PhD Yassine Jaber

PhD Ismail Kabouri

Mohamed Bouzahzah

Mohamed Karim

Economic growth and education in Morocco: Cointegration and Toda Yamamoto Granger Causality

Keywords: ARDL bounds test, Education index, Gross enrolment ratio, Economic growth, Toda-Yamamoto approach JEL Classification: C5 C22, I25, O4 Paper type: Empirical research

This paper aims to investigate the cointegration and causal effect of economic growth and education in the case of Morocco, over the period 1990 to 2019. Education in all of it's forms, considered a fundamental factor of development. Therefore, a country needs to invest more in education. Without a significant investment in education (or human capital), no country can achieve sustainable economic development.

 and the Augmented Granger Causality approach given by Toda and Yamamoto (1995) are applied. To do that, we have used time-series data on economic growth, labor force, physical capital and four separate education indexes from 1990 to 2019. The data are collected from the World Development Indicators (WDI) and Human Development Reports (HDR). Furthermore, we used a global measure of education, called the education index, by adopting the UNDP methodology. The education index is a proxy for education development. The several indicators were used to demonstrate the robustness of the empirical results. Our results of the ARDL-bound test show that there exist cointegration between economic growth and education. The results also show that there is bidirectional causality between education and economic growth. All levels of education Granger cause economic growth, while economic growth causes all indicators of education except gross enrollment ratio, primary in all cases of causality analysis. These findings suggest that the government should invest in gross primary enrollment to accelerate economic growth, which leads to further education and thus economic growth.

Introduction

Morocco has attempted various education reforms since independence in 1956, none of which have yielded the desired outcomes. The education system has been the subject of continuous controversy and debate in a wide range of national forums. There have undoubtedly been successes throughout the years, but dysfunctions persist and the results are still below expectations. In 2019, 66% of Moroccan children aged 10 were unable to read and understand simple text, which was 2.5 points below the regional average for the Middle East & North Africa and 10.7 points below the average for lower-middle-income countries (World Bank, 2021). Morocco has therefore adopted a new strategy, namely the strategic vision 2015-2030, which needs to construct a better school based on equity and equal opportunities, quality for everybody, and the promotion of the individual and society.

The effect of education level on economic growth has always caught the interest of economists. The theoretical analysis has investigated the relationship between human capital, education and economic growth, the term "human capital" was adopted, and a human capital theory was developed. According to this theory education was consider as an essential institutional mechanism of the production, accumulation and diffusion of human capital [START_REF] Becker | Investment in human capital: A theoretical analysis[END_REF][START_REF] Mincer | Family investments in human capital: Earnings of women[END_REF][START_REF] Schultz | Capital formation by education[END_REF][START_REF] Schultz | Investment in human capital[END_REF]. The role of education in economic welfare has historically ebbed and flowed. The theoretical growth literature has long been interested in the role of human capital investment, especially through education and training, on the economic growth and development of a nation. The theoretical approaches defend multiple and different theses and arguments on this subject. The human capital and endogenous growth theories consider education as an essential factor for developing and stimulating growth economic and development. Throughout the first half of the twentieth century, education was prioritized at the expense of physical capital accumulation. However, investment in physical capital is considered the predominant theory of economic growth. It was during the 1960s, that the importance of human capital became increasingly important. The focus was initially on the contribution of human capital to an individual's standard of living (income per person), as well as its contribution to aggregate wealth, later, the focus shifted to its role in contributing to aggregate economic growth. Since the work of [START_REF] Schultz | Capital formation by education[END_REF] and [START_REF] Becker | Investment in human capital: A theoretical analysis[END_REF], has been developed extensive literature that contributed to the abandonment of the accumulation of physical capital and opened the way for a comprehensive analysis of the role of human capital. [START_REF] Schultz | Capital formation by education[END_REF] argues that investment in physical capital is only useful as long as human capital is sufficient to apply new technologies. Education trains young people to understand and deal with the complexities of economic growth. Distinguished economists have developed various theoretical methods and models about the relationship between human capital, education, and economic growth. According to this vast literature, there are mainly three mechanisms through which education may affect economic growth. First, the accumulation of human capital has long been considered as a major factor in economic development. Human capital as presented by education can contribute in the distribution and transmission of knowledge required to understand and process new information as well to successfully utilize new technologies developed by others, promoting economic growth once more [START_REF] Benhabib | The role of human capital in economic development evidence from aggregate cross-country data[END_REF][START_REF] Nelson | Investment in humans, technological diffusion, and economic growth[END_REF]. Second, as described in augmented neoclassical growth theories, education enhance human capital inherent in labor force, which increases labor productivity and thus transitional growth towards a higher equilibrium level of output [START_REF] Mankiw | A contribution to the empirics of economic growth[END_REF]. Third, education may help the economy become more innovative, and new information about new technology, goods, and processes can help the economy develop (as in theories of endogenous growth, Lucas, (1988), Romer, (1990), Howitt & Aghion, (1998)). www.ijafame.org

The current resurrection in interest began with the seminal papers by [START_REF] Barro | Economic growth in a cross section of countries[END_REF] and [START_REF] Mankiw | A contribution to the empirics of economic growth[END_REF] and a vast literature has developed in recent years [START_REF] Benhabib | The role of human capital in economic development evidence from aggregate cross-country data[END_REF][START_REF] Bouzahzah | Quality of the education system and economic growth. Projections in the case of Morocco[END_REF][START_REF] Gemmell | Evaluating the impacts of human capital stocks and accumulation on economic growth: Some new evidence[END_REF][START_REF] Hanushek | Interpreting recent research on schooling in developing countries[END_REF][START_REF] Hanushek | The role of school improvement in economic development[END_REF][START_REF] Krueger | Education for growth: Why and for whom?[END_REF][START_REF] Shobande | Has knowledge improved economic growth? Evidence from Nigeria and South Africa[END_REF][START_REF] Temple | Growth effects of education and social capital in the OECD countries[END_REF]. We can divide the impact of human capital investment on economic growth into quantitative and qualitative dimensions. Education quantity measured by schooling enrolment ratios [START_REF] Barro | Economic growth in a cross section of countries[END_REF][START_REF] Levine | A sensitivity analysis of cross-country growth regressions[END_REF][START_REF] Mankiw | A contribution to the empirics of economic growth[END_REF][START_REF] Romer | Human capital and growth: Theory and evidence[END_REF], the average years of schooling [START_REF] Hanushek | The role of school improvement in economic development[END_REF][START_REF] Krueger | Education for growth: Why and for whom?[END_REF], adult literacy rate [START_REF] Durlauf | Multiple regimes and cross-country growth behaviour[END_REF][START_REF] Romer | Human capital and growth: Theory and evidence[END_REF]) and education spending [START_REF] Baldacci | Social spending, human capital, and growth in developing countries[END_REF]. Various researchers have utilized different proxies for human capital, education is measured by using its proxy as enrollment rate of primary, secondary and tertiary level [START_REF] Chatterji | Tertiary education and economic growth[END_REF]. However, in this study, the quantity dimension of education is proxied by primary, secondary, tertiary enrolment ratios and an education index. The economic advantages of education to improve growth rates appear to be substantial. A better education community leads to higher rates of economic growth and, as a result, the government's capacity to reduce poverty. Education is often regarded as a powerful instrument for reducing poverty, boosting economic growth, empowering individuals, increasing private wages, providing a flexible and healthy environment, and promoting a competitive economy. We pose the following problems: "There is a relationship between education measured by four indicators and economic growth in the long-term?" "Does education Granger cause economic growth or there exist a two-way causality between variables?". Study objectives:

• Is there a long-run relationship between education and economic growth?

• Is it the case that education granger causes economic growth or on the contrary, it is economic growth Granger causes education? This study aims to answer empirically these issues using the ARDL method of cointegration and Toda Yamamoto Granger Causality. We examine these issues at the Moroccan level by using four different variables to proxy for education, labor, and physical capital as a variable of control.

To meet these objectives, we have established a review of the theorical and empirical literature (2), described the methodology and specification adopted to conduct the study (3), presented its results and discussions in section (4) and finally, we have listed some conclusions and policy implications [START_REF] Azariadis | Threshold externalities in economic development[END_REF]. That is the plan for this study.

A Review of the Theoretical and Empirical Literature

Education was generally one of the most important components of human capital, and it is considered a key indicator of economic development [START_REF] Goode | Adding to the stock of physical and human capital[END_REF][START_REF] Schultz | Investment in human capital[END_REF]. The pioneering work of [START_REF] Schultz | Capital formation by education[END_REF] and [START_REF] Becker | Investment in human capital: A theoretical analysis[END_REF], both have developed and analyzed growth models augmented with human capital and found a significant positive association between human capital formation and economic growth. Human capital is directly or indirectly contributing to the determination of economic growth. Since then, human capital theory has been an important part of economic theory. According to human capital theory, education directly helps to growth by improving the knowledge, qualifications, skills, and productive capacity of individuals. Education, as the main component of human capital, improves the socio-economic factors. In general, the influence of human capital investment on growth may be divided into two categories: quantitative and qualitative. The quantitative component is described by the number of available employees and hours worked, which measure the entire labor force available in a particular period. In this case, health is very important since it influences labor force growth as well as available work hours. The qualitative component, on 5 www.ijafame.org the other hand, addresses the problem of job performance quality, which involves both education in the form of individual skills and personal knowledge and, once again, healthy, which has a good impact on productivity. As a result, not only physical capital investments but also human capital investments should be included as growth factors.

