LEPTAB Cristian Ghiaus
email: cristian.ghiaus@usa.net

R Belarbi

F Allard

Univérsité De

La Rochelle

Av M Crépeau

Optimal Settings of Residential Oil Burners

Residential oil burners are capable of almost complete burning of the fuel oil, without visible smoke, when they are operated to deliver approximately 12% CO 2 in the flue gases. The positions of the air damper and of the combustion nozzle are adjusted at start-up and during operation in order to maximize the combustion efficiency. In practice, one factor at a time is varied, starting with the air damper. However, this method fails to detect the interaction between air excess and nozzle position and results in non-optimal settings. Optimal designed experiments allow obtaining local regression models and statistical analysis indicates if experiment augmentation is required. The air damper and combustion nozzle settings are changed in the direction of local gradient until a second order model, that contains the optimal point in its experimental region, is obtained. By using this approach, the gain in combustion efficiency may be 5% higher than by using the classical approach.

Introduction

Oil burner

An oil burner combines oil and air under controlled conditions for combustion; the resulted energy is then delivered to the heating system of the building. Operated with excess air as low as 20%, the oil burners are capable of almost complete burning of the fuel oil, without visible smoke. We try: 1) to maximize the burning efficiency and 2) to limit the pollutant concentration in the flue gases by controlling: 1) the smoke opacity, 2) the flue gases temperature and 3) the CO 2 percentage. Small flue gases opacity is desired but it indicates that air excess may be too high; although low polluting, this operating condition is not energetically efficient. The temperature of the flue gases should be as low as possible but higher than the acid dew point (160°C). The CO 2 percentage in the flue gases is a function of air excess. The combustion efficiency depends on the combination of these three parameters.

Siegart [START_REF] Chardot | Dupont Guide pratique du brûleur fioul[END_REF] gives a simplified relation of the combustion efficiency: The optimal settings of the oil burner, which consist in the positions of the air damper and of the combustion nozzle (Fig. 1), are when the combustion efficiency is maximal and the flue gases concentration in CO 2 is less than 12%. Preliminary adjustments of the air damper, based on the flame color and flue gases opacity, result in a near-optimal operating point. Then, the optimal settings are achieved by measuring the combustion efficiency and the concentration of CO 2 in the flue gases.

        - ⋅ - =
The residential oil burner studied (Fig. 1) presents an on-off combustion control system that starts and stops the fuel burner to satisfy the heat demand. The burner remains on until the controlled temperature of the outlet water reaches 85°C; at this temperature, the burner shuts off. The burner again ignites at 75°C. Although this combustion control system maintains the operating conditions in a limited domain, there is a systematic bias due to the variation of the outlet water temperature (Fig. 2a).

Our purpose is to elucidate the behavior of the burner near the optimal operating point by fitting an approximating function or model. The information needed is to be obtained by using as few experiments as possible.

Operating point optimization

Optimization consists in finding the set of operating conditions for the process variables that results in best performance. Generally, even rigorously formulated models have some unknown parameters, so experiments are used to acquire the data necessary for parameter estimation. In the case of oil burners, two settings are required: the positions of the air damper and of the combustion nozzle.

In practice, the one-factor-at-a-time procedure is used: first, the air damper position is changed and then the nozzle position. But the results obtained are misleading for two reasons: 1) the process parameters change in time with the increase of the outlet water temperature (Fig. 2a), and 2) the optimum point is not correctly localized due to the interaction between input variables. In Fig. 2b, for the nozzle position 0.8, the maximum combustion efficiency found by modifying the air damper position is 86.6%; then, by modifying the combustion nozzle position, the maximum efficiency found is 87.5%. The settings are not optimal. Optimal designed experiments and response surfaces overcome these two problems.

Based on experiments, we may construct the response surfaces of combustion efficiency and CO 2 concentration and use them to maximize the combustion efficiency. Response surface methodology is the most widely employed and successful optimization technique, developed in the early 1950s and initially applied in the chemical and process industries [START_REF] Montgomery | Introduction to Statistical Quality Control[END_REF]. However, research on optimal experimental design, which allows us to obtain the measurements necessary to construct the response surfaces, was started in 1970s. The main idea of this approach is to design experiments intended to estimate the parameters and the adequacy of a given model [START_REF] Atkinson | Optimum Experimental Designs[END_REF]. Optimal experimental design makes optimization techniques applicable to real engineering systems.

Response surface and the design of experiments

The experiments for finding the optimal operating point are sequential and follow the formal description of steepest ascent and second order designs [START_REF] Box | On the experimental attainment of optimum conditions (with discussion)[END_REF]. In the first stage, the experimenter decides on the input variables; in the case presented here, there are two: the positions of the air damper and of the nozzle. Then, based on experiments, local models are constructed. Following experiments are performed sequentially in the direction of steepest ascent. When the operating point is near the optimum, second order models are fitted; they may indicate a maximum in or near the experimental region.

In general, an experimental trial gives the measured values of responses, or output variables, y, and of m factors, or input variables, u. In order to estimate a local model with n parameters, at least N = n trials are necessary at distinct points in the experimental domain. However, if the input-output relationship is blurred by errors, more trials will be required (N > n) and regression will be used to fit the model; larger the degree of freedom,

n N v - =
, better the estimation. Statistical analysis can provide an estimation of both the variance of readings and of the residual sum of squares attributes to lack of fit.

Experimental design domain and scaled variables

Local models are obtained by varying the input variables, u, in a given domain; the variation range is a trade off between model accuracy and experiment efficiency. It is convenient to scale and center the variables to lie between -1 and +1. If the variation range is independent for each of the m factors, the experimental domain, in terms of the scaled variables, is an m-dimensional cube; for m = 2, the domain is a square. We consider hereafter that the input variables are scaled and centered.

Multiple regression models

For a system with p inputs and one output, when we make m experiments, we obtain ] ,..., , [

2 1 m y y y = y
values of the output of interest that correspond to

1 2 1 1 ] ,..., , [ p u u u = u , 2 2 1 2 ] ,..., , [ p u u u = u , …, m p m u u u ] ,..., , [ 2 1
= u values of inputs. The general least square (or linear regression) problem is to find out the unknown parameters β of the linear model [START_REF] Jang | Neuro-Fuzzy and Soft Computing[END_REF]:

) ( ... ) ( ) ( 2 2 1 1 u u u n n f f f y β β β + + + = , ( 1 )
where f 1 , … , f n are known functions of u. By transforming the input variables, we may note ) (

1 1 u x f = , … , ) (u x n n f =
. Writing equation [START_REF] Chardot | Dupont Guide pratique du brûleur fioul[END_REF] for m data pairs {(u i ; y i ), i=1, … , m} by using matrix notation, we obtain:

y X = β , ( 2 )
where

          = ) ( ... ) ( ... ... ... ) ( ... ) ( 1 1 1 1 m n m n f f f f u u u u X ,           = n β β β ... 1 et           = m y y ... 1 y
are respectively the information matrix of dimension n m × , parameters (regression coefficients) vector of dimension 1 × n and the output vector of dimension 1 × m . In order that equations ( 1) and ( 2) account for random noise and modeling error, an error vector e of dimensions [ ]

1 × m is added: y e X = + β . (3)
The least square problem is to find

β = b
that minimizes the sum of squared error,

e e y y y y X y X y x T T T i = - - = - - =         ⋅ - = ∑ ∑ = = ) ( ) ( ) ( ) ( ) ( ) ( 1 2 1 β β β β m i n j j j i f y E (4) where b X y = ˆ
is the estimated output. If the model is linear in parameters and X X T is nonsingular, b is unique and given by:

y X X X T T b 1 ) ( - =
.

