3.5 Results
3.5.1 Results of the bivariate models
We start by presenting the lag structure of the bivariate VAR models focused on variables of the United States and France in Table 3.3. The information used to choose the lag order of each bivariate model is described in detail in Tables 3.8 and 3.9 of appendix 3.A. We considered an interval of lags between 1 and 4 and estimated a VAR model for each one. Then, we obtained information criteria, performed a bootstrap serial correlation test and determined whether the stability condition was satisfied. This was assessed by calculating the maximum modulus of the eigenvalues of the companion matrix and determining whether it was smaller or equal to 1. Finally, we chose the lag for which most criteria pointed to be the optimal,16 conditional on the null hypothesis of no serial correlation not being rejected at a 5% significance level and the stability condition being satisfied. In both countries, the most appropriate lag order is 1, but some models are of order 2, especially in the United States.
15It can be easily be shown thatE(v∗t(v∗t)′) =Ω∗. Considering the definition of the error term as in the text, we have thatE(v∗t(v∗t)′) =B∗cB−1c E(vtv′t) B∗cB−1c ′
=B∗cB−1c Ω B∗cB−1c ′
. Further noting that Ω = B B′ and that Ω = B∗(B∗)′, we have that (v∗(v∗)′) = B∗B−1B B′ B−1′
(B∗)′ =
Table 3.3: Lag order of VAR models focused on different pairs of variables (a) United States
∆incGSU S ∆invU S ∆RGDPU S ∆plU S ∆openU S ∆govU S ∆redistU S
∆invU S 1
∆RGDPU S 2 2
∆plU S 1 1 1
∆openU S 2 1 2 1
∆govU S 1 1 1 1 1
∆redistU S 1 1 1 1 1 1
∆enrU S 1 1 2 1 1 1 1
(b) France
∆incGSF R ∆invF R ∆RGDPF R ∆plF R ∆openF R ∆govF R ∆redistF R
∆invF R 1
∆RGDPF R 1 1
∆plF R 1 1 1
∆openF R 2 1 1 1
∆govF R 1 1 1 1 1
∆redistF R 1 1 1 1 1 1
∆enrF R 1 1 1 1 1 1 1
Source: Authors’ own computations.
Note: Each element of this Table represents the lag order chosen for the VAR model in first differences of a vector process that contains that variables in the corresponding column and row.
We then estimated each model with the chosen lag and conducted bootstrap Granger and instantaneous causality tests. The results are in Table 3.4, of which we highlight some of the most important. Concerning the relationship between inequality and growth, the Granger causality tests reveal that it is not possible to reject the null hypothesis of no Granger causality in both directions. Nonetheless, the instantaneous causality test leads to a rejection of the null hypothesis of no instantaneous causality between both variables at the 5% level. This could result from a contemporaneous relationship or simultaneous effects of other variables. We also highlight that the null hypothesis of no instantaneous causality is rejected at 1% level for several pairs of variables. These include ∆incGSU S and ∆redistU S and also many pairs of variables containing ∆invU S and ∆RGDPU S. The results also suggest that the latter Granger causes other variables in the system, including ∆incGSU S. This reinforces the notion that changes in ∆RGDPU S have important effects that must be considered when performing growth regressions.
Table 3.4: Bootstrap causality tests in bivariate models focused on the US (a) Bootstrap Granger causality test
∆incGSU S ∆invU S ∆RGDPU S ∆plU S ∆openU S ∆govU S ∆redistU S ∆enrU S
∆incGSU S 0.438 6.779∗∗ 1.538 10.947∗∗∗ 0.000 2.835∗ 5.403∗∗
∆invU S 0.296 20.432∗∗∗ 0.598 3.228∗ 0.149 0.013 1.800
∆RGDPU S 0.357 9.301∗∗ 1.527 5.135∗ 0.481 0.013 1.738
∆plU S 1.897 0.668 4.565∗∗ 0.051 7.054∗∗ 0.171 0.001
∆openU S 0.410 1.401 5.749∗ 0.006 1.951 0.700 0.005
∆govU S 1.892 0.207 0.312 0.162 0.561 0.028 0.350
∆redistU S 0.148 4.471∗∗ 3.260∗ 0.881 3.795∗ 0.668 2.912
∆enrU S 0.051 0.195 14.377∗∗∗ 0.159 1.419 1.140 0.006
(b) Bootstrap instantaneous causality test
∆incGSU S ∆invU S ∆RGDPU S ∆plU S ∆openU S ∆govU S ∆redistU S
∆invU S 1.736
∆RGDPU S 5.430∗∗ 20.361∗∗∗
∆plU S 1.092 2.520 0.011
∆openU S 0.002 16.283∗∗∗ 9.679∗∗∗ 1.325
∆govU S 1.706 11.721∗∗∗ 13.460∗∗∗ 0.337 0.908
∆redistU S 11.091∗∗∗ 7.252∗ 2.981 3.847∗ 4.380 3.227
∆enrU S 0.053 14.523∗∗∗ 6.453∗∗ 0.672 9.570∗∗∗ 8.105∗∗ 5.581∗∗
Source: Authors’ own computations.
