## _wp13265

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---

### I. INTRODUCTION
- China’s real GDP per capita increased almost 10-fold since 1978, growing at an average of 8.5 percent per year.
- Urban household survey growth: 7 percent per year since 1978 for urban households; 5 percent per year since 1985 for rural households.
- Paper objective: use a subset of the Urban Household Survey in 1993–2005 to estimate real income growth implied by Engel curves for food consumption, following methods from Nakamura (1997), Costa (2001), Hamilton (2001), and others.
- Conceptual point: under Engel’s law, the food share of household expenditures declines as real income grows; deviations between Engel-curve–implied real expenditure growth and CPI-deflated headline real expenditure growth are attributed to CPI measurement error (bias).
- Main empirical finding preview: urban CPI overstated the true cost of living by about 1 percent per year in flexible specifications (with larger estimates under some parametric specifications); this is broadly similar to bias estimates for the U.S.

### II. EMPIRICAL METHODOLOGY
- Structural starting point: Almost Ideal Demand System demand function for food with household i, region j, period t; w = food budget share; P_F, P_N, P_G = true (unobservable) price indices for food, nonfood, and general index; Y = nominal household expenditure; X = household characteristics; μ = residual.
- CPI measurement notation:
  - Π_G,j,t denotes percent cumulative increase in CPI-measured price.
  - E_G,j,t denotes percent cumulative measurement error from period 0 to t for food, nonfood, or all goods.
- Key estimation features:
  - Estimated equation includes ln(1+Π) terms, regional and time dummies D_j and D_t, and parameters mapped to cumulative CPI bias via expressions involving |E_G,t|.
  - Extensions: allow bias to be linear function of ln(Y_it): E_G,it = a + b·ln(Y_it) and include λ (slope of bias with ln expenditure).
  - Semi-parametric approach: estimate non-parametric term f() using Yatchew (1997) differencing method; order households by CPI-measured real income and use locally-weighted linear regression with quartic kernel weights to estimate bias at different expenditure levels.
- Interpretation note: bias estimates are multiplicative on CPI changes (example: a 10 percent CPI change with 3 percent bias implies true change 6.7 percent since (1-0.03)·1.1=1.067).

### III. DATA
- Data source: subset of the Urban Household Survey (UHS) covering 10 provinces/municipalities for 1993–2005.
  - The 10 provinces/municipalities listed: Anhui, Beijing, Chongqin, Ganshu, Guangdong, Hubei, Jiangsu, Liaoning, Shanxi and Sichuan.
- Coverage reasons: provincial-level CPI by expenditure group starts in 1993 for most provinces; sample limited to those 10 provinces/municipalities in 1993–2005.
- Representativeness: sub-sample fairly representative; provinces slightly richer than others with incomes about 10 percent higher than average.
- Variable definitions and treatment:
  - Food consumption includes food, beverages and tobacco; includes expenditures inside and outside house.
  - Food budget share = ratio of food to total consumption expenditures.
  - Income includes labor income, property income, transfers (social and private), and household sideline production; owner-occupied housing value not captured (reported rental value from 2002 onwards).
  - All variables annualized and nominal values deflated to 2005 prices using province CPI produced by NBS.
- Survey design: probabilistic, stratified; households keep monthly records; sample size increases significantly starting in 2002.

### IV. RESULTS — SUMMARY STATISTICS
- Food budget share declines from 54 percent in 1993 to 44 percent in 2005.
- CPI-measured real consumption per capita increases 78 percent during sample (4.9 percent per year).
- Income per capita increases 99 percent during sample (5.9 percent per year).
- Real per capita food expenditures increase substantially despite declining food share.
- Home ownership rises from 24 percent in 1993 to 86 percent in 2005; 65 percent of home owners in 2005 bought through housing reform.
- Consumption expenditures are on average about 80 percent of income; consumption as share of income declines 7 percentage points in sample.

### IV. RESULTS — PARAMETRIC MODEL FINDINGS
- Engel-curve coefficient estimates:
  - Coefficients on log consumption expenditure range from -0.13 to -0.18.
  - Coefficients on log relative price of food range from 0.30 to 0.37.
- Cumulative CPI bias estimates (parametric):
  - Assuming bias constant across households and using consumption: cumulative bias 34.3 percent in 1993–2005, or 3.4 percent per year (CPI overstating true cost of living).
  - Using instrumental variables (income as instrument): bias increases to 4.0 percent per year.
  - Allowing bias to vary linearly with log real expenditure:
    - Population-weighted average bias: 2.0 percent per year (column 3), increasing to 2.3 percent per year with IV (column 4).
    - Expenditure-weighted averages: 0.9 and 1.1 percent per year in columns (3) and (4) respectively.
- Interpretation:
  - Typical urban household experienced substantially higher purchasing power gains than implied by CPI-deflated expenditures.
  - When weighting by household expenditure (relevant for aggregate CPI comparisons), much of the effect disappears.

### IV. RESULTS — SEMI-PARAMETRIC MODEL FINDINGS
- Non-parametric Engel curves: food share declines with log real expenditure; curves shift downward over time (with occasional upward shifts due to relative food price increases).
- Estimated annual bias by headline real expenditure:
  - Bias higher for poorest households, declines with real expenditure, becomes negative at very upper tail.
  - Annualized bias for average population-weighted household: 1.6 percent per year.
  - Expenditure-weighted aggregate bias: 0.9 percent per year.
- Pattern: poorer households exhibit larger unmeasured gains relative to CPI-deflated measures; richer households have smaller unmeasured gains; very rich may show negative bias.

### V. ROBUSTNESS
- Methods compared: baseline sample and Winsorized sample (values below 5th and above 95th percentiles set to percentile values); semi-parametric preferred estimates shown as thick line in figures.
- Robustness findings:
  - Across methods, population-weighted estimates yield much larger bias than expenditure-weighted ones.
  - Parametric specifications assuming constant bias across households yield much larger bias estimates.
  - Parametric specifications allowing bias to vary linearly with ln real expenditure produce results comparable to semi-parametric estimates.
  - Because bias varies notably across income levels, flexible specifications (semi-parametric or linear-varying) are preferred.
- Confidence intervals:
  - For expenditure-weighted measures, a cumulative bias equal to 1 percent per year lies within the 95 percent confidence interval most of the time and coincides with end-sample point-estimates in the three methods illustrated.

### V. DISTRIBUTIONAL IMPLICATIONS
- CPI-deflated growth (headline CPI):
  - Average value grows ½ percentage point faster per year than median.
  - Consumption growth by quintile (CPI-deflated): richest quintile = 6.1 percent per year; poorest quintile = 2.9 percent per year.
  - Income growth by quintile (CPI-deflated): richest quintile = 6.7 percent per year; poorest quintile = 4.2 percent per year.
- Using Engel-curve–implied true cost of living:
  - Consumption growth reverses: richest quintile = 5.5 percent per year; poorest quintile = 6.4 percent per year.
  - Income growth reverses: richest quintile = 6.2 percent per year; poorest quintile = 7.7 percent per year.
- Averages:
  - Average and median consumption grow at similar rate: around 5.7 percent per year.
  - Average and median income grow at similar rate: 6.9 percent per year.
- Implication: adjusting CPI using Engel-curve methods indicates stronger relative gains for poorer urban households up to 2005.

### CONCLUSION — Main findings on CPI accuracy and real income growth
- The urban CPI provides a fairly accurate measure of the cost of living for the urban economy as a whole.
- Expenditure-weighted estimates point to a discrepancy of "1 percentage point or less per year", comparable to the bias often associated with the U.S. CPI.
- The paper’s estimates use household-level consumption data to infer the path of real income growth compatible with the shrinking participation of food expenditure in total household expenditures in urban China.

### CONCLUSION — Distributional implications (corrected annual growth rates 1993–2005, percent per year)
- Consumption
  - Deflated by CPI: Mean 5.1, Median 4.5, Quintiles 1 2.9, 2 4.0, 3 4.5, 4 5.1, 5 6.1
  - Correcting for Estimated Bias: Mean 5.7, Median 5.8, Quintiles 1 6.4, 2 6.0, 3 5.8, 4 5.7, 5 5.5
- Income
  - Deflated by CPI: Mean 6.1, Median 5.7, Quintiles 1 4.2, 2 5.5, 3 5.9, 4 6.3, 5 6.7
  - Correcting for Estimated Bias: Mean 6.9, Median 6.9, Quintiles 1 7.7, 2 7.5, 3 7.2, 4 6.9, 5 6.2

### Quantified CPI bias estimates (selected parametric cumulative |E_G,t| percent for full sample)
- Cumulative bias 1993-95: 3.99 [1.97]; alternative specification 5.05 [2.12]
- Cumulative bias 1993-97: 12.4 [1.6]; alternative 14.24 [1.71]
- Cumulative bias 1993-99: 20.7 [1.24]; alternative 23.47 [1.32]
- Cumulative bias 1993-2001: 29.14 [1.18]; alternative 32.81 [1.24]
- Cumulative bias 1993-2003: 34.47 [0.92]; alternative 38.31 [0.98]
- Cumulative bias 1993-2005: 34.29 [1.04]; alternative 38.8 [1.1]

### Expenditure-weighted cumulative bias (selected)
- Cumulative bias 1993-95 (%) 4.15 [1.76]; alternative 5.91 [1.94]
- Cumulative bias 1993-97 (%) 7.19 [1.67]; alternative 8.86 [1.89]
- Cumulative bias 1993-99 (%) 8.46 [1.65]; alternative 10.4 [1.93]
- Cumulative bias 1993-2001 (%) 10.68 [1.93]; alternative 13.74 [2.27]
- Cumulative bias 1993-2003 (%) 12.69 [1.78]; alternative 13.88 [2.2]
- Cumulative bias 1993-2005 (%) 10.28 [2.06]; alternative 12.3 [2.51]

### Methodological notes
- Estimates are based on Engel-curve methods (parametric, linear IV, and semi-parametric approaches).
- Notes: Robust standard errors for regression coefficients and bootstrapped standard errors for bias estimates are reported in brackets. Controls include regional dummies. Income is used as an instrument to consumption in the IV regressions. Cumulative bias reported corresponds to |E_G,t|.
- Implied conversions:
  - The implied gross change in the true cost of living is (1-|E_G,t|) times the gross change in the CPI.
  - The resulting gross true real income growth is 1/(1-|E_G,t|) times the gross real income growth obtained by deflating nominal income by the CPI.
- Semi-parametric estimates and non-parametric Engel curves obtained from locally weighted linear regressions using quartic kernel weights.

### Caveats and limitations
- Survey coverage limitations:
  - The survey sample only covers migrants from rural areas starting in 2002 (migrants are among the poorest groups in urban areas).
  - Coverage of very rich households is limited (the 99th percentile of income in the 2005 sample was 120,000 Yuan), making estimates less precise at the upper tail of the distribution.
- Potential confounders:
  - The increasing shift from public to private provision of health and education services could have affected the evolution of the food budget shares.
- With these caveats, the paper’s conclusion is that the evolution of consumption patterns suggests that real income growth in urban China has been more equitable than commonly perceived.

*Source: _wp13265 - References and selected sections from the PDF chapter "5.  Estimated Cumulative Bias and Confidence Intervals under Different Methodologies" and CONCLUSION.*

### References .............................................................................................................

### _wp13265 - References

### Tables
- 1. Summary Statistics .............................................................................................................17
- 2. Parametric Regression Results .......................................................................................... 18
- 3. Implications of Estimated Bias for Consumption and Income Growth  
  in 1993–2005 ..............................................................................................................19

### Figures
- 1.  Non-Parametric Estimates of Relationship between Food Shares and Household  
  Expenditure ................................................................................................................. 20
- 2.  Semi-Parametric Estimates of Relationship between Food Shares and Household 
  Expenditure ................................................................................................................. 20
- 4. Estimated Bias in 1993–2005 as a Function of CPI-Measured Real Expenditure  
  in 2005 .........................................................................................................................21
- 4.  Estimated Cumulative Bias in China since 1993 across Different Methods  
  and samples ................................................................................................................. 22

*Source: _wp13265 - References.*

### 5.  Estimated Cumulative Bias and Confidence Intervals under Different Methodologies .... 23

### 5.  Estimated Cumulative Bias and Confidence Intervals under Different Methodologies .... 23

### I. INTRODUCTION
- China’s real GDP per capita increased almost 10-fold since 1978, growing at an average of 8.5 percent per year.
- Urban household survey growth: 7 percent per year since 1978 for urban households; 5 percent per year since 1985 for rural households.
- Paper objective: use a subset of the Urban Household Survey in 1993–2005 to estimate real income growth implied by Engel curves for food consumption, following methods from Nakamura (1997), Costa (2001), Hamilton (2001), and others.
- Key conceptual point: under Engel’s law, the food share of household expenditures declines as real income grows; deviations between Engel-curve–implied real expenditure growth and CPI-deflated headline real expenditure growth are attributed to CPI measurement error (bias).
- Main empirical finding preview: urban CPI overstated the true cost of living (so actual real income growth is higher than official statistics indicate) by about 1 percent per year in flexible specifications (with larger estimates under some parametric specifications); this is broadly similar to bias estimates for the U.S.

### II. EMPIRICAL METHODOLOGY
- Structural starting point: Almost Ideal Demand System demand function for food with household i, region j, period t:
  - w = food budget share; P_F, P_N, P_G = true (unobservable) price indices for food, nonfood, and general index; Y = nominal household expenditure; X = household characteristics; μ = residual.
- CPI measurement notation:
  - Π_G,j,t denotes percent cumulative increase in CPI-measured price.
  - E_G,j,t denotes percent cumulative measurement error from period 0 to t for food, nonfood, or all goods.
- Key estimated equation (after assumptions and transformations) includes log(1+Π) terms, regional and time dummies D_j and D_t, and yields parameters that map to cumulative CPI bias via expressions such as:
  - If food and nonfood equally biased: ln(1)/Gtt E δβ  (formula as presented in source).
- Extensions:
  - Allow bias to be linear function of log real expenditure: E_G,it = a + b·ln(Y_it) and derive estimation equation that includes λ (slope of bias with log expenditure).
  - Semi-parametric approach: estimate non-parametric term f() using Yatchew (1997) differencing method; order households by CPI-measured real income and use locally-weighted linear regression with quartic kernel weights to estimate bias at different expenditure levels.
- Interpretation: bias estimates are multiplicative on CPI changes (example in text: a 10 percent CPI change with 3 percent bias implies true change 6.7 percent since (1-0.03)·1.1=1.067).

### III. DATA
- Data source: subset of the Urban Household Survey (UHS) covering 10 provinces/municipalities for 1993–2005.
  - The 10 provinces/municipalities listed in source for 1986–1997 databank: Anhui, Beijing, Chongqin, Ganshu, Guangdong, Hubei, Jiangsu, Liaoning, Shanxi and Sichuan.
- Coverage reasons: provincial-level CPI by expenditure group starts in 1993 for most provinces; sample limited to those 10 provinces/municipalities in 1993–2005.
- Representativeness: sub-sample fairly representative; provinces slightly richer than others with incomes about 10 percent higher than average.
- Variable definitions and treatment:
  - Food consumption includes food, beverages and tobacco; includes expenditures inside and outside house.
  - Food budget share computed as ratio of food to total consumption expenditures.
  - Income includes labor income, property income, transfers (social and private), and household sideline production; owner-occupied housing value not captured (reported rental value from 2002 onwards).
  - All variables annualized and nominal values deflated to 2005 prices using province CPI produced by NBS.
- Survey design: probabilistic, stratified; households keep monthly records; sample size increases significantly starting in 2002.

### IV. RESULTS — SUMMARY STATISTICS
- Food budget share declines from 54 percent in 1993 to 44 percent in 2005.
- CPI-measured real consumption per capita increases 78 percent during sample (4.9 percent per year).
- Income per capita increases 99 percent during sample (5.9 percent per year).
- Real per capita food expenditures increase substantially despite declining food share.
- Home ownership rises from 24 percent in 1993 to 86 percent in 2005; 65 percent of home owners in 2005 bought through housing reform.
- Consumption expenditures are on average about 80 percent of income; consumption as share of income declines 7 percentage points in sample.

### IV. RESULTS — PARAMETRIC MODEL FINDINGS
- Engel-curve coefficients:
  - Coefficients on log consumption expenditure range from -0.13 to -0.18.
  - Coefficients on log relative price of food range from 0.30 to 0.37.
- Cumulative CPI bias estimates (parametric):
  - Assuming bias constant across households and using consumption: cumulative bias 34.3 percent in 1993–2005, or 3.4 percent per year (CPI overstating true cost of living).
  - Using instrumental variables (income as instrument): bias increases to 4.0 percent per year.
  - Allowing bias to vary linearly with log real expenditure:
    - Population-weighted average bias: 2.0 percent per year (column 3), increasing to 2.3 percent per year with IV (column 4).
    - Expenditure-weighted averages: 0.9 and 1.1 percent per year in columns (3) and (4) respectively.
- Interpretation:
  - Typical urban household experienced substantially higher purchasing power gains than implied by CPI-deflated expenditures.
  - When weighting by household expenditure (relevant for aggregate CPI comparisons), much of the effect disappears.

### IV. RESULTS — SEMI-PARAMETRIC MODEL FINDINGS
- Non-parametric Engel curves (Figure 1 and Figure 2 in source): food share declines with log real expenditure; curves shift downward over time (with occasional upward shifts due to relative food price increases).
- Estimated annual bias by headline real expenditure (Figure 3):
  - Bias higher for poorest households, declines with real expenditure, becomes negative at very upper tail.
  - Annualized bias for average population-weighted household: 1.6 percent per year.
  - Expenditure-weighted aggregate bias: 0.9 percent per year.
- Pattern: poorer households exhibit larger unmeasured gains relative to CPI-deflated measures; richer households have smaller unmeasured gains, very rich may show negative bias.

### V. ROBUSTNESS
- Methods compared (Figure 4):
  - Baseline sample and Winsorized sample (values below 5th and above 95th percentiles set to percentile values).
  - Semi-parametric preferred estimates shown as thick line.
- Robustness findings:
  - Across methods, population-weighted estimates yield much larger bias than expenditure-weighted ones.
  - Parametric specifications assuming constant bias across households yield much larger bias estimates.
  - Parametric specifications allowing bias to vary linearly with log real expenditure produce results comparable to semi-parametric estimates.
  - Because bias varies notably across income levels, flexible specifications (semi-parametric or linear-varying) are preferred.
- Confidence intervals:
  - Figure 4 plots point estimates and 95 percent confidence interval for parametric specifications with bias varying linearly on log real income and for semi-parametric specification.
  - For expenditure-weighted measures, a cumulative bias equal to 1 percent per year lies within the confidence interval most of the time and coincides with end-sample point-estimates in the three methods illustrated.

### V. DISTRIBUTIONAL IMPLICATIONS
- CPI-deflated growth (headline CPI):
  - Average value grows ½ percentage point faster per year than median.
  - Consumption growth by quintile (CPI-deflated): richest quintile = 6.1 percent per year; poorest quintile = 2.9 percent per year.
  - Income growth by quintile (CPI-deflated): richest quintile = 6.7 percent per year; poorest quintile = 4.2 percent per year.
- Using alternative deflator (Engel-curve–implied true cost of living):
  - Consumption growth reverses: richest quintile = 5.5 percent per year; poorest quintile = 6.4 percent per year.
  - Income growth reverses: richest quintile = 6.2 percent per year; poorest quintile = 7.7 percent per year.
- Averages:
  - Average and median consumption grow at similar rate: around 5.7 percent per year.
  - Average and median income grow at similar rate: 6.9 percent per year.
- Implication: adjusting CPI using Engel-curve methods indicates stronger relative gains for poorer urban households up to 2005.

*Source: Excerpt from the document titled "5.  Estimated Cumulative Bias and Confidence Intervals under Different Methodologies .... 23" (chapter/section content from the provided PDF).*

### CONCLUSION

### CONCLUSION

### Main findings on CPI accuracy and real income growth
- The urban CPI provides a fairly accurate measure of the cost of living for the urban economy as a whole.
- Expenditure-weighted estimates point to a discrepancy of "1 percentage point or less per year", comparable to the bias often associated with the U.S. CPI.
- The paper’s estimates use household-level consumption data to infer the path of real income growth compatible with the shrinking participation of food expenditure in total household expenditures in urban China.

