## wpiea2024067-print-pdf

## Source details

**Canonical URL:** [wpiea2024067-print-pdf](https://www.imf.org/-/media/files/publications/wp/2024/english/wpiea2024067-print-pdf.pdf)

## Other formats

- [Markdown version](/-/media/files/publications/wp/2024/english/wpiea2024067-print-pdf.pdf.md)
- [Structured JSON version](/-/media/files/publications/wp/2024/english/wpiea2024067-print-pdf.pdf.json)

---

### Key findings
- Frequently used country-level capital stock data contain excessive measurement error, rendering them implausible for many uses.
- High dispersion in estimated capital per capita exists even among countries with similar incomes:
  - United Kingdom in 1975: reported capital stock per person was 692 percent higher than Japan, though GDP per capita differed by only 10 percent.
  - Zambia in 1960: capital stock per capita was 1,469 percent higher than Mozambique in one data set.
  - Algeria’s reported capital stock was 673 percent higher than Egypt’s in the same data set.
- Implausible and inconsistent Total Factor Productivity (TFP) growth estimates arise from these capital data:
  - Estimated TFP growth between 1960 and 1985 in prior datasets: Republic of Congo 5.16 percent per annum; Bangladesh 1.92 percent; West Germany 2.12 percent; United States 0.03 percent.
  - Revised estimate in this paper: TFP growth in the United States between 1970 and 2014 is estimated at 2.6 percent per-year using proposed methods.

### Mechanism: Steady State Method (SSMA) and how it generates errors
- Common perpetual inventory starting point:
  - K̇ = I − δK. (0.1)
  - g_k = K̇/K = (I/Y)(Y/K) − δ. (0.2)
- Steady-state assumption often imposed:
  - K̇/K = Ẏ/Y = g_y. (0.3)
  - Leading to initial capital solutions:
    - K(0) = I (1/(g_y+δ)). (0.4)
    - (K/Y)(0) = (I/Y) (1/(g_y+δ)).
- SSMA is implemented with differing auxiliary choices (choice of I, choice of g_y, averaging periods, population growth n), producing inconsistent results across studies.

### Empirical puzzles traced to SSMA
- Two consistent puzzles from SSMA applications:
  - Extremely high dispersion of the capital-output ratio at low levels of GDP per-capita.
  - Inconsistency of capital-output ratio estimates for the same country across different data sets and vintages due to differing auxiliary assumptions.
- Examples of extreme dispersion from SSM applications (selected entries):
  - PWT6.1, 1960 sample: Botswana GDP per-capita (PPP) 958.01; Estimated K/Y 0.10; Index 100. Zimbabwe GDP per-capita (PPP) 1231.78; Estimated K/Y 6.65; Index 6650.
  - PWT7.1, 1970 sample: Gambia GDP per-capita (PPP) 1283.65; Estimated K/Y 0.27; Index 100. Congo, Republic of GDP per-capita (PPP) 1348.29; Estimated K/Y 5.72; Index 2128.
- Industry and partial indicators do not corroborate SSM-implied magnitudes (examples: McKinsey manufacturing comparisons, auto usage, electricity consumption).

### Error-correction effect: mechanism, asymmetry, magnitude, and persistence
- Perpetual inventory solution with constant I and δ and initial percentage error μ:
  - K(t) = K(0)(1 + μ)e−δt + I/δ (1 − e−δt). (Equation (0.6))
- Growth expression:
  - g_k = I / [K(0)(1+μ)e−δt + I/δ (1−e−δt)] − δ. (Equation (0.8))
  - If μ>0 (positive error in initial K), g_k is biased downward for all t; if μ<0, g_k is biased upward.
- Asymmetry and persistence illustrated by simulations:
  - Symmetric μ = ±0.5, I = 25, δ = 0.05, true K(0) = 500 (steady-state K = 500):
    - Year 5: mu=0.5 → -1.45%; mu=-0.5 → 3.44%
    - Year 10: mu=0.5 → -1.20%; mu=-0.5 → 2.30%
    - Year 15: mu=0.5 → -0.98%; mu=-0.5 → 1.61%
    - Year 20: mu=0.5 → -0.79%; mu=-0.5 → 1.16%
    - Year 25: mu=0.5 → -0.64%; mu=-0.5 → 0.85%
    - Year 30: mu=0.5 → -0.51%; mu=-0.5 → 0.64%
  - Country-based example (true K/Y = 1.0):
    - Reported K/Y: Nigeria = 0.19 (μ = -0.81), Zambia = 2.93 (μ = 1.93).
    - Simulated g_k errors:
      - Year 5: mu=1.93 → -3.06%; mu=-0.81 → 9.70%
      - Year 10: mu=1.93 → -2.74%; mu=-0.81 → 5.21%
      - Year 15: mu=1.93 → -2.42%; mu=-0.81 → 3.26%
      - Year 20: mu=1.93 → -2.11%; mu=-0.81 → 2.20%
      - Year 25: mu=1.93 → -1.80%; mu=-0.81 → 1.55%
      - Year 30: mu=1.93 → -1.52%; mu=-0.81 → 1.12%
- TFP implication example:
  - Misestimating Nigeria’s initial K/Y as 0.19 rather than 1.0 yields capital growth estimates 1.12 percentage points too high after 30 years; with a capital coefficient of 1/3 this implies a TFP growth estimate 0.37 percentage points too low after 30 years.
  - TFP growth estimates typically range between 0.5-1.5 percent per year; a 0.37 percentage-point distortion is large.
- Memory statistics:
  - Level-based memory m1(t) = K(0)e−δt / [K(0)e−δt + I/δ (1 − e−δt)]. (Equation (0.9))
    - Example: I = 20, δ = 0.05: with K(0) = 600 memory = 29 percent after 30 years; with K(0) = 200 memory = 12 percent after 30 years.
  - Growth-based memory m2(K(0), μ, δ, t) = g_k(K(0), μ, δ, t) − g_k(K(0), 0, δ, t). (Equation (0.11))
    - Example: μ = 1.93 gives m2(500,1.93,0.05,30) = −1.53.
- Regression evidence on persistence:
  - Partial regression of post-1980 capital growth on 1960 K/Y and average investment since 1980: coefficient for initial estimated K/Y in 1960 = −.037, se = .0025, t = −14.78.
  - Partial regression of post-1990 capital growth on initial-estimated K/Y in 1950: coefficient = −.0043, se = .00126, t = −3.42.
- Empirical illustration: Jamaica
  - Jamaican average investment ratio (1953–2010): 24.39 percent.
  - Assumptions: constant investment rate = 0.2439, δ = 0.06 → implied steady-state K/Y = 4.06.
  - Initial SSMA estimate used: 2.0 → observed K/Y rose toward 4.06 illustrating error-correction-driven capital growth unrelated to rising investment rates.

