Testing for Granger causality in heterogeneous panels with cross-sectional dependence

It is well documented that panel data methods offer more powerful unit root and co-integration tests. The literature indicates the importance of considering cross-sectional heterogeneity and cross-sectional dependence within a panel framework. In parallel to panel unit root and co-integration methods,¹ testing for panel Granger causality has attracted interest and different approaches have recently been developed.

One attempt to test Granger causality in panel data is based on estimating a panel vector autoregressive or vector error correction model by means of the generalized method of moments (GMM)² (see Holtz-Eakin et al. 1988), which does not consider neither heterogeneity nor cross-sectional dependence. The panel Granger causality method proposed by Kónya (2006) accounts for cross-sectional dependence and heterogeneity by employing the seemingly unrelated regressions (SUR) of Zellner (1962). As it is well known, the SUR estimator is only feasible when time dimension (T) is reasonably larger than cross-section dimension (N). The panel Granger causality methods proposed by Dumitrescu and Hurlin (2012) and Emirmahmutoglu and Kose (2011) focus on considering cross-sectional heterogeneity by extending Granger (1969) and Toda and Yamamoto (1995) approaches into the panel framework, which work best when N is smaller than T. In a more recent study, Juodis et al. (2021) propose a novel approach to testing for Granger causality in heterogeneous panels, which can be used in large N cases and works best for cases where N is same to or larger than T.

Early contributions in the literature on panel unit root and co-integration tests benefit from bootstrapping to account for cross-sectional dependence.³ Emirmahmutoglu and Kose (2011) and Dumitrescu and Hurlin (2012) similarly employ bootstrap distributions for the panel statistics. As discussed in Hadri et al. (2015), bootstrapping accommodates cross-sectional dependence of general form and does not provide information on the structure of dependency. Factor models have gained their importance to account for cross-correlations in panel data during the last decades.⁴ The intuition behind using factor models is that the global shocks or common factors can affect each cross section with different weights and intensities, depending on their intrinsic properties (Karabiyik et al. 2019).

This paper focuses on testing for Granger causality in heterogeneous panels by accounting for cross-sectional dependence. We extend Emirmahmutoglu and Kose (2011) approach within the context of factor models. Specifically, we define a panel VAR model with unobserved common factors and apply the PANIC approach of Bai and Ng (2004) to consistently estimate common factors for obtaining the de-factored data. We then estimate the lag augmented (LA)-VAR model for each cross section to obtain the individual Wald statistics and construct the panel statistics based on the meta-analytic approach by combining the p-values of individual Wald statistics.

To examine the small sample performance of proposed tests, we consider different data-generating processes with different unit root and co-integration properties for cross-sectional units. The Monte Carlo simulations indicate that (i) the tests show mild size distortions when N is not smaller than T, which disappear as T grows and N is fixed; (ii) they have good empirical power even in small samples; and (iii) our approach suits the dependent panels where cross-sections may have different unit root or co-integration properties. The simulations also reveal that the proposed tests show better size and power properties than the bootstrap tests of Emirmahmutoglu and Kose (2011) and Dumitrescu and Hurlin (2012) as T grows.

We finally re-investigate Granger causality between export and economic growth for OECD countries within a comprehensive framework by conducting existing and new testing procedures. While the panel results support two-way Granger causality, the country-specific results indicate the importance of accounting for cross-sectional dependence. The approach proposed in this study also unveils less evidence on Granger causality between export and economic growth in in the majority of the European Union (EU) countries.

The rest of the paper is as follows. The next section describes the model and testing procedures. Section 3 is devoted to small sample analysis. Section 4 illustrates an empirical application of the proposed approach compared to the existing methods. The concluding remarks are finally presented in Sect. 5.

We consider panel VAR model with m-dimensional series \({Z}_{it}^{cd}={\left({Z}_{1,it}^{cd},\dots ,{Z}_{m,it}^{cd}\right)}^{\prime}, i=1,\ldots ,N\) and \(t=1,\dots ,T\), given by⁵where \({\mu }_{i}\) is unknown m-dimensional fixed effects, \({F}_{t}=({F}_{1t}, \dots ,{F}_{rt})^{\prime}\) is r-dimensional vector of unobservable common factors that r is same for all i, \({\Lambda }_{i}\) are \(\left(r\times m\right)\) matrix of factor loadings, \({Z}_{it}\) are the idiosyncratic components, \({A}_{i1},\dots ,{A}_{i,{p}_{i}}\) are \(m\times m\) matrices of coefficients, and \({p}_{i}={k}_{i}+{d}_{i}\) where \({k}_{i}\) is the lag order of VAR model and \({d}_{i}\) is the maximum integration order of \({Z}_{it}\).

