Anna Creti

Marc Joëts
email: marc.joets@u-paris10.fr

Valérie Mignon
email: valerie.mignon@u-paris10.fr.

Agnès Bénassy-Quéré

Gunther Capelle-Blancard

On the links between stock and commodity markets' volatility ⇤

Keywords: JEL Classification: C22, G01, G10, Q4 Commodities, stock market, financial crisis, volatility, correlations, DCC-GARCH

This paper investigates the links between price returns for 25 commodities and stocks over the period from January 2001 to November 2011, by paying a particular attention to energy raw materials. Relying on the dynamic conditional correlation (DCC) GARCH methodology, we show that the correlations between commodity and stock markets evolve through time and are highly volatile, particularly since the 2007-2008 financial crisis. The latter has played a key role, emphasizing the links between commodity and stock markets, and underlining the financialization of commodity markets. At the idiosyncratic level, a speculation phenomenon is highlighted for oil, coffee and cocoa, while the safe-haven role of gold is evidenced.

Introduction

Throughout the last decade, commodity prices experienced an exceptional volatility, with simultaneous and alternating phases of rising and falling trends. This evolution can be compared to that of financial markets, as illustrated by Figures 1 and2 representing the Standard and Poor's 500 (S&P 500) and Commodity Research Bureau (CRB) price returns' volatility. As shown in Figure 3-which displays the dynamics of the S&P 500 and CRB price indexescommodity prices have experienced a drop during the 2007-2008 financial crisis, and their link to stock prices seems to have strengthened since that turmoil. At the same time, commodities increasingly become part of portfolio allocation, together with stock classes. Source: Data are extracted from Datastream. Volatility is proxied by the daily squared returns of prices.

At a macroeconomic level, policymakers pay a particular attention to commodity prices and their volatility given their potential to feed inflation pressures. Volatility of commodity prices is thus a central issue for the world economy, as notably illustrated by the G20 which addressed the question of excessive fluctuations and volatility of commodity prices in its September 2009 Pittsburgh summit. Moreover, analyzing the links between commodity and stock markets is of particular interest for financial players as raw materials enter many investment portfolios, together with stock classes [START_REF] Silvennoinen | Financialization, Crisis and Commodity Correlation Dynamics[END_REF][START_REF] Dwyer | Global commodity markets-Price volatility and financialisation[END_REF][START_REF] Vivian | Commodity volatility breaks[END_REF]. Furthermore, as documented by [START_REF] Choi | Volatility behavior of oil, industrial commodity and stock markets in a regime-switching environment[END_REF], commodity traders concurrently look at both stock and commodity markets fluctuations to infer the trend of each market. Comparing the dynamic volatility of raw materials and equities prices provides useful information about possible substitution strategies between commodity and stock classes. In particular, volatility plays a key role regarding hedging possibilities, and impacts asset allocation across raw materials and their risk-return trade-off. Building on the observed links between commodity and stock markets, a recent literature has emerged regarding the impact of investors' behavior in explaining the increase in both level and volatility of commodity prices.1 However, as underlined by [START_REF] Vivian | Commodity volatility breaks[END_REF], no clear-cut conclusion has been reached so far. Source: Data are extracted from Datastream.

In this paper, we contribute to the emerging empirical literature dealing with the relationships between commodity and stock markets. More specifically, we focus on the dynamics of the correlations between both markets, and analyze whether those correlations evolve according to the situation-bullish or bearish-in the stock market. We pay a particular attention to the recent 2007-2008 financial crisis by investigating whether it has strengthened or disrupted the links between stock and commodity markets. From a methodological viewpoint, we follow the dynamic conditional correlation (DCC) GARCH approach introduced by [START_REF] Engle | Dynamic Conditional Correlation -A Simple Class of Multivariate GARCH Models[END_REF] which allows to assess the changes in correlations between commodity and stock returns over time. The DCC-GARCH approach has been followed by [START_REF] Choi | Volatility behavior of oil, industrial commodity and stock markets in a regime-switching environment[END_REF] in a quite similar context, but our study considerably extends the analysis.2 Our sample consists of 25 commodities covering various sectors over the period from January 3, 2001 to November 28, 2011. Relying on a large panel of raw materials (energy, metals, agricultural, food, ...) allows us to study whether commodities constitute an homogenous asset class with regard to their links with stock markets, and whether the crisis has engendered a financialization of commodity markets.3 This kind of relationship has typically been investigated in the case of oil [START_REF] Doyle | Growth in commodity investment: risk and challenges for commodity market participants[END_REF][START_REF] Mouawad | One year after oil's price peak volatility[END_REF], though the cross-effect on oil and stock market volatility remains globally unclear.

Our results show that correlations between commodity and stock markets are time-varying and highly volatile. The impact of the 2007-2008 financial crisis is noticeable, emphasizing the links between commodity and stock markets, and highlighting the financialization of commodity markets. We also show that, while sharing some common features, commodities cannot be considered a homogeneous asset class: a speculation phenomenon4 is for instance highlighted for oil, coffee and cocoa, while the safe-haven role of gold is evidenced.

The rest of the paper is organized as follows. Section 2 briefly reviews the literature about the links between commodity and stock markets. Section 3 presents the data as well as some stylized facts, and Section 4 deals with methodological aspects. Results are displayed in Section 5, and Section 6 concludes the paper.

Literature review

As documented in the introduction, commodity markets share several characteristics with stock markets and financial assets. So far the literature has analyzed this phenomenon mainly by focusing on oil, and looking at the comovements between stock and oil markets. Most of this literature offers substantial evidence on the impact of oil on stock prices, putting forward a negative relationship between oil price and stock market returns.5 For instance, [START_REF] Jones | Oil and the stock markets[END_REF], using a standard cash-flow dividend valuation model, find a significant negative impact of oil price shocks on US and Canadian quarterly stock prices in the postwar period.

Several models, relying on some variants of Vector Autoregressive analysis (VAR), highlight similar findings. [START_REF] Park | Oil Price Shocks and Stock Markets in the U.S. and 13 European Countries[END_REF], performing a multivariate VAR analysis, find statistically significant impact of oil prices shocks on real stock returns for US and 13 European countries over the period from January 1986 to December 2005. [START_REF] Sadorsky | Oil price shocks and stock market activity[END_REF] investigates relationships among monthly oil prices, S&P 500 stock returns, short-term interest rate, and industrial production for the January 1947-April 1996 period by means of an unrestricted VAR model. The author shows that oil prices and oil price volatility both play important roles in affecting S&P 500 stock returns. [START_REF] Papapetrou | Oil price shocks, stock market, economic activity and employment in Greece[END_REF] estimates a vector error-correction model on monthly data for Greece from January 1989 to June 1999, and concludes that oil prices drive stock price dynamics.

Shifting from the study of comovements to volatility analysis, the most recent literature focuses on volatility spillovers between oil/industrial commodity and stock markets. [START_REF] Hammoudeh | Relationships among U.S. oil prices and oil industry equity indices[END_REF] investigate the spillover effects, day effects, and dynamic relationships among five daily S&P oil sector stock indices and five daily oil prices for the US oil markets 6 from July 17, 1995 to October 10, 2001 using cointegration techniques as well as ARCH-type models.

They evidence volatility spillovers from the oil futures market on the stocks of some oil sectors. They also find an oil volatility transmission day effect, Friday having a calming effect on the volatility of oil stocks. [START_REF] Chiou | Jump dynamics and volatility: Oil and the stock markets[END_REF] Our study extends the previous literature by considerably enlarging the sample of commodities analyzed. We consider 25 different strategic commodities, traded in the US and covering various sectors: energy, precious metals, agricultural, non-ferrous metals, food, oleaginous,

6
The US oil industry encompasses companies engaged in various phases of oil production and processing.

The US oil markets include the West Texas Intermediate (WTI), Cushing spot and the New York Mercantile Exchange (NYMEX) for 1 to 4 month futures prices. 7 The following sectors are considered: financials, industrials, consumer services, health care, and technology.

exotic and livestock. The dataset we have built allows us to compare the behavior of each commodity group regarding stock market fluctuations, and to study whether correlations between commodities and equities evolve over time and depend on the situation-bearish or bullish-on the stock market.

