George Giakkoupis

Nicolas Schabanel

Le r ôle de la dimension dans la recherche de chemins optimaux dans les petits mondes

Dans cet article, nous apporterons une réponse exacte (asymptotiquement) et surprenante à une question ouverte depuis plusieurs années dans le domaine de la modélisation des comportements sociaux : est-il possible de naviguer dans un graphe petit-monde de Kleinberg en temps optimal ? C'est-à-dire, est-il possible de suivre un chemin optimal d'un point à un autre en utilisant seulement les informations disponibles localement. Cette question a clairement des applications dans le design de tables de routage et de protocoles pair-à-pair. La réponse que nous apportons est la suivante : si la dimension sous-jacente du graphe petit-monde est 1, la réponse est non, les chemins optimaux sont de longueur Θ(log n) alors que le mieux que l'on puisse faire à partir des informations locales est Θ(log n • log log n) ; lorsque la dimension sous-jacente est 2, on démontre qu'un simple algorithme de parcours en largeur réussit à naviguer optimalement.

Introduction

Milgram revealed in his famous experiment [START_REF] Milgram | The small world problem[END_REF] that not only are individuals a few handshakes away from each other, but they are also able to find such short paths between them, in spite of their extremely local view of the worldwide social network. In 2000, Kleinberg [START_REF] Kleinberg | The small-world phenomenon : an algorithmic perspective[END_REF] proposed a simple random network model that captures this surprising property of social networks. Beyond this natural sociological motivation, his model had an important impact on the design on several peer-to-peer protocols (e.g., [START_REF] Zhang | Using the small-world model to improve Freenet performance[END_REF]), because it addresses the general question of how decentralized algorithms can find short paths in a partially unknown network. Kleinberg's small world model consists of a d-dimensional grid {-n, . . . , n} d (representing local acquaintance between individuals, such as geographic or professional) augmented with one "long-range" directed link per node pointing to a random node at distance r chosen with probability proportional to 1/r s where s is a parameter of the model (this long-range link represents a random acquaintance met in the past for instance). Kleinberg defined a decentralized routing algorithm as an algorithm that tries to route locally a message from a node (the source) to another (the target), that is to say, by visiting only neighbors (local or long-range) of already visited nodes (starting from the source). Kleinberg's most striking result is that no decentralized algorithm can found short paths (i.e., of length polylog(n) where n is the size of the grid) if s = d, even when the diameter of the augmented graph is Θ(log n) as it was shown later on by [START_REF] Martel | Analyzing kleinberg's (and other) small-world models[END_REF][START_REF] Martel | Analyzing and characterizing small-world graphs[END_REF]. Only when d = s, decentralized algorithm may found short paths between random pairs. Indeed, the simple greedy algorithm that simply routes the message to the closest neighbor (local or long-range) of the current message holder computes paths of expected length O(log 2 n) [START_REF] Kleinberg | The small-world phenomenon : an algorithmic perspective[END_REF]. Several decentralized algorithms [START_REF] Fraigniaud | Eclecticism shrinks even small worlds[END_REF][START_REF] Lebhar | Close to optimal decentralized routing in longrange contact networks[END_REF] have been proposed to compute better paths when s = d efficiently (i.e. by visiting at most polylog(n) nodes with high probability) and the best so far [START_REF] Lebhar | Close to optimal decentralized routing in longrange contact networks[END_REF] for Kleinberg's original model computes path of length O(log n(log log n) 2 ) between arbitrary pairs, which is still significantly larger than the diameter Θ(log n) of the graph. The question of the ability of efficient decentralized algorithms to find optimal paths was then stated by Kleinberg in 2006 in its celebrated paper [START_REF] Kleinberg | Complex networks and decentralized search algorithms[END_REF] (open problem n • 3).

We answer precisely to this question by first showing that if d = 1 no efficient decentralized algorithm may found shorter paths than Ω(log n log log n) on expectation, and second, by providing a new efficient decentralized algorithm (largely inspired from the work of [START_REF] Lebhar | Close to optimal decentralized routing in longrange contact networks[END_REF]) which computes optimal paths of expected length O(log n log log n) for d = 1 and O(log n) for d 2, respectively. To our knowledge, this is the first time that such a transition in performance is observed in this model when s = d.

Path length is a critical indicator of performance in routing protocols (speed, fault tolerance,...). A lot of the peer-to-peer protocols (Symphony, Chord,...) so far use an augmented 1D ring, and our paper shows that performance might be improved by considering 2D rings instead, with a modest overhead cost since the degree of each node remains constant (5 for d = 2). We may also wonder if other networks (like the one in [START_REF] Fraigniaud | On the searchability of small-world networks with arbitrary underlying structure[END_REF]) present similar transition where, even if the diameter is small and short paths can be computed by decentralized algorithms, diameter may not be obtained by any efficient decentralized algorithm.

Definitions and Main Results

The network. We consider the d-dimensional Kleinberg's smallworld random network K d with one longrange links per node [START_REF] Kleinberg | The small-world phenomenon : an algorithmic perspective[END_REF] defined as follows. † The network consists of a d-dimensional toric lattice L d = {-n, . . . , 0, . . . , n} d of (2n + 1) d nodes, where each node u has, in addition to its 2d neighbors in the lattice (its local contacts), an extra directed link pointing to a random node v = u (its long-range contact) chosen independently with probability 1/(Z d • β ||u-v|| ), where ||u -v|| denotes the ( 1 -)distance in the un-

derlying toric lattice L d (||u -v|| = ∑ d i=1 |u i -v i |),
Z d = ∑ v =0 1/β ||v|| = Θ(log n).
Efficient decentralized routing algorithms. We study algorithms that compute a path to transmit a message or a file from a source to a target, along the local and (directed) long-range links of the network. Following Kleinberg's definition, such an algorithm is decentralized if it navigates through the network using only local information to compute the path. Precisely, it has the knowledge 1) of the underlying lattice structure (the d-dimensional torus), 2) of the coordinates of the target in the lattice, and 3) of the nodes it has previously visited as well as their long-range contacts. But, crucially, 4) it can only visit nodes that are local or long-range contacts of previously visited nodes, and 5) does not know the long-range contacts of any node that has not yet been visited. However, 6) the algorithm (but not the path it computes) is authorized to travel backwards along any directed links it has already followed. As Kleinberg pointed out in [START_REF] Kleinberg | Small-world phenomena and the dynamics of information[END_REF], this is a crucial component of human ability to find short paths : one can interpret point 6) as a web user pushing the back button, or an individual returning the letter to its previous holder (who wrote his address on the envelope before sending it). We ask furthermore the algorithm to be efficient, in the sense that it has to visit a number of intermediate nodes which is at most poly-logarithmic in the size of the network (polylog(n)) with high probability, when computing a path between a pair of nodes. This is a standard requirement in social network ‡ where the size of the network is typically exponentially larger than the capacity of each individual. Efficient decentralized algorithms are thus likely to model the capacity of individuals to route messages in a social network.

Since each node has a constant degree 2d + 1, the diameter of K d is Ω(log n). It was shown by [START_REF] Martel | Analyzing kleinberg's (and other) small-world models[END_REF] that the diameter of Kleinberg's smallworld is indeed Θ(log n) for all dimension d ; their analysis suggests a decentralized algorithm that would visit Ω( √ n) nodes to compute a path of length O(log n) for each pair of nodes, which is obviously unrealistic in the framework of social network (where n ≈ 10 9 ). [START_REF] Kleinberg | The small-world phenomenon : an algorithmic perspective[END_REF] initially showed that greedy routing (that always passes the message to the neighbor (local or long-range) of the current holder that is the closest to the target) computes path of expected length O(log 2 n) while visiting O(log 2 n) nodes. [START_REF] Singh Manku | Know thy neighbor's neighbor : the power of lookahead in randomized P2P networks[END_REF] showed that all decentralized routing algorithm has indeed to visit at least Ω(log 2 n) nodes on expectation to compute a path between a random pair of nodes. [START_REF] Lebhar | Close to optimal decentralized routing in longrange contact networks[END_REF] proposed an efficient decentralized algorithm that computes near-optimal paths of expected length O(log n(log log n) 2 ) between each pair of nodes while visiting an optimal expected number of nodes O(log 2 n). The question of finding an efficient decentralized algorithm computing optimal paths of length O(log n) between any pair of nodes was finally formulated as the open problem n • 3 by Kleinberg in its celebrated paper [START_REF] Kleinberg | Complex networks and decentralized search algorithms[END_REF]. The following theorem, which sums up our main results, solves precisely this problem. †. Note that our results are still valid with O(1) long-range links per node instead of one.

‡. with the notable exception of [START_REF] Fraigniaud | On the searchability of small-world networks with arbitrary underlying structure[END_REF] whose routing algorithm visits 2 O( √ log n) nodes.

Le rôle de la dimension dans la recherche de chemins optimaux dans les petits mondes

Theorem 1 (Main results)

(1) For any dimension d and any ε > 0, there is an efficient decentralized routing algorithm that computes a path of expected length at most O(log n log log n) between any pair of nodes (s,t), while visiting at most O(log 2+ε n) intermediate nodes with high probability.

(2) For any dimension d 2, there is an efficient decentralized algorithm that computes paths of optimal expected length O(log n) between any pair of nodes, while visiting at most O(log 2 n) nodes with high probability.

(3) For dimension d = 1, for all efficient decentralized routing algorithm A, there exists a constant c > 0 and a pair of nodes (s,t) such that the expected length of the path computed between s and t is at least c log n log log n (showing that (1) is asymptotically optimal for d = 1).

