## Systematizing Macroframework Forecasting: High-Dimensional Conditional Forecasting with Accounting Identities (WP/2022/110) — Section highlights

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### Method overview and objective
- Purpose: provide a two-step, systematized approach to produce internally consistent macroframework forecasts when forecasters supply paths for a subset of variables.
- Two-step procedure:
  - Step 1: Forecast each unknown variable conditionally on forecasters’ supplied known-variable paths using elastic net with time series cross-validation.
  - Step 2: Reconcile (project) the collection of first-step forecasts onto the affine space defined by accounting identities so constraints are satisfied; closed-form solution exists under stated rank conditions.
- Key modeling assumption: constraints are affine C_t r_t = d_t and known-variable paths r_t^k are taken as given inputs.

### Practical forecasting setup and notation
- Data structure:
  - m variables in total; k known variables r_t^k with T+ h samples; m−k unknown variables r_t^u with T samples.
  - Objective: fill NaN forecast cells r̃_t^u for t = T+1, ..., T+ h so that C_t [r̃_t^u; r_t^k] = d_t.
- Constraint structure:
  - C_t is (m−n)×m, d_t is (m−n)×1, number of free variables n < m; focus on non-trivial case k < n.

### Step 1 — Conditional forecasting details
- Estimator: elastic net with time series cross-validation to handle high-dimensional predictor sets and small T relative to number of predictors.
- Data standardization:
  - r̅_it = (r_it − μ_i) / σ_i with μ_i = (1/T) ∑_{τ=1}^T r_iτ and σ_i = sqrt[(1/(T−1)) ∑_{τ=1}^T (r_iτ − μ_i)^2].
- Cross-validation design:
  - Time series cross-validation with successive training sets; number of folds s = 5 (default in scikit-learn).
- Forecast construction:
  - First-step standardized forecast r̅_it^u = β̂_1(i,T) r̅_{1t}^k + ... + β̂_k(i,T) r̅_{kt}^k for t = T+1, ..., T+ h.
  - Transform back: r̂_it^u = μ_i + σ_i r̅_it^u.
  - Special case: if elastic net sets all β̂_k(i,T) = 0 then forecast equals historical mean μ_i.
- Implementation note: elastic net implemented with scikit-learn; constant variables are carried forward.

### Step 2 — Forecast reconciliation and formulas
- Reconciliation objective (weighted projection):
  - r̃_t = argmin_{r̃} (1/2) (r̂_t − r̃)' Ŵ^{-1} (r̂_t − r̃) subject to C_t r̃ = d_t and r̃_k = r_t^k.
  - Ŵ estimates one-step-ahead first-step forecast error covariance W = E[(r̂_{T+1} − r_{T+1}^*) (r̂_{T+1} − r_{T+1}^*)'].
  - Intuition: variables with smaller first-step forecast error are adjusted less in reconciliation and help inform others via constraints.
- Reduction by substituting known variables:
  - Partition C_t = [U_t' K_t']' and focus on unknowns via reduced problem:
    - r̃_t^u = argmin_{r̃_u} (1/2) (r̂_t^u − r̃_u)' W_u^{-1} (r̂_t^u − r̃_u) s.t. U_t' r̃_u + K_t' r_t^k = d_t.
- Estimation of Ŵ_u:
  - Generate out-of-sample first-step forecast errors via rolling subsamples; compute sample covariance S_T.
  - Shrink S_T to obtain Ŵ_u: Ŵ_u = λ ⋅ diag(S_T) + (1 − λ) ⋅ S_T with λ given by Shafer and Strimmer (2005).
- Closed-form reconciled solution (Theorem 1):
  - If U_t and Ŵ_u are full rank:
    - r̃_t^u = r̂_t^u − Ŵ_u U_t (U_t' Ŵ_u U_t)^{−1} (C_t r̂_t − d_t).

### Computational and practical considerations
- The closed-form solution enables reconciliation with large numbers of variables and is tractable in high-dimensional settings.
- The approach assumes no inequality constraints (acknowledges literature on non-negative reconciliation but notes potential bias).
- The formulation assumes first-step forecasts are unbiased; alternative approaches exist for biased first-step forecasts.

