François Jeanjean
email: francois.jeanjean@orange.com

† Georges

Vivien Houngbonon
email: gvivienh@gmail.com

Market Structure and Investment in the Mobile Industry *

Keywords: Market structure, Investment, Mobile Telecommunications. JEL Classification: D21, D22, L13, L40

The impact of market structure, that is the number of firms and asymmetry, on investment is an important topic in the mobile industry. However, previous literature remains ambiguous about the direction of the relationship.

This paper provides an empirical evidence of the impact of market structure on investment in the European mobile industry. The empirical assessment is based on a Salop model with vertical differentiation. Consistently with the prediction of this model, we find that both the number of operators and market share asymmetry have significant effects on investment. In symmetric markets, investment per operator falls with the number of operators, with larger effects for operators that lose market share more than the average. The industry investment rises with the number of operators in the short run, but eventually falls in the long run due to significant adjustment costs of investment in the mobile industry. These findings suggest that investment should be taken into account when analysing the welfare effects of market structure in the mobile industry.

Introduction

The impact of market structure, that is the number of firms and asymmetry, on investment has become an important topic in the mobile industry in the context of market concentration as underscored by the 4-to-3 mergers cleared in Austria. The ex-post effects of these changes in market structure have fuelled an ongoing policy debate, particularly in the European Union resulting in tougher scrutiny on future entry and merger in the industry. 1 As a matter of fact, a 4-to-3 merger was cleared in Germany, whereas others have been blocked in Denmark and the United Kingdom. A key issue in the policy debate is whether or not investment in mobile networks is endogenous, that is whether it is affected by market structure.

Existing theoretical literature predicts an ambiguous effect of market structure on investment. 2 In symmetric markets, [START_REF] Vives | Innovation and Competitive Pressure[END_REF] finds that investment in costreducing innovations falls with the number of firms, provided that demand elasticity is sufficiently small. In asymmetric markets, [START_REF] Schmutzler | Competition and investment -A unified approach[END_REF] shows that the effect of market structure on investment tends to be negative for less efficient firms.

In this paper, we investigate the impact of market structure on investment in the European mobile industry, taking into account adjustment costs of investment. Market structure is defined by the number of operators and their market share asymmetry. This latter is measured as the difference between an operator's market share and the average market share. Investment is measured as the logarithm of capital expenditures, excluding licence fees. We account for the adjustment costs of investment by estimating a dynamic econometric model that links investment to its lagged values. The econometric model is derived from a Salop model with vertical differentiation. Vertical differentiation stems from investment that either lowers marginal costs or increases quality, two key features of investment in mobile networks. The theoretical model predicts that investment in mobile networks depends on the number of operators and their asymmetry in terms of quality. More specifically, in symmetric markets, investment per operator falls with the number of operators, and this effect tends to be larger for lower-quality operators.

In accordance with the predictions of the theoretical model, we formulate a dy-1 See the report by the OECD which highlights a positive relationship between the number of mobile operators and investment in quality (OECD, 2014), and the report by the Centre on Regulation in Europe which suggests a negative relationship between the number of mobile operators and investment [START_REF] Genakos | Evaluating Market Consolidation in Mobile Communications[END_REF].

2 See [START_REF] Schumpeter | Capitalism, Socialism and Democracy[END_REF] and [START_REF] Arrow | Economic Welfare and the Allocation of Ressources for Invention[END_REF] for earlier analyses of the link between competition and innovation.

namic panel econometric model that links investment to the number of operators, market share asymmetry and the lagged values of investment, controlling for operator fixed effects, market characteristics and year fixed effects. The identification of the parameters relies on the Arellano-Bond estimator, a two-step system generalised method of moments estimator. We use 3-to-5 years lagged values of investment as instruments, ensuring that there is no serial correlation. We also use political ideology, mobile termination rates and population size as instruments for the number of mobile operators, market share asymmetry and market size, respectively.

Consistently with the predictions of the theoretical model, we find that investment per operator falls with the number of operators. This negative effect is larger for operators that lose market share more than the average. The industry investment tends to rise in the short run, but eventually falls in the long run due to significant adjustment costs of investment in the mobile industry. The magnitude of the long run effect amplifies the short run effect by a factor of up to 5. These findings are robust to market characteristics such as market size, consumers' income, competition from fixed lines, cash flow and retail prices.

This paper is amongst the first to show that market structure has significant effects on investment in the mobile industry. Theoretical papers such as [START_REF] Vives | Innovation and Competitive Pressure[END_REF] and [START_REF] Schmutzler | Competition and investment -A unified approach[END_REF] find that investment in cost-reducing technologies tends to decrease with the number of firms. [START_REF] Boone | Competitive pressure: the effects on investments in product and process innovation[END_REF] emphasizes the role of cost efficiency gap between firms in determining how competition affects their investment incentives. Few empirical papers provide evidence for these theoretical propositions. The findings of this paper lend support to these theoretical predictions. [START_REF] Sacco | Is there a U-shaped relation between competition and investment[END_REF] show that the relationship between competition and investment can be U-shaped. Our findings suggest a monotone effect of market structure on investment in symmetric markets. However, this relationship strongly depends on asymmetry. [START_REF] Beneito | Competitive Pressure and Innovation at the Firm Level[END_REF] test the effect of competitive pressure, measured by the degree of product differentiation and market size, on investment in a free entry setting, thus excluding the analysis of the effect of market structure. Consistent with their results, we also find a positive effect of market size on investment per operator.

The remaining of the paper is organised as follows. Section 2 presents the theoretical model. Section 3 outlines the empirical framework, and in particular the data, econometric model and estimation strategy. Section 4 reports and discusses the results while section 5 concludes.

Theoretical background

In this section, we start by summarising the relevant findings from [START_REF] Vives | Innovation and Competitive Pressure[END_REF] in symmetric markets and [START_REF] Schmutzler | Competition and investment -A unified approach[END_REF] in asymmetric markets. Then we focus on how market structure affects investment in the mobile industry using a Salop model with vertical differentiation as this provides a good representation of the mobile market. Salop's model is an extension of the Hotelling model often used to describe telecommunication markets characterised by price competition with horizontal and vertical differentiation [START_REF] Laffont | Network Competition: I. Overview and Nondiscriminatory Pricing[END_REF]. The Hotelling model represents a duopoly and the Salop model extends to a more general oligopolistic market structure. In particular, it can be used to compare market outcomes when the number of firms changes. Moreover, the predictions of this theoretical model will be useful in specifying our econometric model. [START_REF] Vives | Innovation and Competitive Pressure[END_REF] shows that, in a symmetric market, the number of firms impacts investment and output in the same way. He shows that, generally, an increasing number of firms decreases both production and investment in each firm. This may occur when the growth of the total market size does not compensate for the decline in per-firm market share. According to Vives, the decrease in market share, called demand effect, is a direct consequence of the rise in the number of firms, while the growth of market size is an indirect outcome (price-pressure effect). The rise in the number of firms tends to reduce prices and thus increase consumer participation and market size. In the case where the market is close to full coverage, the demand of the industry is weakly sensitive to price or market structure changes. In this case, the price-pressure effect is weak and the demand effect would likely dominate. Therefore, an increase in the number of firms has a negative impact on investment.

