Jamal Bouoiyour
email: jamal.bouoiyour@univ-pau.fr

Aviral Kumar Tiwari
email: aviral.eco@gmail.com

Muhammad Shahbaz
email: shahbazmohd@live.com

The Nexus between Oil price and Russia's Real Exchange rate: Better Paths via Unconditional vs Conditional Analysis

Keywords: Oil price, Exchange rate, Unconditional analysis, Conditional analysis, Russia

 technique). The obtained findings are highly suggestive of a sharp causal connection running from oil price to real exchange rate in lower frequencies. The reverse link is not supported. This implies that Russia should effectively tackle with short-run disturbances triggered by oil price and continue to decrease its great dependence on the energy sector via drastic and proactive measures. The economic and fiscal initiatives of Putin administration may help to better cope with sudden shocks, to reach weaker oil dependency and to create the confidence needed for economic recovery. While our research does not say much about the routes through which oil price may affect differently the real exchange rate, it clearly indicates the presence of shortterm nexus conditioning upon potential control variables including GDP, governmental revenues, terms of trade and productivity differential. The unconditional analysis appears as a meaningless exercise to find a clearer relationship between the focal variables.

Introduction

Since the 1973 oil crisis, the world has experienced several oil price jumps and declines. In particular, we wonder how the rise of oil prices affects movements of the dollar. As a result, the possible detrimental effects of excessive oil price fluctuations on real effective exchange rate still have no clear-cut answers despite the huge number of researches on the issue [START_REF] Dibooglu | Real disturbances, relative prices and purchasing power parity[END_REF], [START_REF] Amano | Exchange rates and oil prices[END_REF], [START_REF] Camarero | Oil prices and Spanish competitiveness: A cointegrated panel analysis[END_REF], [START_REF] Akram | Oil wealth and real exchange rates : The FEER of Norway[END_REF], [START_REF] Chen | Oil prices and real exchange rates[END_REF], [START_REF] Bénassy-Quéré | China and the relationship between the oil price and the dollar[END_REF], [START_REF] Narayan | Understanding the oil price -exchange rate nexus for the Fiji islands[END_REF], [START_REF] Lizardo | Oil price fluctuations and U.S. dollar exchange rates[END_REF], Gosh (2011), [START_REF] Mansor | Oil Price and Real Effective Exchange Rate In A Small Oil-Exporting Country: A Recursive-Rolling View[END_REF], [START_REF] Tiwari | Oil price and exchange rates: a wavelet based analysis for India[END_REF], Abulescu et al. (2013), Bouoiyour and Selmi (2014 a), among others). This increases the need to tackle properly and appropriately whether oil price may be considered as a potential source of real exchange rate volatility. . The majority of post-Bretton Woods studies provide evidence that real exchange rate and real oil price are cointegrated and that oil prices may have been the dominant contributor of persistent shocks and sizeable volatility of real exchange rates [START_REF] Krugman | Oil shocks and exchange rate dynamics[END_REF] and [START_REF] Meese | Was it real? the exchange rate-interest differential relation over the modern floating-rate period[END_REF]). There are various routes through which oil price may affect heavily real exchange rate, including fiscal flexibility [START_REF] Zhou | The response of real exchange rates to various economic shocks[END_REF] and [START_REF] Amano | Exchange rates and oil prices[END_REF]), terms of trade [START_REF] Coudert | Oil Price and the Dollar[END_REF], productivity shocks [START_REF] Bénassy-Quéré | China and the relationship between the oil price and the dollar[END_REF]) and the cyclical behavior of international primary commodity prices [START_REF] Benhmad | Modelling nonlinear Granger causality between the oil price and US dollar: A wavelet approach[END_REF] and [START_REF] Bouoiyour | Commodity price uncertainty and manufactured exports in Morocco and Tunisia: Some insights from a novel GARCH model[END_REF]). Intuitively, because an excessive volatile behavior of exchange rate creates uncertainty in the development of macroeconomic policies an accurate investigation of the connection between real effective exchange rate and oil price fluctuations seems substantial for countries which have swapped their exchange regimes from fix to float. Given that the effectiveness of an exchange rate regime is measured through instability, the remarkably excessive fluctuations of oil prices have led to a huge amount of studies analyzing the nexus between the key variables. The empirical survey on the issue suggests the presence of an inconclusive and unclear relationship between oil price and real exchange rate variability. To gauge appropriately the exchange rate-oil price nexus, we attempt to conduct initially a through guidance of the main findings of previous studies. Indeed, the literature on the focal link has followed four main axes:

The first one presents the studies that assess the long-run interaction dynamic between real oil price and real effective exchange rate and figure out whether the reversal connection could be permanent or not. Accordingly, [START_REF] Chen | Oil prices and real exchange rates[END_REF] and [START_REF] Bénassy-Quéré | China and the relationship between the oil price and the dollar[END_REF] argue that there is a causal link between oil price and exchange rate running from oil price to real effective exchange rate and not vice versa. Nevertheless, [START_REF] Sadorsky | The empirical relationship between energy futures prices and exchange rates[END_REF] and [START_REF] Zhang | Spillover effect of US dollar exchange rate on oil prices[END_REF] support the reverse link.

The second strand of the literature on this subject attempts to assess empirically the short-run dynamics between changes in oil prices and those of real exchange rate [START_REF] Narayan | Understanding the oil price -exchange rate nexus for the Fiji islands[END_REF], Gosh (2011), [START_REF] Mansor | Oil Price and Real Effective Exchange Rate In A Small Oil-Exporting Country: A Recursive-Rolling View[END_REF] and [START_REF] Selmi | Another look at the interaction between oil price and exchange rate: Case of small open economies[END_REF]). They propose an indicator that explains the real exchange rate uncertainty in accordance with fluctuations of oil price or the conditional variance between the two variables using generalized autoregressive conditional heteroskedasticity models. More recently and using copula-based GARCH model, [START_REF] Reboredo | Modelling oil prices and exchange rate co-movements[END_REF] examined the co-movements between oil prices and exchange rate in the US economy. This research provides evidence that there is a dependence structure and a significant interdependence between oil price and US dollar exchange rate returns that change substantially within the periods in question.

Furthermore, several studies examine the nexus between oil price and real exchange rate in a linear framework [START_REF] Dibooglu | Real disturbances, relative prices and purchasing power parity[END_REF] and [START_REF] Amano | Exchange rates and oil prices[END_REF]). To distinguish the effects of increases and decreases of oil prices on exchange rates, some researches analyze the focal issue in a nonlinear fashion while trying to detect regimes at which one real exchange rate reacts differently to changes in oil price. For instance, [START_REF] Raymond | Oil and the Macroeconomy: A Markov State-Switching Approach[END_REF] use a Markov switching regime model to evaluate the impacts of rising and falling oil price trends on fluctuations in aggregate U.S. economic performance before and after World War II. At the same context, [START_REF] Akram | Oil wealth and real exchange rates : The FEER of Norway[END_REF] explores the linkage between changes in oil prices and those of the exchange rate in Norway within a nonlinear framework. He shows that the strength of this relationship varies with the trends in oil prices and that Norway's exchange rate policy offers a firmer stance against appreciation pressures rather than depreciation ones.

Intuitively, recent studies have recognized that there are different time periods for decisions-making, whereas assessments are usually restricted to time domain or at least two time horizons, namely, the short-run and the long-run. Although most of the empirical researches have focused on the time domain nexus between exchange rate variability and oil price movements, few works on this field have paid attention to how this nexus evolves over time using discrete wavelet decomposition, continuous wavelets and frequency domain analysis [START_REF] Shahbaz | Analyzing Time-Frequency Relationship between Oil Price and Exchange Rate in Pakistan through Wavelets[END_REF] and [START_REF] Tiwari | Oil price and exchange rates: a wavelet based analysis for India[END_REF] and Bouoiyour and Selmi (2014 a)). The majority of these studies considers the interdependence between each two time series without accounting for the possible interaction with other control variables, which may be ineffective. On the one hand, these methods may lead to confusing outcomes since the occurrence of noise cannot be heavily neglected, disrupting then the relationship under consideration. On the other hand, wavelet decomposition and the frequency domain causality are generally applied to assess respectively the multiple signals and the periodicity that happens over time. Moreover, when we consider only two variables, we generally fall on the problem of simple regression without control variable which is unable to capture appropriately proper results with regard to the focal nexus since it may distort the estimate. In that context, Aguiar-Conraria and Soares (2011) argue that the findings change intensely when we move from an analysis with two variables (unconditional assessment) to an investigation that considers more than two variables (conditional analysis). This highlights insightful evidence that the use of large-scale parameters of each two variables may yield to ambiguous and inconclusive outcomes in terms of the causal connection between oil price and the real exchange rate. To be effective, we need, therefore further investigation of the focal nexus. The contribution of the present study is to apply different models including ARDL bounds testing approach, unconditional cross-wavelet coherency and frequency causality versus conditional ones through various explanatory variables (GDP, governmental revenues, terms of trade and productivity differential) while trying to highlight new insights and to find better paths.

Another contribution of this paper is related to the studied country. An analysis of Russian case may be interesting for at least four main reasons: Firstly, The fact that Russia is very dependent to the ups and down oil price movements can generate serious problems in terms of monetary and exchange rate policies. An increase in the energy commodity price on the world markets encourages more investment in the given sector, which in turn increases sectoral output. However, the need for more labor to produce more output in the energy sector causes an increase in wages across sectors. This leads necessarily to a drop in the competitiveness of the non-oil sector, implying a slowdown in exports. In that context, Egert (2005) argues that countries rich in natural resources and especially those with economic structures relying heavily on oil exports including Russia are usually good candidates for the Dutch disease. Secondly, Russia has experienced a particular evolution of its exchange regimes accompanied with a sharp energy policy evolution during the post-Soviet period [START_REF] Rautava | The role of oil prices and the real exchange rate in Russia's economy cointegration approach[END_REF]. Thirdly, Russian monetary authorities have different objectives determined stochastically over time that lead to evaluate frequently the interactive dynamic between the concerned macroeconomic time series. Lastly, the tendency of a currency to co-move with commodity prices, including oil price instability "the commodity-currency property" [START_REF] Coudert | Oil Price and the Dollar[END_REF] improve the need to be more attentive to oil shocks, sudden disturbances, speculative attacks and price cyclicality that characterize the energy market. All the above elements have reinforced our interest to a succinct description and a proper empirical investigation of the nexus between real effective exchange rate and oil price with special reference to an oil exporting country (i.e. Russia).

