Laurent Gilquin

Antoni Jiménez Rugama

Elise Arnaud

Fred J Hickernell

Hervé Monod

Clémentine Prieur

Iterative construction of replicated designs based on Sobol' sequences

In the perspective of estimating main effects of model inputs, two approaches are studied to iteratively construct replicated designs based on Sobol' sequences. Space-filling properties of the resulting designs are studied based on two criteria.

Introduction

Mathematical models often involve a substantial number of poorly known parameters. The effect of these parameters on the output of the model can be assessed through sensitivity analysis. Global sensitivity Email addresses: laurent.gilquin@inria.fr (Laurent Gilquin), ljimene1@hawk.iit.edu (Lluís Antoni Jiménez Rugama).

analysis methods are useful tools to identify the parameters having the most influence on the output. A well known approach is the variance based method introduced by Sobol' in [START_REF] Sobol | Sensitivity indices for nonlinear mathematical models[END_REF]. This method estimates sensitivity indices called Sobol' indices that summarize the influence of each model input. Among all Sobol' indices, one can distinguish the first-order indices that estimate the main effect of each input.

The procedure to estimate first-order Sobol' indices proposed by Sobol' and its improvements (see Saltelli [START_REF] Saltelli | Making best use of model evaluations to compute sensitivity indices[END_REF] for an exhaustive survey) all suffer from a prohibitive number of model evaluations that grows with respect to the input space dimension. An elegant solution to reduce this number relies on the construction of particular designs of experiments called replicated designs. The notion of replicated designs was first introduced by McKay through its definition of replicated Latin Hypercubes in [START_REF] Mckay | A comparison of three methods for selecting values of input variables in the analysis of output from a computer code[END_REF]. Below we provide this definition in a more general framework: Definition 1.1 Consider x ∈ [0, 1) s , and x u ∈ [0, 1) |u| the subset of elements of x given by u {1, . . . , s}, where |u| is the cardinality of u. Let P = {x i } n-1 i=0 and P = {x i } n-1 i=0 be two point sets in [0, 1) s , and denote by P u = {x i,u } n-1 i=0 (resp. P u ) the subset of elements of points in P (resp. P ) indexed by u. We say that P and P are two replicated designs of order a = 1, . . . , s -1, if for any u {1, . . . , s} with |u| = a, P u and P u are the same point set in [0, 1) a , perhaps in a different order.

The replication procedure described in [START_REF] Mara | Comparison of some efficient methods to evaluate the main effect of computer model factors[END_REF][START_REF] Tissot | A randomized orthogonal array-based procedure for the estimation of first-and second-order Sobol' indices[END_REF] allows the estimation of all first-order Sobol' indices with only two replicated designs of order 1. This procedure has the major advantage of reducing the number of model evaluations, evaluating only on designs P and P regardless of the input space dimension. However, Sobol' indices estimates may still not be accurate enough if designs P and P do not explore the input space properly.

In this note, we propose two different constructions of replicated designs of order 1 based on Sobol' sequences. Both constructions ensure that the input space is properly explored and can be used within the replication procedure to estimate all first-order Sobol' indices. The definition of these constructions is recursive, therefore one can iteratively refine each replicated design by adding the corresponding new set of points. We first provide a brief introduction on digital sequences and then present two iterative approaches to construct the two replicated point sets. We end this note by analyzing the space-filling properties of the two designs constructed.

Digital sequences background

Preliminaries

Digital nets and sequences were first introduced by Niederreiter [START_REF] Niederreiter | Random Number Generation and Quasi-Monte Carlo Methods[END_REF] in the numerical integration framework to define good uniformly distributed points in [0, 1) s . They can also appear in the literature as digital (t, m, s)-nets and digital (t, s)-sequences, or simply (t, m, s)-nets and (t, s)-sequences. Sobol' and Niederreiter-Xing sequences are two examples of digital sequences detailed in [START_REF] Sobol | On the distribution of points in a cube and the approximate evaluation of integrals[END_REF] and [START_REF] Niederreiter | Low-Discrepancy Sequences and Global Function Fields with Many Rational Places[END_REF]. Definition 2.1 Let A be the set of all elementary intervals A ⊂ [0, 1)

s where A = s j=1 [α j b -γj , (α j + 1)b -γj ), with integers s ≥ 1, b ≥ 2, γ j ≥ 0, and b γj > α j ≥ 0. For m ≥ t ≥ 0, the point set P ∈ [0, 1) s with b m points is a (t, m, s) -net in base b if every A with volume b t-m contains b t points of P.
Thus, a (t, m, s)-net is defined such that all elementary intervals of volume b t-m will enclose the same proportion of points of P, namely b t-m |P| points. The most evenly spread nets are (0, m, s)-nets, since each elementary interval of the smallest volume possible, b -m , contains exactly one point. The quality of any (t, m, s)-net is therefore measured by the parameter t, called t-value.

By increasing m, we increase the number of points of the (t, m, s)-net. In the limiting case where m → ∞, we can define the (t, s)-sequence as:

Definition 2.2 For integers s ≥ 1, b ≥ 2, and t ≥ 0, the sequence {x i } i∈N0 is a (t, s)-sequence in base b, if for every set P ,m = {x i } ( +1)b m -1 i= b m
with ≥ 0 and m ≥ t, P ,m is a (t, m, s)-net in base b. The replicated design properties can also apply to digital sequences. Hence, we introduce the following definition, Definition 2.3 Two digital sequences {x i } i∈N0 and {x i } i∈N0 are digitally replicated of order a if for all m ≥ 0,

{x i } b m -1 i=0 and {x i } b m -1 i=0
are two replicated designs of order a.

Sobol' sequences

Sobol' sequences in dimension s are digital sequences in base b = 2 that can be computed using the generating matrices, a set of s full rank infinite dimensional upper triangular matrices over the Galois field F 2 := {0, 1}. These generating matrices are recursively constructed given some primitive polynomials and initial directional numbers. In [START_REF] Joe | Constructing Sobol Sequences with Better Two-Dimensional Projections[END_REF], Kuo and Joe detail this construction and also suggest a particular choice for these matrices that optimize the 2 dimensional projection t-values.

Consider the generating matrices C 1 , . . . , C s , and C m 1 , . . . , C m s their upper left corner blocks of size m × m. Although C 1 , . . . , C s are of infinite size, one only requires the knowledge of C m 1 , . . . , C m s to construct the first 2 m Sobol' points: for each i = 0, . . . , 2 m -1, the point x i = (x i,1 , . . . , x i,s ) of the sequence is obtained dimension-wise by:

(x i,j,1 , . . . , x i,j,m ) = C m j i, j = 1, . . . , s , (1)
where x i,j = m k≥1 x i,j,k 2 -k is the binary expansion of x i,j and i = (i 0 , . . . , i m-1 ) is the vector obtained from the binary expansion of i = m-1 k≥0 i k 2 k . All matrix operations are performed in F 2 . Below we provide an example of how to compute x 11,1 = 0.8125 and x 9,2 = 0.4375,

      x 11,1,1 x 11,1,2 x 11,1,3 x 11,1,4       =       1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1       C 4 1       1 1 0 1       11 =       1 1 0 1       ,       x 9,2,1 x 9,2,2 x 9,2,3 x 9,2,4       =       1 1 1 1 0 1 0 1 0 0 1 1 0 0 0 1       C 4 2       1 0 0 1       9 =       0 1 1 1      
To compute the next 2 m points of the sequence, one can infer from (1) that (x i+2 m ,j,1 , . . . , x i+2 m ,j,m+1 ) = (x i,j,1 , . . . , x i,j,m , 0) ⊕ (c m+1 j

) , i = 0, . . . , 2 m -1, (2)
where c m+1 j is the last column of C m+1 j and ⊕ is the addition in F 2 . In addition, Sobol' sequences have good group structure properties. For any m ≥ 1, the first 2 m points of the sequence form an Abelian group under ⊕:

x = m k=1 x 1,k 2 -k , . . . , m k=1 x s,k 2 -k , z = m k=1 z 1,k 2 -k , . . . , m k=1 z s,k 2 -k , x ⊕ z := m k=1 (x 1,k + z 1,k mod 2)2 -k , . . . , m k=1 (x s,k + z s,k mod 2)2 -k . ( 3
)
From equation ( 1) above, one may see that the first 2 m points of the sequence are elements of (F m 2 ) s . Lemma 2.4 All Sobol' sequences are digitally replicated of order 1. Proof : Consider any two s-dimensional Sobol' sequences generated by C 1 , . . . , C s and C 1 , . . . , C s respectively. Since any two matrices C m j and C m j are square and full rank, the operations i → C m j i and i → C m j i are one-to one and onto for all i = 0, . . . , 2 m -1. Therefore, they generate the same point sets.

Iterative constructions of replicated point sets

In this section we propose two different approaches to iteratively construct two replicated point sets, P and P , based on Sobol' sequences. These two constructions are carried out according to the following recursive scheme :

   P 0 = B 0 P = P -1 ∪ B    P 0 = B 0 P = P -1 ∪ B , ≥ 1,
where B and B are new sets of points added at step to refine P -1 and P -1 . For all ≥ 0, P and P are two replicated designs of order 1.

The first approach is called multiplicative because |P | = 2 while the second one is called additive and |P | = |B 0 |. In the multiplicative case, we will directly use 2 s-dimensional sequences as a result from Lemma 2.4. However, for the additive case, we will consider an initial set of Sobol' points and apply different scramblings and digital shifts to extend the point sets. Additionally, in both cases one can randomize the points using Owen's scrambling [START_REF] Owen | Scrambling Sobol' and Niederreiter-xing points[END_REF] as long as same coordinates of P and P share the same scrambling.

Multiplicative approach

Two replicated point sets of order 1, P and P , can be constructed using two s-dimensional Sobol' sequences. We note C 1 , . . . , C s the generating matrices used to generate P and C 1 , . . . , C s those used to generate P . To ensure that P and P are as uniform as possible, these generating matrices need to be different from each other. We choose C 1 , . . . , C s , C 1 , . . . , C s to be the first 2s generating matrices suggested by Joe and Kuo in [START_REF] Joe | Constructing Sobol Sequences with Better Two-Dimensional Projections[END_REF], not necessarily in this order. In [START_REF] Joe | Constructing Sobol Sequences with Better Two-Dimensional Projections[END_REF], Joe and Kuo minimized the t-value for all 2 dimensional projections.

