4.2 Methodology of Granger Causality
4.2.1 Theory of Granger Non-Causality Test
The Granger causality proposed by Sir Clive W.J. Granger in 1969 is a statistical restriction test to investigate whether a time depended variable is useful in predict-ing another variable. The Granger causality method is a famous practical approach not only in economics, but also in medicine, neuroscience, weather forecasting, military science etc. The idea of Granger causality test is simple: if a variable X with information included in its previous lag values can statistically forecast variable Y while all past information on Y is also present, then variable X Granger cause Y.
Thus, the structure of Granger (non)-causality test builds on two following regres-sion equations:
n is the number of lag lengths chosen to satisfy dynamic structure where the n+1 and the higher lags coefficients are not significant, and the error terms in both equations should be white noise and uncorrelated (Hamilton 1994, Stern 2011). In each equa-tion, if the jointed parameters of β1,m are statistically significant, then e.g. the null hypothesis H0 : β1,1 = β1,2 = ⋯ = β1,m = 0 in the first equation, i.e. variable X does not Granger cause Y, can be rejected. Similarly, in the second equation, if the coefficients of variable Y are significant, then the null hypothesis (H0 : β2,1 = β 2,2= ⋯ = β2,m = 0) of Y does not Granger cause X is rejected. Moreover, if rejection happens in both equa-tions simultaneously, then a bilateral or feedback relaequa-tionship can be defined be-tween variables X and Y.
Granger non-causality test has a few limitations. First, the basic Granger causality formula is used only for linear models. The formulated nonlinear versions, e.g. Ancona et al. (2004), and Chen et al. (2004) - used in neuroscience - are difficult to use with complex statistical
infor-mation. Secondly, in Granger non-causality test it is assumed that the variables are
48
covariance stationary.10 Therefore it is important to make the optimum lag length of VAR system high enough to make sure that variables have not stochastic trends.
Thirdly, the Granger non-causality test is a theoretic because it uses less prior infor-mation (Gujarati, 2004), i.e. it is useful to find the direction of relationship between variables but not for estimating the exact effect coefficients.
Like other econometrics tests there are alternatives for Granger causality test e.g.
Sims causality test proposed by Sims (1972) where the leading values of exogenous variable is added to Granger causality test11. Note that this leads to decrease the degrees of freedom12 (Hamilton, 1994). Hence, in case of health economics where we face to lack of data availability at the time, the degree of freedom is low, the use of Sims non-causality test is not recommended.
Basically, the Granger causality corresponds to idea that when a relationship between two or more time series is statistically significant on some lags and we seek for the direction of effects (causality) between them13. Note that all this means causality only in weak temporal sense, i.e. predictability power. Also, the null hypothesis demands non-causality and its rejection does not say necessarily anything about true causality. Thus, the test name is Granger non-causality test, for evident theoretical and econometrical reasons14. There are several different extended versions of Granger causality test in both time series and panel analysis such as Hsiao (1981), Toda and Yamamoto (1995), Hurlin and Venet (2001), Hurlin (2004a, 2004b), and Dumitrescu and Hurlin (2012).
10 Ding et al. (2000) and Hesse et al. (2003) extended some modified versions of Granger causality ap-proach used in applied physics also with adding non-stationary variable in Granger non-causality test.
See also for the Toda-Yamamoto approach presented in Section 4.2.2.
11 For example, in Sims causality test eq. 1 and 2 is changed to:
Yt= 1,0+ β1,k
12 This also increase the probability of facing singular matrix error when there exists a shortcoming of data availability during time in panel analysis.
13 Routinely, if two variables are co-integrated it means that there is a significant relationship between them in long run, and then there would be at least one unidirectional Granger causal relationship be-tween them.
14When the estimation aims only to find the direction of potential relationships then the exact coeffi-cients values of relationship are not important.