3 Data and stylized facts

We consider daily spot price series extracted from Datastream for a large sample of commodities over the January 3, 2001 -November 28, 2011 period. 8 We investigate 25 different commodities covering the following various sectors: energy, precious metals, agricultural, non-ferrous metals, food, oleaginous, exotic and livestock. All price series are quoted in US dollars. We also consider an aggregate commodity price index, the Commodity Research Bureau (CRB) index. Regarding the equity market, we rely on one of the main US stock market index, namely the S&P 500 index.

Table 1 in Appendix provides some descriptive statistics regarding the returns series, defined as r t = ln(P t /P t 1 ), where P t denotes the price index at time t. The group of energy commodities seems to differ from other groups in terms of volatility: the variance of electricity, gas and to a lesser extent oil price returns is higher than that obtained for the other commodities; 9 being also higher than those of S&P 500 and CRB returns. The electricity series is extremely volatile, as its high kurtosis value shows. This is not surprising given that electricity is not storable and prices reflect the real-time equilibrium between demand and supply, with contingencies that vary greatly from one day to another. 10 Together with high volatility, the group of energy commodities exhibits low returns on average, leading to the lowest benefit-risk trade off compared to the S&P 500 and the CRB indexes, and the group of food and oleaginous commodities which are more profitable on the return-risk basis. Statistics in Table 1 also show that all series are characterized by a time-varying volatility, an ARCH effect being present for almost all returns series. Finally, returns tend to be autocorrelated, especially for the energy and the precious metals groups, indicating some persistence phenomenon.

Methodology

To investigate the time evolution of correlations between the commodity and stock markets, we rely on the dynamic conditional correlation (DCC) GARCH models introduced by [START_REF] Engle | Dynamic Conditional Correlation -A Simple Class of Multivariate GARCH Models[END_REF]. Let r t be the vector composed of two returns series, r t = (r 1t , r 2t ) 0 . Denoting by A(L) the lag polynomial, we have:

8 An alternative would have been to rely on futures prices. However, as highlighted by [START_REF] Vivian | Commodity volatility breaks[END_REF], spot prices are the underlying asset upon which derivatives are based, a fact that is important when analyzing volatility. In addition, relying on spot prices avoids issues related to rollover of futures contracts. 9 The increasing trend in volatility of oil and gas market prices in the USA has also been documented by [START_REF] Pindyck | Volatility in Natural Gas and Oil Markets[END_REF] among others.

10 Though the Commodity Futures Trading Commission provides no data regarding the financialization of electricity (see Table 2 in Appendix), the latter can also be considered as a financial product. An illustration is given by the Nordic financial electricity market, whose liquidity provided by a number of speculators highlights that it is also important for financial trading purposes. More fundamentally, with the creation of electricity spot markets-including various standardized products-pure financial trading has been progressively growing to the point that the Dodd-Frank Act provides that these markets are monitored within the framework of financial stability measures.

A(L)r t = µ + e t

(1) where e t is the error-term vector.

The DCC model is based on the hypothesis that the conditional returns are normally distributed with zero mean and conditional covariance matrix

H t = E h r t r 0 t |I t 1 i .
The covariance matrix is expressed as follows:

H t = D t R t D t (2)
where

D t = diag ⇥p h 1t , p h 2t
⇤ is a diagonal matrix of time-varying standard deviations issued from the estimation of univariate GARCH(1,1) processes:

h t = ↵ 0 + ↵ 1 " 2 t 1 + 1 h t 1 (3)
and R t is the conditional correlation matrix of the standardized disturbances " t , with

" t = D 1 t r t : R t =  1 q 12t q 21t 1 (4)
The matrix R t is decomposed into:

R t = Q ⇤ 1 t Q t Q ⇤ 1 t (5)
where Q t is the positive definite matrix containing the conditional variances-covariances of " t , and Q ⇤ 1 t is the inverted diagonal matrix with the square root of the diagonal elements of Q t :

Q ⇤ 1 t =  1/ p q 11t 0 0 1/ p q 22t (6)
The DCC(1,1) model is then given by:

Q t = ! + ↵" t 1 " 0 t 1 + Q t 1 (7)
where ! = (1 ↵ ) Q, Q being the unconditional covariance of the standardized disturbances " t .

The dynamic conditional correlations are finally given by:

⇢ 12t = q 12t p q 11t q 22t (8)
Note that, following [START_REF] Engle | Dynamic Conditional Correlation -A Simple Class of Multivariate GARCH Models[END_REF], the estimation of this model is done using a two-step maximum likelihood estimation method, the likelihood function being given by:11

L = 1 2 T X t=1 ⇣ 2 log (2⇡) + 2 log |D t | + log |R t | + " 0 t R 1 t " t ⌘ (9)
5 Results

To assess the evolution of correlations between stock and commodity markets over time, Figures A1 in Appendix report the dynamic conditional correlations between each commodity and the S&P 500 returns series. The links between markets during periods of financial stress are clearly underlined,12 putting forward that investment in equities constitutes an alternative to commodities, providing a mechanism for substitution between asset classes. Although there are some specific features for each type of commodity market (as we will explain in detail below), some common characteristics emerge.

First, correlations are highly volatile throughout the period. For many raw materials, this volatility is particularly marked after the 2007-2008 financial crisis. In all cases, there is an increase in volatility during and following the crisis. Second, in most cases, the largest drop in the correlations appears at the time of the 2008 financial crisis. The stock market collapse has loosened the conditional links between stock and commodity price returns, but only in the very short run. This decrease in correlations during times of high financial markets' stress may be linked to a flight-to-quality phenomenon. When risk market rises, the benefits of diversification are most appreciated and investors tend to choose commodities as refuge instruments. This short-run characteristic could thus explain the temporary disrupted link between both markets (see [START_REF] Silvennoinen | Financialization, Crisis and Commodity Correlation Dynamics[END_REF][START_REF] Chong | Conditional correlation and volatility in commodity futures and traditional asset markets[END_REF]. Third, for almost all of the series, the highest correlations are observed after the crisis, at the end of the period under study. Both markets move upward during episodes of growing world demand for industrial commodities, giving an important role to commercial traders who use commodity futures to hedge their business activities. On the whole, the 2007-2008 financial crisis has caused significant changes in the relationship between stock and commodity markets, as well as increased correlation in the volatility. Regarding the long-run trends, correlations are likely to be governed by industrialization and financialization processes, as well as by commercial and non-commercial traders.

Let us now look more specifically at the different types of markets, starting by the energy group. Oil is clearly the commodity the most related to the stock market, confirming previous studies focusing on the oil market [START_REF] Jones | Oil and the stock markets[END_REF][START_REF] Hammoudeh | Relationships among U.S. oil prices and oil industry equity indices[END_REF][START_REF] Filis | Dynamic correlation between stock market and oil prices: The case of oil-importing and oil-exporting countries[END_REF] and references in Section 2). This predominance of oil may be due to the fact that it is one of the most important production factors. From a theoretical viewpoint, the fundamental value of any asset is given by its expected discounted cash flows. Consequently, an oil price increase will generate a rise in production costs, leading to restraining profits and, in turn, to a diminution in shareholders' value. In times of rising stock prices, the correlations between stock and oil markets increase. During periods of declining stock prices, correlations tend to decrease and become negative during the 2007-2008 crisis. This is also consistent with the well documented oil speculation phenomenon, the increase in crude prices being accentuated in times of rising stock market. From this perspective, oil cannot be seen as a means of portfolio diversification. Gas and electricity display a quite similar evolution in terms of dynamic correlations. Correlations tend to increase at the beginning of the period under study and then remain relatively stable, regardless of the situation on the stock market. Correlations are often negative between stock and electricity markets, putting forward that the behavior of the electricity market is mainly driven by its own market fundamentals (i.e. non-storability, inelasticity of the supply,...).