### Alternative methods for estimating initial capital stocks
- Six methods compared (overview):
  - (a) Traditional steady-state method criticized in this paper (SSMA).
  - (b) Second steady-state method that sets initial K/Y equal to long-run equilibrium given observed investment data (SSMB): K/Y = [Ī/Y] / δ. (Equation (0.13))
  - (c) and (d) Feenstra, Inklaar, and Timmer (2015) variants assuming same initial K/Y by type of capital good (FITR with constant-price investment data, FITN with nominal investment data). Assumed K/Y by asset-type (example shown): structures (2.2); transport equipment (0.1); other machinery and assets (0.3); ICT assets (0.0).
  - (e) New proposal using cross-country electricity consumption data as a proxy for parts of the capital stock.
  - (f) New proposal using automobile data as a proxy for parts of the capital stock.
- Proxy-method rationale and procedure (electricity example):
  1. Convert a high-quality country's K/Y (example: US) to capital per-person for a given year T0.
  2. Run cross-sectional regression: log(electricity per capita_j) = α + β log(GDP per capita_j) + ε_j (year with broad coverage; example year 2000).
  3. Assume capital per capita of country i falls relative to US by the same proportion as electricity per capita, using fitted relation.
  4. Convert fitted capital per-capita back to initial K/Y for country i in T0; then evolve K/Y forward via the perpetual inventory equation:
     - KY_i,t+1 = (GDP_i,t / GDP_i,t+1) (INV_i,t + (1−δ) KY_i,t) for t > T0.
- Regression results (year 2000):
  - Log cars per-person:
    - Constant: -4.057
    - log GDP per-person: 1.018
    - N: 162
    - R2: 74%
    - (T-ratio) Constant: (-9.76); (T-ratio) log GDP per-person: (21.18)
  - Log electricity consumption per person:
    - Constant: (-2.508)
    - log GDP per-person: 1.122
    - N: 126
    - R2: 77%
    - (T-ratio) Constant: (-5.20); (T-ratio) log GDP per-person: (20.63)
- Cross-sectional elasticity: a one percent increase in GDP per-person is associated with a 1.12 percent increase in electricity use per-person (1.02 percent for automobiles).

### Comparison of methods: dispersion, plausibility, and robustness
- Main assessment criterion: plausibility measured by dispersion of K/Y both unconditionally and conditional on GDP level.
- Descriptive statistics for capital-output ratio in 1975 for 108 countries with GDP per-capita < 10,000:
  - Electricity data: Mean 1.67; Median 1.58; Standard Deviation 0.98; 90th percentile 2.41; 10th percentile 0.88; 90–10 difference 1.53
  - Automobile data: Mean 1.81; Median 1.71; Standard Deviation 0.99; 90th percentile 2.76; 10th percentile 0.97; 90–10 difference 1.79
  - Same initial KY by asset – FIT nominal: Mean 1.54; Median 1.44; Standard Deviation 0.91; 90th percentile 2.50; 10th percentile 0.65; 90–10 difference 1.85
  - Steady State Method: Mean 2.90; Median 2.76; Standard Deviation 1.23; 90th percentile 3.98; 10th percentile 1.70; 90–10 difference 2.28
  - Long Run Eq of PIE: Mean 2.24; Median 1.92; Standard Deviation 1.33; 90th percentile 3.51; 10th percentile 1.10; 90–10 difference 2.41
  - Same initial KY by asset – FIT real: Mean 2.90; Median 2.42; Standard Deviation 2.03; 90th percentile 4.99; 10th percentile 1.27; 90–10 difference 3.72
- Summary: three methods deliver lower variance than steady-state methods:
  - Electricity-based method
  - Automobile-based method
  - FIT variant using nominal investment data
- Influence of investment data quality noted; extreme investment ratios (examples: Cape Verde 49 percent, Cyprus 47.8 percent) create outliers and may warrant prudential exclusion.

### Country-level comparisons and implications for TFP estimates
- OECD countries: similar capital-output and TFP estimates across methods for many (United States, United Kingdom); Spain an exception where SSMA yields much higher KY and TFP growth not corroborated by OECD data.
- Low- and middle-income countries: large method-dependent differences. Selected comparisons (Initial KY - 1970; TFP growth 1970–2014):
  - United States: Initial KY — Steady State 2.63, Electricity 2.60, Automobile 2.60; TFP growth — Steady State 0.73%, Electricity 0.83%, Automobile 0.83%
  - United Kingdom: Initial KY — 2.64, 2.71, 2.73; TFP growth — 1.14%, 1.22%, 1.23%
  - Spain: Initial KY — 3.57, 1.81, 1.84; TFP growth — 1.62%, 1.06%, 1.07%
  - Jamaica: Initial KY — 5.96, 1.86, 1.92; TFP growth — 0.51%, -0.43%, -0.40%
  - Cameroon: Initial KY — 0.87, 1.62, 1.88; TFP growth — 0.17%, 1.02%, 1.16%
  - Cambodia: Initial KY — 7.79, 1.66, 2.30; TFP growth — 1.83%, 0.67%, 0.97%
- Illustrative revisions relative to prior datasets:
  - Republic of the Congo (Klenow and Rodríguez-Clare 1997 / PWT5.6): prior TFP growth 5.16% (1960–1985) with capital growth −1.81% per year. Using electricity estimates: capital growth 4.35% per year, TFP growth revised to 3.94% per year.
  - Bangladesh (same prior dataset): prior TFP growth 1.92% per year with capital growth −1.11%. Revised: capital growth 0.07% per year, TFP growth 0.15% per year.
  - United States (same prior dataset): prior TFP growth 0.03% per year; revised estimate 1.76% per year.
  - Singapore (Young 1995): prior TFP growth −0.3% per year (1966–1990); revised estimate in this paper 0.9% per year.
- Asian Tigers revised TFP averages:
  - Hong Kong: 2.3 percent per year
  - South Korea: 3.2 percent per year
  - Taiwan: 2.9 percent per year
  - Comparable Young (1995) numbers: Hong Kong 2.3; South Korea 1.6; Taiwan 2.4