In order to iteratively extend designs P and P , one just needs to compute B = {x 2 -1 , . . . , x 2 -1 } and B = {x 2 -1 , . . . , x 2 -1 }, starting with B 0 = B 0 = {0}, with 0 the null vector. Each set B and B can be constructed using (2) applied to P -1 and P -1 .

As a direct consequence of Lemma 2.4, at each step designs P and P are two replicated designs of order 1. Furthermore, they both inherit the space-filling properties of (t, , s)-nets.

Additive Approach

With the multiplicative approach, the size of designs P and P is multiplied by 2 each time is increased by one. This growth rate may be inadequate for some applications. The additive approach presented in this section is attractive due to a slower size growth. Given an initial choice of r ≥ 1 that specifies the size of B 0 and B 0 , only |B 0 | = 2 r points are added to both designs at each step. The main drawback of this approach is that P and P do not inherit the structure of a Sobol' sequence when ≥ 1. Nevertheless, both designs display good space-filling properties as shown in the next section.

Analogously to the multiplicative case, the two replicated point sets P and P constructed with the additive approach are iteratively refined with B and B . First {x i } 2 r -1 i=0 and {x i } 2 r -1 i=0 are set to be the first 2 r points of two s-dimensional Sobol' sequences. The generating matrices of these two sequences are selected as in the multiplicative approach. Then, B (resp. B ), for ≥ 0, is obtained from {x i } 2 r -1 i=0 (resp. {x i } 2 r -1 i=0 ) by carrying out digital shifts and scrambling operations. Therefore, both B and B inherit the (t, r, s)-net structure of these initial sets. 0.001 0.002 0.005 N average L 2 star discrepancy q q q q q q q q q q q q q q q q 256 512 768 1280 2048 3328 q design M design A LH design 0.08 0.10 0.12 0.16 0.20 N average maximin q q q q q q q q q q q q q q q q 256 512 768 1280 2048 3328 q design M design A LH design criterion, design A shows slightly worse results. Concerning the number of steps, design A allows 15 refinement steps from 2 8 to 2 12 points, instead of only 4 steps for design M .

Figure 1 .

 1 Figure 1. Log-log graph of averaged L 2 star discrepancy (left) and maximin (right) for designs M and A over 100 repetitions.

Acknowledgements

This work is supported by the CITiES project funded by the Agence Nationale de la Recherche (grant ANR-12-MONU-0020) and by the United States National Science Foundation (grant DMS-1522687).

At step = 0, B 0 = {x (0) i } 2 r -1 i=0 and B 0 = {x (0) i } 2 r -1 i=0 are generated as follows: for each i = 0, . . . , 2 r -1, points x (0) i = (x (0) i,1 , . . . , x (0) i,s ) and x (0) i = (x (0) i,1 , . . . , x (0) i,s ) are obtained by linearly transforming x i , (x (0) i,j,1 , . . . , x (0) i,j,r ) = L(x i,j,1 , . . . , x i,j,r ) , j = 1, . . . , s ,

i,j,r ) = L (x i,j,1 , . . . , x i,j,r ) , j = 1, . . . , s ,

where

i,j,k 2 -k and L, L are two distinct full rank lower triangular matrices of size r × r over F 2 . Left multiplications by L and L in (4) are called linear scramblings [START_REF] Matoušek | On the L2-discrepancy for anchored boxes[END_REF].

At step ≥ 1, using addition as defined in (3) we construct:

where e ∈ (F r 2 ) s \P -1 and e ∈ (F r 2 ) s \P -1 . Indeed, B 0 is a subgroup of (F r 2 )

s . Therefore, we construct B = B 0 ⊕ e as a coset of B 0 under the right choice of e . The same reasoning applies to B 0 . Additions by vectors e j and e j in ( 5) are referred as digital shifts.

There are exactly 2 rs different choices of e and e in (F r 2 ) s , and 2 r(r-1)/2 choices of L and L . The maximum number of iterations that can be performed equals the number of cosets, 2 r(s-1) . If this maximum is reached, designs P and P both correspond to a full grid design of 2 rs points with 2 r distinct values per input. Lemma 3.1 Under the additive construction P and P are two replicated designs of order 1, ≥ 0. Proof : For each ≥ 1, digital shift and scrambling operations carried out in (4) and ( 5) are bijections from F r 2 and (F r 2 ) s . Hence, the same coordinates of B and B will contain the same set of 2 r values.

Space-filling properties

To analyze the space-filling properties of both approaches we will use two criteria. The first one is the L 2 star discrepancy that measures the uniformity of a point set. This criterion can be easily computed through an analytical expression provided by [START_REF] Morokoff | Quasi-random sequences and their discrepancies[END_REF]. The lower the criterion, the more uniform the point set is. Sobol' sequences were originally constructed to be low discrepancy sequences and therefore, behave well under this criterion. The second criterion is called maximin distance and measures the regularity of a point set. It returns the minimum of all Euclidean distances between pairs of points in the set. The higher the criterion, the more spread the point set is.

We name by design M (resp. A) one of the two replicated designs constructed with the multiplicative (resp. additive) approach, and set the input space dimension s to 6. For design M , we study the properties of P 8 , . . . , P 12 , and for design A, we set the initial value to r = 8 and study the properties of P 0 , . . . , P 15 . For both M and A we generate 100 designs whose average criteria are presented in Figure 1. While each of the 100 designs M is independently randomized applying Owen's scrambling [START_REF] Owen | Scrambling Sobol' and Niederreiter-xing points[END_REF], designs A are constructed by randomly selecting matrix L (uniformly among all 2 r(r-1)/2 choices) and vectors e used in equation [START_REF] Morokoff | Quasi-random sequences and their discrepancies[END_REF]. In addition, we also show in Figure 1 the results obtained with an optimized Latin Hypercube design (LH design) according to each criterion for an equal number of points as in design A. The x-axis corresponds to the number of design points N and the y-axis to the value of the criterion. Although LH designs are the standard designs used in the replication procedure to estimate first-order indices, the replicated design of an optimized LH design is not guaranteed to be optimal.

As expected, design M performs the best and the estimated slope of the discrepancy curve (-0.72) falls within the expected range for Sobol' sequences [START_REF] Morokoff | Quasi-random sequences and their discrepancies[END_REF]. Although design A does not perform as well as design M , it outperforms an optimized LH design for the discrepancy criterion. However for the maximin


Keywords: Cooperation, R&D, Spillovers JEL Classification: O31, O32, D21

We use the French portion of the 2002 Community Innovation Survey to test how spillovers affect the likelihood that firms cooperate in R&D. Unlike most existing empirical studies, our results clearly support well-established theoretical predictions of the industrial organization literature. We find that a firm which benefits from higher spillovers from her rivals is more likely to cooperate horizontally in R&D. Moreover, the impact of incoming spillovers on the likelihood of horizontal R&D cooperation is positive and statistically significant only when they are above a threshold. Both the value, and the precision of the estimates, increase with the information flow which firms report receiving from their competitors.

Introduction

The relationship between technological spillovers and the profitability of R&D cooperation has received much attention in the theoretical industrial organization literature. Most models consider oligopolistic firms which may cooperate in R&D with rivals (i.e., horizontal cooperation), and which focus on costreducing R&D (d'Aspremont and Jacquemin 1988[START_REF] Kamien | Research joint ventures and R&D cartels[END_REF], De Bondt 1997, among others). Although these models may have different setups, they all find that firms earn higher profits when they cooperate in R&D than when they do not. Moreover, the profitability of R&D cooperation increases with the level of R&D spillovers when they are sufficiently high. Another stream of the literature focuses on quality-improving R&D in models with vertical product differentiation [START_REF] Motta | Cooperative R&D and vertical product differentiation[END_REF][START_REF] Rosenkranz | Innovation and cooperation under vertical product differentiation[END_REF]. Although these papers differ from the cost-reducing R&D models in some of their specifications, they also find that R&D cooperation increases firms' profits.

While most analysis assume that R&D spillovers are exogenous, more recent theoretical research allows firms to choose spillovers as a strategic variable [START_REF] Amir | Noncooperative versus cooperative R&D with endogenous spillover rates[END_REF]. The main message of these analysis is that firms which cooperate horizontally in R&D generally find it profitable to maximize intra-industry spillovers. A few papers propose a dynamic model of firms' choice to cooperate in R&D [START_REF] Kesteloot | Stable R&D cooperation with spillovers[END_REF]. When incoming spillovers are distinguished from outgoing ones, it is found that only the latter may result in higher incentives to free ride on R&D partners while incoming spillovers increase the attractiveness of cooperation. This leads to the theoretical prediction that horizontal R&D cooperation is more likely to be observed the higher the spillovers coming from rivals. Surprisingly, these well-established theoretical results have received very little empirical support.

Only a few papers investigate the empirical link between R&D spillovers and R&D cooperation. Those which do so reach mixed results. As most existing analysis use relatively small data sets, they cannot distinguish between cooperative agreements exclusively among competitors in the same industry from other possible types of partners (suppliers, customers, universities, independent labs). [START_REF] Kaiser | An empirical test of models explaining R&D expenditures and R&D cooperation[END_REF] considers three categories of R&D agreements: horizontal, vertical (with customers and /or suppliers), and mixed, i.e. those which are horizontal and vertical. He finds that horizontal spillovers have a weakly significantly positive effect on firms' propensity to cooperate while they do not significantly affect the choice of vertical or mixed cooperation. However, as [START_REF] Kaiser | An empirical test of models explaining R&D expenditures and R&D cooperation[END_REF] analysis concerns only the German service sector it may be difficult to extrapolate the results to the whole population of firms. Moreover, as only few firms (that is, 36 firms) cooperate horizontally, they are included in the mixed agreements category (Kaiser 2002, p. 760). [START_REF] Cassiman | R&D cooperation and spillovers: Some empirical evidence from Belgium[END_REF] use Flemish data to find that firms which benefit the most from public information sources are more likely to cooperate in R&D. However, they cannot investigate horizontal cooperation separately because only 33 firms in their survey cooperated in R&D with competitors [START_REF] Cassiman | R&D cooperation and spillovers: Some empirical evidence from Belgium[END_REF]Veugelers 2002, footnote 10, page 1172).

