Energy Consumption-Economic Growth nexus in Sub-Saharan Countries: what can we learn from

Keywords: JEL Q40, Q49 Energy Economics, Economic Growth and Development, Energy Consumption. I

The relationship between energy consumption and economic growth remains one of the most debated topics in the economics literature. Studies in this field have been carried out in developed countries since the end of the 70s, but they not have led to consensus about the relationship other than finding four causality directions: unidirectional in two directions, bidirectional, or neutral. This lack of consensus remains one of the most relevant findings on energy issues. During the 2010s, the scope of the studies on this relationship broadened since, for example, energy demand in Sub-Saharan Africa has outpaced that in the North and the IEA has forecasted the greatest increase in energy consumption to come from this area. The relationship between energy consumption and economic growth began to be studied in Sub-Saharan Africa in the late 90s, using the same method as for developed countries and with the same lack of consensus on the direction of causality. This paper attempts to clarify this situation through a meta-analysis of fifty articles published since 1996 to 2016. This meta-analysis involves five analytical categories: type of publication, geographical area studied, econometrics method used, energy consumption indicators, and control variables. Each of these dimensions includes many disaggregated variables. Logistic regressions are run on the variables presented above for each of the four causality hypotheses. In research that studies single countries, the likelihood of finding for a given causality

Alexis Vessat

hypothesis is very sensitive to the econometric method implemented. Findings on a panel of countries are then presented; their methods assert the neutrality hypothesis.

Introduction

Energy has emerged as one of the key "drivers" of economic development (Sebri, 2015). In the industrialized countries, the magnitude of energy's influence on economic growth remains a controversial issue (Kraft & Kraft, 1978;Payne, 2010). Nevertheless, macroeconomists agree on the predominant role energy plays. Energy serves not only to improve the productivity of the main factors of production such as capital and labor (technical progress), but also allows a given country to assess whether its own level of development can be considered advanced [START_REF] Jumbe | Cointegration and causality between electricity consumption and GDP: empirical evidence from Malawi[END_REF]. The 2004 International Energy Agency report sheds light on the contribution of energy consumption to economic growth (IEA, 2004). As an explanatory variable, energy consumption is positively related to the level of output in both developed and developing countries (IEA, 2004).

Academics and professional circles have assumed energy to have a predominant role in economic growth (Kraft & Kraft, 1978). However, this relationship has been controversial since the beginning [START_REF] Akarca | Energy and employment: a time series analysis of the causal relationship[END_REF]. Interest in this relationship could be understood as a result of the oil supply shocks of 1973 and 1979, raising concerns about the increase in real energy prices or the scarcity of natural resources. However, the most important finding, repeated through a large number of studies, is the lack of consensus on the direction of causality between energy consumption and economic growth.

In studies of the causality between energy consumption and economic growth in developing countries, the findings are also conflicted, as the empirical findings on the direction are largely inconclusive.

The economic and energy contexts that serve as a framework for investigating the nexus for the Sub-Saharan countries are very singular. Since the early 2000s, the pace of economic growth in Sub-Saharan countries has been higher than that of the world economy, although this trend is not uniform across countries. Already in 1980-2010, the pace of economic growth in Sub-Saharan countries exceeded that of developed countries, with the exception of the last few years. Since the 1990s, the Sub-Saharan countries have embarked on the structural adjustment plans/policies jointly promoted by both the World Bank and the International Monetary Fund. Major international organizations highlight that before these changes, there had been a dramatic slowdown in real incomes, economic growth, investment, and savings in Sub-Saharan countries, often associated with the rapid increase of public debt. Although heterogeneous, this phase of growth follows a very difficult economic context: the World Bank's 1989 report focused on the weakness of economic growth in Sub-Saharan countries, where the annual economic growth rate fell from 7% between 1965-1970 to 2.2% during the period 1980-1987. The impact of this economic slowdown created major concerns for oil-exporting countries.

In terms of energy, despite a large endowment of natural energy resources, Sub-Saharan countries remain characterized by poor energy supply and small quantities delivered [START_REF] Wolde-Rufael | Energy demand and economic growth: The African experience[END_REF][START_REF] Wolde-Rufael | Electricity consumption and economic growth: a time series experience for 17 African countries[END_REF]. Investments in electricity production capacities are insufficient. For instance, between 1973 and 1998, electricity generation in Sub-Saharan Africa grew at a rate of 5.1%, while it grew at a rate of 6% in Latin America, and 7.8% in Asia (8 % in China) (Turkson & Wohlgemuth, 2001). In Sub-Saharan Africa, energy market supply only covers a small part of the needs of the population, even excluding non-market demand, and the gap between electricity demand and supply is continuing to increase. From 2000 to 2012, electricity demand rose by 45%, whereas electricity supply increased by 19% (World Energy Outlook, 2014). At the same time, demand for electricity in Sub-Saharan Africa is extremely fragmented, coming from a limited group of countries, and an even more limited number of people connected to the centralized "on-grid" network. In Africa, the main centers of electricity demand are South Africa and Nigeria, which account for almost 40% of total electricity consumption in the region (World Energy Outlook, 2014).

In this general context, a vast body of published scientific papers on the nexus has accumulated no consensual results and thus cannot inform energy policies although each hypothesis on the direction of causality provides clear policy direction.

 The "growth" hypothesis (H1) sees unidirectional causality running from energy consumption to economic growth. This hypothesis states that energy consumption is a prerequisite for economic growth. In this context, energy consumption is a direct input for achieving economic development and an indirect one that complements the main factors of production [START_REF] Ebohon | Energy, economic growth and causality in developing countries: a case study of Tanzania and Nigeria[END_REF]Templet, 1999;Toman & Jemelkova, 2003). In this scenario, the country's economy is energy dependent [START_REF] Ebohon | Energy, economic growth and causality in developing countries: a case study of Tanzania and Nigeria[END_REF]Templet, 1999;Toman & Jemelkova, 2003;Ozturk & al, 2010). Thus policies favoring energy consumption are likely to accelerate economic growth.

 The "conservation" hypothesis (H2) suggests unidirectional causality running from economic growth to energy consumption: Any increase in wealth necessarily leads to an increase in energy consumption. If this hypothesis holds, policymakers could make significant steps to foster energy efficiency improvement, such as reduction in greenhouse emissions and demand management policies (Payne, 2010), without hampering economic growth. The economy in this scenario is less energy dependent.

 The "feedback" hypothesis (H3) indicates a bidirectional causal relationship between energy consumption and economic growth. If the two variables influence each other, any energy policies introducing limits on energy consumption will negatively impact the increase of country's wealth, and any increase in economic growth will lead to an increase in energy demand (Sebri, 2015).

 The "neutrality" hypothesis (H4): sees an absence of any causal relationship between energy consumption and economic growth (Ozturk, 2010). In this scenario, energy policies can be implemented without any effects on economic growth.

Published outcomes on the relationships between economic growth and energy consumption in the Sub-Saharan countries can be categorized into two types:

The first group includes studies based on individual countries using time-series analysis (Table 1). These studies use the cointegration test [START_REF] Engle | Co-integration and Error Correction: Representation, Estimation, and Testing[END_REF] as well as a maximum likelihood test based on a system-based reduced rank regression model (Johansen, 1988(Johansen, , 1991[START_REF] Johansen | Likelihood-based inference in cointegrated vector autoregressive models[END_REF]Johansen & Juselius, 1990). Other authors have added four significant improvements to the preliminary classification between single-country and panel datasets. These include [1] using a divergent time horizon taking into account the shocks occurring in an economy, [2] employing alternative indicators (using an aggregated energy consumption indicator or a disaggregated one like electricity, gas, coal, or biomass consumption), and [3] specifying the model through various econometric methodologies, with reliance on bivariate analysis (Payne, 2010). Previous studies have adopted bivariate models, in which aggregate energy consumption was considered as the whole input (Wolde-Rufael, 2009). [4] Since the pioneering research of Lutkepohl (1982), however, bivariate models have become outdated due to potential bias of omitted variable. Some authors have thus introduced a multivariate framework, including the main factors of production (labor and capital) (Wolde-Rufael, 2009).

The study of the energy consumption-economic growth nexus and its gradual evolution coincided markedly with improvements in econometrics techniques and tools [START_REF] Karanfil | How many times again will we examine the energy-income nexus using a limited range of traditional econometric tools[END_REF]). Hamilton's studies point out that the explosion of research on the energy consumption-economic growth nexus over the past decades can be explained by the application of time-series econometrics to empirical macroeconomic studies (Hamilton, 1984). [START_REF] Karanfil | How many times again will we examine the energy-income nexus using a limited range of traditional econometric tools[END_REF] shows some surprise at the massive use of The present meta-analysis in the energy consumption-economic growth literature, is based on 50 studies published between 1996 and 2016 about countries in Sub-Saharan Africa. As far as we know, and before the introduction of meta-analysis, the pioneer study to examine the nexus for Sub-Saharan countries was [START_REF] Ebohon | Energy, economic growth and causality in developing countries: a case study of Tanzania and Nigeria[END_REF]. In addition, a meta-multinomial regression is employed to estimate the effects of the potential sources of variation on the different outcomes of existing studies in regard to the relationship between energy consumption and economic growth.

To select the studies for our meta-regression analysis, we primarily used the following keywords: energy consumption AND economic growth, electricity consumption AND economic growth, meta-analysis between energy consumption and economic growth. We found these studies in several bibliographic databases of working papers and journal articles such as Sciencedirect, Taylor & Francis, RePEc & IDEAS. We then applied inclusion and exclusion criteria to these publications: those focusing on Sub-Saharan countries and including simulations of electricity, oil, coal, gas, and biofuel consumption. This paper is organized as follows. After introducing the study, Section 2 provides the detailed methodology adopted in the work. Section 3 describes the models. Section 4 presents the empirical results.

II. Methodology

Data

Publication characteristics

As shown in Figure 1, since the pioneer work of [START_REF] Ebohon | Energy, economic growth and causality in developing countries: a case study of Tanzania and Nigeria[END_REF], a large body of economic literature has been published on the energy consumption-economic growth nexus in Sub-Saharan countries and has dramatically increased over time. However, the second study of the topic did not appear until 2004. The main reasons explaining the gap are the appearance of the new econometrics techniques later applied, and the availability of data used for Sub-Saharan countries. We identify 50 papers, including thirteen articles using econometric panel data analyses and thirty-seven using single-country time series. The studies use different sets of variables in order to explain energy consumption-economic growth nexus. The studies are classified into analytical categories of moderators, using binary variables (1 if the modality is true, 0 otherwise). These definitions, as well as descriptive statistics for all the variables included in our meta-analysis, are presented in Table 3.

