## 1. Estimates of the logistic function—different data frequency

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### Purpose and modeling approach
- Models the relationship between excess reserves and short-term interest rates in a symmetric interest rate corridor using a bivariate logistic function.
- Logistic function defined so that:
  - short-term rates converge to the top of the corridor when excess reserves are increasingly negative;
  - short-term rates converge to the bottom of the corridor when excess reserves increase;
  - short-term rates take the value 0.5 when there are no excess reserves.
- Estimated using Eurosystem data (long time series since 1999) to determine the position of short-term rates in the corridor and to predict their behavior as excess reserves vary.
- Estimates identify ranges of excess reserves where short-term rates are anchored to: higher edge, mid-corridor, lower edge, or become “un-anchored” (fluctuate between an edge and the mid-point or the other edge).

### Key empirical context and data features
- Eurosystem experience provides:
  - equal periods of low and stable excess reserves: 1999–2008;
  - periods of large and volatile excess reserves: 2009–2018.
- Excess reserves stood at the beginning of 2018 above 1 trillion euro.
- Estimation periods referenced: January 1999 to February 2018; October 2008 to December 2011; January 2012 to February 2018.

### Economic interpretation and implications
- “Un-anchoring” of short-term rates is undesirable because:
  - policy stance becomes less precisely defined when short-term rates fluctuate in the corridor;
  - short-term rate volatility and uncertainty increase the liquidity premium, transmitting higher funding costs through the term structure.
- The logistic function can help central banks:
  - predict excess reserve ranges causing un-anchoring;
  - accelerate transitions to states where anchoring occurs (to a corridor edge or mid-point) to avoid negative consequences.
- The shape and parameters of the logistic function and the reserve levels at which volatility occurs vary across countries and must be locally estimated.

### Market segmentation, full allotment, and coordination failure
- Market segmentation reduces efficiency of reallocation of excess reserves, altering the speed of convergence to corridor edges.
- Fixed-rate full allotment can, in a segmented market, lead to episodes of “coordination failure” among banks bidding at main open market operations.
- Coordination failure defined as individual bids that are individually optimal but lead to a sub-optimal system outcome, increasing volatility of short-term rates and excess reserves.
- Counterparty risk compounds coordination failure by hindering redistribution of reserves from banks with excess reserves to banks with shortages.

### Policy operational considerations for steering short-term rates
- Central banks influence short-term rates by:
  - altering the stock of reserves via allotment of open market operations (net reserves supply);
  - setting interest rates for refinancing and remuneration of excess reserves;
  - using reserve requirement averaging and standing facilities to absorb forecast errors.
- Standing facilities create an interest rate corridor:
  - floor = deposit facility rate where banks can deposit unlimited amounts;
  - ceiling = lending facility where borrowing is limited by eligible collateral.
- Corridor width trade-offs:
  - too narrow: reduces interbank trading incentives; central bank intermediates more, losing market signaling;
  - too wide: increases short-term interest rate volatility.
- Under “neutral liquidity allotment” (keep excess reserves low and stable), short-term rates tend to be steered to the middle of the corridor and market transactions are maximized.
- With large, volatile, and unpredictable demand for excess reserves (post-global financial crisis), the market’s ability to redistribute liquidity breaks down, complicating steering.

### Logistic model (stylized and technical)
- Model choice and properties:
  - The logistic function is used because it allows positive and negative explanatory values, asymptotic convergence, and a mid-rate outcome when the explanatory variable is zero.
  - In perfectly functioning money markets demand for excess reserves would be a step function; fragmentation and coordination failures give the excess-reserve demand a logistic shape.
- Four-parameter logistic function (modified Oliver (1969) form):
  - Model form (as used): y = s + (κ − s) / (1 + e^(−α(x − x0)))
  - Main features emphasized:
    - Allows empirical estimation of the lower asymptote s (to account for market-required pick-up above the deposit facility).
    - Reveals horizontal movements along the corridor, enabling computation of short-term rate when excess reserves are null.
- Economic interpretations:
  - Lower asymptote (s): short-term rate toward which EONIA converges at very high excess reserves (reflects counterparty risk, transaction costs, market microstructure).
  - Inflection point: level of excess reserves above which short-term rates decrease and below which they increase (neutral allotment).
  - Second-derivative maxima: benchmark reserve level beyond which marginal effect of additional excess reserves on short-term rates diminishes.