Overview of previous results. Our optimal decentralized routing algorithms (propositions (1) and ( 2) in Theorem 1) rely on simplifying and then improving the approach of [START_REF] Lebhar | Close to optimal decentralized routing in longrange contact networks[END_REF]. The key steps of the results in [START_REF] Lebhar | Close to optimal decentralized routing in longrange contact networks[END_REF] were obtained through rather complicated calculations. We obtain here a simple geometric rereading of their results that allows us to improve them and get an optimal routing algorithms. In our reformulation of [START_REF] Lebhar | Close to optimal decentralized routing in longrange contact networks[END_REF], if we forget about the technical details, their algorithm rely on the fact that in a Kleinberg's network, the long-range link of any node u has an equal probability ∼ 1 log n to fall in each of the rings centered on u and whose distance ranges in [2 i , 2 i+1 ) for i ∈ {1, . . . , log n}. It follows that if u is at distance r ∈ [2 i , 2 i+1 ] from the target t, its long-range links has probability ≈ i log n ∼ log r log n to be closer to t than u. If one neglects the possible overlapping (which is fine for large enough r), the breadth-first search tree rooted in u expands at a rate of ≈ 1 + i log n towards the target (1 because there is always at least one local contact which is closer to t, plus i log n for the probability that the long-range contact of the current node is closer to t). It follows that after h = log n log log n i steps, the BFS tree rooted contains ≈ (1 + i log n ) log n log log n i ∼ log n leaves among which one has a positive constant probability to fall in the ith ring (the one that contains t) and thus has a positive constant probability to fall at distance at most r/2 from t. Routing the message from u to this leaf and repeating the process until reaching t yields a path whose expected length is at most

≈ ∑ log n i=1 log n log log n i = log n • log log n • H log n ≈ log n (log log n) 2 .
In order to get proposition (1), we now fix some ε > 0 and consider concentric rings of radius log ε j n, with j ∈ {1, . . . , log log ε n n}, centered on the target. Consider that the message has reached a node u at distance r ∈ (log εi n, log ε(i+1) n]. As in [START_REF] Lebhar | Close to optimal decentralized routing in longrange contact networks[END_REF], we explore the BFS tree rooted in u but (1 + ε) times deeper, up to depth h = log n (1+ε) log log n log r

. This ensures that we get, with constant probability, ≈ (1+ log r log n )

log n (1+ε) log log n log r ∼ log 1+ε n leaves among which one of them has with constant positive probability its long-range contact log ε n times closer to the target. A constant number of BFS are thus explored in each ring centered on the target on expectation. It follows that the expected length of the computed path between any pair of nodes is now

at most ≈ ∑ log log ε n n i=1 (1+ε) log n log log n log log εi n log n • log log n.
In order to get proposition (2), we improve the analysis of the algorithm above by taking into account that the cardinal of the spheres grows at least linearly with their radius when d 2. It follows from our calculation that the expansion factor of the BFS tree rooted at a node at distance r ∈ (log εi n, log ε(i+1) n] is now at least ≈ 1 + log r log n . Thus, we just need to explore each BFS tree up to depth h ≈ (1 + ε) log log n log n log r to gather, with constant probability, log 1+ε n leaves among which one long-range link points to a node log ε n times closer to the target. It follows that the expected length of the computed path between any pair of nodes is at most

≈ (1 + ε) log log n ∑ log log ε n n i=1 log n log log εi n ≈ log log n √ log n ∑ log log ε n n i=1 1 √ εi log log n ≈ 1 √ ε √ log log n log n log log ε n n = 1 ε log n.
It follows that this exploration based efficient decentralized algorithm computes asymptotically optimal paths on expectation for any pair of nodes.

Our last result, proposition (3), shows that, even if it does not matches the diameter of the graph, no other efficient decentralized algorithms can do better when d = 1. Consider an efficient decentralized routing algorithm which is bounded to visit at most m = O(log c n) nodes, for some constant c > 0. The intuition for this last result is the following. First, in a Kleinberg's network, with high enough probability, none of the long-range contacts of any set of m nodes can be more than O(log c-1 n) times closer to the target than any node in S. It follows that the algorithm has to visit at least one node in each concentric rings of radii m i centered on the target. The first step consists of partitioning the underlying grid into concentric rings of radii m 9i , for i ∈ {1, . . . , log m 9 n}, centered on the target. According to the above, no matter how high the exponent c is in m = O(log c n), the algorithm will need to go through Ω(log m 9 n) = Ω(log n/ log log n) rings. The second step consists of proving by a coupling argument that the algorithm has almost no control over the first nodes it will visit each times it enters a ring. Indeed, we prove that since the algorithm visits at most m nodes in total, we can force the algorithm to enter each ring through a set of at most m 9 nodes (the entry points), which are close to the farthest border of the ring from the target, and chosen randomly and independently of the algorithm. The third and last step consists then of bounding the extent of the BFS trees rooted on these entry points. Since the entry points are independent of the algorithm, we are left with a purely geometric analysis of these trees. This requires a finer analysis than for the routing algorithm we proposed before, since we need to bound precisely the total length of the possibly explored long-range links. By bounding the cumulated length of the long-range links used in these BFS trees, we are able to prove that, with high enough probability, none of these BFS trees reaches the next ring, closer to the target, before depth log n log log n log r

, at least for the range of distances r ∈ [2 log 0.1 n , 2 log 0.9 n ]. We conclude that the expected length of the computed path is at least ∑ log m 9 (2 log 0.9 n ) i=log m 9 (2 log 0.1 n ) log n log log n log m 2i log n log log n 9 log m (H log 0.9 n/2 log m -H log 0.1 n/2 log m )

log n log log n 9c log log n 0.8 log log n = Ω(log n log log n). It follows that no efficient decentralized routing algorithm can do better than our algorithm in proposition 1.

  and β r denotes the size of the balls of radius r in L d (β r = #{u : ||u|| r}), and Z d is the normalizing constant


Zaccaria Ricci

Cristiana Garisto

Isabella Favia

Ulrike Schloderer

Chiara Giorni

Tiziana Fragasso

Sergio Picardo

Cerebral NIRS as a marker of superior vena cava oxygen saturation in neonates with congenital heart disease

Keywords: neonate < Age, congenital heart disease < Cardiac, near infrared spectroscopy, venous oxygen saturation Near Infrared Spectroscopy, Pediatric Cardiac Surgery, venous oxygen saturation, low cardiac output syndrome

Background: Near infrared spectroscopy (NIRS) is a non invasive method to monitor hemoglobin oxygen saturation using non pulsatile oximetry. Methods: We retrospectively analyzed perioperative data from 100 newborn patients who underwent cardiac surgery for CHD. RSO2c, ScvO2 from 24 hours before to 72 hours after surgery were recorded. Results: rSO2c had a fair correlation with ScvO2 (r 0,37; p<0.001). The relationship between rSO2c and ScvO2 did not change when analyzed between patients with cyanotic or acyanotic CHD. During the preoperative period rSO2c levels overestimated ScvO2; in the first 18 postoperative hours rSO2c understimated ScvO2; after that period they showed very close trends. Hypocapnia caused rSO2c to underestimate ScvO2; in normocapnic patients rSO2c-ScvO2 average differences were close to zero; in hypercapnic neonates rSO2c tended to overestimate ScvO2.

The best performance of rSO2c as a surrogate of ScvO2 was found in the venous saturation ranges from 40 to 60% (r 0.3, p: 0.03). Conclusions: rSO2c in newborn patients with cyanotic and acyanotic CHD provides a continuous non invasive information with a fair correlation with ScvO2 %: some predictable variables (i.e. time from surgery, carbon dioxide and venous saturation levels), should guide the operators in order to adjust rSO2c values in terms of ScvO2. Serial measures of ScvO2 seem recommended in order to tailor rSO2c information on actual venous saturation percentage. Cerebral hemoglobin oxygen saturation measured with near-infrared spectroscopy (NIRS) is used to monitor and titrate brain oxygen delivery preoperatively, during cardiopulmonary bypass and postoperatively in the pediatric cardiac intensive care unit (PCICU) [START_REF] Hirsch | Near-infrared spectroscopy: what we know and what we need to know--a systematic review of the congenital heart disease literature[END_REF]. It has also been evaluated as a surrogate measure of superior vena cava oxygen saturation (ScvO2), in order to non-invasively assess adequacy of oxygen delivery and consumption [START_REF] Tortoriello | A noninvasive estimation of mixed venous oxygen saturation using near-infrared spectroscopy by cerebral oximetry in pediatric cardiac surgery patients[END_REF][START_REF] Mcquillen | Regional and central venous oxygen saturation monitoring following pediatric cardiac surgery: concordance and association with clinical variables[END_REF][START_REF] Bhutta | Noninvasive cerebral oximeter as a surrogate for mixed venous saturation in children[END_REF][START_REF] Tyree | Correlation of brain tissue oxygen tension with cerebral near-infrared spectroscopy and mixed venous oxygen saturation during extracorporeal membrane oxygenation[END_REF].

Somatic NIRS has also been proposed, as an estimate of somatic oxygen delivery and as a measure of optimized systemic perfusion, coupled with cerebral NIRS (rSO2c) [6]. Different locations have been described for NIRS sensors: typically, cerebral sensor is positioned on the right forehead and somatic sensor is placed on the flank at the T10-12 vertebrae level, in order to theoretically measure renal oxygen saturation (rSO2r). Initial studies on correlation between rSO2c and ScvO2 saturation have showed conflicting results [7,8], due to wide limits of agreement between the two methods presumably due to individual features (such as age), specific cardiovascular disease (cyanotic or acyanotic), or clinical condition (arterial blood pressure and PaCO2): these considerations lead to the conclusion that rSO2c monitoring might be more suitable to monitor ScvO2 trends than absolute values.

The management of neonates with congenital heart diseases (CHD) who are candidates for early surgery is challenging and perioperative cerebral blood flow and oxygen delivery monitoring is mostly important due to neurological impairment that may occur in these children [9]. Furthermore, anatomical considerations, such as their limited skull thickness and Since these age groups are so different in terms of surgery and cerebral physiopathology, they should probably be examined separately.

