Larefi -Laboratoire D'analyse

Keywords: Output volatility, Inflation, FDI, gravity model, source countries, European Union, MENA JEL codes : F21, F43, F44

Macroeconomic determinants of FDI are seldom analyzed from the perspective of source countries, priority being generally given to host country characteristics. In a gravity set-up, we analyze FDI flows from European Union to MENA economies. We find that European investment to our MENA host countries is higher, the lower the source country output volatility, thereby supporting the existence of an income effect for European Transnational corporations. In the case of MENA economies, source country output volatility's adverse impact on FDI is counterbalanced by the positive attraction effect of domestic swings of activity. We also find that 1995's Barcelona agreement has reinforced MENA countries' vulnerability to European short-and medium-term macroeconomic cycles. The emergence of non-traditional sources of European FDI is, however, a positive evolution since Eastern and Central European investment to MENA countries is less sensitive to host and source country macroeconomic volatility that traditional Western and southern European sources tend to be. Our results are robust to various changes in estimator, sample composition or measurement of instability.

Introduction

Whereas the domestic determinants of foreign direct investment (FDI) attraction have been extensively studied1 , the empirical literature has, until very recently, tended to disregard external macroeconomic conditions. Those conditions need, however, to be considered as crucial determinants of FDI inflows since, every time there is a global financial crisis, economists and policymakers are abruptly reminded of just how volatile financial flows can be. [START_REF] Loayza | Macroeconomic volatility and welfare in developing countries: An introduction[END_REF] have expressed that need, stressing that the role of trade and FDI flows as shock absorbers or amplifiers has not yet been fully established.

Undoubtedly, external shocks tend to trigger Gross Domestic Product (GDP) volatility for developing economies [START_REF] Martin | Globalization and Emerging Markets: With or Without Crash?[END_REF], the impact of those shocks on activity and employment being channelled, inter alia, by huge variations in foreign investment flows (Lensink and Morrissey, 2007). As an illustration, global FDI flows severely dropped in 2009 when western economies and firms were hit by the 2008 financial crisis and the ensuing drastic credit restriction (UNCTAD, 2009). Even though global crises matter to explain brutal swings of FDI inflows to developing economies, we claim in this paper that it would be also fruitful to take the external instability into account at the more disaggregated source country level. Source-related macroeconomic determinants of FDI have, so far, not been studied properly or explicitly in the literature interested in the determinants of FDI inflows to developing economies. Since they generally explain FDI inflows to developing economies without considering their origin, the bulk of the studies that are typically found in the literature are constrained by their framework to use one average measurement of external conditions, thereby failing to address source-related determinants of FDI2 .

In sharp contrast with this literature, the present paper tests the sensitivity of FDI levels to source country GDP instability in a gravity set-up. The gravity model is generally seen as a relevant approach to explaining bilateral trade flows [START_REF] Evenett | On theories explaining the success of the gravity equation[END_REF][START_REF] Mello-Sampayo | Competing-destinations gravity model: an application to the geographic distribution of FDI[END_REF], and this model has also been used recently to explain bilateral flows of FDI [START_REF] Frenkel | A panel analysis of bilateral FDI flows to emerging economies[END_REF][START_REF] Bevan | The determinants of foreign direct investment into European transition economies[END_REF][START_REF] Desbordes | Foreign Direct Investment and bilateral investment treaties: An international political perspective[END_REF][START_REF] Busse | FDI promotion through bilateral investment treaties: more than a bit?[END_REF]. This framework allows the FDI effect of host countries characteristics to be differentiated according to the degree of instability of the source country, while controlling for a series of distance-related factors. A couple of recent papers have used this gravity set-up in order to identify the specific effects on bilateral FDI flows of various sources of macroeconomic volatility. [START_REF] Cavallari | Nominal and real volatility as determinants of FDI[END_REF] have examined the role of host and source country nominal and real volatility, but their study is limited to the bilateral FDI flows among 24 OECD economies. Levy-Yayeti, Panizza and Stein (2007) are closer to our concern since they address more explicitly North-South FDI. They place their focus, however, on the source level, and they do not explicitly address the host country determinants of the corresponding FDI inflows that are received by developing economies because of their high level of output volatility.

Our paper focuses on North-South investment and aims at identifying the simultaneous FDI effect of both source and host country nominal instability by studying the impact of European countries' macroeconomic instability on FDI flowing to Mediterranean economies of Middle East and North Africa (MENA) 3 . There are several reasons why this set-up has been chosen. First, as in many other parts of the world, Mediterranean MENA countries have progressively reformed their institutions and opened their economies up to foreign trade and investment throughout the last two decades [START_REF] Mina | The institutional reforms debate and FDI inflows to the MENA region: The "best" ensemble[END_REF]. However, although MENA countries have enjoyed a steep increase in FDI inflows from the mid-1990s onwards, they have failed to reap the benefits of their efforts in terms of their share of world FDI. They have, equally, suffered from the marked instability of those incoming flows of investment during the last decade (UNCTAD, 2009). Second, FDI sensitivity to macroeconomic outcomes is of vital concern for all those economies that, as highly age-dependent labour-exporting MENA economies, are particularly dependent on sources of foreign investment. As such, they tend to be extremely vulnerable to economic fluctuations in source countries, with dramatic consequences on their own output volatility [START_REF] Koren | Volatility and development[END_REF]. Third, Mediterranean MENA economies have been more closely associated, since the mid-1990s, with the European Union via the bilateral Euro-Med trade and exchange agreements, which has made their 3 We consider Morocco, Algeria, Tunisia, Lybia, Egypt, Jordan, Lebanon, Syria and Turkey as our MENA Mediterranean countries. In the econometric part of our work, we have focused on the five biggest FDI receivers as our host country sample: Morocco, Tunisia, Egypt, Lebanon and Turkey. MENA Gulf countries are not considered by our analysis, neither as host nor as source countries, because they are not as closely associated with European trade and investment than the Mediterranean ones.

economies more exposed to external sources of instability 4 . Table 1 shows that, since 1995, FDI instability has been significantly higher for South and Eastern Mediterranean countries than it has been for other developing regions in the world. The MENA Mediterranean countries have become much more subject to external sources of instability coming from Europe, without benefitting from structural transfers flows like those that sustained Central and East European economies during their accession to the European Union. Fourth, GDP volatility is traditionally strong in MENA economies, thereby having significantly reduced domestic investment (Aysan, Pang and Veganzonès-Varoudakis, 2009) Whereas [START_REF] Cavallari | Nominal and real volatility as determinants of FDI[END_REF] and Levy-Yayeti, Panizza and Stein (2007) recently found that investors tend to substitute foreign to domestic investment when domestic macroeconomic conditions worsen, we find that European FDI to Mediterranean MENA economies is marked by an income effect, i.e. transnational corporations tend to reduce their foreign investment when macroeconomic instability increases on their domestic market. As a consequence, 4 They have also increased regional integration by adopting various free trade agreements (UMA, GAFTA). 5 Aysan, Pang and Veganzonès-Varoudakis (2009) also find a significant adverse effect of external instability, as measured by the debt burden, on private capital formation in the region. They estimate that a more stable external environment would have increased investment decisions on average by 8.4% in Morocco, by 10% in Egypt in the 1980s, by 3.9% in Morocco and by 2.6% in Tunisia in the 1990s. [START_REF] Chan | Impact of foreign direct investment volatility on economic growth of ASEAN-5 countries[END_REF] have also found that the instability of ICRG scores for political, financial and economic risks has an adverse effect on the FDI entering MENA economies, but they do not explicitly address macroeconomic instability.