### Empirical illustration — France (WEO data)
- Data and design:
  - WEO vintages used: seven vintages from 2016 April to 2022 April.
  - Sample start: 1980 in all the vintages.
  - Total variables: 53.
    - Known variables before adding lags: 2 types of GDP, 4 GDP subcomponents related to government, 28 fiscal variables, and 1 external current account balance.
    - Unknown variables: 18 subcomponents of the GDP expenditure approach not directly related to government.
  - Notation: N = national accounts, G = government finance statistics, B = balance of payments.
- Step 1 implementation choices for France:
  - Transformations: most variables normalized by nominal GDP and differenced (year-over-year difference of the share); nominal and real GDP as growth rates.
  - Number of lags included: 2.
  - Number of folds in cross-validation: s = 5.
  - Number of variables used in elastic net: 159 = 53 × 3 (lags included).
    - Unknown variables in elastic net: 18.
    - Known variables in elastic net: 141.
  - Number of elastic net applications (forecasts): 108 = 6 × 18.
  - First-step forecasts were un-differenced (adding previous year’s observation) so constraints remain time-invariant in step 2.
- Accounting identities used:
  - Eleven constraints considered; GDP expenditure identities and related decompositions (investment decomposition, gross national savings identities). Chain-linked volumes excluded from additive identities because they are not additive.

### France — performance metrics and exact results
- Forecast error metric: root mean squared error (RMSE); true value defined as the year t observation in year t+1 vintage.
- Aggregation: each unknown variable’s forecast errors aggregated over vintages using RMSE; methods summarized as mean RMSE over unknown variables.
- Mean RMSE (preserved exactly):
  - Mean RMSE of WEO forecast r_WEO: 0.0154.
  - Mean RMSE of first step forecast r̂: 0.0145.
  - Mean RMSE of second step forecast r̃: 0.0123.
  - Mean RMSE of second step forecast conditional on true values r̃*: 0.0121.
- Performance interpretation:
  - The second step forecast r̃ improves the WEO forecast r_WEO by around 20 percent on average.
  - Improvement is heterogeneous: larger gains for domestic variables (e.g., consumption and investment) than for external variables (e.g., imports of services).
  - Suggestion: include more external-sector known variables (commodity prices, other countries’ GDP growth) to improve external-variable forecasts.
  - Behavior over time: second step and WEO forecast errors behave similarly for most variables and years, with divergence for several variables in 2020.

### Annex country example — Seychelles (key points and exact results)
- Design choices:
  - Smaller variable set; known variables include exports of services (bxs) to capture inbound tourism; fiscal variables replaced by services exports.
  - Forecast object: contributions to nominal GDP defined as r_it = (y_it − y_it−1) / (NGDP_t−1).
  - Accounting identities for contributions remain linear.
- Performance results (preserved exactly):
  - The 2nd step forecast r̃ improves the WEO forecast by around 15 percent on average.
  - Improvements mainly for external-activity-related variables (imports and exports and their goods/services breakdowns).
  - WEO forecasts perform better for domestic-activity-related variables (consumption and investment).

### Diagnostics, caveats, and policy guidance
- Main caveats:
  - Lucas critique: first-step forecasts rely on historical correlations; novel policies or structural breaks can yield large forecast errors.
  - First-step coefficients are not causal estimates; embed known causal policy impacts into the known-variable inputs rather than rely on historical correlations.
  - Forecast accuracy depends on the quality of forecasters’ supplied known-variable paths; biased known-variable forecasts induce biased unknown-variable forecasts when those known variables historically inform unknowns.
  - If priors on some variables are weak, include them among the unknowns to avoid spillovers from poorly specified known-variable paths.
- Robustness measures and uses:
  - Time-series cross-validation in step 1 reduces overfitting; shrinkage in Ŵ_u construction discounts noisy covariance estimates.
  - Scenario analysis: method can be used by supplying nonstandard policy-relevant paths for known variables; uncertainty on known variables can be reflected by specifying distributions, drawing samples, and applying the method to obtain distributions for unknown-variable forecasts.
- Future extensions suggested:
  - Multilateral consistency (cross-country constraints via trade and financial linkages).
  - Mixed-frequency data to incorporate higher-frequency information and nowcasting.
  - Log-linear constraints to handle multiplicative accounting identities.
  - Use the framework conversely to search for important variables.

*IMF Working Paper — Systematizing Macroframework Forecasting: High-Dimensional Conditional Forecasting with Accounting Identities (Working Paper No. WP/2022/110) — Section 1, 3, 4, 5, Annexes*

### 1. Introduction ........................................................................................................

