Marc Gurgand
email: gurgand@pse.ens.fr

Eric Maurin
email: maurin@pse.ens.fr.

A large scale experiment: wages and educational expansion in France

Keywords: education, returns to schooling, natural experiment, signaling

We evaluate the wage impact of the strong and rapid increase in schooling levels experienced by the cohorts born after WWII in France. In order to identify the causal effect of education, we exploit the fact that the small group of people graduating from elite education (Grandes Ecoles) remained stable, while the rest of the system experienced tremendous transformation. This provides a well defined control group. Using large scale labor force surveys for the 1990's, we find that the cohorts that received more education have a lower wage gap, relative to Grandes Ecoles. We show that such a large scale experiment measures a social return to schooling even in the presence of signaling, whereas strategies based on quasi-experiments are not necessarily robust to signaling. Our instrumental variable estimation finds returns to schooling very similar to the rest of the literature, which is a strong case against the signaling hypothesis.

Introduction

As a result of renewed interest for education policies and evaluation methods, significant progress has been made over the past decade in the measure of returns to schooling. Convincing figures mostly confirm the picture based on traditional Mincerian wage regressions: that returns to schooling are about 8% in developped countries, maybe slightly higher (see [START_REF] Card | Estimating the return to schooling: Progress on some persistent econometric problems[END_REF], for a survey). They result from empirical strategies based on quasi-experiments: exploiting a source of exogenous variation in individual schooling levels allows to identify the causal effect of education on wages. Typical examples are Angrist and Krueger (1991) that use quarter of birth as a random event or Oreopoulos (2006) using a regression discontinuity design. group: the male population that receives elite education in the very traditional Grandes écoles system. It hardly changes in size (4%) and composition over this period: we thus assume that the unobserved characteristics of this group remain constant, relative to the 96% remaining population. Using a differencein-difference approach, we can trace out the wage gap between the Grandes écoles graduates and the rest of the population, accross cohorts observed in the same labor market -that of the 1990's. As the latter group becomes more and more educated, the wage gap should decrease if schooling increases wages. Moreover, this cannot result from mere signaling, as it cannot result from relative positions within the 96% "treated".

The rest of the paper is organized as follows. Using a canonical signaling model, Section 2 clarifies the conditions under which schooling returns based on quasi-experiments can measure social returns if the signaling model is relevant, and argues that the current approach is robust in this sense. Section 3 presents educational expansion in France over the period considered and the specific situation of Grandes écoles. The identification strategy based on difference-indifference and instrumental variables is exposed in Section 4 and results are presented in Section 5. Finaly, Section 6 explores the possibility that the results could be driven by the evolution of the minimum wage. Section 7 concludes.

2 Human capital vs. signaling : interpreting returns to schooling

From a policy perspective, the labor market signaling model introduced by [START_REF] Spence | Job market signaling[END_REF] challenges the literature on returns to education. This model argues that wage-education contracts can be rationalized as a separating device that reveals unobserved productivity to employers (see [START_REF] Riley | Silver signals: Twenty-five years of screening and signalling[END_REF], for a general exposition). As a result, the wage-education relationship does not necessarily imply that schooling raises productive capacity. Social and private returns to educational investment then differ. This section clarifies the conditions under which returns to schooling estimated using quasi-experiements are likely to measure social returns, if the signaling model is relevant. It illustrates that an empirical strategy based on a full scale experiment, like this one, is very likely to measure the full social returns to education in all cases.

Consider an economy with one schooling degree and initial productive capacity u continuously distributed in the population, with cumulative F (u). Individuals can undertake education at a cost c(u) that is decreasing in u and education enhances productivity by β ≥ 0. Call w 0 (u) the wage received by an individual with productivity u and no degree, and w 1 (u) her wage if she receives education. The optimal choice is always to choose education if c(u) ≤ w 1 (u)w 0 (u).

We consider two opposite models. In the human capital model, individuals are paid their own marginal productivity, so that:

w 0 (u) = u w 1 (u) = β + u
We specify the signaling model in a way similar to [START_REF] Bedard | Human capital versus signaling models: University access and High School dropouts[END_REF]. If a separating equilibrium exists2 , it is such that:

c(u * ) = w 1 (u * ) -w 0 (u * )
where u * is the marginal type that chooses to receive education, with:

w 0 (u * ) = E(u|u ≤ u * ) (1) w 1 (u * ) = β + E(u|u ≥ u * )
In essence of the signaling model, workers are paid the average productivity in their group. 3The typical quasi-experimental strategy to estimate returns to education does identify private returns to education, holding down unobserved ability, but it may fail to estimate the same parameter in either model. Consider an experiment or an instrumental variable that randomly lowers the schooling cost for a share α of the population: individuals thus treated (T = 1) face a cost e c(u) < c(u). Call u * and e u * the marginal types in the untreated and treated population respectively. The corresponding shares of unschooled individuals are F (u * ) and F (e u * ). The Wald estimator used in the quasi-experimental approach is based on the comparison of treated with untreated, namely:

E(w|T = 1) -E(w|T = 0) [F (u * ) -F (e u * )]
In the human capital model, the reduced form difference is:

E(w|T = 1) -E(w|T = 0) = [1 -F (e u * )] β + E(u) -[1 -F (u * )] β -E(u) = [F (u * ) -F (e u * )] β
so the Wald estimator measures β, which happens to be both the social and the private return in this model.

In the signaling model, provided that the employers cannot identify the treated from the untreated, which would be typically true in a small-scale experiment or if information on wether a person is under treatment is not readily observable, wages are now formed as a mixture of treated and untreated average productivity in each schooling level. Assuming a separating equilibrium exists, the marginal types in each population are such that:

c(u * ) = e c(e u * ) = w 1 (u * , e u * ) -w 0 (u * , e u * )
with equilibrium wages set at:

w 0 (u * , e u * ) = F (e u * )E(u|u ≤ e u * ) + (1 -α) [F (u * ) -F (e u * )] E(u|e u * ≤ u ≤ u * ) F (e u * ) + (1 -α) [F (u * ) -F (e u * )] w 1 (u * , e u * ) = [1 -F (u * )] E(u|u ≥ u * ) + α [F (u * ) -F (e u * )] E(u|e u * ≤ u ≤ u * ) [1 -F (u * )] + α [F (u * ) -F (e u * )] + β
Comparing wages of treated and untreated now gives:

E(w|T = 1) -E(w|T = 0) = F (e u * )w 0 (u * , e u * ) + [1 -F (e u * )] w 1 (u * , e u * ) -F (u * )w 0 (u * , e u * ) -[1 -F (u * )] w 1 (u * , e u * ) = [F (u * ) -F (e u * )] [w 1 (u * , e u * ) -w 0 (u * , e u * )]
and the Wald estimator thus measures:

E(w|T = 1) -E(w|T = 0) [F (u * ) -F (e u * )] = w 1 (u * , e u * ) -w 0 (u * , e u * ) > β
The signaling effect of education is captured altogether with direct productivity effect. In particular, a positive private return can be found, even when β = 0.

Notice also that, in this model, the quasi-experimental approach would be strictly equivalent to ordinary least squares. This could provide another possible interpretation to the fact that instrumental variable estimates are only slightly higher than ordinary least squares in most empirical applications [START_REF] Card | Estimating the return to schooling: Progress on some persistent econometric problems[END_REF]: the small rise could simply be due to measurement error correction in the absence of a more foundamental difference. Our results, however, will rule this out as a serious line of interpretation.

The center piece is that a quasi-experimental or instrumental variable strategy is robust to the signaling hypothesis only if we can assume that employers can sort out treated and untreated. The ideal controlled experiment that would draw randomly a small sample of persons would identify private returns perfectly, but would fail to demonstrate the existence of a social return. In contrast, an empirical approach based on a full scale policy that affects entire generations is more robust to the possibility that there is signaling. In such an experiment, treated and untreated can be sorted out by employers, as they well known that schooling distribution has evolved and that degrees held by younger people may mean less. An interesting implication is that robustness to signaling may come at the cost of a weaker identification strategy.

The experiment considered in this paper can be viewed as a decrease in schooling cost from c(u) to e c(u) < c(u) over two generations, which leads to a general schooling level increase in the second generation. The central point is that generations are sufficiently far appart that employers are very aware of their potential differences, in particular regarding their unobserved ability. The second generation being the treated one (T = 1), it is trivial that the Wald estimator measures β in the human capital model. In the signaling model, the first generation receives w 0 (u * ) and w 1 (u * ) as defined in equation 1 whereas the second generation receives w 0 (e u * ) and w 1 (e u * ). However, the Wald estimator still measures β:

E(w|T = 1) -E(w|T = 0) = F (e u * )w 0 (e u * ) + [1 -F (e u * )] w 1 (e u * ) -F (u * )w 0 (u * ) -[1 -F (u * )] w 1 (u * ) = [F (u * ) -F (e u * )] β
The intuition is very obvious: as long as the mean E(u) is the same in the two populations, an increase in average wage can only result from the real impact of schooling, β. Obviously, large scale experiences bring more information on social returns than do small scale experiments. As a corolary, the signaling model can be ruled out if returns are insensitive to the size of the experiment used for identification.

