## 1. Gross Capital Formation

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### A. Introduction and Context
- Purpose: Estimating potential output to assess the economy’s maximum sustainable production and identify output gaps guiding policy.
- Major recent shocks and structural changes:
  - The pandemic and Russia’s war in Ukraine altered labor markets, investment patterns, and trade dynamics.
  - Structural shifts: movement from agriculture to mining and services, shrinking informal sector, technological advances, demographic changes, climate change, and new trade patterns.
- Reported growth and averages:
  - Strong real GDP growth of 9 percent annually from 2022-2024.
  - CCA average growth during 2022-2024 was 6.1 percent.

### B. Methodology Overview
- Approaches used to estimate potential output:
  - Production function method (Cobb-Douglas) following Denis et al. (2006).
  - State-space models (multivariate and univariate Kalman filters).
  - Univariate filters including the Hodrick-Prescott (HP) filter and Beveridge-Nelson decomposition.
- Data and sample specifics:
  - Quarterly data from 2000 onward; sample for VECM estimation is Q1 2001 to Q4 2023.
  - Capital stock initial benchmark from 1999 data in "The Changing Wealth of Nations 2021".
  - Depreciation rates averaged 3.6 percent (source: Penn World Table Database).
  - Labor input constructed from employment and average hours worked; quarterly labor components Q1 2010 to Q4 2023 from ILOSTAT; extended back to 2000 using NSC employment dynamics.
  - HP filter applied after extending endpoints with four-year ARIMA forecasts through 2027 to mitigate endpoint bias.

### C. Production Function Specification and Inputs
- Cobb-Douglas production function:
  - Y_t = TFP_t * K_t^α * L_t^(1−α)
  - Definitions: Y_t = real GDP; TFP_t = Total Factor Productivity; K_t = capital input; L_t = labor input; α and 1−α are capital and labor elasticities.
- Capital stock via perpetual inventory:
  - K_t = K_{t−1} * (1 − δ_{t−1}) + I_t
  - δ denotes depreciation rate.
- Labor input composite:
  - L_t = LF_t * (1 − U_t) * AHW_t
  - Potential labor input transformed to: L̅_t = LF̅_t * (1 − U_t/NAIRU) * AHW̅_t
  - NAIRU level used: 4.4 percent.
- Output elasticities estimated using a Vector Error-Correction Model (VECM) to account for cointegration between real GDP, labor, and capital.

### D. Key Quantitative Findings (Production Function and Baseline)
- Potential output growth (recent years): 5.3 percent.
- Non-gold potential output growth (recent years): 5.6 percent.
- TFP turned positive in 2022 but remains below historical averages prior to 2015.
- Contribution decomposition:
  - Capital stock accumulation and labor are primary drivers.
  - Gold sector contribution is negative due to plateauing gold production.

### E. Migration Sensitivity Scenario
- Assumption: 70,000 Russian migrants settled permanently in the Kyrgyz Republic.
  - Basis: Department of Population Registration figures: 170 thousand Russian citizens registered between January and October 2022; assumption that 60 percent left and 40 percent stayed.
- With 70,000 migrants:
  - Potential output growth rises to 5.8 percent.
  - Non-gold potential output growth rises to 6.1 percent.
- Rationale: direct labor force expansion, boosted capital stock, and potential TFP gains from young skilled migrants (many noted as IT professionals).

### F. State-Space Models, Univariate Filters, and Robustness
- Alternative methods produce corroborating estimates:
  - Potential output growth estimates range between 5.3 and 5.8 percent.
  - Non-gold potential output growth up to 6.4 percent (reported from state-space/univariate approaches).
- Beveridge-Nelson and AR/ARMA results:
  - Stable potential output growth of 4.4 percent during 2001–2014.
  - Decline to 3.5 percent in 2015–2021.
  - Rebound to 5.3 percent in 2022–2024.

### G. Output Gap Dynamics and Labor Market Links
- Output gap dynamics:
  - Output gap turned persistently positive since 2021, more pronounced for non-gold GDP.
  - Positive output gap reflects strong post-pandemic recovery, capital inflows, and labor migration from Russia.
  - Risks of overheating: inflationary pressures, asset price distortions, macroeconomic imbalances.
- Labor market indicators:
  - Declining unemployment rates and rising real wages point to tightening labor market.
  - Observed correlation between non-gold output gap, growth of real non-public wages, and the unemployment gap (unemployment gap defined as NAIRU minus actual unemployment).
  - Notes on migrants: some work remotely for foreign companies with limited local impact; others add to local labor supply but may not fully ease pressures due to skill mismatches and higher demand for services from migrants.

