Alexandra Belova

Philippe Gagnepain
email: philippe.gagnepain@univ-paris1.fr

Stéphane Gauthier
email: stephane.gauthier@univ-paris1.fr

Philippe Jehiel

Philipp Ketz

David Martimort

Otto Toivanen

Ecopsy Consulting

An assessment of Nash equilibria in the airline industry

Keywords: C51, D21, L13, L40, L93 Rationalizability, Nash equilibrium, Cournot competition, structural model, airline industry, welfare analysis, relevant market JEL classification numbers: C51, D21, L13, L40, L93 Rationalizability, Nash equilibrium, Cournot competition, structural model, airline industry, welfare analysis, relevant market

We study competition in the U.S. airline industry relaxing the Nash equilibrium assumption that airlines are able to predict perfectly the behavior of their competitors. We assess empirically whether an equilibrium is more likely to occur if it is the unique rationalizable outcome. We find that equilibria of short distance routes with high traffic and low concentration are the most fragile, and low-cost companies appear detrimental to their occurrence. Our analysis is applied to the measurement of welfare gains from firms' entry, and to the characterization of the relevant market when some products are unobserved.

Introduction

Economic analysis often refers to the Nash equilibrium concept at the moment of describing strategic interactions between agents. In this situation, every agent is assumed to be able to forecast correctly the behavior of the other agents. The recent literature in Industrial Organization shows that such an assumption is more likely to hold in a stable environment where firms operate in markets that are geographically close to the market of their competitors (Aguirregabiria and Magesan (2020)) or if agents can accumulate experience and gradually learn how the others behave. However, it also suggests that beliefs can loose accuracy in more disturbed environments where participants often change. Then the equilibrium reference is potentially less relevant [START_REF] Doraszelski | Just starting out: Learning and equilibrium in a new market[END_REF]).

While agents may not be able to forecast the behavior of their competitors, it is usually assumed that they are rational in the sense that they maximize their objective given their expectations on what the others do. An important lesson from the concept of rationalizability is that rationality, even pushed at a high degree of sophistication, is not always enough to reach a Nash equilibrium. A player is rational at level k = 1 if she plays a best-response given her beliefs; at level k = 2 she's rational and believes that the other players are rational; at level k = 3 she also believes that the others believe that the others are rational. Reproducing inductively at any level k ≥ 1 this process of higher-order beliefs about rationality eventually yields the set of rationalizable outcomes which always includes but does not necessarily reduce to Nash equilibria [START_REF] Bernheim | Rationalizable strategic behavior[END_REF], [START_REF] Pearce | Rationalizable strategic behavior and the problem of perfection[END_REF], Moulin (1979)).

In this paper we build an empirical index for the likelihood that an equilibrium occurs, based on the postulate that an equilibrium is more likely to occur when it is the only rationalizable outcome [START_REF] Guesnerie | An exploration of the eductive justifications of the rationalexpectations hypothesis[END_REF]). The index uses the characterization of rationalizable outcomes as being those surviving an iterative process of elimination of dominated strategies. The process eliminates every outcome close to an equilibrium, but not the equilibrium itself, if it is locally contracting around this equilibrium. Contraction obtains when the spectral radius of the Jacobian matrix governing locally the process is less than 1. We estimate a proxy for the spectral radius and use it as an index for the likelihood that the observed market will reach an equilibrium: the theory predicts that the market should be in equilibrium if the index is lower than 1.

The value of the index depends on sufficient statistics for supply and demand character-istics. To recover these statistics we construct a structural model applied to the case of the U.S. air transportation industry. Our model generalizes [START_REF] Desgranges | Rationalizability and efficiency in an asymmetric Cournot oligopoly[END_REF] to the case of heterogeneous production facilities, which is known to be a crucial ingredient in the airline industry. We merge several Department of Transportation databases for the period 2003:2016 to estimate supply and demand functions for a large number of air routes.

This information provides us with the sufficient statistics that enter the value of the index on every route. It also allows us to calculate the hypothetical volume of transported passengers by each airline at the equilibrium, and so the difference between the actual observed production levels and those that would prevail in the equilibrium.

Our main result is that our index is a reliable indicator of this difference: we find that a 10% increase of the index is associated with a 7% increase in the difference between observed and Nash quantities. Descriptive statistics show that the Nash equilibrium is less likely to occur on short distance and intense traffic routes linking populated cities. Low cost companies also complicate the convergence toward the equilibrium whereas greater concentration tends to yield a lower value of the index. At this stage of the analysis, we conclude that an equilibrium would be reached on approximately 90% of the markets in our dataset.

We then extend our main analysis into three directions. First, we note that adaptive learning in a dynamic horizon can also explain convergence toward the equilibrium. We propose a simple test that allows us to shed light on whether convergence toward the equilibrium is also facilitated when airlines use past observations to form their anticipations on the decisions that their competitors will take. We find that, on top of rationalizabilitybased deductive arguments, adaptive learning is also potentially relevant as the current observed/Nash spread is significantly reduced by those two and three quarters before.

Our second extension of the baseline analysis illustrates how nonequilibrium outcomes in a particular market may affect consumer surplus. The empirical studies on firms' entry and exit usually assume that the market is in equilibrium both before and after entry/exit and so refer to the corresponding prices and quantities in each period. Here, we view entry/exit as potential perturbations that may lead to a multiplicity of rationalizable outcomes. In an illustration based on the New York-Tampa route, we show how firms may over-estimate other airlines' fares and schedule too many seats following entry: consumer surplus is in this case higher than what one would get in equilibrium.

Our third extension relates to the debate on what competition authorities call the relevant market (Davis and Garcés (2009)). In practice, there are in our data markets where the observed quantity produced differs from Nash even though the index is below 1.

We argue that the relevant threshold should in fact be lower than 1 when the econometrician does not observe the full set of services supplied by the competing airlines due to missing data issues. The higher the share of missing observations, the smaller the value of the relevant threshold. We propose a method based on machine learning to identify the relevant threshold for each market. We obtain an average threshold of 0.80, which implies that almost one-third of the markets in our database could fail to reach an equilibrium. We confront our methodology whith a natural experiment, namely the Wright amendment, which restricted flights from the Dallas Love airport in order to promote the development of the Dallas/Fort Worth airport [START_REF] Ciliberto | Market structure and multiple equilibria in airline markets[END_REF]). We suggest that the estimated relevant threshold in routes from Dallas/Fort Worth is sharply reduced after the repeal of the amendment, as the relevant market that includes the airline services of this airport expands over the period.

The rest of the paper is organized as follows. We build the theoretical model applied to the airline industry in Section 2 and Section 3 discusses the details of the estimation strategy, including data cleaning and descriptive statistics. Section 4 reports the estimation results of the cost and demand functions and uses this information to compute the index that governs the plausibility of the equilibrium. This section shows how the index correlates with observed departures from Nash behavior. In section 5 we introduce adaptive learning and we also compute both the actual and equilibrium welfare difference following a change in the set of competing airlines in a given route. Section 6 extends our analysis to the identification of the relevant market. Finally section 7 concludes.

Theoretical benchmark 2.1 General framework

In the airline industry a market is defined as the set of air services offered by different carriers in a route linking a pair of origin and destination airports or cities. The airlines compete in the route for carrying freight and passengers. They all face the same demand function for transportation services but they typically differ according to their technolog-ical and organizational characteristics summarized by their cost structure. Some airlines may use a few large capacity aircraft to spread the cost of booking airport slots, boarding passengers and operating flights on few departures, whereas others rely on lower capacity aircraft but schedule more departures. Such choices result from medium-run intermittent contractual negotiations between airlines and airports and long-run airlines capacity investment policies. Over a shorter horizon, the fact that Delta Airlines in the route linking Chicago and Atlanta allocates a A320 Airbus to a booked slot at O'Hara airport from 8h00 to 8h30am every Monday is essentially given.

Over this shorter horizon, airlines instead rely on yield management to control prices and quantities of transported passengers given the available aircraft allocated to the route [START_REF] Borenstein | Competition and price dispersion in the US airline industry[END_REF]. Following [START_REF] Ciliberto | Market structure and multiple equilibria in airline markets[END_REF], we allow for a firm f specific cost function c af (q) for transporting q passengers using a type a aircraft that varies with aircraft type as well as additional firm characteristics, e.g., negotiated input (fuel and employees) prices. We assume that the cost function is twice differentiable, increasing and convex with the number of transported passengers. A polar case obtains if the marginal cost for transporting one additional passenger is low except when the total number of passengers approaches the aircraft capacity.

Let A f be the given set of aircraft used by airlines f in the route and n af be the given number of flights operated by the airlines using type a aircraft. The total cost of firm f for transporting q f passengers is

C f (q f ) = min (q af )    a∈A f n af c af (q af ) | a∈A f n af q af ≥ q f    , (1)
which is also an increasing and convex C 2 -function with the number q f of passengers.

Assuming Cournot-Nash behavior, as in e.g., [START_REF] Brander | Market conduct in the airline industry: an empirical investigation[END_REF], [START_REF] Brueckner | Airport congestion when carriers have market power[END_REF] or [START_REF] Basso | Airport deregulation: Effects on pricing and capacity[END_REF], firm f takes as given the number Q -f of passengers transported by the other airlines and produces

q f ∈ arg max q P (Q -f + q)q -C f (q),
where P (Q) is the inverse demand function, and Q = Q -f + q f is the total number of passengers transported in the route. Assuming that the marginal revenue P (Q -f + q)q is decreasing in q, the best choice for firm f is

q f = R f (Q -f ), where the best-response function R f (•) is decreasing. A Cournot-Nash equilibrium is a F component vector q * = (q * f ) such that q * f = R f (Q * -f
) for all f , with Q * -f being the aggregate production of others in the equilibrium. Thus, in an equilibrium, every airline f is assumed to predict correctly the number Q * -f of passengers transported by its competitors.