Note: The test statistic in rowiand columnjis a Wald test statistic of the Null Hypothesis “variable j does not Granger cause variablei”, in the case of Granger causality tests, and “variablej does not instantaneously cause variablei”, in the case of instantaneous causality tests. Both are tested in the context of a bivariate model of a vector process comprised of variables j and i. In the case of the instantaneous causality test, the aforementioned null hypothesis implies that the covariance of the residuals inΩof equations of variablesj andiis zero. Since this matrix is symmetric, it is equivalent to the null hypothesis “variableidoes not instantaneously cause variablej”. The p-values of the tests are based on a bootstrap distribution of 2000 bootstrap values of the corresponding test statistics. For more details, see section 3.4.4.
Significance levels: ∗∗∗p-value<0.01,∗∗p-value<0.05,∗p-value<0.1
In the case of France, the results concerning both Granger and instantaneous causality tests are quite different and indicate a much less evident relationship between all the variables in general. The Granger causality tests indicate that a single null hypothesis of no Granger causality is rejected at the 5% level which is “incGSF R does not Granger cause RGDPF R”. The results of the bootstrap instantaneous causality suggest contemporaneous relationships between pairs of variables which were also found to have a relationship in this time-frame in the United States. However, the number of pairs for which the null hypothesis of no instantaneous causality is not rejected is much higher. Bearing in mind the results of the Granger causality tests, the results suggest that some variables only have a contemporaneous relationship.
Table 3.5: Bootstrap causality tests in bivariate models focused on France (a) Bootstrap Granger causality test
∆incGSF R ∆invF R ∆RGDPF R ∆plF R ∆openF R ∆govF R ∆redistF R ∆enrF R
∆incGSF R 0.297 0.002 1.493 4.862∗ 0.055 0.461 0.445
∆invF R 0.244 0.288 2.618 2.419 7.792∗∗ 0.074 0.085
∆RGDPF R 4.595∗∗ 0.550 0.548 3.683∗ 0.867 0.864 0.240
∆plF R 1.748 0.098 0.044 0.356 1.344 1.040 0.000
∆openF R 7.129∗∗ 0.000 1.243 3.416∗ 0.187 0.757 4.352∗∗
∆govF R 2.443 0.000 2.455 0.027 1.513 0.000 6.861∗∗
∆redistF R 0.171 0.049 0.691 1.064 0.079 0.008 0.001
∆enrF R 0.022 3.737∗ 2.913 0.111 2.629 0.435 2.158
(b) Bootstrap instantaneous causality test
∆incGSF R ∆invF R ∆RGDPF R ∆plF R ∆openF R ∆govF R ∆redistF R
∆invF R 0.054
∆RGDPF R 0.242 12.328∗∗∗
∆plF R 0.374 1.745 0.022
∆openF R 0.519 16.914∗∗∗ 2.929∗ 1.822
∆govF R 0.038 6.240∗∗ 5.981∗∗∗ 0.092 0.546
∆redistF R 16.789∗∗∗ 0.019 0.005 0.221 0.449 0.100
∆enrF R 2.673 0.112 0.560 0.175 0.303 0.055 1.805
Source: Authors’ own computations.
Note: For the See Table 3.4.
Significance levels: ∗∗∗p-value<0.01,∗∗p-value<0.05,∗p-value<0.1