State intervention in the economic system is controversial both in principle and in scope. Several conceptions of the role of the state have marked economic thought. For classical economists, the role of the State must be limited to the exercise of its sovereign functions and the guarantee of the free operation of the market. According to the Keynesian vision, the state can play a decisive role in economic recovery. For Keynesians, the state must pay particular attention to the well-being of the population and, if necessary, intervene in the economy to correct the dysfunctions of the market. Indeed, when the national income falls and private expenditure decreases, the state must support the economy by injecting additional expenses. This injection supports the effective demand which in turn influences the level of production and consequently on employment: it is the Keynesian multiplier. According to Jean-Baptiste Say 1972, any supply creates its own demand, the overall increase in the supply of goods produced leads to an increase in income and therefore to an increase in general demand. In other words, if a producer sells a commodity the money received allows him to buy other goods. So there is a request for an equivalent amount. Say's law, supporte by Ricardo but rejected by Malthus, implies that growth depends on supply (producers) and not on demand (consumers). relies on the role played by public investment, ie the accumulation of public capital, in growth:for certain authors public infrastructures (roads, airports, energy distribution network electrical water distribution network etc.) and purification, publics services, socials infrastructures, schools, hospitals, jails stimulate the productivity of private agents and in turn economic activity [1-2]. But with growth the state will take more taxes and taxes so it can finance new infrastructure. A virtuous circle emerges: public investment promotes growth and growth in turn promotes public investment. It is generally accepted that public investment is a determining factor in a country's economy and is considered a factor of sustainable economic growth. With the recent revival of interest in the role of investment in the economy, a review shows that many studies have been conducted over the past two decades, to determine the role of public investment in the economy. The majority of studies suggest that the effect of public investment on growth is positive. Recognized for leaning toward interventionist state measures, Keynes argues that public investment as well as any other current public spending can enrich the community. However, according to although this link is positive, it is not significant. These reflection will be contradicted very early by many researchers, including who say that, in addition to being positive, the correlation between public investment and growth is strong. The conceptual divergence, relating to the role of public expenditure in economic growth, is also perceptible in the results of empirical work [3]. Various studies, based on chronological or international approaches, aimed at assessing the contribution of public spending to contemporary economic growth. The relationship between public spending and economic growth is analyzed according to different approaches: a first, which attempts to determine the sign of the causality between the growth of the national income and the growth of the State's share in the economy; a second one that looks at the effects of public spending on economic growth; finally, the third, resulting from the new theories of growth, is an extension of the previous approach. The first approach, seeking to establish the causal link between public spending and economic growth, leads to very controversial empirical results. Indeed, arguing that public spending is at the root of economic growth or vice versa is the subject of heated debate and the empirical results vary in one sense, others reciprocal causality. Some empirical work, inspired by the work of the economist Adolph Wagner, explains that public expenditure is a consequence rather than a cause of national income; they oppose works inspired by the Keynesian logic according to which public expenditures would cause economic growth in his study of the impact of public investment on economic growth, has also shown by means of causality tests that it is growth that induces public investment and not investment and not the reverse in ivory coast. Concerning the studies attempting to establish a two-way causality between these two variables, we can cite among others [4]. Who found that there is no two-way causality in the WAEMU countries. Nubukpo shows that, in the short term, total public expenditure has no effect on growth, but taking into account the composition of expenditure has many short-term effects on the economies of WAEMU countries. The second approach to studying the relationship between public spending and growth is to analyze the role of public spending in the process of economic growth. Landau in his search for the factors of growth questioned the link between economic growth and the level of public expenditure. It shows that public consumption negatively influences economic growth. In contrast to Landau, Ram and Aschauer highlight a positive effect of public spending on growth. They explain that an increase in the size of the investments the State contributes to increase the private investments by the externality that this procures. Ram further shows that this externality is stronger in low-income countries. The theorists of endogenous growth drive another approach to the analysis of the link between public spending and economic growth. The point is now to highlight the differential impact of public expenditure components on GDP growth. Lucas's work highlights the role of human capital investment in the growth process. Thus, the role of the state is reinforced by these theorists of endogenous growth. Using the VAR model to analyze the dynamic effects of public investment for six industrial countries, Swrm finds that public investment in infrastructure positively influences growth. This paper seeks to analyze the dynamics of public investment in Benin's economic growth. It relies mainly on public investment in the infrastructure and rural sectors [5].

The work is based in general on a Cobb Douglas production function with constant elasticity of the product with respect to each of the factors. The specification of this function differs, however, as regards the introduction of a capacity utilization rate which take cyclical aspects into account or a temporal trend, according to some authors, those variables would replace infrastructures as explanatory factors. As for the econometric methods, the estimates in first differences question the results obtained in level. Cointegration is a more satisfactory approach, as is the econometrics of panel data, despite being highly sensitive to the specification of individual effects. In fact, R. Barro estimates that public expenditure can go back either to the utility function of consumers (perfect substitution) or to the production function of firms (complementarity between public and private expenditure). The private profit function can increase the marginal profitability of capital, and thus stimulate investment instead of ousting it. In addition to the basic models of Aschauer and Barro, several other studies have used formalization and the econometric tool to model the infrastructure investment sector. Some argue that studies should instead work on cost functions that can disentangle the effects of infrastructure, economies of scale, and the ongoing effects on costs and the relationship between cost and production. Public investment is an area in which, even without a cost-benefit analysis minute examples, one can discover extreme examples of expensive and useless achievements [6]. On the other hand, in countries, overall public investment does not seem to contribute significantly to economic growth, and inefficiency is probably general (IMF, 1992).

From the literature review, analysis of the effect of public investment on economic growth is done using panel data or instantaneous data. However, beside this method of analysis, the traditional method based on time series has been developed, which represents in our study the method of analysis [7]. The latter is achieved while ensuring the stationary series with the test Dicker-Fuller Augmented order to proceed with the estimation of the model parameters while checking the non-violation of the assumptions for model validation. Indeed for the validation of the model three important tests are carried out each by the highlighting of two hypotheses. Thus the null hypothesis is accepted when the probability is greater than 5%. It is Jaques Bera's normality test for errors, which makes it possible to check whether the errors follow a normal distribution. Thus the following hypotheses are highlighted.

H0: Presence of normality of errors

H1: Lack of normality of errors

The autocorrelation test for Breusch-Godfrey errors, which provides information on the actual presence of an autocorrelation of errors according to the following hypothesis:

H0: No autocorrelation of errors

 H1: Presence of autocorrelation of errors

The Breusch-Pagan heteroscedasticity test to check the heteroscedasticity of the errors. It follows the following assumptions:

 H0: Homoscedasticity of errors

 H1: Heteroskedasticity of errors

Two software will be used for this analysis: Eviews and Excel.

After the stationarity test we make sure of the existence of a short-term as well as long-term relationship between the explained variable and the explanatory variables by the Bound test method. Most studies of the effect of public investment on economic growth favor the VAR model based on stationarity of series. This technique has some limitations, especially with the macroeconomic and financial series. To overcome this problem, the ARDL approach defined by Pesaran, Shin and Smith (2001) is important for testing the existence of long-term and short-term relationships. The ARDL is a least squares regression containing explained variable delays and explanatory variables. It is generally specified with the  where p is the number of delays of the explained variable and   

the delays of the explanatory variables. Based on the Co-integration procedure (PSS) we examine the existence of a long-run equilibrium relationship without worrying whether the underlying variables are I (0) or I (1). Pesaran had two sets of critical values ??taking into account the number of variables used in the regression (k), and taking into consideration whether the model contains a constant and / or a trend. One set assumes that all variables are I (0) and another set assumes they are I (1). This provides a band covering all possible classifications of the I (0) and I (1) variables. This is the bounds test. Because of the flexibility it offers, this technique is increasingly being used as an alternative to the usual co-integration tests. This technique is better suited to small samples and offers the opportunity to jointly deal with long-term dynamics and short-term adjustments. The PSS approach is done in three steps. It is first of all an estimation of the MCE error correction model then of carrying out the joint nullity test of the long term multipliers and using the F-test and finally we calculate the Correction Term of Errors (ECT)

Estimation of the Error Correction Model (ECM).

 Where the coefficients  are the long-term multipliers; the parameters  are the short-term dynamic coefficients and  is a white noise that is not self-correlated with the differentiated variables and with the delayed values.  Joint nullity test of long-term multipliers and using the F-test At this level we proceed to the joint significance test of the long-term multipliers    using the Fisher test after checking for the absence of residues [8].

Calculation of ECT.

The ECT (error correction term) can be calculated from the long period relation as follows:

In the presence of a cointegration relationship, the short-run effects will therefore be examined on the basis of the following equation:

On the other hand, note that in the event of rejection of the cointegration hypothesis, the effects will be tested only in the short term and estimates based on VAR modeling.

Model specification The model used is an original Cobb-Douglas growth model that looks like this:

Choice of variables

To estimate the econometric model, while drawing on the literature review, the dependent variable is GDP and the explanatory variables are as follows: Global Public Investment (GPI): This is the total public expenditure to improve or increase production capacity. Private Investment (PI): It is the total expenditure of households or companies to increase the quantity or improve the quality of factors (natural resources, labor and capital) used in the production of goods and services. Active Population (PA): All persons of working age are available on the labor market, whether they are employed or unemployed. Investment in the Infrastructure Sector (ISI): It combines government spending on the infrastructure sector [9]. Investment in the Rural Sector (SRI): it brings together government spending on the rural sector. The data subject of our study are annual and taken from the databases of International Institutions (the IMF and the World Bank), the National Institute of Statistics and Economic Analysis (INSAE), the Central Bank West African States (BCEAO) and the General Directorate of Public Investment Monitoring Programming (DGPSIP). They cover the period 1989-2016 due to the availability of data on public investments.

Descriptive analysis

Variation of gross domestic product

Figure 1 shows the evolution of the GDP over the period from 1989 to 2016. From the observation of this figure is revealed a tendency to rise of the GDP which is explained by a strong growth of the GDP passing of 507.754 million FCFA in 1989 to 5.088.991 million FCFA in 2016. The average growth rate over the period of the analysis is 2%. 

Figure 1: Evolution of GDP from 1989 to 2016.

Variation in Global Public Investment (IPG)

The analysis in Figure 2, showing the change in overall public investment from 1989 to 2016, shows an upward trend. With a strong oscillation, the strongest investment recorded is that of 2015, which 345,902 million FCFA is. On the other hand the lowest investment recorded is that of 1989 that is 37 902 million FCFA.

Figure 2: Evolution of the change in overall public investment over 1989 to 2016.

Variation in public investment in the rural sector (ISR) and in the infrastructure sector (ISI)

Figure 3 below shows the variation in investment in the rural sector and in the infrastructure sector from 1989 to 2016. The observation made on this figure reveals on the one hand an upward trend of investment in the infrastructure sector despite some remarkable falls. The highest investment during this period was CFAF 127,421 million recorded in 2015, while the lowest investment was CFAF 7,168 million in 1989. On the other hand, it reveals a variation in rural investment around the average is 16,379 million FCFA. However, the lowest investment in the rural sector is 116.654 million FCFA in 1989, while the largest investment is 35.483 million FCFA in 2009.

Figure 3: Evolution of the variation of public investments in the rural sector and in the infrastructure sector from 1989 to 2016.

Variation of Private Investments (IP)

The analysis of figure 4, which shows the variation of private investment from 1989 to 2016, shows an upward trend with a peak of 905 400 million FCFA in 2013 observed despite the flickering. The lowest private investment is 21,600 million CFA francs in 1989.

Figure 4: Evolution of the variation of private investment between 1989 and 2016.

Results of the econometric analysis

Augmented Dickey-Fuller Test (ADF)

The Dickey-Fuller test is used to know the order of integration of the different variables. The application of this test shows the results that are presented in the table below. Indeed, it is found that the variables in question, GDP, labor force and overall investment as investment in the infrastructure sector are integrated order 1. On the other hand, private investment and investment in the rural sector are integrated of order 0 (Table 1).

Table 1: Signes about hypothesis verification.

Bound test

The results of the Bound test to ensure the long-term relationship between the variables are summarized in the table below. From this test, it is found that the obtained statistic (8.26) is greater than the upper bound of the Bound of the value interval at the conventional threshold of 5%. It follows from this result that we have enough statistical evidence to accept the presence of a long-term relationship between the variables.

Estimation of the analysis model

The optimal ARDL model selected on the basis of the Akaike Information Criterion (AIC) is an ARDL (1, 1, 1, 1, 1, 1). The probability of the Fisher test, below the 5% threshold, is indicative of the overall significance. In addition to reading this model (see appendix), all the variables are significant at the conventional threshold of 5%. The value of R2 indicates that at 99.54%, economic growth is explained by the variables global public investment, investment in the infrastructure sector, in the rural sector, for private investment and for the labor force. The writing of the model is as follows:

LPIB = - 22.610*** - 0.470*** × (LIPG) + 0.178*** × (LIP)

+ 0.156*** × (LISR) + 0.385*** × (LISI) + 2.358*** × (LPA)

(***) Represente the significant at 1%

Table 2: Test de racine unitaire.

We also try to find the optimum level of infrastructure investment by puting the variable ISI square in the model. We will make the first derivative of LPIB with respect to ISI. We will set the first derivative equal zero. This will help us to find the threshold above which the relation between GDP and investment in infrastructure will become negative. The model estimation is the following one:

LPIB = -26,986 -0,352*** × LING+ 0,216*** × LINVP +0,106*** × LIRUR       

-0,07*** × LISI² + 1,709*** × LISI +2,158*** × LPOA

(***) represente the significativity level at 1%

After making the first derivative, we have this equation

-0,07× LISI² + 1,709 = 0 thus LISI = 12.207142, this means that

ISI = 200. 213.000.000

This help us to know that above 200. 213.000.000 the impact of infrastructure investment on GDP is positive and beyond this level the impact becomes negative.

Model validation test

Following the validation tests of the model presented in Appendix 4 and summarized in the table below, it follows that the Jarque-Bera test for the normality of residues and errors follows a normal distribution. As for the heteroscedasticity and autocorrelation tests, they reveal respectively that the residues are homoscedastic and are not correlated.

Analysis of the results and validation of the hypotheses

Analysis of results

Analysis results obtained show that over the study period from 1989 to 2016, there was economic growth with an average growth rate of 2%. This economic growth is explained by the mixing of several interacting factors. These are global public investment, investment in the infrastructure sector, the rural sector, private investment and the labor force. Indeed, the size of the labor force and private investment have a positive influence on economic growth. A 1% increase in the size of the labor force and private investment led to an increase of 2.35% and 0.17% respectively in economic growth. On the one hand, this positive influence between economic growth and the active population can be explained by the importance of the active population on productivity. Indeed, the increase in the labor force means an increase in human capital. The latter, which boosts productivity, contributes positively to economic growth. On the other hand, the positive influence of private investment on economic growth can be explained by the strong contribution of households and businesses through their productive activities. For public investment in the rural sector like in infrastructure also positively influences economic growth. A 1% increase in investments in the rural sector and infrastructure led to an increase of 0.15% and 0.38% respectively in economic growth (Table 3).

Table 3: F-stat resulting from the test of the existence of a long- term relationship between the variables (bound test).

This can be explained by the high intensity of investments in the agricultural sector, thus making it possible to increase productivity in the rural sector and, secondly, by investing in transport through road construction. In addition, overall public investment negatively influences economic growth. A 1% increase in overall public investment, decreases economic growth by 0.47%. This negative effect of overall public investment on economic growth can be explained by low investment, misallocation of resources and poor investment orientation in the productive sectors, key sectors essential for economic growth. In the long run, the resulting statistic (8.26) is greater than the upper bound of the Bound from the value range to the conventional 5% threshold. There is a long-term relationship between the variables. Moreover, the positive sign of the coefficient of the size of the active population, of private investment, of public investments in the rural sector as well as in infrastructures show that all these variables have a positive impact in the long term on economic growth.

Validation of hypotheses

Hypothesis 1

This first hypothesis states that "an increase in public investment in the infrastructure sector in Benin favors economic growth". The variable public investment in the infrastructure sector must have a positive and significant coefficient. The results show that the elasticity of public investment in the infrastructure sector is positive in the long run. Hypothesis 1 is validated (Table 4).

Table 4: Summary of different validation tests.

 Hypothesis 2

This assumption states that "public investment in the rural sector has a positive and significant effect on economic growth". This means that investment in the rural sector must not only have a positive coefficient, but also must be significant. From our estimates, investing in rural areas has a positive and significant long-term effect. Hypothesis 2 is validated. Public investment has a long terme impact on the infrastructures specially in agricultural sector. Approaches to solutions The study of the effect of public investments on Benin's economic growth should lead the Beninese state to the formulation of policies likely to stimulate or revitalize its economy. To accentuate economic growth in Benin, we make a number of recommendations to know: Strengthen the selection of projects by ensuring the relevance of the investment made. The strengthening of Public Private Partnership for the implementation of public investment projects The reinforcement of the efficiency of public investments. There is a need to increase investment in areas with high growth potential and reduce operating expenses on projects. This will boost economic activity. Given that Benin is a country dependent on agriculture, investments in the rural sector and infrastructure (mainly road infrastructure) should be encouraged [10]. The government must improve the policy of major works started and improve the quality of existing infrastructure

Conditions for implementing solution approaches through some suggestions

Improve the level of public investment spending including transport, health and education infrastructure ; Undertake structural reforms to facilitate access to land and secure public investments ; Improve the quality of teaching and the living conditions of the teacher in order to increase his performance ; Strengthen technical and vocational training for access to quality education

Short-term analysis

These various proposals made to the public power will certainly and certainly have repercussions on the national economy. The effects of these different proposals for solutions are as follows: Improvement of the various road axes; Facilitating the evacuation of agricultural products to the markets; Lower prices for road transport; Increase in producers' income; Increase in production

Long-term analysis:

In the long term, the impacts on the economic structure are as follows: Increase in Gross Domestic Product (GDP); Improvement of the welfare of the producers; Improved purchasing power;

This research has investigated the effect of public investments on Benin's economic growth. A methodology based on ARDL modeling has been adopted, estimating the link between public investment and economic growth. After the estimation and the analysis, the results show that public investments as a whole have a negative effect on economic growth. However, public investment in the infrastructure and rural sectors has positive elasticities. Overall, it appears that public investment policies aimed at improving living conditions must be directed more and more towards sectors with high growth potential (especially rural and infrastructure) and not towards operating expenses that create a loss effect. Indeed, two major conclusions are revealed. First, public investment as a whole has a negative effect on economic growth. These results are not the same with thoses of FMI (1992).For developing countries, it is difficult for them to make publics investments. Second, public investment in the infrastructure and rural sectors has positive elasticities. Thoses results are in the same line with Ram (1986) and Swm (1998) who have analyzed the public expendicture in supply function. Public investment is negatively linked to growth and therefore not enough a priority to reduce poverty. Nevertheless, given the fact that investments in the productive sectors (rural and infrastructure) have positive and significantly different elasticities of zero, they are hopeful. From all this, it emerges that public investment policies aimed at improving living conditions and economic growth must be directed more and more towards sectors with high sectoral growth potential (especially rural and infrastructural) and no to operating expenses that create a loss effect.

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