scenario generation requires the analyst to provide a vector of future values for curve inputs; some inputs may remain unchanged while others require forecasts, ideally with predictive distributions or quantiles.
  - The approach cannot guarantee that scenario quantiles will not cross the mean prediction or other quantiles for values outside the fitting sample; baseline parameterisation contains restrictions to prevent this in-sample, but nonlinear curves can still exhibit crossing out-of-sample.

### 5.2. Stability of Reserve Demand Curve
- Question: Has the reserve demand curve shifted over time, or can a consistent curve estimated on the complete sample be used?
- Empirical approach:
  - Split sample into three consecutive periods and fit bivariate curves to each; contrast with a multivariate fit across all periods that includes additional regressors.
  - Visualization (Figure 14) for the logistic curve shows bivariate solutions suggest a shift over time.
- Key finding:
  - When additional regressors are available, a single curve estimated on the full sample is sufficient.
  - Regressors effectively explain changes in reserve demand, implying the curve has not fundamentally shifted.

### 5.3. Endogeneity
- Concern: Excess reserves and short-term rates can be endogenous; central banks react within days to money market volatility by changing reserve supply.
- Treatment considerations:
  - Including lags of the rate as regressors can cater for endogeneity; endogeneity is critical for inference as an omitted variable can bias estimates and regressor selection.
  - Methodology emphasis is predictive (CV procedure) rather than inferential; nonlinearity limits interpretability of individual coefficients.
  - High-frequency confounders may be unobservable in lower-frequency data and instead be captured by additional regressors.
- Empirical assessment (ECB): include lags of the rate and evaluate out-of-sample performance gains/losses relative to Table 3 results; reported percentage gains (losses are negative) in Table 6:
  - Logistic: -10.75
  - Red. Logistic: 2.19
  - Double Exp.: -8.94
  - Exponential: -3.72
  - Arctangent: 9.14
  - Linear: 19.36
  - Random Forest: -0.98
  - Spline Regr.: -19.94
  - Average: -1.71
  - Average w/o Linear: -4.71
- Interpretation:
  - For the majority of curve specifications, including lags leads to a loss of performance.
  - Linear model improves by almost 20 percent (19.36).
  - Conclusion: Little benefit overall from including lagged rates for nonlinear curves, partly due to nonlinearity and partly due to the presence of other regressors.

### Conclusion (summary of implications and future research directions)
- Contribution: Introduces a complete modeling framework to identify appropriate reserve demand curves, select useful regressors, and provide quantile estimates reflecting reserve demand uncertainty across conditions; contrasts parametric sigmoid curves with non-parametric approaches in Euro area and US case studies.
- Scenario work: Framework facilitates scenario analysis with predictive distributions that incorporate uncertainty from both curve modeling and regressor forecasts.
- Open issues and limitations:
  - Scenario generation methodologies deserve further research; model complexity increases scenario-generation complexity.
  - The approach cannot guarantee absence of quantile crossing in all scenarios due to nonlinear curves; an open question in non-linear quantile models.
  - Nonlinear parametric curves introduce estimation challenges; proposed remedies increase computational cost—parallelisability helps, but alternative estimation procedures (Bayesian or shrinkage estimators) merit exploration.
- Final policy relevance: Reserve demand modeling supports high-impact, frequent decisions in monetary policy implementation and warrants additional research attention.

*Source: Authors’ Calculation (5. Discussion, wpiea2023179-print-pdf)*

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_Source: https://www.imf.org/-/media/files/publications/wp/2023/english/wpiea2023179-print-pdf.pdf_