## Liquidity Forecasting—Part II: The Statistical Component ## Source details **Canonical URL:** [Liquidity Forecasting—Part II: The Statistical Component](https://www.imf.org/-/media/files/publications/miscellaneous/english/2022/mcm-technical-assistance-handbook/liquidity-forecasting-part-ii.pdf) ## Other formats - [Markdown version](/-/media/files/publications/miscellaneous/english/2022/mcm-technical-assistance-handbook/liquidity-forecasting-part-ii.pdf.md) - [Structured JSON version](/-/media/files/publications/miscellaneous/english/2022/mcm-technical-assistance-handbook/liquidity-forecasting-part-ii.pdf.json) --- ### Executive summary: purpose and scope - Purpose: provide outlooks on changes in liquidity‑impacting items of central banks' balance sheets to calibrate open market operations (OMO) and control short‑term interest rates or reserve money targets. - Rationale: statistical forecasts are necessary when qualitative information from counterparties is unavailable or insufficiently accurate. - Autonomous factors requiring statistical forecasting: Currency in Circulation (CIC), Net flows in the Government Account Balance (GAB), Net Foreign Assets (NFA). - Framework goals: - Cross‑validate a family of time series models and identify best‑performing models. - Allow generic yet flexible country‑specific customizations. - Employ reconciliation techniques (hierarchical forecasting) to improve aggregate liquidity supply estimation. - Recommendation: forecast the full predictive distribution (not only the mean) and, where appropriate, target a percentile reflecting risk preference; if forecasts are published, publish the predictive distribution as well. ### Core problem and operational context - Central banks steer short‑term interbank rates by controlling reserves; the net of balance sheet items determines OMO amounts. - Principal liquidity impacts (stylized): - Assets increasing liquidity: Net Foreign Assets (NFA), OMO: Liquidity Supply, Lending Facility, Commercial Banks’ Required Reserves, Commercial Banks’ Excess Reserves. - Liabilities decreasing liquidity: Currency in Circulation (CIC), Net Government Account Balance (GAB), OMO: Liquidity Drain, Deposit Facility. - Aggregate relation (as used): AGG = NFA − CIC − GAB (following Equation (1)/(2) notation in source). - Operational timing: daily data are preferred given central bank operations and OMO schedules; forecast horizons should align with OMO timing (examples: 7-day or 14-day horizons, or 5-day/10-day if weekends are skipped). ### Characteristics of autonomous factors and empirical patterns - Common properties: - All time series (CIC, NFA, GAB, AGG) are non-stationary. - Daily series can exhibit multiple seasonal patterns: day of the week, day of the month, payday, day of the year. - Structural breaks and special events (e.g., COVID-19) can induce permanent or transient level shifts. - Factor-specific behaviors and implications: - CIC: - Canonical time series patterns; amenable to time series models. - Affected by holidays and calendar effects but generally smaller effects relative to GAB and NFA. - NFA: - Mix of standard components and exogenous effects; unsterilized FX intervention is a major liquidity driver. - Rarely exhibits seasonality; mean often follows a random walk while first differences show conditional heteroskedasticity. - GAB: - Can exhibit strong seasonal patterns in expenditures and revenues (e.g., biweekly payrolls, quarterly tax collections). - Considered the most challenging series to forecast by central bankers. ### Proposed statistical framework and methodological features - Objective: generate daily forecasts for all autonomous factors and aggregate liquidity (AGG). - Key elements: - Unified approach across factors to reduce modeling overheads. - Automatic model calibration and selection via cross‑validation of a family of time series models. - Detailed reporting to support analyst interventions. - Hierarchical (reconciliation) forecasting to improve aggregate forecasts. - Flexibility for country‑specific adjustments and operational conditions. - Intended users: central bank staff with strong quantitative background. ### Modeling toolkit: model families and practical encodings - Benchmark and baseline: - Naive method (random walk): Ŝ_t+h = S_t. Seasonal naive: Ŝ_t+h = S_{t−s+h} for seasonal period s. Serves as simple benchmark and upper bound of acceptable forecasting errors. - ETS family: - Decomposes series into level, trend, seasonality within a state‑space framework; up to 30 ETS models exist (selection typically restricted due to potential instability). - Additive level example: level_t = level_{t−1} + α⋅N_{t−1} with α ∈ [0,1]. - ETS can include exogenous regressors. - ARIMA / SARIMA / ARIMAX (RegARIMA): - ARIMA(p,d,q); seasonal SARIMA(p,d,q)(P,D,Q)[m]; ARIMAX includes regressors (holidays, paydays). - Identification follows Hyndman and Khandakar (2008): differencing for stationarity, local search over p and q, selection via AIC. - Relationship: purely additive ETS and ARIMA can be algebraically equivalent in some cases. - TBATS: - Handles multiple seasonalities via trigonometric terms, Box‑Cox, ARMA errors; suitable for multiple seasonal cycles without special treatment. - Multiple seasonalities encoding: - Binary indicators for day‑of‑week; trigonometric (Fourier) encoding for longer seasonalities. - Practical recommendation: use binary encoding for day‑of‑week seasonality; trigonometric encoding (with sparsity via lasso) for other seasonalities. - Preprocess by removing trend using a centered moving average before fitting trigonometric terms. - Level shifts and calendar events: - Encode permanent/transient shifts via binary indicators; parabolic indicators for leading/trailing holiday effects: D_t = max(0, 1 − ((t − t_holiday)/7)^2). - Use regressors parsimoniously; evaluate inclusion using lasso or stepwise AIC. - Volatility models (for NFA and highly volatile series): - GARCH family (standard GARCH, eGARCH, gjr‑GARCH) to capture conditional heteroskedasticity. - Standard GARCH variance: σ_t^2 = ω + Σ_{i=1}^p β_i σ_{t−i}^2 + Σ_{i=1}^p α_i N_{t−i}^2. - eGARCH: log σ_t^2 = ω + Σ β_i log σ_{t−i}^2 + Σ α_i g(ε_{t−i}) with g(ε) = θ ε + λ(|ε| − E|ε|). - gjr‑GARCH: σ_t^2 = ω + δ σ_{t−1}^2 + α ε_{t−1}^2 + φ ε_{t−1}^2 C_{t−1}, where C_{t−1}=0 if ε_{t−1}≥0 else 1. - Procedure: fit several volatility models to obtain probabilistic forecasts and convert to point forecasts via bootstrapping simulation. - Model tiers and complexity ordering: - Tier 1: level and seasonal random walks (benchmarks). - Tier 2: ETS, ARIMA, TBATS (automatic). - Tier 3: ETS and ARIMA augmented with regressors (higher complexity). ### Hierarchical reconciliation methods (aggregation coherence) - Main aggregation: AGG = NFA − CIC − GAB. - Approaches: - Bottom‑up: forecast disaggregates (NFA, CIC, GAB) and aggregate. - Direct: forecast AGG directly. - Reconciliation concept: - Separate forecasts generally do not satisfy AGG identity; reconciliation error Ñ_t captures disagreement. - Reconciliation can reduce total forecast errors by blending information across hierarchy. - Matrix formulation: - Bottom vector b_t = (CIC, GAB, NFA)'; full vector y_t = (AGG, b_t)' with y_t = S b_t. - Summing matrix example: S = [−1 −1 1; 1 0 0; 0 1 0; 0 0 