Economix

Abdoulaye Kané

Abdoulaye Kane

Measurement of total factor productivity: Evidence from French construction firms

Keywords: French construction sector, Production function, Total factor productivity, Parametric estimation, Semi-parametric estimation, Non-parametric estimation, Market structure JEL Classification: C13, C14, C23, D24, D43

teaching and research institutions in France or abroad, or from public or private research centers.

Introduction

Total Factor Productivity (TFP) is generally defined as the portion of output not explained by the amount of inputs used in production. It is crucial in terms of economic fluctuations, economic growth and cross-country per capita income differences 1 insofar as it determines long-term economic growth and is a comprehensive industry-level productivity measure. However, if its theoretical definition seems comprehensive, its empirical implementation is far from being an easy task.

Parametric and semi-parametric methods show decreasing returns to scale in French construction. The underlying economic interpretation would be the imposing weight of large firms on small firms. In other words, the market structure could be likened to an oligopoly situation. Moreover, while the elasticity of capital per worker is low in output per worker (ranging from 0.0395 for the Fixed Effect method to 0.0636 for the Wooldridge method), the elasticity of materials per worker is very high (ranging from 0.776 to 0.819 following the same methods). We also find that the ACF method can be considered a good estimator of TFP in the French construction sector.

The differences between the different parametric and semi-parametric estimators are very small when comparing the estimated TFP. Spearman's correlation coefficients between these TFP measures are generally greater than 0.92 and even 0.99 when comparing the results obtained by the GMM method and the three semi-parametric methods (OP, LP and ACF). However, the correlations between these methods and the non-parametric methods are very low, even negative with the calibration method. The rest of the paper is structured as follows. Section two presents the methods for the theoretical calculation of TFP. We move to the empirical part in section three which will present the data as well as the estimation results. Section 4 provides some concluding remarks.

Theoretical framework

If the authors are unanimous in attributing the term "total factor productivity" to the work of Solow (1957), they are less so as to its measurement. This measure is all the more difficult as the sector under study is fragmented. Usually the literature starts with a production function. Following this idea I assume a Cobb-Douglas production function:

Y it = A it K β k it L β l it M βm it (1)
Where Y it represents gross output of firm i in period t, K it , L it and M it are inputs of physical capital, labor (Total employment) and materials (intermediate inputs), respectively, and A it is the Hicksian neutral efficiency level of firm i in period t. The labor, capital and intermediate inputs elasticities are given by β l , β k and β m respectively. We take natural logarithm of (1) and in order to understand the nature of returns to scale4 in the industry studied, we divide each variable by the labor input. Thus, equation (1) becomes:

ln

Y it L it = β 0 + β k ln K it L it + (β l + β k + β m -1) ln L it + β m ln M it L it + it (2)
Where ln A it = β 0 + it ; β 0 measures the mean efficiency level across firms and over time; it is the error term.

To simplify the writing of (2) , we rewrite it as follows:

y it = β 0 + β k k it + γl it + β m m it + it ( 3
)
where y it = ln Yit Lit ; k it = ln Kit Lit ; m it = ln Mit Lit ; γ = (β l + β k + β m -1). It should be noted that γ provides us with information on the nature of the returns to scale.5 it can be decomposed into an observable and unobservable component:

y it = β 0 + β k k it + γl it + β m m it + υ it + µ it (4)
Where υ it is the observable component, µ it is the unobservable component and β 0 + υ it represents firm-level productivity.

Thus using the ordinary least squares (OLS) method, we can estimate TFP as follows:

tf p it = β0 + υit = y it -βk k it -γl it -βm m it (5)
To obtain the level productivity, we take the exponential of tf p it , i.e., T F P it = exp( tf p it )

However, the OLS estimation method is automatically biased. Indeed, estimating a production function by OLS assumes that the factors of production are exogenous in the production function, i.e. determined independently of the firm's level of efficiency. However, authors such as Marschak and Andrews (1994) have already shown that the factors of production in the production function are not chosen independently, but rather determined by the characteristics of the firm, including its efficiency. We face a simultaneity bias.6

In addition to this bias, we have the selection bias. TFP is typically estimated with a cylindrical panel by omitting all firms that enter and exit during the sample period (Olley and Pakes, 1996). Although some economists believe that firm entry and exit are implicitly taken into account in the analysis (Fariñas and Ruano, 2005), explicitly omitting consideration of the exit decision of firms leads to selection bias. The reason is as follows: Firms' decisions on factor allocation in a particular period are made conditional on its survivors. In sum, the selection bias will cause the error term to be negatively correlated with capital, thus causing the capital coefficient to be biased. For all these reasons, OLS estimation of a production function will provide us with inconsistent coefficients. To overcome these issues, different methods of estimating the TFP have been proposed.

