## _wp1096 - References

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### I. MOTIVATION
- Purpose: examines convergence and spillovers across Indian states using non-stationary panel data econometrics.
- Innovations highlighted:
  - Ranking procedure to group states by performance.
  - Restatement of convergence/divergence across states during 1960-2004.
  - Explanation of per capita income disparities by sector shares, infrastructure, private investment, and spillovers.
  - Study of club convergence and dynamic spillover effects among states (dynamic spillovers claimed as novel).

### II. LITERATURE SURVEY
- Frameworks and concepts:
  - Growth-regression framework and β-convergence concepts (absolute vs conditional convergence).
- Prior findings summarized:
  - Mixed evidence: absolute and conditional convergence in pre-1990s; divergence including post-1990s in several studies.
  - Sector shares and infrastructure, private investment, and institutions often explain growth dispersion.
- Methodological rationale:
  - Non-stationary panel techniques used because they are robust to endogeneity, omitted variables, simultaneity, measurement error, and arbitrary start/end points.
  - Techniques allow pairwise/spatial spillover analysis and combination of individual unit-root tests into panel tests.

### III. DATA AND STYLIZED “FACTS”
- Data:
  - Annual per capita real net state domestic product for 15 major Indian states over 1960-2004.
  - States: Andhra Pradesh, Assam, Bihar, Gujarat, Haryana, Karnataka, Kerala, Madhya Pradesh, Maharashtra, Orissa, Punjab, Rajasthan, Tamil Nadu, Uttar Pradesh, West Bengal.
- Stylized facts and groupings:
  - All states grew, but income gaps widened; by 2003 average per capita income in India was twice as high as in 1970.
  - State groups by per capita income:
    - High-income states: Gujarat, Haryana, Maharashtra, Punjab, Tamil Nadu.
    - Medium-income states: Andhra Pradesh, Karnataka, Kerala, West Bengal, Rajasthan.
    - Low-income states: Assam, Bihar, Orissa, Madhya Pradesh, Uttar Pradesh.
- Growth differentials and structural breaks:
  - High-income states grew at average annual rates of 3-4 percent over 1970-2003.
  - Low-income states grew at only at 1½ -2 percent.
  - By end of period, average per capita income of the five richest states in 2003 was 2½ times as much as that of the poorest states.
  - Structural breaks associated with policy reforms identified at 1980 and 1992:
    - Gap between log real NSDP per capita of five richest and five poorest states increased by around 20 basis points in 1980 and 13 basis points in 1992.

### IV. THE ECONOMETRICS OF NON-STATIONARY PANELS
- Convergence concept:
  - Defined via stationarity of pairwise differences; cointegration used to detect convergence.
- Panel tests employed:
  - Levin, Lin, and Chu (LLC)
  - Im, Pesaran, and Shin (IPS)
  - Maddala and Wu (MW)
- Enhancements:
  - Bootstrap techniques used to improve small-sample performance of IPS and MW.
  - Structural-break and club-convergence complications addressed by subsample and subgroup testing.

### V. CONVERGENCE REVISITED: EMPIRICAL RESULTS

- Overall:
  - All 15 series are I(1).
  - Full-sample (1960-2003): non-convergence in states’ per capita income is not rejected (panel tests indicate divergence overall).

A. Convergence and Structural Change
- Subsample findings:
  - Pre-1980:
    - 7 of 15 states reject non-convergence at less than 5 percent; Assam rejects at around 8 percent.
    - Panel tests (except LLC ADF) reject non-convergence.
  - 1980-1992:
    - Only 3 of 15 states show individual convergence; panel tests firmly reject unit-root null at less than 1 percent significance (conditional convergence).
  - Post-1992:
    - 5 of 15 states individually reject unit-root null; panel tests (LLC, IPS, MW) robustly reject non-convergence (overall income convergence in this subsample).
- Interpretation of growth episodes:
  - Pre-1980: low and volatile growth ≈ ½ percent per annum.
  - 1981-1992: growth ≈ 3 percent.
  - Post-1992: growth ≈ 4 percent on average with differential outcomes.

