Why to use panel unit root tests?

The main advantage of using panel unit root tests is that their power is significantly greater compared to the low power of the standard time-series unit root tests in finite samples against alternative hypotheses with highly persistent deviations from equilibrium. Monte-Carlo simulations have also shown that this problem is particularly severe for small samples like ours (see Campbell and Perron 1991). Since the power of unit root tests depend on the total variation in the data used (both in the number of observations and their variation), panel unit root tests are more powerful than standard time-series unit root tests because the variation across countries adds a great deal of information to the variation across time, resulting in potentially more precise parameter estimates (Taylor and Sarno 1998). Another advantage of panel unit root tests is that their asymptotic distribution is standard normal, in contrast to individual time series unit root tests (such as the DF or ADF) which have non-standard limiting distributions.

The firsts unit root tests are those of Quah (1992, 1994), Breitung and Mayer (1994) and Levin and Lin (1992, 1993). We do not employ Quah’s (1990,1994) random fields approach because it cannot accommodate the serial correlation of error terms which we detected in our data. We find that Breitung and Mayer’s test is not appropriate due to the asymptotic dimensional assumptions not being suitable for datasets in which T and N have the same or larger order of magnitude. Furthermore, their test does not allow for heterogeneous residual distributions and individual-specific effects can have large effects on the appropriate critical values at which to evaluate the t-statistic.

Levin and Lin (1992, 1993) and Levin, Lin and Chu (2002)10 (LLC thereafter) generalise Quah’s procedure to allow for heterogeneity of individual deterministic effects (a constant and/or linear time trend) and heterogeneous auto-correlation structure of the errors assuming homogeneous AR(1) parameters. They assume that both N and T tend to infinity but that T increases at a faster rate, such that N/T→0 .

The structure of the LLC analysis may be specified as follows

Their procedure involves pooling the t-statistic of the estimator to evaluate the hypothesis that every individual time series contains a unit root against the alternative hypothesis that every individual time series is stationary. Thus, LLC assume homogeneous autoregressive coefficients for all individuals, i.e. pi= p for all i , and test the hypotheses:

H0: p = 0

Ha: p < 0

By imposing a cross-equation restriction on the first-order partial serial correlation coefficients under the null, this LLC test has much higher power than performing a separate unit root test for each individual. (Barbieri 2006) However, since we felt that the LLC was bypassed by the IPS (for several reasons discussed later) we have chosen not to present results from our LLC tests in this paper.

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Im-Pesaran-Shin

Im, Pesaran and Shin (1997) propose an alternative testing procedure which uses a standardized t-bar test statistic based on the augmented Dickey-Fuller test statistics averaged across the panels. While they consider the case of serially correlated and uncorrelated errors, we utilise the former since results from our Breush-Godfrey tests for autocorrelation confirms the presence of residual serial correlation. Consider the following ADF regression:

And the null and alternative hypotheses are:

H0: ρi = ρ = 0

Ha: ρi < 0 for i= 1,2,3 … N1

ρi = 0 for i=N1+1, N1+2 …N

An important feature of IPS is that it allows pi to vary across groups under the alternative hypothesis, i.e the test does not assume that all cross-sectional units converge towards the equilibrium value at the same speed, making it much less restrictive test than its predecessor the LLC test. While they shared the same null hypothesis of all series containing a unit root, the LLC offered a homogenous alternative of all series being stationary i.e. pi=p<0, Since IPS allows for pi to differ across groups, the alternative hypothesis is that at least one (but not necessarily all) series to be stationary. Thus we have chosen to use IPS rather than LLC because the alternative hypothesis under IPS is more general and we think this is particularly appropriate for our study because it allows for some, but not all series, to contain a unit root. Formally, for consistency purposes, we are assuming that under the alternative hypothesis the fraction of series which are stationary are non-zero.

where 0 < δ < 1

The t-bar statistic is formed by an average of the individual ADF statistics, tiT gained by running ADF regressions on each individual time series:

When the lag order in the above equation is non-zero for some cross-sections, the test converges to an asymptotic standard normal distribution when they have been properly standardized.