New and sharper results appear in a recent paper by [START_REF] Belderbos | Heterogeneity in R&D cooperation strategies[END_REF], who construct a multivariate probit model to explore differences in the determinants of innovating firms' decisions to participate in three distinct types of cooperation (with competitors, with suppliers or customers, with universities and research institutes).1 With a large Dutch dataset (627 firms with R&D cooperation of some type), it is found that incoming spillovers, which specifically originate from competitors, are statistically significant (at the 10% level) in explaining R&D cooperation with firms from the same industry.

We contribute to this emerging empirical literature by using French data from the 2002 Community Innovation Survey (CIS), which is conducted in member countries of the European Union, to investigate the link between horizontal R&D cooperation and R&D spillovers among competitors. The objective of this note is twofold. First, we test the robustness of existing empirical results to the use of a larger and more recent dataset. Second, we exploit the link between a non-binary detailed measure of information firms receive from competitors and their incentive to engage in horizontal R&D agreements to obtain new insights.

Our first main finding is that firms are more likely to cooperate horizontally the more they benefit from competitors in the industry as their main source of information to innovate. This confirms the findings of [START_REF] Belderbos | Heterogeneity in R&D cooperation strategies[END_REF]. In addition, we establish two completely new results. First, we find that incoming spillovers have a significant positive impact on the likelihood that a firm cooperates horizontally in R&D only when these spillovers exceed a threshold. Second, the estimates of the impact of spillovers on the decision to cooperate, together with their precisions, are increasing in the value of the measure of information flow.

The remainder of the paper is organized as follows. Section 2 describes the empirical model and data. We then discuss the results in section 3 and conclude in section 4. Tables 1 and2 are in the appendix.

Model and data

Denote by m the R&D organizational mode, with m = 0 if a firm does not cooperate horizontally in R&D, and m = 1 if she does. Let π m (s, x) denote a firm's profits as a function of technological spillovers s and a vector of characteristics x. The benefits from cooperation are given by the difference:

∆(s, x) = π 1 (s, x) -π 0 (s, x) + ε, = sγ + xβ + ε, ( 1
)
where ε is a mean-zero normally distributed error term which captures the impact of unobservables on the benefits of cooperation, γ is a parameter and β is a vector of parameters. The second line of ( 1) is a linear approximation of the difference in profits when a firm cooperates versus does not cooperate.

∆(s, x) is not observable but a firm chooses:

m =          1 if ε ≥ -(sγ + xβ) 0 if ε < -(sγ + xβ) (2)
Equation ( 2) is a simple probit model. The null hypothesis of interest is γ = 0 which can be tested by using the 2002 CIS data for France.

The survey includes a representative sample of 7,016 establishments in France. Firms reported whether they had any cooperation arrangements on innovation activities between 1998 and 2000. There are 3,830 firms for which we know whether they cooperated or not.2 This constitutes a larger dataset than the ones used in existing studies on the same topic. We distinguish between cooperative arrangements with any type of partner (1,511 firms) and cooperative arrangements only with competitors and other firms from the same industry (376 firms). This distinction allows us to focus on horizontal cooperation, which is the subject of the theoretical literature of interest. While rivals choose to cooperate in order to maximize their individual profits, this is not necessarily the case for establishments within the same group, where profits are consolidated, or for firm-university agreements.

The relevant summary statistics, differentiated by whether firms do not cooperate, cooperate with any type of partner, or cooperate horizontally, are given in Table 1. Firms which cooperate are on average much bigger, and export more, than those which do not. The share of firms which innovated by cooperating with other firms in the past is highest for those which report cooperating horizontally between 1998 and 2000. There is no significant difference in the percentage of multinationals which do not cooperate or those which choose to cooperate with any type of partner or horizontally. Firms which cooperate horizontally have the highest R&D intensity.

As suggested by [START_REF] Cassiman | R&D cooperation and spillovers: Some empirical evidence from Belgium[END_REF]Veugelers (2002, p. 1169) and [START_REF] Belderbos | Heterogeneity in R&D cooperation strategies[END_REF]Belderbos et al. ( , pp. 1245-6)-6), we distinguish between horizontal firm-level and industry-level incoming spillovers. As in the latter papers, we can proxy R&D spillovers among competitors by using a specific feature of the CIS questionnaire.

Firms were asked to rank from 0 (unimportant) to 3 (highest) if "information from competitors and other enterprises from the same industry" is a source for innovation. 3 We use that variable as a direct measure of firm-level incoming spillovers, and average the scores at the NACE 2-digit sector level for industry-level incoming spillovers. Given our focus on horizontal cooperation, these measures allow us to better capture technological spillovers as commonly understood in the theoretical literature.

We need to control for other factors which potentially incite firms to cooperate, so as to isolate the effect of horizontal incoming spillovers. We draw from the literature mentioned above to identify the explanatory variables, and list them in the first column of Table 2. In particular, we control for firm participation in cooperative agreements with other types of partners. However, we acknowledge that we cannot fully account for firm-specific unobserved characteristics. Doing so would require panel data as in [START_REF] Belderbos | Heterogeneity in R&D cooperation strategies[END_REF]. Unfortunately, given that the questionnaires used in the successive waves of the Community Innovation Survey for France differ significantly, such an option is not possible. 4

Harmonizing future surveys should help address this issue and confirm the extent to which our results are robust. The next section discusses our estimates.

3 Information from competitors and other enterprises from the same industry as a source of innovation does not appear in the Belgian 1993 survey used by [START_REF] Cassiman | R&D cooperation and spillovers: Some empirical evidence from Belgium[END_REF]Veugelers (2002, p. 1171) who refer to three sources of information: (i) patents; (ii) specialist conferences, meetings and publications; (iii) trade shows and seminars. 4 The question on sources of information used in various types of R&D cooperation lists 12 possibilities in the 1997 CIS and only 9 possibilities in the 2002 CIS.

Results

The results in Table 2 are for two types of dependent variables: models 1-A and 1-B concern cooperation with all types of partners; models 2-A and 2-B are for cooperation among rival firms only. For each type of dependent variable we consider two specifications for the source of information for innovation "from competitors and other firms from the same industry": (A) the respondent's score, and (B) a dummy variable for each of the three values of that variable. This distinction allows us to capture potential non-linearity, and threshold effect, in how that variable affects the incentives for firms to cooperate horizontally in R&D.

We find that none of the sources of information which firms used to innovate between 1998 and 2000 reduces their likelihood of cooperating in R&D. Only the sources from competitors, from other establishments within the same group, and from universities, are statistically significant in explaining R&D cooperation at large, i.e. when partners are not discriminated. Sources from other establishments within the same group matter for cooperation only when we do not distinguish among types of partners.

In other words, this type of information is only relevant for cooperation among establishments within the same group and is not useful for horizontal cooperation.

When cooperation is with any type of partner, firm-level incoming spillovers have no statistically positive effect on the likelihood that firms cooperate (model 1-A). However, when we restrict R&D cooperation to competitors only, these spillovers do have a statistically positive effect on the likelihood that firms cooperate in R&D (model 2-A). These results are robust to allowing for potential endogenous spillovers at the firm-level, but the size of the coefficient increases significantly. [START_REF] Cassiman | R&D cooperation and spillovers: Some empirical evidence from Belgium[END_REF]Veugelers (2002, p. 1175) also find that accounting for potential endogeneity increases the size of their estimates.

Our main results are obtained at a finer level of analysis. We find that firm-level incoming spillovers are still not statistically significant in model 1-B. Interestingly, in model 2-B these spillovers are not significant when they are low. However, they have a significant and increasingly positive impact on the likelihood that a firm cooperates in R&D when they exceed a threshold. When the reported level of incoming spillovers is low, their impact on the likelihood to cooperate with rivals is only sligtly positive (+0.098) and not significant. When the level is intermediate, the impact is higher (+0.318) and significant (at the 5% level). When respondents report the highest level of incoming spillovers, the impact is even higher (+0.466) and more significant (at the 1% level). These estimates, together with the associated statistical significativity, are consistent with the theoretical prediction that the profitability of R&D cooperation increases with the level of R&D spillovers when they are sufficiently high.

Concerning the other explanatory variables of cooperation in R&D, we find that firms which benefited from central government or European Union public funding are more likely to cooperate than those which received no funding. It is not surprising that larger firms, and those where a larger share of their personnel do R&D (R&D intensity), are more likely to cooperate. These findings suggest that a threshold level of financial and human resources is necessary for firms to cooperate in R&D. Moreover, firms which successfully innovated by cooperating with other firms or institutions are more likely to cooperate in R&D. This means that conditional on a successful experience, cooperation in R&D is persistent. Finally, the multinational dummy is never statistically significant in all specifications.

Conclusion

The theoretical industrial organization literature finds that firms which benefit from high R&D spillovers are more likely to cooperate horizontally in R&D. We use the large dataset produced by the 2002 Community Innovation Survey for France to test this result. As in [START_REF] Belderbos | Heterogeneity in R&D cooperation strategies[END_REF], who use Dutch data, we find that a firm which benefits from higher spillovers from her rivals is more likely to cooperate in R&D with her competitors. A specific feature of our paper is that it exploits a three-level measurement of the magnitude of incoming spillovers as directly reported by firm respondents. We obtain that incoming spillovers have a positive and significant impact on the likelihood that a firm cooperates horizontally in R&D only when they exceed a threshold. Moreover, we find that both the probability of inter-firm cooperation and the degree of significativeness of estimations increase with the level of a discrete scaling of received information flows from competitors. This offers new support to well-established theoretical predictions. Industry-level spillovers (4) 1.5 1.6 1.6 Industry-level average log of employees 5.0 5.3 5.3

Notes

(1) A score of 0 to 3 is attributed by firms to each one of the 9 sources of information for innovation.