To understand the theoretical debate on the impact of education on economic growth. Education has long been considered a key factor in determining economic welfare. At least three mechanisms through which education can contribute to economic growth are highlighted in the theoretical growth literature. First, according to neoclassical growth theories, [START_REF] Mankiw | A contribution to the empirics of economic growth[END_REF] demonstrate that an augmented Solow model that incorporates both human and physical capital accumulation gives an excellent description of the cross-country data, they found a positive link between economic growth and working-age population in secondary school for 1960-1985 in 121 countries. Second, endogenous growth models, assume that the "human capital" plays a central role in several approaches to growth theory (Lucas, 1988 andRomer, 1990). This works focuses on education, which is considered as one of the factors of education. Education, according to (Lucas, 1988), is a mechanism for the accumulation of human capital and should be evaluated as a factor of production among labor and physical capital. This signifies those improvements in the labor force's educational attainment have a positive impact on productivity, leading to improved overall economic performance. Externalities are also generated by human capital. Third, Aghion & Cohen, (2004) analyzed the impact of education on economic growth, which show that the mechanisms through which education affects economic growth are multiple and well-identified. The authors highlight the essential impact of the level of technological development and accumulation of human capital. Through the accumulation of human capital, an individual cannot become productive without passing through the educational system. On the other hand, a higher level of education allows for the development or adoption of new technologies more easily. [START_REF] Benhabib | Human capital and technology diffusion[END_REF] generalizes the Nelson & Phelps, (1966) model of technology diffusion, Admit the possibility that the technology diffusion model is exponential. This specification predicts that nations will exhibit positive catch-up in growth rates. and this catch-up only works with a critical level of human capital/education. The empirical literature on the relationship between education and economic growth starts with cross-section studies; see for examples [START_REF] Azariadis | Threshold externalities in economic development[END_REF][START_REF] Romer | Human capital and growth: Theory and evidence[END_REF], both have found that literacy is significantly positive with economic growth. They worked on 112 economies data between 1960-1985 and 71 low-and middle-income countries during 1960-1980. [START_REF] Barro | Economic growth in a cross section of countries[END_REF] reveals that the enrollments ratios primary and secondary are positively related to economic growth, while the student-teacher ratio is negatively related to growth in 98 countries during 1960-1980. In general, quantitative studies conducted over the previous three decades reveals that there exists a positive relationship between education and economic growth (Abbas & Foreman-Peck, 2008; Acemoglu & Zilibotti, 1999; [START_REF] Barro | Human capital and grouth in cross-country grouth regressions[END_REF][START_REF] Benhabib | The role of human capital in economic development evidence from aggregate cross-country data[END_REF][START_REF] Bils | Does schooling cause growth?[END_REF]. Some of the most solid studies even suggested that higher levels of education had a stronger influence on economic growth, which is especially true in industrialized nations where research and development are critical. [START_REF] Gylfason | Education, social equality and economic growth: A view of the landscape[END_REF] examined the relationship and interactions between equality, education, and economic growth in a sample of 87 industrial and developing countries during the period 1965-1998. Using seemingly unrelated regression analysis, they found that education and inequality have a significant, independent impact on economic growth, education seems to encourage economic growth. [START_REF] Hassan | Education's contribution to the economic growth of Sub-Saharan Africa[END_REF] found that all five measures of human capital have a positive causal relationship with per capita GDP, using cross-section panel data regression of Sub-Saharan countries. [START_REF] Baldacci | Social spending, human capital, and growth in developing countries[END_REF] using panel data from 118 developing countries during 1971-2000, showed that both education and health have a positive www.ijafame.org and significant impact on economic growth, any increase in education spending by one percentage point of GDP is associated with 3 additional years of schooling on average and increases annual GDP per capita growth by 1.4 percentage points over 15 years. [START_REF] Islam | Relationship between education and GDP growth: A multivariate causality analysis for Bangladesh[END_REF] investigated the direction causal link between education and economic growth. They used a multivariate causality test (Vector Error Correction Model) to examine the relationship between education and economic growth in Bangladesh during 1976-2003. They found bidirectional causality between education and economic growth and the long-run relationship between variables in the presence of labor and capital. In the case of Bangladesh, it appears that education and growth are mutually beneficial. [START_REF] Pradhan | Education and economic growth in India: Using error correction modelling[END_REF] examine the causal link between education and economic growth in India during 1951-2001, the study found that there is a unidirectional causality between education and economic growth running from economic growth to education. In recent studies, (Afzal and al., 2011) used the ARDL Model of cointegration and Augmented Granger Causality given by [START_REF] Toda | Statistical inference in vector autoregressions with possibly integrated processes[END_REF] to test cointegration and causal link between education (measured by different indicators of education) and economic growth. The results showed that there is cointegration among education and growth and also indicate bidirectional causality between education and economic growth. The study proposes more investment in university education to promote economic growth, which leads to further education and hence economic growth. [START_REF] Rahman | Causal relationship among education expenditure, health expenditure and GDP: A case study for Bangladesh[END_REF] examined the causal link between health expenditure, education expenditure, and economic growth in the case of Bangladesh. Including health and education expenditure as an investment in health, the study found the existence of bidirectional causality running from education expenditure to economic growth and from education expenditure to health expenditure and oneway causality from health expenditure to economic growth. [START_REF] Sharma | Human capital and economic growth in India: A cointegration and causality analysis[END_REF] examine the causation between human capital investment (education and health investment) and economic growth in India using time series data from 1991-92 to 2012-13. The study found a long-run link between education investment, health investment, and economic growth. It also found two-way causation between education investment and economic growth, as well as between health and economic growth. Essardi and Razzouk, (2017) in the case of Morocco, investigated the relationship between human capital and economic growth, using two methods of estimation, firstly, Johansen multivariate cointegration test and Granger causality. Secondly, Bayesian Model Averaging. Using four proxies of human capital, the results show that the average years of schooling, life expectancy index and the indicator of the quality of health affect positively GDP per worker. Also, the causality shows that only the indicator of the quality of health causes the GDP per worker. Sarania, (2020) examined the relationship between education expenditure, higher education, and economic growth in India during the period 1971-2015 based on Vector Autoregression (VAR) and Johansen's cointegration procedure. The findings indicate unidirectional causality runs from education expenditure to economic growth and one-way causality runs from economic growth to higher education. Esen & Keçili, (2021) investigate the effects of health expenditure on economic growth in Turkey. The author's results showed unidirectional causality from health expenditure to economic growth in the short term.

Econometric methodology and data

Data

We have used the time-series data on 𝐺𝐷𝑃 𝑡 , real physical capital (𝑅𝑃𝐶 𝑡 ), the labor force (𝐿𝐹 𝑡 ) and four separate education variables in Morocco context from 1990 to 2019. The data are collected from the World Development Indicators (WDI, 2020) and Human Development Reports (HDR). Furthermore, we used a global measure of education, called the education index, adopting the UNDP methodology. The Education Index measures a nation's relative achievement in adult literacy index (ALI) and cumulative total enrollment in primary, secondary, and tertiary education. First, the adult literacy index and the combined gross enrollment index are calculated, these indices are first combined to create the education index, with a two-third weighting for adult literacy rate (ALR) and with one-third for combined primary, secondary, and tertiary gross enrollment ratio index (GEI). The adult literacy indicates the ability to read and write, while the gross enrollment ratio indicates the level of education.

Education index = For the rest of the paper, we will use the following notations: ln(GDP) is the logarithm of gross domestic product per capita (constant 2010 US$), ln(RPC) is the logarithm of the real physical capital, we take also the ln(LF) as the logarithm of the labor force and we have four separate indicators for education, 𝑒𝑑𝑢 1𝑡 is the education index as described above, ln(𝑒𝑑𝑢 2𝑡 ) the Gross enrolment ratio, secondary, ln(𝑒𝑑𝑢 3𝑡 ) is Gross enrolment ratio, tertiary, ln(𝑒𝑑𝑢 4𝑡 ) Gross School enrollment, primary. The figures below show the evolution of these variables between 1990-2019 (Figure 1 in the appendix).

The GDP per capita as can be seen in the figure present an increasing trend between 1990 to 2019. The average value during this period was 2447,91 U.S. dollars, with a minimum of 1704,69 U.S. dollars, in 1993 and a maximum of 3396,06 U.S. dollars, in 2019. Morocco is classified as a lower-middle-income country. The country's annual growth rate is predicted to be between 3 and 5 percent. This expansion is the outcome of a more diverse economy and a commitment to advance the industrial sector. It is believed that advances in the manufacturing sector will aid in reducing persistently high unemployment, which is expected to continue above 9%. Currently, the agricultural sector accounts for 13% of economic growth, but production capacity is heavily dependent on annual rainfall. Nearly 30% of economic growth is ascribed to the industrial sector, while the remaining 60% is assigned to services. With an average annual growth rate of 7.40%, the gross enrollment rate in higher education climbed from 10.35% in 2009 to 38.08% in 2019. Furthermore, the secondary enrollment rate increased significantly from 37.35% to 80.58%, increasing at an average annual rate of 14.51% in 2017, with a 1.20% reduction in 2019. Furthermore, the primary school enrollment rate was 114.8 percent, up from 67.44% in 1990 to 115.35% in 2019, with an average yearly growth rate of 1.13% A graphical representation in logarithmic form as we can see below, shows that all series seems to have an increasing trend over our case study. We can conclude that the series seems to be non-stationary in the level. To avoid the spurious regression problem, we will examine the data series to see if they contain unit-roots.

Econometric Model

This section introduces the econometric model framework used to test of cointegration and causality between education and economic growth. Holding in perspective the theoretical assumptions of the relationship between GDP, labor force, real physical capital, and human capital. We estimate equations similar to that used by [START_REF] Afzal | Education and economic growth in Pakistan: A cointegration and causality analysis[END_REF].

To understand the empirical association between education and economic growth, we consider the two following econometric log-linear models:

𝑙𝑛𝐺𝐷𝑃 𝑡 = 𝛽 0 + 𝛽 1 𝑙𝑛𝑒𝑑𝑢 𝑖𝑡 + 𝛽 2 𝑙𝑛𝑅𝑃𝐶 𝑡 + 𝛽 3 𝑙𝑛𝐿𝐹 𝑡 + 𝑢 𝑡 (1) 𝑙𝑛𝑒𝑑𝑢 𝑖𝑡 = 𝛼 0 + 𝛼 1 𝑙𝑛𝐺𝐷𝑃 𝑡 + 𝛼 2 𝑙𝑛𝑅𝑃𝐶 𝑡 + 𝛼 3 𝑙𝑛𝐿𝐹 𝑡 + 𝑢 𝑡
(2) In those equations, 𝑙𝑛 is the natural logarithm, 𝐺𝐷𝑃 𝑡 the Gross domestic product per capita used as a proxy for economic growth, 𝐿𝐹 𝑡 the labor force comprises people ages 15 and older; it includes people who are currently employed and people who are unemployed but seeking work as well as first-time job-seekers, 𝑅𝑃𝐶 𝑡 the real physical capital measured as gross fixed capital formation deflated by GDP deflator. 𝑒𝑑𝑢 𝑖𝑡 measures education level. In our study education was measured by different indicators such as: 𝑒𝑑𝑢 1𝑡 = Education index proxy for education www.ijafame.org development, 𝑒𝑑𝑢 2𝑡 = Gross enrollment ratio, secondary, 𝑒𝑑𝑢 3𝑡 = Gross enrollment ratio, tertiary, 𝑒𝑑𝑢 4𝑡 = Gross enrollment ratio, primary. Our data and definitions are summarized in Table 1 below. We include summary statistics.