In our model, we assume that the market is fully covered in order to limit the pricepressure effect. This is consistent with the mobile industry where, in most countries, the penetration rate is sufficiently high such that we can expect the demand effect to dominate. 3 In the empirical model, we control for the growth of market size. [START_REF] Schmutzler | Competition and investment -A unified approach[END_REF] shows that in asymmetric markets, smaller firms are more sensitive to changes in market structure. We also verify this in both the theoretical and empirical models.

Settings of the model

All operators are located equidistantly around a circle where consumers are uniformly distributed. It is assumed that product space is totally homogeneous, thus the location of operators does not matter. The perimeter of the circle and the density of consumers are equal to unity and consumers move around the circle with a transportation cost equal to t to purchase one unit of the good from one of the operators.

We consider a restricted entry regime where the number of operators, N > 1, is exogenously determined by regulation. The distance between two operators is 1/N . We also assume that the gross consumer surplus s of each operator is high enough such that the market is fully covered. The demand for operator i's variety is q i and the demand of the industry is Q = N i=1 q i , where q i = Qσ i . Q is a constant since the market is fully covered and σ i is the market share of operator i. We normalise Q = 1 and consider the following two-stage game:

In the first stage, operators choose their investment z, which determines the level of quality of their variety. For operator i, we define the level of quality as d i = s i -c i , where c i represents the constant marginal cost of production and s i is the gross surplus of purchasing from that operator (c i ≤ s i ). This definition reflects the fact that higher quality may result from higher consumer surplus or lower marginal cost of production. In order to choose the quality d i , operator i invests an amount z i (d i ) increasing and convex with d i and incurs a sunk fixed cost F to enter the market.

In the second stage, operators compete in price and operator i sets price p i . The game is solved by backward induction.

The utility of a customer located at a distance x from operator i to purchase from that operator is: U i = s i -tx -p i , and the utility from purchasing from operator

i + 1 is: U i+1 = s i+1 -t( 1 N -x) -p i+1
. An indifferent consumer between operators i and i + 1 is located at:

x = 1 2N + (s i -p i ) -(s i+1 -p i+1 ) 2t
and between operator i and operator i -1 is located at:

-y = 1 2N + (s i -p i ) -(s i-1 -p i-1 ) 2t
The market share of operator i is σ i = x -y (see FIGURE 1), and the corresponding profit is expressed as:

π i = (p i -c i ) q i -z i -F (1)

Incentive to invest

The first order conditions, given the level of quality, leads to (p i -c i ) ∂σ i ∂p i = -σ i , and

∂σ i ∂p i = ∂(x-y) ∂p i
= -1 t , therefore, the Nash equilibrium price is expressed as:

p * i = c i + σ i t (2)
Profit at equilibrium is expressed as:

π * i = σ 2 i t -z i -F (3)
In the first stage, operator i chooses investment z i and the corresponding quality d i .

The first order condition leads to:

∂z i ∂d i = 2σ i ∂σ i ∂d i t (4)
At equilibrium, replacing p * i using equation (2) in σ i = x -y and d i = s i -c i yields the following expression:

σ i = 1 2N + 1 2t d i - d i+1 + d i-1 2 + σ i+1 + σ i-1 4 (5)
Operator i's market share, σ i , can be expressed as a linear combination of the quality of each operator d i , d i+1 , ...d N . The impact of operator j on operator i, (j = i) depends on its distance from operator i. Operator i + j and operator i -j are located at the same distance from operator i. Let β j denotes the coefficient associated with d i+j and d i-j . It represents the impact of operators i + j and i -j on operator i's market share. β 0 is then the coefficient associated with d i . It is expected that coefficients β j are decreasing with j, as the impact of the operators decreases with their distance.

Notice that, for a number of operators N , ∀j ∈ 0, N 2 , operator i + j is also operator i -(N -j). In addition, when N is even, operator i + N/2 is also operator i -N/2 and when N is odd, operator i+(N -1)/2 and operator i-(N -1)/2 are neighbours. Expressions of market shares may be different depending on whether N is even or odd (see FIGURE 2 below). For a single expression whether N is even or odd, we use the floor function . where

N/2 =    N 2 if N is even N -1 2 if N is odd
With this notation, if j ∈ [0, N/2 ] , i + j and i -j can represent all the operators. It is also necessary to add the coefficient γ j defined by:

γ j =    1 2 if j = 0 or j = N 2 1 otherwise
Coefficient γ j is useful when j = 0 or j = N/2, in both cases, operator i + j is also operator i -j. When j = 0, this is operator i, when j = N/2, this is the farthest operator from i. Thus, d i+j = d i-j and d i+j + d i-j = 2d i+j . The coefficient for operator i + j is counted twice. It is then necessary to correct it, this is the reason why, for j = 0 or j = N/2, γ j = 1/2. (see FIGURE 2)

With these notations, j ∈ [0, N/2 ] , the linear combination can be expressed as:

σ i = 1 N + 1 t   N/2 j=0 γ j β j (d i+j + d i-j )   (6)
In order to calculate the values associated with the coefficient denoted by β j , we replace σ i+1 and σ i-1 in equation ( 5) by their expressions calculated using (6). This yields:

σ i = 1 N + 1 4t   2d i -(d i+1 + d i-1 ) + N/2 j=0 γ j β j (d i+j+1 + d i-j+1 + d i+j-1 + d i-j-1 )   (7)
The comparison between equations ( 6) and ( 7) for all operators, i ∈ [1, N ] provides a system of N equations involving N/2 + 1 coefficients β j . Although this system is overidentified, it nevertheless admits a unique solution (see proof in the appendix):

β 0 = 1 - 2 + √ 3 N + 1 √ 3 2 + √ 3 N -1 (8)
and

β j = - 2 + √ 3 N -j + 2 + √ 3 j √ 3 2 + √ 3 N -1 if j = 0 (9)
For example, we can check:

For N = 2, σ i = 1 2 + d i -d i+1 3t
For N = 3,

σ i = 1 3 + 2d i -d i+1 -d i-1 5t For N = 4, σ i = 1 4 + 5d i -2d i+1 -2d i-1 -d i+2 12t
For N = 5,

σ i = 1 5 + 8d i -3d i+1 -3d i-1 -d i+2 -d i-2 19t
As expected, coefficient β 0 is positive and coefficients β j are negative. Operator i's market share increases with its own quality, but decreases with the quality of rivals. This can be illustrated by the positive value of ∂σ i ∂d i = β 0 (N ) t and the negative value

of ∂σ i ∂d i+j = β j (N ) t . β 0 is increasing in N, however it converges quickly towards 1 - √ 3 3
as N tends to infinity.