B. Determinants of Per Capita Income Differentials
- Regression setup: differences in log per capita income between state pairs regressed on differences in factor variables and a spatial distance index.
- Main estimated elasticities (Table 7 summary):
  - Share of the service sector: a 1 percent difference → about 0.02 percent difference in per capita income between states.
  - Development expenditure: a 1 percent difference → approximately 0.07 percent difference in per capita income.
  - Private investment (private sector credit per capita): the largest contributor — 0.5 percent.
- Other variables:
  - Number of telephone lines significant but negligible effect.
  - T&D loss ratio has expected sign but not significant.
  - Literacy rate and employment in private sector statistically insignificant in regressions.
- Spillovers in regressions:
  - Spatial spillover estimates generally not strongly different from zero except for low-low income state pairs.
  - Negative estimates for high-high pairs suggest potential crowding out; positive spillovers in other combinations but economically small.

C. Club Convergence and Dynamic Spillover Effects
- Club convergence evidence:
  - Strong convergence among high-income and low-income state groups across full sample and subsamples:
    - High-income group: full-sample non-convergence rejected by bootstrapped IPS and MW; stronger rejection in sub-periods (less than 1 percent).
    - Low-income group: full-sample rejection with large-sample adjusted IPS and bootstrapped IPS; uniform rejection in sub-samples.
  - Medium-income group: divergence over full sample; convergence pre-1980 and 1980-1992 but not post-1992.
    - Divergence largely driven by Andhra Pradesh and Karnataka (rapid growth rises post-1980s and post-1992, becoming IT leaders).
    - Excluding Andhra Pradesh and Karnataka yields convergence among remaining medium-income states in all periods (robust panel rejections of non-convergence).
- Dynamic spillovers (dynamic panel VAR system):
  - Spillover magnitudes small: a one standard deviation increase in income in one state transfers to only 0.01-0.07 standard deviation increase in other states.
  - Within-group spillovers larger than between-group; high-medium spillovers stronger than high-low and medium-low.
  - Subsample dynamics:
    - 1980-1992 shows some interconnectedness and weak spillovers.
    - Pre-1980 and post-1992 spillovers are very small or negligible; shocks die out quickly.

### VI. CONCLUDING REMARKS
- Empirical conclusions:
  - Over 1960-2003, evidence of overall divergence, but convergence in particular sub-periods and clear club convergence patterns.
  - Strong evidence of club convergence among high- and low-income states; mixed evidence for middle-income states.
  - Dynamic spillover effects among states are small.
- Policy-relevant implications:
  - States that forged ahead benefited from advances in the services sector, better infrastructure, credit availability, and efficient development spending.
  - Lack of strong dynamic spillovers suggests potential need for better infrastructure and connectivity to disseminate benefits of growth across states.

### Appendix 1: Test Procedures for Panel Unit Roots
- Overview: detailed procedures for LLC, IPS, and MW tests; step-down lag selection; and bootstrap implementations.

1. Levin, Lin, and Chu Test
- Nature: Parametric panel unit-root test analogous to the augmented Dickey–Fuller (ADF) test; models serial correlation with an autoregressive of order k specification in lagged differences.
- Key equations and steps preserved from source (ADF regression, time effects extraction, residual variance, auxiliary regressions, normalization, pooled regression and t-test).
- Hypotheses:
  - H0: ρ_i = 0, ∀i (all series contain a unit root)
  - H1: ρ_i < 0, ∀i (no unit root for all series)
- Interpretation: Rejection of H0 implies there is no unit root for all series in the panel.

2. Im, Pesaran, and Shin Test
- Nature: Group-mean between-dimension unit-root test for dynamic heterogeneous panels, analogous to ADF.
- Procedure: estimate univariate ADF for each i, collect t-statistics t_{ρ,i}, compute group-mean t̄_ρ and standardize using IPS tabulated μ and σ.
- Hypotheses:
  - H0: ρ_i = 0, ∀i
  - H1: ρ_i < 0 for some i
- Interpretation: Rejection of H0 indicates the panel does not contain unit roots (i.e., at least some series are stationary).