Turning to the precious metals group, gold is different from the other commodities. Indeed, correlations are mostly negative and diminish in times of declining stock prices, highlighting adverse evolution in the markets. This is consistent with a safe-haven role of gold (see for instance [START_REF] Baur | Is gold a safe haven? International evidence[END_REF]. For the other precious metals, the dynamics are relatively close, with increased correlations' volatility after the 2007-2008 crisis followed by a rise in correlations until mid-2010.

The group of exotic commodities also displays an interesting pattern. While the dynamics of correlations for sugar has no particular link with the US stock market trends, coffee and cocoa show a specific profile. As for oil, the correlations tend to grow in times of rising stock prices, and to diminish in periods of declining equity prices. This is in line with a speculation phenomenon in these commodities (see also [START_REF] Gilbert | Has food price volatility risen?[END_REF].

Regarding the other groups, two main findings can be highlighted: (i) volatility evolves over time, being quite stable before the 2007-2008 crisis and becoming relatively high during the financial turmoil, and (ii) correlations tend to rise during the crisis, showing increased links between stock and commodity markets.

On the whole, our results show that the 2007-2008 crisis has played a key role in the evolution of the links between stock and commodity markets. Indeed, higher correlations between both markets are generally observed during the financial turmoil, reflecting the phenomenon of financialization of commodity markets that starts to be documented by the literature (see [START_REF] Tang | Index Investment and Financialization of Commodities[END_REF][START_REF] Silvennoinen | Financialization, Crisis and Commodity Correlation Dynamics[END_REF]. This growing financialization of commodities can be illustrated by the notional values provided by the Commodity Futures Trading Commission (CFTC): as shown in Table 2 in Appendix, these notional values-and especially short nominal values-have increased for all products between 2007 and 2011. This phenomenon is particularly noticeable for oil, a result which is consistent with the fact that it is the most financiarized commodity according to the CFTC-the long and short notional values being respectively estimated at $69.4bn and $26.7bn at the end of November 2011 (see Table 2 in Appendix). In addition, our findings show that raw materials cannot be aggregated in an homogeneous asset class: they are certainly influenced by common macroeconomic factors but also by their own market determinants.

To complement these figures, Tables 3 to 7 in Appendix report the estimation results of DCC-GARCH(1,1) models for the whole period, as well as for four sub-periods: (i) two bearish stock market sub-periods: January 3, 2001-March 11, 2003and October 13, 2007-March 6, 2009, and (ii) two bullish stock market sub-periods: March 12, 2003-October 12, 2007and March 7, 2009-November 28, 2011. Looking at the sum of the coefficients ↵ + (see Equation ( 7)), our results show that volatility is highly persistent given that this sum is very close to 1 for the majority of commodity series. While being high for all considered periods, persistence tends to be higher during the second, bullish stock market sub-period for 12 commodities, including all precious metals. This result illustrates that the persistence of volatility goes along with the financialization of commodities.

Conclusion

This paper investigates the links between commodity and stock markets. To this end, we rely on the dynamic conditional correlation (DCC) GARCH methodology to establish whether the correlations between both markets evolve over time and depend on the situation-bearish or bullish-on the stock market.

Our main findings can be summarized as follows. In our panel of 25 commodities over the period from January 2001 to November 2011, first, the correlations between commodity and stock returns evolve through time, being highly volatile, particularly since the 2007-2008 financial crisis. While the stock market collapse has loosened the links between both markets on the very short run, the highest correlations are observed during the financial turmoil, showing increased links between stock and commodity markets. Second, some commodities are characterized by a speculation phenomenon, especially oil, coffee and cocoa: while their correlations with S&P 500 returns grow in times of increasing stock prices, they diminish in times of bearish financial markets. Third, the safe-haven role of gold is evidenced, as its correlations with stock returns are mostly negative and diminish in times of declining stock prices. Fourth, while sharing some common features, commodities can not be considered as an homogeneous asset class.

On the whole, our findings show that the 2007-2008 financial crisis has played a key role, emphasizing the links between commodity and stock markets, and highlighting the financialization of commodity markets. This evolution in commodity and stock correlations reduces their potential substitutability in portfolios. At the idiosyncratic level, the main exceptions are gold, coffee and cocoa for which risk management strategies are possible, with increased risk diversification allowed by their adverse evolution compared to the stock market in times of declining equity prices. Live / S&P 500 Note: The grey bands correspond to periods of bearish stock market, the white stripes corresponding to periods of bullish stock market. These periods have been identified on the basis of the evolution of S&P 500 stock returns using the [START_REF] Bai | Computation and Analysis of Multiple Structural Change Models[END_REF] structural break test.

Figure 1 :

 1 Figure 1: S&P 500 stock returns volatility (01/03/2001-11/28/2011)

Figure 2 :

 2 Figure 2: Commodity price returns volatility (01/03/2001-11/28/2011)

Figures

  Figures A1. Dynamic conditional correlations

  Source: Data are extracted from Datastream. Volatility is proxied by the daily squared returns of prices.

	Figure 3: Evolution of S&P 500 and CRB indexes (01/03/2001-11/28/2011)
	0,0014															600
	0,0012															500
	0,001
																400
	0,0008
																300
	0,0006
	0,0004															200
	0,0002															100
	0
	03/01/2001 01/02/2001	03/05/2001	03/09/2001 2/26/2002 03/01/2002	03/05/2002	03/09/2002 4/22/2003 03/01/2003 03/05/2003	03/09/2003	03/01/2004 6/15/2004 03/05/2004	03/09/2004	03/01/2005 08/09/2005 03/05/2005 03/09/2005 SP500 03/01/2006 10/03/2006 03/05/2006 03/09/2006 CRB 03/01/2007	03/05/2007 11/27/2007 03/09/2007 03/01/2008	03/05/2008 1/20/2009 03/09/2008 03/01/2009	03/05/2009	03/09/2009 3/16/2010 03/01/2010 03/05/2010	03/09/2010 05/10/2011 03/01/2011 03/05/2011	03/09/2011	0

Table 1 :