To examine size and power performance of \(P\) and \({P}_{m}\) statistics, we consider different data-generating processes (DGPs) with different unit root and co-integration properties. The general form of DGP is given bywith \(\phi =0.5\), \({u}_{t}\sim iid N(\mathrm{0,1})\), \({\varepsilon }_{it}\sim iid N(0,{\sigma }_{\varepsilon i}^{2})\), and \({\Lambda }_{i}\sim U(\mathrm{0,0.5})\) for all \(i\) where \(U\) denotes the uniform distribution. To allow heterogeneous variances across cross sections, \({\sigma }_{\varepsilon i}^{2}\) are drawn from \(U(\mathrm{0.5,1.5})\) in line with Dumitrescu and Hurlin (2012) and Emirmahmutoglu and Kose (2011). Since our procedure is invariant to \({\mu }_{i}\), we set \({\mu }_{i}=0\) without any loss of generality.¹³ For size analysis, \({a}_{12,i}=0\) for all \(i\) under the null hypothesis. For power analysis, we follow Juodis et al. (2021) and Juodis (2018) and define \({a}_{12,i}=0.05+{\xi }_{i}\) with \({\xi }_{i} \sim U(-0.1, 0.1)\). Following Arsova and Örsal (2018), the number of factors and the lag order are assumed to be known and set to their true values.¹⁴ The data for \(T\in \left\{50, 100, 200, 300\right\}\) and \(N\in \left\{10, 15, 25, \mathrm{50,100,200}\right\}\) are generated with T + 50 observations, where the first 50 observations are disregarded to reduce the initial value effect. Monte Carlo simulations are carried out with 5,000 replications at the 5% level.

We concentrate on the performance of \(P\) and \({P}_{m}\) in the presence of various types of unit root and co-integration in the idiosyncratic components \({Z}_{it}\). In that respect, we consider five distinct experiments. We define a full unit root model with \({a}_{11,i}={a}_{22,i}=1 \forall i\) in experiment 1. We have a co-integrated model with \({a}_{11,i}=0.9\) and \({a}_{22,i}=1 \forall i\) in experiment 2; and \({a}_{11,i}=1\) and \({a}_{22,i}=0.9 \forall i\) in experiment 3. Our interest in experiment 4 is centered on a more practical case with different unit root and co-integration properties for different cross sections, defined by \({a}_{11,i}={a}_{22,i}=1\) for \(i\le N/3\); \({a}_{11,i}=0.9 \mathrm\;{\text{and} }\;{a}_{22,i}=1\) for \(N/3<i\le 2N/3\); and \({a}_{11,i}=1\) and \({a}_{22,i}=0.9\) for \(2N/3<i\le N\). We finally consider the stationary model with \({a}_{11,i}={a}_{22,i}=0.9 \forall i\) in experiment 5.

We are also interested in comparing the small sample properties of \(P\) and \({P}_{m}\) statistics with those of the bootstrap test of Dumitrescu and Hurlin (2012) (hereafter DH test) and Emirmahmutoglu and Kose (2011) (hereafter EK test). The bootstrap distributions of DH and EK tests are derived from 1,000 replications, and the bootstrap critical value is obtained at the 5% level. Since DH test requires stationary variables, we employ the first-differenced data for the DGP having a unit root process.

Table 1 presents the size and the size-adjusted power for experiment 1. Note that PC and CA imply \(P\) and \({P}_{m}\) tests based on the PANIC and PANICCA approaches. The \(P\) statistic tends to have size distortions as T is fixed and N grows, and shows better size properties for fixed N with growing T. The \({P}_{m}\) statistic appears to have similar size properties to the \(P\) statistic, with the advantage of having empirical size close to nominal size as T and N grow together. These results are consistent with the conjecture (outlined above) implying that the \({P}_{m}\) statistic is more suitable than the \(P\) statistic for the large N panels. In contrast to the \(P\) and \({P}_{m}\) statistics, the DH and EK tests suffer from size distortions, especially as T grows. The power of \(P\) and \({P}_{m}\) statistics increases rapidly as N and T grow, which is presumably a reflection of the consistency of the tests. These tests dominate the DH and EK tests, especially in panels where T is large. Finally, while using the PANIC or PANICCA method for de-factorization does not lead to important differences for size properties, the PANIC approach leads to better power properties. The empirical size and the size-adjusted power of the tests for experiments 2–5, reported in Tables 2, 3, 4 and 5, are similar to those for experiment 1.