(2) R&D intensity is the ratio of number of employees doing R&D to the total number of employees.

(4) Industry-level spillovers is the average at 2-digit NACE of information obtained from competitors and other enterprises.

(3) Percentage of firms with headquarters outside France. The first column means that 46% of all firms which do not cooperate are multinationals.

(1-A) (1-B) (2-A) (2-B)

Importance of sources of information for innovation from

Competitors and other enterprises from the same industry 0.027 0.168*** (0.784) (3.617) Dummy equals 1 if firm sources from competitors equals 1 (1) 0.079 0.098 (0.777) (0.651) Dummy equals 1 if firm sources from competitors equals 2 (1) 0 (1) The reference group is composed of firms for which the source of information from competitiors is not relevant.

(2) See footnote 4 of Table 1.

(3) The industry is defined at the NACE 1-digit sector level. The reference industry is NACE 7.

Cooperation with All types of partners

Rivals only (2) The dependent variable equals 1 if a firm cooperates with her rivals, and 0 otherwise. Absolute value of t-statistics are in parentheses. * significant at 10%; ** significant at 5%; *** significant at 1%.

(1) The dependent variable is the score 0,1,2 or 3 which a firm assigns to the importance of sources of information for innovation from competitors and the model is estimated by OLS as in [START_REF] Cassiman | R&D cooperation and spillovers: Some empirical evidence from Belgium[END_REF].

Table 1 :

 1

	Summary statistics
	2002 Community Innovation Survey (CIS) --France
		No cooperation All cooperation Agreements between
		agreements	agreements	competitors only
	Cooperation
	Percentage of firms	61	39	10
	Importance of sources of information for innovation from (1)
	Competitors and other enterprises from the same industry	1.4	1.7	2.0
	Other establishments within the group	0.9	1.4	1.4
	Suppliers of equipment, materials, components or software	1.4	1.6	1.7
	Within the establishment	2.2	2.5	2.6
	Universities	0.4	0.9	1.1
	Government or private non-profit research institute	0.4	0.8	1.0
	Professional conferences, meetings and journals	0.9	1.1	1.3
	Fairs and exhibitions	1.1	1.2	1.3
	Clients or customers	1.9	2.1	2.3
	Percentage which successfully innovated by cooperating with other firms	19	33	39
	Size and R&D personnel
	Average turnover in 1998 (in euros)	71 261	363 036	796 225
	Average exports in 1998 (in euros)	16 947	152 795	349 510
	Average number of employees in 1998	293	1 051	2 156
	R&D intensity in percentage (2)	4.4	6.5	8.2
	Organizational feature
	Percentage of multinationals (3)	46	48	47
	Percentage of establishments which is part of a group	74	87	85
	Percentage

which obtained public funding for innovation from

	Local or regional authorities	6	13	17
	The central government	14	33	39
	The European Union	3	16	26
	Industry-level effects

Table 2 : Probit Estimates of the Determinants of R&D Cooperation

 2 2002 Community Innovation Survey (CIS) --France Dependent variable equals 1 if a firm cooperates in R&D Model

Table for the referees only: Accounting for endogenous firm-level spillovers

	OLS First stage	Probit Second stage
	Dependent variable Firm-level spillovers (1)	Model (2-A BIS) (1)
	Importance of sources of information for innovation from
	Fitted values of Competitors and other enterprises from the same industry		1.920***
			(3.925)
	Other establishments within the group		0.011
			(0.296)
	Suppliers of equipment, materials, components or software		0.029
			(0.696)
	Within the establishment		0.066
			(1.208)
	Universities		0.200***
			(3.191)
	Government or private non-profit research institute		0.068
			(1.083)
	Professional conferences, meetings and journals		0.054
			(0.849)
	Fairs and exhibitions		-0.04
			(0.672)
	Clients or customers		0.093**
			(2.230)
	1998 turnover	0.089***
		(2.699)
	1998 turnover square (divided by 1,000,000)	-2.01*
		(1.723)
	Share of new or improved products to market	0.257*
		(1.901)
	Size and R&D personnel
	Log of 1998 turnover		0.098**
			(2.337)
	Log of 1998 exports		-0.028
			(1.063)
	R&D intensity		0.028***
			(3.158)
	R&D intensity squared (divided by 100)		-0.000***
			(2.644)
	Organizational feature
	Multinational dummy		-0.141
			(1.641)
	Establishment is part of a group	0.282***	-0.449**
		(6.103)	(2.323)
	Firm successfully innovated by cooperating with other firms		0.207**
			(2.488)
	Obtained public funding for innovation from
	Local or regional authorities		0.055
			(0.424)
	The central government		0.229**
			(2.416)
	The European Union		0.250**
			(2.119)
	Industry-level effects
	Industry-level spillovers (2)		0.468
			(1.497)
	Industry-level average log of employees		0.182
			(1.435)
	Industry dummies (3)
	NACE 1	0.297***	-0.845***
		(4.129)	(3.429)
	NACE 2	0.368***	-1.282***
		(5.754)	(4.474)
	NACE 3	0.541***	-1.795***
		(7.694)	(4.635)
	NACE 4	0.409*	-1.798***
		(1.729)	(2.661)
	NACE 5	-0.017	-0.045
		(0.167)	(0.161)
	NACE 6	0.584***	-1.920***
		(5.960)	(4.233)
	Constant	0.938***	-6.011***
		(14.605)	(8.405)
	Number of observations	3 315	2 026
	Notes

We refer the reader to[START_REF] Belderbos | Heterogeneity in R&D cooperation strategies[END_REF] for a thorough survey of the theoretical and empirical literatures on horizontal R&D cooperation.

The survey lists 9 possible partners: Other enterprises within your enterprise group; Suppliers of equipment, materials, components or software; Clients or customers; Competitors and other firms from the same industry; Consultants; Commercial laboratories/R&D enterprises; Universities; and Government or private non-profit research institutes (see section 8.2 of the third CIS harmonized questionnaire).


Robert Gagné

Testing Optimal Punishment Mechanisms under Price Regulation: the Case of the Retail Market for Gasoline By

Keywords: price regulation, oligopoly supergame, Markov switching model, gasoline prices. JEL codes: L13, L81, C32

We analyse the effects of a price floor on price wars (or deep price cuts) in the retail market for gasoline. Bertrand supergame oligopoly models predict that price wars should last longer in the presence of price floors. In 1996, the introduction of a price floor in the Quebec retail market for gasoline serves as a natural experiment with which to test this prediction. We use a Markov Switching Model with two latent states to simultaneously identify the periods of price-collusion/price-war and estimate the parameters characterizing each state. Results support the prediction that price floors reduce the intensity of price wars but increase their expected duration.

Introduction

The behaviour of retail gasoline prices has long been and still is the object of fierce public debate. Given its importance for the consumers and the apparently "suspect" behaviour of the integrated oil companies (the "majors"), many jurisdictions have regulated aspects of gasoline retailing. For example, refiners are forbidden to operate retail outlets in some U.S. states. Different types of price regulations are also enforced in several U.S. states and Canadian provinces. In this paper, we provide evidence of the effects of price floor regulation on the behaviour of retail gasoline prices in Montreal, the largest market in the province of Quebec.

The May 4, 1992 edition of Bloomberg's Oil Buyers Guide reported the conclusions of market studies indicating that "major Canadian oil companies were going to use price wars, new credit terms, and the strategic closure of service stations and refineries to squeeze independent gasoline retailers out of the market in central Canada". 1 A few years later, following a severe price war in the summer of 1996, the Quebec provincial government responded to the lobbying of independent gasoline retailers by establishing a price floor in December 1996. The regulation was motivated by the claim that price wars formed as a discipline device for the implementation of anticompetitive strategies. This was made very clear in a report by the local association of independent gasoline retailers: "The summer 1996 episode was very harmful for suppliers in Quebec. The price war, triggered by an integrated major, resulted in retail prices that were observed well below wholesale prices. It was so severe as to force several independent retailers either to close down temporarily or to exit the market" (pp. 7-8). 2 By limiting the severity of price wars, the floor was thus rationalized as a mean to reduce the ability of firms to punish retailers deviating from a high price strategy. When the latter retailers were small independent suppliers, characterized by less favourable financial conditions than majors, the floor was viewed as a form of protection that would help to maintain competition in the market. The floor was introduced to limit the severity of price wars which, according to the independent retailers, were evidence of predatory behaviour by the integrated majors. The price floor was therefore viewed as a form of market protection for independent retailers which would thereby help to maintain competition in the market.

The floor is computed weekly and regionally as the sum of the wholesale (rack) price, transportation costs and taxes, and is the only type of economic regulation specific to Quebec's retail market for gasoline. The price floor computed in week t is the one which applies in the market during week t+1.

Our objective in this paper is to determine whether the effects of a price floor on retail gasoline markets is consistent with the predictions of a class of models which assume that there is no explicit collusion between firms. In a theoretical section, we review two types of such models which establish that a punishment price below a retailer's marginal cost makes economic sense, and that such severe price wars may be observed in the absence of regulation. A first group of models [START_REF] Lambson | Optimal Penal Codes in Price-setting Supergames with Capacity Constraints[END_REF][START_REF] Lambson | Some results on Optimal Penal Codes in Asymmetric Bertrand Supergames[END_REF][START_REF] Häckner | Optimal Symmetric Punishments in a Bertrand Differentiated Products Duopoly[END_REF] relies on Bertrand oligopoly supergames in which, during collusive periods, firms price between the Nash equilibrium value of the stage game and the joint profit-maximizing price. If a firm deviates from the collusive strategy, all firms in the next period price below the oneshot symmetric Nash equilibrium price (a punishment), before returning to the collusive price to earn above normal profits. When deviation payoffs are relatively high, a price below marginal cost may be required in the punishment phase for tacit collusion to be sustainable. In this framework, by imposing a boundary on the set of available retail prices, a price floor restricts the severity of price wars. It can be viewed as a monkey wrench in the collusive works that contributes to making coordinated pricing strategies less stable. Theory predicts that a longer punishment period can compensate for a higher punishment price. This means that, when a price floor is introduced, the same collusive price can be sustained if the length of punishments is also a control variable of firms.