Fixed effects estimation

Assuming that productivity is firm-specific but time-invariant, it is possible to estimate TFP using the fixed effects estimator (Pavcnik, 2002;[START_REF] Levinsohn | Estimating production functions using inputs to control for unobservables[END_REF]:

y it = β k k it + γl it + β m m it + tf p i + µ it (6)
Where tf p i = β 0 + υ it Equation (6) can be estimated in level using the within-individual estimator or in first difference providing unbiased coefficients as long as the unobserved productivity tf p i does not vary over time. In this respect, the simultaneity bias is eliminated because we have only the within-sector variation in the sample. The same is true for selection bias because exit decisions are made in an invariant time period.

However, in practice, the fixed effects estimator on a production function often leads to unreasonably low estimates of the capital coefficient because it imposes strict exogeneity of inputs conditional on firm heterogeneity [START_REF] Wooldridge | Econometrics: Panel Data Methods[END_REF]. In economic terms, this would mean that factors of production cannot be chosen in response to productivity shocks. This assumption probably does not hold in practice, especially not in the construction sector, which has been facing a decline in the rate of productivity growth in recent years, and thus a variation in productivity over time. Another estimator is often proposed to overcome these problems: the instrumental variables estimator and the Generalized Moment Method (GMM).

Instrumental Variables and Generalized method of moments

An alternative method for achieving consistency of coefficients across a production function is to instrument for the factors that cause the endogeneity problem. Unlike the fixed effects estimator, the instrumental variables (IV) estimator does not rely on strict exogeneity. Nevertheless, this method requires a certain number of conditions, notably on the variable or variables used as external instruments. First, the instruments must be correlated with the endogenous variable(s). Second, the instruments cannot enter directly into the production function. Finally, the instruments must not be correlated with the error term. The last assumption rules out the existence of imperfect competition in the factor market if output or factor prices are used as instruments. Assuming a perfect market, then, factor and output prices are natural choices of instruments for the production function.

Nevertheless, factor prices become valid instruments if and only if the firm does not have market power. Indeed, if the firm has market power, it will set its prices at least partly according to the quantities of factors and its productivity. This makes prices endogenous. This endogeneity problem will always arise even if we use the average wage per worker (reflecting exogenous labor market conditions) because this wage often varies according to the qualification and quality of the employee.

Other instruments can be taken into account, such as weather conditions, exogenous shocks on the labor or capital market, independently of the firm's market power. However, as pointed out by [START_REF] Ackerberg | Econometric tools for analyzing market outcomes[END_REF], even in this case, the IV approach only deals with the endogeneity of factors, but not the endogeneity of firms' outputs. If instrument choices are correlated with firm-level output, endogenous output would invalidate the use of instruments. Some authors lag inputs and then use them as instruments. This biases the capital coefficient, which is often not significant. To remedy this, Blundell and Bond (1999) propose the GMM estimator. For them, the poor performance of the IV estimator is due to the weakness of the instruments used for identification. We must therefore put the instruments in delayed first differences in the level equations and thus we will have good estimates.

Nevertheless, the main drawback of this method is to find a valid instrument to instrument the TFP. Apart from these parametric estimation methods, we have semi-parametric estimates that give us a time variability of the TFP.

The Olley-Pakes (1996) estimation algorithm

This semi-parametric estimation method solves the simultaneity bias by taking the firms' investment decision as a proxy for unobserved productivity shocks. The selection problem is solved by incorporating an exit rule in the model. This is a dynamic model of firm behavior where we have two fundamental assumptions:

1. Only one variable (state variable) is unobserved at the firm level. This is productivity, which is also assumed to evolve as a first-order Markov process.

2. The monotonicity of the "investment" variable so that the demand function for investment is invertible. This means that investment is increasing in productivity. One of the corollaries of this second hypothesis is that only positive values of investment are accepted.

Starting with the Cobb-Douglas production function given by equation (4), the estimation procedure can be described as follows: capital is a state variable, affected only by current and past values of the productivity level tf p it . Investment is described as follows:

I it = K it-1 -(1 -δ)K it
where δ is the capital depreciation. Also, investment decisions at the firm level can be written as a function of capital and productivity: i it = i t (k it , tf p it ) where the lower case notation refers to the logarithmic transformation of the variables. Since investment is an increasing function of productivity, conditional on capital, the investment decision can be inverted, allowing us to write unobserved productivity as a function of observable variables: tf p it = h t (k it , i it ) where h t (.) = i -1 t (.). Using this formula, equation ( 4) is rewritten as follows:

y it = β 0 + β k k it + γl it + β m m it + h t (k it , i it ) + µ it (7) Let's denote ϕ(k it , i it ) = β 0 + β k k it + h t (k it , i it )
The estimation of equation ( 7) is done in two steps. The first step is to estimate, using OLS, the following equation:

y it = γl it + β m m it + ϕ(k it , i it ) + µ it (8)
where ϕ(k it , i it ) is approximated by a third-order polynomial in investment and capital.

In the second step, where we address the attrition bias problem, we identify the capital coefficient by estimating the following equation:

y it -γl it -β m m it = β 0 + β k k it + g(ϕ t-1 -β k k t-1 ) + µ it (9)
where g(.) is an unknown function which is again approximated by a third-order polynomial expression in ϕ t-1 and k t-1 . The probability of survival is normalized to 1. The capital coefficient can then be obtained by applying nonlinear least squares to equation (9).