To sum up, although the \(P\) and \({P}_{m}\) statistics show mild size distortions for small samples, they have good size and high empirical power as T increases.¹⁵ Dumitrescu and Hurlin (2012) intuitively explain the source of size distortion in panel data. Under the null hypothesis, the individual Wald statistics take large values for some cross-sections. For a given value of N, these large values are not compensated by the realizations obtained for other cross sections since the latter only range from zero to the chi-squared critical value. As a result, the panel statistic tends to be larger than the corresponding critical value. As N increases, the probability of obtaining large values for some cross sections increases, and the panel test tends to over-reject the null hypothesis for small T values. In that respect, the solution of Juodis et al. (2021) based on the Split Panel Jackknife procedure leads the Wald statistic to have good size and power properties even with a moderate T dimension in large N panels.

Another insightful consequence of our simulations is that the DH and EK bootstrap-based tests tend to reject the null hypothesis as T increases.¹⁶ Using the PANIC and PANICCA methods seems more convenient than using the bootstrap method in modeling cross-sectional dependence for testing Granger causality in heterogeneous panels with a factor structure.¹⁷

There is a huge and growing empirical literature on causality¹⁸ between export and economic growth. The literature mainly focuses on the export-led growth (ELG) and growth-led export (GLE) hypotheses. The ELG hypothesis implies that export activity leads to economic growth. The GLE hypothesis postulates a reverse direction and implies causality from economic growth to export. On the other hand, if there is bidirectional causality between export and economic growth, this supports the existence of feedback relation.

The ELG and GLE hypotheses are examined by a large number of empirical studies. Giles and Williams (2000) review more than 150 empirical papers published until the late 1990s. Most of these papers use time series techniques and do not find any consistent result for the validity of the ELG and GLE hypotheses. After the 2000s, the empirical literature focuses on using panel causality methods to benefit from the high-power property of such tests in small samples. Initial papers assume homogeneity and independency across countries (Hsiao and Hsiao 2006; Won and Hsiao 2008; Çetintaş and Barişik 2009; Narayan and Smyth 2009; Nasreen and Anwar 2014). Some recent studies allow heterogeneity but do not consider cross-sectional dependence across countries (Sanjuán-López and Dawson 2010; Pradhan and Bagchi 2012; Dawson and Sanjuán-López 2013). Some other recent papers, among others Tekin (2012) and Chang et al. (2013), find out that allowing heterogeneity and cross-sectional dependence (with bootstrapping) is important to examine the direction of causality between export and economic growth. Nonetheless, to the best of our knowledge, no study accounts for cross-sectional dependence based on factor models.

This gap in the literature motivates us to question whether using factor models makes sense for causality between export and growth for OECD countries, in which international trade plays a crucial role in economic structure.¹⁹ For a comparison purpose, we also conduct the existing approaches proposed by Kónya (2006), Dumitrescu and Hurlin (2012), and Emirmahmutoglu and Kose (2011).²⁰ Our data covers the real GDP and export for 20 OECD countries from 1996:Q1 to 2019:Q4.²¹ The real GDP is constructed using the GDP deflator, and the real export is obtained using the consumer price index. The data are retrieved from the International Financial Statistics Database. The natural logarithm of the variables is employed in the estimations.

With respect to the panel causality literature focusing on OECD countries, Kónya (2006) examines causality between real export and real gross domestic product (GDP) in 24 OECD countries for the period from 1960 to 1997. By conducting the panel causality test on the SUR, he finds one-way causality from export to GDP in Belgium, Denmark, Iceland, Ireland, Italy, New Zealand, Spain, and Sweden; one-way causality from GDP to export in Austria, France, Greece, Japan, Mexico, Norway, and Portugal; two-way causality between exports and growth in Canada, Finland, and the Netherlands; no causality in the case of Australia, Luxembourg, S. Korea, Switzerland, the UK, and the USA. Emirmahmutoglu and Kose (2011) use the panel causality test based on the LA-VAR approach to investigate causality between export and growth in 20 OECD countries from 1987 to 2006. Different from Kónya (2006), they show one-way causality from export to growth in Japan, from growth to export in Australia, Germany, Norway, S. Korea, and the USA, and two-way causality in Türkiye. They moreover support the validity of the GLE hypothesis for OECD countries. Our study differs from these studies by accounting for cross-sectional dependence and demonstrates the importance of factor modeling framework for country-specific results.

