## tarea2024002

## Source details

**Canonical URL:** [tarea2024002](https://www.imf.org/-/media/files/publications/tar/2024/english/tarea2024002.pdf)

## Other formats

- [Markdown version](/-/media/files/publications/tar/2024/english/tarea2024002.pdf.md)
- [Structured JSON version](/-/media/files/publications/tar/2024/english/tarea2024002.pdf.json)

---

### Mission and engagement and operational context
- An MCM Department/CAPTAC-DR technical assistance (TA) mission visited Guatemala City from June 12 to 16, 2023, at the request of Banco de Guatemala (Banguat) to assist in liquidity forecasting to calibrate daily deposit operations.
- The mission met with Mr. Vinicio Caceres (Banguat’s Financial Manager), Mr. Marco López (Director, DMFX), Mr. Ariel López (Deputy Director, DMFX), and Mr. Juan Antonio Ibañez (Deputy Director, DMAF).
- CAPTAC-DR donors funded the project under which this TA was delivered.
- Banguat currently conducts daily open market operations offering 24-hour deposits to monetary policy counterparties (banks).
  - Recourse to the deposit facility averaged 20 percent of the daily auctioned amounts.
  - Use of the deposit facility represents one percent of the banks’ total position with Banguat.
  - There are currently 18 banks on operation; the mission provided forecasts for 16 banks with enough data.

### Reserve requirement (RR) design and forecasting challenges
- RR design and operational features:
  - The reserve requirement (RR) in Guatemala is contemporaneous with the base: every day banks must maintain an equivalent in reserves of 14.6 percent of the deposits that they have on the same day.
  - Neither Banguat nor the banks know the exact requirement until the end of the day.
  - Eligible RR fulfillment: banks’ balances at the central bank (excluding the deposit facility) are eligible for RR fulfillment, and up to 25 percent can be fulfilled with the cash vault.
  - Under-fulfillment allowance: banks are allowed to under fulfill the reserve requirement for up to 14 days of a calendar month, provided the sum of deficiencies divided by 14 does not exceed 20 percent of the average monthly reserve requirement.
- Forecasting implications:
  - Banguat needs to forecast banks’ reserve needs to decide how much deposit to offer at daily deposit operations.
  - Banks need forecasts of the requirement to decide participation in Banguat’s daily deposit operations and the interbank market.

### Statistical forecasting framework deployed
- The MCMCO framework introduced includes 12 forecasting models of three types:
  - Exponential smoothing (simple, with exogenous regressors, and seasonal).
  - ARIMA (simple, with exogenous regressors, and seasonal).
  - TBATS and volatility models.
- The framework has out-of-sample performance testing using four performance criteria reflecting accuracy, bias, and confidence intervals.
- The framework can produce individual forecasts and combine them (best single forecast, best three models, or average of all models).

### Key findings on forecasting components
- Reserve requirement forecasting
  - The reserve requirement is the main driver of banks’ demand for reserves and presents forecasting challenges because it is contemporaneous with its base.
  - The mission factored heterogeneity in the RR base by forecasting different banks’ deposits included in the RR base and reconciled forecasts (e.g., using OLS and MinT) to obtain the RR assuming a constant reserve requirement ratio.
  - Statistical reconciliation is more accurate than the aggregated forecast because it captures behavior of depositors.
- Reserve fulfillment (banks’ intra-month preferences)
  - Banks exhibit clear intra-month preferences for on-account amounts relative to the RR and for deposits into the deposit facility depending on the day of the month.
  - The mission forecasted demand for excess reserves for each bank and applied statistical reconciliation; reconciled-bank forecasts are more accurate than aggregate forecasts.
- Autonomous factors
  - Banguat forecasts the government account (STA) at Banguat and Net Foreign Assets (NFA) more accurately than the MCMCO models, reflecting useful institutional information beyond pure statistical models.
  - The MCMCO framework forecasts Currency in Circulation (CiC) slightly better than Banguat’s forecasts due to more statistical model diversity.
  - The framework adds forecasts of Net Other Assets (NOA), Net Liquidity (NL) with reconciliation, and provides information on prediction uncertainty.
- Liquidity calibration
  - A liquidity table combining demand and supply forecasts can be used to calibrate open market operations: autonomous factors determine system liquidity supply while RR and excess reserve forecasts determine banks’ demand; the neutral allotment is the difference between supply and demand.

### Policy and implementation recommendations (from Table 1)
- Liquidity Forecast
  - Complement Banguat’s bank survey with statistical models to produce a daily reserve requirement and demand for excess reserve forecasts (#6 and 57). — DMFX — Short term
  - Complement Banguat’s forecasts with statistical models to produce a daily forecast for autonomous factors (#7 and #37). — DMAF — Short term
  - Evaluate forecast performance periodically (#38). — DMFX & DMAF — Medium term
- Open Market Operations
  - Complement the survey of banks’ demand for daily deposit operations with a calibration based on forecasting autonomous factors and the demand for reserves (#37). — DMFX — Medium term
- Information to the Monetary Counterparties
  - Publish daily the forecasts of autonomous factors, the reserve requirement, and the demand for excess reserves for the next day (#42 & 43). — DMFX & DMAF — Long term
  - Publish daily the forecasts of autonomous factors, the reserve requirement, and the demand for excess reserves at the one-week horizon (#42 & 43). — DMFX & DMAF — Long term
- Timeframe definitions:
  - Short term: < 6 months; Medium term: 6 to 12 months; Long term: more than 12 months.

### The Banguat’s current survey practice and statistical complement
- Institutional forecasting and daily survey practice
  - Banks daily inform the Banguat of the amounts of deposits that they would likely request at the auction via a survey.
  - The Banguat announces an operation of a total size that is equivalent to the sum of banks’ net declared demand.
  - The Banguat allots the same amount or less if the demand turns out lower than expected.
- Development of the statistical component of liquidity forecasting
  - The mission assisted Banguat to develop statistical forecasts of balance-sheet components affecting liquidity to complement institutional information (e.g., weekly government expected outflows; contractual payments and income related to foreign-denominated debt).
  - Statistical forecasts serve as a default benchmark and allow processing of time-series information extracted from data.
  - The mission used MCMCO models to forecast autonomous factors and to statistically reconcile them into one liquidity forecast and compared forecast errors with Banguat’s models.

### Reserve requirement estimation approaches and performance metrics
- Estimating the demand for reserves — overview
  - Demand for reserves is decomposed between regulatory demand arising from the reserve requirement (RR), defined as a percentage of the RR base (demand, saving, term, and other deposits), and demand for excess reserves as reflected in banks’ preferred fulfillment profile.
  - The mission forecasted the different deposits of the RR base (assuming a constant ratio) to predict RR; and the demand for excess reserves of each bank.
  - Statistical reconciliation techniques were used to obtain aggregated forecasts from disaggregated predictions.
- Reserve Requirement — aggregated basis
  - Historical characteristics:
    - Clear undamped upward trend with an average annual growth rate of 10 percent across the year.
    - Seasonality: RR appears significantly lower during Q2 and Q3.
  - Model selection and performance:
    - ARIMA with regressor emerges as the best model to predict the reserve requirement across time horizon and performance metrics (RMSE, MEA, ME, MIS), with seasonal ARIMA performing better only for bias (ME).
    - The framework can present the best single forecast or combinations (best three models or average of all models).
  - Forecast horizon example:
    - Daily forecast produced as of April 30, 2023, up to the 30 days horizon (approximately the under-fulfillment allowance period).

