L Decreusefond

T T Vu

P Martins

Modeling energy consumption in cellular networks

In this paper we present a new analysis of energy consumption in cellular networks. We focus on the distribution of energy consumed by a base station for one isolated cell. We first define the energy consumption model in which the consumed energy is divided into two parts: The additive part and the broadcast part. The broadcast part is the part of energy which is oblivious of the number of mobile stations but depends on the farthest terminal, for instance, the energy effort necessary to maintain the beacon signal. The additive part is due to the communication power which depends on both the positions, mobility and activity of all the users. We evaluate by closed form expressions the mean and variance of the consumed energy. Our analytic evaluation is based on the hypothesis that mobiles are distributed according to a Poisson point process. We show that the two parts of energy are of the same order of magnitude and that substantial gain can be obtained by power control. We then consider the impact of mobility on the energy consumption. We apply our model to two case studies: The first one is to optimize the cell radius from the energetic point of view, the second one is to dimension the battery of a base station in sites that do not have access to permanent power supply.

I. INTRODUCTION

According to the GSM Association, more than 80% of a typical mobile network operator's energy requirements are associated with operating the network. The typical annual CO2 emissions per average GSM subscriber is now about 25kg CO2, which equates to the same emissions created by driving an average European car on the motorway for around one hour. However, the mobile industry continues to look for ways to reduce energy needs. Air conditioning is being replaced by fans or passive air flows whenever possible. Several programs are aiming to deploy solar, wind, or sustainable bio fuels technologies to 118,000 new and existing off-grid base stations in developing countries by 2012. Network optimization upgrades currently can reduce energy consumption by 44% and solar-powered base stations could reduce carbon emissions by 80%. Optimization of the physical network through improved planning and the spectrum allocations for mobile broadband can also contribute to significant energy savings.

As a consequence of the previous statements, it appears clearly that energy consumption must be taken into consideration at the very beginning of the conception of cellular networks. For the development of cellular communications in emerging countries, it is necessary to be as energy conservative as possible by using the least possible number of base stations for a given quality of service. As the size of a cell covered by a given base station depends essentially on the emitting power of its antenna, the smaller the size of a cell, the less the consumed energy. However, when base stations cover a small region, many of them are necessary to cover a given region. There is thus a trade-off between the number and the coverage of each base stations. In order to fix the optimal radius of a cell, one must have quantitative models of the energy consumed at a base station in terms of positions, locations and traffic of the terminals. Furthermore, if we think about the deployment of base stations in low populated regions without power supply, base stations should be energetically autonomous, thus powered by a battery, be it solar or chemical. The energy consumed in such a situation is thus a key parameter in the building of a cellular network. These are the two questions we aim to answer in the following considerations.

Several measurement based analysis pointed out the different aspects in energy consumption (see [START_REF] Fehske | The global footprint of mobile communications: The ecological and economic perspective[END_REF], [START_REF] Fettweis | ICT energy consumption -trends and challenges[END_REF], [3], [START_REF] Etoh | Energy consumption issues on mobile network systems[END_REF] and references therein). In [START_REF] Ullah | Applicability of different models of burstiness to energy consumption estimation[END_REF], some models are proposed for the activity of a single terminal and the resulting energy consumption. In [START_REF] Chinara | Mobility based clustering algorithm and the energy consumption model of dynamic nodes in mobile ad hoc network[END_REF], [START_REF] Heinzelman | Energyefficient communication protocol for wireless microsensor networks[END_REF], the choice of the cluster heads in Ad-Hoc networks integrates energy consumption consideration and take into account the geometry of the terminals by considering the proximity graph. In [START_REF] Wu | Minimum-energy multicast in mobile ad hoc networks using network coding[END_REF], energy saving motivates to use network coding in ad-hoc networks. As a conclusion, there are many investigations about how to save energy but no model seems to emerge in order to evaluate quantitatively the energy consumption. There is however one notable exception which is the paper [START_REF] Xiang | Energy efficiency evaluation of cellular networks based on spatial distributions of traffic load and power consumption[END_REF]. In that work, a stochastic geometry based model for energy consumption in a cellular network is considered. It assumes that mobiles are connected to their closest base stations and introduces an energy consumption model based on the distance between base stations and mobiles, taking into account interference due to the presence of several base stations. We go further in this direction by considering a refined model for the energy consumption in an isolated cell, as it includes the energy devoted to broadcast messages (like the beacon signal) and takes into account both traffic activity and users mobility. Moreover, instead of relying on simulation results, we give as much as possible closed form formulas for different statistics of the energy consumption. This leads to qualitative results showing the importance of the path-loss exponent (see below for its definition).

This paper is organized as follows: In Section II, we recollect basic and advanced facts about Poisson point processes in general spaces. We then present the system model based on a Poisson point process including not only the positions but also the traffic activity and the mobility pattern of each user. In Section IV, we first evaluate the consumed energy for motionless users. We show, in Section V, that mobility does not change the mean value but decreases the variance of the consumed energy. Thus, as far as dimensioning is concerned, it is conservative to consider that users do not move. We apply these considerations in Section VI to two case studies: Finding the optimal radius of a cell under energy constraints and estimating the power of a battery to maintain a functioning network during a given time.

II. A PRIMER ON POISSON POINT PROCESS

A Poisson process N on the real line admits a usual description based on a sequence (S n , n ≥ 1) of independent exponentially distributed random variables. Denote by λ the parameter of S 1 . The atoms of a Poisson process are the sequence (T n = S 1 + . . . + S n , n ≥ 1). Then, one can prove [START_REF] Decreusefond | Stochastic modeling and analysis of telecom networks[END_REF] that the number of points in a domain of Lebesgue measure l is a Poisson random variable of parameter λl. Moreover, given N (D) = n, i.e. given the number of points in D is equal to n, the atoms of N are independently and uniformly distributed over D. This explains the definition of a Poisson process in any dimension.

Definition 1. π µ a measure on the set configuration on E = R d , is a Poisson point process (PPP) of intensity ν if for all sets (C 1 , • • • , C n ) of mutually disjoint compact subsets of E: P(N (C 1 ) = k 1 , • • • , N (C n ) = k n ) = n i=1 e -ν(Ci) (ν(C i )) ki k i ! . (1)
If ν( d z) = λ d z, π ν is called the homogeneous Poisson point process with intensity parameter λ on R d .

Actually, words for words, this definition does not need that E is an R d -like space. For the mathematical details to work, it is sufficient to have E a metric space with some weak topological properties. It is a useful point of view since it is often interesting to add some information to the location of users when these are represented by the realization of Poisson point process. For instance, one may want to add to the position of a customer, the fading and/or the shadowing he is experiencing, his traffic rate, etc. In the simplest case, all these characteristics should be independent from one user to the other and identically distributed. We then say that they are marks of the Poisson process. Under the above mentioned hypothesis of independence and identity in distribution, the process whose particles are couples (x, v) is still a Poisson process on the product space R d × V where V is the space in which the marks "live". The intensity measure of this process is the product of λ d x times the probability distribution of the marks, denoted by dV(v). Many quantities we are longing to compute are expressible as a sum over the points of a realization of a deterministic function:

F = x∈η f (x, m)
where η is a realization of π µ . The calculations of the different moments of such a functional turn to be known and resort to the Bell polynomials. The complete Bell polynomials B n (a 1 , ..., a n ) are defined as follows:

exp ∞ n=1 a n n! θ n = ∞ n=1 B n (a 1 , a 2 , ..., a n ) n! θ n
for all a 1 , ..., a n and θ such that all above terms are correctly defined. The first four Bell complete polynomials are given as:

B 1 (a 1 ) = a 1 B 2 (a 1 , a 2 ) = a 2 1 + a 2 B 3 (a 1 , a 2 , a 3 ) = a 3 1 + 3a 1 a 2 + a 3 B 4 (a 1 , a 2 , a 3 , a 4 ) = a 4 1 + 4a 2 1 a 2 + 4a 1 a 3 + 3a 2 2 + a 4 Theorem 1 (Generalized Campbell's formula). Let ν λ ( d x, d v) = λ d x ⊗ d V(v)
and n be an integer and assume that f ∈ L p (E, ν λ ) for p ≥ n. The moments of F are given by:

E [F n ] = B n E f (z) d ν λ (z), • • • , E f n (z) d ν λ (z)
where z = (x, v).