( 5 )

Confidence intervals on the parameter estimates

Equation ( 5) gives the values of the regression coefficients (or estimated parameters), b. While the true value β is fixed, the estimated value, b, varies randomly from estimation to estimation. In order to find the confidence intervals on b we need the value of the error variance 2 σ ; we may find an estimation, s 2 , of the variance, 2 σ , by using the residual

y y e - = [6]: ∑ = - - = - = n i i i T y y p n p n s 1 2 2 ) ( 1 1 e e . ( 6
)
Then, the variance of the estimated parameter b i is:

∑ = = m j ij i x s 1 2 var β (7)
The )% 1 ( 100 α confidence limits for i β may be estimated by using the t distribution

for n N v - =
degrees of freedom:

i v i i t b β β α var , ± = . ( 8
)

D-optimal design of experiments

Let us consider that we opted for a given model on a given domain. An optimal designed experiment minimizes the variance of parameter estimates for the chosen model on the given domain. Under the assumption of independent errors of constant variance 2 σ ,

0 ) ( = e E and I ee 2 ) ( σ = T E
(Gauss-Markov conditions), the covariance matrix of the least squares estimates is:

1 2 ) ( var - = X X T b σ . ( 9
)
If interest is in the comparison of experimental designs, the value of 2 σ is not relevant since it is the same for all proposed designs for a specific experiment. The variance of b j is proportional to the jth diagonal element of

1 ) ( - X X T . The determinant X X X X T T / 1 ) ( 1 =
is called the generalized variance. Designs that maximize the value of the determinant of the Fischer's information matrix, X X T , are called D-optimal (D for determinant). Maximizing the determinant X X T is equivalent to minimizing the determinant of the parameter covariance matrix [START_REF] Atkinson | Optimum Experimental Designs[END_REF].

The optimal design of experiments is based on the General Equivalence Theorem, which can be viewed as an application of the result that the derivatives are zero at an extremity of a function. The function (the local model chosen) depends on the measurements through the information matrix, X. This theorem provides methods for asserting and checking the optimum location of the experimental points but it states nothing regarding their number. Practically, the number of experiments may be increased until the confidence multidimensional interval on the estimated parameters becomes acceptably small [START_REF] Atkinson | Optimum Experimental Designs[END_REF]. Hereafter, we give a method for testing if the augmentation of experiments is necessary.

The optimum design is found by searching over the design region. For simple problems, an analytical solution is sometimes possible [START_REF] Box | Factorial designs, the F F ' criterion and some related matters[END_REF]. In more complex situations, a sequential algorithm seeks the maximal increase of X X T . The search over the continuous experimental domain is replaced by a search over a list of candidate points. This list, which is a coarse grid in the experimental region, frequently includes the points of the Doptimum continuous design. The problem is then to choose from the available grid the m experimental points which maximize X X T .

The product of the algorithm for optimum experiment design is a list of input combinations. The set of inputs must be sampled in such a way as to avoid systematic bias. Randomizing the sampling in the input multidimensional space results in avoiding the confounding of input effects with those due to omitted variables, which are nevertheless important.

The optimum design experiment is not self-evident (Fig. 3). The classical approach is to vary one factor at a time (while keeping the other unchanged). This design does not minimize the parameter estimation error and does not allow finding the interaction between variables. To illustrate these two aspects, let us consider two designs (Table 1 and Fig. 3) for a linear model with interaction:

2 1 3 2 2 1 1 0 x x b x b x b b y + + + = . ( 10
)
Each measurement is affected by the same error, σ . Considering the theorem of variance applied in the case that all the variances are equal,

n x x σ σ = , (11)
and solving the system of equations, we find for the classical design:

4 ) ( 4 1 4 3 2 1 0 σ ± + + + = v v v v b , 2 ) ( 4 1 1 2 1 σ ± - = v v b , 2 ) ( 4 1 3 4 2 σ ± - = v v b (12)
and for the optimum design:

4 ) ( 4 1 4 3 2 1 0 σ ± + + + = w w w w b , 4 ) ( 4 1 4 3 2 1 1 σ ± + - + - = w w w w b (13) 4 ) ( 4 1 4 3 2 1 2 σ ± - + - - = w w w w b , 4 ) ( 4 1 4 3 2 1 3 σ ± + - - = w w w w b .
The classical design does not reveal the interaction term, b 3 . The optimal design gives the necessary data to obtain the model and a smaller parameter estimation error. The determinant X X T based on the information matrix, X, which is formed by the values of the variables of the model for every experiment (with rows corresponding to experiments and columns to variables in the model), gives us synthetically the same information (Fig. 3).

The analysis of experiments and design augmentation

The results of an experiment may be not sufficient for more reasons: 1. the model is inadequate; 2. the model does not give reproducible predictions.

In these cases, a solution is the augmentation of the number of experiments (Fig. 4). An adequate model should predict the results of new experiments and not change significantly its response surface when the number of experiments is increased (Fig. 5 and Fig. 6).

Inadequacies of the model are revealed during the course statistical analysis. The F value, obtained by the analysis of variance, and the R 2 statistics give a measure of the model adequacy. If the model is inadequate, it means that the range of the experimental domain is too large. Excessive changes in the response suggest that the domain range should be reduced while failure to observe an effect suggests that a larger domain range would be necessary. Ideally, the ellipsoidal contour levels should be circles or circle sectors (Fig. 7).

EXPERIMENTAL RESULTS AND DISCUSSION

We applied response surface methodology to optimize the operating settings of the oil burner described in section 1. Using a D-optimum design augmented from 8 to 13 trials (Fig. 4), we obtained the response surfaces for combustion efficiency and for CO 2 concentration in flue gases. The positions of both the combustion nozzle and of the air damper may be continuously varied on a domain marked from 0 to 10. A first experiment was achieved around the operating point of coordinates [2.5 2.5] in the experimental domain of [-1/2 1/2] and [- 1 1] for the positions of the air damper and of the combustion nozzle, respectively (Table 2 and Fig. 7 a).

When the response surface is evaluated considering the confidence intervals on the parameter estimates, two domain surfaces result, for the upper and lower limits of the confidence interval (Fig. 5). A cross-section through these surfaces is depicted in Fig. 6. The facts that the contour levels do not change significantly and that the confidence interval does not decrease when the experiment is augmented indicate that eleven experiments would suffice.