With generations far appart, however, we cannot exclude systematic differences in mean productive capacity accross cohorts. This would be confounded with the effect of schooling, β. In our empirical setup, therefore, we introduce "Grandes Ecoles" (thereafter GE) to control for these potential changes. In the context of this illustrative model, GE would come as a third level, rationed by a numerus clausus, so that the specific cutoff type u for GE does not change accross generations. The data examined in the next section show that this is a reasonable assumption. As a result, we can interpret changes in average GE wages between generations as a measure of cohort-specific changes rather than changing selectiviy into that level. This provides a control for systematic differences in unobserved ability accross cohorts.

Educational expansion in France

The cohorts born after World War II in France have experienced a strong and rapid increase in schooling levels, only comparable to the generalization of primary education that occurred in the late 19th century [START_REF] Marchand | Formation de la main-d'oeuvre et capital humain en France depuis deux siècles[END_REF]. Two phases can be distinguished. The first one concentrated efforts on vocational education, especially at secondary education level: it implied cohorts born from 1946 until the mid sixties. The second phase witnessed a fast increase in high school graduation for cohorts born from the mid-sixties to the mid-seventies.

In order to document this evolution, we use data from the French labor force surveys (enquêtes Emploi) from 1990 to 2002. This data is a yearly representative sample of 1/300 of the metropolitan population, in which schooling information is very detailed. We restrict our sample to individuals aged 26 to 56 (and out of school) at survey date. Because the survey has a three years rotating panel structure, we use only one out of three years in order to compute the cohort schooling levels, so as to exploit information on a single individual only once. When we get to estimating wages, we will use the same data, but with all available years and make proper adjustment to estimator variances for individual correlation over time.

Two periods of a fast increase in secondary education

As illustrated by table 1, the main target during the first phase was lower secondary school, with specific emphasis on its vocational components 4 : about 30% of men born in 1946 had one such degree and this figure rose to 38% twenty years later. For women, the figures are 21% and 30% respectively. Meanwhile, the share who ended up with no degree at all from initial education decreased from 43% (for men) and 53% (for women), to 30% (for men) and 28% (for women). This trend is due to reforms that have made access to lower secondary education easier and more democratic. During that period, the effort on vocational schooling also took place at a higher education level (grades "Brevet de technicien supérieur " and "Diplôme universitaire technologique", both lasting for two years after High School): the share of a cohort with such degrees increased from 5% to 9% for men and 6% to 12% for women. This resulted from the increased supply (in volume and diversity) that took place in the late 1960's.

The second phase was shorter but faster: over ten years (individuals born from the mid-sixties to the mid-seventies), the share ending up with no degree decreased by an additional 8 percentage points, and the share at lower vocational education also decreased, whereas access to higher vocational education continued to rise, by about 5 percentage points for both men and women. More importantly, the share of the population getting a High School degree ("Baccalauréat ") increased at an unprecedented pace: it doubled from 11 to 22% for men and rose from 15 to 24% for women. To a large extent, this increase was due to technical High School curricula, so that, overall, long term vocational training has been substituted for short term one over this period. It should be noted that the share of a cohort getting a Baccalauréat degree (or higher) is stable since the mid 1990's (that is for cohorts born after 1975 approximately).

To sum up, during the first period (twenty cohorts), average declared schooling duration increased by 1.4 years for men and 1.9 years for women (from 11.3 and 11.1 years, respectively). The second phase was even more impressive: over only ten cohorts, average schooling increased again by 1.6 years for both men and women.

Grandes écoles and higher education

While such strong changes were taking place, little or very little was happening at higher education levels. In table 1, we have broken down non-vocational higher education into three levels: the first level is a University two-years degree, corresponding to the American Colleges; the second level is University Graduate level, up to PhD. The third level is specific to the French two-tier system. Since the early 19th century, a higher education track, distinct from University, developped in parallel, under the label Grandes écoles. They form an elite sector, with rationed entry, ruled by selective examinations. The students from High School considered the most academically performant are accepted into two-years "Classes préparatoires", where they are trained for this examination. Depending on their ranking, they are then received into the clearly stratified GEs, where training lasts two to three years. Those who fail the selective examination join University at the Graduate level. GEs are traditionnally public and train mostly engineers, but private management schools are also present.

From table 1 and figures 1 and 2, it is apparent that the share of a cohort getting any of these three levels as a highest degree hardly changed during the first phase defined above, and this is in sharp contrast with what happened in the rest of the schooling system. This observation will be the basis for our empirical strategy. As access to GE is a rationed process that has not expanded, it is reasonable to assume that, year after year, it was attended by the "same" population: the 5% most successful students. This provides us with a control group, whose unobserved characteristics (specifically "ability") has not changed relative to the rest of the population. In another context, a related identification strategy has been used by [START_REF] Acemoglu | Changes in the wage structure, family income, and children's education[END_REF], who assume rank in the income distribution to be a proxy for the household unobservable characteristics that affect demand for schooling, and could be counfounded with income level.

In a nutshell, if educational expansion affecting the bunch of the population has a causal effect on wages, then the cohort-based wage gap with those who access GEs must have decreased. Such a decrease cannot result from unobserved characteristics.

Strictly speaking, however, the share of GE appears to be constant for men belonging to cohorts 1946-1964 (see tests in table 2), that is during the first phase of educational expansion. The male cohorts 1964-1973, as well as females over the whole period, experienced a slow but statistically significant increase. When we isolate the Top ranked GEs (that represent about 20% of GE students5 ), the share is also constant for the cohorts 1946-1964, and even so for a larger span, 1946-1970. In contrast, if we enlarge the GE group to include Gratudate studies, the share moves significantly with cohorts, even for males, first decreasing, then increasing.

In the following, we will thus consider two definitions of the "control group", GE or Top ranked GE, the latter for both cohorts sets 1946-1964 and 1946-1970. Also, in order to avoid issues related to labor supply and selectivity, we will restrict our wage analysis to males. 6 As illustrated in tables 1 and 2, the wage earners sub-sample shares the features of the whole population.

In order to justify further the hypothesis that the GE group maintained a stable composition relative to the population, we test that its social origins did not change. The labor force surveys have information on the father's profession. Using five social classes, in order to have sufficient data, table 3 regresses the probability to be a GE or Top-GE graduate, conditional on social origin, a cohort trend and a cohort trend interacted with social origin. The social stratification is very clear, but, for GEs, it does not change significantly over time (the same holds for the wage earners sub-sample). [START_REF] Albouy | Les inégalités sociales d'accès aux grandes écoles[END_REF] have made a comparable statement using the same data. The hypothesis seems less robust for Top-GEs: sons of modest origin tend to catch up with sons of managers, whereas the intermediate class tends to loose ground. However, these movements are very small. Furthermore, Euriat and Thelot (1995) have analyzed exhaustive data on the four most prestigious Top-GEs: Ecole polytechnique, Ecole normale supérieure, Ecole nationale d'administration (ENA) and Hautes études commerciales (HEC). They find that, for our period of interest, the students' social origins, relative to the social structure of the whole population, has remained remarkably stable.

4 Empirical model and identification strategy 4.1 The difference-in-difference model Consider the following wage equation:

log w = θ c + γ a + β d + u
where c is an index for birth cohort, a for age, d for schooling degree (starting d = 1 for no degree, up to d = n for the highest) and u unobserved productive capacity. As date effects cannot be separated from age and cohort effects, they are embedded in parameters θ c and γ a , that have thus no direct interpretation. This model is additive in age, following the recent literature (e.g. [START_REF] Duflo | Schooling and labor market consequences of school construction in Indonesia : Evidence from an unusual policy experiment[END_REF][START_REF] Meghir | Educational reform, ability and family background[END_REF], Oreopoulos, 2006, etc.) that relies on cohort analysis to estimate returns to schooling.

Wage expectancy conditional on schooling, cohort and age is:

E(log w|d, c, a) = θ c + γ a + β d + E(u|d, c)
Expectancy E(u|d, c) varies with schooling d both because schooling decisions are endogenous and because the schooling system is selective: this is the so-called "ability bias". This expectancy also varies with cohorts c because, as access to some degrees is enlarged or restricted, the average ability of the population with any level d may change. For that reason, variations in schooling levels accross cohorts cannot be used directly as a source of identification for parameters β d . Yet, the identification strategy pursued in this paper relies on the observation that the share and social structure of the population receiving the elite degree d = n ("Grandes écoles") remained stable over several cohorts, whereas huge changes were taking place in the rest of the schooling system. Therefore, we assume that selectivity between that level and the rest of the population remained constant accross cohorts, in the sense that:

E(u|d < n, c) -E(u|d = n, c) = E(u|d < n) -E(u|d = n), ∀c (2)
This provides a control group for difference-in-difference identification. For the upper level:

E(log w|d = n, c, a) = θ c + γ a + β n + E(u|d = n, c)
Also, for the group that did not get d = n:

E(log w|d < n, c, a) = θ c + γ a + X j<n β j P (d = j|d < n, c) + E(u|d < n, c)
Comparing average GE wages with that of the rest of the population defines ∆(c). Under (2):

∆(c) = E(log w|d < n, c, a) -E(log w|d = n, c, a) = X j<n β j P (s = j|d < n, c) -β n + E(u|d < n) -E(u|d = n)
A difference-in-difference estimator thus identifies:

∆(c 0 ) -∆(c) = X j<n β j [P (d = j|d < n, c 0 ) -P (d = j|d < n, c)]
Because the share getting a given degree rises with cohorts at all levels (except "no degree"), the difference ∆(c 0 )-∆(c) must be positive for c 0 > c, if it is the case that rising education increases wages (namely:

β 1 < β 2 < . . . < β n ).
As such, the wage gap between GE graduates and the rest of the population taken as a whole must decrease, as the general schooling level rises (even though the level of selectivity may change over time for any degree d < n). Note that this does not exclude the possibility that E(u|d = j, c) decreases along cohorts for every j < n, as a result of expanded access of each level to students with lower academic performance.