### Empirical results — Estimation January 1999 to February 2018 (full sample and frequencies)
- Sample: 249 maintenance periods (January 1999 to February 2018).
- Regimes in excess reserves during sample:
  - Neutral allotment: January 1999–October 2008.
  - Moderate excess reserve periods: October 2008–December 2011, and March 2014–March 2015.
  - High excess reserve regimes: January 2012–February 2014 and since April 2015.
- Full-sample maintenance-period estimates:
  - Lower asymptote ߛ = 0.0860*** (t-statistic (6.234))
  - Coefficient ߙ = 1.04e-05*** (t-statistic (11.95))
  - Constant ܿ = 0.115*** (t-statistic (2.523))
  - R-squared = 0.951
  - Observations = 249
  - Predicted mid-corridor intercept = 0.52
  - Excess reserves at inflection point = -11.0 (EUR bn)
  - 2nd-derivative maxima = 115.5 (EUR bn)
  - Excess reserves at minimum bid rate = 7.05 (EUR bn)
- Alternative frequencies — Daily:
  - Lower asymptote = 0.0738*** (22.68)
  - Coefficient = 1.07e-05*** (44.68)
  - Constant = 0.109*** (9.60)
  - R-squared = 0.932
  - Observations = 4935
  - Predicted mid-corridor intercept = 0.51
  - Excess reserves at inflection point = -10.0 (EUR bn)
  - 2nd-derivative maxima = 112.9 (EUR bn)
  - Excess reserves at minimum bid rate = 4.7 (EUR bn)
- Alternative frequencies — Weekly (MRO):
  - Lower asymptote = 0.0938*** (15.09)
  - Coefficient = 1.45e-05*** (21.06)
  - Constant = 0.106*** (4.82)
  - R-squared = 0.953
  - Observations = 965
  - Predicted mid-corridor intercept = 0.52
  - Excess reserves at inflection point = -7.5 (EUR bn)
  - 2nd-derivative maxima = 98.1 (EUR bn)
  - Excess reserves at minimum bid rate = 7.0 (EUR bn)
- Interpretation of full-sample estimates:
  - Lower asymptote estimated at 8.6 percent of the normalized corridor (market requires a pick-up relative to deposit facility).
  - Intercept (mid-corridor with zero excess reserves) ~ 0.52.
  - Interpretation of coefficient: one billion in additional excess reserves leads to a decrease of EONIA by 1.05 percent in the normalized corridor on average (noting nonlinearity).
  - Logistic fit shows short-term rates well-anchored near the MRO minimum bid rate when excess reserves ≈ EUR 7 billion, and anchored to deposit facility at excess reserves > EUR 400 billion; intermediate ranges show larger deviations.

### Market segmentation and subperiod estimates (impact on logistic shape)
- Subperiod split and rationale:
  - Oct. 2008–Dec. 2011 versus Jan. 2012–Feb. 2018 to capture change in market risk sentiment and segmentation (fixed-rate full allotment applies to both).
- Subperiod parameter estimates — Oct. 2008–Dec. 2011:
  - Lower asymptote ߛ = 0.1332*** (7.330)
  - Coefficient ߙ = 4.99e-05** (2.450)
  - Constant ܿ = -0.141 (t-statistic (-0.240))
  - R-squared = 0.855
  - Observations = 41
  - Predicted mid-corridor intercept = 0.60
  - Excess reserves at inflection point = 3.0 (EUR bn)
  - 2nd-derivative maxima = 29.2 (EUR bn)
  - Excess reserves at minimum bid rate = 9.0 (EUR bn)
- Subperiod parameter estimates — Jan. 2012–Feb. 2018:
  - Lower asymptote ߛ = 0.0674*** (4.740)
  - Coefficient ߙ = 9.10e-06*** (7.600)
  - Constant ܿ = -0.344*** (-2.090)
  - R-squared = 0.942
  - Observations = 86
  - Predicted mid-corridor intercept = 0.61
  - Excess reserves at inflection point = 37.5 (EUR bn)
  - 2nd-derivative maxima = 182.4 (EUR bn)
  - Excess reserves at minimum bid rate = 53.6 (EUR bn)
- Interpretation:
  - Convergence to the deposit facility is faster (smaller required excess reserves) in Oct. 2008–Dec. 2011; the market behaved closer to a step-function.
  - From Jan. 2012–Feb. 2018, increased segmentation implies slower convergence: inflection point moves to EUR 37.5 billion, 2nd-derivative maxima to EUR 182.4 billion, and excess reserves at minimum bid rate to EUR 53.6 billion, indicating higher demand for excess reserves and a more sluggish response of rates to reserve injections.