The Levinsohn-Petrin (2003) estimation algorithm

Like the Olley-Pakes (OP) method, the Levinsohn-Petrin (LP) method is also a semi-parametric estimate, but intermediate inputs are used as a proxy. Indeed, the monotonicity condition of OP requires that investment is strictly increasing in productivity. This implies that only observations with positive investment can be used when estimating equations ( 8) and ( 9), and this may lead to a significant loss of efficiency depending on the data available. Moreover, if firms report zero investment in a significant number of cases, this casts doubt on the validity of the monotonicity condition. To do this, the LP estimation method uses intermediate inputs as a proxy for unobserved productivity. Since firms generally report positive material and energy use each year, it is possible to retain most observations; this also implies that the monotonicity condition is more likely to hold. The LP estimation algorithm differs from the OP algorithm in two ways. First, it use intermediate inputs as a proxy for unobserved productivity, rather than investment. This implies that intermediate inputs are expressed in terms of capital and productivity, i.e. , :

m it = i t (k it , tf p it )
Using the monotonicity condition, intermediate inputs are strictly increasing in productivity and are also invertible: tf p it = s t (k it , m it ) where s t (.) = m -1 t (.). Equation (4) becomes:

y it = β 0 + β k k it + γl it + β m m it + s t (k it , m it ) + µ it (10)
The second difference is related to the correction for selection bias. Although the OP approach allows for both a non-cylindrical panel and the incorporation of the probability of survival in the second stage of the estimation algorithm, the LP method does not incorporate the probability of survival in the second stage.

Recently, new and even more robust production function estimation techniques have emerged in an effort to correct the LP method. These include the [START_REF] Wooldridge | Econometrics: Panel Data Methods[END_REF] and [START_REF] Ackerberg | Identification properties of recent production function estimators[END_REF] methods. [START_REF] Wooldridge | Econometrics: Panel Data Methods[END_REF] proposed an alternative implementation of OP/LP moments that involves the simultaneous minimization of first and second stage moments. Using the LP model, he suggested estimating all the parameters simultaneously using the moment conditions :

Wooldridge method (2009)

E[mu it |I it ξ it + µ it |I it-1 ] = E[ y it -γl it -Φ t (k it , m it )|I it y it -β 0 -γl it -β k k it -g(Φ t-1 (k it-1 , m it-1 ) -β 0 -β k k it-1 )|I it-1 ] = 0 (11)
As pointed out by Wooldridge, there are several advantages to this First, the joint approach avoids the functional dependence issue above. Even if l it is functionally dependent on m it , k it and t, γ might be identified by the second set of moments. Other benefits of the Wooldridge approach are potential efficiency gains, and simpler standard error calculations. There are also disadvantages of the joint approach; in particular, the joint approach nonlinear search over β 0 , β k , γ and the parameters representing the two unknown functions Φ t and g. This method is more time-consuming and probably more error-prone than the two-step approach, which can often be obtained by a nonlinear search on only β k , γ.

The Ackerberg-Caves-Frazer (2015) method

The main argument of the Ackerberg, Caves, and Frazer (ACF) method is that the labor coefficient may not be identified in the estimation procedures proposed by OP and LP. This is what the authors call the "functional dependence problem". Indeed, the authors believe that labor can be an argument of the demand function of the proxy variable and, consequently, of the unobserved productivity function.

Thus, the authors start from the same point of view as the LP method by taking the same assumptions of monotonicity of the proxy function but including labor: m it = f t (k it , l it , tf p it ). One interpretation of this assumption is that the gross output production function is Leontief in intermediate inputs. 7Given the assumption of strict monotonicity, we can invert the intermediate input demand : tf p it = f -1 t (k it , l it , m it ). Therefore, equation (4) becomes :

y it = β 0 + β k k it + γl it + β m m it + f -1 t (k it , l it , m it ) + µ it = Φ t (k it , l it , m it ) + µ it ( 12
)
where

Φ t (k it , l it , m it ) = β 0 + β k k it + γl it + β m m it + f -1 t (k it , l it , m it )
Using the first stage moment condition, we have:

E[µ it |I it ] = E[y it -Φ t (k it , l it , m it )|I it ] = 0 ( 13
)
Where I it represents a set of information. We note that unlike the OP and LP methods, no coefficient is identified in the first step. In short, all the coefficients are estimated in the second step using the following second stage moment condition:

E[ξ it + µ it |I it-1 ] = E[y it -β 0 -γl it -β k k it -β m m it -g(Φ t-1 (k it-1 , l it-1 , m it-1 ) -β 0 -β k k it-1 -γl it-1 -β m m it-1 )|I it-1 ] = 0 ( 14
)
Where Φ t-1 is replaced by its estimate from the first stage. The coefficients γ, β k , β m are estimated through a first order Markov process.