1]. - Optimal reconciliation (minimizing reconciliation errors): ŷ_t^r = S (S' W^{−1} S)^{−1} S' W^{−1} ŷ_t, where W is forecast error variance‑covariance matrix. - Practical W approximations: - (i) OLS: W = I. - (ii) STR (structural): W diagonal with row‑wise sum of S. - (iii) WLS: W diagonal with each element = MSE of corresponding forecasting model. - (iv) MinT: estimate full variance‑covariance matrix with shrinkage of off‑diagonals. - Benchmarks for reconciled AGG: - Direct forecast of AGG (unreconciled). - Direct aggregation from the three autonomous factors (bottom‑up). ### Forecast evaluation, selection, and probabilistic scoring - Point forecast metrics: - RMSE = sqrt{ (1/n) Σ_{i=1}^n (S_i − Ŝ_i)^2 } (preferred for mean). - MAE = (1/n) Σ_{i=1}^n |S_i − Ŝ_i| (preferred for median, robust to extremes). - ME = (1/n) Σ_{i=1}^n (S_i − Ŝ_i) (bias). - Scaled metrics: divide by in‑sample RMSE or MAE of the random walk when needed. - Probabilistic metrics: - Recommend forecasting full predictive distribution and publishing it. - Mean Interval Score (MIS) for prediction intervals (95 percent intervals preferred): - MIS = (1/n) Σ_{i=1}^n [ (U − L) + (2/α) (L − Y_i) 1_{Y_i < L} + (2/α) (Y_i − U) 1_{Y_i > U} ] with α the significance level. - Pinball loss for specific quantiles. - Evaluation scheme: - Rolling origin out‑of‑sample cross‑validation: use q observations in‑sample, generate h‑step‑ahead forecasts, expand in‑sample by one period and repeat to obtain S − h + 1 rolling forecasts. - Prefer simpler models when errors are similar. - Statistical testing for multiple comparisons: - Friedman test to detect differences among methods. - Post‑hoc Nemenyi test to group methods by mean rank and a critical distance; choose simplest within best group. - Heuristics for analysts: - For mean forecasts consult RMSE, then MAE, then ME. - For predictive distributions consult MIS first. - Weight metrics: RMSE > MAE > other metrics; use Friedman/Nemenyi to support selection. ### The forecasting process: combining statistical models and expert judgment - Framework: baseline statistical forecasts + expert adjustments from stakeholder intelligence. - Best practices for expert judgment: - Evaluate expert adjustments separately to provide feedback. - Limit expert input to unstructured information not easily encoded as regressors. - Decompose expert adjustments into constituent factors so adjustments are traceable. - Implementation and automation: - Toolkit implemented in R; analysts prepare data, codify holidays/level shifts, run model automations, obtain rolling origin errors, generate reconciled forecasts, and produce reports. - Reconfiguration of models not required every cycle but recommended periodically. - Reporting and visual workflow: - Process schematic: statistical forecasts + historical liquidity + known events + trend/season patterns + expert input → evaluation → final forecast. - Workflow: data preparation → model automations → model selection → forecasting → reconciliation → final output. ### Limitations, extensions, and machine learning stance - Forecast pooling: - Static and dynamic forecast combinations are supported but optional; hierarchical reconciliation already provides combination benefits. - Dynamic combinations were useful during COVID‑19 when models failed in complementary ways. - Machine learning / AI: - Toolkit does not automate ML/AI; trials with shallow neural networks and random forests showed no consistent forecasting gains relative to cost. - Global forecasting methods (training across countries) show promise but raise data collection challenges. - Temporal hierarchies: - Temporal Hierarchies and cross‑temporal reconciliation can bridge short‑ and long‑term forecasting and are useful extensions. ### Technical Assistance (TA) approach, mission timeline, and priorities - Recommended TA timeline: - T‑6 (≥ six weeks before mission): - Request at least three years of daily central bank balance sheet data converted into a liquidity table. - Share software (R and RStudio) and staffing requirements. - Prepare data, run models, and a PowerPoint covering: descriptive analysis and model configuration; model selection statistics; point forecasts with predictive intervals; reconciliation enhancements; alignment with liquidity table. - T‑2: - Present standardized results virtually; Q&A; prepare participants for in‑person workshop. - T‑0 (mission start): - Workshop: set up packages, R coding essentials, customize model configurations, run forecasting codes, interpret results, align with liquidity table and OMO calibration, discuss qualitative inputs and scenarios. - Share draft report; participants to provide comments by end of mission. - T+1 to T+3: - Circulate draft for internal review; finalize report incorporating comments and aide‑mémoire. - TA priorities: - Skill development via hands‑on workshops using central bank data and mission code. - Ensure central bank staff can produce forecasts independently before mission end. - Common challenges and solutions: 1. Inter/intra‑departmental data sharing: assess availability before mission; consider rescheduling or flexible horizons if data delayed. 2. IT/hardware restrictions: coordinate with IT to install RStudio and packages in advance. 3. Package version updates causing bugs: advise forecasters not to update packages related to the forecasting exercise; mission team may need to update code if package changes occur. - TA advocacy: - Encourage publishing forecasts only if forecast uncertainty (predictive distributions) is disclosed. ### Appendix II: standardized output and example metrics - Time series descriptive report example (CIC): - Period used: 2022-12-29 to 2023-07-20. - Data unit: millions of Guatemalan quetzals (GTQ). - Seasonal profiles plotted as distributions showing minimum, 10 percent, 25 percent, median (50 percent), 75 percent, 90 percent, maximum. - Cross‑validation and comparison report example (CIC): - Models fit over 2022-12-29 to 2023-07-20; GTQ millions. - Accuracy metrics reported up to one, two, and four weeks ahead: RMSE, MAE; bias: ME. - Predictive distribution evaluation: MIS for 95 percent intervals. - Statistical testing: Friedman and Nemenyi at 95 percent significance; models ranked by mean rank. - Forecasting results example (AGG): - Forecast origin date: 2023-08-31. - Selected model for AGG in example: ARIMA with Regression. - Forecast plots show 80 percent (darker) and 95 percent (lighter) prediction intervals. - Model averages: equally weighted average provided; all models’ forecasts shown. - Reconciliation example: - Models fit period: 2023-01-05 to 2023-07-27. - Selected models (configurable): - State Account Balance: ETS with Regression. - Net Foreign Accounts: ETS with Regression. - Net Other Assets: ARIMA with Regression. - Currency in Circulation: ARIMA with Regression. - Aggregate: ARIMA with Regression. - Forecast origin: 2023-08-31; one‑step‑ahead forecasts tabulated reconciled vs unreconciled. - Method selected in example: STR (shaded); best past‑performance method by RMSE: OLS (highlighted). - Reconciliation comparisons reported across RMSE, MAE, ME, MIS with STR chosen in the report and OLS/MinT/Base identified as best on different metrics. - Enhanced AGG forecasts: - Changes in Net Liquidity provided