### By-deposit type approach (RR base) — heterogeneity and reconciliation gains
- Method:
  - Forecast the different types of deposits (demand, term, saving, other) and apply the RR coefficient, 14.6 percent, to obtain the RR forecast.
- Descriptive findings:
  - Significant heterogeneity across deposit types:
    - Saving deposits more widely distributed in sizes than term deposits.
    - Term deposits tend to be larger; demand deposits bi-modal; other deposits smaller and concentrated.
- Selected models by deposit type (Average in GTQ million; Selected Model, RMSE Horizon = 30):
  - Monetary Deposits: 76,564.25 — ARIMA with Regression
  - Saving Deposits: 56,205.56 — TBATS
  - Term Deposits: 96,734.47 — ETS with Regression
  - Others: 801.71 — ARIMA with Regression
- Reconciliation and accuracy:
  - Model aggregation (reconciliation) improves forecast quality: RMSE at 1, 2, and 4-week horizons are lower for deposit-base reconciled RR forecasts than unreconciled aggregated forecasts.
  - Across specifications: by-deposit ARIMA with regression reconciled with OLS performs best at the 1-week horizon; deposit-base ARIMA with regression bottom up performs best for the 2 and 4-week horizons.
- RMSE (Million GTQ) — selected values from Table 3:
  - Unreconciled RR Aggregate (ARIMA with Regression): Week 1 = 148.90; Week 2 = 176.30; Week 4 = 219.10
  - Reconciled RR Aggregate by Base (ARIMA with Regression, OLS): Week 1 = 142.77; Week 2 = 163.65; Week 4 = 201.89
  - Reconciled RR Aggregate by Base (ARIMA with Regression, Bottom Up): Week 1 = 142.81; Week 2 = 163.43; Week 4 = 201.78
  - Reconciled RR Aggregate by Base (ARIMA with Regression, Min T): Week 1 = 143.93; Week 2 = 166.39; Week 4 = 204.52

### Reserve requirement fulfillment profile — patterns, models, and by-bank heterogeneity
- Aggregated intra-month pattern:
  - Excess reserves show a clear monthly pattern: banks tend to front-load (exceed requirement early in the month) and then reduce unremunerated balances later (placing reserves in the deposit facility).
  - Policy implication:
    - Banguat should accommodate excess reserve demand to stabilize short-term interest rates:
      - Reduce OMO allotment when banks keep more reserves early in the period to avoid low participation and rate increases.
      - Issue more at the end of the period to prevent downward pressure on daily deposit rates and increased recourse to the deposit facility.
- Model performance (excess reserves):
  - ARIMA with regressor is the best model across out-of-sample metrics for forecasting excess reserves, except bias (ME) where simple ARIMA performs better.
  - Example: 30-day forecast as of April 30, 2023, produced using ARIMA with Regression.
- Reserve requirement fulfillment profile — by bank
  - Heterogeneity:
    - Significant heterogeneity across banks in average excess reserve behavior and volatility; some banks show wide and bimodal distributions while others keep lower, less volatile excess reserves.
    - Heterogeneity stems from each bank treasurer’s preferred fulfillment profile (front-loaded, linear, backloaded).
  - Selected models for RR fulfillment by bank (Average RF Million GTQ, 2017-01-01 to 2023-04-30; Selected Model, RMSE Horizon = 30) — examples:
    - Bank 1: Average RF 28.67 — Seasonal ARIMA
    - Bank 2: Average RF 44.40 — ARIMA with Regression
    - Bank 3: Average RF 51.87 — ARIMA with Regression
    - Bank 4: Average RF 17.63 — Seasonal ARIMA
    - Bank 6: Average RF 55.47 — TBATS
    - Bank 7: Average RF 7.22 — Seasonal ARIMA
    - Bank 8: Average RF 7.22 — TBATS
    - Bank 11: Average RF 1.65 — ARIMA with Regression
    - Bank 12: Average RF 0.36 — ETS
    - Bank 19: Average RF 1.33 — TBATS
  - Model choice summary:
    - Four different models best forecast behaviors across 19 banks; TBATS performs best for 9 out of 19 banks.
  - Reconciliation gains for RF:
    - Reconciled by-bank forecasts improve RMSE at 1, 2, and 4-week horizons compared to aggregated unreconciled forecasts.
    - RMSE for RR Fulfillment Profile (Million GTQ) — Table 5:
      - Unreconciled RF Aggregated (ARIMA with Regression): Week 1 = 1464.5; Week 2 = 1516.0; Week 4 = 1521.8
      - Reconciled RF Aggregate by Bank (ARIMA with Regression, OLS): Week 1 = 1264.9; Week 2 = 1308.8; Week 4 = 1313.6
      - RF Reconciled by bank (ARIMA with Regression, Bottom up): Week 1 = 1368.8; Week 2 = 1423.2; Week 4 = 1442.9
      - RF Reconciled by bank (ARIMA with Regression, Min T): Week 1 = 1439.3; Week 2 = 1472.2; Week 4 = 1453.7

### Forecasting autonomous factors and Net Liquidity (NL)
- Objectives of adopting the Liquidity Forecasting Framework:
  - Benchmark Banguat’s current models against pure statistical estimations from MCMCO Framework.
  - Introduce reconciliation techniques to obtain an aggregated forecast of autonomous factors’ impact on NL and provide forecast uncertainty (confidence intervals).
- Comparative performance (T tests on absolute errors — p-values, Table 6):
  - CiC: Unpaired = 0.11; Paired = 0.10
  - NFA: Unpaired = 0.12; Paired = 0.04*
  - STA: Unpaired = 0.00*; Paired = 0.00*
  - Note: * indicates significant difference at 5 percent.
  - Interpretation:
    - Banguat’s current models outperform pure statistical MCMCO forecasts for NFA and STA due to qualitative institutional inputs.
    - MCMCO forecasts CiC slightly better.
- Currency in Circulation (CiC)
  - Time-series features:
    - Clear upward trend and seasonal pattern.
    - Strong weekly seasonality with increasing demand towards the end of the week and seasonal jumps around year-end/new year.
  - Model selection:
    - ARIMA with Regression selected for 1-week, 2-week, and 4-week forecasts.
    - Seven models tested including Naïve and Naïve seasonal; ARIMA with Regression and ETS with Regression ranked top in accuracy.
    - Regressors used include trigonometric terms, weekly seasonality, and holidays (New Year, Assumption Day, Revolution Day, All Saints Day, Good Friday, Christmas).
    - Selected ARIMA with Regression shows ME closer to 0 (relatively unbiased).
- Net Foreign Assets (NFA)
  - Historical behavior:
    - NFA series ascending before 2022 and then levels off with four historical drastic jumps and drops attributed to activities such as Eurobond issuance and FX purchase.
    - Structural-break dummy experiments were attempted but produced unsatisfactory forecasts; mission removed these jumps and reconnected the series prior to modeling.
  - Model choice:
    - Volatility models (e.g., GARCH) were tested but did not outperform the simple Naïve benchmark.
    - ETS with Regression is the recommended forecasting method for NFA: lowest averaged RMSE and closest-to-0 ME across horizons; trigonometric seasonality used as regressors.
- State Treasury Account (STA)
  - Seasonality and forecasting:
    - Monthly and quarterly seasonality is obvious in STA data; intraweek pattern is weak.
    - ARIMA with regression ranks best for STA; Fourier terms and weekly dummies were incorporated.
    - STA does not show solid jumps or drops during New Year holiday periods but transitory drops with fluctuations; Fourier terms are preferred over holiday dummies.
- Net Other Assets (NOA)
  - Time-series features:
    - NOA does not exhibit strong volatility; before 2021 stock declined then climbs after 2021.
    - Intraweek seasonality not evident, monthly and quarterly patterns exist.
  - Model choice:
    - A more parsimonious ARIMA model selected; ARIMA performs better for 1 and 4-week horizons; seasonal ARIMA and ARIMA with Regression follow for 2-week horizon.
- Net Liquidity (NL)
  - Construction:
    - NL = NFA + NOA – CiC – STA.
  - Model selection and reconciliation:
    - TBATS selected based on lowest RMSE score for NL forecasts overall; TBATS produces lower error on average for 1 and 4-week horizons.
    - ARIMA with Regression ranks better for the two-week horizon.
    - Reconciliation recommendations:
      - OLS recommended for longer forecast horizons and Bottom-Up for one-week forecast.
      - OLS has lower ME scores across horizons, indicating lower bias.
  - Predictive accuracy and bias (Table 7 — scores in Million GTQ):
    - OLS: RMSE +1 week 1873.8, +2 weeks 2214.0, +4 weeks 2597.9; MAE +1 week 1461.6, +2 weeks 1795.7, +4 weeks 2166.1; ME +1 week 65.0, +2 weeks -10.3, +4 weeks -56.7
    - Base (Unreconciled): RMSE +1 week 1959.7, +2 weeks 2312.7, +4 weeks 2687.9; MAE +1 week 1553.6, +2 weeks 1850.7, +4 weeks 2210.4; ME +1 week 110.0, +2 weeks 62.6, +4 weeks -66.8
    - Bottom Up: RMSE +1 week 1814.8, +2 weeks 2300.0, +4 weeks 2732.1; MAE +1 week 1378.6, +2 weeks 1827.0, +4 weeks 2244.9; ME +1 week -114.8, +2 weeks -301.9, +4 weeks -550.9
    - MinT: RMSE +1 week 2135.9, +2 weeks 2574.2, +4 weeks 2952.2; MAE +1 week 1704.6, +2 weeks 2021.2, +4 weeks 2382.9; ME +1 week 145.7, +2 weeks 180.1, +4 weeks 286.2