The remainder of the article proceeds as follows: We begin with a brief overview of exchange policy and oil price evolution in the Russian case (section 2). Then, we describe our data followed by methodology in section 3. Section 4 reports our main results and discusses them. Section 5 deals with some economic implications and offers concluding remarks.

An overview of exchange policy and oil price evolution: Russian context

Since the dismantlement of the Soviet Union, the Russian economy has experienced several events. As a result, Russia's exchange rate policy has passed through multiple regimes in order to better cope with external shocks. This research presents a succinct description of the Russian exchange rate development and oil price evolution.

The Russian oil stock is the seventh largest oil reserves in the world after Saudi Arabia, Venezuela, Iran, Iraq, Kuwait and UAE (Galkovskaya, 2010). Russia plays an important role as a supplier of energy in the world. In this country, oil exports account for almost 40% of total country's exports. In the period from 1994 to 2009, the oil exports generated almost 50% of export revenues and therefore Russian economy depends substantially on this sector, which increase almost continuously from 2000 to 2009 [START_REF] Bouoiyour | Commodity price uncertainty and manufactured exports in Morocco and Tunisia: Some insights from a novel GARCH model[END_REF]. However, oil production declined in 2008 for the first time since 2000 to 2009 and then increases slightly in 2009 (statistical review of world energy, 2010). Given these circumstances, Russia has played a major role in setting oil prices in the world market. It is therefore a "price maker".

Being a country highly dependent on energy's exports, Russia has to deal with sizable volatility in export earnings relating to commodity price developments -in particular oil price fluctuations-in international markets. As a response, the Russian monetary authorities have given greater preference to massive currency interventions [START_REF] Merlevede | Russia From Bust to Boom and back: Oil Price, Dutch Disease and Stabilisation Fund[END_REF]. The question here is whether the adopted exchange rate policy seems effective to overcome oil shocks, the possible detrimental effects of the cyclical behavior of international commodity prices, speculative attacks and possible sudden disturbances that obviously characterize energy markets.

From 1992 to 1995, the real effective exchange rate is based on stabilization program. Despite this policy reform, the tension between price of oil and exchange rate stills. [START_REF] Mallick | Does inflation or currency depreciation drive monetary policy in Russia?[END_REF] attribute this unexpected outcome to the authorities' use of macroeconomic policy as a direct instrument of social welfare provision. Then and particularly in 1995 M06, the Central Bank of Russia (CBR) has consistently limited the flexibility of the exchange rate by introducing the corridor system that allows the nominal exchange rate to move in a band within upper and lower limits. This exchange rate policy was instrumental in stabilizing the ruble and succeeded to mitigate the violent inflation that distinguish this period [START_REF] Rautava | The role of oil prices and the real exchange rate in Russia's economy cointegration approach[END_REF]. However, the financial crisis of 1998 affected heavily Russia's external position, i.e. the exchange rate uncertainty and the positive boost from rising energy prices put the Bank of Russia in an unfavorable position in managing monetary and exchange rate policies [START_REF] Antczak | The Russian financial crisis of 1998[END_REF]. Besides, the aftermath of the Asian crisis puts the ruble under much greater pressure since Asia is one of the main trading partners of Russia1. To act appropriately to this event, the CBR began operating under a managed float exchange regime in order to keep the depreciation of its currency [START_REF] Galkovskaya | Does nominal exchange rate have a predictive power for oil and gas prices?" The case of Russia[END_REF].

Furthermore, the ruble exhibited sharp instability from the end of 1998 to 2003, reflecting the excessive volatility of both the U.S. dollar and the euro, to which the ruble was pegged. The exchange market underwent September 2001 events, leading to a depreciation of ruble in relation to the U.S. dollar. Fortunately, the fiscal discipline, the strength of the oil price and the position of Russia as a price maker in the international oil market has succeeded to sustain budget surpluses since 2002. For the period through 2002 to 2003, the ruble remained tightly managed. However, in 2004, the Russian policy makers have chosen to adopt less restrictive capital control regulations, making a switch from an authorization-based system to flow control. As a result, the ruble has undergone steady upward pressure and the Bank of Russia intervened in the exchange market in order to contain the ruble's appreciation. Given this consequence, the Bank of Russia introduced a dual-currency basket from the end of 2005 in order to smooth the excessive volatility of ruble in relation to the major currencies. The basket composition was fixed at 55% for the US dollar and 45% for the euro. In 2008, the current economic crisis has led to remarkable decline in oil prices affecting significantly the current account balance putting again the ruble under downward pressure. In response, Russian monetary authorities changed the exchange rate policy focus towards moderating the ruble's depreciation. Additionally, the Central Bank of Russia has conducted large scale interventions in exchange markets in order to improve the economic adjustments [START_REF] Beck | Towards a Flexible Exchange Rate Policy in Russia[END_REF]. There are also some studies showing that the Putin administration has succeeded, to some extent, to better cope with oil price shocks and the current economic crisis. More precisely, the political and economic stabilization brought by president Putin and the initiation of a number of proactive fiscal policy reforms reduced economic uncertainty and political risk, which created the confidence obviously needed for economic recovery [START_REF] Merlevede | Russia From Bust to Boom and back: Oil Price, Dutch Disease and Stabilisation Fund[END_REF]. From the second quarter 2009, the great flexibility of nominal effective exchange rate lets the CBR to change its interest rate policy. During this period, the interventions have steadily decreased, allowing monetary policy to focus squarely on inflation. This has helped to reduce the volatile behavior of ruble and has strengthened the interest rate channel policy transmission mechanism.

Figure-1 confirms the above descriptions. It is well seen that both oil price and real effective exchange rate have exhibited a great instability over the period 1995-2009. The graphs display the evolution of real exchange rate and oil price as well as their returns over the period in question. The crisis of Russia in 1998 proved to be a "purifying event" [START_REF] Merlevede | Russia From Bust to Boom and back: Oil Price, Dutch Disease and Stabilisation Fund[END_REF]) since the real exchange rate sharply decreased from 1999. The oil price increased rapidly from this last date. As largest oil producer and second largest oil exporter of the world, Russia should benefit from the gradually observed increase of the price of crude oil. This does not mean that this economy is not vulnerable to oil price shocks. There are clearly periods of small fluctuations (for example 2002-2004 and 2006-2007) and periods of large fluctuations (for example 1997-1999). These observations reinforce the utmost importance of studying the frequency causal relationship between the two considered variables in order to effectively elucidate the understanding of practitioners and help Russian authorities in their decisions making and policy formulation. Notes: lnOil: the logarithm of oil price; lnREER: the logarithm of real effective exchange rate.

Analytical and Methodological framework

Analytical methodology

In this article, we aim at better capturing the nexus between oil price and real exchange rate by paying particular attention to the evolution of this connection over time and at different frequencies unconditionally and conditioning upon GDP, governmental revenues, terms of trade and productivity differential. These additional control variables have been selected based on the following procedure.

In present real exchange rate model, [START_REF] Meese | Was it real? the exchange rate-interest differential relation over the modern floating-rate period[END_REF] : the prices of traded and non-traded goods in the home and foreign countries; α and * α : weights associated to the expenditure shares on traded goods for the home and foreign countries.

The real exchange rate can be expressed then as follows:

) )( 1 ( ) )( 1 ( ) ( * * * * NT t T t NT t T t T t T t t t p p p p p p e REER - - - - - + - + = α α (4)
Thereafter, McDonald (1998) follows his proposal by dividing the real exchange rate fundamentals into various components, including essentially the real oil price as a potential source of real exchange rate variability; the effect of oil price shock depends substantially on the source of oil price fluctuations. A positive oil shock depends substantially on the source of oil price fluctuations. Theoretically, a positive oil shock may prompt a real exchange rate appreciation (depreciation) in oil exporters (importers). Several researches have provided that there are various factors that may attenuate the power of this link or more accurately the possible detrimental effects of oil shocks on real exchange rate, including economic growth, governmental revenues, terms of trade and productivity differential [START_REF] Corden | Booming sector and de-industrialisation in a small open economy[END_REF], [START_REF] Chen | Commodity Currencies[END_REF], [START_REF] Cashin | Commodity currencies and the real exchange rate[END_REF], [START_REF] Tokarick | Commodity currencies and the real exchange rate[END_REF], among others). The equation that links real exchange rate with real oil prices and its further determinants may be expressed as follows:

) , , , , ( prod te gov gdp Oil f REER t = (5)
where gdp: the growth of output; prod: the difference between the domestic and the foreign productivity; gov: the governmental revenues; Oil: the real price of oil; te: the terms of trade .

-The economic growth (gdp): The GDP has been considered in some studies as the main determinant of the dynamic interaction between oil price and real exchange rate [START_REF] Rautava | The role of oil prices and the real exchange rate in Russia's economy cointegration approach[END_REF]. Conventional wisdom shows that an increase in oil price reduces the economic growth in oil importers and enhances it in oil exporting countries. In our case of study (Russian case), this evidence is not absolutely observed, especially when we clearly see that the oil price has increased remarkably in 2006 compared with 2005 (for example). The oil price expands by 1.62 per cent in 2005 and 1.87 per cent in 2006.2 This implies that oil price may be a significant contributor of economic growth in some economies, while it may possibly explain only partially the changes in the growth of GDP in other ones. This highlights -to some extent-that oil price may affect significantly the real exchange rate in both oil importers and exporters but slightly through the economic growth.

