Robustness checks

We conduct robustness checks of the estimated VAR models, with the results in appendix 3.B. In Table 3.10, we present the results of diagnostic tests of the residuals of all the estimated VAR models. They indicate for all models that the null hypothesis that the residuals are not serial correlated and do not have autoregressive conditional heteroskedasticity cannot be rejected at any significance level. We consider this as evidence that the models are well-specified. Moreover, the stability condition is satisfied for all models, which is an expected result considering that all the corresponding bivariate models are also stable.

We also analyze the possible existence of structural breaks. This is necessary to assess the stability of the results in the context of VAR models focused on relatively long periods, as in our case. Moreover, in the particular case of this paper, this analysis is motivated additionally by the evidence found in the context of the analysis of individual time series that suggest the existence of structural breaks in many.

This analysis is accomplished with the sample split Chow test, which is implemented following L¨utkepohl and Kr¨atzig (2004, p. 135). It implies selecting a particular year where a structural break is suspected of having occurred, estimating VAR models by subperiods defined according to the location of the potential structural break and determining if the estimated variance-covariance matrix has suffered structural changes. This matrix can change not only due to changes in parameters but also in the correlation of the residuals due to the effects of variables outside the model that occur specifically in one of the periods. This is relevant because the analysis period encompasses crisis periods, such as the global financial crisis of 2007-2008. These events simultaneously affected inequality and growth, but their effects are not necessarily captured by the variables we consider.

We conduct this test individually for a range of years, determine if there is evidence of structural breaks in any and, if there is, choose the location of the structural break as

Figure 3.2: Selected orthogonal impulse response functions with 95% bootstrap Equal- tailed confidence intervals for the VAR models of y^∗_{1,U S} and y^∗_{2,U S}

(a) y^∗_{1,U S}

(b)y^∗_{2,U S}

Source: Authors’ own computations.

Figure 3.3: Selected orthogonal impulse response functions with 95% bootstrap Equal- tailed confidence intervals for the VAR models of y^∗_{1,F R} and y^∗_{2,F R}

(a)y^∗_{1,F R}

(b) y^∗_{2,F R}

Source: Authors’ own computations.

Note: See Figure 3.2.

the year associated with the lowest p-value. We consider that this approach is the most adequate because it is exhaustive since it considers all plausible possibilities and permits considering theoretical or empirical reasons for selecting the years to which apply this test. Additionally, it avoids subjective considerations in the selection of the location of the structural break, in case of evidence supporting its existence, by considering an objective criterion.

We choose to analyze breaks in the years between 1982 and 1995. This permits considering subsamples periods sufficiently long to estimate adequate VAR models and is a plausible range considering the results of the individual structural change tests and the plots in Figure 3.1, which suggest the existence of breaks in this period.¹⁷ The results are in Table 3.11. In the case of the VAR models of y^∗_{1,F R} and y^∗_{2,F R}, the null hypothesis of no structural breaks is not rejected for any of the considered years for any significance level. By contrast, in the case of the VAR models of y^∗_{1,U S} and y^∗_{2,U S}, the null hypothesis of no structural breaks is rejected at 5% level for years in the range, respectively, 1989-1990 and 1988-1992. Moreover, the largest test statistic is associated with 1989 in both. Therefore, we conclude that there are no structural breaks in the models concerning France but only in the United States. Moreover, in the latter case, the structural break most likely occurred in 1989.

In the case of France, these results are surprising considering the presence of structural breaks in individuals series. We interpret this as evidence that the structural shifts found in individual series do not represent structural changes in the relation between variables. On the other hand, in the case of the United States, we interpret the results as evidence that the structural factors underlying, for example, the structural shift in inequality in the 1980s (e.g., Bound and Johnson 1992; Piketty and Saez 2003) lead to significantly shifts in the relationship between inequality, growth and the transmission variables between them that only materialized completely in 1989.

Considering this, we now perform the previous impulse response analysis in the subperiods 1962-1989 and 1990-2018 for the United States, with the corresponding results in Figures 3.4 and 3.5 of appendix 3.B. We find some results that are qualitatively the same across both periods, namely the positive and statistically significant effect of inequality on redistribution and the non-statistically significant impact of shocks of the

latter on economic growth and investment. However, the first is substantially weaker in relation to the effect found in the analysis of the entire period. We also obtain other results that are different in both periods. In particular, we obtain a non-statistically significant effect of inequality on growth in the first period that becomes negative and statistically significant in the second, with and without controlling for transmission variables. Moreover, we find contemporaneous effects of inequality on enrollment rates in each subperiod that are negative in the first and positive in the second. These contrary effects can explain the non-statistically significant results found in the context of the entire analysis period.

Comparing these results to those obtained for the whole sample, we obtain more non-statistically significant or weaker results. This is an expected outcome because an analysis by subperiods implies a substantial loss in degrees of freedom and, consequently, much wider confidence intervals for the impulse response estimates. Nonetheless, we obtained qualitatively similar results to those obtained for the entire analysis period, with the differences suggesting that the negative effects of inequality on growth are mostly present after 1989.

Finally, we also conduct a robustness check of the identification approach, by considering the opposite assumption of (A7). This is also plausible because changes in Real GDP per capita can influence investment decisions and, in turn, the investment share. Bearing this in mind, we consider the corresponding alternative specification to B₂, and B^∗₂, where instead of the baseline specification where γ_I,G = 0 and γ_G,I is unrestricted,γ_G,I = 0 andγ_I,G is unrestricted. The results concerning y^∗_{2,U S} and y^∗_{2,F R} are, respectively, in Figures 3.6 and 3.7 and it is apparent that the main effects of interest remain qualitatively the same.