Poisson point processes enjoy a lot of useful properties for thinning, superposition and displacement which roughly say that whatever one of these transformations we apply to a Poisson point process, the resulting process is still a Poisson point process with a tractable intensity measure (see [START_REF] Baccelli | Stochastic geometry and wireless networks, volume 1: Theory[END_REF] for complete references). In the forthcoming computations, we need a more recently established property. From [START_REF] Coutin | Stein's method for Brownian approximations[END_REF], [START_REF] Decreusefond | Robust methods for lte and wimax dimensioning[END_REF], we have the following theorem.

Theorem 2. Let ν λ ( d x, d v) = λ d x ⊗ d V(v) and π ν λ be a Poisson process of intensity measure ν λ . For some function f sufficiently integrable, let F (η) = x∈η f (x, v) and F (η) = F (η) -f (x, v)ν λ ( d x, d v) f (x, v) 2 ν λ ( d x, d v) 1/2 • For any p ≥ 1, let m(p, λ) = ( C f 2 (x, v)ν λ (dx, d v)) -p/2 × ( C |f (x, v)| p ν λ ( d x, d v).
Let µ be the standard Gaussian measure on R and µ 3 , the measure given by

dµ 3 (x) = (1 + 1 6 m(3, λ)H 3 (x)) d µ(x),
where H 3 (x) = 8x 3 -12x is the third Hermite polynomial. Then,

sup ψ C 3 b ≤1 E ψ( F ) - R ψ d µ 3 ≤ E λ where E λ = m(3, 1) 2 6 + m(4, 1) 9 2 π λ -1 .
This means that for E λ small, the distribution of F is in some sense very close to µ 3 .

III. SYSTEM MODEL

For the application we have in mind, i.e. deployment of a cellular network in low populated region, one can consider an isolated cell, neglecting the interference from adjacent base stations. To simplify the computations, we consider that the region covered by the base station, hereafter called the cell and denoted by C, is circular of radius R, centered at the base station o. The forthcoming analysis can be extended to any bounded domain of coverage but the integrals would have to be numerically evaluated. The terminals are identified to a cloud of points, which we denote by η, whose elements are the positions of each terminal in a domain larger than C.

The power consumed by the battery of the base station o can be divided into two parts:

• The power dedicated to transmit, receive, decode and encode the signal of any active user. The cumulative power over the whole configuration is then the sum over all terminals of the energy consumed for each one. • The power dedicated to broadcast messages. In order to guarantee that all active users receive these messages, the power must be such that the farthest user in the cell is within the reception range (if the system performs power control) or all the cell is within reception range (if the system does not perform power control). Thus, the power is a function of the maximum distance between the base station and the terminals or it is constant if power control is not performed. For a very simple propagation model (without fading and shadowing), the Shannon's formula states that for a receiver located at x, the transmission rate is given by

W log 2 (1 + P e l(x)),
where W is the bandwidth, P e is the transmitted power and l(x) is the path-loss function. This implies that in order to guarantee a minimum rate at position x, P e must be proportional to 1/l(x). Usual choices of path-loss functions are of the form l(x) = |x| -γ (singular path-loss model) or

l(x) = (r 0 ∨ |x|) -γ or l(x) = (1 + |x| -γ ) -1
. The forthcoming analysis does not depend on a particular choice, so we keep it generic. It follows that for a user configuration η, the total consumed power, in presence of power control, is given by:

P (η) = β A x∈η∩C,x active l -1 (x) + β B max x∈η∩C l -1 (x) := P A (η) + P B (η),
where β A and β B are multiplicative factors defined below. The subscript A stands for "additive" and B stands for "broadcast". The term x∈η l -1 (x) means that we add over all points x of the configuration η, the value of l -1 (x). Now, if the terminals are moving, we denote by η t the configuration at time t which represents the locations of all terminals at this instant. Since the energy is the integral of the power over time, the total consumed energy between time 0 and time T is given by

J T := J T (η, T ) = T 0 P A (η s ) d s + T 0 P B (η s ) d s = J A (η, T ) + J B (η, T ).
We should also add a constant part for the energy associated to operate the network but it doesn't alter the statistical aspects we aim to analyze. For years, models for the locations of users in cellular networks were left aside considering a sort of diffuse ether from which a density of calls per unit of surface and unit of time would emerge. After [START_REF] Baccelli | Stochastic geometry and wireless networks, volume 1: Theory[END_REF], [START_REF]Stochastic geometry and wireless networks, volume 2: Applications[END_REF], we know how to represent users locations by a Poisson point process. Note that according to the Mecke formula [START_REF] Fettweis | ICT energy consumption -trends and challenges[END_REF], the earlier fluid model can be viewed as a space average of this refined description. As is, one cannot expect to compute variances and higher order statistics from this model. We hereby consider that terminals are initially located according to a Poisson point process in the plane, of intensity λ: for two disjoint bounded subsets of the plane, the random variables counting the number of users in each subset are independent and Poisson distributed with parameter λ times the surface of the subset. We enrich the Poisson point process description by adding traffic and mobility characteristics. The traffic of the user initially located at x, is an ON/OFF process, denoted by A x , independent of the position of the user. We assume, as usual, that all the traffic processes of all users are independent and identically distributed. Moreover, at the beginning of the time observation window, they have all reached they stationary state (supposed to exist). We denote by π ON the probability for a given traffic process to be in its ON phase at any given time. One simple example of such a process is the exponential ON/OFF model, in which exponentially distributed ON periods alternate with exponentially distributed OFF periods. If we denote by µ ON and µ OFF the parameters of the exponential distributions, then π ON = µ OFF /(µ ON + µ OFF ). The choice of a traffic model boils down to choose a probability measure on the space T of piece-wise, two valued, functions. We denote by T this probability measure, hence T ( d a) is the probability to have a traffic process close to the process a and

H(a) T ( d a) = E [H(A)] ,
means that we compute the mean value of a function H with respect to all possible values of the generic traffic process A. We also envision the impact of mobility on energy consumption. We just assume that users move independently and are statistically indistinguishable: If M x denote the movement of user initially located at x, so that its position at time t is x + M x (t), then we assume that the collection of processes (M x , x ∈ η 0 ) are independent and identically distributed. Besides the motionless situation where M x (t) = o for any t and any x, the simplest model is that constant speed movement: M x (t) = v x t where the vectors (v x , x ∈ η 0 ) are independent and identically distributed over R 2 . Choosing a mobility model boils down to determine a probability measure M on the space C of continuous functions on R 2 , starting at o. Putting the pieces together means that we consider a Poisson process of the product space R 2 ×T×C with intensity λ d x ⊗ T ( d a) ⊗ M( d m). In plain words, this means that a user, say located at x, is equipped with a traffic process A x and a mobility process M x such that all these processes are independent and identically distributed among all users.

Moreover, the so-called Mecke formula stands that

E x∈η ζ(x, A x , M x ) = ζ(x, a, m)λ d x T ( d a) M( d m), (2)
for any bounded function ζ. The configuration of users at time t is

η t = x∈η0 δ x+Mx(t) ,
while the configuration of active users is

x∈η0 A x (t) δ x+Mx(t) .
In particular, the additive part of consumed energy can be rewritten as:

J A (η, T ) = x∈η0 T 0 A x (t)l -1 (x + M x (t))1 x+Mx(t)∈C d t
(3) and the broadcast part is:

J B (η, T ) = T 0 max x∈ηt∩C l -1 (x + M x (t)) d t. (4)
IV. MOTIONLESS USERS When users do not move, from (3), we get

J A (η, T ) = β A x∈η0 1 x∈C T 0 A x (t) d t l -1 (x),
so that J A (η, T ) appears as a shot noise process. In view of Theorem 1, we can compute easily the moments of any order of the additive part.