### H. Conclusion and Policy Recommendations
- Summary assessment:
  - Potential output growth estimated to have increased to 5.3 percent since the pandemic (from about 4.4 percent before).
  - Current actual growth has outpaced estimated potential, producing a persistently positive output gap.
  - TFP remains below early 2010s levels despite recent positive trend.
- Policy recommendations to mitigate overheating risks and support sustainable growth:
  - Adopt counter-cyclical fiscal and monetary policies.
  - Advance structural reforms to attract investment and improve labor markets.
  - Foster productivity through:
    - A business-friendly environment.
    - Improved access to finance.
    - Strengthening property rights and rule of law.
    - Enhancing education and skills development.

### I. Annex — Model Specifications (high-level)
- Production function estimation steps:
  1. Calculate equilibrium levels of capital and labor inputs.
  2. Estimate output elasticities of capital and labor (VECM).
  3. Derive TFP dynamics linked to potential output growth.
- Labor input construction and trend treatment: HP filter with ARIMA-based endpoint extensions described.
- VECM includes dummy variables to capture shocks (e.g., political instability in 2010, onset of Covid-19 in 2020) and outputs long-term coefficients β1 and β2 as elasticities.

### Output elasticity of capital — dual-model approach and estimation results
- Purpose: Two models estimated—one using capital stock and the other using investment flows—to capture long-term and short-term dynamics and serve as a robustness check.
- Pretesting and cointegration: Johansen Cointegration test used; variables checked for unit roots.
- Kyrgyz Republic: Results for Long-Term Equation (Original coefficients / Standard errors in parentheses; Normalized coefficients)
  - Capital stock:
    - Original coefficient: 0.281** (0.110)
    - Normalized coefficient: 0.311** (0.123)
  - Investments:
    - Original coefficient: 0.293* (0.046)
    - Normalized coefficient: 0.294* (0.046)
  - Labor input:
    - Original coefficients and standard errors shown as: 0.624** (0.274); 0.689** (0.303)
    - Normalized coefficients and standard errors shown as: 0.703* (0.226); 0.706* (0.227)
  - Significance levels: *p<0.01; **p<0.05; *** p<0.1; ^ insignificant with p>0.1
- Normalization:
  - Original coefficients normalized so their sum equaled one to align with constant returns to scale.
  - Specification using investments inherently satisfied constant-returns assumption as original coefficients naturally summed to one.
  - Resulting estimates closely align with Kudabaeva (2010): output elasticity of capital at 0.26 and labor at 0.74.

### Total Factor Productivity (TFP) — measurement and incorporation into potential output
- TFP measured as residual:
  - ΔTFP_t = (ΔY_t − (0.3*ΔK_t +0.7*ΔL_t))
- Adjustment and smoothing:
  - TFP adjusted for shocks that may distort true productivity trends and smoothed using the HP filter.
- Integration into potential output growth:
  - ΔȲ_t = (ΔTFP̅_t +0.3*ΔK̅_t +0.7*ΔL̅_t)

### Multivariate Kalman Filter (Multivariate Filter methodology)
- Methodology:
  - Multivariate Filter of Blagrave et al. (2015) incorporating real GDP growth, inflation (core inflation), and the unemployment rate.
  - Potential output modeled as a latent variable and estimated alongside the output gap using the Kalman filter with Bayesian estimation for parameters and shock variances.
- Data and sample extension:
  - Annual data for GDP growth, inflation, and unemployment rate from 2000 to 2023.
  - Sample extended to 2026 by including estimates for 2024 and projections for 2025–2026 to address endpoint problem.
- Model structure (selected equations and definitions):
  - y = Y − Ȳ
  - Ȳ_t = Ȳ_t−1 + G_t + ε^Ȳ_t
  - G_t = θ G_SS + (1−θ) G_t−1 + ε^G_t
  - y_t = φ y_t−1 + ε^y_t
  - Phillips Curve: π_t = λ π_t+1 + (1−λ) π_t−1 + β y_t + ε^π_t
  - Unemployment gap and NAIRU dynamics specified with Ū_t, gŪ_t and associated shock terms.
  - Okun’s Law: u_t = τ^2 u_t−1 + τ^1 y_t + ε^u_t