Using a Cobb Douglass production function with neutral productivity differences in the sense of Hicks does not allow for the identification of factor biases in technological change, especially since value added or sales are generally used as a measure of output. Non-parametric techniques -Elasticity calibration and Data Envelopment Analysis (DEA) -solve this problem.

Calibration method

Introduced by Solow (1956Solow ( , 1957)), the basic idea of this method is to calibrate the different elasticities of the factors of production using two crucial assumptions8 :

• The constant returns to scale assumption • The perfect market assumption in both the labor and goods and services markets Thus, with these two assumptions and as long as we have information on the different factors of production as well as on output, we are able to calculate equation (5). By calibration, the elasticities of labor (β l ) and intermediate inputs (β m ) are obtained by the ratio between the labor cost and the production value, the ratio between intermediate inputs and the production value respectively. Assuming constant returns, the elasticity of the capital stock becomes

β k = 1 -β l -β m .
Finally, we use the average of each elasticity over time.

The calibration method is particularly interesting in that it calculates TFP in the simplest possible way (no estimation is required). However, the assumption of perfect competition in product and input markets assumption is a strong assumption, especially in the construction sector, which tends towards an imperfect market structure.

Data Envelopment Analysis approach (DEA)

The DEA approach, also called non-parametric frontier estimation, constructs for each observation a linear combination of all other observations (normalized by production) for explicit comparison. Here, no particular production function is assumed. Efficiency (productivity) is defined as a linear combination of output over a linear combination of factors of production. The weights on the factors (u l , u k ) and output (v q ) are chosen directly by maximizing the efficiency (productivity) denoted by θ. Observations that are not dominated are labeled 100% efficient. Dominance occurs when another firm, or a linear combination of other firms, produces more of all output with the same aggregate of factors, using the same weights to aggregate the factors. A linear maximization program is solved separately for each observation. For unit 1 (firm-year), in the case of a single production, the problem is :

max vq,u l ,u k θ 1 = v q Q 1 + v * u l L 1 + u k K 1 (15)
subject to :

v q Q i + v * u l L i + u k K i ≤ 1; i = 1, . . . , N. v q , u l + u k > 0; u l , u k > 0 v * ≥ 0
Where v * = 0 when returns to scale are constant.

The efficiency of all firms cannot exceed 100% when the same weights are applied. A normalization is necessary to properly define the problem: u l L 1 + u k K 1 = 1 and v * is an additional game variable to allow for varying returns to scale. When we have constant returns to scale (v * = 0), the production frontier is a ray through the origin in the aggregate production-factor space. Under varying returns to scale, this frontier is the piece-wise linear envelope of all production plans.

The efficiency measure θ i can be interpreted as the productivity difference between unit i and the most productive unit. The efficiency score ranges from 0 to 1. An efficiency score of 1 means that the firm is fully efficient. The level and growth rate productivity estimates are defined as follows, respectively:

logA DEA it -logA t DEA = logθ it - 1 N t N j=1 logθ jt (16) logA DEA it -logA DEA it-1 = logθ it -logθ it-1 (17)
The main advantage of the DEA method is the absence of functional form or behavioral assumptions of firms and no distributional assumption is imposed on the inefficiency term. The underlying technology is completely indeterminate and allowed to vary across firms. However, it has serious limitations in that, first, it interprets any deviation from the set production potential as inefficiency. This means that all factors that affect the firm's performance are considered to be under the firm's control. The consequence is that the estimated inefficiency is biased due to exogenous factors such as measurement error [START_REF] Kumbhakar | Stochastic frontier analysis[END_REF]. The second limitation is that it measures inefficiency relative to the best performing unit among observations, making its result susceptible to outlier bias [START_REF] Coelli | Additional topics on data envelopment analysis. An introduction to efficiency and productivity analysis[END_REF]. Based on this review of the TFP, we present a summary result of each method.

Methods

Advantages Drawbacks Fixed effects

Resolution of the simultaneity bias when we assume strict exogeneity of the inputs.

Underestimation of the capital elasticity.

Instrumental variables and GMM (Blundell and Bond, 1999) Resolution of the simultaneity bias and these methods do not impose strict exogeneity.

The challenge is to find valid instruments. Olley and Pakes (1996) Resolution of simultaneity and attrition biases.

The monotonicity condition no longer holds when the investment value is zero and the functional dependence problem. [START_REF] Levinsohn | Estimating production functions using inputs to control for unobservables[END_REF] Resolution of simultaneity and attrition biases

The functional dependence problem. [START_REF] Wooldridge | Econometrics: Panel Data Methods[END_REF] Resolution of simultaneity and attrition biases by performing a one-step system GMM.

This method is more timeconsuming and probably more error-prone than the two-step approach. [START_REF] Ackerberg | Identification properties of recent production function estimators[END_REF] Resolve simultaneity and attrition biases by addressing the functional dependence problem suffered by the Olley and Pakes and Levinsohn and Petrin methods.

Like the previous methods, the problem of this method is the specification of a functional form of the production function.

