Vivien Georges
email: gvivienh@gmail.com

Houngbonon

François Jeanjean
email: francois.jeanjean@orange.com

Jérôme Pouyet

Marc Lebourges

Bruno Jullien

Wilfried Sand- Zantman

Marc Petulowa

Maude Hasbi

Julienne Liang

Olivia Bertelli

What Level of Competition Intensity Maximises Investment in the Wireless Industry? *

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

This paper investigates the relationship between competition and investment in the wireless industry from a dynamic perspective. Using firm level data and instrumental variable estimation strategy, it finds that the relationship is inverted-U shaped. The investment maximising intensity of competition is reached when operators' gross profits represent 37 or 40 percent of their revenues, depending on whether capital expenditures are normalised by the number of subscribers. This finding means that investment increases with competition as long as operators' profits are above the thresholds of 37 or 40 percent of their revenues. Under these thresholds, there is a tradeoff between competition and investment. The paper also finds a significant long run effect of competition on investment which amplifies the short run effect by a factor of 3 to 4.

Introduction

The relationship between competition and investment is the topic of a vast body of economic literature. Although the debate on the issue has been lively for many decades, no clear-cut conclusion has been reached so far. There are two main opposing views about the nature of this relationship. The Schumpeterian view highlights that large firms in more concentrated markets are more likely to invest, while the Arrow's view emphasises the virtues of competition, which stimulates investment to escape competition. 1 Theoretically, the relationship between competition and investment is ambiguous. For instance, [START_REF] Sacco | Is there a U-shaped relation between competition and investment[END_REF] find a U-shaped relationship, whereas [START_REF] Belleflamme | Incentives to innovate in oligopolies[END_REF] predicts an inverted-U shaped relationship. [START_REF] Schmutzler | Competition and investment -a unified approach[END_REF] argues that: "The relationship between competition and investment is not affected in an unambiguous way by the level of pre-existing competition". As the theory is not conclusive, empirical evidence is needed for each specific industry. This paper takes advantage of firm level data on wireless operators worldwide to estimate the relationship between competition and investment in the wireless industry. The intensity of competition is measured at the firm level on the basis of the Lerner index of monopoly power, and investment is measured as total or per subscriber capital expenditures. Some empirical papers investigate the relationship between competition and investment in the wireless industry, but they all assume a linear relationship and do not correct for the endogeneity of competition.2 This paper employs non-parametric identification strategies, in particular, a locally weighted scatterplot smoothing and a local polynomial smoothing to identify the shape of the relationship without making any a priori assumption. It corrects for endogeneity by using an instrumental variable estimation strategy and a Generalised Method of Moments estimator. The causal impact of competition on investment stems from cross-operators differences in competition intensity due to their date of entry into the market and lags of competition intensity. The paper also accounts for the speed of adjustment of investment in the wireless industry by formulating a dynamic model linking investment to a quadratic function of competition, controlling for the lagged values of investment as well as market and operators' characteristics.

The non-parametric methods suggest an inverted-U shaped relationship between competition and investment in the wireless industry. This finding is confirmed by the econometric estimation and is robust to alternative measures of investment, modelling specifications and estimators. The relationship is also robust to heterogeneity in terms of demand and cost parameters such as market size and network deployment cost. More specifically, a wireless operator's investment increases with competition as long as its profit is above the thresholds of 37 or 40 percent of its total revenue. These thresholds depend on whether or not capital expenditures are normalised by the number of subscribers. Under these thresholds, there is a tradeoff between competition and investment as more competition lowers operators' investment in wireless networks. In addition, the estimates suggest a significant long run effect of competition on investment which amplifies the short run effect by a factor of 3 to 4.

The rest of the paper is organised as follows. Section 2 summarises the findings from the literature on the relationship between competition and investment. Section 3 provides some background information to motivate the analysis of the link between competition and investment in the wireless industry. Section 4 presents the dataset and the variables used in the empirical analysis. Section 5 describes the econometric model and the estimation strategy. Section 6 presents and discusses the results along with their policy implications. Section 7 summarises the main findings and concludes.

Related literature

As reviewed by [START_REF] Mathis | Competition and investment: What do we know from the literature? Working papers[END_REF], most papers dealing with the relationship between competition and investment focus on investment in process or product innovation. 3 According to [START_REF] Boone | Competitive pressure: the effects on investments in product and process innovation[END_REF], investment in process innovation lowers the marginal cost of production for a given product, whereas investment in product innovation affects consumers' preferences. This latter introduces new products without necessarily affecting the quality or the marginal cost of production of existing products. Investment in wireless networks can be viewed as expenditures in both process and product innovation. Not only, it reduces the marginal cost of each unit of information due to technological progress, but also it introduces new product into the market. 4 Therefore, the findings from the literature on the relationship between competition and innovation apply to the wireless industries.

The relationship between competition and innovation is the subject of a long standing debate initiated by [START_REF] Schumpeter | Capitalism, Socialism and Democracy[END_REF] and [START_REF] Arrow | Economic welfare and the allocation of ressources for invention. Nber chapters[END_REF]. Two conflicting views lie at the heart of this debate. On the one hand, Schumpeter points out the role of firm's size, financial constraints and more specifically the incentive of a monopoly to invest more than a firm in a competitive market [START_REF] Schumpeter | Capitalism, Socialism and Democracy[END_REF]. 5 This view is driven by the efficiency effect as epitomised by the findings of [START_REF] Gilbert | Preemptive patentive and the persistence of monopoly[END_REF]. The efficiency effect refers to the theoretical finding that an efficient monopolist always gets bigger profit than two uncoordinated duopolists altogether. 6 The argument made by Gilbert and Newbery can be summarised as follows: due to the efficiency effect, a monopolist would lose more than a potential entrant by not innovating. 7

On the other hand, Arrow's view emphasises that a firm in a competitive market has more incentive to innovate than a monopoly due to the escape-competition effect [START_REF] Arrow | Economic welfare and the allocation of ressources for invention. Nber chapters[END_REF]. The escape-competition effect corresponds to the profit advantage of being the technological leader in a competitive market, compared to a monopoly. Indeed, when investment in a cost-reducing technology yields an exclusive and unlimited right, the difference between ex-ante and ex-post profits is larger in a competitive market than in a monopoly. This is because a firm in a competitive market earns lower profit ex-ante compared to a monopoly. However, the Arrow's view can also be interpreted as a replacement effect because the monopoly has lower incentive to invest in a new technology due to the opportunity cost of replacing its old technology [START_REF] Tirole | The theory of industrial organization[END_REF].

In a general equilibrium framework, the empirical findings are mixed. For instance, [START_REF] Blundell | Market share, market value and innovation in a panel of british manufacturing firms[END_REF] find that larger firms invest more than smaller ones, while those in more concentrated markets invest less. Likewise, [START_REF] Kraft | Market structure, firm characteristics and innovative activity[END_REF] finds a negative relationship between the number of competitors and the percentage of sales attributable to new products, whereas [START_REF] Nickell | Competition and corporate performance[END_REF] identifies a positive relationship between the number of competitors and total factor productivity growth for price. However, from the firm's perspective, they actually correspond to process innovations.

5 He also emphasises the role of competition in the form of a threat of entry.

6 See [START_REF] Tirole | The theory of industrial organization[END_REF], chap. 8, section 8.6.2.1 for an exposition of this effect.

7 Consider a race for innovation between an incumbent monopoly an a potential entrant. Winning the race amounts to a reduction in the marginal cost of production from c to c. If the entrant wins the race, it gains the asymmetric duopoly profit denoted by Π d (c), whereas the incumbent's profit becomes Π d (c). In case the monopoly wins the race, it gains the efficient monopoly profit Π m (c). The monopolist incentive to win the race is the difference

Π m (c) -Π d (c), whereas the entrant's incentive to invest is Π d (c). Because of the efficiency effect, Π m (c) -Π d (c) > Π d (c).
Therefore, the monopoly has more incentive to invest in order to win the race than the entrant. British companies. [START_REF] Aghion | Competition and innovation: An inverted-u relationship[END_REF] show, both theoretically and empirically, an inverted-U shaped relationship between competition and innovation. In their general equilibrium model, the central force driving the inverted-U shaped relationship is the technological gap between firms within each industry. As the technological gap increases, more competition lowers incentives to innovate.

In a partial equilibrium framework, [START_REF] Schmutzler | Competition and investment -a unified approach[END_REF] relies on a single parameter of competition, in the tradition of [START_REF] Boone | A new way to measure competition[END_REF], and shows that the relationship between competition and investment can take various shapes. He emphasises four channels through which competition affects investment in cost-reducing technologies. The margin effect is negative as more competition reduces the marginal profit from investment. The output sensitivity effect is positive as more competition reallocates output towards the more efficient firms. The output effect depends on cost asymmetry between firms. It is negative for less efficient firms and positive for the more efficient ones. Finally, the cost pass-through effect characterises the extent to which cost reduction is passed to the consumers in terms of lower prices. It is positive for Cournot competition and negative for Bertrand competition. The overall effect of competition on investment is the sum of these four effects, the sign of which is indeterminate. Specific theoretical case studies include [START_REF] Belleflamme | Incentives to innovate in oligopolies[END_REF] who study an oligopolistic market where firms invest in the acquisition of a new technology that provides an exclusive cost advantage. They find that the relationship between competition and investment is inverted-U shaped. On the contrary, [START_REF] Sacco | Is there a U-shaped relation between competition and investment[END_REF] analyse a duopoly model in which an increase in competition is defined as a switch from Cournot to Bertrand competition. They find a U-shaped relationship between competition and investment. Furthermore, [START_REF] Boone | Competitive pressure: the effects on investments in product and process innovation[END_REF] argues that the relationship depends on how a firm's profit is affected by competition according to its initial level of cost efficiency.