Mediterranean MENA economies are highly vulnerable to macroeconomic fluctuations in European source countries, via the FDI channel.

The remainder of the paper is organized in four sections. Section 2 discusses the effects of several kinds of macroeconomic instability (internal/external) on FDI inflows. Section 3 presents the indicators of macroeconomic stability (internal/external) retained in the empirical study, as well as the model and the estimation strategy. In Section 4, we first present and then discuss the results of our gravitational model panel data estimation, paying specific attention to source countries and the possibility of cumulative effects. Section 5 shows our estimation results when source countries are differentiated according to their development level. Section 6 develops a series of robustness checks, and Section 7 concludes.

Fdi and macroeconomic volatility : Overview of the literature

Though the nominal sources of volatility, essentially price and exchange rate fluctuations, have been frequently analyzed since the early nineties 6 , real output volatility, especially external, has been somewhat downplayed by empirical studies macroeconomic stability. The theoretical effect of output volatility on FDI flowing to developing countries is far from being straightforward, especially when it comes to considering the different sources of macroeconomic instability. As for host country volatility, it may, undoubtedly, increase the entry costs associated with investing abroad, thereby leading to a FDI reduction. This is particularly true when the recorded high levels of real instability indicate host country's excessive macroeconomic vulnerability to domestic or external shocks. Uctum and Uctum (2011) have shown for example that, in the case of Turkey, FDI reacts significantly and adversely to the occurrence of crises in the host country, whereas portfolio investment are most affected by source country instability. But FDI may also be attracted towards highly volatile economies insofar as they generally tend to be also rapidly growing countries. In that case, foreign investment is driven toward highly volatile countries by market opportunities. A marked depreciation of the exchange rate, or a fall in the value of stocks in host economies, may, equally, increase investment by those foreign firms that look for merger opportunities [START_REF] Krugman | Fire-sale FDI[END_REF][START_REF] Aguiar | Fire-sale foreign direct investment and liquidity crises[END_REF]. A recent illustration can be provided by the waves of sector-based consolidations that were reported in the oil and gas sectors, in mining, in the automobile industries or in financial services (UNCTAD 2009), in years of sharp decrease in global FDI flows in 2008 (-15 %) and 2009 (-30%). Hence, host country volatility may theoretically have either a negative or positive on FDI inflows. In the case of European investment in MENA economies, we may reasonably expect that the positive effects will be prevalent, insofar as vertical green-field FDI forms, that are less sensitive to output volatility than to exchange rate instability, tend be dominant7 .

When it comes to source country instability, it may have either a negative or a positive effect on FDI flows. When business cycles in the source and host countries are not synchronized, the foreign country undergoing a bust while the domestic country is benefiting from a boom, substituting FDI for domestic investment may become attractive for the firms undergoing high volatility on their origin market. In that case, a positive substitution effect explains that more volatility on the source country may increase FDI flows to other countries. In a gravity model of foreign investment between OECD countries, Cavalleri and D'Addona (2012) have found that FDI tends to increase when host country has higher output volatility, in sharp contrast with the adverse FDI effect of source volatility.

They argue that such asymmetry in the behavior of investments in host and source countries is consistent with the view that investors choose between investment options at home and abroad, in relation to the differential of volatility between economies. Likewise, Levy Yeyati, Panniza and Stein (2007) have found on a gravity model of North-South FDI that local and foreign investments tend to move in opposite directions relative to the cycle in the source country.

When business cycles in the source and host country are coordinated, transnational corporations may react to the anticipated fall in their profits and to rising financing costs by holding back on investment projects, and by disinvesting abroad [START_REF] Aizenman | Volatility, Employment and the Patterns of FDI in Emerging Markets[END_REF][START_REF] Wang | Foreign direct investment outflows and businesscycle fluctuations[END_REF].

Hence, source country volatility may reduce FDI outflow because of an adverse revenue effect undergone by firms. The effect of an external shock on FDI inflows might well worsen by bad internal macroeconomic outcomes in the host country (Eichengreen 2000, Kose, Prasad, Rogoff and[START_REF] Kose | Financial Development: A reappraisal[END_REF]. If growth turns out to be less stable, it is likely that the exchange rate instability 8 will produce even greater negative effects on FDI attraction [START_REF] Kaminsky | Leading indicators of currency crises[END_REF]. Accordingly, there are only very few chances of a country's recovering its FDI flows when an external shock is combined with bad domestic macroeconomic outcomes [START_REF] Ishii | Capital account liberalization and financial sector stability[END_REF] 9 . [START_REF] Méon | FDI Waves, waves of neglect of political risk[END_REF] have found that FDI flows to developing economies tend to be reduced in times of global symmetric economic crisis, with generalized bad external macroeconomic conditions tending to outweigh the domestic institutional or structural factors of FDI attraction or repulsion. FDI flows tend, however, to react quite differently to increased macroeconomic risks, depending on whether those investments are sourced in countries whose firms traditionally invest abroad or in economies where foreign investment is less common. Levy Yeyati, Panizza and Stein (2007) had also found that FDI flows tend to be countercyclical when they are sourced in Europe and US, and more pro-cyclical when they are sourced in Japan. Andrès, [START_REF] Andrés | What drives FDI from nontraditional sources ? A comparative analysis of the determinants of bilateral flows[END_REF] have also found that non-traditional sources of FDI are less risk-adverse than traditional ones.

Accordingly, real macroeconomic instability in source country may have both a negative and a positive effect on FDI flows to developing economies. The net effect of source country output volatility depends on the relative magnitude of those income and substitution effects [START_REF] Cavallari | Nominal and real volatility as determinants of FDI[END_REF]. The next section explains our approach for identifying which one of these two effects dominates in the case of Euro-Mediterranean FDI flows.

Model, variables and methods

In order to identify the sensitivity of FDI inflows to the host and source country's instability, we use a gravity model that links thirty-two source countries of investments to the five largest receiver countries of the MENA region (Egypt, Lebanon, Morocco, Tunisia and Turkey), during the 8 For a recent study of the contrasted effects of exchange rate instability on FDI inflows, see Schmidt and Broll (2009). 9 As the decision to set up a subsidiary abroad is generally justified by the intention to obtain long-term profits from the selected localization (Caves, 1996), the odds are, however, that the corresponding sunk costs (especially for greenfield projects) will make the FDI less reactive than other forms of capital to a transitory deterioration of macroeconomic conditions. That latter assumption has received empirical support in Fernandez-Arias and Hausman ( 2001), [START_REF] Levchenko | Do some forms of financial flows help protect against "sudden stops[END_REF] and [START_REF] Levy Yeyati | The cyclical nature of north-south FDI flows[END_REF].

period 1987-2009 10 . The standard expression of the gravity model adapted to bilateral flows of FDI can be written as:

1 2 3 i j ij ij A Y Y FDI D β β β = ( 1
)
where FDIij is the annual flow of FDI from source country i in host country j, Yi and Yj are the annual GDP levels of source and host countries, Dij is an indicator of the distance between the two countries, and A, β1, β2 and β3 are the parameters to be estimated. When Equation ( 1) is loglinearized, it gives the following equation:

Ln(FDI ijt ) = α + β 1 Ln(GDP it ) + β 2 Ln(GDP jt ) -β 3 Ln(D ijt ) + β 4 Macro ijt + β 5 Controls jt + u ij +v t +ε ijt (2)
where FDI ijt represents the value in dollars of the inflows of FDI from a country i (country of origin) entering the country j (host country) at time t. So as to control for the multilateral resistance terms identified by [START_REF] Anderson | Gravity with gravitas: A solution to the border puzzle[END_REF], we have chosen to introduce bilateral country-pair fixed effects u ij since, as stressed by [START_REF] Bergstrand | A knowledge-and-physical-capital model of international trade flows, foreign direct investment, and multinational enterprises[END_REF], sources of multilateral resistance appear to move only slowly, and country-pair fixed effects reduce the omitted variable bias associated with unobserved time-invariant pair-specific heterogeneity11 . As the dependent variable is censored, the model has been estimated by using the Tobit method, which includes temporal random effects, in order to circumvent the problem of zeros corresponding to null FDI flows without excluding them12 .

If we now consider the right-hand side of the equation (2), Ln(GDP it ) and Ln(GDP jt )

respectively stand for the natural logarithm of GDP levels of the source and origin countries, and β 1 and β 2 take a positive sign if there is a "mass" effect at work in determining bilateral direct investment flows. By extension, higher host country GDP is generally considered to increase horizontal FDI, as the size of the local market is worth being served by a multinational firm's production subsidiary. The Difference in GDP per capita (in log) between the two countries is used as a proxy for the differences in factor endowments or in the level of economic and technological development of each country. The coefficient takes a positive sign if FDI is pulled by low labour costs, and a negative sign if the FDI-related labour requirements are more skill-intensive. D ijt is the vector of the various concepts of distance controlling for the most typical sources of transaction and transport costs involved in an investment moving from one country to another. The physical bilateral distance (Distance) corresponds to the distance between the countries' capitals 13 ; FDI is generally taken as lower, the greater the distance is between the two countries involved. When, however, the host country shares a common border, language, or a former colonial link with the origin country, it is generally considered that FDI is higher. Adjacency and Common language take the value 1 if the origin and host countries respectively share a common border or have a common language; they take the value 0 in the contrary case 14 . The variable Past colonial links takes value 1 if the source country had colonized the host country, and 0 otherwise 15 .

Macro ijt is a vector of macroeconomic determinants of FDI inflows regarding either source country i or host country j, including source and host country characteristics in terms of output volatility (respectively Instability source and Instability host). The first indicator focuses on output volatility since foreign investment tends to be reduced when the short-and medium-run growth potential of the host economy is afflicted by instability. The point is particularly true of market-led FDI, but is also the case for other categories of foreign investment, every time investors seek out a stable macroeconomic environment. On the contrary, financial difficulties imposed on host country domestic firms by a high degree of growth instability can favour opportunity-led investments of a fire-sale type. In that case, FDI inflows should increase with the extent of host country's growth volatility. From what precedes, we can anticipate that the coefficient for Instability host can be either negative or positive, but the opportunity-driven positive effect seems to be plausible for MENA economies, given the nature of the foreign investments they tend to attract. As foreign firms' 13 Stein and [START_REF] Stein | Longitude matters: time zones and the location of foreign direct investment[END_REF] have also recently shown that differences in time zones have an adverse effect on the location of investment because time distance increases the transaction costs associated with activities necessitating real-time communication and interaction. Given the concentration of our sample on a limited array of time zones, that aspect has not been included in our model. 14 Former colonization links, as they have influenced the institutional, linguistic or cultural proximities that ease the building of international network ties, are considered by [START_REF] Bénassy-Quéré | Institutional Determinants of Foreign Direct Investment[END_REF] and [START_REF] Abderrezak | Foreign direct investment in the MENA region: Is there a colonisation memory effect?[END_REF] to be key determinants in explaining trade or FDI flows. 15 It should be noted that Past colonial links is a good proxy for legal origin, which appears to be significant in explaining bilateral portfolio investment flows [START_REF] Lane | International Investment Patterns[END_REF] as well as bilateral FDI flows [START_REF] Stein | Longitude matters: time zones and the location of foreign direct investment[END_REF].

investments are conditioned by the extent of their revenue in their origin markets, we also assume that source country GDP growth instability can influence FDI levels. More volatility on European source-markets may well reduce FDI outflows, thereby depressing growth prospect for periphery economies relying on external sources of finance. In that case, foreign investment can possibly act as a transmission channel for growth instability, and source-country higher volatility may reduce FDI inflows to MENA economies.

For each time period, a pseudo-coefficient of variation has been calculated, for both host and source country, using the following formula:

GDP GDP Cv Mean jt jt jt σ = (3)
where σ GDPjt is the GDP growth rate's standard deviation for the country i at the period t, and

Mean GDPjt is the average rate of growth of the GDP for country i over the same period t. Standard deviation and average values at time t have been computed as a three-year moving average over t-2, t-1 and t. We have supposed that investors observe short-term past volatility and compare it across different potential destinations. In order to avoid a null average value, we have chosen to compute it in terms of absolute value and to express it in logarithmic form 16 .

Controls jt is a vector of additional determinants of bilateral FDI flows. As it is expected that trade openness will alternatively tend to increase inflows of efficiency-led FDI by lowering export costs for TNCs, and tend to decrease market-led FDI motivated by tariff-jumping, we control for the degree of trade openness (Openness jt ), measured by the ratio of exports to GDP at time t for the host country j. Dummies for membership in a common regional trade agreement are also included, since integration spurs FDI [START_REF] Daude | Regional Integration and the Location of FDI[END_REF], especially that of a North-South nature [START_REF] Stein | Longitude matters: time zones and the location of foreign direct investment[END_REF]. As our study uses a sample of both MENA and European countries, we explicitly introduce controls for membership in three regional trade agreements (GAFTA, UMA and Euro-Med, noted as MED). [START_REF] Medvedev | Beyond trade: The impact of Preferential Trade Agreements on FDI inflows[END_REF] has provided evidence that there is a positive correlation between the size of any two economies bound by a Free Trade Agreement and their 16 Note that the huge fall in GDP growth rate for Lebanon due to the war has been controlled in estimations by the introduction of a dummy variable taking the value 1 for the war years of our sample (1987 to 1991), and 0 from 1992 onwards. A post-war dummy, taking 1 for the years 1992 to 1995 has also been introduced to control for the ambiguous trends in FDI associated with the first years of reconstruction at the end of the war.

mutual FDI flows. Similarly, it has also been found that bilateral investment treaties have positive effects on FDI inflows to developing economies in general [START_REF] Desbordes | Foreign Direct Investment and bilateral investment treaties: An international political perspective[END_REF], and to MENA economies in particular [START_REF] Mina | The institutional reforms debate and FDI inflows to the MENA region: The "best" ensemble[END_REF]. Here, we focus exclusively on bilateral trade agreements, as they are often associated with increased FDI flows of export processing by TNCs.

Equally, since the investment decisions of transnational corporations generally use a global evaluation of host country property rights (Ali, Fiess and MacDonald, 2012), any empirical assessment of FDI flows requires the introduction of a variable to control for institutional quality17 .