Theorem 3. For motionless users, the moments of J A (η, T ) are given by:

E [J A (η, T ) n ] = B n (α 1 , • • • , α n )
where

m k (A, T ) = E T   T 0 A(s) d s k   for k ≥ 2.
and

α k = λβ k A m k (A, T ) C |l -1 (x)| k d x.
In particular, for the singular path-loss model,

E [J A (η, T )] = 2β A γ + 2 ρ R γ T ( 5
)
where n = λπR 2 is the mean number of terminals into the cell and ρ = n π ON is the mean number of active customers in the cell C of radius R.

We now show how to determine β A . If P e denotes the power emitted by a mobile located at x from the base station, since we do not take into account interference, shadowing and fading, the received power at the base station is P r = P e K l(x) where K = (c/4πf d ref ) 2 d γ ref , f is the frequency of the radio transmission, c is the light celerity and d ref is the so-called reference distance. Since the base station can detect a signal of power greater than P r min , this requires that P e is greater than P r min /Kl -1 (x). One can consider that the same considerations hold for the downlink channel so that β A = 2P r min /K. In practical situation, P r min is of the order of 10 -9 mW and f around 2 GHz, hence β A varies between 2.10 -10 for γ = 5 to 2.10 -8 for γ = 3.

On the other hand, without power control, let P ′ r the power sufficient to ensure a reception at any point of the cell. If we denote by P b min the minimum power for the beacon to be detected by a mobile at distance R from the base station, we should have

KP ′ r R γ ≥ P b min .
This amounts to say that we can take β B = P b min /K. Usually, P b min is around 10 -8 mW. In the usual frequency bands, we obtain β B varying from 10 -9 for γ = 5 to 10 -7 for γ = 3. If there is no power control, the energy consumed for the beacon is thus equal to

J 0 B (η, T ) = β B R γ T. ( 6
)
If power control is used, the power should be adjusted for the farthest terminal to be able to receive the beacon signal:

P b min K max x∈η∩C l -1 (x) = β B max x∈η∩C l -1 (x).
Hence, in absence of movement,

J B (η, T ) = β B T max x∈η∩C l -1 (x).
Since all the usual path-loss functions depend only on the distance between x and o, let L be defined as L( x ) = l(x).

From the remark that

( max x∈η∩C x ≤ u) = (η(B(o, R)\B(o, u)) = 0),
it follows from (1) that the random variable δ = max x∈η∩C x has probability density function:

f δ (u) = 2λπe -λπR 2 u e λπu 2 .
The next result follows.

Theorem 4. For motionless users, with power control, the consumed energy to maintain the beacon signal, denoted by J p B has moments given by:

E (J p B (η, T )) k = (β B T ) k R 0 (L -1 (u)) k f δ (u) d u, for any k ≥ 1. (7)
For the singular path-loss function, we obtain

E [J B (η, T )] = e -n n -γ/2 n 0 v γ/2 e v d v J 0 B (η, T ).
Thus the gain of power control does depend only on the mean number of terminals whatever the radius, be it a few meters or some kilometers. Figure 2 shows that, as expected, the gain is higher for lower load. In view of ( 5) and ( 6), we have

κ := E [J A (η, T )] J 0 B (η, T ) = 2 γ + 2 ρβ A β B = 2 γ + 2 ρP r min P b min • (8)
Since ρ, the mean number of active users is of the order of 10 to 50, in view of the values of β A and β B , (8) says that κ depends essentially of the ratio between P r min and P b min . For the values we considered, this means that J A and J B are of the same order of magnitude.

V. IMPACT OF MOBILITY

For the sake of simplicity for the proofs, we now assume that J B is constant, equal to J 0 B given by [START_REF] Chinara | Mobility based clustering algorithm and the energy consumption model of dynamic nodes in mobile ad hoc network[END_REF]. We now evaluate the impact of mobility on energy consumption.

According to the displacement theorem for Poisson processes, we known that for each time t, the point process

η M t = (x + M x (t), x ∈ η 0
) is a still a Poisson point process with intensity λ d x. It follows that the expectation of consumed energy does not depend on the mobility model and is equal to the value for motionless users.

However, for higher order moments, the correlations between positions at different instants are to be taken into account so that the variance and other moments are different for truly mobile users. Let

̥ M n (f, T ) = R 2 E T , M T 0 f (x + M (t))A(t)1 x+M(t)∈C d t n d x.
The following theorem uses the same techniques of proof as before but in a more involved fashion.

Theorem 5. The moments of J A (η, T ) with mobility are given by

E T J A (η M , T ) n = B n (λ̥ M 1 (β A l -1 , T ), λ̥ M 2 (β A l -1 , T ), • • • , λ̥ M n (β A l -1 , T )).

It follows that mobility reduces moments of J

A ; i.e E T , M [J A (η, T ) n ] ≤ E T , 0 [J A (η, T ) n ] . (9)
Proof: The first part of the proof follows from Theorem 1.

Let f (x) = β A l(x) 1 x∈C . Since the Lebesgue measure is translation invariant, R 2 f (x + y) d x = R 2 f (x) d x.
If we combine that remark with Hölder inequality, we get that for y

1 , • • • , y n in R 2 , R 2 n j=1 f (x + y j ) d x ≤ n j=1 ( R 2 f (x + y j ) n d x) 1/n = R 2 f (x) n d x. (10)
For the sake of presentation, we denote by d t the product measure ⊗ n j=1 d t j According to [START_REF] Baccelli | Stochastic geometry and wireless networks, volume 1: Theory[END_REF], we get

̥ M n (f, T ) = R 2 [0, T ] n E T ,M n i=1 (f (x + M (t i ))A(t i )) d t d x = R 2 [0, T ] n E M n i=1 f (x + M (t i )) E T n i=1 A(t i ) d t d x = [0, T ] n E T n i=1 A(t i ) R 2 E M n i=1 f (x + M (t i )) d x d t ≤ [0, T ] n E n i=1 A(t i ) R d f (x) n d x d t = E T 0 A(t) d t n R 2 f (x) n d x = m n (A, T ) R 2 f (x) n d x.
Since the coefficients of Bell polynomials are non-negative, (9) follows.

We now study the limiting variance when the speed of particles is large (for instance, terminals in a high speed train). We say that a movement distribution M has property T whenever P(M (s) = M (t)) = 0 for all s = t. We denote by M ǫ the accelerated version of M (and M ǫ the corresponding probability distribution on C): M ǫ (t) = M (t)/ǫ. The full proof of the next result is given in [START_REF] Vu | Spatial models for cellular network planning[END_REF]. Theorem 6. If M has the property T then in high mobility regime, the variance of J A tends to 0, i.e V J A (η M ǫ , T ) → 0 as ǫ → 0.

The above results say that, when users move the total consumed energy by a base station does not change in average, and the moments of the additive part are reduced. Moreover, when users move very fast, the consumed energy during a time period is almost constant. We can see this as a consequence of weak central limit theorem. When users move faster, the configuration of users is more mixing during a same period of time, thus converge faster to the mean.

VI. APPLICATIONS

A. Dimensioning optimal cell size

Consider an operator aiming to design the optimal cell radius R to cover a region of total area S ⊂ R 2 . We assume that the cells are circular. The average total cost of the network is assumed to be the sum of the operating cost during the life time of the network (say T ) and the cost of facilities (base stations). We assume a fixed deployment cost for any base station regardless of its transmission range. The number of base stations is then roughly equal to S R -2 so the installation cost of base stations is c 1 SR -2 with c 1 > 0. The operating cost is assumed to be proportional to the consumed energy.

We assume that l is the singular path-loss function, i.e. l(x) = x -γ . From the previous results, the mean energy consumed by the network during its operating time is:

S R 2 (1 + κ)β B R γ T.
This is an increasing function of R, which means that small cell systems will consume less energy than larger cell systems. The average total cost for the network is then

Cost(R) = S(1 + κ)β B T R γ-2 + c 1 S R 2 . (11)
Note that κ implicitly depends on R, as the larger the cell, the higher the mean number of active customers. Equation [START_REF]Stochastic geometry and wireless networks, volume 2: Applications[END_REF] shows that there are two antagonist trends: large radius cells minimize the deployment cost whereas they increase the operating cost.