### Conclusion and policy implications
- Improving initial capital stock estimates by incorporating electricity consumption or automobile stock data (or by harmonizing initial K/Y by asset class) materially improves plausibility and consistency of country-level capital and TFP data.
- Preferred methods reduce implausible dispersion in capital-output ratios, mitigate the error-correction bias in capital growth and TFP estimates, and align better with theory and partial indicators.
- Practical implications:
  - Estimate initial capital conservatively to avoid producing highly variable cross-country estimates.
  - Use correlated proxy data (electricity, automobiles) to yield more credible initial capital estimates and more reliable TFP growth measures.
  - Avoid reliance on SSMA without careful auxiliary choices, as it can lead to misleading empirical conclusions about countries’ TFP and growth performance.

*Source: Executive Summary and selected sections, "A Proposal to Improve Country-level Data on Total Factor Productivity Growth" (IMF Working Paper).*

### Executive Summary ......................................................................................................

### Executive Summary

### Key findings
- Frequently used country-level capital stock data contain excessive measurement error, rendering them implausible for many uses.
- High dispersion in estimated capital per capita exists even among countries with similar incomes:
  - In one data set, reported capital stock per person in the United Kingdom in 1975 was 692 percent higher than that of Japan, even though GDP per capita differed by only 10 percent.
  - In another data set, the capital stock per capita for Zambia in 1960 was 1,469 percent higher than that of Mozambique.
  - Algeria’s reported capital stock was 673 percent higher than Egypt’s in the same data set.
- Implausible and inconsistent Total Factor Productivity (TFP) growth estimates arise from these capital data:
  - Estimated TFP growth between 1960 and 1985: Republic of Congo 5.16 percent per annum; Bangladesh 1.92 percent; West Germany 2.12 percent; United States 0.03 percent.
  - Revised estimate: TFP growth in the United States between 1970 and 2014 is estimated at 2.6 percent per-year using methods proposed in this paper.

### Mechanism: Steady State Method (SSMA) and how it generates errors
- Common procedure starts from the perpetual inventory equation:
  - 퐾̇ = 퐼 − 훿퐾. (0.1)
  - Transform to capital growth: 푔푘 = 퐾̇/퐾 = (퐼/푌)(푌/퐾) − 훿. (0.2)
- Frequently used assumption: country is in a steady state so capital stock growth equals GDP growth:
  - 퐾̇/퐾 = 푌̇/푌 = 푔푦. (0.3)
- Solving for initial capital or initial capital-output ratio yields:
  - 퐾(0) = 퐼 (1/(푔푦+훿)). (0.4)
  - 퐾/푌 (0) = (퐼/푌) (1/(푔푦+훿)).
- The steady-state assumption (SSMA) is often used for computational convenience and implemented with differing auxiliary choices (e.g., choice of I, choice of g, averaging periods, population growth n), producing inconsistent results across studies.

### Empirical puzzles traced to SSMA
- Two consistent empirical puzzles from applying SSMA:
  - Extremely high dispersion of the capital-output ratio at low levels of GDP per-capita.
  - Inconsistency of capital-output ratio estimates for the same country across different data sets and vintages due to different auxiliary assumptions (e.g., differing start years, choices of I and 푔푦).
- The mathematics of the perpetual inventory equation ensures that errors in initial capital estimates transmit to errors in subsequent capital growth estimates; these are inversely related and can produce large, persistent bias in capital growth and thus TFP growth.

### Properties of the bias from SSMA
- Bias is asymmetric: negative errors in initial capital stocks impart larger errors to the growth rate of capital than positive errors of equal magnitude.
- Bias can be large for some countries and does not necessarily decay rapidly over time; errors in initial capital can significantly affect TFP growth data 40 years after the initial estimate.
- The error-correction effect embedded in the perpetual inventory equation mechanically generates an automatic negative relation between errors in estimating the initial capital stock and errors in subsequent growth rates of the capital stock.

### Proposed alternatives and their effects
- Supplementing or replacing SSMA with data-grounded proxies reduces implausible dispersion and mitigates biased capital growth estimates:
  - Use of electricity consumption time series as a grounding variable for capital stock estimation (motivated by historically high correlations between energy consumption and capital stock).
  - Use of automobile stock data as a proxy for capital.
  - Imposition of constant capital-output ratios by asset class at the beginning (same initial K/Y by asset for all countries) as another approach.
- Results from these alternatives:
  - Greatly reduced dispersion in capital-output ratios across countries of similar incomes.
  - Reduced error-correction effect that biases capital growth and TFP estimates.
  - Many extreme TFP growth estimates are revised toward more plausible values and closer alignment across countries with similar incomes.

### Conclusion and policy implication
- Improving initial capital stock estimates by incorporating electricity consumption or automobile stock data (or by harmonizing initial K/Y by asset class) materially improves the plausibility and consistency of country-level capital and TFP data.
- Better capital stock data are crucial because TFP growth estimates derived from capital series are widely used to evaluate policy efficacy and the determinants of long-term prosperity; reducing measurement error in capital stocks strengthens the empirical basis for such analyses.

