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---

### 3.1 Growth Decomposition — Methodology, Data, and Findings
- Methodology
  - Framework: semi-endogenous growth model and growth accounting decomposition following Jones (2022).
  - Aggregate production function: Y = K^α (A h L)^(1−α).
  - Per capita expression and decomposition: y ≡ Y/P = (K/Y)^(α/(1−α)) A h (L/P); with labor force LF introduced: y = (K/Y)^(α/(1−α)) A h (L/LF) (LF/P).
  - Decomposed contributions: capital deepening (K/Y), total factor productivity (A), human capital (h), employment-population ratio (L/P), and demographic factor (LF/P).
- Data
  - Economies covered: Australia, Bangladesh, Brunei Darussalam, Cambodia, China, Fiji, Hong Kong SAR, India, Indonesia, Japan, Korea, Lao P.D.R., Malaysia, Maldives, Mongolia, Nepal, New Zealand, Philippines, Singapore, Sri Lanka, Taiwan Province of China, Thailand, Vietnam.
  - Sources: Penn World Tables (real GDP 2017 prices, capital stock 2017 prices, human capital index, employment, population); International Labour Organisation (average labor income shares); United Nations (working-age labor force).
- Key findings (1991–2019 averages and decadal dynamics)
  - Cross-group average growth differences:
    - Advanced Economies (AEs): slowest average economic growth.
    - Emerging Market and Middle Income Economies (EMMIEs): intermediate growth.
    - Low-Income Developing Countries (LIDCs): fastest growth.
  - Drivers by group (average annual 1991-2019 decomposition components: k/y = capital-output ratio, tfp = TFP growth, hk = human capital, emp_pop = employment-population ratio, gdp_pc = GDP per capita growth):
    - AEs:
      - Growth mainly driven by increases in TFP and human capital.
      - Demographic factors contributed to a lesser extent.
      - Capital-to-output ratios remained essentially unchanged and marginally detracted from growth.
    - EMMIEs and LIDCs:
      - Capital deepening accounted for roughly one-third of total growth.
      - TFP, human capital, and demographic factors also contributed positively.
  - Decadal dynamics (1991-1999, 2000-2009, 2010-2019):
    - AEs: strongest performance in the 1990s; growth declined thereafter; capital-to-output ratios accounted for most of the decline in AE growth.
    - EMMIEs: uptick in TFP contributions and softening of capital deepening contributions.
    - LIDCs: capital deepening was the main factor for acceleration in the last two decades.
  - Demographics:
    - All Asia-Pacific economies benefited from demographic tailwinds as the share of active workers in the population increased over time.
    - Mechanisms: (i) higher labor force participation (e.g., increased female labor force participation); (ii) demographic dividends via a relatively young population increasing the working-age share (inverse dependency ratio).
    - Sustainability concern: participation increases are bounded and population ageing is already underway; demographic tailwinds unlikely to be sustained.
  - Employment-population ratio contributions (average annual 1991-2019):
    - AEs: average annual 0.5 percentage point increase in employment-to-population ratio driven almost exclusively by higher labor force participation.
    - EMMIEs and LIDCs: increases mainly from improved inverse dependency ratios as younger cohorts entered working age.
    - Outlook: recent trends point to a reversal as populations age and retiree shares grow relative to working-age individuals.

### 3.2 Tree-Based Approach — Methodology, Data, and Findings
- Methodology
  - Models: tree-based ensemble machine learning models—random forests and XGBoost.
  - Target variable: average real GDP growth in the next five years (from year t+1 to t+5) for each country-year observation.
  - Predictor contributions explained with Shapley values (Strumbelj and Kononenko, 2010; Lundberg, 2017).
  - Additional technical details in Appendix A.
- Data and predictors
  - Sample: 22 Asia-Pacific economies over 1970-2019; predictor values taken from year t to predict t+1 to t+5.
  - Predictors: total of 40 predictors grouped into development and demographics; structural conditions; macroeconomics; global factors and openness.
  - Note: some determinants excluded due to collinearity (e.g., level of governance highly correlated with nominal GDP per capita—level excluded but percentage change included).
- Main predictive drivers (Shapley-value-based; variables accounting for 70 percent of total contributions reported)
  - Most important predictors:
    - Nominal GDP per capita (income level): higher nominal GDP per capita observations tend to reduce five-year average growth relative to sample average; lower values increase it (consistent with convergence).
    - Working-age population share growth: higher growth associated with higher medium-term growth.
    - Trade openness and change in trade openness: higher degree and increases associated with higher medium-term growth.
    - Growth in human capital: higher growth associated with higher medium-term growth.
    - Improvement in governance: associated with higher medium-term growth.
    - Import partners’ growth and FDI inflows: important external drivers associated with higher medium-term growth.
  - Nature of associations:
    - Many drivers show approximately linear associations with medium-term growth (nominal GDP per capita, working-age population share growth, trade openness, human capital growth, governance, import partners’ growth, FDI).
    - Domestic macroeconomic variables exhibit nonlinear associations:
      - External debt: economies with external debt more than 50 percent of GDP found to have (slightly) higher medium-term growth in some cases, though high growth also observed with much lower external debt.
      - Public debt: when public debt is lower than about 60 percent of GDP, lower public debt is associated with higher medium-term growth; relationship less strong above 60 percent of GDP.
      - Fiscal balance: fiscal deficits larger than 5 percent of GDP are associated with low medium-term growth; some weak evidence suggests larger fiscal balances could be associated with lower future growth (possibly reflecting effects of fiscal consolidation).
- Evolution over time by income group (contribution to five-year average growth in excess of entire Asia 1970-2019 average; period 1990-2019 in five-year windows)
  - Average growth rates (five-year windows, 1990-2019):
    - Advanced Asia: average growth of 3.3 percent.
    - Emerging Asia: average growth of 4.2 percent.
    - Low-Income Developing Countries (LIDCs) in Asia: average growth of 5.5 percent.
  - Advanced Asia:
    - Development level and demographics became the main growth drag since the late 1990s; negative contribution increased over time, lowering AE growth by as much as 2 percentage points below the regional average in the last decade.
    - Macroeconomic conditions contributed positively historically (especially early 2000s) but have not contributed positively to medium-term growth since the Global Financial Crisis (except recovery episode).
    - Global factors have become a growth drag in the last decade; importers’ growth had the largest negative contributions in the last five years (likely linked to China’s slowdown).
    - Structural conditions (governance, human capital, labor force participation) continued to improve and contributed positively.
  - Emerging Asia (EMMIEs):
    - Gaps from the development frontier and growing working-age populations contributed positively but these contributions have been diminishing.
    - Since the early 2010s, development and demographic factors began to become a growth drag relative to the regional average.
    - Structural conditions improved significantly since the 2000s, contributing positively.
    - Lower global growth after the Global Financial Crisis, including China’s slowdown, contributed less to medium-term growth in the last decade.
    - Higher public debt since the Global Financial Crisis has been one main negative macroeconomic contributor.
  - LIDCs:
    - Development level and demographics have been the most important positive growth drivers relative to the regional average; positive contributions decreased in the last decade but remained positive.
    - Structural improvements contributed positively, especially in the 2000s; governance index increased by about 50 percent in the second half of the 2000s for LIDCs.
    - Macroeconomic conditions contributed positively before the last decade but have become a growth drag since the 2010s due to large fiscal deficits.