### Liquidity table, calibration and publication guidance
- Liquidity table and calibration
  - The liquidity table summarizes reserves available and demand for reserves to calibrate daily deposit operations.
  - The first four lines under Autonomous Factors determine reserves available at each forecasted date.
  - Under reserve requirement: predicted demand due to regulation (the RR) plus banks’ preference regarding reserve requirement fulfillment.
  - Neutral allotment = available reserves (from autonomous factors) minus how much banks want to keep for regulatory reasons or predictable excess reserve preferences.
  - Facilities are expected to capture forecast errors under neutral allotment; ex-ante OMOs are calibrated to leave no excess or shortage, so recourse to facilities should be null unless forecast errors or market frictions occur.
  - Liquidity tables can be prepared in flows (focused) or levels (comprehensive); levels translate full central bank balance sheet but are more cumbersome.
- Suggested items to publish to inform banks’ bidding:
  - Opening banks’ balances at Banguat.
  - Liquidity forecast including:
    - The total autonomous factors.
    - The daily reserve requirement objective.
    - The demand for excess reserves.
- Publication recommendation:
  - Banguat should gradually increase the published forecast horizon once forecast quality is vetted.
  - Start with publishing forecast for the next day and consider publishing 1-week horizon when quality is sufficient.
- Operational recommendations and implementation
  - Banguat should complement institutional forecasts and the bank’s survey with the proposed statistical models to produce forecasts for autonomous factors and the RR and to calibrate daily auctions.
  - The mission provided Banguat with codes to estimate all models for each variable, the reconciliation for liquidity, and RR.
  - With the codes, Banguat staff can test model performance and do out-of-sample forecast evaluation to select models.
  - Banguat should periodically evaluate selected models (e.g., quarterly, semiannually) because the best model can change due to new information or structural changes.
  - MCM can provide technical support.

### Appendix I — statistical methods summary (models, seasonality, reconciliation)
- Four time series model types in the MCMCO Liquidity Forecasting Framework: Naïve, Exponential Smoothing (ETS), ARIMA, and TBATS.
- Key methodological points:
  - ETS models decompose time series into level, trend, and seasonality; smoothing parameter α (0<α<1) updates level via α times last error.
  - A smoothing parameter value of 0 implies no update; value of 1 implies fully updated by last observation. Low α → long-weighted moving averages; high α → very reactive components.
  - Model selection via information criteria (AIC/AICc) balances fit and complexity.
  - ARIMA model selection follows the stepwise algorithm: determine d with KPSS, estimate initial models, expand candidates, iterate; selection criterion AICc; auto.arima implementation referenced.
  - Seasonality encoding uses indicator variables for short/unsmooth seasonal lengths or trigonometric pairs to encode multiple seasonal cycles; lasso regression can be used for parsimony.
  - TBATS handles multiple seasonalities, Box-Cox transforms, ARIMA innovations, and allows seasonality to change over time.
  - Volatility models (GARCH, eGARCH, GJR-GARCH) are described for high-volatility series though often simple time-series models suffice.
- Forecast aggregation and reconciliation
  - Approaches: Bottom-up, Direct forecasting, Hybrid.
  - OLS reconciliation forces additivity: ŷ = M(M′M)−1M′ŷ.
  - MinT exploits covariance of forecast errors: ŷ = M(M′Σ−1M)−1M′Σ−1ŷ; Σ is covariance matrix of one-step-ahead forecasting errors.
  - OLS recommended for longer horizons; Bottom-Up sufficient for one-week horizon; MinT preferred when exploiting cross-series error correlations.

### Comparative country practices (CCB and BOM summaries)
- CCB (central bank example)
  - Forecast approach:
    - CCB obtains NL by subtracting expected liquidity demand from expected liquidity supply using a demand-supply approach for the RR period.
  - Regulatory demand:
    - Individual RRs estimated applying relevant RR coefficient: 0.11 or 0.45.
    - CCB incorporates an estimate of the precautionary level of reserves based on historical over-compliance.
  - Liquidity supply forecast includes approved permanent monetary flows, FX operations, government projected cash flow, CCB net income statement flows, and OMOs’ maturities.
  - Publication and OMO announcement practice:
    - CCB does not publish either forecast but announces the daily OMO amount for the following day, close to the average approved amount.
    - Calibration uses data collected from OMO market participants and constant communication with the government about STA flows.
- Bank of Mexico (BOM) practices
  - Institutional framework:
    - BOM formally adopted an IT regime in 2001 and transitioned to an interest rate target in January 2008.
    - Current monetary policy target: 11.25 percent.
  - Operational details for one-day horizon:
    - Neutral liquidity on a one-day horizon implies an aggregate zero balance of banks’ accounts (no reserve requirement).
    - Positive balances are not remunerated; overdrafts are charged twice the market’s overnight rate.
    - BOM auctions a predetermined amount; banks bid interest rates with multiple price allotment.
    - Liquidity injection via collateralized credit operations (overnight up to an average of 30 days) with the policy rate as minimum accepted bid rate; liquidity absorption via overnight deposit auctions with policy rate as maximum acceptance rate.
    - Institutional arrangements provide high certainty about autonomous factors at session opening (FX operations settled T+2; BOM law requires a one-day preannouncement for STA credits/debits).
    - BOM conducts a fine-tuning auction before payments system closure to address forecast errors and publishes a daily liquidity forecast before morning auctions (around 7:30 a.m.), including expected daily liquidity movement and the day’s net OMOs amount.
  - Medium and medium-to-long-term sterilization and forecasting:
    - BOM pre-sterilizes excess liquidity for the medium horizon to achieve a net creditor position.
    - Banks must participate in BOM’s liquidity-providing OMOs if system has a short-term shortage; participation in deposit-taking OMOs is not mandatory when system has excess liquidity.
    - Quarterly BOM announces medium-term sterilization policy and calibrates medium-to-long-term operations based on a forecast for up to two calendar years (annual → monthly → daily breakdowns updated with latest information).
  - BOM forecasting of autonomous factors:
    - CiC: start with annual flow forecast via econometric models using expected macro variables (GDP growth, inflation, interest rates); distributed into monthly/daily flows using statistical models and events.
    - STA: annual flows forecast by category linked to Congress’s approved figures; monthly calendar published by Ministry of Finances used; daily distribution based on historical data and operational rules (e.g., income taxes due after the seventeen each month).
    - NFA: FX interventions usually under pre-announced programs with known rules; expected annual flows from such programs incorporated; Pemex historical sales were main source of NFA accumulation and forecasted using oil price and production estimates.