Elasticity calibration

It calculates the TFP in the simplest way possible.

The condition of a perfect market, both in the labor market and in the market for goods and services, is a strong assumption. Data Envelopment Analysis (DEA)

The main advantage of the DEA method is the absence of functional form or behavioral assumptions of firms and no distributional assumption is imposed on the inefficiency term.

As it is a deterministic method, it is vulnerable to measurement errors.

Table 1: Summary of methods

The following section will empirically test these different methods.

Empirical Application

Data

Our data come from the Esane approximate results file (FARE), which contains accounting information from tax returns that are consistent with information from the Annual Sector Survey. The FARE system aims to build up a coherent set of business statistics. It combines administrative data (obtained from the annual profit declarations that companies make to the tax authorities, and from annual social data that provide information on employees) and data obtained from a sample of companies surveyed by a specific questionnaire to produce structural business statistics.

We mobilize data from the construction sector in metropolitan France from 2009 to 2018. Of the three subsectors in the construction sector (building construction including real estate development, civil engineering, and specialized construction), we focus only on the building construction. Specifically, this is the "Residential and Non-Residential Building Construction (Sector 4120)" sub-sector, which includes general construction or "all crafts" firms with overall responsibility for the construction of a building. It also includes the conversion or renovation of existing residential structures. This sub-sector has the advantage of being highly focused on on-site work and is also representative of the construction sector as shown in the estimation results in the Appendix. Moreover, unlike specialized construction, which is a highly atomized sector, building construction contains more large companies. Firms are reported for at least 3 years and our data do not show a temporal break for the same firm. Furthermore, to overcome selection bias, we need to work on an unbalanced panel data set. Indeed, the attrition bias is lower the more the panel is unbalanced with a large number of samples [START_REF] Ackerberg | Identification properties of recent production function estimators[END_REF].

We use total gross production as the output variable. Three inputs have been mobilized: labor, capital and intermediate inputs. The labor input is measured by the number of full-time equivalent employees. The capital stock is approximated by gross tangible fixed assets. The gross investment is also given by FARE. We measure intermediate inputs by the difference between total gross production and value added at factor costs. Because we have nominal values, we deflate them using price indices in the French construction sector obtained from the STAN 2020 edition database (constant price 2015) to obtain real values. These deflators cover added value, output, investment, capital and intermediate inputs. The table clearly shows that the average number of employees in the sample does not reach 20 (average employment amounts to 19.195 employees). This situation is specific to the sector, which has many small firms. Second, these data reveal strong heterogeneity across firms. While the minimum number of employees is 1, the maximum number is 4,635. For all these reasons, it is interesting to examine the structure of our sample through the size of the firms. The INSEE size classification (classification according to the number of employees, turnover and balance sheet) will be difficult to apply to a sector composed essentially of micro-enterprises. Thus, we assume that firm size depends only on the number of employees. To do this, we follow [START_REF] Baldwin | The trend to smaller producers in manufacturing: A Canada/US comparison[END_REF] classification that considers a firm to be :

• Micro if the number of employees is < 20;

• Small if the number of employees is between 20 and 99;

• Medium if the number of employees is between 100 and 499;

• Big if the number of employees is ≥ 500.

The following table shows that 84.10% of our sample has less than 20 employees and 13.23% are small businesses. Medium-sized companies represent 2.23%, while large companies are very few (0.44%). However, although they are few in number, the large companies in the French construction sector have enormous weight. This is explained by the following graph on average total labor productivity 9 : Figure 1: Average labor productivity by firm size Figure 1 shows that, on average, the labor productivity of large firms is the highest, followed by intermediate-sized firms. However, the gap between medium-sized firms and micro-firms is not large. Average labor productivity is lowest at the small firm level.

Firm size

Estimation results

In the following table we present the estimation results of the production function based on the methods presented in section 2. All reported estimates are obtained through an unbalanced panel of firms (allowing for implicit entry and exit). [START_REF] Wooldridge | Econometrics: Panel Data Methods[END_REF] estimator is used as the GMM method. Its advantage is that it provides consistent and efficient parameter estimation using the one-step system-GMM approach. It solves potential serial correlation, heteroscedasticity as well as the endogeneity due to simultaneity and 9 Labor productivity is measured as the ratio of real value added to the number of full-time equivalent employees.