The confidence interval indicates the precision to be used in the positioning of the optimum (Fig. 8). We notice that the combustion efficiency may be estimated with a precision of ±1% and the CO 2 concentration with ±0.5%. These values should be considered when the contour lines of Fig. 7 are interpreted; two contour lines that differ with 1% in efficiency may not be distinct.

The gradient of the local quadratic model of the combustion efficiency

, ] 52 . 1 67 . 0 [ ) ( grad - - = η
, gives the direction for changing the operating point (Fig. 7a). The difference between the elements of the gradient vector indicates that the unitary variation for the second input variable should be reduced. A second trial is achieved in the point [3.0 2.75] by defining the input variables in the domain [-1/4 1/4] and [-1 1] for air damper and combustion nozzle, respectively. The resulted response surface is shown in Fig. 7b.

The model obtained based on the second experiment (Fig. 7b) indicates the optimum, which in the same time satisfies the imposed condition (CO 2 concentration less than 12%). By changing the position of the combustion nozzle to 2.75 and of the air damper to 2.75, the measured combustion efficiency becomes 90% and the CO 2 concentration 11.7%. This is the optimal operating point attainable by varying the input variables and represents a gain of 2.5% as compared with the classical approach that consists in varying one variable at a time (Fig. 2b). Nevertheless, by changing the operating conditions (such as clearing up the boiler or using another type of fuel), the optimal point might change. In this case, the procedure for optimal settings should be applied again.

CONCLUSION

The experimental finding of optimal settings by using the response surfaces is very suitable to residential oil burners. The combustion efficiency depends on the positions of the nozzle and of the air damper upon a smooth function. This type of dependence makes the second order models appropriate; however, the slight curvature of the function, corroborated with the confidence intervals on the parameter estimates, introduces uncertainty in the exact optimum location. Practically, the optimal operating point may be situated anywhere in the region delimitated by the contour line corresponding to the maximum efficiency minus 1%. For the same experimental effort, the gain in combustion efficiency may be 5% higher than the value obtained by using the classical approach that consists in varying one input at a time. -1

1 -1 0 1 (1) (2) (4)
(3)

x 1 x 2 -1 1 -1 0 1 (1) (2)
(4) (3)

x 1 x 2 2 1 2 1 x x x x I 2 1 2 1 x x x x I             + + - + + + - + = 0 1 0 1 0 1 0 1 0 0 1 1 0 0 1 1 X             + + + + - + - + - - + + + - - + = 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 X 0 = X X T 4 4 = X X T Experiment number (1) (2)
(3) (4) -1

Experiment number

1 1 0 1 x 1 x 2 -1 1 -1 0 1 x 1 x 2 -1 1 -1 0 1 x 1 x 2 -1 1 1 0 1 x 1 x 2 -1 1 -1 0 1 x 1 x 2 -1 1 -1 0 1 x 1 x 2 N=11 N=12 N=13 N=8 N=9 N=10

  air temperature [C]; CO 2 CO 2 concentration in flue gases [%]; f coefficient depending on air excess ( 56 . 0 ≅ f for air excess of 20%).

Fig. 1

 1 Fig. 1 Oil burner (adapted from [1]).

Fig. 2

 2 Fig. 2 One-factor-at-a-time method: a) Combustion efficiency changes with the outlet water temperature; b) Optimization.

Fig. 3

 3 Fig. 3 Experiment design for a linear model of the form y=b0 + b 1 x 1 + b 2 x 2 + b 3 x 1 x 2 .a) One-factor-at-a-time method. b) Optimum designed experiment.

Fig. 4 Fig. 5 Fig. 6 Fig. 7 Fig. 8

 45678 Fig. 4 Augmentation of D-optimal designed experiments for second order response surface from 8 to 13 trials. The trial number is indicated next to the points.

Table 1 Two designs of experiments.

 1

		Classical design			Optimal design
	x 1 -1 +1 0 0	x 2 0 0 -1 +1	y v 1 σ ± v 2 σ ± v 3 σ ± v 4 σ ±	x 1 -1 +1 -1 +1	x 2 -1 -1 +1 +1	y w 1 σ ± w 2 σ ± w 3 σ ± w 4 σ ±

Table 2 Experimental results obtained by varying the air damper and the combustion nozzle positions around the point [2.5 2.5] with ½ and 1 division respectively.

 2

			Inputs		Outputs
	Unscaled	Scaled
	Nozzle Damper Nozzle Damper	Efficiency	CO 2
	div.	div.	-	-	%	%
	2.0	1.5	-1	-1	92.8	13.4
	2.0	2.5	-1	0	90.5	12.8
	3.0	1.5	1	-1	90.8	12.7
	3.0	3.5	1	1	88.3	11
	3.0	2.5	1	0	89.4	12.1
	2.5	1.5	0	-1	90.7	13
	2.5	2.5	0	0	89.6	12.2
	2.0	3.5	-1	1	89.2	12.1
	2.5	3.5	0	1	87.7	11.3


Pierpaolo Parrotta

Dario Pozzoli
email: dpozzoli@asb.dk.

Tor Eriksson

Lars Geerdsen

Sanne Hiller

Hans Kongsted

Joep Konings

Simone Moriconi

Michael Rosholm

Davide Sala

Valerie Smeets

Måns Soderbom

The Effect of Learning by Hiring on Productivity

Keywords: JEL Classification: J24, J60, L20 Knowledge transfer, knowledge carrier, labor mobility, learning-by-hiring

published or not. The documents may come   L'archive ouverte pluridisciplinaire

Introduction

Knowledge can be defined as a fundamental resource employed in all valuable human activities. Its identification as an economic good has, for a long time, commanded considerable attention in research efforts (Antonelli, 1995). A group of models belonging to New Growth Theory suggested different frameworks to understand and describe the role of knowledge in driving productivity and thereby fostering economic growth. Several macroeconomists (Romer, 1990; Grossman and Helpman, 1991; Aghion and Howitt, 1992) have identified the transmission of knowledge as main driver of sustained economic growth.

A potential channel of knowledge transfer is inter-firm labor mobility. Most of a firm's knowledge set is not codified but is characterized by a tacit nature. A valuable part of such a tacit, or less articulated, knowledge is likely to be embodied in highly-educated or skilled workers, a knowledge acquired through job tenure.

Thus, moving from one firm to another, these individuals carry the acquired knowledge and apply it to new contexts. It can hardly be transmitted in a different way, as it is not codified but is inherent in the routines and procedures of a firm's production. Moreover, the transfer of this acquired knowledge, which informally belongs to the firm, cannot be effectively prevented by patents or other forms of intellectual property rights, which can only protect forms of codified or, at least, articulated knowledge (Petit and Tolwinski, 1996; Gilson, 1999). This process of knowledge transmission via inter-firm labor mobility is known as learning-by-hiring (LbH).