However, the coordination mechanisms of these models lack realism in the sense that they make collusion sustainable with no punishment at equilibrium. But price wars do occur on real-world markets. In a second category of papers we consider (Porter 1983a[START_REF] Green | Noncooperative Collusion under Imperfect Price Information[END_REF][START_REF] Slade | Price Wars in Price-setting Supergames[END_REF][START_REF] Slade | Vancouver's Gasoline-Price Wars: An Empirical Exercise in Uncovering Supergame Strategies[END_REF], unobserved random shocks in demand may induce price wars to appear in equilibrium. Taken together, these models lead to a simple prediction: the duration of price wars (spotted out as deep, irregular, unpredictable price cuts) should increase when a price floor is introduced. We believe the introduction of a price floor on retail gasoline markets in Quebec provides a natural experiment in which this prediction can be tested.

In this study, we do not attempt to integrate the many other models of price formation in gasoline markets. Their relevance to the present analysis of the impact of a price floor regulation is somewhat limited, as they do not allow for the possibility of pricing below marginal costs (not even out of equilibrium), nor rationalize the occurrence of price wars (in equilibrium). A prominent example of such a model is that of [START_REF] Rotemberg | A Supergame-Theoretic Model of price Wars during Booms[END_REF], who focus on the impact of observable independently and identically distributed changes in demand on the level of tacitly collusive prices. In a price-setting oligopoly supergame with infinite horizon and undifferentiated products, a deviation from the collusive path triggers a punishment mechanism, i.e. competitive pricing, in all subsequent periods. 3 The basic argument is that the incentives to deviate from collusion increase when demand is relatively high, since by undercutting its rivals a firm may capture a larger market share. Firms can sustain collusive profits in that case by lowering the collusive price and thereby making deviations less profitable. In equilibrium, we therefore only observe a "tuned" collusive price, which gets closer to the marginal cost during periods of high demand; it is not a punishment price. [START_REF] Haltiwanger | The Impact of Cyclical Demand Movements on Collusive Behavior[END_REF] extend this model, assuming that demand moves cyclically in some deterministic way, and that firms' expectations of future demand change over time. Although their analysis confirms that the profitability of deviation from collusion is large when demand is high, periods of decline in demand are the most difficult times for firms to sustain collusion. A direct implication is that prices will be closest to the competitive level -and thus questionably interpreted as a price war -in early periods of recession, that is when demand is not only high but also falling. Again, these prices are charged above marginal costs -a feature that does not fit our empirical data in the episode that motivated the introduction of a price-floor -and remain of a tacitly collusive nature, in all equilibria. 4,5 The model is generalized by [START_REF] Bagwell | Collusion over the Business Cycle[END_REF], who assume that turning points are unpredictable. However, as noticed by the authors (see footnote 4, p. 82), the relevance of this assumption is questionable for the analysis of markets subject to seasonal demand fluctuations such as gasoline markets.

Finally, recent studies by [START_REF] Eckert | Retail Price Cycles and Response Asymmetry[END_REF] and [START_REF] Noel | Edgeworth Price Cycles, Cost-based pricing and Sticky Pricing in Retail Gasoline Markets[END_REF] also use price data for Canadian cities in order to explain features of retail gasoline prices. These two papers use the same alternating-move price-setting duopoly model, adapted from [START_REF] Maskin | A Theory of Dynamic Oligopoly II: price Competition, Kinked Demand Curves and Edgeworth Cycles[END_REF], in which a class of equilibria can be constructed that consists of price cycles. In these cycles, firms repeatedly undercut one another in the downward portion of the cycle.

When the price reaches the competitive value, a war of attrition begins, until one firm raises its price back to the top of the cycle (i.e. above the monopoly price) and then the cycle is repeated. [START_REF] Eckert | Retail Price Cycles and Response Asymmetry[END_REF] formulates an econometric version of the model of price cycles, and estimates it with daily data that describe prices in the city of Windsor (Ontario) from 1989 to 1994. He finds results consistent with the theoretical predictions that price reductions result when firms undercut rivals, and that the size of price restorations depends on the wholesale price. [START_REF] Noel | Edgeworth Price Cycles, Cost-based pricing and Sticky Pricing in Retail Gasoline Markets[END_REF] develops an empirical framework to separate out different pricing phenomena, and applies it to the analysis of dynamic pricing behaviour in nineteen Canadian retail gasoline markets from 1989 to 1999. He finds that the most frequent pattern is one in which prices cycle rapidly, beginning with a large increase of short duration, before declining gradually over a longer time period. The evolution of prices in Montreal, in the period that precedes the 1996 price war, appears less easily classifiable in a given category of pricing phenomena than the patterns observed in other Canadian marketplaces. 6 In both papers, price increases are initiated when the retail price is close (but not lower than) the wholesale price, as suggested by the underlying theory. 7 By contrast, our reference to models of collusion, in which short price wars alternate with longer collusive episodes, is justified by the specific feature of the data we use, that is the occurrence of several and relatively large negative margins, in early 1995 and mid-1996. 8 To test for collusive pricing and estimate the length of price wars, we use a Markov Switching Regression framework with two latent states to simultaneously identify the periods of price-collusion/price-war and estimate the parameters characterizing each state. Consistent with the theory, we allow regulation to influence both the stateconditional prices and the expected duration of each state. The switching regression is then estimated on weekly data for retail gasoline price margins in Montreal from 1994 to 2001. In summary, we find that the introduction of a price floor significantly increases margins during price wars, and also increases the persistence of those price wars. The net effect on average margins is near zero, as the increase in margins during price wars is offset by the rise in the average duration of price wars.

Section 2, below, motivates the specific prediction tested in this paper, by exploiting some insights from theoretical contributions on the possible impact of a price floor on firms' ability to sustain collusion. The empirical model and data are described in Section 3. The results are discussed in Section 4, and Section 5 presents concluding remarks.

Severe Price Wars and Binding Price Floors

It is well known that a price-setting oligopolist may find it profitable to undercut the price that maximizes industry profits. This may not be the case if deviations from the collusive price are credibly "punished" via lower industry prices for a period of time. When the punishment is sufficiently severe, the immediate gains to deviation are smaller than what is lost in the punishment phase. In a legal context which prohibits price cartels, price wars can thus be used as an enforcement device to maintain a tacit cooperative agreement.

Accordingly, by imposing a constraint on firms' ability to dissuade deviations from a tacitly cooperative arrangement on the retail gasoline markets in Quebec, in 1996, the provincial government aimed at limiting the severity of price wars and firms' ability to sustain anti-competitive prices. Since then, the floor has been computed weekly and regionally as the sum of the rack price, transportation costs, and taxes. (We refer to it hereafter as retailer's marginal cost.) However, for this mechanism to be effective, it must be the case that a punishment price below a retailer's marginal cost makes economic sense, and that such severe price wars are observed in the absence of a price floor.

The literature on optimal penal codes in supergame oligopoly models provides a rationale for punishment prices which are below marginal cost. The specifications of these models, in their Bertrand version and with complete information, reflect the main properties of gasoline markets: sales are repeated, retailers use price as a strategic variable, prices are observable by all, rival sales can be monitored, price adjustment can be small and occur at low cost. In particular, [START_REF] Lambson | Optimal Penal Codes in Price-setting Supergames with Capacity Constraints[END_REF][START_REF] Lambson | Some results on Optimal Penal Codes in Asymmetric Bertrand Supergames[END_REF] characterizes optimal punishments for a large class of infinitely-repeated games, with price-setting sellers of a homogenous good and a constant marginal (and average) cost c. 9 Häckner (1996) builds on this by demonstrating that a symmetric two-phase "stick-and-carrot" structure is an optimal price path in a specific supergame duopoly model with differentiated products. 10 This means that, in a collusive period, all firms sell at a price that yields the highest sustainable level of profits. 11 This price, say p*, is between the one-shot Nash equilibrium value p N and the joint profit-maximizing price p m . When firms detect a deviation from the collusive strategy p*, they both switch to a lower price p in a one-period severe punishment phase (the "stick"), where p is strictly less than p N , before returning to p* to earn collusive profits again (the "carrot"). 12 The higher the deviation payoffs, the lower the optimal punishment price must be for collusion to remain sustainable. This price may therefore need to fall below marginal cost c, in which case the price floor constraint p ≥ c would be binding. 13 When this occurs, one-period punishments are not sufficiently large to outweigh the gain from undercutting the collusive price. However, Häckner shows that collusion sustainability can be maintained by a τ-period price war (τ > 1), with a punishment price equal to c in the first τ -1 periods, and to p' in the τ-th period, with c ≤ p'< p*. Then firms return to p* in the next period onward. By constraining the punishment price to remain above the marginal cost, it is possible that a prolonged price war can sustain the same collusive price as in the unrestricted case. 14 Note that the relatively large number of geographically close outlets that are available to retail gasoline customers in an urban area contributes to a high level of product substitutability among consumers. In this theoretical framework, such substitutability implies larger market share gains (and related profits) for those who deviate from collusive pricing. Large incentives for deviations from collusive pricing in turn can call for a severe (below marginal cost) punishment price. This may motivate competition authorities to establish a price floor, as occurred in the province of Quebec.