 1 Summary statistics (whole sample)

		Mean	S. dev.	Var	Kurtosis Skewness	LB test	ARCH test
	Energy group
	Oil	0.0004	0.025	0.0006	8.29	0.01	56.77 (0.00) 299.7 (0.00)
	Gas	-0.0004	0.043	0.0019	9.62	0.34	155.0 (0.00) 325.1 (0.00)
	Electricity	-0.0002	0.112	0.0126	16.75	0.12	138.0 (0.00) 159.7 (0.00)
	Precious Metals group
	Gold	0.0006	0.011	0.0001	7.10	-0.34	68.55 (0.00) 122.7 (0.00)
	Silver	0.0006	0.022	0.0005	12.27	-0.57	30.23 (0.40) 241.2 (0.00)
	Platinium	0.0002	0.015	0.0002	16.61	-0.48	58.76 (0.00) 137.4 (0.00)
	Palladium	-0.0001	0.023	0.0005	7.91	-0.36	53.96 (0.00) 79.26 (0.00)
	Agricultural group
	Cotton	0.0001	0.021	0.0004	15.74	-0.58	35.33 (0.23)	0.48 (0.48)
	Lumber	4.66E-05	0.021	0.0004	5.95	0.75	33.99 (0.28)	8.96 (0.00)
	Non-ferrous Metals group
	Aluminium	8.74E-05	0.014	0.0002	5.34	0.33	26.28 (0.61) 98.85 (0.00)
	Copper	0.0005	0.019	0.0003	6.66	-0.15	44.43 (0.03) 408.9 (0.00)
	Zinc	0.0002	0.021	0.0004	5.24	-0.23	32.71 (0.33) 220.6 (0.00)
	Tin	0.0004	0.019	0.0003	8.65	-0.24	32.22 (0.35) 156.64 (0.00)
	Lead	0.0005	0.023	0.0005	5.68	-0.22	30.14 (0.45) 100.4 (0.00)
	Nickel	0.0003	0.026	0.0006	6.08	-0.11	25.69 (0.69) 193.2 (0.00)
	Food group
	Corn	0.0003	0.019	0.0003	5.39	0.17	61.33 (0.00) 59.09 (0.00)
	Wheat	0.0003	0.021	0.0004	5.20	0.17	44.59 (0.04) 64.03 (0.00)
	Oleaginous group
	Palm oil	0.0004	0.020	0.0004	9.94	0.44	69.22 (0.00) 39.99 (0.00)
	Soybean oil	0.0004	0.016	0.0002	5.14	0.13	28.41 (0.54) 108.4 (0.00)
	Exotic group
	Cocoa	0.0003	0.020	0.0004	5.83	-0.30	43.89 (0.04)	2.14 (0.14)
	Coffee	0.0004	0.020	0.0004	5.76	-0.23	47.33 (0.02) 31.38 (0.00)
	Sugar	6.34E-06	0.023	0.0005	7.55	-0.63	30.97 (0.41) 88.63 (0.00)
	Livestock group
	Lean hogs	0.0001	0.020	0.0004	25.68	0.43	11.98 (0.99)	0.14 (0.70)
	Feeder cattle	0.0001	0.009	8.64E-05	6.54	-0.35	54.60 (0.00) 72.59 (0.00)
	Live cattle	0.0001	0.010	0.0001	8.81	0.12	52.02 (0.00) 20.83 (0.00)
	Standard & Poor's	-7.05E-06	0.013	0.0001	10.98	-0.16	59.56 (0.00) 786.8 (0.00)
	CRB index	0.0002	0.004	2.50E-05	8.05	-0.60	125.1 (0.00) 239.8 (0.00)

Notes: Between parentheses: p-values. Ljung-Box statistics correspond to a test of the null of no autocorrelation with r = 30. ARCH Lagrange multiplier statistics correspond to a test of the null of no ARCH effect.

Table 2 :

 2 Index investment data: notional values (in Billions U.S. dollars)

		December 31, 2007 November 30, 2011
		Long	Short	Long	Short
	Oil (WTI)	46.7	7.0	69.4	26.7
	Gas	13.2	1.8	16.2	4.1
	Gold	8.4	1.1	30.0	9.5
	Silver	2.4	0.3	6.9	1.6
	Cotton	3.2	0.6	4.3	1.5
	Copper	3.1	0.3	7.9	2.2
	Corn	9.5	1.9	19.2	7.2
	Wheat	10.2	2.1	10.7	4.8
	Soybean oil	2.5	0.3	4.0	1.3
	Cocoa	0.5	0.1	1.1	0.4
	Coffee	2.8	0.6	5.3	2.0
	Sugar	3.9	0.7	8.4	2.4
	Lean hogs	3.0	0.9	5.2	1.7
	Feeder cattle	0.6	0.1	0.8	0.2
	Live cattle	5.9	1.3	9.1	2.9

Notes: Source: CFTC, http://www.cftc.gov/MarketReports/IndexInvestmentData/index.htm. Short (resp. long): denotes the gross short (resp. long) notional value and refers to the case where investors are short (resp. long) a commodity index.

Table 3 :

 3 Estimation results of DCC-GARCH(1,1) models (whole sample)

		Oil	Gas	Elec	Gold	Silver
	µ	8.2e-04 (3.6e-04) ⇤	-7.2e-04 (5.5e-04) 2.7e-03 (1.2e-03) ⇤	5.3e-04 (1.5e-04) ⇤ 3.8e-04 (2.3e-04)
	!	1.2e-05 (6.9e-07) ⇤	3.1e-05 (2.5e-06) ⇤ 1.0e-03 (2.2e-05) ⇤	1.8e-06 (7.8e-08) ⇤ 5.1e-06 (4.3e-07) ⇤
	↵	0.061 (1.6e-03) ⇤	0.111 (2.4e-03) ⇤	0.230 (4.7e-03) ⇤	0.042 (9.9e-04) ⇤	0.088 (1.4e-03) ⇤
		0.917 (1.5e-03) ⇤	0.876 (1.9e-03) ⇤	0.683 (3.2e-03) ⇤	0.943 (7.8e-04) ⇤	0.906 (1.2e-03) ⇤
	↵ +	0.978	0.987	0.913	0.985	0.994
		Platinum	Palladium	Cotton	Lumber	Aluminium
	µ	-3.6e-04 (1.6e-04) ⇤ 7.2e-05 (2.4e-04)	3.0e-04 (3.0e-04)	1.8e-04 (3.5e-04)	4.5e-05 (2.0e-05) ⇤
	!	4.1e-06 (2.3e-07) ⇤	1.7e-05 (3.1e-07) ⇤ 8.0e-06 (3.5e-07) ⇤	1.1e-06 (1.0e-07) ⇤ 1.3e-06 (1.0e-07) ⇤
	↵	0.107 (1.7e-03) ⇤	0.079 (1.4e-03) ⇤	0.054 (7.8e-04) ⇤	0.010 (2.2e-04) ⇤	0.033 (5.7e-04) ⇤
		0.879 (1.4e-03) ⇤	0.889 (9.4e-04) ⇤	0.931 (6.4e-04) ⇤	0.987 (2.0e-04) ⇤	0.960 (4.9e-04) ⇤
	↵ +	0.986	0.968	0.985	0.997	0.993
		Copper	Zinc	Tin	Lead	Nickel
	µ	4.7e-04 (1.5e-04) ⇤	1.5e-04 (1.8e-04)	5.3e-04 (2.3e-04) ⇤	3.9e-04 (2.5e-04)	4.5e-04 (3.0e-04)
	!	3.5e-06 (2.2e-07) ⇤	2.5e-06 (1.9e-07) ⇤ 1.4e-05 (4.0e-07) ⇤	5.1e-06 (2.6e-07) ⇤ 1.0e-05 (5.7e-07) ⇤
	↵	0.047 (8.8e-04) ⇤	0.037 (6.8e-04) ⇤	0.109 (0.002) ⇤	0.038 (7.8e-04) ⇤	0.046 (0.001) ⇤
		0.941 (8.1e-04) ⇤	0.956 (5.9e-04) ⇤	0.856 (0.001) ⇤	0.951 (6.5e-04) ⇤	0.936 (9.7e-04) ⇤
	↵ +	0.988	0.993	0.965	0.989	0.982
		Corn	Wheat	Palm oil	Soybean oil	Cocoa
	µ	6.3e-04 (1.4e-04) ⇤	4.5e-04 (2.8e-04)	7.2e-04 (2.9e-04) ⇤	7.1e-04 (2.5e-04)	2.9e-04 (3.3e-04)
	!	5.1e-06 (2.3e-07) ⇤	5.0e-06 (3.8e-07) ⇤ 5.9e-06 (2.1e-07) ⇤	3.8e-06 (2.1e-07) ⇤ 3.4e-06 (1.6e-07) ⇤
	↵	0.038 (8.9e-04) ⇤	0.039 (9.5e-04) ⇤	0.063 (1.1e-03) ⇤	0.035 (1.0e-03) ⇤	0.017 (4.4e-04) ⇤
		0.947 (7.0e-04) ⇤	0.949 (8.8e-04) ⇤	0.923 (8.0e-04) ⇤	0.948 (9.1e-04) ⇤	0.974 (4.0e-04) ⇤
	↵ +	0.985	0.985	0.986	0.983	0.991
		Coffee	Sugar	Lean	Feeder	Live
	µ	5.0e-04 (3.4e-04)	2.1e-04 (3.7e-04)	1.6e-04 (3.7e-04)	1.5e-04 (1.2e-04)	2.0e-04 (2.0e-04)
	!	2.1e-06 (1.4e-07) ⇤	1.0e-05 (6.2e-07) ⇤ 8.6e-05 (6.5e-07) ⇤	1.0e-06 (3.5e-08) ⇤ 1.2e-06 (1.7e-06) ⇤
	↵	0.013 (3.2e-04) ⇤	0.059 (0.001) ⇤	-4.0e-03 (1.4e-03) ⇤ 0.012 (3.3e-04) ⇤	0.062 (8.6e-03) ⇤
		0.981 (3.1e-03) ⇤	0.923 (0.001) ⇤	0.804 (1.4e-03) ⇤	0.975 (2.8e-04) ⇤	0.164 (0.015) ⇤
	↵ +	0.994	0.982	0.808	0.987	0.226
	Notes: Standard errors are in parentheses.* denotes rejection of the null hypothesis at 1%,
			5% or 10% significance level.

