3.3 Time series analysis of panel data
3.3.3 Cointegration tests
The possible cointegration between inequality and GDP is tested with panel cointegration test developed by Pedroni (2004), which consist of 11 different test statistics.17 To allow for possible cross-sectional dependence present in the panel, cointegration is also tested with a test developed by Banerjee and
16First differenced series are stationary according to all panel unit root tests. GDP and inequality thus seem to be I(1).
17There are 7 different test statistics, but Eviews 6 gives also the results of weighted test statistics on the first four tests. Tests statistics include the panel versions of P P and ADF tests, a form of the average of the Phillips and Ouliaris (1990) test statistics (ρ), and panel variance ratio statistics (v).
Carrion-i-Silvestre (2006). A more detailed discussion about these tests is provided in the appendix.
Pedroni’s panel cointegration test
The model for testing for cointegration between inequality and GDP is:
log(GDPit) = αi+ δit+ γilog(inequalityit) + βilog(investmentsit) + ϵit, (3.1) where the changes in GDP are explained by the changes in inequality and on the level of investments, and (1, −γi) is the individual cointegration vector between inequality and GDP. Results of Pedroni’s panel cointegration tests on equation (3.1) are presented in table 3.3.18
According to all of the 11 test statistics presented in table 3.3, the series of inequality and GDP are cointegrated at the 5% level. The test is also conducted using only the inequality as an explanatory variable for GDP.
In this case, 9 of the 11 test statistics find the GDP and inequality to be cointegrated.19
If the test is conducted using only investments as an explanatory variable, all of the 11 test statistics find the investments and GDP to be cointegrated.
This indicates that there might be cross-sectional cointegration relations in the panel of investments, which may have affected on the results of panel unit root tests. Cross-unit cointegration can bias the results of panel unit root tests towards type I error, i.e., that hypothesis of unit root is rejected far too often (Banerjee et al. 2005; Breitung and Pesaran 2008). If series are cross-sectionally cointegrated, the common trends present in the data may be identified as common factors in unit root tests that model the cross-sectional correlation through common factors, like the Pesaran’s test, and removed from the analysis (Breitung and Pesaran 2008). In this case, if the remaining idiosyncratic component is stationary, the panel unit root test has a ten-dency to present the time series as stationary when panel units are actually nonstationary. So, although all panel unit root tests found the investments
18The test was conducted with Eviews 6.
19If the original data of 60 countries is used, 8 of the 11 tests find the GDP and inequality to be cointegrated.
Table 3.3: Pedroni’s panel cointegration test statistics for log(GDP) and log(inequality)
Within-dimension
statistic prob. weight. statistic prob.
panel v-statistic 49.309 <.0001 44.793 <.0001 panel ρ-statistic 7.152 <.0001 7.329 <.0001 panel PP-statistic 2.888 0.0062 3.494 0.0009 panel ADF-statistic 2.489 0.0180 3.039 0.0039 Between-dimension
statistic prob.
group ρ-statistic 9.417 <.0001 group PP-statistic 4.555 <.0001 group ADF-statistic 2.313 0.0275
countries 53
observations 1961
The null hypothesis is that there is no cointegration between variables. Within-dimension tests presuppose common AR coefficients among cross sections. Between-dimension tests presuppose individual AR coefficients. Lag lengths were determined with Schwarz infor-mation criterion. Spectral estiinfor-mation was done with Bartlett method and bandwidth was selected with Newey-West method.
to be I(0), the possibility that investments is actually I(1) process that is cointegrated with GDP has to be taken into account in estimation.20 Banerjee & Carrion-i-Silvestre’s cointegration test
As with panel unit root tests, the presence of cross-sectional dependency may have affected the results of cointegration tests. There may also be structural breaks in the relation between inequality and GDP. To account for possi-ble cross-sectional dependence and structural breaks in the relation between inequality and GDP, cointegration is also tested with the panel cointegra-tion test developed by Banerjee and Carrion-i-Silvestre (2006). Banerjee and Carrion-i-Silvestre’s test allows for cross-sectional dependence by introducing common factors in the estimated model.
Table 3.4 reports the results of Banerjee and Carrion-i-Silvestre’s panel cointegration test between inequality and GDP using the dataset of 38 coun-tries with 25 yearly observations. The test allows for level and cointegration vector shifs.21
According to the basic model allowing just time trend in the tested series, inequality and GDP would be cointegrated. If level and slope trend shift are allowed, only ρ test finds the variables to be cointegrated at the 5% level. If both time trend and cointegration vector shifts are allowed, ρ test finds the variables to be cointegrated at 0.01% level and the t test finds the variables to be cointegrated at the 10%level.
Thus, inequality and GDP seem to be cointegrated even when possible structural breaks in the relation and the possible cross-sectional correlation present in the panel are taken into account. When cointegration relation-ship includes structural breaks, cointegration tests tend to be biased towards accepting the null hypothesis of no cointegration, whereas cross-sectional correlation tends to bias the results towards rejecting the null (Banerjee and
20The cointegration between log(GDP) and log(inequality), and log(GDP) and log(investments) were also tested with Johansen’s combined Fisher panel cointegration test developed by Maddala and Wu (1999). According to it, both GDP and inequality and GDP and investments are cointegrated of order one. Detailed results are available upon request.
21Estimation done with Gauss. We are grateful to Carrion-i-Silvestre for providing the program code.
Table 3.4: Banerjee & Carrion-i-Silvestre’s cointegration test for log(GDP) and log(inequality)
Pedroni model with a time trend
statistic p-value
ZˆtN T(ˆλ) -6.434 <.0001
ZρˆN T(ˆλ) -7.122 <.0001
Model with level shift
statistic p-value
ZˆtN T(ˆλ) -1.256 0.1046
ZρˆN T(ˆλ) -3.913 <.0001
Model with coint. vector shift
statistic p-value
ZˆtN T(ˆλ) -1.619 0.0527
ZρˆN T(ˆλ) -5.992 <.0001
countries 38
observations 950
Model with level shift includes time trend and a level and slope trend shift. Model with a cointegrating vector shift includes time trend and cointegration vector shifts.
Carrion-i-Silvestre 2006; Banerjee et al. 2004). Results presented in table G.2 imply that cross-sectional correlation and/or structural breaks in the relation between inequality and GDP have not biased the result of Pedroni’s panel cointegration tests presented in table 3.3.