### Appendix A (model training, testing, Shapley explanation)
- Cross-validation and hyperparameter tuning (Appendix A.2)
  - Block time-series cross-validation with five training–validation pairs (training windows ending 2004/2006/2008/2010/2012; corresponding validations 2005–2006, 2007–2008, 2009–2010, 2011–2012, 2013–2014).
  - For each algorithm: 1000 trees constructed; hyperparameters tuned (Random Forest: maximum tree depth, subsample ratio of columns; XGBoost: L2 regularization, learning rate) via Bayesian optimization over 100 iterations.
  - Final model selected as algorithm with lowest average RMSE across validation sets.
- Model explanation (Appendix A.3)
  - Shapley values used to unpack predictor contributions; for linear models Shapley value = estimated coefficient × predictor value; for non-linear models Shapley value computed as average marginal contribution across coalitions of predictors.

### Policy-relevant implications (from tree-based findings)
- Key actionable drivers of medium-term growth: policies that raise working-age population participation, deepen trade openness, improve human capital, strengthen governance, attract FDI, and foster strong external demand.
- Macro-fiscal constraints: fiscal deficits larger than 5 percent of GDP and rising public debt above about 60 percent of GDP are associated with weaker medium-term growth—underscoring fiscal sustainability considerations.

### 3.3 Comparing the Two Approaches — Complementarities and Transition to Forecasting
- Complementarities
  - Both growth decomposition and tree-based approaches identify demographics and structural transformation as central to past and future growth patterns.
  - The two approaches provide consistent historical diagnostics and inform forecasting methods: a production-function-based approach and a pattern-matching approach (DTW).
- Historical drivers by country group (summary)
  - AEs: demographic factors became a drag since late-1990s/early-2000s; human capital and labor force participation became important growth drivers.
  - EMMIEs: productivity growth and increased human capital have been main drivers since 2000s; gains linked to technology transfer and trade openness.
  - LIDCs: capital deepening has been the most important growth driver since 2000s; governance improvements supported investment and FDI attraction.
- Transition to forecasting
  - Growth accounting projects fundamental factors (K, A, h, L) mostly via linear regressions.
  - Pattern-matching (DTW) leverages historical analogues to identify likely development paths, capturing non-linearities and structural transformations.

### Growth accounting methodology and assumptions (key structure and explicit assumptions)
- Production function growth-rate form: ˆY = α ˆK + (1−α)(ˆA + ˆh + ˆL).
- Sample for forecasting: 20 regional economies (Brunei Darussalam, Macao SAR, Maldives, and Myanmar drop out due to continuous negative TFP growth).
- Key data and assumptions:
  - Capital stock K projected using the perpetual inventory method; data and depreciation from Penn World Table (PWT) version 10.01. Linear regressions based on 2009-19 imply increasing depreciation rates for most economies.
  - TFP growth projected from country-specific linear regressions of TFP levels during 2009-19; historical TFP derived as residual from semi-endogenous growth decomposition.
  - Employment L modeled by age cohort-gender groups Li,j = Pi,j × LFPRi,j with Pi,j from UN World Population Prospects medium-fertility scenario; LFPRi,j forecast by country-specific linear regressions based on 2000-22; projected linear trends bound by regional maxima/minima and logistic growth regression applied if bounds hit.
  - Human capital assumed to continue its linear trend (PWT human capital measure: average years of schooling and assumed rate of return from Mincer equations).
  - Income shares (1−α labor share and α capital share) assumed constant based on ILO data.

### Main growth accounting forecast findings (numeric projections preserved)
- Aggregate potential growth across the region projected to decline:
  - Around 3.1 percent by 2030 and 2.2 percent by 2040, compared to 3.7 percent pre-pandemic growth.
- EMDEs projections:
  - Growth projected to fall to 3.8 percent in 2030 and 2.8 percent in 2040, compared to pre-pandemic growth of 4.8 percent.
- Drivers of slowdown:
  - Labor: lower contribution from labor due to ageing and reversal of demographic dividend; partially offset by positive net migration and higher LFPRs among females and the elderly.
  - Capital: declining contribution from capital projected to lower potential growth for AEs and more so for EMMIEs.
  - TFP: contributions from TFP will slow.
  - Human capital: positive contributions will continue.
- Per capita considerations:
  - Improvements in per capita metrics (human capital, employment ratios) can support sustainable long-term growth.
  - In per capita terms, slowing capital-output ratios (especially for LIDCs) and falling TFP contributions (in particular for EMMIEs) explain downward growth trend.

### Policy implications from growth accounting forecasts
- Address demographic headwinds by raising labor force participation (especially female and elderly cohorts), and by enabling productive migration where feasible.
- Support human capital accumulation to sustain per capita growth.
- Reassess policies to sustain capital accumulation and productivity growth, including policies facilitating structural transformation and technology adoption.
- Maintain fiscal sustainability to preserve macroeconomic space for growth-supporting policies.