In the literature, very little evidence exists of the effect of LbH on firm performance. Previous work has found a positive impact of specific categories of recruited workers on firms' patenting activity. However, the distribution of patenting firms is strongly concentrated and knowledge transfers may or may not influence patenting activity. But because acquired knowledge can affect firm productivity and it is a broader measure of performance, it seems opportune to investigate the effects of LbH on such a measure. This allows us to take a large number of firms into account and thereby include a representative sample of both production and service sectors. This article broadens earlier investigations to include the variable LbH, and studies how workers newlyrecruited from other firms affect productivity in the hiring firm. Specifically, the analysis focuses on the role of employees with a tertiary (Bachelor's, Master's or post-graduate degree) or vocational/technical education. Furthermore, we impose that these knowledge carriers receive a higher wage than that received in their previous job. These higher wages (i.e. premium) signal the firm's willingness to acquire the knowledge carried by the recruited worker.

However, the receipt of knowledge transfer does not occur simply with the recruitment of a knowledge carrier. A successful transfer may depend on two relevant features: intensity and proximity. Intensity is defined as a concave function of the number of workers coming from a given donor firm, and proximity weights each worker's hire by the technological closeness between the donor and hiring firm pair (Jaffe, 1986; Adams, 1990; Inkmann and Pohlmeier, 1995; Cincera, 2005). Proximity is computed in a metric space generated by vectors, the elements of which are firms' shares of patent applications in the eight main technological areas defined by the International Patent Classification (IPC). [START_REF] Ackerberg | Structural Identification of Production Functions[END_REF] The production function parameters are estimated in accordance with the algorithm proposed by Ackerberg et al. (2006), who address both collinearity and endogeneity issues. Specifically, firms are allowed to observe productivity shocks before hiring knowledge carriers, which ensures that the contribution of this input to productivity is estimated properly.

Findings from our empirical analysis support our definition of knowledge carriers, and are confirmed by several robustness checks. The effect we are interested in, i.e. how knowledge transfers enhance firm productivity, is significantly positive and the inclusion of intensity and proximity in the LbH measure seems to better explain productivity. Our findings also show that the LbH contribution to productivity is relevant: a unitary increase induces on average a growth between one and two percent in firms' productivity.

Overall, the results are consistent with New Growth theoretical models, which place a strong emphasis on knowledge diffusion. In particular, our findings suggest improvements in the modeling of the process of knowledge transfer at the micro level, stimulating further analyses on the additional effects associated with inter-firm labor mobility.

The structure of the article is as follows: Section 2 briefly reviews the relevant literature; Section 3 describes the data; Section 4 provides details on the empirical strategy; Section 5 presents the main results; Section 6 presents the sensitivity analysis; and Section 7 presents our conclusions and suggestions for further research.

Literature background

LbH refers to valuable knowledge transferred via inter-firm labor mobility. It is defined as the acquisition of knowledge from other firms through the hiring of experts (Song et al., 2003). In accordance with this definition, the benefits of LbH should not be confused with either the usual high labor productivity engendered by newly-employed experts or potential externalities associated with scale effects. As we have defined here, inflows of knowledge carried by technicians and highly-educated workers are typically forms of developed but not codified knowledge. As a consequence, what is carried by these mobile workers cannot spill over freely.

Hence, the impact of LbH can only be precisely quantified if a donor and a recipient firm are identified.

Without a matched employer-employee register, then, it is not possible to estimate the contribution of this knowledge transfer through any kind of firm performance indicator. Though such registers are now available, the use of inter-firm labor mobility as a learning mechanism has not, until now, received particular attention.

Although their study is not conducted at the firm level, Almeida and Kogut (1999) find that the mobility of engineers holding major patents affects the intra-and inter-regional pattern of patent citations, which is considered a proxy of knowledge flow. Mapping US semiconductor plant clusters through the use of county level establishment and employee data, they illustrate how the mobility of engineers partially explains innovations occurring among regional clusters. Rosenkopf and Almeida (2003) take this one step further.

Tracking both inter-firm mobility of engineers and patent citations in the US semiconductor industry, their analysis concludes that the effectiveness of labor mobility in terms of knowledge flows increases with the degree of technological distance between firms. These findings are confirmed and reinforced by Song et al. (2003).

In this work they investigate the conditions under which labor mobility is more likely to facilitate knowledge transfers. Results seem to support the hypotheses that LbH is more effective (in terms of contribution to patent production) when (i) the recipient firm is less path dependent;2 (ii) mobile engineers carry knowledge distant from that characterizing the recipient firm; and (iii) knowledge carriers work in non-core technological areas in the recipient firm. Thus, inter-firm labor mobility may mitigate difficulties of learning in firms characterized by research in distinct technological areas and increase the possibility of benefiting from external knowledge.

A different perspective is investigated by Kim and Marschke (2005). Using a sample of US firms, they analyze how the risk of a key employee's departure reduces the firm's R&D expenditure and/or increases its patenting propensity. Their findings are consistent with the statement that firms use patents to minimize the negative effect associated with the departure of an engineer or a scientist and helps to explain different propensities among firms to patent. Maliranta et al. (2010) provide further evidence that R&D workers transmit valuable knowledge. Using Finnish employer-employee data, the authors find that firms hiring workers previously engaged in R&D activities may increase productivity and profitability in the hiring firm even if these workers do not perform R&D activity in their new job. This finding is interpreted as evidence that mobile R&D workers may transmit knowledge that can be readily copied and implemented even when not engaged in a strong research effort.

A crucial study for the work presented in this article is Kaiser et al. (2008). They constructed a data set in which the number of patent applications sent by Danish firms to the European Patent Office (EPO) was matched to the Danish employer-employee register. They looked at how labor mobility affects innovation in Danish firms, and specifically how the composition and past patenting experience of labor inflows influence firm-level patenting activity. They split the firm workforce into "R&D workers" and "non-R&D workers".

Whereas the former class identifies employees with Bachelor's or Master's degrees in the natural sciences and technical fields, the latter one identifies individuals with the same level of education in the humanities.

The results of their analysis support the idea that mobile R&D workers contribute more to firm patenting activity than immobile R&D workers do. This effect is stronger in the case where the R&D worker has been previously employed in a patenting firm.

Although Kaiser et al. (2008) more formally assess the quantitative effects of labor mobility on innovation, they only focus on patenting activity as a proxy for innovation efforts. Because they constrain the amount of knowledge transferred to patent applications or grants only, the entire fruit of a knowledge transfer is only related to patents. Moreover, since the patenting activity is particularly concentrated among firms (this holds even more strongly in Denmark), the number of firms observed in the sample may not be representative of the entirety of production activities in which relevant, valuable knowledge is inherent.