### 1. Introduction

### Overview
- Forecasting a macroframework (a set of macroeconomic variables linked by accounting identities) is useful to present an underlying economic narrative and check the narrative’s consistency.
- Common practice: specify paths for a subset of variables (e.g., public investment and GDP) and adjust others (private investment, consumption, imports) using assumptions (crowd-in/out, fiscal multiplier, import leakage); check consistency via accounting identities (e.g., gross national saving and current account balance).

### Practical challenges identified
- Forecasters often have information only on a subset of variables and must use that to forecast less-informed variables.
- Ensuring all accounting identities are satisfied is difficult.
- Typical ad-hoc practice: extend forecasts using rules of thumb (stable GDP ratios, hand-picked regressions), then choose a residual variable to absorb inconsistencies and iterate manually.
- This ad-hoc process is error-prone and resource-intensive because changes to known-variable forecasts cascade through the macroframework.

### Proposed solution (high level)
- The paper proposes a two-step method to systematize macroframework forecasting:
  1. Forecasters supply forecasts for a subset of variables (known variables) plus historical data for all variables.
  2. Step 1: Forecast each unknown variable conditionally on known variables using elastic net with time series cross-validation (to handle cases with small T relative to number of predictors).
  3. Step 2: Project (reconcile) the collection of unknown-variable forecasts onto the affine space defined by the accounting identities so the constraints are satisfied; the projection problem admits a closed-form solution under stated rank conditions.

### Positioning in related literature
- Integrates two strands:
  - High-dimensional conditional forecasting (penalized regression; cites Uematsu and Tanaka (2018); Banbura et. al. (2015)).
  - Forecast reconciliation for hierarchical time series (cites Wickramasuriya et. al. (2019); Athanasopoulos et. al. (2020); Capistran et. al. (2010)).
- Extension: allows conditional forecasting (fixing known variables to forecasters’ supplied paths) combined with reconciliation.

### Empirical illustration and performance
- Country example: France (World Economic Outlook data).
  - Forecasted 18 real sector variables in the GDP expenditure approach using forecasts of 35 variables (GDP, fiscal variables, and current account balance) and historical data of all 53 variables.
  - Evaluation uses five April vintages from 2016 to 2020.
  - Result: the method generates smaller root mean squared forecast errors than WEO forecasts by 20 percent on average.
- Additional example: Seychelles (Annex 2) for a tourism-dependent economy.

### Key caveats
- The method relies on historical correlations; subject to the Lucas critique (Lucas, 1976): novel policies that move variables in unobserved ways can lead to large forecast errors. Forecasters should embed known policy impacts in the known-variable inputs rather than rely solely on historical correlation.
- The method uses forecasters’ supplied forecasts of known variables as inputs; inaccurate inputs produce inaccurate forecasts of unknown variables. If priors on some variables are weak, include them among the unknown variables to avoid spillover.
- The method can be used for scenario analysis by supplying nonstandard but policy-relevant paths for known variables.

### Notation and problem setup (summary of framework)
- Data contain m>0 variables; k are known variables (r_t^k) with T+ h samples; m−k are unknown variables (r_t^u) with T samples.
- Vector form: r_t^u: (m−k) × 1, r_t^k: k × 1, r_t = [r_t^u; r_t^k]: m × 1.
- Constraints are affine: C_t r_t = d_t where C_t is (m−n)×m and d_t is (m−n)×1; the number of free variables is n < m and the paper focuses on the non-trivial case k < n.
- Objective: fill NaN forecast cells r̃_t^u for t = T+1, ..., T+ h so that C_t [r̃_t^u; r_t^k] = d_t (equation (3)).

### Step 1: Forecasting each unknown variable (details and key numeric choices)
- Use elastic net with time series cross-validation (accommodates cases where T is smaller than number of predictors).
- Data standardization:
  - r̅_it = (r_it − μ_i) / σ_i, with μ_i = (1/T) ∑_{τ=1}^T r_iτ and σ_i = sqrt[(1/(T−1)) ∑_{τ=1}^T (r_iτ − μ_i)^2] (equation (4)).
- Elastic net optimization (cross-validated MSE with L1 and L2 penalties):
  - For each i ∈ u, estimate (β̂(i,T), λ̂1(i,T), λ̂2(i,T)) by minimizing the cross-validated objective shown in equation (5).
- Time series cross-validation:
  - Use successive training sets that are supersets of previous ones; number of folds set to five (the default in scikit-learn). The last five observations rotate as test sets.
- First-step standardized forecast (equation (6)):
  - r̅_it^u := β̂_1(i,T) r̅_{1t}^k + ... + β̂_k(i,T) r̅_{kt}^k for t = T+1, ..., T+ h.
- Transform back to original scale (equation (7)):
  - r̂_it^u = μ_i + σ_i r̅_it^u.
- Special cases:
  - If elastic net sets all β̂_k(i,T) = 0, forecast equals historical mean μ_i.
  - Known variable forecasts remain r̂_t^k = r_t^k.
- Note: first-step forecasts do not necessarily satisfy the accounting constraints (3).