Table 3. List of variables

III.2. Cointegration tests 2

Johansen & Juselius

(1990) JJ =1 if JJ is applied, =0 otherwise
First generation 2 The cointegration tests were introduced by Granger (1981). "An important property of I(1) variables is that a linear combination of these two variables that is I(0) may exist. If this is the case, these variables are said to be cointegrated".

The first generation tests have several limitations: [1] asymptotic properties and sensitive to specification errors in limited samples [2] it cannot identify the cointegrating vectors where there are multiple cointegrating relations [3] cannot be applied when one cointegrating vector of different orders exists; and are replaced by the second generation tests. 3.)

Gregory & Hansen

The results are grouped by hypothesis supported.

III. Models

For modeling the contribution of variables, two approaches are investigated: the first one, based on a Logit regression on each causality options, aims to identify what are the influent variables to each independent hypothesis. The second one, based on a Multinomial Logit Model, aims to establish a relation between hypothesis and identify the most explicative variables.

Logit on causality options with independent hypothesis

Implementing a qualitative econometric regression or using a binary variable is a major problem in econometrics [START_REF] Cameron | Microeconometrics using stata[END_REF]. Given a dependent variable coded Y, an appropriate choice of econometric model is necessary. If Y is a continuous variable, then it is appropriate for time-series or econometric panel data analyses to use the ordinary least squares with a best linear unbiased estimator (BLUE) that respects the Gauss-Markov assumptions (the model is well specified: E(ε) = 0; the errors are homoscedastic so the variance is constant equal to VAR(ε) = σ²; the error terms are uncorrelated each other cov(εt; εt') = 0; and the errors are linearly independent of the exogenous variables with cov (ε;X) = 0). On the other hand, what if the dependent variable is a discrete random or unobserved variable? Three types of discrete random variables are identified in the econometric literature: the binary discrete random variable, ordered or unordered multiple discrete random variables, and accounting models. Our dependent variables fall under four hypotheses: H1, unidirectional causality running from energy consumption to economic growth, H2, unidirectional causality from economic growth to energy consumption, H3, bidirectional causality, and H4 neutrality. For each testable hypothesis, we can build econometric models; the final decision is linked to a sample of factors. We respect the general framework of probability models with Prob (event j occurs) = Prob (Y = j ) = F [parameters] as Y takes the value 0 with p probability and 1 with 1-p probability. The logit and probit models correspond to different regression models for p. The choice of adopting a logit or probit model is determined by the error charts (Chart 4) for each hypothesis (h1, h2, h3, h4), the minimization of the log-likelihood of each model, and the minimization of the Akaike information with AIC=-2LnL + 2k and the Schwartz criteria as SC=-2LnL + KLnN. All this indicates that the null hypothesis of accepting the normal law is rejected.

We use a logit model for each type of option. Those logit models are controlled by a contingency table in order to compare the observations and the predicted values to determine the concurring cases in the logit models.

Multinomial logit model (H2 is the reference category)

Multinomial logit model is implemented in order to establishing a relation between hypothesis and identifying the most explicative variables. This model is applied in response to the no significant results obtained for the logit models.

Multinomial choice implies a choice that provides the greatest utility among two or more alternatives; For the causal relationships between gross domestic product (GDP) and energy consumption (EC), MNL is implemented on four directions of causality in an analysis of the 50 references about the Sub-Saharan countries. Each option chosen by a consumer is relative to the degree of utility it provides.

The utility for the four choices:

U i1 = Pa i1 β 1 + Yp i1 β 2 + Ts i1 β 3 + Gri1β4 + Gryi1β5 + Vcem i1 β 6 + α + γI i + ε i1 (1) U i2 = Pa i2 β 1 + Yp i2 β 2 + Ts i2 β 3 + Gri2β4 + Gryi2β5 + Vcem i2 β 6 + α + γI i + ε i2 (2) U i3 = Pa i3 β 1 + Yp i3 β 2 + Ts i3 β 3 + Gri3β4 + Gryi3β5 + Vcem i3 β 6 + α + γI i + ε i3 (3) U i4 = Pa i4 β 1 + Yp i4 β 2 + Ts i4 β 3 + Gri4β4 + Gryi4β5 + Vcem i4 β 6 + α + γI i + ε i4 (4) Corresponding to the following matrix of attributes and characteristics:

                                             0 0 0 0 Vcemi4 4 4 4 34 4 Ii 0 0 0 Vcemi3 3 3 3 3 3 0 1 0 Vcemi2 2 2 2 2 2 0 0 Ii 1 Vcemi1 1 1 1 1 1 Gryi Gri Tsi Ypi Pai Gryi Gri Tsi Ypi Pai Ii Gryi Gri Tsi Ypi Pai Gryi Gri Tsi Ypi Pai
Zi MNL is primarily based on rejecting the hypothesis of independent irrelevant alternatives (IIA). This assumption supports the independence of log-odd ratios (P ij /P im ) relative to the remaining probabilities.

Log-odd rations are described as the effects of one unit of change in X on the predicted odds ratio with other variables in the model held constant.

We apply MNL in our meta-analysis as follows:

Prob(Yi = j) = exp (z'ijθ)/ Σ l=1 6 exp(z ′ ijθ) (1
where:

 pa: represents econometric panel data analysis;

 yp: Publication year;

 ts: Time span;

 gr: Granger causality test;

 gry: Granger modified Toda and Yamamoto test;

 vcem: Error correction model.

Then, the calculation of marginal effects, based on the value of the parameters (the increase of one unit of a given estimator depends first on the value of other estimators and then on the starting value, meaning a difference between marginal effects and simple coefficients) helps identify the impact of significant variables on the assessment of each hypothesis. Three types of marginal effects are identified: the average marginal effect (AME), the marginal effect at mean (MEM), and the marginal effect at representative value (MER). The empirical results of the multinomial logit model are displayed in Table 6 . The "Conservation" hypothesis is the reference category as [1] it is representative of the total sample and [2] it contains more information about the different integration levels of the non-stationary variables. For the following regressions, we create a hypo variable that takes the values h1, h2, h3, h4 by taking into account the causality results from our literature review. If the hypo takes the value h1 ("Growth hypothesis"), the use of econometric panel data analysis can be identified as a significant variable. As for the marginal effects, the use of econometric panel data analysis decreases the probability of asserting unidirectional causality from energy consumption to economic growth by 0.53. If the hypo variable takes the value h3, two significant variables are presented: the error correction model (VCEM) and the use of the Toda and Yamamoto causality test (GRY). The use of the error correction model increases the probability of asserting the "Feedback" hypothesis by 0.16, whereas the use of a Toda and Yamamoto Causality test decreases the probability of asserting the "Feedback"' hypothesis by 0.21. If the hypo takes the h4 value, one significant variable is identified: econometric panel data analysis, which increases the probability of asserting the "Neutrality hypothesis" by 0.58.

IV. EMPIRICAL RESULTS

V. CONCLUSION

In order to investigate the controversial issue of the relationship between economic growth and energy consumption for Sub-Saharan Africa, this contribution performs a meta-analysis, using 50 articles for the period 1996-2016, focusing on the main components of the literature studying the nexus.

For modeling the contribution of variables, two methods are applied: First, in order to identify what are the influent variables to each independent hypothesis, a Logit regression on each causality options is applied. Second, in order to establish a relation between hypothesis and to identify the most explicative variables, a Multinomial Logit Model is implemented.

The findings are twofold. The first one is that logit model does not provide any useful information on causality options to determine what are the influent variables to each independant hypothesis. Logit regression is not significant to inform on the nexus.

The second one is that Multinomial Logit Model, implemented in order to establish a relation between hypothesis and identifying the most explicative variables, with the Conservation hypothesis as the reference category, shows that the panel variable is the most decisive one as it establishes a relation between the hypothesis. This result provides interesting insights, for decision-makers and academics, to reinterpret the nexus for further energy policies implemented.

Nevertheless, any statistical inference of the results obtained by the investigation of the energy consumption-economic growth nexus for the Sub-Saharan countries should be interpreted with care. All studies are related to the specific context on Sub-Saharan countries. The economic growth and energy consumption measurement indicators are thus not constructed as well as they could be, and studies undertaken on the nexus suffer from potential estimation bias. Examples of mismeasurement include the proportion of the informal sector in economic activities in Africa, which is still substantial and thus difficult to capture in a single economic growth indicator. Similarly, when the electricity consumption indicator is employed, only a small part (reliant on traditional biomass) is captured in mainly gridsupplied demand.

APPENDICES BIBLIOGRAPHY

META-ANALYSIS:

  various econometric techniques in the fields of energy economics and environmental economics. These studies first investigate the properties of time series by testing the presence of unit roots to determine the non-stationary nature and testing the first differences of the variables (Dickey & Fuller, 1981; Philipps & Perron, 1988). These econometrics techniques thus capture the dynamics of the energy consumption-economic growth nexus. Two major econometric advances in time-series econometrics have emerged in the form of cointegration and causality tests. Studies employ cointegration by estimating an error correction model to capture the long-run common stochastic trend among the variables (Payne, 2010). Due to major advances in econometrics methods, the Engle-Granger cointegration procedure has been improved to infer the causal relationship between energy consumption and economic growth. The main reasons are low power and size properties of small samples (Harris & Sollis, 2003; Payne, 2010). Approaches based on Pesaran's Autoregressive Distributed Lag bounds test and model (ARDL) have overcome the hypothesis of the existence of a unit root or cointegration among the variables (Akinlo, 2008). Studies that employ the ARDL bounds test and model overcome the endogeneity bias estimation of some regressors (Odhiambo, 2009). The Toda & Yamamoto causality test advocates the first differences hypothesis by using Wald statistics, which corresponds to the use of the chi-square test statistic. The Wolde-Rufael panel research, using the ARDL bounds test and the Toda & Yamamoto causality test, found there is a causal relationship between per capita electricity consumption and GDP in 17 countries over the period 1971-2001 (Wolde-Rufael, 2006). To complete the panel research study (Wolde-Rufael, 2006), Wolde-Rufael (2009) investigates the nexus for seventeen Sub-Saharan countries over the period 1971-2004 by using the Toda & Yamamoto causality test and the Pesaran & Shin variance decomposition to evaluate the impact factor of energy consumption variation on economic growth in a multivariate framework.The purpose of this study is to quantitatively synthesize the empirical literature on the energy consumption-economic growth nexus for Sub-Saharan countries using a meta-analysis method. A metaanalysis is a set of statistical methods applied to a collection of previous research studies related to a given topic(Stanley, 2001). In the literature, meta-analysis is also called "quantitative research synthesis" (Hunter & Schmidt), meaning "an analysis of the analyses"(Glass, 1976; Borenstein & al, 2009). One of the main goals of meta-analysis is often to estimate the combined effects of the set of studies. The more specific information a study contains, the more it represents an important part of the information captured in a meta-analysis(Glass, 1976; Borenstein & al, 2009). The main advantages of a meta-analysis on the topic: [1] It goes beyond the limit of classical surveys or narrative reviews (Payne, 2010) which identify no relation between variables; [2] it provides an analysis of the links between variables; [3] it allows for selection of variables to understand the factors that have contributed most to the different outcomes obtained in existing studies.