Data

Data sources

The data set used in our empirical analysis has been constructed by merging information from three different main sources. The first data source is the "Integrated Database for Labor Market Research" (IDA) provided by Statistics Denmark. IDA is a longitudinal employer-employee register containing valuable information (age, demographic characteristics, education, labor market experience, work experience and earnings) on each individual employed in the recorded population of Danish firms during the period 1980-2005. Apart from deaths and permanent migration, there is no attrition in the data set. The labor market status of each person as of November 30 is recorded as the relevant datum for each person for that year. The retrieved information has been aggregated at the firm level and consequently merged with variables such as firm location (county), size and related industry. [START_REF] Aghion | A Model of Growth through Creative Destruction[END_REF] The second data source is "Regnskab", which lists firms' business accounts, as provided and compiled by Statistics Denmark. It covers the construction industry from 1994, manufacturing from 1995, wholesale trade from 1998, and the remaining part of the service industry from 1999. Data in "Regnskab" are aggregates of yearly financial items, which are crucial for the estimation of the production function. In particular, it is possible to retrieve information on sales, intermediate goods or materials, fixed assets and profits. [START_REF] Almeida | Localization of Knowledge and the Mobility of Engineers in Regional Networks[END_REF] The third data source identifies patent applications and grants awarded to Danish firms by the EPO during the period 1978-2003. Access to such data has been made possible by the Centre for Economic and Business Research (CEBR). [START_REF] Antonelli | The Economics of Localized Technological Change[END_REF] A total of 12,109 patent applications has been recorded, and includes 2,822 Danish non-person patent applicants. [START_REF] Blundell | GMM estimation with persistent panel data: an application to production functions[END_REF] This allows us to compute measures of technological proximity based on the number and areas of firms' patent applications.

Variables

This section describes the variables used in the empirical analysis. As the purpose of this article is to provide evidence concerning the impact of LbH on firm productivity, attention is devoted primarily to measures indicating inter-firm knowledge transfers via labor mobility.

Combining information from these sources, the final data set allows for the mapping of all mobile workers aged 18 to 60. To identify knowledge carriers, we focus only on employees holding either a tertiary (at least, Bachelor's degree) or a vocational/technical education. In addition, we require that i) the mobile worker has experienced a real [START_REF] Cincera | Firms' Productivity Growth and R&D Spillovers: An Analysis of Alternative Technological Proximity Measures[END_REF] wage increase of at least five percent after moving; ii) the real annual wage of the current job is higher than the average of real wages received in the previous three years; [START_REF] Gilson | The Legal Infrastructure of High Technology Industrial Districts: Silicon Valley, Route 128, and Covenants not to Compete[END_REF] iii) the incomer's wage is greater than the average wage recorded in the recipient firm; iv) the donor firm is not downscaling the labor force; 9 and v) the period of unemployment preceding the start of the new job is less than 3 months. These conditions can be interpreted as a signal of a precise and intentional recruitment strategy implemented by the recipient firm. In fact, it is assumed that every recipient firm, aware of the expected benefit deriving from the recruitment of knowledge carriers, is willing to pay an opportune wage premium.

Three measures of inter-firm knowledge transfers have been defined for this article. The first measure is constructed as the sum (count) of all knowledge carriers moving from donor firm d to recipient r:

LbH_count r = D d=1 k dr ,
where k dr is the total number of knowledge carriers moving from a donor firm d to a recipient firm r and D represents the total number of donor firms.

Though quite similar to the first measure, the second one accounts for the hiring of more than one knowledge carrier from a single donor firm and weights the number of knowledge carriers using a concave function.

facilitate responses, questions are formulated similarly to those in the Danish annual accounts legislation. [START_REF] Antonelli | The Economics of Localized Technological Change[END_REF] An independent research centre affiliated with the Copenhagen Business School (CBS). [START_REF] Blundell | GMM estimation with persistent panel data: an application to production functions[END_REF] Patenting firms are here considered for the period 1995-2003 and belong to the following industries: manufacturing of food, beverages and tobacco; manufacturing of textiles and leather; manufacturing of wood products and printing; manufacturing of chemicals, plastic products; manufacturing of other non-metallic mineral products; construction; wholesale, except motor vehicles; transport; finance and insurance; rentals and sale of real estate; and business activities. More detail concerning the construction and composition of the dataset can be found in Kaiser et al. (2005). [START_REF] Cincera | Firms' Productivity Growth and R&D Spillovers: An Analysis of Alternative Technological Proximity Measures[END_REF] The annual wage is deflated using the price deflator for the year 2000. [START_REF] Gilson | The Legal Infrastructure of High Technology Industrial Districts: Silicon Valley, Route 128, and Covenants not to Compete[END_REF] Or two, in the case the third year is missing. [START_REF] Grossman | Innovation and Growth in the Global Economy[END_REF] The yearly reduction in the total labor force is imposed to be smaller than one percent.

6

Concavity is ensured by applying a concave transformation to the total of qualified workers (technicians and highly-educated employees) coming from donor firm, d:

LbH_concave r = D d=1 k [(k dr -1)/2k dr ] dr .
This implies that LbH_concave r increases less than proportionally with respect to the total number of qualified workers leaving a given donor firm, d, and being employed in the same recipient firm, r. The first and second measures coincide in the case that only one worker moves between a specific donor-recipient firm pair. This means that the concave version adjusts according to the intensity of each inter-firm knowledge inflow: two workers from the same donor firm will together transfer less than double the amount of knowledge a single worker transfers. Hence, a larger amount of embodied knowledge is brought by the first carrier from a given donor firm. The second measure also attributes more public good features to the knowledge inflow, even though it is still characterized by partial-excludability10 and a low degree of appropriability.

The third and most complete measure introduces a further element represented by the technological proximity index ψ:

LbH_prox r = D d=1 ψ (d, r) k [(k dr -1)/2k dr ] dr .
This allows us to correct the intensity of inter-firm labor mobility for the technological proximity between donor and recipient firms (Jaffe, 1986; Adams, 1990; Inkmann and Pohlmeier, 1995; Cincera, 2005). Thus, the level of exploitability associated with labor mobility could depend on the degree of similarity in the technological knowledge characterizing the donor-recipient firm pair. To measure the proximity between firms' technological capabilities, a technological space needs to be defined. In the present work, the generator vector f is composed of elements reflecting firms' stocks of patent applications in the eight main IPC technological areas noted earlier. Technological proximity here is computed using the uncentered correlation suggested by Jaffe (1986):

ψ (d, r) = f d f r f d f d (f r f r ) 1/2 .
If the donor and recipient firms coincide perfectly in the generated technological space, then ψ (d, r) = 1.

Conversely, if they do not overlap at all, the weight takes on the value 0. However, it is worth underlining that technological proximity can only be used for patenting firms.

In addition to these three described measures, the empirical analysis includes the following variables: valued added [START_REF] Jaffe | Technological Opportunity and Spillovers of R&D[END_REF] , materials (intermediates), capital stock (fixed assets), labor force, dummies for counties, [START_REF] Kaiser | Labor Mobility and Patenting Activity[END_REF] industries 13 and years. Using the price deflator for the year 2000, the monetary values were converted in real terms. Additional explanatory variables include shares of specific categories of employees.