In the LA-VAR approach, we need to determine the maximum integration degree (d) of the variables for each cross section. To test for unit root in panel data, one important issue is to consider cross-sectional dependence. As a preliminary analysis, we first test for cross-sectional dependence by using LM (Lagrange Multiplier) test proposed by Breusch and Pagan (1980), and \(C{D}_{LM}\) and \(CD\) tests developed by Pesaran (2021). The results in Table 6 indicate that the null hypothesis of cross-sectional independence is strongly rejected for both export and GDP, supporting the evidence on cross-sectional dependence. We benefit from the PANIC approach of Bai and Ng (2004) and the PANICCA approach of Reese and Westerlund (2016) to investigate the unit root properties of the series. The results from the unit root analysis,²² reported in Table 7 indicate that the maximum order of integration is determined as one for all countries.

We estimate common factors by means of the PANIC and the PANICCA approaches for de-factoring the data. To determine the number of common factors in the PANIC approach, we benefit from three different information criteria, \(IC\) panel information criterion of Bai and Ng (2002), and ER (eigenvalue ratio) and GR (growth ratio) criteria of Ahn and Horenstein (2013). These criteria are based on PCA that often requires standardizing panel data before estimating unobserved common factors. Greenaway-McGrevy et al. (2012) investigate the importance of standardization in eigenvalue-based decompositions by particularly focusing on PCA. They find that the validity of standardization is dependent on the main source of heteroscedasticity. If heteroscedasticity arises from idiosyncratic errors, standardization is required to consistently estimate PCA factors and factor numbers. If heteroscedasticity comes from factor loadings, standardization may lead to inconsistent estimation of PCA factors and over-estimating factor numbers. Since it is difficult to know the source of heteroscedasticity in practice, Greenaway-McGrevy et al. (2012) suggest using a minimum rule, whereby the factor number is estimated as \({\widehat{r}}_{min}={\text{min}}[{\widehat{r}}_{std},{\widehat{r}}_{no-std}]\) where \({\widehat{r}}_{std}\) and \({\widehat{r}}_{no-std}\), respectively, denote the estimated number of factors with and without standardization.

Table 8 reports the results from the factor number estimations from the panel comprising export and GDP for all units. The number of factors is estimated by setting the maximum number \({{\text{r}}}_{{\text{max}}}=3\). Since the variables have the unit process, we conduct PCA on the first-differenced data to avoid spurious results arising from non-stationarity. While the IC, ER, and GR criteria select different factor numbers (3, 1, and 2) for the non-standardized data, they chose one common factor for the standardized data. Accordingly, the minimum rule of Greenaway-McGrevy et al. (2012) determines one factor based on all three criteria.²³ Figure 1 plots the estimated factor from both PCA and CA. At the first glance, the factor estimates from both approaches almost coincide, and there is a downfall midway through the sample, which can be attributed to the Global Financial Crisis between mid 2007 and early 2009. Except for the downfall, both PCA and CA estimates look like to be circling on a constant mean. Hence, the estimated factor from either approach captures fluctuations in the data.

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The results from the panel LA-VAR causality test with the PANIC and PANICCA approaches are presented in Tables 9 and 10, respectively.²⁴ The panel statistics, P and P_m, indicate that the null hypothesis of Granger non-causality is rejected, supporting the existence of both the ELG and GLE hypotheses in OECD countries. The rejection of the joint null hypothesis for the panel does not mean that all countries have causality since the alternative hypothesis allows some cross sections not to have non-causality.

We proceed with the cross-section specific results to examine the countries having causality and non-causality. We should, at this point, note that imposing the null hypothesis for 20 countries means 40 tests for causality, which naturally raises a concern about mass significance. Using a p-value correction method may be of critical importance in the case of relatively large numbers of tests,²⁵ and we adjust p-values using the Holm’s (1979) procedure. Our discussion hereafter is based on the adjusted p-value at 5%, that an adjusted p-value less than 0.05 rejects the null hypothesis of non-causality. The country-specific results from the PANIC and PANICCA approaches support the ELG hypothesis for USA; the GLE hypothesis for Australia, France, and Norway; and the neutrality for the majority of countries (Austria, Canada, Denmark, Finland, Germany, Italy, Japan, Mexico, Netherlands, New Zealand, Portugal, S. Korea, Spain, Sweden, Türkiye, and UK).