The market share of operator i depends on the difference between the quality of operator i, d i and the quality of the other operators weighted by the coefficients -β j /β 0 .

In the following, we denote this weighted quality:

d -i = -1 β 0 N/2 j=1 γ j β j (d i+j + d i-j ).
With this notation, the market share is expressed as:

σ i = 1 N + β 0 t (d i -d -i ) (10)
Market share depends only on the number of operators N and on the quality difference d i -d -i between operators. Equation (4) can thus be reexpressed as:

∂z i ∂d i = 2β 0 N + 2β 2 0 t (d i -d -i ) (11)
This equation shows that the incentive to invest depends on the number of operators, and the quality difference between operators (d i -d -i ). This equation will be useful for the specification of the econometric model.

In symmetric markets, equation ( 11) becomes:

∂z i ∂d i = 2β 0 N (12)
The marginal profit from quality investment 2β 0 N is decreasing in N , therefore, investment in quality falls with the number of operators. In asymmetric markets, this negative effect is larger for operators with relatively lower quality, where d i < d -i .

Consumer surplus

This fall in quality negatively impacts consumer and social surplus. Meanwhile, an increase in the number of operators reduces market power, through the transportation cost. This highlights a static-dynamic efficiency trade-off.

Formally, the following equations correspond to the changes in consumer surplus CS and social surplus W following a change in market structure in symmetric markets. The details of the calculation are provided in the appendix.

∂CS ∂N = ∂d ∂N + 5t 4N 2 (13) ∂W ∂N = ∂d ∂N (1 -2β 0 ) -z -F + t 4N 2 (14)
We know that

β 0 < 1 - √ 3 3 , thus 1 -2β 0 > 0.
In addition, in symmetric markets, quality falls with the number of operators, therefore, the effect of the number of operators on the dynamic terms is negative:

∂d ∂N (1 -2β 0 ) -z -F < 0
Equations ( 13) and ( 14) suggest that the optimal market structure in the mobile industry involves a trade-off between static and dynamic efficiencies if investment in quality falls with the number of operators. In the next section we test whether this trade-off actually exists by investigating the empirical relationship between market structure and investment in the mobile industry.

3 Empirical Evidence

Data and summary statistics

The empirical estimation relies on an unbalanced panel of 50 mobile operators from 17 Western European markets, observed over 10 years, from 2006 to 2015. The dataset comprises operator-level information regarding their capital expenditures, subscribers market share, average revenue per subscriber, earnings before interest, taxes, depreciation and amortisation (Ebitda), and their mobile termination rates.

At the market level, we also have information about the number of mobile operators, the total number of subscribers, the penetration rate of fixed lines, socio-demographic characteristics such as gross domestic product (GDP) per capita and population size, and a political variable that aims to capture the position of the government towards the welfare-state [START_REF] Volkens | The Manifesto Data Collection[END_REF]. The definitions and sources of these variables are presented in TABLE 1 in the appendix.

We use this information to construct the main variables of the econometric model, namely investment, the number of mobile operators and their market share asymmetry. Investment is measured as the natural logarithm of capital expenditures in order to make it easier to interpret the estimates, and reduce variance in the residuals of the model. Capital expenditures are limited to the mobile network, but include license fees. We include year fixed effects into the econometric model in order to capture, among other things, the component of capital expenditures that stems from license fees. The number of mobile operators only includes network's owners. Mobile virtual network operators (MVNO) are not taken into account because they typically do not invest in network elements that allow traffic transportation. In particular, they do not own passive network elements such as towers and antennas, or invest in active network elements that involve frequency spectrum.

Market share asymmetry is measured as the difference between an operator's market share and the average market share, that is the inverse of the number of operators.

From the theoretical model, asymmetry refers to difference in quality. However, given that quality is not observed, we rely on market shares that provide a more practical way of assessing asymmetry. This measure of asymmetry is consistent with the quality difference obtained from equation ( 10). As we discuss in section 3.2, defining asymmetry in this way provides a simple method for interpreting the coefficient of the number of operators. This is because, in a symmetric market the difference between an operator's market share and the average market share is zero. Market share is measured in terms of operators' number of subscribers. It includes the subscribers of MVNOs hosted by mobile network operators.

The total number of subscribers, which encompasses both prepaid and postpaid subscribers, is used as a proxy for market size. Subscribers' income is proxied by GDP per capita while competition from fixed lines is proxied by the household penetration rate of fixed lines. Operators' cash flow is proxied by their accounting profit (Ebitda) and retail prices are proxied by the average revenue per user.

TABLE 2 provides summary statistics of these variables. The operators' market shares are very heterogeneous, ranging from 3 to 62 percent. The number of operators ranges from 3 to 5. TABLE 3 in the appendix presents the evolution of the number of operators per market.

Econometric model

Consistent with equation ( 11), operators' incentive to invest depends on the number of operators and their asymmetry in terms of quality. The econometric model will, therefore, link operators' investment to the number of operators and the measure of asymmetry.

In addition, there are several potential sources of adjustment costs of investment in the mobile industry. For instance, given technology, increasing marginal cost of network deployment implies that operators first invest in least costly areas before moving into more costly ones. As a result, the remaining gap to the long run desired stock of capital that is closed each period gets smaller and smaller. Moreover, decreasing marginal cost of production, due to strong technological progress, makes any deviation from the steady-state level of investment costly to catch up. 4Typically, when a change in market structure lowers investment, marginal cost of production is higher and next period investment, which depends on this marginal cost, tends to be lower. We model these sources of adjustment costs with a dynamic model that links investment to its one-year lag.