### Step 2: Forecast reconciliation (details and key formulas)
- Second-step reconciles first-step forecasts r̂_t to satisfy constraints by solving a weighted projection problem (equation (8)):
  - r̃_t = argmin_{r̃} (1/2) (r̂_t − r̃)' Ŵ^{-1} (r̂_t − r̃)
    s.t. C_t r̃ = d_t and r̃_k = r_t^k.
  - Ŵ is an estimate of the one-step-ahead first-step forecast error covariance (W = E[(r̂_{T+1} − r_{T+1}^*) (r̂_{T+1} − r_{T+1}^*)']).
  - Intuition: smaller first-step forecast error → second-step forecast stays closer to first-step forecast for that variable; variables with small forecast errors help inform others via constraints.
- Reduce problem by substituting known variables:
  - Partition constraint matrix C_t = [U_t' K_t']', with U_t (m−k)×(m−k) corresponding to unknowns and K_t corresponding to knowns; let Ŵ_u be submatrix of Ŵ for unknown variables.
  - Reduced optimization (equation (9)):
    - r̃_t^u = argmin_{r̃_u} (1/2) (r̂_t^u − r̃_u)' W_u^{-1} (r̂_t^u − r̃_u)
      s.t. U_t' r̃_u + K_t' r_t^k = d_t.
- Construction of Ŵ_u via shrinkage:
  - Generate out-of-sample first-step forecast errors using rolling subsamples: r̂_it^u (equation (10)) for t = T − T_iio + 1, ..., T, where T_iio denotes number of forecast errors.
  - Sample covariance matrix of forecast errors S_T (equation (11)):
    - S_T = (1/(T_iio − 1)) (ê − ē)' (ê − ē) where ê stacks the out-of-sample errors; ē is the mean error vector.
  - Shrink S_T to obtain Ŵ_u (equation (12)):
    - Ŵ_u = λ ⋅ diag(S_T) + (1 − λ) ⋅ S_T, with λ given by Shafer and Strimmer (2005) formula:
      - λ = [∑_{i≠j} S_T(ρ̂_{ij})] / [∑_{i≠j} ρ̂_{ij}^2], where ρ̂ and S_T(ρ̂) are defined via z and ρ formulas (equation (13) and accompanying definitions).
- Closed-form solution (Theorem 1):
  - If U_t and Ŵ_u are full rank, the reconciled forecast for unknowns is (equation (14)):
    - r̃_t^u = r̂_t^u − Ŵ_u U_t (U_t' Ŵ_u U_t)^{−1} (C_t r̂_t − d_t).
  - The closed-form solution enables reconciliation with large numbers of variables and makes the method tractable in high-dimensional settings.

### Implementation and computational notes
- Elastic net implementation: scikit-learn package in Python (time series cross-validation with five folds).
- The method treats constant variables by carrying forward the constant value as forecast.
- The approach assumes no inequality constraints; references note optimal non-negative reconciliation can be done (Wickramasuriya et. al. (2020)) but may incur bias.
- The formulation assumes the first-step forecast is unbiased (comment referencing Taieb and Koo (2019) for alternative approaches).

*IMF WORKING PAPERS Systematizing Macroframework Forecasting*

### 3. Country Example

### 3. Country Example

### 3.1 Data
- Source: French data retrieved from World Economic Outlook (WEO).  
- WEO vintages used: seven vintages from 2016 April to 2022 April.  
- Sample start: 1980 in all the vintages.  
- As of 2022, French quarterly national accounts publication lags: two-month lag for the first estimate and a three-month lag for detailed figures.  
- Variables used in the exercise: 53 total variables.  
  - Known variables before adding lags: 2 types of GDP, 4 GDP subcomponents related to government, 28 fiscal variables, and 1 external current account balance.  
  - Unknown variables: 18 subcomponents of the GDP expenditure approach not directly related to government.  
  - Notation: variables starting with N denote national accounts, G denote government finance statistics, and B denote balance of payments.  
- Rationale for variable selection: choose France because (i) a rich set of variables is available, and (ii) the size of the government is the largest among economies with rich variable sets, so fiscal variables are expected to be informative.  
- Aggregation note: the list contains both highly disaggregated variables and their aggregates, reflecting the observation that aggregated data may have a higher signal-to-noise ratio than highly disaggregated data.