3. Maddala and Wu Test
- Nature: Combines p-values of individual ADF t-tests across panel members using Fisher’s method; parameters heterogeneous across members.
- Procedure: collect p-values π_i from individual ADFs (π_i obtained by simulation); statistic −2 ∑_{i=1}^N ln(π_i) follows χ^2 with 2N degrees of freedom under independence.
- Hypotheses:
  - H0: ρ_i = 0, ∀i
  - H1: ρ_i < 0 for some i

4. Step-Down Procedure for Choosing Lag Truncation
- Importance: correct lag truncation crucial to avoid misspecification and size/power problems.
- Adopted procedure (Pedroni and Yao (2006) "step down"):
  - Start with initial lags = nearest integer of 1/5 of the sample length.
  - Perform ADF regression; if largest lag significant, stop; otherwise eliminate largest lag sequentially until significance achieved.
  - Number of lags allowed to differ across states.

5. Bootstrap Procedures
- Purpose: simulate IPS adjustment terms and MW p-values to account for finite-sample behavior, serial correlation, lag truncation sensitivity, and cross-sectional dependence.
- Implementation summary:
  - Estimate serial correlation properties by running ADF for each i.
  - Draw 10,000 realizations of a pure random walk of length T+100 for ε_it.
  - Fix parameters α_i, φ_{iL}, ρ_i and replicate serially correlated processes using pseudo-innovations to generate y*_it.
  - Re-estimate ADF on pseudo-samples, discard first 100 observations, collect pseudo-distributions to compute means, variances, and p-values.

*Source: Appendix 1: Test Procedures for Panel Unit Roots (contained in _wp1096 - Appendix 1: Test Procedures for Panel Unit Roots).*

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

### _wp1096 - References

### Tables
- 1. Indian States: Growth and Convergence Literature—A Survey of Results ..........23
- 2. Rank of Indian States by Income, 1970-2003 ........................................................24
- 3. India: Survey of Structural Breaks .........................................................................25
- 4. Panel Unit Root Test Results for All Indian States ...............................................26
- 5.         Club         Convergence         ..................................................................................................27
- 6. Medium Income States without Andhra Pradesh and Karnataka ..........................28
- 7. Income Divergence across Indian States: Regression Analysis ............................29

### Figures
- 1. Indian States Growth: Explanatory Variables ........................................................30
- 2. Spillovers among Indian states (full sample) .........................................................31
- 3. Spillovers among Indian states (pre-1980)  ...........................................................32
- 4. Spillovers among Indian states (1980-1992) .........................................................33

*Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2010/_wp1096.pdf*

### 5. Spillovers among Indian states (post-1992) ...........................................................34

### 5. Spillovers among Indian states (post-1992) ...........................................................34

### I. MOTIVATION
- Examines convergence and spillovers across Indian states using non-stationary panel data econometrics.
- Innovations of the paper:
  - Ranking procedure to group states by performance.
  - Restatement of convergence/divergence across states during 1960-2004.
  - Explanation of per capita income disparities by sector shares, infrastructure, private investment, and spillovers.
  - Study of club convergence and dynamic spillover effects among states (dynamic spillovers claimed as novel).

### II. LITERATURE SURVEY
- Growth-regression framework and β-convergence concepts summarized (absolute vs conditional convergence).
- Prior findings summarized:
  - Mixed evidence: absolute and conditional convergence in pre-1990s; divergence including post-1990s in several studies.
  - Sector shares and infrastructure, private investment, and institutions often explain growth dispersion.
- Non-stationary panel techniques motivate alternative testing strategy:
  - Robust to endogeneity, omitted variables, simultaneity, measurement error, and arbitrary start/end points.
  - Allows pairwise/spatial spillover analysis and combination of individual unit-root tests into panel tests.