49 4.2.2 Toda-Yamamoto Test
Toda and Yamamoto (1995) formulated a modified version of Granger non-causality procedure based on modified Wald (MWALD) tests. Toda-Yamamoto test removes the possible inference mistakes associated with ordinary version of Granger non-causality test with respect to series non-stationarity and co-integration (Mavrotas and Kelly 2001, Zapata and Rambaldi 1997). Toda and Yamamoto start with a vector autoregression (VAR) model to levels of variables while selecting the optimal series lag length under non-stationarity and co-integration (Wolde-Rufael, 2004, 2005). To explain Toda-Yamamoto test we summarize the following VAR sys-tem as a practical bivariate model:
Yt= 0+ β1,k
The idea is to extend the VAR lag order, k, with maximum order of (co)-integration between X and Y, called dmax. Then dmax selects the optimum lag length of VAR where the error terms of both equations 7 and 8 are white noise. In equation 7, if the null hypothesis of MWALD test for all k and j, γ1,K = γ2,j = 0 can be rejected signifi-cantly, then there is a Granger causal relationship from X to Y. Similarly, if the null hypothesis ϑ1,k = ϑ2,j = 0 in equation 8 can be rejected, then there is a causal relation-ship from Y to X. For more explanation of Toda-Yamamoto test, see Rambaldi and Doran (1996), Rambaldi, 1997), Kuzozumi and Yamamoto (2000), Wolde-Rufael (2004, 2005, 2006), and Amiri and Ventelou (2012).
4.2.3 Hsiao’s Version of Granger Non-Causality Test
Hsiao’s version of Granger non-causality test solves the optimal leg length choice differently as it assumes series stationarity. In Hsiao method the main shortcoming and deficiency of ordinary Granger non-causality tests, the assumption of predetermined lag lengths of test model variables, is removed. By using the final prediction error (FPE) criterion, we are able to determine optimal lag length for each variable separately. As Granger non-causality test is sensitive to the choice of length of involved series, result with Hsiao’s test would be more exact and supportive.
50
The procedure of Hsiao test is following: in the first step the optimum lag length of variable Y in a restricted equation 9 is determined with calculating the sum of squared errors (SSE) for each number of lags from 1 to maximum order of lags P15 and computing FPE for each lag in formula 10. The smallest FPE verifies the opti-mum lag length of variable Y called p*.
% = 0+ α1i
Next step is to calculate FPE for unrestricted equation 11 using formula 12 includ-ing variable X. In equation 11, k* is calculated from the smallest FPE(p*, k). The con-clusion of Granger non-causality test here is that if the smallest FPE of unrestricted equation 11 called FPE(p*, k*) is smaller than the smallest FPE of restricted equation 9 called FPE(p*), then the null hypothesis that X Granger-non-cause Y is rejected.
There is same procedure for testing Granger causality from Y to X.
% = 0+ β1i
As the maximum order of lag length which should be tested in Hsiao method is at least 10, the degree of freedom of regressions in Granger test structure decreases considerably. Therefore, a high number of time observations is needed to avoid miss-specification error in Hsiao’s method. For detailed explanation of Hsiao’s ver-sion of Granger non-causality test, see Cheng and Lai (1997).
4.2.4 Panel Fixed Effect Method
The panel fixed effect model is practically a useful approach for analysing pooled data with short time dimension but large number of cross section units. Fixed effect panel methods are proposed by Dumitrescu and Hurlin (2012), Hurlin and Venet (2001), Hurlin (2004a, b), and Hansen and Rand (2004).
15 Maximum order of lag is depended to matter of analysis and data availability. As Cheng and Lai (1997) state, at least 10 lags should be tested.
51 In a bivariate panel Granger non-causality test model, variable Xit with lags is Granger-cause variable in each individual cross unit if variable Xit is an important in predicting variable Yit, while all other information of Yit is also included with its lags on test model.
Following time-stationary VAR system is introduced for a panel system for i = 1,2,…,N cross sections:
% = 1 + 1. ,%−.