### Distributional implications
- The cost of living for poorer households has increased slower than the CPI, thus contributing to reduce real expenditure inequality.
- Corrected (semi-parametric) annual growth rates in 1993–2005 (percent per year), by quintile (from poorer to richer), are:
  - Consumption
    - Deflated by CPI: Mean 5.1, Median 4.5, Quintiles 1 2.9, 2 4.0, 3 4.5, 4 5.1, 5 6.1
    - Correcting for Estimated Bias: Mean 5.7, Median 5.8, Quintiles 1 6.4, 2 6.0, 3 5.8, 4 5.7, 5 5.5
  - Income
    - Deflated by CPI: Mean 6.1, Median 5.7, Quintiles 1 4.2, 2 5.5, 3 5.9, 4 6.3, 5 6.7
    - Correcting for Estimated Bias: Mean 6.9, Median 6.9, Quintiles 1 7.7, 2 7.5, 3 7.2, 4 6.9, 5 6.2

### Quantified CPI bias estimates (selected)
- Parametric (Table 2) cumulative bias estimates |E_G,t| (percent) for the full sample (columns shown):
  - Cumulative bias 1993-95: 3.99 [1.97]; alternative specification 5.05 [2.12]
  - Cumulative bias 1993-97: 12.4 [1.6]; alternative 14.24 [1.71]
  - Cumulative bias 1993-99: 20.7 [1.24]; alternative 23.47 [1.32]
  - Cumulative bias 1993-2001: 29.14 [1.18]; alternative 32.81 [1.24]
  - Cumulative bias 1993-2003: 34.47 [0.92]; alternative 38.31 [0.98]
  - Cumulative bias 1993-2005: 34.29 [1.04]; alternative 38.8 [1.1]
- Expenditure-weighted cumulative bias (Table 2, expenditure weighted bias rows):
  - Cumulative bias 1993-95 (%) 4.15 [1.76]; alternative 5.91 [1.94]
  - Cumulative bias 1993-97 (%) 7.19 [1.67]; alternative 8.86 [1.89]
  - Cumulative bias 1993-99 (%) 8.46 [1.65]; alternative 10.4 [1.93]
  - Cumulative bias 1993-2001 (%) 10.68 [1.93]; alternative 13.74 [2.27]
  - Cumulative bias 1993-2003 (%) 12.69 [1.78]; alternative 13.88 [2.2]
  - Cumulative bias 1993-2005 (%) 10.28 [2.06]; alternative 12.3 [2.51]

### Methodological notes from the study
- Estimates are based on Engel-curve methods (parametric, linear IV, and semi-parametric approaches).
- Notes in Table 2: Robust standard errors for regression coefficients and bootstrapped standard errors for bias estimates are reported in brackets. Controls include regional dummies. Income is used as an instrument to consumption in the IV regressions. Cumulative bias reported corresponds to |E_G,t|. The implied gross change in the true cost of living is (1-|E_G,t|) times the gross change in the CPI, and the resulting gross true real income growth is 1/(1-|E_G,t|) times the gross real income growth obtained by deflating nominal income by the CPI.
- Semi-parametric bias estimates and non-parametric Engel curves are obtained from locally weighted linear regressions using quartic kernel weights (Figures notes).

### Caveats and limitations highlighted by the authors
- Survey coverage limitations:
  - The survey sample only covers migrants from rural areas starting in 2002 (migrants are among the poorest groups in urban areas).
  - Coverage of very rich households is limited (the 99th percentile of income in the 2005 sample was 120,000 Yuan), making estimates less precise at the upper tail of the distribution.
- Potential confounders:
  - The increasing shift from public to private provision of health and education services could have affected the evolution of the food budget shares.
- With these caveats, the paper’s conclusion is that the evolution of consumption patterns suggests that real income growth in urban China has been more equitable than commonly perceived.

*Source: _wp13265 - CONCLUSION*

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_Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2013/_wp13265.pdf_