### 4.2 Dynamic Time Warping (DTW) Forecasts — Algorithm, PatternSync, Backtesting, and Forward Forecasts
- DTW algorithm and intuition
  - DTW measures similarity between time series that may vary in speed/timing by constructing pair-wise distances (Euclidean distance for multivariate series) and a cumulative cost matrix; final DTW distance = C[n][m].
  - Multivariate distances use q features; variables normalized to minimize scaling impact.
  - Intuition: DTW ‘warps’ time to align shapes (e.g., growth spurts at different times) to identify similar episodes despite timing differences.
- Pattern matching data (multivariate indicators)
  - Six indicators from Penn World Table 10.01: GDP per capita; capital-output ratio; consumption-output ratio; labor productivity; real GDP growth; population growth.
  - Dataset spans 183 economies from 1950 to 2019.
  - Robustness: dropping one variable degrades performance but method remains functional.
- PatternSync forecasting procedure
  - Target period: k = 5 years (most recent available base period [t−4,t]).
  - Compute DTW distances between target base period and all historical periods; select top x percent most similar country-periods; baseline x = 2.
  - Forecast horizon y set to 10 years in backtesting; aggregate subsequent growth trajectories of selected similar periods using median (primary), simple average, weighted average (weights ∝ inverse DTW distance), and percentile values for scenarios.
- Backtesting results (26 Asia-Pacific economies, rolling 5-year base periods 1995–1999 to 2015–2019; forecast horizon 10 years)
  - Evaluation metrics: average error (AE), root mean squared error (RMSE), hit rate (HR; percentage of time forecast falls within 25% band of actual).
  - Key backtesting observations:
    - Median Absolute Error (MAE) stabilizes around 0.9 percentage points for 5-year-ahead forecasts by end of sample.
    - Hit rate improves over time; around 60% by end of sample for 5-year-ahead forecasts.
    - DTW maintains relatively stable accuracy between short-term (Years 1-3) and medium-term (Years 4-7) horizons; decay in accuracy is smaller than for conventional forecasts.
    - Weighted average aggregation (inverse distance weights) performs slightly better for longer horizons.
  - Comparison with conventional forecasts (WEO and EIU) for 2011-2019:
    - Short-term (Years 1-2): WEO and EIU outperform DTW on hit rates (WEO and EIU hit rates 68-79% in Year 1; DTW 65%).
    - Year 1 RMSE: EIU 0.80; WEO 1.41; DTW 1.49.
    - Medium-term (Years 4-7): DTW MAE remains around 0.7-0.8 percentage points while WEO and EIU show larger increases; by Year 5 DTW achieves comparable or better performance across most metrics.
    - Directional bias: WEO and EIU exhibit optimistic bias (positive average errors growing with horizon); DTW shows a mild negative bias that remains relatively stable (DTW average errors: -0.33, -0.32, -0.32, -0.30, -0.31, -0.33, -0.31, -0.33, -0.36 for Years 1–9).
  - Table 1 numeric excerpts (Asia-Pacific, 2011-2019):
    - Median absolute error (in pp):
      - WEO: 0.64, 0.86, 0.97, 1.03, 1.12 (Years 1–5 partial listing)
      - EIU: 0.56, 0.58, 0.68, 0.76, 0.76, 0.69, 0.63, 0.62, 0.57 (Years 1–9)
      - DTW: 0.82, 0.81, 0.81, 0.78, 0.79, 0.69, 0.74, 0.78, 0.72 (Years 1–9)
    - RMSE (in pp):
      - WEO: 1.41, 1.75, 2.03, 2.06, 1.88 (Years 1–5 partial listing)
      - EIU: 0.80, 0.91, 1.22, 1.38, 1.39, 1.15, 1.20, 1.21, 1.08 (Years 1–9)
      - DTW: 1.49, 1.37, 1.29, 1.27, 1.25, 1.17, 1.10, 1.03, 1.00 (Years 1–9)
    - Average error (in pp):
      - WEO: 0.30, 0.44, 0.58, 0.65, 0.69 (Years 1–5 partial listing)
      - EIU: 0.11, 0.27, 0.40, 0.63, 0.51, 0.35, 0.23, 0.18, 0.17 (Years 1–9)
      - DTW: -0.33, -0.32, -0.32, -0.30, -0.31, -0.33, -0.31, -0.33, -0.36 (Years 1–9)
    - Hit rate:
      - WEO: 68%, 59%, 55%, 50%, 41% (Years 1–5 partial listing)
      - EIU: 79%, 71%, 68%, 59%, 63%, 61%, 63%, 59%, 56% (Years 1–9)
      - DTW: 65%, 64%, 64%, 63%, 62%, 65%, 65%, 65%, 65% (Years 1–9)
- Applicability to Advanced Economies (AEs)
  - AEs have fewer historical analogues near the technological frontier, but DTW remains applicable by matching across phases rather than levels; backtesting suggests reasonable accuracy though higher variance for AEs.
- Forward-looking DTW forecasts (long-term through 2040 using 2015-2019 base period)
  - Median forecast patterns for 2030–2040 (Figure 14 summary):
    - Advanced Economies (AEs): around 1.8-2.0% on average over 2030-2040.
    - Emerging Market and Middle-Income Economies (EMMIEs): 3.2-3.6%.
    - Low-Income Developing Countries (LIDCs): 4.5-5.3%.
    - Growth differentials narrow over time (convergence).
    - Fast-growing EMMIEs/LIDCs (e.g., Cambodia, Vietnam, Bangladesh) projected to maintain growth above 4% through 2040 with some moderation.
    - China: expected moderation from around 4.2% in 2030 to 3.3% by 2040 based on 75th percentile forecasts.
    - India: growth around 3.3% through the forecast horizon.
  - Interpretation of percentiles and mean reversion:
    - High percentile forecasts reflect recent outperformance; historical evidence indicates mean reversion in cross-country growth rankings.
    - Regression evidence on non-overlapping 10-year data (183 economies, 1959-2019) with country fixed effects shows a negative and significant coefficient for current ranking predicting future ranking (mean reversion).
    - Example without fixed effects: coefficient for current growth ranking ≈ 0.16 and constant ≈ 0.42 implies a country at 75th–80th percentile now expected to drop to 54th–55th percentile in 10 years.
  - Caveats:
    - Forecasts assume continuation of relatively stable global technological and productivity growth as in the post-World War II sample.
    - Accuracy could decline under disruptive changes (e.g., uneven AI-driven productivity gains, major geopolitical shifts).

### 4.3 Comparing DTW and Growth Decomposition — Differences, Complementarity, and Limitations
- Shared broad features
  - Both methods forecast moderation of growth over time across income groups and narrowing differentials (convergence).
  - Both highlight demographics, slower TFP growth, and less capital deepening as important explanatory factors for projected moderation.
- Key differences
  - DTW almost consistently forecasts higher growth than growth decomposition for many economies.
  - Reasons:
    - DTW leverages a large pool of historical analogues and captures non-linear structural transformations that linear projections in growth accounting may miss.
    - Growth accounting projects each factor linearly (capital, TFP, human capital, labor); DTW searches for historical analogues matching multivariate structural patterns and momentum.
    - DTW can imply higher or lower growth paths for economies undergoing substantial structural shifts.
- Shared limitations and interpretation
  - Both rely on historical patterns and observed relationships; neither can predict structural changes not observed in the past (e.g., disruptive technologies or major reforms).
  - Projections are baseline scenarios conditional on historical persistence and are methodological contributions rather than operational IMF desk forecasts.
  - Usefulness: complementary analytical tools—growth accounting provides theory-grounded factor decomposition; DTW provides history-based pattern-matching and scenario distribution (median/percentiles).

### Appendix C — DTW Case Study: Cambodia (policy-relevant insights)
- Method and sample selection
  - DTW PatternSync used to identify historically similar country-periods to Cambodia’s 2015-19 base period.
  - Similarity selection: top 2 percent most similar episodes; number of similar episodes identified: 156 (concentrated among East and Southeast Asian economies).
  - Examples of top similar country-periods: India 2001–2005; Indonesia 2002–2006; Lao P.D.R 2008–2012; Sri Lanka 1983–1987; Japan 1951–1955; China 1986–1990; Vietnam 2010–2014; Korea 1973–1977; Philippines 2004–2008; Thailand 1985–1989.
- Medium-term forecast results for Cambodia (2030)
  - Forecasted potential growth for 2030:
    - 5.8 percent (average forecast).
    - 6.3 percent (medium forecast).
  - Cambodia’s trend growth during the target period ranked at the 75th to 80th percentile of the top 2 percent most similar cohort.
  - Aspirational percentile forecasts if Cambodia maintains 75th–80th percentile performance:
    - Potential growth of 7.5 to 7.9 percent for 2030.
- Mean reversion and peer subsequent performance
  - Subset peers matching Cambodia’s base-period growth performance (75th–80th percentile):
    - Number of such country-periods: 7.
    - Average growth rate 10 years after the base period: 6.4 percent.
    - Median growth rate 10 years after the base period: 5.9 percent.
    - In all except one instance (China 1996–2000), growth performance dropped out of the 75th–80th percentile range after a decade.
  - Broader subgroup (75th–100th percentile):
    - Number of instances: 40.
    - Average growth rate a decade after base period: 5.8 percent.
    - Median growth rate a decade after base period: 6.4 percent.
- Outlier sustained-high-growth cases and commonalities
  - Outliers sustaining high growth: Japan (1956–1967 narrative), South Korea (1971–1986 narrative), China (1996–2002 narrative).
  - Common factors associated with sustained outperformance: export-oriented industrialization and integration into global value chains; high investments in human capital; institutional reforms improving governance and property rights.
  - Comparative observation: Cambodia lags these outliers in human capital and labor productivity during their base periods.
- Analytical value and policy implications for Cambodia and similar economies
  - DTW PatternSync offers structured identification of historical parallels, quantitative baseline and aspirational forecasts, and statistical assessment of likelihood of sustained outperformance.
  - Policy priorities suggested by historical successful peers: export orientation and global integration, human capital investment, and institutional reforms to improve governance and attract investment.