for one, two, and four weeks ahead; plotted using chosen method (STR). *Source: Liquidity Forecasting—Part II: The Statistical Component (IMF | MCM Technical Assistance Handbook)* ### Part II: The Statistical ### Part II: The Statistical Component ### Executive summary: purpose and scope - Liquidity forecasting provides an outlook on changes in liquidity‑impacting items of central banks' balance sheets to calibrate open market operations (OMO) and control short‑term interest rates or reserve money targets. - Statistical forecasts are necessary when qualitative information from counterparties is unavailable or insufficiently accurate. - Some liquidity factors are predetermined or known in advance by central banks; others require systematic statistical forecasting—most notably: - Currency in Circulation (CIC) - Net flows in the Government Account Balance (GAB) - Net Foreign Assets (NFA) - The handbook proposes a framework to: - Cross‑validate a family of time series models and identify best‑performing models for forecasts. - Allow generic yet flexible country‑specific customizations. - Employ reconciliation techniques (hierarchical forecasting) to improve aggregate liquidity supply estimation. - Recommendation: central banks should forecast the full predictive distribution (not only the mean point forecast) and, where appropriate, target a percentile of that distribution that reflects their risk preference. If forecasts are published, publish the predictive distribution as well. ### Core problem and operational context - Central banks steer short‑term interbank rates by controlling reserves provided to commercial banks; the net of changes in balance sheet items determines the amount to absorb or inject through OMO. - Table 1 (Stylized Central Bank Balance Sheet) highlights principal items affecting liquidity, with signs indicating their liquidity impact: - Assets increasing liquidity: Net Foreign Assets (NFA), OMO: Liquidity Supply, Lending Facility, Commercial Banks’ Required Reserves, Commercial Banks’ Excess Reserves. - Liabilities decreasing liquidity: Currency in Circulation (CIC), Net Government Account Balance (GAB), OMO: Liquidity Drain, Deposit Facility. - The equilibrium equation between reserve demand and supply is presented as equation (1) in the chapter (see source). ### Characteristics of autonomous factors and modeling needs - Autonomous factors are typically outside direct central bank control and include CIC, GAB, and NFA. - Some accounts are straightforward to forecast when predetermined by policy (e.g., required reserve ratio), but autonomous factors need statistical modeling because: - They are controlled by non‑monetary policy institutions or the public. - They can exhibit regular patterns (e.g., CIC seasonality) or volatile behavior (e.g., NFA). - Existing literature and central bank practice often focus narrowly on CIC with bespoke, hard‑to‑maintain models; a unified, systematic methodology is needed for all autonomous factors. ### Proposed statistical framework and methodological features - Objective: develop daily forecasts for all autonomous factors and the aggregate liquidity supply. - Key elements of the framework: - Unified approach across autonomous factors to reduce modeling overheads. - Automatic model calibration and selection via cross‑validation of a family of time series models. - Detailed reporting to support individualized analyst interventions. - Use of hierarchical (reconciliation) forecasting to improve aggregate forecasts. - Flexibility for country‑specific adjustments and operational conditions. - Intended audience: central bank staff with a strong quantitative background. ### Practical implications for liquidity management and OMO calibration - Estimating changes in liquidity supply is essential for deciding OMO amounts to return short‑term rates to policy targets or to inject liquidity when needed. - Statistical forecasts are particularly important when: - Counterparty forecasts are unavailable (e.g., public demand for banknotes). - Settlement lags allow exact knowledge only up to short horizons (for example, foreign exchange transactions settling on a T+2 can be known exactly at a two‑day horizon). - The framework supports central banks in moving beyond point forecasts to probabilistic forecasting, enabling operations that reflect risk preferences via choice of distribution percentile. ### Key benefits summarized - A single, maintainable framework for all autonomous factors. - Reduced dependency on manual bespoke specifications and analyst‑specific model knowledge. - Improved forecasting accuracy through automated model selection and hierarchical reconciliation. - Promotion of transparent risk management by emphasizing predictive distributions. *Source: Liquidity Forecasting—Part II: The Statistical Component (IMF | MCM Technical Assistance Handbook)* ### Section 6 elaborates on the technical assistance (TA) approach, explaining the liquidity forecasting ### liquidity-forecasting-part-ii - Section 6: Technical assistance (TA) approach and supporting sections ### Technical assistance (TA) approach and mission process - Section 6 elaborates on the technical assistance (TA) approach, explaining the liquidity forecasting mission process and briefly describing the implementation of the framework into a toolbox. - Section 7 discusses limitations and future avenues for research, followed by concluding remarks. ### II. Literature Review — key findings - Literature on forecasting aggregate liquidity or autonomous factors is scarce. - Most quantitative modeling on the autonomous factors centers on specifying an Autoregressive Integrated Moving Average (ARIMA) to forecast CIC. - Non-linear approaches to quantify calendar and seasonal effects follow Bell et al. (1983) and Harvey et al. (1997). - Cabrero et al. (2009) used ARIMA and Structural Time Series models to forecast CIC in the euro area, finding that a combination forecast was best. - El Hamiani-Khatat (2018) investigated short- and long-term money demand forecasting models: - For long-term money demand, macroeconomic factors were important. - Short-term demand is dominated by seasonal patterns or recurring events relevant for calibrating OMO. - ARIMA with additional regressors (RegARIMA) is widely used to forecast the demand for currency and has been applied in many countries, including Poland (Koziński et al., 2015), Ghana (Nasiru et al., 2013), and Nigeria (Ikoku, 2014). - Regressors are often used to model recurring events, seasonal patterns, and holidays. - Hlaváček et al. (2005) compared a neural network model with a RegARIMA to predict Fed CIC: - The neural network model was slightly better. - Neural network models can overlearn sparse holiday events. - Existing literature provides little on detailed quantitative modeling for the GBA and NFA. - Gray (2008) elaborated on factors influencing aggregate liquidity and autonomous factors but did not discuss quantitative models in detail. - Williams (2010) highlighted challenges in obtaining good cash flow forecasting for GAB and stressed intelligence input from spending/revenue departments to improve forecasts of government cash balance. - Iskandar et al. (2018) investigated ARIMA, neural networks, and hybrid models to forecast Indonesian government