  Since the energy consumption -economic growth nexus represents a vigorous field of research, numerous meta-analyses of this scope have recently been published (Sebri, 2015; Kalimeris & al 2014; Chen & al, 2012). This paper is meant to provide the first such contribution on Sub-Saharan African countries.

Figure 1 .

 1 Figure 1. Number of papers published on the nexus for Sub-Saharan countries

Figure 2 .

 2 Figure 2. Heterogeneity of results on the energy consumption-economic growth nexus

1 .

 1 Unit root tests

24 [ 1 ][ 2 ]

 2412 The Growth hypothesis-based (H1) studies supporting unidirectional causality from energy consumption to economic growth are long-run time-series analyses that use the Granger causality test without an error correction model. Their models are based on univariate indicators as single aggregated energy consumption indicators or more specific indicators like electricity consumption by including control variables like inflation rate or CO2 emissions. This type of study is not representative in our sample because it concerns only six countries. The Conservation hypothesis-based (H2) studies validating unidirectional causality running from economic growth to energy consumption are medium-run panel data analyses that apply the Toda and Yamamoto causality procedure without error correction models. They use aggregated energy consumption or specified indicators (diesel consumption, gas consumption, coal consumption, or energy efficiency) and include control variables as proxies in a univariate framework. This type of study is representative in our sample because it concerns 27 countries. "The Conservation hypothesis" will be take as reference category for the multinomial logit model. [3] The Feedback hypothesis-based (H3) studies that support bidirectional causality are both shortrun and medium-run time-series analyses. These studies use Toda and Yamamoto causality procedure without error correction model. Aggregated energy consumption is excluded, and a multivariate framework is employed as specific indicators are used (oil consumption, coal consumption, and electricity consumption) without control variables. This type of study is representative in our sample because it concerns 27 countries [4] The Neutrality hypothesis-based (H4) studies supporting the absence of causality are panel data analyses that use a bivariate model and specify energy consumption indicators in a multivariate as oil or gas consumption). This type of study is not representative in our sample because it only concerns 15 countries

Z

  ij . z ij includes both individual and choice characteristics. If we note z ij = [x ij , w i ] where x ij represents the choice characteristics and w i the individual characteristics, in the case of a Multinomial Logit Model, the individual characteristics prevail on the choice characteristics (if the reverse is true, the conditional logit model is required). McFadden (1974) bases Multinomial Logit Model on the following test statistic, Prob(Yi = j) = p ij = exp(x i B j ) / Σ l=1 m [exp(x i B j )], with the hypothesis j = 1 to m and where 0 < pij < 1 et Pml=1 [pij = 1)] (Cameron and Trivedi., 2010), if the J disturbances are independent and identically distributed with Gumbel (type 1 extreme value) distribution as: F(εij)= exp(-exp(-εij)).

4

  the error correction model (VCEM) (P<0.05) and the use of the Granger causality test (gr)(P<0.05).Looking at the marginal effects reveals that the use of the Granger causality test increases the probability of finding unidirectional causality running from energy consumption to economic growth by 0.23. In contrast, the use of an error correction model in the econometric analysis decreases the probability of asserting the "Growth hypothesis" by 0.39. For the "Conservation hypothesis" (h2), the logit model identifies only one significant variable: the use of the Toda and Yamamoto causality test. The use of the Toda and Yamamoto test increases by 0.69 the probability of supporting unidirectional causality running from economic growth to energy consumption. The logit model for "Feedback" (h3) identifies two significant variables: The error correction model (VCEM) (P<0.01) and the publication year (ap) (P<0.05). The use of the error correction model increases the probability of asserting the "Feedback" hypothesis by 0.42. On the other hand, the publication year decreases the probability of asserting the hypothesis by 0.04. The logit model for the "Neutrality" hypothesis (h4) admits only one variable: the use of an econometric panel data analysis, which increases by 0.45 the probability of supporting the "Neutrality hypothesis". The logit model is insufficient to inform on the nexus. Thus, Multinomial Logit Model is implemented in order to establishing a relation between hypothesis and identifying the most explicative variables. The choose of H2 as the reference category is justified by two arguments: the first one is that the hypothesis is representative of the total sample. The second one is that this hypothesis concentrates more informations about the different level of integration of the non stationary variables.

Table 1 . Summary of the nexus in individual countries in SSA

 1

	Author(s)	Countries	Reviews	Methodology	Findings
					Tanzania:
		Tanzania
	1. Ebohon	(1960-1981),	Energy Policy	Sims and	H3
	(1996)	Nigeria		Engle-Granger
		(1960-1984)		causality tests	Nigeria:
					H3
	2.Jumbe	Malawi	Energy	Sims and	Malawi:
	(2004)	(1970-1999)	Economics	Engle-Granger
				causality tests	H3
				Sims and	Ivory Coast:
				Engle-Granger
	3.Kouakou	Ivory Coast	Energy Policy	causality tests	H3
	(2011)	(1971-2008)		within an	Short-run:
				ECM
					H1
					Vs Long-run:
					H2

Table 2 . Summary of the nexus for a panel of countries in SSA

 2

	Author(s)	Countries	Reviews	Methodology	Findings
					H1

Zimbabwe 2. Wolde- Rufael (2006)

		Algeria, Benin, Cameroon,		Pesaran &
		Ivory Coast, Egypt, Gabon,			H2
	3.Wolde-	Ghana, Kenya, Morocco,	Energy	Shin variance	Ivory Coast, Egypt, Ghana,
		Nigeria, Senegal, South			Morocco, Nigeria, Senegal,
	Rufael	Africa, Sudan, Togo,	Economics	decomposition	Sudan, Tunisia, Zambia
	(2009)	Tunisia, Zambia, Zimbabwe		& Toda &	H3
		(1971-2004)		Yamamoto	Gabon, Togo, Zimbabwe
				causality test	H4
					Cameroon, Kenya
		Benin, Botswana,
		Burkina
		Faso, Burundi,
		Cameroon, Cape-
		Verde, Central
	4. Kahsai &	African Republic, Chad, Comoros,
	al (2012)	Congo, Ivory Coast,			H2 Cameroon,
		Algeria, Benin,			Ghana, Nigeria,
		Cameroon, RDC,			Senegal, Zambia,
		Congo, Egypt,			Zimbabwe
					H1
		Gabon, Ghana, Kenya, Morocco,	Energy Policy	ARDL &	Benin, DRC
		Nigeria,		Toda &	H2
		Senegal, South			Cameroon,
		Africa, Sudan,		Yamamoto	Ghana, Nigeria,
		Tunisia, Zambia,			Senegal, Zambia,
		Zimbabwe		causality test	Zimbabwe
		(1971-2001)
					H3
					Egypt, Gabon,
					Morocco
					H4
					Algeria, Congo,
					Kenya, South
					Africa, Sudan
					H1
					Algeria, Benin, South
					Africa

Table 4 . Descriptive statistics

 4

	Type and	Code	H1	H2	H3	H4	Mean Std.dev
	nature of
	variable
	Number of
	articles
	(Numeric)	-	6	27	27	15	-	-
	Data circs	PA=1	0	1	1	1	0.32	0.47
	(Dichotomic)	TS=0
	Time span	<30 years=0					0.24	0.43
	(Categorial)	=30 years=1	2	0&1	1	1&2	0.52	0.50

Table 5 . Logit on causality options

 5

	VARIABLES	H1	H2	H3	H4
	Panel Dataset (pa)	0.156	1.294	0.821	2.115
		(1.03)	(0.915)	(1.060)	(0.905) ***

Table 5

 5 shows the significant variables identified for the logit model, corrected for heteroskedasticity.As indicated by the results of the residual forms of each causality option and the minimization of the Akaike and Schwartz criteria, we draw conclusions only from this model's results. The logit of the first hypothesis (h1 or "Growth hypothesis") of unidirectional causality running from energy consumption to economic growth identifies two significant variables at close to 5 % (Student t-test for the coefficients):

Table 6 . MNL reference categories: H2

 6

		(1)	(1)	(1)
	VARIABLES
		H1	H3	H4
	Panel Dataset	17.99	-0.937	2.637

A unit root is defined as a stochastic trend in a time series (also called a "random walk with drift"). If a time series contains a unit root, it means that the systematic pattern is unpredictable. Unit root tests are implemented to test the stationarity in a time series. The hypothesis is: "if a shift in time doesn't cause a change in the shape of the distribution". The first generation tests have several limitations:[1] low power in small sample sizes;[2] unable to distinguish nonstationary series from stationary series; and are replaced by the second generation tests.

The causality tests were introduced byGranger (1981).[1] The detects the structural changes in a variable that has a considerable impact on another variable.[2] Two natures of causality are distinguished: long and short-run (a specific test is required to test the joint significance of the lagged explanatory variables by using an F-statistics or Wald test[3] causality test are often associated with the use of an error correction model (VCEM). The first generation tests have several limitations:[1] It requires pre-testing cointegration before causality[2] it cannot be applied in the case of multivariate causality (impossible to identify the explanatory variable causing the causality through the error correction model; and are replaced by the second generation tests.

