It is now well know that the original unit root tests often suffer from low power when applied to series of only moderate length, and it has been proposed that pooling the data across individual members of a panel helps increase power. Panel cointegration techniques are intended to allow researchers to selectively pool information regarding common long-run relationship from across the panel while allowing the associated
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short-run dynamics and fixed effects to be heterogeneous across different members of the panel (Banerjee, 1999; Maddala and Wu, 1999). Given the properties of our data, we utilize both panel unit root tests and panel cointegration test techniques.
As is now standard practice, before testing for cointegration we conduct panel unit root tests to consider the order of integration and common unit root properties of the data. Three kinds of panel unit root tests, Levine et al. (2002, thereafter LLC), Im et al. (2003, thereafter IPS) and Hadri (2000), are provided in this study. Each has different assumptions, constraints and statistical power. LLC propose an ADF test with a panel setting that restricts parameters γi by keeping them identical across cross -sections (in our case cities) as follow:
(4-2)
∑
the panel data are non-stationary while the alternative hypothesis is<0
=
2 =
1=γ γ
γ " . This test is based on the statistics, ty =γˆ/s.e.(γˆ). The IPS (2003) relaxes this assumption of LLC by allowing γ to vary across units (cities) under the alternative hypothesis. The null hypothesis of the IPS test is that γi =0 for all i, while the alternative hypothesis is γi <0 for all i. This IPS test uses the mean-group approach and obtains the average of to compute the following statistic: ty
(4-3) ~ ( ( ))/ var( )
converges to a Normal distribution, and we can compute 131
the significance level in a simple way. By contrast, Hadri (2000) argues that the null hypothesis should be reversed to be a stationary hypothesis in order to increase the power of the test. His Lagrange Multiplier (LM) statistics is given by the follow expression:
Where is the consistent Newey-West (1987, 1994) estimate of the long-run variance of disturbance terms (
ˆε2
σ
εij).
In many circumstances it is hard to judge which panel unit root tests is best as most of time we do not know the properties of the price series. Some authors prefer some types, while others prefer other types of tests. For example, Hlouskova and Wagner (2006) found that the Breitung (2000) panel unit root test generally had the highest power and smallest size distortions of any of the so-called first generation panel unit root tests and therefore Narayan and Smyth (2007) employed this test in their paper. However, this test assumes a common unit root process, which may not reflect the reality, especially for this empirical study, which covers 35 city markets located in 31 provinces in China. Thus, to obtain more robust results, this study uses six panel unit root tests to determine whether the panel dataset is stationary. In addition, three other panel unit root tests: Breitung, Fisher ADF, and Fisher PP are considered. The null hypothesis for LLC and Breitung is a common unit root process;
for IPS, Fisher ADF and Fisher PP individual unit root processes are assumed and for Hadri stationarity is the null. Conclusions may (and do) vary in the main because of varying assumptions and restrictions made when testing the underlying data set whose
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true properties are unknown. We will attempt to explain inconsistencies as and when they arise.
Using these panel unit root tests, we proceed to test for cointegration in the data.
Using the heterogeneous panel cointegration test developed by Pedroni (1999) allows for cross-sectional interdependence with different individual effects. If the panel data follow an I(1) series, the Pedroni (1999 and 2004) panel cointegration model is applied to find whether a cointegratiing relationship exists. Pedroni (1999) suggests the following time series panel expression:
(4-5) yit =αit +γitt+Xiβi+eit
Where and are the observable variables with dimension of and , respectively. He develops asymptotic and finite-sample properties of test statistics to examine the null hypothesis of non-cointegration in a panel. The tests allow for heterogeneity among individual member of panel, including heterogeneity in both the long-run cointegration vectors and in the dynamics, for there is no reason to believe that all parameters are the same across cities.
yit
×m )
Xit (N∗T)×1
T N∗ (
There are two types of residual-based tests (Pedroni, 1999). The first type is distributed as being standard Normal asymptotically and is based on pooling the residuals of the regression for the within-group. It includes the panel υ -statistic, panel
ρ -statistic, panel PP-statistic (or t -statistic, non-parametric) and the panel
ADF-statistic (or t -ADF-statistic, parametric). The second type is also distributed as standard Normal asymptotically, but is based on pooling the residuals for the between-group. It includes the group ρ-statistic, group PP-statistic (or t -statistic, non-parametric) and
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the group ADF-statistic (or t -statistic, parametric). Pedroni (1999) presents the following the heterogeneous panel cointegration statistics:
Panel υ -statistic:
Panel -statistic (non-parametric): t
(4-8) (ˆ ˆ ˆ ) ˆ (ˆ 1 ˆ ˆ)
Panel -statistic (parametric): t
(4-9) it it
and the following heterogeneous group-mean panel cointegration statistics:
Group ρ-statistic:
Group -statistic (non-parametric): t
(4-11)
∑ ∑
− −Group -statistic (parametric): t
(4-12)
∑ ∑ ∑
Where is the estimated residual from equation (4-5) above and is the estimated long-run covariance matrix for . Similarly, and ( ) are,
respectively, the long-run and contemporaneous variances for individual i. The other terms are defined in Pedroni (1999) with the appropriate lag length determined by the Newey-West method. All seven tests are distributed as standard Normal asymptotically. This requires the standardization based on the moments of the underlying Brownian motion function. The panel υ -statistic is one-sided test where large positive values reject the null of no cointegration. The remaining statistics diverge to negative infinitely, which means that large negative values reject the null.
The critical values are tabulated in Pedroni (1999).
The statistics above are based on estimators that simply average the individually estimated coefficients for each member, and each of these tests is able to accommodate individual specific short-run dynamics, individual specific fixed effects and deterministic trends, as well as individual specific slope coefficients (Pedroni, 2004). The number of observations available is greatly increased in a panel framework and this can substantially increase the power of the cointegration tests (Rapach andWohar, 2004).
It is easy to form a conclusion if all seven tests reject the null of no cointegration.
It is, unfortunately, not always the case that all of them reject the null hypothesis simultaneously. Under this circumstance, therefore, we need to decide which version of the available tests has the greatest power for the panel cointegartion tests. As discussed in Pedroni (2004), in terms of monthly data, with little more than 20 years of data it may be possible to distinguish even the most extreme cases from the null of no cointegration when the data are pooled across members of panels with these
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dimensions. This condition appears to have been met in our case since we have 36 observations each year or 3 observations each month. Furthermore, if the panel is fairly large so that size distortion is less of an issue, the panel υ -statistic tends to have
the best power relative to the other statistics and can be most useful when the alternative is potentially very close to the null. In very small panels, however, if the group-rho statistic rejects the null of no cointegration, we can be relatively confident of the conclusion because it is slightly undersized and empirically the most conservative of the tests. The other statistics tend to lie somewhere in between these two extremes and have minor comparative advantages over different ranges of the sample size. In this study, we choose the panel-υ statistic as the basic panel cointegration test. In other words, even other tests reject the null of no cointegration, we will still accept the null if the panel-υ statistic does so.
Finally, the ADF test statistics are biased toward the non-rejection of a unit root when there are structural breaks in the data (Nelson and Plosser, 1982; Perron, 1989;
Enders, 1995). Thus, testing for the presence of structural breaks is necessary.
Moreover, energy economic reforms have been carried out since the early 1990s.
Therefore, energy reform could likely produce some structural changes of price series.