Notice that P (d = j|d < n, c) is observable, so that in principle, all β j 's could be identified separately. Unfortunately, over the period considered, those probabilities move in parallel, so that the contribution of each schooling level is not distinguishable empirically. The parameter that we can actually estimate instead provides a general assessement of twenty years of educational expansion, which is not disaggregated, but is very robust under hypothesis (2).

The model is estimated based on the following specification: 1946)] that must be positive if there is a return to education. Parameter α estimates E(u|d < n) -E(u|d = n)β n , that is the baseline wage gap between GE and the rest of the population, including the effect of unobserved characteristics, which cannot be distinguished from the causal effect β n .

log w = θ c + γ a + α1(d < n) + b c 1(d < n) + ε (

Instrumental variable estimation

The core hypothesis in this paper is that a change in the differential wage between GE graduates and the rest of the population accross cohorts can only be explained by educational expansion. In order to quantify this effect in terms of return and compare orders of magnitude with the literature, we can correlate the drift in b c with the increase in average schooling years (below level n) by cohort. More formally, we can switch this approach to an instrumental variable strategy. Consider the equation:

log w = θ c + γ a + α1(d < n) + rS + ε (4)
where S is an individually defined variable that measures schooling time. 7 The difference-in-difference model assumes that the interaction between cohort dummies and no-GE dummy, explains education but has no direct effect on wages, once no-GE dummy and cohort dummy are included additively in the model. As a result, the set of interactions provides a vector of instrumental variables for S. This approach does not imply that we have to assume a constant marginal return to schooling time, r, accross education levels or accross individuals. Rather, the IV estimate can be interpreted as a weighted average of possibly heterogenous returns, what [START_REF] Angrist | Two-stage least squares estimation of average causal effects models with variable treatment intensity[END_REF] call an average causal response (ACR). To be explicit, take cohort 1946 as a reference, consider cohort c and the instrument

Z c = 1(d < n) × 1(cohort= c). Specify log w = θ c + γ a + α1(d < n) + R S + u,
where R S is a counterfactual individual specific shift associated with S years of education. For IV to estimate the ACR, two assumptions are needed. First, the monotonicity assumption that moving from cohort 1946 to cohort c would never imply reduced schooling for anybody. Second, the independance assumption that the instrument is orthogonal not only to counterfactual wages at any level S, but also to the sequence of schooling responses to the instrument itself. Under these conditions, the IV parameter r (based on the two cohorts only in this example) estimates:

X k ω k E [R k -R k-1 |S 1 ≥ k > S 0 ]
where S 0 is counterfactual schooling for Z c = 0 and S 1 is counterfactual schooling for Z c = 1 and:

ω k = P (S 1 ≥ k > S 0 ) P j P (S 1 ≥ j > S 0
) is a weighting factor. R k -R k-1 captures the local return at schooling duration k. The condition S 1 ≥ k > S 0 implies that only individuals who would increase their education under the effect of the instrument contribute to the estimation. Moreover, the education levels at which changes are more likely make a stronger contribution. The monotonicity assumption, that allows such an interpretation, is reasonable in this context: it implies that the increase should take place at all education levels (except GE) along cohorts, which we actually observe. The independance assumption basically requires here that, if cohort 1946 had been exposed to the cohort c schooling system, it would have had the same schooling levels on average.

When more cohorts are included, the estimator measures a weighted sum of these effects, where cohorts that contribute a more efficient estimation receive more weight (see Angrist and Imbens, 1995, theorem 2).

Estimation results

Difference-in-difference estimation

Equation ( 3) is estimated based on data on males aged 26 to 56, from the 1990 to 2002 labor force surveys enquête Emploi, using the hourly log-wage rate as an explained variable. As individuals are surveyed three years in a row -following a rotating panel structure -we allow for panel individual autocorrelation. Figure 3 presents the series of estimates of the cohort wage gap between GE and non-GE, (α + b c ) defined in equation ( 3), for several definitions of the control group over the samples 1946-1964 and 1946-1970. It shows very clearly that the wage gap between GE graduates and the rest of the population is decreasing along cohorts, for both control groups and samples. Assuming that the differences in unobserved ability accross the two groups did not change, we can interpret this as the result of the strong increase in the schooling level of the "treated" group.

Figure 3 also shows that the evolution of cohort effects is reasonably approximated by a linear trend. The corresponding specification is presented in table 4 for the two samples and GE or Top-GE when applicable. Specification (1) implies that the wage gap between GE students and the rest of the population has decreased by about 0.35% per generation, or about 6.3% in all 18 generations.8 During this first shooling expansion period, the share of the male population with some degree increased by 14% (table 1). Therefore, the average degree was worth a 45% wage increase (0.063/0.14).

The effect is somewhat higher when we consider Top-GEs and when we extend the cohort range. The wage gap reduction rises up to 1% per generation. This trend in wage gap reduction is all the more striking, as school duration increased slightly for GE graduates over the period (see table 1). An obvious reason for these higher figures is the accelerated increase of schooling level that took place during the second period of educational expansion. In addition, as we extend the cohort range, the very nature of schooling progress changes. Indeed, during the second period, about 22% of the population has shifted from no degree or lower vocational degree to High school graduation or tertiary vocational degree (table 1). These maybe more rewarding degrees, or the population concerned by these very changes may have higher latent returns to schooling than the population affected by the increase at lower secondary education, that characterized the first period. Generally, one must bear in mind that this is an overall evaluation of a set of actions that resulted in increased levels but included many qualitative changes and a complex intrication of schooling policies.

Instrumental variable estimation

Relating the above cohort trends with the corresponding increase in schooling duration provides a measure of the returns to schooling that would make this results comparable to the literature. Equivalently, instrumental variable estimation of the standart returns to schooling model are presented in Table 5, using the declared age at completed initial education to measure schooling time. Ordinary least squares are also reported. The OLS returns are around 6%, which is very much in line with what is usually found in developped countries.

The IV estimates use either GE dummy interacted with cohort dummies or Top-GE dummy interacted accordingly. For the base sample, 1946-1964, they are slightly lower, between 5 and 6%, depending on the set of instruments. This should not result from weak instruments that would bias the IV estimator towards OLS, because the excluded instruments explain schooling duration very well. The estimation based on the extended cohorts generates a somewhat higher return (9.8%), but it is not significantly different from the OLS at the 5% level. As discussed in the above section, the nature of the educational expansion for the additional cohorts 1965-1970 is specific: the reduced form has shown that, when these cohorts are included, the cohort effect is somewhat steeper. Using different identification strategies, Maurin and McNally (2005) and Maurin and Xenogiani ( 2007), find very comparable results for France. 9In theory, there is a possibility that selectivity could drive those results: the younger low education cohorts could be more often unemployed, so that the productivity of the working sample could be higher than for older cohorts, if the unemployed are on average less productive. Yet, the 26-56 age range is not the most sensitive to unemployment. Moreover, selectivity correction can be implemented using the GE×cohort interactions as instruments for both schooling and selection probability, because there are as many as there are cohorts. When we introduce the selection probability as a control function in various ways (as an inverse Mills ratio or as a quartic function), returns to schooling remain very robust.

Overall, these results are very much in line with the previous litterature that uses IV based on quasi-experiments to estimate the returns to schooling. In most of the available estimates, OLS returns are most often around 6 or 7% and incrase slightly -but not always -when IV is implemented (cf. Card, 2001, table II). In most cases, the difference between OLS and IV are not significant. As argued, our estimation based on a large scale experiment should be more robust to the possibility that the underlying model involves signaling, so that this 5 to 9% coefficient can be considered a measure of the social return to schooling. There is no evidence that this return is very different from either OLS in France or estimates elsewhere based on strategies that would measure the private rather than the social return if they were distinct. In this context, there is little evidence in favor of the signaling hypothesis.

6 The minimum wage impact on wage equalization

In assumption (2), E(u|d < n, c) and E(u|d = n, c) are understood to describe the evolution of the unobserved characteristics of individuals, that may affect simultaneously school performance and the wage rate. We have argued that it is a reasonable hypothesis, in the French context, to consider that E(u|d < n, c) -E(u|d = n, c) is constant accross cohorts 1946-1964 or 1946-1970 (depending on the definition of n). As a result, we attribute changes in the wage gap, as depicted in figure 3, to the increased schooling level of the treated group. But we may consider that labor market or other events also contained in the residual u, could equalize wages along the cohort dimension. In this section, we explore the possibility that the minimum wage could have such an equalizing effect, if workers at lower schooling levels have benefitted from minimum wage increases.

Figure 4 shows the strong movements in the (net) minimum wage that occurred since WWII, whether expressed in real terms or relative to the average wage. First, notice that the current level of the minimum wage is not a source of concern. As cohorts get more educated on average, they should be less affected by the minimum wage truncature, conditional on age: by itself, this would predict an increase in the wage gap, rather than the observed decrease. However, conditional on age, different cohorts are not observed at the same date, therefore at the same level of the minimum wage. Yet, as illustrated in figure 4, the minimum wage was fairly stable during the 1990's, our period of observation. All in all, current minimum wage is not an obvious alternative explanation to our findings.