*Source: tarea2024002 - IMF staff.*

### Preface.................................................................................................................

### PREFACE

### Mission and engagement
- An MCM Department/CAPTAC-DR technical assistance (TA) mission visited Guatemala City from June 12 to 16, 2023, at the request of Banco de Guatemala (Banguat) to assist in liquidity forecasting to calibrate daily deposit operations.
- The mission met with Mr. Vinicio Caceres (Banguat’s Financial Manager), Mr. Marco López (Director, DMFX), Mr. Ariel López (Deputy Director, DMFX), and Mr. Juan Antonio Ibañez (Deputy Director, DMAF).
- CAPTAC-DR donors funded the project under which this TA was delivered.

### Statistical component and operational context
- Banguat conducts daily open market operations offering 24-hour deposits to monetary policy counterparties (banks).
- Current operation characteristics and usage:
  - Recourse to the deposit facility averaged 20 percent of the daily auctioned amounts.
  - Use of the deposit facility represents one percent of the banks’ total position with Banguat.
  - There are currently 18 banks on operation; the mission provided forecasts for 16 banks with enough data.

### Reserve requirement and forecasting challenges
- The reserve requirement (RR) in Guatemala is contemporaneous with the base: every day banks must maintain an equivalent in reserves of 14.6 percent of the deposits that they have on the same day.
- Neither Banguat nor the banks know the exact requirement until the end of the day.
- Eligible RR fulfillment: banks’ balances at the central bank (excluding the deposit facility) are eligible for RR fulfillment, and up to 25 percent can be fulfilled with the cash vault.
- Under-fulfillment allowance: banks are allowed to under fulfill the reserve requirement for up to 14 days of a calendar month, provided the sum of deficiencies divided by 14 does not exceed 20 percent of the average monthly reserve requirement.
- Forecasting implications:
  - Banguat needs to forecast banks’ reserve needs to decide how much deposit to offer at daily deposit operations.
  - Banks need forecasts of the requirement to decide participation in Banguat’s daily deposit operations and the interbank market.

### Statistical forecasting framework deployed
- The MCMCO framework introduced includes 12 forecasting models of three types:
  - Exponential smoothing (simple, with exogenous regressors, and seasonal).
  - ARIMA (simple, with exogenous regressors, and seasonal).
  - TBATS and volatility models.
- The framework has out-of-sample performance testing using four performance criteria reflecting accuracy, bias, and confidence intervals.
- The framework can produce individual forecasts and combine them.

### Key findings on forecasting components
- Reserve requirement forecasting:
  - The reserve requirement is the main driver of banks’ demand for reserves and presents forecasting challenges because it is contemporaneous with its base.
  - The mission factored heterogeneity in the RR base by forecasting different banks’ deposits included in the RR base and reconciled forecasts (e.g., using OLS and MinT) to obtain the RR assuming a constant reserve requirement ratio.
  - Statistical reconciliation is more accurate than the aggregated forecast because it captures behavior of depositors.
- Reserve fulfillment (banks’ intra-month preferences):
  - Banks exhibit clear intra-month preferences for on-account amounts relative to the RR and for deposits into the deposit facility depending on the day of the month.
  - The mission forecasted demand for excess reserves for each bank and applied statistical reconciliation; reconciled-bank forecasts are more accurate than aggregate forecasts.
- Autonomous factors:
  - Banguat forecasts the government account (STA) at Banguat and Net Foreign Assets (NFA) more accurately than the MCMCO models, reflecting useful institutional information beyond pure statistical models.
  - The MCMCO framework forecasts Currency in Circulation (CiC) slightly better than Banguat’s forecasts due to more statistical model diversity.
  - The framework adds forecasts of Net Other Assets (NOA), Net Liquidity (NL) with reconciliation, and provides information on prediction uncertainty.
- Liquidity calibration:
  - A liquidity table combining demand and supply forecasts can be used to calibrate open market operations: autonomous factors determine system liquidity supply while RR and excess reserve forecasts determine banks’ demand; the neutral allotment is the difference between supply and demand.

### Policy and implementation recommendations (from Table 1)
- Liquidity Forecast
  - Complement Banguat’s bank survey with statistical models to produce a daily reserve requirement and demand for excess reserve forecasts (#6 and 57). — DMFX — Short term
  - Complement Banguat’s forecasts with statistical models to produce a daily forecast for autonomous factors (#7 and #37). — DMAF — Short term
  - Evaluate forecast performance periodically (#38). — DMFX & DMAF — Medium term
- Open Market Operations
  - Complement the survey of banks’ demand for daily deposit operations with a calibration based on forecasting autonomous factors and the demand for reserves (#37). — DMFX — Medium term
- Information to the Monetary Counterparties
  - Publish daily the forecasts of autonomous factors, the reserve requirement, and the demand for excess reserves for the next day (#42 & 43). — DMFX & DMAF — Long term
  - Publish daily the forecasts of autonomous factors, the reserve requirement, and the demand for excess reserves at the one-week horizon (#42 & 43). — DMFX & DMAF — Long term

- Timeframe definitions:
  - Short term: < 6 months; Medium term: 6 to 12 months; Long term: more than 12 months.

*Source: IMF staff.*

### 4. The Banguat calibrates its operations based on a daily survey of banks.

### 4. The Banguat calibrates its operations based on a daily survey of banks.

### Institutional forecasting and daily survey practice
- Banks daily inform the Banguat of the amounts of deposits that they would likely request at the auction via a survey.
- The Banguat announces an operation of a total size that is equivalent to the sum of banks’ net declared demand.
- The Banguat allots the same amount or less if the demand turns out lower than expected.

### Development of the statistical component of liquidity forecasting
- The mission assisted Banguat to develop statistical forecasts of balance-sheet components affecting liquidity to complement institutional information (e.g., weekly government expected outflows; contractual payments and income related to foreign-denominated debt).
- Statistical forecasts serve as a default benchmark and allow processing of time-series information extracted from data.
- The mission used MCMCO models to forecast autonomous factors and to statistically reconcile them into one liquidity forecast and compared forecast errors with Banguat’s models.

### Estimating the demand for reserves — overview
- Demand for reserves is decomposed between:
  - Regulatory demand arising from the reserve requirement (RR), defined as a percentage of the RR base (demand, saving, term, and other deposits).
  - Demand for excess reserves as reflected in banks’ preferred fulfillment profile.
- The mission forecasted:
  - The different deposits of the RR base (assuming a constant ratio) to predict RR; and
  - The demand for excess reserves of each bank.
- Statistical reconciliation techniques were used to obtain aggregated forecasts from disaggregated predictions.