To establish an error correction mechanism in this equation, we add the first lag of the level of each variable. The ECM is expected to have a negative sign, which implies that the model is converging toward the equilibrium after a certain threshold level. The ECM of ARDL model ( 5) can be written as follows:

ΔlnEdu 𝑖𝑡 =

Co Integration Tests

First, we estimate the equations ( 3) and ( 4) to examine the existence of a long-run relationship between our variables in the model. In this case, we are using the ARDL approach to cointegration, by using the F-statistic. We choose the different lag lengths, 1,2,3, and 4 as the maximum order of lags in the ARDL and estimate for the period 1990-2019. Tables 3 and4 shows the cointegration results, when 𝐺𝐷𝑃 𝑡 serves as a dependent variable, and when 𝑒𝑑𝑢 𝑖𝑡 serves as a dependent variable respectively. [START_REF] Pesaran | Bounds testing approaches to the analysis of level relationships[END_REF] , 2001). This implies that the null hypothesis of no cointegration cannot be accepted. Similarly, the Table 4 show that there exists a long-run relationship between education and all variables (𝐺𝐷𝑃 𝑡 , 𝑅𝑃𝐶 𝑡 , and 𝐿𝐹 𝑡 ) when education is a dependent variable. As we described our variable education 𝑒𝑑𝑢 𝑖𝑡 is presented by four separate variables (𝑒𝑑𝑢 1𝑡 ,𝑒𝑑𝑢 2𝑡 , 𝑒𝑑𝑢 3𝑡 , 𝑒𝑑𝑢 4𝑡 ). According to both Narayan, 2005 and Pesaran and all, 2001 critical values, we reject the null hypothesis of no cointegration between education and economic growth and we confirmed that there is a cointegration between our variables. In conclusion, the results at this stage confirm that the cointegration between education and economic growth exists. We can then go ahead to estimate the ARDL for both models. In our case, we limit our analysis in the cointegration test.

Toda-Yamamoto Augmented Granger Causality (TYAGC) test

Granger Causality is the traditional method for testing causality between variables as proposed by [START_REF] Granger | Investigating Causal Relations by Econometric Models and Crossspectral Methods[END_REF]. The Granger causality test without considering other variables because of their effect was considered a possibility of specification bias. It has been shown in the literature that the Granger causality test is extremely sensitive to model specification and lags. In dealing with these problems, TYAGC overcoming the problem of asymptotic critical values when causality is done in the presence and absence of stationarity and cointegration [START_REF] Ziramba | Wagner's law: An econometric test for South Africa, 1960-2006[END_REF]. To use the Tada Yamamoto causality approach, we must first identify the lag length 𝑝 and the maximal order of integration of a series, denoted by 𝑑 𝑚𝑎𝑥 , and estimate (𝑝 + 𝑑 𝑚𝑎𝑥 ) 𝑡ℎ order of VAR model using Seemingly Unrelated Regression. Secondly, the hypothesis is tested by www.ijafame.org conducting a standard Wald statistic test which has an asymptotic Chi-square distribution with m degrees of freedom. According to [START_REF] Toda | Statistical inference in vector autoregressions with possibly integrated processes[END_REF], the modified Wald test is valid for integrated and co-integrated variables, for series I (0), I(1), or I(2). Let consider the bivariate model with p lag. In this case, the relevant VAR model is based on the following equations:

𝑙𝑛𝐺𝐷𝑃 𝑡 = 𝜇 + ∑ 𝑝+𝑚 𝑖=1 𝛼 𝑖 𝑙𝑛𝐺𝐷𝑃 𝑡-𝑖 + ∑ 𝑝+𝑚 𝑖=1 𝛽 𝑖 𝑙𝑛𝑒𝑑𝑢 𝑖𝑡-𝑖 + 𝜇 1𝑡 (7)
𝑙𝑛𝑒𝑑𝑢 𝑖𝑡 = 𝜇 + ∑ 𝑝+𝑚 𝑖=1 𝛾 𝑖 𝑙𝑛𝑒𝑑𝑢 𝑖𝑡-𝑖 + ∑ 𝑝+𝑚 𝑖=1 𝛿 𝑖 𝑙𝑛𝐺𝐷𝑃 𝑡-𝑖 + 𝜇 2𝑡 (8)
Where 𝑚 is the maximal order of integration, 𝑝 is the optimal lag length of 𝑙𝑛𝐺𝐷𝑃 𝑡 and 𝑙𝑛𝑒𝑑𝑢 𝑖𝑡 , and 𝜇 𝑖𝑡 are the error terms are assumed to be white noise, ∼ (0, 𝜎 2 ).

The null hypothesis of non-causality between variables, against the alternative hypothesis, is given by: 𝐻 0 : 𝑙𝑛𝑒𝑑𝑢 𝑖𝑡 does not Granger cause 𝑙𝑛𝐺𝐷𝑃 𝑡 , 𝑖𝑓 𝛽 𝑖 = 0 𝐻 1 : 𝑙𝑛𝑒𝑑𝑢 𝑖𝑡 does not Granger cause 𝑙𝑛𝐺𝐷𝑃 𝑡 , 𝑖𝑓 𝛽 𝑖 ≠ 0

Before employing the TYAGC test, it is necessary to determine the optimal lag length 𝑝 and the maximum order of integration 𝑚. In our case, the order of integration is determined using the PP and ADF unit root test. The maximum integration for our VAR model is shown as, 𝑚 = 1, we specify the model by determining the lag length based on the Akaike Information Criterion. The optimal lag length selected by AIC for our study is shown as 3. The TYAGC test can be performed by constructing the Vector Autoregressive (VAR) model of (𝑚 + 𝑝 = 4).

Bivariate TYAGC Framework

In this section, the study investigates the bivariate Toda Yamamoto Granger Causality. Between economic growth and education model ( 1) and ( 2), between GDP 𝑡 and RPC 𝑡 model ( 3) and ( 4), between GDP 𝑡 and LF 𝑡 model ( 5) and [START_REF] Baldacci | Social spending, human capital, and growth in developing countries[END_REF], between education and RPC 𝑡 model ( 7) and [START_REF] Barro | Human capital and grouth in cross-country grouth regressions[END_REF], between education and LF 𝑡 Model ( 9) and [START_REF] Benhabib | The role of human capital in economic development evidence from aggregate cross-country data[END_REF]. We estimate all equations, the results are reported in summary in table 5. Throughout, results that are mentioned but not presented in detail are available from the authors. All equations are reported in appendix 1.

𝑙𝑛𝐺𝐷𝑃 𝑡 = 𝛼 1 + ∑ 4 𝑖=1 𝛽 1𝑖 𝑙𝑛𝐺𝐷𝑃 𝑡-𝑖 + ∑ 4 𝑖=1 𝛾 1𝑖 𝑙𝑛𝐸𝑑𝑢 𝑖𝑡-𝑖 + 𝑢 1𝑡 (1) lnedu 𝑖𝑡 = 𝛼 2 + ∑ 4 𝑖=1 𝛽 2𝑖 𝑙𝑛𝐺𝐷𝑃 𝑡-𝑖 + ∑ 4 𝑖=1 𝛾 2𝑖 𝑙𝑛𝐸𝑑𝑢 𝑖𝑡-𝑖 + 𝑢 2𝑡
(2) The main results of the bivariate analysis. As indicated in detail, the null hypothesis of noncausality between all measures of education and economic growth was tested using the modified Wald test. The results of bivariate TYAGC reveal that the null hypothesis was rejected. At the same moment, the null hypothesis that the economic growth does not Granger cause education was rejected except gross enrollment ratio, in primary. Therefore, we can conclude that there is bidirectional causality (feedback) between education and economic growth which validates the theory as well as with the past empirical research. The hypothesis that economic growth does not Granger cause gross enrollment ratio, in primary, indicate that economic growth fails to cause significatively gross enrollment ratio, primary. The bivariate TYAGC result also indicates from model ( 5) and ( 6) that there is bidirectional causality between economic growth and labor force. From model (3) and (4) the result shows that there is only one-way causality between economic growth and real physical capital, run from economic growth to real physical capital. According to model [START_REF] Barro | Economic growth in a cross section of countries[END_REF] and [START_REF] Barro | Human capital and grouth in cross-country grouth regressions[END_REF] in table 4, education index proxy for education development and gross enrollment ratio, secondary does not Granger cause real physical capital. Also, the result indicates that real physical capital does not granger cause gross enrollment ratio, primary. At the same time, we can conclude that gross 13 www.ijafame.org enrollment ratio, secondary and gross enrollment ratio, in primary Granger cause real physical capital, and there is causality from real physical capital to education index, gross enrollment ratio, secondary and gross enrollment ratio, tertiary. From model ( 9) and ( 10), the result indicates that there is bidirectional causal nexus between gross enrollment ratio, tertiary and labor force. At the same time, there is unidirectional causality between education index, gross enrollment ratio, secondary and gross enrollment ratio, in primary and labor force. To summarize, the bivariate TYAGC shows that there is a feedback causality between education and economic growth.

Multivariate TYAGC with three variables

After estimating bivariate TYAGC, in this section, we proceed to estimate a multivariate test using three variables in our model. We estimate equations ( 11), [START_REF] Bils | Does schooling cause growth?[END_REF], and [START_REF] Bouzahzah | Quality of the education system and economic growth. Projections in the case of Morocco[END_REF] to capture the causality between economic growth, real physical capital and education and among economic growth, LF 𝑡 and Edu 𝑖𝑡 Eqs. ( 14), [START_REF] Durlauf | Multiple regimes and cross-country growth behaviour[END_REF], and [START_REF] Esen | Economic Growth and Health Expenditure Analysis for Turkey: Evidence from Time Series[END_REF] [START_REF] Bils | Does schooling cause growth?[END_REF] The causality between economic growth and all indicators of education is tested and the results are summarized in detail available from the authors. The result shows that the null hypothesis of no Granger causality between education and economic growth except gross enrollment ratio, primary is rejected. Overall, the result supports our previous bivariate argument confirming that there is bidirectional causality between education and economic growth. Our results also show a feedback relationship (two-way Granger causality) between RPC 𝑡 and economic growth, and between education and RPC 𝑡 . The TYAGC in detail indicates that there is two-way causality between education and economic growth when RPC 𝑡 and LF 𝑡 as 3 rd variable except gross enrollment ratio, primary. There is oneway causality between economic growth and gross enrollment ratio, primary running from GDP 𝑡 to gross enrollment ratio, primary. The results also indicate that there is one-way causality between LF 𝑡 and education flowing from education to LF 𝑡 . At the same moment, the null hypothesis that GDP 𝑡 does not granger cause LF 𝑡 when education serves as 3 rd variable is rejected. We can conclude that there is bidirectional causality between economic growth and the labor force. So, it is clear that LF 𝑡 appears to be a very important variable that influences two main variables education and economic growth. In general, the result of multivariate TYAGC supports our previous bivariate causality that there is two-way causality between education and economic growth.