As they can account for punishment prices below marginal costs, the above models shed some valuable light on the possible impact of a price floor on observed prices. However, these models lack realism in that they never produce price wars in equilibrium. This outcome is rooted in the simplifying assumption of complete information. On the grounds that price wars are frequently observed in real-world markets, Porter (1983a) and [START_REF] Green | Noncooperative Collusion under Imperfect Price Information[END_REF] challenge the relevance of such models. They construct a dynamic model in which firms observe only their own production levels and a common market price. As the market demand has an unobservable stochastic component, firms face an inference problem. An unexpectedly low price may signal either a deviation from the collusive path by rival firms or a demand-reducing shock. In this set-up, firms produce a collusive total output until price drops below a trigger value, in which case they revert to the one-shot Nash equilibrium for a number of periods -the price war -before returning to the collusive strategy. Since this result is obtained in a quantity-setting oligopoly model, it should be invoked cautiously for the rationalization of observed price wars on gasoline markets. 15 For this reason, the proposition that unobserved random shocks in demand may induce price wars in equilibrium has been extended by [START_REF] Slade | Price Wars in Price-setting Supergames[END_REF][START_REF] Slade | Vancouver's Gasoline-Price Wars: An Empirical Exercise in Uncovering Supergame Strategies[END_REF] to a price-setting supergame model, which conforms more closely to the characteristics of a retail gasoline market. The main idea is that, although all prices are observable and rival sales can be monitored, when a shock has occurred firms do not know the true demand conditions unless they change their price and thereby precipitate a price war. In the model, price is measured above constant marginal costs, and pricing strategies are formulated as intertemporal continuous reaction functions which express a firm's price as a function of the previous-period prices charged by competitors. 16 A Nash equilibrium in prices can be computed for stationary demand. When an unanticipated change in demand occurs, a firm's price is no longer the best response to other firms' prices and a new equilibrium must be found. Each firm then uses the prices charged by competitors as signals to update its expectations about new demand conditions in a Bayesian fashion.

Price wars are then rationalized as a learning process for parameter identification. 17 By introducing an element of uncertainty, models with price wars in equilibrium gain in realism compared to studies with complete information. However, this comes at a cost, since the mentioned price-setting models in which price wars occur rule out pricing below marginal cost by construction. 18 In the empirical work that follows, we draw on the lessons obtained from the two categories of models discussed above to investigate the effects of price floor regulation on the behaviour of retail gasoline prices in Montreal. In a nutshell, if punishment prices below marginal costs can be rationally justified, and if price wars can occur in the absence of complete information, we expect that the introduction of a price floor should lengthen the duration of price wars. This is the empirical prediction we test in what follows.

Econometric Implementation

Data and Variables

Retail price data (P t ) have been provided by M.J. Ervin Inc., a Calgary-based firm which conducts a weekly survey on gasoline retail prices in all major Canadian markets. In each market surveyed, retail prices are collected by gasoline grade using a sample of selfservice gas stations. Whenever possible, the same stations are surveyed each week. In the Montreal market, the survey covers approximately 20 stations. For our analysis, we use the average retail price for unleaded regular gasoline computed from all stations in the Montreal survey. The data cover the 1994-2001 period (416 weekly observations). Our analysis is limited to the price of unleaded regular gasoline as retail prices for all other grades follow closely unleaded regular gasoline prices.

Wholesale prices (W t ) have also been provided by M.J. Ervin. Those prices are "rack" prices (excluding taxes) posted everyday at wholesale distribution outlets. As for retail prices, we used the average (unweighted) weekly wholesale price computed from posted prices. No transportation costs are considered given the proximity of the retail market to the different wholesale distribution outlets.

In our empirical analysis, we use retail margins (M t ) rather than prices in order to eliminate effects coming from the wholesale market and thereby concentrate on retail market effects. Retail margins are computed as retail prices in week t minus wholesale prices observed during the same week. Recall that the price floor for week t is computed as the sum of the wholesale price observed during week t-1, corresponding taxes and transportation costs.

Finally, our empirical analysis includes a regulation dummy (R t ) equals to 0 until the price floor regulation was introduced during the last week of December 1996, and equal to 1 thereafter.

Figure 1 shows the evolution of retail margins over time. The pattern of fluctuations of the retail margins seems similar before and after the introduction of the price floor regulation; of course, large negative margins are no longer observed (as in early 1995 and mid-1996). (Small negative margins are still observed under regulation because the price floor is computed from the wholesale price observed the week before.) Deep price cuts can be observed in addition to variations of more limited magnitude, or cycles. These cuts are irregular, and seem unpredictable. At first glance, periods in which margins are either negative or very close to zero never last more than a few weeks. However, given the weekly nature of the data, each point of the graph may represent a constant price for up to seven days, implying that price wars may take place over tens of days. To see that, focus for example on the circumstances that motivated the introduction of a price floor, in mid-1996 about week 26 (when the margin is -5.875, the lowest value of our data). If a price war refers to circumstances in which the margin falls rapidly down to a level measured below a given threshold, say 1¢/litre, before returning to higher positive values, we

Econometric Model

To test the prediction from Section 2, we need to allow for the structural relationships determining margins to vary depending on whether the industry is in a period of collusion or price war (competition). We further need to investigate how the duration of these different states may be influenced by changes in the regulatory environment. Estimation and inference is complicated by the fact that, while regulatory changes are directly observed, the presence or absence of price wars must be inferred indirectly. 20 Of course, one could simply construct a binary variable to indicate which observations appear to correspond to price wars, then use traditional methods (e.g. Ordinary Least Squares) separately on the two distinct subsets of observations (for example, see [START_REF] Borenstein | Selling Costs and Switching Costs: Explaining Retail Gasoline Margins[END_REF][START_REF] Borenstein | Dynamic Pricing in Retail Gasoline Markets[END_REF]). While intuitive, this approach has serious problems. First, since the separation into price war/collusion is somewhat uncertain, some observations will be misclassified. That means this approach will produce biased and inconsistent estimates of the underlying relationships. Second, standard errors for the resulting regressions will ignore the contribution of uncertainty about the sample separation, making inference unreliable as well. The approach we adopt avoids both of these problems. Porter (1983) and [START_REF] Lee | Switching Regression Models With Imperfect Sample Separation Information -With An Application on Cartel Stability[END_REF] address the same kind of problem using regime switching techniques, which estimate the structural parameters of each pricing regime together with probability that each observation may have been produced by a price war.

We use an extension of their approach, inspired by Hamilton (1993)'s Markov Switching Models with time-varying transition probabilities. 21 As our baseline model, we estimate the following system of equations by maximum likelihood

t t i t i t i i t W R M M ε γ β ρ α + + + + = -1 (1) ) ( ) Pr( 1 t i t i i t t W R i S i S ω θ ϕ + + Λ = = = - (2)
where M t , W t and R t are defined as earlier, i = 1 for price wars and 0 otherwise, t S is the market state (1 for price wars and 0 otherwise) at time t, (.) Λ is the logit cumulative distribution function, t ε is a an i.i.d. mean-zero normally-distributed error term with a standard deviation of ε σ , and } , , , ,

0 1 0 1 0 1 0 1 0 1 0 1 0 ε σ ω ω θ θ ϕ ϕ γ γ β β ρ ρ α α , , , , , , , , , , { 1
is the vector of unknown parameters to be estimated by maximum likelihood. We performed the usual diagnostics tests suggested by [START_REF] Hamilton | Specification Testing in Markov-Switching Time-series Models[END_REF] for the fit of such models.

Since the two states in this model follow a first-order Markov chain, we can calculate the half-life of a regime (the length of time over which the probability of remaining in the same regime has fallen to 50%) as ln 0.5/ln a, where a is the probability given by equation ( 2). Similarly, we can calculate the expected duration of the regime (in periods) as 1/(1a). Furthermore, if {a 0 ,a 1 } are the probabilities of remaining in regimes 0 and 1 for one more period, then on average the market will spend fraction (1 -a 1 )/(2a 0a 1 ) of the time in regime 0 (price collusion) and the remainder in regime 1 (price war).

The sign and significance of i β determine whether the introduction of a price floor has an impact on margins either during price wars (i = 1) or collusion (i = 0). Due to the dynamic nature of the model, the long-run impact of the regulation on margins in each regime will be ) 1 /(

i i ρ β -
. The parameter i θ tells us whether the price floor raises the probability of being in the same regime the following period. To determine the average margin before and after the introduction of a price floor, we therefore need to take account of the floor's effect on average margins in each state (e.g. making price wars less intense) as well as on the average fraction of the time the market will spend in that regime (e.g. price wars last longer.) This average will be given by

) ( ) ( 2 ) ( 1 1 ) ( ) ( 2 ) ( 1 1 1 1 0 0 0 0 1 1 1 1 1 0 0 1 1 0 0 0 W W W W W W W W ω ϕ ω ϕ ω ϕ ρ γ α ω ϕ ω ϕ ω ϕ ρ γ α + Λ - + Λ - + Λ - ⋅ - + + + Λ - + Λ - + Λ - ⋅ - + (3)
before introduction of the price floor and by

+ + + Λ - + + Λ - + + Λ - ⋅ - + + ) ( ) ( 2 ) ( 1 1 1 1 1 0 0 0 1 1 1 0 0 0 0 W W W W ω θ ϕ ω θ ϕ ω θ ϕ ρ γ β α ) ( ) ( 2 ) ( 1 1 1 1 1 0 0 0 1 1 1 1 1 1 1 W W W W ω θ ϕ ω θ ϕ ω θ ϕ ρ γ β α + + Λ - + + Λ - + + Λ - ⋅ - + + (4)
after, where W is the average wholesale price. 22

Empirical Results

The results of the estimation of equations ( 1) and ( 2) by maximum likelihood are presented in Table 3. The first two columns present estimates under the restriction that 0 1 0

= = ω ω

; the last two columns present estimates of the unrestricted model. We test the fit of the above models using the score-based tests proposed by [START_REF] Hamilton | Specification Testing in Markov-Switching Time-series Models[END_REF].

Results are shown in Table 4. These tests have power against unmodeled serial correlation and heteroscedasticity in equation (1). They also test for omitted higher-order Markov dependence in equation (2). Since a higher-order Markov chain may be rewritten as a first-order chain with a larger number of states, the latter test also has power against omitted states. We test for each effect in each regime and also present a test for the joint null hypothesis of none of these forms of misspecification in any of the model's equations.