The Student test tests if a given variable has an influence on the Y dependent variable. The null hypothesis is the H0=bj=0; otherwise the hypothesis is H1 = bj ≠ 0. Conserving the estimated coefficient and the standard deviation are necessary to compare the test statistic with t*. If t (absolute value) ≥ t*; we reject the null hypothesis of a significant variable.


Guillaume Chevillon
email: chevillon@essec.edu

Alain Hecq
email: a.hecq@maastrichtuniversity.nl

Sébastien Laurent
email: sebastien.laurent@univ-amu.fr

Karim Abadir

Richard Baillie

Niels Haldrup

David Hendry

Jurgen Doornik

Sophocles Mavroeidis

Bent Nielsen

Anders Rahbek

Peter Robinson

Paolo Zaffaroni We

Generating Univariate Fractional Integration within a Large VAR(1) *

Keywords: Long memory, Vector Autoregressive Model, Marginalization, Final Equation Representation JEL: C10, C32, C55

This paper shows that a large dimensional vector autoregressive model (VAR) of finite order can generate fractional integration in the marginalized univariate series. We derive high-level assumptions under which the final equation representation of a VAR(1) leads to univariate fractional white noises and verify the validity of these assumptions for two specific models.

Introduction and motivations

Long memory is commonly observed in many fields, dating back at least to [START_REF] Smith | An empirical law describing heterogeneity in the yields of agricultural crops[END_REF], [START_REF] Cox | The use of the correlogram in measuring yarn irregularities[END_REF] and [START_REF] Hurst | Long-term storage capacity of reservoirs[END_REF]. However, as argued by [START_REF] Cox | Scaling. Symposium presentation and video[END_REF], the origin of long memory is unclear. [START_REF] Müller | Testing models of low-frequency variability[END_REF] show that this may be due to the fact that very large samples are needed to discriminate between the various models generating strong dependence at low frequencies.

Hence several competing models of long range dependence have been proposed in the literature, see the recent overview by Haldrup and Vera-Valdés (2017) and references therein. For a covariance stationary process z t , long memory of degree d is often defined, as in [START_REF] Beran | Statistics for Long-Memory Processes[END_REF] or [START_REF] Baillie | Long memory processes and fractional integration in econometrics[END_REF], through the behavior of its spectral density f z (ω) about the origin: f z (ω) ∼ c f ω -2d , as ω → 0 + , for some positive c f . Since [START_REF] Granger | An introduction to long-memory time series models and fractional differencing[END_REF], fractional integration of order d, denoted I(d), has proved the most pervasive example of long memory processes in econometrics and statistics. The prototypical example of an I(d) process is the fractional white noise z t = (1 -L) -d t , where L denotes the lag operator and t is a white noise sequence. The class of fractionally integrated processes extends to cases where t admits a covariance stationary ARMA representation.

To the best of our knowledge five reasons have been put forward in the literature so far to explain the presence of long range dependence: (i) aggregation across heterogeneous series, frequencies or economic agents (inter alia [START_REF] Granger | Long memory relationships and the aggregation of dynamic models[END_REF], Gonçalves and Gouriéroux, 1988[START_REF] Chambers | Long memory and aggregation in macroeconomic time series[END_REF][START_REF] Abadir | Aggregation, persistence and volatility in a macro model[END_REF][START_REF] Zaffaroni | Contemporaneous aggregation of linear dynamic models in large economies[END_REF]; (ii) linear modeling of a nonlinear underlying process (e.g., [START_REF] Davidson | Generating schemes for long memory processes: regimes, aggregation and linearity[END_REF]Sibbertsen, 2005, Miller and[START_REF] Miller | Nonlinearity, nonstationarity, and thick tails: How they interact to generate persistence in memory[END_REF]; (iii) structural change [START_REF] Parke | What is fractional integration?[END_REF][START_REF] Diebold | Long memory and regime switching[END_REF][START_REF] Perron | An analytical evaluation of the log-periodogram estimate in the presence of level shifts[END_REF]; (iv) learning by economic agents in forward looking models of expectations [START_REF] Chevillon | Learning can generate long memory[END_REF] and (v) network effects [START_REF] Schennach | Long memory via networking[END_REF].

The contribution of this paper is to provide conditions under which fractional integration arises as the result of the marginalization of a large dimensional linear multivariate system, hence being caused by hidden dependence across variables within a system. Our result is about the existence of such conditions and we do not aim to give a general framework for the conditions under which marginalization generates fractional integration. More specifically, we provide an asymptotic para-metric framework under which the variables entering an n-dimensional vector autoregressive process of finite order (here a VAR(1)) can be individually modelled as fractional white noises as n → ∞.

Long memory may therefore be a feature of univariate or low dimensional models that vanishes when considering larger systems. The source of long memory identified here differs distinctly from the five sources listed above, and in particular, from the aggregation mechanism à la [START_REF] Granger | Long memory relationships and the aggregation of dynamic models[END_REF]. Indeed, the mechanism that leads here to long memory does not rely on heterogeneity assumptions, nor does it involve aggregation.

The intuition behind our theoretical result is the following. We consider a simple VAR(1) model x t = A n x t-1 + t , for t = 1, ..., T, where (A n ) denotes a sequence of n-dimensional square Toeplitz matrices. 1 We use the final equation representation of this model (as proposed by [START_REF] Zellner | Time series analysis and simultaneous equation econometric models[END_REF] to prove that some of the marginalized processes x jt belonging to x t (for j = 1, ..., n) may converge weakly, as n → ∞ but holding j fixed, to a long memory process of order δ ∈ (0, 1). Our result is not about empirical inference about the process but relates to the population properties of the marginalized series. We introduce a high-level assumption concerning (A n ). Under this assumption, the moving average lag polynomial associated with x jt is asymptotically (throughout the paper, this means as n → ∞) proportional to the ratio of det (I n-1 -A n-1 L) over det (I n -A n L). We parameterize A n by defining a scalar sequence (δ n ) with δ n ∈ (0, 1) such that lim n→∞ δ n = δ ∈ (0, 1) , and a circulant matrix C n such that the polynomials det (I n -A n z) ∼ det (I n -C n z) as n → ∞. C n is assumed to possess close to a fraction nδ of unit eigenvalues. We then use the first theorem of [START_REF] Szegö | Ein genzwertsatz über die toeplitzschen determinanten einer reellen positiven funktion[END_REF] to prove that under our high-level assumptions, as n → ∞, det (

I n-1 -A n-1 z) / det (I n -A n z) → (1 -z) -δ .
From this, we prove weak convergence of the process x jt to the fractional white noise (1 -L)

-δ t .
We then show that the high level assumption is satisfied for at least two specific examples of VAR(1) models. In the first parameterization, A n denotes a Toeplitz matrix with diagonal elements converging to δ = 1/2 as n → ∞, and with vanishing off-diagonal elements. Importantly, the off-

1
The class of Toeplitz matrices is chosen for the technical reason that we use in our proofs the well-established First Theorem of [START_REF] Szegö | Ein genzwertsatz über die toeplitzschen determinanten einer reellen positiven funktion[END_REF]. The theory of Toeplitz matrices also has the advantage that it provides a simple way to reduce the dimensionality of the VAR system, from n 2 parameters in a generic matrix An to n in the Toeplitz case.

diagonal elements decrease at an O n -1 rate and the sum of each row equals 1 at all n. We show that as n → ∞, each series x jt of this system behaves as an ARFIMA(0, 1/2, 0). In the second example, we consider a similar setting but with limiting value δ ∈ (0, 1) on the main diagonal of A n and with the additional assumption that one innovation (say jt ) dominates the others in terms of magnitude. In this case, we prove that the dominant series j tends to an ARFIMA(0, δ, 0) for δ ∈ (0, 1). Our results exemplify that vanishing interaction coefficients in a multivariate system can give rise to long memory in individual series.

This phenomenon can be referred to as "hidden cross-section dependence" for two reasons. First, long memory appears through the marginalization mechanism and therefore in the univariate series or by extension, when estimating the model on a small subpart of a large system. The cross-section dependence appearing in the large system is therefore hidden in the univariate models. Second, because the off-diagonal elements of the VAR(1) may be so small that, in finite samples, the process is likely to be indistinguishable from a diagonal VAR(1) on the sole basis of the parameter estimates.

The hidden dependencies induce modeling issues that were pointed out, inter alia, in [START_REF] Hendry | The methodology of empirical econometric modeling: Applied econometrics through the looking-glass[END_REF].

The rest of this paper is organized as follows. Section 2 provides our main theoretical results and we give two examples where they hold. Section 3 presents some Monte Carlo simulations. Section 4 concludes. The appendix contains the proofs of the main result, and an online supplement provides the proofs corresponding to the examples as well as an additional technical proof.

In the paper, we use the following notation. N, R and C denote the sets of, respectively, natural integers, real and complex scalars. For any x ∈ R, x and x denote the floor and ceiling of x.

For any sequences a n , b n and c n of real-valued scalars

a n = O (b n ), b n = o (c n ) , and a n ∼ b n imply, respectively, that as n → ∞, |a n /b n | is bounded, b n /c n → 0, and a n /b n → 1. For stochastic processes,
⇒ denotes weak convergence of the associated probability measures. For any complex-valued square matrix A, det (A) is the determinant of A, tr (A) its trace, A its adjugate matrix, A its conjugate transpose and |A| = tr A A its weak (Frobenius) norm. For two sequences (A n ) and (B n ) of square matrices with bounded maximal eigenvalues,

A n ∼ B n means that |A n -B n | → 0 as n → ∞.
1 {•} is the indicator function which takes value one if {•} is true and zero otherwise.

Theory

Set-up

We consider the n-vector x t = (x 1t , ..., x nt ) which admits a vector autoregressive, VAR(1), representation:

x t = A n x t-1 + t , t = 1, . . . , T (1a)
t = ( 1t , ..., nt ) ∼ NID (0,Σ n ) , (1b)
where IID (0,Σ n ) denotes a process that is identically and independently distributed with zero expectation and variance-covariance matrix Σ n . Σ n is not necessarily diagonal, although it is in the examples considered below, but we denote its diagonal by σ 2 n = σ 2 n,1 , ..., σ 2 n,n such that σ n,k > 0 for k = 1, ..., n. We also assume throughout that A n does not admit eigenvalues with modulus strictly greater than unity.