### 3.2 Accounting Identities
- Eleven constraints are considered; constraints are time-invariant and the time subscript is dropped.  
- The GDP expenditure approach gives seven constraints. Examples (preserve form as in source):
  - GDP expenditure approach equation (symbolic, see equation (15) in source).  
  - Investment decomposition: NFI = NFIP + NFIG, and NIG = NIP + NIG (symbolic, see equation (16) in source).  
  - Gross national savings identities: NGSP = NFI + bca, and NGSP = NGP + NGSPG (symbolic, see equation (17) in source).  
- Real chain-linked volumes are not included in additive identities because chain-linked volumes are not additive.

### 3.3 Variables to Forecast
- Forecast target: shares of GDP for the unknown variables (forecasting shares in GDP rather than levels).  
- Rationale: normalized series are typical for cross-country comparison and transformations to stationarity are recommended.  
- Caution: performance of forecasting methods can depend on transformation; spurious regression risks if nonstationary series used.

### 3.4 Step 1: Forecasting Each Unknown Variable
- Transformations applied:
  - Most variables: normalized by nominal GDP and differenced (year-over-year difference of the share).  
  - Nominal and real GDP: transformed into growth rates.  
  - Growth rates not used for variables that can be zero or negative (e.g., current account balance).  
- Cross-validation and model settings:
  - Number of folds in cross-validation: s = 5.  
  - Number of lags included: 2 (chosen following format of selected economic indicators table in France Article IV consultation staff report).  
  - Number of variables used in the elastic net: 159 = 53 × 3 (lags included).  
    - Unknown variables in elastic net: 18.  
    - Known variables in elastic net: 141.  
  - Number of forecasts (elastic net applications): 108 = 6 × 18 times in total.  
- Forecast construction:
  - Elastic net applied with time-series cross-validation to the differenced series; first step forecast r̂ obtained by un-differencing (adding previous year’s observation) to revert differencing.  
  - Un-differencing simplifies constraints for step 2 by making them time-invariant (alternatively one could keep differencing and impose time-variant constraints).

### 3.5 Step 2: Forecast Reconciliation
- Purpose: obtain second step forecast r̃ by projecting the first step forecast r̂ of unknown variables’ shares of GDP onto the affine, time-invariant constraint space so that forecasts satisfy accounting identities and incorporate information from known variables rk.  
- Example constraint for shares (GDP expenditure approach): the shares sum to 1 (symbolic, see equation (18) in source).  
- Result: r̃ reflects known variables rk and satisfies all accounting identities.

### 3.6 Performance Assessment
- Forecast error metric: root mean squared error (RMSE).  
- True value definition: the year t observation in year t+1 vintage (one-year ahead vintage). Example: true value of 2016 is the 2016 observation in 2017 vintage.  
- Aggregation and comparison:
  - Each unknown variable’s forecast errors aggregated over vintages using RMSE; each forecasting method summarized as the mean RMSE over unknown variables.  
  - Three comparison methods:
    1. WEO forecast r_WEO (benchmark).  
    2. First step forecast r̂ (to assess improvement due to reconciliation).  
    3. In-sample forecast using true values r̃* (lower bound of forecast error, uses information unavailable to forecasters).  
- Quantitative results (mean RMSE, preserved exactly):
  - Mean RMSE of WEO forecast r_WEO: 0.0154.  
  - Mean RMSE of first step forecast r̂: 0.0145.  
  - Mean RMSE of second step forecast r̃: 0.0123.  
  - Mean RMSE of second step forecast conditional on true values r̃*: 0.0121.  
- Performance interpretation:
  - The second step forecast r̃ improves the WEO forecast r_WEO by around 20 percent on average.  
  - Improvement is not uniform across variables: greater improvement for domestic variables (e.g., consumption and investment) than for external variables (e.g., imports of services).  
  - Suggestion: include more external sector known variables (commodity prices, other countries’ GDP growth, etc.) to improve external-variable forecasts.  
  - Behavior over time: second step and WEO forecast errors behave similarly for most variables and years, with divergence for several variables in 2020.

### 4. Discussion (implications and caveats)
- Scalability:
  - Method can be scaled to other accounting identities (production and income approaches) and the broader macroframework (price, fiscal, external, monetary statistics).  
  - Can potentially scale up to multilateral consistency and the macroframework of the entire world given ample data (trade and financial linkages impose cross-country constraints).  
- Caveats and limitations:
  - Lucas critique applies: if a new policy moves variables in unprecedented ways, historical-correlation-based first-step forecasts can yield large errors.  
  - First-step coefficients should not be interpreted as causal effects; if forecasters know causal impacts, embed them into known variables rather than relying on historical correlation.  
  - Structural breaks and unprecedented non-policy shocks can cause large forecast errors due to overfitting; time-series cross-validation in the first step helps subsequently by discounting historical data via the weight matrix.  
  - Performance depends on accuracy of forecasts of known variables: biased known-variable forecasts will bias unknown-variable forecasts when known variables were historically informative.  
  - If forecasters have weak priors on some variables, they should include those variables among the unknowns to avoid spillovers.  
  - Scenario analysis: forecasters can specify policy-relevant but unlikely known-variable forecasts; uncertainty on known variables can be reflected by specifying distributions, drawing samples, and applying the method to obtain distributions for unknown-variable forecasts.

### 5. Conclusion and future research directions
- Summary of contribution:
  - Proposed a method combining high-dimensional conditional forecasting (elastic net with time-series cross-validation) and forecast reconciliation (projection onto accounting-identity constraint space) to systematize macroframework forecasting.  
  - Application to France using WEO data suggests the method can accommodate a large number of variables and improve WEO forecasts (mean RMSE improvement around 20 percent).  
- Future research directions:
  - Extend the framework to multilateral consistency.  
  - Apply the framework to mixed-frequency data to incorporate higher-frequency information and nowcasting insights.  
  - Extend the framework to allow log-linear constraints (to handle multiplicative accounting identities).  
  - Use the framework conversely to search for important variables.  

### Annex: Alternate country example (Seychelles) — key points
- Seychelles is a tourism-dependent economy; the example uses the same WEO vintages as France and a smaller variable set (Table 4 in source).  
- Known variables for Seychelles include exports of services (bxs) to capture inbound tourism; fiscal variables were replaced by services exports.  
- Forecast object: contributions to nominal GDP, defined as
  - r_it = (y_it − y_it−1) / (NGDP_t−1) (see equation (20) in source); applied to all variables including nominal GDP.  
- Accounting identities for contributions remain linear (see equation (21) in source).  
- Performance results for Seychelles (preserved exactly as reported):
  - The 2nd step forecast r̃ improves the WEO forecast by around 15 percent on average.  
  - Improvements mainly for external-activity-related variables (imports and exports and their goods/services breakdowns).  
  - WEO forecasts perform better for domestic-activity-related variables (consumption and investment).

*IMF Working Paper — Section 3: Country Example (from wpiea2022110-print-pdf)*

### Annex 3.  Forecast Error of Unknown Variables

### Annex 3.  Forecast Error of Unknown Variables

### One-year ahead forecast error of unknown variable — year 1/3
- WEO overestimated consumption in 2020....
- WEO overestimated private consumption in 2020....
- WEO overestimated foreign balance in 2016 ....
- WEO forecast is lower than 2nd step forecast for gross fixed capital formation....
- Forecast errors of private gross fixed capital formation look similar to NFI...
- Both WEO and 2nd step forecasts underestimated gross national saving....

### One-year ahead forecast error of unknown variable — year 2/3
- Both WEO and 2nd step forecasts underestimated private gross national saving....
- 2nd step forecast error is bigger in 2021....
- WEO overestimates changes in inventories in 2021....
- Forecast errors of private gross capital formation look similar to gross capital formation....
- 2nd step forecast error in imports of goods and services is bigger in 2019...
- Forecast errors of imports of goods trace each other....

### One-year ahead forecast error of unknown variable — year 3/3
- Forecast errors of imports of services have the opposite trend....
- 2nd step forecast generates a more volatile error in statistical discrepancies....
- WEO forecast error is smaller in 2021....
- 2nd step forecast overestimated exports of goods and services in 2020....
- 2nd step forecast overestimated exports of goods in 2020...
- 2nd step forecast overestimated exports of services in 2020....

*Source: Systematizing Macroframework Forecasting: High-Dimensional Conditional Forecasting with Accounting Identities, Working Paper No. WP/2022/110 — Annex 3. Forecast Error of Unknown Variables*

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