If we keep κ constant, i.e. we may have larger cell providing that the mean number of customers per unit of surface is decreasing in such a way λR 2 is constant; the optimization problem has a solution obtained by differentiation:

R opt = 2c 1 (γ + 2)(1 + κ)β B T 1/γ •
As expected, the optimal radius depends heavily on the value of γ which is linked to the density of obstacles in the path of radio waves.

B. Dimensioning cell battery

The proposed model can be used to dimension sites that do not have access to power supply facilities. In this situation, operators have to replace or reload base station's battery at each period T . We want to determine the energy level α of battery so that the probability of running out of energy before replacement (or reloading) be smaller than some given threshold ǫ. We use results derived in the previous sections to find α. The problem is to find α such that:

P J T (η M , T ) > α < ǫ.
In order to simplify the problem, we assume J B to be constant equal to K B = β B R γ T and that the users are motionless. In view of Theorem 4, it may be thought as a pessimistic point of view without the economy due to power control. Hence, J T (η M , T ) = J A (η M , T ) + K B . One could resort to the Bienaymé-Tcebycev inequality and use Theorem 3 to bound the variance of J T but this approach is known to give imprecise results. Otherwise, one can use the Gaussian approximation of Theorem 2. We can apply this theorem to the function f (x, a) = β A l -1 (x) Since the process A is supposed to be ergodic, it is well known that m k (A, T ) ∼ (π ON T ) k as T goes to infinity or at least if T is large compared to the cycle duration of A, i.e. the time between two successive communications plus the length of a communication. Since that is usually so, we get:

m(k, λ) = γ + 1 γk/2 + 1 n 1-k/2 .
Note that m(k, λ) depends weakly on the geometric properties of the domain which are summarized by γ and on the traffic pattern but mainly on the mean number of customers in the cell. The procedure is then the following: We first verify that E λ is negligible compared to ǫ. Then, we solve the equation in α, µ 3 ([α, +∞)) = ǫ and take

ζ = m 1 (A, T ) β A R γ γ/2 + 1 n + α m 2 (A, T ) β A R γ √ γ + 1 √ n ∼ 1 γ/2 + 1 + α √ γ + 1 √ n β A ρR γ T. (12)
With this procedure, we get a simple way to determine the threshold ζ which guarantees that the real value of P J T (η M , T ) > α is smaller than ǫ. In Figure 3, we analyze the variations of ζ with respect to γ. Once again, we see that the provisioning of resources, i.e. energy for the time being, is exponentially dependent of γ, the path-loss exponent.

VII. CONCLUSION

We have shown how to model energy consumption in a cellular network taking into account both communication and signaling traffic. The closed form formulas we obtained may be used for several purposes mainly in order to dimension cells or batteries under energy constraints. We pointed out the great importance of the domain geometry which is summarized by the path-loss parameter γ.

Figure 1 .

 1 Figure1. Illustration of the model, each user is associated with a ON-OFF process and a mobility process.

GainFigure 2 .

 2 Figure 2. The energy gain with power control.

T 0 a

 0 (t) d t,and hence,C T f (x, a) k ν λ ( d x, d a) = m k (A, T ) (β A R γ ) k γk/2 + 1 n.It follows that for any k ≥ 3,m(k, λ) = γ + 1 γk/2 + 1 m k (A, T ) m 2 (A, T ) k/2 n 1-k/2 .


Antoine Auberger
email: antoine.auberger@wanadoo.fr

Eric Dubois
email: eric.dubois@univ-paris1.fr

The influence of local and national economic conditions on French legislative elections

This article's aim consists in building and estimating a model which explains and forecasts the outcomes of the French legislative elections by department. This model, which constitutes the first attempt for such a geographical level, emphasises the role of the economic and political factors in the explanation of the legislative vote. The model seems to be very accurate in forecasting the elections of the past at the local and national level. Furthermore, its behaviour for the 2002 election was very satisfactory. This model is therefore a reliable alternative to the vote intention polls as an electoral forecasting instrument.

Introduction

Since the beginning of the 1970s and Kramer (1971)'s pioneer study for the United States, numerous studies put in evidence the influence of economic conditions on elections outcomes 1 .

The earlier econometric vote models explain the election results of the past by using time series analysis methods applied to data aggregated at the national level. This kind of models offers forecasts which are not always accurate due to the instability of the estimated vote-functions. This instability which can result from a badly adapted modelling and/or from statistical problems like the weak number of observations. Two ways have been explored to remedy this patent instability. Abrams (1980), Budge and Farlie (1983), Rosenstone (1983) and Abrams and Butkiewitz (1995) analysed the influence of the economic and political factors on the vote by state for some American presidential elections by using cross sectional data. This method is very useful to study the national and local context of a particular election but fails to catch the difference from an election to another one since the time dimension is missing.

The second way is the building of econometric vote models which use pooled data (see e.g. Holbrook, 1991, Campbell, 1992). These models allow to explain and to forecast the results at the local and national level for several American presidential elections. This method uses at the same time cross sectional analysis and time series analysis. Compared to national model, the number of degrees of freedom is larger and a greater number of independent variables can be introduced into the vote-function 2 .

In France, models using pooled data have been developed since the beginning of the 1990's and Jérôme et al. (1993)'s pioneer study 3 . This paper covers four legislative elections between wing parties at the first round is explained by the preceding elections' results, the change in the President's popularity, and the change in the regional unemployment rate. A seat-function allows to transform the vote share received in each region in seats won in each region. The sum of these local forecasts leads to a national forecast. This model was successful in announcing for the rightwing parties results in metropolitan France: 453 seats in 1993 (against 470 actual seats) 5 and 242 seats in 1997 (against 255 actual seats) 6 . Jérôme et al. (1999) built a vote-function which explains the vote for the ruling majority at the first round of the six legislative elections between 1978 and 1997 at the regional level. Four independent variables are used: the vote at the previous presidential elections, an "ideology" variable showing a persistence to the Right or to the Left in the orientation of the regional vote, an "instability" variable for the regions which changed several times in the orientation of their vote and the change of the regional unemployment rate during the year preceding the election year 7 . In their seat-function, the dependent variable is the percentage of seats obtained by the ruling majority and the dependent variables are the percentage of vote going to the ruling majority, the vote abstention propensity in the first round and a dummy variable which takes into account the "Front National" (the main Extreme Right party) strength 8 . This model allowed them to forecast a posteriori 263 seats for the right-wing parties in 1997 which obtained 238 seats.

A last model is developed by Fauvelle-Aymar et al. (2000). This model has the capacity to explain the results of several types of elections. The dependent variable is the first round vote going to the left-wing parties 9 . Among the independent variables, there are the difference between the left-wing parties' level of popularity and the right-wing parties' one, the votes going to the left parties for the previous elections of the same type, the rate of turnout for the previous elections of the same type (as well as the same rate when the Left is the ruling majority) and dummy variables to take into account the specificity of every type of elections. This model was not used to forecast the legislative elections 10 . All these three models share the same level of data, that is regional data. We think that it is better to use departmental data. Indeed, the department appears to be a more homogenous electoral entity than the region (see Bon and Cheylan, 1988: 7) 11 . This homogeneity can be explained by an historical point of view. Departments have been created in 1790 on sociological criteria (that is a structural criterion) whereas the division into regions dates back to 1960 and has been mainly motivated by national planning considerations (that is a short-term criterion). Furthermore, except the region "Alsace", there is no region totally rooted to the Right or to the Left. A good example is the region "Bretagne" which votes always in majority for the right-wing parties. On four departments which compose this region, three departments always vote in majority for the right-wing parties and one department always votes in majority for the left-wing parties. These behaviour differences may be partly explained by the fact that economic conditions vary from a department to another in the same region. For example, in December 2000, the regional unemployment rate was 13.0 % in "Languedoc-Roussillon" and within this region, the departmental rate ranged from 5.5 % in "Lozère" to 14.3 % in "Hérault" 12 .

Our paper is organised as follows: we begin by describing the French political system (section 2), we present the empirical model used (section 3) and we detail the estimates' results (section 4). Finally, we study the behaviour of the model for the 2002 French legislative elections (section 5).