*Source: Executive Summary, "A Proposal to Improve Country-level Data on Total Factor Productivity Growth" (IMF Working Paper).*

### 1960.   For each country, the initial capital stock was calculated for the first year in which the investment series

### Estimated Capital-Output Ratios and Measurement Issues (excerpt)

### Key findings on estimated capital-output ratios (1960 and 1970)
- Applying the Steady-State Method (SSM) to PWT6.1 data for 1960 yields highly dispersed capital-output ratios across countries with similar GDP per-capita (PPP). Selected results (Table 1):
  - Madagascar: GDP per-capita (PPP) 1239.57; Estimated Capital-Output Ratio 0.33; Index (Botswana=100) 330
  - Zimbabwe: GDP per-capita (PPP) 1231.78; Estimated Capital-Output Ratio 6.65; Index (Botswana=100) 6650
  - Chad: GDP per-capita (PPP) 1212.39; Estimated Capital-Output Ratio 1.50; Index (Botswana=100) 1500
  - Zambia: GDP per-capita (PPP) 1206.58; Estimated Capital-Output Ratio 2.93; Index (Botswana=100) 2930
  - Thailand: GDP per-capita (PPP) 1091.12; Estimated Capital-Output Ratio 1.15; Index (Botswana=100) 1150
  - Bangladesh: GDP per-capita (PPP) 1057.28; Estimated Capital-Output Ratio 0.33; Index (Botswana=100) 330
  - Nigeria: GDP per-capita (PPP) 1032.72; Estimated Capital-Output Ratio 0.19; Index (Botswana=100) 190
  - Cape Verde: GDP per-capita (PPP) 994.47; Estimated Capital-Output Ratio 2.32; Index (Botswana=100) 2320
  - Mali: GDP per-capita (PPP) 982.62; Estimated Capital-Output Ratio 0.27; Index (Botswana=100) 270
  - Zaire: GDP per-capita (PPP) 979.89; Estimated Capital-Output Ratio 0.56; Index (Botswana=100) 560
  - Botswana: GDP per-capita (PPP) 958.01; Estimated Capital-Output Ratio 0.10; Index (Botswana=100) 100
  - Rwanda: GDP per-capita (PPP) 937.83; Estimated Capital-Output Ratio 0.13; Index (Botswana=100) 130
  - Indonesia: GDP per-capita (PPP) 936.08; Estimated Capital-Output Ratio 0.30; Index (Botswana=100) 300
- Applying SSM to PWT7.1 data for 1970 again yields highly variable K/Y for countries with similar GDP per-capita. Selected results (Table 2):
  - Cote d`Ivoire: GDP per-capita (PPP) 1396.29; Estimated Capital-Output Ratio 0.98; Index (The Gambia=100) 364
  - Botswana: GDP per-capita (PPP) 1383.36; Estimated Capital-Output Ratio 1.90; Index (The Gambia=100) 706
  - Congo, Republic of: GDP per-capita (PPP) 1348.29; Estimated Capital-Output Ratio 5.72; Index (The Gambia=100) 2128
  - Haiti: GDP per-capita (PPP) 1310.59; Estimated Capital-Output Ratio 0.51; Index (The Gambia=100) 188
  - Gambia, The: GDP per-capita (PPP) 1283.65; Estimated Capital-Output Ratio 0.27; Index (The Gambia=100) 100
  - Senegal: GDP per-capita (PPP) 1255.10; Estimated Capital-Output Ratio 0.81; Index (The Gambia=100) 302
  - Comoros: GDP per-capita (PPP) 1212.04; Estimated Capital-Output Ratio 2.37; Index (The Gambia=100) 881
  - Guinea-Bissau: GDP per-capita (PPP) 1202.00; Estimated Capital-Output Ratio 3.33; Index (The Gambia=100) 1239
  - Egypt: GDP per-capita (PPP) 1170.71; Estimated Capital-Output Ratio 0.86; Index (The Gambia=100) 318
  - Madagascar: GDP per-capita (PPP) 1160.89; Estimated Capital-Output Ratio 1.22; Index (The Gambia=100) 452
  - Togo: GDP per-capita (PPP) 1157.28; Estimated Capital-Output Ratio 1.12; Index (The Gambia=100) 418
  - Cape Verde: GDP per-capita (PPP) 1117.18; Estimated Capital-Output Ratio 4.51; Index (The Gambia=100) 1678
  - Kenya: GDP per-capita (PPP) 1026.14; Estimated Capital-Output Ratio 1.67; Index (The Gambia=100) 620

### Theoretical expectations versus empirical patterns
- Benchmark economic theory (Cobb-Douglass production, constant returns to scale, competitive conditions) predicts:
  - Richer countries should have higher capital-output ratios.
  - Higher w/r ratios imply higher capital-labor ratios, higher GDP per worker, and higher capital-output ratios (holding technology constant).
  - Associations between capital-output ratio and GDP per-worker (and GDP per-capita, absent huge differences in labor force participation) should be positive.
- Empirical results from these SSM applications instead show a triangular-shaped scatter with no strong positive slope in Figures 1 and 2: high dispersion in K/Y for given GDP per-capita, especially among lower-GDP countries.

### Lack of corroboration from firm- and industry-level evidence
- Industry or firm-level evidence does not support the extreme cross-country differences implied by SSM estimates:
  - Table 1’s estimates would imply, for example, that structures and machinery in Zambia are 29 times those of Botswana, or Zimbabwe is 35 times (6.65/0.19) that of Nigeria; these magnitudes are not supported by firm/industry evidence.
  - McKinsey manufacturing productivity comparisons (1993) did not cite capital stocks or the capital-labor ratio as important explanations for labor productivity differences; in an auto assembly example (US vs Japan) the capital-labor ratio in the US was only 4 percent higher.
  - Partial indicators (automobile usage, electricity consumption) do not corroborate cross-country differences on the order of 2000-3000 percent.