>
.=1
+ 1. ,%−. + 1 %
>
.=1
1 % ~ 0(0, @A21,) (13)
% = 2 + 2. ,%−.
>
.=1
+ 2. ,%−.+ 2 %
>
.=1
2 % ~ 0(0, @A22,) (14)
z is the number of lag length and u’s are normally distributed errors. In standard fixed effect panel model, all αkpi = αjp and all βkpi = βjp, (k = 1, 2), i.e. cross unit slope coefficients are equal at different lag values of p. However, the lack of homogeneity across the coefficients may lead to biased estimates in non-causality testing. Gener-ally, there are two main sources of heterogeneity in cross units included: permanent cross-sectional disparities and heterogeneous slope coefficients of regression. As Hurlin (2004a) state, fallacious regression may be caused by non-homogenous in-tercepts δ1i and δ1i as the first heterogeneity source. The non-homogenous coeffi-cients αkpi and βkpi, (k = 1, 2) at different lags are the second source.
There are many test procedures for analyzing heterogeneity in the panel fixed effect Granger non-causality test. The first one is called homogenous non-causality hypothesis (GC-1) which tests if the coefficients of “cause” variable for all z lags have a value instead of zero in the standard fixed effect model. If there is at least one coefficient with a non-zero value then the null hypothesis of GC-1 is rejected.
The null hypothesis of GC-1 for equation 13 is followed as:
B0: 1.= 0 , ∀. ∈ [1, >] #0 B1: 1.≠ 0
(15)
The second model allows for the heterogeneous non-causality hypothesis for each cross section. Hence, the validity of all cross-unit coefficients β1pi should be tested for each country. This called GC-2 test (see Erdil and Yetkiner 2009). The null hy-pothesis of GC-2 for equation 14 is following:
B0: 1. = 0 ∀ G[ , H], ∀. ∈ [1, >] #0 B1: 1. ≠ 0 (16)
52
If the null hypothesis of each i cross units is rejected, then there is a significant Granger cause relationship from Xit to Yit. For more explanation, see Studies 3 and 4.
4.2.5 Panel Pairwise and Pairwise Dumitrescu-Hurlin Tests
Two alternative panel forms of Granger non-causality tests are Panel Pairwise and Pairwise Dumitrescu-Hurlin tests. The main assumptions of homogeneity between cross units coefficients are different in these two tests but are close to tests in preceding section.
Panel Pairwise method in the first step investigates pooled data as a large stacked data set and then investigates the standard Granger non-causality test. The Granger null hypothesis test is conducted under restrictions that all cross sections coefficients are same to each other at different lag p values. That is, e.g. for test equation 13 assumes that for all i and j we have
1i 1j
δ = δ
,1. = 1.1, 1. = 1.1 ∀ , 1G[ , H], with each . ∈ [1, >]
(17)
Second approach proposed by Dumitrescu and Hurlin (2012) assumes an extreme opposite hypothesis which all cross-section unit’s coefficients are different to each other. Thus, Pairwise Dumitrescu-Hurlin non-causality test is constructed under following assumption for equation 18:
1i 1j
δ ≠ δ
,1. ≠ 1.1, 1. ≠ 1.1 ∀ , 1G[ , H], with each . ∈ [1, >]
(18)
The procedure of this test is to perform Granger causality regressions for each cross-sectional unit separately, and then investigate the average of these test statis-tics called Wbar statistic and Zbar which is its normal standardized version (Eviews, 2012).
4.2.6 Problem of Choosing Appropriate Lag Length in Granger Non-Causality Test
Based on above test alternatives and data availability limitations in health economics we can ask following questions: What version of Granger non-causality test would be the most appropriate one in different settings? To answer this question, we first highlight the current limitations of different versions of Granger non-causality tests. Different versions of Granger’s test have different options for
53 choosing the optimum lag length of VAR or VECM models. The panel data approach leads also to different options, i.e. tests for time series for each cross-unit individually or pooled cross-unit tests. Irrespectively of the data format, the main limitation in tests is the selection of the equal number of lag lengths for test series.