expenditures — the only time series model investigation for non-CIC autonomous factors identified in the review. ### III. Characteristics of the Autonomous Factors — empirical patterns and implications - Purpose: provide exploratory analysis of typical CIC, GAB, NFA, and their aggregate (AGG), which captures the total contribution of the autonomous factors on liquidity. - Aggregate definition (following Equation (1)): - 푁푁퐺퐺퐺퐺=푁푁푁푁푁푁−퐶퐶퐶퐶퐶퐶 −퐺퐺𝑁𝑁𝑮𝑮. (2) - Data and frequency: - Daily data are used given typical central bank operations and OMO modeling. - Some central banks use a five-day week data organization; others use complete seven-day weeks. - Time series properties: - All time series (CIC, NFA, GAB, AGG) are non-stationary. - Daily series can exhibit multiple seasonal patterns: day of the week, day of the month, dominant events like payday, and day of the year. - Seasonal cycles for CIC are shown after trend removal using classical decomposition; for each seasonal period the minimum, 10 percent, 25 percent, median (50 percent), 75 percent, 90 percent empirical quantiles, and maximum are plotted. - Because days per month vary, it is more convenient to consider seasonal cycles of days in a quarter. - Factor-specific behaviors and forecasting implications: - CIC: - Exhibits fairly canonical time series patterns; amenable to time series models. - Affected by holidays and other calendar effects but generally smaller effects relative to GAB and NFA. - NFA: - Mix of standard time series components and exogenous effects. - Most significant liquidity-impacting factor is unsterilized foreign exchange intervention. - In floating exchange rate regimes, FX operations are less common, leading to less drastic NFA changes. - NFA rarely exhibits seasonality. - GAB: - Sometimes exhibits strong seasonal patterns in both expenditure and revenue decomposition (e.g., biweekly payrolls for civil servants, quarterly corporate tax collections). - Net government balance often considered the most challenging series to forecast by central bankers. - Structural breaks and special events: - Structural breaks can arise from policy effects or long-lasting extraordinary events (for example, the COVID-19 pandemic). ### IV. Forecasting Models — framework choices and operational alignment - Objective: provide forecasts of short-term liquidity movements to support OMO. - Modeling scope: - Exogenous macroeconomic variables are not considered for short-term OMO-focused forecasting as short-term changes are dominated by stochastic trends, seasonal components, and calendar events (payroll dates, subsidy dates, regular FX auctions, holidays). - Macroeconomic variables change slower and are not typically sampled at high frequencies; they may nevertheless impact model quality and are discussed further in other sections. - Forecast horizon and OMO schedule alignment: - Forecast horizon should align with the actual OMO schedule. - Examples: - If OMO is scheduled on Wednesday every week, perform the forecasting exercise on Tuesday, generating 7-day (or 5-day if the weekend is skipped) forecasts ahead from Wednesday to next Tuesday. - If OMO happens biweekly, the forecast horizon should be 14 days (or 10 days if weekends are skipped). - Forecasters must be flexible with forecast horizon due to special factors such as delayed delivery of data or ad-hoc fine-tuning at the end of the reserve maintenance period. - Forecasting errors increase over time; horizon-setting should account for that. - Model groups provided by the framework: - (i) Extrapolative time series models. - (ii) Volatility models. - Forecast outputs and use: - OMO requires point forecasts, but the framework provides predictive distributions in all cases to help analysts assess uncertainty. - Communicating predictive distributions to counterparties can lead to better decisions where costs may be asymmetric. - Volatility models capture structure in the predictive distribution beyond the mean, particularly useful when the mean exhibits minimal or no structure (common for NFA). - Reconciliation and coherence: - Equation (2) (connection between autonomous factors) holds observationally. - When forecasting autonomous factors separately, forecasts are not expected to satisfy Equation (2), and reconciliation errors (deviations) will occur. - Reconciliation errors are connected with forecast errors and can be exploited to improve forecasts further using forecast reconciliation methods (Athanasopoulos et al., 2023). - Additional guidance: - Readers are recommended to consult Ord et al. (2017) and Hyndman and Athanasopoulos (2021) for more thorough treatment of various models. *Source: liquidity-forecasting-part-ii - Section 6 elaborates on the technical assistance (TA) approach, explaining the liquidity forecasting* ### 4.1 The Naive Method (Random Walk) ### 4.1 The Naive Method (Random Walk) ### Naive method (random walk) — concept and role - Forecast rule: Ŝt+h = St, where St is the observation of period N, Ŝt+h is the forecast for period N+h, and h is the forecast horizon. - Underlying model: yt+1 = St + εt, with εt = yt+1 − St and εt assumed i.i.d. normally distributed innovations. - Forecasting assumption: future innovations are unknown and set to zero, so all forecasted values equal the last observation. - Role in framework: - Serves as a simple benchmark and useful upper bound of acceptable forecasting errors. - Guards against overfit models: any more complex model must outperform the naive in forecasting performance to be considered predictively valuable. - Emphasizes preference for simpler, more transparent models. ### Seasonal naive modification - Seasonal naive forecast: Ŝt+h = St−s+h, where s is the seasonal period (for example, 5 when weekends are excluded). - Use case: useful benchmark for highly seasonal time series. ### Exponential Smoothing (ETS) family — decomposition and mechanics - ETS conceptualization: model the time series as collection of patterns — level, trend (slope), and seasonality — typically within a state-space framework. - General state-space formulation: St = f(μt, εt) and μt = g(levelt, seast, …) where f(⋅) and g(⋅) can be additive, multiplicative, or mixed. - Fully additive form: - St = μt + εt - μt = levelt + seast + … - Level update example (additive level): - levelt = levelt−1 + α⋅Nt−1, where α is a smoothing parameter between 0 and 1, and Nt−1 is the previous period error. - Interpretation: α decides how much to update the previous level with the last observed error. Example: if α= 0.2, the last estimated level is updated by 20 percent of the last observation. - Model space and selection: - There are 30 possible ETS models (though selection is typically restricted to a subset because some models can be unstable). - Examples of instability: combining an additive trend and multiplicative seasonality can result in calculation issues when the additive trend pushes observations to zero or negative values. - Parameter estimation via maximum likelihood, with information criteria (e.g., AIC) used for automatic model selection. - Extensions: - ETS models can include exogenous regressors to capture factors like calendar events. ### Autoregressive Integrated Moving Average (ARIMA) models - General ARIMA form: - (1 − φ(G))(1 − G)^d St = (1 + θ(G)) εt - G is the backshift operator (G St = St−1, G^2 St = St−2, etc.). - d (order of differencing) typically equals 1 (or rarely 2) for non-stationary series and 0 for stationary series. - Autoregressive polynomial: (1 − φ(G)) = 1 − φ1 G − φ2 G^2 − … − φp G^p. - Moving average polynomial: (1 + θ(G)) = 1 + θ1 G + θ2 G^2 + … + θq G^q. - Nomenclature: ARIMA(p,d,q). Example: ARIMA(2,1,2) refers to p=2, d=1, q=2. - Seasonal ARIMA (SARIMA): - General SARIMA form: - (1 − φ(G))(1 − Φ(G^m))(1 − G)^d (1 − G^m)^D St = (1 + θ(G))(1 + Θ(G^m)) εt - P, D, and Q are seasonal orders; m is the seasonal periodicity. - Nomenclature: ARIMA(p,d,q)(P,D,Q)[m]. - SARIMA explicitly captures one seasonality of periodicity m; additional seasonal cycles increase complexity and data requirements. - ARIMAX / RegARIMA: - ARIMA models augmented with regressors (ARIMAX/RegARIMA) to incorporate holidays, paydays, and other exogenous information relevant for liquidity forecasting. - Identification and estimation: - Follow Hyndman and Khandakar (2008) methodology: make series stationary by testing for appropriate differencing, then perform local search over p and q by iterating ±1 and selecting models with lowest AIC. - Relationship with ETS: - Purely additive ETS models and ARIMA are algebraically equivalent in some cases (Hyndman et al., 2008). - Multiplicative forms can be obtained by applying ARIMA on log-transformed data. - Mixed-form ETS models are not encompassed by ARIMA. - Both families are used in the liquidity forecasting framework. ### TBATS (Trigonometric Seasonality, Box-Cox, ARMA errors, Trend, Seasonal components) - TBATS components: - Handles seasonality and trend via exponential smoothing using trigonometric terms for seasonality. - Uses Box-Cox transformation. - Incorporates ARIMA errors. - Key advantage: ability to handle multiple seasonalities without special treatment. ### Modeling multiple seasonalities with trigonometric encoding - Binary indicators approach for day-of-week: e.g., D_t(SUNDAY) = 1 if day t is a Sunday, 0 otherwise; similar dummies for Mon, Tue, Wed, Thur (five-day week example). - Trigonometric encoding for a season of F periods: - Construct F/2 pairs of trigonometric variables with C = 1,...,F/2: - (cosine and sine terms defined as functions of C, π, N, and F as in the source). - When F is odd, F/2 is rounded up to the closest integer. - Trigonometric encoding is mathematically equivalent to using F binary indicators (with one corresponding to a constant, so F−1 informative indicators). - Parsimony and sparsity: - Trigonometric encoding behaves better under sparsity: removing terms produces a smoother, approximate representation of the full seasonality, enabling more parsimonious models. - Binary encoding with sparsity captures only the explicit periods for retained indicators and uses the mean elsewhere. - Trade-offs: - Binary indicators are directly interpretable (vertical shifts from the constant). - Trigonometric coefficients are interpretable as a Fourier decomposition and are less directly interpretable per-period. - Practical recommendation for liquidity forecasting: - Use binary encoding for day-of-the-week seasonality (minimal additional terms and direct interpretation). - Use trigonometric representation for remaining seasonalities, made sparse via regression (preferably lasso regression) or stepwise selection with AIC. - Lasso is preferred because it balances fit and model complexity better than stepwise AIC. - Preprocessing for parsimony: - Remove trend first by subtracting a centered moving average (the centered moving average calculates the average of all values within a season to model the trend), then model residuals with trigonometric indicators. - Deterministic vs. stochastic seasonality: - Seasonality modeled with binary or trigonometric indicators is deterministic (shape does not evolve over time). - Seasonality modeled via ARMA/ETS/ARIMA/TBATS is stochastic (shape can evolve over time). - For liquidity forecasting: longer period seasonalities (e.g., annual) assumed to evolve slower and can be modeled as deterministic; day-of-the-week seasonality is preferably treated as stochastic by models directly. ### Level shifts - Causes: economic, policy, or event-driven changes (example cited: Covid-19 caused strong level shifts). - Types: - Permanent or transient level shifts. - Encoding in models with regressors (ETS and ARIMA): - Binary indicator: D_t = 1 if t occurs after the level shift, 0 otherwise. - Transient shift options: - Two binary indicators (one for start, one for termination), or - A single binary indicator equal to 1 only for the duration of the transient shift. - Autoregressive models: AR terms tend to smooth level shifts over multiple periods and can model gradual shifts without complex encoding. ### Calendar and special events (regressors and encoding) - Common deterministic effects (example: currency in circulation (CIC) series structure per Cabrero et al. (2009)): i. Fixed and Moving Festivals (holidays with fixed dates or moving dates). ii. Intramonthly Effects Related to Payrolls. iii. Trading Day (Intraweek) Effects. - Modeling options: - Restrict effect solely to the period of the special event. - Spread effect into following periods via autoregressive terms. - Model leading and trailing effects via multiple binary dummies for full control. - Use parabolic indicators for economical encoding of leading/trailing behavior: - Parabolic indicator formula: Dt = max(0, 1 − ((t − tholiday)/7)^2), spreading effect across seven periods. - Optionally use two parabolic indicators centered across consecutive periods to capture complex responses. - Behavioral rationale: e.g., public withdraws cash ahead of holidays and demand wanes after events; payroll timing affects weekend withdrawals. ### Augmented ETS and ARIMA models — application and model selection - ETS and ARIMA can be augmented with regressors described (seasonality indicators, level shifts, special events). - Risk: oversaturating models with regressors may produce cumbersome, hard-to-estimate models and potential overfitting. - Recommended approach: - Evaluate inclusion of regressors using lasso or stepwise selection based on AIC. - In the framework, all regressors (except level shifts) are evaluated before being introduced into the models. - Seasonality indicators and special event regressors are evaluated in two consecutive steps. - For ARIMA specifically: - Consider alternatives with and without seasonality when adding regressors (i.e., model day-of-week seasonality as stochastic, deterministic, or mixed). - Resolve choice by comparing resulting AIC values. *Source: IMF staff calculations (excerpt from Liquidity Forecasting—Part II: The Statistical Component).