*Source: wpiea2025168-print-pdf — Authors’ calculations based on data from PWT, ILO, and UN.*

### 3.1  Growth Decomposition

### 3.1  Growth Decomposition

### 3.1.1  Methodology
- Framework: semi-endogenous growth model and growth accounting decomposition following Jones (2022).
- Aggregate production function: Y = K^α (A h L)^(1−α).
- Per capita expression: y ≡ Y/P = (K/Y)^(α/(1−α)) A h (L/P).
- Introducing labor force LF gives: y = (K/Y)^(α/(1−α)) A h (L/LF) (LF/P).
- Interpretation of terms:
  - K: capital stock.
  - A: total factor productivity (TFP).
  - h: human capital.
  - L: employment.
  - LF: labor force.
  - P: population.
  - L/LF: employment rate (employment-to-labor-force ratio).
  - LF/P: inverse of the dependency ratio (share of working-age population in total population).
- Approach: differentiate the per capita expression with respect to time to decompose GDP per capita growth into contributions from capital deepening (K/Y), TFP (A), human capital (h), employment-population ratio (L/P), and demographic factors (LF/P).

### 3.1.2  Data
- Economies covered: Australia, Bangladesh, Brunei Darussalam, Cambodia, China, Fiji, Hong Kong SAR, India, Indonesia, Japan, Korea, Lao P.D.R., Malaysia, Maldives, Mongolia, Nepal, New Zealand, Philippines, Singapore, Sri Lanka, Taiwan Province of China, Thailand, Vietnam.
- Data sources:
  - Penn World Tables (real GDP in 2017 prices, capital stock at 2017 prices, human capital index, employment, population).
  - International Labour Organisation (average labor income shares).
  - United Nations (working-age labor force).

### 3.1.3  Findings
- Cross-group average growth differences:
  - Advanced Economies (AEs) exhibited the slowest average economic growth.
  - Emerging Market and Middle Income Economies (EMMIEs) had intermediate growth.
  - Low-Income Developing Countries (LIDCs) grew the fastest.
- Drivers by group (average annual 1991-2019 decomposition components: k/y = capital-output ratio, tfp = TFP growth, hk = human capital, emp_pop = employment-population ratio, gdp_pc = GDP per capita growth):
  - AEs:
    - Growth mainly driven by increases in TFP and human capital.
    - Demographic factors contributed to a lesser extent.
    - Capital-to-output ratios (capital deepening) remained essentially unchanged and marginally detracted from growth.
  - EMMIEs and LIDCs:
    - Capital deepening was relatively the most important factor, accounting for roughly one-third of total growth.
    - TFP, human capital, and demographic factors also contributed positively.
- Decadal dynamics (1991-1999, 2000-2009, 2010-2019):
  - AEs: strongest performance in the 1990s; growth declined thereafter.
  - EMMIEs and LIDCs: growth accelerated over time, especially in LIDCs.
  - Capital-to-output ratios accounted for most of the decline in AE growth (possible shift to sectors with lower capital intensity).
  - EMMIEs showed an uptick in TFP contributions and a softening of capital deepening contributions.
  - LIDCs: capital deepening was the main factor for acceleration in the last two decades.
- Demographics:
  - All Asia-Pacific economies benefited from demographic tailwinds as the share of active workers in the population increased over time.
  - Mechanisms:
    - (i) Higher labor force participation (e.g., increased female labor force participation).
    - (ii) Demographic dividends via a relatively young population increasing the working-age share (inverse dependency ratio).
  - Sustainability concern: these growth-enhancing demographic factors are unlikely to be sustained—participation increases are bounded and population ageing is already underway.
- Employment-population ratio contributions (average annual 1991-2019):
  - AEs: average annual 0.5 percentage point increase in employment-to-population ratio driven almost exclusively by higher labor force participation.
  - EMMIEs and LIDCs: increases mainly resulted from improved inverse dependency ratios as younger cohorts entered working age.
  - Outlook: recent trends point to a reversal as populations age and retiree shares grow relative to working-age individuals (see Section 4.1).

---

### 3.2  Tree-Based Approach

### 3.2.1  Methodology
- Models used: tree-based ensemble machine learning models—random forests and XGBoost.
- Purpose: non-parametric prediction of medium-term growth drivers beyond traditional production factors; capture nonlinearities and interactions.
- Target variable: average real GDP growth in the next five years (from year t+1 to t+5) for each country-year observation.
- Explanation of predictor contributions: Shapley values (Strumbelj and Kononenko, 2010; Lundberg, 2017) measure additive marginal contribution of each predictor relative to the sample average.
- Additional technical details (model training, testing, hyperparameter tuning, model explanation) provided in Appendix A.

### 3.2.2  Data
- Sample: 22 Asia-Pacific economies over 1970-2019; each observation is a country-year pair.
- Predictor timing: predictor values taken from year t to predict five-year average growth from t+1 to t+5, addressing endogeneity concerns from contemporaneous growth.
- Predictors: total of 40 predictors chosen for interpretability and literature relevance.
- Predictor grouping: development and demographics; structural conditions; macroeconomics; global factors and openness.
- Note on collinearity: some potential determinants excluded due to collinearity (e.g., level of governance highly correlated with nominal GDP per capita—level excluded but percentage change included).