The results from the panel SUR causality test of Kónya (2006)²⁶ are given in Table 11. In this approach, testing for the cross-sectional dependence is crucial for selecting the appropriate estimator since the SUR estimator is more efficient than the OLS only if there is cross-sectional dependence in the system. To investigate the existence of cross-sectional dependence, we carry out the cross-sectional dependence tests on the residuals from the individual OLS regressions. The cross-sectional dependence tests reject the null of cross-sectional independence, implying that the SUR estimator is appropriate rather than country-by-country OLS estimations. The results support the ELG hypothesis for Netherlands; the GLE hypothesis for Mexico and UK; and the neutrality for Australia, Austria, Canada, Denmark, Finland, France, Germany, Italy, Japan, New Zealand, Norway, Portugal, S. Korea, Spain, Sweden, Türkiye and USA.

The results from the panel VAR causality test of Dumitrescu and Hurlin (2012)²⁷ are reported in Table 12. Testing for the cross-sectional dependence is also required to decide whether one needs to use asymptotic or bootstrap distribution since the asymptotic distribution is only valid under cross-sectional independence. The cross-sectional dependence tests indicate that the null of cross-sectional independence is strongly rejected, implying that we need to use bootstrap distribution. The results show that the panel statistic is larger than the bootstrap critical values; hence, the null hypothesis of non-causality for the panel is rejected. This finding supports two-way causality between export and GDP, implying the existence of feedback relation. The country-specific results provide evidence on the validity of the ELG hypothesis for Austria; the GLE hypothesis for Denmark, Finland, Italy, and Sweden; the feedback relation in S. Korea; and the neutrality for Australia, Canada, France, Germany, Japan, Mexico, Netherlands, New Zealand, Norway, Portugal, Spain, Türkiye, UK, and USA.

The results from the panel LA-VAR causality test of Emirmahmutoglu and Kose (2011)²⁸ are reported in Table 13. We test the cross-sectional dependence since the limiting distribution of the panel statistic is no longer valid in the existence of cross-sectional dependence. The results from the cross-sectional dependence tests show that the null of cross-sectional independence is strongly rejected, implying that we need to use the bootstrap critical values. The panel statistic is greater than the bootstrap critical values and rejects the null hypothesis of non-causality for the panel, supporting the existence of both the ELG and GLE hypotheses. The country-specific results imply the existence of the ELG hypothesis for S. Korea; the GLE hypothesis for Mexico and Sweden; the feedback relation for Finland; and the neutrality for Australia, Austria, Canada, Denmark, France, Germany, Italy, Japan, Netherlands, New Zealand, Norway, Portugal, Spain, Türkiye, UK, and USA.

To succinctly compare the findings from all the panel causality tests carried out so far, we summarize the results in Table 14. The panel results support the validity of both the ELG and GLE hypotheses, implying feedback causality between export and economic growth in OECD countries. This finding is consistent with the view in Bhagwati (1988) that higher export leads to more income, which triggers more trade. The country-specific results signal the importance of using various methods to better understand the nature of causality. The different approaches lead to different country-specific results as it is expected.

The results in Table 14 overall throw light on the importance of factor modeling to account for cross-sectional dependence. Using the LA-VAR approach with factor model framework leads the country-specific results to differ substantially. Specifically, the LA-VAR with PANIC and PANICCA approaches compared to the LA-VAR leads to change from causality to non-causality for Finland, Mexico, S. Korea, and Sweden; and change from non-causality to causality for Australia, France, Norway, and USA. Furthermore, the causality approach based on PANIC and PANICCA procedures indicates that the EU countries (with two exceptions, France and Norway) do not have any causality between export and economic growth while other methods provide mixed results for the direction of causality. This finding can plausibly be explained by some arguments. Hagemejer and Mućk (2019) claim that the continued investment needs of a growing economy are likely to drag down the overall contribution of net exports to growth in a highly export-oriented developing country with fast export growth. The other possibility is about trade barriers. Some studies argue that a rise in trade barriers would have the negative effects on growth (see among others, Head and Mayer 2016; Dhingra et al. 2017). Especially, trade policy regulations in USA and the Brexit negotiations have affected the trade in the EU (Altomonte et al. 2018). Finally, such a finding can broadly be associated with competitiveness. Turégano and Marschinski (2020) show that the EU´s global share diminished in last two decades due to an overall structural shift in the composition of final demand away from manufacturing goods. They also find that participation losses significantly declined the EU´s share in global manufacturing value chains.

We examine the small sample properties of the proposed tests under different data-generating processes with different unit root and co-integration properties for cross-sectional units. The simulations show that although the test statistics show mild size distortions for small samples, they have good size properties with an increasing time dimension. Moreover, the test statistics have high empirical power even in small samples.

The empirical application re-investigates Granger causality between export and economic growth in OECD countries. The findings uncover a crucial role of considering common factors for country-specific inferences and reveal less evidence on causality in the majority of the EU countries.