Formally, the econometric model is expressed as:

z ijt = α + ρz ijt-1 + θN jt + δ∆ ijt (N jt ) + λX ijt + µ i + ν t + ijt (15)
Where z ijt denotes operator i's investment in market j in year t. N jt is the number of operators, which may change over time within some markets due to entry or mergers. ∆ ijt represents operator i's level of asymmetry with respect to its rivals. Market structure is defined as the combination of the number of operators N jt and their market share asymmetry ∆ ijt . This latter is expressed as:

∆ ijt (N jt ) = σ ijt - 1 N jt
Where σ ijt is operator i's market share. This expression of asymmetry is consistent with the quality difference in equation ( 10) of the theoretical model. By construc-tion, ∆ ijt = 0 in symmetric markets, a feature that will be useful for the interpretation of coefficients θ and δ. The marginal short run effect of the number of operators on investment is expressed as:

∂z ijt ∂N jt = θ + δ∆ ijt (N jt )
Coefficient θ corresponds to the effect of the number of operators on investment in symmetric markets, and more generally to the marginal effect of market structure for an operator that lose market share as the average. Coefficient δ corresponds to the additional effect due to the change in market share asymmetry. Negative ∆ ijt (N jt ) means that the operator's market share falls more than the average following an entry. This feature characterises smaller or lower-quality operators. When δ is positive, then the effect of the number of operators on investment tends to be more negative for lower-quality operators.

The magnitude of the adjustment costs is captured by the coefficient ρ. It provides a way to obtain the long run effect by multiplying the short run effect by the ratio 1 1-ρ . 5 ρ should be between 0 and 1 to ensure long run convergence.

X jt is a control variable representing market size. We introduce additional controls such income, competition from fixed lines, cash flow and retail prices, in the robustness analysis. µ i are time-invariant operators fixed effects of investment and ν t are year-specific shocks to investment, such as spectrum purchase. ijt represents unobserved idiosyncratic shocks affecting operators' investment that might be correlated with some explanatory variables. In the following section, we present the estimation strategy that helps overcome the bias that could stem from this correlation.

Estimation strategy

OLS estimation of equation ( 15) is likely to yield biased estimates due to several sources of endogeneity, including unobserved cost or demand parameters that jointly determine market structure and investment. For instance, regulators may allow more operators in markets where consumers have high valuation for mobile services. In addition more operators would be willing to enter a market if they are more efficient.

5 The long run effect is determined by summing all the contemporaneous effects over an infinite period. Typically, the marginal effect of market structure on investment is θ + δ∆ ijt (N jt ) at date t, ρ[θ + δ∆ ijt (N jt )] at date t + 1, ρ 2 [θ + δ∆ ijt (N jt )] at date t + 2, and so on. Thus, the long run effect is:

∞ k=0 ρ k [θ + δ∆ ijt (N jt )] = 1 1-ρ [θ + δ∆ ijt (N jt )].
Both mechanisms will tend to raise investment per operator and, as a result, OLS estimates of equation ( 15) will underestimate the magnitude of the effect of market structure.

The number of mobile operators can be deemed exogenous because entry and exit in mobile markets are not free. On top of the large fixed cost of network deployment, entry into mobile markets requires spectrum licences from government. This requirement constitutes a legal barrier to entry which restricts the number of mobile operators in a given market. Unlike mobile network operators, MVNOs can enter the market without purchasing a spectrum license, but they are out of the scope of this paper as they do not own networks. Exit of mobile operators is also not free due to merger control. Exit from the mobile market is typically carried out through a merger. However, most mergers require approval by competition authorities.

Nonetheless, as shown by [START_REF] Grajek | Regulation and Investment in Networks Industries: Evidence from European Telecoms[END_REF], regulatory decisions may be endogenous, in which case the number of operators may fail to be exogenous. Unfortunately, the time dimension of our panel data (10 years with gaps) is not long enough to apply the exact Granger non-causality test in heterogeneous panels, as proposed by [START_REF] Dumitrescu | Testing for Granger noncausality in heterogeneous panels[END_REF]. 6 To overcome this, we test any reverse causality from investment to the number of operators on the basis of three-stage least squares estimator. The system of simultaneous equation involves two equations, one for the number of operators and another for investment. In the tradition of the Granger non-causality test, each dependent variable is regressed over its lags, the other dependent variable and controls. The system of equations is expressed as:

   N jt = α 1 + ρ 11 N jt-1 + ρ 12 N jt-2 + β 1 z ijt + γ 1 X 1 + µ 1i + ν 1t + 1ijt z ijt = α 2 + ρ 21 z ijt-1 + ρ 22 z ijt-2 + β 2 N jt + δ 2 ∆ ijt (N jt ) + γ 2 X 2 + µ 2i + ν 2t + 2ijt
To further deal with a potential endogeneity of the number of operators, we use government political ideology towards the welfare-state as an instrument. This instrument, already used by [START_REF] Grajek | Regulation and Investment in Networks Industries: Evidence from European Telecoms[END_REF], is measured by an index provided by the Manifesto Project [START_REF] Volkens | The Manifesto Data Collection[END_REF] that characterises political parties' position towards the welfare state. For instance, the index is smaller for right-wing parties. We compute the weighted average of this index at the market-year level, using the percentage vote obtained by each political party during the most recent legislative election.

Additional endogenous explanatory variables include market share asymmetry, mar-ket size and the lagged investment. The endogeneity of market share asymmetry and market size stems from the fact that they are determined by unobserved demand, cost and quality parameters. We use population size as an instrument for market size. We expect larger number of subscribers in more populated markets, while population size is predetermined. Population size affects investment only through market size because it is orthogonal to other parameters of demand such as price elasticity or quality valuation. In addition, it does not directly affect investment because it is independent from production costs.

We use mobile termination rates (MTR) as instrument for market share asymmetry.7 MTRs are regulated wholesale prices that operators pay each other to terminate calls on their rivals' networks. In some markets, termination rates are not regulated, but freely negotiated by network operators, or operators use a bill-and-keep approach, in which case termination rates are zero. In Western European markets, MTRs are set by a regulator. They can differ from one operator to another within the same market. This asymmetry in MTRs is typically implemented in favour of the smaller operators. They are determined by an already asymmetric state of the market, due to late entry or persistent size difference between operators. They are expected to be exogenous and lower asymmetry. However, we check any reverse causality from market share asymmetry to MTR by implementing again a three-stage least squares estimator.