Recent references include[START_REF] Eckaus | The Oil Price Really Is A Speculative Bubble[END_REF],[START_REF] Khan | The 2008 oil price 'bubble[END_REF],[START_REF] Masters | The 2008 commodities bubble: assessing the damage to the United States and its Citizens[END_REF], Capelle-Blancard and Coulibaly (2011),[START_REF] Du | Speculation and volatility spillover in the crude oil and agricultural commodity markets: A Bayesian analysis[END_REF][START_REF] Stout | Risk, speculation, and OTC derivatives: an inaugural essay for convivium[END_REF][START_REF] Valiante | Commodity Price Formation in Boom-and-Bust Cycles[END_REF],[START_REF] Büyükşahin | Commodities and equities: a 'Market of One[END_REF]

2011),[START_REF] Irwin | Testing the masters hypothesis in commodity futures markets[END_REF],[START_REF] Vivian | Commodity volatility breaks[END_REF], and[START_REF] Manera | Returns in commodities futures markets and financial speculation: A multivariate GARCH approach[END_REF] for a review.

Only five commodities were considered in[START_REF] Choi | Volatility behavior of oil, industrial commodity and stock markets in a regime-switching environment[END_REF], instead of 25 in our case.

The financialization process refers to a situation in which the price of an individual commodity is not only determined by its primary supply and demand, but also by several financial factors and investors' behavior in derivative markets.

We use the term "speculation" for simplifying purposes to refer to a situation in which investors (i) engage in transactions to profit from short-term fluctuations in the market value of the considered asset or product, and (ii) focus only on price movements rather than on the fundamentals linked to the considered asset or product. Empirically, speculation is assessed here through the dynamics of correlations between oil and commodity markets: increasing correlations in times of rising oil prices, and decreasing-and even negative-correlations during periods of declining stock prices.

For an extensive review of the literature on this topic, see[START_REF] Filis | Dynamic correlation between stock market and oil prices: The case of oil-importing and oil-exporting countries[END_REF].

See Engle (2002).

The grey bands correspond to periods of bearish stock market, the white stripes corresponding to periods of bullish stock market.


Christian Bidard
email: christian.bidard@u-paris10.fr

The Ricardo-Lemke parametric algorithm on oddity and uniqueness

Keywords: Oddity, parametric Lemke algorithm, Ricardo, uniqueness JEL classi…cation. B12, C61, C63

The parametric Lemke algorithm …nds an odd number of solutions to the linear complementarity problem LCP (q, M ), for a matrix M with zero blocks on the diagonal and vector q within a certain domain. A criterion for monotonicity and uniqueness is given. The algorithm applies to the determination of a long-run equilibrium in the presence of scarce resources, and its …rst description can be traced back to the nineteenth century economist

Introduction

For a given matrix M , the parametric Lemke algorithm aims to solve the linear complementarity problem LCP(q ; M ) by transferring a known solution of LCP(q(0); M ) along a curve joining vector q(0) to vector q . Most of the time, in a neighborhood of a point q(t), the transfer involves a smooth adaptation of the solution obtained at q(t); from time to time, some positive component of a solution vanishes and a more dramatic change is required to avoid it becomes negative. This change is mechanically determined by the algorithm, but it cannot be excluded that it leads to a U-turn on the path (antitone move), and then a second solution is obtained for some q(t). Under which circumstances does the algorithm …nd a solution associated with q ? Assume that the algorithm has the following properties: it works everywhere in a neighborhood of a point on the curve (U-turns being admitted), the solution for q(0) is unique, there are …nitely many solutions (if any) for any vector on the path and, …nally, the algorithm never returns to a previously examined solution (that last property follows from a reversibility property of the algorithm). Then, starting from the unique solution attached to q(0), transfers along the path de…ne a unique sequence of transformed solutions, and the algorithm must eventually reach q : a solution of LCP(q ; M ) is obtained. If the curve is prolongated beyond q and goes to a point q(1) for which the solution is also unique, it may be the case that other solutions are found at q , because U-turns may lead the algorithm to return to that point. However, since the trajectory starts from q(0) and reaches q(1), it goes an odd number of times to q (except if q is itself a point of U-turn); s times in the direct way from q(0) to q(1), s 1 in the opposite way. This suggests that the working of the parametric Lemke is intrinsically linked with an oddity property of the number of solutions. Uniqueness of the solution at any point of the path goes hand in hand with the absence of antitone move, and a global uniqueness result can be expected under some additional hypothesis.

These ideas are applied to a speci…c type of matrix M , with two blocks of zeroes on the diagonal. Section 2 describes the algorithm and Section 3 studies its properties when q belongs to some domain. Section 4 shows that the algorithm …nds an odd number of solutions. Section 5 states a criterion for monotonicity, local and global uniqueness. The problem here examined stems from economic theory and can be traced back to the English economist David Ricardo (1772-1823), who studied how, in the presence of a scarce resource (land), a long-term equilibrium is a¤ected by increasing demand. Ricardo may be considered as a precursor of the parametric Lemke algorithm (Section 6), which we therefore dub 'Ricardo-Lemke algorithm'.

Notations: For a real vector x, notation x >> 0 means that it is positive (all its components are positive), x > 0 that it is semipositive (x nonnegative and nonzero) and x 0 that it is nonnegative. The notation x 0 [y] means x and y are nonnegative and componentwise complementary. For a vector x, x T denotes the transpose of x, and the same for matrices.

The Ricardo-Lemke algorithm

Let matrices A and B be given m n real matrices, while q 1 2 R n and q 2 2 R m are given vectors. For q = q 1 q 2 and M = 0 A T B 0 , the problem LCP(q; M ) consists in …nding nonnegative vectors

z 1 2 R m ; z 2 2 R n such that A T z 2 + q 1 0 [z 1 ] (1) Bz 1 + q 2 0 [z 2 ] (2)
Let us assume: (H 1 ) A + B 0 Lemma 1 Under assumption (H 1 ), the unique solution to (1)-( 2) for q >> 0 is z 1 = z 2 = 0.