### 3.2.3  Findings
- Top predictive drivers of five-year average growth (measured by Shapley values; variables accounting for 70 percent of total contributions reported):
  - Nominal GDP per capita (income level) — the most important predictor; consistent with convergence: higher nominal GDP per capita observations tend to reduce five-year average growth relative to sample average, lower values increase it.
  - Working-age population share (demographics) — critical role; higher growth of working-age share associated with higher medium-term growth.
  - Trade openness and change in trade openness — higher degree and increases in trade openness associated with higher medium-term growth.
  - Growth in human capital — higher growth associated with higher medium-term growth.
  - Improvement in governance — associated with higher medium-term growth.
  - Import partners’ growth and FDI inflows — important global/external drivers associated with higher medium-term growth.
- Nature of associations:
  - Many drivers show approximately linear associations with medium-term growth: lower nominal GDP per capita, higher working-age population share growth, higher trade openness, higher human capital growth, governance improvements, higher import partners’ growth, and more FDI inflows → higher medium-term growth.
  - Domestic macroeconomic variables exhibit nonlinear associations:
    - External debt: economies with external debt more than 50 percent of GDP found to have (slightly) higher medium-term growth in some cases, though high growth also observed with much lower external debt.
    - Public debt: when public debt is lower than about 60 percent of GDP, lower public debt is associated with higher medium-term growth; relationship is less strong above 60 percent of GDP.
    - Fiscal balance: fiscal deficits larger than 5 percent of GDP are associated with low medium-term growth, suggesting fiscal sustainability concerns. Some weak evidence suggests larger fiscal balances could be associated with lower future growth, possibly reflecting effects of fiscal consolidation.
- Evolution over time by income group (contribution to five-year average growth in excess of entire Asia 1970-2019 average; period 1990-2019 examined in five-year windows):
  - Average growth rates (five-year windows, 1990-2019):
    - Advanced Asia: average growth of 3.3 percent.
    - Emerging Asia: average growth of 4.2 percent.
    - Low-Income Developing Countries (LIDCs) in Asia: average growth of 5.5 percent.
  - Advanced Asia (AEs):
    - Development level and demographics (higher GDP per capita and lower growth rate of working-age population share) have been the main growth drag since the late 1990s.
    - Negative contribution from these factors increased over time, lowering AE growth by as much as 2 percentage points below the regional average in the last decade.
    - Macroeconomic conditions contributed positively historically (especially early 2000s with stronger fiscal positions and declining public debt) but have not contributed positively to medium-term growth since the Global Financial Crisis (except for the recovery episode).
    - Global factors have become a growth drag in the last decade; importers’ growth had the largest negative contributions in the last five years (likely linked to China’s slowdown).
    - Structural conditions (governance, human capital, labor force participation) continued to improve and contributed positively to medium-term growth.
  - Emerging Asia (EMMIEs):
    - Gaps from the development frontier and growing working-age populations contributed positively but these positive contributions have been diminishing as GDP per capita increases and population aging progresses.
    - Since the early 2010s, development and demographic factors began to become a growth drag relative to the regional average.
    - Structural conditions improved significantly since the 2000s (human capital, reduced informality), contributing positively over the last two decades.
    - Lower global growth after the Global Financial Crisis, including China’s slowdown, contributed less to medium-term growth in the last decade.
    - Higher public debt since the Global Financial Crisis has been one main negative macroeconomic contributor to emerging Asia’s growth.
  - Low-Income Developing Countries (LIDCs) in Asia:
    - Development level and demographics have been the most important positive growth drivers relative to the regional average, due to lower development levels and faster-growing populations and working-age shares.
    - These positive contributions decreased in the last decade but remained positive as GDP per capita increased and working-age population share growth slowed.
    - Structural improvements contributed positively, especially in the 2000s, a period of governance improvements and economic reforms; governance index increased by about 50 percent in the second half of the 2000s for LIDCs.
    - Macroeconomic conditions contributed positively before the last decade but have become a growth drag since the 2010s, when many low-income economies faced fiscal challenges with large fiscal deficits.

*Source: Authors’ calculations based on data from PWT, ILO, and UN.*

### 3.3  Comparing the Two Approaches

### 3.3  Comparing the Two Approaches

### Summary of approaches and complementarities
- Both approaches—the growth decomposition and the tree-based approach—yield insights into the main drivers behind the growth performance in the Asia-Pacific region in recent decades.
- The two approaches (see Figures 2 and 7) also show how trends have evolved over time.
- Any forward looking approach will thus have to take developments in demographics and structural transformation into account.
- Having identified the key growth drivers, the study moves to forecast the evolution of growth through two complementary frameworks: a production-function-based approach and a novel pattern-matching approach (DTW).

### Historical drivers by country group (as found by the approaches)
- Advanced Economies (AEs)
  - Demographic factors became a drag on growth since late-1990s/early-2000s.
  - Increases in human capital and labor force participation became important growth drivers since then.
- Emerging Market and Middle-Income Economies (EMMIEs)
  - Productivity growth and an increase in human capital have been the most important drivers for growth since 2000s.
  - Productivity gains are likely linked to technology transfer and knowledge spillovers from increasing trade openness.
- Low-Income Developing Countries (LIDCs)
  - Capital deepening has been the most important growth driver since 2000s.
  - Governance improved significantly during this period, likely helping boost domestic investments and attract foreign investments, accelerating capital deepening.

### Transition to forecasting
- The text introduces two complementary forecasting approaches:
  - A traditional growth accounting framework (production-function-based).
  - A pattern-matching methodology based on Dynamic Time Warping (DTW).
- Rationale:
  - Growth accounting provides theoretically-grounded projections based on fundamental factors.
  - Pattern matching leverages historical growth experiences to identify likely development paths.

### Growth accounting methodology (key structure)
- Production function: Y = K^α (A h L)^(1−α) with growth-rate log-linearization leading to:
  - ˆY = α ˆK + (1−α)(ˆA + ˆh + ˆL)
- Forecast targets: ˆK, ˆA, ˆh, ˆL mostly via linear regressions.
- Sample includes 20 regional economies: Australia, Bangladesh, Cambodia, China, Fiji, Hong Kong SAR, India, Indonesia, Japan, Korea, Lao P.D.R., Malaysia, Mongolia, Nepal, New Zealand, Philippines, Singapore, Sri Lanka, Thailand, and Vietnam.
- Note: Several economies drop out of the sample due to continuous negative TFP growth in past years (Brunei Darussalam, Macao SAR, Maldives, and Myanmar).

### Data and key assumptions (explicit)
- Capital stock K projected using the perpetual inventory method; data and depreciation from Penn World Table (PWT) version 10.01. Linear regressions based on 2009-19; implies increasing depreciation rates for most economies.
- TFP growth projected from country-specific linear regressions of TFP levels during 2009-19; historical TFP derived as residual from the semi-endogenous growth decomposition.
- Employment L modeled by age cohort-gender groups Li,j = Pi,j × LFPRi,j:
  - Pi,j from UN World Population Prospects medium-fertility scenario.
  - Age groups considered: 15-24, 25-54, 55-64, and 65+.
  - LFPRi,j forecast by country-specific linear regressions based on 2000-22; historical LFPRs from ILO.
  - Projected linear trends are bound by regional maxima or minima; if a projection hits the bound a logistic growth regression with that regional bound is used.
  - Employment forecast: L = Σi Σj (Pi,j × LFPRi,j).
  - Assumes economies operate at full employment with constant natural rates of unemployment (frictional and structural unemployment included in the natural rate and abstracted from growth-rate calculations).
- Human capital assumed to continue its linear trend:
  - Human capital in PWT measured by average years of schooling and an assumed rate of return to education estimated via Mincer equations.
  - Projections assume human capital follows its linear trend over 2000-19, improving across all economies.
- Income shares: The share of labor income in GDP, 1−α, based on ILO data, is assumed constant; same assumption for capital income share α.

### Main findings from growth accounting forecasts (numeric projections preserved)
- Aggregate potential growth across the region is projected to decline over the medium- and long-term:
  - Around 3.1 percent by 2030 and 2.2 percent by 2040, compared to 3.7 percent pre-pandemic growth.
- Emerging market and developing economies (EMDEs) projections:
  - Growth projected to fall to 3.8 percent in 2030 and 2.8 percent in 2040, compared to pre-pandemic growth of 4.8 percent.
- A slight downward trend is also observed for AEs.
- Drivers of the projected slowdown:
  - Labor: A significant factor behind the long-term slowdown is a lower contribution from labor due to rapidly ageing societies and the reversal of Asia’s demographic dividend.
    - Some demographic effects will be dampened by positive net migration and increases in labor force participation rates, especially among females and the elderly.
    - Old populations (those with a shrinking working-age population) will see labor inputs continuously subtract from growth over the projection period; younger populations will experience labor inputs that fall more rapidly but may still contribute positively overall.
  - Capital: A declining contribution from capital is projected to lower potential growth for AEs and more so for EMMIEs.
  - TFP: Contributions from TFP will slow over time.
  - Human capital: Positive contributions from human capital improvements will continue.
- Per capita considerations:
  - While labor contributes significantly to aggregate growth, improvements in per capita metrics—such as human capital and employment ratios—can lead to sustainable long-term growth.
  - In per capita terms, slowing capital-output ratios—especially for LIDCs—and falling TFP contributions—in particular for EMMIEs—explain the downward growth trend.