### Robustness checks and alternative calculations
- Using longer spans for growth and investment (example: calculate real GDP growth over 1960-1985, mean I/Y over 1965-1985, depreciation d = 0.06, formula K/Y = (I/Y)/(g+d), results shown for 1960 with PWT6.1) still yields high dispersion:
  - Example: Uganda K/Y 0.27; Zambia K/Y 2.88; Guyana an outlier with K/Y above 4.0.
- Directly reported capital and GDP per worker in PWT5.6 (1970) also show extreme dispersion:
  - Sierra Leone (per-capita GDP PPP $1435 in 1970) had K/Y 0.034; Madagascar (per-capita GDP PPP $1146) had K/Y 0.727 — 21 times higher.

### Rising capital-output ratios and measurement discrepancies
- The Steady-State Method for choosing initial capital can produce dramatically rising K/Y over time even when investment rates are not high or rising — the rise can be an artifact of the initial capital assumption.
- This produces an increasing discrepancy between two measures of capital input:
  - Investment ratio (I/Y)
  - Growth in the capital stock (ΔK/K)
- Jamaica example:
  - Capital-output ratio rises substantially (Figure 8).
  - Investment ratios show no upward trend and remain relatively normal (Figure 9).
  - Over time the discrepancy expands; by 2020 the investment ratio is about 18 percent while data on capital stock growth show negative growth (Figures 9–10).

### Graphical evidence across multiple data vintages and studies
- High dispersion in K/Y for given GDP per-capita is present across:
  - Caselli’s SSM with PWT6.1 (Figure 1)
  - Caselli’s SSM with PWT7.1 (Figure 2)
  - SSM with long-period averages and d = 0.06 (Figure 3)
  - PWT5.6 implicit data (Figure 4)
  - Nehru and Dhareshwar (1993) (Figure 5)
  - Klenow and Rodriguez-Clare (1997, PWT5.6) (Figure 6)
  - PWT9.0 (Figure 7)

### Implications highlighted in the excerpt
- The method for choosing initial capital (Steady-State Method variants) can generate implausible cross-country differences in capital-output ratios and produce internal inconsistencies between investment-based and capital-stock-based measures of capital input.
- These measurement issues can bias estimates of Total Factor Productivity growth and lead to misleading cross-country comparisons of capital intensity.

*IMF WORKING PAPERS — A Proposal to Improve Country-level Data on Total Factor Productivity Growth*

### section is rooted in an error-correction effect that is triggered whenever the initial capital estimate is out of line

### section is rooted in an error-correction effect that is triggered whenever the initial capital estimate is out of line

### Error-correction effect: mechanism and formal expression
- Perpetual inventory equation (stable differential equation): K̇ = I − δK. (Equation (0.5))
- Solution with constant I and δ, and initial percentage error μ in K(0):  
  K(t) = K(0)(1 + μ)e−δt + I/δ (1 − e−δt). (Equation (0.6))
- Implication: observed capital evolves from K(0)(1 + μ) at t = 0 to steady state I/δ as t → ∞. Dividing by Y gives the relation in terms of K/Y and I/Y.
- If the initial value is lower than the steady state, capital will grow; if higher, capital will decline. This is the error-correction effect: the deviation between initial capital-output ratio and steady-state capital-output ratio implied by investment data determines trending in K/Y.
- Rewriting growth in capital: g_k = K̇/K = I/K − δ. (Equation (0.7))
- Substituting K(t) yields:  
  g_k = I / [K(0)(1+μ)e−δt + I/δ (1−e−δt)] − δ. (Equation (0.8))
- Sign and bias: if μ>0 (positive error in initial capital), growth in capital g_k is biased downward for all t; if μ<0, g_k is biased upward. A positive μ imparts a negative bias in capital growth and a positive bias in TFP growth (given the usual decomposition).

### Empirical illustration: Jamaica
- Jamaican average investment ratio (1953–2010): 24.39 percent.
- Illustration assumptions: constant investment rate = 0.2439, constant depreciation rate = 0.06.
- Implied steady-state capital-output ratio for Jamaica: 4.06.
- Initial estimate using steady-state method: 2.0 (far lower than 4.06), producing prolonged adjustment with growth in capital stock.
- Observed capital-output ratio in Jamaica rose from around 2.0 to close to 4.0, demonstrating the error-correction effect can introduce significant differences between capital growth rates and investment rates.

### Simulations: asymmetry, magnitude, and persistence of bias
- Asymmetry: downside errors in K(0) translate into larger errors in g_k than upside errors of equal magnitude.
- Simplified simulation setup (illustrative): μ = ±0.5 (i.e., 50 percent too low or 50 percent too high), I = 25, δ = 0.05 → steady-state K = 500. Base true K(0) = 500 so true growth = 0.
- Table 3 (simulated errors in capital growth from symmetric +/-50 percent errors):
  - Year 5: mu=0.5 → -1.45%; mu=-0.5 → 3.44%
  - Year 10: mu=0.5 → -1.20%; mu=-0.5 → 2.30%
  - Year 15: mu=0.5 → -0.98%; mu=-0.5 → 1.61%
  - Year 20: mu=0.5 → -0.79%; mu=-0.5 → 1.16%
  - Year 25: mu=0.5 → -0.64%; mu=-0.5 → 0.85%
  - Year 30: mu=0.5 → -0.51%; mu=-0.5 → 0.64%
- Illustration with country-based errors (Table 4) assuming true K/Y = 1.0 for both Nigeria and Zambia:
  - Reported K/Y: Nigeria = 0.19 (implies μ = -0.81), Zambia = 2.93 (implies μ = 1.93).
  - Table 4 (simulated errors in capital growth from these μ values):
    - Year 5: mu=1.93 → -3.06%; mu=-0.81 → 9.70%
    - Year 10: mu=1.93 → -2.74%; mu=-0.81 → 5.21%
    - Year 15: mu=1.93 → -2.42%; mu=-0.81 → 3.26%
    - Year 20: mu=1.93 → -2.11%; mu=-0.81 → 2.20%
    - Year 25: mu=1.93 → -1.80%; mu=-0.81 → 1.55%
    - Year 30: mu=1.93 → -1.52%; mu=-0.81 → 1.12%
- Magnitude and TFP implication: misestimating Nigeria’s initial K/Y as 0.19 rather than 1.0 yields capital growth estimates 1.12 percentage points too high after 30 years; assuming a capital coefficient of 1/3, this implies a TFP growth estimate 0.37 percentage points too low after 30 years. TFP growth estimates typically range between 0.5-1.5 percent per year, so 0.37 percentage points is large.