### Coordination failures and market functioning
- Coordination failures mechanism:
  - Under fixed-rate full allotment, counterparties must estimate their reserve needs and coordinate bids; misestimation and segmentation (credit limits, opacity of others’ positions) can lead to aggregate under- or over-borrowing at the central bank relative to the “coordinated” optimum (Bcb*), generating spikes in short-term rates or unnecessary use of central bank funding.
  - Examples:
    - Aggregate Bcb < Bcb* → BL1 > BL1* → short-term market jumps to Rh > Rc.
    - Aggregate Bcb > Bcb* → BL1 < BL1* → market rates revert toward Rd but counterparties paid premia (Bcb − Bcb*)*(Rc − Rd).
- Empirical evidence of coordination failures:
  - Nine episodes where the weighted average of MRO refinancing and EONIA exceeded the MRO rate, showing that counterparties collectively paid more than they could have by coordinating bids fully at the MRO.
  - Absolute deviations between predicted and actual short-term rates are largest in the intermediate excess-reserve range (EUR 7 billion to EUR 400 billion), indicating un-anchored rates and the influence of variables beyond aggregate excess reserves.

### Other contributing factors
- Regulatory impacts that may affect the shape of the logistic function and demand for excess reserves:
  - Liquidity Coverage Ratio (LCR) can increase banks’ demand for excess reserves (as HQLA) once inflow caps are reached, reducing willingness to extend interbank loans.
  - Basel III capital requirements and the Capital Requirement Directive increase the cost of interbank lending via capital opportunity costs, tending to raise spreads between interbank rates and the deposit facility.

### Policy implications and recommendations
- When unwinding excess liquidity, the ECB should monitor interest-rate volatility for intermediate levels of excess reserves and assess instruments to minimize undesirable volatility (for example, accelerate transition to a new targeted steady state).
- Market de-segmentation would reduce the premium short-term rates command over the deposit rate at low excess reserves; if segmentation persists, higher levels of excess reserves may be required to keep rates anchored unless alternative instruments are used.
- Actions to stabilize short-term rates under fixed-rate full allotment:
  - Continue publishing autonomous factor forecasts to facilitate counterparties’ coordination and informed bidding as excess reserves decline.
  - Consider higher reserve requirements than historic lows to provide more averaging room and accelerate reduction of excess reserves, aiding return to neutral allotment (maintenance period extended from four to six weeks).
  - Maintain a negative deposit facility rate, especially when increasing the MRO rate (i.e., a negative deposit facility associated with an increase in the interest rate corridor) to create incentives for market de-segmentation.
- Usefulness of the logistic function:
  - A practical tool for central banks steering short-term rates in an interest rate corridor and for other frameworks (reserve money, exchange rate) to estimate the responsiveness of short-term rates to excess reserves and monitor changes in market functioning and risk perception.

*Source: wp1880 - 1. Estimates of the logistic function—different data frequency*

### 1. Estimates of the logistic function—different data frequency _______________________24

### 1. Estimates of the logistic function—different data frequency _______________________24

### Purpose and modeling approach
- Models the relationship between excess reserves and short-term interest rates in a symmetric interest rate corridor using a bivariate logistic function.
- Logistic function defined so that:
  - short-term rates converge to the top of the corridor when excess reserves are increasingly negative;
  - short-term rates converge to the bottom of the corridor when excess reserves increase;
  - short-term rates take the value 0.5 when there are no excess reserves.
- Estimated using Eurosystem data (long time series since 1999) to determine the position of short-term rates in the corridor and to predict their behavior as excess reserves vary.
- Estimates identify ranges of excess reserves where short-term rates are anchored to:
  - higher edge,
  - mid-corridor,
  - lower edge,
  - or become “un-anchored” (fluctuate between an edge and the mid-point or the other edge).