The impact of past minimum wage could be more relevant, to the extent that entry wages have lasting effects. [START_REF] Baker | The wage policy of a firm[END_REF] argue that this is the case in the United States. In the same line, [START_REF] Beaudry | The effect of implicit contract on the movement of wages over the business cycle: evidence from microdata[END_REF] find a link between past unemployment and current wage, although von [START_REF] Wachter | In the right place at the wrong time. The role of firms and luck in young workers' careers[END_REF] argue the evidence is not so strong. Still, there is no available evidence for the minimum wage impact specifically. If there is persistency, then a wage increase (relative to GE) accross cohorts could be explained by a trend in the minimum wage at the time of entry in the labor market. For cohorts 1946-1964, the relevant minimum wages are those prevailing between 1959 and 1990. Over this period, the minimum wage was first stable (or decreased relative to the average wage), then increased strongly, then was stable again (figure 4).

To test whether the minimum wage is an omitted variable in the previous estimations, we include it, evaluated at time of entry on the labor market. Our initial equation is thus extended to:

log w = θ c + γ a + α1(d < n) + rS + λw c+6+S + ε 0
where w t is the minimum wage at date t and c + 6 + S measures age at entry on the labor market. It is important to note that, because schooling is endogenous, so is w c+6+S . A higher minimum wage due to longer schooling and thus later entry, may be confounded with unobserved ability. We can instrument this variable using the same instruments as previously, because there are as many instruments as there are cohorts. Identification is made possible because the schooling variable is increasing steadily with cohorts, whereas the minimum wage is stable for early and late cohorts.

Results are presented in table 6, with the real minimum wage included in levels. Based on ordinary least squares, entry minimum wage seems to have some effect on future wages, but this has no impact on the estimated returns to schooling in any specification. Once schooling and minimum wage are instrumented, there is no longer any evidence that the latter has some impact, whereas the IV returns to schooling remain similar to previous results, or even slightly higher but also less precisely estimated.

Another approach is to leave out the workers with low education, say less than S = 10, thus those most affected by the minimum wage. 10 If minimum wage changes were driving the results, the returns to schooling should drop. If we do this, however, the "treated" population is no longer stable accross cohorts: as general education levels increase, individuals with a given low schooling level represent a smaller and smaller share of the population. Should they be in the lower tail of productive capacity, the remaining group (S ≥ 10) would have decreasing average unobserved productivity. Therefore, if anything, we would underestimate the returns to schooling. The corresonding IV estimates are presented in table 7. Coefficients are similar to baseline IV, with only an increase for cohorts 1946-1964 with GE as a control group, but not a significant one, given confidence intervals. Overall, we can find no convincing evidence that the minimum wage does explain the constant trend in wage gap reduction.

Conclusion

This paper uses a difference in difference strategy to evaluate a large scale policy: the significant and fast increase in schooling levels that benefited the cohorts born after World War II in France. It exploits the fact that the group that had access to the elite segment of the educational system, the Grandes écoles, has remained stable in size and social composition. As a result, the evolution of the wage gap between GE graduates and the rest of the population, cohort after cohort, should reflect the fact that the schooling level of the "treated" group has increased steadily relative to the "control", and should not result from changes in their relative unobserved characteristics. Using labor market data for the 1990's, thus observing every one in the same economic environment, we find indeed that the wage gap has become regularly and significantly smaller cohort after cohort. Quantifying this effect implies wage returns to schooling between 5 and 9%, in the usual range for developped countries.

Naturaly, there can be other events that, for some cohorts and not others, affected GE graduates differentialy from the rest of the population. In particular, we explore the possibility that increased minimum wage over much of the period when these cohorts entered the labor market, could explain the gap reduction. Our results are robust to this possibility.

Although our identification strategy is not based on a quasi-random feature, it offers an evaluation of a large and long lasting experiment taken as a whole. Moreover, the interpretation of returns to schooling is straightforward in such a setup: steady wage increases of entire cohorts is not compatible with the view that there are private but no social returns to schooling. As our estimated returns are similar both to ordinary least squares and to returns estimated elsewhere, it is likely that there is little gap between private and social returns in general. This is a strong case against the signaling hypothesis. Minimum wage relative to mean wage

Female schooling shares by cohort

  3) with 1(d < n) a dummy for non-GE graduate. Normalizing b 1946 = 0, for instance, b c would estimate P j<n β j [P (d = j|d < n, c) -P (d = j|d < n,

Figure 3 :

 3 Figure 3: Cohort specific relative wage -Males born 1946-1964

  Figure 4

  Source: EnquêtesEmploi, 1990Emploi, /1993Emploi, /1996Emploi, /1999Emploi, /2002;; 26 to 56 years old men. Variances are adjusted for survey year-wise effects. Based on 119,720 observations for1946-1964 (89,126 for wage earners) and144,958 for 1946-1970 (108,967 for wage earners).

	Sub-sample
		F-test	p-value	F-test	p-value	F-test	p-value
	Born 1946-1964	4.63	0.0000	0.95	0.5140	1.48	0.0878
	Born 1946-1964, earning a wage	3.42	0.0000	0.97	0.4867	1.20	0.2473
	Born 1946-1970	4.23	0.0000	2.00	0.0026	1.27	0.1723
	Born 1946-1970, earning a wage	3.71	0.0000	2.23	0.0005	1.02	0.4283

Table 2 :

 2 Effect of birth year on the proportion of men graduating from elite education

	Top GE

Source: Enquêtes

Emploi, 1990Emploi, /1993Emploi, /1996Emploi, /1999Emploi, /2002;; 26

to 56 years old men. Linear probability model. Standard errors are adjusted for survey year-wise effects. The cohort variable is standardized

(original mean :1957.78 and s.e.: 7.34

).

Table 3 :

 3 Probability to access elite education, conditional on social origin

	Grandes écoles : Males born 1946-64	Top Grandes écoles : Males born 1946-64

Source: Enquêtes

Emploi, 1990Emploi, to 2002;; 26

to 56 years old men. Explained variable is log-wage. Each couple of coefficients is from a separate regression. Standard errors account for the panel autocorrelation structure. The cohort variable is standardized

(original mean :1957.78 and s.e.: 7.34

).

Table 4 :

 4 Difference-in-difference wage regression Enquêtes Emploi, 1990[START_REF] Th | Analyse économique des politiques éducatives: l'augmentation de la scolarisation en France de 1982 à 1993[END_REF] 26 to 56 years old men. Explained variable is log-wage. Each coefficients is from a separate regression. Standard errors account for the panel autocorrelation structure. Excluded instruments for schooling time are Non-GE or Non-Top-GE dummy interacted with cohort dummies.

	Males born 1946-1964	Males born 1946-1970

Table 5 :

 5 OLS and IV returns to education Enquêtes Emploi, 1990 to 2002; 26 to 56 years old men. Explained variable is log-wage. Minimum wage is expressed in real terms. Standard errors account for the panel autocorrelation structure. Excluded instruments for schooling time and minimum wage are Non-GE or Non-Top-GE dummy interacted with cohort dummies.

	Males born 1946-1964		Males born 1946-1970
	OLS	IV	OLS	IV

Table 6 :

 6 Impact of minimum wage at labor market entry EnquêtesEmploi, 1990Emploi, to 2002;; 26 to 56 years old men, with more than 10 years of schooling. Explained variable is log-wage. Each coefficients is from a separate regression. Standard errors account for the panel autocorrelation structure. Excluded instruments for schooling time are Non-GE or Non-Top-GE dummy interacted with cohort dummies.

	IV

Table 7 :

 7 IV returns to education, restricted to individuals with 10 years of schooling or more

	IV	IV

In a very different setup,[START_REF] Acemoglu | How large are human capital externalities? Evidence from compulsory schooling laws[END_REF] cannot find positive externalities deriving from average schooling in the population, which implies similarly that private returns are no different from social returns.

We do not discuss here the conditions for existence and unicity. We simply consider the possibility that there is signaling in the economy and the implications this would have for estimation methods.

Empirical wage heterogeneity within schooling levels is not modelled here and could result from job heterogeneity for instance.

Grades: "Certificat d'aptitudes professionnelles " and "Brevets d'études professionnelles".

The list of Top GEs is imposed to us by enquête Emploi.

If anything, the fact that women access to GE has increased over cohorts would imply that male became even more selected. We are thus on the safe side.

Notice that, as we condition on age rather than on potential experience, the return that we estimate takes into account the fact that an additional schooling year reduces potential experience by one year. In that sense we compute net returns. They should be lower than the returns based on traditional Mincer equations, but are more policy relevant and are comparable to much of the returns estimated in the recent literature.

This figure is not directly apparent in the table because the cohort variable has been normalized.

More precisely, those two papers report returns to the order of 13-14%. But this paper measures years of education as actual duration spent studying, including grade repetition, whereas the above mentioned papers use years implied by attained levels. Using the latter measure here would generate returns around

10-13%. The difference between our baseline 6-9% and this 10-13% indicates how much is spared by frequent repetition in the French system.