The French political system

The fifth Republic began on October 4, 1958, after the adoption of a new constitution subjected to referendum on September 28, 1958. The French political system became then a presidential system keeping the fundamental principles of a parliamentary system (we can describe it as a semi-presidential regime). Traditionally, the President of the Republic and the Prime Minister belong to the same political side. When the opposite occurs, we speak about cohabitation 14 . Except in periods of cohabitation, the President of the Republic determines the nation's policy, while during the periods of cohabitation, the Prime Minister is fully the head of government.

The

Since the 1958 legislative election, the legislative ballot is an "unominal system" 15 in two rounds by electoral districts. If no candidate obtains the absolute majority of the expressed votes and a number of votes equal at least to the quarter of the registered voters in the first round, a second round is organised in which the candidates having obtained 12.5 % of the registered votes are in competition. For the 1986 legislative election, a one-round proportional scheme was adopted 16 before returning then to an election with a majority basis by electoral districts in 1988.

There is a second assembly, the Senate, consisted in 321 senators 17 , whom role is to discuss bills subjected by the government or the deputies. They can also make their own bills.

The National Assembly and the Senate together constitute the Parliament, in charge of certain constitutional changes.

The deputies belong generally to political parties distributed on a Left-Right political spectrum. The left-wing parties are traditionally progressive parties and the right-wing parties are conservative ones.

The right-wing parties dominated the French political life from 1958 to 1981 until the victory of the socialist candidate, François Mitterrand, at the 1981 presidential election. Over the period 1981-2002, the left-wing is formed by the PC and the PS which constitute the moderate left-wing, and the Extreme Left 18 . The ecological parties can be classified to the Left because the main ecological party, the Greens, got closer to the PS after the 1994 European elections and was a part of the Left majority from 1997 to 2002. Over this period, the Right is formed mainly by the RPR and the UDF 19 for the moderate Right, and the FN for the Extreme Right. We precise that, during the periods 1986-1988 and 1993-1997, when the Right was the majority in the National Assembly, the Extreme Right was not a part of this majority.

After the 1981 presidential election, François Mitterrand, the President of the Republic, dissolved the National Assembly and the Left won easily the 1981 legislative election, the PS obtaining even the absolute majority in seats. Several communists participated in the government (1981-1984). The 1986 legislative election was won by the moderate Right, which led, for the first time in France, to a period of cohabitation. In May 1988, François Mitterrand was reelected President of the Republic and dissolved then the National Assembly, which led to a legislative election which the Left won with difficulty. The majority in the National Assembly was only relative because the PC did not support the government officially. The Right won then the 1993 legislative election, which led to a second period of cohabitation. In 1995, it is a Right candidate, Jacques Chirac, who is elected President of the Republic. He decided to dissolve the National The table below summarises the formation and the strength of the different majorities following the six elections from 1981.

[Table 1 about here]

The model

Our model consists in a vote-function and in a seat-function 20 .

On one hand, we have the following vote-function:

Vote = f (economic conditions, political factors)
The vote is explained by the economic conditions and the political factors. In order to build the vote-function, we suppose that voters behave accordingly to the responsibility hypothesis, that is they reward (respectively, punish) the ruling majority for good (respectively, bad) economic performances (Key, 1966). More precisely, we suppose that voters take into account the recent and future results of the economic policy led by the government. Voters have then a retrospective and a prospective behaviour. The dependent variable is the share of vote received at the first or the second round 21 by the political parties of the ruling majority in the large sense 22 in the department i in the elections at the date t (VOTE it ).

We retained six independent variables. Two variables are used to take into account the impact of the economic conditions on the vote 23 : the rate of real growth of GDP at the national level (variable noted DGDP) and the growth rate of the number of job-seekers at the departmental level (variable noted UNEM). The first variable accounts for the national economic conditions and the second one accounts for the local economic conditions. When the economic growth increases and the number of job-seekers falls, the vote for the ruling majority is going to increase and it is going to fall otherwise. The expected signs for the coefficients are thus:

0 1   and 0 2   .
The third variable is the popularity of the Prime Minister (variable noted POP). It's a political variable which allows to take into account factors like personal image, wearing effect of being in power, scandals, etc. Higher the popularity is, higher the vote for the ruling majority will be.

Then, we are expecting a positive sign for this variable (

0 3   ).
Our fourth variable is a political variable constituted by the results of the previous elections (variable noted PREV) 24 . This variable thus translates the short-term electoral strength of the ruling majority (

0 4    25 .
The fifth variable is a partisan one which takes into account the persistence in the orientation of the vote (variable noted VPAR). Contrary to the previous variable, the VPAR variable translates the electoral medium-term strength of the ruling majority. When a department votes distinctly more for the current ruling majority in the previous legislative election, we may think that it will vote in favour of the ruling majority in the following legislative election. We are then expecting a positive coefficient for VPAR:

0 5   .
Finally, our sixth variable is supposed to take into account the influence on the vote of the electoral districts where the ruling majority is defeated after the first round (variable noted ELI).

The coefficient of the variable ELI indicates that when the ruling majority is defeated in more electoral districts than the opposition, there is, obviously, a loss in vote for the ruling majority

( 0 6   ).
On the other hand, we have the following seat-function:

Seats = f (vote, political factors)
The number of seats is explained by the vote and political factors. More precisely, we suppose that the percentage of seats obtained by the parties of the ruling majority in the strict sense 26 in the department i in the elections at the date t (SEATS it ) is explained by two independent variables. The first variable is VOTE, as defined previously. The expected sign is positive since generally more votes lead to more seats (

0 1  
). The second one, noted TRI, takes into account the extreme right-wing parties' power of nuisance on the moderate right-wing parties when there is a triangular contest. Indeed, we suppose that the extreme right-wing parties harm more in the moderate right-wing parties because a majority of extreme right-wing voters are closer to the moderate right-wing parties than to the left-wing ones. In a triangular contest between a left-wing candidate, a moderate right-wing candidate, and an extreme right-wing candidate, the vote for the whole Right is shared between the moderate right-wing candidate and the extreme right-wing candidate whereas when there is a classical opposition between a left-wing candidate and a moderate right-wing candidate, a majority of extreme right-wing voters at the first round report their vote in favour of the moderate right-wing candidate at the second round. We are then expecting a negative sign for the coefficient of TRI (

0 2   ) 27 .

Estimation method, estimates' results and model's performances

When one wants to estimate a system of equations, one has to choose the estimation method.

Since we estimate a system with an endogenous variable (VOTE in seat equation), we are inclined to move the OLS apart. Other methods are available like 2SLS, 3SLS or FIML. 3SLS provides more efficient estimates than 2SLS by taking into account the possibility of a correlation between the independent variables and the error terms. The choice between 3SLS and FIML is less clear. If FIML is efficient among all estimators, Greene (2000: 695) indicates "3SLS dominate FIML nonetheless". Due to these unclear prescriptions, we have chosen to present the estimates' results and to show the forecasting performances for the more efficient method, the FIML one.

As we estimate a pooled data model, we have to choose the correct specification between a model without effect, a model with fixed effects, or a model with random effects regarding the cross-sectional units. First of all, we have moved the random effects specification apart. As pointed out by Greene (2000: 567), random effects are appropriate in studies for which the sample is not composed of all the cross-sectional units. In our study, the set of cross-sectional units (the departments) is exhaustive. In over words, our study covers all the metropolitan French departments (96 units). So we can expect that a random effects specification is not necessary and that a fixed effects model or a model without effect is more appropriate. To choose between a fixed effects model or a model without effect, we have performed a Fisher's test for each of the candidate methods we have retained (see Greene, 2000, for details on this test). The null hypothesis is, broadly speaking, "the model without effect is better". In our case, we conclude that the fixed effects model is preferable to the model without effect for the vote equation as for the seat equation 28 .