In order to allow for ease of standardization of the t-bar statistic, IPS (2003) present tables of the values of the mean and the variance which have been computed via Monte Carlo methods for different values of T and p’s The critical values used for interpreting test results in this paper were taken directly from IPS (1997). While these critical values are easy to use because they are directly implementable in Stata we must remember that this test is parametric and the critical values are only applicable when the ADF is used for the individual regressions. Although IPS tabulate E(ti,T) and V(ti,T) for different lag lengths, in practice, to make use of their tables, we are implicitly limited to using the same lag length for all ADF regressions for each individual time series. The IPS test is claimed to be a generalization of the LL tests. However, it is better viewed as a way of combining the evidence of several independent unit root tests.

IPS is particularly appropriate for a data-set like ours with a moderate number of cross-sectional units over a relatively long period of time. It is also appropriate for our dynamic heterogeneous panel data because it allows for heterogeneity across countries such as individual-specific effects and unique patterns of residual serial correlations

IPS’s (1999) simulations show that, When the disturbances in the dynamic panel are auto-correlated, LLC test tends to over-reject null hypothesis as N is allowed to increase and that the size and power of the IPS test are reasonably satisfactory, given that T and N are sufficiently large. It is also critically important not to under-estimate the order of the underlying ADF regressions: they find that if a large enough lag order has been selected for the underlying ADF regressions, then the finite sample properties of the t-bar test is satisfactory and generally superior to that of the LLC test. Another important feature lies in the fact that the power of the t-bar test is much more favourably affected by a rise in T than by an equivalent rise in N. However it must be remembered that the LLC test is based on pooled regressions while IPS test considers a combination of different independent tests and does not pool the data as LLC test does. Therefore, when making power comparisons, the worse performances of LLC test may be because this test has to use a panel estimation method which is invalid if there is no necessity for pooling.

Special care needs to be exercised when interpreting the results of the IPS panel unit root test. Due to the heterogeneous nature of the alternative hypothesis, rejection of the null hypothesis does not necessarily imply that the unit root null is rejected for all i, but only that the null hypothesis is rejected for N1<N members of the group such that as as N→ ∞, N1/N → δ. Moreover, the test does not provide any guidance as to the magnitude of δ, or the identity of the particular panel members which the null hypothesis is rejected.

Fisher

Maddala and Wu (1999) and Choi (2001) consider the weaknesses of both the LLC and IPS tests and offer an alternative procedure for performing unit root tests on panel data. They suggest using non parametric Fisher-type tests which approach panel-data unit root testing from a meta-analysis perspective. More specifically, these tests conduct unit-root tests for each time series individually, and then combine the p-values from these tests to produce an overall test. Both IPS and Fisher tests combine information based on individual unit root tests and allow for a heterogenous alternative hypothesis where ρi can vary across countries.

Choi (2001) considers the model:

where

where eit is integrated of order zero. Note that the observed data uit are composed of a non-stochastic process dit and a stochastic process xi . Each time series uit can be of a different length and have different specification of non-stochastic and stochastic component depending on i.

The null hypothesis and alternative hypothesis are:

H0 : ρi =1 for all i .

Ha : ρi <1 for at least one i for finite N

The null implies that all time series in the penl data set contain a unit root and are non-stationary, whereas the alterative indicated that at least one time series is stationary. This makes the Fisher test uni allows for the scenario that some time series are non-stationary while the others are not

Let pi be the p-value of a unit root test for a cross-sectional unit Then the proposed Fisher test involves the following statistic:

which combines the p-values from unit root tests for each cross-sectional unit. Under null hypothesis of unit root, P is distributed as χ2 (2N) 2 as T i®¥ for all N.

