Spillovers at the Firm Level:
Some Evidence from Econometric Count Models for Panel Data
SUMMARY
This chapter analyses the relationship between the main determinants of technological activity and patent applications. To this end, an original panel of 181 international manufacturing firms investing substantial amounts in R&D during the late 1980s has been constructed. The number of patent applications by firms is explained by current and lagged levels of R&D expenditures and technological spillovers. Technological and geographical opportunities are also taken into account as additional determinants. In order to examine this relationship, several econometric models for count panel data are estimated. These models deal with the discrete nature of patents and firm specific unobservables arising from the panel data context. The main findings of the analysis are, first, a high sensitivity of results to the specification of patent distribution. Second, the estimates of the preferred GMM panel data method suggest decreasing returns to scale in technological activity and finally a positive impact of technological spillovers on firm’s own innovation.
“The patent system, which grants a legal monopoly for a limited period, conflicts with antitrust laws, which promote competition. At a fundamental level, however, this conflict vanishes. The patent system grants a temporary legal monopoly as a way of promoting dynamic efficiency - efficiency in the production of technological advance. Antitrust laws promote competition as a way of obtaining static efficiency - efficiency in the production and allocation of resources using existing technology. Both policies aim to improve market performance.”
Ward S. Bowman (1973: p.1)
4.1. INTRODUCTION48
Recent economic studies on Research and Development (R&D) activities indicate an increasing interest in the relationship between firms’ R&D investment and patent applications.
Though patents are not a perfect measure of R&D output (Griliches, 1990), they constitute a relevant measure of the technological effectiveness of R&D activities. Over the past years, several authors have examined the dynamic structure of the patent-R&D relationship49 by considering the number of patent applications as a function of present and lagged levels of R&D expenditures. The purpose of this chapter is to further explore the link between patent applications and firms’ R&D activities by applying recently developed econometric techniques on a new international dataset of R&D firms over the period 1983-91. Besides the international feature of the dataset, the chapter extends the framework of previous studies on the patent-R&D relationship by taking into account additional determinants of patenting.
These determinants are a measure of technological spillovers, i.e. technological knowledge borrowed by one firm from other firms, as well as technological and geographical opportunities. Despite all difficulties encountered when measuring technological spillovers, evidence of their importance has been found in many empirical studies50. Moreover, such effects take time to be expressed in new patents and it is worth giving attention to their precise timing.
48 This chapter is an extension of Cincera (1997) and Capron and Cincera (1995b).
49 For example, see Hausman, Hall and Griliches (1984), Hall, Griliches and Hausman (1986), henceforth HHG, Montalvo (1993), Blundell, Griffith and Windmeijer, henceforth BGW, (1995), and Crépon and Duguet (1993, 1995) for studies measuring the effects of R&D on patents with firms’ panel data. In Blundell, Griffith and Van Reenen, henceforth BGVR, (1995), the dependent variable is the count of the number of innovations.
50 For a review, see Griliches (1992). It should be noted that with the exception of Jaffe (1986), Crépon and Duguet (1993) and Capron and Cincera (1995b) who estimated the impact of technological spillovers on patents with a simpler model (no distributed lag in R&D and spillovers), none of the cited papers above consider this additional variable.
In order to treat appropriately specific issues arising from the discreteness of patent counts in the context of panel data, ad hoc econometric models for count panel data have to be implemented51. For instance, the discrete non-negative nature of the dependent patent variable generates non-linearities that make the usual linear regression models inappropriate.
Moreover, in panel data, the presence of firm-specific unobservables or unobserved
‘heterogeneity’ such as the aptitude of engineers to invent new products are not uncommon and these unobservables influence the way by which firms decide to apply for patents. It is well known from the analysis of panel data that the treatment of these firm unobserved specific effects leads to the so-called ‘fixed’ and ‘random’ effects models. Although the question of whether to treat the effects as fixed or random is not an obvious one52, when firm specific effects are correlated with some right-hand-side explanatory variables, the random effects model is no longer consistent. In the context of the patent-R&D relationship, there are reasons to believe that the unobservables are not independent of the regressors. For instance, if the aptitude to invent is high, then R&D investments will be higher. This (positive) correlation shows itself in upward-biased estimates and the random specification is no longer valid. In order to get around this problem, one possibility is to consider the conditional maximum likelihood estimator developed by HHG (1984). However, this fixed effects approach relies on the assumption of strong exogeneity of the right-hand-side variables. As it will be discussed below, this assumption is hard to maintain in the patent-R&D relationship and hence a more general approach allowing for both correlated effects and predetermined variables is also estimated. Finally, some issues related to the specification of the explanatory variables are examined. In particular, it is ascertained that these variables are not reflecting neglected serial correlation and dynamic misspecification.