Multivariate TYAGC with all variables

Know we investigates the multivariate TYAGC with all variables. Considering each variable as a dependent variable. The causality between GDP 𝑡 , RPC 𝑡 , LF 𝑡 and Edu 𝑖𝑡 was tested using TYAGC. The main results detail is available from the authors. All equations are reported in appendix 3.

𝑙𝑛𝐺𝐷𝑃

The results presented support our previous bivariate causality and multivariate causality (with only three variables in the equation) that there is causality between education and economic growth. Thus, we can conclude that there exists two-way causality between education and economic growth. While economic growth does not granger cause 𝑒𝑑𝑢 4𝑡 (Gross enrollment ratio, primary). The TYAGC results also indicate that we cannot reject the null hypothesis that economic growth does not granger cause 𝐿𝐹 𝑡 . Our results also show that we cannot reject the null hypothesis that economic growth does not granger cause RPC 𝑡 . This implies that there exists a feedback relationship (two-way Granger causality) between LF 𝑡 , RPC 𝑡 and GDP 𝑡 in the presence of all indicators of education. The null hypothesis of no Granger causality between LF 𝑡 and all indicators of education is rejected, concluding that there is feedback between LF 𝑡 and education. In case of testing causality between RPC 𝑡 and education, the result reveals that RPC 𝑡 only Granger cause education index and gross enrollment ratio, secondary and does not Granger cause gross enrollment ratio, tertiary and gross enrollment ratio, primary. However, all indicators of education Granger cause RPC 𝑡 . The causality between education and economic growth is tested and results from bivariate, multivariate (with three and all variables) are summarized in Table 5.

Source: Author's calculations

From Table 5, it can be observed that the null hypothesis of no Granger causality between all indicators of education and economic growth was rejected. There is bidirectional causality between education and economic growth. all indicators of education Granger cause economic growth in the case of bivariate and multivariate TYAGC. Economic growth does Granger cause education in all causality cases except gross enrollment ratio, primary. Economic growth fails to cause gross enrollment ratio, primary in all causality cases. In general, our results confirm the importance of education and its contribution to economic growth. it is documented in the literature that economic growth and education have a long-run relationship and both have the causal link either one or two-way. Our results are consistent with. [START_REF] Afzal | Education and economic growth in Pakistan: A cointegration and causality analysis[END_REF], investigate the cointegration and causality between education and economic growth in the case of Pakistan, they showed that there is a cointegration among education and www.ijafame.org growth, and they suggest more investments in university education in order the promote economic growth. The government should focus more on education because its acts as input for higher education and to stimulate economic growth.

Conclusions

In this paper we investigate the cointegration and causal link between education and economic growth in Morocco. Education, in all of it forms, is one of the fundamental factors of development. Hence, a country needs to invest more in education. Without significant investments in human capital, no country can achieve sustainable economic development.

Based on the results of unit root tests (ADF, PP), we found that none of the variables are integrated of order 2, the variables are a mix of I (0) and I (1). According to unit root tests, the ARDL bounds test approach was examined followed by Toda Yamamoto Granger Causality. Education and economic growth cointegration were examined using the ADRL approach and the causal relationship between education (measured by four indicators) and economic growth was investigated in bivariate, multivariate (with three variables) in the presence and absence of the labor force, and real physical capital as a third and fourth variable and multivariate (with all variables). Several indicators were used to demonstrate the robustness of the empirical results.

Our results indicate that there exists two-way causality between economic growth and education except for the gross enrollment ratio in the primary. Education and all levels of education cause economic growth in the presence of real physical capital in bivariate, multivariate (with all variables), and multivariate (with three variables), whereas education and levels of education cause economic growth in the presence of labor force, except education index proxy for education development. While economic growth granger causes all metrics of education except the gross enrollment ratio, primary. As a result, we can conclude that there is a two-way causality between education and economic growth. While GDP 𝑡 does not granger cause Edu 4𝑡 (Gross enrollment ratio, primary) in all cases of bivariate, multivariate (with three variables), and multivariate (with all variables). The results from bounds test cointegration confirm the long-run relationship between education and economic growth, when 𝐺𝐷𝑃 𝑡 serves as a dependent variable, and when 𝑒𝑑𝑢 𝑖𝑡 serves as a dependent variable respectively. According to the findings of the study, the government should pay attention more on primary education. The primary schooling should be a priority because it acts as an input for all higher education. To stimulate economic growth, more funding is needed for education, especially at the primary and university levels. It is also suggested that the relationship between education and economic growth be investigated and broadened by incorporating other variables than physical capital and labor force, such as education expenditures.

Table 1 : Variable definitions and sources

 1

	Variables	Definition and source	Obs Mean
	Ln(GDP)	Gross Domestic per capita (constant 2010 US$); World Development Indicators.	30	7.77
	Labor Force (lnLF)	Labor force comprises people ages 15 and older; World Development Indicators.	30 16.12
	Real physical capital (lnRPC)	Gross fixed capital formation (constant 2010 US$); World Development Indicators.	30 23.55
		Education index is an average of mean years of
		schooling (of adults) and expected years of
	Education index	schooling (of children). HDRO calculations based	30	0.39
		on expected years of schooling and mean years of
		schooling from UNESCO Institute for Statistics.
		Total enrollment in secondary education,
	Gross enrollment ratio, secondary	regardless of age, expressed as a percentage of the population of official secondary education age;	30	3.93
		UNESCO Institute for Statistics.
		Expressed as a percentage of the total population
	Gross enrollment ratio, tertiary	of the five-year age group following on from secondary school leaving; UNESCO Institute for	30	2.68
		Statistics.
		Total enrollment in primary education, regardless
	Gross enrollment ratio, primary	of age, expressed as a percentage of the population of official primary education age; UNESCO	30	4.54
		Institute for Statistics.

Source:

Authors 4. Empirical results 4.1. Unit root tests The

  autoregressive distributed lag method for cointegration and Toda-Yamamoto Augmented Granger Causality (TYAGC) test are used to test the cointegration and causality between education and economic growth. Stata 17.0 was used to generate all results. The first step in the Bounds Testing methodology of[START_REF] Pesaran | Bounds testing approaches to the analysis of level relationships[END_REF] and[START_REF] Pesaran | An autoregressive distributed lag modelling approach to cointegration analysis[END_REF] is to make sure that none of the variables is integrated of order 2 or > I (2). This study uses two-unit root tests such as Augmented Dickey-Fuller Test (ADF), Phillips-Perron Test (PP) to ensure that none of the variables is of I (2) or higher order. The descriptive statistics of unit root tests regarding the order of integration are presented in Table2. According to the given estimations results in Table[START_REF] Abbas | Human capital and economic growth: Pakistan 1960-2003[END_REF], 𝐺𝐷𝑃 𝑡 , 𝑅𝑃𝐶 𝑡 , 𝑒𝑑𝑢 1𝑡 , 𝑒𝑑𝑢 1𝑡 , 𝑒𝑑𝑢 3𝑡 and 𝑒𝑑𝑢 4𝑡 , all these variables are I (1) with constant under the ADF and PP, while 𝐿𝐹 𝑡 is of I (0) with constant under ADF and PP. While the order of integration for all variables is I (1) with intercept and trend according to the PP. According to all criteria, none of the variables are I (2), then the ARDL model to cointegration is indeed the appropriate technique for cointegration.

Table 2 : Unit root tests

 2 1Edu 𝑖𝑡 ln Edu 𝑖𝑡-1 + 𝛿 2Edu 𝑖𝑡 ln RPC 𝑡-1 + 𝛿 3Edu 𝑖𝑡 ln GDP 𝑡-1 + 𝛿 4Edu 𝑖𝑡 ln LF 𝑡-1 www.ijafame.orgThe four terms (𝛽, 𝛾, 𝜃, and 𝜃 ) with summation signs in equations 3 and 4 reflect the error correction dynamics, whereas the second part (terms with 𝛿 i ) represents the long-run relationship. ∆ is the first difference operator. To investigate the long-run relationship between the variables of interest, the first step in the ARDL model was to compute the well-known F-statistic on the differenced variables components of the error correction mechanism (ECM) model for the joint significance of the variables lagged level coefficients. In this first step, the ARDL equation when 𝑒𝑑𝑢 𝑖𝑡 serves as dependent variable is modeled as follows:

						ADF	PP
	Variables				Intercept	Intercept and trend	Intercept	Intercept and trend	Decision
	GDP ∆GDP: 1 st difference				-0.144 (0.9449) -10.790 (0.0000) *	-3.126 (0.1001) -10.885 (0.0000) *	0.113 (0.9669) -9.529 (0.0000) *	-3.049 (0.1190) -9.615 (0.0000) *	I (1)
	RPC ∆RPC: 1 st difference				-0.496 (0.8927) -3.835 (0.0026) *	-1.319 (0.8833) -3.765 (0.0184) **	-0.553 (0.8812) -3.758 (0.0034) *	-1.632 (0.7796) -3.679 (0.0238) **	I (1)
	LF ∆LF: 1 st difference				-6.316 (0.0000) *	0.294 (0.9962) -3.692 (0.0229) **	-4.705 (0.0001) *	0.135 (0.9954) -3.681 (0.0236) **	I (0)
	𝒆𝒅𝒖 𝟏𝒕 ∆𝒆𝒅𝒖 𝟏𝒕 : 1 st difference			0.010 (0.9594) -3.017 (0.0334) **	-2.185 (0.4982)	-0.063 (0.9530) -3.062 (0.0295) **	-2.578 (0.2903)	I (1)
	𝒆𝒅𝒖 𝟐𝒕 ∆𝒆𝒅𝒖 𝟐𝒕 : 1 st difference			1.068 (0.9949) -3.453 (0.0093) *	-2.791 (0.2001)	0.514 (0.9853) -3.513 (0.0077) *	-2.554 (0.3013) -3.134 (0.0984) ***	I (1)
	𝒆𝒅𝒖 𝟑𝒕 ∆𝒆𝒅𝒖 𝟑𝒕 : 1 st difference			2.173 (0.9989) -3.644 (0.0050) *	-0.616 (0.9781)	1.788 (0.9983) -3.688 (0.0043) *	-0.688 (0.9739) -4.530 (0.0013) *	I (1)
	𝒆𝒅𝒖 𝟒𝒕 ∆𝒆𝒅𝒖 𝟒𝒕 : 1 st difference			-1.869 (0.3467) -3.621 (0.0054) *	-0.274 (0.9901)	-1.557 (0.5050) -3.683 (0.0044) *	-0.696 (0.9734) -5.109 (0.0001) *	I (1)
	Figures in parentheses are p-values; *, **, *** indicates significance at 1%, 5% and 10% respectively.
							Source: Authors
			𝑛			𝑛	𝑛
	∆ln𝐺𝐷𝑃 𝑡 = 𝛼 0GDP 𝑡 + ∑	𝛽 𝑖GDP 𝑡 ΔlnGDP 𝑡-𝑖 + ∑	𝛾 𝑖GDP 𝑡 Δln𝑅𝑃𝐶 𝑡-𝑖 + ∑	𝜃 𝑖GDP 𝑡 Δln𝐿𝐹 𝑡-𝑖
	𝑛	𝑖=1		𝑖=1	𝑖=1
	+ ∑	𝜗 𝑖GDP 𝑡 ΔlnEdu 𝑖𝑡-𝑖 + 𝛿 1GDP 𝑡 lnGDP 𝑡-1 + 𝛿 2GDP 𝑡 lnRPC 𝑡-1 + 𝛿 3GDP 𝑡 lnLF 𝑡-1	(3)
	𝑖=1
	+𝛿 4GDP 𝑡 lnEdu 𝑖𝑡-1
	Equation if education serves as dependent variable is: 𝑛 𝑛
	ΔlnEdu 𝑖𝑡 = 𝛼 0Edu 𝑖𝑡 + ∑	𝛽 𝑖𝐸𝑑𝑢 𝑖𝑡 ΔlnEdu 𝑖𝑡-𝑖 + ∑	𝛾 𝑖𝐸𝑑𝑢 𝑖𝑡 ΔlnRPC 𝑡-𝑖
		𝑛	𝑖=1		𝑛	𝑖=1
	+ ∑	𝜃 𝑖𝐸𝑑𝑢 𝑖𝑡 ΔlnGDP 𝑡-𝑖 + ∑	𝜗 𝑖𝐸𝑑𝑢 𝑖𝑡 ΔlnLF 𝑡-𝑖
		𝑖=1			𝑖=1

To analyze our questions, this study use the error-correction version of the ARDL method of equation (

1

) and (2) to investigate the long run (LR) relationship between 𝐺𝐷𝑃 𝑡 , 𝑅𝑃𝐶 𝑡 , 𝐿𝐹 𝑡 and 𝑒𝑑𝑢 𝑖𝑡 by following Pesaran & Pesaran, (1997) and

[START_REF] Pesaran | An autoregressive distributed lag modelling approach to cointegration analysis[END_REF]

, as: +𝛿

  𝐻 0 ∶ 𝛿 1Edu 𝑖𝑡 = 𝛿 2Edu 𝑖𝑡 = 𝛿 3Edu 𝑖𝑡 = 𝛿 4Edu 𝑖𝑡 = 0 against the alternative hypothesis 𝐻 1 ∶ 𝛿 1Edu 𝑖𝑡 ≠ 𝛿 2Edu 𝑖𝑡 ≠ 𝛿 3Edu 𝑖𝑡 ≠ 𝛿 4Edu 𝑖𝑡 ≠ 0 The test can be denoted as: 𝐹 𝑒𝑑𝑢 𝑖𝑡 (𝑒𝑑𝑢 𝑖𝑡 /𝐺𝐷𝑃 𝑡 , 𝑅𝑃𝐶 𝑡 , 𝐿𝐹 𝑡 ). A similar test was conducted for the equation when 𝐺𝐷𝑃 𝑡 serves as a dependent variable, this can also be denoted as follows: 𝐹 𝑒𝑑𝑢 𝑖𝑡 (𝐺𝐷𝑃 𝑡 /𝑒𝑑𝑢 𝑖𝑡 , 𝑅𝑃𝐶 𝑡 , 𝐿𝐹 𝑡 ). If the computed value of the Fstatistic is larger than the upper critical value (upper critical bound), given by Pesaran et al.(2001) and[START_REF] Narayan | Reformulating Critical Values for the Bounds F-statistics Approach to Cointegration: An Application to the Tourism Demand Model for Fiji[END_REF] the variables are cointegrated. Then the null hypothesis of no cointegration is rejected, indicating that there is a long-run relationship among the variables when 𝑒𝑑𝑢 𝑖𝑡 serves as a dependent variable. If the calculated value of the F-statistic is less than the lower bounds critical value, then the variables are not cointegrated in the long term, and the hypothesis of no cointegration cannot be rejected. Finally, if the calculated value falls between lower and upper bounds critical values, then the cointegration test becomes inconclusive.

		𝑛		𝑛
	𝑎 0Edu 𝑖𝑡 + ∑	𝑏 𝑖𝐸𝑑𝑢 𝑖𝑡 ΔlnEdu 𝑖𝑡-𝑖 + ∑	𝑐 𝑖𝐸𝑑𝑢 𝑖𝑡 ΔlnRPC 𝑡-𝑖
	𝑛	𝑖=1	𝑛	𝑖=1
	+ ∑	𝑑 𝑖𝐸𝑑𝑢 𝑖𝑡 ΔlnRGDP 𝑡-𝑖 + ∑	𝑒 𝑖𝐸𝑑𝑢 𝑖𝑡 ΔlnLF 𝑡-𝑖	(6)
	𝑖=1		𝑖=1
		+𝛿 1Edu 𝑖𝑡 lnEdu 𝑖𝑡-1 + 𝛿 2Edu 𝑖𝑡 lnRPC 𝑡-1 + 𝛿 3Edu 𝑖𝑡 lnRGDP 𝑡-1
		+𝛿 4Edu 𝑖𝑡 lnLF 𝑡-1 + 𝜀 1𝑡

To test the cointegration among variables. The F-test was used to examine whether or not longrun correlations exist. The null hypothesis for no cointegration among the variables is given as:

  [START_REF] Narayan | Reformulating Critical Values for the Bounds F-statistics Approach to Cointegration: An Application to the Tourism Demand Model for Fiji[END_REF], lower and upper critical values for bounds testing ARDL for 1%, 5%, and 10% significance levels are[4.61-5.96],[3.27-4.30] and [2.67-3.58] respectively. All the value of the calculated F-statistic exceeds the upper bound even at the 1%, 5%, and 10% significance level in both the[START_REF] Narayan | Reformulating Critical Values for the Bounds F-statistics Approach to Cointegration: An Application to the Tourism Demand Model for Fiji[END_REF][START_REF] Pesaran | Bounds testing approaches to the analysis of level relationships[END_REF], we can conclude that there is evidence of a long-run relationship between our variables in the model. When 𝐺𝐷𝑃 𝑡 serves as a dependent variable, all the indicators of education, indicate that there exists a cointegration relationship among 𝐺𝐷𝑃 𝑡 , 𝑅𝑃𝐶 𝑡 , 𝐿𝐹 𝑡 and 𝑒𝑑𝑢 𝑖𝑡 according to both[START_REF] Narayan | Reformulating Critical Values for the Bounds F-statistics Approach to Cointegration: An Application to the Tourism Demand Model for Fiji[END_REF]; Pesaran et al.

, lower and upper critical values for bounds testing ARDL for 1%, 5%, and 10% significance levels are

[4.29-5

.61],

[3.23- 4.35]

, and [2.72-3.77] respectively.

Table 3 : Cointegration: When 𝑮𝑫𝑷 𝒕 is a dependent variable

 3 ∆𝐺𝐷𝑃 𝑡 [𝐹 𝐺𝐷𝑃𝑡 (𝐺𝐷𝑃 𝑡 /𝑅𝑃𝐶 𝑡 , 𝑒𝑑𝑢 1𝑡 , 𝐿𝐹 𝑡 )] 9.14 22.65 25.66 11.42 Cointegration Cointegration ∆𝐺𝐷𝑃 𝑡 [𝐹 𝐺𝐷𝑃𝑡 (𝐺𝐷𝑃 𝑡 /𝑅𝑃𝐶 𝑡 , 𝐿𝐹 𝑡 , 𝑒𝑑𝑢 2𝑡 )] 9.78 13.20 26.10 22.69 Cointegration Cointegration ∆𝐺𝐷𝑃 𝑡 [𝐹 𝐺𝐷𝑃𝑡 (𝐺𝐷𝑃 𝑡 /𝑅𝑃𝐶 𝑡 , 𝐿𝐹 𝑡 , 𝑒𝑑𝑢 3𝑡 )] 4.83 7.91 32.83 28.92 Cointegration Cointegration ∆𝐺𝐷𝑃 𝑡 [𝐹 𝐺𝐷𝑃𝑡 (𝐺𝐷𝑃 𝑡 /𝑅𝑃𝐶 𝑡 , 𝐿𝐹 𝑡 , 𝑒𝑑𝑢 4𝑡 )]

			Lag length (1,2,3,4)		Decision results
	Cointegration	1	2	3	4	Pesaran et al. (2001)	Narayan (2005)
		3.17 6.54 10.31 12.90 Cointegration	Cointegration
						Source: Author's calculations

Table 4 : Cointegration: When 𝒆𝒅𝒖 𝒊𝒕 is a dependent variable?