* ### 4.2 Volatility Models ### 4.2 Volatility Models ### Purpose and applicability - Volatility models are appropriate for forecasting series with high volatility, normally applied to forecast NFA. - NFA series: - Usually volatile with little structural information to be modeled. - The mean seems to follow a random walk, while the first difference exhibits conditional heteroskedasticity. - Conditional heteroskedasticity indicates that generalized autoregressive conditional heteroskedasticity (GARCH)-type models are likely more appropriate for NFA forecasts. - Several conditional volatility models are fitted to obtain probabilistic forecasts of NFA, which will be later transformed into point forecasts using bootstrapping simulation. - Recommendation: use the random walk as a benchmark for the volatility models. ### GARCH-family specifications - Standard GARCH variance specification (as presented): - 휎휎_t^2 = 휔휔 + Σ_{i=1}^p 훽훽_i 휎휎_{t−i}^2 + Σ_{i=1}^p 휶휶_i N_{t−i}^2 - Two asymmetric variations provided: - Exponential GARCH (eGARCH): - log 휎휎_t^2 = 휔휔 + Σ_{i=1}^p 훽훽_i log 휎휎_{t−i}^2 + Σ_{i=1}^q 휶휶_i g(휖휖_{t−i}) - where g(휖휖_t) = 휽휽 휖휖_t + 휆휆 (|휖휖_t| − E(|휖휖_t|)). Advantage: asymmetry — sign and magnitude of innovations have different effects on the variance. - gjr (Glosten-Jagannathan-Runkle)-GARCH: - 휎휎_t^2 = 휔휔 + 휹휹 휎휎_{t−1}^2 + 휶휶 휖휖_{t−1}^2 + 휙휙 휖휖_{t−1}^2 C_{t−1} - where C_{t−1} = 0 if 휖휖_{t−1} ≥ 0 and C_{t−1} = 1 if 휖휖_{t−1} < 0. Like eGARCH, this allows for asymmetric effects. - For more details on the volatility models the reader is referred to Tsay (2005). ### Forecasting procedure and point conversion - Fit several conditional volatility models to produce probabilistic forecasts of NFA. - Transform probabilistic forecasts into point forecasts using bootstrapping simulation. --- ### 4.3 Hierarchical Reconciliation Methods ### Objective - Main quantity of interest for OMO: AGG = NFA − CIC − GAB. - Two approaches: - Forecast each autonomous factor (NFA, CIC, GAB) separately and aggregate (bottom-up). - Directly forecast AGG. ### Forecast errors and reconciliation - Forecasts are imperfect; denote forecasts with hats and forecast errors (examples shown as equations): - NFA_hat_t = NFA_t + N_AAA,t (Equation (3)) - CIC_hat_t = CIC_t + N_CIC,t (Equation (4)) - GAB_hat_t = GAB_t + N_AAAG,t (Equation (5)) - NNNN_hat_t = NNNN_t + N_NNAA,t (Equation (6)) - NFA_agg_hat_t = NNNN_hat_t − CIC_hat_t − GAB_hat_t + Ñ_t (Equation (7)) - Ñ_t is the reconciliation error: how much forecasting the disaggregate autonomous factors disagrees with forecasting directly the AGG. - Eliminating the reconciliation error can reduce forecast errors because reconciliation blends incomplete information from disaggregate forecasts. - Panagiotelis et al. (2021): total forecast errors across the hierarchy, after reconciliation, will be reduced, irrespective of how reconciliation is done. This does not guarantee AGG forecast errors will always be better — empirical evaluation remains important. ### Matrix formulation and optimal reconciliation - Define bottom-level vector b_t = (CIC, GAB, NNNN)' and full vector y_t = (NFA_agg_t, b_t)'. - Summing matrix S satisfies y_t = S b_t with S shown as: - S = [−1 −1 1; 1 0 0; 0 1 0; 0 0 1] (matrix rows correspond to aggregation equation and bottom-level pointers). - Forecast relation compactly: y_hat_t = S b_hat_t + e_hat_t. - Optimal reconciliation that minimizes reconciliation errors (Wickramasuriya et al. (2019)): - y_hat_t^r = S (S' W^{−1} S)^{−1} S' W^{−1} y_hat_t (Equation (8)) - W is the variance-covariance matrix of the forecast errors of y_hat. - In practice, estimating W is challenging. Recommended approximations: - (i) OLS approximation: W = I (identity matrix). - (ii) Structural (STR): W is diagonal with each element being the row-wise sum of S. - (iii) Weighted least squares (WLS): W is diagonal with each element being the MSE of the corresponding forecasting model for that row. - (iv) MinT: estimates the complete variance-covariance matrix, shrinking off-diagonal elements toward zero. - Equation (8) can be written as y_hat_t^r = S G y_hat_t where G = (S' W^{−1} S)^{−1} S' W^{−1}. - Interpretation: forecasts across the hierarchy are combined using weights in G to construct reconciled bottom-level forecasts; multiplying by S provides reconciled forecasts for the complete hierarchy. - Forecasts y_hat_t are model independent and can originate from any models (Sections 4.1 and 4.2) or be adjusted by expert judgment before reconciliation. ### Benchmarks for reconciled AGG forecasts - Recommended benchmarks: - Direct forecast of AGG without reconciliation. - Direct aggregation from the three autonomous factors to AGG (bottom-up). --- ### V. Forecast Evaluation and Selection ### Evaluation metrics (5.1) - Central practice: rely on point forecasts of the mean; proper score is Mean Squared Error (MSE). - Preferred reporting metric for scale: Root Mean Squared Error (RMSE). - Robust alternative metric: Mean Absolute Error (MAE) — proper score for the median and more robust to extreme errors. - Bias metric: Mean Error (ME). - Formal definitions (as provided): - RMSE = sqrt{ (1/n) Σ_{i=1}^n (S_i − Ŝ_i)^2 } - MAE = (1/n) Σ_{i=1}^n |S_i − Ŝ_i| - ME = (1/n) Σ_{i=1}^n (S_i − Ŝ_i) - where S_i is the actual value, Ŝ_i is the forecast for the same period, and n is the number of periods considered. - Scale-independent metrics: - If required, divide metrics by the in-sample RMSE or MAE (for ME as well) of the random walk (these are called scaled metrics). - Probabilistic evaluation: - Recommend forecasting the complete predictive distribution and publishing it; the predictive distribution represents forecasting risk (modeling risk and inherent stochasticity). - Target percentiles other than the mean/median when under- and over-allotting have different costs. - Mean Interval Score (MIS) for prediction interval width (95 percent intervals preferred): - MIS = (1/n) Σ_{i=1}^n [ (U − L) + (2/α) (L − Y_i) 1_{Y_i < L} + (2/α) (Y_i − U) 1_{Y_i > U} ] - α corresponds to desired confidence level (e.g., 95 percent), U and L are the upper and lower prediction intervals matching α, and 1(⋅) is the indicator function. - For specific quantiles, use Pinball loss (similar to MIS but for one side). - Note: well-calibrated models are assumed when interpreting interval widths; overfit models can produce unreasonably narrow intervals. ### Evaluation scheme (5.2) - Organize forecasting models in tiers of complexity; prefer simpler models when forecasting errors are similar. - Tier 1: level and seasonal random walks (upper bound on acceptable forecasting performance). - Tier 2: ETS, ARIMA (level/seasonal/fully automatic), TBATS. - Tier 3: ETS and ARIMA augmented with regressors (higher complexity, potential to overfit). - For volatility forecasts, use the Naive as a benchmark over more complex GARCH models. - Use rolling origin out-of-sample evaluation (cross-validation over time): - Use q observations as in-sample to specify models; generate h-step-ahead forecasts for out-of-sample periods. - Increase in-sample by one period and repeat until out-of-sample is exhausted. - If S is size of out-of-sample set, generate S − h + 1 rolling forecasts and respective errors. - Advantages: averaging errors across multiple origins helps separate structure from noise and reduces randomness in performance assessment. ### Testing forecasting performance (5.3) - Use non-parametric statistical tests for multiple comparisons: - Friedman test: evaluates whether at least one method is statistically different. - Post-hoc Nemenyi test: groups methods into subgroups using mean ranks and a critical distance; methods within mean rank ± critical distance are grouped together. - Prefer the simplest method within the group with the lowest mean rank. - Non-parametric tests are distribution-free