Our paper provides an empirical assessment of this assumption appealing to rationalizability. Rationalizable outcomes can be characterized by referring to an iterative process of elimination of dominated strategies (see, e.g., [START_REF] Osborne | A course in game theory[END_REF]. The process starts from the assumption of common knowledge among airlines that at some initial step τ = 0 production satisfies

q f ∈ [q inf f (0), q sup f (0)] (2)
for every f , with q inf f (0) ≤ q * f ≤ q sup f (0). Assuming the equilibrium amounts to require q inf f (0) = q * f = q sup f (0) for all f . In the sequel we assume that firms restrict attention to a neighborhood of the equilibrium, with (q inf f (0), q sup f (0)) is close to, but different from (q * f , q * f ). By individual rationality airlines only select volumes of transported passengers that are best-response to decisions consistent with [START_REF] Allen | The Wright Amendment: The Constitutionality and Propriety of the Restrictions on Dallas Love Field[END_REF]. Airlines f thus chooses some production in a new interval [q inf f (1), q sup f (1)] at step τ = 1 of the process, with 1

q inf f (1) = R f k =f q sup k (0) , q sup f (1) = R f k =f q inf k (0) .
This reasoning applies to every airlines. Thus, in the immediate vicinity of an equilibrium, we have

q inf f (1) -q * f = R f (Q * -f ) k =f [q sup k (0) -q * k ] (3)
and

q sup f (1) -q * f = R f (Q * -f ) k =f q inf k (0) -q * k ( 4
)
1 If q inf f (1) ≤ q inf f (0) then we set q inf f (1) = q inf f (0). Similarly, if q sup f (1) ≥ q sup f (0) then we set q sup f (1) = q sup f (0). The process therefore remains at (2) if both q inf f (1) ≤ q inf f (0) and q sup f (1) ≥ q sup f (0). No strategy is eliminated. The same procedure applies to every step τ ≥ 1 of elimination. for all f . Under common knowledge of rationality and the slopes R f (Q * -f ) of the firms' best-response functions, one can iterate the above argument. The iterative process of elimination of dominated strategies defined by ( 3) and ( 4) is governed by the F × F matrix B whose every entry in the f -th row equals R f (Q * -f ) except the diagonal entry (in the f -th column) which is 0. It has the Cournot-Nash equilibrium q * is a fixed point.

Level-k thinking popularized by [START_REF] Crawford | Level-k auctions: Can a nonequilibrium model of strategic thinking explain the winner's curse and overbidding in private-value auctions?[END_REF] would iterate the process k times, for some finite number k. If iterated ad infinitum, the process eventually pins down the equilibrium if and only if the spectral radius of B is less than 1. If the Nash equilibrium is locally the only rationalizable outcome, one can argue that firms should eventually convince themselves that their competitors will behave according to Nash. In this case we say that the equilibrium is locally 'stable'. Otherwise, if the spectral radius is greater than 1, there are multiple rationalizable outcomes and the iterative process can no longer justify that firms eventually pin down their Nash productions. We say that the equilibrium then is 'unstable'. The following proposition gives a condition for local stability.

Proposition 1. The Nash equilibrium is locally stable if and only if

S(q * ) = f R f (Q * -f ) R f (Q * -f ) -1 < 1 (5)
where

R f (Q -f ) = - P (Q) q f + P (Q) P (Q) q f + 2P (Q) -C f (q f ) , (6)
and

C f (q f ) a∈A f n af c af (q af ) = 1.
Proof. See Appendix A Proposition 1 forms the basis of our empirical illustration by providing us with a simple criterion for the plausibility of the occurrence of the Nash equilibrium. It predicts that the spread between the theoretical Nash equilibrium productions q * and the actual observed productions should be magnified if the 'stability index' S(q * ) defined in [START_REF] Basso | Airport deregulation: Effects on pricing and capacity[END_REF] is greater than a threshold of 1.

Condition [START_REF] Basso | Airport deregulation: Effects on pricing and capacity[END_REF] shows that local stability of the Nash equilibrium obtains if firms are not too sensitive to the production of others, i.e., R f (Q * -f ) is close to 0, which accords with the early insights developed by [START_REF] Guesnerie | An exploration of the eductive justifications of the rationalexpectations hypothesis[END_REF] for the competitive case. The intuition is that firms find it difficult to understand the behavior of others when others are sensitive to their beliefs.

Linear-quadratic specification

One can derive simple comparative static properties for the stability index S(q * ) in the particular case where demand is linear and cost is quadratic. Then the slope of the bestreaction function, and so the value of the stability index, no longer depends on the number q * of transported passengers in the equilibrium. With a linear demand function,

P (Q) = δ 0 -δQ, δ 0 > 0, δ > 0, (7)
and a quadratic cost,

c af (q) = q 2 2σ af ,
where σ af > 0 is a technological parameter that is specific to aircraft × airlines, the cost function solution to the program (1) is

C f (q) = q 2 2σ f , σ f = a∈A f n af σ af . ( 8
)
The parameter σ f plays a central role in our model. It can be interpreted by noticing that both the marginal cost C f (q) associated with [START_REF] Blundell | Initial conditions and moment restrictions in dynamic panel data models[END_REF] and its derivative C f (q) are decreasing with σ f for any given production q. Since C f (0) = 0 a higher value of σ f implies a production efficiency gain (a lower production cost), which is made possible thanks to an increase in the individual σ af or because the capacity n af goes up. This efficiency gain however comes with greater flexibility captured by dampened marginal costs. This makes firms more sensitive to expected changes in the production of others: the slope of the best-response function of firm f

R f (Q * -f ) = - δσ f 2δσ f + 1 (9)
is decreasing with σ f . This bundle of efficiency gains and greater flexibility drives a tradeoff between surplus maximization in the Nash equilibrium and stability of this equilibrium illustrated by Proposition 2.

Proposition 2. The transfer of an additional aircraft to some airlines in the linearquadratic setup increases the aggregate equilibrium production Q * but it locally destabilizes the equilibrium, i.e., it leads to an increase in the index S(q * ).

Proof. See Appendix B.

An additional aircraft allocated to the route corresponds to a higher transportation seat capacity, and so corresponds to an increase in the σ f parameter. By Proposition 2 one should consequently observe in the data that routes with high traffic display a higher spread between the theoretical Nash equilibrium and the actual production.

The next result controls for route size by considering a transfer of aircraft between two airlines in the same route. This allows us to highlight the impact of the distribution of transportation capacities across airlines.

Proposition 3. An aircraft reallocation from airlines f to airlines f in the linear-quadratic setup increases the aggregate equilibrium production Q * if and only if σ f > σ f . This reallocation locally destabilizes the equilibrium, i.e., it leads to an increase in the index S(q * ), if and only if σ f > σ f .

Proof. See Appendix C.

The trade-off between efficiency and stability illustrated in Proposition 2 is still valid.

However, Proposition 2 would not allow us to discuss the impact of the transfer considered in Proposition 3 since the contributing airlines f entails an efficiency loss and a stability gain whereas airlines f , which enjoys the additional aircraft, is associated with an efficiency gain and a stability loss. Proposition 3 actually obtains by comparing the magnitudes of these two changes. It leads to the new testable prediction that some asymmetry in the airlines capacity in a given route, with large seat capacity firms competing against smaller ones, should be associated with a theoretical Nash equilibrium production closer to the actual one.

Empirical illustration to the airline industry

Our theoretical analysis predicts that the Nash equilibrium production should stand far from the observed production when the stability index S(q * ) is high (Proposition 1), the total production capacity is high (Proposition 2), and the total production capacity is distributed evenly across firms (Proposition 3). We assess these predictions in the U.S.

domestic airline industry over the period 2003:2016 using data from the Bureau of Transportation Statistics to estimate the demand for airlines tickets and aircraft cost functions fitting the linear-quadratic setup developed in Section 2.2. These data allow us to compute the stability index S(q * ) and the volumes of transported passengers in the Nash equilibrium q * by each airline, which can then be compared to the actual observed number of transported passengers.

Data

A market consists of all the flights between two endpoint cities, identified by their City Our analysis of the supply side of transportation services exploits information contained in schedule P-5. the ticket level for every quarter. In order to match the segment perspective used on the supply side, we restrict our attention to markets with a high enough proportion of direct flights. We have chosen a threshold that eliminates routes with less than 60 percent of direct non-stop flight tickets. As shown in Table 2 there remains 3, 337 routes in the DB1B database. We disregard routes with one airlines in a monopoly situation, to which the theoretical part does not apply, and we only keep routes matching a segment existing in our final T-100 dataset. Then we use a cleaning process that mirrors the one applied to the T-100 dataset: we remove routes with distance below 100 miles and get rid of the routes with few passengers. We also discard tickets with extreme reported prices in the bottom and top 5% quantiles of the price per mile distribution, and we remove routes observed during less than 12 years in our 14-year sample window. We are eventually left with 379 routes. Compared to the original sample, our dataset tends to be biased toward routes with greater passenger traffic. To estimate demand we compute the average quarterly number of passengers booking a flight on the selected segments and the corresponding average fare for each of these 379 routes. Fares are adjusted using the transportation sector price index of the Bureau of Labor Statistics. The database is enlarged with temperature and population of origin and destination cities. 3 Descriptive statistics are presented in Table 3.