-The governmental revenues (gov): The oil price may be able to lead to great appreciations of real exchange rate via the governmental revenues. The latter can yield to the Dutch disease phenomenon [START_REF] Corden | Booming sector and de-industrialisation in a small open economy[END_REF].

-Terms of trade (te): The terms of trade show the interaction between export and import prices. It is therefore closely linked to the international commodity prices and their cyclical behavior. A positive shock in terms of trade may heavily increase the price of the non-traded products and then the real exchange rate [START_REF] Camarero | Oil prices and Spanish competitiveness: A cointegrated panel analysis[END_REF]. The latter is the relative price of a basket of traded and non-traded goods between the domestic and the foreign country.

-The productivity differential (prod): The oil price may affect the real exchange rate via the productivity differential between the traded and non-traded good sectors. Accordingly, [START_REF] Tokarick | Commodity currencies and the real exchange rate[END_REF] argues that as the productivity is a potential source of terms of trade, it can reinforce the impact of terms of trade on the interconnection between oil price and the real exchange rate.

For empirical purpose, we have collected quarterly frequency data on oil prices and real effective exchange rate over the period of 1995q1-2009q4. The exchange rate is proxies by real effective exchange rate and collected from international financial statistics (CD-ROM, 2012). The crude oil price variable is expressed in real terms, i.e. deflated by U.S. consumer price index following [START_REF] Faria | The effect of oil price on China's exports[END_REF]. The data on crude oil prices are the spot prices and collected from the U.S. Department of Energy: Energy Information Administration. The data on GDP, governmental expenditure revenues, terms of trade and productivity differential are available at Econstats TM .

3.2.Motivation and Introduction to Methodology

Existing economic literature provides various studies applying various approaches to appropriately investigate the relationship between oil prices and exchange rate change and to reach a robust conclusion. Despite the large strand of literature on this issue, it is striking to note that the researchers have not paid proper attention to frequency transformations when examining the nexus between the two variables under consideration.

The majority of the previous studies use OLS, VECM and VAR methods to assess the focal relationship. These techniques were largely criticized for their perceived lack of robustness [START_REF] Bouoiyour | Exchange Rate Impact on Russia's Exports: Some evidence from an evolutionary co-spectral analysis[END_REF], highlighting the importance of the use of more parsimonious methods. While econometric modeling often focuses on average, it seems more fruitful to assess the considered nexus under different and well-defined time horizons. We try in this research to apply different methods which are able to investigate the oil price-real exchange rate nexus under different time scales and to see whether there is substantial to account for nonlinearity when assessing this link.

Importantly, the use of various techniques may have different implications and provide conceptual background for the adequate econometric methods at an economic level in relation particularly to the focal issue with special reference to Russian case. To this end, we apply dynamic method (ARDL bounds testing approach), wavelet coherency and frequency domain. It will be important here to mention that the results of ARDL bounds testing approach may be vulnerable since this technique seems unable to detect possible structural breaks stemming in the variables. This drawback highlights the need to carry out more sophisticated methods that consider the time varying dynamics in the concerned series such as wavelet decomposition and frequency domain analysis. These last methods seem appropriate to depict signal transformations. The first one captures a signal changes over time-frequency, whereas a frequency-domain analysis indicates how the signal moves among given frequency bands. Basically, the time scale is the ability to highlight precisely when a variation happens by identifying specific time horizons. Wavelet transform represents an engineering tool for multi-resolution decomposition of signals that has attracted serious attention from several economists. From a computational point of view, it evaluates the time scale details of data whose spectral content varies over time (non-stationary series). There are no one-reached wavelet package for accurately assessing two time series together (for analyzing if regions in time-frequency space have a consistent phase relationship and thus are highly suggestive of a causality connection between oil price and the real exchange rate). Nevertheless, the frequency band is a component that allows us to determine effectively the degree of a certain variation. These econometric techniques enable to assess the causal relationships between the two key variables in the short, medium and long-run. Various researches have criticized the unconditional causality or the causality between the two series without accounting for other control variables that may have great influence on the studied linkage. When we consider only two series in the empirical investigation, it will be difficult to reach robust and unambiguous results with regard to the relationship in question since it may distort the estimate. This heavily indicates the great importance of deeper analysis of a causal relationship while accounting for additional variables that may play determinant roles in explaining the focal linkage.

Unconditional versus conditional causality

Considering X and Y as variables of interest and Z as the conditional variable, the bivariate equation that allow us to rely the focal variables may be expressed as follows:

t i t i i t i i t Z b X a X 0 1 0 1 1 0 ε + + = - ∞ = - ∞ = ∑ ∑ (6) t i t i i t i i t Z b X a Z 0 1 1 1 1 1 µ + + = - ∞ = - ∞ = ∑ ∑ (7)
where t 0 ε and t 0 µ are independent and identically distributed with zero mean and variance- covariance matrix written as follows:

        = ∑ ∑ ∑ ∑ ∑ 0 0 0 0 , , 1 µ ε µ µ ε ε à (8) t i t i i t i i t Z d Y c Y 0 1 0 1 1 0 ν + + = - ∞ = - ∞ = ∑ ∑ (9) t i t i i t i i t Z d Y c Z 1 1 1 1 1 1 µ + + = - ∞ = - ∞ = ∑ ∑ (10)
where t v 0 and t 1 µ are independent and identically distributed with zero mean and variance- covariance matrix written as follows:

        = ∑ ∑ ∑ ∑ ∑ 1 0 1 1 0 , , 2 µ ν µ µ ν ν à (11)
Then, we can explore the link that runs from X to Y and inversely by accounting for the conditional variable by estimating trivariate equations expressed as follows:

t i t i i i t i t i i t Z Y X X 1 1 1 1 1 1 1 ξ δ β α + + + = - ∞ = - - ∞ = ∑ ∑ ∑ (12) t i t i i t i i i t i i t Z Y X Y 1 1 2 1 1 2 1 2 ς δ β α + + + = - ∞ = - ∞ = - ∞ = ∑ ∑ ∑ (13) t i t i i t i i i t i i t Z Y X Z 1 1 3 1 1 " 1 3 η δ β α + + + = - ∞ = - ∞ = - ∞ = ∑ ∑ ∑ (14)
Where t 1 ξ , t 1 ς and t 1 η are independent and identically distributed with zero mean and variance- covariance matrix can be expressed as following:

            = ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ 1 , , , , , , 3 η ς η ξ η η ς ς ξ ς η ξ ς ξ ξ (15)
The causalities between X and Y conditionally to Z are measured based on Geweke (1984)'s study by:

        = ∑ ∑ → 1 0 / ln ξ ε Z Y X F ( 16
)
and

        = ∑ ∑ → 1 0 / ln ς µ Z X Y F (17)
The effect of the additional control variables on the nexus between X and Y may be detected properly from the above equations ( 16) and ( 17 Both unconditional and conditional causalities are estimated within wavelet coherency and phase difference and frequency domain framework. More precisely, we test in the first case the interconnection between oil price and real exchange rate (unconditional causality). Then, we re-explore the wavelet coherency and phase difference and the frequency causality test, but conditional to GDP, governmental revenues, terms of trade and productivity differential.

Wavelet coherency and phase difference

To properly capture the oil price-exchange rate nexus, the use of sophisticated models able to depict this relationship at different periods seems substantial for several reasons: (i) the excessive volatility of energy supply leads to changes in demand conditions and thereby to multiple energy price regimes; (ii) the massive currency intervention of Russian monetary authorities prompts us to account for nonlinearity; (iii) there may be a relationship between the time series under consideration at different frequencies such as oil price may act like a supply shock at lower and medium frequencies rather than long-run time frequencies [START_REF] Naccache | Oil price cycles and wavelets[END_REF] and in resulting it affects frequently real effective exchange rate. For instance, the real effective exchange rate affects the oil prices following demand-effect in a short time scale (high frequency); (iv) considering low and high frequencies, we can differentiate between time horizons and help then policy makers in their decision as well defined time-horizons.

The wavelet analysis may provide a fresh look into the connection between oil price and Russia's oil price by evaluating it under different and precise time periods that may help policy makers to better cope with oil shocks. The wavelet coherency (WTC) and phase difference may allow us to effectively deal with the time-frequency dependencies between the two variables and to find thus significant coherence. Strictly speaking, these techniques have been applied quite successfully to large amount data and to extract the information relevant to nonlinear dynamic interaction between two variables [START_REF] Tiwari | Oil price and exchange rates: a wavelet based analysis for India[END_REF] and Shahbacz et al. ( 2013)). They exhibit the time contribution of the different frequencies of the signal, to obtain then temporal frequency dependence. This can reflect structural changes that can happen over time.

Fourier analysis highlights an effective quantity, which is the coherence. It is defined as the square of the cross-spectrum normalized by the individual power spectra. This gives a quantity between 0 and 1. Indeed, the Wavelet Coherency presents a high resemblance and a significant nexus between the key time series if coherence is near to 1 otherwise no relationship is found between the time series. It enables us to determine the cross-correlation between the two variables under consideration as a function of time and frequency. Given two time series X and Y, one can define the complex wavelet coherency ( )

( 2 s Q n ) written as follows: , ) ( ) ( )) ( ( ) ( 2 1 2 1 1 2       ⋅       = - - - s W s S s W s S s W s S s Q Y n X n XY n n (18)
where smoothing operator is shown by S . It is a traditional correlation coefficient definition which is helpful in thinking of the Wavelet Coherence as a localized correlation coefficient following time frequency space.