but typically weaker (require more samples); liquidity forecasting often has abundant data. - Example guidance for analysts (heuristics): - If focus is on point forecast of the mean: consult RMSE first; if no clear winner, consult MAE and then ME. MIS can provide additional evidence on uncertainty. - If focus is on the predictive distribution: consult MIS first; MIS is also important for volatility forecasts. - Weight recommendations: give more weight to RMSE, then MAE, with other metrics following. - Use statistical tests (Friedman/Nemenyi) to support the choice of the appropriate forecast. *Liquidity Forecasting—Part II: The Statistical Component, IMF | MCM Technical Assistance Handbook* ### 5.4 The Forecasting Process: Statistical Forecasts and Expert Judgment ### 5.4 The Forecasting Process: Statistical Forecasts and Expert Judgment ### Statistical + Expert framework - A complete short-term liquidity forecasting framework combines statistical modeling and expert adjustments originating from communications with related stakeholders. - Statistical forecasts provide baseline forecasts using well-structured information: historical observations and indicator variables for holidays and special events. - Additional information from counterparties (budget revenues, gold purchase plans, etc.) can be added to reduce forecast uncertainty. - At each layer of forecasting, it is critical to evaluate performance to guide improvements. ### Role and evaluation of expert judgment - Literature finds experts are influenced by behavioral biases that can produce inconsistent adjustments. - Recommendation: separately evaluate the performance of expert adjustments to provide feedback and improve future adjustments. - Sroginis et al. (2023) find experts can be overwhelmed by contextual information leading to erroneous adjustments; mitigate by: - Using augmented ETS and ARIMA to incorporate all structured information. - Limiting expert adjustments to unstructured information that cannot be easily incorporated into regression models. - Decompose expert-judged adjustments into constituent factors so experts must account for each factor—helps mitigate overly optimistic adjustments. ### Implementation and automation - The proposed framework is implemented in R. - Workflow elements: - Analysts prepare data of autonomous factors and codify holidays, level shifts, and other events. - Implementation automations obtain rolling origin errors for the described forecasting models. - Reports aid analysts in choosing the best forecasts for each time series. - Forecasts are generated and reconciled; reports guide selection of hierarchical reconciliation, producing the final forecast output. - Reconfiguration of models is not required at every forecasting cycle but is strongly advised periodically; the reconfiguration frequency should balance analyst workload and computational resources. ### Figures referenced (schematic/process) - Figure 10: schematic combining statistical forecasts, historical liquidity, known special events, trend & season patterns, expert knowledge, policies, decisions by actors (e.g., banks), news, evaluation, leading to final forecast. - Figure 11: workflow for statistical modeling of short-term liquidity (data preparation → model automations → model selection → forecasting → reconciliation → final output). ### Forecast evaluation and selection - Models are compared using rolling origin errors and cross-validation techniques so the best-performing model can be identified. - Hierarchical reconciliation methods are leveraged to reduce modeling uncertainty and improve forecast accuracy. --- ### 5.5 Limitations and Extensions - Framework incorporates static and dynamic forecast combinations of pools of models; forecast pooling helps mitigate modeling uncertainty and improve accuracy but is not part of the main workflow. - Practical experience: forecast combinations often offer marginal improvements relative to added complexity; hierarchical reconciliation already provides a form of forecast combination. - Diminishing returns observed from using two “layers” of forecast combination; consequently, pooling options are optional. - During the COVID-19 pandemic, dynamic forecast combination was helpful as models failed in complementary ways. - Machine learning / AI: - Toolkit does not automate machine learning or artificial intelligence methods. - Trials with shallow neural networks and random forests did not demonstrate forecasting performance benefits given additional computational cost. - Global forecasting methods show promise: training models across liquidity data from multiple countries could leverage global learning but entails data collection complications. - Temporal Hierarchies: - Temporal Hierarchies (Athanasopoulos et al., 2017) can enable reconciliation over time scales and help bridge short- and long-term liquidity forecasting. - Cross-temporal formulations (Kourentzes and Athanasopoulos, 2019) could incorporate reconciliation with temporal hierarchies in a single modeling step and are a useful extension. --- ### VI. Technical Assistance (TA) Approach — recommended process and timeline - i. T-6 (at least six weeks before mission) - Request time series; ideally at least three years of central bank daily balance sheet data converted into a liquidity table showing autonomous factors, banks’ reserves at the central bank, and monetary operations. - If extensive data absent, use the standardized data request in Appendix I. - Share mission requirements for software (R and RStudio) and staffing (staff with coding capacity). - At T-6: prepare data for modeling, follow up on issues, run models, prepare a PowerPoint including: - (i) time series descriptive analysis and model configuration; - (ii) model selection statistics; - (iii) point forecasts output with predictive intervals; - (iv) possible forecast enhancement with reconciliation; - (v) alignment with existing liquidity table. - Standardized output template is presented in Appendix II. - ii. T-2 - Present standardized results to TA recipient virtually; organize Q&A. - Explain that in-person mission will consist mainly of a workshop with staff running the model on their devices. - Remind TA recipient of staffing and software requirements. - iii. From T-2 to T-0 - Prepare draft report and circulate internally in MCMCO for preliminary comments. - iv. T-0 (mission start) - Share draft report with TA recipient for comments (due before end of mission). - Workshop agenda: - (i) set-up of infrastructure and working environment with relevant packages; - (ii) R coding essentials; - (iii) customization of model configurations; - (iv) execution of forecasting codes; - (v) interpretation of results; - (vi) alignment with existing liquidity table and OMO calibration; - (vii) discussion of qualitative input and other liquidity impacting scenarios. - At T+1 (end of mission), authorities provide comments on draft reports; draft circulated for formal review at IMF Headquarters. - v. T+3 - Include comments with aide-mémoire and finalize the report. - TA should advocate publishing forecasts if forecast uncertainty is disclosed; publish distribution forecasts so users can factor in forecasting risk. ### TA priorities and common challenges - Prioritize skill development within central banks; hands-on workshop training using central bank data and mission code is critical. - TA should ensure central bank staff can prepare forecasts independently before end of mission; one approach is to request central bank colleagues prepare and present forecasts to senior management at end-of-mission meeting. - Common challenges and solutions: 1. Difficulties in inter- and/or intra-departmental data sharing: - Assess data availability before mission; emphasize data communication channels and the integral role of expert judgment. - If data delivery delayed, advise rescheduling forecasting date or adopting flexible forecasting horizons. 