### The Reserve Requirement — aggregated basis
- Historical characteristics:
  - Clear undamped upward trend with an average annual growth rate of 10 percent across the year.
  - Seasonality: RR appears significantly lower during Q2 and Q3.
- Model selection and performance:
  - ARIMA with regressor emerges as the best model to predict the reserve requirement across time horizon and performance metrics (RMSE, MEA, ME, MIS), with seasonal ARIMA performing better only for bias (ME).
  - The framework can present the best single forecast or combinations (best three models or average of all models).
- Forecast horizon example:
  - Daily forecast produced as of April 30, 2023, up to the 30 days horizon (approximately the under-fulfillment allowance period).

### By-deposit type approach (RR base)
- Method:
  - Forecast the different types of deposits (demand, term, saving, other) and apply the RR coefficient, 14.6 percent, to obtain the RR forecast.
- Descriptive findings:
  - Significant heterogeneity across deposit types:
    - Saving deposits more widely distributed in sizes than term deposits.
    - Term deposits tend to be larger; demand deposits bi-modal; other deposits smaller and concentrated.
- Selected models by deposit type (Average in GTQ million; Selected Model, RMSE Horizon = 30):
  - Monetary Deposits: 76,564.25 — ARIMA with Regression
  - Saving Deposits: 56,205.56 — TBATS
  - Term Deposits: 96,734.47 — ETS with Regression
  - Others: 801.71 — ARIMA with Regression
- Reconciliation and accuracy:
  - Model aggregation (reconciliation) improves forecast quality: RMSE at 1, 2, and 4-week horizons are lower for deposit-base reconciled RR forecasts than unreconciled aggregated forecasts.
  - Across specifications: by-deposit ARIMA with regression reconciled with OLS performs best at the 1-week horizon; deposit-base ARIMA with regression bottom up performs best for the 2 and 4-week horizons.
- RMSE (Million GTQ) — selected values from Table 3:
  - Unreconciled RR Aggregate (ARIMA with Regression): Week 1 = 148.90; Week 2 = 176.30; Week 4 = 219.10
  - Reconciled RR Aggregate by Base (ARIMA with Regression, OLS): Week 1 = 142.77; Week 2 = 163.65; Week 4 = 201.89
  - Reconciled RR Aggregate by Base (ARIMA with Regression, Bottom Up): Week 1 = 142.81; Week 2 = 163.43; Week 4 = 201.78
  - Reconciled RR Aggregate by Base (ARIMA with Regression, Min T): Week 1 = 143.93; Week 2 = 166.39; Week 4 = 204.52

### Reserve requirement fulfillment profile — aggregated basis
- Intra-month pattern:
  - Excess reserves show a clear monthly pattern: banks tend to front-load (exceed requirement early in the month) and then reduce unremunerated balances later (placing reserves in the deposit facility).
- Policy implication:
  - Banguat should accommodate excess reserve demand to stabilize short-term interest rates:
    - Reduce OMO allotment when banks keep more reserves early in the period to avoid low participation and rate increases.
    - Issue more at the end of the period to prevent downward pressure on daily deposit rates and increased recourse to the deposit facility.
- Model performance:
  - ARIMA with regressor is the best model across out-of-sample metrics for forecasting excess reserves, except bias (ME) where simple ARIMA performs better.
  - Example: 30-day forecast as of April 30, 2023, produced using ARIMA with Regression.

### Reserve requirement fulfillment profile — by bank
- Heterogeneity:
  - Significant heterogeneity across banks in average excess reserve behavior and volatility; some banks show wide and bimodal distributions while others keep lower, less volatile excess reserves.
  - Heterogeneity stems from each bank treasurer’s preferred fulfillment profile (front-loaded, linear, backloaded).
- Selected models for RR fulfillment by bank (Average RF Million GTQ, 2017-01-01 to 2023-04-30; Selected Model, RMSE Horizon = 30) — examples from Table 4:
  - Bank 1: Average RF 28.67 — Seasonal ARIMA
  - Bank 2: Average RF 44.40 — ARIMA with Regression
  - Bank 3: Average RF 51.87 — ARIMA with Regression
  - Bank 4: Average RF 17.63 — Seasonal ARIMA
  - Bank 6: Average RF 55.47 — TBATS
  - Bank 7: Average RF 7.22 — Seasonal ARIMA
  - Bank 8: Average RF 7.22 — TBATS
  - Bank 11: Average RF 1.65 — ARIMA with Regression
  - Bank 12: Average RF 0.36 — ETS
  - Bank 19: Average RF 1.33 — TBATS
- Model choice summary:
  - Four different models best forecast behaviors across 19 banks; TBATS performs best for 9 out of 19 banks.
- Reconciliation gains:
  - Reconciled by-bank forecasts improve RMSE at 1, 2, and 4-week horizons compared to aggregated unreconciled forecasts.
  - RMSE for RR Fulfillment Profile (Million GTQ) — Table 5:
    - Unreconciled RF Aggregated (ARIMA with Regression): Week 1 = 1464.5; Week 2 = 1516.0; Week 4 = 1521.8
    - Reconciled RF Aggregate by Bank (ARIMA with Regression, OLS): Week 1 = 1264.9; Week 2 = 1308.8; Week 4 = 1313.6
    - RF Reconciled by bank (ARIMA with Regression, Bottom up): Week 1 = 1368.8; Week 2 = 1423.2; Week 4 = 1442.9
    - RF Reconciled by bank (ARIMA with Regression, Min T): Week 1 = 1439.3; Week 2 = 1472.2; Week 4 = 1453.7

### Forecasting autonomous factors and net liquidity (NL)
- Objectives of adopting the Liquidity Forecasting Framework:
  - Benchmark Banguat’s current models against pure statistical estimations from MCMCO Framework.
  - Introduce reconciliation techniques to obtain an aggregated forecast of autonomous factors’ impact on NL and provide forecast uncertainty (confidence intervals).
- Comparative performance:
  - Banguat’s current models outperform pure statistical MCMCO forecasts for NFA and STA, benefiting from qualitative inputs (e.g., settlement dates of FX operations, prior notification of large government transactions).
  - T tests on absolute errors (Table 6) — p-values:
    - CiC: Unpaired = 0.11; Paired = 0.10
    - NFA: Unpaired = 0.12; Paired = 0.04*
    - STA: Unpaired = 0.00*; Paired = 0.00*
    - Note: * indicates significant difference at 5 percent.
- Extension:
  - The framework adds a forecast of Net Other Assets (NOA) to complete autonomous factors and reconciles the sum with a forecast of total NL.

### Currency in Circulation (CiC)
- Time-series features:
  - Clear upward trend and seasonal pattern.
  - Strong weekly seasonality with increasing demand towards the end of the week and seasonal jumps around year-end/new year.
- Model selection:
  - ARIMA with Regression selected for 1-week, 2-week, and 4-week forecasts.
  - Seven models tested including Naïve and Naïve seasonal; ARIMA with Regression and ETS with Regression ranked top in accuracy.
  - Regressors used include trigonometric terms, weekly seasonality, and holidays (New Year, Assumption Day, Revolution Day, All Saints Day, Good Friday, Christmas).
  - Selected ARIMA with Regression shows ME closer to 0 (relatively unbiased).