attrition using lagged values as instrument [START_REF] Ackerberg | Identification properties of recent production function estimators[END_REF]). The Olley and Pakes (1996), [START_REF] Levinsohn | Estimating production functions using inputs to control for unobservables[END_REF], [START_REF] Wooldridge | Econometrics: Panel Data Methods[END_REF]Ackerberg, Caves, and[START_REF] Ackerberg | Identification properties of recent production function estimators[END_REF] methods are estimated using the "prodest" command10 developed by Mollisi and Rovigatti (2017), which has the advantage of correcting for attrition and simultaneity bias and adding control variables. We denote the Fixed Effects, Olley and Pakes, Levinsohn and Petrin, Wooldridge and Ackerberg, Caves and Frazer estimators by (FE), (OP), (LP), (WRDG) and (ACF) respectively. The calibration method is designated by (CM). To better interpret these results, we will make some general comments and comparisons between the different methods. With respect to the estimated elasticities, there are two important points in the general comments. First, whatever the estimator, the value of γ is negative, which means that returns to scale are slightly decreasing in the French construction. In other words, output varies less than the variation in the inputs used. This result shows that the more we produce, the more expensive it is to produce an additional unit in French construction. Moreover, we can link this dis-economy of scale to the organizational structure of the construction market. Indeed, in microeconomics, diseconomies of scale are the cost disadvantages that firms accumulate as a result of increasing their size or output, leading to the production of goods and services at higher unit costs. For this reason, large companies, although few in number, are imposing themselves on the French construction market. We are therefore in a market where a few companies (large or medium-sized) have a certain amount of market power and where a large number of micro or small companies are under the weight of these giants: the French construction market is similar to an oligopoly situation. This result is supported by [START_REF] Lowe | The measurement of productivity in the construction industry[END_REF] conclusion that the construction industry deviates significantly from the perfect competition model.

Second, the elasticity of capital per worker in the production process is very low ranging from 0.0395 (FE method) to 0.0636 (Wooldridge method). This low elasticity of capital is not a surprise insofar as construction, retaining its manual character, is very labor intensive. Physical assets will inevitably be less noticeable in a sector composed mainly of micro-firms. The weakness of investment spending in the French construction industry has been noted through the study of [START_REF] Ferrand | Les mutations de l'investissement dans le bâtiment[END_REF]. The author shows that, since the beginning of the 2000s, the rate of investment in French construction, on average, does not exceed 10% of GDP. This rate seems to be the norm when an international comparison is made (comparison with the United States or with the euro zone). The elasticity of per capita intermediate goods in per capita output is more than 0.77% in each of the estimates.

Comparing the estimation methods, we find that the elasticity of capital intensity provided by the FE method is always lower, while the elasticity of intermediate inputs per worker and the absolute value of the scale effect (γ) are still high. These results are consistent with the findings of Van Beveren (2012) who showed that the FE method leads to a low capital coefficient. We add that the fixed effects estimator overestimates the effect of intermediate inputs and the scale effect in our sample.

The estimation results of the other 4 methods (WRDG, OP, LP and ACF) are quite similar especially between the OP and LP methods. However, the absolute value of the scale effect obtained by the ACF method is the lowest (0.0146%). This result is probably linked to the problem of functional dependence (non-dynamic labor input) that is blamed on the OP and LP methods. The ACF method can be a good estimator of TFP in the construction sector. Not only does it correct for the OP and LP methods by making the labor coefficient dynamic, but it also highlights the crucial role of intermediate inputs in the production process. Using intermediate goods as a proxy for unobserved productivity is particularly important for the construction industry, where expenditures on equipment and machinery rentals are very large. Moreover, no amount of labor can replace the concrete, asphalt, wood and other materials needed to build.

The [START_REF] Wooldridge | Econometrics: Panel Data Methods[END_REF] estimator provides the highest coefficient of capital per worker (0.0636%). The author states that this method has the advantage of easily obtaining robust standard deviations and makes effective use of the moment conditions implied by the OP and LP assumptions. [START_REF] Wooldridge | Econometrics: Panel Data Methods[END_REF] argues that two-step estimators (OP and LP in this case) are inefficient for two reasons. First, they ignore the contemporaneous correlation of errors between two equations. Second, they do not effectively account for serial correlation or error heteroscedasticity. However, we have a loss of information (4,784 fewer observations) with the [START_REF] Wooldridge | Econometrics: Panel Data Methods[END_REF] estimator due to the use of lagged instruments which is not the case with the ACF method which estimates elasticities in one step as well (second step).

Elasticities can also be obtained by calibration or by the DEA method. (sections 2.6 and 2.7). This means that no estimation is required to obtain them.

Non-parametric results

In this subsection, we present the results related to the calculation of TFP by the elasticity calibration method and by the DEA method. The following Table 5 The calibration method has the advantage of showing that labor (21%) explains output better than capital (14%) in the construction sector. This result is consistent with the idea that the construction industry in general and the French construction industry in particular is labor-intensive. Materials (65%) contribute more to output, consistent with estimation methods. Although the calibration method requires strong assumptions, in this case the assumption of perfect markets, it provides results consistent with the reality of the sector. Based on these results, we represent the relationship between output and each factor of production (labor, capital and materials). The most striking observation is that the capital stock explains 50.37% of output, while labor and materials explain 70.55% and 97.86% of output respectively. The DEA method is used through the BCC model [START_REF] Banker | Some models for estimating technical and scale inefficiencies in data envelopment analysis[END_REF]) which assumes variable returns to scale (VRS model). Indeed, previous estimation results have shown that the constant returns assumption cannot be applied to the French construction sector. We adopt the input-oriented DEA approach because it allows us to determine the extent to which a firm's input use could be reduced if they were used efficiently to achieve the same output level. The effective decision units (DMUs) are represented by the different legal units. Since it is not possible to display the efficiency score or productivity (θ) of each of the 4,784 legal units in a given year, we will present the average efficiency by size. The average efficiency scores are given in the following 6, only large companies have an average score equal to about unity. In other words, all big firms have total technical efficiency. It also means that, on average, all big firms in the French construction sector operate on or very close to the production frontier from 2009 to 2018. This is a sign that big firms are using resources to their full potential. The efficiency score decreases with firm size. Mediumsized and small firms have higher average scores than micro-firms. These results confirm once again that the large companies, despite their paltry numbers, have total influence in the sector.