Proof. Consider a solution (z 1 ; z 2 ) of LCP(q; M ) with q >> 0. By summing up the equalities

0 = z T 1 A T z 2 + z T 1 q 1 and 0 = z T 2 Bz 1 + z T 2 q 2 , one obtains 0 = z T 2 (A + B)z 1 + z T 1 q 1 + z T 2 q 2 ,
and assumption (H 1 ) implies z 1 = z 2 = 0: It is convenient to transform inequalities (1)-(2) into a more tractable system. Let vectors

q 2 = q 2 0 2 R m+n + ; z 2 = z 2 ! 2 R m+n +
be obtained by extending the previous vectors q 2 and z 2 with n additional components (! 0). Similarly, let I n be the identity matrix of dimension n and let

A = A I n ; B = B I n
be extended matrices of dimension (m + n) n. Clearly enough, any solution (z 1 ; z 2 ) gives birth to a solution (z 1 ; z 2 ) of the problem 3

A T z 2 + q 1 = 0 (3) Bz 1 + q 2 0 [z 2 ] (4)
where the last n components of vector z 2 are ! = A T z 2 + q 1 0, and viceversa.

The assumptions on matrices A and B are better expressed on their extensions A and B. Consider a continuous curve (C) q 1 = q 1 (t) in R n (t 2 [0; 1]) with unchanged q 2 . The curve is oriented according to increasing values of t and is assumed to have the following properties (H 2 ) and (H 3 ). For any q 1 on (C) and any solution to (3)-( 4):

(H 2 ) The equality (Bz 1 + q 2 ) i = 0 holds for at most n components; then the corresponding n rows of A are independent, and the same for B.

(H 3 ) The curve (C) cuts the cones positively generated by n 1 rows of A at …nitely many points, and no point of (C) belongs a cone generated by n 2 rows.

Conditions (H 2 ) and (H 3 ) avoid degeneracies in the working of the Ricardo-Lemke algorithm (or parametric Lemke algorithm) we now describe.

At a given step of the algorithm, let the starting point be a value b t 2 [0; 1] which is not one of the intersection points mentioned in (H 3 ), a direction of change of t (say, increasing values), and a known solution (z 1 ( b t); z 2 ( b t)) of (3)-(4) for q 1 ( b t) 2(C) such that z 2 ( b t) has exactly n positive components I f1; :::; m + ng. By the complementarity condition (4) and (H 2 ), equality (Bz 1 + q 2 ) i = 0 holds exactly for these components i 2 I, and the knowledge of the set I su¢ ces to identify the solution (z 1 ( b t); z 2 ( b t)). With q 1 (t) moving on the curve, the problem is to …nd a solution such that z 2 (t) varies continuously. The algorithm describes the unique answer, if any, to that problem. For small variations of t around b t, the second condition in (H 2 ) implies the existence of z 2 (t) 0 such that equality A T z 2 (t) + q 1 (t) = 0 holds, with the same positive components I as for z 2 ( b t). Then conditions (3) and ( 4) are met for an unchanged vector z 1 , therefore the solution is extended by continuity to a neighborhood of b t. The process works in an interval [t 0 ; t 1 ] containing b t until t reaches a value t 1 (t 1 > b t, given the above assumption that t is increasing) such that some positive component j 2 I of z 2 (t 1 ) vanishes and would become negative for a higher value of t. The point q 1 (t 1 ) 2(C) is one of the points referred to in condition (H 3 ), and that condition ensures that the components of z 2 (t 1 ) other than j remain positive. The question is to extend the solution beyond that limit, with a continuous change in z 2 (t). Then the complementarity conditions (4) imply that equalities (Bz 1 + q 2 ) i = 0 hold in a neighborhood of t 1 for the n 1 components i 2 In fjg. According to the third condition in (H 2 ), the general solution z 1 (t) of these n 1 a¢ ne equalities is of the type z 1 (t) = z 1 ( b t)+ z 0 1 where z 1 ( b t) is the previous solution, z 0 1 is a nonzero solution of (Bz 0 1 ) i = 0 for i 2 In fjg and is an arbitrary scalar. z 0 1 is unique up to a factor that we choose in order that (Bz 0 1 ) j > 0. Then inequality (4) for z 1 = z 1 (t) holds as an equality for any i 2 In fjg, and as a strict inequality for component j if and only is positive.

Assume that the following condition (D) holds (otherwise, there is no solution of LCP(q(t); M ) with positive components in In fjg, and a fortiori no continuous extension of the previous solution):

(D) Bz

0 1 0 Condition (D) means that b k z 0 1 < 0 for some row k of B (then k = 2 I).
For any row of that type, the initial inequality (Bz 1 (t) + q 2 ) k 0 is transformed into an equality for z 1 (t) = z 1 (t) + z 0 1 and some positive . Let us pick up the component k corresponding to the minimum positive value of for which that equality occurs (that minimum rule might alternatively be described in geometric terms as the choice of a new facet). For that minimum value, the vector inequality (4) holds with exactly n equalities for the components in the n-set J = fkg [ In fjg. According to (H 2 ), there exists a vector z 2 = z J2 (t), with null components outside J, such that equality (3) holds. At t = t 1 ; z J2 (t 1 ) is a strictly positive combination of the n 1 rows of B belonging to In fjg = Jn fkg. By the second condition in assumption (H 2 ), a unique decomposition (a i is the ith row of A)

q 1 (t) = X i2Jnfkg [z J2 (t)] i a i + [z J2 (t)] k a k (5)
is obtained in a neighborhood of t 1 , with [z J2 (t)] i close to [z J2 (t 1 )] i and therefore positive, while 4) is thus found on that half-interval. A new starting point t 1

[z J2 (t)] k is close to [z J2 (t 1 )] k = 0. According to (H 3 ), [z J2 (t)] k is nonzero for t close but di¤erent from t 1 , therefore [z I2 (t)] k is positive either on an interval ]t 1 ; t 1 + [ (monotone move) or on ]t 1 ; t 1 [ (antitone move). A solution of (3)-(
, a new orientation for t and a new set J of positive components for z 2 (t 1

) are thus de…ned, so that the algorithm works locally.

Properties of the algorithm

The oriented curve (C) from q 1 (0) to q 1 (1) being de…nitely given, consider a pair (I; [t 0 ; t 1 ]), where I is an n-subset of f1; :::; m + ng sustaining a solution of LCP(q(t); M ) with positive components of z 2 (t) in I for t in [t 0 ; t 1 ], and [t 0 ; t 1 ] is an oriented interval (t moves towards t 1 ). The knowledge of I su¢ ces to identify the interval [t 0 ; t 1 ] or [t 1 ; t 0 ], not its orientation, as well as the associated solution of LCP(q(t); M ) for any t in that interval. The set S made of all pairs of that type is …nite. If the algorithm works locally, the knowledge of (I; [t 0 ; t 1 ]) also su¢ ces to identify the pair (J; [t 1 ; t 2 ]) which succeeds it, which is denoted suc(I; [t 0 ; t 1 ]) (however, the function suc : S ! S is not de…ned if t 1 = 0 or 1, as the algorithm stops if either q 1 (0) or q 1 (1) is reached). The algorithm can therefore be represented as a …nite directed graph whose nodes are the elements of S. The algorithm, which starts from a given solution (I 0 ; [0; ]) at q 1 (0), …nds a solution at q 1 (1) if the successor of each node is well de…ned and if it admits no loops. Lemma 2 deals with the …rst condition, while the second condition results from the reversibility property (Lemma 3).

Let D R n be the open and convex set, which contains the strictly positive orthant, de…ned as

D = q 1 ; 9y > 0 q 1 >> B T y R n (6)
Lemma 2 means that the algorithm transfers a solution from interval

[t 0 ; t 1 ] to the next [t 1 ; t 2 ].
Lemma 2 Let the curve (C) lie in D. Each element (I; [t 0 ; t 1 ]) of S admits a unique successor, except if t 1 = 0 or 1.