The third source of endogeneity is the one stemming from lagged investment, a standard issue in dynamic panel models.8 This arises due to potential correlation between the lagged investment and operators' fixed effects. It generates the "Nickel bias", following [START_REF] Nickell | Biases in Dynamic Models with Fixed Effects[END_REF] who shows that the adjustment costs parameter ρ in equation ( 15) is respectively under and over-estimated by OLS estimation with and without operators' fixed effects, when the number of time periods is small. [START_REF] Arellano | Some tests of specification for panel data: Monte Carlo evidence and an application to employment equations[END_REF] suggest implementing the Generalised Method of Moments estimator (GMM) of [START_REF] Hansen | Large sample properties of Generalized Method of Moments Estimators[END_REF] on equation ( 15) in the first difference, using the lagged dependent variables as instruments as long as they are not serially correlated. This difference GMM estimator can suffer from a loss in sample size with unbalanced panel data. In this case, [START_REF] Arellano | Another look at the instrumental variables estimation of error components models[END_REF] suggest a forward orthogonal deviations transformation to preserve sample size. Under this transformation, the average of all future available observations of investment is subtracted from its contemporaneous observation.

To improve the efficiency of the estimates, a two-step approach is implemented. It involves estimating the variance-covariance matrix of the residuals and then using it as a weighting matrix in the GMM estimation. Efficiency can be further improved by using the first differences of investment as instruments for the lagged investment [START_REF] Blundell | Initial conditions and moment restrictions in dynamic panel data models[END_REF]. This is done with equation ( 15) without operators' fixed effects, assuming that first differences of investment are not correlated with operators' fixed effects. This approach, on top of the difference GMM, yields a system of two equations, one in level and another in first difference. [START_REF] Blundell | Initial conditions and moment restrictions in dynamic panel data models[END_REF] show that this "two-step system GMM" estimator provides more efficient estimates of the parameters of the dynamic model ( 15).

Results

Main results

The main estimation results are presented in TABLE 4. Specification ( 1) is an OLS estimation of equation ( 15) without operators fixed effects. The coefficient of N is negative and significant, suggesting that investment per operator falls with the number of operators in symmetric markets. The positive and significant coefficient of asymmetry variable means that the effect of market structure is positively correlated with changes in asymmetry. The coefficient of the lagged investment is positive and significant, suggesting large and significant adjustment costs in the mobile industry. Market size is positively correlated with investment per operator. Specification (2) replaces the discrete variable N with dummies for the number of operators in order to test the monotonicity of the relationship between market structure and investment. Three dummies N i (i=3,4,5) have been constructed respectively for markets with 3, 4 and 5 operators. N i takes on the value 1 when a market has i operators, and 0 otherwise. Markets with 3 operators are used as the reference group. The outcome of this specification lends support to a monotonous and downward sloping relationship between market structure and investment. Investment per operator is predicted to be 14% and 23% lower in 4 and 5-operators markets, respectively compared to markets with 3 symmetric operators. Specification (3) introduces operators fixed effects (FE) in order to account for unobserved and time-invariant cost and demand parameters. The main results still hold, and the magnitude of the effect of market structure becomes larger. However, there are significant changes in the coefficients of asymmetry, lagged investment and market size. As discussed in section 3.3, these variables are likely to be endogenous. In particular, the sharp fall in the coefficient of the lagged investment reflects the Nickel bias whereby this coefficient tends to be underestimated in fixed effects estimation.

Specifications ( 4) and ( 5) account for these issues by implementing respectively Difference and System Generalised Method of Moments estimators (DGMM and SGMM) with instruments for asymmetry, lagged investment, and market size. The outcomes of these specifications accord well with a negative relationship between market structure and investment per operator, with a stronger effect for operators that lose market shares more than the average and a significant adjustment costs of investment in the mobile industry. Note that the estimation results do not include a constant term because it is differenced-out. Specifications ( 6) and ( 7) account for the endogeneity of market structure by using government political ideology towards the welfare-state as instrument for the number of operators. The negative relationship between market structure and investment still holds, but the system GMM, that is specification (7), yields better outcomes than the difference GMM (specification ( 6)). Indeed, as predicted by [START_REF] Blundell | Initial conditions and moment restrictions in dynamic panel data models[END_REF], the coefficients of the SGMM are more precisely estimated. In addition, the p-values of the Sargan and Hansen statistics of specification ( 7) are greater than 10%, lending support to the validity of the set of instruments. Moreover, the p-value of the difference in Hansen test for this specification suggests that the additional instruments in the level equation are exogenous. Therefore, specification (7) provides the least biased causal effects of market structure on investment. This specification further suggests that market size is a positive determinant of investment per operator, in accordance with the finding by [START_REF] Beneito | Competitive Pressure and Innovation at the Firm Level[END_REF].

Robustness checks

TABLE 5 presents some complementary tests in support of the main results. Specification ( 1) is a non-causality test between investment and the number of operators. It estimates a system of two equations, one for investment and another for the number of operators as dependent variables, on the basis of three-stage least squares estimator. It shows that investment have a significant effect on the number of operators, but only at 5% level. Likewise, there is a negative and significant effect of the number of operators on investment. These findings do not rule out reverse causality from investment to the number of operators. However, the use of an instrument for the number of operators in specification (7) reduces any bias stemming from reverse causality. Specification (2) tests reverse causality from market share asymmetry to mobile termination rates and finds that market share asymmetry does not determine mobile termination rates, but the reverse holds. Specifications (3), ( 4) and ( 5) are first-stage estimates of the correlation between the instruments and the endogenous explanatory variables. Specification (3) shows a negative and significant correlation between mobile termination rates and market share asymmetry. In specification (4), market size is positively and significantly correlated with population size. Likewise, the political ideology variable is a significant predictor of the number of operators implying that countries governed by pro-welfare-state are predicted to have fewer mobile operators. The F-statistics provided at the bottom of the table are all larger than 10, suggesting that the instruments are not weak. TABLE 6 presents results corresponding to equation ( 15) with additional controls such as income, competition from fixed lines, cash flow, and retail prices. We account for the endogeneity of the last three controls by using their 2-years lags as internal instruments. Estimation relies on the SGMMIV of specification (7) of TABLE 4, that is system GMM with instrument for the number of operators. It turns out that none of these additional controls has a significant effect on investment. Note that the Sargan test cannot reject the exogeneity of the instruments, unlike the Hansen test. The failure of the Hansen over-identifying test is probably due to the large number of instruments, though they do not outnumber observations. The main findings remain valid.