Comparison of TFP methods

In this subsection, we compare the different firm-level TFP calculation methods that have been obtained and focus on the correlation between them. In order to make the methods comparable, we normalize the parametric, stochastic and calibration methods:

z it = xit-min(x) max(x)-min(x)
Where x = (x 1t , ..., x nt ) and z it is the i th normalized value during the period t. 7 clearly shows that the TFPs obtained by the TFP estimation methods (FE, WRDG, OP, LP, and ACF) are very similar to each other. The average TFP for these 5 estimators is between 0.103 (FE estimator) and 0.131 (WRDG estimator). The different standard deviations are also very close, about 0.04 in each case. The FE, stochastic and calibration methods range from 0 to 1 because they have been normalized. The non-parametric methods (calibration and DEA methods) have slightly higher means. The calibration method provides a average TFP equal to 0.165 and the DEA method provides a mean equal to 0.507.

Finally, using the productivity levels obtained, it is possible to calculate the overall average industry productivity for each year based on each estimator. shows that the TFP of the "residential and nonresidential building construction" industry displays a rather stable trend regardless of the method. We also add that the estimation methods are quite similar, especially the semi-parametric methods (OP, LP and ACF). The DEA method has a higher average evolution than the others. The calibration method is slightly above the rest. A similar graph by firm size for each method is also available in the Appendix. But what about the correlation between these methods? Table 8 below shows the Spearman rank correlation between the TFP calculation methods. Since the relationship between the TFPs is not necessarily linear, it is appropriate to use the Spearman correlation. Not surprisingly, there is a strong positive relationship between TFP estimation methods (more than 92%), which is consistent with the work of Van Beveren (2012). The strongest is the correlation between the OP and WRDG methods (0.9942) followed by the correlation between LP and ACF (0.9937). The correlation between the OP and LP methods is 0.9930. However, the correlations between the non-parametric methods (DEA and calibration methods) and the other methods are relatively low, even negative with the calibration method. We have a negative correlation between the calibration method and the other methods. However, this negative correlation is less strong. The highest and lowest correlation (in absolute value) is between the CM and FE methods (0.3569) and between the CM and WRDG methods (0.1165), respectively. The DEA method is positively correlated with the WRDG, OP, LP, and ACF methods at over 32%. The correlation between the DEA and FE methods is less significant (0.2379). The strongest correlation -positive -(0.4190) is between the two non-parametric methods (DEA and CM).

Conclusion

This paper reviewed some methods for calculating TFP. The first part of the paper theoretically explored the strengths and weaknesses of each TFP estimation method, ranging from parametric, semi-parametric to non-parametric techniques. In the second part, we mobilized firm-level data from 2009 to 2018 from the FARE database (Statistique structurelle annuelle d'entreprises issue du dispositif ESANE) in the French construction sector, particularly in the construction of residential and non-residential buildings. Our estimation results (especially the semi-parametric methods and the Wooldridge method) reveal few differences in input elasticities among the methods used. However, some remarks can be made.

In each of the estimates (FE, [START_REF] Wooldridge | Econometrics: Panel Data Methods[END_REF] or semi-parametric methods), returns to scale are decreasing. This result may be related to the organizational structure of the sector in which big firms dominate the market. We are close to an oligopolistic market. [START_REF] Wooldridge | Econometrics: Panel Data Methods[END_REF] estimator produces a statistically significant estimate of capital relative to other approaches. Consistent with the literature, the fixed-effects estimator provides a lower coefficient on capital than the other methods. On the other hand, we have evidence that the same estimator overestimates both the value of the scale effect (in absolute value) and the coefficient on intermediate inputs.

Because the [START_REF] Ackerberg | Identification properties of recent production function estimators[END_REF] methodology treats labor as a dynamic input whose choice affects future profits, the scale effect (in absolute value) is lower. Also, the capital elasticity obtained by the ACF method is lower than that obtained by the OP, LP and WRDG methods. However, it is likely to provide the most plausible estimates among the TFP estimation methods because it corrects for the OP and LP methods and does not lose information compared to the WRDG method.

The non-parametric TFP methods reveal two major results. First, based on the calibration method, the building sector in France is labor-intensive. Second, using the DEA method, we show that on average, all big firms operate on or very near the production frontier from 2009 to 2018. The TFP estimation methods are highly correlated with each other, except for the correlations between the non-parametric methods (DEA and calibration methods) and the other methods. They are very weak, even negative with the calibration method.