*Source: tarea2024002 - 4. The Banguat calibrates its operations based on a daily survey of banks.*

### 29. The NFA series is ascending before 2022 and then levels off with four historical

### 29. The NFA series is ascending before 2022 and then levels off with four historical

### NFA historical behavior and preprocessing
- The NFA series is ascending before 2022 and then levels off with four historical drastic jumps and drops.
- Four sharp changes in trend are attributed to activities such as Eurobond issuance and FX purchase (Figure 18).
- Structural-break dummy experiments were attempted but produced unsatisfactory forecasts.
- The mission removed these jumps and reconnected the series prior to inputting it into forecasting models.

### Model selection for NFA
- Volatility models (e.g., GARCH) were tested but did not outperform the simple Naïve benchmark.
- Times series models were therefore employed instead of volatility models.
- ETS with Regression is the recommended forecasting method for NFA (Figure 19).
  - Inclusion of regression improves predictive accuracy.
  - For NFA, only trigonometric seasonality is picked out as regressors.
  - Across three different horizons, ETS with Regression has the lowest averaged RMSE and closet-to-0 ME among all models, making it most accurate and unbiased.

### State Treasury Account (STA): seasonality and forecasting
- Monthly and quarterly seasonality is obvious in STA data; intraweek pattern is weak.
  - Mean within a week indicates deposits are higher midweek than at the beginning, suggesting possible intraweek seasonality (Figure 20).
- ARIMA with regression ranks best for STA in terms of predictive performance (Figure 21).
  - Fourier terms (to gauge multiple seasonalities) and weekly dummies were incorporated with both ARIMA and ETS models.
  - STA does not show solid jumps or drops during the New Year holiday periods but transitory drops with fluctuations (Figure 22); Fourier terms are more appropriate than holiday dummies.

### Net Other Assets (NOA): trend, seasonality, and model choice
- NOA series does not exhibit strong volatility; volatility models are not best fit.
- Before 2021, the NOA stock is on a downward trend but starts to climb after 2021 (Figure 23).
- Intraweek seasonality is not evident, but monthly and quarterly patterns exist.
- A more parsimonious ARIMA model was selected to forecast NOA.
  - ARIMA performs better for 1 and 4-week horizons.
  - Seasonal ARIMA and ARIMA with Regression follow but only produce more accurate forecasts on average at the two-week horizon.
  - Statistical tests showed the simpler ARIMA has no significant differences from more complex models; thus ARIMA is recommended for NOA.

### Net Liquidity (NL): construction, models, and reconciliation
- NL is constructed as NL = NFA + NOA – CiC – STA.
- TBATS is selected based on lowest RMSE score for NL forecasts.
  - For 1 and 4-week horizons, TBATS produces lower error on average.
  - ARIMA with Regression ranks better for the two-week horizon (Figure 25).
  - If MAE is examined, TBATS is constantly the best; RMSE sensitivity to outliers may explain differences in ranking.
- Forecast reconciliation:
  - Forecasts for autonomous factors and NL are reconciled to produce a more coherent NL forecast.
  - OLS is recommended for longer forecast horizons and Bottom-Up for one-week forecast.
  - Table 7 (Predictive Accuracy and Bias of Forecasting Models for NL) reports exact scores (Million GTQ):
    - OLS: RMSE +1 week 1873.8, +2 weeks 2214.0, +4 weeks 2597.9; MAE +1 week 1461.6, +2 weeks 1795.7, +4 weeks 2166.1; ME +1 week 65.0, +2 weeks -10.3, +4 weeks -56.7
    - Base (Unreconciled): RMSE +1 week 1959.7, +2 weeks 2312.7, +4 weeks 2687.9; MAE +1 week 1553.6, +2 weeks 1850.7, +4 weeks 2210.4; ME +1 week 110.0, +2 weeks 62.6, +4 weeks -66.8
    - Bottom Up: RMSE +1 week 1814.8, +2 weeks 2300.0, +4 weeks 2732.1; MAE +1 week 1378.6, +2 weeks 1827.0, +4 weeks 2244.9; ME +1 week -114.8, +2 weeks -301.9, +4 weeks -550.9
    - MinT: RMSE +1 week 2135.9, +2 weeks 2574.2, +4 weeks 2952.2; MAE +1 week 1704.6, +2 weeks 2021.2, +4 weeks 2382.9; ME +1 week 145.7, +2 weeks 180.1, +4 weeks 286.2
  - For longer horizons (two and four weeks), OLS provides gains over unreconciled NL forecasts.
  - For one-week horizon, Bottom-Up (forecast components separately then aggregate) is sufficient.
  - OLS has lower ME scores across all horizons, indicating lower bias.

### Operational recommendations and implementation
- Banguat should complement institutional forecasts and the bank’s survey with the proposed statistical models to produce forecasts for autonomous factors and the RR and to calibrate daily auctions.
  - The mission provided Banguat with codes to estimate all models for each variable, the reconciliation for liquidity, and RR.
  - With the codes, Banguat staff can test model performance and do out-of-sample forecast evaluation to select models.
- Banguat should periodically evaluate selected models (e.g., quarterly, semiannually) because the best model can change due to new information or structural changes.
  - MCM can provide technical support.

### Liquidity table and calibration
- The liquidity table summarizes reserves available and demand for reserves to calibrate daily deposit operations.
  - The first four lines under Autonomous Factors determine reserves available at each forecasted date.
  - Under reserve requirement: predicted demand due to regulation (the RR) plus banks’ preference regarding reserve requirement fulfillment.
  - Neutral allotment = available reserves (from autonomous factors) minus how much banks want to keep for regulatory reasons or predictable excess reserve preferences.
- Facilities are expected to capture forecast errors under neutral allotment; ex-ante OMOs are calibrated to leave no excess or shortage, so recourse to facilities should be null unless forecast errors or market frictions occur.
- Liquidity tables can be prepared in flows (focused) or levels (comprehensive); levels translate full central bank balance sheet but are more cumbersome.
- Suggested items to publish to inform banks’ bidding:
  - Opening banks’ balances at Banguat.
  - Liquidity forecast including:
    - The total autonomous factors.
    - The daily reserve requirement objective.
    - The demand for excess reserves.
- Banguat should gradually increase the published forecast horizon once forecast quality is vetted.
  - Start with publishing forecast for the next day and consider publishing 1-week horizon when quality is sufficient.
- Table 8 (illustrative liquidity table) contains actuals and forecasts with specific line items and example flows (values preserved in original formatting).

### Appendix I — statistical methods summary
- Four time series model types in the MCMCO Liquidity Forecasting Framework: Naïve, Exponential Smoothing (ETS), ARIMA, and TBATS.
- Naïve model: random walk benchmark with seasonal counterpart; forecasts defined as ŷ(t+h) = y(t) or seasonal ŷ(t+h) = y(t−s+h), where s is seasonal period.
- Forecasting separates structure μi and noise εi: yi = μi + εi with εi ~ N(0, σ2).
- ETS models decompose time series into level, trend, and seasonality within a state-space formulation; smoothing parameter α (0<α<1) updates level via α times last error, producing exponentially weighted moving averages.

*Source: Banguat and staff calculation; IMF staff calculation.*

### 6. The smoothing parameter for each component defines how reactive that component

### 6. The smoothing parameter for each component defines how reactive that component is to new information

### Interpretation of smoothing parameters
- A smoothing parameter value of 0 suggests that the component (e.g., level) is not updated at all by the observed data.
- A smoothing parameter value of 1 suggests that the component is fully updated by the last observation and does not retain any underlying structure.
- Low smoothing parameters can be interpreted as long-weighted moving averages that are resilient to increased noise and outliers.
- High smoothing parameters produce very reactive components.

