Description

approximation of the distribution of a quantity of interest than alternatives based on asymptotic theory (Horowitz, 2019) as a result of making no or weak assumptions, at most, regarding the Data Generating Process of the original data (L¨utkepohl, 2005, p.

127). Therefore, they are particularly important for time series analysis focused on annual data due to the relatively small sizes and the large number of variables that are included. In this context, these methods are commonly used to obtain confidence intervals for impulse responses (L¨utkepohl, 2005, p. 377). We innovate by using them for this purpose and conducting statistical inference regarding statistical tests.

Moreover, in the process, we also design a novel bootstrap method to be applied in the context of instantaneous causality tests, as explained in detail in subsection 3.4.4.

• In the case of France, we retrieved them from the World Bank Development Indicators Database (World Bank, 2020), hereafter WBDI. This source had missing values in 1962-1970 and 1984. In the case of the first period, we considered the values in Table V from Radica (1975), while the single missing observation in 1984 was imputed using linear interpolation between the actual observations of the surrounding years.³

• In the case of the United States, we calculated the gross enrollment rate directly from values obtained from Digest Education Statistics (DES) using the definition used by the WBDI database.⁴ Specifically, to calculate the gross enrollment ratio, we obtained (a) estimates for the resident population in the United States between 18 and 24 years old (the official age group of tertiary education in this country) and (b) total fall enrollment in degree-granting post-secondary institutions for the period of analysis (1962-2018) and divided (b) by (a) for each year. To obtain updated figures for both variables, the estimates of (a) for the entire period of analysis were retrieved from Table 303.10 of the 2021 edition (NCES, 2022). For (b), we were able to obtain figures from 1970 to 2021 from Table 101.10 from this edition as well. The values for 1963 to 1969 were obtained from Table 14 from the edition of 1996 (Snyder, Hoffman, & Geddes, 1996). We could not obtain a value for (b) for the year 1962 and therefore imputed it using linear interpolation from surrounding years.

We acknowledge that the gross enrollment rate has limitations. For example, a high indicator might result from additional factors other than a good education system, such as the presence of overaged individuals enrolled to repeat academic years. This problem could be mitigated by considering instead the corresponding net enrollment rate, which only measures the percentage of the population group corresponding to tertiary education enrolled in a higher education system instead of all individuals.

However, we could not find a time series with observations for all the years of the

3We must note that there is a substantial increase in the enrollment rate from 12.92% in 1967 to 16.90% in 1968 due to the creation in the latter year of the University Institutes of Technology (IUC) (Radica, 1975, p. 13). For the years 1971-1973 we obtained enrollment rates of 18.55%, 19.57% and 20.78% in Radica (1975) and of 18.56%, 19.64% and 21.40% in WBDI, which are very similar. For these years, we opted for using the values from the WBDI database.

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period of analysis for both countries.

Moreover, we also found that the alternative of using the human capital index time series available at PWT posed several problems related to how this variable is constructed, which is by linear interpolation from the corresponding five-year series from Barro and Lee (2013). First, most of the yearly observations of the human capital index time series are imputed rather than real values, with these only existing for years that are multiples of five. In contrast, it is possible to obtain real values of the gross enrollment rate for almost all years in the analysis period. Second, these values are obtained using linear interpolation. This assumes a linear evolution between existing data points, which might not be realistic. Third, this method leads to sudden changes in the slope of the human capital index that leads to artificial discontinuities in the data that create technical difficulties in the analysis.

Finally, we also include variables that can theoretically affect both inequality and growth since, as explained in detail further ahead, this is necessary in the context of our analysis. Bearing in mind the availability of data and the theoretical and empirical evidence in the literature, we consider in our analysis inflation (Afonso & Lima, 2022), education (Huang et al., 2009; Lundberg & Squire, 2003), government expenditures and openness (Huang et al., 2009).⁵ The proxy for education was already described.

In the case of the remaining variables, as shown in Table 3.1, we use the conventional proxies in the literature (e.g., Gr¨undler & Scheuermeyer, 2018), namely a price index from PWT, the share of government expenditures and the sum of the shares of exports and imports.

5Another variable that theoretically also affects inequality and growth is financial development (e.g., Clarke et al., 2006; Huang et al., 2009). However, we do not consider it here because both countries enjoyed a stable high level of financial development throughout the analysis period, as two

Table 3.1: Description of the data

Variable Formula Description of Unit of Measurement Source rgdp^(a) 100×log(Real GDP

per capita) GDP per capita at current PPP (in millions of 2017 international dollars).

PWT

gini^(b) 100×Gini coefficient of the income distribution

Gini of the pre-tax national income

of the entire distribution of income. SWIID redist^(c) 100×(Pre-tax Gini -

Post-tax Gini) Difference between the estimate of the Gini index of inequality in equivalized household market (post-tax) and disposable (pre-tax) incomes.

SWIID

enr^(c)(d) 100×Gross

Enrollment Rate Ratio of the resident population enrolled in tertiary education to the corresponding official group.

France:

RA;WBDI US: DES inv^(c) 100×Investment

Shares Share of investment in Real GDP at current PPP (in millions of 2017 international dollars).

PWT

pl^(d) 100 × log (Price

Index) Price level of household

consumption, with the benchmark being the price level in the US in 2017.

PWT

open^(d) 100×Investment

Shares Share of government consumption in Real GDP at current PPP (in millions of 2017 international dollars).

PWT

gov^(d) 100×(|Import Shares|

+ |Export Shares|) Sum of the absolute values of the shares of merchandise exports and imports in Real GDP at current PPP (in millions of 2017 international dollars).

PWT

Note: (a) Proxy for economic productivity; (b) Proxy for aggregate inequality; (c) Proxies for the intermediate variables of transmission channels; (d) Proxies for variables that can simultaneously affect inequality and growth.

Figure 3.1: Plots of all the variables concerning the United States and France during the period 1962-2018

Source: See Table 3.1.