Costs

The estimation of an aircraft cost function is based on the quadratic specification

c af st = q 2 af st 2σ af st , (10)
which applies to a given flight operated by airline f with a type a aircraft during period t on segment s. The T-100 dataset provides information on the number of passengers q af st transported on segment s by airlines f using aircraft type a during quarter t. However we only observe in the schedule P-5.1 the aggregate cost over all the segments served by airlines, namely

C af t = s n af st c af st (11)
where n af st denotes a number of departures. In order to estimate the parameter σ af st that enters the stability index at the segment level using the aggregate cost in [START_REF] Brandenburger | Identification of Reasoning about Rationality[END_REF], we express this parameter as

1 2σ af st = β 0 ξ f t µ s ν a (12)
where β 0 is a constant term, ξ f t varies across firms and time periods, while µ s and ν a are segment and aircraft fixed effects, respectively. The variable ξ f t accounts for unobserved characteristics of airlines productive efficiency, e.g., managerial effort or marketing strategies, each of which plausibly varies over time. In [START_REF] Brandenburger | Intrinsic correlation in games[END_REF], segment and aircraft fixed effects are restricted to be time invariant, but our final cost specification includes time fixed effects common to segments and aircraft. Using [START_REF] Borenstein | Competition and price dispersion in the US airline industry[END_REF] and ( 12) the aggregate cost C af t given in [START_REF] Brandenburger | Identification of Reasoning about Rationality[END_REF] becomes

C af t = β 0 ξ f t ν a s µ s n af st q 2 af st . (13)
We argue that unobserved managerial efforts and/or marketing strategies in ξ f t are correlated with the input prices that airlines bargain with input providers. The contribution ξ f t is modeled as a linear function of wages and fuel prices faced by each airline f during period t; in other words,

log ξ f t = b log Wage f t + (1 -b) log PFuel f t + ξ f + Quarter t + Year t , (14)
where ξ f is a carrier fixed effect, and Quarter t and Year t are quarter and year time dummies. The property of linear homogeneity of degree one in input prices guarantees that the corresponding coefficients sum to 1.

The 3, 575 observation data used to estimate cost only entails 1, 298 different segments (see Table 1), which prevents us from estimating reliable individual segment fixed effects µ s for every segment. To circumvent this difficulty we assume that

µ s = d 0 + d 1 Distance s + d 2 Temperature s , (15)
where Distance s is the segment length measured as the geographical distance between two cities, and Temperature s is the average temperature at the departure and arrival cities over the whole sample window.

The cost function to be estimated obtains by reintroducing ( 14) and ( 15) into [START_REF] Brander | Market conduct in the airline industry: an empirical investigation[END_REF]. The final expression of this function is

log C af t = log b 0 + b log Wage f t + (1 -b) log PFuel f t + log s ( 1 + d * 1 Distance s + d * 2 Temperature s ) n af st q 2 af st + ξ f + ν g + Quarter t + Year t + ε af t , (16)
where ε af t is an error term. To get this expression, we have made three main simplifications.

First the constant log b 0 replaces the sum of the two constants log β 0 and log d 0 that cannot be estimated separately. Second, we normalize the coefficients

d * 1 = d 1 /d 0 and d * 2 = d 2 /d 0 .
Third, in the data airlines have preferences for specific aircraft types that gives rise to a high correlation between the airlines and aircraft fixed effects ξ f and ν a . This prevents us to keep track of both airlines and aircraft unobserved heterogeneity at the detailed level of the aircraft type. We therefore work with a more aggregated aircraft group by clustering the 29 different aircraft types into 12 groups referring to model characteristics and carrier 4 . This provides us with an aircraft group g fixed effect ν g that replaces the original aircraft fixed effect ν a . Note: * * * (resp., * * and * ) Significant at the 1 (resp., 5 and 10) percent level.

Table 4 reports the maximum likelihood estimates of the parameters of the cost function ( 16) for three variants. The results are very similar in each case, except from the impact of distance between the two cities located at the endpoints of the segment. Costs increase with input prices; they are also higher on cold weather segments and during the colder first and fourth quarters. There is a positive significant correlation between route distance and temperature in our data that makes the impact of distance on cost in variant (2) magnified when temperature is omitted. The likelihood of the variant (1), which only differs from the one in (3) by excluding aircraft group fixed effects on the segment, shows the importance of taking into account the aircraft type in airline costs. In the absence of control for the aircraft type, we find that a greater distance does not involve higher costs for the operating airlines. The expected positive impact is recovered once the control is introduced, which reflects the fact that airlines allocate specific aircraft types conditionally on the length of each segment.

Demand

We start from the linear demand specification [START_REF] Bernheim | Rationalizable strategic behavior[END_REF],

Q st = γ 0 st + γ st P st + ζ s + Quarter t + Year t + ν st , (17)
where P st and Q st are respectively the average price level and the aggregate quantity of passengers transported in segment s during period t. Both are computed from the DB1B dataset. We have added segment fixed effects ζ s and quarter and year fixed effects, Quarter t and Year t . The intercept γ 0 st and the slope γ st are two parameters to be estimated. We assume that they depend on route distance and population size of endpoint cities, Pop 1 st and Pop 2 st (with Pop 1 st ≤ Pop 2 st ). We also assume that the slope γ st may directly depend on time. The expressions of the intercept and the slope rewrite as

γ 0 st = α 0 + α 1 Pop 1 st + α 2 Pop 2 st , (18)
and

γ st = α 3 + α 4 Distance s + α 5 Pop 1 st + α 6 Pop 2 st + Quarter t + Year t . (19)
The demand equation that we estimate is [START_REF] Ciliberto | Does multimarket contact facilitate tacit collusion? Inference on conduct parameters in the airline industry[END_REF] with intercept and slope given by ( 18) and [START_REF] Coibion | How do firms form their expectations? New survey evidence[END_REF], respectively.

The (one quarter) lagged price or input prices are two potential candidates that can be used to deal with the joint determination of the price P st and the quantity Q st . As the input prices also enter the cost expression [START_REF] Ciliberto | Market structure and multiple equilibria in airline markets[END_REF], we prefer to use lagged prices in what follows. Table 5 presents the demand function estimated from the 379 selected routes. The price variable is the route average ticket price in columns 1 and 2 and the median price in columns 3 and 4. The specification in columns 2 and 4 fits the specification described in ( 7), ( 18) and [START_REF] Coibion | How do firms form their expectations? New survey evidence[END_REF]. For comparison purposes, we report in columns 1 and 3 the results in the case where the slope of the demand function is assumed to be independent from route characteristics.

We observe a higher demand for routes linking densely populated cities. In such markets demand displays lower price sensitivity. The greater distance between origin and destination points makes substitution with alternative transport facilities more difficult as α 4 is positive.

Moreover, α 5 and α 6 are both positive, which suggests that consumers departing from or 16 Note: fixed-effects (within) IV regression * * * (resp., * * and * ) Significant at the 1 (resp., 5 and 10) percent level.

(1), (2) -mean price is employed, (3), (4) -median price is employed arriving to cities with a larger population (i.e., hubs) are less sensitive to price fluctuations.

The contribution of demand to market stability captured by the index S(q * ) for route s at time t only relies on the slope δ = δ st = -1/γ st in Table 5. The results are very similar when we use average and median lagged prices as instruments. In the sequel we shall use the specification in column 2 referring to the average price. To provide the reader with an order of magnitude for this slope from this specification, we obtain γ st = -0.986 + 0.168 × 0.783 + 0.004 × 1.785 + 0.051 × 6.419 -0.519 plus quarter and year fixed effects in the average segment in Table 3. Including the time fixed effects one gets γ st = -0.621: a 10 USD increase in the fare yields 6, 210 passengers less in the route (see [START_REF] Oum | Concepts of price elasticities of transport demand and recent empirical estimates: an interpretative survey[END_REF], for an overview of transport demand elasticity estimates).

Empirical results

The cost estimation procedure is based on 1, 298 segments of the T-100 dataset while we estimate demand from 379 routes of the DB1B dataset. In order to match supply and demand information, all the subsequent results are based on the 301 common segments that appear in the two samples.

Stability index

The cost efficiency parameter σafst obtains by replacing the parameters that appear in [START_REF] Brandenburger | Intrinsic correlation in games[END_REF] with their estimated values reported in Table 4. The constant term β 0 is set to the estimate b0 = β 0 + d 0 and the statistics σafst can only be recovered at the coarse level of aggregation of the aircraft group g rather than the fine level of the aircraft type a. We use [START_REF] Blundell | Initial conditions and moment restrictions in dynamic panel data models[END_REF] to aggregate σafst over aircraft groups to get the sufficient statistics σfst for the contribution of market supply to the stability of the Nash equilibrium. On the demand side, the results in Table 5 give the estimated slopes of the demand function γst = -1/δ st that provide us with the relevant summary statistics for the contribution of market demand to stability. We therefore have all the information that is needed to compute the slope of the best-response function [START_REF] Boguslaski | Entry patterns in the southwest airlines route system[END_REF] in the linear-quadratic specification of the theoretical model, and so the stability index S(q * ) in [START_REF] Davis | Quantitative techniques for competition and antitrust analysis[END_REF]. Figure 2 in Appendix D depicts the distribution of σfst , δst , and the best-response slope.

We use formula [START_REF] Basso | Airport deregulation: Effects on pricing and capacity[END_REF] to recover the stability index S(q * ) at each segment × period level from the estimated slopes R f st (q * ). Table 6 shows that S(q * ) is lower than 1 for most segments: it exceeds 1 for only 9 percent of the segment × quarter observations. Our theoretical model thus predicts successful coordination in most markets, conditional on the fact that the set of the observed services × airlines coincides with the complete set of the services × airlines that constitute the relevant market.

The variability in the stability index comes from within and between segment heterogeneity. Although our sample window includes the 2007 crisis and the Great Recession we find that time only explains 5 percent of the variance of the stability index whereas segments contribute to 81 percent of this variance. Table 7 provides us with a more detailed picture of the stability index by reporting the results of the regression log(S st ) = X β +period t +ε st where the right-hand side variables in X consist of exogenous route characteristics as well as potentially endogenous competition indicators.

