Panel Unit Root Tests

The term Panel Data in this study refers to the pooling of time series data for all of the countries in the study. For example, the exports variable consists of Australia’s exports to China over thirty-eight years from 1963 to year 2000 and Australia’s exports to Hong Kong

for the same period of time and so on until all ten countries in this study are included. Each variable in the panel data set consist of 380 observations (10 cross-section countries times 38 years). Each variable in the panel data set inherits the characteristics of all individual

series that made up the variable. As pointed out by Levin et al (2002), if the time dimension

of the panel is very large, the unit root testing procedure discussed in Section 7.4.3.1 will be sufficient and powerful to test the individual data series in the panel. If the time dimension is very small compared to the cross-section dimension, the non-stationarity problem has little impact on the regression results, and the usual panel data regression procedure will be appropriate. However, with a moderate size panel (between 20 and 250 cross-sections and 25 to 250 time periods per cross-section), the traditional unit root tests may not be sufficient or powerful.

Panel unit root tests are mostly similar to the unit root tests on single time series in Equation (7.1), but modified as:

1 1 1

it i it it i it

Y =ρY₋ +X ′δ +u − ≤ ≤ρ (7.10)

where i = 1, 2, 3,…N cross-section series. Within the range of − ≤ ≤1 ρ 1, if ρ<1, the

series of Y_i is said to be stationary, if ρ=1, Y_i is said to be a unit root nonstationary series.

A number of panel data nonstationary tests have been developed in recent years. Those tests can be classified into two broad categories, (i) those cross-section data that can be assumed independent, and (ii) those cross-section data that cannot be assumed independent. For the category of cross-sectional independent panel unit root tests, the popular method of tests are Levin, Lin and Chu test (2002), the Im, Pesaran and Shin test (2003), the Breitung test (2000) and the Hadri test (2000). For the category of cross-section dependent panel unit root tests, the Pesaran’s (2004) cross-section dependence (CD) test, Moon and Perron’s (2004) dynamic factor model test to capture the cross-section correlation, Bai and Ng’s (2004) ADF test, Choi’s (2002) error component model, and Pesaran’s (2003) cross-

sectional augmented Dickey-Fuller (CADF) test are available other tests.64 _{The panel unit}

root test techniques assuming cross-section independence are better established.

In this study, we assume that the cross-sections of data in our sample are independent. The variable GDP, per capita GDP and price level are independent between countries since different countries have their own business cycles and stages of economic development. Strictly speaking, if in a three-country trade model, Country A is the exporter country and Countries B and C are importers, then exports from Country A to Country B are not independent from the exports from Country A to Country C. They are private goods that have the characteristics of rivalry and mutual exclusivity. In statistics terminology, if two events, A and B, are mutually exclusive events, then Event A and Event B are not independent. In fact, Event A and Event B are inter-dependent events because if one event happens, the other event must not happen, to be able to term “mutual exclusive” events. Bringing this concept of “mutual exclusive” in our exports example, for a particular item of goods, which is exported to one country, it will not be able to be exported to another country at the same time. One item of the goods exported to one country “excludes” the other country to import the same goods. In addition, if Country A’s export capacity is fully utilised, increasing exports to Country B will reduce exports to Country C to be able to free the goods to export to Country B. In this situation, exports to Country B and exports to Country C are cross-section dependent. If cross-section dependence exists, the cross-section dependent panel unit root tests of Moon and Perron’s (2004), Bai and Ng’s (2004), Choi’s (2002) and Pesaran’s (2003) are more appropriate.

However, in this study, our sample of ten trade partners is a relatively small in a large population of all Australia’s trade partners. Increasing exports to one of the Asian trade partner has very little impact on reducing exports to all other trade partners. Hence the effect of interdependence between the trade partners can be assumed quite small. Thus, we use the cross-section independent panel unit root tests of Levin, Lin and Chu (2002), Im, Pesaran and Shin (2003), Breitung (2000) and Hadri (2000).

The following subsections focus on the panel data unit root tests, which assume cross- section independence of the panel data. This category can be further classified into two subcategories: a more restrictive assumption of a common unit roots for all cross-sections or less restrictive assumption of individual unit roots for each cross-section units.