The comprehensive indicator that is used here, Investprofil, is particularly suitable for capturing the impact of business regulation and property rights enforcement on FDI, since it combines ratings of contract viability, risks of expropriation, profits repatriation and payment delays18 . A dummy variable, War, has been added to control for the effect of the Lebanese war on the FDI inflows received by Lebanon. Equally, a dummy variable post-war was introduced to account for the possible specific effect of post-war reconstruction on FDI inflows to Lebanon.

FDI and source country volatility

Results of the regressions for FDI levels have been reported in Table 2. The baseline regression (Column 2.1.) is in accordance with the results of the existing literature. GDP source, GDP host have the expected positive influence on the FDI levels of our five host countries. Of the estimated coefficients for the variables which represent standard proxies for the transaction and transport costs, only one, that of Common language, has the expected signs and is significant.

Adjacency and Distance, and Past colonial links, on the contrary, are not significant, suggesting that geographical distance does not explain the FDI localization concerning the five MENA economies in our sample. Difference in GDP positively explains FDI for our five MENA host countries, meaning that the greater the technological distance, the greater the investment flow. Trade openness also increases FDI levels to MENA economies, with that positive effect of trade being reinforced by regional integration, since MED and AMU both show a positive and significant coefficient. The significant effect of the Euro-Med dummy (MED), moreover, confirms that the reduction in institutional distance increases FDI flows, for European firms, irrespective of the particular colonial history of the origin and destination countries. The positive and significant impact of the quality of institutions (InvestProfil) in host countries confirms that minimizing transaction costs is an objective of European firms when they invest in the Mediterranean periphery. It should be noted, however, that participation in regional trade agreements is not always a factor of attraction for FDI: whereas MED and AMU significantly increase FDI, GAFTA has a no effect. This means that trade economic integration of MENA economies with the European Union (MED), or the economic integration between Tunisia and Morocco (AMU) has contributed significantly to the increase in FDI that each of those countries receives from the others. Having first examined the standard determinants invoked in the empirical analyses of bilateral FDI flows, we can now proceed to consider the simultaneous effects of our specific variables that focus on macroeconomic instability: host and source country output instability. Column 2.3 shows that host countries undergoing more GDP instability attract significantly higher FDI levels.

: FDI levels baseline regressions

When all the other determinants of localization are equal, output volatility is not an obstacle to FDI attraction. On the contrary, FDI flows to those economies whose GDP is the most unstable, signalling that those investments are probably more export-oriented than market-led in the case of MENA economies. In what concerns GDP instability in source countries, we find that it significantly reduces FDI flows to our sample of MENA economies (Column 2.2). This result suggests that an increase in output volatility in source country reduces the resources available to potential investors. Our result, therefore, supports the existence of an income effect for European Transnational corporations and of a vulnerability of MENA FDI inflows to European macroeconomic fluctuations. The estimated coefficient suggests that a 1% increase in source country volatility reduces the average amount of FDI outflows by 0.84%, whereas it the same increase in host country volatility tends to increase the average amount of FDI inflows by 0.87%. As results are unchanged when both source and host volatility are included in the regression, we can conclude that source country volatility decreases FDI holding host country volatility fixed, and that of host country volatility increases FDI holding source country volatility fixed. When instability increases for European source countries, investment to the MENA region slows down. When, on the contrary, real instability on their domestic market is lowered, European firms tend to increase their investment toward the most volatile MENA economies.

Our results therefore show that European investors don't arbitrate between investment options at home and abroad, thereby bringing contradiction to the substitution view of Levy-Yayeti, Panizza and Stein (2007) or [START_REF] Cavallari | Nominal and real volatility as determinants of FDI[END_REF]. As our indicator of output instability does not allow differentiating booms and busts, FDI cannot be considered as being strictly counter-cyclical, relative to the source country, and pro-cyclical, relative to the host country 19 . These results can be explained by the nature of FDI in the zone, but also by the fact that European firms have been engaged in long-term North-African localization. Our estimations also show that real exchange rate levels have no effect on the FDI levels received by our five MENA economies (Column 2.4), suggesting that fire sales, or Mergers & Acquisitions driven by price-opportunities, are not the dominant form of foreign investment in the countries of our sample. This result is consistent with the significantly higher share of green-field investment in total FDI, which differentiates MENA economies from those in other developing regions. However, their dependence on European FDI makes MENA economies particularly vulnerable to European macroeconomic volatility.

Volatility, traditional and non-traditional FDI sources

At this point, it might be objected that the estimation of the coefficient for source country instability could be driven by a selection bias. Firms from more stable countries might invest more in MENA because they are more internationalized than those from more unstable economies. One reason for this could be that those more unstable economies are less developed, which means that their firms are further from the technological frontier than those from more stable and developed economies, which limits their capacity to invest at an international level. The negative coefficient of Instability source might then simply reflect the fact that more developed economies are also more stable, and that their firms invest more in less developed countries than is the case for firms from less stable economies. In order to test this hypothesis, the set of source countries has been separated into two subsets in terms of average GDP level. The first subset, labelled Stable, is exclusively composed of developed Western European economies 20 . The second group, labelled Unstable, includes less developed and more unstable countries Eastern and Central European, Mediterranean and MENA countries 21 . The average GDP coefficient of variation, computed over the whole period for the Stable group, is about twice as low (.001435) as that of the Unstable group (.002887), with this mean value being statistically different. Two dummies, Stable and Unstable, have been generated, with Stable taking the value 1 for the countries belonging to the Stable group, and 0 otherwise, and Unstable taking the value 1 for the countries belonging to the Unstable group, and 0 otherwise. Then, Stable (or alternatively Unstable) is successively interacted with Instability source and Instability host, with those interactive variables being introduced into estimated models of FDI levels. In accordance with Lensink and Morissey (2006) and Choong and Liew (2009) who have both pointed out that FDI instability has consistent adverse effects on GDP per capita increase, whereas 20 Those of the European Union before Eastern and Central European members' accession, plus Switzerland and Norway. 21 Eastern and Central European (Hungary), Mediterranean (Malta) and MENA countries (Egypt, Morocco, Tunisia, Turkey, Lebanon, Syrian Arab Republic, Jordan, Mauritania).

the alleged positive effect of FDI levels is far less robust, we have also tested our gravity equation with standard deviations of FDI inflows as the dependent variable, computed by a three-year moving average. Results are reported in Table 4.

: FDI levels & volatility and alternative FDI sources

Table 3. Table 3 shows, first, that the coefficient for Instability source does not lose its statistical significance, and also keeps its negative sign. Second, the adverse effect of Instability source on FDI levels is magnified when limited to the more stable group, which is composed of European economies (Column 3.1). That adverse effect is, however, smoothed when restricted to the more unstable group of less developed Eastern European and MENA source countries. Column 3.2 shows, moreover, that the Instability source significantly reduces FDI volatility when it is restricted to the more stable European economies. Thus, FDI flows from more stable Western European economies are more likely to be reduced when volatility increases in host-countries, and their volatility is more likely to increase in that case. Conversely, FDI from more unstable source economies increases in response to host country output volatility. It should also be noted that FDI instability is higher for those MENA host economies that are more open to trade, and for those which are involved in regional trade agreements with Europe (Med) or with other MENA economies (AMU, GAFTA). Those countries which have the most attractive institutional environment for foreign investments are also those that experience more FDI instability.