Higher values of the stability index apply to segments linking two distant and densely populated cities. Population matters even after controlling for the number of passengers. In accordance with Proposition 2, routes with high traffic display a higher stability index. Such segments are likely to involve high competition intensity. We have looked at two measures of competition. A higher number of competitors does not influence the value of the stability index. However we find that a lower Herfindahl index, which reflects greater similarity of passenger transportation market shares among the competing airlines, is associated with higher values of the stability index. This fits Proposition 3, using market shares as proxies for airlines seat capacity.

The bottom of Table 7 delineates the types of interactions among airlines that give rise to higher indexes by clustering routes based on the operating airlines. Unlike the clustering relying on exogenous route characteristics that was used to estimate the cost function, the decision to enter is probably correlated with components of the stability index. We apply 19 the Fanny clustering by [START_REF] Kaufman | Partitioning around medoids (program pam). Finding groups in data: an introduction to cluster analysis[END_REF] to the four airlines with the greatest market share in our whole sample: American Airlines (AA), Delta Airlines (DL), United Airlines (UA) and Southwest Airlines (WN). The clusters and an associated measure of within cluster similarity are reported in Table 8 (the highest within cluster similarity, which is normalized to 1, obtains when all the observations in the cluster are identical). Columns 2 to 5 of Table 8 give the percentage of observations classified in the clusters where the four main airlines is active. In cluster 1 the low-cost airlines WN is present in every segment and quarter, occupying a quasi-monopoly position by competing against airlines with low market shares in the whole sample. Cluster 2 only consists of segments × quarter operated by low market share airlines. In cluster 10, UA appears in 17% of observations, and it always compete against WN and DL. The lower within cluster similarity found for cluster 10 shows that the competition structure changes over segments and time: some observations correspond to a duopoly with WN and DL possibly competing against smallest airlines, while other observations in cluster 10 involve WN and DL competing against UA and/or AA as well as smaller airlines.

Assessing the Nash equilibrium

We are now in a position to assess whether the spread between the actual and the theoretical Nash numbers of transported passengers is positively correlated with the indexes S st over segments s and quarters t. We start from the squared spread between actual and Nash productions as

q st -q * st 2 = f ∈Fst q f st -q * f st 2 ,
where F st represents the set of airlines active in segment s during quarter t. However, in order to avoid a mechanical bias with a higher spread in segments with intense traffic, the above spread is normalized by the squared number of passengers

q * st 2 = f ∈Fst q * 2 f st .
Our normalized measure for the spread between Nash and actual production (at the segment

× period level) is therefore ∆ st = q st -q * st q * st .
We refer to the linear-quadratic specification to get the equilibrium production q * f of airlines f ,

q * f = σ f δ 0 -δQ * 1 + σδ ,
where the aggregate equilibrium production Q * obtains by summation over airlines. The estimate of q * f in segment s during quarter t obtains by replacing δ with the estimate of -1/γ st and σ f with the estimate of σ f st used in Section 4.1, and by using the results in Table 5 to replace δ 0 with the estimate of -γ 0 st /γ st satisfying [START_REF] Coibion | Information rigidity and the expectations formation process: A simple framework and new facts[END_REF]. Some details about the resulting spread density distribution are given in Appendix E.

Figure 1 shows the high positive correlation between the stability index S st and the spread ∆ st . The figure displays a smooth rise in the spread from 0 (where actual and Nash number of transported passengers coincide) when the index increases. The spread reaches its highest level around 0.8: the departure from Nash then represents 80% of the Nash production. The density of S st × ∆ st observations depicted in dotted red shows that this pattern applies to most observations, with a stability index between exp(-1.5) 0.2 and exp(-0.25) 0.8. Unlike the theoretical prediction of Proposition 1, the spread is not The results of this regression are reported in Table 9.

All the three variants reported in Table 9 display a positive high significant relation between the theoretical stability index and departures from Nash outcomes. We conclude that the stability index can be considered as a reliable predictor of the occurrence of the Nash equilibrium.

The variants differ according to fixed effects that are controlled for. The inclusion of time fixed effects does not yield substantial changes in the results. Instead, unobserved heterogeneity across segments explains an important large part of Nash departures. Our preferred specification in the third column includes both segments and quarter fixed effects: Notes: * * * (resp., * * and * ), significant at the 1 (resp., 5 and 10) percent level We report robust standard errors at the segment level.

a 1 percent increase in the stability index is associated with a 0.713 percent rise in the spread between actual and Nash numbers of transported passengers.

The recent literature provides mixed evidence about whether firm size asymmetry helps stability by facilitating equilibrium coordination. In [START_REF] Kumar | Inflation targeting does not anchor inflation expectations: Evidence from firms in New Zealand[END_REF], competitive pressure improves firms' attention to global macroeconomic indicators and so makes forecasts about current market outcomes more accurate. In Byrne and de Roos (2019), the dominant firm BP in the retail gasoline industry is able to orient price strategies of the smaller retailers toward some focal point. The results reported in Tables 8 and9 show that coordination failures on Nash outcomes are more likely in competitive markets with high traffic where competitors have similar market shares. The presence of Delta Airlines (DL), which completed its merger with Northwest (NW) from 2010, favors non-Nash behavior whereas Southwest (WN) complicates convergence toward a Nash equilibrium. This echoes the empirical literature that documents the ability of Southwest to trigger fierce reactions by its competitors (see [START_REF] Morrison | Actual, adjacent, and potential competition estimating the full effect of Southwest Airlines[END_REF] and Goolsbee and Syverson (2008)).

Time and welfare

In this section we first depart from the strictly static perspective we took so far and test whether firms accumulate any experience over time. In a second step, we shed light on potential welfare biases in the analysis of the effect of changes in competition, in situations where the entry and exit of firms creates instability and impedes firms from reaching a Nash equilibrium.

Adaptive learning

The stability index governs a process where airlines iteratively eliminate dominated choices of volumes of passenger-carrying services that can be viewed as taking place within every period t. Empirical evidence from surveys about expectations documents an extrapolative nature of expectations based on past observations and forecasts (see Ma, Sraer and Thesmar (2018) for recent examples). In our context, it seems plausible that airlines also process past information, e.g., past capacity choices of their competitors, to form/revise their beliefs about period t choices of their competitors. The existing literature has actually shown close links between the two approaches (Nachbar (1990)): a unique rationalizable outcome often implies convergence of adaptive learning toward the Nash equilibrium. One can therefore interpret the results reported in Table 9 as a signal that airlines process past information using some ad hoc myopic learning rule which eventually yields the equilibrium if the stability index is below 1.

Here we test whether the role of the stability index only transits through adaptive learning. We restrict our attention to the special case where airlines hold identical beliefs over the individual productions of their competitors. Then, a reduced form for the actual realization of the spread ∆ st that fits the local stability properties of our theoretical setup is ∆ st = S st ∆ e st where ∆ e st stands for the common forecast about the (normalized) spread between the actual and Nash volume of passengers in segment s during quarter t. We introduce adaptive learning by assuming that the belief ∆ e st is determined by a log-linear function of past forecast errors,

∆ e st = i>0 ∆ γ i s(t-i) ,
where γ i is a parameter that weights the realized error i quarters before the formation of the forecast ∆ e st . With this rule, the actual dynamics of the spread writes

∆ st = S st i>0 ∆ γ i s(t-i) , or equivalently log ∆ st = log S st + i>0 γ i log ∆ s(t-i) .
This specification highlights that log ∆ st should be in the same proportion as log S st if the role played by the stability index S st only transits through adaptive learning. Hence we consider the following econometric specification:

log ∆ st = b 1 log S st + i>0 γ i log ∆ s(t-i) + Segment s + Quarter t + ε st .
If the estimated parameter b1 differs from 1, we can reject the null hypothesis that only adaptive learning matters in the determination of the current departure from Nash.

The model is estimated using the extended linear GMM estimator based upon lagged differences of ∆ st in addition to lagged levels of ∆ st as instruments [START_REF] Blundell | Initial conditions and moment restrictions in dynamic panel data models[END_REF]. The results are provided in Table 10 together with several specification tests. The Sargan test is not completely convincing as the statistic shown presents significant evidence against the null hypothesis that the overidentifying restrictions are valid. The latter implies that we need in principle to reconsider our model or our instruments, unless we attribute the rejection to heteroskedasticity in the data-generating process. The presence of heteroskedasticity is a realistic assumption, as suggested by [START_REF] Arellano | Some tests of specification for panel data: Monte Carlo evidence and an application to employment equations[END_REF].

Another important assumption is that, when the errors ε are independently and identically distributed, the first difference errors are first-order serially correlated, which is confirmed here: first, the test statistic presents strong evidence against the null hypothesis of zero autocorrelation in the first-differenced errors at order 1; second, it presents no significant evidence of serial correlation in the first-differenced errors at order 2, which strengthens the validity of our results.

The results reported in Table 10 show that past forecasting errors do indeed matter.

The estimate b1 is usually significantly different from 1. One can thus reject the null hypothesis that the role played by the stability index is only adaptive learning based. The specification used in Column 1 shows that errors made by airlines two and three quarters earlier have a significant impact on the current Nash spread, but not the errors made in the last quarter; the short-run stickiness of seat capacity adjustments makes the last quarter seat capacity the best proxy for the current one, but this information may not be publicly available when airlines form their forecasts. In every specification airlines put less weight on far distance past errors,

|γ 2 | > |γ 3 |.
The full sample of observations is used in the first two columns of Table 10. The impact b1 of the stability index on the current departure from Nash remains very similar to the one found when neglecting the effect of past departures from Nash (in Column 3

of Table 9). The results in the last two columns apply to sub-samples that only consist of segments where there is no major airlines among the four companies selected in the clustering made in Table 8 (Column 3), or segments where only these airlines are active (Column 4). The specifications in these last two columns suggest that large airlines display a greater confidence in past strategic choices of their competitors than small airlines to form their current forecasts as the difference between b1 and 1 seems to loose significance in column 4.