*Source: wpiea2025168-print-pdf - 3.3  Comparing the Two Approaches*

### 4.2  Dynamic Time Warping Forecasts

### 4.2  Dynamic Time Warping Forecasts

### The DTW Algorithm
- DTW measures similarity between time series that may vary in speed or timing.
- For two time series X={x1,x2,...,xn} and Y={y1,y2,...,ym}:
  - Initialize cost matrix D of size n×m (in the application, n=m= 5 years). Set D[0][0]=0 and first row and column to infinity.
  - Pair-wise distances D[i][j] computed for multivariate data using Euclidean distance:
    - D[i][j] = √(∑_{p=1}^{q} (x_{ip} − y_{jp})^2) where q is the number of features.
    - Variables are normalized to minimize scaling impact.
  - Cumulative cost matrix C constructed recursively:
    - C[1][1] = D[1][1]
    - C[i][1] = C[i−1][1] + D[i][1], for i = 2,...,n
    - C[1][j] = C[1][j−1] + D[1][j], for j = 2,...,m
    - C[i][j] = D[i][j] + min(C[i−1][j], C[i][j−1], C[i−1][j−1])
  - Final DTW distance = C[n][m], the minimum cumulative distance aligning the sequences.
- Intuition: DTW ‘warps’ time to align shapes (e.g., growth spurts occurring at different times) to identify similar episodes despite timing differences.

### Pattern Matching Data
- Six fundamental indicators from Penn World Table 10.01 are used:
  - GDP per capita
  - Capital-output ratio
  - Consumption-output ratio
  - Labor productivity
  - Real GDP growth
  - Population growth
- Dataset spans 183 economies from 1950 to 2019.
- Notes on measurement and robustness:
  - Robustness tests dropping one variable at a time show the method remains functional with five variables but with degraded performance (higher prediction errors in backtesting).
  - GDP per capita and labor productivity measured using chained PPPs in 2017 USD.
  - Capital-output and consumption-output ratios use current price PPPs.

### The PatternSync Forecasting Procedure
- Target period defined as [t−k+1, t] with k = 5 years; t is the most recent year available.
- Compute DTW distances between target country base period and all historical periods across the dataset, using the multivariate indicators.
- Select most relevant historical patterns by choosing the top x percent of country-periods with shortest DTW distances:
  - Baseline x = 2 (selecting the most similar two percents of all available country-periods).
- Forecast horizon y set to 10 years in backtesting for medium-term projections.
- Aggregate subsequent growth trajectories of selected similar periods using multiple methods:
  - Median of subsequent growth rates across similar periods (primary).
  - Simple average.
  - Weighted average with weights proportional to inverse of DTW distances (gives greater importance to more similar episodes).
  - Percentile values of the growth distribution to indicate upside/downside scenarios.
- Multivariate DTW enables capturing similarity across development, productive capacity, demand structure, technology, momentum, and demographics.

### Backtesting Results
- Backtest sample: 26 Asia-Pacific economies using rolling 5-year base periods from 1995-1999 to 2015-2019; forecast horizon 10 years.
- All 183 economies used as potential pattern sources to identify matches.
- Evaluation metrics: average error (AE), root mean squared error (RMSE), and hit rate (HR) calculated for each forecast year; hit rate defined as percentage of time forecast falls within a 25% band of actual value.
- Backtesting observations:
  - Median Absolute Error (MAE) shows downward trend over time; MAE stabilizes at around 0.9 percentage points for 5-year-ahead forecasts by end of sample.
  - Hit rate shows upward trend over time, with a notable reversal during 2005-10 (Global Financial Crisis period).
  - DTW PatternSync maintains relatively stable accuracy between short-term (Years 1-3) and medium-term (Years 4-7) horizons, unlike typical methods that deteriorate with horizon.
  - Hit rate improves over time, reaching approximately 60% by end of sample for 5-year-ahead forecasts.
  - Weighted average aggregation (inverse distance weights) performs slightly better than simple averages or medians, particularly for longer horizons; performance differences across aggregation methods are modest.
- Comparison with conventional forecasts (WEO and EIU) for 2011-2019 Asia-Pacific economies:
  - Short-term (Years 1-2): WEO and EIU show stronger performance and higher hit rates (WEO and EIU achieve hit rates of 68-79% in Year 1; DTW 65%).
  - Year 1 RMSE: EIU 0.80; WEO 1.41; DTW 1.49.
  - Medium-term (Years 4-7): DTW decay in accuracy is smaller; DTW median absolute error remains around 0.7-0.8 percentage points while WEO and EIU show larger increases.
  - By Year 5, DTW achieves comparable or better performance across most metrics; DTW hit rate remains around 62-65% through Year 9, while conventional forecasts’ hit rates decline to the 40-60% range.
  - Directional bias:
    - WEO and EIU exhibit optimistic bias (positive average errors that grow with horizon).
    - DTW shows a mild negative bias that remains relatively stable (DTW average errors reported as: -0.33, -0.32, -0.32, -0.30, -0.31, -0.33, -0.31, -0.33, -0.36 for Years 1–9 respectively).
- Table 1 key numeric entries (Asia-Pacific, 2011-2019):
  - Median absolute error (in pp):
    - WEO: 0.64, 0.86, 0.97, 1.03, 1.12 (Years 1–5 partial listing)
    - EIU: 0.56, 0.58, 0.68, 0.76, 0.76, 0.69, 0.63, 0.62, 0.57 (Years 1–9)
    - DTW: 0.82, 0.81, 0.81, 0.78, 0.79, 0.69, 0.74, 0.78, 0.72 (Years 1–9)
  - RMSE (in pp):
    - WEO: 1.41, 1.75, 2.03, 2.06, 1.88 (Years 1–5 partial listing)
    - EIU: 0.80, 0.91, 1.22, 1.38, 1.39, 1.15, 1.20, 1.21, 1.08 (Years 1–9)
    - DTW: 1.49, 1.37, 1.29, 1.27, 1.25, 1.17, 1.10, 1.03, 1.00 (Years 1–9)
  - Average error (in pp):
    - WEO: 0.30, 0.44, 0.58, 0.65, 0.69 (Years 1–5 partial listing)
    - EIU: 0.11, 0.27, 0.40, 0.63, 0.51, 0.35, 0.23, 0.18, 0.17 (Years 1–9)
    - DTW: -0.33, -0.32, -0.32, -0.30, -0.31, -0.33, -0.31, -0.33, -0.36 (Years 1–9)
  - Hit rate:
    - WEO: 68%, 59%, 55%, 50%, 41% (Years 1–5 partial listing)
    - EIU: 79%, 71%, 68%, 59%, 63%, 61%, 63%, 59%, 56% (Years 1–9)
    - DTW: 65%, 64%, 64%, 63%, 62%, 65%, 65%, 65%, 65% (Years 1–9)
- Applicability to Advanced Economies (AEs):
  - AEs have fewer historical analogues near the technological frontier, but the method remains applicable because:
    - AEs still experience demographic, productivity, and structural patterns with parallels.
    - Six-dimensional matching can identify similarity across phases rather than levels.
    - AEs can match to their own historical periods or other AEs.
  - Backtesting suggests reasonable accuracy for AEs, though smaller analogue pool implies higher variance in projections for AEs.