### Memory of initial capital errors: levels versus growth rates
- Common memory statistic for level influence with constant I and δ:  
  m1(t) = K(0)e−δt / [K(0)e−δt + I/δ (1 − e−δt)]. (Equation (0.9))
- Level-based memory examples (simulations with I = 20, δ = 0.05):
  - With K(0) = 600: memory statistic is 29 percent after 30 years.
  - With K(0) = 200: memory statistic is 12 percent after 30 years.
- But relevance for TFP is the effect on growth rates. Define growth-based memory statistic:  
  m2(K(0), μ, δ, t) = g_k(K(0), μ, δ, t) − g_k(K(0), 0, δ, t). (Equation (0.11))
- Using prior simulations: with μ = 1.93, m2(500,1.93,0.05,30) = −1.53 (i.e., measured capital growth would be −1.53 percent rather than the true value after 30 years).
- Regression evidence on persistence:
  - Partial regression of post-1980 capital growth on 1960 capital-output ratio and average investment since 1980 yields coefficient for initial estimated K/Y in 1960: coef = −.037, se = .0025, t = −14.78 (Figure 13).
  - Partial regression of post-1990 capital growth on initial-estimated K/Y in 1950 yields coefficient: coef = −.0043, se = .00126, t = −3.42 (Figure 14).
- Conclusion: estimates of initial capital decades earlier still strongly influence subsequent capital growth estimates.

### Alternative methods for estimating initial capital stocks (overview)
- Six methods to be described and compared in the remaining paper (listed here as presented):
  - (a) Traditional steady-state method criticized in this paper (SSMA).
  - (b) Second steady-state method that sets initial K/Y equal to long-run equilibrium given observed investment data (SSMB).
  - (c) and (d) Two variants proposed by Feenstra, Inklaar, and Timmer (2015) assuming same initial K/Y by type of capital good (FIT series: FITR with constant-price investment data, FITN with nominal investment data).
  - (e) New proposal using cross-country electricity consumption data as a proxy for parts of the capital stock.
  - (f) New proposal using automobile data as a proxy for parts of the capital stock.
- SSMB rationale and formula:
  - Problem with SSMA: reliance on an arbitrarily chosen GDP growth rate g introduces unavoidable arbitrariness in SSMA estimates.
  - SSMA formula summary: K/Y = [Ī/Y] / [g_y + δ]. (Equation (0.12))
  - SSMB proposal: use mean investment so that K/Y = [Ī/Y] / δ. (Equation (0.13))
  - SSMB is expected to greatly reduce (but not eliminate) the error-correction influence by choosing an initial K/Y closer to the long-run equilibrium implied by observed investment.
- FIT constant capital-output ratios by capital-good type (assumptions shown in parentheses):
  - structures (2.2); transport equipment (0.1); other machinery and assets (0.3); ICT assets (0.0).
- Use of proxies (autos, electricity):
  - Proxy logic: component-level data may be more accurately measured and could be viable proxies if capital components are used in fixed proportions.
  - Both cars per-person and electricity consumption per-person exhibit a strong positive log-linear relation with GDP per-capita (Figures 15 and 16).
  - Regression estimates (Table 5, referenced) show a cross-section elasticity slightly above 1.0 for both variables for the year 2000.

_italic Source: wpiea2024067-print-pdf - section is rooted in an error-correction effect that is triggered whenever the initial capital estimate is out of line_

### 2000.  Other years reveal similar results.  A one percent increase in GDP per-person is associated with a 1.12

### wpiea2024067-print-pdf - 2000.  Other years reveal similar results.  A one percent increase in GDP per-person is associated with a 1.12

### Empirical findings from cross-country regressions (year 2000)
- A one percent increase in GDP per-person is associated with a 1.12 percent increase in electricity use per-person (1.02 percent for automobiles).
- Outliers have identifiable explanations:
  - Zimbabwe: hyperinflation and economic crisis.
  - Equatorial Guinea and Gabon: oil production inflates GDP relative to auto consumption.
  - Singapore and Hong Kong: congested urban areas, extensive public transport, low auto consumption given GDP per-capita.
  - Mozambique: Africa’s largest hydropower plant enabling exports to South Africa and Zimbabwe and boosting domestic urban service, hence high electricity consumption given GDP.
- Table 5 — Regressions of capital stock proxies on GDP (year 2000):
  - Dependence Variable: Log Cars per-person
    - Estimated Coefficient (Constant): -4.057
    - Estimated Coefficient (log GDP per-person): 1.018
    - N: 162
    - R2: 74%
    - (T-ratio) for Constant: (-9.76)
    - (T-ratio) for log GDP per-person: (21.18)
  - Dependence Variable: Log electricity consumption per person
    - Estimated Coefficient (Constant): (-2.508)
    - Estimated Coefficient (log GDP per-person): 1.122
    - N: 126
    - R2: 77%
    - (T-ratio) for Constant: (-5.20)
    - (T-ratio) for log GDP per-person: (20.63)