The panel unit root test will be conducted to test variables for the export model and the import model. Some variables are used in the Export model only, they are: Australia’s exports, Australia’s GDP (treatment discussed in Section 7.3) and foreign country’s GDP (treatment discussed in Section 7.3). Some variables are only used in the Import model, they are: Australia’s imports, Australia’s GDP (different values from the Australia’s GDP variable in the exports model, see Section 7.3) and foreign country’s GDP (also different values from the foreign country’s GDP in the exports model, see Section 7.3). Some variables are used for both export model and the import model, they are: per capita GDP for foreign countries, price level for foreign countries, foreign countries Openness and Australia’s migrant intake level. Some variables do not need to have panel unit root test because they are invariant across cross-sections, e.g. Australia’s per capita GDP, Australia’s price level and Australia’s openness variable. Unit root tests for those variables have been conducted as individual time series unit root tests in Section 7.4.3.1.

Levin, Lin and Chu (LLC) Test

Levin, Lin and Chu (2002) assume a common unit root across the cross-sections. LLC started with a basic ADF specification for panel data:

1 1 i p it it iL it L it it L y αy ₋ β y ₋ X δ u = ′ Δ = +

∑

Δ + + (7.11)

where 1α ρ= − , 1, 2,3,...,i= Ncross-section units and L is the number of lag terms. p_i, the

optimal lag order terms are allowed to vary across the cross-sections. If the null of α =0 is

rejected in favour of the alternative of α <0, then the panel data has no unit root. Since p_i

is allowed to vary for different cross-sections, the first step in the LLC test is to select the

optimal p_i for each cross-section by using one ADF regression for each cross-section.

Based on the optimal p_i and the appropriate exogenous (deterministic) variables of X_it the

orthogonalised residuals ( ˆe_it and vˆit−1) are generated by two auxiliary regressions (regress

it y

Δ on Δy_{it L−} and X_it, and yit−1 on Δyit L− andXit). The residuals are then standardised as

it

e and v_it−1by the regression standard error from Equation (7.11). The second step is to find

ratio is then used to adjust the t-statistics in step 3. The final step is to obtain the panel test statistics by pooling all cross-section and time series observations to estimate:

1 it it it

e =αv ₋ +ε (7.12)

With the conventional hypothesis t test for α =0, if α =0 cannot be rejected, the panel

variable has a common unit root. If the null is rejected, then the test concludes that each of

the individual series is stationary.65

The LLC test has its limitations, as acknowledged in Levin et al (2002). They are as the

restrictive alternative hypothesis of identical first order autoregressive coefficient and the test’s reliance crucially upon the “no-cross-section-dependence” assumptions.

To implement the LLC test, the specification of the number of lags used in the individual ADF regression is required. The kernel choice and the exogenous variables are also required to be determined. We performed the LLC test on all pooled variables for our export and import models allowing the Eviews to select the maximum lag length using Schwarz information criterion, and the Bartlett kernel method to obtain the average standard deviation ratio from cross-sections. We tested three exogenous variable models with intercept but no trend, intercept and trend, and no intercept no trend. The three LLC test results are presented in Table 7.4.

For the intercepts but no trend model, for eight out ten variables, the test rejects the null hypothesis of unit root. Only the variables of foreign price level and foreign openness are found to be with a unit root. When the intercepts and linear trends models are applied, the exports and imports variables are found to have no unit root. When the no intercept no trend model is used, all variables are found to have a unit root. When a unit root is found in the panel data, we should be cautious for the non-stationarity properties of the panel data.

Table 7.4 Levin, Lin and Chu Panel Unit Root Tests

Variables Intercept, no trend Intercept and trend No intercept, no trend

t-statistic Probability t-statistic Probability t-statistic Probability For Export model

Australia’s Exports -5.5230 0.0000 -7.4943 0.0000 4.9934 1.0000

Australia’s GDP -4.5319 0.0000 -0.5978 0.2750 3.9378 1.0000

Foreign country’s GDP -3.8868 0.0001 -0.9872 0.1618 8.5281 1.0000

For Import Model

Australia’s Imports -3.5946 0.0002 -1.7467 0.0403 5.6752 1.0000

Australia’s GDP -4.5409 0.0000 -0.6073 0.2718 3.9264 1.0000

Foreign country’s GDP -3.4283 0.0003 -0.8878 0.1873 7.1651 1.0000

For Export and Import

Foreign per capita GDP -2.0245 0.0215 0.1463 0.5582 12.3314 1.0000

Foreign Price level -1.2792 0.1004 -0.9277 0.1768 0.0059 0.5024

Foreign Openness -0.9584 0.1689 -1.6331 0.0512 7.8341 1.0000

Australia’s Immigrant Intake -2.9808 0.0014 -0.0058 0.4977 3.8213 0.9999

The values of Australia’s per capita GDP, Australia’s price level and Australia’s openness variables change over time but do not change in cross-sections. Levin, Lin and Chu panel unit root tests do not apply to these variables.