No studies are reported, so far, on correlation between rSO2c and ScvO2 saturation in a large cohort of exclusively newborn patients with CHD. This retrospective study had two objectives: a) to evaluate the relationship of changes in rSO2c and rSO2r with changes in ScvO2 in this specific subgroup of children with CHD and eventually confirm NIRS reliability as a surrogate non-invasive measure of ScvO2; b) to examine the impact of different clinical variables (type of CHD, perioperative period, venous saturation level and carbon dioxide partial pressure) on such correlation.

Patients and methods.

We retrospectively reviewed data from newborn patients who underwent cardiac surgery for Central venous access was achieved by a 4 Fr bilumen catheter. If, after chest X ray verification, CVC was not correctly positioned in the superior vena cava, patients were excluded from the study. Data collection started 24 hours before transport to the operatory room (T0 and T1, after 12 hours) if the patient required intubation before the scheduled surgery. Thereafter, data collection was achieved after anesthesia induction (T2), at ICU admission (T3) and successively after 6, (T4), 12 (T5), 18 (T6), 24 (T7), 48 (T8) and 72 hours (T9). At each data point, rSO2c, rSO2r, ScvO2 were recorded; ScvO2 was measured by co-oximetry and arterial carbon dioxide partial pressure (PaCO2) was also registered together with all blood gas analysis parameters.

Patient age, diagnosis and surgical procedure were reported.

Statistical analysis.

Kolmogorv-Smirnov test was used to test sample deviation from Gaussian distribution. Pearson test was chosen for correlation estimation. Bland Altman analysis was used to verify bias and agreement of correlated variables. Results are expressed as mean (standard deviation). A p value less than 0.05 was considered significant. Statistical analysis was performed with the GraphPad Prism 4.00 software package (GraphPad Software, San Diego, CA, USA).

Institutional review board approved the study and waived the need for informed consent due to the observational nature of the research. One hundred and sixty-two neonates with CHD were admitted to our intensive care unit within the studied period. Of these, 12 were not surgical patients, in 10 cases CVC was not correctly positioned in superior vena cava and 40 had a femoral vascular access. One hundred neonates with superior vena cava vascular access and NIRS monitoring were examined. Ages, diagnoses and surgical procedures are summarized in table 1. We retrieved 890 of 1000 expected data points and in 110 cases data have been missed: 30 patients were not intubated before surgery and were not monitorized in the T0-T1 period (60 data points); in 10 cases 72 hours of full monitoring were not reached (15 data points); in few occasions it was not possible to achieve simultaneous NIRS and ScvO2 data (35 data points). The analyzed data had Gaussian distribution. Bland Altman plot confirmed a fair correlation between rSO2c and ScvO2 with a bias of 0.049 and a 95% agreement of 25% to -25% (figure 1). Correlation between ScvO2 and rSO2c resulted significant (p< 0.001) with an r of 0,37 (figure 2). RSO2C/ScvO2 correlation achieved an r of 0.31 (p<0.001) in TGA patients, an r 0.37 of (p<0.001) in HLHS patients and an r of 0.30 (p 0.03) in TOF patients (table 2). Correlations of subgroups divided per preoperative (T0-T2), early postoperative (T3-T6) and late postoperative periods (T7-T9) resulted, respectively, of 0.24 (p: 0.05), 0.34 (p < 0.01) and 0.50 (p<0.001) (table 2). Interestingly, Bland Altman analysis revealed that surgical operation significantly affected precision of ScvO2 and rSO2c correlation: during the preoperative period bias was 1.5, indicating that average rSO2c tended to overestimate ScvO2; during the early postoperative period bias was -1.2 because average rSO2c levels underestimated ScvO2; finally, in the late postoperative period bias reduced to 0.1 meaning that rSO2c and ScvO2 average difference tended to normalization. However, trends of average values of rSO2c and ScvO2 % showed a good correlation (figure 3): when the differences of average rSO2c values between consecutive time points (T X+1 -T X ) were calculated and correlated with the same differences of average ScvO2 values, an r of 0.8 was found (p<0.005). Correlations of subgroups divided per PaCO2 < 31 mmHg, PaCO2 >31 and < 41 mmHg and PacO2 > 41 mmHg resulted, respectively, of 0.34 (p: 0.02), 0.39 (p < 0.001) and 0.39 (p<0.001) (table 2). Again, Bland Altman analysis revealed that PaCO2 altered mean error of ScvO2 and rSO2c correlation: in the hypocapnic group bias was -5, indicating that rSO2c tended to underestimate ScvO2; in the normocapnic group bias was -0.15 because rSO2c-ScvO2 average differences were likely close to zero; in the in the hypercapnic group bias was 2, because rSO2c overestimated ScvO2. We also tried to explore the correlation existing between high (90 to 61 %), medium (60 to 41 %) and low (< 40 %) ScvO2 saturations and rSO2c: the high group showed an r of 0.1 (p 0.8), the medium level group showed an r of 0.3 (p 0.03) and the low group showed an r of 0.25 (p 0.05) (table 2).

Discussion.

Near-infrared spectroscopy is a noninvasive, continuous monitoring technique using nonpulsatile oximetry to determine a venous-weighted oxyhemoglobin saturation of the underlying regional tissue. NIRS emits photons into the underlying capillary vascular bed, which readily transmit through skin, bone, and fat. Chromophores such as hemoglobin absorb some photons, causing wavelength-dependent changes in those photons to be reflected back to the detector. By analyzing this spectral absorption pattern, the oxygenation of the underlying tissue is determined [9]. Being primarily venous, the regional oxygen saturation drops -and NIRS % accordingly falls-when regional oxygen delivery is impaired and/or regional oxygen extraction whereas tissue with little metabolic demand (kidney) has a higher baseline value [6]. NIRS sensors placement has not been clearly standardized yet: whereas cerebral cortex capillary vascular bed lies few millimeters beyond forehead skin, making a strong rationale for rSO2c evaluation, especially in neonates, it is less clear which is the position and the indication for the somatic sensor [11,12]. In our institution two site NIRS (cerebral-renal) monitoring has become a standard since three years: the aim of this retrospective analysis was to examine the clinical variables that mostly influence correlation of NIRS saturation with systemic oxygen delivery in a large homogeneous cohort of newborn patients. To our knowledge, this study is the first specifically addressing NIRS utilization in neonates with CHD: for anatomical reason, this is an ideal population where to validate rSO2c/ScvO2 correlation since cerebral blood flow is relatively high with respect to other tissue beds drained by superior vena cava. Furthermore, finding a non-invasive surrogate to central venous blood withdrawal has great clinical and practical implications.

Our results are in line with previous descriptions, remarking that also in newborn patients a significant correlation between rSO2c and ScvO2 exists, even if with wide limits of agreement.

Our analysis also showed that performance of rSO2c is not apparently affected by the CHD diagnosis, whether it is a cyanotic or an acyanotic syndrome. Cerebral NIRS monitoring showed an optimal correlation to ScvO2 in the crucial venous saturation range from 40 to 60%, where timely diagnosis of low oxygen delivery might rapidly improve clinical conditions by early therapy. Furthermore, our data suggest that before cardiac surgery venous cerebral blood flow is less desaturated than ScvO2, maybe due to relative redundancy of blood flow, especially when the patient is sedated and in stable hemodynamic conditions: this might explain rSO2c overestimation of ScvO2 in this phase. After surgery, especially when prolonged cardiopulmonary bypass times and/or deep hypothermic circulatory arrest are required, cerebral oxygen consumption to blood flow ratio is probably less favorable, and ScvO2 tends to be higher than rSO2c. This phenomenon reaches an equilibrium after 18 hours where rSO2c/ScvO2 difference is no more so evident and rSO2c/ScvO2 correlation highest. PaCO2 is well known to affect cerebral vascular resistances and flow: in case of hypocapnia and cerebral vasoconstriction, a reduction of cerebral oxygen delivery might decrease brain venous saturation due to constant consumption in the face of decreased supply (low rSO2c saturations). Probably for this reason low rSO2c values underestimated ScvO2 in neonates with a PaCO2 level below 31 mmHg, whereas in normocapnic patients rSO2c/ScvO2 seemed to be optimized. Hypercapnia (and relative cerebro-vascular vasodilation), on the other side, induced a rSO2c tendency to overestimate ScvO2. As a matter of fact, however, rSO2c is only a fair surrogate of ScvO2 values and its measures are probably more suitable as a trend for single cases and not reliable for pooled evaluation of a large cohort of subjects and, differently from ScvO2, a rSO2c value considered low for somebody might appear adequate in another patient:

inter-individual values are widely variable, also when age and diagnosis are considered as confounding variables. Nevertheless, rSO2c and ScvO2 trends showed to have a significant correlation, confirming that, in our neonates with CHD, rSO2c is at least able to show ScvO2 tendency. From our analysis, moreover, it was not possible to find specific diagnostic indications from rSO2r (data not shown). Its correlation to ScvO2 is similar to rSO2c, but not better. Our study has several limitations. This is a retrospective analysis of data collected during a relatively wide perioperative period. Time points were pre-defined without any specific rationale: such "raw" information registration of a continuous signal might have missed specific perioperative moments when significant hemodynamic events occurred (i.e delayed sternal closure, vasopressors dose modifications, periods of hypotension, etc) where oxygen delivery monitoring was mostly important. Furthermore, wrong NIRS values coming from artifacts and partial sensor detachment might have occurred in our routine clinical NIRS utilization. Finally, missing data might have affected our results. Nevertheless, ten time points should adequately cover a 96 hours perioperative time lapse and more then 800 data points should reliably minimize any hypothetical wrong information. We did not examine the impact of different ventilation modalities, sedation and hemoglobin variations that may modify rSO2c value independently or differently from ScvO2 %. However, these have been already illustrated [START_REF] Mcquillen | Regional and central venous oxygen saturation monitoring following pediatric cardiac surgery: concordance and association with clinical variables[END_REF] and the analysis of confounding factors secondary to non-hemodynamic therapy modifications was beyond the scope of our analysis that approached NIRS as a surrogate of ScvO2.