The model to be estimated is then:

it it 6 it 5 it 4 t 3 it 2 t 1 i it ε ELI VPAR PREV POP UNEM DGNP c VOTE               it it 2 it 1 i it u TRI VOTE d SEATS      
Our study concerns the 96 departments of metropolitan France over the period 1981-1997 29 , that is five legislative elections : 1981, 1986, 1988, 1993, and 1997. The tables below show the estimated parameters of our two equations 30 :

[Table 2 about here]

[Table 3 about here] The results of our estimation provide strong support for our model. The adjusted R-squared indicates that it accounts 93 % of the variance in the departmental vote and 66 % of the variance in the percentage of seats by department. All the coefficients have the expected sign and are significant at the 99 % confidence level 31 . Furthermore, the values of the Durbin-Watson statistics (DW) suggest that the residuals are not correlated at least for the vote equation 32 .

The results of the vote estimate show that the vote in the legislative election for the ruling majority (in the large sense) depends partially on the national and local economic conditions. The coefficient of the variable DGDP suggests that if the economic growth during election year is superior by 1 point to what there was in the previous year, then the vote in the legislative election increases by 0.83 point. The coefficient of the variable UNEM indicates that if the growth rate of the job-seekers in a department decreases by 5 points then the departmental vote increases by 0.5 point. The vote in the legislative election also depends on political factors. The Prime Minister's popularity plays an important role. Indeed, an increase in the popularity ratio by 0.25 leads to an increase in the vote by 1.13 point. The outcomes of the preceding legislative elections have an influence on the current ones. The coefficient of the variable VPAR suggests that in a department with a vote 5 point higher than the national average at the previous legislative election, the gain in vote is equal to 1.55 point. The coefficient of the variable ELI indicates that when the ruling majority is defeated in 10 % of the electoral districts in a department, there is a loss in vote for the ruling majority by 2.9 points in this department.

The percentage of seats obtained by the ruling majority (in the strict sense) depends mainly on the percentage of vote obtained by the ruling majority (in the large sense). An increase by 1 point of the percentage of departmental vote leads to an increase by 1.89 point of the percentage of seats by department. The variable TRI plays also an important role: an increase by 50 % of the triangular contests with the Extreme Right in a department penalises the moderate Right by 6 points of vote in this department.

From the equations above, we can calculate, for the 96 departments of our sample, the ex post forecasts for the 1981, 1986, 1988, on 1993 and 1997 elections 33 . These forecasts allow to analyse the predictive capacity of the model by studying the forecasts in vote and in seats at the departmental and national levels.

The following table shows that for the departmental ex post forecasts in vote, the difference between the actual value and the predicted value is lower than 2 points in 60 % of departments and lower than 3 points in almost 80 % of departments 34 .

[Table 4 about here]

The table is read as follows: in 1981, the model made a forecast error included between 0 and 1 point in 33 departments. We notice that, on our whole sample, the forecast error in vote is weak:

1.97 points on average for 480 forecasts 35 .

For the ex post forecasts in seats at the local level, the mean absolute error is 0.48 seat by department 36 . The model makes a less-than-two-seats-error in near 95 % of departments.

[Table 5 about here]

In order to build forecasts at the national level, we compute the mean of the forecasts at the departmental level. Indeed, the table below shows that the figures are very close.

[Table 6 about here]

To check that the mean of the departmental vote and the national vote are not statistically different, we performed the Wijvekate's test resumed in Kanji (1993). Our conclusion is that the difference between the national vote and the mean departmental vote is not statistically significant at the 99 % confidence level. In other words, the mean of the departmental vote and the national vote are, statistically speaking, equal. , for 1981, 1986, 1988, 1993, and 1997, the national ex post forecasts in vote for the ruling majority:

The table below gives

[Table 7 about here] The errors are very low. The mean absolute error on five elections is less than 0.2 point. The national forecasts in seats are computed by summing the local forecasts in seats. The following table gives the national forecasts in seats for the five elections of our study.

[Table 8 about here]

We notice that by predicting the systematic defeat of the ruling majority, the model always gives the majority in the right way. The mean absolute error is 14.8 seats, essentially due to the strong error in 1988 37 .

In 1997, before the first round, the model predicted ex ante 50.36 % for the whole Right and 254 or 241 seats for the moderate Right 38 . After the first round, it predicted ex ante 51.38 % for the whole Right and 249 seats for the moderate Right [START_REF]As a comparison[END_REF] . The actual figures were 51.97 % and 247 seats. The model caught a unique situation in the history of the fifth Republic: the whole Right was majority in vote but the moderate Right was minority in seats. It allows to think that the seat equation takes well into account the disruptive role of the extreme right-wing parties. The following table displays the ex ante forecast errors after the first round by departments (vote and seats).

[Table 9 about here] As we can see, the mean absolute error in vote is lower than three points in more than half of departments and the mean absolute error in seats is lower than two seats in more than 80 % of departments. Even if the ex ante performances of the model are not as good as the ex post one, we can say that they are acceptable.

Behaviour of the model during the 2002 legislative election 40

Before presenting the forecasts obtained from the model for the 2002 election, it is necessary to bring a precision on the variable PREV defined as the percentage of vote obtained by the ruling majority in the second round at the previous elections. The 2002 legislative election was preceded by the presidential election, election for which the ruling majority, the Left, was missing in the second round. To overcome this problem, we chose to retain the results of the first round [START_REF] Besancenot | This classification can be discussed[END_REF] .

The table below gives the forecast vote and the actual vote for the whole France (metropolitan France).

[Table 10 about here] The table below gives the forecasts and the actual figures in seats for the whole France (metropolitan France). As for the 1997 election, to make forecasts before the first round, several scenarios were drawn concerning the presence of the Extreme Right in triangular contests in the second round. In the hypothesis 1, there are no triangular contests with the Extreme Right. In the hypothesis 2, the Extreme Right's presence is comparable to what it was in the 1997 legislative election 42 . Finally, in the hypothesis 3, the Extreme Right is present in the second round in 237 triangular contests. This scenario, built from the figures of the newspaper Le Monde [START_REF] Roger | Comment le FN peut faire chuter la droite aux législatives[END_REF] , tries to redraw the ascent of the Extreme Right in the first round of the 2002 presidential election.

[Table 11 about here] We notice that the majority in seats is clearly predicted (the moderate Right becomes sharply majority). After the first round, the model generates a perfect forecast in seats. In the evening of the first round, poll institutes gave ranges with the following average point: 154 for CSA, 155 for SOFRES, 167 for IPSOS 44 . So they quite underestimated the results of the Left whereas our model did not. Furthermore, the only other vote equation used to forecast the 2002 legislative election outcomes gave only 129 seats for the Left 45 .

The following table resumes the preceding tables for the forecasts by department after the first round.

[Table 12 about here]

We notice that the model behaves ex ante as during the 1997 election in vote but the errors in seats are more sensitive with an error larger than one seat in about a third part of departments and a mean error of 1.31 seats by department. We are clearly in a scenario where the forecast errors by department counterbalanced to give a national forecast close to the actual value.

The tables below show the estimated parameters of our two updated equations.

[Table 13 about here]

[Table 14 about here] The quality of the estimate is always good 46 . We can simply notice that the adjusted Rsquared of the seat-function has decreased a little bit. However, the coefficients of the independent variables are rather stable. This is particularly satisfactory regarding the seatfunction because of the big variability of the number of triangular contests with the Extreme Right from an election to another one 47 .

The following tables show the ex post performances of the model in vote and seats.

[Table 15 about here]

[Table 16 about here] As we can see, the ex post forecast error in vote for the 2002 legislative election is the largest of the sample (3.01 points). Similar finding applies to the ex post performances in seats.

Conclusion

The national and local economic situation plays an important role on the outcomes of the French legislative elections. In our model, the national economic conditions are taken into account by the real growth of the GDP while the departmental economic conditions are integrated in a relative way by the growth rate of the number of job-seekers (in every department). The national and local economic conditions are also included in political variables as the popularity of the Prime Minister and the outcomes of the previous elections.

This model seems to be very accurate in forecasting ex post as well as ex ante elections' outcomes. This is particularly true for the 2002 legislative election when the model has supplanted all other forecasting instruments (other vote-functions as well as vote intention polls). This good behaviour does not prevent us from finding some improvements. Future researches may focus on the seat-function which is clearly the weak part of our model (its adjusted Rsquared is not very high and the estimate leads large ex ante errors by department). The relationship between vote and seats is probably not linear: in an electoral district, if you obtain 51 % of the vote or 80 % of the vote, you still obtain one seat. Another track is to look more precisely at the 1986 election. For this election, we have to pay more attention to the modelling of the Extreme Right's power of nuisance. A last way is to take into account more precisely some local specificities as for example the electoral heterogeneousness within some departments.