Welfare impact of entry

A frequent exercise proposed by competition authorities consists in assessing the impact of a new firm entry (or exit) on the consumer surplus. The empirical framework usually assumes that a Nash equilibrium is obtained before and after entry (Scenario 1). Hence, the change in consumers' surplus is

∆ * Surplus = - 1 2 (p * 1 Q * 1 -p * 0 Q * 0 ) ,
where Q * 0 and Q * 1 are the equilibrium productions before and after entry, respectively. Here, we relax this hypothesis and assume instead that higher competition following entry possibly entails instability so that firms may not be able to play Nash quantities. We are thus interested in a scenario where airlines play a Nash equilibrium before entry while the equilibrium becomes unstable after entry and entails a production level Q 1 which is different from the Nash quantity and is directly observed in our sample (Scenario 2). The change in consumers' surplus after entry can then be computed as

∆ Surplus = - 1 2 (p 1 Q 1 -p * 0 Q * 0 ) .
To evaluate the difference between ∆ * Surplus and ∆ Surplus , we pick up a specific market, namely New-York/Tampa, where a change in the number of the competitors makes the value of the stability index jump above 1. In this market, Continental (CO), Southwest (WN), Delta (DL), and Jetblue (B6) compete against each other in 2011:4, while the next quarter witnesses the exit of Continental, and the simultaneous entry of American Airlines and United Airlines. We compute the equilibrium productions Q * 0 and Q * 1 as suggested in Section 4.2. As shown in Table 11, the corresponding prices p * 0 and p * 1 obtain using the estimated demand slope and intercept on segment s at time t from δ0

st and δst in Table 5. We find that the Nash quantity Q * 1 is greater than Q * 0 while the Nash price decreases from 336.2 USD to 321.2 USD. Thus, the increase in total surplus ∆ * Surplus per passenger in Scenario 1 is equal to 19.5 USD. In Scenario 2, where carriers make prediction mistakes in 2012:1, the increase in total surplus ∆ Surplus per passenger is equal to 34.2 USD. Hence assuming Nash behavior after entry in this particular market leads to an under-evaluation of the surplus gain.

The change in total production from one period to another can be decomposed at the airline level, as illustrated in Table 12. We find that Delta (DL) behaves very much in line with Nash while JetBlue (B6) and American Airlines (AA) produce much more than what the Nash equilibrium would predict. 6 Nash departures and the relevant market

The theoretical prediction derived from Proposition 1 is that large departures from Nash equilibria should occur in routes with a stability index above 1. This prediction does not always fit the actual pattern reproduced in Figure 1 where the spread is sometimes positive for values of the stability index falling below 1. We now explore a potential explanation for this particular feature. We show that a greater scope for unobserved competition translates into a lower value of the stability index threshold S * s above which large departures from Nash equilibria can occur. That is, S * s = 1 if the full set of products which constitute the relevant market is accounted for by the econometrician. Otherwise, the stability index threshold S * s should fall below 1. In this section, we assess the relevant threshold in every route of our dataset and comment upon the size of the relevant market.

Stability from sub-market observations

There are many circumstances in which the relevant market, i.e., the market that includes all the services that are relevant substitutes, cannot be considered in isolation. One may think of multimarket contacts through common endpoints or transportation services as a composite good that consists of differentiated items, e.g., economy versus business class services, or non-stop direct versus indirect flights. Also, on short distance routes, alternative products could entail other modes of transportation such as car or railway. Neglecting part of the potential goods or services in the relevant market leads to underestimate the overall transportation capacity that the airlines observed in our sample have actually to predict. In order to highlight the possible biases associated with missing competition, we first examine a simple variant of our theoretical setup where the relevant market consists of two substitutable items. The demand for item m (m = 1, 2) is P m (Q m , Q -m ), where Q m and Q -m stand for the aggregate demands for the two items. On the supply side we assume that the set of firms producing the two items are disjoint. This is a strong assumption if the missing information about the relevant market mostly consists of indirect flights since traditional carriers often offer both direct and indirect flights. The assumption could be better suited on short distance routes which involve few alternatives to direct flights, with airline services competing against rail transportation or other inland transport modes for instance. In addition, we simplify the exposition by working under two symmetry assumptions:

Assumption A1. Symmetry of demand: P -m (Q -m , Q m ) = P m (Q -m , Q m ) for all (Q -m , Q m )
and all m. Assumption A2. Symmetry of supply: every firm f producing item 1 has a mirror firm F + f producing item 2, i.e., σ f = σ F +f for every f = 1, . . . , F , with F being the total number of airlines producing any given item.

Firm f in market m produces q f that maximizes its profit given the aggregate production Q m -f of the other firms in market m and the aggregate production Q -m in the other market. Its best-response can be written

q f = R m f (Q m -f , Q -m
), which is decreasing with its two arguments if the two items are substitutes. In view of Assumptions A1 and A2, we focus attention on a symmetric Nash equilibrium where q * f = q * F +f for every f ≤ F . In equilibrium the aggregate output is Q * in each sub-market. The iterated process of elimination of dominated strategies in the neighborhood of such an equilibrium is now driven by the partial derivatives

R f 1 (q * ) = ∂R m f ∂Q m -f Q * -f , Q * , and R f 2 (q * ) = ∂R m f ∂Q -m Q * -f , Q *
of the best-response function of firms f and F + f with respect to its first and second argument, respectively. In this setup the stability condition given in Proposition 1 becomes: Proposition 4. A symmetric Nash equilibrium is locally the unique rationalizable outcome if and only if

f ≤F R f 1 (q * ) + R f 2 (q * ) R f 1 (q * ) -1 < 1 2 . (20)
Proof. See Appendix F.

Proposition 4 departs from Proposition 1 in two respects. First, the new stability index that appears in the left-hand side of (20), computed using information from sub-market m only, now accounts for the cross derivative R f 2 (q * ) since firms active in sub-market m need to predict the behavior of firms in the other sub-market. With two substitutable items, R f 2 (q * ) ≤ 0 and so the new index tends to be higher than the one derived in Proposition 1.

The need of predicting the behavior of competitors in the other sub-market makes stability more difficult to achieve.

Second, in the right-hand side, there is a reduction from 1 to 1/2 in the route threshold S * s above which stability is lost. In a route s where S * s < S s < 1, the equilibrium is unstable even though the stability index S s computed in Proposition 1 is below 1. A threshold S * s below 1 can be used as a signal that the relevant market of route s is larger than the observed one.

In practice, in the situation where the econometrician has no a priori knowledge about how far the relevant market goes beyond the observed one, the derivative R f 2 (q * ) cannot be computed. In the empirical illustration below we will neglect this derivative by setting R f 2 (q * ) equal to 0, and continue to refer to the stability index given in Proposition 1.

Rewriting [START_REF] Crawford | Structural models of nonequilibrium strategic thinking: Theory, evidence, and applications[END_REF] as

f ≤F R f 1 (q * ) R f 1 (q * ) -1 < 1 2 - f ≤F R f 2 (q * ) R f 1 (q * ) -1 (21)
shows that we underestimate the true right-hand side of (20) when we set R f 2 (q * ) to 0.

Stability index threshold

To identify the S * s threshold, we exploit the fact that a route s with a high spread ∆ st is more likely to originate from a high spread regime, i.e., a regime where S st ≥ S * s . Instead a low spread is more likely to be associated with a low spread regime where S st < S * s . Our estimation strategy works as follows. In a first step, we compute the empirical probability π st that route s falls in the low spread regime during quarter t using the machine learning EM algorithm developed by [START_REF] Dempster | Maximum Likelihood from Incomplete Data via the EM Algorithm[END_REF]. The details of the estimation procedure are provided in Appendix G. In a second step, we obtain individual estimates of the threshold Ŝ * st by matching π st to the probability Pr(S st ≤ Ŝ * st ) referring to the empirical distribution of the estimated stability index, where the stability index S st for all s and t is estimated from [START_REF] Basso | Airport deregulation: Effects on pricing and capacity[END_REF].

To switch from Ŝ * st to one single statistic Ŝ * s for each segment s, we set Ŝ * s = max t Ŝ * st . This should be seen as a conservative choice that classifies segment s as in equilibrium if it is in equilibrium during at least one period t. In most routes this choice is without loss of great generality since the time dispersion of Ŝ * st within routes is low (see Figure 5 in Appendix).5 Table 13 provides summary statistics about the estimated Ŝ * s at the route level. All values fall below 1, ranging from 0.18 to 0.87, which suggests that all routes in our dataset suffer to a certain extend from unobserved competition. The scope of unobserved competition is however limited since most routes involve a threshold above 0.8. With these new estimates we find that one-third of the routes can be regarded as falling in the high spread regime, while our initial threshold equal to 1 led to significantly more optimistic conclusions (less than 10% of the markets were supposed to be out of an equilibrium).

Table 14 illustrates how the estimated statistics Ŝ * s interacts with several explanatory variables. In column 1 we regress Ŝ * s on variables that can be considered as exogenous. A high value of the threshold (a narrower scope for unobserved competition) is associated with routes linking less populated cities, which probably indicates that the share of indirect services supplied by competing airlines is lower in this case. The second column, which introduces additional regressors, sheds light on the fact that more concentrated markets are associated with lower Ŝ * s , which is not completely surprising given that these markets are generally operated by big traditional airlines supplying large menus of indirect services in addition to direct flights.