The chapter is organized as follows. Section 4.2 presents the specification of the patent-R&D relationship to be estimated as well as some econometric count models for panel data and their properties. To begin, the basic Poisson model is introduced as a benchmark model. Then, the more General Event Count model and a semi-parametric estimator both based on a random effects specification of the unobservables are presented. In order to allow for correlated firm specific effects, a conditional maximum likelihood estimator and a non- linear GMM estimator are discussed. These estimators are based on fixed effects specifications and the later model relaxes the strict-exogeneity assumption of regressors. Finally, the specification issues raised above are investigated by imposing restrictions on serial correlation in the previous GMM estimator and by considering an alternative one based on a dynamic specification of the patenting process. A review of selected studies that focus on the patent- R&D relationship at the micro level is exposed in Section 4.3. Section 4.4 summarizes the
51 For a discussion of count data models, see Gouriéroux, Monfort and Trognon, henceforth GMT, (1984b), Cameron and Trivedi (1986) and Winkelmann and Zimmermann (1991, 1995).
52See Hsiao (1986, p.41).
main empirical findings. Section 4.5 concludes with the economic and methodological implications of the chapter.
4.2. PATENT-R&D SPECIFICATION AND COUNT MODELS FOR PANEL DATA
4.2.1. The Knowledge Production Function and the Basic Poisson Model
Several authors have examined the dynamics of the firms’ patenting process. For instance, in order to learn something about lags in the productivity of R&D, Pakes and Griliches (1984), HHG (1984, 1986), Montalvo (1993) and BGW (1995) have considered a specification in which patents, the dependent variable, is a function of contemporary and lagged flow of the firms’ annual R&D expenditures. According to Pakes and Griliches (1984:
p.56-57), the motivation for considering such a relationship rests “on the underlying notion of a research project whose success depends stochastically on both the amount of resources devoted to it and the amount of time that such resources have been deployed”. As far as the use of patent counts rather than other indicators of the output of technological activities is concerned, the authors argue that patents have the advantage that they are registered “at an intermediate stage in the process of transforming research input into benefits from knowledge output. They can be used, therefore, to separate the lags that occur in that process into two parts: one which produces patents from current and past research investments, and another which transforms patents, with the possible addition of more research expenditures, into benefits.”
In the present analysis, we are concerned only with the first part. Moreover, the knowledge-production function has been extended by considering, besides current and past levels of R&D investments, three additional technological determinants. These variables are the annual flow of technological spillovers and the technological and geographical opportunities. As it has already been discussed in Chapter 2, technological opportunity and spillovers are often described as technology-push forces, i.e. the exogenous technological factors which exercise pressures on the innovative activity (Rosenberg, 1983; Griliches, 1979, 1992). The technological opportunity represents the costs or the difficulties linked with technological activity. Such difficulties vary with technological areas because of the physical properties inherent to technology and because of the stock of scientific knowledge available at a certain time. Also, if the costs of doing R&D vary among countries, then the geographical opportunities can be expected to be important. Important variations in technological and geographical opportunity effects should be reflected in different propensities to patent among
technological areas and countries. Technological spillovers are also an important determinant of R&D activities. Technological spillovers are often divided into competitive and diffusion spillovers. In the theoretical literature of patent races models (Loury, 1979), competitive spillovers have a negative rivalry effect on a firm’s likelihood to apply for a patent to the extent that the more competitors invest in R&D, the less a firm is likely to invent a new product first. In Aghion and Howitt (1992), the probability to discover an invention depends positively from the level of research carried out53. The expected rents associated with a monopoly position in the good invented give firms incentives to engage in R&D. Yet, the introduction by competitors of new goods that are technologically more advanced reduce the sales of existing goods become obsolete. Furthermore, firms anticipating an increase of rivals’
R&D view their potential rents under threat, and as a results are less incited to invest in R&D (negative externalities). However, each innovation adds to the stock of existing knowledge, since firms can not appropriate all the benefits resulting from their discoveries. Hence, firms can free ride from this increased stock, so that the costs associated with new innovations are less important (positive externalities). Griliches (1992) defines these externalities as diffusion spillovers, i.e. the potential benefits of the research activity of other firms for a particular firm.