Proof. From Section 2, we know that the successor of (I; [t 0 ; t 1 ]) is well de-…ned if condition (D) holds, where z

T 1 q 1 (t 1 ) = z 0 T 1 A T z 2 (t 1 ) z 0 T 1 B T z 2 (t 1 )
, and the last term is zero because B T z 2 (t 1 ) is a combination of the columns of matrix B belonging to the set In fjg, which are all orthogonal to z 0 1 . According to de…nition (6), there exists y > 0 such that z

0 T 1 B T y < z 0 T 1 q 1 (t 1 ) 0, therefore condition (D) is met.
The reversibility property considers the e¤ects of a reverse move of t:

Lemma 3 For t 1 6 = 0; 1, let suc(I; [t 0 ; t 1 ]) = (J; [t 1 ; t 2 ]). Then suc(J; [t 2 ; t 1 ]) = (I; [t 1 ; t 0 ]).

Proof. (J; [t 1 ; t 2 ]) being constructed as the successor of (I; [t 0 ; t 1 ]), let us now start from (J; [t 2 ; t 1 ]). The limit of the solution sustained by J is reached at value t 1 when the kth component of z J2 (t) vanishes. Let T be the n-set which succeeds J. According to the construction examined in Section 2, the new vector z T 1 di¤ers from z J1 by y 0 1 , where y 0 1 is a solution of the equations (By 0 1 ) i = 0 for i 2 Jn fkg. As Jn fkg = In fjg, these equations are the same as those determining z 0 1 in the passage from (I; [t 0 ; t 1 ]) to (J, [t 1 ; t 2 ]), so that y 0 1 is proportional to z 0 1 . We have already noticed that (Bz ). The scalars and are both de…ned as the minimum value for which equality (Bz J1 + q 2 ) i = 0 , which holds for the n 1 components in Jn fkg, also holds for one more component. Therefore = , T = I, and the successor of (J; [t 2 ; t 1 ]) is (I; [t 1 ; t 0 ]).

Let rev be the function which associates with each pair (I; [t 0 ; t 1 ]) of S its reverse (I; [t 1 ; t 0 ]). Lemma 3 applied to (I; [t 1 ; t 0 ]) states that the function suc rev suc rev is the identity on S: Lemma 4 For t 0 6 = 0, each element (I; [t 0 ; t 1 ]) of S admits a unique predecessor.

Proof. The function pre = rev suc rev de…nes the unique predecessor of an element of S.

Lemma 5 From a given starting point, the algorithm does not reach a node and its reverse.

Proof. It the algorithm reaches (I; [t 0 ; t 1 ]) at step and (I; [t 1 ; t 0 ]) at step , with > , it reaches suc(I; [t 0 ; t 1 ]) at step + 1 and pre(I; [t 1 ; t 0 ]) at step 1. By Lemma 3 these nodes are also reverse from each other but the gap is reduced by two. But cannot be reduced either to zero (this would mean t 0 = t 1 ) or to one (this would mean that (I; [t 1 ; t 0 ]) succeeds (I; [t 0 ; t 1 ]), when two consecutive sets di¤er by one element).

Lemma 6 If the solution of LCP(q(0); M ) is unique, the algorithm never returns either in a neighborhood of t = 0 or on a previously examined solution at q(t):

Proof. Let (I 0 ; [0; ]) be the starting point in S. If the solution at q(0) is unique and the algorithm returns at step in a neighborhood of t = 0, the set I of positive components of z 2 (t) must be the same by the uniqueness of the solution, and the values of t are decreasing, therefore (I ; [t ; t +1 ]) = rev(I 0 ; [0; ]), which is excluded by Lemma 5. The same Lemma also implies that if the algorithm returns at steps and to a previously examined solution (then I = I ), the direction must be the same in both cases, therefore (I ; [t ; t +1 ]) = (I ; [t ; t +1 ]). That identity also holds for their predecessors at steps 1 and 1, and a contradiction is obtained by considering the …rst pair ( , ) for which a return occurs.

Oddity property

The …rst part of Theorem 1 considers an oriented curve from q 1 (0) >> 0 to q 1 (in the literature on the parametric LCP, it is generally the segment joining these points), the second part prolongates that curve, which returns into the positive orthant.

Theorem 1 Let (C) q 1 = q 1 (t) be a curve in D, joining from q 1 (0) >> 0 to q 1 . Under assumptions (H 1 );(H 2 ) and (H 3 ), the Ricardo-Lemke algorithm …nds a solution of the linear complementarity problem (1)-( 2) at (q 1 ; q 2 ). If the curve (C) returns to q 1 (1) >> 0, the algorithm …nds an odd number of solutions (…nitely many points on the curve apart), the solutions corresponding to a monotone move exceeding by one those corresponding to an antitone move.

Proof. The algorithm admits a representation in terms of a …nite directed graph, with the (I ; [t ; t +1 ])'s as nodes. The starting point is unique (Lemma 1), each node has a unique successor (except if t +1 = 1) and a unique predecessor (except if t = 0), and loops are excluded. Therefore, the trajectory starting from q 1 (0) must reach q 1 and de…nes a solution at that point. Assume moreover that the curve (C) continues and goes to q 1 (1) >> 0. The unique solution at q 1 (0) is transferred along the path and transformed into one or more solutions at q 1 , because of possible antitone moves, then is transferred to q 1 (1). Among the solutions thus generated at q 1 , those corresponding to monotone moves exceed by one those corresponding to antitone moves, except if q 1 is a point of U-turn.

Theorem 2 For q 1 in D, the number of solutions of LCP(q ; M ) is generically odd.

Proof. Take an arbitrary solution at q 1 as starting point, with an initial direction towards q 1 (1). If the trajectory (T) generated by the algorithm reaches either q 1 (0) or q 1 (1), it is a sub-trajectory of the unique trajectory joining q 1 (0) to q 1 (1). These solutions are those examined in Theorem 1 and their number is odd. It may also be the case that (T) never reaches q 1 (0) or q 1 (1) because the graph admits a loop. Then, as the successor and the predecessor of a node are unique, that disconnected part of the graph is itself is a loop. The trajectory oscillates around q 1 , with alternate monotone and antitone moves, and a new solution is found everytime it reaches q 1 , until it comes back to the solution one started with. The number of solutions thus generated at q 1 is even, with an equal number of monotone and antitone moves. If the starting point is anyone of these new solutions, the same set of solutions is found. A partition of those additional equilibria is thus obtained, with an even number of solutions in each subset.

Monotonicity and Uniqueness

Let (J; [t 1 ; t 2 ]) succeed (I; [t 0 ; t 1 ]):By assumption (H 2 ), the four n n submatrices A I , A J , B I and B J made of the rows I and J of A and B are regular. The following criterion determines if the move of t is the same for the consecutive nodes (t is either increasing or decreasing in both cases) or if it changes (monotone and antitone moves, therefore the values of t overlap).

Theorem 3 When J succeeds I, the direction of the move of t is unchanged if and only if det A I = det B I and det A J = det B J are the same sign. Otherwise, a U-turn on the curve (C) occurs.

Proof. We have Jn fkg = In fjg. By (2), inequality Bz I1 + q 2 0 holds for the vector z I1 associated with I, and since it holds as an equality for the I-components, assumption (H 2 ) implies that the inequality is strict for component

k: k = b T k z I1 + q 2k > 0. As b T k z J1 + q 2k = 0, we have k = b T k (z I1 z J1 )> 0. Similarly, j = b T j (z J1 z I1 ) > 0. Vector z J1 z I1 is orthogonal to the row-vector k b T j + j b T k . By construction, z J1 z I1 = z 0 1
is also orthogonal to the n 1 rows of B belonging to In fjg = Jn fkg. Thus matrix k B J + j B K transforms z J1 z I1 into the null vector, therefore k det B I + j det B J = 0 and det B I and det B J have opposite signs.