This formula (18) can be expressed in polar form:

xy i n n e s Q s Q φ ) ( ) ( 2 2 = (19)
The absolute value of complex wavelet coherence of two time series is named the

wavelet coherency ( ) ( 2 s R n
) and is denoted by: ( )

, ) ( ) ( ) ( ) ( 2 1 2 1 2 1 2       ⋅       = - - - s W s S s W s S s W s S s R Y n X n XY n n (20)
According to [START_REF] Aguiar-Conraria | Using wavelets to decompose the time-frequency effects of monetary policy[END_REF], pp. 2872), the wavelet coherency is "the ratio of the cross-spectrum to the product of the spectrum of each series, and can be thought of as the local (both in time and frequency) correlation between two time-series". The equation ( 15) can be then re-written once smoothing operator is equal to 1 and smoothing operator S as a convolution in time and scale: The time of convolution is determined using Gaussian and scale convolution is done by regular window [START_REF] Torrence | A practical guide to wavelet analysis[END_REF]. The functional form of smoothing power following Morlet wavelet is articulated as:

(

)

s s t n s time c s W W S | ) ( | ) ( 2 2 2 / 1 - * = (22) ( ) n n n scale s c s W W S | 6 . 0 ( ) ( | ) ( 2 Π * = (23)
where normalized constants are 1 c and 2 c while Π denotes is the rectangle function. The scale de-correlation length for Morlet wavelet is empirical determined by 0.6. Practically, we determine convolutions directly, but the normalized coefficients are determined indirectly. The Monte Carlo simulation approach is used to assess the performance of wavelet coherency based on its theoretical distributions. It seems the most convenient in depicting the interdependence between two variables. Arguably, Aguiar-Conraria and Soares, (2011, p. 649) show that this technique is more appropriate than cross wavelet for two main reasons: "(1) the wavelet coherency has the advantage of being normalized by the power spectrum of the two time-series, and (2) that the wavelet cross spectrum can show strong peaks even for the realization of independent processes suggesting the possibility of spurious significance tests".

The Monte Carlo methods are used to find statistical level of significance of the wavelet coherence, we need therefore to determine the confidence interval of the phase difference or the phase relation between two time series. Based on equation ( 19), the angle (

Y X XY φ φ φ - =
) is called the phase-difference. The main advantage of the phase difference is the fact that it cannot be affected by the smoothing choice (Aguiar-Conraria and Soares, 2011). It is written as following:

] , [ , } { } { tan 1 π π φ φ - ∈ = - XY XY n XY n XY W R W I (24)
The imaginary and real parts are indicated by I and R respectively, of the smooth power spectrum. The phase relationship between the two time series is characterized using path difference which is considered useful. The time series moves together with specified frequency if value of phase difference ranges to zero (Figure -2 .

Figure-2: Phase difference circle

Note: For more details, we can refer to Aguiar-Conraria and Joana Soares (2011).

Frequency domain approach

The frequency domain analysis tests the causality between two variables as the Granger causality test. However, the latter seems not suited to distinguish between the short and long-run effects. To get a new look at the relationship and more proper analysis depending to frequency transformations, we apply a frequency domain Granger causality test [START_REF] Breitung | Testing for short and long-run causality: a frequency domain approach[END_REF]. To this end, we begin by testing Granger causality "A variable Y t is said to Granger cause X t , if Y t contains information to predict X t that is not available otherwise" (Lütkepohl 2005, pp.41). This method is restrictive since it cannot depict the concerned relationships at different horizons. It can be written as matrix notation:

t t t t t Y X L L L L Y X L ε θ θ θ θ =                 =         = Θ ) ( ) ( ) ( ) ( ) ( 22 21 12 11 (25) Y t does not Granger cause X t if ( 0 ) ( 12 = Θ L
), indicating that the past values of Y t seem not closely related to X t . This can be tested by using an F-Test for the coefficients the causality test can be performed under different frequencies without loss of explanatory power, implying that causality measure ( X Y F → ) can be decomposed as follows:

ω ω π d f F X Y X Y ) ( 0 ∫ → → = (26)
Or with a consideration of conditional variable, meaning that causality measure ( X Y F → ) can be decomposed as follows:

ω ω π d f F Z X Y Z X Y ) ( 0 / / ∫ → → = (27)
Various tests have been proposed to analyze the frequency domain nexus between two variables. In this study, we refer to [START_REF] Breitung | Testing for short and long-run causality: a frequency domain approach[END_REF] study. They construct an F-test for the coefficients ) (L Θ at different frequencies. To do so, the above equations (Equations ( 26) and ( 27)) should be re-written in matrix form: Ultimately, the null hypothesis of no Granger Causality at frequency ω can be written based on matrix notations ( 28) and ( 29) respectively as follows:

        =                 = Ψ t t t t t Y X L L L L L 2 1 22 21 12 11 ) ( ) ( ) ( ) ( ) ( υ υ ψ ψ ψ ψ υ (
= = = = = t t υ υ ψ ψ ψ ψ [ ] 1 ) ( - Θ = Ψ G L t ; G
0 ) ( ) ( : 0 = Θ L R H ω (30) 0 ) ( ) ( : 0 = Φ L R H ω (31) where       = ) sin( )... 2 sin( ) sin( ) cos( )... 2 cos( ) cos( ) ( ω ω ω ω ω ω ω p p R
Throughout the rest of this research, we gauge empirically the nexus between oil price and Russian real exchange rate unconditionally and conditioning upon potential control variables. By doing so, we try to identify which technique among the above different methods is more suitable in this case and then to reach clearer and "one sided" relationship.

Results and discussions

ARDL findings

We have converted all series into logarithm to lessen fluctuations in the data set. The results of descriptive statistics and correlation matrices are reported in Table-1 containing log levels as well as returns. The sample mean of log of real effective exchange rate. Similarly, the sample mean of oil prices is positive and higher as compared to data for real effective exchange rate. Both variables of interest are positively correlated (Table -2). We clearly note that the skewness is negative for REER, gdp, gov, prod, while it is positive for Oil and te. This implies that the asymmetrical distribution is plausible for all series (except Oil and te). The Kurtosis is less important than 3 for all variables under consideration, meaning that their distributions are less flattened than the Gaussian ones. We have applied [START_REF] Ng | Lag Length Selection and the Construction of Unit Root Tests with Good Size and Power[END_REF] unit root test to examine whether the variables are stationary in the level form or first difference form. The obtained outcomes indicate that almost all considered variables show unit root behavior at level and found stationary at 1 st difference with intercept and trend (Table -3). The findings of Ng-Perron unit root tests may be biased due not having information about structural breaks stemming in the series. To solve this issue, we have applied de-trended [START_REF] Zivot | Further evidence of great crash, the oil price shock and unit root hypothesis[END_REF]'s structural break unit test to examine the integrating orders of the variables in the presence of structural breaks.

We find from our results that real effective exchange rate is non-stationary at a level showing structural breaks in 1996q4. The variable of oil prices is stationary at level with structural break in 1998q2. Likewise, gdp, te and prod are stationary with respective breaks in 1997q4, 1996q4 and 1997q3, while gov seems non-stationary at level with structural break in 1999q1. These variables have mixed order of integration i.e., I(1)/I(0). This mixed order of integration of the variables leads us to apply the ARDL bounds testing approach to examine whether cointegration exists between the variables in the presence of structural breaks in the series. The appropriate lag order is a prerequisite to apply bounds testing to examine the long-run relationship between the variables. In doing, we chose Akaike information criteria (AIC) to select lag order of the variables . It is suggested by [START_REF] Lütkepohl | Structural vector autoregressive analysis for cointegrated variables[END_REF] that we should apply AIC considered as the more appropriate method in choosing the lag length compared to sequential modified LR test statistic (LR), final prediction error (FPE); Schwarz information criterion (SIC) and Hannan-Quinn information criterion (HQ). When treating REER as dependent variable, the lag order 1 is chosen following the minimum value of AIC either unconditionally or conditionally on control variables (gdp, gov, te and prod). However, when Oil is treated as dependent, the selected lag order is 2 unconditionally and 3 conditioning upon GDP, government revenues, terms of trade and productivity differential. We then calculated the ARDL F-statistic to decide whether cointegration exists between the real effective exchange rate and oil price. We have used the critical bounds generated by [START_REF] Pesaran | Bounds testing approaches to the analysis of level relationships[END_REF]. Our results reported in Table-5 reveal that the calculated ARDL F-statistic exceeds upper critical bound (UCB) i.e., 5.236 > 5.23 at the 1 percent level once we treated real effective exchange rate as the dependent variable. We may reject the hypothesis of no cointegration and conclude that there is one cointegrating confirming a long-run relationship between real effective exchange rate and oil prices in the Russian economy during the period 1995q1-2009q3. When accounting for the above mentioned control variables, the results change slightly, reinforcing the above outcome. Accurately, we note that when the real exchange rate is treated as the dependent variable, the F-statistic exceeds UCB at the 1 percent level either unconditionally or conditionally to gdp, gov, te and prod. Nevertheless, the F-statistic seems less important than the lower critical bound (LCB) i.e., 2.189 < 2.97 at the 10 percent level, indicating the absence of long-run cointegration. The consideration of further control variables has led to significant cointegration. In that case, we clearly find that the F-statistic exceeds upper critical bound (UCB) i.e. 4.391 > 4.23 at the 5 percent level. Summing up, we can argue that bidirectional long term causality (i.e. cointegration) between oil price and Russia' real exchange rate is deeply supported when accounting for additional explanatory variables, while we only find a unidirectional interaction dynamic running from real exchange rate to oil price when neglecting the possible "pulling" effects of control variables in explaining the focal nexus. This link that runs from REER to Oil appears very logical, but the reverse relationship seems less intuitive.

Table-3: Unit Root Analysis

This highlights the need of applying more parsimonious techniques on the one hand and of accounting for additional variables when investigating oil price-real exchange rate nexus on the other hand. While trying to confirm this assertion, we attempt in the following to compare between unconditional and conditional estimations via the use of "sophisticated" methods able to accurately capture nonlinearity in the focal causal connection (i.e., to analyze the relationship in question under different frequencies by applying wavelet coherency and frequency domain). ; *, **, *** denotes the significance at 1%, 5% and 10% levels, respectively. The optimal lag structure is determined by AIC. Critical values bounds are from Pesaran et al. (2005).