Descriptive statistics

With data and variables described, descriptive statistics can be discussed. In the following and the main analysis that follows, we exclude (a) firms with imputed accounting variables and (b) values referring to the last (99th) percentile of the LbH distribution. This exclusion strengthens the reliability of our empirical investigation. However, firms with imputed accounting variables are later taken into account in the sensitivity section, to check their actual influence on the main results.

Table 1 reports number of observations, median and mean values, and standard deviations of the accounting, workforce and sector variables, for the whole sample and by firm size. Size is classified by three categories: Size1, Size2 and Size3, referring to firms with less than 50, between 50 and 99 and more than 99 employees, respectively. Despite the large number of observations, more than 95 percent are accounted for by firms with less than 50 employees. This size distribution of firms is consistent with the structure of the Danish industrial private sector, which is largely dominated by small-sized firms. [START_REF] Kim | Labor mobility of scientists, technological diffusion, and the firm's patenting decision[END_REF] Although not all industries were recorded during the first years of the sample period, percentages reported among industries approximately reflect aggregate data for the same time span. Whereas observations for the construction and manufacturing industries are over-represented, the opposite is true for the financial and business sectors. However, that is entirely explained by the unbalanced structure of the data set: construction and manufacturing were the only industries recorded in the first years, while financial and business were included only in the last ones.

The distance between median and mean values for the accounting variables (reported in thousands of real DKK) and workforce variables is considerable, indicating a certain degree of skewness (and polarization)

among firms in the sample. This is also confirmed by the notable size of the standard deviations. Whereas Size1 firms account for the highest shares of workers with vocational education, Size2 firms account for the largest proportions of managers and younger employees, and Size3 firms employ the highest proportion of [START_REF] Jaffe | Technological Opportunity and Spillovers of R&D[END_REF] It is computed as the difference between total sales and materials. [START_REF] Kaiser | Labor Mobility and Patenting Activity[END_REF] Copenhagen, Frederiksborg, Roskilde, West Zealand, Storstrom, Bornholm, Funen, South Jutland, Ribe, Vejle, Ringkobing, Aarhus, Viborg, North Jutland. [START_REF] Kaiser | Patenting Activity in Denmark[END_REF] Manufacturing of food, beverages and tobacco; manufacturing of textiles and leather; manufacturing of wood products and printing; manufacturing of chemicals and plastic products; manufacturing of other non-metallic mineral products; manufacturing of basic metals and fabricated metal products; manufacturing of furniture; manufacturing n.e.c.; construction; sale and repair of motor vehicles, sale of automotive fuel; wholesale except for motor vehicles; retail trade of food; department stores; retail sale of pharmaceutical goods and cosmetic articles; retail sale of clothing and footwear; other retail sale, repair work; hotels and restaurants; land transport and transport via pipelines; water transport; air transport; supporting transport activities; post and telecommunications; finance; insurance; activities auxiliary to finance; real estate activities; renting of transport equipment and machinery; computer and related activities; research and development; consultancy activities; and cleaning activities. [START_REF] Kim | Labor mobility of scientists, technological diffusion, and the firm's patenting decision[END_REF] The structure of the Danish firm population is mainly composed of small and medium-sized companies. Enterprises with less than 50 employees account for 97 per cent of the total number of firms and represent 42 per cent of the total employment in manufacturing and services. Thus, it seems reasonable to distinguish between SME and larger enterprises using a limit of 100 employees. However, compared to the other Nordic countries, Denmark presents fewer micro firms (1-9 employees): these represent 87 per cent of firms against 92 and 94 percent recorded in Finland and Sweden, respectively (OECD, 2005).

highly-educated workers. The defined variables change in value as expected among the definitions of inter-firm labor mobility. The count measure presents the higher mean value, and both count and concave measures increase their standard deviation with respect to firm-size. LbH_prox is not reported in Table 1 because it refers only to firms that applied for a patent at least once during the observation period.

Table 2 reports the median, mean and standard deviation of the main variables according to firms with and without patents. Patenting firms typically exhibit a larger size and significantly higher accounting values than non-patenting ones. The value of average capital stock is about 25.8 times that of non-patenting firms.

Significant differences in worker skill levels are also noted for the two firm groups. Larger shares of highlyeducated employees are recorded for the patenting firms; larger shares of technicians or vocational workers are recorded for the non-patenting firms.

Empirical strategy

Issues on production function estimation

The literature on firm-level production functions has identified a number of problems associated with the estimation of input productivity. The most common issue with the estimation of such parameters concerns simultaneity, i.e. the endogeneity of inputs. This occurs when productivity shocks, unobserved by econometricians, are observed by firms and allow management to reallocate resources in the production process accordingly. If that is the case, such shocks may be taken into account by firms to maximize their profits or minimize their costs. Specifically, positive (negative) productivity shocks induce firms to expand (reduce) output, which requires a higher (lower) quantity/quality of inputs for production. This implies that OLS estimates of coefficients on inputs observed by econometricians are biased and inconsistent, and error terms and regressors are correlated. Moreover, it is widely acknowledged that whereas fixed-effects (FE) estimation techniques (Mundlak, 1961) take account of firm heterogeneity, they do not solve the simultaneity problem when productivity shocks fluctuate over time.

Several remedies have been proposed to cope with simultaneity, including the recent structural approach advocated by Olley and Pakes (1996) (OP henceforth) and Levinsohn and Petrin (2003) (LP henceforth). [START_REF] Levinsohn | Estimating Production Functions Using Inputs to Control for Unobservables[END_REF] Both OP and LP suggest semi-parametric methods based on (i) the identification of a proxy variable, which is assumed to be a function of time-varying productivity shocks (total factor productivity) and (ii) the definition of conditions under which this function is invertible. The idea behind this is to infer total factor productivity by using observed firms' input choices.

OP present a two-step estimation method: in the first step, semiparametric methods are used to estimate the coefficients on the variable inputs along with the nonparametric function linking productivity to capital [START_REF] Levinsohn | Estimating Production Functions Using Inputs to Control for Unobservables[END_REF] See Ackerberg et al. (2006) for a survey. and investment; in the second step, parameters on capital inputs are identified under the assumptions of the dynamics of the productivity process [START_REF] Maliranta | Is Inter-Firm Labor Mobility a Channel of Knowledge spillovers? Evidence from a Linked Employer-Employee Panel[END_REF] (Wooldridge, 2009). However, OP's estimation method presents two major drawbacks. First, because adjustment costs create lumpiness in investment levels, these levels may not respond smoothly to productivity shocks. Second, OP approach excludes firms reporting zero investment levels: it induces a de facto truncation bias. To overcome these drawbacks, LP use a measure of intermediate inputs as a proxy for investment levels. Many benefits are associated with this choice. First, changes in intermediate inputs do not typically involve adjustment costs; intermediate inputs therefore respond better to productivity shocks than investments do. Second, intermediate inputs provide a simple link between estimation strategy and economic theory because they do not typically represent state variables. Third, because intermediate inputs are almost always used in production, LP's approach circumvents the abovementioned data truncation problem. Moreover, LP suggest three specification tests for evaluating the proxy's performance (Petrin et al., 2004). However, the coefficient on the proxy is recovered at the second stage rather than in the first, as in the OP approach.

