The dataset selected for the study is from the World Bank's latest education and economic development indicators (annually) released in 2021. The dataset (1970–2021) belongs to the time series data, which covers all the valid data of Sweden, Denmark and France since 1970 in the four aspects of "proportion of basic education expenditure", "proportion of secondary education expenditure", "proportion of higher education expenditure" and "annual GDP growth rate".
3.1 Data and variable description
Although the World Bank has been collecting relevant data on education and economic themes since 1970, due to the influence of many objective factors, there are still certain problems in the integrity and continuity of statistics. In order to ensure the completeness of statistical data content and the continuity of time, this study roughly selects three relatively close periods according to the actual situation. Among them, Sweden (1998–2016) and Denmark (1998–2014) were chose the same period, while France (1970–1996) intercepted the period relatively early. Therefore, when doing a comparative analysis between countries, the focus of the discussion will be on the first two countries.
Clarifying the variables is a prerequisite for in-depth exploration of the Granger causal relationship between the structure of public education expenditure and economic development. This study takes the change in the rate of GDP development (annual basis), the annual change in the proportion of basic, secondary and higher education (all these indicators are on annual basis) as the main variables, using Sweden, Denmark and France as the foothold, and tests the Granger causal relationship and the correlativity that exist between them, with the help of two specific econometric methods. Additionally, the proportion of expenditure on primary/secondary/tertiary education (in percentage) should be taken as a percentage of total expenditure on primary/secondary/tertiary education of a country respectively.
A comprehensive understanding of the basic situation of each variable is an essential prerequisite for the successful completion of this survey. So in the beginning, we investigated the basic situation of the proportion of essential education expenditure, the proportion of secondary education expenditure, the proportion of higher education expenditure and the GDP development rate over the years, and found their respective maximums, minimum values and standard deviations (as shown in Table 1).
3.2 Specific methods
This study mainly used the following two research methods: the Granger causality test and Pearson correlation coefficient. The Granger causality test is to verify whether there is a significant causal relationship between the hierarchical structure of public education expenditure and economic development, while the Pearson correlation coefficient is to find out what type of correlation exists in the proportion of expenditure at each level within the public education expenditure. The ADF test is a kind of unit root test and it is the mainstream method to detect the stationarity of time series data, and it is also the basic premise of the Granger causality test. Considering the database is on annual basis (low number of observations), a safer unit root test could be KPSS. So, this article implemented the KPSS test primarily.
Different from the causal relationship in reality, the Granger causality test belongs to a statistical significance estimation, which is mainly used to analyze the causal relationship between economic variables (Granger, 1980). Granger (1980) once pointed out that due to the lack of collection and arrangement of the original literature, many researchers had many inappropriate applications in the process of applying the Granger causality test, which led to some rather absurd conclusions (Granger, 1980). For example, some studies have proved that the Shanghai Composite Index and the SZSE Component Index contain the forecast information for each of them (Cao, 2006) (Pang & Chen, 1999). But in fact, this conclusion may be false, because theoretically, the transmission channel of mutual influence between them is difficult to be reasonably explained, and if we take the supply and demand of funds into account, then the so-called causal relationship between them is likely to disappear (Cao, 2006) (Pang & Chen, 1999). Therefore, before using the Granger test, it is necessary to reasonably define the information set and clarify the essential variables. In addition, although the original definition of Granger causality does not explicitly specify the stationarity of the variables, if the Granger test is performed stubbornly on non-stationary data sets, problems can be found easily, and skewed results can be generated (Cao, 2006) (Pang & Chen, 1999) (Granger, 1980). The causality test between the money supply and GDP is a typical case (Pang & Chen, 1999). In addition, the reason for using the Pearson correlation coefficient is mainly to measure the linear relationship between the hierarchical structure of public education expenditure and to estimate the strength of this linear relationship.
Unlike the natural sciences, the study of economics usually has a large uncertainty, so the method of using probability theory to infer causality has become a common method for economic researchers (Pang & Chen, 1999). The Granger causality test is a familiar approach among them. Besides the theoretical analysis, variable-controlling, cross-correlation coefficient, regression analysis, etc. can also be used to deduce the causal relationship between two variables (Mallick et al., 2016) (Si, 2011) (Sobiech, 2019) (Tian, 2014) (Si, 2011) (Bai et al., 2015). Among these methods of using probability theory to detect causality, the simplest is the detection method proposed by Suppes (Reiss, 2016), that is, if the occurrence of event A increases the probability of event B’s presence, so event A constitutes the cause of event B. An obvious shortcoming of Suppes' approach is that it lacks of an exploration of the sequence of events (Reiss, 2016). Therefore, the causal relationship tested by this method can only be proved to be a causal relationship supported by prima facie evidence (Cao, 2006) (Pang & Chen, 1999) (Granger, 1980). In order to solve the problem of the order of occurrence of events, Granger creatively proposed a new causal relationship test method by introducing a series of concepts and methods, such as information sets, that is, Granger causality test (Cao, 2006) (Pang & Chen, 1999) (Granger, 1980). For the operation of the Granger causality test, it is important to accurately define the information set and determine the stationarity of the related variables. Because the absence of important variable information and the non-stationary nature of related variables can lead to false causal relationships or weaken the credibility of conclusions (Cao, 2006) (Pang & Chen, 1999) (Granger, 1980).
In the scope of economics, causation is a relatively important concept (Granger, 1980). However, the confirmation of causality tends to be a difficult problem (Cao, 2006) (Pang & Chen, 1999). While statistical methods can be used to estimate Granger causation in observational data, it is worth noting that the Granger causality test has many problems (Cao, 2006) (Pang & Chen, 1999) (Granger, 1980). For example, the transformation of variables, the pre-whitening of residuals, and metrical errors can all distort the causal relationship between the original variables (Cao, 2006) (Pang & Chen, 1999) (Granger, 1980). In addition, the characteristics of the Granger test determine that it is only suitable for causality tests of time series data (with stationarity), and it is difficult to use it to test the causality or nonlinear causality between cross-sectional data (Cao, 2006) (Pang & Chen, 1999) (Granger, 1980).
The conclusion of the Granger causation test is only statistical causality, not necessarily true causation (Cao, 2006) (Pang & Chen, 1999) (Granger, 1980). While it can be used as a support for true causation, and it cannot be used as a basis for affirming or denying causation (Cao, 2006) (Pang & Chen, 1999). Even if Granger causation is not equal to actual causation, it does not hinder its reference value. Because the causal relationship in the statistical sense is also meaningful, it can still play a important role in economic forecasting.