Conclusion.

We examined rSO2c/ScvO2 correlation in the largest cohort of neonates evaluated so far. Our study confirmed existing data and allowed to refine some information in the specific setting of CHD: rSO2C monitoring in CHD newborn patients during the perioperative period did not provide an absolute oxygen saturation value that could reliably be considered a marker of ScvO2 and serial measures of ScvO2 seem recommended in order to tailor rSO2c information on actual venous saturation percentage. The continuous rSO2c monitoring in neonates, Figure 2 Correlation and linear regression analysis of Cerebral Near Infrared Spectroscopy (rSO2c) and Superior Vena Cava Venous Oxygen Saturation (ScvO2): r 0.37, p< 0.001.

  investigate the correlation between cerebral NIRS (rSO2c) and superior vena cava venous oxygen saturation (ScvO2) in newborn patients with congenital heart disease (CHD).

  cerebral blood flow, make rSO2c monitoring potentially applicable in this population[9]. Finally, previous studies pooled in the same analysis neonates and infants [10].

  CHD from January 2007 to December 2009. Enrolled patients diagnoses were transposition of the great arteries (TGA), hypoplastic left heart syndrome (HLHS), tetralogy of Fallot (TOF) and other less frequent CHD (atrioventricular canal, aortic arch interruption, truncus arteriosus, aortic coarctation with ventricular septal defect). According to our institutional protocol on newborn patients' monitoring, after induction of anesthesia, neonatal sensors for the INVOS System (Model 5100C, Somanetics, Troy, Michigan, USA) for rSO2c (right forehead) and rSO2r (left flank) were placed and an arterial cannula was inserted for the measurement of invasive

  pressure. A central venous catheter (CVC) was positioned in the superior vena cava.

  beds with high oxygen consumption (brain) have a lower baseline saturation,

  with ScvO2, and is best with normocapnia, in ScvO2 ranges of 40-60%, and in the later postoperative period. Furthermore it showed to be consistent both in cyanotic and acyanotic CHD.

  SJ, Berens RJ, Havens PL, Kim MK, Hoffman GM. Effect of volume resuscitation on regional perfusion in dehydrated pediatric patients as measured by two-site near-infrared spectroscopy. Pediatr Emerg Care 2009; 25:150-3. 7 ) Dullenkopf A, Baulig W, Weiss M, Schmid ER. Cerebral near-infrared spectroscopy in adult patients after cardiac surgery is not useful for monitoring absolute values but may reflect trends in venous oxygenation under clinical conditions. J Cardiothorac Vasc Anesth 2007; 21: 535-9 8 ) Knirsch W, Stutz K, Kretschmar O, Tomaske M, Balmer C, Schmitz A, Berger F, Bauersfeld U, Weiss M; Working Group on Non-Invasive Haemodynamic Monitoring in Paediatrics. Regional cerebral oxygenation by NIRS does not correlate with central or jugular venous oxygen saturation during interventional catheterisation in children. Acta Anaesthesiol Scand 2008; 52: 1370-4. 9 ) Chakravarti S, Srivastava S, Mittnacht AJ. Near infrared spectroscopy (NIRS) in children. Semin Cardiothorac Vasc Anesth 2008; 12: 70-9. 10 ) Weiss M, Dullenkopf A, Kolarova A, Schulz G, Frey B, Baenziger O. Near-infrared spectroscopic cerebral oxygenation reading in neonates and infants is associated with central venous oxygen saturation. Paediatr Anaesth 2005; 15: 102-9. 11 ) Chakravarti SB, Mittnacht AJ, Katz JC, Nguyen K, Joashi U, Srivastava S. Multisite nearinfrared spectroscopy predicts elevated blood lactate level in children after cardiac surgery. J Cardiothorac Vasc Anesth 2009; 23:663-7.

  J, Almodovar MC, Zuk J, Friesen RH. Correlation of abdominal site near-infrared spectroscopy with gastric tonometry in infants following surgery for congenital heart disease. Pediatr Crit Care Med 2008; 9:62-8. TABLES Table 1 Demographic characteristics of evaluated sample. Age, weight and RACHS are expressed as mean (standard deviation). TGA: transposition of the great arteries, HLHS: hypoplastic left heart syndrome, TOF: tetralogy of Fallot, RACHS: Risk Adjusted classification for Congenital Heart Surgery. *) at the time of surgical procedure. §) other diagnoses: Truncus arteriosus (3 pts), anomalous left coronary artery from pulmonary artery (1 pt), Aortic coarctation and interventricular septal defect (4 pts), aortopulmonary window (1pt), aortic arch interruption (2 pts). £) other procedures: correction of truncus arteriosus (3 pts), correction of TOF (16 pts), aortic decoarctation without interventricular septum closure (3 pts), aortic decoarctation with interventricular septum closure (1 pt), correction of anomalous left coronary artery from pulmonary artery (1 pt), aortopulmonary window (1pt) and aortic arch interruption (2 pts).

Figure 1 Bland

 1 FIGURES

Figure 3 Mean

 3 Figure 3

  figure 1 144x94mm (150 x 150 DPI)

Table 2

 2 Correlation (r) between cerebral Near Infra Red Spectroscopy and Superior Vena Cava VenousOxygen Saturation (ScvO2). TGA: transposition of the great arteries, HLHS: hypoplastic left heart syndrome, TOF: tetralogy of Fallot. T0-T2 indicate preoperative phase, T3-T5 indicate early

Table 1

 1

	Diagnosis	TGA: 39 pts
		HLHS and UVH: 26 pts
		TOF: 24 pts
		Other § : 11 pts
		Total: 100 pts
	Age* (days)	13 (10)
	Sex	Males 62
		Females 38
	Weight (kg)	3.3 (0.9)
	RACHS	4.3 (1.4)
	Surgical procedure	Arterial Switch operation: 39 pts
		Norwood I stage: 26 pts
		Blalock Taussig shunt: 8 pts
		Other £ : 27 pts

Table 2

 2 Group definition

	Number of
	r	p
	examined pairs


Keywords: JEL classification: C14, D21, G14, J63, L25 Workforce reduction, downsizing, layoff, financial performance, Return on Equity, selection bias

Using an exhaustive database with labour, accounting and financial market information on French firms (1994)(1995)(1996)(1997)(1998)(1999)(2000), the authors analyse the causes and the consequences of a workforce reduction in 1996 -the year chosen as reference -on firms' performance, in a long term perspective. One important contribution to the topic consists in comparing the estimates of publicly-listed and non-listed companies. As far as we know, the comparative method had not be used before. A logistic model shows that in both groups, headcount reduction occurs in less-productive and financially distressed firms, resorting to downsizing as a defensive response to an adverse economic shock. However, the former anticipates better than the latter the decision to eliminate jobs. An econometric model that captures the initial characteristics of the firms, suggests the major performance indicators are significantly improved only for non-listed companies. Yet, there is no net gain on the full period studied.

Introduction

In the 1980's, downsizing -"the planned elimination of positions or jobs" designed to improve productivity and competivity (Cascio 1993: 96) -became a pervasive phenomenon throughout the American business world [START_REF] Cascio | Downsizing: What do we Know? What Have we Learned?[END_REF][START_REF] Freeman | Organizational Downsizing : A convergence and Reorientation Framework[END_REF][START_REF] Budros | A Conceptual Framework for Analyzing why Organizations Downsize[END_REF]) and increasingly from the 1990's, in Europe as well [START_REF] Segalla | Cultural Influences on Employee Termination Decisions: Firing the Good, Average or the Old?[END_REF]. This practice quickly became one of the world's leading management fads. Despite its prevalence, the relation between downsizing and corporate performance has not been studied thoroughly.

According to Freeman & Cameron (1993: 10), "the study of organizational downsizing is still in its infancy." Cameron considered downsizing as the "most pervasive -but yet understudied phenomenon in the business world" (Cameron 1994: 183).

Ten years after, [START_REF] Baumol | Downsizing in America[END_REF] pointed out the lack of empirical studies.

Currently the downsizing issue continues to be a topic of interest for firms of the "new capitalism". The few empirical studies highlight their "conflicting findings" (Wayan & Wermer 2000: 343-344). Some papers conclude that downsizing improve economic performance, reducing the labour costs and increasing the speed of decision-making. Others report either equivoqual outcomes or mixed results [START_REF] Cascio | Downsizing: What do we Know? What Have we Learned?[END_REF]Cappelli, Bassi, Katz, Knoke, Osterman & Useem 1997;Baumol, Blinder & Wolff 2004: 194-197) or negative effects [START_REF] Cascio | Downsizing: What do we Know? What Have we Learned?[END_REF]: 100, Cameron 1994;[START_REF] Mishra | Explaining How Survivors Respond to Downsizing: The Roles of Trust, Empowerment, Justice, and Work Redesign[END_REF]. So, it had not been clear whether downsizing does indeed improve economic performance. As a result, firms have either a hazy idea of the consequences of downsizing or think that the impact of downsizing is generally negative. The paradox rests on the fact that firms continue to downsize. Some scholars such as [START_REF] Budros | A Conceptual Framework for Analyzing why Organizations Downsize[END_REF] and Palmer, Kabananoff & Dunford (1997) have attempted to solve the paradox. [START_REF] Budros | A Conceptual Framework for Analyzing why Organizations Downsize[END_REF] outlined another rationality, while Palmer, managerial field. The former deals with the relationship between downsizing and strategy, the latter with implementation techniques. In that perspective, the diversity of outcomes may reflect the diversity of the effectiveness of the firms' strategy. There are better or worse ways to downsize with respect to organizational performance. Well-managed companies are more likely than others do to downsize successfully (Delanay & Huselid, 1996;[START_REF] Palmer | Managerial Accounts of Downsizing[END_REF]Kats de Vries & Balazs, 1997;[START_REF] Bowman | When Does Restructuring Improve Economic Performance?[END_REF]. This paper adopts a complementary approach, focusing on the statistical methods that have been underestimated. They may be another source of diversity of outcomes. In actual fact, several statistical shortcommings entache their fiability. First, the choice of samples in the previous studies are somewhat so limited in their size (the largest is the Standard and Poors index) that no serious conclusions may be drawn. Second, in order to analyse the impact of downsizing on firms' performance, some papers usually compare the average profitability growth, depending whether companies have or not reduced their workforce. They implicitly assume that the consequences of a workforce reduction are not contingent upon the initial characteristics of the firms that downsize. Doing so, these studies entail an important selection bias. Third, most of the researchs deals with big firms. They never draw the distinction between publicly listed and non-listed companies. If we assume that downsizing policy belongs to the gouvernance of the firms that depends on the the balance of powers of shareholders and of stokholders, the causes of downsizing and -as a consequence -the economic performance of firms should be different.