We consider sequences of square Toeplitz matrices (T n ) n∈N that are n-dimensional with parameters

given by the sequence t

(n) k , for -(n -1) ≤ k ≤ n -1, such that coefficients t (n) k and t (n) -k (with k > 0)
denote, respectively, the values of the kth upper and lower sub-diagonals, i.e.

T n =             t (n) 0 t (n) 1 • • • t (n) n-1 t (n) -1 . . . . . . . . . . . . . . . . . . t (n) 1 t (n) -(n-1) • • • t (n) -1 t (n) 0             . ( 2
)
Let δ denote a parameter and (T δ,n ) a sequence of n-dimensional Toeplitz matrices whose entries, t δ,k , do not depend on n (so only the dimension of T δ,n does). To (T δ,n ) we can associate a symbol, defined as the function

g δ (•) : z ∈ C → g (δ, z) = ∞ k=-∞ t δ,k z k . The function ω ∈ R : ω → g δ e iω
is often referred to as "spectral density" of T δ,n (whose entries are obtained by the Fourier transform of the spectral density) but to avoid confusion with the spectral density of the processes x jt , we do not use this terminology. Yet with a slight abuse of notation, we refer to g δ e iω as the function

ω → g δ e iω .
We consider in this paper two examples which constitute applications of our main theorem and provide primitive conditions for its high-level assumptions.

Example (1). The vector x t is generated by Model (1) with A n = T * n + η n D n , where T * n is the Toeplitz matrix with entries

t * (n) k = Re   1 n n-1 j=0 g δ n , e i 2πj n e -2iπjk n   , (3)
where g (•, •) is defined for δ ∈ (0, 1) and ω ≥ 0 as

g δ, e iω = 1 {0≤u<πδ} + 1 {π( 3 2 -δ)<u≤ 3π 2 } , ω = u mod 2π, (4)
and where ω → g δ, e iω is even; the sequence δ n satisfies δ n = 1 2 + o n -2 , with δ n < 1 2 ; η n is a real scalar sequence that satisfies η n = o n -2 , D n is a real antisymmetric Toeplitz matrix with absolutely summable entries. All innovations jt have finite variance:

Σ n = σ 2 I n with σ 2 > 0. 2
Example (2). The vector x t is generated by Model (1) with A n = T * n where T * n and g are defined as in (3) and (4) where δ n ≡ δ ∈ (0, 1) . Σ n is diagonal and there exists j ∈ N such that the variances of the innovations satisfy

       σ 2 n,j = σ 2 j > 0, σ 2 n,k = o n -1 , k = j,
In both examples A n is Toeplitz. In Example (1), the system is such that all processes entering the VAR present similar dynamics (i.e., the results are invariant by rotations of A n ). This example stresses the fact that asymmetry, or heterogeneity, is not necessary for long memory to arise. We show in the online appendix that the entries t * (n) k of T * n satisfy, as n → ∞, and for (n -1) /4 ∈ N,

t * (n) 0 = 1/2 + o n -1 , (5a) t * (n) k = O n -1 , k = 0. (5b)
Hence, as n → ∞, the off-diagonal entries of A n individually tend to zero. Yet the convergence is slow enough to ensure that for all n, A n induces persistence that differs from that induced by a diagonal matrix. Indeed the off-diagonal elements of each row admit a nonzero sum:

n-1 k=1 t * k (n) = 1/2 + O n -1 .
Hence, although the dependence across individual units x jt and x kt vanishes, the system retains sufficient dependence to differ from the one where A n is diagonal for finite n. In Example (2), by contrast, the variance of one innovation jt dominates the others so the system is asymmetric. The off-diagonal entries of A n do not necessarily tend to zero. This constitutes an heterogenous case where the results are not symmetric for all x jt and the distinction with the case of long memory arising from aggregation is clear.

We provide next a preliminary result relating to Toeplitz matrix theory that we use in our subsequent analysis.

A preliminary lemma

Consider sequences of square Toeplitz matrices (T n ) n∈N that are n-dimensional with parameters given as in (2) by the sequence t

(n) k , for -(n -1) ≤ k ≤ n-1.
We make a set of assumptions on the elements

t (n) k
of the Toeplitz sequence that we collect in Assumption T (T stands for Toeplitz).

Assumption T.

(i) There exists a real-valued function g (•, •), defined on (0, 1) × {ζ ∈ C, |ζ| ≤ 1} which is continuous with respect to its first argument and such that

(i.a) g (•, •) ≤ 1; (i.b) 2π 0 g •, e iω dω < ∞; (i.c) ∀ (δ, z) ∈ (0, 1) × (-∞, 1) , 1 2π 2π 0 log 1 -g δ, e iω z dω = δ log (1 -z) .
(ii) ∀δ ∈ (0, 1) , defining t

(n) δ,k = 1 n n-1 j=0 g δ, e i 2πj n e -2iπjk/n , the sequence (t δ,k ) , with t δ,k = lim n→∞ t (n) δ,k , satisfies ∞ k=-∞ |t δ,k | < ∞;
(iii) There exists a nondecreasing sequence δ n ∈ (0, 1) N → δ ∈ (0, 1) as n → ∞ such that (iii.a) the parameters of the matrix sequence (T n ) satisfy

t (n) k = 1 n n-1 j=0 g δ n , e i 2πj n e -2iπjk/n ; (iii.b) n 2 t (n) δ,k -t (n) k → 0 as n → ∞.
Assumption T(i) characterizes the symbol of the Toeplitz matrix sequence T δ,n with entries t δ,k .

Since the identity matrix I n is Toeplitz with symbol g I (•) = 1, hence for all z < 1 and δ ∈ (0, 1) ,

the matrix I n -T δ,n z is Toeplitz with symbol 1 -g δ (•) z. Under T(ii), T δ,n belongs to the so-called
Wiener class of Toeplitz matrices with absolutely summable entries, so we can use associated results [START_REF] Gray | Toeplitz and circulant matrices: A review[END_REF]

n) k to t δ,k is sufficient to ensure that det(In-T δ,n+1 z)
det(In-T δ,n z) and det(In-Tn+1z) det(In-Tnz) admit the same limit as shown in the following lemma, whose proof is in the appendix, Section 5.1.

Lemma 1. Under Assumption T and for all z < 1, the sequence of Toeplitz matrices

(T n ) n∈N with elements t (n) k satisfies, as n → ∞, det (I n-1 -T n-1 z) det (I n -T n z) → (1 -z) -δ .
The lemma will be used to study the properties of marginalized series within vector autoregressive processes. In particular, Lemma 1 gives rise to the polynomial (1 -z) δ , δ ∈ (0, 1) , that is pervasive when considering the dynamics of fractionally integrated processes. We will use this result in relation to lag polynomials in moving average representations.

Main result

In order to link Model (1) to its marginalized univariate components, we use the final equation representation (FER) as studied by [START_REF] Zellner | Time series analysis and simultaneous equation econometric models[END_REF]. They show how the elements of vector processes can be marginalized to yield univariate ARMA representations; see also [START_REF] Cubadda | Studying co-movements in large multivariate data prior to multivariate modelling[END_REF] in the context of factor models and [START_REF] Hecq | On the univariate representation of BEKK models with common factors[END_REF] for an application to multivariate volatility processes. The FER of Model ( 1) is

det (B n (L)) x t = B n (L) t , (6)
where B n (L) = I n -A n L, with L the lag operator and, in our notation, B n (L) denotes the adjugate of B n (L). If A n admits unitary eigenvalues, we implicitly assume that t = 0 for t ≤ 0 and x 0 = 0.

In the spirit of [START_REF] Johansen | The role of initial values in conditional sum-of-squares estimation of nonstationary fractional time series models[END_REF], we therefore consider truncations of lag polynomials and convergence to fractional processes with a fixed start (hence of type II). For t > 0, and for any polynomial P (z) , we let [P (z)] + denote the truncation of P (z) for degrees of z k strictly less than t so that P (L) x t only involves x k at dates k > 0. 3 Expression (6) shows that elements x jt (for j = 1 . . . , n)

obtained by marginalizing the n-dimensional VAR(1), admit a finite ARMA(n, n -1) representation with a common AR lag polynomial. Hence, as n → ∞, the univariate process x jt without roots cancellation follows an ARMA(∞, ∞). Root cancellation may yield an ARMA representation that remains of finite order as n → ∞ (Zellner and Palm, 204) but our paper explicitly considers the case where this does not arise, see below. For clarity of the exposition, consider the following trivariate example:

Example. x t is a trivariate VAR(1) specified as follows:

        x 1t x 2t x 3t         =         a b 0 b a b 0 b a                 x 1t-1 x 2t-1 x 3t-1         +         1t 2t 2t         , 3 [P (z)
] + in effect depends on t, but we fix the latter throughout so the relation is implicit to avoid being cumbersome.

For z 0 such that P (z 0 ) = 0, we define P (z

) -1 + z=z 0 = limz→z 0 P (z) -1 + , such that P (z) -1 + z=z 0 < ∞.
where Denote the jth,

E [ jt kt ] = 0 for j = k. The FER of x t is det (B n (L)) x t = B n (L) t , where det (B n (L)) = 1 (a 2 -2b 2 ) (1 -aL) 1 -a - √ 2b L 1 -a + √ 2b L , B n (L) =         (1 -aL) 2 -b 2 L 2 bL (1 -aL) b 2 L 2 bL (1 -aL) (1 -aL) 2 bL (1 -aL) b 2 L 2 bL (1 -aL) (1 -aL) 2 -b 2 L 2         . Hence x jt ∼ ARM A (3, 2) for j = 1,
1 ≤ j ≤ n, row of B n (L) by B n (L) j. such that B n (L) j• = B n (L) j1 B n (L) j2 ... B n (L) jn ,
where, for any matrix D, we denote by D jk its entry at the intersection of the jth row and kth column.

Hence x jt admits the ARMA representation det (B n (L))

x jt = B n (L) j• t = n k=1
B n (L) jk kt .

The process x jt therefore admits the representation

x jt = n k=1 det (B n (L)) -1 B n (L) jk kt . (7)
Expression ( 7) constitutes the basis of our theoretical argument. To keep the generality of the exposition, we implicitly consider that (7) is replaced by

x jt = n k=1 det (B n (L)) -1 B n (L) jk + kt when det (B n (1)) = 0. 4
We make Assumption P for the parameters of the model (P stands for parameters).

Assumption P. There exists j ∈ N for which the parameters of Model (1) satisfy for all finite t, and as n → ∞,

(i) E   n k=1, k =j det (B n (L)) -1 B n (L) jk kt 2   → 0;
4 The degree of the polynomial det (Bn (z)) -1 Bn (z) jk

+ is finite (at most t -1). Therefore det (Bn (z)) -1 Bn (z) jk + < ∞ for all z. (ii) B n (z) jj ∼ det (B n-1 (z)) ; (iii) σ 2 n,j → σ 2 j > 0;
(iv) there exists a sequence of Toeplitz matrices (T n ) that satisfies Assumption T and such that for

any real z < 1, det (B n-1 (z)) det (B n (z)) ∼ det (I n-1 -T n-1 z) det (I n -T n z) .
Assumption P(i) ensures that in the moving average representation for x jt , the only innovation that retains a contribution as n → ∞ is jt , the others play no role asymptotically. In this sense, P(i) is the key assumption that ensures that the mechanism generating long memory here differs completely from that of aggregation, see the discussion that follows Theorem 1. P(i) may for instance hold when the off-diagonal elements of B n (z) are negligible compared to those on the main diagonal (Example (1)), or when the the variance of one error jt dominates all others (Example ( 2)). Assumption P(ii) provides a recursive formula on the dimension of the system in the FER for x jt . It relates B n (z) jj -which expresses the contribution relative to jt in the MA lag polynomial of the n-dimensional system, see Expression (6) -to the AR lag polynomial of the n -1 dimensional system.

Assumption P(iii) ensures that the distribution of jt is nondegenerate as n → ∞, so that of x jt is nondegenerate too. Finally, Assumption P(iv) ensures that we can make use of Lemma 1: as n → ∞,

[det (B n (L))] -1 det (B n-1 (L)) → (1 -L) -δ where δ ∈ (0, 1) is defined in Assumption T(iii) .
The main result of the paper resides in the following theorem, whose proof is in the appendix, Section 5.2. In order to emphasize that our results involve Type II fractional integration, we introduce the truncated fractional difference operator ∆ c + = [∆ c ] + for any c ∈ R (where, as usual, ∆ = 1 -L).

Theorem 1. Let the real vector x t of dimension n be defined by Model (1). Then under Assumption P, and as n → ∞, the process (x jt ) t∈N satisfies

(x jt ) t∈N ⇒ ∆ -δ + jt t∈N .
In Theorem 1, the marginalized univariate process x jt tends weakly to an I(δ) type II fractional white noise as the dimension n of the system diverges to infinity.

Remark 1. In Theorem 1, the limit of x jt is only driven by one innovation, the others' impacts appear to vanish. Hence, this is in particular distinctly different from the heuristic example of Granger (1980, Section 4) where he generalizes his argument about long memory via aggregation of heterogenous microunits to a large scale dynamic model similar to our VAR(1) model B n (L) x t = t . He notices that B -1 n (L) is given by Expression (7) above, i.e., the sum of n moving averages of the innovations kt , k = 1, ..., n, and he argues that there is a resemblance with his model of aggregation. It may indeed be possible that some specific assumptions about the parameters may lead to long memory by aggregating heterogenous processes in such a setting. Our framework differs here and, in fact, precludes it: we specify in Assumption P(i) that the contribution of the moving averages of kt , for k = j, vanishes as n → ∞, so they do not play a role in the long memory of x jt . Theorem 1 shows clearly this distinction since the limiting fractional white noise is ∆ -δ + jt , so it is entirely driven by the innovation sequence jt without any contribution from kt , for k = j.

Remark 2. Theorem 1 can be used to generate richer short-run dynamics. Consider the process y t defined as F n (L) y t = x t where F n (L) is a matrix lag polynomial of dimension n and degree k, and x t satisfies the assumptions of Theorem 1. Then F n (L) (I n -A n L) y t = t , so y t follows a VAR(k + 1).

There exist conditions on F n (L) such that α (L) y jt = β (L) x jt , for some polynomials α, β of fixed degrees, say p and q, i.e., y jt follows a classical autoregressive distributed lag (ADL) model in x jt , see, e.g., [START_REF] Hendry | Dynamic specification[END_REF]. Then, under the assumptions of Theorem 1 and as n → ∞, y jt tends to a type II ARFIMA(p, δ, q) that satisfies α (L) (1 -L) δ y jt = β (L) jt . For instance, if the elements of F n (L) are diagonal then y jt is asymptotically ARFIMA(p, δ, 0) ; and if the elements of F n (L) are of reduced rank, and with zero coefficients for the impact of y kt-to y jt , where k = j and ≥ 0, then an ARFIMA(p, δ, q) structure may asymptotically arise, see [START_REF] Cubadda | Studying co-movements in large multivariate data prior to multivariate modelling[END_REF] and the examples provided in the online supplement.

Theorem 1 implies that fractional integration arises in Examples (1) and (2) as follows. Proofs are provided in the Supplementary Appendix where we show that both Assumptions T and P hold for n such that (n -1) /4 ∈ N.

Example (1 continued). Theorem 1 implies that, as n → ∞, (n -1) /4 ∈ N, for all k ∈ N, processes

x kt defined for t ≥ 0 tend weakly to uncorrelated type II fractional white noises of order 1/2 : for all

k ∈ N (x kt ) ⇒ ∆ -1/2 + kt . (8)
In this example, all marginalized series within an n-dimensional system present the exact same fractional degree of integration as n → ∞. Since the entries of A n -1 2 I n tend to zero as n → ∞, the cross section dependence between series x jt vanishes asymptotically. Yet, as

n-1 k=1 t * k (n) remains
nonzero, the dependence across individual series is sufficient to generate long memory in each of the marginal processes.

Example (2 continued). Theorem 1 implies that, as n → ∞, (n -1) /4 ∈ N, for all δ ∈ (0, 1), the process x jt tends to a type II fractional white noise:

(x jt ) ⇒ ∆ -δ + jt . (9)
In Example (2), only one innovation jt with nonzero variance remains in the system as n → ∞.

Following [START_REF] Diebold | Measuring financial asset return and volatility spillovers, with application to global equity markets[END_REF], if we interpret our VAR setting as a network, then there are n nodes, k = 1, ..., n, in the system which are in state x kt-1 at the end of any period t -1. At time t, each node k combines x t-1 with an additional idiosyncratic signal kt to produce x t . Since, for k = j, the ratio σ 2 j /σ 2 k → ∞ as n → ∞, we can interpret our framework as providing an example of impact of a relatively "large" innovation sequence within a network. Albeit different, such a setting bears some resemblance with the famous "Noah effect" of [START_REF] Mandelbrot | Noah, Joseph and operational hydrology[END_REF] and [START_REF] Mandelbrot | Fractals and Scaling in Finance: Discontinuity, Concentration, Risk[END_REF] where they show that the existence of outliers arising from heavy tailed distributions can be a source of self-similarity (usually in the marginal distribution). Because here self-similarity takes the form of long memory (the "Joseph effect"), this interpretation of dynamic networks constitutes a bridge between the Noah and Joseph effects which are considered different forms of non-similarity in the literature.5

Simulation evidence

In this section, we evaluate our key theoretical results via a Monte Carlo simulation. We provide simulations that examine the validity of our theoretical asymptotic results when the dimensions of the system and of the sample become large.

An n-dimensional VAR(1), as defined in Equations ( 1a)-(1b), is used to generate data for different choices of T , n and model parameters. We first compare the simulated data and estimated degree of fractional integration. To save space, we only report in Figures 1 to 5 the results for n = 201 series and T = 4, 000 observations.

As a benchmark, we consider in our first experiment the case of a diagonal matrix,

A n = dI n ,
where the parameter d is set to 0.499. The first panel of Figure 1 shows the value of the elements of the first row of A n , denoted a

(for k = 0, . . . , n -1), i.e., when

A n = dI n , a (n) k
= 0.499 for k = 0 and 0 otherwise. In this simple setting, the derived univariate processes have short memory and follow a stationary AR(1) model with an autoregressive parameter of 0.499 for each series.

Panel 2 of Figure 1 plots the empirical distribution (over 1,000 replications) of the long memory parameter of series x 1t estimated using three popular estimation methods, i.e., the log periodogram regression (GPH) of [START_REF] Geweke | The estimation and application of long memory time series models[END_REF], the Local Whittle Likelihood Estimator (LWLE)

of [START_REF] Robinson | Gaussian semiparametric estimation of long range dependence[END_REF], both with bandwidth T /2 and the MLE of an ARFIMA(1, d, 0). We deliberately choose a large bandwidth, as implemented by default in [START_REF] Doornik | A Package for Estimating, Forecasting and Simulating Arfima Models: Arfima Package 1.01 for Ox. Package manual[END_REF] to reduce the variability of the estimators. As expected the estimated long memory parameters are concentrated around 0 suggesting that there is no evidence of long memory in the individual series. This is confirmed by the third panel of Figure 1 which reports the ACF of x 1t for the first replication.

In the next two experiments, we consider a symmetric Toeplitz matrix A n = T * n , under the assumptions of Example 1 with η n = 0. We denote by d the value taken by δ n : we choose two values single input that transits through a network (Schennach's process is scalar). Contrary to Schennach, all the coefficients of An are strictly less than unity in absolute value, so the signal is transmitted from x kt-1 to x jt with attenuation (she requires that some nodes transmit without attenuation, see her Section 3).

of d close to 1/2, i.e., respectively d = 0.499 in Figure 2, and d = 0.45 in Figure 3. The structure of these figures is similar to that of Figure 1 except that now, since d is close to 1/2, i.e., to the nonstationary region of an I (d) process, we follow the approach of [START_REF] Abadir | Nonstationarity-extended local Whittle estimation[END_REF] and apply the three long memory estimators to (1 -L) d x 1t (for the values we report, we have added d ex-post to the estimate). 6 The first panel of these figures emphasizes that the diagonal elements are near d while the off-diagonal elements of A n are small for d = 0.45 and very small for d = 0.499.

The sum of each row of T * n is unity by construction and therefore although the off-diagonal elements of A n can be very small when d is close to 1/2, they are nonzero. Unlike in Figure 1, long memory is detected in x 1t , with a Monte Carlo mean (over the 1,000 replications) of 0.444, 0.484 and 0.488 respectively for the GPH, LWLE and ARFIMA(0, d, 0) methods for d = 0.499 and 0.417, 0.451 and 0.465 for d = 0.45. The ACF of x 1t in the first replication also suggests the presence of long memory.