The Fisher and IPS test are directly comparable because both tests are a combination of different independent tests and both seek to verify the same hypothesis. The main difference between the two tests is that the Fisher test is conducted by combining the significance levels of the different tests, whereas the IPS test is conducted by combining the test statistics. Furthermore, the Fisher test is a non-parametric test, whereas the IPS test is a parametric test. The distribution of the t-bar statistic used in IPS requires the mean and variance of the t-statistics used. Even though IPS have provided directly implementable critical values (for different lag lengths and sample sizes), these are only valid if the ADF test is used for the individual regressions. In comparison, the Fisher test has the advantage of being compatible with the use of any unit root test – and even if we choose to use the ADF test, we can choose a different lag length for each sample can be separately determined. Lastly, in the IPS test the length of the time series must be the same for all samples so we require a balanced panel while Fisher imposes no such restriction of the sample sizes for different samples,

The Fisher test is an exact test while the IPS test is an asymptotic test, however this does not lead to a great difference in finite sample results: the adjustment terms in the IPS test and the p-values in the Fisher test are all derived from simulations. However, the asymptotic validity of the IPS test depends on N ®¥€ while for the Fisher test it depends on T ®¥. Since our dataset is considerably larger in T than in N, Fisher test may be more appropriate.

Maddala and Wu (1999) conduct simulations (not size-corrected) to compare their Fisher test, LLC test and IPS test and show that the Fisher test has the highest power in all cases. In fact, the relative advantage grows stronger with the number of stationary processes included. Thus we can recognise that if only part of our panel is stationary, the Fisher test is the most likely to point it out. Therefore, while both tests offer alternative hypotheses which allow for a mixture of stationary and non-stationary series in the group, the Fisher test is strictly preferred because it has the highest power in distinguishing the null and the alternative. They also find that while both tests can take care of heteroskedasticity and serial correlation of the error terms, when there is cross-sectional dependence neither can handle this problem well. The Monte Carlo evidence suggests that the problem is more severe with the IPS test than the Fisher test. Specifically, when T is large but N is not very large (such as in our case), the size distortion is smallest with the Fisher test. Choi’s simulations (2001) find that in terms of size-adjusted power, Fisher test is superior to the t-bar.

Pesaran’s CADF

Both of the previously presented panel unit root tests made the crucial assumption that the individual time series in the panel were cross-sectionally independent of each other. This condition is required in order to satisfy the Lindberg-Levy central limit theorem and to acquire asymptotically normal test statistics.

However, some of the recent literature (i.e. Backus and Kehoe, 1992) which has provided evidence on the strong co-movements between economic variables caused us to pause to consider whether the assumption of cross-sectional independence is too restrictive given the economic, political, cultural, and other linkages between the different countries in our panel data-set. If any such cross-sectional correlation existed in our panel, it would adversely affect the finite sample properties of our panel unit root tests. (O’Connell, 1998). This realisation spurred us to conduct Pesaran’s (2004) CD test, and the results showed that [Burak please insert your results here

Pesaran (2007) proposes a simple augmentation of the standard ADF regression which involves adding the cross sectional averages of lagged levels and first-differences of the individual series. He considers the following modification to the basic IPS equation

where

H0: ρi = ρ = 0

Ha: ρi = 0 for i= 1,2,3 … N1

ρi=<0 for i=N1+1, N1+2 …N

Since this test is a direct generalization of the panel unit-root test of Im et al. (2003), the corresponding test statistics is denoted as a cross section IPS (CIPS) and is based on a simple average of individual CADF t-ratios

Due to the presence of the lagged level of the cross-sectional average, the limiting distribution of both the CADF statistics and the CIPS statistic do not follow a standard Dickey-Fuller distribution. However, Pesaran (2007) provides critical values based on simulations for the CADF and CIPS distributions.

The Pesaran test comes with the advantage that it controls for cross-sectional dependence of the errors. It is particularly appropriate for the case of our data because it is reasonable to assume that economic, political, cultural inter-relationships may result in cross-country correlations which will influence our results. However we must also consider several of its weaknesses before we begin to interpret test results. The Pesaran (2003) test assumes that under the null hypothesis the non-stationarity of the data comes from common and idiosyncratic stochastic sources. It also assumes that the idiosyncratic and the common component of the data share the same order of integration. Pesaran (2003) also does not allow for the possibility of co-integration among the uit as well as between the observed data and the common factors. As a result, in the case where the observed non-stationarity arises only due to a non-stationary common factor (i.e. if the individual time series are pair-wise co-integrated in the cross sectional dimension) the test will not enable us to detect this situation. On the contrary, the Pesaran (2003) test tends to systematically reject the non-stationarity of the series (Barbieri 2006)