The S = 10 threeshold is used because a large share of increased schooling under the period resulted from a shift from S = 8 to S = 10 to 12.

the CEPREMAP Public finance and redistributive policies program. We thank seminar participants at Crest, Université


Jonathan Tireau

Laetitia Van Schoors

Karim Benzarti

Xavier Colin

Consequences of thermo-and photo-oxidation on end-use properties of pure PE

Keywords: Polyethylene, thermo-oxidation, photo-oxidation, embrittlement PACS: 82.35.-x

Thin films of polyethylene have been exposed to thermal and photochemical accelerated agings at different temperatures and different UV intensities. In both cases, mechanical tests performed on aged samples revealed a severe embrittlement of the PE material. However, a multi-scale analysis showed that i) under thermal oxidative conditions, such embrittlement mainly resulted from a reduction of molecular mobility of the amorphous phase due to annealing phenomena, while ii) it was rather associated to a competition between chain scission and cross-linking processes under photo-oxidative conditions. As regards the oxidation mechanisms, it was found that thermo-oxidation is clearly initiated by a bimolecular decomposition of hydroperoxides, while photo-oxidation seems rather initiated by the photolysis of ketones and hydroperoxides. Over a critical threshold of the hydroperoxide concentration, thermolysis of hydroperoxides seems to predominate and even to control the overall photo-oxidation kinetics.

INTRODUCTION

Polyethylene (PE) is commonly used in many civil engineering applications (geomembranes, protective sheaths of cable stayed bridges, gas and water pipes...) because of its low cost, easy processability, good impermeability and chemical resistance. However, in external environment, several factors such as ultraviolet radiations, seasonal temperature variations or washing by rain water can lead ultimately to a deterioration of the PE material [START_REF] Colin | Lifetime Prediction of Plastics[END_REF] . It is usually admitted that photo-oxidation is the main cause of degradation for PE [START_REF] Gijsman | The long term stability of polyolefins[END_REF] . Nevertheless, other physical and chemical phenomena, such as physical loss and chemical consumption of antioxidants [START_REF] Colin | [END_REF]4 , thermo-oxidation 5 or annealing phenomena 6 can also contribute to an overall degradation of the material properties.

The objective of this study was to investigate the respective effects of these various phenomena on the durability of pure PE. With this intention, thin PE films (200-300 µm thick) were exposed to two types of ageing conditions: thermal ageing in a ventilated oven at different temperatures (40, 60, 90, 100 and 110°C) and photochemical ageing in a UV chamber at different intensities (0.51W.m -²; 0.65W.m -² and 0.75W.m -²) at 38°C. Exposed samples were then characterized using a multi-scale approach: i) infrared spectrometry (IR) at the molecular level, ii) rheology in the melt state and dynamic mechanical analysis (DMA) at the macromolecular level, iii) differential scanning calorimetry (DSC) , X-ray diffraction (XRD) and density measurements at the microstructural level and iv) tensile tests at the macroscopic scale.

RESULTS AND DISCUSSION

The main results obtained for exposed specimens are presented and discussed in this section.

At the molecular scale, IR spectroscopic analysis coupled with chemical derivatization (NH 3 treatment) made it possible to identify and quantify the different oxidation products (Figure 1). As regards thermo-oxidation, the initiation stage consisted mainly in the bimolecular decomposition of hydroperoxides. Differently, for photooxidation, photolysis of hydroperoxides and ketones (by Norrish II) were identified as the main initiation reactions. Beyond a critical threshold of the hydroperoxide concentration, thermolysis of hydroperoxides seemed to predominate over the two other initiation reactions and even to control the overall kinetics of oxidation. At the macromolecular scale, in the case of thermo-oxidation, a large part of the carbonyl species resulted from the β-scission of alkoxyl radicals. Chain scissions (s) lead to a sharp decrease in the weight average molecular mass (Figure 2a). As regards photo-oxidation, rheology in the melt state revealed, on the one hand, the occurrence of chains scissions (s) probably due to the photolysis of hydroperoxides and ketones, and on the other hand, the appearance of a cross-linking phenomenon (x) from the early period of exposure (Figure 2b). At the microstructural scale, DSC analysis revealed microstructural changes due to annealing phenomena during the first moments of the thermo-oxidative treatment. Due to an increase in the molecular mobility, the smaller crystalline lamellae (lc) seemed to coalesce, hence forming thicker lamellae exhibiting a melting temperature higher than the ageing temperature (Figure 3a). Thus, annealing could lead to a more uniform crystalline structure in PE. After the oxidation induction time, chain scissions destroyed the entanglement network located in the rubbery amorphous phase and thus liberated small macromolecular segments. These segments were able to diffuse toward interfaces between amorphous and crystalline regions, leading to a global increase in the lamellae thickness (Lc) and crystallinity ratio 7 . Such effects were even more pronounced in the case of photo-oxidation due to chimicrystallisation phenomena (Figure 3b). At a macroscopic scale, ultimate tensile elongation of PE decreased sharply during thermal ageing (Figure 4a). Embrittlement appeared before the oxidation induction time, hence before the onset of chain scission. Thus, embrittlement might result mainly from a reduction of molecular mobility of the amorphous phase, which is itself a consequence of the increasing crystalline ratio. Differently, for photochemical ageing, mechanical properties of PE seemed directly affected by changes in the weight average molecular mass. A competition between chain scission and cross-linking processes led rapidly to the embrittlement of the material (Figure 4b).

FIGURE 1 .

 1 FIGURE 1. Changes in the absorbance spectra after NH 3 treatment on films of pure PE (200 µm) exposed a) in a ventilated oven at 100°C for periods up to 384h, b) in a Weather-o-meter chamber at 0.75W.m -² and 38°C in atmospheric air for periods up to 42 days.

FIGURE 2 .

 2 FIGURE 2. Evolutions of the rheological behaviours for pure PE films (200µm) aged a) in a ventilated oven at 100°C, b) in a Weather-o-meter chamber at 0.75W.m -² and 38°C in atmospheric air.

From

FIGURE 3 .

 3 FIGURE 3. DSC derivative-thermograms for samples aged a) in a ventilated oven at 100°C, b) in Weather-o-meter chamber at 0.75W.m -² and 38°C in atmospheric air (heating rate of 10°C/min).

FIGURE 4 .

 4 FIGURE 4. Tensile behaviours of pure PE films (200µm thick) exposed for various periods of time a) in a ventilated oven at 100°C, b) in Weather-o-meter chamber at 0.75W.m -² and 38°C in atmospheric air.


Dulin Chen

Laurent Bako

Stéphane Lecoeuche

A Recursive Sparse Learning Method: Application to Jump Markov Linear Systems

Keywords: Sparse estimation, Recursive identification, Error correction, Robust identification, Jump Markov Linear Systems (JMLS)

This paper addresses the problem of identifying linear multi-variable models from the inputoutput data which is corrupted by an unknown, non-centered, and sparse vector error sequence. This problem is sometimes referred to as error correcting problem in coding theory and robust estimation problem in statistics. By taking advantage of some recent developments in sparse optimization theory, we present here a recursive approach. We then show that the proposed identification method can be adapted to estimate parameter matrices of Jump Markov Linear Systems (JMLS), that is, switched linear systems in which the discrete state sequence is a stationary Markov chain. Some numerical simulation results illustrate the potential of the new method.

INTRODUCTION

In this paper, we consider the problem of identifying a linear multi-variable model, subject to sparse vector errors. More specifically, given a collection D N = {x (t) , y (t)} N t=1 (with N possibly infinite) of data generated by a model of the following form, y (t) = Hx (t) + f (t) + e (t) , (1) the problem is to estimate the matrix H ∈ R m×n . Here, y (t) ∈ R m is the output of the model, x (t) ∈ R n is referred to as an explanatory variable and sometimes as a regressor vector; {e (t)} is an unknown i.i.d noise sequence with a normal distribution N (0, σ 2 e I m ). In equation [START_REF] Candès | Highly robust error correction by convex programming[END_REF], the sequence {f (t)} ⊂ R m is assumed to be an unknown but sparse sequence of possibly gross error vectors. By sparse sequence of vectors, it is meant here that the cardinality of the set {t : f (t) = 0}, is very large compared to the total number N of data. In other words, only a few number of vectors f (t), t = 1, . . . , N , are nonzero vectors. Those nonzero vectors in the sequence {f (t)} can in principle have arbitrarily large magnitudes. Typically, the sequence {f (t)} can be thought of as occasional errors occurring in data measuring devices. For example, it can be viewed as accounting for data transmission/reception errors over communication networks [START_REF] Candès | Highly robust error correction by convex programming[END_REF] or sensor/actuator faults in linear dynamical systems. A particularly interesting application of the above problem concerns, as will be suggested later in the paper, the identification of switched linear models. In this case, if one focuses on the problem of recovering a specific linear multi-variable submodel, then the data generated by the other submodels can be regarded as a sparse sequence of errors [START_REF] Ozay | A sparsification approach to set membership identification of a class of affine hybrid systems[END_REF][START_REF] Bako | Identification of switched linear systems via sparse optimization[END_REF][START_REF] Bako | A recursive identification algorithm for switched linear/affine models[END_REF]. We will elaborate more on this point in Section 3.