On general level both panel data and time series tests would provide more efficient and consistent results if the optimum lag length choices were based on each test time series or cross-unit individually. In principle this could be done, but lag length tests for each separate series condition the main non-causality test, and test distributions would be non-standard.
Furthermore, in the context of time series Granger non-causality tests, the basic tests criterions – e.g. Akaike’s or Schwarz’s information criterion – have the limitation of selecting equal optimum lag lengths of VAR model for both unrestricted and restricted equations in Granger non-causality tests. The more recent tests like Toda-Yamamoto and Hsiao versions of Granger causality tests have less limitations on selecting the optimum lag length in restricted and unrestricted equations, i.e. they are more efficient and can provide more reliable results compared to classical time series tests.
On the other hand, the short length of series has a significant effect on reducing the degree of freedom (df) and efficiency in Granger non-causality test procedure, especially in methods which use a high range of optimum lag lengths like Hsiao test. In response to this, the panel data tests can remove the df problem if the num-ber of cross-units is large enough16. From the statistical point of view, df has an in-verse relationship with autocorrelation in time series analysis and lower df increas-es the probability of autocorrelation (i.e. lags) in Granger non-causality tincreas-ests. Litera-ture provides some results on the relationship between df to total observations, df/N, with one lag model. Results show that increasing df/N will reduce quite effec-tively inconsistent results in time series regression models (for more details, see Hartmann 2016). Hence, with health economic data, where we face still the lack of data availability, economists should consider the panel Granger non-causality test or select appropriate time series versions of test that lean on some methods of se-lecting optimum test lag length.
16 Similarly, when we have few cross-units but quite many series observations, the validity of tests can be maintained.
54
5 RESULTS
5.1 TODA-YAMAMOTO VERSION OF GRANGER
NON-CAUSALITY TEST RESULTS BETWEEN HCE AND GDP IN 20 OECD COUNTRIES (1970-2009)
Presenting a visual explanation of sample is the first step of empirical analysis, Fig.
3 shows the histogram, Kernel fit and confidence ellipse (in 95% level17) in paths between logarithm of HCEc and GDPc (LnHCEc and LnGDPc) for a sample of 20 OECD countries over the period 1970-2009.
Figure 3: Cross plot of HCEc (lnHCEc) and GDPc (lnGDPc) in 20 OECD countries 1970-2009
As can be seen health expenditures have a positive growth sustained with increas-ing GDPc. Obviously, by the end of path in 2009, US has the highest amount of health spending compared to other OECD countries, while Turkey keep the last place.
Toda-Yamamoto version of Granger non-causality allows for possible co-integration between series in the model. However, in order to calculate the number of lag length used in Toda-Yamamoto equations (see Eqs. 12 and 13 in Section 4.2.2), the unit root information of series and lag order of VAR of variables are needed.
Result of Toda-Yamamoto version of Granger non-causality test between HCEc and
GDPc in 20 OECD countries supports the existence of feedback theory as bilateral
17 Probabilities computed with F-distribution.
55 causal relationship is predominant in OECD. Moreover, in the case of unidirection-al significant relationships, in less than hunidirection-alf of totunidirection-al countries the relationship from GDP to HCE is predominated over the HCE→GDP relationship. The strategy of Toda-Yamamoto test and summary of its result is explained in Fig. 4 and result for individual countries is presented in Table 11.