### Key empirical context and data features
- Eurosystem experience provides:
  - equal periods of low and stable excess reserves: 1999–2008;
  - periods of large and volatile excess reserves: 2009–2018.
- Excess reserves stood at the beginning of 2018 above 1 trillion euro.
- Estimation periods referenced: January 1999 to February 2018; October 2008 to December 2011; January 2012 to February 2018.

### Economic interpretation and implications
- “Un-anchoring” of short-term rates is undesirable because:
  - policy stance becomes less precisely defined when short-term rates fluctuate in the corridor;
  - short-term rate volatility and uncertainty increase the liquidity premium, transmitting higher funding costs through the term structure.
- The logistic function can help central banks:
  - predict excess reserve ranges causing un-anchoring;
  - accelerate transitions to states where anchoring occurs (to a corridor edge or mid-point) to avoid negative consequences.
- The shape and parameters of the logistic function and the reserve levels at which volatility occurs vary across countries and must be locally estimated.

### Market segmentation, full allotment, and coordination failure
- Market segmentation reduces efficiency of reallocation of excess reserves, altering the speed of convergence to corridor edges.
- Fixed-rate full allotment (central bank fulfills all banks’ bids at refinancing operations at a fixed rate if sufficient eligible collateral exists) can, in a segmented market, lead to episodes of “coordination failure” among banks bidding at main open market operations.
- Coordination failure defined as individual bids that are individually optimal but lead to a sub-optimal system outcome, increasing volatility of short-term rates and excess reserves.
- Counterparty risk compounds coordination failure by hindering redistribution of reserves from banks with excess reserves to banks with shortages.

### Policy operational considerations for steering short-term rates
- Central banks influence short-term rates by:
  - altering the stock of reserves via allotment of open market operations (net reserves supply);
  - setting interest rates for refinancing and remuneration of excess reserves;
  - using reserve requirement averaging and standing facilities to absorb forecast errors.
- Standing facilities create an interest rate corridor:
  - floor = deposit facility rate (deposit facility) where banks can deposit unlimited amounts;
  - ceiling = lending facility (lender of last resort) where borrowing is limited by eligible collateral.
- Corridor width trade-offs:
  - too narrow: reduces interbank trading incentives; central bank intermediates more, losing market signaling;
  - too wide: increases short-term interest rate volatility.
- Under “neutral liquidity allotment” (keep excess reserves low and stable), short-term rates tend to be steered to the middle of the corridor and market transactions are maximized.
- With large, volatile, and unpredictable demand for excess reserves (post-global financial crisis), the market’s ability to redistribute liquidity breaks down, complicating steering.

### Literature links and methodological precedents
- Prior uses of logistic representations: Valimaki (2001 and 2008), Bindseil (2017) (did not estimate empirically).
- Related empirical work: Bech and Monnet (2015) showed relationship between excess reserves and EONIA but did not estimate state-dependent relationships.
- Models addressing fragmentation and excess liquidity: Vari (2016) shows counterparty risk engenders market fragmentation and endogenous excess liquidity.
- Studies on fixed-rate versus variable-rate tenders and full allotment: Bindseil (2002); Catalão-Lopes (2010) included full allotment considerations.
- Studies on short-term rate spread dynamics and MRO interactions: Moschitz (2004), Nautz and Offermanns (2007), Wurtz (2003).
- Analysis of corridor width and market transaction trade-offs: Bindseil and Jablecki (2011).

*Source: wp1880 - 1. Estimates of the logistic function—different data frequency _______________________24*

### conclusion is that wider corridors between standing facilities are associated with greater

### wp1880 - conclusion is that wider corridors between standing facilities are associated with greater

### Key findings
- Wider corridors between standing facilities are associated with greater interbank trading volumes and greater volatility of overnight rates.
- Fixed-rate full allotment can anchor overnight rates to the deposit facility rate, but coordination failures among counterparties and market segmentation can lead to deviations from the deposit facility rate.
- Market segmentation raises short-term rates for a given level of excess reserves by reducing the tradable portion of excess reserves (credit limits, collateral eligibility, precautionary demand).
- The logistic-function representation of the short-term rate as a function of excess reserves captures: two asymptotes (deposit and lending facility rates), an inflection point (neutral liquidity allotment), diminishing sensitivity of rates as excess reserves grow, and potential horizontal shifts (empirically estimated lower asymptote).