The objective of this paper is to provide a recursive estimate of the matrix H in model [START_REF] Candès | Highly robust error correction by convex programming[END_REF] as the data {x (t) , y (t)} are getting sequentially acquired in time. In doing so, no additional information other than that of sparsity is available on the vector sequence {f (t)}. As argued earlier in the paper, this problem has numerous applications in various engineering fields [START_REF] Candès | Highly robust error correction by convex programming[END_REF][START_REF] Candès | Compressive sampling[END_REF].While batch sparse estimation has been extensively treated in the existing literature [START_REF] Candès | Enhancing sparsity by reweighted l 1 minimization[END_REF][START_REF] Donoho | For most large underdetermined systems of linear equations, the minimal ℓ 1 -norm solution is also the sparsest solution[END_REF], recursive sparse estimation is a problem that has not received much attention so far. And yet there are many practical situations where a recursive processing of the measurements is desirable. For example, when we are interested in monitoring on-line a physical process with the objective of detecting abrupt changes, the data are sequentially sampled, hence calling for a recursive treatment. Or when the number of data available for parameter identification is too large, it may be computationally difficult to process them all at once using batch algorithms. In such a situation, recursive estimation is an attractive alternative. Despite the relevance of the problem of recursive sparse estimation, only a few recent works brought it up [START_REF] Blackhall | Recursive sparse estimation using a gaussian sum filter[END_REF][START_REF] Angelosante | RLS-weighted lasso for adaptive estimation of sparse signals[END_REF]. Note however that the solutions proposed in those papers appeared to be a recursive extension of the well-known LASSO algorithm [START_REF] Tibshirani | Regression shrinkage and selection via the lasso[END_REF] which applies to a different framework from the one of the present work. To emphasize the differences, note first that the data-generating model in ( 1) is a multi-variable one so that the unknown variable to be computed is a parameter matrix instead of a vector as is usually the case in related works [START_REF] Candès | Highly robust error correction by convex programming[END_REF][START_REF] Candès | Compressive sampling[END_REF][START_REF] Candès | Enhancing sparsity by reweighted l 1 minimization[END_REF][START_REF] Donoho | For most large underdetermined systems of linear equations, the minimal ℓ 1 -norm solution is also the sparsest solution[END_REF]. Second, while the LASSO seeks to estimate a parsimonious vector of parameters, the sparsity constraint is imposed here on a sequence of error vectors. This comes with an additional challenge specially when the data are also corrupted by zero-mean noise as in model [START_REF] Candès | Highly robust error correction by convex programming[END_REF]. To the best of our knowledge, our work is the first one to address the problem of identifying a multi-variable model such as (1) (involving a sparse error sequence) in a recursive framework.

The contribution of this paper is twofold. In the first part, a new method is presented for solving the recursive sparse estimation problem stated above. Our approach consists in the optimization of an ℓ 2 -norm based cost function. For the sake of easy recursive update, we approximate the solution through a Weighted Recursive Least Squares (W-RLS) algorithm. The way the weights are selected is inspired by the idea used in the re-weighting ℓ 1 -minimization algorithm of [START_REF] Candès | Enhancing sparsity by reweighted l 1 minimization[END_REF] which, recall, applies to a different setting from the one treated here. The second contribution of this work is an application of the method developed in the first part to the identification of Jump Markov Linear Systems (JMLS). If s denotes the number of subsystems composing the JMLS under consideration, we show that the submodels can, following a similar procedure as in [START_REF] Bako | A recursive identification algorithm for switched linear/affine models[END_REF], be recursively identified using s W-RLS identifiers operating in parallel.

The rest of this paper is organized as follows. The novel method is outlined in Section 2. In Section 3, we show that this method can be applied for the identification of JMLS from input-output data. Numerical simulation results are reported in Section 4. All of the simulation results, on linear models as well as on JMLS, reveal an excellent behavior of our algorithm. In the end, the conclusion and future work are discussed in Section 5.

THE RECURSIVE SPARSE LEARNING METHOD

In this section we propose a solution to the identification problem stated in the previous section. Recalling the multi-variable model (1), our goal is to identify the parameter matrix H recursively, from the experimental measurements {x (t) , y (t)}, which are corrupted by unknown noise and sparse error sequences, {e(t)} and {f (t)}. For the sake of clarity in the presentation, we will assume that the noise sequence {e (t)} is identically equal to zero1 .

The main idea

We start by emphasizing that no other prior information is available, except the sparsity condition on the error sequence {f (t)}. Standing on this point, the solution to our problem is to minimize the number of non-zero vectors in the error sequence {f (t)}, subject to [START_REF] Candès | Highly robust error correction by convex programming[END_REF]. It is formulated as, min

H |{t : f (t) = 0}| , subject to y (t) = Hx (t) + f (t) , t = 1, . . . , N (2)
where the notation |S| with S representing a set, stands for the cardinality of the set S. The optimization problem (2) attempts to find the matrix H fitting the data record {(x(t), y(t)} to model [START_REF] Candès | Highly robust error correction by convex programming[END_REF] and also making the error sequence {f (t)} as sparse as possible. Sparsity is reinforced here by minimizing the cardinality of the set {t : f (t) = 0}. Noting that f (t) = 0 if and only if f (t) ℓ = 0, where ℓ stands for any norm, problem (2) can be reformulated as min

H |{t : f (t) ℓ = 0}| subject to y (t) = Hx (t) + f (t) , t = 1, ..., N. (3)
In principle, ℓ can be any type of norm. For example, one can take ℓ to be the 2-norm or the ∞-norm. But in practice, the type of norm is of critical importance in the quality of the results one can hope for. By now defining

f = ( f (1) ℓ . . . f (N ) ℓ ) ⊤ , (4) problem (3) is equivalent to, min H f 0 subject to y (t) = Hx (t) + f (t) , t = 1, ..., N. (5)
Here and in the following, z 0 refers to the ℓ 0 -norm of z defined by z 0 = |{i : z (i) = 0}| for any vector z = z (1) . . . z (N ) ⊤ . It is well known that an ℓ 0 -minimization problem as above is nonconvex and generally NP-hard, because its solution requires an intractable combinatorial search [START_REF] Candès | Enhancing sparsity by reweighted l 1 minimization[END_REF].

The idea of recasting this nonconvex optimization problem into a convex one has been developed in the past several decades [START_REF] Candès | Enhancing sparsity by reweighted l 1 minimization[END_REF][START_REF] Donoho | For most large underdetermined systems of linear equations, the minimal ℓ 1 -norm solution is also the sparsest solution[END_REF], by changing the ℓ 0 norm into the ℓ 1 norm in the objective function [START_REF] Candès | Compressive sampling[END_REF]. Here, we pursue the same approach to obtain, min

H f 1 subject to y (t) = Hx (t) + f (t) , t = 1, ..., N (6)
where

f 1 = N t=1 f (t) ℓ is the ℓ 1 -norm of f
. From now on, we assume that ℓ = 2. Note that we can then drop the constraints in the above problem and view it as the one of computing the parameter matrix H by minimizing the cost function

J (H) = N t=1 y(t) -Hx (t) 2 . ( 7
)
It is important to notice that the cost function [START_REF] Donoho | For most large underdetermined systems of linear equations, the minimal ℓ 1 -norm solution is also the sparsest solution[END_REF] consists in a sum of norms (on the fitting errors) instead of the usual sum of squared norms 2 . This detail is crucial in promoting a solution H such that the vector sequence {f (t) = y(t) -Hx(t)} is sparse [START_REF] Ohlsson | Segmentation of arxmodels using sum-of-norms regularization[END_REF]. To see why, note that ( 7) can be rephrased as

J (H) = N t=1 ω(t) y(t) -Hx(t) 2 2 , (8)
where the weight

ω (t) = 1/ y (t) -Hx(t) 2 has been intro- duced. In fact, ω(t) = 1/ f (t) 2 if f (t) = 0 and ω(t) = ∞ if f (t) = 0.
Here the cost function [START_REF] Blackhall | Recursive sparse estimation using a gaussian sum filter[END_REF] shows the exact behavior of the objective function [START_REF] Donoho | For most large underdetermined systems of linear equations, the minimal ℓ 1 -norm solution is also the sparsest solution[END_REF]. Comparing [START_REF] Blackhall | Recursive sparse estimation using a gaussian sum filter[END_REF] with the standard least squares, we can see that the objective (7) (or, ( 8)) behaves as if infinite weights were assigned to the fitting errors for which f (t) = 0. Hence this cost function tends to assign an infinitely high contribution to the errors for which f (t) = 0 and to annihilate the effects of the errors for which f (t) = 0. As a consequence, provided the cardinality of the set {t : f (t) = 0} is reasonably small, minimizing [START_REF] Donoho | For most large underdetermined systems of linear equations, the minimal ℓ 1 -norm solution is also the sparsest solution[END_REF] may result in the exact matrix H, despite the presence of the sparse error vector sequence.

In practice, one can compute a minimizer of ( 7) by considering the objective [START_REF] Blackhall | Recursive sparse estimation using a gaussian sum filter[END_REF]. This latter cost function [START_REF] Blackhall | Recursive sparse estimation using a gaussian sum filter[END_REF] would correspond simply to a weighted least squares problem if ω (t) were known for any t. Unfortunately, this is not the case here since the weights ω(t) depend on the unknown matrix H. In the following, an approximate solution is computed in a recursive framework.