The paper is organized as follows. Section 2 explains the emergence of several meanings of downsinzing. Section 3 describes the database we built. Section 4 outlines the econometric strategy. Section 5 analyses the determinants of the workforce reduction in 1996, the reference year chosen for this research. Section 6 is devoted to estimate the impact of the workforce reduction using a corrected Difference in Differences estimator. Some concluding remarks are offered in section 7.

2 The Emergence of Several Meanings of Downsizing : a Lesson of Two Decades of Downsizing Policies in the U.S.

Although the subtleties of defining organizational downsizing we are aware of, we adopt the notion commonly accepted by scholars. Cascio, 1993, Freeman andCameron (1994: 12) and most of the academics [START_REF] Cascio | Downsizing: What do we Know? What Have we Learned?[END_REF][START_REF] Dewitt | Firm, Industry, and Strategy Influences on Choice of Downsizing Approach[END_REF][START_REF] De Meuse | Announced Layoffs: Their Effect on Corporate Financial Performance[END_REF]Budros, 1999: 70), define downsizing as an intentional policy involving a reduction in the workforce in an attempt to improve the firms' efficiency. Since in France a layoff has no institutional diference with a downsizing -the separation with workers is permanent -we will use alternatively workforce reduction, downsizing or layoff.1

According to the literature, firms downsize for two opposite reasons that have historically prevailed, depending of the period of their adoption. First, in the 1980's, downsizings were a common strategic response to save failing companies who were losing market share to foreign firms, or had large drops in demand for their products (Cameron & Whetten, 1987;Freeman & Cameron, 1993). However, the merger wave of the 1980's influenced the evaluation of firms and, as a consequence, the downsizing strategy. On the one hand, any company trading at a price-earnings multiple lower than the industry-wide multiple was viewed as undervalued, or a poor-performer, and ripe for a takeover. On the other hand, shareholders considered that CEOs and executive management ought to be theirs servants. These above statements explain the change of the meaning of downsizing that characterized a second period in the mid of the 1990's. Urged by financial markets to increase their return on equity, even though they already enjoyed strong profits, large corporation embraced internal workouts, which consisted in both reducing employment and financial actions such as selling non-performing assets or refocusing.

These two reasons corresponding to two business periods (the eighties and the nineties) define two meaning of downsizing.

On the conceptual side, [START_REF] Cameron | Downsizing and Redesigning Organizations[END_REF] have clearly defined an offensive downsizing in three steps: (1) an intentional plan (2) which means a reduction of a company's size, that is to say either it's workforce or it's assets (3) and whose purpose is mainly an increase in it's profitability. So offensive downsizing appears as a well-prepared strategy from the managers.2

On the contrary, a defensive downsizing signals an effort to stymie, or lessen the depth of a profitability decline. Thus, layoffs are driven by changes in the product market and product demand, and constitute a defensive answer of firms, in order to avoid bankruptcy. [START_REF] Cappelli | Examining the Incidence of Downsizing and its Effect on Establishment Performance[END_REF] accurately sums-up these findings: The distinctiveness of an offensive downsizing, as opposed to a defensive downsizing, is that the job cuts do not necessarily appear to be driven by shortfalls in demand but instead are the consequence of a well-prepared strategy from the managers.

On the empirical side, a first set of studies have examined the main predictors of downsizing. [START_REF] Budros | A Conceptual Framework for Analyzing why Organizations Downsize[END_REF] outlined a general framework with both sociological and economic causes of downsizing. The latter are classified in two categories: internal inefficiency (oversized firms) or external pressures (shareholder value, deregulation). However, few studies tried to measure explicitly the importance of these factors. Based on a sample of 297 firms, the González' and Vicente-Lorente study (2000) concluded that for the period 1989-1994, a defensive downsizing occurred among the largest Spanish firms, with low productivity levels, financial difficulties, and decreasing scale of activity. A second set of economic studies looks at the consequences of downsizing. The seminal work comes from De [START_REF] De Meuse | Announced Layoffs: Their Effect on Corporate Financial Performance[END_REF], but their sample of 57 companies is so small that their results are hardly significant. [START_REF] Cascio | Financial Consequences of Employment-Change Decisions in Major U.S. Corporations[END_REF] finded some positive relationships between reduction in employment and financial performance.

Interestingly, companies that combine employment downsizing with asset restructuring, generate a higher return on assets. Albeit still scarce, studies on French accounting data also tended to suggest a positive outcome of restructuring. Using a sample of 90 large companies whose workforce has been reduced by more than 10 percent, [START_REF] Sentis | Perfomances à long terme et caractéristiques financières des entreprises qui réduisent leurs effectifs[END_REF] showed that indebtness decreased after a large workforce reduction. For D 'arcimoles and FakhFakh (1997), layoffs are profitable as they affect, not only the workforce level, but also its structure. However, there are important statistical drawbacks in the previous estimates. Accounting standards used in these studies are not always satisfactory for evaluating the change in economic performance.

For instance, one should compute the labour productivity per hour, not per capita, since the ratio of the yearly flow of output on a final reduced stock of workers overestimates the change in labour productivity.

The data: accounting, labour and financial variables

The data we use originates from three data sets, thanks to which we get labour, economic and financial information on French companies over the period 1994-2000. Note that these data sets are exhaustive, and are supposed to cover all companies in their field of interest, as it is compulsory for French firms to provide this information. The BRN database (Bénéfices Réels Normaux) provides extensive accounting and fiscal data on operating profit, debt and equity for any company with sales turnover above 530,000 euros. More than 500,000 companies are included in the data set each year. The DADS database (Déclarations Annuelles de Données Sociales) gives information on the labour structure (wages, qualifications), and covers more than 80% of employees. Financial Market data is provided by Euronext, for all listed companies3 that were quoted at least one year between 1994 and 2000.

The DADS data set is based on the plant level, from which we reconstitute the data at the firm level. Firms keep the same ID number, called Siren, throughout their economic life, allowing us to merge the BRN and DADS, and to follow companies along the period of study.

Finally, the Euronext dataset track the traded securities of listed companies, each stock being registered under a unique ID code, called Sicovam. We identify for each traded security the firm it represents and build the link between the Sicovam and the Siren identifiers. Whenever several securities are related to the same firm, we only keep the most traded stock.

Our measure of workforce and employment is based on the average number of employees over the year 4 . We thus avoid the important accounting bias induced by a measure of labour exclusively based on the end of the fiscal year. We then require that firms in our sample have a workforce of at least 20 employees in 1995. Doing so, we eliminate small companies for which the purpose of this study may be irrelevant. After filtering for influential data 5 and eliminating specific sectors6 , the final sample, named general set of companies or dataset A, has 62,798 observations, which have the same distribution over industrial sectors as the complete BRN data set. Note that one observation corresponds to one firm for which statistical information is available at least one year between 1994 and 2000. Only one fifth of these firms have available information each year from 1994 to 2000, due to a high rate of bankruptcies, creations and mergers7 Though most of them are already included in dataset A, we also analyse publicly-listed companies in a separate database, called dataset B. We include large French companies (quoted at the "Réglement Mensuel" and the "Marché au comptant") and medium sized companies with a good record in accounting practices and financial key figures (quoted at the "Second Marché"). The final dataset records 417 observations in our reference year (1996).

We focus on the change in workforce between 1995 and 1996, this being our key variable used to distinguish between two groups: employment downsizers and employment upsizers.

Our variable includes both full-time and part-time jobs. However, firms could reclassify fulltime positions into part-time positions, while our variable would fail to measure a decrease in plants of the same firm as a decrease in workforce.

5 We exclude observations corresponding to the first and last percentiles of economic and financial ratios.

workforce. We first address this concern: the share of part-time jobs in the total workforce for employment upsizers is actually decreased from 7.18% to 6.55%, and from 7.31% to 6.48% for employer downsizers. A Tukey's Studentized Range Test indicates that the difference between the two groups is not statistically different at a 5% threshold. More generally, firms are allowed to reduce the working time instead of cutting the workforce down. In such a case, our measure would be biased. We look at the change in the average number of hours worked per worker between 1995 and 1996. Firms were not reducing the number of hours instead of downsizing;

on the contrary, in the case of listed companies the two working policies were jointly used.

• The Profitability Ratios tell us whether a business is making profits -and if so whether at an acceptable rate.

-Return on Assets: ROA = Net Profit before tax, interest and dividend (EBIT)

Assets

-Return on Equity: ROE = Net Profit before tax, interest and dividend (EBIT) + Financial result Equity + Long-term debt

• The Operating Ratios give us an insight into how efficiently the business is employing those resources invested in fixed assets and working capital.