In short, we can conclude that Western European investment to MENA is less likely to trigger volatility than FDI from countries further removed from the technological frontier. Hence, our results, like those of Andrès, [START_REF] Andrés | What drives FDI from nontraditional sources ? A comparative analysis of the determinants of bilateral flows[END_REF], confirm that FDI from traditional sources (developed economies) exhibits different sensitivity to host country conditions than FDI from non-traditional sources.

Robustness checks

In this section, we address several issues of robustness of our results.

A first issue is that focusing on a country-level indicator of macroeconomic instability such as the GDP's coefficient of variation could entail a misspecification bias arising from the fact that the real volatility that reduces FDI flows to MENA economies may be global instead of being related to source country characteristics. [START_REF] Méon | FDI Waves, waves of neglect of political risk[END_REF] have recently provided convincing evidence that global waves of FDI increase can outweigh the most prominent domestic determinants of FDI attraction. In our initial model, global FDI waves were not controlled for. However, as that might lead to overestimating the impact of source country macroeconomic characteristics, an indicator of global waves of FDI, similar to that of [START_REF] Méon | FDI Waves, waves of neglect of political risk[END_REF] was subsequently introduced, leaving the coefficients for source country and host country instability unchanged (Table 4).

A second issue concerns the choice of our estimation procedure. The tobit Random Effect model, based on maximum likelihood estimation, is considered as a robust approach when dealing with censored data concerning the dependent variable (Eaton and Tamura, 1994;[START_REF] Wei | How taxing is corruption on international investors[END_REF]. An alternative method, the one most commonly used in the literature, circumvents the problem of the zero for the dependent variables by using a simple transformation of the form ln(1+FDI), which enables the coefficient to be interpreted as an elasticity when the value of ln( 1+FDI) is approximately equal to the ln(FDI), which is accepted as a reasonable assumption [START_REF] Eichengreen | The role of history in bilateral trade flows[END_REF]

Explicitly accounting for zero FDI flows increases the variation of the dependent variable, thereby producing higher values and significance for the estimated coefficients of the various determinants of FDI since it.

: FDI levels and FDI waves Table 4. Consequently, if applying fixed effects estimator to FDI flows corrected à la [START_REF] Eichengreen | The role of history in bilateral trade flows[END_REF] does not change our results, then we can consider that they are robust and not overvalued by the use of a Tobit estimator explicitly accounting for the zero flows. A first round of estimations was made on Equation 2, with time fixed effects to control for temporal heterogeneity.

Then, Equation 2 was estimated with random effects, since unobserved heterogeneity could affect 22 See Stein and Daude (2007: 100-101) for a discussion.

the variance-covariance matrix and, consequently, bias estimations. The χ 2 statistic indicates that the fixed effects model should be preferred to the random effects model at 1%. Because of the presence of country dummies among explaining variables, however, the RE was preferred to the FE. The results reported in Table 5 show that our main findings are robust to a change of estimator.

: FDI levels, Eichengreen's correction and RE estimator Table 5. A third issue is that focusing on three-year moving averages might lead to overestimating the coefficients used to measure the effect of host and source country GDP volatility on FDI. Table 6 shows that when an alternative computation of GDP instability (five year-moving average-based variation coefficients) is used, the results reported in Table 2 are not changed. The coefficients estimated for source instability and host instability are, thus, robust to a change in the measurement of output volatility.

A fourth source of bias in estimation could result from the likelihood of Lebanon being an outlier because of the war and post-war periods. The war period itself necessarily exacerbated GDP instability in Lebanon, while FDI nose-dived, thereby possibly producing an adverse relation between host GDP volatility and FDI inflows. As for post-war reconstruction from 1991 onwards, with its context of high GDP growth variation springing from growth and FDI recovery that may have induced a positive relation between GDP host and FDI inflows. As the net effect of the Lebanese war on the estimated impact of GDP volatility on FDI might be contradictory, we introduced a war dummy, taking the value 1 for the Lebanon between 1987 and 1991 and 0 otherwise, and a post-war dummy taking value 1 for the Lebanon from 1992 to 1995, and 0 otherwise. The introduction of the two dummies did not change the coefficient estimated for GDP host volatility, which remained positive and significant in explaining FDI levels. We also checked whether our estimations are robust to change in the sample composition. Estimations of Table 2 were run without incorporating Lebanon, as that country is the most unstable of the five host countries of our sample. As the result regarding source country volatility remained unchanged, this suggests that the estimation based on the whole sample was not driven by the presence of a highly unstable outlier.

: FDI levels and 5 year-moving averages Table 6. A fifth issue is whether our findings are driven by a specific sub-period. Chow tests found a structural break for 1995 in the empirical model used to explain FDI levels. That particular year 1995 corresponds to the Barcelona agreement between European Union and the MENA countries. The results reported in Table 7 show that our main results are driven by the post-1995 period, and not by the pre-1995 period, which includes both the Lebanese war and post-war reconstruction periods.

Equally, our results show that prior to 1995, FDI volatility was driven by source country economic conditions, and more specifically by the investment climate ad measured by the quality of economic governance. After that structural break, exchange rate crises become highly significant in explaining FDI volatility, as well as source and host country GDP volatility. After the Barcelona agreement, economic governance became less significant, whereas trade openness and regional trade agreements simultaneously became more significant in explaining FDI levels and volatility. FDI inflows proved to be increasingly attracted by price-opportunities, which were probably related to nominal or real volatility in the five MENA economies. Institutional distance started to fall as a consequence of the Euro-Med agreement, while short-tem macroeconomic factors of attraction started to outweigh the structural factors that had predominated before 1995. Although regional integration increases FDI levels to developing economies via the reduction of microeconomic transaction costs, it also tends to raise the macroeconomic costs associated with volatility.

Conclusion

In the present paper, we analyze the impact of various combinations of macroeconomic instabilities on FDI localization in Middle East and North African countries within the framework of a gravity model. As such a model allows the reduction of risks and costs associated with distance (geographical, linguistic and legal) to be controlled for, this enables the impact on FDI inflows of macroeconomic sources of risks and cost to be differentiated from the other sources of risks and costs. As a gravity model also allows the impact of various determinants concerning FDI flow origins to be assessed, this enables our paper to accord special attention to source countries and to pinpoint the specific influence of their macroeconomic characteristics on FDI flows.

We consider two indicators of macroeconomic instability: output volatility in both source and host countries, and find evidence of contradictory impact on FDI flows for these two variables. When controlling for bilateral and host country characteristics such as economic and political risk, trade openness and distance, we find that macroeconomic instability is a major determinant for European firms' location decisions in MENA economies. We show that (1) FDI levels decrease with the extent of output instability in the origin country; (2) countries with a higher level of output instability tend to attract more FDI; and (3) FDI from those peripheral non-traditional sources tends to be less reactive to economic fluctuation in the source country than FDI from core traditional sources.