Im, Pesaran and Shin (IPS) Test

Im, Pesaran and Shin (2003) provide a less restrictive test of panel unit root, allowing the panel data set to have individual unit roots. IPS specifies one ADF regression for each cross-section: 1 1 i p it i it iL it L it it L y α y ₋ β y ₋ X δ u = ′ Δ = +

∑

Δ + + (7.13)

The only difference between Equation (7.11) and Equation (7.13) is the termα_i. The null

hypothesis becomes α_i =0 where i=1, 2,3,...,N cross-sections rather than α =0 in the LLC

specification. The IPS testing procedure is based on the averaging of individual unit root test statistics: 1 1 i N i t t N = α =

∑

(7.14) where the i

t_α is the individual t-statistics for testing the null of α_i =0 for all i in Equation

(7.13) and it generally has an asymptotic standard normal distribution if not all lag order p_i

simulated critical values for given cross-section N and time series length T with intercepts

only or intercepts and linear trends.66

To test the stationarity of the variables in our export and import models, we allowed the

Eviews to choose the maximum lag length p_i using the Schwarz information criteria. We

tested both models of intercepts only, and intercepts and linear trends as exogenous variables for the ADF regression. The test results are provided in Table 7.5. For the intercepts only model, three out of ten variables are found stationary: they are exports and Australia’s GDP in the export model, and Australia’s GDP in the import model. For the intercepts and linear trends model, two variables, exports and foreign price level are found stationary.

Although the LLC test and the IPS test are not directly comparable since the LLC test assumes a common unit root while the IPS test allows individual unit roots in the alternative hypothesis, the test results from the two tests revealed that, with our data set, the IPS test tends to be more conservative in rejecting a unit root.

Table 7.5 Im, Pesaran and Shin Panel Unit Root Tests

Variables Intercept, no trend Intercept and trend

t-statistic Probability t-statistic Probability

For Export model

Australia’s Exports -5.7217 0.0000 -6.7153 0.0000

Australia’s GDP -3.3468 0.0004 -0.5697 0.2845

Foreign country’s GDP -0.1663 0.4340 -0.6404 0.2610

For Import Model

Australia’s Imports -1.4342 0.0758 -1.0039 0.1577

Australia’s GDP -3.4048 0.0003 -0.5297 0.2982

Foreign country’s GDP 0.1599 0.5635 -0.5334 0.2969

For Export and Import

Foreign per capita GDP 1.7953 0.9637 0.9909 0.8391

Foreign Price level -0.8291 0.2035 -3.0981 0.0010

Foreign Openness 2.6030 0.9954 -0.3479 0.3640

Australia’s Immigrant Intake -0.6164 0.2688 0.6553 0.7439

The values of Australia’s per capita GDP, Australia’s price level and Australia’s openness variables change over time but do not change in cross-sections. Im, Pesaran and Shin panel unit root tests do not apply to these variables.

Breitung’s Test

Breitung (2000) test uses the same ADF regression as in LLC test of Equation (7.11), assuming a common unit root. However, the Breitung approach differs from the LLC test in that the Breitung test does not employ a bias adjustment (Baltagi, 2005, p.243) whereas such an adjustment is employed by the LLC test to correct for cross-sections specific variances in the panel data set. By using the bias adjustment, the LLC test suffers the loss of power when individual trends are included in the exogenous variable specification. Breitung uses three steps approach as in the LLC test but it differs from the LLC test in the following two

ways: first, after selecting the appropriate lag length p_i for each cross-section by using one

ADF regression for each cross-section in Step 1 of the LLC test, to obtain ˆe_it and vˆ_it−1,

Breitung regresses Δy_it on Δy_{it L−} to find the residual ˆe_it and regresses y_it−1 on Δy_{it L−} to find

the residual vˆ_it−1. That is, Breitung dropped the deterministic variableX_it. Then the same

standardising procedure as in LLC is applied to ˆe_it and vˆ_it−1 to obtain e_it andv_it−1. Second,

Breitung detrends the residuals of e_it and v_it−1 to obtain *

it

e and *

1 it

v ₋ . Step 3 replaces

Equation (7.12) with the detrended residuals of *

it e and * 1 it v ₋ as: * * * 1 it it it e =αv ₋ +ε (7.15)

The Breitung Test results are presented in Table 7.6. The intercepts and linear trends specification found that four variables are stationary and other two models found that all variables have a unit root.