[Tables 3 and4 here] The joint misspecification test in Table 4 finds some evidence of misspecification only in the case where 0 1 0 = = ω ω . However, this evidence appears to be entirely confined to higher-order Markov dependence in the collusive regime; the test statistics are more than three times larger than those for any other single test and are the only ones significant at the 5% level. There is no significant evidence of misspecification in the unrestricted model. Although we have provided no theoretical justification for the inclusion of wholesale prices in the regime switching model, we include these results to check whether our conclusions are robust to minor changes in specification, and whether the limited misspecification detected in our restricted model seriously affects the reliability of our statistical inferences. We also find that the effects of wholesale price changes on margins have intuitive interpretations in the context of price wars. 23

The Effects of Regulation on Price Wars

As far as the effects of regulation are concerned, results are similar for both specifications shown in Table 3. Significant effects of regulation are found in the margin equation as well as in the state transition probability equation, but only in the price war regime. The introduction of a price floor significantly increases margins during price wars and significantly increased the persistence of price war periods.

As already mentioned, in both specifications and in both sets of equations (margin and regime selection) considered, the parameters associated with the regulation dummy (R t ) are positive and significant only during price wars. This result is consistent with the theoretical predictions where a price floor regulation has no effect on prices (or margins) during collusive periods since a price close to the monopoly level is assumed to be charged during those periods. Moreover, our results show that during price war regimes, the price floor regulation increases both margins and probabilities of continuing the price war. Table 5 reports estimated regime-dependent conditional probabilities (using equation 2), expected durations (using 1/(1 -q), where q is the regime dependent conditional probability) and margins (using equations 3 and 4), all computed from the estimates presented in Table 3.

[Table 5 here]

During collusive regimes, the transition probabilities are similar before and after regulation and their estimates are quite robust to the inclusion/exclusion of the wholesale price. However, during price war regimes, transition probabilities are significantly higher after regulation again, regardless of the specification considered. For example, in the unrestricted model the transition probabilities increased from 0.34 to 0.90 after regulation during price wars while the same probabilities stayed at around 0.90 during collusive periods.

Regulation therefore significantly increased the expected duration of price war periods, while reducing sharply the expected duration of collusive periods (although not significantly.) Before regulation, the expected duration of price wars was a little less than two weeks, which rises to ten weeks after regulation.

Results on estimated margins are fully consistent with the theory. On one hand, during collusive regimes, estimated margins are about the same magnitudes with and without a price floor regulation. On the other hand, regulation increases significantly the margins during price wars: from approximately -1.5 cents to 3.5 cents with a price floor.

However, and unsurprisingly, we verify that regulation did not raise price war margins to the level of collusive margins. Furthermore, since the price floor raises margins in the low-margin state, but increases the time spent in this state, its net effect on the average margin (i.e. its unconditional expectation) in this market is potentially ambiguous. To answer this question, Table 6 presents estimated unconditional probabilities of the collusive regime (this is 1 -the corresponding probability for the price war regime) as well as the unconditional expected margin. From those figures, it appears that the increase in margins during price wars has been almost exactly offset by the increase in the average duration of a price war, resulting in no significant change in the average margins in the industry. In other words, the price floor regulation had little or no effect on average margins (and therefore prices) even if margins are now higher during price wars, simply because those wars now last longer.

[Table 6 here]

Concluding Remarks

In theory, price wars should last longer under a price floor regulation. This prediction is entirely supported by our empirical findings. The results obtained with a Markov Switching Model using data on the Montreal retail market for gasoline show that the introduction of a price floor regulation reduces the intensity of price wars but raises their expected duration.

Our results support three important conclusions. First, given the coherence between the estimated results and the theoretical predictions, it appears that a model of tacit collusion between firms captures well some important features of the evolution of prices on the retail market for gasoline in Montreal in the period 1994-2001. Second, the government regulation succeeded in reducing the severity of price-wars; we see far less pricing below marginal cost after the introduction of regulation than before. Third, the net effect of the regulation on the competitiveness of the industry is ambiguous. There is no demonstrable change in the average price margin after regulation. This would appear to have benefited gasoline consumers only if one assumes that margins would have risen had regulation not been introduced. ¢/litre 1996-24 1996-28 1996-25 1996-26 1996-27 3 In [START_REF] Rotemberg | A Supergame-Theoretic Model of price Wars during Booms[END_REF], the punishment price is thus exactly equal to the price floor we consider in this paper, and never appears in equilibrium. Note that all models of collusion which adopt [START_REF] Friedman | A Non-cooperative Equilibrium for Supergames[END_REF]'s formulation of "grim" strategies (i.e. punishments of infinite duration) rule out the possibility of punishment prices below marginal costs. 4 An empirical test by [START_REF] Ellison | Theories of Cartel Stability and the Joint Executive Committee[END_REF], in a reexamination of the Joint Executive Committee, gives weak support to Rotember and Saloner (1986)'s theory. [START_REF] Borenstein | Dynamic Pricing in Retail Gasoline Markets[END_REF], by using panel data on retail margins on gasoline markets in a sample of U.S. cities, offer evidence in support of [START_REF] Haltiwanger | The Impact of Cyclical Demand Movements on Collusive Behavior[END_REF]'s theoretical prediction. 5 In this paper, we follow Slade (1992, footnote 4, p. 260) by using the term "price war" to denote a punishment strategy of some kind in which prices are below their collusive level. 6 In a second price pattern, retail prices remain "fixed" for months at a time. In a third, less common, and so-called "normal" pattern, retail prices follow closely wholesale prices. In [START_REF] Noel | Edgeworth Price Cycles, Cost-based pricing and Sticky Pricing in Retail Gasoline Markets[END_REF], retail gasoline prices in Montreal are reported to cycle 66.5% of the time, to be fixed 17.9% of the time, and to display a normal behavior in the remaining 15.6%. These figures (in Interestingly, in a related paper which also examines the Montreal market, [START_REF] Eckert | Retail Price Cycles and the presence of Small Firms[END_REF] observes that "prices are not easily classified as being rigid over the entire period, or as displaying a price cycle" (p. 157), by using weekly data over the period 1990-1995 that precedes the introduction of a price floor. 7 In this theoretical context, although the formal model assumes a wholesale price that remains constant over time, one can intuitively evoke an unexpected cost change in order to explain that retail prices may be observed below marginal costs, for some short time period, on real-world markets for gasoline. See Eckert (2002, p. 64).

8 Following Slade (1989, footnote 3, p. 309), we distinguish oligopoly supergame models of collusion from alternating-move models of price cycles adapted from [START_REF] Maskin | A Theory of Dynamic Oligopoly II: price Competition, Kinked Demand Curves and Edgeworth Cycles[END_REF], on the grounds that stable periods of high prices are absent in the latter model. This distinction is compatible with Eckert (2002)'s remark that cities in which price cycles are observed may be subject to the exercise of less market power than marketplaces in which cycles are not observed. 9 In [START_REF] Lambson | Some results on Optimal Penal Codes in Asymmetric Bertrand Supergames[END_REF], firm-specific capacities are introduced to make firms dissimilar, with no particular reference to retail gasoline markets. [START_REF] Eckert | Retail Price Cycles and the presence of Small Firms[END_REF] and [START_REF] Noel | Edgeworth Price Cycles, Cost-based pricing and Sticky Pricing in Retail Gasoline Markets[END_REF] mention the assumption that gasoline retailers are capacity constrained, but do not formalize it. 10 In [START_REF] Häckner | Optimal Symmetric Punishments in a Bertrand Differentiated Products Duopoly[END_REF], the chosen formulation of the differentiation assumption fits well with the fact that, in retail gasoline markets, product varieties are not only differentiated by brand or service, but also (most importantly) by location. This is emphasized in an empirical analysis by Borenstein and Shepard (1996, p. 430), who argue that retailers compete with local rivals only, and thereby justify the use of oligopolistic settings in urban areas where stations are many. 11 The most severe punishments are defined as "optimal" in the sense that they lead firms to obtain the highest level of sustainable collusive profits. This follows [START_REF] Abreu | Extremal Equilibria of Oligopolistic Supergames[END_REF]. 12 The optimal symmetric punishment must satisfy two conditions. First, the one-period gain from a deviation from the collusive price must be less than (or equal to) what is lost when the punishment price is charged in the next period. Second, the one-period gain from such a deviation must be less than (or equal to) what is lost by prolonging the punishment by one more period. When these two conditions hold with equality, and for a given discount factor δ, they constitute a system of two equations which is solved by the pair (p*, p). In the absence of restriction on δ, all firms would charge the joint profit-maximizing price as a collusive price, that is p* = p m . [START_REF] Häckner | Optimal Symmetric Punishments in a Bertrand Differentiated Products Duopoly[END_REF] demonstrates that there exists a threshold value δ' such that δ < δ' implies p* < p m . 13 In [START_REF] Häckner | Optimal Symmetric Punishments in a Bertrand Differentiated Products Duopoly[END_REF], the constant marginal (and average) cost is normalized to zero (i.e. c = 0) for simplicity, and the case of positive marginal costs is treated as a generalization (see [START_REF] Häckner | Optimal Symmetric Punishments in a Bertrand Differentiated Products Duopoly[END_REF], footnote 13, p. 623).

14 This claim is not limited to the model specifications chosen by [START_REF] Häckner | Optimal Symmetric Punishments in a Bertrand Differentiated Products Duopoly[END_REF]; it also applies (with a change in the formalization of the differentiation assumption) in a recent contribution by [START_REF] Lambertini | Non-negative Quantity Constraints and the Duration of Punishment[END_REF]. Those authors establish that the strictest punishment is optimal because it requires the lowest critical threshold for the discount factor in order to sustain the joint profit-maximizing collusive price.

Using this reinterpretation of the definition of optimal penal codes, they focus on collusion at the monopoly level. In that context, when a price floor constraint that binds is introduced, the unchanged monopolistic price can be sustained (i.e., p* = p m ) by a multiple-period price war. 15 Porter (1983a) and [START_REF] Green | Noncooperative Collusion under Imperfect Price Information[END_REF] do not examine gasoline markets. Porter (1983b[START_REF] Porter | On the Incidence and Duration of Price Wars[END_REF] uses data on the Joint Executive Committee, a railroad cartel organized in 1879 to coordinate prices for transport between Chicago and the East Coast. [START_REF] Porter | On the Incidence and Duration of Price Wars[END_REF] recognizes that railroads chose price, not quantities, and that the distinction has some implication for the econometric analysis, but reckons that "the basic ideas from the theory still apply" (p. 415). This point is also discussed in Ellison (1994, pp. 38-39).