Discussion of the results

In this section, we discuss the marginal effect of market structure on investment from different perspectives: investment per operator, industry investment, short and long run effects. As emphasised in section 3.2, the short run effect of market structure on investment per operator is a linear combination of the coefficients of the number of operators and asymmetry:

∂z ijt ∂N jt = θ + δ∆ ijt (N jt ) ∆ ijt (N jt ) = σ (N ) + 1 N 2
, the variation in asymmetry, represents how much an operator's market share changes, compared to the variation in the average market share. Using equation ( 10) from the theoretical model, ∆ ijt (N jt ) can also be interpreted as the change in the quality of an operator with respect to its rivals.

We use the point estimates from specification (7) in TABLE 4 to evaluate the total short run marginal effect of market structure on investment per operator as a function of change in asymmetry. The outcome of this evaluation is presented in FIGURE 3. The upward sloping line in this figure has a slope of δ = 1.27 and intercept of θ = -0.16.

Investment per operator falls by 16% as long as change in market share is identical to the variation in the average market share: σ (N ) = -1 N 2 . This result is particularly relevant in symmetric markets where operators are identical. The magnitude of this negative marginal effect becomes larger for operators who lose more than the average:

|σ (N )| > 1 N 2 .
As illustrated by FIGURE 3, the marginal effect of market structure is still negative for operators who lose slightly less than the average. More precisely, investment per operator still falls with entry when the loss in market share is less than 6 percentage points (pp) compared to the average: 0 < ∆ ijt (N jt ) * 100 < 6. However, the effect of market structure is no longer significant when the loss in market share is far less than the average, typically when 6 ≤ ∆ ijt (N jt ) * 100 ≤ 28.

For instance, consider the effect of a fourth entrant on the investment of an operator with 33% market share in a 3-operators market. Given the marginal effect, investment is expected to fall by 16% if market share falls to 25% with the entry. In this case, its market share loss is 8 pp, just as the average.9 However, if market share falls to, say 32%, the entry of the fourth operator is not predicted to have a significant effect on its investment. Its market share loss is 1 pp, compared to 8 pp for the average market share. This gap is 7 pp which is larger than 6 pp.

These marginal effects are predicted to be stronger in the long run due to significant adjustment costs. The coefficient of the lagged investment in specification (7) in TABLE 4 is 0.8, suggesting that the short run effect is multiplied by a factor of 5 in the long run.10 This implies that the relationship between the marginal effect of N and change in asymmetry, as depicted in FIGURE 3, becomes steeper in the long run. The curve rotates counterclockwise in the long run with respect to the point where it crosses the x-axis.

The effect of market structure on industry investment, that is the total investment by all operators, strongly depends on market share asymmetry. In symmetric markets, simple calculations show that industry investment increases with the number of operators in the short run, but eventually falls in the long run.

Conclusion

Consistently with the prediction of the theoretical model, this paper finds that the effect of market structure on investment strongly depends on asymmetry. In particular, investment per operator falls with the number of operators in symmetric markets. This negative effect is larger for operators who lose market share more than the average. In symmetric markets, the industry investment increases with the number of operators in the short run but eventually falls in the long run.

These results are consistent with the theoretical predictions of [START_REF] Vives | Innovation and Competitive Pressure[END_REF] in symmetric markets and Schmutzler (2013) in asymmetric markets. They are also consistent with [START_REF] Genakos | Evaluating Market Consolidation in Mobile Communications[END_REF] who find a negative relationship between the number of mobile operators and their investment. The positive effect of market size on investment accords well with the findings of [START_REF] Beneito | Competitive Pressure and Innovation at the Firm Level[END_REF]. However, our findings do not lend support to a non-monotonic relationship between competition and investment as found by [START_REF] Sacco | Is there a U-shaped relation between competition and investment[END_REF]. They are also not in line with the conclusions of a recent OECD report advocating for more mobile operators as a means to improve the quality of mobile telecommunications services (OECD, 2014).

To the extent that investment lowers marginal cost of mobile services, our findings suggest that raising the number of mobile operators could lower dynamic efficiencies.

In the mobile industry, the magnitude of these dynamic efficiencies losses can be huge due to an exceptionally high rate of technological progress. According to [START_REF] Amaya | The progress in Wireless Data Transport and its Role in the Evolving Internet[END_REF], the mobile industry is experiencing an exponential rate of technological progress of almost 50% for mobile data, which means the performance doubles every 1.4 years. Therefore, a change in the market structure of the mobile industry entails a trade-off between static and dynamic efficiencies. Merger analyses in the mobile industry should assess the magnitude of these efficiencies in order to determine the socially optimal market structure.

The empirical analysis uses capital expenditure in mobile networks as a proxy for investment in cost-reducing or quality improving technologies. To the extent that network quality is positively correlated with capital expenditures, our results remain valid. Otherwise, a fall in capital expenditure might not correspond to a fall in quality or in investment in cost-reducing technologies. Such cases may occur for instance when operators overinvest in more concentrated markets. In these cases, increasing the number of mobile operators may not entail a trade-off between static and dynamic efficiencies. Moreover, this paper has tested the hypothesis that there may be a trade-off between static and dynamic efficiencies in the mobile industry.

While the strong rate of technological progress tends to suggest that dynamic efficiencies outweigh static efficiencies, it remains an empirical question to determine the exact magnitude of these efficiencies in order to characterise the optimal market structure in the mobile industry. Future work shall address these issues.

Remember that when N is even, operator i + (N/2 + 1) is also the operator i -

(N/2 -1)
The comparison between equation ( 6) and ( 7) provides four equations: 16)

β 0 = β 1 + 1 2 (
β 1 = β 0 + β 2 -1 4 (17) ∀j ∈ [2, N/2 -1]; β j = β j-1 + β j+1 4 (18) if N is odd βN-1 2 = β (N -3)/2 + β (N -1)/2 4 and if N is even βN 2 = β N/2-1 2 (19)
For j ∈ [0, N/2 -3], Equation ( 18) leads to:

β N 2 -(j+2) = 4β N 2 -(j+1) -β N 2 -j (20)
Let us denote two sequences W j and X j such that, for j ≤ N -3 2 , β N -1 2 -j = βN-1 2 W j when N is odd and βN 2 -j = βN 2 X j when N is even. (This means that for j ≥ 1; 20) leads to W j+2 = 4W j+1 -W j and X j+2 = 4X j+1 -X j . Sequences W j and X j are both defined by the same linear recurrence relation whose characteristic equation is:

β j = βN-1 2 W N -1 2 -j if N is odd and β j = βN 2 X N 2 -j if N is even). Equation (
x 2 -4x + 1 = 0. This equation has two positive roots: a = 2 + √ 3 and b = 2 -√ 3. It is noteworthy that ab = 1.