Hence, the results we obtain for MENA economies confirm recent evidence by Méon and Sekkhat (2012) that FDI varies as much because of what happens in the world economy (waves) as it does because of the of host country risk characteristics. We have found that developing economies, like MENA, that are dependent to foreign investment, can finally suffer from external sources of macroeconomic instability that are not global but, instead, related to the countries where FDI is sourced. However, FDI tends to be less pro-cyclical when it comes from non-traditional sources and, conversely, tends to be more responsive to any source of volatility when it comes from more traditional developed sources. Our results challenge the now common idea that, because FDI is less pro-cyclical than the other forms of financial inflows, it is a more stable source of external financing for developing economies [START_REF] Fernandez-Arias | Is Foreign direct investment a safer form of financing?[END_REF][START_REF] Calderón | Will FDI be Resilient in this Crisis? Mimeo[END_REF]. It also suggests that there is a possible trade-off, for developing economies, between technology and skillintensive but unstable FDI sourced from more developed economies, and stable but less technologyintensive investments sourced from middle-income peripheral countries. Further analyses of the relative costs-benefits of those alternative sources of FDI need to be undertaken in order to improve the economic and social efficiency of FDI attraction policies that are financially costly for developing economies.

  1. Introduction ................................................................................................................................. 2. Fdi and macroeconomic volatility : Overview of the literature .................................................. 3. Model, variables and methods .................................................................................................. 4. FDI and source country volatility ............................................................................................. 5. Volatility, traditional and non-traditional FDI sources ............................................................. 6. Robustness checks .................................................................................................................... 7. Conclusion ............................................................................................................................ REFERENCES ............................................................................................................................. Annexes......................................................................................................................................... List of tables : FDI flows and dispersion: before and after 1995 ......................................................

  , **, *** significant at 10%, 5% and 1% risk. Number of observations: 3220; Number of years: 23; Number of country-pairs: 28*5=140

  , **, *** significant at 10%, 5% and 1% risk. Number of observations: 3220; Number of years: 23; Number of country-pairs: 28*5=140

  , **, *** significant at 10%, 5% and 1% risk. Number of observations: 3220; Number of years: 23; Number of country-pairs: 28*5=140

Table 7 .

 7 . Table 1. : FDI levels baseline regressions ................................................................................ Table 2. : FDI levels & volatility and alternative FDI sources ................................................. Table 3. : FDI levels and FDI waves ........................................................................................ Table 4. : FDI levels, Eichengreen's correction and RE estimator .......................................... Table 5. : FDI levels and 5 year-moving averages ................................................................... Table 6. : FDI levels and Chow tests ........................................................................................

flows and dispersion: before and after 1995Table 1 .

 1

	FDI inflows (US	1987-1995	1996-2009
	millions dollars)
	Countries	Annual average	Coefficient of	Annual average	Coefficient of
		levels	variation	levels	variation
	Egypt	863.221	0.425	3,689.557	1.112
	Morocco	287.908	1.602	1,540.046	2.663
	Tunisia	301.774	0.085	1,086.767	1.102
	Turkey	622.111	0.020	6,666.500	0.211
	Lebanon	10.452	9.530	2,281.927	0.728
	Southern and Eastern Mediterranean	2,613.595	0.316	25,062.096	0.857
	Countries
	Asia	36,913.603	0.670	196,046.140	0.545
	Latin America and	14,889.762	0.589	102,751.060	0.455
	Caribbean

5 .

: FDI

Table 2 .

 2

		2.1	2.2	2.3	2.4
	Openness MED AMU GAFTA InvestProfil War Post War Instability source Instability host Real exchange rate GDP source GDP host Difference in GDP per capita Distance Adjacency Past colonial links Common language Constant	6.434 (10.21)*** 1.981 (5.04)*** 6.135 (9.08)*** 0.290 (0.47) 3.036 (4.95)*** -6.810 (-4.28)*** -3.996 (-3.54)*** ---3.520 (25.44)*** 1.801 (9.29)*** 0.739 (4.27)*** -0.240 (-0.94) 0.296 (0.91) -1.044 (-1.55) 3.525 (8.77)*** -132.244 (-18.98)***	6.473 (10.25)*** 2.034 (5.18)*** 6.062 (9.01)*** 0.319 (0.52) 2.932 (4.79)*** -6.721 (-4.27)*** -4.229 (-3.73)*** -0.841 (-3.94)*** --3.378 (24.08)*** 1.803 (9.30)*** 0.644 (3.71)*** -0.186 (-0.73) 0.344 (1.06) -0.882 (-1.31) 3.450 (8.60)*** -135.016 (-19.10)***	6.231 (9.84)*** 1.878 (4.76)*** 6.072 (9.00)*** 0.245 (0.40) 3.163 (5.14)*** -8.200 (-4.86)*** -4.712 (-4.05)*** -0.870 (2.62)*** 3.521 (25.38)*** 1.750 (9.02)*** 0.734 (4.25)*** -0.230 (-0.90) 0.301 (0.93) -1.068 (-1.59) 3.525 (8.77)*** -125.362 (-17.13)***	6.455 (10.17)*** 1.983 (5.03)*** 6.130 (9.07)*** 0.304 (0.49) 2.995 (4.66)*** -6.797 (-4.22)*** -4.021 (-3.55)*** --0.053 (0.08) 3.519 (25.42)*** 1.804 (9.29)*** 0.740 (4.27)*** -0.238 (-0.93) 0.296 (0.91) -1.037 (-1.54) 3.521 (8.75)*** -132.325 (-17.79)***
	Log likelihood	-5040.3203	-5034.3057	-5037.8804	-5041.8735
		Wald χ 2 (14)=1434.66 Prob > χ 2 = 0.0000	Waldχ 2 (15)=1449.98 Prob > χ 2 = 0.000	Waldχ 2 (15)=1423.45 Prob > χ 2 = 0.0000	Wald χ 2 (15)= 1426.39 Prob > χ 2 = 0.0000
	Notes: *, **, *** significant at 10%, 5% and 1% risk.
	Number of observations: 3220; Number of years: 23; Number of country-pairs: 28*5=140

Table 7 .

 7

		7.1	7.2	7.3	7.4
	N=3220	First group 1987-1995	Second group 1996-2009	First group Core/Stable	Second group Peripheral/Instable

Trade

intensification with Europe has opened new channels of volatility transmission for MENA economies, with FDI being one of those channels.

: FDI levels and Chow tests

  country belongs to the European core (UE15) in 2009, and takes null otherwise. Unstable Dummy variable takes the value 1 if the country does not belong to the European core (UE15) in 2009, and null otherwise.

See Bloningen (2005) for an overview of that literature.

A recent illustration is[START_REF] Méon | FDI Waves, waves of neglect of political risk[END_REF] who proxy external macroeconomic volatility by an aggregate ratio of world FDI to world GDP.

[START_REF] Frenkel | A panel analysis of bilateral FDI flows to emerging economies[END_REF],[START_REF] Apergis | Does inflation uncertainty matter in foreign direct investment decisions ? An empirical investigation for Portugal, Spain and Greece[END_REF] orGaribaldi et al. (2001) have studied the effects of domectic inflation on FDI inflows. As for exchange rate volatility effects on FDI inflows, see Froot andStein (1991),[START_REF] Bénassy-Quéré | Exchange-rate strategies in the competition for attracting foreign direct investment[END_REF],[START_REF] Bechri | Institutional obstacles, reform uncertainty and Tunisia's integration with the European Union[END_REF],[START_REF] Kiyota | Exchange rate, exchange rate volatility and foreign direct investment[END_REF],[START_REF] Frenkel | A panel analysis of bilateral FDI flows to emerging economies[END_REF] Stadtmann (2004), Schmidt anjd Broll (2009),[START_REF] Calderón | Will FDI be Resilient in this Crisis? Mimeo[END_REF],[START_REF] Lederman | Large devaluations, foreign direct investment and exports: A speculative note[END_REF] and[START_REF] Takagi | Exchange rate movements and foreign direct investment: Japanese investment in Asia, 1987-2008[END_REF].