 4

			Lag length (1,2,3,4)		Decision results
	Cointegration	1	2	3	4	Pesaran et al. (2001)	Narayan (2005)
	∆𝑒𝑑𝑢 1𝑡 [𝐹 𝑒𝑑𝑢1𝑡 (𝑒𝑑𝑢 1𝑡 𝐺𝐷𝑃 𝑡, 𝑅𝑃𝐶 𝑡 , 𝐿𝐹 𝑡 )]	6.21 6.48 9.41 9.41 Cointegration	Cointegration
	∆𝑒𝑑𝑢 2𝑡 [𝐹 𝑒𝑑𝑢2𝑡 (𝑒𝑑𝑢 2𝑡 𝐺𝐷𝑃 𝑡, 𝑅𝑃𝐶 𝑡 , 𝐿𝐹 𝑡 )]	6.57 5.67 6.02 10.58 Cointegration	Cointegration
	∆𝑒𝑑𝑢 3𝑡 [𝐹 𝑒𝑑𝑢3𝑡 (𝑒𝑑𝑢 3𝑡 𝐺𝐷𝑃 𝑡, 𝑅𝑃𝐶 𝑡 , 𝐿𝐹 𝑡 )]	3.74 3.74 7.23 7.23 Cointegration	Cointegration
	∆𝑒𝑑𝑢 4𝑡 [𝐹 𝑒𝑑𝑢4𝑡 (𝑒𝑑𝑢 4𝑡 𝐺𝐷𝑃 𝑡, 𝑅𝑃𝐶 𝑡 , 𝐿𝐹 𝑡 )]	22.55 8.07 8.07 8.07 Cointegration	Cointegration
						Source:

Author's calculations

  . The following equations are estimated, considering each of the variables as the dependent variable. All equations are reported in appendix 2.

	𝑙𝑛𝐺𝐷𝑃 𝑡 = 𝛼 11 + ∑	4 𝑖=1 𝛽 11𝑖 𝑙𝑛𝐺𝐷𝑃 𝑡-𝑖 + ∑	4 𝑖=1 𝜆 11𝑖 𝑙𝑛𝑅𝑃𝐶 𝑡-𝑖 + ∑	4 𝑖=1 𝛾 11𝑖 𝑙𝑛𝐸𝑑𝑢 𝑖𝑡-𝑖 + 𝑢 11𝑡
	(11)
	𝑙𝑛𝐸𝑑𝑢 𝑖𝑡 = 𝛼 12 + ∑	4 𝑖=1 𝛽 12𝑖 𝑙𝑛𝐺𝐷𝑃 𝑡-𝑖 + ∑	4 𝑖=1 𝜆 12𝑖 𝑙𝑛𝑅𝑃𝐶 𝑡-𝑖 + ∑	4 𝑖=1 𝛾 12𝑖 𝑙𝑛𝐸𝑑𝑢 𝑖𝑡-𝑖 + 𝑢 12𝑡

  𝑡 = 𝛼 17 + ∑

			4		4
	𝑙𝑛𝐸𝑑𝑢 𝑖𝑡 = 𝛼 18 + ∑	𝛽 18𝑖 𝑙𝑛𝐺𝐷𝑃 𝑡-𝑖 + ∑	𝛾 18𝑖 𝑙𝑛𝑅𝑃𝐶 𝑡-𝑖
	+ ∑	𝑖=1 𝑖=1 𝜆 18𝑖 𝑙𝑛𝐿𝐹 𝑡-𝑖 + ∑ 4	𝑖=1 𝑖=1 𝛿 18𝑖 𝑙𝑛𝐸𝑑𝑢 𝑖𝑡-𝑖 + 𝑢 18𝑡 4
			4		4
				𝛽 17𝑖 𝑙𝑛𝐺𝐷𝑃 𝑡-𝑖 + ∑	𝛾 17𝑖 𝑙𝑛𝑅𝑃𝐶 𝑡-𝑖
		+ ∑	𝑖=1 𝑖=1 𝜆 17𝑖 𝑙𝑛𝐿𝐹 𝑡-𝑖 + ∑ 4	𝑖=1 𝑖=1 𝛿 17𝑖 𝑙𝑛𝐸𝑑𝑢 𝑖𝑡-𝑖 + 𝑢 17𝑡 4	(17)
			www.ijafame.org

Table 5 :

 5 Summary causal nexus between education and growth

			Summary of TYAGC tests
	Hypothesis testing	Bivariate	Multivariate with three variables	Multivariate with all variables
		𝑮𝑫𝑷 𝒕 and 𝒆𝒅𝒖 𝒊𝒕	𝑮𝑫𝑷 𝒕 , 𝑹𝑷𝑪 𝒕 and 𝒆𝒅𝒖 𝒊𝒕	𝑮𝑫𝑷 𝒕 , 𝑳𝑭 𝒕 and 𝒆𝒅𝒖 𝒊𝒕	𝑮𝑫𝑷 𝒕 ,𝑹𝑷𝑪 𝒕 , 𝑳𝑭 𝒕 and 𝒆𝒅𝒖 𝒊𝒕
	Education (𝑒𝑑𝑢 1𝑡 )) does granger cause 𝐺𝐷𝑃 𝑡	(0.094)	(0.018)	(0.430)	(0.018)
	𝐺𝐷𝑃 𝑡 does Granger cause education (𝑒𝑑𝑢 1𝑡 )	(0.033)	(0.003)	(0.003)	(0.000)
	Education (𝑒𝑑𝑢 2𝑡 )) does granger cause 𝐺𝐷𝑃 𝑡	(0.000)	(0.000)	(0.000)	(0.000)
	𝐺𝐷𝑃 𝑡 does Granger cause education (𝑒𝑑𝑢 2𝑡 )	(0.000)	(0.000)	(0.000)	(0.000)
	Education (𝑒𝑑𝑢 3𝑡 )) does granger cause 𝐺𝐷𝑃 𝑡	(0.000)	(0.001)	(0.016)	(0.006)
	𝐺𝐷𝑃 𝑡 does Granger cause education 𝑒𝑑𝑢 3𝑡 )	(0.000)	(0.006)	(0.018)	(0.034)
	Education (𝑒𝑑𝑢 4𝑡 )) does granger cause 𝐺𝐷𝑃 𝑡	(0.000)	(0.000)	(0.005)	(0.000)
	𝐺𝐷𝑃 𝑡 does Granger cause education (𝑒𝑑𝑢 4𝑡 )	(0.828)	(0.733)	(0.122)	(0.550)
	Figures in brackets are p-value. *, **, *** indicates significance at 1%, 5% and 10% respectively.

Appendix Appendix 1

Figure 1. The evolution of our variables between 1990 to 2019. [START_REF] Granger | Investigating Causal Relations by Econometric Models and Crossspectral Methods[END_REF]

Source: Authors


domain: spi phys

text:
Florent Feriol
email: florent.feriol@isae-supaero.frdamien.vivet@isae-supaero.fr

Yoko Watanabe

Damien Vivet

GNSS-based environmental context detection for navigation

Environmental context detection is a topic of interest for the navigation community since it enables to build a context-adaptive solution. Indeed if the type of environment is known it is then possible to choose the proper data processing algorithm or to select the sensors to be used to dynamically adapt the navigation solution design itself. This paper proposes to build a supervised machine learning model which can robustly classify multiple contexts such as urban canyons, urban, trees and open-sky areas using GNSS data only. A training and test database have been built with four datasets acquired at different times in order to prove the relevance of the solution. These datasets are made available to the community for research purpose. The choices of features and classifier are also discussed and compared to others papers. Our solution achieved an average 82.40% of classification accuracy.

I. INTRODUCTION

In the recent years, with the addition of new constellations, hardware improvements and the development of new robustification algorithms, Global Navigation Satellite System (GNSS) based positioning has become more efficient than ever. The main issue with the latter is that very often those algorithms are dedicated to mitigate a specific limitation of GNSS which have a tendency to appear in a specific environment (also called context). For example, multipaths mitigation and Doppler aiding algorithms are mainly used in urban areas, shadow matching is exclusively used in urban canyon and the modification of the coherent integration time is useful in indoor or under canopy (see [START_REF] Feriol | A Review of Environmental Context Detection for Navigation Based on Multiple Sensors[END_REF] for more details). This notion of environmental context is therefore considered as a key of future navigation solution since it could help to choose a suitable algorithm and improve the general performance of the solution [START_REF] Groves | Context Detection, Categorization and Connectivity for Advanced Adaptive Integrated Navigation[END_REF]. This context-adaptive idea can also be extended to the navigation filter design itself by, for example, swapping between lose and tight coupling depending on the situation. Besides, such context information could also be useful for a path planning task in robotic applications, in order to ensure the navigation safety by predicting possible degradations of the localization precision.

Different methods already exist to extract the environmental context but require sensors which are not commonly used for navigation applications (cellular, Wifi, magnetometer, light detector) leading to more power consumption, bulkiness and cost [START_REF] Ali | SenseIO: Realistic Ubiquitous Indoor Outdoor Detection System Using Smartphones[END_REF], [START_REF] Zhou | IODetector: A Generic Service for Indoor/Outdoor Detection[END_REF]. Even if multiple visual based approaches for context detection exist, the use of LiDAR or cameras require a huge processing power that should be dedicated to the navigation task by itself. In our opinion, there is a lack of simple and efficient environmental context classifier using GNSS in a kinematic case. Several articles tried to characterize GNSS signals in various environment and showed an evolution of the degradations depending on the context [START_REF] Macgougan | Degraded GPS Signal Measurements With A Stand-Alone High Sensitivity Receiver[END_REF]- [START_REF] Satyanarayana | GNSS channel characterization and enhanced weak signal processing[END_REF]. Hence, it is natural to think of using those degradations as a context detector. As introduced in [START_REF] Feriol | A Review of Environmental Context Detection for Navigation Based on Multiple Sensors[END_REF] the semantic contexts of interest are: urban canyons, urban areas, forest/trees and open-sky. Note that indoor is excluded from our context of interest due to the fact that in deep indoor situation there is not enough visible satellites to compute a position and therefore a lot of features are missing for the classification step. If indoor is a key context for an application, a binary indoor/outdoor classification can be performed before applying the outdoor context detection approach proposed in this paper.

This paper is structured as follows. Section II introduces some existing works on GNSS-based context detection in order to showcase their limitations and the improvements we want to bring. Next, section III presents the strategy used to build the training/testing dataset as well as the acquisition setup. In Section IV, the choice of features used for the classification is discussed. Classifier, pre-processing and post-processing of data are explained in section V. Finally, experiments and results are presented in section VI.

II. STATE OF THE ART

Even if environmental context detection is a well studied problem with visual sensor or multisensor fusion approaches, few works have focus on this task using GNSS data. In [START_REF] Gao | Environmental Context Detection for Adaptive Navigation using GNSS Measurements from a Smartphone[END_REF] both GPS and GLONASS constellations are used to classify the context between indoor, urban and open-sky thanks to a two step processing. First, a Hidden Markov Model (HMM) predicts the indoor/outdoor context probability from means and variances of C/N0 (Carrier to noise ratio) features. Second, if the HMM output is unclear, a Fuzzy Inference System is used to separate open-sky samples from urban ones. Regarding the database, a unique dynamic test trajectory recorded at 1Hz with a smartphone is used. It contains approximately 100 samples. Results mainly focus on the detection capability during transitions. Thus there is no certitude on the reliability of the method for different constellations or localization since only a single test dataset was used.

[6] tries to classify indoor and outdoor using machine learning (ML) algorithms. 36 statistical features are extracted from azimuth/elevation, C/N0, Position/Horizontal/Vertical Dilution of precision (P/H/V-DOP), number of visible satellites (NVS). The database is composed of 195,861 samples recorded at different times with diverse smartphones and meteorological conditions. Results with multiple classifiers are shown and the stacking one, composed of several models, is the most efficient one. They also used a HMM in a post-processing step in order to improve the performances. However, the database is not available making us unable to know what trajectory the user did and at which speed.

[11] also use smartphones for pedestrian application but goes further than the previous paper by classifying four contexts: deep indoor, shallow indoor, semi-outdoor and outdoor. To do so the authors used a Recurrent Neural Network (RNN) model with an input vector of 11 statistical features based on the number of visible satellite and the signal to noise ratio. They showed that the use of multiple constellations could improve the classification results. However the dynamic test dataset is again very limited (only 322 samples) and therefore results cannot be extended to other locations and times.

More recently [START_REF] Liu | Environment Classification for Global Navigation Satellite Systems Using Attention-Based Recurrent Neural Networks[END_REF] proposed a method based on a fusion of RNNs and Fully Convolutional Networks (FCN) to classify in real-time three road navigation contexts: urban canyon, trees and open sky. Note that tree class is a road bordered by trees. The presented results show good performance. Nevertheless, a lot of information is missing. First, there is no information on the receiver and the antenna used. Second, the exact structure of the features vector is not mentioned and is mixed with features extracted with RNNs (which need a 256*11 historical data matrix as input -representing 50 seconds of signal). The architecture and the hidden size of the proposed RNN is not provided. Lastly, no information on the learning phase, the training and testing database (training epoch, batch size) are shared making the reproduction of the work impossible. More over, the system is based on the analysis of 50 seconds of signal making it more suitable for static system or with slow movement than for real navigation cases.

Finally, [START_REF] Wang | Urban environment recognition based on the GNSS signal characteristics[END_REF] tries to recognize six urban contexts: urban canyon, semi-urban, suburb, viaduct-up, viaduct-down and boulevard (roads bordered by trees). The definition of those contexts seems unclear since some classes look very similar to others. The authors proposed to use a five dimensional vector as input of a Support-Vector Machine (SVM) model. The vector is made of the mean µ and standard deviation σ of signal strength attenuation A i (t) defined in (1), blockage coefficient α (2) , GDOP expansion ratio λ (3) and the strength fluctuation coefficient β which are all extracted from the GPS L1 band.

A i (t) = P std i (θ) -P i (t) (1)
where P i (t) is the measured signal strength of satellite i at time t, and P std i (θ) is the standard strength with no attenuation for each elevation θ.

α(t) = 1 - N visible (t) N all (t) (2)
where N visible is the number of visible satellites and N all is the number of all satellites for the same constellation in the sky which can be obtained from the almanac.

λ(t) = GDOP (t) GDOP std (t) (3)
where GDOP (t) is the value computed by the receiver and GDOP std (t) is the GDOP value calculated for all the satellites in the sky. Finally, the strength fluctuation coefficient β is the standard deviation of the strength attenuation (1) in 1 second. The training data have been acquired in Shanghai (100 000 samples) at 5 Hz using an Ublox-M8N placed on the top of a vehicle. The test dataset have been recorded in Nanjing with the same setup. Data are pre-processed thanks to a standardization in order to assure a similar scale for the different features. A one-vs-all SVM model is then trained using the Shanghai data. The last step is to perform a Bayesian filtering at the output of the SVM classifier in order to improve the results by avoiding the random missclassifications. The results are impressive but the ground truth shared is not very precise and the database is not available. Also, in order to compute the feature vector it is mandatory to obtain a position at each epoch and to have access to the almanac which is not always possible. In summary, there are only few works dealing with GNSS signal based context detection. Even if existing works seems to give good classification results, they are based on small private datasets often obtained from smartphones held by pedestrians. It is important to note the relevance of having a large dataset with different conditions since it is the only way to ensure a generalizable solution. Also, the database are built in major chinese cities with contexts that can really differ from those found in Europe. None of these works can be reproduced as there is an obvious lack of public GNSS signal database. Therefore, contributions of this paper are twofold: first to propose a new GNSS-based environmental context detection methodology, and second to provide to public a GNSS signal database with labeled contexts1 .

III. DATASET

In order to acquire the database, a Traxxas rover (Fig. 1) equipped with a Ublox EVK-M8T receiver and a NovAtel VEXXIS GNSS-804 antenna was used. Data are collected at 5Hz thanks to a ROS platform. Compared to previous works dealing at 1Hz in static or low speed situations, we choose to increase the recording frequency to be able to deal with high speed situations. Moreover, as proven in [START_REF] Xia | Recurrent neural network based scenario recognition with Multi-constellation GNSS measurements on a smartphone[END_REF], using multi-constellations could improve the detection capability thus the receiver has been configured to track GPS and Galileo signals. Given the rover's average speed of 4km/h the dynamic platform model of the GNSS receiver is set to portable.

To build a database, four experiments have been conducted on the ISAE-SUPAERO campus in Toulouse (France) on different days/hours and under sunny/cloudy conditions. A summary of the tests can be found in Table I. The training database has only been labeled when the context was obvious I is obtained after the filtering. Finally, to simulate a spatial independence for the tests some areas have been deleted for the training step (see Fig. 2). As a result there is no overlapped data point in training/testing subsets (Table VII).

IV. FEATURES OF INTEREST

In order for the proposed approach to be widely applicable, only the classical outputs which are accessible to end-users of standard off-the-shelf GNSS receiver are used to construct the feature vector. For instance, it excludes the use of correlators information. As using information from multiples constellations can improve the detection by reducing random outliers, features will be declined in their GPS and Galileo version at each epoch, except for the features based on the solution (such as the DOP feature). The first feature selected is the number of visible satellites (NVS) as done in [START_REF] Zhu | A Fast Indoor/Outdoor Transition Detection Algorithm Based on Machine Learning[END_REF], [START_REF] Xia | Recurrent neural network based scenario recognition with Multi-constellation GNSS measurements on a smartphone[END_REF], [START_REF] Wang | Urban environment recognition based on the GNSS signal characteristics[END_REF]. It gives a lot of information on the environment since its value will be very different in open-sky situations compared to urban canyon ones. However the NVS stays high in situations other than opensky like under canopy, making this feature not sufficient to distinguish every context. In such areas the satellite signal quality should be impacted by multi-paths and/or attenuation due to branches and leaves. That is why a second feature based on the NVS is used but with a threshold on the C/N0 as done in [START_REF] Gao | Environmental Context Detection for Adaptive Navigation using GNSS Measurements from a Smartphone[END_REF]:

M ask(i) = 1, if C/N 0 i > ξ 0, otherwise (4)
N V S f iltered = i∈Sv M ask(i) (5)
with i the satellite ID, S v a set of IDs of the visible satellites (i.e. i ∈ S v if the i-th satellite is visible), and ξ the threshold value. In our case we found empirically that a proper value for the threshold was 30dB.Hz. This threshold has been obtained by looking at the evolution of the signal quality when coming from an open-sky environment to under trees. As mentioned before it seems mandatory to track the evolution of the C/N0 since it gives an idea on the signal quality. However in kinematic situation it is prone to a lot of variance (Fig. 3a). This problem is surprisingly not explained in the different papers that try to classify the context in real time [START_REF] Liu | Environment Classification for Global Navigation Satellite Systems Using Attention-Based Recurrent Neural Networks[END_REF] [13]. To resolve this matter we proposed to smooth the signal by using a filtering window on the last 10 samples (2s in our case) in which the mean and the variance are computed (6, 7) for each satellite (Fig. 3b). In addition, since an information on the whole constellation is needed, the mean of both the indicators is computed over all the satellites available [START_REF] Macgougan | Degraded GPS Signal Measurements With A Stand-Alone High Sensitivity Receiver[END_REF][START_REF] Ma | Gps signal degradation modeling[END_REF].

µ C/N 0i (t) = 1 10
9 k=0 C/N 0 i (t -k) (6)
σ 2 C/N 0i (t) = 1 10 9 k=0 C/N 0 i (t -k) -µ C/N 0i (t) 2 (7)
µ C/N 0 (t) = 1 N V S(t) i∈Sv µ C/N 0i (t) (8)
σ 2 C/N 0 (t) = 1 N V S(t) i∈Sv σ 2 C/N 0i (t) (9)
After relying on the signal quality to extract information about the context, the receiver position solution could also give additional information. Indeed it is possible to estimate the error on the position which should be larger in cluttered environment. From the Ublox-M8T we can extract the horizontal/vertical accuracy and the position dilution of precision PDOP. For commercial reason, there is no access on how those values are calculated but they seem to be related to the position estimation error covariance. With the same idea, the pseudorange residuals which enlighten errors on pseudoranges (that can be due to multipaths among others) is also chosen as a feature:

res(t) = i∈Sv |ρ m,i (t) -ρ e,i (t)|, (10)
with ρ m the measured pseudorange, ρ e the estimated pseudorange. Here we decided to use the sum over the mean in order to avoid an over-smoothing of the values.

To have a better understanding of the constellation we can also use the elevation of each satellite as proposed in [START_REF] Zhu | A Fast Indoor/Outdoor Transition Detection Algorithm Based on Machine Learning[END_REF]. In cluttered environment the mean elevation will tend to be higher since tall structures will occlude the signal of low elevation satellites:

elev(t) = 1 N V S(t) i∈Sv e i (t), (11)
with e i the elevation of the i-th satellite. Finally, a 15 dimensional feature vector is obtained for each epoch:

v(t) =[N V Sgps, N V S gal , N V S f ilteredgps, N V S f
V. PRE-PROCESSING, CLASSIFIER AND POST-PROCESSING

A. Pre-processing

Once the feature vector is created, it still has to be preprocessed before using it as input of a classifier. To guarantee a similar scale for each feature, they need to be normalized since an SVM classifier will be used as justified later. To do so, the Z-score standardization ( 13) is used under an assumption of a normal distribution.