Derivation of the recursive estimator

As already emphasized before, our goal here is to obtain the matrix H in a recursive fashion. To this end, denote with H (0) the initial value assigned to the estimate of H and let P(0) ∈ R n×n be a positive-definite matrix. Define H (t) to be the estimate of H at time t based on the data pairs (x(k), y(k)) sampled up to time t. H (t) can be viewed as the matrix H(t) that minimizes the cost function

J H(t) = t k=1 λ t-k ω (k) y (k) -H (t) x (k) 2 2 + λ t tr (H(t) -H(0)) ⊤ P -1 (0)(H(t) -H(0)) (9)
2 The sum of squared norms correspond to the ordinary least squares.

where λ ∈]0, 1] is a forgetting factor. Here, the weights are approximated as

ω (k) = 1 y (k) -H (k -1) x (k) 2 + η , (10)
with η being a small positive number. With this approximation, the weight sequence {ω (k)} t k=1 is now completely known. As a result, the criterion defined in ( 9)-( 10) can be regarded as a Weighted-Recursive Least Squares (W-RLS) criterion. As is well known, the matrix H (t) that minimizes (9) can be computed from H (t -1) by applying the following update rules [START_REF] Kay | Fundamentals of Statistical Signal Processing: Estimation Theory[END_REF]:

K (t) = ω (t) P (t -1) x (t) λ + ω (t) x ⊤ (t) P (t -1) x (t) (11)
H (t) = H (t -1) + y (t) -H (t -1) x (t) K ⊤ (t) (12)
P (t) = 1 λ P (t -1) -K (t) x ⊤ (t) P (t -1) , (13)
where ω (t) is defined as in [START_REF] Tibshirani | Regression shrinkage and selection via the lasso[END_REF], the initial values H (0) can be chosen randomly, and P (0) = αI n with α ≫ 1, I n is an identity matrix with the dimension of n.

APPLICATION TO JUMP MARKOV LINEAR SYSTEMS

In the previous section, we presented a method for identifying the parameter matrix recursively from a collection of measurements which are corrupted by unknown sparse error and white noise sequences. In this section, we will show that this method can be interestingly applied to the identification of Jump Markov Linear Systems (JMLS).

A JMLS corresponds to the behavior that results from switchings among a set of linear subsystems, the transitions between subsystems being determined by a finite state Markov chain. JMLS can be represented by

x (t + 1) = A σ(t) x (t) + B σ(t) u (t) y (t) = C σ(t) x (t) + D σ(t) u (t) , (14)
where x (t) ∈ R nx stands for the continuous state, with n x representing the order of the system (i.e, the dimension of the state space). This order is assumed to be the same for all of the submodels. u (t) ∈ R nu , y (t) ∈ R ny , σ(t) ∈ S = {1, . . . , s} are respectively the input, the output and the discrete state (which will be also called the mode).

A σ(t) ∈ R nx×nx , B σ(t) ∈ R nx×nu , C σ(t) ∈ R ny×nx , D σ(t) ∈ R ny×nu are
the parameter matrices associated with the discrete state σ (t).

It is assumed that σ (t), t ≥ 1, is a discrete-time, first-order stationary Markov chain with constant transition probabilities defined as,

π ij = P {σ (t) = i|σ (t -1) = j} (
15) for any i, j ∈ S = {1, 2, . . . , s}. Hence, the transition probability matrix π is an s × s constant matrix, with the components satisfying π ij ≥ 0 and s i=1 π ij = 1, for each j ∈ S. The initial probability distributions are denoted as,

π 0 j = P {σ (0) = j} (16)
for all j ∈ S and are such that π 0 j ≥ 0, and

s j=1 π 0 j = 1.
Given the measurements sequence3 {x (t) , u (t) , y (t)} N t=1 , our objective is to identify the parameter matrices A j , B j , C j , D j , j ∈ S, as well as the transition probabilities defined in (15). Moreover, all of the identification process will be on-line in our work.

Let us set

x (t) = x (t) ⊤ u (t) ⊤ ⊤ (17) H σ(t) = A σ(t) B σ(t) C σ(t) D σ(t) (18)
ȳ (t) = x (t + 1) ⊤ y (t) ⊤ ⊤ .
(19) Then we can rewrite the model (14) as, ȳ (t) = H σ(t) x (t) .

(20) The application of the previous method to the estimation of the matrices {H i } s i=1 from model ( 20) is based on the following observation. For any i ∈ S, model (20) can be rewritten as

ȳ (t) = H i x (t) + H σ(t) -H i x (t) = H i x (t) + f ′ i (t) , (21)
with

f ′ i (t) = H σ(t) -H i x(t)
. Now by assuming that the discrete state i is visited by the JMLS frequently enough, we can see that the vector sequence f ′ i (t) in ( 21) is sparse since f ′ i (t) = 0 whenever σ(t) = i. Therefore, if we focus on the identification of a specific submodel with index i, we get a similar model as (1), under noise-free assumption. Standing on this point, we can adapt the method proposed in the previous section to identify the parameter matrices of the JMLS model (20). In fact, if there exists one submodel i of the JMLS that is activated very often over time, then running directly the W-RLS identifier of Section 2 on the data generated by model (20) may yield the corresponding matrix H i . However, this may not be the case in practice. For example, when the number of submodels is relatively large, it may happen that none of the submodels is frequently activated.

To deal with this type of situation, the recursive sparse estimation method can be applied efficiently as follows. We estimate all the submodels simultaneously by running s different recursive identifiers in parallel, one identifier for each submodel. At each time t, the discrete state σ(t) is estimated as the index of the submodel which, based on the prior estimates of the different parameter matrices, appears to be the most consistent with the data pair x(t), ȳ(t) . Then the submodel associated with the estimated discrete state has to be updated. To see why this procedure is likely to yield the sought matrices H i , i ∈ S, note that each of the s recursive sparse identifiers has to correct only wrong mode assignments (which are expected to be rare). Therefore, thanks to the error correction ability of the identifiers, this algorithm is, as will be confirmed by simulations results, able to recover the JMLS parameters.

In more details, we follow a similar idea as in [START_REF] Bako | A recursive identification algorithm for switched linear/affine models[END_REF][START_REF] Bako | Identification of switched linear state space models without minimum dwell time[END_REF] and select σ(t) to coincide with the index of the matrix H i (t -1), i ∈ S that minimizes the relative fitting error i.e,

σ (t) = arg min i∈S ȳ (t) -H i (t -1) x (t) 2 I nx+ny H i (t -1) 2 , (22)
where H i (t -1) is the estimated matrix associated with the discrete state i ∈ S at t -1; I nx+ny is an identity matrix of dimension n x + n y .

In the following part of this section, we are going to focus on how to estimate the transition probabilities on-line. Define two sets Z ij (t) and Z i (t) as follows: Z ij (t) records the time indices when the system switched from submodel j to submodel i while Z i (t) records the time indices when the system is switched into submodel i from any submodel. More specifically, let

Z ij (t) = {1 < k ≤ t| σ (k) = i, σ (k -1) = j} , (23)
Z i (t) = {1 < k ≤ t| σ (k) = i} .
(24) Then, the transition probability from state j to state i can be evaluated as

π ij (t) = Z ij (t) Z i (t) . ( 25
)
Here π ij (t) is the estimated transition probability at time t; Z ij (t) and Z i (t) represent the cardinality of the sets Z ij (t) and Z i (t) respectively. Furthermore, we can derive the following update rule for the transition probability,

π ij (t) = Z i (t -1) Z i (t -1) + δ i (t) π ij (t -1) + + δ i (t) δ j (t -1) Z i (t -1) + δ i (t) (26)
where

δ q (t) = 1 if σ (t) = q 0 if σ (t) = q. ( 27
)
Finally, we summarize in Table 1 the complete algorithm for recursively identifying the parameter matrices H i , i ∈ S, along with the transition probabilities π ij (t) of the JMLS model.

Table 1. Algorithm for Identifying JMLS.

Algorithm for Identifying JMLS Initialization:

Provide the matrices H i (0), the covariance matrices P i (0), i ∈ S, as well as the transition probabilities π ij (0), (i, j) ∈ S 2 .

For any incoming data pair x(t), ȳ(t) :

(1) Select the activated discrete state σ (t) as in ( 22).

(2) Update the parameter matrix associated with the activated discrete state σ (t) by ( 11)- [START_REF] Bako | Identification of switched linear state space models without minimum dwell time[END_REF]. For any i such that i = σ (t), H i (t) = H i (t -1).

(3) Update the transition probability matrix using (26).

NUMERICAL RESULTS

In this section we illustrate the whole identification procedure on synthetic data generated by a model with the form (1) and a JMLS model respectively.

Identification results for a model of the form (1)

Firstly, we test our identification method on the data generated by a linear multi-variable model of the type [START_REF] Candès | Highly robust error correction by convex programming[END_REF]. With the dimensions m = 3 and n = 3, the "true" model that generates the data is defined as follows:

y (t) =
1.5847 0.3944 -0.8044 -0.6355 0.6342 -0.2397 -0.1957 -0.9011 -0.7516 x (t) + f (t) + e (t) .