### Forward-Looking Forecasts
- Long-term forecasts for Asia-Pacific economies through 2040 using 2015-2019 base period.
- Forecast outputs include median and percentile-based forecasts (percentile chosen based on recent performance relative to peers).
- Median forecast patterns (Figure 14 summary):
  - Clear divergence across groups for 2030-2040:
    - Advanced Economies (AEs): around 1.8-2.0% on average over 2030-2040.
    - Emerging Market and Middle-Income Economies (EMMIEs): 3.2-3.6%.
    - Low-Income Developing Countries (LIDCs): 4.5-5.3%.
  - Growth differentials narrow over time, consistent with convergence theory.
  - Fast-growing EMMIEs/LIDCs (e.g., Cambodia, Vietnam, Bangladesh) projected to maintain growth above 4% through 2040 with some moderation.
  - China: expected moderation from around 4.2% in 2030 to 3.3% by 2040 based on 75th percentile forecasts.
  - India: growth around 3.3% through the forecast horizon.
- Interpretation of percentile forecasts:
  - High percentile (e.g., 75th for China, Singapore, Malaysia) signals recent outperformance relative to peers; projects continuation of that outperformance.
  - Historical evidence indicates mean reversion in cross-country growth performance rankings; economies rarely maintain very high or very low percentile rankings over multiple decades.
  - Regression evidence (non-overlapping 10-year data on 183 economies, 1959-2019) with country fixed effects shows a negative and significant coefficient for current ranking predicting future ranking (indicating likely mean reversion).
  - Example without fixed effects: coefficient for current growth ranking ≈ 0.16 and constant ≈ 0.42 implies a country at 75th–80th percentile now expected to drop to 54th–55th percentile in 10 years.
- Caveats and limitations:
  - Forecasts implicitly assume continuation of relatively stable global technological and productivity growth as in the post-World War II sample.
  - Accuracy could decline if disruptive changes occur (e.g., uneven productivity gains from AI, major geopolitical shifts).
  - Limitations inherent to history-based forecasting methods are particularly relevant for long-term projections in a rapidly evolving global economy.

*Source: Authors’ calculations.*

### 4.3  Comparing the Two Approaches

### 4.3  Comparing the Two Approaches

### Shared broad features
- Growth differs by income group, with the forecast for the lower income group to continue to see higher growth (see Figure 14).
- Growth differentials are forecast to narrow over time, in line with economic convergence.
- Both approaches forecast a moderation of growth over time across all income groups.
- Growth decomposition suggests demographics might be an important explanatory factor, reinforced by slower TFP growth and less capital deepening.

### Key differences between DTW (pattern-matching) and growth decomposition
- DTW almost consistently forecasts higher growth than the growth decomposition.
- Reasons for differences:
  - The machine learning methodology (DTW) allows inclusion of an extensive pool of historical data for comparison, showing higher forecasting accuracy over longer horizons than the traditional growth decomposition.
  - Long-term structural factors such as growth convergence and structural transformation may be implicitly taken into account in DTW while not incorporated in the growth accounting framework.
  - Unlike the production-function approach that projects each factor linearly (capital, TFP, etc.), DTW searches for historical analogues that match current structural and macroeconomic patterns, naturally accounting for complex non-linearities (e.g., rapid structural transformation).
  - As a result, DTW forecasts can imply slightly higher [or lower] growth paths for specific economies, especially those still undergoing substantial structural shifts.
- DTW can help assess whether outlier ‘growth miracles’ are replicable by identifying parallels in demographics and sectoral shifts, thereby informing country-specific strategies to maintain or accelerate convergence.

### Shared limitations and interpretation of projections
- Both methods project growth based on historical data—either the country’s own observed ongoing trends or historical patterns of other economies—and therefore lack the ability to forecast structural changes not observed in the past (for example, significant technological advances or ambitious reforms).
- Consequently, the projections can be considered baseline projections if current trends persist and economies follow historical patterns of their own and others.
- The projections in this paper are methodological contributions designed to understand long-term structural patterns through historical analysis, rather than operational forecasts for policy guidance.
  - Unlike IMF country desk forecasts that integrate real-time data and expert judgment on country-specific circumstances, these projections are derived purely from historical patterns and econometric relationships to illustrate forecasting methodologies.
  - Readers should view results as complementary analytical tools that suggest potential growth paths under historical development patterns, rather than as predictions of most likely outcomes given current policy settings.

*Source: Authors’ calculations, as presented in the chapter "4.3 Comparing the Two Approaches" of the provided IMF document.*

### Appendix A.2  Model Training, Testing, and Hyperparameter

### Appendix A.2  Model Training, Testing, and Hyperparameter Tuning

### Sample and prediction framework
- Sample: 22 Asia-Pacific economies over the period of 1970-2019; each observation is a country-year pair.
- Target variable: the average real GDP growth in the next five year, i.e., from year t+1 to t+5.
- Predictors: values chosen from year t to avoid contemporaneous influence of predictors by the target.
- Evaluation metric: root mean squared error (RMSE).

### Cross-validation and hyperparameter tuning procedure
- Method: block time-series cross-validation based on Burman et al. (1994) and Racine (2000) to account for cross-sectional dependence in the panel.
- Step 1: Construct 5 training–validation pairs using year cutoffs:
  - Training 1970–2004; validation 2005–2006
  - Training 1970–2006; validation 2007–2008
  - Training 1970–2008; validation 2009–2010
  - Training 1970–2010; validation 2011–2012
  - Training 1970–2012; validation 2013–2014
- Step 2: For each ML algorithm:
  - Start with a random set of hyperparameters; train on each training set and evaluate RMSE on corresponding validation set to obtain five RMSEs.
  - In each tree-based algorithm, 1000 trees are constructed.
  - Hyperparameters tuned:
    - Random Forest: maximum tree depth for base learners; subsample ratio of columns when constructing each tree.
    - XGBoost: L2 regularization term on weights; boosting learning rate.
  - Hyperparameter search: Bayesian optimization over 100 iterations.
  - Selection criterion: set of hyperparameters with the lowest average RMSE across the five validation sets.
- Step 3: Final model: the ML algorithm (with its optimal hyperparameters) that yields the lowest average RMSE is chosen as the final selected ML algorithm and corresponding hyperparameters.

---

### Appendix A.3  Model Explanation: Shapley Values

### Purpose and interpretation
- Goal: unpack ML black box to understand contributions of predictors; results reported as Shapley values (Strumbelj and Kononenko, 2010; Lundberg, 2017), based on cooperative game theory (Shapley, 1953; Young, 1985).
- Shapley values measure the additive contribution of each predictor to the deviation of an observation’s prediction from the sample-average prediction.

