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

1. Levin, Lin, and Chu Test

Levin, Lin, and Chu (2002) propose a parametric test analogous to the augmented Dickey Fuller test. They model serial correlation dynamics using autoregressive of order k specification in lagged differences. To perform the test, first estimate ADF regression by OLS for each member:

The time effects for cross sectional dependence can be extracted by replacing *y*_{it} by

Next, use the estimated residuals,

Then, run two auxiliary regressions to generate orthogonalized residuals by estimating the following for each member i:

and

After that, normalize

Now, compute the panel test statistics by pooling all cross sectional and time series observations to estimate

and so apply the t-test for *δ* = 0 where t-statistic is given by

The above t-statistic would require some adjustment to have a standard normal limiting distribution as follows:

where the mean adjustment

The hypotheses for unit root test are as follows:

The rejection of the null hypothesis implies there is no unit root for all series in consideration.

2. Im, Pesaran, and Shin Test

Im, Pesaran, and Shin (2003) propose a unit root test for dynamic heterogeneous panels based on the group mean between-dimension test. The test is also analogous to the augmented Dickey-Fuller test. The univariate ADF test is estimated for each member:

Again, to extract the time effects, *y*_{it} can be replaced by

Then, collect t-statistics of each *ρ*_{i}, and calculate the group-mean value of t-statistic for panel:

To obtain the asymptotically standardized normal distribution, the t-bar statistic requires some adjustments as follow:

where *µ* and *σ* are tabulated mean and variance from IPS paper.

Unlike LLC test, the IPS hypotheses take into account the null hypothesis that all series in panel are unit-root against the alternative that at least one of them is stationary.

The rejection of the null again indicates the panel does not contain unit root.

3. Maddala and Wu Test

Maddala and Wu (1999) propose somewhat different test. Similar to IPS test, they treat all parameters as heterogeneous among members and base their model on ADF type of regression. Instead, they combine significance values of ADF t-tests across members of panel to get the Fischer’s test statistic. The procedure is as follows:

The first step is to estimate ADF regression for each member i.

or replacing *y*_{it} by

Then, collect the t-statistic, *H*_{0}: ρ_{i =} 0 for each member i and compute the corresponding p-values, π_{i} (*i* = 1,2,.., *N*). In particular, π_{i} value must be obtained by simulation since _{i} (*i =* 1,2,.., *N*) are independent uniform (0,1) variables, and − 2ln π_{i} has a χ^{2} distribution with two degrees of freedom. With additive property of the χ^{2} variables, the MW panel unit-root test statistic is ^{2} distribution with degrees of freedom.

The MW test considers the same unit-root test hypotheses as IPS test.

There is evidence of unit root in panel if the null cannot be rejected.

4. Step-Down Procedure for Choosing Lag Truncation

The number of lags is crucial. The lag truncation must be large enough to ensure the ADF residual is white noise. If the number of lags is too small, tests will be misspecified and potentially lead to serious size distortion. Nonetheless, if the number of lags is too large, tests are inefficient. Power of the test is gradually loss.

The time series literature often uses Akaike Information Criterion (AIC) and Schwarz Bayesian information criterion (SBIC). Pedroni and Yao (2006) argue that these are not sufficiently conservative for panel unit root tests as the two tests tend to undertruncate. On the other hand, panel unit root and panel cointegration procedures do best with less conservative “step down procedure”.

This paper employs the procedure as in Pedroni and Yao (2006) to the above three tests. We first start with a sufficiently large number of lags. Specifically, we take the nearest integer of 1/5 of the sample length for an arbitrary initial starting number of lags relative to sample size. The ADF regression is then performed. If the largest lag is significance, we can stop the process and choose this truncation. If not, the number of lags is sequentially eliminated one at a time and continues the above process until significance. Additionally, the number of lags is allowed to be different across states.

5. Bootstrap Procedures

This paper applies bootstrap technique to simulate the IPS adjustment terms as well as MW p-value. The IPS test requires the appropriate adjustment values for mean and variance. Theoretically, values of the adjustment terms are asymptotically invariant to lag truncation, provided that the number of lags is large enough to ensure that the ADF residuals are white noise. For finite sample, however, the use of asymptotic value of adjustment terms can lead to substantial size distortion. The specific adjustment terms to each panel’s members depend on its serial correlation nature, and so are very sensitive to the choice of lag truncation. Moreover, if the data are cross-sectional dependent, the test statistics are no longer the same as their asymptotic version. The MW test, too, needs simulation for non-standard

Following Pedroni and Yao (2006), we use a bootstrap to condition the Monte Carlo simulation on both the sample size and the specific number of fitted lags. If the number of lag truncation in ADF regression is chosen so that its residuals are white noise, the ADF limiting distribution is asymptotically the same as DF distribution. One can then simulate the DF distribution using pure random walk and use this distribution to map

We draw 10,000 realizations of pure random walk of length T+100 for *ε*_{it}. We fix parameters, *ρ*_{i}, *ø*_{iL} ,*α*_{i}, and replicate the serial correlated process. We repeat the ADF regression using these pseudo-innovations,

We then discard the first 100 results to eliminate an arbitrary initial condition. We then collect the parameters of interest and generate the pseudo distribution. Finally, we compute corresponding mean and variance, as well as the probability distribution.

## References

Aiyar, S., 2001, Growth Theory and Convergence across Indian States, in

*India at the Crossroads: Sustaining Growth and Reducing Poverty*, ed. by Tim Callen, Patricia Reynolds, and Christopher Towe (Washington: International Monetary Fund), pp. 143-69.Baddeley, M.C., et. al., 2006, Divergence in India: Income differentials at the state level, 1970-97,

*Journal of Development Studies*, Vol. 42, 6, pp. 1000-22.Bajpai, N., and J. Sachs, 1996, Trends in Inter-State Inequalities of Income in India,

*Harvard Institute for International Development, Development Discussion Paper No. 528*.Balakrishnan, P., 2005, Macroeconomic Policy and Economic Growth in the 1990s,

*Economic and Political Weekly*, September.Bandyopadhyay, S., 2003, Convergence Club Empirics: Some Dynamics and Explanations of Unequal Growth across Indian States,

*STICERD—Distribution Analysis Research Program Papers 69*.Barro, R., and X. Sala-i-Martin, 1991, Convergence Across States and Regions,

*Brookings Papers on Economic Activity*, Vol. 1, No. 1, pp. 107-82.Barro, R., and X. Sala-i-Martin, 1999, Economic Growth (Cambridge, Massachusetts: MIT Press).

Cashin, P., and R. Sahay, 1996, Internal Migration, Centre-State Grants, and Economic Growth in the States of India,

*IMF Staff Papers*, Vol. 43, No. 1, pp. 123-71.Choi, C-Y., 2004, A Reexamination of Output Convergence in the U.S. States: Toward Which Level(s) are they Converging?

*Journal of Regional Science*, Vol. 44(4), pp. 713-41.Evans, P., 1998, Income Dynamics in Regions and Countries,

*Department of Economics Working Paper*, Ohio State University.Evans, P. and G. Karras, 1996, Do Economies converge? Evidence from a panel of U.S. states,

*Review of Economics and Statistics*, 78, 384-388.Fleissig A. and J. Strauss, 2001, Unit Root Tests of OECD Stochastic Convergence,

*Review of International Economics*, 9(1), 153-162.Hausmann, R., L. Pritchett, and D. Rodrik, 2004, Growth Accelerations,

*NBER Working Paper No. 10566*, National Bureau of Economic Research, Cambridge, Massachusetts.Im, K.-S., H. Pesaran, and Y. Shin, 2003, Testing for unit roots in heterogeneous panels.

*Journal of Econometrics*, 115, 53–74.Kochhar, K., U. Kumar, R. Rajan, A. Subramanian, and I. Tokatlidis, 2006, India’s Pattern of Development: What Happened, What Follows,

*IMF Working Paper No. 06/22*.Kohli, A., 2006a, Politics of Economic Growth in India, 1980-2005,

*Part I: The 1980s, Economic and Political Weekly*, Vol. 41, No. 13, April 1.Kohli, A., 2006b, Politics of Economic Growth in India, 1980-2005,

*Part II: The 1990s and Beyond, Economic and Political Weekly*, Vol. 41, No. 14, April 8.Levin, A., C-F Lin, and C. Chu, 2002, Unit root test in panel data: Asymptotic and finite sample results.