### Numerical optimization and loss functions
- Numerical optimization is typically preferable to manual parameter setting, especially for models with more parameters.
- Appropriate loss function: typically based on quadratic errors.
- Quadratic errors are summarized in the Mean Squared Error (MSE), which tracks the mean of a time series.
- Numerical optimization provides smoothing parameters—one corresponding to each state of the model—that minimize the number of errors for the in-sample data used to fit the model.

### Model selection and information criteria
- The appropriate exponential smoothing model form can be identified using an information criterion such as the Akaike Information Criterion (AIC).
- Intuition: information criteria balance model fit against complexity (number of parameters).
- Underfitting: model without appropriate states (level, slope, seasonality) will provide poor forecasts.
- Overfitting: overly complex models can model noise and produce substantially inaccurate forecasts.
- Simplified AIC expression from the source:
  - 퐴퐴퐴퐴퐴퐴  = 2
    √
    푀푀푀푀푀푀  + 2푘푘, 
    - where 푘푘 is the number of model parameters.
  - For exponential smoothing, k is connected with the number of states/components in the model.
- The model with the lowest AIC is preferable because it balances fit and complexity and results in selecting models that can forecast well.

### Including regressors
- Regressors can be included by augmenting the model description of 훍훍
  풕풕 in the same fashion as conventional regression modelling.
- Additional details referenced: Hyndman et al. (2008) and Ord, Fildes, and Kourentzes (2017).

### ARIMA and SARIMA: definitions and estimation
- General ARIMA form from the source:
  - (1−휙휙(퐵퐵))(1−퐵퐵)
    푑푑
    푦푦
    푡푡
    =(1 +휃휃(퐵퐵))휖휖
    푡푡
- Backshift operator 퐵퐵: 퐵퐵푦푦
  푡푡 = 푦푦
  푡푡−1, 퐵퐵푦푦
  푡푡 = 푦푦
  푡푡−2, etc.
- Differencing order 푑푑: typically equal to 1 (or in rare cases 2) for nonstationary series and 0 for stationary series.
- AR polynomial: �1−휙휙(퐵퐵)�=1−휙휙
  1
  퐵퐵−휙휙
  2
  퐵퐵
  2
  −⋯−휙휙
  푝푝
  퐵퐵
  푝푝
- MA polynomial: �1 +휃휃(퐵퐵)�=1 +휃휃
  1
  퐵퐵+휃휃
  2
  퐵퐵
  2
  +⋯+휃휃
  푝푝
  퐵퐵
  푝푝
- Nomenclature: ARIMA(p,d,q). Example: ARIMA(2,1,2) for 푠푠=2, 푑푑=1, 푞푞=2.
- Steps for ARIMA order selection (stepwise algorithm of Hyndman and Khandakar (2008)):
  i. Find 푑푑 using the KPSS (Kwiatkowski–Phillips–Schmidt–Shi) test.
  ii. Estimating four initial models and choose the best.
  iii. Expand the candidate model set by considering models that have 푠푠 or 푞푞 differing from the current best by 1.
  iv. Iterate until no improvement is made.
- Selection criterion: Akaike Information Criterion corrected for small sample size (AICc).
- The algorithm is implemented in the auto.arima function within the forecast package in the R software environment.
- SARIMA extension for seasonal patterns of length 푚푚:
  - (1−휙휙(퐵퐵))(1−훷훷(퐵퐵
    푚푚
    ))(1−퐵퐵)
    푑푑
    (1−퐵퐵
    푚푚
    )
    퐷퐷
    푦푦
    푡푡
    =(1 +휃휃(퐵퐵))(1 +휃휃(퐵퐵
    푚푚
    ))휖휖
    푡푡
  - Nomenclature: ARIMA(p,d,q)(P,D,Q)[m]. Example: ARIMA(1,0,0)(0,1,1)[5] example equivalence:
    - 푦푦
      푡푡 = 푦푦
      푡푡−5 + 휙휙(푦푦
      푡푡−1 − 푦푦
      푡푡−6) + 휖휖
      푡푡 − 휖휖
      푡푡−5
  - SARIMA models explicitly capture one form of seasonality; covariates can be incorporated by extending the equation as in regression.

### Seasonality encoding: indicator variables and multiple seasonalities
- Indicator (dummy) variables are well suited when seasonal length is short and patterns are not smooth.
  - Example: flexible day-of-week effects can be modelled using variables such as:
    - 퐷퐷
      푡푡
      (푆푆푆푆푆푆) = 1 if day t is a Sunday, 0 otherwise.
  - Similar dummies defined for Mon, Tue, Wed, Thur.
  - Indicators included in a covariate vector 푥푥′
    푡푡, and ARIMA specified with 푦푦
    푡푡 replaced by 푦푦
    푡푡 − 푥푥′
    푡푡
    훽훽; similar modification for ETS.
- Structural breaks encoded with continuous indicator:
  - D
    푡푡 = 1 if t occurs after the structural break, 0 otherwise.

- Daily series can exhibit multiple seasonal cycles: day in the week, day in the month, day in the year.
  - Many models typically incorporate a single seasonal periodicity; multiple seasonalities complicate forecasting.
  - For day-in-month seasonality variability, quarterly seasonality can be used since a quarter contains a fixed number of weeks and days.

### Trigonometric encoding of multiple seasonal cycles
- For a season of length 푠푠 periods, construct 푠푠/2 pairs of trigonometric variables with i=1,...,푠푠/2:
  - 푑푑
    푖푖 = 푎푎푠푠푠푠
      �2푖푖π푡푡
      푠푠
      �,
    - 푑푑
      푖푖+푠푠/2 = 푠푠푖푖푛푛
      �2푖푖푖푖푡푡
      푠푠
      �,
  - where 푡푡 = 1,...,푛푛 (푛푛 sample size).
  - When 푠푠 is odd, 푠푠/2 is rounded up to the closest integer.
- Trigonometric encoding is mathematically equivalent to 푠푠 binary indicator variables; in some cases one indicator corresponds to a constant, resulting in 푠푠−1 informative indicators.
- Advantage: trigonometric representation can encode complex seasonal effects (e.g., leap years) which binary encoding cannot.
- Example numeric: for day in the year in a five-day week year with 260 days, use 260/2 cosines and 260/2 sines.

### Parsimony and selection of seasonal indicators
- To obtain parsimony, filter redundant seasonal indicators:
  - Remove trend using a centered moving average (Ord, Fildes, and Kourentzes 2017).
  - Subtract trend, model residuals with different trigonometric indicators via regression.
  - Use lasso regression to eliminate less informative inputs and obtain a sparse representation; lasso balances fit against complexity (number of parameters).
- Binary encoding vs trigonometric when eliminating terms:
  - Binary encoding omits all seasonal information for that period.
  - Trigonometric encoding provides a smoother approximation of the seasonal profile.