### Logistic model (stylized and technical)
- Model choice and properties:
  - The logistic function is used because it allows positive and negative explanatory values, asymptotic convergence, and a mid-rate outcome when the explanatory variable is zero.
  - In perfectly functioning money markets demand for excess reserves would be a step function; segmentation and coordination failures give the excess-reserve demand a logistic shape.
- Four-parameter logistic function (modified Oliver (1969) form):
  - Model form (as used): y = s + (κ − s) / (1 + e^(−α(x − x0)))
  - Main features emphasized:
    - Allows empirical estimation of the lower asymptote s (to account for market-required pick-up above the deposit facility).
    - Reveals horizontal movements along the corridor, enabling computation of short-term rate when excess reserves are null.
  - Economic interpretations:
    - Lower asymptote (s): short-term rate toward which EONIA converges at very high excess reserves (reflects counterparty risk, transaction costs, market microstructure).
    - Inflection point: level of excess reserves above which short-term rates decrease and below which they increase (neutral allotment).
    - Second-derivative maxima: benchmark reserve level beyond which marginal effect of additional excess reserves on short-term rates diminishes.

### Empirical results—Estimation January 1999 to February 2018 (full sample and frequencies)
- Sample: 249 maintenance periods (January 1999 to February 2018).
- Regimes in excess reserves during sample:
  - Neutral allotment: January 1999–October 2008.
  - Moderate excess reserve periods: October 2008–December 2011, and March 2014–March 2015.
  - High excess reserve regimes: January 2012–February 2014 and since April 2015.
- Full-sample maintenance-period estimates (Table 1):
  - Lower asymptote ߛ = 0.0860*** (t-statistic (6.234))
  - Coefficient ߙ = 1.04e-05*** (t-statistic (11.95))
  - Constant ܿ = 0.115*** (t-statistic (2.523))
  - R-squared = 0.951
  - Observations = 249
  - Predicted mid-corridor intercept = 0.52
  - Excess reserves at inflection point = -11.0 (EUR bn)
  - 2nd-derivative maxima = 115.5 (EUR bn)
  - Excess reserves at minimum bid rate = 7.05 (EUR bn)
- Alternative frequencies (Table 1):
  - Daily frequency:
    - Lower asymptote = 0.0738*** (22.68)
    - Coefficient = 1.07e-05*** (44.68)
    - Constant = 0.109*** (9.60)
    - R-squared = 0.932
    - Observations = 4935
    - Predicted mid-corridor intercept = 0.51
    - Excess reserves at inflection point = -10.0 (EUR bn)
    - 2nd-derivative maxima = 112.9 (EUR bn)
    - Excess reserves at minimum bid rate = 4.7 (EUR bn)
  - Weekly (MRO) frequency:
    - Lower asymptote = 0.0938*** (15.09)
    - Coefficient = 1.45e-05*** (21.06)
    - Constant = 0.106*** (4.82)
    - R-squared = 0.953
    - Observations = 965
    - Predicted mid-corridor intercept = 0.52
    - Excess reserves at inflection point = -7.5 (EUR bn)
    - 2nd-derivative maxima = 98.1 (EUR bn)
    - Excess reserves at minimum bid rate = 7.0 (EUR bn)
- Interpretation of full-sample estimates:
  - Lower asymptote estimated at 8.6 percent of the normalized corridor (market requires a pick-up relative to deposit facility).
  - Intercept (mid-corridor with zero excess reserves) ~ 0.52 (slightly above corridor midpoint).
  - Interpretation of coefficient: one billion in additional excess reserves leads to a decrease of EONIA by 1.05 percent in the normalized corridor on average (noting nonlinearity).
  - The logistic fit shows short-term rates well-anchored near the MRO minimum bid rate when excess reserves ≈ EUR 7 billion, and anchored to deposit facility at excess reserves > EUR 400 billion; intermediate ranges show larger deviations.