Notes

1 See Lewis-Beck and Stegmaier (2000) for a survey on economic determinants of electoral outcomes.

2 And also in the seat-function for parliamentary elections.

3 For a survey on the early vote-functions in France, see Dubois (2003) (in French). 4 France is subdivided in regions, each of them subdivided in several departments. Each department is composed by several electoral districts. There are 22 regions, 96 departments and 555 electoral districts in metropolitan France. 5 See Lafay (1993). We note that the actual value given here and thereafter depends on author's personal computation and then may differ in a study from each other. 6 See Lafay (1998). 7 The building of the ideology and instability variables raises questions because all the elections' outcomes over the period 1973-1993 are used to construct these variables for each election. distinction between ruling left-wing majority and ruling right-wing one while the "Front National" harms more in the moderate Right than in the Left. 9 This model contains another equation explaining simultaneously to the vote the rate of turnout in the elections. 10 In this survey, we have only considered models for legislative elections (except Fauvelle-Aymar et al., 2002 which include legislative elections among different types of elections).

11 For a comparison between national, regional and departmental models, see Dubois and Fauvelle-Aymar (2003). 12 The figures are from Institut National de la Statistique et des Etudes Economiques.

13 555 in metropolitan France and 22 for the overseas departments and territories.

14 This political configuration, also called "dual executive", is close to the divided government one in the United States. 15 It means that voters have to vote for a candidate and not for several candidates or for a list. 16 It was a ballot in one round with several lists of candidates in every department. 17 The senators are indirectly elected in every department for nine years and renewed by third every three years. 18 See appendix 1 for the signification of the acronyms used in this section. 19 The UDF was formed by the centrists and the liberals. Before the 2002 election, a part of centrists, the liberals, and the RPR grouped together to form a new party, the UMP. 20 Precise definition and descriptive statistics of the variables and sources of the data are shown in appendixes 2 and 3. 21 In the French electoral system, legislative elections take place as follows: in each electoral district, the candidate who obtains more than 50 % of vote in the first round is elected. If no candidate obtains more than 50 % of vote in the first round, there is a second round with candidates who have obtained at least 12.5 % of the registered voters in the first round. The candidate who obtains the biggest share of vote is elected. To take into account the fact that you can be elected either in the first round or in the second one, we have chosen to retain, for each department, the sum of vote in the first round in electoral districts where the representative was elected in the first round and of vote in the second round in electoral districts where the representative was elected in the second round. respectively 0.89 and 0.93, which leads to a Fisher statistics of 1.73. For the seat equation, the R-squared of the without effect model and the R-squared of the fixed effects model are respectively 0.52 and 0.73, which leads to a Fisher statistics of 2.33. So in both cases, we can reject the null hypothesis since the critical value is 1.30 at 5 %. 29 We have excluded overseas departments and territories because of the lack of local economic data before 1986. 30 The 96 values of i c ˆand i d ˆare not reproduced here and are available on request. Notice that estimations with other methods (OLS, 2SLS, 3SLS) lead to very close outputs (available on request too). 31 We checked the normality of the residuals by performing the Jarque-Bera test. The null hypothesis is: "the residuals are normally distributed". Under the null, the Jarque-Bera statistic has a Chi-square distribution with two degrees of freedom. Performed on stacked data with no panel dimension, the value of the Jarque-Bera statistics is 3.52 for the vote equation and 4.50 for the seat equation. Since the critical value is 5.99 at 5 %, the Jarque-Bera test leads to the acceptance of the null in both cases. Computed for each of our cross-sectional units, the Jarque-Bera statistic never exceeds 1.56 for the vote equation and 1.65 for the seat equation (not reported here, available upon request). So normality is accepted in each case. 32 For samples as ours, the lower bond (dL) is 1.76 and the upper bound (dU) is 1.94 at 5 %. So the Durbin-Watson statistics lies in an inconclusive region (between dL and dU) for the seat equation. For the vote equation, we can accept the null hypothesis of no correlation in the residuals since the statistics lies between dU and 4-dU.

33 An ex post forecast is computed on the basis of the estimated coefficients for the whole sample whereas an ex ante forecast is computed from the estimated coefficients for a reduced sample including all the preceding elections and by using only the data available at the moment the forecast is made. 34 In this table as in the followings, we report only the MAE because this indicator of accuracy is more tractable since it can be directly interpreted in point of vote or in seats. Moreover, when the predicted values were below 0 or up to 100, we have retained 0 and 100. This explains the difference between the MAE reported in table 2 (computed with the residuals) and the MAE reported in table 4. 35 Estimates obtained with alternative methods (OLS, 2SLS, 3SLS) lead to similar forecasts. Same feature applies for the forecasts in seats. 36 For the seats, we have computed the forecasts directly in seats and not in percentage of seats. Moreover, the same remarks as stated in note 34 apply. 37 For this election, we notice that in the departments where the ex post forecast errors are the most important (two seats or more), the results in several electoral districts were closed in favour of the ruling majority: Loire-Atlantique, Moselle, Paris, and Hauts-de-Seine. In these departments, we can also notice that there were at least one electoral district very much in favour of the ruling majority and one electoral district very much in favour of the opposition. With a simple fictive example, it is easy to see how this configuration can lead to different outcomes. In a department where there are five electoral districts with respectively 32 %, 51.5 %, 51 %, 50.5 %, and 60 % in vote, the outcome is four seats for the incumbent majority. But, with a mean of 49 % at the departmental level, the forecast could be two seats according to the coefficients of the seat equation.

  President of the Republic is the head of State and, since the 1965 presidential election, he is elected by direct universal suffrage in a two-ballot majority poll. He appoints the Prime Minister, presides over the Council of Ministers and takes charge mainly of the Foreign Policy and the Defence. Until the 2002 presidential election, the president's term of office was seven years. Since the constitutional revision of October 2, 2000, and the 2002 presidential election, the president's term of office is five years. The President of the Republic can also dissolve the National Assembly once a year. Since the 1986 legislative election, the National Assembly consists in 577 deputies 13 elected for five years. It votes for the laws proposed by the government or the parliamentarians. The government, led by the Prime Minister, decides on the policy of France and is responsible to the National Assembly. The National Assembly may cause the resignation of the Prime Minister by voting a motion de censure.

  Assembly in May 1997 by thinking that the Right would win the legislative election but it is the Left which obtained the majority in seats in the National Assembly. After a period of cohabitation of five years, Jacques Chirac is reelected President of the Republic in May 2002 and the Right won then the 2002 legislative election easily, the UMP obtaining the absolute majority in seats.

Descriptive statistics of variables (1981-1997)

  There were also local agreements between the moderate Right and the National Front which could favour the moderate Right in two departments (Alpes-Maritimes, Share of vote received at the first or the second round by the political parties of the ruling majority in the large sense Ministère de l'Intérieur DGNP Difference between the rate of real growth of GDP in the election year (expected in December of the year preceding the election year) and the rate of real growth of GDP in the year preceding the election year OECD -Economic Outlook UNEM Difference between the growth rate of the number of job-seekers at the departmental level and at the national level over three months, two months before the elections DARES -Bulletin Mensuel des Statistiques du Travail (monthly data) Note : The miss of data published between June 1995 and November 1996 prevents the consideration of a growth over a period longer than three months POP Prime Minister's popularity: percentage of people relying on the Prime Minister / percentage of people not relying on the Prime Minister in the last survey published before the elections SOFRES -Le Figaro Magazine PREV Percentage of vote received by the ruling majority (in the large sense) at the elections preceding the legislative election Ministère de l'Intérieur VPAR Difference between the local vote and the national vote for the parties of the ruling majority in the large sense at the previous legislative election