Our contribution to the literature on productivity measurement is obvious. Using several methods, we measure TFP in an economically important sector of the French economy that has faced a productivity gap in recent years. We show that returns to scale are decreasing in the sector and that the semi-parametric estimation methods and the [START_REF] Wooldridge | Econometrics: Panel Data Methods[END_REF] method used on firm-level data are quite similar. However, as the choice of the appropriate method is strongly conditioned by the research question, the ACF method, to our knowledge, can be a good estimator of TFP in the French construction sector. The estimation results in Table 8 are in perfect agreement with those obtained in the body of the paper. All of these results emphasize that our sample is in harmony with the full sample.

Appendix Production function estimation using total construction

  0420*** -0.0200*** -0.0249*** -0.0201*** -0.0146*

Figure 2 :

 2 Figure 2: Relationship between output and inputs

Figure 3 :

 3 Figure 3: Total average evolution of total factor productivity

  0520*** -0.0239*** -0.0296*** -0.0281*** -0.0107*

Table 2

 2

	presents summary statistics

Table 2 :

 2 Summary statistics of production variables

Table 3 :

 3 Proportion by size

		Number of observations Proportion (%)
	Micro	23,164	84.10
	Small	3,645	13.23
	Medium	615	2.23
	Big	120	0.44

Table 4 :

 4 Production function estimates Where γ=Scale effect=(β l + β k + β m -1); ln(Y/L)=Production per worker in logarithm; β k is the elasticity of capital per worker; β m is the elasticity of intermediate inputs per worker; N is the number of observations; ID is the Number of legal units; Firm FE and Year FE are the individual and time fixed effects respectively.

	**

Table 5 :

 5 presents the results of the calibration of the different elasticities. TFP calculation using calibration method

	Observation (N )	Labor (β l ) Capital (β k ) Materials (β m )
	27,544	0.21	0.14	0.65

table :

 :

	Size	Year 2009 2010 2011 2012 2013 2014 2015 2016 2017 2018
	Micro		0.49 0.49 0.50 0.49 0.49 0.48 0.47 0.50 0.51 0.50
	Small		0.55 0.55 0.56 0.55 0.54 0.53 0.53 0.53 0.54 0.55
	Medium	0.78 0.76 0.78 0.77 0.77 0.75 0.76 0.76 0.76 0.78
	Big		0.90 0.89 0.90 0.91 0.90 0.89 0.89 0.88 0.88 0.89

Table 6 :

 6 Average efficiency by sizeAccording to Table

Table 7 :

 7 The following table provides a descriptive statistic of the TFP methods. Summary statistics of TFP methods Table

	Methods	N	Mean	Sd	Min	Max
	Fixed effects	27,544 0.103 0.035	0	1
	Wooldridge (2009)	27,544 0.131	0.04	0	1
	Olley and Pakes (1996)	27,544 0.125 0.038	0	1
	Levinsohn and Petrin (2003)	27,544 0.117 0.038	0	1
	Ackerberg, Caves and Frazer (2015) 27,544 0.115 0.037	0	1
	Calibration method	27,544 0.165 0.092	0	1
	Data Analysis Envelopment	27,544 0.507	0.2	0.132	1

Table 9 :

 9 Production function estimates

	**

See Lipsey and Carlaw (2000, 2004)

Statista, March 2021

In its simple definition, returns to scale represent the increase in efficiency as a result of the increase in production factors.

If γ = 0 then the returns are constant. If γ > 0 then the returns are increasing. If γ < 0 then the returns are decreasing.

The choice of production factors is not under the control of the econometrician, but determined by the individual choices of firms[START_REF] Griliches | Production functions: the search for identification[END_REF]. According to De Loecker (2007), this simultaneity bias is defined as the correlation between the level of production factors and the unobserved productivity shock.

There is a technology characterized by a linear relationship between intermediate inputs and output, and these intermediate inputs are proportional to output[START_REF] Gandhi | Does value added overstate productivity dispersion? Identification and estimation of the gross output production function[END_REF]).

These assumptions can be relaxed. However, we will need other assumptions to calculate a user cost of capital.

50 replicates are performed for the semi-parametric and[START_REF] Wooldridge | Econometrics: Panel Data Methods[END_REF] methods. We use investment as a proxy in[START_REF] Wooldridge | Econometrics: Panel Data Methods[END_REF].

Powered by TCPDF (www.tcpdf.org)

Average change in total factor productivity by size

The following graph shows the average evolution of the TFP obtained by each method according to the size of the firm. Regardless of the method used, large firms consistently show a high average variation in TFP. For FE, CM and DEA methods, the larger the size of the company, the higher the average variation in TFP. However, with both the LP and ACF methods, micro-firms have a higher average TFP change than medium-sized firms in recent years, and even the same as large firms in 2018 with the ACF method.