In non-oil MENA countries, the ratio of the number of greenfield operations over the number of M&A is four to five times higher than the world average, and three times higher than that of developing economies [Authors' calculations based onUNCTAD (2009)].

The country list is given in Appendix. Algeria is excluded because of the size of the FDI related to energy. Turkey is also treated as an origin country for FDI because of its proximity with the European Union.

Random effects enable time-invariant factors such as distance or common border to be controlled for. Time dimension is accounted for by random effects because it enables all kinds of dyadic events, not specified in equation (2), such as trade or investment bilateral treaties, to be controlled for.

When there is a significant proportion of zeros for the bilateral FDI flows, using Ordinary Least Squares would result in highly biased estimates and a non-linear model like the Tobit is most appropriate(Eaton and Tamura, 1994;[START_REF] Wei | How taxing is corruption on international investors[END_REF][START_REF] Head | FDI as an outcome of the market for corporate control: Theory and evidence[END_REF]. This is especially so when the model does not suffer from heteroskedasticity (Santos[START_REF] Silva | The log of gravity[END_REF], which, because of the extensive size of our sample, is our case.

[START_REF] Anderson | Insecurity and the pattern of trade: An empirical investigation[END_REF] have previously shown that omitting indexes of institutional quality biases typical gravity model estimates of trade.

The definition, source and descriptive statistics of the variables are reported in the Appendix.

Moreover, Host volatility and Source volatility do not have cumulative effect on FDI since their interactive term was not significant (Result not reported).

Annexes


Outline 1. Weibull tail-distributions.

2. Kernel estimators of the Weibull tail-coefficient.

3. Bias-reduced estimator of the Weibull tail-coefficient. 4. Estimation of extreme quantiles.

5. Simulation study.

Nidd river data.

Stphane Girard

1. Weibull tail-distributions.

Let X 1 , X 2 , . . . , X n be a sequence of independent and identically distributed random variables with cumulative distribution function F such that (A1) 1 -F (x) = exp(-H(x)), H -1 (t) = inf{x, H(x) ≥ t} = t θ ℓ(t), where • θ > 0 is the Weibull tail-coefficient,

• ℓ is a slowly varying function i.e. ℓ(λx)/ℓ(x) → 1 as x → ∞ for all λ > 0.

The inverse failure rate function H -1 is said to be regularly varying at infinity with index θ and this property is denoted by H -1 ∈ R θ .

In the following, we also assume a second order condition on ℓ:

(A2) There exist ρ ≤ 0 and b(x) → 0 such that uniformly locally on λ ≥ 1 log ℓ(λx) ℓ(x) ∼ b(x)K ρ (λ), when x → ∞, with

K ρ (λ) = λ 1 u ρ-1 du.
It can be shown that necessarily |b| ∈ R ρ . The second order parameter ρ ≤ 0 tunes the rate of convergence of ℓ(λx)/ℓ(x) to 1. The closer ρ is to 0, the slower is the convergence.

Examples: Related work: Berred (1991), Broniatowski (1993), Beirlant, Broniatowski, Teugels, Vynckier, (1995), Beirlant, Bouquiaux, Werker (2006).

θ ℓ(x) b(x) ρ Absolute Gaussian |N |(µ, σ 2 ) 1/2 2 1/2 σ - σ 2 3/2 log x x + O(1/x) 1 4 log x x -1 Gamma Γ(α = 1, β) 1 1 β + α -1 β log x x + O(1/x) (1 -α) log x x -1 Weibull W(α, λ) 1/α λ 0 -∞
2. Kernel estimators of the Weibull tail-coefficient.

Principle: Our approach is based on the following approximation. Denoting by q(t) the quantile function

q(t) = F -1 (1 -t) = H -1 (log(1/t)) = (log(1/t)) θ ℓ(log(1/t)),
we obtain for t and s small:

log(q(t)) -log(q(s)) = θ(log 2 (1/t) -log 2 (1/s)) + log ℓ(log(1/t)) ℓ(log(1/s)) ≃ θ(log 2 (1/t) -log 2 (1/s)), (1)
where log 2 (x) = log log(x) and since ℓ ∈ R 0 . Considering t = i/n, s = k n /n and replacing F by its empirical counterpart yield

log(X n-i+1,n ) -log(X n-k n +1,n ) ≃ θ(log 2 (n/i) -log 2 (n/k n )),
for i = 1, . . . , k n -1 and where k n is an intermediate sequence.

Our method: Estimation via linear combination of upper order statistics.

θn (α) = k n -1 i=1 α i,n (log(X n-i+1,n ) -log(X n-k n +1,n )) k n -1 i=1 α i,n (log 2 (n/i) -log 2 (n/k n )) ,
where

• α i,n = W (i/k n ) + ε i,n ,
• ε i,n , i = 1, . . . , k n -1 is a non-random sequence, and

• W : [0, 1] → R is a smooth score function, verifying (
A3) W has a continuous derivative W ′ on (0, 1), (A4) There exist M > 0, 0 ≤ q < 1/2 and p < 1 such that, for all x ∈ (0, 1):

|W (x)| ≤ M x -q and |W ′ (x)| ≤ M x -p-q .
Asymptotic normality: Suppose (A1)-(A4) hold. Then

k 1/2 n ( θn (α) -θ) d → N (0, σ 2 (θ, W )),
for any sequence (k n ) satisfying k n → ∞ and

k 1/2 n max{b(log(n/k n )), 1/ log(n/k n ), ε n,∞ } → 0
, where we have defined:

ε n,∞ = max i=1,...,k n -1 |ε i,n |, µ(W ) = 1 0 W (x) log(1/x)dx, σ 2 (W ) = 1 0 1 0 W (x)W (y) min(x, y)(1 -max(x, y)) xy dxdy, σ(θ, W ) = θ σ(W ) µ(W ) .
Example 1. Constant weights α i,n = 1 for all i = 1, . . . , k n -1 yield an existing estimator [START_REF] Girard | A Hill type estimate of the Weibull tail-coefficient[END_REF]:

θG n = k n -1 i=1 (log(X n-i+1,n ) -log(X n-k n +1,n )) k n -1 i=1 (log 2 (n/i) -log 2 (n/k n )) .
We found back the same limiting result: If (A1) and (A2) hold then

k 1/2 n ( θG n -θ) d → N (0, θ 2 ),
for any sequence (k n ) satisfying k n → ∞ and

k 1/2 n max{b(log(n/k n )), 1/ log(n/k n )} → 0.
Example 2: A new estimator of the Weibull tail-coefficient based on a QQ-plot.

• Drawing the pairs (log 2 (n/i) , log (X n-i+1,n )) for i = 1, . . . , k n gives a graph which is approximatively linear (with slope θ).