### III. DATA AND STYLIZED “FACTS”
- Data:
  - Annual per capita real net state domestic product for 15 major Indian states over 1960-2004.
  - States: Andhra Pradesh, Assam, Bihar, Gujarat, Haryana, Karnataka, Kerala, Madhya Pradesh, Maharashtra, Orissa, Punjab, Rajasthan, Tamil Nadu, Uttar Pradesh, West Bengal.
- Stylized facts:
  - All states grew, but income gaps widened; by 2003 average per capita income in India was twice as high as in 1970.
  - States grouped by per capita income:
    - High-income states: Gujarat, Haryana, Maharashtra, Punjab, Tamil Nadu.
    - Medium-income states: Andhra Pradesh, Karnataka, Kerala, West Bengal, Rajasthan.
    - Low-income states: Assam, Bihar, Orissa, Madhya Pradesh, Uttar Pradesh.
  - Growth differentials:
    - High-income states grew at average annual rates of 3-4 percent over 1970-2003.
    - Low-income states grew at only at 1½ -2 percent.
    - By end of period, average per capita income of the five richest states in 2003 was 2½ times as much as that of the poorest states.
  - Structural breaks associated with policy reforms identified at 1980 and 1992:
    - Gap between log real NSDP per capita of five richest and five poorest states increased by around 20 basis points in 1980 and 13 basis points in 1992.

### IV. THE ECONOMETRICS OF NON-STATIONARY PANELS
- Convergence defined via stationarity of pairwise differences; cointegration used to detect convergence.
- Panel unit-root/convergence tests employed:
  - Levin, Lin, and Chu (LLC)
  - Im, Pesaran, and Shin (IPS)
  - Maddala and Wu (MW)
- Bootstrap techniques used to improve small-sample performance of IPS and MW.
- Structural-break and club-convergence complications addressed by subsample and subgroup testing.

### V. CONVERGENCE REVISITED: EMPIRICAL RESULTS
A. Convergence and Structural Change
- All 15 series are I(1).
- Full-sample (1960-2003): non-convergence in states’ per capita income is not rejected (panel tests indicate divergence overall).
- Subsample results:
  - Pre-1980:
    - 7 of 15 states reject non-convergence at less than 5 percent; Assam rejects at around 8 percent.
    - Panel tests (except LLC ADF) reject non-convergence.
  - 1980-1992:
    - Only 3 of 15 states show individual convergence; panel tests firmly reject unit-root null at less than 1 percent significance (conditional convergence).
  - Post-1992:
    - 5 of 15 states individually reject unit-root null; panel tests (LLC, IPS, MW) robustly reject non-convergence (overall income convergence in this subsample).
- Interpretation:
  - Three growth episodes: pre-1980 (low and volatile growth ≈ ½ percent per annum), 1981-1992 (growth ≈ 3 percent), post-1992 (growth ≈ 4 percent on average with differential outcomes).

B. Determinants of Per Capita Income Differentials
- Regression specification: differences in log per capita income between state pairs regressed on differences in factor variables and a spatial distance index.
- Main determinants (Table 7 summary):
  - Share of the service sector: a 1 percent difference → about 0.02 percent difference in per capita income between states.
  - Development expenditure: a 1 percent difference → approximately 0.07 percent difference in per capita income.
  - Private investment (private sector credit per capita): the largest contributor — 0.5 percent.
- Other variables:
  - Number of telephone lines significant but negligible effect.
  - T&D loss ratio has expected sign but not significant.
  - Literacy rate and employment in private sector statistically insignificant in regressions.
- Spillovers:
  - Spatial spillover estimates generally not strongly different from zero except for low-low income state pairs.
  - Negative estimates for high-high pairs suggest potential crowding out; positive spillovers in other combinations but economically small.

C. Club Convergence and Dynamic Spillover Effects
- Club convergence evidence:
  - Strong convergence among high-income and low-income state groups across full sample and subsamples.
    - High-income group: full-sample non-convergence rejected by bootstrapped IPS and MW; stronger rejection in sub-periods (less than 1 percent).
    - Low-income group: full-sample rejection with large-sample adjusted IPS and bootstrapped IPS; uniform rejection in sub-samples.
  - Medium-income group: divergence over full sample; convergence pre-1980 and 1980-1992 but not post-1992.
    - Divergence largely driven by Andhra Pradesh and Karnataka (rapid growth rises post-1980s and post-1992, becoming IT leaders).
    - Excluding Andhra Pradesh and Karnataka yields convergence among remaining medium-income states in all periods (robust panel rejections of non-convergence).
- Dynamic spillovers (dynamic panel VAR system):
  - Spillover magnitudes small: a one standard deviation increase in income in one state transfers to only 0.01-0.07 standard deviation increase in other states.
  - Within-group spillovers larger than between-group; high-medium spillovers stronger than high-low and medium-low.
  - Subsample dynamics:
    - 1980-1992 shows some interconnectedness and weak spillovers.
    - Pre-1980 and post-1992 spillovers are very small or negligible; shocks die out quickly.