X = X -µ σ , (13)
To further improve the uniformity between small and high values, the sigmoid function define as follows is applied:

f ( X) = 1 1 + exp -X , (14)
Those standardization steps are important due to the fact that SVM classifiers try to find the optimal hyperplane between two classes and therefore are sensible to magnitude differences. At this point, we applied Principal Component Analysis (PCA) in order to verify the segmentability of our data (Fig. 4). It can be seen that except for the urban/open-sky case the different contexts are classifiable. It is important to mention that only the two main components are plotted on Fig. 4 and that the classifier will not be in this space. To compare the segmentability of our feature vector with the one proposed in [START_REF] Wang | Urban environment recognition based on the GNSS signal characteristics[END_REF], Fig. 5 shows the PCA plot for the feature vector which relies only on the five features listed earlier in Section II. It can be seen that the data are way more mixed up and that the resulting classification should be less efficient since 72.45% of the variance is represented here.

B. SVM classifier

As mentioned before, we chose to use an SVM classifier for the following reasons. The first is to avoid the randomness of algorithms such as boosting and bagging. Second, it allows to estimate posterior probabilities which can be useful for filtering/smoothing. Finally its fine tuning is more understandable than a black box such as deep learning solutions (SVM has only few hyperparameters). For multi-class problem, it has been shown that error-correcting output codes (ECOC) model can improve classification [START_REF] Dietterich | Solving Multiclass Learning Problems via Error-Correcting Output Codes[END_REF]. Indeed it reduces the possibility of wrong classification because each learner will find a single bit of the code whereas in other methods each learners define a class. Therefore this type of model will be used with SVM learners. The one-vs-one classifier is adopted due to the fact that it is less computationally-expensive than one-vs-all for training and performs better most of the time.

We then obtain K * (K -1)/2 (K being the number of where 1 means that the class will be positive, -1 negative and 0 ignored during the training. It is possible to fine tune the type of kernel as well as the soft-margin and kernel scale hyper-parameter of the SVM learners. After multiple tests, it has been found that the gaussian kernel is the most efficient one with γ = 15.9631. The best soft margin parameter C is 4.0364. Those parameter have been found thanks to a Bayesian optimization.

1 1 1 0 0 0 Open-sky -1 0 0 1 1 0 Trees 0 -1 0 -1 0 1 Urban 0 0 -1 0 -1 -1

C. Post-processing

Since the outputs of an SVM classifier are probabilities, it is possible to filter the result by applying a Bayesian approach. The prior probabilities are initialized uniformly for each context to 0.25 in order to avoid to favour a class over the others:

P (c 0 |m 0 ) = [0.25, 0.25, 0.25, 0.25] (15)
The posteriors of the next epoch can be computed using the previous probabilities as priors by Bayes' rule (16):

P (c t |m t ) = P (m|c) * (P trans • P (c t-1 |m t-1 )) P (m) , (16)
with P (m|c) provided by the SVM, P trans is the transition matrix between the contexts defined as follows (Canyon, Open-sky, Trees, Urban)(17). In order to define this transition matrix, we analyse the existing transitions in the dataset to get the transitions probabilities:

P trans =    
0.9 0.01 0.01 0.08 0.01 0.9 0.045 0.045 0.01 0.045 0.9 0.045 0.1/3 0.1/3 0.1/3 0.9

    (17)
The context giving the highest probability of P (c t |m t ) is then considered as our predicted label.

VI. EXPERIMENTS AND RESULTS

A. Setting and performance metrics

To show the relevance of our feature vector choice, different input vectors and pre-processing have been tried. Firstly, only GPS data are used in the input vector and the standard Z-score normalization [START_REF] Wang | Urban environment recognition based on the GNSS signal characteristics[END_REF] is the only pre-processing (SVM-G). Then the Galileo constellation has been added into the input vector as shown in [START_REF] Liu | Environment Classification for Global Navigation Satellite Systems Using Attention-Based Recurrent Neural Networks[END_REF] (SVM-GGa). In a third time the normalization is improved by the use of the sigmoid function (14) in addition (SVM-GGaS). Finally, the post processing Bayesian filtering presented in ( 16) is performed (SVM-GGaSB).The model has been trained using three of the four trajectories and using the last one as a test. From each input vector we can extract the recall (18), the precision (19) and the F1-score (20) to demonstrate the performance of our solution. This evaluation summary is 4-folded.

Recall =

T rue positive T rue positive + F alse negative

B. Results

A comparison of the performances of the classifier for the four different input vectors is available in Table III where the best classifier is the one using GPS/Galileo features, sigmoid standardization and Bayesian filtering. The arctan function has also be tested instead of sigmoid but obtained inferior results. We also tried to fuse GPS and Galileo features as done in [START_REF] Liu | Environment Classification for Global Navigation Satellite Systems Using Attention-Based Recurrent Neural Networks[END_REF] but ended up with worse results. Because each constellation has its own characteristics it is better to keep their features independent. This result confirms the importance of using multiple constellations as mentioned in [START_REF] Xia | Recurrent neural network based scenario recognition with Multi-constellation GNSS measurements on a smartphone[END_REF]. We also tested different training/testing datasets folding in order to validate the reliability and robustness of our solution with reference to: different constellation geometries (time of acquisition), different weathers. The F1-score for each test is available in Table IV. To see if the spatial distribution of the constellation has an impact on the classification results, the model is now trained using only the Dataset 1 and each of the remaining trajectories (Dataset 2, Dataset 3 and Dataset 4) are used as test (Table V). This table shows that F1-score is very close to those in Table III which is a promising result considering that a single dataset has been used to build the training database. This result shows that our solution is robust to a difference in time and that the constellation has a limited impact in a In order to check if a specific context is leading to larger number of wrong classifications, the confusion matrix for the SVM-GGaSB classifier is given in Table VI. It can be seen from this table that every context has a correct classification rate of at least 76.5%. The high classification rate of the canyon samples can be explained by its limited spatial distribution. To visualize where the errors are located, the predicted labels are plotted on a satellite view (Fig. 6). It can be seen that many missclassifications are coming from transitions or areas with a mix of contexts (such as trees close to buildings). Scores can also be plotted to have a better understanding of the Bayesian filtering (Fig. 7). Those values provide a confidence level of the context detection results, and hence could be very useful for decision-making of navigation filter adaptation or path planning application. As a final test, testing samples have been filtered depending on their position to prove that our solution is spatially generalizable. If a sample has a position at a distance of at least 3 meters from the closest point in the database the sample is kept otherwise it is filtered. We can then extract the confusion matrix built only on those samples (Table VII). This table shows that similar results than the classifier without filtering (Table VI) are achieved which demonstrates the spatial robustness of our solution.

VII. CONCLUSION

In this paper we proposed to used classical GNSS features provided by any off-the-shelves receiver and additional signal processing for context classification. We showed the possibility of using GNSS signals to extract the environmental context information with a mobile platform with an average F1-score of 0.8240 (with a single dataset as training). We also showed that the methodology is robust to different constellation geometries and weathers. Considering the dataset, there is no GNSS dataset available in the communities making any comparison with state-of-the art solutions very complex. We decided to provide publicly this first set of experiments to the navigation research community. This dataset is going to be enriched and completed with on-coming experiments from bigger area and towns. In future works, we plan to provide additional synchronized sky-facing fisheye images. We think that the context detection capability can be improved with additional information coming from such sensor with few

Fig. 1 :

 1 Fig. 1: Our rover setup at ISAE-SUPAERO. The robot is equipped with GNSS receiver and cameras for context ground truth labelling and visual confirmation.

Fig. 2 :

 2 Fig. 2: Satellite view of the training database ( :Urban, :Canyon, :Trees, :Open-Sky) and the ground truth trajectory in yellow

Fig. 3 :

 3 Fig. 3: C/N0 analysis in static and dynamic situation (a), and with the results of our proposed smoothing strategy (b)

Fig. 4 :Fig. 5 :

 45 Fig. 4: PCA plot of the two main components (60.66 % of the variance) with our set of features

F 1 -

 1 score = 2 * P recision * Recall P recision + Recall (20)

Fig. 6 :

 6 Fig. 6: Satellite view with the predicted labels ( :Urban, :Canyon, :Trees, :Open-Sky)

Fig. 7 :

 7 Fig. 7: Confidence score of the context prediction for each points of the tested trajectory

TABLE I :

 I List of the different recordings (SoR:Start of Recording)

  iltered gal , Hacc, V acc, P DOP, µ C/N 0gps , µ C/N 0 gal , σ 2 C/N 0gps , σ 2 C/N 0 gal , elevgps, elev gal , resgps, res gal ]

TABLE II :

 II Coding matrix of our ECOC model classes) binary learners and the coding matrix (Table. II)

TABLE III :

 III Average recall, precision and F1-score for each classifier (G:GPS, Ga:Galileo, S:Sigmoid, B:Bayesian filtering)

	Classifier	Test D 1	Test D 2	Test D 3	Test D 4
	SVM-G	0.8535	0.8568	0.8012	0.7662
	SVM-GGa	0.8621	0.8458	0.8372	0.7903
	SVM-GGaS	0.8516	0.8612	0.8385	0.8057
	SVM-GGaSB	0.8625	0.8707	0.8410	0.8124

TABLE IV

 IV

	: F1-score for each classifier (G:GPS, Ga:Galileo,
	S:Sigmoid, B:Bayesian filtering) and each dataset

TABLE V :

 V Recall, precision and F1-score for each classifier (G:GPS, Ga:Galileo, S:Sigmoid, B:Bayesian filtering) using a single trajectory as training dataset local environment, which is an unexpected result. Effectively there is a time difference of 1h30 between Dataset 1 & Dataset 2 and of 3h30 between Dataset 1 & Dataset 3\4. As a reminder the constellation period is about 12 hours. The same test has been conducted but with respectively the Dataset 2, Dataset 3 and Dataset 4 as training database and the remaining trajectories as test. The average F1-score is 0.8176 (respectively 0.8165, 0.8122 and 0.8240 for the three others training/testing configurations).

TABLE VI :

 VI Confusion matrix for the SVM-GGaSB classifier

TABLE VII :

 VII Confusion matrix in the filtered areas

Y. WATANABE is from ONERA (DTIS), Toulouse, France yoko.watanabe@onera.fr

The database used in this paper is available at: https: //gitlab.isae-supaero.fr/d.vivet/GNSS_based_ context_detection and to be completed with new experiments

ACKNOWLEDGMENT This work was supported by the Defense Innovation Agency (AID) of the French Ministry of Defense (research project CONCORDE N°2019 65 0090004707501).


Tim Harrison

Pascal Chanez

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Giorgio Walter Canonica

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