-Profit Margin: Pmarg = Net Profit before tax, interest and dividend (EBIT)

Sales

-Labour Productivity: Lprod =

Econometric strategy

Our variable of interest is the economic performance induced by the downsizing policy. Let Y 1 it be one of our main economic indicators (for instance, the level of ROE in one year), where the superscript stands for the treatment status (1 if a downsizing program has been adopted in 1995-96, 0 otherwise), and the subscripts i and t identify respectively the firm and the time period. Let also T be a dummy variable with value T i = 1 when the firm belong to the group of downsizers. At time t after 1996, the average treatment effect over the treated population 8 Adjustment coefficients were provided by Euronext.

is:

τ | true T =1 = E(τ i |T i = 1) = E(Y 1 it |T i = 1) -E(Y 0 it |T i = 1)
The problem of unobservability is summarized by the fact that we can estimate E(Y 1 it |T i = 1),

but not E(Y 0 it |T i = 1). A natural way to cope with this problem is to use a Difference In Differences (DID) estimator, whenever panel data on firms both before and after the treatment are available (at date t and t ):

τ | simple T =1 = E(Y 1 it -Y 1 it |T i = 1) -E(Y 0 it -Y 0 it |T i = 0).
This DID estimator is the one usually used in the management literature. However, we argue this estimator is potentially biased when it does not include the characteristic of firms that influence their participation. In the simple case where the treatment effect is homogenous among participating firms, assume that Y it = g(X i ) + τ T i + φ i + it , where φ i is an individualspecific fixed effect and it a temporary individual-specific fixed effect. Whenever the selection treatment is correlated with it the DID estimator is inconsistent and approximates to

τ | simple T =1 = τ | true T =1 + E( it -it |T i = 1) -E( it -it |T i = 0)
This bias has been illustrated by the so-called Ashenfelter's dip in the case of earnings gain and training programme. In our case, firms are more likely to adopt a downsizing treatment in 1996 if a temporary dip in profitability occurs the year before (for instance, if Y it falls below a threshold Y ). Then a faster growth in indicators such as ROE and ROA is expected among the treated.

Our main contribution is first to consider observable variables that affect employment policy, as the initial structural characteristics of the firms (in the economic, financial, labour and stock-market fields), and then to assess the importance of this temporary dip between 1995 and 1996. Conditioning on a large set of observable covariates X, we then assume that the remaining unobservable variables affecting employment policy (T ) do not affect the change in economic performance (Y ), and hence are not present in :

Y i1 , Y i0 ⊥ T i |X i , ∀i. Intuitively,
this assumes that conditioning on observable covariates, we can take assignment to treatment as having been random. If we define

τ i | corrected X i ,T i =1 = E(Y 1 it -Y 1 it |X i , T = 1) -E(Y 0 it -Y 0 it |X i , T = 0)
Then using the distribution of covariates X, an unbiased estimator of τ | true T =1 is given by

τ | corrected T =1 = E X i τ i | corrected X i ,T i =1
We proceed in two steps. The first step consists in estimating a logit model explaining the probability that a firm is involved in a workforce reduction, both for the publicly-listed Y it -Y it is explained by the employment policy T and the set of variables X we included in the logistic estimate. We then eliminate insignificant variables in an iterative procedure, using a threshold of 10%, and we report the coefficient of T whenever the variable is significant.

The OLS estimation is equivalent to a controlled Difference In Differences estimator. We use two different starting points for our estimates: 1995 and 1996, and we look for short-term , medium term (1996-97) and long term (1996)(1997)(1998)(1999)(2000) paths of performance variables.

The short term differences are used to assess the dip in economic performance before the treatment. The medium and long term differences give a gross measure of the treatment effect.

Finally, the net change in economic performance can be approximated as a difference between the gross change and the dip. Note that these estimates give the impact of employment policy on the gap between downsizing firms and other firms for each variable of interest, rather than on the level of these variables.

Our econometric strategy requires that labour, accounting and financial-market information is available both before and after the year when headcount reduction occurs 10 . That is to say, we only keep firms that are still registered in the database from 1995 to 2000. Due to a high rate of bankruptcy and turnover, only 13,615 companies from the general dataset are used for both the logistic and OLS estimates, and 222 companies for the listed companies dataset.

5 Determinants of Workforce Reduction Differ in Listed and

Non-Listed Companies

We run two different specifications for the logit model (Table 1). The first one includes a complete set of characteristics of firms, as well as information on workers and on financial markets (for the dataset B). Specifically, this estimate includes the size of the firm, different accounting data (profitability ratios, operating ratios, liquidity ratios, investment ratio), labour 10 We discuss at the end of section 5 the problems raised by missing data and bankrupted firms and how this may affect our estimates.

characteristics (gender, qualification, age, weight of part time work in the firm), and sectorial dummies. However, a large number of control variables can generate biases. In order to tackle this problem, the second estimate selects the more robust model. The procedure consists in dropping, one after one, the less significant variables, using a threshold of 10%.

Reducing Workforce as a Defensive Strategy for the Non-Listed Companies

We now focus on the results of the logit model for the general set of firms, using the second specification (column 2 of Table 1 reports the coefficients estimated).

First, it appears that the probability for a firm to be involved in a workforce reduction in 1996 increases with some structural parameters which are : 1/ the size of the firms (more than 500 employees in 1995: 0.6355); 2/ The proportion of old workers (more than 50 years old in 1995: 1.8823); 3/ The proportion of part-time workers in 1995 (0.2880). A large share of part-time workers is indeed a signal that the firm is using precarious jobs. 4/ The high level of the Herfindal index 11 (hi_95 : 1.2167). Firms who were initially facing less competitive 11 We first compute a Herfindhal index for each of the 600 sectorial activities in which companies are classified.

For activity k, this index is computed as a ratio between the sum of squared sales done at the companies' level, and the square of the whole sales in this activity.

Hk = i (SALES i k ) 2 ( i SALES i k ) 2
This index has a value between 1/Nk (where Nk is the number of companies in activity k) and 1. We then compute a index of the average competite pressure the firm has to face for the markets she is implemented in, as a weighted average of sectorial indexes H k :

H i = k SALES i k SALES i Hk where SALES i = k SALES i k
pressure were possibly oversized and had to adjust their workforce in 1996.

Second, the workforce reduction is correlated with a financial structure on the verge of bankruptcy. As expected, the probability of reducing workforce is higher in companies characterised by a low level of Return on Equity (ROE :-0.4696), an increase of long term debt pressure (Lpress: 0.2771) and of insolvency (interest cover : 2.9983). Notice that this insolvency increases despite the leverage effect implied by a reduction of firms assets (∆Assets_95 : -0.7519) in 1994-1995.

Third, the cost of labour of unskilled workers is not a significant predictor of workforce reductions. This suggests that downsizing stems from factors outside the firms, on the demandside, such as the decline of sales (sales turnover : -1.2436). Hence, the employment reduction appears as a flexible and defensive response to a fall in sales and profitability.

Note that due to the gloomy economic outlook, firms reduce employment in spite of a fostered productivity (d1_lpht = 0.2616). The rise in labour productivity is a necessary step before reducing workforce without a disorganization of the production. However, a higher cost of the highly qualified workers decreases the probability of downsizing. Indeed, this variable acts as a dummy variable for the firms making enough profits to share them with the top management. This means that firms do not analyse the wage of the highly qualified workers as a cost that should be reduced. Finally, the logistic regression does not show that publiclylisted, or group-owned companies have a higher probability to reduce the employment, which contradicts the hypothesis of shareholder-driven downsizing. However, this issue has to be raised in a separate logit estimation on publicly-listed firms.

Reducing Workforce as a Way to Improve Financial Stance for Listed

Companies

We turn now to estimate the probability for a publicly-listed firm to be involved in a workforce reduction. The columns 3 and 4 in Table 1 display the results of a logit model that includes nearly the same dependent variables as those used in the previous model. However, two exceptions must be noted. First, we exclude the variable group because all the publiclylisted companies belong to a group. Second, we include stock market-based variables: the change in capitalization, a dummy variable indicating whether the firm's stock outperformed the CAC40 index12 , and a dummy variable adjust which indicates whether the stock price has been adjusted by the firm. Column 3 shows the estimate of the benchmark model while column 4 reports only the variables that are significant (threshold of 10%).

The share price adjustment is negatively correlated with a workforce reduction (adjust_95:-0.6365), as this variable may indicate important restructuring the year before, such as a merger or an acquisition, which usually leads to a change in the number of shares and their price.

Workforce reductions in 1996, primarily, are more likely to occur in firms whose competitiveness is undermined by the high labour cost of unskilled workers (lchq1_95 : 1.7465 ). Both a low share and low wages of highly skilled workers (q3_95 : -1.9903, lchq3_95 :-1.5967)

indicate that firms where earnings before interest and taxes are too low to be shared among the managers are more inclined to shed jobs.

Second, listed-companies are more likely to be involved in a workforce reduction when they have to struggle in a more competitive sector (d1_Hi95 =-0,3440), with an inadequate skilled structure. For example, a low proportion of skilled workers (q3_95 :-1.9903) at low cost (lchq3_9 5:-1.5967) increases the probability to layoff.

Finally, the workforce reduction seems to be a strategic response to a poor financial stance, rather than an economic one. Downsizers do face financial difficulties, like a high level of the debt-pressure and a low ROE (respectively cper_tendt95 : 0.6978, ROE_95 : -4.7143). They deal with a high level of debt through a decrease of their assets in 1994-1995 (d1_assets_95 :-1.7709). However, and this is a major difference with the general set of firms, the demand side, as measured by sales turnover, is not a significant predictor.

Estimating the Impact of a Workforce Reduction

In tables 2, 3 and 4, we report the impact of a workforce reduction (dummy variable T), for both the simple estimator (second column) and the corrected OLS estimator (third column).

Each row indicates a different regression, where one of the economic indicators (first column) is explained by the dummy variable T for the simple estimator, and also a complete set of control variables for the corrected estimator13 .