From ( 19), when N is odd, we have 3βN-1 2 = βN-3 2 and when N is even 2βN 2 = βN 2 -1 . As a result, the first terms of the sequences are W 0 = 1 ; W 1 = 3 and X 0 = 1 ; X 1 = 2. This allows us to calculate the general term of the sequences which are written in the form: W j = λa j + µb j and X j = λ a j + µ b j

The initial conditions are W 0 = λ+µ; W 1 = λa+µb,. Therefore,

λ = √ 3 + 1 /2 √ 3; µ = √ 3 -1 /2 √ 3;
In the same manner, λ = µ = 1/2. The general terms of the sequence are thus expressed as:

W j = ( √ 3+1)a j +( √ 3-1)b j 2 √ 3
and X j = a j +b j 2 When the market is symmetrical, quality d and market shares are the same for all operators: φ = 1/N ; thus from equation ( 6) we can write β 0 = -2

(N -1)/2 j=1 β j if N is odd and β 0 = -2 N/2-1 j=1 β j -βN 2
if N is even. Using the sequences W and X, this can be rewritten

β 0 = -2βN-1 2 (N -3)/2 j=0 W j if N is odd and β 0 = -βN 2 1 + 2 N/2-1 j=1 X j
if N is even. Using the sum of geometric sequences, the expression of β 0 according to N is then:

β 0 = b (N -1)/2 -a (N -1)/2 √ 3 β (N -1)/2 if N is odd and β 0 = (b-1)a N/2 +(a-1)b N/2 2 β N/2 if N is even. We know that β 1 = β (N -1)/2 W (N -1)/2-1 if N is odd and β 1 = βN 2 X N 2 -1 if N is even. Replacing those expressions in equation (16) leads to 2(b (N -1)/2 -a (N -1)/2 ) √ 3 β (N -1)/2 - β (N -1)/2 ( √ 3+1)a (N -3)/2 +( √ 3-1)b (N -3)/2 2 √ 3 = 1 if N is odd and (b -1) a N/2 +(a -1) b N/2 β N/2 - β N/2 a N/2-1 +b N/2-1 2 = 1 if N is even. This allows us to express β (N -1)/2 and βN 2 in function of N. Using b = 1/a yields: β (N -1)/2 = 2 √ 3a (N -3)/2 (9-5 √ 3)-(9+5 √ 3)a N -3 if N is odd and βN 2 = 2a N/2 (1-2a)a N -1 +(a-2)
Those expressions can be used to calculate the general term of β 0 and β j for j ≥ 1 that no longer depend on N being odd or even because b (N -1)/2 -a (N -1)/2 √ 3

β (N -1)/2 = (b-1)a N/2 +(a-1)b N/2 2
β N/2 and β (N -1)/2 W (N -1)/2-j = β N/2 X N/2-j . General terms are:

β 0 = 1 - a N + 1 √ 3 (a N -1) β j = - a N -j + a j √ 3 (a N -1) if j = 0
Using a = 2 + √ 3 leads to the result.

Calculation of consumer surplus and social welfare

Consumer surplus is the sum of utility for all consumers. The utility of a consumer located at x between operator i and operator i + 1, purchasing i's offer is

U i = s i -tx-p i . At equilibrium, p * i = c i +σ i t, thus U i = s i -tx-c i -σ i t or U i = d i -σ i t-tx.
In a symmetric market σ i = 1/N, moreover, indifferent consumer is located at x = 1/2N. Half of the customers purchasing i s offer are located between operator i and operator i + 1, the other half is located between operator i and operator i -

1 thus CS i = 2 1/2N 0 d i -t N -tx dx = d i N -5t 4N 2
In a symmetric market all operators have an equal quality. ∀ i, j ∈ N,

d i = d j = d, therefore, CS = N i=1 CS i = N.CS i
Consumer surplus at the market level is thus:

CS = d - 5t 4N
Welfare is the sum of consumer surplus and profit. Profit at the market level is expressed as π = t N -N (z +F ),therefore, the welfare at the market level is expressed as:

W = d - t 4N -N (z + F )
Derivative of these equations according to N yields equations ( 13) and ( 14). Specifications (1) OLS, (2) OLS with dummies N i equal 1 in markets with i operators. We use markets with 3 operators as the reference group.

(3) Fixed-effects estimator, (4) 2-step Difference GMM, (5) 2-step System GMM, ( 6) and (7) 2-step Difference and System GMM with instrument for N . L. is the lag operator. P-value (Level) and (Diff) are the p-values for the level and difference equations.

In specifications (4) and ( 6), observations drop from 391 to 342 due to the differentiation. Significant at 1% (***), 5% (**) and 10% (*). Standard errors in parentheses.

Standard errors robust to heteroskedasticity and autocorrelation, clustered at operator level. Specifications: (1) 3-stage least squares with N and lninv as dependent variables, (2) 3-stage least squares with mtr and asym as dependent variables, (3), ( 4) and ( 5) are correlations between the instruments and the endogenous explanatory variables. gdppc as control in all specifications.

The number of observations is smaller in specifications (1) and (2) due to the lagged explanatory variables. Significant at 1% (***), 5% (**) and 10% (*). Standard errors in parentheses.

Standard errors robust to heteroskedasticity and autocorrelation, clustered at operator level.

Figure 1 :

 1 Figure 1: Market share calculation

Figure 2 :

 2 Figure 2: Location of operators

Figure 3 :

 3 Figure 3: Short-run marginal effect of market structure on investment per operator

Table 1 :

 1 Variables Description

	Variable	Name	Definition	Source
	Capital expenditures	capex	mobile capital expenditures	Yankee Group
			in millions 2015 $US
	Investment	lninv or z logarithm of capex	constructed
	Market share	mshare	prepaid and postpaid	Yankee Group
		or σ	subscriber market share
	Asymmetry	asym	market share asymmetry	constructed
	Average revenue per subscriber	arpu	ratio of revenue to	Yankee Group
			subscribers, in 2015 $US
	Earning before taxes interest	cashf low	measure of profit	Yankee Group
	amortization and depreciation		millions of 2015 $US
	Mobile termination rate	mtr	US dollars ppp	Ovum
	Number of operators	N	virtual operators excluded	GSMA
	Dummy for the number of operators N i	equals 1 in markets
			with i operators
			and 0 otherwise
	Market size	msize	number of subscribers	Yankee Group
	Fixed telephony	f tel	household penetration rate	ITU
	GDP per capita	gdppc	constant thousands	World Bank
			$US ppp
	Population	pop	millions	World Bank
	Political ideology	welf are	government position towards Manifesto Project
			the welfare-state
	Note: Yearly observations from 2006 to 2015
	ITU: International Telecommunications Union