These theoretical studies focus on cases where gross profit is higher than investment. [START_REF] Jeanjean | Incentives to invest in improving quality in the telecommunications industry[END_REF] studies the Hotelling model and shows that for high intensity of competition, firms underinvest with respect to the level which would maximise profit. This situation arises because their profit is not sufficient to sustain a profit maximising level of investment. The underinvestment behaviour arises beyond a certain threshold of competition intensity. It is highlighted in the Hotelling framework, but it also shows up in other types of models. 8 In cases where investment is increasing with competition below the threshold, the overall relationship may be inverted-U shaped due to underinvestment beyond the threshold. [START_REF] Vives | Innovation and competitive pressure[END_REF] depart from the use of a single parameter of competition and investigate how specific parameters of competitive pressure such as the number of firms, the magnitude of barriers to entry and the degree of product substitutability affect investment. He shows that the relationship between competition and investment in cost-reducing innovation is negative when competition is parameterised by the number of firms, but positive when it is parameterised by the magnitude of barriers to entry or the degree of product substitutability. [START_REF] Beneito | Competitive pressure and innovation at the firm level[END_REF] provide empirical support for these predictions using data from Spanish manufacturing firms. However, this sector-wide analysis may suffer from cross-industry difference in the rate of innovation, as emphasised by [START_REF] Kamien | Market structure and innovation: A survey[END_REF]. In addition, given that the relationship depends on the nature of investment and the mode of competition, industry-specific analyses are needed.

Several attempts have been made to uncover the relationship between competition and investment within specific industries.9 In the telecommunications industry, most papers have analysed the impact of competition on investment in the fixed broadband market. [START_REF] Cambini | Broadband investment and regulation: A literature review[END_REF] review this literature and find that the impact is rather ambiguous. Likewise, [START_REF] Grajek | Regulation and investment in networks industries: Evidence from european telecoms[END_REF] find that competition through access regulation negatively affects investment in fixed broadband networks, whereas [START_REF] Bacache | Dynamic entry and investment in new infrastructures: Empirical evidence from the fixed broadband industry[END_REF] find that access regulation has no effect on new entrants' investment in fixed broadband networks.

In the wireless industry, the empirical investigation of the relationship between competition and investment is still nascent. Most papers assume a linear relationship and find that competition has an ambiguous impact on investment. In particular, [START_REF] Garrone | Seeking the links between competition and telecommunications investments[END_REF] find no correlation between competition and investment in wireless networks. [START_REF] Kang | Competition and mobile network investment in China's telecommunications industry[END_REF] highlight a positive relationship between market concentration and investment in China's wireless industry. [START_REF] Kim | Access regulation and infrastructure investment in the mobile telecommunications industry[END_REF] find a negative relationship between competition, defined as the entry of Mobile Virtual Network Operators (MVNO), and investment by Mobile Network Operators (MNO). Furthermore, [START_REF] Lestage | Competition and investment in telecommunications: Does competition have the same impact on investment by private and state-owned firms?[END_REF] find that the impact of competition on investment depends on whether the firm is private (negative impact) or state-owned (positive impact).

Overall, the findings from the literature suffer from some important limitations. First, the speed of adjustment of investment in the wireless networks is still missing from both theoretical and econometric models. The theoretical models discussed above typically considers two-stage games in which firms first invest in cost-reducing technologies and then compete on price or quantity. These models do not distinguish between short and long run effects of competition on investment. Second, theoretical and econometric models implicitly assume a linear relationship between competition and investment. Yet, the ambiguity of the theoretical predictions and empirical findings may stem from a nonlinear relationship between competition and investment. Finally, most empirical papers do not address the endogeneity of competition probably due to the lack of good quality data from which instruments can be derived.

Background on competition and investment in the wireless industry

This section provides some background information to motivate the analysis of the effect of competition on investment in the wireless industry. It starts with an overview of the policy debate and then emphasises the significance of innovation in the wireless industry due to a strong rate of technological progress.

The policy debate on the relationship between competition and investment in the wireless industry

The policy debate about the relationship between competition and investment in the wireless industry arises in the context of merger and entry waves worldwide and particularly in the European Union. Recently, some national regulators have granted spectrum licenses to new entrants like Free Mobile in France in 2012. These decisions partly reflect the view of the regulators about the market structure that would lower prices and raise investment in the wireless industry. To the extent that competition typically leads to lower price, the debate hinges on the nature of the relationship between competition and investment and more specifically whether investment falls with competition. The answer to this question would guide the regulator in deciding when an entry is more desirable than a merger, and inversely. Yet, there is no consensus about the shape of the relationship.

As a matter of fact, a recent report by the OECD argues that more competition, through more wireless operators, lowers prices and increases the quality of wireless services (OECD, 2014). In the USA, the Department of Justice (DoJ) considers that the wireless market is not competitive enough to require consolidation. It thus challenged the proposed merger between AT&T andT-Mobile in 2011 (DoJ, 2011). Unlike these positions, [START_REF] Faulhaber | Assessing competition in US wireless markets: Review of the FCC's competition reports[END_REF] argue that the USA wireless market is effectively competitive and that market structure, measured by the Herfindahl-Hirschman index, is not correlated with monthly bills. More recently, a report by the Centre on Regulation in Europe suggests that market consolidation increases prices, but it also raises investment in wireless networks [START_REF] Genakos | Evaluating market consolidation in mobile communications[END_REF]. Thus, the policy debate about the effect of competition on investment remains open.

Technological progress as a driver of investment in the wireless industry

The literature on the relationship between competition and innovation applies to innovative markets such as the wireless industry. Indeed, this industry experiences a significant rate of technological progress which drives regular investment in the adoption of new technologies. According to the estimates by [START_REF] Koh | A functional approach for studying technological progress: Application to information technology[END_REF] and [START_REF] Koh | A functional approach for studying technological progress: Extension to energy technology[END_REF], the rate of technological progress in the information technologies typically surpasses that of other sectors such as the energy industry.

Investment in wireless networks is driven by the new products that equipment manufacturers release every year. These new products can bring about a radical change in the provision of wireless communications services or simply an improvement in the transmission of communication traffic over the network. Radical innovations are characterised by a shift in data transmission speeds or protocols. Over the last two decades, there have been four overlapping generations of mobile telecommunications networks (1G, 2G, 3G and 4G), with several variants in between. The overall outcome is a continual enhancement of performances of wireless networks.

The rate of technological progress is even greater for wireless data. [START_REF] Amaya | The progress in wireless data transport and its role in the evolving internet[END_REF] estimate an exponential rate of technological progress for wireless data transportation of around 50 per cent since 1970. This corresponds to download speeds doubling almost every year, at the same cost.10 Thus current investment affects subsequent investment through its effect on marginal cost. Therefore, investment should be analysed with a dynamic model in order to disentangle between short run and long run effect of competition on investment.

Dataset and variables

Dataset

The dataset is built by assembling information from four databases: the World Cellular Information Services (WCIS) from Ovum, the GSMA Intelligence database, the online International Telecommunications Union (ITU) database, and the World Development Indicators (WDI) database managed by the World Bank.

The WCIS database provides quarterly accounting information on wireless operators' revenues, subscribers, profits and capital expenditures (CAPEX). The GSMA database provides information on operators' characteristics, namely, their year of entry and whether or not they host MVNO. The WCIS and the GSMA databases are proprietary, but widely used in academic research as in [START_REF] Whalley | From global aspirations to regional dominance: The case of Deutsche Telekom[END_REF], [START_REF] Kim | Access regulation and infrastructure investment in the mobile telecommunications industry[END_REF] and [START_REF] Hazlett | Natural experiments in mobile phone regulation: Estimated effects of prohibiting handset bundling in Finland and Belgium[END_REF]. The ITU database provides yearly information on the penetration rate of fixed telephony. Finally, the WDI database contains yearly information on market characteristics such as population density and working-age population (ages 15 to 64).

Information retrieved from these four sources has been assembled to obtain an unbalanced panel of 110 wireless operators observed over 25.2 quarters on average, from the first quarter of 2005 to the last quarter of 2012. The final sample is made of 2770 observations. Most of the missing observations are concentrated in the earlier periods. They are due to smaller sized wireless operators not disclosing their accounting figures as they are not listed on the stock market. Very few are due to entry or exit. Table 3 in the appendix provides more details about the sources of data used in this paper.

Variables

The definitions of the main variables follow: Dependent variables: Investment in network infrastructure Investment in network infrastructure is measured using two variables: the logarithm of CAPEX, and the CAPEX per subscriber. The CAPEX includes expenditures on the acquisition of tangible assets, maintenance of the existing tangible assets with a useful lifetime that extends beyond the fiscal year, and license fees. It also includes spectrum license fees. A dummy variable will be included in the econometric model to account for license fees.

Policy variables: competition intensity

Competition intensity is calculated on the basis of the Lerner index of monopoly power at the firm level.11 It can be expressed as:

Θ i = 1 - P i -c i P i
Θ i denotes the competition intensity experienced by firm i. Calculating competition intensity requires information on price (P i ) and marginal cost (c i ) per minute of voice and per megabyte of wireless data. However, these variables are not observed. Under the assumptions of homogenous good and constant marginal cost of production, the ratio of a firm's operating profit to its total revenue is a valid approximation of the Lerner index of market power.12 Therefore, the competition intensity is calculated as:

Θ i = 1 - Ebitda i Revenue i
Ebitda i , the operating profit, is measured using earnings before interest, tax, depreciation and amortisation (EBITDA). It is the difference between total revenues and operating expenditures, excluding portions arising through tax, interest payment, depreciation and amortisation of assets. Revenue i is the sum of revenue generated through the provision of wireless communications services and the revenue generated from the sales of mobile devices or other equipment sold by the operator during a quarter. Revenue from the provision of wireless communications services includes revenue from voice and data services over the wireless networks, roaming charges, revenue from international direct dialling and interconnection revenue.

Instruments and control variables:

As further discussed in section 5.3 on identification strategy, the lagged values of competition intensity and the year of entry into the market are the two instruments for competition intensity.

The estimation controls for potential market size and network deployment costs using respectively the logarithm of working-age population and population density as proxies. The number of subscribers is used in the robustness analysis as an alternative proxy for market size as it is potentially endogenous. For simplicity, we refer to the working-age population as a proxy for market size in the remainder of the paper.

The penetration rate of fixed telephony is used as a proxy for competition from fixed lines markets. The penetration rate of fixed telephony is more exogenous than the penetration rate of fixed broadband, a more recent technology whose diffusion may be influenced by investment in wireless networks. The estimation also controls for firms' characteristics by building a dummy variable characterising a wireless operator which hosts MVNO. Additional controls include market and quarter dummies as well as a dummy variable for wireless operators that purchase a frequency license during a given year. This latter is intended to retrieve spectrum license fees from total capital expenditures.

Empirical methodology

Given that the theoretical literature does not provide any clear prediction for the relationship between competition and investment in wireless networks, this paper starts by assuming a flexible relationship as expressed in the following equation:

Y iq = f (Θ iq ) + ν iq (1)
Where Y iq denotes investment by firm i during quarter q. Θ iq denotes the intensity of competition experienced by firm i during quarter q. f (.) is a function that characterises the relationship between competition and investment.