The effect of the observed market size when one accounts for unobserved competition can be put in perspective with our previous results in Table 7. We report in Table 15 information about the 10 routes with respectively the largest and smallest average number of passengers per quarter. The 10 largest routes display a larger average spread ∆ s and these departures from Nash behavior appear to reflect the combination of both higher stability index S s and lower route threshold Ŝ * s . No largest route in our sample eventually reaches equilibrium (S s > Ŝ * s ) and instability here comes with high share of unobserved products ( Ŝ * s is further away from 1); the two routes Atlanta-Miami and Boston-Washington are detected to be especially subject to high unobserved competition.

The Wright amendment repeal

The Wright amendment (WA) investigated in [START_REF] Ciliberto | Market structure and multiple equilibria in airline markets[END_REF] serves as an interesting natural experiment which can be exploited in order to provide additional feedback about the validity of our methodology. The aim of the WA was to restrict airline services out of the Dallas Love airport (DAL) in order to stimulate the activity of the Dallas/Fort Worth airport (DFW). In 1980, the WA gets effective and states that airline services in DAL using large aircraft could be provided only to airports within Texas and its four neighboring U.S. states, namely Arkansas, Louisiana, New Mexico and Oklahoma [START_REF] Allen | The Wright Amendment: The Constitutionality and Propriety of the Restrictions on Dallas Love Field[END_REF]). Flights to other states are allowed only on small aircraft. Airlines could not offer connecting flights, through service on another airline, or through ticketing beyond the five-state region. In October 2006 a partial repeal is decided and the full repeal gets effective in 2014.

The abrogation of airline service restrictions from DAL in a Southwest stronghold area implies greater competitive pressure on DFW, where American Airlines operates direct non-stop long-haul flights. The abrogation of service restrictions affects the expansion of the size of the relevant market of services including the Dallas/Fort Worth area as an origin or destination point, depending on whether the point of destination of origin belongs to the so-called five-state region or not. Consider the case of the Dallas-Washington market for instance: under the WA, all flights had to go through DFW since no services were allowed from/to DAL; all of the airline services of the relevant market Dallas-Washington would therefore be products operated from/to DFW. After the abrogation of the WA, the same relevant market would typically include all airline services from both DAL and DFW. If the econometrician has only data on airline services from/to DFW, she does not suffer from any missing information as long as the WA is effective (in which case the stability index threshold should be close to 1); after the abrogation of the WA however, a significant share of information would be missing, and this should be reflected in a fall in the stability index threshold. The results reported in Table 17 are largely consistent with these predictions.

We propose to test empirically this prediction with our data, using only information on services from/to DFW. We consider three sub-periods, namely 2003:1-2006:2 (before the announcement of the repeal of the WA), 2006:3-2014:2 (from the announcement to the repeal of the WA) and 2014:2-2016:4 (after the repeal of the WA). Our subsample contains 18 routes that include DFW at some endpoints. Table 17 in the Appendix shows that the stability index threshold is stable across the three periods in every market that makes a connection between the Dallas/Fort Worth area and a city market in the WA zone (i.e., a city market located in Texas, Arkansas, Louisiana, New Mexico or Oklahoma). We do not detect any systematic change in the stability index or the spread between actual and Nash volumes of transported passengers in these routes.

The situation is however quite different for markets connecting the Dallas/Fort Worth area to cities outside the Wright zone. Indeed we find that the threshold is sharply reduced after the repeal of the amendment (i.e., from 2006:3-2014:2 to 2014:2-2016:4), while the announcement effect seems to be not significant (there is no significant difference between 2003:1-2006:2 and 2006:3-2014:2). We also find a larger spread after the repeal as we observe higher stability indexes. This probably suggests that the repeal of the WA introduced some instability in each market from/to the Dallas/Fort Worth area.

Conclusion

Greater competition is often viewed as driving welfare gains from lower equilibrium prices;

our paper shows that it may also compromise the occurrence of an equilibrium. Thus, in markets where the usual indicators of high competitive pressure are present, i.e., those where several airlines with similar market shares or competitive low cost companies are present, the traditional equilibrium welfare analysis has to be worked out carefully. Eventually the equilibrium would be a reliable reference in only 70 -90% of the routes.

Our analysis is subject to a number of potential limitations that could be analyzed in future work.

1. Data from the U.S. Department of Transportation provide us with information about airlines costs at the non-stop flight segment level. We therefore estimate demand at the same level, i.e., we restrict ourselves to routes where the share of direct flights is high enough. Our identification procedure for the scope of the relevant market however suggests that indirect flights matter, especially in routes with large flows of passengers, and so should be explicitly taken into account. Any initiative that could ease the combination of the two types of information is obviously welcome.

2. Our analysis abstracts from dynamic aspects that are certainly important in shaping the regular interactions between the airline companies that compete in a route. The insights from our robustness check suggest that firms certainly retrieve valuable information from bad prediction in the past. Brandenburger, Danieli and Friedenberg (2019) makes progress toward identification of the level of rationality in this context.

An empirical application on the airline industry may be more challenging to implement as medium and long-run strategies also encompass both slot portfolios, which requires introducing airports into the analysis, as well as capacity choices. We hope to investigate some of those issues in future research.

For online publication A Proof Proposition 1

By Lemma 3 in Desgranges and Gauthier (2016) the matrix B has a spectral radius less than 1 if and only if ( 5) is satisfied. The slope in [START_REF] Basu | A characterization of the class of rationalizable equilibria of oligopoly games[END_REF] obtains by differentiating the first-order condition for firm f profit maximization. The expression of C f (q * f ) obtains by applying the generalized envelope theorem to the cost minimization program (1). This yields C f (q f ) = λ, where λ is the non-negative Lagrange multiplier associated with the production constraint in [START_REF] Aguirregabiria | Identification and estimation of dynamic games when players' beliefs are not in equilibrium[END_REF]. Therefore,

C f (q f ) = dλ dq f .
The first-order conditions for the cost minimization problem (1) are λ = c af (q af ) for all a ∈ A f and a∈A f n af q af = q f for every f . Differentiating these first-order conditions yields dλ = c af (q af )dq af for all a ∈ A f and a∈A f n af dq af = dq f for every f . Reintroducing the expression of dq af into the last equality we finally obtain

dλ a∈A f n af c af (q af ) = dq f .
This completes the proof.

B Proof Proposition 2

In the linear-quadratic specification, the stability index

S(q * ) = f δσ f 1 + 3δσ f . (22)
does not depend on production. The equilibrium production of firm f satisfies

q * f = σ f 1 + σ f δ (δ 0 -δQ * ), (23)
Summing gives the aggregate equilibrium production

Q * = f δ 0 σ f 1 + δσ f 1 + f δ σ f 1 + δσ f . ( 24
)
Both the index S(q * ) and the production Q * are increasing with σ f (which itself increases with σ af ). An additional marginal aircraft transfer corresponds to dσ f > 0. The result follows.

C Proof Proposition 3

It is similar to the proof of proposition 2 but now exploits second-order derivatives of S(q * ) and Q * . In the linear-quadratic specification, S(q * ) and Q * are increasing concave functions of σ f (which is itself increasing linear in σ af ). A marginal aircraft transfer dn af = -dn af = -1 yields dσ f = -σ af < 0 and dσ f = -dσ f > 0. The result follows.

D Stability index estimate

The left-panel of Figure 2 depicts the density of σfst at the airlines × segment × quarter level. Its distribution is positively skewed with a long right tail (Table 6 shows that the median estimated value of this parameter is 0.515 while its average of 0.762). The statistics σfst is inversely related to the marginal cost. Hence Figure 2 shows that the marginal cost for transporting one additional passenger quickly increases with the total number of transported passengers. The small subset succeeding to contain marginal costs consists of the largest airlines: we find that the average number of transported passengers is 3.6 times higher for observations in the last quartile of the σfst distribution (σ f st ≥ 0.965).

The shape of the density of δst shown in the middle panel in Figure 2 is quite different. The homogeneity of demand behavior (the low variability of δst ) across segments makes the distribution of the slope of the best-response R f st (q * ) in ( 9) at the airlines × segment × quarter level to be mostly driven by cost heterogeneity (the high variability of σfst ). It is reported in the right panel in Figure 2. Airlines are found to display significant inertia in how they react to the production of others. The average slope equals -0.299, i.e., an average airlines would only transport 30 passengers less when its competitors are expected to transport 100 additional passengers. Short-run inertia is plausible in the airline industry since the number of passengers can be adjusted within the limit of the route transportation seat capacity, which is mostly fixed over a quarter. The theory predicts that this inertia favors stability of the Nash equilibrium, but we know from (5) that stability relies on the sum of all the slopes of the best-response functions of the airlines active on the segment.

Despite individual inertia, stability would be lost with more than 4 identical competing airlines characterized by a -0.299 average best-response slope.

E Spread density distribution

F Proof Proposition 4

The new B matrix that governs the iterated process of elimination of dominated strategies is a 2F ×2F matrix whose f -th row, f ≤ F , is 0 at f -th column, is R f 2 (q * ) in every column j > F and R f 1 (q * ) otherwise. Similarly, its (F + f )-th row is 0 at F + f -th column, is R f 2 (q * ) in every column j ≤ F and R f 1 (q * ) otherwise. Matrix B is contracting if and only if the spectral radius of the positive matrix -B is lower than 1. Let e some eigenvalue e of -B and v be the associated 2F -eigenvector (v 1 , . . . , v 2F ). From ev = -Bv, we have: for all f ≤ F ,

ev f -R f 1 v f = - k≤F R f 1 v k - F <k≤2F R f 2 v k ⇔ v f = - R f 1 e -R f 1 k≤F v k - R f 2 e -R f
v F +f = - R f 2 e -R f 1 k≤F v k - R f 1 e -R f 1 F <k≤2F v k .
All the derivatives of the best-response functions are evaluated at q * . Summing over firms yields

f ≤F v f = - f ≤F R f 1 e -R f 1 k≤F v k - f ≤F R f 2 e -R f 1 F <k≤2F v k and F <f ≤2F v F +f = - F <f ≤2F R f 2 e -R f 1 k≤F v k - F <f ≤2F R f 1 e -R f 1 F <k≤2F v k .
The symmetry properties of B imply that the eigenvectors are such that v f = v F +f for all f ≤ F . Hence the two previous equations reduce to

f ≤2F v f = - f ≤2F R f 1 + R f 2 e -R f 1 k≤2F v k .
Eigenvalues e of -B thus are solutions to G(e) ≡ -

f ≤2F R f 1 + R f 2 e -R f 1 -1 = 0
The function G is continuous decreasing for all e ≥ 0, with G(0) > 0 > -1 = G(+∞).