### Proposed method to estimate initial capital stocks using capital proxies
- Key assumption: the empirical relation between total capital and GDP is the same as that between the capital proxy (electricity or automobiles per-capita) and GDP per-capita.
- Steps using electricity data (illustrative):
  1. Start with the KY ratio of a country believed to have high-quality measurement of capital (example: the US). Convert the US’s capital-output ratio for a given year to capital per-person.
  2. Run a cross-sectional regression of log electricity per capita on log GDP per capita for a year with broad country coverage (example: year 2000; see Figure 16 and Table 5).
  3. Assume capital per capita of a given country falls below the US by the same proportion as electricity consumption per capita, using the fitted relation from step 2 (use fitted relation, not country-level datapoints).
  4. Calculate estimated initial capital-output ratio for the country by multiplying the fitted-value for capital per-capita by GDP per-capita in the first year investment data are available; this becomes the country's initial capital-output ratio.
- Regression specification presented:
  - log(electricity per capita_j) = α + β log(GDP per capita_j) + ε_j
- Implementation for country i with first investment year T0:
  - Observe KY_US(T0); compute log capital per person for US: Lllog(KY_US,T0).
  - Estimate log capital per person for country i:
    - Lllog(KY_i,T0) = Lllog(KY_US,T0) + β̂ (llog(GDP per capita_i,T0) − llog(GDP per capita_US,T0))
  - Convert back by exponentiating and divide by GDP per-person to obtain initial capital-output ratio.
- Evolution over time via discrete-time perpetual inventory equation:
  - KY_i,t+1 = (GDP_i,t / GDP_i,t+1) (INV_i,t + (1−δ) KY_i,t) for t > T0
  - Where KY is capital-output ratio, INV is investment as a fraction of GDP, and δ is depreciation.

### Advantages versus Steady State Method (SSMA)
- Three main arguments in favor of the capital-proxy approach:
  1. Based on an empirically supported relationship between observable correlates of the capital stock and GDP.
  2. Consistent with a straightforward production function framework.
  3. Does not require the more dubious assumption that all countries are in steady-state where output growth equals capital growth.

### Comparison of alternative methods (dispersion and plausibility)
- Main assessment criterion: plausibility of capital-output ratio estimates, judged by dispersion both unconditionally and conditional on GDP level.
- Visual comparisons: Figures 18–23 underline lower dispersion for methods using electricity data, automobile data, and the FIT (2015) method using nominal investment data compared with steady-state methods.
- Influence of investment data quality: unusually high investment ratios (e.g., Cape Verde 49%, Cyprus 47.8%) create outliers; prudential exclusion of extreme investment observations is argued.
- Descriptive statistics for capital-output ratio in 1975 for 108 countries with GDP per-capita < 10,000 (N=108):
  - Electricity data: Mean 1.67; Median 1.58; Standard Deviation 0.98; 90th percentile 2.41; 10th percentile 0.88; Difference between 90th and 10th percentile 1.53
  - Automobile data: Mean 1.81; Median 1.71; Standard Deviation 0.99; 90th percentile 2.76; 10th percentile 0.97; Difference 1.79
  - Same initial KY by asset – FIT nominal: Mean 1.54; Median 1.44; Standard Deviation 0.91; 90th percentile 2.50; 10th percentile 0.65; Difference 1.85
  - Steady State Method: Mean 2.90; Median 2.76; Standard Deviation 1.23; 90th percentile 3.98; 10th percentile 1.70; Difference 2.28
  - Long Run Eq of PIE: Mean 2.24; Median 1.92; Standard Deviation 1.33; 90th percentile 3.51; 10th percentile 1.10; Difference 2.41
  - Same initial KY by asset – FIT real: Mean 2.90; Median 2.42; Standard Deviation 2.03; 90th percentile 4.99; 10th percentile 1.27; Difference 3.72
- Summary conclusion: three methods deliver lower variance than steady-state methods:
  - Electricity-based method
  - Automobile-based method
  - FIT variant using nominal investment data

### Country-level comparisons and implications for TFP estimates
- OECD countries: generally similar capital-output and TFP estimates across methods (United States, United Kingdom). Exception: Spain under steady-state method shows much higher KY and TFP growth relative to US; OECD data do not corroborate that high TFP growth.
- Low- and middle-income countries: methods differ sharply. Examples from Table 7 (Initial KY - 1970; TFP growth 1970-2014):
  - United States: Initial KY — Steady State 2.63, Electricity 2.60, Automobile 2.60; TFP growth — Steady State 0.73%, Electricity 0.83%, Automobile 0.83%
  - United Kingdom: Initial KY — 2.64, 2.71, 2.73; TFP growth — 1.14%, 1.22%, 1.23%
  - Spain: Initial KY — 3.57, 1.81, 1.84; TFP growth — 1.62%, 1.06%, 1.07%
  - Jamaica: Initial KY — 5.96, 1.86, 1.92; TFP growth — 0.51%, -0.43%, -0.40%
  - Cameroon: Initial KY — 0.87, 1.62, 1.88; TFP growth — 0.17%, 1.02%, 1.16%
  - Cambodia: Initial KY — 7.79, 1.66, 2.30; TFP growth — 1.83%, 0.67%, 0.97%
- Mechanism: steady-state method often produces high initial KY for some low/middle-income countries, leading to negative capital growth during an error-correction phase and artificially high TFP growth estimates. The electricity and automobile proxy methods mitigate this error-correction distortion.
- Illustrative revisions relative to prior datasets:
  - Republic of the Congo (in Klenow and Rodríguez-Clare 1997 / PWT5.6): prior TFP growth 5.16% (1960–1985) with capital growth −1.81% per year. Using electricity estimates: capital growth 4.35% per year, TFP growth revised to 3.94% per year.
  - Bangladesh (same prior dataset): prior TFP growth 1.92% per year with capital growth −1.11%. Using revised estimates: capital growth 0.07% per year, TFP growth revised to 0.15% per year.
  - United States (same prior dataset): prior TFP growth 0.03% per year; revised estimate 1.76% per year.
  - Singapore (Young 1995): prior TFP growth −0.3% per year (1966–1990); revised estimate in this paper 0.9% per year driven by revised higher GDP growth (9.7 percent rather than ...).