These figures show that although A n is near diagonal, the very small off-diagonal elements play a crucial role in the apparition of long memory.

Next, we evaluate the robustness of the previous result using the asymmetric Toeplitz matrix

A n = T * n + η n D n , with d = 0.499, η n = 1 n 2 log(n)
, and where the elements of D n in the upper triangle are drawn independently from a standard normal distribution. Figure 4 suggests that results are qualitatively the same as in the case of the symmetric Toeplitz matrix in the sense that long memory is detected in x 1t with a parameter estimate close to d.

Theorem 1 states that not only x 1t but all variables belonging to x t should tend to fractional white noises when n → ∞. Our next experiment illustrates this finding for the case of a symmetric Toeplitz matrix with d = 0.499, as investigated in Figure 2. Figure 5 plots the empirical distribution of the long memory parameter estimated on all series, i.e., on x 1t , . . . , x 201t , for the three estimation methods. We only report the results for four replications, each row in the figure corresponding to a replication. Results suggest that the estimated long memory parameters do not vary much across series and are all concentrated in a region close to 1/2, especially for the LWLE and MLE of the ARFIMA(0, d, 0).

Finally, following the suggestion of an anonymous referee, we apply two tests allowing to discriminate between exact fractional integration and spurious long memory since Theorem 1 gives rise to the former as n → ∞. The test of [START_REF] Shimotsu | Simple (but effective) tests of long memory versus structural breaks[END_REF] builds upon the idea that if an observed time series is generated by an I(d) process, then it is also true on every subsample with the same value of d and its dth differences is I(0). [START_REF] Haldrup | Discriminating between fractional integration and spurious long memory[END_REF] propose an alternative test whose principle is that nonlinear transformations of stationary fractionally integrated Gaussian processes decrease the order of memory in a specific way which is determined by the Hermite rank of the transformation.

They rely on the Wald test of [START_REF] Shimotsu | Gaussian semiparametric estimation of multivariate fractionally integrated processes[END_REF] to test the null hypothesis that a vector time series of properly transformed variables is I(0). Their test has reasonable power against various models generating spurious long memory.

Therefore, we simulate again T = 4, 000 observations of a VAR(1) with

A n = T * n ,
x t = A n x t-1 + t , with A n = T * n + η n D n , where T * n is given in Example 1, with δ n = d = 0.499, η n = 1/(n 2 log(n))
, D n is an antisymmetric Toeplitz matrix with above diagonal elements drawn from a standard normal distribution, t iid ∼ N (0, I n ), n = 201 and t = 1, . . . , 4000. The panels report respectively, (a) the value of the elements of the first row of A n , denoted a

x t = A n x t-1 + t , with A n = T * n , where T * n is given in Example 1, with δ n = 0.499, t iid ∼ N (0, I n ), n = 1001, 2001, .
. . , 15001 and t = 1, . . . , 4000. The top panel reports the empirical distribution, over 1000 replications, of the estimated long memory parameter of x 1t obtained by the LWLE method estimated on four nonoverlapping samples of 1000 observations. The bottom panel corresponds to the rejection frequencies of the null hypothesis of fractional integration (vs. spurious long memory) of x 1t at the nominal level of 5% for the test of [START_REF] Haldrup | Discriminating between fractional integration and spurious long memory[END_REF] and the test of [START_REF] Shimotsu | Simple (but effective) tests of long memory versus structural breaks[END_REF] for 2 and 4 sub-samples.

Conclusion

Our paper contributes to the time series literature investigating the mechanisms generating slowly decaying autocorrelations and low frequency variability, in particular those leading to long memory processes. We show that an n-dimensional vector autoregressive model of order 1 can generate long memory in the marginalized univariate series. To achieve this goal, we consider the Final Equation Representation of this model and obtain the n univariate implied ARMA(n, n -1) models whose MA lag polynomial is expressed as a sum that is derived from the determinant and the adjugate of the matrix lag polynomial of the VAR. We then develop high-level assumptions ensuring that at least one of the elements of the vector process converges in probability to a fractional white noise of degree δ ∈ (0, 1). We show that this assumption is satisfied for two specific examples of an n-dimensional VAR(1) model where either (i) all univariate processes tend in probability to an I 1 2 fractional white noise as n → ∞, or (ii) one univariate process tends to an I (δ) fractional white noise.

Our approach is general enough to allow extending it to groups of time series sharing within each group the properties we study in this paper and where each group is orthogonal to others. This would be the case for instance in a large dimensional block-diagonal VAR or in a GVAR, see [START_REF] Pesaran | Modeling regional interdependencies using a global error-correcting macroeconometric model[END_REF]. Another route for extending our results would consist in replacing

A n , the matrix parameter of the VAR(1) , with V n A n V -1 n where V n denotes a sequence of matrices.

This would modify the adjugate matrices but not the determinant of B n (L).

Appendix

Throughout the proofs, κ denotes a generic positive real number and N a generic integer. Entries of a Toeplitz matrix Ψ n of dimension n, at the intersection of row k = 1, ..., n and column = 1, ..., n, 20 are denoted by ψ -k such that

Ψ n =             ψ (n) 0 ψ (n) 1 • • • ψ (n) n-1 ψ (n) -1 . . . . . . . . . . . . . . . . . . ψ (n) 1 ψ (n) -(n-1) • • • ψ (n) -1 ψ (n) 0            
.

We present below the proofs of Lemma 1 and Theorem 1. Proofs related to the validity of Assumptions T and P for Examples 1 and 2, together with additional technical proofs, are placed in the online supplementary material.

Proof of Lemma 1

To prove the lemma, we first need to show that, for z = 1 and as n → ∞,

det (I n-1 -T δ,n-1 z) det (I n -T δ,n z) → (1 -z) -δ . (10)
For this, we start by recalling the First Theorem of Szegö, as stated in Grenander and Szegö (1958, reprinted F f e ix dx.

7 The original definition is for the integral over [-π, π] . To match the setting of this paper, we shift it to [0, 2π] .

8 Grenander and Szegö let f denote the spectral density, here it denotes the symbol of (Ψn) for notational consistency.

The spectral density evaluated at ω coincides with the symbol evaluated at e iω . In Grenander and Szegö, the eigenvalues of Ψ n+1 are denoted λ We now show that the same results hold, replacing T δ,n with T n . For this, we show that

det (I n-1 -T n-1 z) det (I n -T n z) - det (I n-1 -T δ,n-1 z) det (I n -T δ,n z) → 0. (12)
Assumption T(i) states that g (•, •) is real-valued, which implies that for d ∈ (0, 1) , t d,k = 2π 0 g d, e iω e -ikω dω = t d,-k , i.e., T d,n is Hermitian. This entails in particular that t (Gray, 2006, p. 40).

Let T δn,n denote the matrix with entries t δn,k defined as t d,k evaluated at d = δ n . Under Assumption T(iii), T n and T δn,n are asymptotically equivalent under the weak norm, which is denoted Gray, 2006, Section 2.3 for the definition of equivalent matrices).

T n ∼ T δn,n (see
To any Toeplitz matrix T δ,n within the Wiener class, we can associate a Circulant matrix C δ,n such that C δ,n ∼ T δ,n defined as

C δ,n =             c (n) δ,0 c (n) δ,1 • • • c (n) δ,n-1 c (n) δ,n-1 . . . . . . . . . . . . . . . . . . c (n) δ,1 c (n) δ,1 • • • c (n) δ,n-1 c (n) δ,0             . ( 13
)
The sequence (C δ,n ) is not uniquely defined, see Grenander and Szegö (1958, Section 7.6). Gray (2006, Lemma 4.6, p. 52) shows for instance that choosing t δ,-k + t δ,n-k for the entries c

(n) δ,k yields a matrix which is asymptotically equivalent to T δ,n but this is not how we define c such that, respectively,

c (n) δ,k = t (n) δ,-k + t (n) δ,n-k (14a) c (n) k = t (n) -k + t (n) n-k (14b) with t (n) δ,k = 1 n n-1 j=0 g δ, e i 2πj n e -2iπjk/
-zC n )] 1/n ∼ [det (I n-1 -zT n-1 )] 1/n . Now [det (I n -zC n )] 1/n ∼ [det (I n-1 -zC n-1 )] 1/(n-1) . As in Expression (11), it follows that det (I n -zC n ) det (I n-1 -zC n-1 ) ∼ [det (I n -zC n )] 1 n . (15)
Similarly, replacing C n with T n , we obtain, by transitivity of the equivalence, that

det (I n -zC n ) det (I n-1 -zC n-1 ) ∼ det (I n -zT n ) det (I n-1 -zT n-1 ) . (16)
Notice that C δ,n is constructed so that Gray (2006, Lemma 4.6) implies that C δ,n ∼ T δ,n (the notation for C δ,n in Gray's lemma is C fn ). Hence, using the same arguments as in Expression ( 15), together with Expression (10),

det (I n-1 -zC δ,n-1 ) det (I n -zC δ,n ) ∼ det (I n -zT δ,n ) det (I n-1 -zT δ,n-1 ) . ( 17
)
We prove next that as n → ∞,

det (I n-1 -zC δ,n-1 ) det (I n -zC δ,n ) - det (I n-1 -zC n-1 ) det (I n -zC n ) → 0. (18)
Expressions ( 16), ( 18) and ( 17) yield ( 12) which, together, prove the lemma since

det(In-zT δ,n ) det(In-1-zT δ,n-1 ) → n→∞ (1 -z) -δ .
There only remains to provide the proof of (18), which is as follows. First notice that det (

I n -zC n ) = det (I n -zC δ,n + z (C δ,n -C n )), so det (I n -zC n ) = det (I n -zC δ,n ) det I n + z (I n -zC δ,n ) -1 (C δ,n -C n ) and det (I n-1 -zC n-1 ) det (I n -zC n ) = det (I n-1 -zC δ,n-1 ) det (I n -zC δ,n ) (19) × det I n-1 + z (I n-1 -zC δ,n-1 ) -1 (C δ,n-1 -C n-1 ) det I n + z (I n -zC δ,n ) -1 (C δ,n -C n ) .
We now consider the limit of the expression on the second line of ( 19).