The estimation of the relationship between competition and investment proceeds in two steps. In the first step, the nature of the relationship between competition and investment is identified using a non-parametric identification strategy. This strategy does not estimate the parameters of the function f , but it is useful for identifying the best functional specification of an econometric model without making any a priori assumption about the relationship between the variables of the model. In the second step, parameters of the function f have been estimated using instrumental variable strategy.

Non-parametric identification of the relationship between competition and investment

Market size is an exogenous determinant of demand for wireless services and, therefore, of operators' investment. As a result, it is included in equation ( 1). The new equation writes:

Y iq = f (Θ iq ) + αS jq + µ iq (2)
S jq denotes the size of market j and µ iq corresponds to the residuals of the model. Under the assumption that f is linear in its parameters and using the Frish-Waugh Theorem [START_REF] Frisch | Partial time regressions as compared with individual trends[END_REF], equation ( 2) can be written as:

Y r iq = f (Θ r iq ) + µ iq (3)
Where Y r iq and Θ r iq are the residuals of the Ordinary Least Squares (OLS) regression of investment and competition on market size. The locally weighted scatterplot smoothing (Lowess), a non-parametric algorithm proposed by [START_REF] Cleveland | Robust locally weighted regression and smoothing scatterplots[END_REF], is implemented on the set of data points (Y r k , Θ r k ) to determine the nature of f . The integer k is an index that ranks the observations on competition intensity in ascending order.

For a given competition intensity Θ r k , let's define a bandwidth b around this point. This bandwidth determines a subset of pairs (

Y r l , Θ r l ) such that Θ r k-b 2 ≤ Θ r l ≤ Θ r k+ b 2 .
The corresponding Lowess smoother of the investment Y r k is the predicted value of the following weighted OLS regression:

Y r l = β * w l * Θ r l + µ l (4) l ∈ [k -b 2 ; k + b 2 ].
w l is a weight attached to the observations indexed by l. Several kernel weighting functions can be used. The non-parametric smoothing relies in particular on the tricube weighting function, a robust kernel widely used in the literature on nonparametric modelling. 13The Lowess smoother of investment associated with the intensity of competition Θ r k is determined as:

Ŷ r k = β * w l * Θ r k
The same procedure is replicated for all k, that is, for all observations of the intensity of competition. The Lowess smoother is a graphical representation of the set of points ( Ŷ r k , Θ r k ). Fig. 1 below presents the Lowess smoother. It highlights an inverted-U relationship between competition and investment.

Econometric model

The non-parametric relationship between competition and investment identified from the section 5.1 lends support to a quadratic specification of the econometric model. However, the quadratic form should not be interpreted as a causal relationship due to three main issues: speed of adjustment of investment in wireless networks, omitted variables that jointly determine both competition and investment,

K(u) = 70 81 1 -u 3 ) 3 (5)
For all u such that |u| ≤ 1, u = l -k.

and finally reverse causality from investment to competition. The econometric model will address these issues.

As discussed in section 3.2, the strong rate of technological progress in wireless industries means that investment in new wireless technologies significantly reduces marginal cost of production. This dynamic feature of investment can be modelled by assuming that a wireless operator invests y q in quarter q in order to reduce marginal cost of production at quarter q + 1: c q+1 = c(y q ). As y q depends on current marginal cost of production, the previous assumption implies that y q = g(c q ) = h(y q-1 ). Therefore, a contemporaneous shock on investment has a lasting effect on the subsequent level of investment. This dynamic feature will be modelled as a first-order autoregressive process.

In addition, the econometric model controls for potential omitted variables characterising the demand for wireless services and the cost of network deployment. To guarantee the exogeneity of the controls variables, the model includes the logarithm of the working-age population and the population density as proxies for demand and cost side variables.14

More formally, the econometric model can be expressed as:

Y iq = γ 0 + γ 1 Θ iq + γ 2 Θ 2 iq + ρY iq-1 + γX iq + λX jq + µ q + µ i + µ iq (6)
This equation links investment by firm i during quarter q, Y iq , to a quadratic function of competition intensity Θ iq , controlling for operators' characteristics X iq and market characteristics X jq . Variables µ q , µ i and µ iq correspond to unobserved investment shocks that are respectively quarter-specific, operator-specific and quarterly idiosyncratic. In particular, µ iq corresponds to the residuals, assumed to be heteroskedastic and autocorrelated.

Given that the panel is unbalanced and that sufficient operator's level instrumental variables are not available, equation ( 6) cannot be identified with the operators' fixed effects µ i . The sample is an unbalanced panel with several observations missing for some operators. In addition, at least two instruments which vary across operators and quarters are necessary to identify the causal impact of competition on investment at the firm level. Without these instruments, the identification can still be made for a representative wireless operator in a given market. This requires the assumption that operators fixed effects are the same within markets. Under this assumption, operators fixed effects can be replaced by markets fixed effects denoted by µ j . The econometric model thus writes:

Y iq = γ 0 + γ 1 Θ iq + γ 2 Θ 2 iq + ρY iq-1 + γX iq + λX jq + µ q + µ j + µ iq (7)
Coefficients γ 1 and γ 2 characterise the short-run effect of competition on investment.

If the inverted-U relationship suggested by the non-parametric curve holds, these coefficients should be statistically significant and respectively positive and negative.

The short run effect of competition on investment depends on the initial level of competition intensity. It is expressed as:

SRE(Θ) = dE(Y /Θ, X i , X j ) dΘ = γ 1 + 2γ 2 Θ
The short run effect is positive as long as competition intensity Θ is lower than the thresholdγ 1 2γ 2 , and negative otherwise. This threshold corresponds to the investment maximising intensity of competition, denoted by Θ:

Θ = - γ 1 2γ 2
The standard error of Θ is estimated using the delta method. 15

Coefficient ρ characterises the speed of adjustment of investment. It should be comprised between 0 and 1, reflecting the convergence of the long run effect of competition on investment. The long run effect of competition on investment is determined by dividing the short run effect by 1 -ρ: 16

LRE(Θ) = 1 1 -ρ γ 1 + 2γ 2 Θ = 1 1 -ρ SRE(Θ)
The long run effect of competition on investment is larger than the short run effect 15 The delta method is a way to approximate the variance of a non-linear estimator. It basically linearises the estimator using the Taylor's expansion around the mean of the estimate, and then calculate the variance. See [START_REF] Oehlert | A note on the delta method[END_REF] for a detailed presentation of the delta method. An estimation algorithm is developed in Stata by [START_REF] Hole | WTP: Stata module to estimate the confidence intervals for willingness to pay measures[END_REF]. 16 The long run effect is determined by summing all the contemporaneous effects over an infinite period. Typically, the impact of competition on investment is SRE(Θ) at date t, ρSRE(Θ) at date t + 1, ρ 2 SRE(Θ) at date t + 2, and so on. Thus, the long run effect is:

∞ k=0 ρ k SRE(Θ) = 1 1-ρ SRE(Θ).
as long as 0 < ρ < 1, that is, 1 1-ρ > 1. The magnitude of this difference increases with the autocorrelation parameter ρ. The investment maximising intensity of competition Θ remains the same in the long run.

OLS estimates of coefficients γ 1 and γ 2 would suffer from a simultaneity bias if both investment and competition are determined simultaneously.17 Therefore, competition intensity and its square are potentially endogenous. Typically, more efficient firms, which invest more than their rivals, will have higher price-cost margins and thus experience lower competition intensity. In this case, OLS estimate would suggest a downward sloping relationship between competition and investment. As a result, the investment maximising intensity of competition will be underestimated. Another issue is that in a dynamic model, the lagged dependent variable Y iq-1 is endogenous. Therefore, OLS estimate of coefficient ρ will be biased.

Causal identification and estimation strategies

The discussion of the econometric model emphasises two potential sources of estimation bias. The first is related to the endogeneity of the intensity of competition, and the second is related to the endogeneity of the lagged dependent variable. This section presents the identification strategies that will be employed to overcome these biases.

The bias stemming from the endogeneity of the intensity of competition is mitigated using an instrumental variable estimation strategy. Due to the presence of the quadratic term of competition intensity, the estimation relies on two instruments, namely, the two-quarter lagged intensity of competition and the year of entry into the market. In the econometric literature, a lagged variable is a standard and natural instrument for a contemporaneous endogenous variable when the sample includes a time dimension. 18From a theoretical perspective, competition and investment in the wireless market can be viewed as a two-stage game in which a regulator first determines the intensity of competition through parameters such as the number of firms and the year of entry into the market. In the next stage, wireless operators invest and compete on price or quantity. In this framework, a firm's year of entry is given and only affects investment through the intensity of competition. As emphasised by [START_REF] Grajek | Regulation and investment in networks industries: Evidence from european telecoms[END_REF], regulatory decisions can be endogenous. For instance, a regulator can choose the year of entry in order to provide an incentive for investment. However, endogenous regulation of the year of entry should not hold in the wireless industry if the regulator does not have perfect information about the operators' production costs. In addition, two wireless operators may enter into an identical market at different dates due to the randomness of administrative processes.

Regarding the endogeneity of the lagged dependent variable (Y iq-1 ), [START_REF] Bond | Dynamic panel data models: A guide to micro data methods and practice[END_REF] reviews the literature on dynamic panel models and suggests estimating model ( 7) in first-difference using the lags of the dependent variables as instruments. However, the unbalanced nature of the panel data does not allow this identification strategy. [START_REF] Nickell | Biases in dynamic models with fixed effects[END_REF] shows that the coefficient ρ is underestimated by OLS estimation of model ( 7) including the market-specific fixed effects (FE). On the contrary, ρ is overestimated when model ( 7) is estimated by OLS without the market-specific fixed effects (Pooled). Therefore, fixed effects and pooled estimates provide lower and upper bounds to the true coefficients. The paper provides both estimates using two and three-quarters lagged variables (Y iq-2 and Y iq-3 ) as instruments for Y iq-1 .

The econometric model ( 7) is estimated using two definitions of investment, namely, the natural logarithm of capital expenditures and capital expenditures per subscriber. Using the natural logarithm has the advantage of providing a better fit and more precise estimates as the variance of the residuals is smaller with the logarithm. 19 Regarding the capital expenditures per subscriber, it accounts for the size of the operator. Typically, an operator's investment may change with competition due to a net loss or gain of customers.