### Beveridge–Nelson (BN) decomposition
- Purpose: Separates permanent and transitory components in economic time series to identify long-term trends versus short-term fluctuations.
- ARIMA-based specification and decomposition:
  - Y_t = φ(B) θ(B)^{-1} ε_t
  - Y_t = Y_t^P + Y_t^T
  - Y_t^P = lim_{h→∞} E[Y_{t+h} | Y_t]
  - Y_t^T = Y_t − Y_t^P

### Hodrick–Prescott (HP) filter application
- Data and parameterization:
  - Applied to quarterly data spanning from 2000Q1 to 2024Q3, incorporating estimates and projections for 2024Q4 to 2027Q4 to address the end-point problem.
  - Smoothing parameter (lambda) set to 1600.

*International Monetary Fund — sipea2025081 (Selected Issues Paper chapter content provided)*

### 1. Gross Capital Formation ______________________________________________________________ 3

### 1. Gross Capital Formation

### A. Introduction and Context
- Estimating potential output is used to assess the economy’s maximum sustainable level of production and to identify output gaps that guide policy decisions.
- Major recent shocks and structural changes: the pandemic and Russia’s war in Ukraine; these altered labor markets, investment patterns, and trade dynamics.
- Reported strong real GDP growth of 9 percent annually from 2022-2024.
- CCA average growth during 2022-2024 was 6.1 percent.
- Notable structural shifts: movement from agriculture to mining and services, shrinking informal sector, technological advances, demographic changes, climate change, and new trade patterns.

### B. Methodology Overview
- Three complementary approaches used to estimate potential output:
  - Production function method (Cobb-Douglas) following Denis et al. (2006).
  - State-space models (multivariate and univariate Kalman filters).
  - Univariate filters including the Hodrick-Prescott (HP) filter and Beveridge-Nelson decomposition.
- Data and sample:
  - Quarterly data from 2000 onward; sample for VECM estimation is Q1 2001 to Q4 2023.
  - Capital stock initial benchmark from 1999 data in "The Changing Wealth of Nations 2021".
  - Depreciation rates averaged 3.6 percent (source: Penn World Table Database).
  - Labor input constructed from employment and average hours worked; quarterly labor components Q1 2010 to Q4 2023 from ILOSTAT; extended back to 2000 using NSC employment dynamics.
  - HP filter applied to series after extending endpoints with four-year ARIMA forecasts through 2027 to mitigate endpoint bias.

### C. Production Function Specification and Inputs
- Cobb-Douglas production function:
  - Y_t = TFP_t * K_t^α * L_t^(1−α)
  - Y_t = real GDP; TFP_t = Total Factor Productivity; K_t = capital input; L_t = labor input; α and 1−α are capital and labor elasticities.
- Capital stock via perpetual inventory:
  - K_t = K_{t−1} * (1 − δ_{t−1}) + I_t
  - δ denotes depreciation rate.
- Labor input composite:
  - L_t = LF_t * (1 − U_t) * AHW_t
  - Potential labor input transformed to: L̅_t = LF̅_t * (1 − U_t/NAIRU) * AHW̅_t
  - NAIRU level used: 4.4 percent.
- Output elasticities estimated using a Vector Error-Correction Model (VECM) to account for cointegration between real GDP, labor, and capital.

### D. Key Quantitative Findings (Production Function and Baseline)
- Estimated potential output growth (recent years): 5.3 percent.
- Non-gold potential output growth (recent years): 5.6 percent.
- TFP turned positive in 2022 but remains below historical averages prior to 2015.
- Contribution decomposition highlights capital stock accumulation and labor as primary drivers; gold sector contribution is negative due to plateauing gold production.

### E. Migration Sensitivity Scenario
- Assumption: 70,000 Russian migrants settled permanently in the Kyrgyz Republic.
  - Basis: Department of Population Registration figures: 170 thousand Russian citizens registered between January and October 2022; assumption that 60 percent left and 40 percent stayed.
- With 70,000 migrants:
  - Potential output growth rises to 5.8 percent.
  - Non-gold potential output growth rises to 6.1 percent.
- Rationale: direct labor force expansion, boosted capital stock, and potential TFP gains from young skilled migrants (noted as many being IT professionals).