### TBATS model and volatility models
- TBATS model features:
  - Handles seasonality and trend via exponential smoothing (using trigonometric terms for seasonality).
  - Uses Box-Cox transformation.
  - Incorporates ARIMA innovations.
  - Allows seasonality to change over time and can handle multiple calendars.
- Volatility models appropriate for high-volatility series (typically used for Net Foreign Assets, though times series models may be more suitable in specific missions).
- Conditional volatility models overview:
  - GARCH variance specification from the source:
    - 휎휎
      푡푡
      2 = 휔휔 + �훽훽
      푖푖
      푝푝
      푖푖=1
      휎휎
      푡푡−푖푖
      2 + �훼훼
      푖푖
      푝푝
      푖푖=1
      푙푙
      푡푡−푖푖
      2
  - eGARCH specification:
    - log휎휎
      푡푡
      2 = 휔휔 + �훽훽
      푖푖
      푝푝
      푖푖=1
      log휎휎
      푡푡−푖푖
      2 + �훼훼
      푖푖
      푞푞
      푖푖=1
      푔푔(휖휖
      푡푡−푖푖)
    - 푔푔(휖휖
      푡푡) = 휃휃휖휖
      푡푡 + 휆휆�|휖휖
      푡푡| − 푀푀(|휖휖
      푡푡|)
      � (asymmetric effects of sign and magnitude)
  - GJR-GARCH specification:
    - 휎휎
      푡푡
      2 = 휔휔 + 훿훿휎휎
      푡푡−1
      2 + 훼훼휖휖
      푡푡−1
      2 + 휙휙휖휖
      푡푡−1
      2
      퐴퐴
      푡푡−1
    - 퐴퐴
      푡푡−1 = 0 if 휖휖
      푡푡−1 ≥ 0 and 퐴퐴
      푡푡−1 = 1 if 휖휖
      푡푡−1 < 0.
    - Allows for asymmetric effects like eGARCH.

### Forecast aggregation and reconciliation
- Forecasting approaches for net liquidity (net foreign assets, currency in circulation, and state account balance):
  - Bottom-up: add up forecasts of autonomous factors.
  - Direct forecasting: develop a model for the total of autonomous factors.
  - Hybrid: forecast each autonomous factor and the total to hedge against model misspecification.
- Downside of forecasting components separately: forecasts may not add up correctly.
- OLS reconciliation (to force additivity):
  - Let vector of four point forecasts that do not add up be given by 푦푦� and summing matrix 푀푀 =
    �
    1 1 1
    1 0 0
    0 1 0
    0 0 1
    �
  - Reconciled forecasts guaranteed to add up:
    - 푦푦� = 푀푀(푀푀′푀푀)
      −1
      푀푀′푦푦�
  - Referred to as OLS reconciliation.
- MinT method (exploits correlation between forecast errors) — preferable:
  - 푦푦� = 푀푀(푀푀′ 훴훴
    −1
    푀푀)
    −1
    푀푀′ 훴훴
    −1
    푦푦�
  - 훴훴 is the covariance matrix of one-step-ahead forecasting errors.

*Source: IMF staff computations from the provided content.*

### 10. The CCB does a forecast with a demand-supply approach for the RR period. The

### 10. The CCB does a forecast with a demand-supply approach for the RR period. The

### CCB overall forecast approach
- The CCB obtains a projection of the NL by subtracting the expected liquidity demand from the expected liquidity supply.
- The forecast is done with a demand-supply approach for the RR period.

### Regulatory demand for liquidity (CCB)
- The CCB forecasts the monetary base using time series models conditioned to seasonal factors.
- Each institution’s RR is estimated with the deposits for the base period applying the relevant RR coefficient: 0.11 or 0.45.
- Individual RR forecasts are aggregated to project the system’s demand.
- The CCB incorporates an estimate of the precautionary level of reserves based on historical over-compliance for the RR.

### Liquidity supply forecast (CCB)
- The CCB forecasts liquidity supply for the RR period and includes:
  - Approved permanent monetary flows (government securities purchases and sales).
  - FX operations.
  - The government’s projected cash flow.
  - CCB net income statement flows that affect liquidity (interest payments/collections on monetary policy operations, payroll, operative expenses, among others).
  - Previous OMOs’ maturity to project liquidity before new OMOs.

### Publication and OMO announcement practice (CCB)
- The CCB does not publish either forecast but announces the daily OMO amount for the following day, close to the average approved amount.
- Calibration incorporates data points collected from OMOs’ market participants.
- The CCB constantly communicates with the government about near-term expected flows on the STA.

### Bank of Mexico (BOM) institutional and operational framework
- BOM formally adopted an IT regime in 2001 and transitioned to an interest rate target in January 2008.
- Current monetary policy target: 11.25 percent.
- BOM uses liquidity forecasts to calibrate OMOs and manage short- and medium-term liquidity.
- BOM follows an active sterilization strategy, withdrawing excess liquidity via OMOs in both short-term and medium-to-long-term operations.
- BOM steers the market’s overnight interest rate towards the policy target through OMOs.

### One-day horizon operational details (BOM)
- Neutral liquidity on a one-day horizon implies an aggregate zero balance of banks’ accounts (no reserve requirement).
- Positive balances are not remunerated; overdrafts are charged twice the market’s overnight rate.
- Daily operations are calibrated based on a one-day forecast to offset expected daily liquidity movement.
- BOM auctions a predetermined amount; banks bid interest rates with multiple price allotment.
- Liquidity injection: collateralized credit operations (overnight up to an average of 30 days) with the policy rate as the minimum accepted bid rate.
- Liquidity absorption: overnight deposit auctions with the policy rate as the maximum acceptance rate.
- Institutional arrangements provide a high degree of certainty about each day’s autonomous factors at session opening:
  - FX operations affecting liquidity are settled on a T+2 basis.
  - BOM law requires a one-day preannouncement for credits and debits on the Sigle Treasury Account (STA).
- BOM conducts a fine-tuning auction before the closure of the payments system to address forecast errors.
- An in-house developed system expedites the daily forecast calculation.
- BOM publishes on its website a daily liquidity forecast before the morning auctions (around 7:30 a.m.), including the expected daily liquidity movement and the day’s net OMOs amount to achieve a balance.

### Medium and medium-to-long-term sterilization and forecasting (BOM)
- BOM pre-sterilizes excess liquidity for the medium horizon to achieve a net creditor position in the money market.
- Banks must participate in BOM’s liquidity-providing OMOs if the system has a short-term liquidity shortage; participation in deposit-taking OMOs is not mandatory when the system has excess liquidity.
- Quarterly, BOM decides and announces its medium-term sterilization policy.
- BOM calibrates medium-to-long-term liquidity operations based on a liquidity forecast for up to two calendar years.
- Forecasting process:
  - Start with an annual forecast for each liquidity driver.
  - Break down the annual forecast into monthly and daily forecasts using each factor’s detailed information (seasonality, rules, predetermined arrangement).
  - Monthly projections are constantly updated with the latest daily information.

### BOM forecasting of autonomous factors
- CiC:
  - Start with an annual flow forecasted with econometric models considering expected values for macroeconomic variables (GDP growth, inflation, interest rates).
  - Annual flow is distributed into monthly and daily flows using statistical models that consider money demand’s seasonality across the year and the week and extraordinary events.
- STA:
  - Annual flows are forecasted by category, income, expenses, and debt, linked to Congress’s approved figures.
  - After Congress approves the budget, the Ministry of Finances publishes an expected monthly calendar for each category; BOM uses historical information for monthly distribution.
  - Daily distribution is based on historical data and operational rules (e.g., income taxes are due after the seventeen each month).
  - BOM maintains constant communication with the national treasury about expected flows.
- Net Foreign Assets (NFA):
  - BOM’s FX interventions are usually under pre-announced programs with known rules to accumulate reserves or sell dollars; expected annual flows from such programs are incorporated into liquidity forecasts.
  - Monthly and daily breakdowns can be incorporated according to the program rules and FX historical volatility.
  - Historically, Pemex’s sales to BOM were the main source of NFA accumulation.
  - Pemex’s domestic currency needs are forecasted with estimations for oil prices and production volume, along with the tax rate applied.

*IMF | Guatemala  The Statistical Component of Liquidity Forecasting | 49–51*

---


_Source: https://www.imf.org/-/media/files/publications/tar/2024/english/tarea2024002.pdf_