The model is estimated using both OLS and the Generalized Method of Moment (IV-GMM) estimators. 20 All estimations correct for arbitrary heteroskedasticity and autocorrelation using the Bartlett Kernel of bandwidth 4 in order to maximise the efficiency of the estimates. 21 The Kelibergen-Paap Wald rank F-statistics have 19 The R-squared of the model is greater with logarithm than without: 0.87 vs. 0.85 respectively. 20 The GMM estimator is better than the two-stage least square estimator in correcting for the heteroskedasticity and autocorrelation of the residuals. See [START_REF] Hansen | Large sample properties of Generalized Method of Moments estimators[END_REF] for the asymptotic properties of the GMM estimator.

21 The Bartlett Kernel is a weighting function used to estimate the asymptotic covariance matrix of the moment conditions. The bandwidth minus one represents the number of time lags used to compute the asymptotic covariance matrix. A bandwidth of 4 corresponds to three lags used been compared to the critical values tabulated by [START_REF] Stock | Testing for weak instruments in linear IV regression[END_REF] to test the weakness of the instruments. Moreover, with more instruments than endogenous variables, the Hansen J test of over-identification is implemented in order to check the validity of the exclusion restriction assumptions, that is, the exogeneity of the instruments. 22 6 Results

Main findings: An inverted-U relationship between competition and investment

Table 2 below presents three outcomes of the estimation of the relationship between competition and investment in wireless networks. The first two outcomes are respectively the OLS and IV-GMM estimates of model ( 7) using the logarithm of capital expenditures as a measure of investment. The third outcome relies on IV-GMM estimator but rather uses capital expenditures per subscriber as a measure of investment. For each outcome, table 2 presents both the results of the econometric model with and without market-fixed effects (FE and Pooled respectively). Two main findings emerge from this table.

First, there is an inverted-U relationship between competition and investment in wireless networks. This relationship holds irrespective of the measure of investment, the estimators (OLS or IV-GMM) and the econometric model (Pooled or Fixed effects). It provides a statistical support to the non-parametric finding of an inverted-U shaped relationship. The coefficients of the two competition variables have the expected signs and are significant at 5 or 10 per cent level. The investment maximising intensity of competition Θ is lower for the OLS estimates such that the relationship is almost downward sloping. This suggests that an OLS estimator underestimates the level of competition intensity that maximises investment, due to reverse causality. As discussed in section 5.3, reverse causality may be explained by more efficient firms investing more and experiencing less competition. This may explain why [START_REF] Kang | Competition and mobile network investment in China's telecommunications industry[END_REF], who do not correct for endogeneity, find a negative in the estimation in order to correct for autocorrelation. See [START_REF] Baum | Enhanced routines for instrumental variables/Generalized Method of Moments estimation and testing[END_REF] for a detailed explanation of the GMM estimator along with the statistical corrections and tests. 22 The null hypothesis of the Hansen J test states that all instruments, taken together, are not correlated with the residuals. More formally, this assumption writes: E(Zµ) = 0, where Z is a matrix of instruments and µ denotes the residuals.

relationship between competition and investment in wireless networks.

Correcting for endogeneity yields an investment maximising intensity of competition between 57 and 65 per cent, depending on the measure of investment: 59 or 60 per cent when investment is measured by the logarithm of capital expenditures, and 61 or 63 per cent when investment is measured by capital expenditures per subscriber. Therefore, as long as the intensity of competition remains below 57 per cent, an increase in the intensity of competition raises investment. Above 65 per cent, a further increase in competition reduces investment.

Second, there is a significant long run effect of competition on investment. The coefficient of the lagged dependent variable is statistically significant in all regressions and comprised between 0 and 1 as expected. In comparison to the magnitude of the short run effect, the long run effect is 4 or 17 times larger, when investment is measured as the logarithm of capital expenditures, and 3 or 8 times larger, when investment is measured by capital expenditures per subscriber. 23As predicted by [START_REF] Nickell | Biases in dynamic models with fixed effects[END_REF], the speed of adjustment of investment tends to be overestimated by the pooled model, but underestimated by the fixed effects model. In addition, the investment maximising intensity of competition is much higher for the fixed effects model than for the pooled model. Therefore, a conservative approach would focus on the outcome of the fixed effects model. In consequence, the investment maximising intensity of competition stands at 60 or 63 percent depending on whether or not capital expenditures are normalised by the number of subscribers. Respectively, the magnitude of the long run effect is 3 or 4 times larger than the magnitude of the short run effect.

Regarding the controls variables, it turns out that investment is positively affected by market size, but negatively affected by the cost of network deployment. When market fixed effects are included in the regression, countries surfaces are fixed and a variation in density corresponds to a proportional variation in population. Therefore, working-age population is removed from specifications with market fixed effects. It turns out that the coefficient of population density is positive, which reflects a positive effect of market size on investment.

Competition from fixed line markets, as measured by the penetration rate of fixed telephony, is positively correlated with investment in wireless networks. However, this correlation is no longer significant when market fixed effects are included in the regression. This variable is not significant in the other regressions and has been removed to obtain more precise estimates.

The regression also controls for whether or not a wireless operator hosts an MVNO. The estimates suggest that wireless operators which host an MVNO invest more than their rivals. This superior investment reflects the additional expenditures needed to provide upstream network access to the downstream MVNO. The control for licenses fees is not significant as frequency bands are not so frequently released and variations in investment due to licence fees are subsumed into the residuals as measurement errors. Estimates of quarterly fixed effects not reported here suggest a yearly seasonality such that investment in wireless networks rises every year from the first to the last quarter.

Finally, all tests statistics support the significance of the model and estimates. The R-squared of the OLS estimates is larger than 85 per cent, suggesting that the econometric model accounts for most of the determinants of investment in wireless networks. The Ficher test confirms the overall significance of all regressions. Regarding the IV-GMM estimates, the under-identification test rejects the null hypothesis that the instruments are not correlated with the endogenous variables. In addition, the p-values of the Hansen's over-identification tests are all greater than 10 per cent, implying that the null hypothesis of exogeneity of the instruments cannot be rejected. Note: Significant at 1% (***), 5% (**) and 10% (*). Standard errors robust to arbitrary heteroskedasticity and autocorrelation are in parentheses. (1 -Lerner) represents the intensity of competition. Y _(t -1) refers to the one-quarter lagged of the dependent variable. CAP EX stands for capital expenditures. The instrumental variables estimates based on the generalised method of moments estimators (IV-GMM) use the two-quarters lagged intensity of competition, and the year of a firm's entry into the market a instruments for the contemporaneous intensity of competition. Two and three-quarters lagged of the dependent variable is used as instruments for the one-quarter lagged explanatory variable Y _(t -1). The Lagrangian Multiplier (LM) statistic of under-identification is greater than its critical value, meaning that the instruments are good at predicting the intensity of competition.

Robustness checks

This section presents the outcome of the first-stage instrumental variable estimation procedure. It also discusses the outcomes of the estimation when operators' fixed effects are substituted for market fixed effects, the performance of the model with linear specification, as well as the use of the number of subscribers as a proxy for market size.

Correlation of the instruments with the endogenous variables

The first three columns of table 5 in the appendix present the first-stage estimates of the main specification in column (4) of table 2. This specification corresponds to the IV-GMM estimation of the econometric model ( 7) with market fixed effects.

As expected, the two-quarter lagged intensity of competition as well as the year of entry into the market have positive and significant effect on competition intensity. The one-quarter lagged investment is significantly affected by all four instruments. It is negatively affected by the two instruments for competition and positively by the two and three-quarters lagged investment. These correlations are in line with the outcome of the under-identification test which rejects the hypothesis that the instruments are not significant determinants of the endogenous variables.

Introducing operators fixed effects

Column (4) of table 5 in the appendix displays the OLS estimate of the econometric model ( 6), that is, when operators fixed effects are included in the model. It turns out that the relationship between competition and investment is still inverted-U, but the estimates are no longer significant. The investment maximising intensity of competition stands at 63 per cent, closer to the level found with the market fixed effects. This paper does not address the endogeneity issue when operators fixed effects are included since this identification requires at least two instruments which vary across operators and quarters. Using two lagged variables of the intensity of competition as instruments fail to pass the under-identification, the weak instrument and the over-identification tests.

Linear specification of the relationship between competition and investment Columns ( 5) and (6) of table 5 in the appendix present respectively the OLS and IV estimates of the relationship between competition and investment using a linear specification. The OLS estimate yields a negative and significant relationship between competition and investment consistently with the outcome of the estimation with a quadratic specification. The investment maximising intensity of competition was found at a very low level of the intensity of competition, 40 per cent more precisely.

However, the effect of competition on investment is no longer significant when endogeneity is accounted for in column ( 6). The instruments are the two-quarter lagged intensity of competition and the two and three-quarters lagged investment. This result contrasts with the significant effect of competition that emerges when a quadratic specification is estimated. It provides a further support for the non-monotone and inverted-U shaped relationship suggested by the non-parametric curve.

Number of subscribers as a proxy for market size Column (7) of table 5 in the appendix presents the IV estimates of the relationship between competition and investment using the total number of subscribers as a proxy for market size in replacement of the working-age population. The inverted-U still holds, but it is no longer significant.

Discussion and policy implications

A number of insights provided by the specificity of the wireless industry lend support to an inverted-U shaped relationship between competition and investment. As indicated in the literature review, the inverted-U shaped relationship may stem from the combination of two effects: The escape-competition effect whereby competition increases investment and the efficiency effect whereby competition reduces investment. In the case of the wireless industry, the escape-competition effect is weaker with intense competition because operators have the opportunity to buy the same equipment from manufacturers at the same time.

In fact, innovation in wireless infrastructure is mainly designed by equipment manufacturers which provide their facilities to all rival operators at the same time. Opportunities to escape competition by investment are, therefore, temporary, in particular when competition is intense. When only one rival has the opportunity to invest, as highlighted by [START_REF] Arrow | Economic welfare and the allocation of ressources for invention. Nber chapters[END_REF] in the case of a symmetric duopoly, investment allows it to gain a longer lasting monopoly power and thus escape competition. However, if both firms invest, neither of them durably succeeds in escaping competition.