### Market segmentation and subperiod estimates (impact on logistic shape)
- Subperiod split and rationale:
  - Oct. 2008–Dec. 2011 versus Jan. 2012–Feb. 2018 to capture change in market risk sentiment and segmentation (fixed-rate full allotment applies to both).
- Subperiod parameter estimates (Table 2):
  - Oct. 2008–Dec. 2011:
    - Lower asymptote ߛ = 0.1332*** (7.330)
    - Coefficient ߙ = 4.99e-05** (2.450)
    - Constant ܿ = -0.141 (t-statistic (-0.240))
    - R-squared = 0.855
    - Observations = 41
    - Predicted mid-corridor intercept = 0.60
    - Excess reserves at inflection point = 3.0 (EUR bn)
    - 2nd-derivative maxima = 29.2 (EUR bn)
    - Excess reserves at minimum bid rate = 9.0 (EUR bn)
  - Jan. 2012–Feb. 2018:
    - Lower asymptote ߛ = 0.0674*** (4.740)
    - Coefficient ߙ = 9.10e-06*** (7.600)
    - Constant ܿ = -0.344*** (-2.090)
    - R-squared = 0.942
    - Observations = 86
    - Predicted mid-corridor intercept = 0.61
    - Excess reserves at inflection point = 37.5 (EUR bn)
    - 2nd-derivative maxima = 182.4 (EUR bn)
    - Excess reserves at minimum bid rate = 53.6 (EUR bn)
- Interpretation:
  - Convergence to the deposit facility is faster (smaller required excess reserves) in Oct. 2008–Dec. 2011; the market behaved closer to a step-function.
  - From Jan. 2012–Feb. 2018, increased segmentation implies slower convergence: the inflection point moves to EUR 37.5 billion, 2nd-derivative maxima to EUR 182.4 billion, and excess reserves at minimum bid rate to EUR 53.6 billion, indicating higher demand for excess reserves and a more sluggish response of rates to reserve injections.

### Coordination failures and market functioning
- Coordination failures mechanism:
  - Under fixed-rate full allotment, counterparties must estimate their reserve needs and coordinate bids; misestimation and segmentation (credit limits, opacity of others’ positions) can lead to aggregate under- or over-borrowing at the central bank relative to the “coordinated” optimum (Bcb*), generating spikes in short-term rates or unnecessary use of central bank funding.
  - Examples:
    - Aggregate Bcb < Bcb* → BL1 > BL1* → short-term market jumps to Rh > Rc (higher rates).
    - Aggregate Bcb > Bcb* → BL1 < BL1* → market rates revert toward Rd but counterparties paid premia (Bcb − Bcb*)*(Rc − Rd).
- Empirical evidence of coordination failures:
  - Nine episodes where the weighted average of MRO refinancing and EONIA exceeded the MRO rate, showing that counterparties collectively paid more than they could have by coordinating bids fully at the MRO.
  - Absolute deviations between predicted and actual short-term rates are largest in the intermediate excess-reserve range (EUR 7 billion to EUR 400 billion), indicating un-anchored rates and the influence of variables beyond aggregate excess reserves.

### Other contributing factors
- Regulatory impacts that may affect the shape of the logistic function and demand for excess reserves:
  - Liquidity Coverage Ratio (LCR) can increase banks’ demand for excess reserves (as HQLA) once inflow caps are reached, reducing willingness to extend interbank loans.
  - Basel III capital requirements and the Capital Requirement Directive increase the cost of interbank lending via capital opportunity costs, tending to raise spreads between interbank rates and the deposit facility.

### Policy implications and recommendations
- When unwinding excess liquidity, the ECB should monitor interest-rate volatility for intermediate levels of excess reserves and assess instruments to minimize undesirable volatility (for example, accelerate transition to a new targeted steady state).
- Market de-segmentation would reduce the premium short-term rates command over the deposit rate at low excess reserves; if segmentation persists, higher levels of excess reserves may be required to keep rates anchored unless alternative instruments are used.
- Actions to stabilize short-term rates under fixed-rate full allotment:
  - Continue publishing autonomous factor forecasts to facilitate counterparties’ coordination and informed bidding as excess reserves decline.
  - Consider higher reserve requirements than historic lows to provide more averaging room and accelerate reduction of excess reserves, aiding return to neutral allotment (maintenance period extended from four to six weeks).
  - Maintain a negative deposit facility rate, especially when increasing the MRO rate (i.e., a negative deposit facility associated with an increase in the interest rate corridor) to create incentives for market de-segmentation.
- Usefulness of the logistic function:
  - A practical tool for central banks steering short-term rates in an interest rate corridor and for other frameworks (reserve money, exchange rate) to estimate the responsiveness of short-term rates to excess reserves and monitor changes in market functioning and risk perception.