Because the returns from R&D are not entirely appropriable, the fruits of a firm’s research activity may benefit or spill over to other firms. Hence, diffusion spillovers have a positive impact on own R&D and, as a result, on patenting.
In order to assess the impacts of these determinants on the number of patent applications, the discreteness of this variable has to be taken into account. For instance, because of difficulties and uncertainty inherent to R&D activities, firms do not always apply for patents and hence a zero value is a natural outcome of this variable. Because of this property, the use of conventional linear regression models may be inappropriate. The reasons are that some basic assumptions such as the normality of residuals or the linear adjustment of data are no longer fulfilled. The usual way to deal with the discrete non-negative nature of the patent dependent variable is to consider the simple Poisson regression model. Let Pit be this variable which represents the number of patent applications by firm i at time t, where i = 1,...,N indexes firms and t = 1,..., T indexes time periods. The Pit are assumed to be independent and have Poisson distributions with parameters λit. Parameters λit depend on a set of explanatory variables which are in this case the determinants of the knowledge production function:
( ) ( ) ( )
λ β β τ τ β δ γ
τ τ τ
it = it = 0+ ∑ 1t it- + ∑τ 2t it- +∑ m m+ ∑ n n
−
= −
exp x β exp log k = log s TD GD
0
4 4
0
(4.1)
53 For Romer (1990) too, the stock of general knowledge represents a collection of ideas and methods that will be useful for generating later innovations.
where xit54 represents the set of explanatory variables, β is the vector of parameters to be estimated, kit is the annual flow of R&D investment, sit is the annual flow of spillovers,
TD and GD are respectively technological and geographical time-invariant dummies which are intended to pick up technological and geographical opportunities55.
The dependent patent variable is related to this function through the conditional mean of the Poisson model. An advantage of such a specification is that when variables xit are expressed in logarithms, parameters βk are elasticity. The Poisson distribution is given by:
P P P
P
( ) exp( )
it it! itit
it
= −λ λ
(4.2)
The βk are estimated by the maximum likelihood method and the log-likelihood is:
( ) ( )
l P P P x P x
t T i
N
t T i
N
t T i
N
( ; )β =∑ ∑ = −∑ ∑exp β +∑ ∑ β
=
= it = = it = = it it
1
1 1 1 1 1
(4.3)
This function is β globally concave, hence unicity of the global maximum is ensured.
An important property of the Poisson model is the equality between its first two conditional moments:
( ) ( )
E Pit|xit,β =V P xit| it,β =λit (4.4)
For panel data such as patents, the failure to include individual specific effects may lead to ‘overdispersion’, i.e. conditional variance exceeds conditional mean, when estimating a cross-section model such as Poisson. For instance, in the patent-R&D relationship the presence of firms unobserved effects like the uncertainty inherent to R&D activities, the ability of engineers to discover new products or the commercial risk of selling an invention, find expression in the fact that only a few successful firms are likely to apply for a large number of patents in a given time period while for a majority of firms the importance of patenting may be limited or even nil. As Winkelmann and Zimmermann (1995) stressed, overdispersion can arise for reasons such as unobserved heterogeneity and this situation is not well suited by the Poisson model given the property of equality between its two first conditional moments.
54 Because of data constraints and in order to allow for comparison with previous studies, a four lag period has been considered for explanatory variables.
55The construction of these variables was the purpose of Section 3.4 of Chapter 3.
Therefore, more general econometric models have to be considered.
4.2.2. Negative Binomial and GEC Models
In order to take into account the unobserved heterogeneity, one possible extension of the Poisson model is to include a firm unobserved specific effect εi into the λit parameters.
This firm-specific effect which is assumed to be invariant over time can be treated as random or as fixed. In the case of random effects, the Poisson’s parameters become:
( )
~ exp
λit = xitβ ε+ i (4.5)
The random terms εi take into account possible specification errors of ~
λit. These misspecifications may result from the omission of non observable explanatory variables or from measurement errors of these variables. The precise form of the distribution of the compound Poisson model depends upon the specific choice of the probability distribution of exp(εi):
( ) ( ( ))( ( )) ( )
P P x x
P g d
P it
it i it i
it it
i i
= − + +
−∞∫
+∞exp exp exp
!
β ε β ε
ε ε (4.6)
where g(εi) indicates the probability distribution of εi.