While the theoretical framework provided by [START_REF] Schmutzler | Competition and investment -a unified approach[END_REF] is consistent with the wireless industry, as it allows firms to invest simultaneously, the examples which illustrate this framework do not show an inverted-U relationship. This may be because the framework only focuses on cases where profit (gross profit minus investment) is positive. In the case of a symmetric duopoly, firms invest expecting to increase their profit. They choose the amount of investment that maximises their profit. Both firms invest the same amount because of the symmetry and therefore cannot escape competition nor increase their profit. It is possible that their gross profit is lower than their profit maximising investment. In that case, they cannot invest the profit maximising amount because they do not have it and cannot expect to earn it. Then firms change their behaviour by limiting their investments so that their profit is nil [START_REF] Jeanjean | Incentives to invest in improving quality in the telecommunications industry[END_REF]. In this case, investments, like gross profits, tend to decrease with competition.

Whatever the shape of the relationship for low levels of competition, investment decreases since competition intensity is higher than the level at which profits are nil. As a result, when the relationship is increasing for lower intensity of competition it becomes decreasing for higher intensity. 24 The magnitude of the impact of investment on quality or cost reduction increases the probability that this change in firms' behaviour will occur. Appendix A.4 shows, using Schmutzler's examples, that a decrease in the cost of investment tends to reduce the threshold of competition intensity at which profit is nil. As a result, increasing competition beyond this threshold pushes operators to reduce their investment to offset the decline in gross profit. As highlighted in section 3, the wireless industry is characterised by a very high rate of technological progress. This means that the cost of each additional megabyte is decreasing and the impact of investment on cost reduction is correspondingly very significant. It is, therefore, not surprising that competition has a negative impact on investment when competition is very intense.

The inverted-U relationship suggests that policymakers should consider wireless operators' profit margins, that is, the ratio of EBITDA to revenue, before allowing a new market entrant or merger in the industry. The inverted-U shaped means that there is an intermediate level of profit margin that maximises investment. This does not imply that welfare is maximised at the same level of competition.

However, in the wireless industry, investment drives traffic and the fall in megabyte prices [START_REF] Jeanjean | What causes the megabyte price drop in the mobile industry[END_REF]. Strong technical progress amplifies the impact of investment on quality as measured by the bit rate offered to consumers. Investment is decisive for traffic growth and megabyte prices drop for consumers. The socially optimal intensity of competition should then be close to the level which maximises investment. Targeting maximum investment thus appears to be a reasonable option for policymakers.

Another issue is how policymakers should target this level. When profit margins are below the lower bound of the confidence interval (35 per cent), a merger is actually more desirable than a new entrant. On the other hand, when profit margins are well above the upper bound of the confidence interval (42 per cent), a new entrant may generate greater incentives to invest than would a merger. When profit margins are between these two thresholds, then merger analysis has to be done on a case-bycase basis. Alternatively, policymakers can adjust profit margin levels by allowing the entry of an MVNO or lowering consumers' switching costs (number portability, changes to the commitment period, etc.).

Summary and Conclusion

This paper uses firm level data and an instrumental variable estimation strategy to identify the relationship between competition and investment in the wireless industry. It finds a robust inverted-U relationship between competition intensity and investment, where competition is measured on the basis of the Lerner index and investment by the logarithm of capital expenditures or the capital expenditures per subscriber. The intensity of competition which maximises investment in wireless networks stands at 60 or 63 per cent when investment is respectively measured as capital expenditures or capital expenditures per subscriber. In addition, the estimates suggest a significant long run effect of competition on investment which amplifies the short run effect by a factor of 3 to 4.

These findings provide empirical evidence on the impact of competition on investment in the wireless industry. They complement the findings of [START_REF] Garrone | Seeking the links between competition and telecommunications investments[END_REF] by focusing specifically on investment in the wireless networks rather than combining investment in fixed and mobile networks. Furthermore, the results of this paper extend those of [START_REF] Kang | Competition and mobile network investment in China's telecommunications industry[END_REF] by providing a causal estimate of the impact of competition on investment. It also investigates the long run effect of competition on investment. In addition, the geographical scope of this paper is much broader, encompassing cases where investment rises and falls with competition.

The findings of this paper suggest that policymakers should consider wireless operators' profit margin levels before allowing a new entrant or a merger in the wireless industry. When the profit margin is below 35 per cent, a merger may be a better way to raise social surplus than a new entry. On the other hand, when it is above 42 per cent, an additional entrant may increase social surplus more than a merger due to higher incentives to invest. Between these two thresholds, a case-by-case analysis of merger proposals is recommended.

The results of this paper should to be interpreted with caution as they apply to a representative wireless operator in a given market. This is due to the assumption behind the inclusion of market fixed effects instead of operators fixed effects in the econometric model. Further investigations need to identify relevant instrumental variable or identification strategy to account for heterogeneity across wireless operators within a given market. Another limitation is the definition of the dependent variable. A better dependent variable which is much more correlated with consumer surplus is the quality of wireless services. To the extent that more investment in wireless networks does not necessarily translate into higher quality of services, fu-ture work needs to examine the relationship between competition and quality in the wireless industry.

Regarding the main explanatory variable, the analysis focuses on the intensity of competition measured on the basis of the Lerner index. However, a much more direct way to influence the intensity of competition is through the market structure and particularly the number of wireless operators. Future research will address how the market structure affects investment incentives and quality and how it depends on asymmetry between firms. Finally, the analysis implicitly assumes that wireless operators compete in the upstream market by investing in their own network. However, network sharing is becoming prevalent. Future works should control for network sharing and investigate its effects on the quality of wireless services.

A Appendix

A.1 Description of the dataset (3) and (4) in table 2). Specification ( 4) is an OLS estimate of the econometric model with operator fixed effects. Specifications ( 5) and ( 6) are respectively OLS and IV-GMM estimates of the main econometric model, but with linear specification. Specification (7) replaces the working-age population by the number of subscribers as a proxy for market size.

A.4 Replication of the findings by [START_REF] Schmutzler | Competition and investment -a unified approach[END_REF] Consider a duopoly in which firm i incurs a marginal cost c i that depends on the investment that rises the efficiency by y i = c 0 i -c i . The corresponding cost of investment is I i = ay 2 i . a is a parameter proportional to the cost of investment. y 0 i is the initial efficiency of firm i when y i = 0. Firm i chooses investment y i in order to maximize π i (y i , y j , θ). Maximum profit is written for j = i: π * i (y j , θ) = max(π i (y i , y j , θ) -ay 2 i )

Example E5 (Competition à la Hotelling: θ = 1 -t)

π i = [(y 0 j + y j ) -(y 0 i + y i ) -3(1 -θ)] 2 18(1 -θ)
In symmetric duopoly y * i = y * j = 1 6a and π * i (y * i , y * j , θ) = 1-θ 2 Condition of positive profit: π * i (y * i , y * j , θ) -ay 2 i > 0 yields θ < 1 -1 18a

The value above which profit is zero is then θ = 1 -1 18a θ is increasing in a.

This property is checked for all the examples proposed by Schmutzler in symmetric duopoly. p i is the price set by firm i and q i the output of firm i.

Example E1: Price competition, demand p i = 1 -q i -θq j

Example E2: Quantity competition, demand p i = 1 -q i -θq j

Example E3: Price competition, demand p i = 1 -1 1+θ q i -θ 1+θ θq j

Example E4: Quantity competition, demand p i = 1 -1 1+θ q i -θ 1+θ θq j

Example E5 is simple, the other examples require more tedious calculations: Fig. 3 presents the increasing relationship between θ and a for all the examples.

  Others have cleared proposed mergers particularly in the European Union where recent cases involve Orange and 3G in Austria in 2012, E-Plus and Telefonica in Germany in 2014, Orange and T-Mobile in The Netherlands in 2007, O 2 Ireland and H3G in Ireland in 2014. Meanwhile, some proposed mergers have been challenged like in the case of TeliaSonera and Telenor in Denmark in 2015, and AT&T and T-Mobile in the United States of America (USA) in 2011.

Figure 1 :

 1 Figure 1: Non-parametric curve between competition and investment

Figure 2 :

 2 Figure 2: Non-parametric curve between competition and investment

Note:

  Significant at 1% (***), 5% (**) and 10% (*). Standard errors robust to arbitrary heteroskedasticity and autocorrelation are in parentheses. Specifications (1) to (3) correspond to the first-stage of the instrumental variable estimates of the main econometric model (specifications

Figure 3 :

 3 Figure 3: The relationship between the cost of investment and positive profit bound of competition

Table 1

 1 below displays summary statistics of the main variables. The list of countries is provided in table 4 in the appendix.

Table 1 :

 1 Summary statistics

	Variable	Observations	Mean Std. Deviation	Min	Max
	Log of CAPEX	2770	3.94	1.51	-1.11	7.62
	Capex per subscriber	2770	10.11	7.63	0.15	39.73
	1-Lerner	2770	0.59	0.12	0.19	1.00
	Log. Working-age Pop.	2770	16.53	1.43	12.93	19.16
	Log. Pop. Density	2770	4.47	1.33	0.99	8.93
	Household penetration of
	fixed telephony	2770	31.94	18.64	0.25	69.51
	Host an MVNO	2770	0.65	0.48	0	1
	Year of entry	2770	1995.1	5.08	1979	2005
	Purchase a license	2770	0.16	0.37	0	1

Table 2 :

 2 The impact of competition on investment in wireless networks

	Dependent variable:	Logarithm of CAPEX (investment)	Capex per subscriber
	Estimator:	OLS		IV-GMM	IV-GMM
	Model	Pooled	FE	Pooled	FE	Pooled	FE
	Specification	(1)	(2)	(3)	(4)	(5)	(6)
	1-Lerner	0.86	0.95	5.02*	19.13**	89.77**	161.27*
		(0.67)	(0.75)	(3.04)	(9.23)	(37.11)	(93.18)
	(1-Lerner)^2	-1.08*	-1.19*	-4.22* -15.88**	-73.77**	-128.30*
		(0.57)	(0.63)	(2.56)	(7.51)	(31.33)	(75.68)
	Y_(t-1)	0.63*** 0.38***	0.94***	0.75***	0.88***	0.72***
		(0.02)	(0.03)	(0.02)	(0.06)	(0.03)	(0.06)
	Log of working-age population	0.33***		0.05***		0.09
		(0.02)		(0.02)		(0.07)
	Log of population density	-0.02** 2.13***	-0.00	1.05*	-0.03	-0.54
		(0.01)	(0.50)	(0.01)	(0.57)	(0.07)	(5.41)
	Fixed telephony penetration	0.01***	0.00
		(0.00)	(0.00)
	Hosting MVNO	0.09*** 0.17***	-0.02	0.09*	-0.39	0.21
		(0.03)	(0.06)	(0.02)	(0.05)	(0.30)	(0.38)
	Having purchased a license	-0.00	0.01	-0.00	0.01	-0.12	-0.11
		(0.03)	(0.03)	(0.03)	(0.03)	(0.40)	(0.41)
	Quarter fixed effects
	Market fixed effects
	Constant	-4.01***	-4.44**	-1.67	-7.45***	-23.69**	-36.91
		(0.37)	(1.76)	(1.03)	(2.81)	(11.15)	(26.96)
	Observations	2,770	2,770	2,586	2,586	2,595	2,595
	R-squared	0.864	0.887
	F-Statistic	519.9	248.1	1029.6	323.57	88.97	53.31
	Investment Maximizing Intensity of Competition: (%)
	Lower bound	19	18	57	58	59	60
	Estimate	40	40	59	60	61	63
	Upper bound	61	62	62	63	63	65
	Under-identification test (H0: the instruments do not significantly affect the intensity of competition)
	LM statistic			30.4	10.5	40.1	15.0
	Weak instruments test (H0: instruments are weak)
	Wald F statistic			18.0	3.64	21.1	4.92
	J-Hansen overidentification test (p-value)		0.24	0.33	0.95	0.96