For the general set of companies, headcount reduction improves labour productivity in the long run, up by +2.21% according to the simple estimator, though the OLS estimator gives a lower figure: only +1.21%. However, the net gain is a more accurate index because it includes the contemporary effects of 1995-1996. Consequently, while the simple analysis estimates the net gain of +2.21% between 1995 and 2000, most of the increase has vanished according to OLS estimator (+0.16%). Furthermore, improved productivity does not imply improved profitability, as the labour cost gap is significantly widened, though the increase is small in magnitude (the labour cost gap between downsizing firms and other firms is +0.39% or +0.50% higher in the short run).

Estimators provide very different results when it comes to the analysis of the financial stance of firms. Basically, the simple estimator exhibits, after 1996, a buoyant growth in profitability (Return on Assets: +5.77 points, Return on Equity: +2.29 points in the long run) and a heightened Profit Margin (+0.74 points). Positive effects of headcount reduction are longlasting: investment efforts are increasing (+2.72 points), meaning that firms become more confident about future prospects. However, there is no significant change in the long-term debt pressure or debt rate. Overall, the simple estimator would tip in favour of a successful offensive downsizing, with a positive impact of cutbacks on financial figures both in the short and the long run.

Conclusions drawn by the corrected estimator are definitely different. Between 1995 and 1996, while there is no significant effect on profitability according to the simple estimator, the OLS estimator gives an opposite picture, in which main financial indicators are strongly deteriorated, especially Return on Assets (-1.86 points) and Return on Equity (-1.87 points). Such a downturn should be attributed to a fall in demand. Therefore cutbacks are consistent with a defensive model of downsizing. Once selection bias is corrected, most of the net gain in ROE and Profit Margin vanishes, meaning that firms that make large layoffs are no longer those that will have a higher profitability growth.

Yet, over the whole period studied from 1995 to 2000, the change in the ROA gap between the two groups of firms is significant, with a net increase of 1.87 points. Such a net gain is not consistent with a defensive downsizing, and calls for further investigation on the change in assets and sales. First, both the simple and corrected estimators indicate that the gap in assets increases in the long run, showing that downsizing firms follow a different pattern of asset capitalization. One may have in mind a so-called "asset-lite" strategy, which calls for the company to slow down its investments. Secondly, the gap in assets is widened by more than the gap in sales (simple estimator: -8.57% vs -4.55%, OLS estimator: -5.26% vs -3.06%).

Hence, employment downsizers managed to increase their average amount of sales per unit of capital more than employment upsizers. This gain in productive efficiency, three times smaller with the OLS estimator than with the simple estimator, is at the core of a net increase in ROA in the long-run.

For listed-companies, the main result is that workforce reduction has no effect on future economic performance (ROE and ROA). However, the heightened labour productivity that the simple and OLS estimators exhibit, continues it's ascending trend in 1997 and 2000 (respectively: +12.37% and +14,53%, OLS estimators), while labour costs are growing more slowly (+9.14% in 1996-2000). This gap suggests that some profitability gains exist but are not yet transformed into an increase of the ROE. Finally, the changes in the structure of the qualifications suggest that firms are always involved in a restructuring process which does not allow to conclude to significant and positive results, possibly because of a high variance in the results of our sample.

The results that refer to listed companies can be compared to the ones found in [START_REF] Wayhan | The Impact of Workforce Reductions on Financial Performance: a Longitudinal Perspective[END_REF] on a set of the largest 250 U.S. corporations. Basing their estimates on the change in capitalization and sales, the authors insist that the pressure stockholders place on listed firms is a cause for downsizing. In our database, according to the logistic estimate, listed-firms that downsized are also characterised by a low return on equity. However, their stocks did not significantly under-perform in the year previous to the workforce reduction.

Finding a positive but fragile impact of workforce reduction on financial performance in the short run, Wayhan and Werner argue that workforce reduction could lead to a lower cost structure, which is leveraged into a competitive advantage by the firm's management. On the contrary, our OLS estimates show an increase in the labour cost and consequently no positive change in profitability.

So far our corrected estimates are based on firms for which data is available until 2000, one question arises about the meaning of missing data points: among companies that were present in our general data set in 1995 with at least 20 employees, 5.89% are missing in 1997 and 13.41% in 2000. First, we test the logical relationship between disappearing from dataset A, and the initial characteristics of companies (Table 5). As expected, a higher probability of being missing is linked with poor financial performances, such as a low level of return on equity and a decreasing profit margin. Poor profitability combined with a gloomy outlook have driven these firms into financial distress and bankruptcy. Listed companies included in dataset A are less likely to be driven into bankruptcy, which is consistent with our results in section 4. Notice that firms belonging to a group are more likely to disappear from the dataset. This might indicate that firms have merged with some other entities within the group. In this case one cannot easily assess the bias that might be generated on our previous estimates as these firms could be either in good or bad financial shape.

We thus leave aside the discussion of mergers and focus on the more severe problem of bankruptcy for missing firms that do not belong to a group. As firms that reduced their employment are twice more likely to face bankruptcy according to our data, our previous OLS estimates based on surviving firm may be upward biased 14 . Therefore, for the three key profitability ratios (ROA, ROE and Profit Margin), we estimate a lower-bound for the impact of workforce reduction by including reconstructed datapoints that were previously missing.

That is, each year datapoints are missing we input values that firms would have been likely

to exhibit, should they have survived 15 . OLS estimates are then run on the corrected dataset.

The results provide some reinsurance that these measurement concerns do not have a serious impact on our results. As in our previous estimate, there is still a positive net gain in ROA and Profit Margin between 1995 and 2000, though it is lower (ROA: +1.02 points, Profit margin: +0.02 points). While the net change in ROE was previously insignificant, corrected estimates show that the profitability gap is now worsened in the long run, but by only 2.3 points.

Concluding Remarks

This paper provides the first comparison of the relationship between workforce reduction and firms performance in listed and non-listed companies. It gives evidence that the nature of performance deterioration that triggered workforce reduction differs between the two group of firms. The former deals with a financial distress, while the latter struggles with a poor economic stance, close to bankruptcy, and use headcount reductions as a defensive response to a fall in sales. Moreover, the downsizing decison is made at different stage of firms' per-

14 We can also argue that given initial characteristics, a downsizing policy may reduce the occurence of bankruptcy, a positive effect not included in our previous OLS estimates. Future research could address this question, looking at a survival model. 15 As we are interested in a lower-bound estimate, when data is missing we input the two first percentiles of the observed distribution.

formance downturn. The reason may rest on the structure of governance. Listed firms, urged by shareholders, downsize before being close to bankrupcy. Defensive layoffs are purported to improve the financial stance before it becomes severe. On the contrary, other firms employ layoffs as the last strategy to avoid bankruptcy. This result gives evidence of a defensive downsizing, rather than the offensive strategy presented the management literature. Moreover, in the general set of firms downsizing is not shareholder driven. Second, if we do not correct for selection bias, our results reach to the same conclusion as the management researches: that is a positive effect of downsizing upon firms performance. Thirdly, after correcting for selection bias, our estimates do not support the management thesis. According to the corrected Difference in Differences estimates, for the general set of firms, the productive efficiency (ROA) is increased but at a slow rate: +1.8% between 1995-2000. It is three times smaller than with the simple estimate. The reason comes from a higher increase in the sales per unit of capital among employment downsizers, than among employment upsizers. Finally, the paper provides evidence for both groups, listed and non-listed companies, that downsizing policy does not foster financial performance (ROE). For non-listed companies, the reason rests on the priority given to the economic ratios over the financial ones. Further research should explain why the listed-companies do not improve their financial ratio, although it is a priority of their strategic plan. An important caveat needs to be made about our findings, as we do not control for unobservable variables. While the OLS method analyses the causal impact of exogenous "treatment", we focus on an endogenous decision chosen by the firms themselves (reducing or not reducing the workforce). A possible avenue of future research would be instrumental variable estimation, especially in the case of listed companies where the shareholder structure may be a variable that does affect downsizing probability, without directly affecting the future path of performance variables. In the same spirit that the Ahmadjian and Robinson's work (2001) on Japon, further research should also examine the spread of downsizing, focusing on economic and institutional pressures on firms. It would give an institutional dimension to the downsizing policies. a standard deviation are given in brackets; ** p-value < 0.05; * p-value < 0.10 NS stands for Non-significant at a 10% threshold. Endogenous variables are given in the first column. Each row corresponds to a specific regression, where the economic indicator (say, change in ROE between 1996 and 1997) is explained by the dummy variable T of the employment policy (simple DID estimator) and control variables (corrected OLS estimator). Only the coefficient of the dummy variable T is reported, if significant. a standard deviation are given in brackets; ** p-value < 0.05; * p-value < 0.10. NS stands for Non-significant at a 10% threshold. Endogenous variables are given in the first column. Each row corresponds to a specific regression, where the economic indicator (say, change in ROE between 1996 and2000) is explained by the dummy variable T of the employment policy (simple DID estimator) and control variables (corrected OLS estimator). Only the coefficient of the dummy variable T is reported, if significant.