### F. State-Space Models, Univariate Filters, and Robustness
- Multivariate and univariate Kalman filters and the HP filter produce corroborating estimates:
  - Potential output growth estimates range between 5.3 and 5.8 percent.
  - Non-gold potential output growth up to 6.4 percent (reported from state-space/univariate approaches).
- Beveridge-Nelson decomposition and AR/ARMA models:
  - Stable potential output growth of 4.4 percent during 2001–2014.
  - Decline to 3.5 percent in 2015–2021.
  - Rebound to 5.3 percent in 2022–2024.

### G. Output Gap Dynamics and Labor Market Links
- Output gap turned persistently positive since 2021, more pronounced for non-gold GDP.
- Positive output gap reflects strong post-pandemic recovery, capital inflows, and labor migration from Russia.
- Risks of overheating: inflationary pressures, asset price distortions, macroeconomic imbalances.
- Labor market indicators:
  - Declining unemployment rates and rising real wages point to tightening labor market.
  - Correlation observed between non-gold output gap, growth of real non-public wages, and the unemployment gap (unemployment gap defined as NAIRU minus actual unemployment).
- Note: Some migrants work remotely for foreign companies and have limited impact on local labor markets; others add to local labor supply but may not fully ease pressures due to skill mismatches and higher demand for services from migrants.

### H. Conclusion and Policy Recommendations
- Summary assessment:
  - Potential output growth estimated to have increased to 5.3 percent since the pandemic (from about 4.4 percent before).
  - Current actual growth has outpaced estimated potential, producing a persistently positive output gap.
  - TFP remains below early 2010s levels despite recent positive trend.
- Policy recommendations to mitigate overheating risks and support sustainable growth:
  - Adopt counter-cyclical fiscal and monetary policies.
  - Advance structural reforms to attract investment and improve labor markets.
  - Foster productivity through:
    - A business-friendly environment.
    - Improved access to finance.
    - Strengthening property rights and rule of law.
    - Enhancing education and skills development.

### I. Annex — Model Specifications (high-level)
- Production function form and steps to estimate potential output:
  1. Calculate equilibrium levels of capital and labor inputs.
  2. Estimate output elasticities of capital and labor (VECM).
  3. Derive TFP dynamics linked to potential output growth.
- Labor input construction and treatment of trends and seasonal adjustments described; HP filter used for trend extraction with ARIMA-based endpoint extensions.
- VECM includes dummy variables to capture shocks (e.g., political instability in 2010, onset of Covid-19 in 2020) and outputs long-term coefficients β1 and β2 as elasticities.

*Source: IMF staff calculations.*

### 5.      Given that capital stock represents the long-term productive capacity of an economy, while

### sipea2025081 - 5.      Given that capital stock represents the long-term productive capacity of an economy, while

### Output elasticity of capital — dual-model approach and estimation results
- Purpose:
  - Two models estimated to evaluate the output elasticity of capital input: one using capital stock and the other using investment flows, to capture long-term and short-term dynamics and serve as a robustness check.
- Pretesting and cointegration:
  - The presence of cointegration among the variables of interest was evaluated using the Johansen Cointegration test.
  - Prior to the test, variables were examined for the presence of a unit root to confirm their non-stationary nature.
- Table 1. Kyrgyz Republic: Results for Long-Term Equation (Original coefficients / Standard errors in parentheses; Normalized coefficients)
  - Capital stock:
    - Original coefficient: 0.281** (0.110)
    - Normalized coefficient: 0.311** (0.123)
  - Investments:
    - Original coefficient: 0.293* (0.046)
    - Normalized coefficient: 0.294* (0.046)
  - Labor input:
    - Original coefficients and standard errors shown as: 0.624** (0.274); 0.689** (0.303)
    - Normalized coefficients and standard errors shown as: 0.703* (0.226); 0.706* (0.227)
  - Standard errors are shown in parentheses.
  - Significance levels: *p<0.01; **p<0.05; *** p<0.1; ^ insignificant with p>0.1
- Normalization:
  - The original coefficients were normalized so their sum equaled one, aligning with the constant returns to scale assumption in the Cobb-Douglas production function.
  - The specification using investments inherently satisfied the constant-returns assumption as the original coefficients naturally summed to one.
  - The resulting estimates closely align with Kudabaeva (2010), who reported an output elasticity of capital at 0.26 and labor at 0.74.