Table 3 :

 3 List of variables used in the estimations

	Original Variables	Constructed Variables	Unit of observation	Time dimension	Data Source
	Total and service revenues	Revenue	Firm level	quarterly	WCIS, Ovum
	Earning Before Interest, Tax,
	Depreciation and Amortization	Profit	Firm level	quarterly	WCIS, Ovum
	(Ebitda)
	Subscribers	Capex per subscriber	Firm level	quarterly	WCIS, Ovum
	Capital expenditures including license fees	Investment	Firm level	quarterly	WCIS, Ovum
	Purchase of licenses	Dummy for license purchase	Firm level		WCIS, Ovum
	Household penetration of fixed telephony		Market level	yearly	ITU
	Year of entry	Year of entry into the market	Firm level		Wireless Intelligence, GSMA
	Mobile Virtual Network Operators (MVNO)	Hosting an MVNO	Firm level		Wireless Intelligence, GSMA
	Date of launch of the mobile network		Firm level		Wireless Intelligence, GSMA
	Population density	Proxy for cost of network deployment	Market level	yearly	World Development Indicator, World Bank Group
	Population between 15 and 64 years old	Market size	Market level	yearly	World Development Indicator, World Bank Group

WCIS: World cellular Information Services, WBIS: World Broadband Information Services

Table 4 :

 4 List of countries within the regions Belarus, Bulgaria, Croatia, Czech Republic, Estonia, Hungary, Kazakhstan, Latvia, Lithuania, Montenegro, Poland, Russia, Serbia, Slovenia, Ukraine, Uzbekistan Western Europe Austria, Belgium, Denmark, Finland, France, Germany, Greece, Ireland, Israel, Italy, Netherlands, Norway, Portugal, Spain, Sweden, Switzerland, Turkey, United Kingdom

	Region	Countries
	Africa	Algeria, Cameroon, Egypt, Ghana, Kenya, Lesotho, Nigeria, South Africa, Sudan, Tanzania,
		Tunisia
	Latin America	Argentina, Brazil, Chile, Ecuador, Mexico, Uruguay, Venezuela
	Asia Pacific	Australia, Bangladesh, Hong Kong, Indonesia, Japan, Korea, Malaysia, Pakistan, Singapore,
		Thailand
	Eastern Europe Armenia, Middle East Iraq, Jordan, Kuwait, Syria
	USA/Canada	Canada, USA

A.2 Local polynomial smoothing of the relationship between competition and investment

Table 5 :

 5 Robustness checks results

			(1-	logcapex_
		1-Lerner	Lerner)^2	(t-1)	Log. of capital expenditures
		(1)	(2)	(3)	(4)	(5)	(6)	(7)
	1-Lerner				0.332	-0.466***	-0.282	3.900
					(0.817)	(0.127)	(0.229)	(2.668)
	(1-Lerner)^2				-0.316			-3.301
					(0.694)			(2.253)
	(1-Lerner)_(t-2)	0.514***	0.625***	-0.453***
		(0.027)	(0.034)	(0.138)
	Year of entry	0.004***	0.005***	-0.015***
		(0.001)	(0.001)	(0.004)
	logcapex_(t-1)				0.316***	0.382*** 0.792*** 0.943***
					(0.029)	(0.029)	(0.046)	(0.017)
	logcapex_(t-2)	-0.003	-0.004	0.307***
		(0.003)	(0.003)	(0.026)
	logcapex_(t-3)	-0.003	-0.004	0.171***
		(0.002)	(0.003)	(0.026)
	Log of working-age
	population				1.390***
					(0.401)
	Log of subscribers							0.051***
								(0.018)
	Log of population
	density	0.150**	0.165**	1.631***	0.173***	2.146***	1.269***	-0.004
		(0.073)	(0.080)	(0.502)	(0.053)	(0.504)	(0.425)	(0.006)
	Hosting MVNO	-0.001	0.001	0.169***	-1.146	0.171***	0.056	-0.031
		(0.008)	(0.010)	(0.051)	(1.204)	(0.056)	(0.038)	(0.023)
	Having purchased a
	license	0.003	0.004	0.050*	-0.003	0.010	-0.004	0.001
		(0.004)	(0.005)	(0.028)	(0.030)	(0.031)	(0.026)	(0.027)
	Quarter fixed effects
	Market fixed effects
	Operator fixed
	effects
	Constant	-7.305*** -10.257*** 26.778***	-20.739***	-4.069**	-2.948**	-0.667
		(1.084)	(1.410)	(8.047)	(6.650)	(1.734)	(1.400)	(0.772)
	Observations	2,586	2,586	2,586	2,770	2,770	2,586	2,586
	R-squared	0.720	0.713	0.890	0.893	0.887
	F-statistics	73.04	70.65	282.30	283.18	250.06	550.27	1053.17

See Schumpeter (1942) and[START_REF] Arrow | Economic welfare and the allocation of ressources for invention. Nber chapters[END_REF].

See Garrone and Zaccagnino (2015) who find no correlation between competition and investment in fixed and mobile networks, and[START_REF] Kang | Competition and mobile network investment in China's telecommunications industry[END_REF] who find a positive correlation between market concentration and investment in the Chinese wireless market.

Note that some papers like[START_REF] Tang | Competition and innovation behaviour[END_REF] argue that firms' perceptions of the intensity of competition are also important determinants of innovation.

 4 In practice, wireless services based on new technologies are often labelled as a product innovation. This is true from consumers' perspective as they purchase greater allowance at the same

Annex A.4 provides the threshold above which firms underinvest for different cases studied by[START_REF] Schmutzler | Competition and investment -a unified approach[END_REF].

In the automobile industry,[START_REF] Hashmi | Market structure and innovation: A dynamic analysis of the global automobile industry[END_REF] finds a weakly positive relationship between a firm's innovation and its price-cost margin and no relationship at the industry level. In the railways industry,[START_REF] Coublucq | Static-dynamic efficiency trade-off in an open access policy: Application to the US rail freight industry[END_REF] finds that competition lowers investment in the network infrastructure.

With 50% exponential rate of progress, download speeds double every 1.4 year ( ln(2) 0.5 ).

This ratio is generally used in empirical papers on competition. See for instance[START_REF] Aghion | Competition and innovation: An inverted-u relationship[END_REF] in a general equilibrium setting and[START_REF] Hashmi | Market structure and innovation: A dynamic analysis of the global automobile industry[END_REF] in the automobile industry.

Another issue analysed by[START_REF] Boone | A new way to measure competition[END_REF] is whether the Lerner index actually measures the intensity of competition. His analyses emphasise that the Lerner index may not be decreasing with the intensity of competition. He proposes the elasticity of profit with respect to marginal cost as an alternative and much more robust way of measuring competition. Empirically, this elasticity can be estimated by regressing the logarithm of profit over the logarithm of cost across the firms within a given market. However, with less than five wireless operators per market, this elasticity cannot be estimated.

The tricube weighting function is defined as follows:

Potential controls such as number of subscribers, penetration rate of smartphones and operators' cash flow are typically endogenous.

The bias remains in a two-stage model where investment is chosen first, followed by a choice of output or price. In this setting, the first-stage investment has an impact on the intensity of competition through the price and the marginal cost.

See Bond (2002) for example. This paper does not use the one-quarter lagged intensity of competition because the intensity of competition does not change significantly from one quarter to the next. As such, a one-quarter lagged intensity of competition would still affect directly investment. In addition, it is simultaneously determined with the one-quarter lagged investment which is an explanatory variable in the dynamic model.

The magnitude of the difference between the short and long run effects is calculated as: 1 1-ρ , where ρ is the first-order autoregressive coefficient.

Examples E3 and E4 in Schmutzler (2013).


PhD Annabelle Couillard

PhD Emilie Tremey

Prof. Christian Prefaut

Prof. Alain Varray

PhD Nelly Heraud
email: nelly.heraud@5-sante.fr

Keywords: Chronic Obstructive Pulmonary Disease, Ventilatory threshold, 6MWT, Breathlessness, Exercise heart rate

may come

The self-perception of dyspnoea threshold during the 6-min walk test

INTRODUCTION

Exercise training is an essential component of pulmonary respiratory rehabilitation for chronic obstructive pulmonary disease (COPD) patients. The efficiency of exercise training programs depends on three parameters: the number of sessions, the duration or frequency of sessions and the exercise intensity. According to the recommendations, exercise intensity could be defined as a percentage of the maximal aerobic power (50 to 80%) or the target heart rate (HR) equivalent to the ventilatory threshold (V th ) obtained during cardiopulmonary exercise testing (CPET) (American [START_REF] Thoracic | ATS/ACCP Statement on cardiopulmonary exercise testing[END_REF][START_REF] Vallet | Comparison of two training programmes in chronic airway limitation patients: standardized versus individualized protocols[END_REF]. The interest of individualization of exercise training at ventilatory threshold is to allow for patients to increase effort tolerance while ensuring better ventilatory comfort [START_REF] Vallet | Comparison of two training programmes in chronic airway limitation patients: standardized versus individualized protocols[END_REF].

CPET is defined as the gold standard to determine ventilatory threshold. However, when CPET is not accessible or feasible, it is difficult to individualised exercise training for all patients and/or follow their evolution when they came back to home.