Table-5: The ARDL Bounds Testing Cointegration Analysis

Estimated Model (1) (2) (3) (4) Optimal Lag Length 3, 3 3,3,1,2,0,1 3, 3 3,3,1,2,0,1 Structural Break 1996q2 2001q2 1998q1 1999q4 F-

Wavelet coherency findings

The squared WTC of a return series of oil prices and real effective exchange rate is shown in Figure -3. From blue (low coherency) to red (high coherency), it seems difficult to make sense of the picture that depicts the interdependence between REER and Oil (the upper left graph, Figure -3). Regions of high frequency are very scarce. This is particularly evident when we focus on the period spanning between 2003 and 2005. This implies that there are significant regions where the interdependence between the two variables seems stronger and other time-frequency spaces where the co-movement between series appears sharply unclear. When looking at the phase-difference, we worthy note that the real exchange rate and oil price move together at a specified frequency band since the phase relationship appears situated between -π/2 and 0. In that case, REER is lagging. The 5-quarter cycles are in phase from 1995q1 to 2009q3, except the period spanning between 2003q4 to 2004 q3 (the blue line in the lower left graph, Figure -3). This implies that for almost all the period under consideration, the focal variables seem positively correlated, while the studied connection appears negative in 2004. In that period, there is an anti-phase relationship between the two variable where oil price is leading (the phase difference is between -π and π/2. This observation may be owing to the adoption of less restrictive capital control regulations, making a switch from an authorization-based system to flow control, leading to a steady upward pressure on the ruble. Nevertheless, when additional variables are controlled for, the relationship between the two key series changes remarkably (the blue line in the lower right graph, Figure -3). We clearly show that the real exchange rate and oil price move together conditioning upon GDP, governmental expenditure revenues, terms of trade and the productivity differential of welldefined frequency. Indeed, the phase difference is situated between -π/2 and 0, at which the Russia's real exchange rate is lagging. The 6-quarter cycles are in phase between 1995q1 and 2009q3, whereas there is an anti-phase linkage between key variables and the additional control series from 1998q4 to 2001q4 (the blue line in the lower right graph, Figure -3). This means that REER is positively correlated with Oil conditionally on potential control variables, while this interdependence becomes clearly negative from 1998 to 2001. During that period, the phase difference is between -π and π/2 and therefore there is a conditional anti-phase nexus where oil price is leading. We cannot attribute this last outcome to specific factors since Russia has experienced several purifying events in this period; For instance, the violent inflation, the Asian crisis and thus the drop in external demand (Asia is one of the main trade partners of Russia) and the positive boost from increasing energy prices. In addition, on the upper right graph, we can see that there are regions of high coherency between 1998 and 2009. More precisely, there is a noticeable interaction dynamic between REER and Oil conditional on GDP, governmental revenues, terms of trade and productivity differential. Unsurprisingly, the studied period (in particular from 1998 to 2009) was marked by failing stabilization and disappointing macroeconomic performance and the aftermath of the current economic crisis that may affect widely the external energy demand, especially when considering that Europe seems a potential trading partner of Russia 3 . The fiscal policy within the Putin administration framework aiming at changing oil price elasticities of revenues and expenditures to avoid the possible harmful effects of sudden disturbances, speculative attacks and negative oil shocks [START_REF] Merlevede | Russia From Bust to Boom and back: Oil Price, Dutch Disease and Stabilisation Fund[END_REF]) may be also advanced as a possible explanation of these observations. Importantly, the conditional continuous wavelet transform approach has allowed us to highlight the time scale interdependence between the real exchange rate and oil price. This seems highly difficult when not accounting for control variables. It is also hard to capture proper nexus between the considered series at specified time horizons with conventional methods as ARDL bounds testing approach.

Figure-3: Conditional vs unconditional wavelet coherency: The REER-Oil nexus

Notes: The thick black contour designates the 5% significance level against red noise which is estimated from Monte Carlo simulations using phase randomized surrogate series. The cone of influence, which indicates the region affected by edge effects, is also shown with a light black line. The color code for coherency ranges from blue (low coherency) to red (high coherency). . This outcome may be due to the fact that the energy market is a large market relative to other commodities and the assumption of financial speculation may be evident. The demand and supply shocks in the global oil market often entailed offsetting changes in oil inventories to reinforce oil price instability that may affect intensely real exchange rate in low frequency bands through differential price [START_REF] Alquist | What do we learn from the price of crude oil futures?[END_REF] and [START_REF] Bouoiyour | Exchange Rate Impact on Russia's Exports: Some evidence from an evolutionary co-spectral analysis[END_REF]). Furthermore, we should mention that only the interaction dynamic that runs from oil price to real exchange rate changes among the different frequencies involved. The reverse interconnection remains insignificant. This implies that Russia should take proper attention to disturbances, mainly caused by oil price and continue to decrease its oil dependency. Thus, there is a great need of solid fiscal, economic and political measures to appropriately avoid external shocks, mitigate oil dependency and then create the confidence needed for economic recovery. Interestingly, wavelet coherency and frequency domain via [START_REF] Breitung | Testing for short and long-run causality: a frequency domain approach[END_REF] method seems more suitable to properly test the dynamic between oil price and Russian real exchange rate than ARDL Bounds testing approach. Unsurprisingly, the last technique seems unable to detect whether the strength of co-movement changes substantially within the time scales over the period in question and if there are some differences with respect to the focal nexus among the different frequencies involved. Beyond this computational comparison, the observed outcomes are rather substitutable than complementary. ARDL outcomes and those of wavelet coherency appear heavily different. More precisely, bidirectional long term causality is supported when using ARDL, while a unidirectional connection is sharply observed when carrying out wavelet method and frequency domain analysis. These new methods seem more intuitive. It is noted that conventional and "sophisticated methods" show the utmost importance to add control variables when assessing the connection between our variables of interest (Oil and REER). With special reference to Russian case, oil price cannot always be regarded as a significant contributor of changes in real exchange rate, since it has only a significant effect on lower frequencies, conditioning upon to GDP, governmental expenditure revenues, terms of trade and productivity differential. This, partially, highlights the effectiveness of fiscal policy within the Putin administration framework aiming at changing oil price elasticities of revenues and expenditures [START_REF] Merlevede | Russia From Bust to Boom and back: Oil Price, Dutch Disease and Stabilisation Fund[END_REF] to avoid the possible harmful effects of excessive oil price fluctuations on the whole economy. But an effective risk management strategy should be able to better cope with oil shocks, to avoid short-run disturbances and to mitigate the detrimental effects of excessive ups and downs of oil price on real exchange rate. This should seriously be consolidated through the development of effective counter-cyclical fiscal policy and compensatory financial system.

Concluding Remarks and Policy Implications

This research contributes to the existing literature on the oil price-real exchange rate nexus by assessing the long-run cointegration, the interdependence and the lead-lag relationship unconditionally and conditionally to GDP, governmental revenues, terms of trade and productivity differential. To this end, we decompose the connection between these variables into short-run and long-run relationships (ARDL bounds testing approach), among different time scales (wavelet coherency) and into various frequency components (frequency domain through [START_REF] Breitung | Testing for short and long-run causality: a frequency domain approach[END_REF] method). Our findings provide insightful evidences that may be summarized as follows:

(i) Using ARDL bounds testing approach, we show a long-run cointegration between oil price and real exchange rate conditionally to the additional control variables either when treating Oil or REER as dependent variables. (ii)

By applying wavelet coherency, we observe a significant and meaningful nexus between variables in regions of high coherency. This outcome is likely to be observed when following conditional analysis rather than unconditional evaluation. Russia's real exchange rate is lagging. The studied nexus seems in phase for almost all the period, while there is an anti-phase connection between the two focal variables and the additional time series from 1998 to 2001 where oil price is leading. Indeed, REER is positively correlated with Oil conditionally on control variables, while this interdependence becomes clearly negative from 1998 to 2001. During that period, the phase difference is between -π and π/2 and therefore there is a conditional anti-phase nexus. (iii) By carrying out [START_REF] Breitung | Testing for short and long-run causality: a frequency domain approach[END_REF] approach, there is a causal relationship that runs from oil price to real exchange rate in lower frequencies. The reverse link is not supported. The oil price cannot always be regarded as a potential contributor, since it has only a significant effect on REER in lower frequencies, conditioning upon to potential control variables. (iv)

The followed methods are likely to be substituted rather than complementary. However, the use of econometric techniques that take into account frequency transformations appear more useful and intuitive. They all reveal the importance to consider nonlinearity and control variables when investigating oil price-real exchange rate relationship. Indeed, the wavelet coherency and the frequency domain technique appear more suitable and fruitful than ARDL bounds testing approach since they depict the nexus in question at well defined time-scales and specified-frequencies, respectively. At this stage, we can say that the lack of studies reaching robust nexus and conclusive results about how behave real exchange rate after the sizeable volatility of oil prices and vice versa may be mainly due to (1) the neglect of control variables that play a potential role in explaining the connection between the two key variables in the majority of works, and to (2) the ineffectiveness or the inadequacy of models used to properly depict this linkage. This research puts in evidence that the consideration of nonlinearity (time scale or frequency decomposition) is needed in the focal issue.

The observed outcomes may have important implications for Russian policymakers, practitioners and advisors on exchange and energy markets.

The significant causal link running from oil price to real exchange rate highlight the need to diversify their products to cope with cyclicality of prices and speculation, without focusing on massive Bank currency interventions in the face of an oil price shock as they will most likely to fail [START_REF] Bouoiyour | Exchange Rate Impact on Russia's Exports: Some evidence from an evolutionary co-spectral analysis[END_REF]. This energy exporting country should tackle appropriately with excessive oil price fluctuations, especially because these prices are triggered by the interplay between supply and unstable demand. The external demand itself seems to be generated by cyclicality of international commodity prices and seasonality. This country should also continue the policy reform towards weaker energy dependency. To be able to act appropriately to oil shocks, the Central bank of Russia should implement efficient monetary policy tools (i.e., robust banking system, deep financial markets and effective institutional quality) and improve the role of interest rates to effectively control the exchange market and inflation dynamics and then to better cope with sudden shocks and disturbances.