Table 2 :

 2 Summary statistics

	Variable	Obs.	Mean Std. dev.	Min	Max
	capex	391	394.69	352.57	9.30 2441.96
	lninv	391	5.58	0.94	2.23	7.80
	mshare	391	0.31	0.11	0.03	0.62
	asym	391	0.06	0.12	-0.22	0.41
	arpu	391	349.49	149.23	100.90	921.97
	ebitda	391 1200.53	1207.56 -101.93 5521.24
	mtr	391	5.49	4.22	0.82	16.73
	N	391	4.12	0.74	3	5
	msize	391	42.84	37.23	4.42	115.23
	ftel	348	46.36	10.62	11.74	67.12
	gdppc	391	34.48	8.50	12.35	67.67
	pop	391	33.37	28.40	4.22	82.31
	welfare	391	13.38	4.69	6.15	25.49
	year	391 2010.58	2.93	2006	2015

Table 4 :

 4 Estimation results

	lninv

Table 5 :

 5 Complementary estimation results

		N	lninv	mtr	asym	asym	msize	N
		(1)		(2)		(3)	(4)	(5)
	lninv	0.393 * *
		(0.186)
	L.lninv		0.194 * * *
			(0.055)
	L2.lninv		0.039
			(0.050)
	N		-0.193 * * *
			(0.074)
	L.N	0.722 * * *
		(0.060)
	L2.N	-0.068
		(0.054)
	asym		3.332 * * *	0.065
			(0.591)	(1.869)
	L.asym				0.849 * * *
					(0.055)
	L2.asym				-0.104 *
					(0.060)
	mtr				-0.004 *	-0.007 * * *	0.575 * * *	0.011
					(0.002)	(0.002)	(0.132)	(0.018)
	L.mtr			0.448 * * *
				(0.048)
	L2.mtr			0.113 * * *
				(0.039)
	pop	0.175 * * *		-0.181		0.015 * * *	1.001 * * *	0.210 * * *
		(0.045)		(0.119)		(0.004)	(0.347)	(0.048)
	welf are					-0.003 * * *	0.157 *	-0.033 * * *
						(0.001)	0.081	(0.011)
	msize		-0.001		0.001
			(0.005)		(0.001)
	year FE
	operators FE
	constant	-11.814 * * *	5.619 * * *	-9.407 * * *	-0.035	-1.047 * * *	-14.851	-7.068 * *
		(3.051)	(0.707)	(3.020)	(0.042)	(0.266)	(22.192)	(3.040)
	obs.	336		336		391	391	391
	F-stat					46.75	769.39	10.96

Table 6 :

 6 Robustness checks SGMMIV including gdppc as additional control and 2-to-4 year lags as internal instruments for L.lninv. (2) SGMMIV including f tel as additional control and 2 years lags as internal instruments for f tel and L.lninv. (3) SGMMIV including cashf low as additional control and 2 years lags as internal instruments for L.lninv and cashf low. (4) SGMMIV with arpu as additional control and 2 years lags as instruments for L.lninv and arpu. Significant at 1% (***), 5% (**) and 10% (*). Standard errors in parentheses are robust to heteroskedasticity and autocorrelation, clustered at operator level. Smaller observations in specifications (2) and (3) due to missing observations for f tel and cashf low in some markets.

				lninv
		(1)	(2)	(3)	(4)
	N	-0.161 * * *	-0.138 *	-0.112 *	-0.114 * *
		(0.059)	(0.074)	(0.059)	(0.047)
	asym	1.272 * * *	0.833 * *	1.114 * *	1.149 * *
		(0.395)	(0.347)	(0.531)	(0.470)
	L.lninv	0.811 * * *	0.779 * * *	0.799 * * *	0.698 * * *
		(0.115)	(0.085)	(0.086)	(0.053)
	msize	0.005 *	0.005 * * *	0.006 * * *	0.007 * * *
		(0.003)	(0.001)	(0.002)	(0.001)
	gdppc	-0.001
		(0.004)
	f tel		-0.001
			(0.002)
	cashf low			-0.00005
				(0.00003)
	arpu				-0.0001
					(0.0001)
	year FE
	obs.	391	348	389	391
	# groups	49	47	49	49
	# instruments	43	43	49	48
	P-value (Sargan)	0.13	0.18	0.42	0.18
	P-value (Hansen)	0.10	0.00	0.00	0.00
	P-value (AR2)	0.16	0.92	0.18	0.20
	P-value (AR3)	0.33	0.34	0.34	0.33
	P-value (AR4)	0.18	0.39	0.18	0.16
	Specification (1)

Worldwide subscription penetration rate in 2015 is close to 100% and user penetration rate is over 65%. (Worldwide cellular user forecast 2015-2020, Strategy Analytics).

See Koh & Magee (2006) and[START_REF] Amaya | The progress in Wireless Data Transport and its Role in the Evolving Internet[END_REF] for the strong rate of technological progression in the Information and Communications Technologies sector.

This test implements the Granger causality test on each individual in the panel and then takes the average of the Wald statistics.[START_REF] Dumitrescu | Testing for Granger noncausality in heterogeneous panels[END_REF] show that this average converge sequentially to a normal distribution.

[START_REF] Genakos | Evaluating Market Consolidation in Mobile Communications[END_REF] initially use MTR as an instrument for mobile market shares.

See Bond (2002) for issues related to the estimation of dynamic panel models and their solutions.

For clarity of the illustration, we consider discrete variation in market shares.

This factor is obtained as 1 1-0.8 = 5.

Appendix Calculation of coefficients β j

The comparison between equations ( 6) and ( 7) for all operators, provides a system whose solutions are the coefficients β j . We will start with relatively simple cases N = 2 and N = 3, to understand how the system works before moving on to the general case.

For N = 2, the comparison between ( 6) and ( 7) is expressed as:

Notice that operator i -1 is also the operator i + 1 and operator i + 2 is also the operator i -2 and the operator i.

The system of equation is expressed as:

and the solution is

For N = 3, the system is expressed as:

and the solution is

Now the general case: For N > 3, equations ( 6) and ( 7) are expressed as:

if N is odd:

if N is even: