Back Matter

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:

(1.1)Δyi,t=ρiyi,t1+L=1piϕiLyi,tL+αmidmt+εi,t,  m=1,2,3; i=1,2,,N; t=1,2,,Td1t=ϕ (the empty set);d2t={1};d3t={1,t}

The time effects for cross sectional dependence can be extracted by replacing yit by yit=yity¯t where y¯t=1Ni=1Nyit

Next, use the estimated residuals, ε^it, to compute the estimated residual variance for each i:

(1.2)σ^εi2=1Tt=1+KiTε^it2

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

(1.3)e^it=ΔyitL=1Piπ^iLΔyitLα^midmt

and

(1.4)v^it1=yit1L=1Piπ˜iLΔyitLα˜midmt

After that, normalize e^it and v^it1 by regression standard error from (2) and get

(1.5)e˜it=e^itσ^εi and v˜it1=v^it1σ^εi

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

(1.6)e˜it=ρv˜it1+ε˜it

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

(1.7)tδ=ρ^STD(ρ^)
(1.8)ρ^=i=1Nt=2+piiTv˜it1e˜iti=1Nt=2+piiTv˜it12;STD(ρ^)=σ^ε˜ [i=1Nt=2+piTv˜it12]1/2

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

(1.9)tρ*=tρNT˜S^Nσ^ε˜2STD(ρ^)μmT˜*σmT˜*dN(0,1), under the null ρi=0

where the mean adjustment μmT˜* and standard deviation adjustment σmT˜* are from tables depending on cases.

The hypotheses for unit root test are as follows:

H0:ρi=ρ=0,iH1:ρi=ρ<0,i

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:

(2.1)Δyi,t=ρiyi,t1+L=1piϕiLyi,tL+αi+εi,t i=1,2,,N; t=1,2,,T

Again, to extract the time effects, yit can be replaced by y˜it=yity¯t where y¯t=1Ni=1Nyit.

Then, collect t-statistics of each ρi, and calculate the group-mean value of t-statistic for panel:

(2.2)t¯ρ=1Ni=1Ntρi

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

(2.3)t˜ρ=N(t¯ρμ)σdN(0,1),under the null ρi=0

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.

H0:ρi=0,i,H1:ρi<0, i

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.

(3.1)Δyi,t=ρiyi,t1+L=1piϕiLyi,tL+αi+εi,ti=1,2,,N; t=1,2,,T

or replacing yit by y˜it=yity¯t where y¯t=1Ni=1Nyit to extract time effects.

Then, collect the t-statistic, tρi, for H0: ρi = 0 for each member i and compute the corresponding p-values, πi (i = 1,2,.., N). In particular, πi value must be obtained by simulation since tρi distribution is non-standard. If the test statistics are continuous, the significance levels πi (i = 1,2,.., N) are independent uniform (0,1) variables, and − 2ln πi has a χ2 distribution with two degrees of freedom. With additive property of the χ2 variables, the MW panel unit-root test statistic is λ=2i=1Nlnπi has a χ2 distribution with degrees of freedom.

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

H0:ρi=0,i,H1:ρi<0, i

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 tρi distribution to map between each member’s tρi value and corresponding p-value.

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 tρi values into corresponding p-values. Specifically, we estimate serial correlation properties by running ADF regression for each member i.

(5.1)Δyi,t=ρiyi,t1+L=1piϕiLyi,tL+αi+εi,t

We draw 10,000 realizations of pure random walk of length T+100 for εit. We fix parameters, ρi, øiL ,αi, and replicate the serial correlated process. We repeat the ADF regression using these pseudo-innovations, yit*.

(5.2)Δyi,t*=ρiyi,t1*+L=1piϕiLyi,tL*+αi+εi,t

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.

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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 (yityjt) ~ I (0) for all i, j pairs, then 1Nj=1N(yityjt)I(0)

But 1Nj=1N(yityjt)=1Nj=1Nyit1Nj=1Nyjt=yity¯t

So (yityjt)I(0)i,j(yity¯t)I(0)i and similarly for the case with fixed effects.

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.

8

Detailed results are available on request.

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.

10

Development expenditure includes spending on education, public health, family planning, water supply, and relief after natural calamities.

11

We estimate the generalized impulses as in Pesaran and Shin (1998) which constructs an orthogonal set of innovations that does not depend on the VAR ordering.

Growth Convergence and Spillovers among Indian States: What Matters? What Does Not?
Author: Piyaporn Sodsriwiboon and Sanjay Kalra