For t in the interval associated with I and close to t 1 , equality A T I z I2 (t) + q 1 (t) = 0 holds with z I2 (t) positive, but the jth component of z I2 (t) vanishes at t = t 1 and would become negative beyond that limit. When row j is replaced by another row k, a second algebraic decomposition of vector q 1 (t) is obtained: Proof. In a neighborhood of the starting point q 1 (0) >> 0 the move is monotone, A I and B I are the identity matrix and the ratio of their determinants is positive. In the next steps, a change of orientation coincides with a change in the sign of det A I = det( B I ).

q 1 (t) = X i2Infjg [z I2 (t)] i a i + [z I2 (t)] j a j = X i2Jnfkg [z J2 (t)] i a i + [z J2 (t)] k a k (7) At t = t 1 both decompositions coincide, therefore [z J2 (t)] i , which is equal to [z I2 (t 1 )] i > 0 at t = t 1 ,
Corollary 2 For q 1 in D, and ‡ukes appart, the number of solutions of LCP(q ; M ) for which det A I = det( B I ) is positive exceeds by one that for which it is negative.

Proof. By Theorem 1 and Corollary 1, the solutions which are reached by the algorithm have the property mentioned in the Corollary. The additional solutions studied in the proof of Theorem 2 have as many monotone as antitone moves.

Corollary 3 For q 1 in D, and ‡ukes apart, if det A I = det( B I ) is positive at any associated equilibrium, that equilibrium is unique.

Proof. This follows from Corollary 2.

If det A I = det( B I ) is everywhere positive in D, the transformation along a path is monotone, and the unique solution in D is reached by the algorithm. algorithm, a variant of Lemke's (1965) initial algorithm. What Ricardo and his followers did not see, however, is that the method k selected on the basis of a pro…tability criterion may not be productive enough, i.e. they did not see the possibility of an antitone move.

For the type of matrix M here considered, an existence result for q 1 in D was established by [START_REF] Dantzig | A complementarity algorithm for an optimal capital path with invariant proportions[END_REF]. Their proof makes reference to another variant of the Lemke algorithm. [START_REF] Salvadori | Land and choice of techniques within the Sra¤a framework[END_REF] applied that result to the existence of a long-term equilibrium. Starting from geometrical considerations, [START_REF] Erreygers | Terre, Rente et Choix de Techniques[END_REF][START_REF] Erreygers | On the uniqueness of cost-minimizing systems[END_REF] stated the local uniqueness condition on the signs of the determinants and assumed that the global uniqueness problem can be reduced to that of local uniqueness. None of these studies refers to the parametric Lemke algorithm. Corollary 2 generalizes Bidard and Erreygers's (1998) result on the oddity of the number of equilibria in the absence of lands (see also [START_REF] Lemke | Equilibrium points of bimatrix games[END_REF] on oddity). Bidard (2012) criticized Ricardo's attempt to get rid of rent in his analysis.

From a formal point of view, the initial formalization (1)-( 2) is symmetric in the indices 1 and 2 but the equivalent formulation (3)-( 4) introduces an asymmetry. A dual treatment of the same problem is therefore possible, with no clear economic interpretation.

Conclusion

The algorithmic Lemke algorithm is usually used to …nd a solution of an LCP by transferring a given solution along a segment. The transfer along a curve which starts from and comes back into a domain in which the uniqueness of the solution is ensured suggests that the working of the algorithm is intrinsically connected with the possibility to assign an orientation to each of these equilibria, with one more solution with a direct orientation. An open problem connected with the possibility to …nd all solutions concerns the distinction between the equilibria which are reached by the parametric method along a curve from those which are not.

0 1 is de…ned by the conditions b T i z 0 1 = 0 for i 2

 02 In fjg and b T j z 0 1 > 0. If z 0 1 has a negative component, then the condition is met by the submatrix I n of B. Assume that z 0 1 is semipositive. It follows from equality (3) and hypothesis (H 1 ) that z 0

0 1 ) k = b k z 0 11

 10 up to a positive factor, and z T 1 = z J1 + ( z 0 1

  remains positive in a neighborhood. [z J2 (t)] k is zero at t = t 1 . The set J which succeeds I makes no U-turn if and only if [z J2 (t)] k is positive whereas [z I2 (t)] j has become negative. Equality[START_REF] Lemke | Bimatrix equilibrium points and mathematical programming[END_REF] shows that the vectors a i for i 2 In fjg = Jn fkg and vector [zI2 (t)] j a j [z J2 (t)] k a k are linearly dependent, therefore det([z I2 (t)] j A I [z I2 (t)] k A J ) = 0. [z I2 (t)] jand [z I2 (t)] k have opposite signs if and only if it is the case for det A I and det A J . This amounts to saying that det A I = det B I and det A J = det B J have the same sign.

Corollary 1 Let I be the set of positive components of z 2 (t). The move is monotone or antitone according as det A I = det( B I ) is positive or negative.

Ricardo as a precursor

David Ricardo (1817) studied the long-run dynamics of a capitalist economy and was especially interested in the evolution of the distribution of national income, which he thought to be unfavourable to capitalists because, when the demand for corn increases, landowners reap higher rents. [START_REF] Sra¤a | Production of Commodities by Means of Commodities[END_REF] proposed a formalization of the notion of long-run equilibrium that post-Sra¢ an authors completed. Let there be n goods and lands, while labor is homogenous. A method of production i is described by a vector e i 2 R n + of material inputs and lands, a quantity of labor input q 2i 2 R + and a vector f i 2 R n + of outputs. The net product of that method is a i = f i e i . When each method i operates at activity level z 2i ;vector A T z 2 represents the net product of goods and the negative of the quantity of lands used in the economy. With vector q 1 representing the negative of an exogenously given …nal demand vector and the available quantities of lands, inequality (1) means that …nal demand and the scarcity constraints on lands are met. The complementary vector z 1 represents the prices of goods and the rents on lands: the price of a good in excess supply or the rent on a partially cultivated land is zero. Let r be a given, nonnegative and uniform rate of pro…t. With b i = (1 + r)e i f i , inequality (2) means that no method yields more than the ruling rate of pro…ts at the price-and-rent vector z 1 when the nominal wage is equal to unity by convention. The complementarity relationships (2) with the vector of activity levels z 2 mean that any operated method yields the normal rate of pro…t. Conditions (1) and (2) de…ne respectively the quantity side and the value side of a long-term equilibrium. Condition (H 1 ) holds. If one introduces free disposal and fallowing as methods of production (these methods dispose at no cost of any surplus of goods and lands), the quantity side of an equilibrium is equivalently written as equality [START_REF] Dantzig | A complementarity algorithm for an optimal capital path with invariant proportions[END_REF]. In Chapter 2 of the Principles (1817), Ricardo studied the e¤ects of an increase of demand, therefore of a change q 1 = q 1 (t) on a given equilibrium. He stressed that, most of the time, only activity levels z 2 (t) need to be adjusted, with no in ‡uence on the price-and-rent vector z 1 . But, spasmodically, the adjustment is no longer possible, for instance because some quality of land becomes fully cultivated. Then the price of corn must rise, and the rents too. Even if Ricardo did not mention in Chapter 2 the e¤ects on the other prices, he was fully aware that the rise in the price of corn triggers changes in those of all commodities whose production requires corn. The process suggested by Ricardo thus corresponds to the choice of a new price-and-rent vector as described in Section 2, the new operated method k being the …rst that yields the normal rate of pro…t r when the price of rice increases (minimum rule for ). On the whole, that process may be identi…ed with what is known today as the parametric Lemke