• Example : |N |(0, 1) distribution, n = 500, k n = 100. • θZ n is the least square estimator of θ based on the k n largest observations:

θZ n = k n -1 i=1 (log 2 (n/i) -τ n ) log(X n-i+1,n ) k n -1 i=1 (log 2 (n/i) -τ n ) log 2 (n/i) ,
where

τ n = 1 k n -1 k n -1 i=1 log 2 (n/i) .
• Similar to the Zipf estimator for the extreme value index proposed by Kratz, Resnick (1996) and Schultze, Steinebach (1996).

• Particular case of θn (α) with W (x) = -(log(x) + 1). Thus, under (A1) and (A2),

k 1/2 n ( θZ n -θ) d → N (0, 2θ 2 ), for any sequence (k n ) satisfying k n → ∞ and k 1/2 n max{b(log(n/k n )), log 2 (k n )/ log(n/k n )} → 0.
3. Bias-reduced estimator of the Weibull tail-coefficient.

Principle: We focus on the case where the convergence in (A2) is slow, i.e.

(A5) x|b(x)| → ∞ as x → ∞.
Let us note that this condition implies ρ ≥ -1. Gamma and (absolute) Gaussian distribution fulfill (A5) whereas Weibull distribution do not. Condition (A2) can be used to precise approximation (1):

log(q(t)) -log(q(s)) = θ(log 2 (1/t) -log 2 (1/s)) + b(log(1/s))K ρ log(1/t) log(1/s) (1 + o(1)). (2)
Exponential regression models:

• Define Z i = i log(n/i)(log X n-i+1,n -log X n-i,n ), i = 1, . . . , k n . Then, under (A1), (A2), (A5), sup 1≤i≤k n Z i -θ + b(log(n/k n )) log(n/i) log(n/k n ) ρ f i = o P (b(log(n/k n ))) , (3)
for any sequence (k n ) such that k n → ∞ and log k n / log n → 0, and where (f 1 , ..., f k n ) is a vector of independent and standard exponentially distributed random variables.

• Similar to the ones proposed by Beirlant, Dierckx, Goegebeur, Matthys (1999), Feuerverger, Hall (1999) and Beirlant, Dierckx, Guillou, Starica (2002) in the case of Pareto-type distributions.

• One can plug the canonical choice ρ = -1 in the regression model (3) without perturbing the approximation so that sup

1≤i≤k n Z i -θ + b(log(n/k n )) log(n/k n ) log(n/i) f i = o P (b(log(n/k n ))) . (4)
Application 1: Bias-reduced estimator. Estimation of θ and b(log(n/k n )) by a Least-Square method after substituting ρ with the value -1:

θR n = Z k n -b(log(n/k n ))x k n , b(log(n/k n )) = k n i=1 (x -x k n )Z i k n i=1 (x i -x k n ) 2 , ( 5
)
where

x i = log(n/k n )/ log(n/i), x k n = 1 k n k n i=1 x i and Z k n = 1 k n k n i=1 Z i .
Asymptotic normality under (A1), (A2), (A5):

k 1/2 n log(n/k n ) ( θR n -θ) d → N (0, θ 2 )
for any sequence (k n ) such that k n → ∞ and

k 1/2 n b(log(n/k n ))/ log(n/k n ) → Λ = 0.
Application 2: Adaptive selection of k n . Adapted from Matthys, Beirlant (2003) in the context of extreme-value index estimation. Neglecting the bias correction in (5) yields the Maximum Likelihood estimator:

θML n = Z k n = 1 k n k n i=1 i log(n/i)(log X n-i+1,n -log X n-i,n ).
From (4), the Asymptotic Mean Squared Error (AMSE) associated to θML n is given by

AM SE( θML n ) = θ 2 k n + b(log(n/k n )) 1 k n k n i=1 log(n/k n ) log(n/i) 2 ,
and can be estimated by

AM SE( θML n ) = ( θR n ) 2 k n + b(log(n/k n )) 1 k n k n i=1 log(n/k n ) log(n/i) 2 .
The minimization of AM SE( θML n ) with respect to k n gives rise to an adaptive selection procedure.

Estimation of extreme quantiles.

Principle: An extreme quantile x p n of order p n < 1/n is defined by:

1 -F (x p n ) = p n .
Recall that, for small t and s

q(t) q(s) = H -1 (log(1/t)) H -1 (log(1/s)) = log(1/t) log(1/s) θ ℓ(log(1/t)) ℓ(log(1/s)) ≃ log(1/t) log(1/s) θ .
Considering t = p n , s = k n /n, replacing F by its empirical counterpart and θ by an estimator θn yield the following estimator

xp n ( θn ) = X n-k n +1,n log(1/p n ) log(n/k n ) θn = X n-k n +1,n exp θn log τ n ,
where we have defined τ n = log(1/p n )/log(n/k n ).

Asymptotic normality: Based on the following result [START_REF] Gardes | Estimating extreme quantiles of Weibull tail-distributions[END_REF]. Suppose (A1) and (A2) hold. If morever

1 ≤ lim inf τ n ≤ lim sup τ n < ∞,
and

k 1/2 n ( θn -θ) d → N (0, σ 2 ), then log(n/k n )k 1/2 n log(k n /(np n )) xp n ( θn ) x p n -1 d → N (0, σ 2 ),
for any sequence (k n ) satisfying k n → ∞, k n /n → 0 and

k 1/2 n b(log(n/k n )) → 0.
= log(n/k n ) X n-k n +1,n 1 k n -1 k n -1 i=1 (X n-i+1,n -X n-k n +1,n )
is the estimator introduced by Beirlant, Broniatowski, Teugels, Vinckier (1995).

Stphane Girard

Bias-reduced estimator: Basing on the refined approximation (2) of log q(t)log q(s), it is natural to introduce

xR p n = X n-k n +1,n exp θR n log τ n + b(log(n/k n ))K -1 (τ n ) .
Asymptotic normality under (A1), ( A2), (A5):

k 1/2 n log(n/k n ) xR p n x p n -1 d → N (Λµ(τ ), θ 2 σ 2 (τ ))
for any sequence (k n ) such that k n → ∞, τ n → τ > 1 and

k 1/2 n b(log(n/k n ))/ log(n/k n ) → Λ = 0, where µ(τ ) = (K -1 (τ ) -K ρ (τ )) and σ 2 (τ ) = (K -1 (τ ) -log(τ )) 2 .
5. Simulation study.

• Comparison of θG n (in black), θZ n (in red), θML n (in green) and θR n (in blue) to the true θ (black horizontal line).

• Simulated distributions : Absolute Gaussian |N |(0, 1), Gamma Γ(0.25, 1), Γ(4, 1), and Weibull W(4, 4), W(0.25, 0.25).

• Sample size n = 500, k n ∈ {2, . . . , 360}, 100 replications.

• Computation of the mean estimate (Hill plot, left pannel) and of the Mean Square Error (MSE, right pannel). 6. Nidd river data.

• 154 exceedances of the level 65 m 3 s -1 by the river Nidd (Yorkshire, England) during the period 1934-1969 (35 years).

• Widely used in extreme value studies i.e. Hosking, Wallis, Wood (1985) and Davison, Smith (1990).

• The N -year return level is the water level which is exceeded on average once in N years.

Gamma

  |N |(0, 1) -→ θ = 1/2, b

  100-year return level:xp n ( θML n ) = 366m 3 s -1

  As a consequence, we obtain the asymptotic normality of xp n ( θG

n ), xp n ( θZ n ) and xp n ( θBBTV n ) where θBBTV n

Stphane Girard