-

  Labour Cost: Lcost = Wages + social contributions Total hours worked • The Liquidity Ratios indicate how capable a business is of meeting its short-term obligations as they fall due. -Debt rate: Debt= Long-term debt Equity + long-term debt -Long-term debt pressure: Lpres = Long-term debt Total debt -Interest Cover: Icov = Interests Sales • The Investment Ratio concentrates on the long-term health of a business. -Investment effort: Effo = Investments Fixed assets Furthermore, we include financial market information for publicly-listed companies, such as the change in stock price and the change in capitalization (computed in consecutive years). Stock prices have been adjusted, taking into account the change in the total number of shares 8 . We also use labour information regarding workforce, namely gender, qualification (divided into four categories from the less qualified to the most qualified), age (four categories) and the weight of part-time work in total workforce. Depending on their change in workforce in 1996, companies are then divided into employment downsizers and employment upsizers.

  and non-listed companies. The logit model allows us to characterise the nature of the workforce reduction. The second step estimates the specific effect of such a strategy upon different performance indicators Y , using standard OLS 9 where the change in economic performance

Table 2 :

 2 Estimated Impact of Workforce Reduction.Short-term Analysis (1995-1996)

	Variables a	Dataset A	Dataset B
	SHORT-TERM

Table 3 :

 3 Estimated Impact of Workforce Reduction.Medium-term Analysis (1996-1997)

	Variables a	Dataset A	Dataset B
	MEDIUM-TERM
	DIFFERENCES
	(1996-1997)	Simple DID Corrected OLS	Simple DID Corrected OLS
	PROFITABILITY RATIOS
	Return on assets
	∆ROA	0,02639**	0,01906**	0,03736*	NS*
		(0,00458)	(0,00460)	(0,02125)
	Return on Equity
	∆ROE	0,02541**	0,01747**	NS	NS
		(0,00449)	(0,00449)
	OPERATING RATIOS
	profit margin
	∆PMARG	0,00413**	0,00300**	NS	NS
		(0.00074)	(0.00074)
	labour productivity
	∆LPROD (log)	0,01267**	0,00918**	0,12117*	0,12374*
		(0.00426)	(0,00433)	(0,07191)	(0,07228)
	labour costs
	∆LCOST (total) (log)	NS	NS	NS	NS
	∆LCOST Q1 (log)	NS	NS	0,04962*	0,05726**
				(0,02523)	(0,02422)
	∆LCOST Q2 (log)	NS	NS	NS	NS
	∆LCOST Q3 (log)	NS	NS	NS	NS
	LIQUIDITY RATIOS
	Long-term debt pressure
	∆LPRES	-0,00808**	-0,00636**	-0,03846*	NS
		(0.00196)	(0,00196)	(0,02106)
	Debt rate
	∆DEBT	0,04416**	0,03138**	NS	NS
		(0,01372)	(0,01382)
	INVESTMENT EFFORT
	∆EFFO	0,01646**	0,01394**	NS	NS
		(0,00241)	(0,00242)
	SALES, ASSETS AND EQUITY
	Sales
	∆Lsales (log)	-0,02629**	-0,02255**	NS	NS
		(0,00311)	(0,00315)
	Assets
	∆Lassets (log)	-0,02964**	-0,02112**	-0,04595**	NS
		(0,00306)	(0,00308)	(0,02103)
	EMPLOYMENT
	Workforce level
	∆LABOUR (log)	NS	0,00692*	NS	NS
			(0,00399)
	Qualifications
	∆Q1	NS	NS	NS	NS
	∆Q2	NS	NS	NS	NS
	∆Q3	NS	NS	0,02733*	NS
				(0,01096)

Table 4 :

 4 Estimated Impact of Workforce Reduction.Long-term Analysis (1996-2000)

	Variables a	Dataset A	Dataset B
	LONG-TERM
	DIFFERENCES
	(1996-2000)	Simple DID Corrected OLS	Simple DID Corrected OLS
	PROFITABILITY RATIOS
	Return on assets
	∆ROA	0,05775**	0,03741**	NS	NS
		(0,00690)	(0,00686)
	Return on Equity
	∆ROE	0,02299**	0,01726**	NS	NS
		(0,00795)	(0,00794)
	OPERATING RATIOS
	profit margin
	∆PMARG	0,00744**	0,00469**	NS	NS
		(0,00109)	(0,00109)
	labour productivity
	∆LPROD (log)	0,022060**	0,01212**	0,15300*	0,14532*
		(0,00597)	(0,00604)	(0,07654)	(0,07566)
	labour costs
	∆LCOST (total) (log)	-0,00395*	NS	NS	0,09152*
		(0,00229)			(0,04559)
	∆LCOST Q1 (log)	NS	NS	NS	NS
	∆LCOST Q2 (log)	-0,00638**	NS	0,03912*	NS
		(0,00246)		(0,02093)
	∆LCOST Q3 (log)	NS	NS	NS	NS
	LIQUIDITY RATIOS
	Long-term debt pressure
	∆LPRES	NS	NS	-0,08874**	NS
				(0,04288)
	Debt rate
	∆DEBT	NS	NS	NS	NS
	INVESTMENT EFFORT
	∆EFFO	0,02724**	0,02160**	NS	NS
		(0,00267)	(0,00270)
	SALES, ASSETS AND EQUITY
	Sales
	∆Lsales (log)	-0,04555**	-0,03062**	NS	NS
		(0,00620)	(0,00620)
	Assets
	∆Lassets (log)	-0,08575**	-0,05264**	-0,12236*	NS
		(0,00675)	(0,00666)	(0,07064)
	EMPLOYMENT
	Workforce level
	∆LABOUR (log)	-0,02982**	NS	-0,16410*	NS
		(0,00940)		(0,09087)
	Qualifications
	∆Q1	NS	NS	NS	-0,02933
					(0,01611)
	∆Q2	NS	-0,00494**	NS	NS
			(0,00236)
	∆Q3	NS	NS	0,05050**	0,05396**
				(0,02248)	(0,02187)

In the U.S., in a layoff the worker is asked to temporarily leave during periods of weak demand, but will be asked back when business picks up.

It also the Cascio'perspective. As it is explained at he beginning of the paper,[START_REF] Cascio | Downsizing: What do we Know? What Have we Learned?[END_REF] asserted that downsizing is essentially a purposive strategy defined as "the planned eliminations of positions or jobs" while[START_REF] Cameron | Strategies for Successful Organizational Downsizing[END_REF] stressed these positive impacts of downsizing on efficiency and productivity.

We do not include foreign companies when they do not have a regular economic activity in France, and do not exist in the DADS data set.

Arithmetical average of the total number of employees at the end of each quarter, from the BRN. Note that the aggregate measure of employment we obtain does not record a displacement of employees between

Companies from specific sectors, such as Agriculture, Energy, Real-Estate Property, Financial services, Government, Associations, are put aside, as they either do not fit with the purpose of this study or with the traditional accounting analysis.

Among the 62,798 observations, some firms are only present at the very beginning of the period and then disappear; other firms only appear after our reference year1995-1996.

We have also used matching estimators, following the work of[START_REF] Rosenbaum | The central role of the propensity score in observational studies for causal effect[END_REF]; the propensity score of downsizing is then computed by the logistic estimate. The results are very similar to the OLS estimates, showing that we do not face a problem on heterogeneity or non-linearity. For a comparison of several evaluation methods, cf.[START_REF] Duflo | Empirical Methods[END_REF].

This index is made up of 40 shares, selected from the one hundred biggest companies listed on Euronext Paris, measured in terms of market capitalization. As the CAC40 is the benchmark for Euronext Paris, it is widely used by portfolio managers to measure performance. In 1996, the CAC40 index fell by 9.09%. The dummy variable is computed as 1(∆ Share Price > ∆ CAC40)

the set of variables X we included is the same as those used in the logistic estimate. We then eliminate insignificant variables in an iterative procedure, using a threshold of 10%.

Acknowlegements

We wish to thank the Institut National de la Statistique et des Etudes Economiques IN-SEE for its technical and statistical support, which permitted this study. Thanks to Sébastien Roux (INSEE), Anne Demartini (Autorité des Marchés financiers, AMF) for their helpful comments on datasets used in this research and Julien Grenet (EHESS-PSE) for his continuing help. We are grateful for thoughtful comments from Philippe Askenazy (CNRS-PSE), Allan Manning (CEP-LSE), Barbara Petrongolo (CEP-LSE), Thomas Piketty (EHESS-PSE) and Muriel Roger (INRA-LEA-PSE) for valuable comments and suggestions. All caveats apply. DIFFERENCES (1995-1996) Simple DID Corrected OLS Simple DID Corrected OLS PROFITABILITY RATIOS Return on assets ∆ROA NS -0,01867** NS NS (0,00465) Return on Equity ∆ROE NS -0,01874** NS NS (0,00405) OPERATING RATIOS profit margin ∆PMARG NS -0,00285** NS NS (0,00069) labour productivity ∆LPROD (log) NS -0,01047** NS NS (0,00379) labour costs ∆LCOST (total) (log) 0,00390** 0,00502** NS NS (0,00150) (0,00143) ∆LCOST Q1 (log) NS NS NS NS ∆LCOST Q2 (log) NS 0,00292* NS NS (0,00169) ∆LCOST Q3 (log) NS NS 0,03759* NS (0,02134) LIQUIDITY RATIOS Long-term debt pressure ∆LPRES -0,00349* NS NS NS (0,00211) Debt rate ∆DEBT NS -0,05789** NS NS (0,01366) INVESTMENT EFFORT ∆EFFO -0,01880** -0,02784** NS NS (0,00249) (0,00194) SALES, ASSETS AND EQUITY Sales ∆Lsales -0,08118** -0,07974** -0,26673** NS (0,00309) (0,00305) (0,09723) Assets ∆Lassets -0,04898** -0,03739** NS NS (0,00288) (0,00286) EMPLOYMENT Workforce level ∆LABOUR (log) -0,19416** -0,18840** -0,20499** -0,16979** (0,00273) (0,00275) (0,02955) (0,03052) Qualifications ∆Q1 -0,00482** -0,00456** NS NS (0,00123) (0,00122) ∆Q2 0,00227* NS NS NS (0,00130) ∆Q3 0,00343** 0,00397** NS NS (0,00091) (0,00089)

Sources: BRN, DADS, Euronext. Endogenous variable: being missing in the year of reference (Y=1) or not (Y=0). Coefficients with a * are significant with a threshold of 10%; coefficients with a ** are significant with a threshold of 5%. N stands for variables that were not included because they are no longer relevant for the dataset considered. Sectorial variables are included in both estimations.