Although, the OP and LP approaches have been broadly used for identification of the production function, each approach is vulnerable to collinearity and identification problems. As pointed out by Söderbom and Bond (2005) (SB henceforth), both approaches can perform poorly as they do not fully control for unobserved productivity shocks. SB provide evidence showing the impossibility of identifying the parameters from crosssection variation if all inputs are perfectly flexible and factor prices are constant. Similarly to SB, Ackerberg et al. ( 2006) (ACF henceforth) explain how collinearity between labor and non-parametric terms (capital and proxy variable) interferes with the identification of the labor coefficient. To address these collinearity and identification issues, ACF suggest an alternative estimation procedure in which labor coefficients are estimated in the second stage. [START_REF] Mundlak | Empirical Production Function Free of Management Bias[END_REF] A key point of the ACF procedure is that it allows the estimation of production functions in which all inputs are assumed not to be perfectly variable. In this case, a first stage is needed just to net out the error component in the production function (Ackerberg et al., 2006).

Empirical model

In our analysis, productivity is obtained from a Cobb-Douglas production function with additive firm heterogeneity, where the real value added Y is the dependent variable. The production factors are labor force L, capital K , our central variable, LbH, additional variable inputs X (shares of managers, wage earners, males, highly-educated workers, technicians, and differently aged workers belonging to the employees' age distribution quartiles). We also specify control variables, C, identified by partial/total foreign ownership, year, 3-digit industry classification and regional dummies because they can potentially affect productivity. For the sake of simplicity, we omit these controls, C, from the following log-linear production function definition:

lnY it = cons + αlnL it + βlnK it + γLbH it + δX it + u it .
The variable L represents the total number of workers in a given firm, and includes potential knowledge carrier inflow, LbH. The coefficient on LbH, then, should not be interpreted as (confused with) human capital or partial labor force contribution to firm productivity: it is the effect of the valuable knowledge transferred via labor mobility to the hiring firm. The error term u it represents a time-varying firm-specific effect, v it (unobserved by econometricians), and an idiosyncratic component, ε it . Specifically, we assume that inputs are uncorrelated with ε it :

E (ε it | l it , k it , LbH it , X it , m it , l it-1 , k it-1, LbH it-1 , X it-1 , m it-1 , ..., l i1 , k i1 , LbH i1 , X i1 , m i1 ) = 0 ,
with t = 1, 2, ..., T, and where m refers to our proxy variable (materials) and lower-case letters to log-variables.

Because past values of ε it are not accounted for in the conditioning set, we allow for serial dependence in the pure shock term. However, we need to restrict the dynamics in the productivity process:

E (v it | v it-1 , v it-2 , ..., v i1 ) = E (v it | v it-1 ) ≡ g (v it-1 )
with t = 1, 2, ..., T . As in ACF's approach, we assume material input to be chosen after labor input. In addition, we assume that LbH and the additional variable inputs, X , are set before or at the same time as material input is chosen. As a result, material demand will not only be a function of capital and productivity, but also of l, LbH and X:

m it = f (k it , v it , l it , LbH it , X it )
and assuming that the material demand function is strictly increasing in productivity shock v it , we get

v it = f -1 (k it , m it , , l it , LbH it , X it ) .
The key advantage of this approach is that it allows, for example, our key variable, LbH, to have dynamic implications or to depend on unobserved input price shock that could potentially be serially correlated over time. In fact, it seems reasonable to assume that the hiring and firing costs for labor or the fixed costs of setting up the recruitment and hiring of a knowledge carrier can last longer than a period. Plugging the inverse material demand into the production function, we get the first-stage equation, which here serves only to separate v it from ε it ,

y it = cons + αl it + βk it + γLbH it + X it + f -1 (k it , m it , l it , LbH it , X it ) + ε it .
The function f -1 (•) is proxied with a polynomial in materials, capital, labor, LbH it and X it . Thus, the estimated output net of the idiosyncratic component is used to identify parameters on inputs in the second stage. Recalling that v it is a first-order Markov process, we define a it as an innovation that can be correlated with current values of the proxy variable m it and inputs l it , LbH it and X it :

a it = v it -g (v it-1 ) ,
where a it is mean independent of all information known at t -1 and g (•, •) is proxied also with a low-degree polynomial in dependent variables. Given our timing assumption, we suggest using the moments:

E          a it | k it l it-1 LbH it-1 X it-1          = 0
to identify coefficients on k ,l , LbH , and X.

Results

The main findings from the empirical analysis are reported in Table 3. Columns 1 and 4 show capital and labor elasticities for non-patenting and patenting firms, respectively, for specifications that include l, k, and control variables, C, only. These elasticities do not substantially differ from elasticities reported for specifications in columns 2, 3, 5, 6 and 7, all of which include a measure of LbH and X. [START_REF] Olley | The dynamics of productivity in the telecommunications equipment industry[END_REF] A higher (lower) capital (labor) contribution is recorded for patenting firms.

Coefficients associated with the LbH measures in columns 2, 3, 5, 6, and 7 represent semi-elasticities since our variables are not in logarithms. This choice is founded on the fact that LbH takes on the value 0 for the majority of observations. Thus, any possible transformation of LbH measures could possibly affect the associated estimated parameters. As expected, LbH affects firm productivity positively, and its contribution is larger once intensity and proximity of transfers are taken into account. LbH semi-elasticity is always significant: it varies between one and two percent for the count and concave indexes. The increase in the coefficient on LbH is particularly high when technological proximity is introduced, which, however, implies the exclusion of non-patenting firms. The comparison of the magnitudes of the coefficients on the LbH measures across different specifications is not informative for deciding which measure better explains the effect of learning-by-hiring on productivity. To determine the best measure, we need to compare the goodnessof-fit for each reported specification that includes an LbH component. Following Windmeijer (1995), the goodness-of-fit here is computed as the correlation between predicted and actual values of the dependent variable. Comparing columns 4, 5, 6 and 7, we find slightly larger improvements in the goodness-of-fit when the technological proximity between donor and recipient firm is accounted for. That supports the relevance of the technological proximity weight ψ (d, r) when calculating the LbH index.

However, the relevance of our findings crucially depends on whether the donor firms experience eventual productivity losses due to the outflow of qualified employees. In fact, gains from LbH may simply compensate such losses, describing only one side of a zero-sum phenomenon. Results reported in columns 8, 9, and 10 of Table 3 are not consistent with a hypothesis of productivity losses for donor firms. Specifically, the parameter on the variable measuring the outflow [START_REF]Economic Survey of Denmark[END_REF] of qualified workers is close to zero and never statistically significant.