(28) Here, the input data sequence {x (t)} is generated randomly following a normal distribution, N (0, I n ). The sparse error sequence {f (t)} N t=1 , with N = 1500, is chosen such that 1000 vectors out of 1500 are strictly equal to zero. The remaining error vectors are generated from a non-centered normal distribution, N (μ 1 , σ 2 1 I m ), μ1 = 5(1, . . . , 1) ⊤ , σ 2 1 = 100. For an overview on the temporal distribution of this sparse error sequence, we plot its ℓ 2 -norm over 1500 samples in Fig. 1.(a) and over the first 200 samples in Fig. 1.(b). Notice that the ℓ 2 norm of non-zero sparse error, as reported in Fig. 1, can be arbitrarily large as stated before. The noise {e(t)} in (28) is generated as a Gaussian white noise so as to achieve a Signalto-Noise-Ratio (SNR) of 30 dB.

Our learning method is implemented in MATLAB. Related parameters are given as follows. The forgetting factor is set to be 0.95. The positive number η appearing in [START_REF] Tibshirani | Regression shrinkage and selection via the lasso[END_REF] should, in principle, be extremely small as its role consists only in preventing division by zero. We set η to be 1e-6. The initial covariance matrix is chosen as, P (0) = αI 3 , with α = 10. And H (0) is a zero matrix.

The mean and the standard deviation of estimated values for 100 independent simulations with SNR=30 dB, are shown in Table 2. Compared to the true value, these numerical results show that our method achieves a good estimate for the parameter matrix of model (28). To further analyze the properties of the proposed method, we define a set of relative errors {ε (t)} N t=1 to assess the quality of the estimates over time,

ε (t) = H (t) -H F H F , (29)
where • F represents the Frobenius norm. We apply both the standard RLS algorithm and the new method proposed in Section 2 to solve the identification problem. During all the varying over time are reported in Fig. 2 for the RLS and in Fig. 3 for our method. In these figures, the red bold lines are the averages of the relative errors over the 100 simulations. Fig. 2 shows clearly that the standard RLS fails to provide correct estimates. Note that convergence occurs for that algorithm only over the time intervals on which the error sequence {f (t)} is equal to zero. In contrast, we can see from Fig. 3 that, despite the presence of the error {f (t)}, the method of this paper achieves accurate estimates. In all the simulations carried out, the relative errors drop off rapidly during the first dozens of steps, and the algorithm almost converges after 80 steps. Hence, the proposed method successfully recovers the parameter matrix, although the sparse error sequence is completely unknown and has a very large magnitude. Note that the small bias which is visible in Fig. 3 is induced only by the noise {e(t)}. This is confirmed by the results of Table 3, where the final estimation error ε(N ) is shown to be strictly equal to zero when {e(t)} is identically null.

In order to analyze the robustness of our method with respect to noise, we test it on model (28) with different noise levels. In this experiment, the sparse error sequence {f (t)} has the profile given in Fig. 1. For each noise level, 100 simulations with N = 1500 data pairs each, are carried out. The average of the final relative errors (that is, the error after convergence) over all of the simulations are given in Table 3. As already mentioned, when there is no noise (that is, when SNR = ∞), the relative error is equal to zero. And the relative errors logically increase when the SNR decreases. Now, we propose to verify up to what extent of sparsity of the sequence {f (t)} our method is able to identify model (28). To this end, we test it for different sparsity levels of {f (t)}. Here, the sparsity ρ is defined as the number of the non-zero vectors divided by the number of data, i.e.,

ρ = f 0 N × 100%, ( 30
)
where f is defined as in [START_REF] Bako | A recursive identification algorithm for switched linear/affine models[END_REF]. For this sparsity measure to make sense in the present recursive context, the temporal distribution of nonzero vectors in the sequence {f (t)} needs to be uniform.

The conditions of our experiment are set so as to meet this requirement. The non-zero vectors in the sequence {f (t)} are generated randomly from the non-centered normal distribution N (μ 2 , σ 2 2 I m ), μ2 = 5(1, . . . , 1) ⊤ , σ 2 2 = 100. For every sparsity level, 100 simulations are carried out with N = 1500 pairs of noisy data (the SNR being equal to 30 dB). The average of the final relative errors over all of the simulations for each sparsity level are reported in Table 4. It shows that our method can perform dramatically well even for a level of sparsity as high as 90%.

Judging from all of the results displayed in this subsection, we are prompted to conclude that the proposed method is reliable for identifying a model of the form (1), which is subject to occasional gross errors.

Identification results for a JMLS model

Finally, we apply our algorithm to identify JMLS parameter matrices, as well as the transition probabilities. We take an example of JMLS composed of three submodels, i.e., the discrete state σ(t) lives in S = {1, 2, 3}. With the dimensions n x = 2, n u = 1 and n y = 1, all of the "true" parameter matrices are presented in Table 5, together with the transition probability matrix. The matrices H i are defined as in (18) for all i ∈ S. The input sequence {u (t)} N t=1 , with N = 1500 for each simulation here, is generated randomly following the normal distribution, N (0, I nu ). x(0) is also given as a random value. A certain amount of Gaussian white noise is added to the output data, so that we get an SNR of 30 dB. The user-defined parameters of the learning method are chosen to be the same as in the previous subsection. And for i S, the initial parameter matrices H i (0), are drawn at random; P i (0) = α i I 3 , with α i = 10. The numerical average and standard deviation of the estimated parameter matrices over 100 simulations are reported in Table 6. It can be noticed that there is a little bias appearing in the result.This is due to the fact that the data are affected by the noise, as argued in Subsection 4.1.

Moreover, all of the simulation examples have already shown the effectiveness of the proposed Weighted-RLS. The results reveal that the method proposed in this paper is feasible and efficient in solving identification problem of JMLS.

CONCLUSION

In this paper we treated the problem of identifying on-line a linear multi-variable model, subject to a sparse vector error sequence. This is an important and generic problem having applications in many engineering fields. For example, as illustrated in the paper, the identification problem for switched linear systems can be viewed as a particular instance of that problem. We showed that by appropriately setting some weights, a simple recursive weighted least squares algorithm can be used to efficiently address this problem. Although the effectiveness of the proposed method has been extensively verified on some numerical examples, convergence analysis is still lacking in this first work. This will be considered in future research.

Figure 1 .

 1 Figure 1. ℓ 2 norm f (t) 2 of the sparse error sequence in model (28): (a) over 1500 samples; (b) over the first 200 samples.

Figure 2 .Figure 3 .

 23 Figure 2. Trajectories of the relative errors ε (t) over time when standard RLS is applied to model (28): SNR=30 dB.

Table 2 .

 2 Empirical mean and standard deviation of the estimated H for model (28), with an SNR=30 dB.

		Mean		Standard Deviation
	1.5846	0.3942 -0.8046		0.0051 0.0056 0.0051
	-0.6355 0.6337 -0.2392	±	0.0061 0.0044 0.0043
	-0.1956 -0.9009 -0.7516		0.0047 0.006 0.0047

Table 5 .

 5 "True" JMLS parameter matrices.

	Matrix	"True" Value
		-0.7901 -0.0458 -1.0834
	H 1	-0.0458 -0.7544 0.4504
		-2.8025	0	0
		-0.5742 -0.3092 0.6379
	H 2	-0.3092 0.3868 -2.3072
		-1.5833 -1.0135 -1.0293
		-0.6389 -0.1516 0.9488
	H 3	-0.1516 -0.5814 0.7786
		0.0005	0	0
		0.2818 0.2594 0.4589
	π	0.2500 0.2606 0.4867
		0.2687 0.2410 0.4917

Table 3 .

 3 Relative Errors when our method is applied to model (28) with different noise levels.

	SNR (dB) ∞	50	40	30	20	10	5
	ε (N )	0	0.0012 0.0045 0.0144 0.0449 0.1293 0.2592

Table 4 .

 4 Relative Errors when our method is applied to model (28) with different sparsity levels-SNR=30 dB.

	sparsity ρ (%)	20	30	40	50	70	80	90	95	96
	ε (N )	0.0088 0.0092 0.0102 0.0108 0.0133 0.0165 0.0236 0.1806 13.1917

Table 6 .

 6 Empirical mean and standard deviation of estimated values for JMLS: SNR=30 dB.

	Estimated Matrix		Mean	Standard Deviation
		-0.7901 -0.0458 -1.0834		0.0000 0.0000 0.0000
	H 1 :	-0.0458 -0.7544 0.4504	± (1.0e-0) ×	0.0000 0.0000 0.0000
		-2.8051 0.0034 -0.0052		0.0543 0.0305 0.0318
		-0.5744 -0.3013 0.6243		0.0428 0.0554 0.1512
	H 2 :	-0.3041 0.3636 -2.2492	± (1.0e-0) ×	0.0506 0.1627 0.4068
		-1.5617 -0.9812 -1.0114		0.1564 0.2324 0.1352
		-0.6389 -0.1516 0.9488		0.0000 0.0000 0.0000
	H 3 :	-0.1516 -0.5814 0.7786	± (1.0e-5) ×	0.0000 0.0000 0.0000
		0.0005	0.0000 -0.0000		0.6261 0.2900 0.1770
		0.301 0.2129 0.4854		0.0593 0.0844 0.0864
	π :	0.2688 0.2635 0.4654	± (1.0e-0) ×	0.0764 0.0552 0.0899
		0.2939 0.2347 0.471		0.0802 0.0713 0.0651

However the method to be presented still works in the presence of noise. This is illustrated in Section 4.

The continuous state x(t) is assumed completely measurable while the discrete state σ(t) is unknown.