16 [START_REF] Slade | Price Wars in Price-setting Supergames[END_REF] models the process of adjustment to new stationary-equilibrium prices. [START_REF] Slade | Vancouver's Gasoline-Price Wars: An Empirical Exercise in Uncovering Supergame Strategies[END_REF] builds on this using price-war data to estimate different specifications of the inter-temporal reaction function. In [START_REF] Slade | Price Wars in Price-setting Supergames[END_REF][START_REF] Slade | Vancouver's Gasoline-Price Wars: An Empirical Exercise in Uncovering Supergame Strategies[END_REF], continuous strategies are distinguished from discontinuous strategies, such as reversion to the one-shot Nash equilibrium in [START_REF] Green | Noncooperative Collusion under Imperfect Price Information[END_REF], in that large (small) deviations lead to large (small) punishments. This formulation is justified by [START_REF] Slade | Interfirm Rivalry in a Repeated Game: An Empirical Test of Tacit Collusion[END_REF], who produces an empirical test of tacit collusion in the Vancouver retail gasoline market. It is found that continuous strategies provide a better model of the price-war dynamics than discontinuous price-reversions to the one-shot Nash equilibrium. In [START_REF] Slade | Vancouver's Gasoline-Price Wars: An Empirical Exercise in Uncovering Supergame Strategies[END_REF], again with data on retail gasoline activity in a region of Vancouver, reaction functions that are piecewise-linear in the previous-period prices of rivals are given empirical support. Pricing below marginal cost is not discussed. 17 A process of adjustment is constructed in which prices converge to a new stationary Nash equilibrium for a game with new demand conditions. Following [START_REF] Slade | Price Wars in Price-setting Supergames[END_REF][START_REF] Slade | Vancouver's Gasoline-Price Wars: An Empirical Exercise in Uncovering Supergame Strategies[END_REF], here we may keep characterizing firms' reactions as a "punishment", although there is only myopic behaviour and no "crime". 18 Note that several models produce price wars at equilibrium with more severe punishments than Nash reversion. For example, Abreu, Pearce, and Stachetti (1986) modify the model of [START_REF] Green | Noncooperative Collusion under Imperfect Price Information[END_REF] by replacing the continuum of possible output levels with a discrete production set for each firm. Another example is [START_REF] Segerstrom | Demons and Repentance[END_REF], who introduces a small probability that a player will deviate, and derives conditions under which the deviating firm will find it optimal to "repent" by reducing its own output level for a certain number of subsequent periods. However, some of their specifications, including the assumption that firms are quantity-setting players with undifferentiated products, do not fit some features of retail gasoline markets. Interestingly, Slade (1989, p. 304) notes that in her price-setting supergame model of collusion, it is possible to substitute a stick-and-carrot price structure for the continuous intertemporal reaction function she considers. 19 In Figure 2, the stepwise pattern of margins emphasizes the fact that each observation in our dataset may account for a pricing policy that remains constant up to seven days. 20 Slade (1992) addresses this issue using Kalman filtering methods for unobserved component models. However, those are not quite appropriate here since they are designed for unobserved variables that are continuous, not dichotomous. See [START_REF] Hamilton | Time Series Analysis[END_REF] or [START_REF] Kim | State-Space Models with Regime Switching: Classical and Gibbs-Sampling Approaches with Applications[END_REF] for a discussion of the relationship between Kalman filtering approach and the regime switching techniques we use. 21 See also [START_REF] Filardo | Business-Cycle Phases and Their Transitional Dynamics[END_REF] and [START_REF] Filardo | Business Cycle Durations[END_REF]. 22 This average should correspond closely to the sample average of margins before or after regulation. 23 It is also interesting to note in table 3 that the parameters associated with the wholesale price are negative and significant only during the collusive regimes. The reduction in margins in the face of increased costs during collusive periods is consistent with monopolistic pricing behaviour. It is likely that similar behaviour is not observed during price wars because margins cannot be reduced further. Increasing cost during collusive periods also decreased the persistence of these periods. 24 Each individual test statistic has an asymptotic χ 2 (1) distribution under the null hypothesis of no misspecification, while the joint test is asymptotically distributed as a χ 2 (5) under the null.

  Figure 1 Retail Margins, Montreal 1994-2001

Table 1 Descriptive Statistics

 1

	Variable	Mean	Minimum	Maximum	Standard Deviation
	Price : P t (¢/litre)	28.7738	15.4000	49.7000	7.1932
	Wholesale price: W t (¢/litre)	24.3781	13.7250	42.8250	6.8470
	Margin : M t (¢/litre)	4.3957	-5.8750	8.8000	2.2452
	Margin before regulation	4.3504	-5.8750	8.7200	2.6105
	Margin after regulation	4.4229	-1.2000	8.8000	1.9993
	Regulation : R t (0,1)	0.6274	0	1	0.4841

Table 2 Percentage of time with margins below or above mean margin

 2

		Before regulation (156 weeks)	After regulation (260 weeks)	Total (416 weeks)
	Margin <Mean Margin	43%	48%	46%
	Margin > Mean Margin	57%	52%	54%

Table 3 Parameter Estimates (Dependent variable is M t )

 3

								AR(1) without W	AR(1) with W
	Parameter	Variable	Estimated	Standard	Estimated	Standard
								value	error	value	error
	α	collusion	constant	Margin Equation 3.7034* 0.5926	4.0761*	0.3621
	ρ	collusion	M t-1	0.2546*	0.0947	0.2088*	0.0598
	β	collusion	R t	0.1679	0.5060	0.7352	0.4501
	γ	collusion	W t	0.0067	0.0324
			α		pw	constant	-1.3040	0.9118	-1.1864*	0.5928
			ρ		pw	M t-1	0.1242	0.1107	0.2591*	0.0739
			β		pw	R t	4.4870*	1.1564	3.7059*	0.6684
			γ	pw	W t	-0.0684*	0.0214
	collusion φ	constant	Regime Selection Equation 1.6296* 0.3169	1.1344*	0.2617
	θ	collusion	R t	-0.3362	0.6831	-0.1476	0.4610
	ω	collusion	W t	-0.1078*	0.0343
			φ	pw	constant	-0.0950	0.3983	-0.4228	0.3535
			θ	pw	R t	1.3906*	0.7181	1.7070*	0.4707
				ω	pw	W t	-0.1451*	0.0511
			σ	ε		Error Variance 1.8386* 0.1233	1.6395*	0.0817
	* : Statistically significant at the 5% confidence level; "collusion" stands for collusive
	regime and "pw" stands for price war or competitive regime.

Table 4

 4

	Diagnostic Tests
	Test for 24		Without W	With W
			Statistic	p-value	Statistic	p-value
	Serial Correlation -		0.13	0.718	3.45	0.063
	Collusion Regime
	Serial Correlation -		0.20	0.655	0.00	0.976
	Price War Regime
	ARCH			5.76	0.016	2.59	0.107
	Higher-order Markov		18.17	0.000	3.56	0.059
	Dependence -Collusion
	Regime
	Higher-order Markov		3.21	0.073	0.04	0.842
	Dependence -Price War
	Regime
	Joint Test			46.75	0.000	10.50	0.062
	Table 5 Regime Dependent Statistics
	Regime	P(S t =i|S t-1 =i)	E(duration)	Estimated Margins
	without W	with W	without W	with W	without W	with W
	Collusive	0.94841	0.87170	19.382	7.794	4.9683	5.1520
	(R t =0)
	Collusive	0.90206	0.83814	10.210	6.178	5.1936	6.0813
	(R t =1)
	Price War	0.46217	0.33624	1.859	1.507	-1.4889	-1.6013
	(R t =0)
	Price War	0.90246	0.90047	10.252	10.048	3.6344	3.4007
	(R t =1)

Table 6 Unconditional Probabilities and Margins

 6 Corresponding author. Address : Institut d'économie appliquée, HEC Montréal, 3000 chemin de la Côte-Sainte-Catherine, Montréal, Québec, Canada H3T 2A7. E-mail : robert.gagne@hec.ca. Quoted from the Report of the Liberal Committee on Gasoline Pricing in Canada, June 1998 (p. 5). 2 Translated from the Mémoire de l'Association Québécoise des Indépendants du Pétrole, June 1998,

	State	P(S t =collusive)	Estimated Margins	Sample Margins
		without W	with W	without W	with W
	R t =0	0.91247	0.83801	4.4031	4.0581	4.3504
	R t =1	0.49899	0.38077	4.4124	4.4214	4.4229

* 1

Table 6

 6

	, p. 35) appear to be intermediate with

obtain a three-week episode from week 25 to week 27, as made clear in Figure 2. 19 Another example can be observed in late-1996, shortly before the introduction of a pricefloor, about week 37 (the margin is -1.1). If one keeps 1¢/litre as a working criterion, again one obtains a three-week price war, from 1996-36 to 1996-38.

[Figures 1 and 2 here]

Table 1 presents some descriptive statistics. We see that average margins are roughly the same before and after regulation but significantly less dispersed after regulation (the standard deviation is 2.61 before regulation and 2.00 after). Given that peaks are approximately of the same magnitude before and after regulation (see Figure 1) and that average margins are about the same size, small margins (i.e. deep price cuts) should be more prevalent after regulation, in accordance with theoretical predictions. This higher prevalence of small margins is not obvious in Figure 1, which calls for a more elaborate data analysis.

Toward this aim, in Table 2 we present the number of weeks (in %) where the margins are below and above average margins, before and after regulation. Before regulation, margins were below average 43% of the time while after regulation this number increased at 48%. Again, given that average margins were nearly the same before and after regulation, this result is also a sign (albeit a weak one) that periods of small margins are more prevalent since regulation. This provides some modest albeit encouraging indirect support for our theoretical prediction. We therefore now turn to a more formal statistical analysis of retail margins using a Markov Switching regression approach.

[Tables 1 and2 here]