### VI. CONCLUDING REMARKS
- Summary conclusions:
  - Over 1960-2003, evidence of overall divergence, but convergence in particular sub-periods and clear club convergence patterns.
  - Strong evidence of club convergence among high- and low-income states; mixed evidence for middle-income states.
  - Dynamic spillover effects among states are small.
- Policy-relevant findings:
  - States that forged ahead were those benefiting from advances in the services sector, better infrastructure, credit availability, and efficient development spending.
  - Lack of strong dynamic spillovers suggests potential need for better infrastructure and connectivity to disseminate benefits of growth across states.

*Source: IMF working paper section "5. Spillovers among Indian states (post-1992)".*

### Appendix 1: Test Procedures for Panel Unit Roots

### Appendix 1: Test Procedures for Panel Unit Roots

### 1. Levin, Lin, and Chu Test
- Nature: Parametric panel unit-root test analogous to the augmented Dickey–Fuller (ADF) test that models serial correlation with an autoregressive of order k specification in lagged differences.
- ADF regression estimated by OLS for each member i:
  - (1.1) timtmiLti p L iLtiiti dyyy i , , 1 1, , εαφρ+++=Δ − = − ∑ , m=1,2,3; i=1,2,...,N; t=1,2,...,T
  - where φ = t d 1 (the empty set); d 2t = {1}; d 3t = {1,t}
- Time effects extraction: replace y_it by y~_it where y~_it = y_it − y_t and y_t = (1/N) ∑_{i=1}^N y_it.
- Residual variance for each i from estimated residuals ε̂_it:
  - (1.2) σ̂_i^2 = (1/(T+K)) ∑_{t=1}^{T} ε̂_it^2
- Auxiliary regressions to generate orthogonalized residuals for each i:
  - (1.3) ê_it = ŷ_it − ∑_{m=1}^{P} π̂_{im} Δ ŷ_{i,t−m}
  - (1.4) v̂_{i,t−1} = ỹ̂_{i,t−1} − ∑_{m=1}^{P} π̂_{im} Δ ỹ̂_{i,t−m}
- Normalization by regression standard errors:
  - (1.5) ẽ_it = ê_it / σ̂_i and ṽ_{i,t−1} = v̂_{i,t−1} / σ̂_{i,−1}
- Panel pooled regression and t-test:
  - (1.6) ẽ_it = ρ̃ ṽ_{i,t−1} + ε̃_it
  - Test H0: δ = 0 using t-statistic t = (ρ̂)/(STD(ρ̂)) as in (1.7)
  - Variance aggregation formulas given in (1.8) (explicit summation structure preserved in source).
- Asymptotic adjustment to obtain standard normal limiting distribution:
  - (1.9) T_m^* = (T_m − μ̃_{T_m}) / σ̃_{T_m} → (1,0) N under the null ρ_i = 0, where μ̃_{T_m} and σ̃_{T_m} are tabulated adjustments depending on cases.
- Hypotheses:
  - H0: ρ_i = 0, ∀i (all series contain a unit root)
  - H1: ρ_i < 0, ∀i (no unit root for all series)
- Interpretation: Rejection of H0 implies there is no unit root for all series in the panel.