Figure 4: Strategy of Toda-Yamamoto test and summary of result
Table 11: Summary result of Toda-Yamamoto version of Granger non-causality test between HCEc and GDPc 20 OECD countries 1970-2009
Number of
countries Percentage of
total Name of countries
Bilateral 10 50% Ireland, Japan, New Zealand, Norway, Portugal, Spain, Switzerland, Turkey, UK, US
HCE→GDP 0 0% -
GDP→HCE 9 45% Australia, Belgium, Canada, Denmark, Finland, Germany, Iceland, Netherland, Sweden No
signifi-cant 1 5% Austria
56
5.2 PANEL AND HSIAO’S VERSION OF GRANGER NON-CAUSALITY TEST RESULTS BETWEEN HCE AND GDP, THEIR GROWTH RATES AND DE-TRENDED VALUES IN 34 OECD COUNTRIES (1970-2012)
Here, bilateral Granger non-causality is tested between HCEc and GDPc, their growth rates and their de-trended18 values. Visual explanation of data for a sample of 34 OECD countries over the period 1970-2012 is presented in Fig. 5.
In Fig. 5, variation in HCE is larger than in GDP when sample outliers are excluded. Czech Republic and Slovakia are two negative outlier in GDP and their de-trended values. Slovakia is the only negative and positive outlier in GDP growth observations. On the other hand, South Korea is a negative outlier in lnHCEc, Turkey and Israel are negative and Portugal is positive outliers in HCE growth.
South Korea and Portugal are negative outliers in de-trended HCE while South Korea is the only positive outlier in lnHCEc_de-trend.
As the first essential step in analysing time series variables is stationarity tests, various panel unit root tests were calculated between lnHCEc and lnGDPc, their growth rates, and their de-trended values. Results confirm that panel of these series are stationary with individual mean and trend components.
In the next step, Panel Pairwise and Pairwise Dumitrescu-Hurlin Granger non-causality test were conducted. Results of tests verify that bilateral causal relationship is dominant between lnHCEc and lnGDPc series. In growth series (DlnHCEc and DlnGDPc) the causal direction is just from GDP to HCE, while the results of de-trended series are different: Pairwise test confirms the existence of lnGDPc_de-trend→lnHCEc_de-trend, while Pairwise Dumitrescu-Hurlin approach verifies that this relationship is bilateral. For the reason that the result of two panel Granger non-causality tests for de-trended values of HCE and GDP were different, a closer analyses with the individual countries using Hsiao’s version of Granger non-causality was done to investigate the direction of causality between de-trended series.
18 De-trended series is a useful stochastic measure when the variable contains long term information which is important in econometrics analysis.
57 Figure 5: Cross plot of lnHCEc and lnGDPc, their growth rates (DlnHCEc and DlnGDPc) and de-trended values (lnHCEc_de-trend and lnGDPc_de-trend) 34 OECD countries 1970-2012 Results of Hsiao’s version of Granger non-causality test indicate that the dominant direction of relationship between these variables is bilateral. Bidirectional relation-ships are observed in more than half of OECD countries (53%). In 35% of countries there exist unidirectional relationships where GDP→HCE is more considerable than HCE→GDP. Interestingly, the optimal lag length of potential impact between HCE and GDP calculated by Hsiao method is higher than previous empirical studies and is around 8 years. The strategy of panel and Hsiao’s version of Granger no-causality tests and summary of their results are explained in Fig. 6 and result of Hsiao test in individual countries is presented in Table 12.
58
Figure 6: Strategy of Pairwise, Pairwise Dumitrescu-Hurlin and Hsiao’s version of Granger no-causality tests and summary of result
Table 12: Summary result of Hsiao’s version of Granger causality test between de-trended values of HCEc and GDPc in 34 OECD countries 1970-2012
Number of
countries Percentage of
total Name of countries
Bilateral 18 53%
Estonia, Finland, France, Germany, Greece, Iceland, Ireland, Israel, Italy, Netherlands, New Zealand, Slovenia, South Korea, Sweden,
Swit-zerland, UK, US
HCE→GDP 3 9% Belgium, Chile, Poland
GDP→HCE 9 26% Austria, Canada, Czech Republic, Denmark,
Greece, Hungry, Japan, Norway, Spain No
signifi-cant 3 12% Luxemburg, Mexico, Slovakia, Turkey