### Total Factor Productivity (TFP) — measurement and incorporation into potential output
- TFP as residual:
  - ΔTFP_t = (ΔY_t − (0.3*ΔK_t +0.7*ΔL_t))
- Adjustment and smoothing:
  - After adjusting TFP for shocks that may distort true productivity trends, the refined TFP series was smoothed using the HP filter.
- Integration into potential output growth:
  - ΔȲ_t = (ΔTFP̅_t +0.3*ΔK̅_t +0.7*ΔL̅_t)

### Multivariate Kalman Filter (Multivariate Filter methodology)
- Methodology:
  - Employs the Multivariate Filter of Blagrave et al. (2015), incorporating real GDP growth, inflation (core inflation to account for global food and energy prices), and the unemployment rate.
  - Models potential output as a latent variable and estimates it alongside the output gap.
  - Kalman filter used to estimate unobservable variables; parameter values and variances of shock terms determined using Bayesian estimation techniques.
- Data and sample extension:
  - Model estimation is based on annual data for GDP growth, inflation, and the unemployment rate from 2000 to 2023.
  - To address the endpoint problem, the sample size was extended to 2026 by including estimates for 2024 and projections for 2025–2026.
- Model structure and equations:
  - Output equations:
    - y = Y − Ȳ
    - Ȳ_t = Ȳ_t−1 + G_t + ε^Ȳ_t
    - G_t = θ G_SS + (1−θ) G_t−1 + ε^G_t
    - y_t = φ y_t−1 + ε^y_t
    - Definitions: y denotes output gap; Y is real GDP in log terms; Ȳ is potential real GDP; G_t is potential output growth; ε^Ȳ_t – potential output shock term; G_SS - potential output steady-state path; ε^G_t – potential output growth shock term; ε^y_t – output gap shock.
  - Phillips Curve equation:
    - π_t = λ π_t+1 + (1−λ) π_t−1 + β y_t + ε^π_t
    - Definitions: π_t denotes inflation; ε^π_t inflation shock.
  - Unemployment equations:
    - u_t = Ū_t − U_t
    - Ū_t = (τ^4 Ū_SS + (1−τ^4) Ū_t−1) + gŪ_t + ε^Ū_t
    - gŪ_t = (1−τ^3) gŪ_t−1 + ε^gŪ_t
    - Definitions: u_t - gap between actual unemployment and its equilibrium process; Ū_t denotes Non-Accelerating Level of Inflation; U_t - actual unemployment; Ū_SS is steady state inflation rate; gŪ_t is variation in the NAIRU; ε^Ū_t – NAIRU shock term; ε^gŪ_t is shock term of the variation in the NAIRU.
  - Okun’s Law equation:
    - u_t = τ^2 u_t−1 + τ^1 y_t + ε^u_t
    - Definitions: ε^u_t denotes unemployment gap shock term.

### Beveridge–Nelson (BN) decomposition
- Purpose and application:
  - Separates permanent and transitory components in economic time series to identify long-term trends versus short-term fluctuations; useful for potential output analysis and distinguishing structural shifts from temporary shocks.
  - Applied within an ARIMA model to isolate economic disturbances.
- Equations and definitions:
  - ARIMA model specification:
    - Y_t = φ(B) θ(B)^{-1} ε_t
    - Where Y_t is the time series, φ(B) is the AR polynomial, θ(B) is the MA polynomial, and ε_t is a white noise error term.
  - Permanent and transitory decomposition:
    - Y_t = Y_t^P + Y_t^T
    - Where Y_t^P is the permanent component, Y_t^T is the transitory component.
  - Estimation of permanent component:
    - Y_t^P = lim_{h→∞} E[Y_{t+h} | Y_t]
    - The permanent component is the long-run forecast from the ARIMA model as the forecast horizon approaches infinity.
  - Estimation of transitory component:
    - Y_t^T = Y_t − Y_t^P

### Hodrick–Prescott (HP) filter application
- Data and parameterization:
  - Applied to quarterly data spanning from 2000Q1 to 2024Q3, incorporating estimates and projections for 2024Q4 to 2027Q4 to address the end-point problem.
  - The smoothing parameter (lambda) is set to 1600.

*International Monetary Fund — sipea2025081 (Selected Issues Paper chapter content provided)*

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_Source: https://www.imf.org/-/media/files/publications/selected-issues-papers/2025/english/sipea2025081.pdf_