There is consequently a unique e ≥ 0 solution to G(e) = 0. This is the spectral radius.

Since G is decreasing, this eigenvalue is lower than 1 if and only if G(1) < 0, or equivalently,

- f ≤2F R f 1 + R f 2 1 -R f 1 -1 < 0.
The result follows from R f m = R (F +f )m for every f ≤ F and m = 1, 2.

G Spread regimes from the EM algorithm

The EM algorithm is unsupervised as it is designed to cluster points (the various spreads in our setup) that do not come with any specific label (a low or high spread regime). We assume that the distribution of the spread ∆ st reproduced in plain black in Figure 4 arises from a mixture of two Gaussian distributions: the first distribution is associated with a low spread, in which case the Nash equilibrium is a plausible outcome of competition; the second distribution is associated with a high spread, which corresponds to a more unstable regime. The EM algorithm aims at generating the probability that a spread point originates from any of the two regimes. This probability is then used to derive the individual market threshold we are interested in.

Market

  ID number assigned by the U.S. Department of Transportation (DOT). To estimate market supply and demand functions, we combine demographic and climate information with three publicly available databases released by the Bureau of Transportation Statistics of the U.S. DOT: the Air Carrier Financial Reports, the Air Carrier Statistics and the Airline Origin and Destination Survey (DB1B).

Figure 1 :

 1 Figure 1: Density and average spread per range 2e-02 stability indexes

3 .

 3 A relevant choice for the measure of the discrepancy between the actual and equilibrium strategies regarding volumes of transported passengers needs a suitable equilibrium reference. Our paper refers to a version of a non-cooperative Cournot game with linear passengers demand and quadratic airline costs, but a positive spread could also come from an inadequate reference. There are two main sources of model misspecification in our context. The first one relates to the restrictive linear-quadratic modeling of market fundamentals that precludes multiplicity of Nash equilibria. In the presence of multiple Nash equilibria, local rationalizability can still be employed as a selection device to eliminate unstable equilibria from the set of empirically relevant outcomes. Still multiplicity of locally stable equilibria is possible, in which case there is no longer any obvious equilibrium reference. The second source of misspecification relates to the short-run strategies assigned to airlines. Volumes of passengers is only part of yield management strategies: individual price setting certainly also enters the airlines choice set; and explicit alliances or collusive behaviors weaken the status of the non-cooperative equilibrium reference.

Figure 2 :

 2 Figure 2: Probability density functions

Figure 3

 3 Figure 3 depicts in solid black the density of the spread between actual and Nash production at the more disaggregated segment × airlines × period level. This distribution combines situations where airlines produce below the equilibrium number of passengers and situations where they instead produce above it. The first (resp. last) type of situations reflects an underestimated (resp., overestimated) expected price. Figure3highlights some symmetry between under and overestimated prices when forecast errors are small (the spread stands below 0.25). For larger values of the spread, the density corresponding to an overestimated production of others / an underestimated price is depicted in dashed blue.

Figure 3

 3 Figure 3 clearly shows limited errors in this case, compared to the case where airlines underestimate the production of others / overestimate prices (depicted in dashed red). If one considers a predicted low price as a symptom of firms' pessimism, Figure 3 suggests limited forecasting errors from pessimistic airlines. This would accord with the recent findings based on surveys about financial analysts, CEO or Chief Financial Officers expectations or proxies for managerial expectations that errors relate to overconfidence and optimism (see, e.g., Ma, Sraer and Thesmar (2018)).

Figure 3 :

 3 Figure 3: Spread density distribution

Figure 4 :

 4 Figure 4: Spread Gaussian mixture from the EM algorithm

Table 1 :

 1 observations at the carrier × segment level that contains 22 carriers operating on 1, 298 segments and carrying 80 percent of the passengers transported in the U.S. domestic market. Some descriptive statistics for the average carrier are provided in Table1. Carrier cost descriptive statistics

	1 of the Air Carrier Financial Reports and the Air Carrier Statistics T-100
	Domestic Segment. Schedule P-5.1 includes cost information, namely, data on input prices,
	maintenance expenses, equipment depreciation, rental costs, and total operating expenses
	disaggregated by airlines × aircraft type. Costs are available for each aircraft type, but the
	data does not include cost broken out by airline route. This data limitation obliges us to
	refer in the cost function to an output defined at the aircraft level as well. Since airlines
	produce passenger transportation services on non-stop flights using a single aircraft, the
	aircraft level coincides with the segment level, i.e., direct non-stop flights. A flight from
	city A to city C that entails a stop at city B consists of the two segments AB and BC, and
	we will estimate in section 3.2 separate costs for each of the two segments. At the segment
	level we can ultimately recover the costs at airlines × segment × aircraft type level that

1. Prices and costs are adjusted using the transportation sector price index of the Bureau of Labor Statistics, http://www.bls.gov/cpi/ Demand is estimated from the Airline Origin and Destination Survey (DB1B) database over the 2003:2016 period. The DB1B sample contains more than 4 million observations at

Table 2 :

 2 Route sub-sample selection from DB1B dataset

		Number of routes Mean Standard deviation Min	Max
	Original base	11141
	Passengers per route (in thousands)	18.146	65.839	0.010 2986.0
	Share of direct tickets per route	0.46	0.39	0	1
	Direct routes (share of direct flights > 0.6)	3337
	Passengers per route (in thousands)	38.118	95.359	0.050 2986.0
	Share of direct tickets per route	0.87	0.12	0.60	1
	Routes in the final sample	379
	Passengers per route (in thousands)	265.9	263.8	9.9 2986.0
	Share of direct tickets per route	0.90	0.09	0.60	1

Table 3 :

 3 Route descriptive statistics

	Mean Standard deviation Min	Max

* Corrected by the consumer price index for transportation sector

3

This information is obtained from ggweather.com and citypopulation.de.

Table 4 :

 4 Estimated cost function C af t

			log C af t
		(1)	(2)	(3)
	Constant (log b 0 )	2.513*** 1.127***	2.182***
		(0.34)	(0.31)	(0.45)
	log Wage (b)	0.558*** 0.597***	0.602***
		(0.08)	(0.07)	(0.07)
	Distance (d * 1 )	0.025	0.709***	0.205*
		(0.02)	(0.37)	(0.11)
	Temperature (d * 2 )	-0.006***		-0.006***
		(0.00)		(0.00)
	Quarter 2	-0.170*** -0.157***	-0.164***
		(0.01)	(0.01)	(0.01)
	Quarter 3	-0.199*** -0.180***	-0.193***
		(0.01)	(0.01)	(0.01)
	Quarter 4	-0.096*** -0.086***	-0.089***
		(0.01)	(0.01)	(0.01)
	Aircraft group f.e. (ν g )	No	Yes	Yes
	Airlines f.e. (ξ f )	Yes	Yes	Yes
	Year f.e. (Year t )	Yes	Yes	Yes
	Number of observations	3,575	3,575	3,575
	Log-Likelihood	-1010	-453	-407

Table 5 :

 5 Estimated demand function

			Number of passengers
		(1)	(2)	(3)	(4)
	Intercept γ 0 st -eq. (18)
	Constant α 0	133.646*** 207.210*** 126.531*** 196.328***
		(16.24)	(16.58)	(16.26)	(16.62)
	Pop 1 (α 1 )	18.116*** 19.037*** 18.829*** 18.518***
		(2.45)	(2.45)	(2.46)	(2.46)
	Pop 2 (α 2 )	47.003*** 22.234*** 46.288*** 17.078***
		(3.02)	(3.70)	(3.03)	(3.82)
	Slope γ st -eq. (19)
	Constant (α 3 )	-0.505***	-0.986***	-0.583***	-1.060***
		(0.02)	(0.04)	(0.02)	(0.04)
	Distance (α 4 )		0.168***		0.148***
			(0.02)		(0.053)
	Pop 1 (α 5 )		0.004**		0.013***
			(0.00)		(0.00)
	Pop 2 (α 6 )		0.051***		0.078***
			(0.01)		(0.01)
	Year f.e.	Yes	Yes	Yes	Yes
	Quarter f.e.	Yes	Yes	Yes	Yes
	R 2	0.176	0.204	0.170	0.198
	Obs.	20808	20808	20808	20808
	F test (route f.e.)	1003.8	982.9	1008.7	978.1

Table 6 :

 6 Sufficient statistics for stability

		Min.	1st Qu. Median	Mean	3rd Qu.	Max.
	Cost efficiency σ f st	0.002	0.28	0.515	0.762	0.965	5.935
	Inverse demand slope δ st	0.915	1.28	1.495	1.609	1.816	5.113
	Best-response slope R (q * ) -0.482	-0.375	-0.309	-0.299	-0.231	-0.002
	Stability index S(q * )	0.167	0.458	0.568	0.629	0.774	1.794

Table 7 :

 7 A picture of the stability index

	log(S st )

Note: robust standard errors clustered by route. * * * (resp., * * and * ) Significant at the 1 (resp., 5 and 10) percent level.