*IMF Working Papers — A Proposal to Improve Country-level Data on Total Factor Productivity Growth*

### 8.5 percent) rather than major differences in estimated capital growth, which in fact go the other way (12.1

### A Proposal to Improve Country-level Data on Total Factor Productivity Growth

### Revised TFP estimates for the Asian Tigers
- Revised estimates show average TFP growth rates of:
  - Hong Kong: 2.3 percent per year
  - South Korea: 3.2 percent per year
  - Taiwan: 2.9 percent per year
- Comparable numbers from Young (1995) were:
  - Hong Kong: 2.3
  - South Korea: 1.6
  - Taiwan: 2.4
- The paper emphasizes that revised capital growth estimates do not account for these differences (12.1 percent rather than 10.8 percent noted for capital growth in the text).

### Empirical figures and observed patterns
- Multiple figures illustrate capital-output ratios under different methods:
  - Figure 18: Capital output ratio – Steady State Method
  - Figure 19: Capital output ratio – Long Run Eq. of PIE
  - Figure 20: Capital output ratio – Electricity data
  - Figure 21: Capital output ratio - same initial K/Y by asset for all countries, real data
  - Figure 22: Capital output ratio - same initial K/Y by asset for all countries, nominal data
  - Figure 23: Capital output ratio – Automobile data
- Graph axes reported in the figures include:
  - KY Ratio (values shown from 0 to 10 on vertical axis)
  - GDP per-person (USD, PPP) with tick labels shown as 0 5000 10000 15000 20000 25000 in the figures.

### Table 8 — TFP growth estimates comparison (Young 1995 vs. revised)
- Time period and variables:
  - Output variable: "rgdpna"; aggregate labor growth: "emp*hc". Source: PWT9.0.
  - Capital growth: electricity-based capital stock estimates from this paper.
- Estimates from Young (1995):
  - Hong Kong 66-91: Output 0.073; Aggregate Capital 0.077; Weighted Capital 0.080; Aggregate Labor 0.026; Weighted Labor 0.032; TFP 0.023; Labor Share 0.628 (Table V)
  - Singapore 66-90: Output 0.085; Aggregate Capital 0.108; Weighted Capital 0.115; Aggregate Labor 0.045; Weighted Labor 0.057; TFP -0.003; Labor Share 0.470 (Table VI)
  - South Korea 66-90: Output 0.104; Aggregate Capital 0.129; Weighted Capital 0.137; Aggregate Labor 0.054; Weighted Labor 0.064; TFP 0.016; Labor Share 0.680 (Table VII)
  - Taiwan 66-90: Output 0.096; Aggregate Capital 0.118; Weighted Capital 0.123; Aggregate Labor 0.046; Weighted Labor 0.051; TFP 0.024; Labor Share 0.710 (Table VII)
- Revised estimates using electricity-based capital growth and PWT 9.0 output and labor force growth:
  - Hong Kong 66-91: Output 0.073; Aggregate Capital 0.069; Weighted Labor 0.045; TFP 0.023; Labor Share 0.628
  - Singapore 66-90: Output 0.088; Aggregate Capital 0.121; Weighted Labor 0.051; TFP 0.009; Labor Share 0.470
  - South Korea 66-90: Output 0.091; Aggregate Capital 0.127; Weighted Labor 0.052; TFP 0.032; Labor Share 0.680
  - Taiwan 66-90: Output 0.087; Aggregate Capital 0.116; Weighted Labor 0.049; TFP 0.029; Labor Share 0.710

### Critique of the steady-state method (SSM) for initial capital estimation
- The paper argues the SSM:
  - Is unavoidably arbitrary due to the steady state assumption and silence on auxiliary assumptions (which data and what period to use).
  - Produces highly variable initial capital estimates across data sets and across time even for the same country.
  - Triggers an error-correction effect from the perpetual inventory equation, causing the capital-output ratio to revert to its long-run equilibrium of (I/Y)/δ (notation in text: (I/Y)/ . ), which:
    - Distorts capital growth estimates.
    - Produces asymmetric, error-laden capital growth data that may persist for decades (empirical result: initial capital-output data still affected capital growth data 40 years after initial estimates).
    - Likely biases TFP estimates: rapidly growing countries are penalized (low initial K/Y → high measured capital growth → low TFP growth), and slow-growing countries are biased upward.
- The paper notes that Feenstra, Inklaar, and Timmer (2015) address some concerns by assuming the same initial capital-output ratio by asset class for all countries.

### Proxy-variable approach: electricity and automobile usage per-capita
- Rationale and properties:
  - Electricity use per-capita and auto use per-capita have an empirically robust cross-country log-linear relationship with GDP per-capita.
  - These proxies are likely correlated with the unobserved capital stock.
  - Drawbacks: the data capture both final consumption and investment.
  - Advantages: grounded in empirical evidence; greatly reduces dispersion in estimated K/Y compared to the steady-state method; minimizes the distorting error-correction effect; makes capital growth over time driven primarily by observed investment data.
- Method summary:
  - Use regression estimates from the cross-country relationship between electricity (or automobile) use per-capita and GDP per-capita to estimate capital stock per-capita for a given GDP per-capita.
  - Compute capital-output ratio for the year corresponding to the start of investment data.
  - Apply the perpetual inventory method together with investment data to generate a full time series of the capital stock and K/Y.

### Comparative conclusion and policy implications for data practice
- The paper compares six methods for estimating initial K/Y and concludes:
  - The automobile- and electricity-based methods, and one method in Feenstra, Inklaar, and Timmer (2015) (equal initial K/Y by asset class across countries), produce more plausible capital stock and TFP growth data.
  - These preferred methods show lower dispersion in the capital-output ratio, aligning better with theory and independent evidence.
  - The key benefit is minimizing the problematic error-correction effect that plagues some existing data sets.
- Implications:
  - There is a case for estimating initial capital conservatively to avoid producing highly variable cross-country estimates.
  - Taking advantage of correlated proxy data (electricity, automobiles) can yield more credible initial capital estimates and more reliable TFP growth measures.
  - Use of SSM without careful auxiliary choices can lead to misleading empirical conclusions about countries’ TFP and growth performance.

*IMF Working Papers — A Proposal to Improve Country-level Data on Total Factor Productivity Growth*

---


_Source: https://www.imf.org/-/media/files/publications/wp/2024/english/wpiea2024067-print-pdf.pdf_