2. IT or hardware restrictions preventing installation of RStudio and packages: - Instruct central bank counterparts on infrastructure installation; advise consulting IT departments in advance to clear restrictions. 3. Open-source package version updates causing code bugs: - Hidden underlying code can break if packages remove functions; mission team may need to update underlying code. - Advise forecasters not to update any package or RStudio related to the forecasting exercise. --- ### VII. Conclusions - Proposed a comprehensive forecasting framework to predict autonomous factors and aggregate liquidity, designed to be generic, adaptable, and highly automated. - Framework is a holistic process: statistical modeling, a modeling process for selection and generation of resilient forecasts, and incorporation of expert judgment. - Two distinct families of models are detailed to accommodate characteristics of autonomous factors; models are rigorously tested and compared using cross-validation to identify best performers. - Forecast reconciliation methods are leveraged to reduce modeling uncertainty and improve accuracy. - Expected impacts: facilitate decision-making, optimize liquidity interventions, and contribute to a more stable financial environment. --- *Source: liquidity-forecasting-part-ii - 5.4 The Forecasting Process: Statistical Forecasts and Expert Judgment* ### Appendix II. Standardized Output ### Appendix II. Standardized Output ### A. Time Series Descriptive Statistics Report: An Example of CIC - Models fit for Currency in Circulation over the period 2022-12-29 to 2023-07-20. - Data measured in millions of Guatemalan quetzals (GTQ). - The period used for testing the forecasting models is highlighted in the report. - Long-term trend captured using a centered moving average and plotted in red. - Daily time series seasonal patterns investigated: - (i) day of the week; - (ii) day of the month; - (iii) day of the year. - Seasonal profile within a quarter investigated due to variable number of days in a month. - Seasonality visualization approach: - De-trend the time series and plot seasons as distributions showing minimum, 10 percent, 25 percent, the median, 75 percent, 90 percent, and the maximum values. - Seasonality identified as a clear pattern across periods of the season. ### B. Forecasting Models Cross-Validation and Comparison Report: An Example of CIC - Models fit to data from 2022-12-29 to 2023-07-20. - Data measured in GTQ millions. Results summarized for Currency in Circulation. Key accuracy and diagnostic metrics reported: - Accuracy metrics up to one, two, and four weeks ahead: - Root Mean Squared Error (RMSE). - Mean Absolute Error (MAE). - Note: RMSE is more sensitive to extreme values than MAE. Lower values indicate more accurate results. Models are sorted by one-week accuracy value. - Bias metric: - Mean Error (ME). Unbiased forecasts should have ME close to zero. Sorted by the one-week bias. - Predictive distribution evaluation: - Mean Interval Score (MIS) for 95 percent prediction intervals. Lower values indicate better performance. Sorted by the 1-week horizon. - Visual summary: - Models' performance plotted and ordered from worst to best according to each criterion for one-week ahead. - Best forecast for each horizon highlighted with a green circle. - When a Naive forecast is available, errors are provided relative to it; any forecast less accurate than the Naive (performance equal to 1) should not be considered. - Statistical testing: - Friedman and Nemenyi non-parametric tests reported at a 95 percent significance level. - Models ranked by mean rank (the lower, the better). Models connected by a vertical line are grouped with no statistical evidence of difference. - Note: As more test periods are accumulated, results will become clearer; for a limited number of periods these tests have limited power. ### C. Forecasting Results: An Example of Aggregated Liquidity (AGG) - All models were forecasted on 2023-08-31. - Latest forecasts (selected model): - The model selected for AGG is ARIMA with Regression. (Selection configurable in the configuration file.) - Forecast plots include prediction intervals: darker bands indicate an 80 percent prediction interval; lighter bands indicate a 95 percent prediction interval. Observed data shown in black. - Latest forecasts (model averages): - Equally weighted average of all models provided. - Forecasts for all models: - Chart shows forecasts for all models. ### D. Enhancement of Aggregated Liquidity Forecast through Reconciliation Report - Models fit over the period 2023-01-05 to 2023-07-27. - Models chosen for each autonomous factor (configurable in the configuration file): - State Account Balance: ETS with Regression. - Net Foreign Accounts: ETS with Regression. - Net Other Assets: ARIMA with Regression. - Currency in Circulation: ARIMA with Regression. - Aggregate: ARIMA with Regression. - Most recent forecast origin date: 2023-08-31. - One-step-ahead forecasts tabulated for reconciled and unreconciled forecasts. - Method selected for reconciliation: STR (shaded in the report). - Best method based on past performance (RMSE): OLS (highlighted in yellow in the report). - Forecast evaluation: - Validation period used: 2023-01-05 to 2023-07-27. - Reconciliation method comparisons summarized for multiple metrics: - RMSE (Accuracy): table summarizes RMSE across reconciliation methods; STR is chosen; OLS is best on average RMSE across horizons. - MAE (Accuracy): table summarizes MAE across reconciliation methods; STR is chosen; OLS is best on average MAE across horizons. - ME (Bias): table summarizes ME across reconciliation methods; STR is chosen; MinT is best on average ME across horizons. - MIS (Predictive Distribution): table summarizes MIS across reconciliation methods; STR is chosen; Base (Unreconciled) is best on average MIS across horizons. - Visual summary: - Performance across metrics plotted; forecasts ordered from worst to best for one-week ahead; best forecast for each horizon highlighted with a green circle. - All errors presented relative to the unreconciled (base) forecast. ### E. Enhanced Forecasts for Aggregated Liquidity Report - Changes in Net Liquidity provided for one, two, and four weeks ahead. - Forecasts of the net liquidity position by the chosen method (STR) plotted. *Source: Appendix II. Standardized Output, Liquidity Forecasting—Part II: The Statistical Component, IMF | MCM Technical Assistance Handbook* --- _Source: https://www.imf.org/-/media/files/publications/miscellaneous/english/2022/mcm-technical-assistance-handbook/liquidity-forecasting-part-ii.pdf_