An alternative to the prescription of V th might be the use of a subjective parameter. Indeed, [START_REF] Nelson | Physical activity and public health in older adults: recommendation from the American College of Sports Medicine and the American Heart Association[END_REF] offers an original approach of prescribing exercise intensity useable by all subjects. The interest of this approach is to replace the intensity defined by V th during CPET by a self-assessment of the difficulty of exercise [START_REF] Nelson | Physical activity and public health in older adults: recommendation from the American College of Sports Medicine and the American Heart Association[END_REF]. In COPD, given that dyspnoea is the major criterion of limiting effort and that is defined as a subjective breathing discomfort felt by patients, the use of dyspnoea threshold could be an alternative to V th . Indeed, V th correspond to the highest exercise intensity before an exponential increase of ventilation. On the other hand, dyspnoea correspond to a disharmony between pulmonary effectors and control of respiratory centres, constrain patient to increase their ventilatory level. In that case, it can be quite simple to say that the perception of that moment by the patient should be fairly spontaneous and precise.

Furthermore, it would be interesting to rely on this subjective criterion to a field test close to daily life activity. Currently, the six-minute walk test (6MWT) is recommended in COPD to assess exercise tolerance, diagnose exercise-induced desaturation, prescribe oxygen therapy for ambulation, and quantify therapeutic effects [START_REF] Spruit | An official American Thoracic Society/European Respiratory Society statement: key concepts and advances in pulmonary rehabilitation[END_REF]. It is a simple and reliable field test. In addition, it gives a more accurate vision of the individual's functional capacity to carry out daily living activities.

We hypothesized that patients with COPD would be able to identify during 6MWT the critical threshold when breathing becomes uncomfortable and that the HR corresponding to this self-perceived dyspnoea threshold (SDT6) could be used to estimate ventilatory threshold.

The objective of this study was therefore to determine the feasibility and reproducibility of SDT6 and to determine if a useful equation to estimate ventilatory threshold from SDT6 could be derived.

METHODS

Subjects

Eighty-two patients with COPD admitted to a pulmonary rehabilitation centre (Clinique du Souffle® La Vallonie, Lodève, France) were included in this study (Figure 1). COPD was diagnosed according to standard criteria, and post-bronchodilator forced expiratory volume in one second (FEV 1 ) was used to distinguish the stage of bronchial obstruction severity [START_REF] Vestbo | The 2011 revision of the global strategy for the diagnosis, management and prevention of COPD (GOLD)--why and what?[END_REF].

We excluded unstable patients (exacerbation < 4 weeks) and/or those with restrictive or mixed respiratory syndrome, angina, recent heart attack (< 1 month) or progressive coronary pathology, and cognitive or motor problems that would significantly limit comprehension or SDT6 evaluation; patients taking beta-blockers were also excluded. This project received the approval of the Internal Ethics Committee (Research Committee, Fontalvie Group, France).

Study design

In order to screen COPD patients, anthropometric and clinical measures were collected. Respiratory function was evaluated by plethysmography (Body Box 5550, Medisoft, Belgium). Vital capacity (VC) and FEV 1 were collected and the Tiffenau ratio (FEV 1 /VC) was calculated. The normal European values were the references values [START_REF] Laszlo | European standards for lung function testing: 1993 update[END_REF]. The BODE index, a grading system predicting the risk of death and, was calculated [START_REF] Celli | The body-mass index, airflow obstruction, dyspnea, and exercise capacity index in chronic obstructive pulmonary disease[END_REF]. Once the inclusion criteria validated, patients performed a 6MWTs familiarization in order to avoid learning effect. 24h after, they performed a CPET and 24h after that they performed two experimental 6MWTs and the HR SDT6 was collected.

Evaluations:

The 6MWTs were performed indoors along an unobstructed corridor. Patients were instructed to cover as much distance as possible in six minutes. Standardized encouragement was given during the test. In order to minimize methodological bias related to the experimenter, the same clinician trained in this evaluation monitored the two 6MWTs for each patient and followed the current international recommendations [START_REF] Brooks | ATS statement on six-minute walk test[END_REF]. At least one hour separated the two 6MWTs. We ensured that HR values and dyspnoea at rest were identical for both 6MWTfor all patients. For each test, oxygen saturation and HR values were recorded every minute through a digital pulse oximeter (Nonin Onyx, Nonin Medical, Inc.Minneapolis, MN, USA).

CPET was conducted by an investigator blinded to the 6MWT. CPET was performed on an electronic ergocycle using an individualized and validated protocol. After a 3 min warm-up period pedalling corresponding to 20% of estimated power, the work rate was increased by 8% of estimated power every minute. The exercise were stopped when patients was unable to maintain the imposed pedalling rhythm. A 12 lead electrocardiogram was continuously monitored. Gas exchange and O 2 saturation were measured continuously. Two blinded and independent evaluators determined the first 6 V th from the changes in the O 2 and CO 2 curves using the V-Slope method of Sue et al. [START_REF] Sue | Impact of integrative cardiopulmonary exercise testing on clinical decision making[END_REF]. HR values corresponding to the first V th was noted. If HR Vth differed between the two evaluators (± 5 bpm), the mean of these two values determined V th .

Self-perceived Dyspnoea threshold during 6MWT (SDT6)

Before 6MWT, each patient was interviewed and the evaluator systematically asked three questions. Question 1: In your opinion, which daily physical activities typically leave you feeling breathless? Question 2: When you perform these activities do you recognize a critical moment where your breathlessness becomes uncomfortable? Question 3: In your own words, could you describe your sensations when this critical moment appears? As recommended by [START_REF] Williams | The language of breathlessness differentiates between patients with COPD and age-matched adults[END_REF][START_REF] Williams | The language of breathlessness differentiates between patients with COPD and age-matched adults[END_REF], the evaluator did not propose or prompt responses, but simply noted the patient's words and clarified when it was deemed necessary. In addition to the standard instructions relative to 6MWT, the evaluators gave all patients the following instructions: "You have described your sensation when breathing becomes uncomfortable in carrying out daily activities. This is a critical moment that you are able to identify. During the 6MWT, we ask you to simply raise your hand as soon as you feel that this critical moment is reached. All you have to do is raise your hand as you continue walking." During each 6MWT, the evaluator noted the corresponding HR. This moment, indicated by the patient's raised hand, was called the "self-perceived dyspnoea threshold of the 6minute walking test" or "SDT6."

Statistics

Statistical analysis was performed using STATISTICA Version 10.0. Data were expressed as mean (± SD). Normal Gaussian distributions of the data were verified by the Shapiro-Wilk test. In case of violation of normality hypothesis, nonparametric tests were used. The feasibility was analysed by descriptive statistics.

The reproducibility of HR SDT6 was analysed by a means comparison with a paired Student t test (or Wilcoxon signed rank test if normality was not verified), by calculating the variation coefficient and an intraclass correlation coefficient (ICC) for repeated measurements. In addition, Spearman's correlation and Bland & Altman plots were performed between the first and the second walk test.

The relationship between HR SDT6 and HR Vth was analysed by a means comparison with a paired Student t test.

A stepwise regression analysis was performed to obtain a predictive equation of HR Vth estimated from HR SDT6 . The HR Vth was used as variable to explain and a set of potential explanatory variables was used (age, height, weight, BMI, FEV 1 and HR SDT6 ) after verification of absence of collinearity between these variables.

The results were considered as statistically significant when p values were ≤ 0.05.

RESULTS

Baseline

Baseline subject characteristics are shown in Table 1. FEV 1 was 37.4% ± 16.4 with 5% at stage I of the GOLD standard, 18% at stage II, 42.5% at stage III and 34.5% at stage IV. On average, the BODE index was 4 ± 2, with 20% patients in quartile 1, 42% in quartile 2, 27% in quartile 3 and 11% in quartile 4.

Feasibility

Language of dyspnoea: All patients were able to describe their sensations when breathing became uncomfortable during activities of daily living. Three word categories were distinguished: 44% of patients described a feeling of "suffocation," 19% a feeling of "chest tightness," and 16% a feeling of "increased respiratory rate." In addition, 11% described at least two of these three feelings ("mixed") and 10% confused dyspnoea with general fatigue or discomfort due to phlegm ("other").

Identification of SDT6:

In the whole study population, 80% (n=66/82) of patients were able to recognize SDT6 during 6MWT and raised their hand when it occurred. Of these 66 patients, 91% (n=60) signalled a SDT6 during both 6MWTs and 9% (n=6) during one of the two 6MWT (Figure 1). For patients who did not identify a dyspnoea threshold, the results reveal a lesser increase of heart rate compared to patients with SDT6 (Table 1).

Reproducibility of SDT6

Reproducibility was assessed in 60 patients (Figure 1).

On average, the absolute HR SDT6 during the first and second tests were statistically similar: 110 ± 15 bpm and 110 ± 14 bpm (p=0.84) (Table 2). The means of paired differences were 0.15 ± 6.08 bpm. The coefficient of variation was 2.8% and the ICC (ICC=0.96) indicated very good reproducibility and concordance (Table 2). Furthermore, a strong correlation between HR SDT6-1 and HR SDT6-2 was observed (r=0.90, r 2 =0.81; p<0.0001) (Figure 2). Bland & Altman plots between HR SDT6-1 and HR SDT6-2

showing a bias at 0.06 bpm and confidence interval at 12.43 bpm (Figure 3).

Relationship between HR Vth and HR SDT6

The relationship between HR Vth and HR SDT6 was evaluated in a sample of 27 patients (Figure 1). The HR revealed a significant difference (107 ± 16 and 115 ± 16 bpm respectively between ventilatory and dyspnoea thresholds respectively; p<0.01). Mean of differences between HR SDT6 and HR Vth was 8 ± 11 bpm.

Estimation of ventilatory threshold from SDT6

Multiple linear regression analysis was performed to estimate HR vth from HR SDT6 in a sample of 27 patients (Figure 1). The stepwise regression analysis selected the following explanatory variables: HR SDT6 (bpm), FEV 1 (L) and age (years). The equation obtained was the following:

Estimated HR !"# = 46.21 + (0.607 × HR !"#$ ) + ( 8.93 × FEV ! ) - (0.3 × age)
Its predictive value was 63% (adjusted r=0.79; r 2 =0.63) and the standard error of estimated was 9.8 bpm.