More importantly, a more flexible exchange rate would better absorb external shocks. This may be reinforced by a policy in favor of a focus on inflation target in order to counterbalance the sizable exchange rate instability. Intuitively, policy makers should be aware that the connection between oil price and the real exchange rate depends on different time scales and the frequencies involved. Since there is a causal link running from Oil to REER and not vice versa unconditionally and conditionally to appropriate variables (GDP, governmental revenues, terms of trade and productivity differential), a considerable increase in oil price may rise heavily federal revenues. This may delay reforms implemented to diversify exports and trade partner destinations and to lessen the great oil dependency.

Beyond our results and if we look more closely at the current situation in Russia, we can say that this economy faces several challenges including the heavily drop in external demand that may have substantial influence on the evolution of commodity prices, public finances and trade performance without forgetting the unpleasant economic sanctions against Russia because of its support for Ukrainian independentists. The economic and fiscal initiatives of Putin administration may help to better cope with shocks and to mitigate the dependence on energy sector, since we believe that Russia is aware of the very prime importance of diversification (in terms of products or sectors and trade destinations) to overcome the possible detrimental effects of the decrease in external demand and the cyclical behavior of international commodity prices. While our research does not say much about the routes through which oil price may affect differently the real exchange rate, it clearly indicates the presence of short-term nexus conditioning upon potential control variables. The substantial question here is whether we can affirm that Russian monetary authorities have won a first round. The real exchange rate depends significantly to oil price fluctuations, but this link in only valuable in the short-run. The drastic and appropriate efforts pursued by Russian authorities have allowed to reach a progressive connection between both focal variables. It remains now a second round to be won by the acceleration of deeper structural reforms to mitigate the dependence of the Russian economy on oil price movements. This needs an impactful strategic development via an increase of investment in human capital, an improvement of innovative capacity and a fight against government corruption.

Figure

  Figure-1: The evolution of real exchange rate and oil prices

  ). If Z has no influence on the studied relationship, this implies that , if the conditional variables under consideration lead to a modification on the studied relationships, this means that

  the wavelet scale axis and time S indicates smoothing in time.

  p. To overcome the[START_REF] Granger | Investigation causal relations by econometric models and crossspectral methods[END_REF]'s test restrictions, we use the frequency-domain Granger causality as a decomposition of the total spectral interdependence between variables into a sum of instantaneous and the bidirectional causality terms.Geweke (1984) argues that

  the lower triangular matrix of the Cholesky decomposition.

  Wavelet Partial Coherency(REER, Oil | gdp, gov, te, prod) the nexus between oil price and the real exchange rate. It is well shown that Oil Granger-cause BPI in the lower frequencies (i.e. long-run time scale) or more precisely when link is not supported at any frequency. These outcomes change when considering the aforementioned control variables. More precisely, we find that the causal relationship running from Oil to REER stills significant at low frequency band but not in large frequency or accurately [ ]

Figure

  Figure-4: Conditional vs unconditional frequency domain: The REER-Oil nexus

  has linked real exchange rate to real interest rate differential while trying to mitigate problems related to

	monetary models. The real exchange rate	(	) REER is defined as follows: t
			t REER	=	t e	+	p	t	-	p	* t	(1)
	where t
			t p	=	α	T t p	+	1 ( α -	)	NT t p	(2)
			* t p	=	α	*	T t p	*	+	1 (	-	α	*	)	NT t p	*	(3)
	where	p	( * T t t p T	)			and	NT t p	( * NT t p	)

e : the price of a unit of foreign currency in terms of domestic currency; t p : the logarithm of domestic price; * t p : the logarithm of foreign price.

Suppose that traded and non-traded goods are produced in the home and foreign countries, respectively.

Table - 1: Descriptive Statistics

 -

														ln	prod	t
	Mean			4.501413	5.176710	7.657595	5.867372 -0.568013	3.915802
	Median		4.522875	5.014018	7.816417	5.953243 -0.585051	3.864302
	Maximum		4.824306	6.524685	9.355912	7.593374 -0.218863	5.660179
	Minimum		4.048301	4.181183	5.389072	3.583519 -0.809386	2.063312
	Std. Dev.		0.214080	0.601103	1.132440	1.110010	0.140820	1.051965
	Skewness	-0.528728	0.411982 -0.279685 -0.179176	0.379927 -0.015771
	Kurtosis		2.445241	2.215188	1.868124	1.838564	2.621271	1.959960
	Jarque-Bera		3.505513	3.183161	3.918679	3.631819	1.771999	2.661586
	Probability		0.173296	0.203604	0.140952	0.162690	0.412302	0.264268
							Table-2: Correlations
				ln	t REER	ln	t Oil	ln	t gdp	ln	t gov	ln	t te	ln	prod	t
	ln	t REER		1
	ln	t Oil		0.6826637		1
	ln	t gdp		0.5654454 0.8858752		1
	ln	t gov		0.6214614 0.8879732 0.99203015		1
	ln	t te		-0.2367291 0.13121513 0.24743089 0.18235737	1
	ln	prod	t	0.7463875 0.88805956 0.85951042 0.88023320 0.07560403	1

t REER ln t Oil ln t gdp ln t gov ln t te ln

Table - 4: Lag Length Selection

 -

				VAR Lag Order Selection Criteria
	Lag	LogL		LR												FPE	AIC	SC	HQ
					(1):	REER F	(ln	t REER	/	ln	t Oil	)
	0	73.94200		NA									0.004603 -2.543346 -2.397358 -2.486891
	1	77.02781		5.610559* 0.004268* -2.619193* -2.436708* -2.548625*
	2	77.21090		0.326228							0.004398 -2.589487 -2.370505 -2.504805
	3	77.26023		0.086117							0.004556 -2.554918 -2.299439 -2.456122
		(2):	F REER	(ln	REER	t	/	ln	Oil	,	ln	gdp	,	ln	gov	,	ln	te	,	ln	prod	)
	0	68.16289		NA									0.005613 -2.345770 -2.122718 -2.259995
	1	71.35393		5.539152* 0.005171* -2.428450* -2.168223* -2.328379*
	2	71.48850		0.228529							0.005346 -2.395793 -2.098390 -2.281426
	3	71.92033		0.716988							0.005468 -2.374352 -2.039774 -2.245689
					(3):		Oil F	(ln	t Oil		/	ln	t REER	)
	0	23.82324		NA									0.028483 -0.720845 -0.574857 -0.664390
	1	25.36429		2.801911							0.027934 -0.740520 -0.558035 -0.669951
	2	29.39411		7.180403* 0.025029* -0.850695* -0.631713* -0.766013*
	3	29.63182		0.414911							0.025746 -0.822975 -0.567497 -0.724180
		(4):	F Oil	(ln	Oil	t	/	ln		REER	,	ln	gdp	,	ln	gov	,	ln	, te	ln	prod	)
	0	32.04563		NA									0.028448 -0.728932 -0.290968 -0.559568
	1	36.34905		6.572487* 0.025278 -0.849056 -0.374596 -0.665578
	2	38.43088		3.103833							0.024360 -0.888396 -0.377438* -0.690804
	3	40.26022		2.660847						0.023700* -0.918553* -0.371099 -0.706848*
	Notes: * indicates lag order selected by the criterion; LR: sequential modified LR test statistic (each test at 5%

level); FPE: Final prediction error; AIC: Akaike information criterion; SC: Schwarz information criterion.

China, for example, is the second export partner (7.0%) and import partner (14.0%) for Russia. Japan is the fourth import partner (4.4%) and Korea appears as the seventh import partner (3.8%). For details about the main trade partners of Russia, we can refer to : http://atlas.media.mit.edu/profile/country/rus/

For more details, you can refer to the following link: http://www.thehindubusinessline.com/features/investment-world/macro-view/do-oil-prices-affect-real-gdpgrowth/article3459463.ece

For details about the trade partners of Russia, see the Observatory of Economic Complexity : http://atlas.media.mit.edu/country/rus/


Mohamed Hadded

Jean-Marc Lasgouttes

Fawzi Nashashibi

Ilias Xydias

Platoon Route Optimization for Picking up Automated Vehicles in an Urban Network †

Keywords: Automated Vehicles, Parking, Platoon, Multicriteria optimization, NSGA-II

In this paper, we consider the problem of vehicle collection assisted by a fleet manager where parked vehicles are collected and guided by fleet managers. Each platoon follows a calculated and optimized route to collect and guide the parked vehicles to their final destinations. The Platoon Route Optimization for Picking up Automated Vehicles problem, called PROPAV, consists in minimizing the collection duration, the number of platoons and the total energy required by the platoon leaders. We propose a formal definition of PROPAV as an integer linear programming problem, and then we show how to use the Non-dominated Sorting Genetic Algorithm II (NSGA-II), to deal with this multi-criteria optimization problem. Results in various configurations are presented to demonstrate the capabilities of NSGA-II to provide well-distributed Paretofront solutions.

I. INTRODUCTION

The rapid increase in the population of big cities (e.g. more than twelve million inhabitants in Paris urban area) has lead to increase in the number of cars, while the average daily use of each car is only 4 km [START_REF]World streets: the politics of transport in cities, project information[END_REF]. In order to reduce greenhouse gas emissions, there is a need to reduce the number of privately owned vehicles. Carsharing services [START_REF] Zhou | Opportunities for and Impacts of Carsharing: A Survey of the Austin, Texas Market[END_REF] are an alternative for people who may want to use their vehicles only once in a while, which can reduce transportation problems and lead toward environmentally friendly cities.

We distinguish between two types of car rental services, namely one-way and two-way systems [START_REF] Nourinejad | Carsharing operations policies: a comparison between one-way and two-way systems[END_REF]. In two-way systems, the user takes a vehicle at a rental station and returns it to its original station. In contrast, one-way systems allow the users to return the cars to any desired location, as long as it is indicated in advance. The vehicles are parked arbitrarily on the urban network by users, which is problematic since collecting vehicles using trucks is not often feasible in cities.