*Source: Authors’ calculations and analysis in the provided IMF working paper content.*

### REFERENCES

### REFERENCES

### Interbank market models and dynamics
- Bech, M., and C. Monnet, 2015. “A search-based model of the interbank money market and monetary policy implementation,” BIS Working Paper No. 529.
- Heifer, F., M. Hoerova, and C. Holthausen, 2009. “Liquidity Hoarding and Interbank Market Spreads: the Role of Counterparty Risk,” ECB Working Paper No. 1126.
- Würtz, F., 2003. “A Comprehensive Model on the Euro Overnight Rate,” ECB Working Paper No. 207.
- Moschitz, J., 2004. “The Determinants of the Overnight Interest Rate in the Euro Area,” ECB Working Paper No. 393.
- Nautz, D., C.J. Offermanns, 2007. “The dynamic relationship between the euro overnight rate, the ECB’s policy rate and the term spread, International Journal of Finance and Economics 12, 287–300.
- Linzert, T., S. Schmidt, 2008. “What Explains the Spread Between the Euro Overnight Rate and the ECB’s Policy Rate,” ECB Working Paper No. 983.
- Beirne, J., 2012. “The EONIA spread before and during the crisis of 2007–2009: The role of liquidity and credit risk,” Journal of International Money and Finance 31, 534–51.

### Eurosystem / ECB liquidity management and policy implementation
- Bindseil, U., 2002. “Equilibrium Bidding in the Eurosystem’s Open Market Operation,” ECB Working Paper No. 137.
- Bindseil, U., and J. Jablecki, 2011. “The Optimal Width of the Central Bank Standing Facilities Corridor and Banks’ Day-to-Day Liquidity Management,” ECB Working Paper No. 1350.
- Bindseil, U., and J. Lamoot, 2011. “The Basel III Framework for Liquidity Standards and Monetary Policy Implementation,” SFB 649 Discussion Paper 2001-041.
- Bindseil, U., 2017. “Monetary Policy Operations and the Financial System.” Oxford University Press.
- Cassola, N., and M. Huetl, 2010. “The Euro overnight interbank market and ECB’s liquidity management policy during tranquil and turbulent times,” ECB Occasional Paper No. 1247.
- Catalão-Lopes, M., 2010. “Fixed- and Variable-Rate Tenders in the Management of Liquidity by the Eurosystem: Implications of the Recent Credit Crisis,” International Journal of Central Banking, Vol. 6(2), 199-230, June.
- Durre, A., A. Maddaloni, and F. Mongelli, 2014. “The ECB’s Experience of Monetary Policy on a Financially Fragmented Euro Area,” Comparative Economic Studies, No. 56, 396-423.
- ECB Bulletin, January 2014. “Recent Developments in Excess Liquidity and Money Market Rates.”

### Tender design, reserves, and pricing in money markets
- Valimaki, T., 2001. “Fixed rate tenders and the overnight money market equilibrium,” Bank of Finland Discussion Paper, August.
- Valimaki, T., 2008. “Why the Effective Price for Money Exceeds the Policy Rate in the ECB Tenders?,” ECB Working Paper Series, No. 981.
- Catalão-Lopes, M., 2010. “Fixed- and Variable-Rate Tenders in the Management of Liquidity by the Eurosystem: Implications of the Recent Credit Crisis,” International Journal of Central Banking, Vol. 6(2), 199-230, June.
- Bindseil, U., 2002. “Equilibrium Bidding in the Eurosystem’s Open Market Operation,” ECB Working Paper No. 137.

### Theoretical and methodological foundations
- Poole, W., 1968. “Commercial Bank Reserve Management in a Stochastic Model: Implications for Monetary Policy,” Journal of Finance, 23(5), pp. 769-791.
- Oliver, F.R., 1969. “Another Generalisation of the Logistic Growth Function,” Econometrica, Vol. 37, No. 1, 144.
- Vari, M., 2016. “Monetary policy transmission with interbank market fragmentation,” CEPREMAP Working Paper No. 1516.

*wp1880 - REFERENCES*

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_Source: https://www.imf.org/-/media/files/publications/wp/2018/wp1880.pdf_