Comparing the overall unit root test results from the LLC test, the IPS test, and the Breitung test, both the IPS test and the Breitung test found unit root for the same variables. That is, both test methods identify the same variables as nonstationary, although under different exogenous variable specifications and different assumptions about the behaviour of the underlying stationarity. Under the LLC test method, fewer variables are identified as having a unit root.

Table 7.6 Breitung Panel Unit Root Tests

Variables Intercepts, no trend Intercepts and trends No intercept, no trend

t-statistic Probability t-statistic Probability t-statistic Probability For Export model

Australia’s Exports 2.2795 0.9887 -3.2324 0.0006 3.41174 0.9997

Australia’s GDP 1.4125 0.9211 -3.0782 0.0010 4.12754 1.0000

Foreign country’s GDP 1.7935 0.9636 -1.2386 0.1077 2.61171 0.9955

For Import Model

Australia’s Imports 0.4618 0.6779 0.0571 0.5228 1.18163 0.8813

Australia’s GDP 1.4358 0.9245 -3.0215 0.0013 4.09732 1.0000

Foreign country’s GDP 1.7849 0.9629 -1.2951 0.0977 2.53898 0.9944

For Export and Import

Foreign per capita GDP 1.4331 0.9241 -1.1377 0.1276 1.95967 0.9750

Foreign Price level -0.6158 0.2690 -1.8046 0.0356 1.88586 0.9703

Foreign Openness -0.0057 0.4977 0.7090 0.7608 -0.99081 0.1609

Australia’s Immigrant Intake 0.3764 0.6467 0.1779 0.5706 0.98012 0.8365

The values of Australia’s per capita GDP, Australia’s price level and Australia’s openness variables change over time but do not change in cross-sections. Breitung panel unit root tests do not apply to these variables.

Some comments can be made about the design of the hypotheses of those three unit root test methods. For the LLC test, its null hypothesis is a common unit root. That is, all the time series of each cross-section unit converged to one single unit root for the whole panel data set. The test rejects the null hypothesis of unit root if all time-series are not unit root. However, it also rejects the null hypothesis if individual time series are unit root, as long as those individual unit roots do not converge to a single unit root for the whole panel data set. For the IPS test method, it acknowledges individual unit roots. As long as unit roots exist in each individual series, the panel data set is said to have a unit root problem. However, since the testing procedure is based on the averaging of the individual unit root test statistics, it could reject the null hypothesis if some of the individual series in the panel data set are not unit root. The Breitung test also pays attention to a common unit root, but reduces the drawback of losing power of the test when individual linear trend is included in the test regression of the LLC and IPS tests (Baltagi, 2005, p.243). The drawback of loss of power when linear trend is included in the test regression is due to the cross-section bias adjustment procedure employed by the LLC and the IPS tests.

For the above three methods of testing, we also experimented with different information

the test results changed with these alternations, the conclusion of rejecting/not rejecting the null for each variable is not affected.

Hadri Test

The LLC, IPS and Breitung unit root tests assume unit root as the null and no unit root as the alternative. Hadri (2000) reversed the hypothesis by assuming that the panel data have a common stationary process with an alternative of nonstationary process. As long as the stationarity null is rejected, the panel data is assumed having the nonstationary problem. Similar to the KPSS test for individual series unit root test, Hadri developed a Lagrange

Multiplier (LM) z-test on the residual of the individual OLS regression:

it it it

y = X ′δ +u . (7.16)

Same as the KPSS test, the exogenous term of X_it′δ can be a constant only:

In document THE IMPACT OF INTERNATIONAL MIGRATION ON INTERNATIONAL TRADE: AN EMPIRICAL STUDY OF AUSTRALIAN MIGRANT INTAKE FROM ASIAN COUNTRIES (Page 161-174)