Table 8 :

 8 Airlines network from the main airlines

	cluster WN UA AA DL Within homogeneity Nb of segments Nb of observations
	1	100	0	0	0	1	83	1, 448
	2	0	0	0	0	1	71	631
	5	0	0	0	100	1	62	1, 071
	8	100	0	100	0	1	57	738
	10	100	17	8	100	0.580	57	719
	9	100	100	0	0	1	56	957
	4	0	100	0	15	0.740	50	486
	7	100	100 100	5	0.920	43	614
	3	0	0	100	32	0.570	30	490
	6	0	100 100	36	0.590	21	268

Table 7

 7 delineates three groups of airlines clusters. A first group with low stability indexes is formed by routes where UA and AA hold a dominant position and routes where they compete against each other. This group also includes routes where no major airlines is active. There is a second set of routes with intermediate values of the index, where WN compete against AA and UA. The presence of the low-cost company WN thus tends to be associated with higher values of the stability index. In the last group, with the highest values of the index, we find routes where WN and the major airlines DL interact.

Table 9 :

 9 Nash departure and stability index

			Nash spread log ∆ st
		(1)	(2)	(3)
	Stability index log S st (b 1 )	0.539 * * *	0.574 * * *	0.713 * * *
		(0.131)	(0.145)	(0.089)
	Constant (b 0 )	-0.829 * * *	-0.731 * * *	-0.560 * * *
		(0.077)	(0.117)	(0.085)
	Quarter fixed effect	No	Yes	Yes
	Segment fixed effect	No	No	Yes
	Number of observations	7,422	7,422	7,422
	R 2	0.096	0.117	0.596
	Adjusted R 2	0.096	0.110	0.575
	Residual Standard Error	0.632 (df = 7420)	0.627 (df = 7365)	0.433 (df = 7065)
	F -statistic	788.090

* * * (df = 1; 7420) 17.459 * * * (df = 56; 7365) 29.230 * * * (df = 356; 7065)

Table 10 :

 10 Adaptive versus rationalizability-based justifications of the Nash equilibrium

				log ∆ st
		(1)	(2)	(3)	(4)
	log ∆ s(t-1)	0.258
		(0.577)
	log ∆ s(t-2)	-0.382 * * *	-0.320 * * *	-0.348 * * *	-0.301 * * *
		(0.142)	(0.057)	(0.074)	(0.080)
	log ∆ s(t-3)	-0.175	-0.227 * * *	-0.247 * * *	-0.221 * * *
		(0.137)	(0.033)	(0.050)	(0.038)
	log S st	0.705 * * *	0.774 * * *	0.705 * * *	0.799 * * *
		(0.200)	(0.089)	(0.123)	(0.125)
	Nb of segments	301	301	178	123
	Quarters used (over 56)	1-51	1-51	1-47	2-51
	Nb of observations	7422	7422	3510	3912
	Sargan test (p-value)	0.01	0.05	0.151	0.331
	Autocorrelation test (1) (p-value)	0.287	1.77e-07	1.71e-04	5.92e-05
	Autocorrelation test (2) (p-value)	0.519	0.676	0.241	0.320
	Wald test for coefficients (p-value)	≤ 2.22e-16 ≤ 2.22e-16	5.01e-12	≤ 2.22e-16
	Wald test for time dummies (p-value)	4.55e-11	5.54e-09	1.44e-05	≤ 2.22e-16
	Notes:

* * * (resp., * * and * ), significant at the 1 (resp., 5 and 10) percent level We use the robust covariance matrix proposed by Windmeijer (2005).

Table 11 :

 11 Welfare in the New-York/Tampa segment

	Period	2011:4	2012:1
	Competing careers	CO, WN, DL, B6	UA, WN, DL, B6, AA
	Stability index S st	0.974	1.055
	Nash Quantity (thousands)	405.5	417.1
	Observed Quantity (thousands)	405.5	426,5
	Nash price (in USD)	336.2	321.2
	Observed price (in USD)	336.2	305.9
	Nash Surplus (in USD)	133,498.6	141,640.4
	∆ * Surplus (in USD)		+8,141.8
	∆ * Surplus per passenger (in USD)		+19.5
	Real Surplus (in USD)	133,498.6	148,088.9
	∆ Surplus (in USD)		+14,590.3
	∆ Surplus per passenger (in USD)		+34.2

Table 12 :

 12 Actual and Nash passengers in the New-York/Tampa segment

	Airlines Nash Quantity 2011:4 Nash Quantity 2012:1 Observed Quantity 2012:1
	CO	105,9	-	-
	NW	58.6	51.6	36.9
	DL	101.8	94.9	89.7
	B6	118.8	104.6	141.7
	AA	-	28.5	19.7
	UA	-	109.4	138.2

Table 13 :

 13 Estimated stability index threshold distribution

		Ŝ * s
	Min.	1st Qu. Median Mean 3rd Qu. Max.
	0.1862 0.8360 0.8698 0.8097 0.8714 0.8715
	Number of routes: 301

Table 15 :

 15 Relevant market for 20 airlines routes

	Origin	Destination	Passengers	Distance	Spread	Stability index	Threshold S * s
	(US DOT city market)	(US DOT city market) (per quarter -thousand)	(km)	(within route average) (within route average) (upper bound)
	10 routes with the greatest average traffic per quarter
	Boston, MA	Washington, DC	1117	358	0.70	1.52	0.45
	Orlando, FL	New York City, NY	1029	937	0.58	1.29	0.54
	San Francisco, CA	Los Angeles, CA	948	332	0.46	1.40	0.60
	Atlanta, GA	New York City, NY	924	748	0.81	1.02	0.54
	Atlanta, GA	Miami, FL	905	584	0.90	0.95	0.31
	Washington, DC	Chicago, IL	854	596	0.65	1.03	0.79
	Atlanta, GA	Washington, DC	839	551	0.65	0.62	0.58
	Las Vegas, NV	San Francisco, CA	761	399	0.59	1.06	0.84
	Denver, CO	Los Angeles, CA	740	846	0.56	1.01	0.73
	Phoenix, AZ	Los Angeles, CA	739	350	0.50	0.76	0.65
	10 routes with the least average traffic per quarter
	Burlington, VT	Philadelphia, PA	44	335	0.27	0.29	0.87
	Chicago, IL	Fort Wayne, IN	41	157	0.49	0.35	0.79
	Denver, CO	Sioux Falls, SD	40	483	0.20	0.27	0.87
	Jackson, WY	Salt Lake City, UT	39	205	0.79	0.25	0.85
	Philadelphia, PA	Syracuse, NY	36	228	0.54	0.26	0.35
	Philadelphia, PA	Richmond, VA	36	198	0.64	0.26	0.30
	Chicago, IL	Sioux Falls, SD	34	462	0.35	0.25	0.82
	Philadelphia, PA	Rochester, NY	33	257	0.35	0.24	0.84
	Greensboro/High Point, NC Philadelphia, PA	32	365	0.44	0.26	0.64
	Denver, CO	Jackson, WY	32	406	0.16	0.21	0.87
	Denver, CO	Rapid City, SD	32	300	0.23	0.19	0.85

The T-100 dataset has been frequently used in the economic literature interested in airline competition. For further discussion on data selection, see for instance[START_REF] Ciliberto | Market structure and multiple equilibria in airline markets[END_REF].

Aircraft in the same cluster belong to the same generation of models and have similar size. For example, Boeing 737-300, Boeing 737-400 and Boeing 737-500 are allocated to the same cluster while next generation larger Boeing 737-800 and Boeing 737-900 are in another cluster. There remain small clusters with rare aircraft types like Aerospatiale/Aeritalia ATR-72 or Saab-Fairchild 340/B.

Our illustration on the Wright amendment in Section

6.3 weakens this criterion by considering several sub-periods in a given route.

regime.

The k-means clustering yields an average probability of π st (0) = 0.65 that a specific spread drawn randomly in the sample originates from the low spread regime. Given this probability and the mean and standard error in each regime, we can compute from Bayes's rule the probability π st (1) that ∆ st is actually drawn from the low spread regime for all s and t. Then, given these a posteriori probabilities, we can compute the maximum likelihood estimators for the means and standard errors of the two regimes. The new Gaussian distributions are used to revise π st (1) into π st (2) for every s and t according to Bayes's rule, which allows us to initiate another step of estimation for the two moments of the two Gaussian distributions. The EM algorithm repeats these steps until convergence. 6 In Table 16 we report the moments of the two Gaussian distributions for two variants. 7 In both cases, the spread distribution for the subsample of routes with a stability index above 1, that we know are part of the high spread regime, is the same as the spread distribution in the high spread regime. The mean spread in the low spread regime is 0.25;

it is twice as high in the high spread regime. The dispersion of the spread is also twice as high in the high spread regime (0.11 versus 0. The final probability π st that ∆ st is drawn from the low spread regime distribution ranges from 0 to 0.83. In Figure 5 we plot the within route average probability and its standard error for each of our 301 route sample. In the horizontal axis routes are ranked in the order of increasing within route average probabilities. The routes with low or high 6 The EM algorithm stabilizes in a local maximum for the likelihood. In our setup it always converges to the same outcome. 7 In the first variant, we a priori require that the spread mean and standard error in the high spread regime are respectively equal to the empirical mean and standard error of the spread among the subset of points ∆ st with a stability index S st above 1. That is, we only apply the EM algorithm to estimate the first two moments of the Gaussian distribution of the spread in the low spread regime. This variant thus a priori imposes the theoretical consistency requirement that observations with a stability index above 1 are drawn from the same probability distribution as those falling in the high spread regime, even though the associated stability index is below 1. In the second variant, we impose no constraint on the moments of the Gaussian distribution of the spread in the high spread regime.