In these regressions, we treat employee outflow as an exogenous variable given that the knowledge carrier decision to be recruited by another firm is not affected by donor firm actions. Parameters on the LbH measures remain significant and stable. In most cases, the parameters increase slightly in value. Table 4 reports findings from regressions at the 1-digit industry level. Again, we evaluate the effects on productivity using the three LbH measures, though the estimation of the parameter on LbH_prox can only be carried out for manufacturing and financial & business service sectors because of the low number of observations of patenting firms in the other industries. It is found that the LbH contribution to productivity varies as expected across industry sectors. Specifically, the contribution is particularly high for the financial & business service sector, but not always significant for the construction and transport sectors. Positive and slightly below the overall average, LbH effects are recorded also for manufacturing and wholesale & retail trade sectors. Overall, the relevance of the LbH contribution to firm productivity is confirmed in this analysis at the industry level.

Sensitivity analysis

Several, precise conditions have been imposed on the definition of knowledge carriers in order to capture as precisely as possible the real contribution of knowledge transfer to firm productivity. Aware that the criteria [START_REF]Economic Survey of Denmark[END_REF] Obviously, the concavity and proximity features do not apply for outflows. Therefore, employee outflow is a count measure. and conditions have been reasonably assumed, but assumed nonetheless, in this section we refine or relax these criteria and conditions, and then check whether such modifications influence the size and significance of the LbH effects.

We implement five refinements: (i) we impose a yearly wage premium of at least 10 percent for knowledge carriers-this increases the strength of the wage increase condition (previously set at five percent) on the willingness to be recruited; (ii) a zero-downsizing of the firm labor force is held at zero. In the main analysis downsizing of the firm labor force is limited to lower than one percent; (iii) we exclude all knowledge carriers with vocational education as well as those with degrees in the humanities and social sciences; (iv) knowledge carriers moving from part-time to full-time jobs are disregarded; and (8) there is no unemployment period between jobs, in contrast to the maximum period of three months allowed in the original analysis. The results of these refinements are presented in Table 5. In refinements iii-v, the coefficients on the LbH effects improve.

In refinement i, LbH_prox increases substantially but is not statistically significant. Refinement ii does not result in any substantial change in parametric value nor statistical significance.

We implement only one but crucial relaxation: the wage criteria are excluded. This allows us to check whether the increase in the carriers' wages, and the positive differentials between them and the average wages in the recipient firm, are the only drivers of higher firm productivity. Our findings reject such a hypothesis.

Although knowledge carriers typically are highly productive, their contribution to firm productivity in terms of knowledge transfers may not be imputed to their labor productivity, which is often associated with the earned wage.

As a further check, we evaluate how parameters on LbH measures change if we do not exclude firms with imputed values in for the accounting variables. The first three columns of Table 6 show that estimated semi-elasticities on LbH measures are very stable. In the fourth column, we test the weighting function, attributing more weight to technological distance rather than closeness. Specifically, we redefine LbH_dist, simply replacing ψ (d, r) with 1-ψ (d, r). We find no evidence to favor technological distance as an appropriate weighting function, confirming our original assumption.

Discussion and conclusions

This article investigates the consequences of LbH on firm productivity. Specifically, the impact associated with the transfer of knowledge through inter-firm labor mobility is analyzed. In such a perspective, technicians and highly-educated workers are viewed as potential knowledge carriers. Their role allows enterprises to benefit from knowledge generated in other firms. However, knowledge transfers may not be pure externalities if recipient firms need to pay a wage premium to induce each inter-firm flow. To assess these learning effects, we estimate firm productivity following the algorithm suggested by Ackerberg et al. (2006).

Our findings support the assumption that LbH enhances productivity at the firm level, confirming the importance of external knowledge as a production factor. Specifically, we find that a unitary change in LbH can induce an average growth in firm productivity between one and two percent. The recruitment of a knowledge carrier can increase a firm's value added on average between real DKK 56,360 and DKK 112,710 (about USD 6,988-13,977 in the base year). In median terms, the same effect translates into a value added increase between real DKK 15,965 and DKK 31,930 (about USD 1,980-3,960 in the base year). This effect is greater when we allow for technological proximity between donor and recipient firms when measuring the exploitability of knowledge flows. It turns out that on average a unitary change in LbH_prox is related to more than a five percent increase in patenting firms' productivity. However, it is worth remembering that a unitary change in the most complete measure (with weights for both intensity and proximity) may potentially correspond to the recruitment of several knowledge carriers.

Our empirical analysis sheds light on the existence of knowledge exchange via labor mobility between any pair of enterprises, rejecting the hypothesis that this phenomenon involves only strongly research-based firms, as stressed in previous studies. We provide evidence that firms focusing on this aspect of the recruitment may gain external knowledge and exploit procedures and routines previously acquired in other enterprises with similar technological profiles. In fact, unlike the results reported in Rosenkopf and Almeida (2003) and Song et al. (2003), we find that closeness in firms' technological profiles is more important than distance, at least in a short-run perspective. Moreover, we devote particular attention not only to the hiring of workers holding degrees in natural sciences, but also to the hiring of those with a vocational education or with degrees in the social sciences and the humanities. Our findings suggest that knowledge transfers are not determined exclusively by workers having a tertiary education in natural or applied sciences, as assumed in the majority of articles in the literature. Several robustness checks corroborate our results and reinforce the reliability of these conclusions.

Human necessities, performing operations/transporting, chemistry/metallurgy, textiles/paper, fixed construction, mechanical engineering/lighting/heating/weapons/blasting, physics and electricity.

Path dependence is measured as the number of self-cited patents. Self-citing occurs when a patent filed by a firm cites another patent from the same firm[START_REF] Song | Learning-by-Hiring: When Is Mobility More Likely to Facilitate Inter-firm Knowledge Transfer?[END_REF].

In our empirical analysis we exclude the following sectors: i) agriculture, fishing and quarrying; ii) electricity, gas and water supply, and iii) public services.

Statistics published in "Regnskab" have been gathered in several ways. All firms with more than

employees or profits higher than a given threshold have been surveyed directly. The other firms are recorded in accordance with a stratified sample strategy. The surveyed firms can choose whether to submit their annual accounts and other specifications or fill out a questionnaire. To

Non-excludability means that once a good has been produced, it is not possible to prevent other people from gaining access to it (more realistically, it is costly for the provider to exclude unauthorized users).

Productivity is assumed to follow a first order Markov process.

If a linear AR(1) model for productivity is implemented, then the second stage in ACF would literally be the dynamic COMFAC model adopted by[START_REF] Blundell | GMM estimation with persistent panel data: an application to production functions[END_REF] [START_REF] Söderbom | ERSA Training Workshop. Lecture 4: Estimation of Production Functions with Micro Data[END_REF].[START_REF] Wooldridge | On estimating firm-level production functions using proxy variables to control for unobservables[END_REF] proposes an alternative implementation of OP/LP that involves estimating the first and second stages simultaneously. This can easily be extended to ACF methodology by simply adding labour input to the first stage non-parametric function.

Findings for the sample of patenting firms are obtained by using a parsimonious specification: X only includes the share of highly educated workers and technicians.