The computation of the compound Poisson’s distribution may be a difficult task - at least from an analytic point of view - because of the integral arising in equation 4.6. However, when it is assumed that exp(εi) follow a gamma distribution with parameters (1,δ)56 and are independent and identically distributed, the computation of the last formula leads to the well known negative binomial model. The probability distribution of this model is given by:
( ) ( ( ) )[ ] ( )
( ( )) ( )[ ] ( )
P P x P
x P
x
x P
it
it it it
it it it it
= +
+ + +
Γ
Γ Γ
exp exp
exp exp
β δ
β δ
β
1 1 β
(4.7)
The log-likelihood of this model is:
( ) ( ( ) ) ( ( )) ( )
( ) [ ] ( ( ) ) [ ]
l P P P
x P x P
x x P
t T i
N
t T i
N
( ; )
ln exp ln exp ln
exp ln exp ln
β β β
β δ β δ
= ∑∑ = + − − +
+ − + +
∑
= ∑
= it = =
it it it it
it it it
1
1 1 1
1 1
Γ Γ Γ
(4.8)
56If the set of explanatory variables contains a constant term, this assumption is not too restrictive.
The conditional mean and variance of this distribution are:
( )
E Pit / xit = λit and V P( it /xit)= + it
1 δ
δ λ (4.9)
provided that εi are independent. The mean-variance relationship allows for overdispersion.
Furthermore the Poisson model is nested in the negative binomial model, that is when parameter δ tends to infinity, the negative binomial model converges to the Poisson one.
Actually, different negative binomial models can be generated according to the way the parameters of the gamma distribution are linked to the xit. For instance, the formulation of Winkelmann and Zimmermann (1991) allows for an even more flexible conditional mean- variance relationship. The authors developed the General Event Count Model (GEC) which is based on a new parametrization of the Katz family. The log-likelihood of the model is equal to:
( ) [ ( )] [( ( ) ) (] )
l P; = ln f 0 / +
i=1
β N λ σ λ σ λ
σ λ
it
it it
it
, ² , itln ²
²
k j
j
k j k
P t
T + − −
− +
∑
∑
∑ = =
1 1
1 1
1 1
(4.10)
where f 0 /( λ σit, ² ,k is the distribution for P) it = 0.
The variance-mean relationship of the GEC model57 is defined as:
( ) ( ) ( ) ( )
V P xit| it = σ2−1 E P xit| it k+1+E P xit| it (4.11) where σ² and k, which are independent of β, represent respectively the dispersion parameter and the non-linearity in the variance-mean relationship. This more general full parametric specification allows for overdispersion. Furthermore, it embraces the Poisson model (for σ² = 1) and negative binomial models such as the so-called Negbin I (for σ² > 1 and k = 0)58 and Negbin II59 (for σ² > 1 and k = 1) as special cases. Using the estimated value of σ² and k, it is possible to discriminate between the Poisson and both negative binomial models or to reject them rather than to choose one of them a priori.
57A similar variance function is given by Cameron and Trivedi (1986, p.33).
58 The probability distribution of this model is given in equation 4.7.
59For a discussion of these models, see Cameron and Trivedi (1986).