Following up on the work in [START_REF] Marouf | Automatic parallel parking and platooning to redistribute electric vehicles in a car-sharing application[END_REF], the French ANR VALET project proposes a novel approach to solve this redistribution problem using platoons of automated electric vehicles whose leaders are driven by professional drivers. An optimal routing algorithm is in charge of defining the platoon drivers' routes to the parking areas where the following cars are parked in a completely automated mode. The main idea of VALET is to retrieve vehicles parked randomly on the urban parking network by users. The vehicles may be parked in electric charging stations, car parks dedicated to car sharing vehicles or in regular parking places. Once the vehicles have been collected and guided in a platooning mode, the objective is then to guide them to their allocated parking areas or to their respective parking lots. Then each vehicle is assigned a parking place into which it has to park in an automated mode.

In this work, we tackle the part of this approach that concerns the optimal allocation of vehicles to platoon tours. Each platoon follows an optimized route to collect and guide the parked vehicles back to their allocated parking area or to their respective original stations. The optimization takes into account 3 performance measures for this use case. First, the time needed to collect and guide all the vehicles to their original station (i.e. the longest platoon tour duration) should be minimized. Second, due to our assumptions on the energy costs, the total energy required by the platoon leaders and the parked vehicles to reach the rental station should be minimized. Third, the number of human drivers should also be minimized. Some other choices would have been possible, but the point of this paper is to show how the optimization can be implemented, and how it scales when the system becomes large.

The remainder of this paper starts with a short overview of related work. An Integer Linear Program formulation of the PROPAV problem is then presented in Section III. It can be solved using the multi-criteria optimization algorithm NSGA-II, as described in Section IV. Section V is devoted to extensive experimental results and performance evaluation. Finally, Section VI summarizes the main contribution of the paper and presents future directions.

II. RELATED WORK

As mentioned in the previous section, traditional vehicle sharing services are based on the two-way scheme, where the user picks up and returns a car to the same parking station. In recent years, services such as Car2Go [START_REF]Car2go[END_REF] or Autolib [6] also permit the one-way scheme, where the user can return the vehicle to a different station. This scheme is much more attractive for users, but the service provider has to develop strategies to relocate the vehicles and restore an optimal distribution of the fleet. Currently, the vehicles are transported using trucks or human drivers. It should be mentioned that, in the carsharing relocation literature, the platooning of the vehicles has also been considered [START_REF] Daviet | Platooning techniques for empty vehicle distribution in the Praxitéle project[END_REF], where the platoon is composed of a chain of technologically innovative vehicles, led by a vehicle head. Furthermore, the problem becomes harder when the vehicles are electric.

In [START_REF] Dror | Redistribution of self-service electric cars: A case of pickup and delivery[END_REF], the authors propose an algorithm to manage auto transport trucks which are used to collect the vehicles, based Fig. 1. An example solution to the PROPAV problem on a Tabu search approach. In [START_REF] Hafez | Management of a pool of selfservice cars[END_REF], the authors extend the previous work and, among the other things, minimize the total travel time of relocation, through three different heuristics. In [START_REF] Kek | A decision support system for vehicle relocation operations in carsharing systems[END_REF] the authors design a system which is based on a threestep optimization-trend simulation for supporting vehicle sharing operators in relocating the vehicles. In [START_REF] Febbraro | One-way carsharing: Solving the relocation problem[END_REF], the authors consider that the relocation can be made by both users and staff, with a twofold objective: reducing the number of staff required and minimizing the number of shared vehicles to satisfy the demand. In [START_REF] Bruglieri | The vehicle relocation problem for the one-way electric vehicle sharing, an application to the Milan Networks[END_REF], the authors consider the electric vehicle relocation problem as a pickup and delivery problem where the routes have to start from and end in a single depot without exceeding a given maximum time duration. In this paper, we consider the problem of parked vehicle collection assisted by multi-platoons, which presents many similarities to the Vehicle Routing Problem (VRP) [START_REF] Baker | A genetic algorithm for the vehicle routing problem[END_REF]: platoon leaders correspond to vehicles, while parked vehicles that send requests to be picked up correspond to the customers to be served. However, there are differences in the objectives to optimize as well as the constraints, as we will see in the next section.

III. FORMALIZATION OF THE PROPAV PROBLEM

As shown in Fig. 1, given a set of vehicles parked somewhere in a city, one rental station and a set of human drivers, the PROPAV problem's output comprises the optimal routes to be followed by the human drivers to pick up the vehicles localized by their GPS positions. To define the PROPAV problem in a formal way, in this section we specify the data, the objective functions and the constraints.

Using the notations below, the PROPAV problem can be modeled as a graph G = (V, E) where V is the set of vertexes representing the positions of the parked vehicles to be picked up by the head of a platoon, and E is the set of edges representing the shortest path between vehicle positions. In our study, we have used the Open Source Routing Machine (OSRM [START_REF] Osrm | Open Source Routing Machine[END_REF]) to determine the set E.

Let {1, . . . , N } be the set of parked vehicles to be retrieved by human drivers, where N is the number of parked vehicles plus the rental station. Vehicle i is geographically located at coordinates (x i , y i ). By convention, vertex 0 is the rental station located at position (x 0 , y 0 ). Let D ≥ 1 be the number of available drivers. The problem is to design a set of K tours (r 1 , . . . , r k ), one platoon per tour with 1 ≤ k ≤ D, that minimizes the longest platoon tour duration, the number of platoons, and the total energy consumed by the platoon leaders and the picked up vehicles under the constraints, that the vehicles have limited energy resources, the tour duration is limited by the energy constraint, each tour begins and ends at a rental station, each parked vehicle should be retrieved by exactly one platoon leader.

The inputs of the model are:

• Q gives the maximum number of cars that can be in the platoon; • d i,j is the shortest distance required to travel from location i to location j; • e i,j is the amount of energy (travel cost) required to travel between positions i and j (KWh). Most electric cars cover 100 km over 10 kWh; • R i is the remaining range for a vehicle parked at location i, which estimates the distance that can still be covered with the energy remaining in the battery of the parked vehicle (see e.g. [START_REF] Bolovinou | Online prediction of an electric vehicle remaining range based on regression analysis[END_REF] for the evaluation of this value); • ϑ k is the average speed of the vehicle k. and the decision variables (with a value of 0 or 1) are:

• X k ij is equal to 1 if the vehicle parked at position j is retrieved by the platoon leader k immediately after the vehicle parked at position i; • Y k i is equal to 1 if the vehicle parked at position j is retrieved by the platoon leader k; • Z t ij is equal to 1 if the vehicle that was parked at position t visited the location j immediately after location i. Let D k be the tour duration of the platoon leader k

D k = N i=0 N j=0 d i,j X k ij /ϑ k
and E be the amount of energy consumed by vehicle to reach its final destination.

E = D k=0 N i=0 N j=0 e i,j Z l ij Y k
Objectives the following values should be minimized:

• longest platoon tour duration Minimize T T = max k∈[0,D] D k , (1)
• total energy required for collection

Minimize

T E = N i=0 E i , (2)
• number of human drivers

Minimize N D = K. (3)
Constraints:

• all platoons have the same starting depot

D k=1 Y k i = 1 i = 1 (4)
Y k i = 1, ∀i ∈ [1, N ] (5)
• residual stored energy constraint

D k=1 N i=0 N j=0 d i,j Z t ij Y k t ≤ R i ∀t ∈ [1, N ] (6)
• all platoons have at most

Q vehicles N i=1 Y k i ≤ Q ∀k ∈ [1, D] (7)
• correct flow of cars through the edges

N i=0 N l=0 X k il = N l=0 N j=0 X k lj ∀k ∈ [1, D] (8)
• sub-tours are eliminated

N i=0 N j=0 X k ij = N i=1 Y k i ∀k ∈ [1, D] (9)
Therefore, the PROPAV problem is defined as follows: minimize (1)-(3) under the constraints (4)-( 9) described above. However, the design of platoon-assisted parked vehicle pickup systems raises a problem of conflicting objective functions. For example, Fig. 2-a and Fig. 2-b illustrate two possible routes that can be used by the platoon leader (PL) to pick up the parked vehicles and guide them to the rental station. The first and the second weight value in each edge respectively give the link duration in min and the amount of energy required in kWh. The first route (Fig. 2-a) is the shortest tour duration, being 38 min, but it is not optimal in terms of total energy consumed 87 kWh, while the second route is the better in terms of total energy 73 kWh but its longest platoon tour duration reaches 48 min. Due to the high number of possible solutions in the PROPAV problem and the conflicting nature of its objectives, there exists no unique optimal solution, but rather a set of solutions that form a Pareto front [START_REF] Deb | A fast and elitist multi-objective genetic algorithm: NSGA-II[END_REF]. The next section describes this front and presents an algorithm able to produce it.

IV. MULTI-CRITERIA APPROACHES FOR PROPAV OPTIMIZATION A. Overview of multi-objective optimization

Multi-objective optimization, also known as multi-criteria optimization, is part of multiple criteria decision-making (MCDM), which considers optimization problems involving more than one criterion to be optimized simultaneously. The general Multi-objective Optimization Problem (MOP) can be written as follows:

min x∈D f i (x), i ∈ [1, m] (MOP)
where the vector x = (x 1 , . . . , x n ) T ∈ D contains the set of n decision variables, f i (x) is the objective function, m is the number of objectives, the vector y = (y 1 , y 2 , . . . , y m ) is a solution, with y i = f i (x), and D is the feasible solution space. In MOP minimization, we say that solution x ∈ D is dominated by solution x ∈ D, denoted by x ≺ x, if and only if the following two conditions are satisfied: i)

f i (x ) ≤ f i (x), ∀i ∈ [1, m] ii) ∃i ∈ [1, m] such that f i (x ) < f i (x)
The set of the non-dominated solutions, often called the Pareto front, is denoted by P F :

P F = {z ∈ D | z ∈ D, z ≺ z}.
In other words, the Pareto front provides the best trade off values for the objective functions considered. NSGA-II [START_REF] Deb | A fast and elitist multi-objective genetic algorithm: NSGA-II[END_REF] and MOPSO [START_REF] Ali | Energy-efficient clustering in mobile adhoc networks using multi-objective particle swarm optimization[END_REF] are used to find the Pareto front for multi-objective optimization problems. These methods are respectively the multi-objective version of the genetic algorithm and particle swarm optimization in which the solutions explored are classified into Pareto-optimal fronts.