	Variable	Definition and source
	Var). Appendix 3. Variable VOTE Ministère de l'Intérieur ELI Difference between the percentage of electoral districts where the ruling majority is defeated and the percentage of electoral districts where the opposition is defeated Ministère de l'Intérieur SEATS Percentage of seats obtained by the parties of the ruling majority in the strict sense Ministère de l'Intérieur TRI Percentage of electoral districts in every department where there is a "triangular contest" with a candidate of an extreme right-wing party multiplied by a dummy variable equal to 1 when the right-wing parties are ruling, and -1 otherwise. For the 1986 election, we chose to normalise the variable TRI with -100 % for all the departments because we can consider that the Extreme Right damaged more the moderate Right than the Left in all the departments because there was only one round. Ministère de l'Intérieur No. obs. Mean Std. Dev. Min Max VOTE 480 45.15 10.43 0.00 91.66 DGNP 480 0.60 0.94 -0.60 2.20 UNEM 480 -0.90 8.75 -27.04 19.21 POP 480 0.68 0.27 0.36 1.02 PREV 480 45.05 8.42 11.10 65.48 VPAR 480 0.05 8.20 -31.95 35.63 ELI 480 4.23 16.80 -88.89 100.00 SEATS 480 34.40 29.01 0.00 100.00 TRI 480 -17.75 42.61 -100.00 100.00 Table 1 . Legislative elections and parties in government Seat share in National Assembly (%) 14-21 June 1981 PS + PC 67.8 16 March 1986 UDF + RPR 50.4 5-12 June 1988 PS 48.5 21-28 March 1993 UDF + RPR 83.9 25 May-1 June 1997 PS + PC + Verts 55.5 9-16 June 2002 UMP + UDF 69.2 Date of elections Winning party or coalition

Source:

Lancelot (1998)

. The figures of the seats share are for the whole France.

Table 2 .

 2 Estimates Significant at 0.01 level. ** Significant at 0.05 level. * Significant at 0.10 level.

	of VOTE (1981-1997)
	DGNP UNEM POP PREV VPAR ELI N Adj. R² DW SER RMSE MAE Thiel's Inequality Coef. *** Variable	Coefficient (t-statistics) 0.83 (3.45)*** -0.10 (-3.52)*** 4.50 (5.18)*** 0.53 (16.27)*** 0.31 (12.31)*** -0.29 (-23.02)*** 480 0.93 1.95 2.83 2.51 1.97 0.03

Table 3 .

 3 Estimates of SEATS (1981-1997) Variable T a b le 4 . F o re c a s t e rro rs in vo te 1 9 8 1 -1 9 9 7

		Coefficient
		(t-statistics)
	VOTE	1.89
		(29.26)***
	TRI	-0.12
		(-4.81)***
	N	480
	Adj. R²	0.66
	DW	1.88
	SER	16.90
	RMSE	15.06
	MAE	11.85
	Thiel's Inequality Coef.	0.17
	*** Significant at 0.01 level.
	** Significant at 0.05 level.
	* Significant at 0.10 level.

Table 10 .

 10 Ex ante national forecast for 2002(vote)

		Forecast	Actual	Error
	before the first round
	scenario 1	183	171	12
	scenario 2	191	171	20
	scenario 3	211	171	40
	after the first round	171	171	0

Table 11 .

 11 Ex ante national forecast for 2002 (seats)T a b le 1 2 . E x a n te fo re c a s t e rro rs fo r 2 0 0 2

	V o te	0 -1 p o in t	1 -2 p o in ts	2 -3 p o in ts	> 3 p o in ts	M A E
		13	20	14	49	3 .7 4
	S e a ts	0 s e a t	1 s e a t	> 1 s e a t	M A E
		25	36	35		1 .3 1
		Table 13. Estimates of VOTE (1981-2002)
			Variable	Coefficient (t-statistics)
			DGNP	0.63
				(2.58)***
			UNEM	-0.11
				(-4.59)***
			POP	4.12
				(4.21)***
			PREV	0.44
				(14.85)***
			VPAR	0.35
				(15.45)***
			ELI	-0.32
				(-28.89)***
			N	576
			Adj. R²	0.90
			DW	1.87
			SER	3.23
			RMSE	2.93
			MAE	2.24
		Thiel's Inequality Coef.	0.03
		*** Significant at 0.01 level.
		** Significant at 0.05 level.
		* Significant at 0.10 level.

Table 14 .

 14 Estimates of SEATS (1981-2002) Significant at 0.01 level. ** Significant at 0.05 level. * Significant at 0.10 level.

	VOTE TRI N Adj. R² DW SER RMSE MAE Thiel's Inequality Coef. *** Variable	Coeffcient (t-statistics) 1.95 (28.32)*** -0.13 (-4.08)*** 576 0.58 1.93 19.06 17.34 13.69 0.20

Table 15 .

 15 Ex post performances of the model in vote(1981-2002)

		Predited	Actual	Error	Error	Error	Error
	Election							MAE
		value	value	0-1 point	1-2 points 2-3 points > 3 points
	1981	44.74	44.46	36	24	15	21	1.94
	1986	45.65	45.30	30	20	18	28	2.17
	1988	48.96	49.34	39	23	13	21	1.76
	1993	35.46	35.60	31	23	13	29	2.19
	1997	51.58	51.97	24	30	16	26	2.31
	2002	43.58	44.15	15	25	18	38	3.01

Table 16 .

 16 Ex post performances of the model in seats(1981-2002)

		Predited	Actual	Error	Error	Error
	Election						MAE
		value	value	0 seat	1 seat	> 1 seat
	1981	162	151	53	41	2	0.47
	1986	258	245	61	31	4	0.41
	1988	233	262	55	33	8	0.51
	1993	95	82	53	37	6	0.57
	1997	245	247	42	41	13	0.71
	2002	177	171	38	41	17	0.85

We can criticise the construction of the variable taking into account the force of the "Front National" because this variable is built with electoral outcomes over the period 1986-1993. Furthermore, the authors do not make the

"Large sense" means whole Right (that is moderate right-wing parties plus extreme right-wing ones) when Right is ruling. The converse is "strict sense" that means only moderate Right when Right is ruling.

Considering the definition of our dependent variable, the model partially attributes the economic assessment to the extreme right-wing parties when the whole Right is ruling while the extreme right-wing parties do not govern. A possible justification is that voters tend to vote more likely for the extreme right-wing parties when the ruling majority presents a bad assessment (and conversely) but that they remain generally closer to the moderate right-wing parties than to the left-wing ones. An estimate of the percentage of vote obtained by the ruling majority in the strict sense shows that voters grant less importance for the economic conditions.

Whatever is their type: cantonal, regional, etc. Here, it is only about presidential or cantonal elections. The municipal elections were put apart because of their high specificity (important local dimension, problems of aggregation of the data).

It may seem debatable in the case of the 1997 legislative election, preceded by the 1995 presidential election which holds two years previously. However we shall note that it is the only case in the five elections of our sample that the preceding elections didn't announce the future change of majority.

The extreme right-wing deputies are not a part of the ruling majority formed by the parties of the moderate Right.

According to the definition of this variable, the influence is negative if the moderate Right is the ruling majority and positive if the Left is the ruling majority.

For the vote equation, the R-squared of the without effect model and the R-squared of the fixed effects model are

Ranges were the following ones: 135-175 for SOFRES, 141-192 for IPSOS and 127-181 for CSA. In these forecasts, it was again necessary to remove several seats because these forecasts included the overseas departments and territories.

This forecast was available three weeks before the second round on the website www.lexpansion.com. See[START_REF] Jérôme | Reordering the French Election Calendar: Forecasting the Consequences for 2002[END_REF] for a presentation of this model.

Regarding the normality of the residuals, even though the Jarque-Bera statistics is still not significative when the test is performed in each of cross-sectional unit, we have to precise that it is no longer the case when the stacked data sample is used (for the vote as for the seats). We have then to be cautious by interpreting the results. The explanation lies in the presence of one large error up to ten points which increases the skewness and the kurtosis and consequently raises the Jarque-Bera statistics. This large error occurs in Haute-Corse and it may be explained by the political context in Corsica and particularly the so-called "processus de Matignon".

* The authors would like to thank the participants of the seminar of public economy (LAEP, University of Paris 1), the participants of the