DISCUSSION

The aims of this study were to test the ability of patients to identify a dyspnoea threshold during 6MWT, evaluate its reproducibility and determine a useful generalized equation allowing to estimate a ventilatory threshold. Our principal results show that: 1) A large majority of patients (80%) were able to identify SDT6 during the 6MWT, 2) the heart rate at SDT6 was highly reproducible. In addition, we defined a linear equation integrating HR SDT6 , FEV 1 and age allowing to estimate closely HR Vth.

As a first step, our aim was to appreciate the feasibility. As recommended in literature, we based our approach on the self-description of dyspnoea [START_REF] Mahler | Descriptors of breathlessness in cardiorespiratory diseases[END_REF][START_REF] Noseda | Dyspnoea and perception of airway obstruction[END_REF][START_REF] O'donnell | Qualitative aspects of exertional breathlessness in chronic airflow limitation: pathophysiologic mechanisms[END_REF][START_REF] Williams | The language of breathlessness differentiates between patients with COPD and age-matched adults[END_REF]. We identified the words that each patient had associated with breathing discomfort and have provided them with guidelines for identifying the SDT6. This procedure allowed the patients to freely express their sensations without prior orientation or influence from the standardized words of dyspnoea categorization (predefined list of words or sentences). In this context, we obtained three main lexical fields: "increased respiratory rate," "chest tightness" and "suffocation," with the highest prevalence of responses for this last ("suffocation"). These results were comparable to those reported in previous studies, in which the authors used different interview strategies and multivariate statistical analyses [START_REF] Mahler | Descriptors of breathlessness in cardiorespiratory diseases[END_REF][START_REF] Noseda | Dyspnoea and perception of airway obstruction[END_REF][START_REF] O'donnell | Qualitative aspects of exertional breathlessness in chronic airflow limitation: pathophysiologic mechanisms[END_REF][START_REF] Williams | The language of breathlessness differentiates between patients with COPD and age-matched adults[END_REF]. We therefore concluded that the ability of patients to define a SDT6 is easily detectable when the good procedure were applied.

The 6MWT is described in the literature as an easy and reproducible field test, and it was therefore important in this study to ensure that the definition of SDT6 during this test met these same criteria [START_REF] Troosters | Six minute walking distance in healthy elderly subjects[END_REF][START_REF] Troosters | Physiological responses to the 6-min walk test in patients with chronic obstructive pulmonary disease[END_REF]. Concerning feasibility, our results showed that 80% of the patients raised a hand during the 6MWT to signal SDT6. The prevalence is comparable to the highest results reported in literature for V th (66 to 79% in cardiac disease [START_REF] Marzolini | The feasibility of cardiopulmonary exercise testing for prescribing exercise to people after stroke[END_REF] and 77% in pulmonary disease [START_REF] Ong | Pulmonary function and exercise capacity in survivors of severe acute respiratory syndrome[END_REF]).

Concerning the 20% of patients who did not identify a SDT6 during 6MWT, we could note that, for comparable distances and a maximal theoretical heart rate similar between groups, maximum heart rate was significantly less than for patients having a SDT6. These results, associated with a higher FEV1, highlight either a smaller relative intensity of effort or either a better cardiorespiratory adaptation during exercise in this group and could explain why they have not reached their dyspnoea threshold.

Concerning reproducibility, our results showed that SDT6 occurred at comparable HR in the two experimental tests. Indeed, all statistical results were consistent and support the good reproducibility of the measurements. Moreover, they were consistent to the results obtained with different tests (6MWT, CPET) and study populations (pulmonary, cardiac and healthy subjects) [START_REF] Meyer | Ventilatory and lactate threshold determinations in healthy normals and cardiac patients: methodological problems[END_REF][START_REF] Poulain | 6-minute walk testing is more sensitive than maximal incremental cycle testing for detecting oxygen desaturation in patients with COPD[END_REF]. It means that COPD patients can detect in a highly reproducible way their dyspnoea threshold, when it is explained with their own words.

In a subgroup of patients, we tested the relationship between HRSDT6 and HRVth. Our results revealed that the mean HR for these thresholds exhibited a bias of 8 bpm between both thresholds. This could be explained by the difference between these tests (nature and modalities). Indeed, CPET is an incremental test and were administrated on ergocycle rather than 6MWT is auto-calibrated test and were realized by walking. On COPD population, it has been shown that Vth occurred at a lower VO2 during cycling than walking [START_REF] Mahler | Mechanism of greater oxygen desaturation during walking compared with cycling in patients with COPD[END_REF]. This confirms that Vth determine during cycling test appears before walking test. In the same vein, [START_REF] Cunha | Influence of exercise modality on agreement between gas exchange and heart rate variability thresholds[END_REF] showed that maximal HR obtained during walking was higher than maximal HR during cycling and with a difference of 7 bpm [START_REF] Cunha | Influence of exercise modality on agreement between gas exchange and heart rate variability thresholds[END_REF]. This supports the idea that HR SDT6 (determined by auto-calibrated walking test) was consistent with HR Vth (determined during incremental exercise test on ergocycle) and with to a fluctuation of HR between both tests close to previous studies [START_REF] Cunha | Influence of exercise modality on agreement between gas exchange and heart rate variability thresholds[END_REF][START_REF] Mahler | Mechanism of greater oxygen desaturation during walking compared with cycling in patients with COPD[END_REF].

Another interest of this study was to determine a useful equation allowing to estimate the heart rate at ventilatory threshold. Using Stepwise linear regression, we were able to account for 63% of the variance in HR Vth using the HR SDT6 when combined with patients FEV 1 and age according to the following equation: 𝐸𝑠𝑡𝑖𝑚𝑎𝑡𝑒𝑑 HR !"# = 46.21 + (0.607 × HR !"#$ ) + ( 8.93 × FEV ! ) - (0.3 × age). From clinical point of view, this equation allows also to estimate a heart rate close to HR Vth in order to individualized exercise training when CPET is not accessible. The difference of 9.8 bpm between estimated HR Vth and HR Vth from CPET is in the same amplitude that the difference observed between HR SD6 and HR Vth from CPET (8 bpm). The hypothesis described in the above paragraph and the related nature and modalities both tests could explain this difference.

To the best of our knowledge, this study is the first to demonstrate that dyspnoea threshold determined during 6MWT is a relevant alternative to predict exercise intensity close to ventilatory threshold. Contrary to ventilatory threshold determined by CPET which needs cost materials and specific staff, SDT6 could be evaluated by the patient himself, and consequently encourage patients in his self-management, which is one of the most important aim to reach in respiratory rehabilitation.

Study limitations

The determination of SDT6 is based on a subjective approach (a hand signal of the critical breathing discomfort during the 6MWT) and is therefore closely linked to the health status and psychological state of the patients. For this reason, patients in exacerbation or an unstable state were not included in our study, and SDT6 determination by our methods is relevant only for patients in steady state. Similarly, even though the 6MWT and HR SDT6 are reproducible, using HR SDT6 to evaluate therapeutic regimens cannot be considered at this point. From a clinical point of view, the positive effect of a therapeutic is expressed either by a decreased HR for a given intensity or a delay in the threshold. However, as the 6MWT is a self-calibrated test (i.e., patients automatically adjust their effort to their sensations), this type of interpretation cannot be made. The SDT6 cannot replace the determination of ventilatory threshold to evaluate the effects of exercise training, but it remains an interesting alternative tool for predicted this threshold in order to individualizing the exercise training intensity.

Conclusion

This study shows that a large majority of patients with COPD are able to identify a selfperceived dyspnoea threshold during the 6MWT. This heart rate associated to this dyspnoea threshold is highly reproducible and allow to estimate heart rate at ventilatory threshold. These findings suggest that SDT6 may be used as a new clinical tool for predicting heart rate at ventilatory threshold in COPD patients in order to individualizing exercise training intensity when CPET is not accessible and/or to follow in ambulatory rehabilitation.

FIGURE CAPTION

Figure 1 :

 1 Figure 1: Study flowchart. SDT6: Dyspnoea threshold in 6MWT; 6MWT: 6 minute walk test;

Figure 2 :

 2 Figure 2: Relationship between heart rate of Self-perceived Dyspnoea Threshold (SDT6)

Figure 3 :

 3 Figure 3: Bland & Altman Plots of inter-reproducibility of HR of SDT6 measurement between

Figure 3 :

 3 Figure 3:

Table 1 : Characteristics of the study population. 1

 1

				All patients	Patients without SDT6	Patients with SDT6
				(n=82)	(n=16)	(n=66)
				Mean ± SD	Mean ± SD	Mean ± SD	p
	Age			63,7 ± 10,3	63,7 ± 8,7	63,7 ± 10,7	ns
	Gender		26 ♀ 56 ♂	5 ♀ 11 ♂	21 ♀ 45 ♂	ns
	BMI (Kg/m	2 )	24,3 ± 4,5	24,2 ± 3,6	24,3 ± 4,7	ns
	PaO2 (mmHg)	72,3 ± 11,9	73,4 ± 10,4	71,9 ± 12,3	ns
	FEV1 (%pred.)	38,3 ± 17,2	48,0 ± 20,8	35,9 ± 15,4	<0.05
	BODE Index	3,9 ± 2,0	3,0 ± 2,4	4,1 ± 1,9	ns
	6MWT distance (m)	396,7 ± 97,5	403,8 ± 77,1	395,0 ± 102,3	ns
	Heart rate at rest (bpm)	90,2 ± 14,3	89,1 ± 19,4	90,4 ± 13,0	ns
	Heart rate max (bpm)	119,5 ± 17,9	111,8 ± 16,8	121,4 ± 17,7	<0.05
	4
	5	significant (p>0.05)
	6
	7

Data presented as mean ± standard deviation, except gender (absolute number). BMI: Body mass 2 index; FEV 1 : Force Expiratory Volume in 1 s expressed in percentage of predicted value; 6MWT: 6 3 minute walk test; p values between patients with and without SDT6; bpm: beats per minute; ns: no

Table 2 : Comparison of heart rate of SDT6 during both 6MWTs

 2

		6MWT-1	6MWT-2	p	CV (%)	ICC
	Heart rate SDT6 (bpm)	110 ± 15	110 ± 14	0.84	2.8 %	0.96
	Data presented as mean ± standard deviation; SDT6: dyspnoea threshold during 6 minute walk test;
	bpm: beats per minute; CV: coefficient of variation and ICC: IntraClass Coefficient

Acknowledgements

The authors wish to thank the team of La Vallonie, pulmonary Rehabilitation Centre for their contribution to this work, Dr Dominique Bourgouin and Dr Jacques Desplan for their support and their fruitful discussion and Cathy Carmeni for her English review.