## Smooth Forecast Reconciliation (wpiea2024066-print-pdf)

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### Problem statement and contribution
- Forecasting multiple macroeconomic time series requires simultaneously:
  - Imposing linear constraints (accounting identities).
  - Ensuring smoothness over the forecast horizon (e.g., seasonally adjusted quarterly GDP aggregates to annual GDP and evolves smoothly).
- Shortcomings of common practice:
  - Fixing one quarter as the residual and manual quarterly adjustments ensure identities but create kinks and require resource-intensive manual re-adjustments.
  - Separate literatures handle (i) reconciliation to satisfy linear constraints and (ii) smoothing, but existing methods do not jointly guarantee both; iterative application can fail to converge.
- Method overview:
  - Integrates minimum-trace (min-T) reconciliation and the Hodrick-Prescott (HP) filter into a single quadratic programming problem.
  - Intuition: min-T allocates adjustments across series and horizons to satisfy constraints; HP imposes series-specific smoothness across horizons.
  - Key capabilities:
    - Accommodates any number of time series with any frequencies.
    - Allows any linear constraints across or within series.
    - Permits any smoothness parameters of choice.
    - Closed-form solution suitable for high-dimensional environments.
  - Positioned as a post-forecasting step; requires reasonable first-step forecasts and sensible constraints.

### Theory and closed-form solution
- Min-T reconciliation (summary):
  - Projects first-step forecast ŷ onto the constraint space C y = d using forecast-error covariance W = Var(ŷ − y*).
  - Second-step forecast ỹ solves argmin (y − ŷ)' W−1 (y − ŷ) s.t. C y = d.
  - Properties: oblique projection; idempotent under repeated application.
- HP filter (summary):
  - Minimizes sum of squared deviations plus λ times sum of squared second differences: solution ŷ = (I_T + λ F)−1 ỹ, where F is a degenerate penta-diagonal T×T matrix.
  - Typical λ values:
    - quarterly: λ = 1600
    - annual: λ = 100
    - monthly: λ = 14400
    - Ravn and Uhlig (2002) alternatives: annual λ = 6.25 and monthly λ = 129600
  - Interpreted here strictly as a smoothing device.
- Integrated optimization (equation (9)):
  - For M series stacked into y of length N, given ŷ ∈ ℝ^N, W = Var(ŷ − y*), constraints (C, d), and symmetric smoothness matrix Φ (block-diagonal with λ_m F_m), the method solves:
    - ỹ = argmin_{y ∈ ℝ^N} (y − ŷ)' W−1 (y − ŷ) + y' Φ y  s.t.  C y = d.
  - Special cases:
    - λ_m = 0 for all m reduces to min-T reconciliation.
    - W = I_N and (C,d) = (0,0) reduces to stacked HP filters.
- Closed-form solution (Theorem; equation (11)):
  - ỹ expressed as a combination of a weighted shrinkage of ŷ involving (W−1 + Φ)−1 W−1 ŷ and a constraint-imposed term (W−1 + Φ)−1 C' {C (W−1 + Φ)−1 C'}−1 d.
  - Nests min-T and HP as limiting cases and is computationally convenient for large N.
- Important property:
  - The integrated solution is not an oblique projection; applying it twice may yield a different result (inherits non-projection property from HP).

### Choosing smoothness parameters when W ≠ I_N (Section 2.2)
- Reduction to HP filter:
  - Problem (9) reduces to HP when W is diagonal with constant diagonals and (C,d) = (0,0).
  - Objective decomposes as Σ_{m=1}^M [ (1/(W_m)_{11}) (y_m − ŷ_m)' (y_m − ŷ_m) + λ_m^* y_m' F_m y_m ] in that special case.
- Baseline alignment (equation (13)):
  - λ_m = λ_m^* σ_m^2,  where σ_m^2 = min_{t=1,...,T_m} V( ŷ_{mt} − y_{mt}^* ) = min_{t=1,...,T_m} (W_m)_{tt}, m = 1, ..., M.
  - λ_m^* is the typical smoothness parameter for the frequency of series m (e.g., λ_m^* = 1600 for quarterly).
- Interpretation with expanding fan charts:
  - If (W_m)_{11} < ⋯ < (W_m)_{T_m T_m}, the objective for series m becomes (equation (15)):
    - (1/(W_m)_{11}) Σ_{t=1}^{T_m} (y_{mt} − ŷ_{mt})^2 + λ_m^* (W_m)_{11} Σ_{t=2}^{T_m−1} { (y_{m,t+1} − y_{mt}) − (y_{mt} − y_{m,t−1}) }^2.
  - Minimum-variance normalization produces stronger smoothing in more uncertain (farther) horizons.
- Alternatives and caveats:
  - Alternatives to minimum variance include median, average, or maximum variance; they yield the same HP-like expression in the special case.
  - Estimating {λ_m} empirically is possible but computationally challenging for large M.
  - Calibrating λ to historical-data smoothness may be suboptimal if forecasting dynamics differ (e.g., random walk DGP).
  - Forecast-error–minimizing λ may not be visually smooth enough for communication; forecasters may prefer smoother paths unless kinks are intentionally introduced.
- Implementation choice in the paper:
  - Uses minimum variance σ_m^2 as baseline normalization.
  - Example parameter used: Typical smoothness parameter for quarterly data: 1600.
  - Smoothness matrix specification for an 11‑period block (equation (20)):
    - Φ = 1600 / σ̂^2 F,  F: 11 × 11,  σ̂^2 = min_i (Ŵ)_{ii}.
  - Covariance estimation uses OASD shrinkage (equation (17)):
    - Ŵ = ρ̂ Σ̂ + (1 − ρ̂) diag(Σ̂),  with ρ̂ = min(1, 1/(n φ̂)), φ̂ defined as in (17).

### Application: Forecast of Annual and Quarterly GDP (Section 3.3)
- Inputs:
  - Forecast horizon:
    - Annual GDP: 2023 to 2028 (extended to 2022-2029 to mitigate HP filter end point problem).
    - Quarterly GDP: 2023Q3 to 2028Q4 (extended to 2023Q2 to 2029Q4).
  - First-step forecast ŷ:
    - Naïve forecast based on the growth rate of the last observation for both annual and quarterly GDP.
    - ŷ is a 35×1 vector stacking annual and quarterly forecasts (equation (22)).
  - Covariance estimation:
    - The 35×35 covariance matrix Ŵ estimated by OASD (17).
  - Constraints (equation (23)):
    - ỹ_t = y_t^WEO, t = 2022, 2023Q2.
    - For 2023: 4 ỹ_2023 − ỹ_2023 Q3 − ỹ_2023 Q4 = y_2023 Q1^WEO + y_2023 Q2^WEO.
    - For t = 2024,...,2029: 4 ỹ_t − ỹ_tQ1 − ỹ_tQ2 − ỹ_tQ3 − ỹ_tQ4 = 0.
    - Constraints in matrix form: 9 × 35 matrix C and 9 × 1 vector d.
  - Smoothness matrix Φ (equation (24)):
    - Block-diagonal:
      - Annual block: 100 / σ_A^2 · F_A (8×8 F_A).
      - Quarterly block: 1600 / σ_Q^2 · F_Q (27×27 F_Q).
    - σ_A^2 = min_{i=1,...,8} Ŵ_{iiii}.
    - σ_Q^2 = min_{i=9,...,35} Ŵ_{iiii}.
  - Second-step ỹ computed via equation (11).
- Findings:
  - ỹ is smooth and satisfies accounting identities.
  - ỹ can deviate substantially from WEO forecasts that incorporate future information (e.g., a recession and recovery in 2024–2025) when only historical data are used.
  - Reconciled-but-not-smoothed series are volatile and difficult to explain; smoothing suppresses unintended noise but may move forecasts away from WEO absent ad-hoc information.
- Incorporating ad-hoc information (equation (25)):
  - Imposed constraints:
    - ỹ_t = y_t^WEO, t = 2023, 2024, 2025 (short-run WEO levels).
    - For t = 2028,2029: ỹ_t − y_2028^WEO − y_2027^WEO · ỹ_{t−1} = 0 (long-run growth conditions).
  - Effect:
    - Ad-hoc constraints bring ỹ close to WEO forecasts.
    - Reconciled-but-not-smoothed forecast coincides with ỹ and y^WEO up to 2025 due to those constraints; remains more volatile in other years.
    - Quarterly GDP path tracks WEO well once annual GDP is conditioned.

### Cross-sectional and temporal constraints with subcomponents (Section 3.4)
- Expanded system inputs:
  - Annual variables: foreign balance FB_t, domestic demand DD_t, statistical discrepancy SD_t, plus annual and quarterly GDP.
  - Forecast horizon:
    - Annual variables: 2023 to 2028 (extended to 2022-2029).
    - Quarterly GDP: 2023Q3 to 2028Q4 (extended to 2023Q2 to 2029Q4).
  - First-step ŷ:
    - Naïve growth-rate forecast for quarterly GDP.
    - Annual variables use differences to allow positive/negative FB and SD.
    - ŷ is a 59×1 vector (equation (26)).
  - Covariance estimation:
    - 59×59 Ŵ estimated by OASD (17).
- Constraints (equation (27)):
  - Initial-year equalities: FB_2022 = FB_2022^WEO, DD_2022 = DD_2022^WEO, GDP_2022 = GDP_2022^WEO.
  - Quarterly-to-annual aggregation for GDP.
  - GDP expenditure identity for t = 2022,...,2029: GDP_t − FB_t − DD_t − SD_t = 0.
- Ad-hoc information (equation (28)):
  - GDP_t = GDP_t^WEO for t = 2023, 2024, 2025.
  - Medium/long-run constraints for GDP, FB, DD at t = 2028,2029 imposing convergence or links to prior levels.
  - Constraints represented by a 28 × 59 matrix C and 28 × 1 vector d.
- Smoothness matrix Φ for expanded system (equation (29)):
  - Block-diagonal with blocks for GDP annual, FB annual, DD annual, SD annual, and quarterly GDP:
    - Each annual block scaled by 100 / corresponding σ^2 and penta-diagonal F_A.
    - Quarterly block scaled by 1600 / σ_Q^2 and penta-diagonal F_Q.
  - Variance definitions (equation (30)):
    - σ_A^2 = min_{i=1,...,8} Ŵ_{iiii}.
    - σ_FB^2 = min_{i=9,...,16} Ŵ_{iiii}.
    - σ_DD^2 = min_{i=17,...,24} Ŵ_{iiii}.
    - σ_SD^2 = min_{i=25,...,32} Ŵ_{iiii}.
    - σ_Q^2 = min_{i=33,...,59} Ŵ_{iiii}.
- Output (section 3.4.2) findings:
  - Second-step forecast ỹ is smooth and traces WEO better than first-step and reconciled-but-not-smoothed forecasts.
  - Reconciled-but-not-smoothed forecast is more volatile and exhibits large kinks.
  - Differences between WEO and the proposed method are almost invisible except for the statistical discrepancy.
  - Foreign balance and domestic demand trace WEO satisfactorily with smoother paths even when only initial and terminal ad-hoc information is imposed.
  - Growth rates reported in figures are in percentage; ratios are not multiplied by 100.

### Comparisons, robustness, and implementation notes
- Comparison with Modified Denton (Annex I):
  - Modified Denton is a quadratic projection without explicit smoothness; yields less smooth series, similar to reconciliation-only results.
- Alternative first-step methods (Annex II):
  - Replacing naïve first step with AR(4) or AutoETS changes magnitudes but preserves qualitative results.
  - Back testing (Figures 9 and 10) compares average root mean squared percentage error across methods when first step is AR(4) or AutoETS.
- Practical implementation guidance:
  - Use baseline minimum-variance normalization for λ_m.
  - Covariance estimation via OASD shrinkage: Ŵ = ρ̂ Σ̂ + (1 − ρ̂) diag(Σ̂), ρ̂ = min(1, 1/(n φ̂)).
  - Forecasters may tune λ for communication needs or to incorporate ad-hoc information through linear constraints.

### Conclusions (section 4)
- The proposed smooth reconciliation method:
  - Produces forecasts that satisfy linear constraints and yield smooth time series.
  - Applies to multivariate mixed-frequency time series and accommodates high-dimensional settings via a closed-form solution.
  - Is flexible enough to incorporate accounting identities (within or across time) and ad-hoc forecaster information.
  - Imposing smoothness does not incur loss in forecast performance and improves practical usefulness by highlighting intentional kinks and suppressing unintended noise.

*Smooth Forecast Reconciliation, Working Paper No. WP/2024/066*

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

### 1. Introduction

### Problem statement
- Forecasting multiple macroeconomic time series often requires imposing linear constraints (accounting identities) while also ensuring smoothness over the forecast horizon (e.g., seasonally adjusted quarterly GDP must aggregate to annual GDP and evolve smoothly).
- Common practice (e.g., fixing Q4 as the residual and manual quarterly adjustments) ensures accounting identities but can create unwanted kinks and resource-intensive manual re-adjustments as the number of series increases.
- Two separate literatures address (i) reconciliation to satisfy linear constraints and (ii) smoothing time series, but existing methods do not jointly guarantee both properties; applying them iteratively can lead to non-convergence (a “whack-a-mole” problem).

### Method overview and contribution
- Proposes a new forecast reconciliation method that integrates minimum-trace (min-T) reconciliation and the Hodrick-Prescott (HP) filter to impose both constraints and smoothness in a single quadratic programming problem.
- Intuition: min-T determines how much each forecast horizon of each series should be adjusted to satisfy constraints; HP controls each series’ smoothness over the forecast horizon.
- The integrated method:
  - Accommodates any number of time series with any frequencies.
  - Allows any linear constraints across or within series.
  - Permits any smoothness parameters of choice.
  - Has a closed-form solution suitable for high-dimensional environments.
- The method is positioned as a post-forecasting process: reasonable first-step forecasts and sensible constraints remain crucial.

### Applications illustrated in the paper (overview)
- Three examples compare the proposed method to:
  - IMF World Economic Outlook (WEO) forecasts (treated as benchmark).
  - min-T reconciliation (reconciled-but-not-smoothed).
- Example 1: Annual US GDP fixed, forecast seasonally adjusted quarterly GDP subject to aggregation constraint. Result: the proposed method replicates WEO smoothness with slightly smaller forecast errors than min-T.
- Example 2: Forecast both annual and quarterly US GDP. Demonstrates incorporation of ad-hoc short-run and long-run constraints to reflect expert information (recessions, recoveries, potential growth), bringing data-driven forecasts closer to WEO.
- Example 3: Mixed-frequency multivariate series including subcomponents, showing ability to handle cross-sectional, temporal, and ad-hoc constraints simultaneously.

### Practical role
- The proposed method complements existing forecasting methods and can reduce the need for ex-post ad-hoc manual adjustments that many macroeconomists use to enforce internal consistency and smoothness.

---

### 2. Theory

### 2.1. Minimum-Trace Reconciliation and Hodrick-Prescott Filter (summary)
- min-T reconciliation (Wickramasuriya et al., 2019):
  - Projects a first-step forecast ŷ onto the space spanned by linear constraints (C, d).
  - Uses the covariance matrix of first-step forecast errors W = Var(ŷ − y*) as weighting; optimal when first-step forecast is unbiased.
  - Second-step forecast ỹ solves argmin (y − ŷ)' W−1 (y − ŷ) s.t. C y = d, with closed-form adjustment proportional to W and constraint residuals.
  - Properties: oblique projection; idempotent in the sense that applying reconciliation twice equals applying it once.

- HP filter (Hodrick and Prescott, 1997):
  - Smooths a single time series ỹ by minimizing sum of squared deviations plus λ times sum of squared second differences.
  - Solution: ŷ = (I_T + λ F)−1 ỹ, where F is a T×T degenerate penta-diagonal matrix.
  - Typical smoothness parameters:
    - quarterly: λ = 1600
    - annual: λ = 100 (Hodrick and Prescott, 1997)
    - monthly: λ = 14400 (Hodrick and Prescott, 1997)
    - alternative suggestions by Ravn and Uhlig (2002): annual λ = 6.25 and monthly λ = 129600
  - Interpreted here strictly as a smoothing device (paper abstracts from debate on HP’s trend-estimation accuracy).

- Similarities enabling integration:
  - Both take some data as exogenous inputs and adjust them.
  - Both are quadratic programming problems, allowing a combined constrained minimization with a closed-form solution.

- Key differences addressed:
  - min-T handles multiple series stacked into a vector and weights variables via W; HP handles a single series and does not weight across variables. The integration allows multiple smoothness parameters with appropriate units.

### 2.2. Closed-Form Solution with Given Smoothness Parameters
- Setup:
  - M time series {y_m}_{m=1}^M with lengths {T_m}_{m=1}^M are stacked into a vector y of length N = sum_{m=1}^M T_m.
  - Exogenous inputs:
    - First-step forecast ŷ ∈ ℝ^N and associated covariance matrix of forecast errors W = Var(ŷ − y*), assumed invertible (a feasible estimate Ŵ may replace the infeasible true W).
    - Linear constraints (C, d) with C being K×N and full rank: C y* = d.
    - Symmetric smoothness matrix Φ: block-diagonal with blocks λ_m F_m for m = 1,...,M, where each F_m is the T_m×T_m penta-diagonal matrix analogous to (4).
- Optimization problem:
  - ỹ = argmin_{y ∈ ℝ^N} (y − ŷ)' W−1 (y − ŷ) + y' Φ y  s.t.  C y = d.  (equation (9) in source)
- Special cases:
  - Reduces to min-T reconciliation when λ_1 = ... = λ_M = 0.
  - Reduces to stacked HP filters when W = I_N and (C, d) = (0, 0).
- Closed-form solution (Theorem; equation (11) in source):
  - ỹ equals a sum of two terms:
    - an adjusted-weighted shrinkage of the first-step forecast involving (W−1 + Φ)−1 W−1 ŷ, pre- and post-multiplied by a projection-like matrix, and
    - a constraint-imposed term involving (W−1 + Φ)−1 C' {C (W−1 + Φ)−1 C'}−1 d.
  - The formula nests (2) and (5) as limiting cases and yields a computationally convenient expression for high-dimensional N.
- Important property:
  - The integrated solution is not an oblique projection; applying it twice may yield a different result than applying it once (inherits non-projection property from HP filter).

### 2.3. Choice of smoothness parameters (note)
- The choice of smoothness parameters must reflect the units of the covariance matrix of forecast errors W.

### Section 2.2 demonstrates that problem (9) reduces to the HP filter when the weight is an identity matrix 푊푊=퐼퐼

### wpiea2024066-print-pdf - Section 2.2 demonstrates that problem (9) reduces to the HP filter when the weight is an identity matrix W = I_N

### Reduction to the HP filter (special case)
- Problem (9) reduces to the HP filter when the weight matrix W is diagonal with constant diagonals and constraints (C,d) = (0,0).
- In that special case the objective decomposes into a sum across m = 1, ..., M as in equation (14):
  - (y − ŷ)′ W−1 (y − ŷ) + y′ Φ y = Σ_{m=1}^M [ (1/(W_m)_{11}) (y_m − ŷ_m)′ (y_m − ŷ_m) + λ_m^* y_m′ F_m y_m ].
- Intuition: this corresponds to forecast errors that are uncorrelated with constant variances (white noise). It is more general than W = I_N since each submatrix W_m can take different values (but each W_m must have constant diagonals).

### Aligning smoothness parameter units when W ≠ I_N
- Typical smoothness parameters cannot be applied directly when W is not identity because forecast-error variances often expand over the horizon (fan charts).
- Baseline alignment uses the minimum variance of forecast errors (equation (13)):
  - λ_m = λ_m^* σ_m^2,  where σ_m^2 = min_{t=1,...,T_m} V( ŷ_{mt} − y_{mt}^* ) = min_{t=1,...,T_m} (W_m)_{tt}, m = 1, ..., M.
  - λ_m^* is the typical smoothness parameter for the frequency of the m-th time series (e.g., λ_m^* = 1600 if the m-th series is quarterly), and W_m is the submatrix of W corresponding to the m-th series.
- This normalization yields the HP-filter-like expression (14) in the special case noted above.

### Behavior when fan charts expand and interpretation
- Using the minimum variance σ_m^2 gives stronger smoothing force on farther forecast horizons when the weight matrix is diagonal with increasing diagonals:
  - If (W_m)_{11} < ⋯ < (W_m)_{T_m T_m}, the objective for series m becomes (equation (15)):
    - (1/(W_m)_{11}) Σ_{t=1}^{T_m} (y_{mt} − ŷ_{mt})^2 + λ_m^* (W_m)_{11} Σ_{t=2}^{T_m−1} { (y_{m,t+1} − y_{mt}) − (y_{mt} − y_{m,t−1}) }^2.
  - Interpretation: the weight on the first (fit-to-forecast) term becomes relatively smaller as t increases, while the smoothness penalty weight remains constant, so the resulting series is smoother where forecasters have relatively weaker confidence.
- Thus, minimum-variance normalization tends to produce smoother paths in more uncertain horizons.

### Alternatives for choosing smoothness parameters and caveats
- Alternatives to the minimum include using the median, average, or maximum variance of forecast errors; these alternatives produce the same expression (14) in the special case.
- Empirical estimation of {λ_m}_{m=1}^M is possible but can be computationally challenging when optimizing M parameters.
- Caveats when estimating λ:
  - Targeting historical-data smoothness may be suboptimal (Section 3.2.2). Example: if the data-generating process is a random walk, the best forecast is constant, so historical volatility should not necessarily determine forecast smoothness.
  - Estimated λ that minimize forecast-error variance may still produce forecasts that are not visually smooth enough for communication purposes; forecasters may prefer smoother paths for public explanation unless kinks are intentionally introduced with an accompanying narrative.

### Implementation choice in the paper and practical guidance
- The paper uses the minimum variance of forecast errors (equation (13)) as the baseline smoothness parameter.
- Practical recommendation: forecasters can choose alternative values depending on their objectives (communication vs. strictly minimizing forecast-error variance).

### Connection to applications (previewed)
- The baseline normalization is used in applied examples (Section 3) that illustrate replication of expert-generated smoothness, slight RM S PE improvements relative to min‑T reconciliation, and the ability to incorporate constraints.
- Example parameters and values used in applications:
  - Typical smoothness parameter for quarterly data: 1600.
  - Smoothness matrix specification for a 11‑period block (equation (20)):
    - Φ = 1600 / σ̂^2 F,  F: 11 × 11,  σ̂^2 = min_i (Ŵ)_{ii}.
  - Estimation of W uses OASD shrinkage (equation (17)) to produce Ŵ:
    - Ŵ = ρ̂ Σ̂ + (1 − ρ̂) diag(Σ̂),  with ρ̂ = min(1, 1/(n φ̂)), φ̂ defined as in (17).

*Italic source attribution: Section 2.2 of wpiea2024066-print-pdf (IMF Working Paper) as provided in the content unit.*

### Section 3.2 illustrates the forecast of quarterly GDP. This section extends the example to the forecast of both

### Section 3.3 Forecast of Annual and Quarterly GDP

### Input
- Forecast horizon:
  - Annual GDP: 2023 to 2028 (extended to 2022-2029 to mitigate HP filter end point problem).
  - Quarterly GDP: 2023Q3 to 2028Q4 (extended to 2023Q2 to 2029Q4).
- First-step forecast ŷ:
  - Naïve forecast based on the growth rate of the last observation for both annual and quarterly GDP.
  - ŷ is a 35×1 vector stacking annual and quarterly forecasts (equation (22)).
- Covariance estimation:
  - The 35×35 covariance matrix of forecast errors Ŵ is estimated by OASD (17).
- Constraints for the second-step forecast ỹ (equation (23)):
  - ỹ_t = y_t^WEO, t = 2022, 2023Q2.
  - For 2023: 4 ỹ_2023 − ỹ_2023 Q3 − ỹ_2023 Q4 = y_2023 Q1^WEO + y_2023 Q2^WEO (constraint uses quarterly WEO values; annual WEO 2023 not used).
  - For t = 2024,...,2029: 4 ỹ_t − ỹ_tQ1 − ỹ_tQ2 − ỹ_tQ3 − ỹ_tQ4 = 0 (accounting identities).
  - Constraints expressed in matrix form using a 9 × 35 matrix C and 9 × 1 vector d.
- Smoothness matrix Φ (equation (24)):
  - Φ is block-diagonal with blocks:
    - Annual block: 100 / σ_A^2 · F_A (8×8 penta-diagonal F_A).
    - Quarterly block: 1600 / σ_Q^2 · F_Q (27×27 penta-diagonal F_Q).
  - σ_A^2 = min_{i=1,...,8} Ŵ_{iiii}.
  - σ_Q^2 = min_{i=9,...,35} Ŵ_{iiii}.
- Second-step forecast ỹ is obtained by combining updated first-step forecast and covariance (ŷ, Ŵ), constraints (C, d), and smoothness matrix Φ via equation (11).

### Output
- Main findings:
  - The second-step forecast ỹ is smooth and satisfies accounting identities.
  - However, ỹ does not trace the WEO forecast closely when only historical data are used; it can deviate substantially from WEO forecasts that incorporate future information (e.g., a recession and recovery in 2024–2025).
  - The reconciled-but-not-smoothed time series are volatile and difficult to explain economically; smoothing suppresses unintended noise but may move forecasts away from WEO if no ad-hoc information is imposed.
- Illustration:
  - Figure 5: shows US annual GDP growth (left) and US quarterly GDP quarter-on-quarter growth rate (right) with only accounting identities imposed.

### Incorporating Ad-Hoc Information
- Rationale:
  - Forecasters may have ad-hoc future information (e.g., planned budget cuts, targeted potential growth rates) they wish to impose and to smooth connections across periods to produce an explainable economic story.
- Constraints to incorporate WEO short-run and long-run information (equation (25)):
  - ỹ_t = y_t^WEO, t = 2023, 2024, 2025 (impose short-run WEO levels).
  - For t = 2028,2029: ỹ_t − y_2028^WEO − y_2027^WEO · ỹ_{t−1} = 0 (long-run growth conditions as specified in (25)).
  - These are imposed in addition to the constraints in (23).
- Effect:
  - Figure 6: the ad-hoc constraints (25) bring ỹ close to the WEO forecast.
  - The reconciled-but-not-smoothed forecast coincides with ỹ and y^WEO up to 2025 due to those constraints, though it remains more volatile in other years.
  - Quarterly GDP path tracks the WEO forecast well once annual GDP is conditioned, indicating limited quarter-specific dynamics beyond the annual conditioning.

### Cross-Sectional and Temporal Constraints: Adding Subcomponents
- Expanded input (section 3.4.1):
  - Additional annual variables: foreign balance FB_t, domestic demand DD_t, statistical discrepancy SD_t, together with annual and quarterly GDP.
  - Forecast horizon:
    - Annual variables: 2023 to 2028 (extended to 2022-2029 for HP filter).
    - Quarterly GDP: 2023Q3 to 2028Q4 (extended to 2023Q2 to 2029Q4).
  - First-step forecast ŷ:
    - Naïve growth-rate forecast for quarterly GDP.
    - For annual variables, the first-step uses differences to allow positive and negative values for FB and SD.
    - ŷ is a 59×1 vector stacking annual and quarterly forecasts (equation (26)).
  - Covariance estimation:
    - The 59×59 covariance matrix of forecast errors Ŵ is estimated by OASD (17).
- Constraints (equation (27)) include:
  - Initial-year equalities: FB_2022 = FB_2022^WEO, DD_2022 = DD_2022^WEO, GDP_2022 = GDP_2022^WEO.
  - Quarterly-to-annual aggregation constraints analogous to (23) for GDP.
  - GDP expenditure identity for t = 2022,...,2029: GDP_t − FB_t − DD_t − SD_t = 0.
- Ad-hoc information (equation (28)) analogous to (25):
  - GDP_t = GDP_t^WEO for t = 2023, 2024, 2025.
  - Medium/long-run constraints for GDP, FB, and DD at t = 2028,2029 imposing convergence to 2028 levels or linking levels to prior period (as specified in (28)).
  - Note: FB and DD are not required to coincide with WEO for 2023–2025 to observe how subcomponents adjust when annual GDP exhibits a recession.
  - Constraints expressed in matrix form using a 28 × 59 matrix C and 28 × 1 vector d.
- Smoothness matrix Φ for the expanded system (equation (29)):
  - Block-diagonal with blocks for GDP annual, FB annual, DD annual, SD annual, and quarterly GDP:
    - Each annual block scaled by 100 / corresponding σ^2 and associated penta-diagonal F_A.
    - Quarterly block scaled by 1600 / σ_Q^2 and penta-diagonal F_Q.
  - Variance definitions (equation (30)):
    - σ_A^2 = min_{i=1,...,8} Ŵ_{iiii}.
    - σ_FB^2 = min_{i=9,...,16} Ŵ_{iiii}.
    - σ_DD^2 = min_{i=17,...,24} Ŵ_{iiii}.
    - σ_SD^2 = min_{i=25,...,32} Ŵ_{iiii}.
    - σ_Q^2 = min_{i=33,...,59} Ŵ_{iiii}.
- Output (section 3.4.2):
  - Figure 7 shows that the second-step forecast ỹ is smooth and traces the WEO forecast better than the first-step and reconciled-but-not-smoothed forecasts.
  - The reconciled-but-not-smoothed forecast is more volatile and exhibits large kinks.
  - Differences between the WEO forecast and the proposed method are almost invisible except for the statistical discrepancy.
  - Foreign balance and domestic demand trace WEO satisfactorily with smoother paths even when only initial and terminal ad-hoc information is imposed.
  - Note: Growth rates are in percentage; ratios are not multiplied by 100.

### Comparison and Robustness (Annexes)
- Comparison with Modified Denton (Annex I):
  - Modified Denton (Cholette 1984) applied to benchmarking is a quadratic projection without an explicit smoothness mechanism.
  - Problem can be written in quadratic form (equation (31) and (32)) with weight matrix Ω = L L′ and diagonal scaling by 1 / ŷ_t.
  - Result: modified Denton yields less smooth time series (similar to reconciliation-only), consistent with lack of smoothness constraint.
- Alternative first-step methods (Annex II):
  - Replacing the naïve first step with AR(4) or AutoETS changes quantitative magnitudes but preserves qualitative results (Figures 9 and 10).
  - Figure 9: back testing when first step is AR(4) — average root mean squared percentage error comparison across four methods.
  - Figure 10: back testing when first step is AutoETS — average root mean squared percentage error comparison across four methods.

### Conclusions (section 4)
- The paper proposes a smooth reconciliation method that:
  - Produces forecasts satisfying linear constraints and smooth time series.
  - Applies to multivariate mixed-frequency time series and is amenable to high-dimensional settings via a closed-form solution.
  - Is flexible enough to incorporate accounting identities (within or across time) and ad-hoc forecaster information.
  - Imposing smoothness does not incur loss in forecast performance and improves practical usefulness by highlighting intentional kinks and suppressing unintended noise.

*Smooth Forecast Reconciliation, Working Paper No. WP/2024/066*

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_Source: https://www.imf.org/-/media/files/publications/wp/2024/english/wpiea2024066-print-pdf.pdf_