*Journal of Econometrics*, 108, 1–24.Maddala, G. S. and S. Wu, 1999, A comparative study of unit root tests with panel data and a new simple test.

*Oxford Bulletin of Economics and Statistics*, 61(4), 631–652.McCoskey, S. K., 2002. Convergence in Sub-Saharan Africa: A Nonstationary Panel Data Approach,

*Applied Economics*, Vol. 34(7), pp. 819-29.Misra, B. S., Regional Growth Dynamics in India in the Post-Economic Reform Period, Palgrave Macmillan, 2007.

Nagaraj, P. et. al., 2000, Long-run Growth Trends and Convergence across Indian States,

*Journal of International Development*, Vol. 12, No. 1, pp. 45-70.Pedroni, P. and J. Yao, 2006, Regional Income Divergence in China,

*Journal of Asian Economics*, Vol. 17, 2, pp. 294-315.Purfield, C., 2006, Mind the Gap—Is Economic Growth in India Leaving Some States Behind?

*IMF Working Paper No. 06/103*.Quah, D., 1993. Exploiting Cross Section Variation for Unit Root Inference in Dynamic Data,

*International Economic Studies Paper 549*, Stockholm.National Sample Survey Organization, 2001, Migration in India, 1999-2000 (New Delhi: Ministry of Statistics and Program Implementation).

Rao, M. G., R. T. Shand, and K.P. Kalirajan, 1999, Convergence of Incomes across Indian States: A Divergent View,

*Economic and Political Weekly*, Vol. 34, No. 13, pp. 769-78.Rodrik, D. and A. Subramaniam, 2004, From “Hindu Growth” to Productivity Surge: The Mystery of the Indian Growth Transition,”

*IMF Working Paper No. 04/77*, (Washington: International Monetary Fund).Sachs, J., N. Bajpai and A. Ramiah, 2002, Understanding Regional Economic Growth in India,

*Asian Economic Papers*, Vol. 1, 3.Virmani, A., 2005, India’s Economic Growth History: Fluctuations, Trend, Break Points and Phases,

*ICRIER Occasional Paper*.Wallack, J., 2003, Structural Breaks in Indian Macroeconomic Data,

*Economic and Political Weekly*, October.

^{}1

Kochhar et. al. (2006) and others use the ratio of transmission and distribution (T&D) losses to generating capacity of state level electricity boards to indicate the quality infrastructure and institutions.

^{}2

For an excellent discussion of data sources, see Purfield (2006).

^{}3

Existing studies examine structural breaks using aggregative data for India, such as the growth rate of real GDP, real per capita GNP, and international trade (Table 3). Wallack (2003), Rodrik and Subramaniam (2004), Hausmann et. al. (2005), Virmani (2005), and Kohli (2006) find evidence of a break around 1980. Wallack (2003) alone records another break in 1993, although the evidence is weak.

^{}4

In addition, the possibility of structural breaks in state level data is tested, following Andrew (1993) and Bai (1994) structural break tests at unknown date, using the difference between the growth paths of high and low income groups. Two possible break dates are found, 1980 and 1992.

^{}5

If _{(}*y*_{it} − *y*_{jt}) ~ *I* (0) for all i, j pairs, then

But

So

^{}6

The common parameter hypothesis of LLC test is restrictive. The IPS and MW tests are considered generalizations of the LLC test. The IPS test is at least as flexible as MW test, and both are more flexible than LLC test as they do not require any parameter commonality. Computationally, the IPS test is easier than the LLC and MW tests. The IPS test simply averages individual ADF tests and use adjustment values to render the asymptotic standard normal distribution. MW test, on the other hand, requires simulation of the p-value. The distributions for individual ADF based unit root tests are nonstandard and depend on Brownian motion functions, and the simulation is non-trivial. The MW test is more conservative in which it is invariant to cross-sectional dependencies than the other two tests.

^{}7

The test might fail to capture the transition property toward stochastic convergence although the actual path is converging.

^{}9

Although the set-up of the convergence tests allows for time and cross-sectional fixed effect, the instruments are not enough to filter out structural changes.