B. NSGA-II algorithm for the PROPAV problem

As shown in Fig. 3, NSGA-II starts from an initial population (P) of randomly generated solution vectors, called individuals. Then, a new population (Q i ) is generated at each iteration by using the crossover and mutation operators. After that, the two populations (P i ) and (Q i ) are merged together to form a single population R i , which will be sorted based on the non-dominance principle F 1 , F 2 , . . . , F n . Finally, only the best solutions passing the selection step are included in the population of the next iteration (P i+1 ) while the other individuals are deleted. If only some solutions of the last front F l should be added to P i+1 , these solutions are chosen such that they have the best crowding distance. The aforementioned steps are repeated until the stop criterion is met.

In PROPAV, an individual represents a set of platoon tours where the i-th human driver follows the i-th tour. Each platoon tour is defined by the sequence of numbers of parked vehicles that will be retrieved by the same platoon leader. Each individual is divided into two parts, namely the vehiclepart and platoon-part. The first part is composed of a set of integers where each integer represents a parked vehicle retrieved by a platoon leader, and the second part contains platoon leader information. In the platoon-part, the i-th integer represents the number of parked vehicles retrieved by the i-th platoon leader (see Fig. 4 for an example). An individual is considered valid if and only if (1) each vehicle number occurs exactly once in the vehicle-part, (2) the sum of the integers included in the platoon-part is equal to the total number of parked vehicles to retrieve and (3) the size of the platoon-part is less than or equal to the number of available human drivers.

The crossover is one of the main operators used in the production step of NSGA-II to create a new population. It constructs two new child individuals by combining parts from a pair of parents (i.e. two solution vectors). After recombination, each new individual undergoes mutation operator which is applied with a probability less than 0.1. This operator is used to change the values of some genes randomly, and serves as a search strategy to prevent the algorithm from being trapped in local optima. An example of mutation is illustrated in Fig. 5.

V. SIMULATION RESULTS AND PERFORMANCE

EVALUATION

In this section, we present and discuss the simulation results obtained by NSGA-II and we compare its performance to MOPSO [START_REF] Ali | Energy-efficient clustering in mobile adhoc networks using multi-objective particle swarm optimization[END_REF], a popular algorithm for multi-criteria optimization problems. We implemented both NSGA-II and MOPSO using the Java programming language. Moreover, all our simulations were conducted using two Intel Core i7 computers running under Windows 10. We evaluated the performance of these algorithms using 4 configurations (see Table I) with different numbers of vehicles {50, 100, 150, 200} located Fig. 5. An example of a mutation operator: The 4th gene of the vehicle-part of the individual suffers a mutation which corresponds to vehicle number 6 belonging to the second platoon. After mutation, vehicle 6 belongs to the first platoon (i.e. vehicle number 6 is retrieved by platoon leader 1). In order to make the mutated individual correct, the platoon-part of the individual is updated (-1 in the number of vehicles retrieved by platoon 2 and +1 for platoon 1). inside Paris, as shown in Fig. 6. As in [START_REF] López-Ibáñez | Automatically improving the anytime behaviour of optimisation algorithms[END_REF], we conducted 15 independent simulation runs for each scenario using randomly selected initial populations. Table II shows the simulation parameters used in our experiments. We used the web API of the Open Source Routing Machine (OSRM [START_REF] Osrm | Open Source Routing Machine[END_REF]) to calculate the distances of the shortest paths between the different positions of the parked vehicles. For the all tests carried out, we fixed the computational time for the two algorithms to 10 min for all configurations under consideration.

Our goal is to determine the most suitable algorithm for the PROPAV problem among those that are very widely used in the area of multi-criteria optimization (e.g. NSGA-II and MOPSO). To do so, we evaluate the solutions provided by both NSGA-II and MOPSO in terms of the three fitness functions considered in the PROPAV problem. Each simulation produces a set of non-dominated solutions (Pareto front). Then, we build the final Pareto front of each scenario by The two figures show that the distribution of non-dominated solutions in the Pareto front found by NSGA-II is more localized than with MOPSO, which makes the choice of the suitable solution easier. In addition, we can note that NSGA-II provided the best approximation of the Pareto front.

In order to better compare the performance of these two algorithms, we evaluate the Pareto fronts (PF) obtained by NSGA-II and MOPSO in terms of the Ratio of Non-dominated Solutions (RNS), spacing [START_REF] Schott | Fault Tolerant Design Using Single and Multicriteria Genetic Algorithm Optimization[END_REF] and spread [START_REF] Ranjithan | Constraint Method-Based Evolutionary Algorithm (CMEA) for Multi-objective Optimization[END_REF]. The goal of this comparison is to demonstrate the effectiveness of NSGA-II on different configurations making it the best solution for the PROPAV problem. The third column of Table III shows the RNS ratio for the two algorithms. NSGA-II provides a Pareto front with more non-dominated solutions than MOPSO. For instance, in configuration U3, NSGA-II achieves an RNS of 19.03% in contrast to MOPSO which shows a rate of 6.84% (i.e. approximately 64.06% lower than NSGA-II). Thus, the proposed approach gives a wide range of non-dominated solutions, whereas the MOPSO algorithm produces a Pareto front with a small number of solutions.

Table III also shows the average spacing and spread for NSGA-II and MOPSO, respectively. We can note from these two figures that the Pareto fronts obtained by NSGA-II are the best on all the scenarios considered in our experiments. For instance, for configuration U4, NSGA-II gives an average spacing of 51.513, whereas it reaches 7028.7 in MOPSO. Thus, the set of non-dominated solutions given by NSGA-II are closely spaced, which can simplify the choice of the best solution from the Pareto front obtained. Moreover, the results presented in Table III prove that NSGA-II is able to provide a Pareto optimal set with a widely spread distribution of solutions.

Table IV compares the best results in the Pareto Front sets given by NSGA-II and MOPSO for each configuration. 15 independent runs are performed for each algorithm. Then, we gathered the final optimal Pareto fronts to identify the global non-dominated solutions set. The best solution is the one that has the smallest normalized Euclidean distance to the ideal objective vector 1 . The table clearly shows that the NSGA-II algorithm gives better results regarding the number of platoons, platoon tour duration and the total consumed energy for all configurations under consideration. Figure 9 shows the generational distance [START_REF] Veldhuizen | Multiobjective Evolutionary Algorithm Research: A History and Analysis[END_REF] for the two algorithms as a function of computational time, where the number of parked vehicles is equal to 200. We can note from this figure that NSGA-II converges very quickly (the generational distance decreases rapidly with time) towards the true Pareto front within 20 min compared to the MOPSO algorithm that does not really converge. Therefore, based on these results, we can conclude that the convergence of NSGA-II is better than MOPSO.

VI. CONCLUSION

In this work, we have proposed an optimal routing algorithm for picking up automated vehicles based on NSGA-II. The experiments results presented in this paper show that the solution proposed is able to find near-optimal routes within reasonable computation time, even for a high number of parked vehicles. This routing algorithm can easily be adapted 1 Ideal objective vector gives the best value in the Pareto front for each separate objective.

for a variety of other cases and to solve other constraints related to vehicle redistribution problem. For instance, as the urban traffic flow is often interrupted by the presence of signalized intersections, which can lead to the presence of vehicles outside platoons as well as platoon dispersion, the number of road junctions in the calculated routes should be minimized. Moreover, a parallel implementation of NSGA-II could make it more optimized to run under a shorter duration.

Fig. 2 .

 2 Fig. 2. Conflicting objective functions in the PROPAV problem

Fig. 3 .

 3 Fig. 3. Overview of the NSGA-II algorithm

Fig. 6 .Fig. 7 .

 67 Fig. 6. A configuration with one rental station (white) and 150 parked vehicles (red)

Fig. 8 .

 8 Fig. 8. Pareto front obtained by NSGA-II and MOPSO for the configuration U4.

Fig. 9 .

 9 Fig. 9. Generational Distance (GD): It measures how far the Pareto front (PF) is from the true Pareto front (PF*) obtained by gathering all the nondominated solutions given by the two algorithms in 15 independent runs.

TABLE I

 I

		SIMULATION SCENARIOS
	Configuration	# of vehicles Platoon size # of platoons
	Low (U1)	50	5	15
	Medium (U2)	100	6	25
	High (U3)	150	7	30
	Very High (U4)	200	8	40

TABLE III PERFORMANCE

 III COMPARISON BETWEEN NSGA-II AND MOPSO IN TERMS OF RNS, SPACING AND SPREAD METRICS.

	Scenario	MOEA	Ratio of non-dominated Spacing Spread
			solutions
	U1	NSGA-II	0.14	12.26	216.98
		MOPSO	0.07	15.89	143.49
	U2	NSGA-II	0.16	33.49	620.62
		MOPSO	0.07	43.12	373.27
	U3	NSGA-II	0.19	32.23	861.27
		MOPSO	0.07	53.48	530.94
	U4	NSGA-II	0.28	51.51	1656.6
		MOPSO	0.1	70.29	690

TABLE IV PERFORMANCE

 IV COMPARISON BETWEEN NSGA-II AND MOPSO FOR THE PROPAV PROBLEM (A BOLD FONT INDICATES THE BEST VALUE)

	Scenario	MOEA	Number	Longest tour	Energy
			of platoons duration (min) (kWh)
	U1	NSGA-II	12	27.80	4.45
		MOPSO	12	32.81	4.88
	U2	NSGA-II	20	48.74	15.00
		MOPSO	21	51.44	15.93
	U3	NSGA-II	23	58.40	27.89
		MOPSO	27	58.99	29.55
	U4	NSGA-II	29	65.30	39.11
		MOPSO	32	69.13	44.95

This research was supported by the VALET grant from French research agency (ANR-15-CE22-0013)