### 2. Im, Pesaran, and Shin Test
- Nature: Group-mean between-dimension unit-root test for dynamic heterogeneous panels, analogous to ADF.
- Univariate ADF estimated for each member i:
  - (2.1) tiiLti p L iLtiiti yyy i , , 1 1, , εαφρ+++=Δ − = − ∑ ; i=1,2,...,N; t=1,2,...,T
- Time effects extraction: y~_it = y_it − y_t with y_t = (1/N) ∑_{i=1}^N y_it.
- Collect individual t-statistics t_{ρ,i} and compute group-mean t-bar:
  - (2.2) t̄_ρ = (1/N) ∑_{i=1}^N t_{ρ,i}
- Standardization to asymptotic standard normal:
  - (2.3) (t̄_ρ − μ)/σ → (1,0) N under the null ρ_i = 0, where μ and σ are tabulated from IPS.
- Hypotheses (heterogeneous alternative):
  - H0: ρ_i = 0, ∀i (all series have unit roots)
  - H1: ρ_i < 0 for some i (at least one series is stationary)
- Interpretation: Rejection of H0 indicates the panel does not contain unit roots (i.e., at least some series are stationary).

### 3. Maddala and Wu Test
- Nature: Combines significance values (p-values) of individual ADF t-tests across panel members using Fisher’s method; treats parameters as heterogeneous across members.
- Individual ADF estimated for each member i:
  - (3.1) tiiLti p L iLtiiti yyy i , , 1 1, , εαφρ+++=Δ − = − ∑ ; i=1,2,...,N; t=1,2,...,T
- Time effects extraction: y~_it = y_it − y_t with y_t = (1/N) ∑_{i=1}^N y_it.
- Procedure:
  - Collect t-statistic t_{ρ,i} for H0: ρ_i = 0, compute corresponding p-values π_i for i = 1,..,N.
  - Note: π_i must be obtained by simulation because t_{ρ,i} distribution is non-standard.
  - Under continuity and independence, −2 ∑_{i=1}^N ln(π_i) has a χ^2 distribution with 2N degrees of freedom (Fisher’s combination).
- Hypotheses (same as IPS):
  - H0: ρ_i = 0, ∀i
  - H1: ρ_i < 0 for some i
- Interpretation: Failure to reject H0 is evidence of unit root presence in the panel.

### 4. Step-Down Procedure for Choosing Lag Truncation
- Importance: Correct lag truncation is crucial—too few lags lead to misspecification and size distortion; too many lags reduce efficiency and power.
- Common criteria: AIC and SBIC often undertruncate for panel unit-root tests (Pedroni and Yao (2006) caution).
- Adopted procedure (Pedroni and Yao (2006) "step down"):
  - Start with a sufficiently large number of lags: nearest integer of 1/5 of the sample length as initial starting number of lags.
  - Perform ADF regression. If the largest lag is significant, stop and choose this truncation.
  - If not significant, eliminate the largest lag sequentially one at a time and repeat until significance is achieved.
  - Number of lags allowed to differ across states.

### 5. Bootstrap Procedures
- Purpose: Simulate IPS adjustment terms and MW p-values to account for finite-sample behavior, serial correlation, lag truncation sensitivity, and cross-sectional dependence.
- Rationale:
  - IPS adjustment terms (mean and variance) are asymptotically invariant to lag truncation if residuals are white noise, but finite-sample use of asymptotic values can cause substantial size distortion.
  - MW requires simulation because individual t_{ρ,i} distributions are non-standard.
- Bootstrap implementation (following Pedroni and Yao (2006)):
  - Estimate serial correlation properties by running ADF regression for each member i:
    - (5.1) tiiLti p L iLtiiti yyy i , , 1 1, , εαφρ+++=Δ − = − ∑
  - Draw 10,000 realizations of a pure random walk of length T+100 for ε_it.
  - Fix parameters α_i, φ_{iL}, ρ_i and replicate the serially correlated process using pseudo-innovations to generate y*_it.
  - Re-estimate ADF regressions on pseudo-samples:
    - (5.2) tii p L iLi Lti i titi yyy , * 1 ** ,1, , εαφρ+++=Δ −− = ∑
  - Discard the first 100 observations to eliminate arbitrary initial conditions.
  - Collect parameters of interest across realizations to generate pseudo-distributions; compute corresponding mean, variance, and probability distributions for adjustment terms and p-values.

*Source: Appendix 1: Test Procedures for Panel Unit Roots (contained in _wp1096 - Appendix 1: Test Procedures for Panel Unit Roots).*

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_Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2010/_wp1096.pdf_