Let

H n (z) = z (I n-1 -zC δ,n-1 ) -1 (C δ,n-1 -C n-1 ) . For all z ∈ (-∞, 1), by Assumption T(iii.b) , t (n) δ,k -t (n) k = o n -2 so c (n) δ,k -c (n) k = o n -2 and the diagonal elements of H n (z) are o n -1 so tr H n (z) = o (1). Let µ (n) k , k = 1, ..., n, denote the eigenvalues of H n (z). Then, as n → ∞, µ (n)
k → 0 since all the elements of H n (z) tend to zero, hence for n large enough, log det (I n + H n (z)) exists and we may use Jacobi's formula (Abadir and Magnus, 2005, results 12.30 p. 339 and 13.36 This proves Expression (18) and concludes the proof of the lemma.

Proof of Theorem 1

We first prove that under Assumption P, for all η > 0, as n → ∞, This establishes pointwise convergence in probability for every finite t. Hence joint convergence in probability holds for (x jt ) t∈T and any finite subset T ∈ N. Uniform convergence of (x jt ) t∈T to ∆ -δ + jt t∈T also holds since pointwise convergence over a finite set implies uniform convergence (in probability and hence weakly). Now let T T = N∩ [1, T ] the subset of integers no greater than T. Pollard (1984, Definition 22 and Theorem 23, p. 108) shows that uniform convergence of the finite-dimensional distributions (x jt ) t∈T T , for all T ∈ N, implies weak convergence of the process: 9 (x jt ) t∈N ⇒ ∆ -δ + jt t∈N .

Pr

  particular, Section 4.4 and Theorem 4.2) regarding the convergence of functions of t δ,k . As shown in the appendix, Assumptions T(i.a-b) and T(ii) ensure that the First Theorem of Szegö (1915) holds for I n -T δ,n z, with z = 1, i.e. det (I n -T δ,n+1 z) det (I n -T δ,n z) g δ e iω z dω . The latter limit is, under Assumption T(i.c) equal to (1 -z) δ . Assumption T(iii.a) defines the sequence t (n) k , and hence the sequence of Toeplitz matrices T n . Assumption T(iii.b) ensures that the proximity of t

(

  2, 3 when a = 0 and b = 0, while x jt ∼ ARM A (2, 1) when a = 0 and b = 0 and x jt ∼ AR (1) when a = 0 and b = 0.

Figure 1 :

 1 Figure 1: Simulation results for a n-dimensional diagonal VAR(1) x t = A n x t-1 + t , with A n = dI n , where d = 0.499, t iid ∼ N (0, I n ), n = 201 and t = 1, . . . , 4000. The panels report respectively, (a) the value of the elements of the first row of A n , denoted a (n) k (for k = 0, . . . , n -1); (b) the empirical distribution, over 1000 replications, of the estimated long memory parameter of x 1t obtained by the GPH, LWLE and ARFIMA(1, d, 0) methods; (c) the empirical ACF of x 1t for the first replication.

Figure 2 :

 2 Figure 2: Simulation results for a n-dimensional diagonal VAR(1) x t = A n x t-1 + t , where A n = T * n

Figure 3 :

 3 Figure 3: Simulation results for a n-dimensional diagonal VAR(1) x t = A n x t-1 + t , where A n = T * n

Figure 4 :

 4 Figure 4: Simulation results for a n-dimensional diagonal VAR(1)x t = A n x t-1 + t , with A n = T * n + η n D n , where T * n is given in Example 1, with δ n = d = 0.499, η n = 1/(n 2 log(n)), D n is an antisymmetric Toeplitz matrix with above diagonal elements drawn from a standard normal distribution, t

  (n) k(for k = 0, . . . , n -1); (b) the empirical distribution, over 1000 replications, of the estimated long memory parameter of x 1t obtained by the GPH, LWLE and ARFIMA(0, d, 0) methods; (c) the empirical ACF of x 1t for the first replication.

Figure 5 :Figure 6 :

 56 Figure 5: Simulation results for a n-dimensional diagonal VAR(1) x t = A n x t-1 + t , where A n = T * n

  in 2001, Section 5.2 p. 64).Grenander and Szegö's class L (2001, Section 1.2, p. 4) denotes the set of all complex-valued functions F which are measurable in the Lebesgue sense and for which 7 2π 0 |F (u)| du exists. We denote by λ (n+1) k , 1 ≤ k ≤ n + 1, the eigenvalues associated with the (n + 1)-dimensional Toeplitz Ψ n+1 matrix with entries ψ k , where ψ k = 1 2π 2π 0 e -ikx f e iu du and f (•) denotes the symbol of (Ψ n ) .8 We are ready to state the required theorem.Theorem 2(Grenander and Szegö, 2001, Sec. 5.2). Let x → f e ix be a real-valued function of the class L. We denote by m and M the 'essential' lower and upper bounds of x → f e ix , respectively, and assume that m and M are finite. If F (λ) is any continuous function defined in the finite interval m ≤ λ ≤ M , we have

  notational consistency, we denote them by λ (n+1) k here.As a corollary, using F (•) = log (•) , the determinant of Ψ n+1 is det (Ψ n+1 ) = are strictly positive, lim n→∞ log det (Ψ n+1 ) g.,Gray (2006, p. 8). Hence lim n→∞ det (Ψ n+1 ) / det (Ψ n ) = exp 1 2π 2π 0 log f e iu du . Now we need to show that the symbol associated with Ψ n = I n -T δ,n z is of class L with m > 0 for all z ∈ (-∞, 1) . For this, we notice that the required symbol is 1 -g δ (•) z which is of class L if g δ is of class L too and its minimum is 1 -z > 0. Hence the eigenvalues of Ψ n are strictly positive. Now exp 1 2π 2π 0 log 1 -g δ e iω z dω = (1 -z)δ hence, as n → ∞, Expression (10) follows.

  R with the notations of Assumption T. Also g d (•) = g (d, •) being bounded ensures (T d,n ) and the associated matrices below are uniformly bounded in strong norm. Assumption T(ii) ensures that n-1 k=-(n-1) t d,k e ikω converges uniformly to g d e iω which characterizes T d,n as belonging to the Wiener class

  here. Instead, we define two Hermitian circulant matrices C δ,n and C n with entries c

  p. 372) that implies det (I n + H n (z)) ≤ exp tr H n (z) → 1. Since det (I n + H n (zthere exists N such that for n > N, det (I n + H n (z)) > 0. Therefore, there exists κ ∈ (0, ∞) such that det (I n + H n (z)) → κ and, for all η > 0, there exists a value N such that for n > N,det I n-1 + z (I n-1 -zC δ,n-1 ) -1 (C δ,n-1 -C n-1 ) det I n + z (I n -zC δ,n ) -1 (C δ,n -C n ) -1 < η implying that det (I n-1 -zC n-1 ) det (I n -zC n ) -det (I n-1 -zC δ,n-1 ) det (I n -zC δ,n ) < det (I n-1 -zC δ,n-1 ) det (I n -zC δ,n ) η.Since lim n→∞ det(In-1-zC δ,n-1 ) det(In-zC δ,n )= (1 -z) -δ , it follows that ∀ζ > 0 and z ∈ (-∞, 1), there existη = ζ (1 -z)δ and N such that for n > N det (I n-1 -zC n-1 ) det (I n -zC n ) -det (I n-1 -zC δ,n-1 ) det (I n -zC δ,n ) < ζ.

∞. 2 E

 2 det (B n (1)) = 0 and det (B n (1)) → 0 as n → ∞. Then for n large enough, Chebyshev's inequality implies that, for η > 0,Pr x jt -det (B n-1 (L)) det (B n (L)) L) jj -det (B n-1 (L)) det (B n (L)) jt + k =j B n (L) jk det (B n (L)) follows from Expression (7). Minkowski's inequality then implies thatPr x jt -det (B n-1 (L)) det (B n (L)) Assumption P(ii) implies that, as n → ∞, Bn(z) jj -det(Bn-1(z)) det(Bn(z))→ 0 for all z such that the denominator remains bounded away from zero. Hence, since det (B n (1)) → 0, E Bn(L) jj -det(Bn-1(L)) the polynomial det (B n (1)) = 0 or det (B n (1)) → 0, then when t is held fixed, det (B n (1)) assumption P(iv), Expression (20) implies that, for all η > 0,Pr x jt -det (I n-1 -T n-1 L) det (I n -T n L) jt > η → x jt -∆ -δ jt > η ≤ Pr x jt -det(In-1k=0 θ n,k z k , then det (I n-1 -T n-1 L) det (I n -T n L) -∆ -δ jt ≤ t max 0≤k≤t |θ n,k | max 0≤k≤t | jk | .Lemma 1 shows that as n → ∞, θ n,k → 0 for all k. Hence, by Chebyshev's inequality,Pr det (I n-1 -T n-1 L) det (I n -T n L) -∆ -δ jt > η ≤ Pr max 0≤k≤t | jk | ≥ η t max 0≤k≤t |θ n,kThere exists κ such that, for fixed t,t η (max 0≤k≤t | jk |) 2 ≤ κ yet (max 0≤k≤t |θ n,k |) 2 → 0 as n → ∞. Hence Pr det(In-1-Tn-1L) det(In-TnL) -∆ -δ jt > η → n→∞ 0.Together with (21), the latter expression implies that, as n → ∞, Pr x jt -∆ -δ jt > η → 0.

  n . Since t , the matrix C n is by construction asymptotically equivalent to that with entries t δn,-k + t δn,n-k .Asymptotic equivalence is transitive (see[START_REF] Gray | Toeplitz and circulant matrices: A review[END_REF], Theorem 2.1), hence C n ∼ T δn,n and C n ∼ T n .It also holds that I n -C n z ∼ I n -T n z and, since they are Hermitian, and with positive eigenvalues for

	(n) k , defined in Assumption T(iii.a), converges to
	t δ,k

z ∈ (-∞, 1) (by Assumption T(i.a)), Corollary 2.4 of

Gray (2006, p. 23) implies [det (I n

We assume that all the diagonal elements of Σn are equal for notational ease but this has no incidence on our results. We can relax the assumption to heterogenous and positive diagonal elements.

The mechanism also bears some similarity with that considered by[START_REF] Schennach | Long memory via networking[END_REF], in the case of the impact of a

Throughout the simulations, we use the fast fractional difference algorithm of[START_REF] Jensen | A fast fractional difference algorithm[END_REF].

Louis Bachelier is gratefully acknowledged by the three authors. Chevillon also benefited from support by CREST and the ESSEC Foundation.