### Linear models
- For linear models, the Shapley value of predictor k for observation xi is the estimated coefficient multiplied by the observation’s predictor value:
  - f̂(xi) = φ0(f̂) + Σk=1^n φk(xi; f̂) = β̂0 + Σk=1^n β̂k xi,k
  - Shapley value for predictor k: β̂k xi,k
  - Sum of Shapley values across predictors for xi equals f̂(xi) minus the average prediction in the training sample.

### Model-agnostic Shapley decomposition for non-linear models
- Views a model prediction as a cooperative game where each predictor value is a “player” and the prediction is the “payout.”
- Shapley value ΦS_k(xi; f̂) is the average marginal contribution of predictor k across all coalitions of other predictors:
  - f̂(xi) = ΦS_0(f̂) + Σk=1^n ΦS_k(xi; f̂)
  - ΦS_k(xi; f̂) = Σ_{x′ ⊆ {x1,...,xn}\{xk}} |x′|!(n−|x′|−1)! / n! × ( f̂(xi | x′ ∪ {xk}) − f̂(xi | x′) )
- Interpretation: difference between model predictions with and without predictor k, averaged over all coalitions, weighted by coalition size.

---

### Appendix B  Identifying Growth Drivers: List of Variables (selected details)

- Notes on transformations: level, 5avg, 5pch, and 5fd indicate the variable itself, five-year average, five-year average percentage change, and five-year average level change.
- Variable categories: Development and Demographics; Global and Openness; Macroeconomics; Structural Conditions.
- Selected variables, transformations, categories, and sources (as listed in the table):
  - GDP per capita — level — Development and Demographics — World Economic Outlook Database
  - Population Growth — 5avg — Development and Demographics — World Development Indicator Database
  - Working-Age Population Share — 5pch — Development and Demographics — World Development Indicator Database
  - ToT %Change — level — Global and Openness — World Economic Outlook Database
  - Export Partners’ Growth — 5avg — Global and Openness — World Economic Outlook Database
  - Fed Funds Rate (Shadow) — 5avg+5fd — Global and Openness — Wu and Xia (2016)
  - Import Partners’ Growth — 5avg — Global and Openness — World Economic Outlook Database
  - Inward FDI — 5avg — Global and Openness — IFS and BoP Databases
  - Trade Openness — 5avg+5fd — Global and Openness — World Economic Outlook Database
  - Country Uncertainty — level+5avg+5pch — Macroeconomics — World Uncertainty Index Database
  - NEER %Change — level+5avg — Macroeconomics — IFS and BoP Databases
  - REER %Change — level+5avg — Macroeconomics — IFS and BoP Databases
  - External Debt — 5avg+5fd — Macroeconomics — World Economic Outlook Database
  - Fiscal Balance — 5avg+5fd — Macroeconomics — World Economic Outlook Database
  - Inflation — 5avg+5fd — Macroeconomics — World Economic Outlook Database
  - M2 Growth — 5avg — Macroeconomics — World Economic Outlook Database
  - Private Credit — 5avg+5fd — Macroeconomics — World Economic Outlook Database
  - Public Debt — 5avg+5fd — Macroeconomics — World Economic Outlook Database
  - Real Interest Rate — 5avg+5fd — Macroeconomics — World Economic Outlook Database
  - Reserves Coverage — 5avg+5fd — Macroeconomics — World Economic Outlook Database
  - Unemployment Rate — 5avg+5fd — Macroeconomics — World Economic Outlook Database
  - Governance — 5pch — Structural Conditions — Worldwide Governance Indicator Database
  - Human Capital — 5pch — Structural Conditions — Penn World Table; Feenstra et al. (2015)
  - Labor Force Participation — 5pch — Structural Conditions — World Development Indicator Database
  - Political Stability — 5pch — Structural Conditions — Worldwide Governance Indicator Database
  - Share of Wage and Salaried Workers — 5pch — Structural Conditions — World Development Indicator Database

---

### Appendix C  DTW Country Case Study: Cambodia

### Method and sample selection
- Method: DTW PatternSync to identify historically similar country-periods to Cambodia’s 2015-19 target period.
- Similarity selection: top 2 percent most similar episodes to Cambodia’s 2015-19 period.
- Number of historically similar episodes identified: 156, concentrated among East and Southeast Asian economies at different development stages.
- Examples of top similar country-periods (from Table 3):
  - India 2001–2005
  - Indonesia 2002–2006
  - Lao P.D.R 2008–2012
  - Sri Lanka 1983–1987
  - Japan 1951–1955
  - China 1986–1990
  - Vietnam 2010–2014
  - Korea 1973–1977
  - Philippines 2004–2008
  - Thailand 1985–1989

### Medium-term growth forecast results and percentiles
- Aggregation methods used: median, average, and weighted average across similar country-periods.
- Forecasted potential growth for 2030:
  - 5.8 percent (average forecast)
  - 6.3 percent (medium forecast)
- Cambodia’s trend growth during the target period ranked at the 75th to 80th percentile of the top 2 percent most similar cohort.
- Percentile-based aspirational forecasts if Cambodia maintains 75th–80th percentile performance:
  - Potential growth of 7.5 to 7.9 percent for 2030.

### Mean reversion and subsequent performance of peers
- Mean-reverting pattern observed: economies that are significant outperformers in a given period tend to see their ranking drop over time, and vice versa.
- Subset analysis — peers matching Cambodia’s base-period growth performance (75th–80th percentile):
  - Number of such country-periods: 7
  - Average growth rate 10 years after the base period: 6.4 percent
  - Median growth rate 10 years after the base period: 5.9 percent
  - In all except one instance (China 1996–2000), growth performance dropped out of the 75th–80th percentile range after a decade.
- Broader subgroup — peers that ranked equal to or higher than Cambodia in the base period (75th–100th percentile):
  - Number of country-period instances: 40
  - Average growth rate a decade after base period: 5.8 percent
  - Median growth rate a decade after base period: 6.4 percent
  - Interpretation: strong base-period performance did not, on average, translate to a persistent advantage over the medium term.

### Outlier sustained-high-growth cases and commonalities
- Identified outliers that sustained high growth over an extended period:
  - Japan (base period shown as 1956–1967 in narrative)
  - South Korea (base period shown as 1971–1986 in narrative)
  - China (base period shown as 1996–2002 in narrative)
- Common factors associated with these outliers cited in the literature:
  - Export-oriented industrialization and strategic integration into global value chains
  - High investments in human capital through education and health infrastructure
  - Institutional reforms improving governance and protecting property rights, enabling private sector development
- Comparative observation: data shows Cambodia lags these outliers during their base periods in human capital and labor productivity.

### Analytical value and policy implications
- DTW PatternSync provides:
  - Structured identification of historical parallels for country-specific medium-term growth analysis.
  - Quantitative forecasts (baseline and aspirational percentile scenarios) and a statistical assessment of the likelihood of sustained outperformance.
  - A framework to identify critical success factors by examining historical peers that maintained exceptional growth.
- For policymakers: the framework highlights policy priorities associated with historically sustained high growth—export orientation and global integration, human capital investment, and institutional reforms—as areas to consider for improving medium-term growth sustainability.

*Source: Appendix A.2–C, wpiea2025168-print-pdf — Dissecting Medium-Term Growth Prospects for Asia, Working Paper No. WP/2025/168*

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_Source: https://www.imf.org/-/media/files/publications/wp/2025/english/wpiea2025168-print-pdf.pdf_