4.2.3. PML, QGPML and SML estimators
The models presented so far can all be estimated by maximum likelihood (ML) techniques. However, the knowledge production function 4.1 has also been estimated by pseudo-maximum likelihood (PML) and quasi-generalized pseudo-maximum likelihood (QGPML) estimators developed by GMT (1984a, 1984b). Indeed, these authors show that when the true process that generates the data admits the same conditional moments of the two first orders as those given by the GEC model for k = 1 and σ² > 0, consistent estimated β can be performed with the pseudo maximum likelihood method. An advantage of such a semi- parametric approach is that it requires fewer distributional assumptions regarding exp(εi). On the other hand, it leads generally to less accurate estimates of the parameters than those obtained by the ML method if the chosen model is the true one. More specifically, it has been shown in the preceding random Poisson model, that there was no loss of generality in assuming that:
[ ( )]
E expεi =1 and V exp[ ( )εi ]=δ (4.12)
With such an assumption, the conditional moments of order one and of order two of the dependent variable are independent of the random term εi60:
( ) ( )
E Pit /xit =exp xitβ and V P( it /xit)=exp(xitβ)+δexp(2xitβ) (4.13)
The pseudo-maximum likelihood method consists in taking a distribution which accepts these conditional moments. Though this distribution may not belong to the true one, when a linear exponential distribution is taken and provided that the mean is correctly specified, consistent and asymptotically normal estimates can be achieved. The log-pseudo-likelihood depends on the chosen pseudo distribution. For a gamma one, the objective function to be maximized is:
60 E P( it /xit)=EE(Pit /εi,xit)=Eexp(xitβ ε+ i)
( ) ( )
=exp xitβ Eexp εi
( )
=exp xitβ
( )
V Pit / xit =E V[ (Pit /εi,xit)]+V[E(Pit /εi,xit)]
( )
[ ] [ ( )]
=E exp xitβ ε+ i +V exp xitβ ε+ i
( ) ( )
=exp xitβ δ+ exp 2xitβ
( )
[− − − ]
∑
∑= = x P x
t T i
N
itβ itexp itβ
1 1
(4.14)
The asymptotic variance-covariance matrix of this PML estimator is given by J IJ−1 −1 where:
( ) ( ( ))
J =E
∂ β −
∂β
∂ β
∂β exp x exp x '
Σ01 (4.15)
( ) ( ( ))
I=E
− −
∂ β
∂β
∂ β
∂β
exp x exp x '
Σ Ω Σ01 0 01 (4.16)
where Σ0 is the variance of the gamma distribution, and Ω0 is the variance of the ‘true’ distribution.
In this case, the asymptotic variance-covariance matrix61 is:
( ) ( ) [ ( )]( )
δ Ex' x −1+ Ex' x −1E x' xexp −xβ Ex' x −1 (4.17) This matrix can be estimated in different ways. When a consistent estimate of δ is available, one way is to replace expectations with sample means in the above formula. It should be noted that this matrix is ‘bigger’ than the one that would have been reached with the true distribution. Nevertheless, GMT (1984a; 1984b) developed a two-step method which provides better asymptotic estimators than all those which could have been obtained by pseudo- maximum likelihood when the conditional moment of second order is known. In the first step, a consistent estimator of δ is computed by:
d
b b b
b
∧
∧ ∧ ∧
=
=
= ∧
−
−
∑
∑
P P P x
x
t T i
N
it it it it
it
exp exp exp
exp
2 1
1
2
4
(4.18)
where b
∧
are estimated by pseudo-maximum likelihood.
In the second step, estimates of parameters β are again performed by the pseudo maximum likelihood method but this time the second order moment is taken into account using the estimated consistent value of δ. For the gamma pseudo-distribution, the objective function
61Actually, it is the variance-covariance matrix of n b( )$ .
to be maximized is:
( )
( )
exp
exp
exp x
x
x P x
t T i
N it
it
it it it
b
d b
∧
∧ ∧
=
=
+
− − −
∑
∑
1 1
1
β β (4.19)
and the asymptotic variance-covariance matrix is equal to:
E
x x x
x
x
it it it
it
' exp exp
b
d b
∧
∧ ∧
−
+
1
1
(4.20)
The matrix of this QGPML estimator is ‘smaller’ than all those which can be computed by the pseudo-maximum likelihood method.
In the previous two methods, it has been possible to get asymptotically efficient and consistent estimates of the β making assumptions only on the moments of the heterogeneity random term εi. Another approach is to replace the distribution of εi by a consistent estimator.
For instance, in the simulated maximum likelihood method, the distribution of the random εi is assumed to be associated with a random vector which is generated by a known distribution.
Without loss of generality, it can be assumed that the εi obey to the relationship:
( )
εi =F u ,i θ (4.21)
where θ is a vector of parameters and the distribution of the ui is given. Gouriéroux and Monfort (1991a, 1991b) show that when the εi represent an univariate continuous variable, the function F .,( )θ can be chosen as the inverse of the cumulated distribution function of the εi and in such a case the distribution of the ui is the uniform distribution on [0,1]. With these assumptions the log-likelihood of the Pit is:
( ) ( ( ))( ( ))
l P x u x u
P dP u
t
T P
i N
; ln exp exp exp
! ( )
β β β
= ∑ − + +
∫
∑ = −∞
+∞
=1 1
it i it i it
it
i (4.22)
The integral of this log-likelihood function can be computed by simulation on the ui and by taking the sample mean. These simulations are quite simple given that the distribution of the
ui is known. Formally, we have:
( ) ( ( ))( ( ))
l P x
H
x u x u
P
s
P h
H t
T i
N
it it
it ih it ih
it it
/ , , ln exp exp exp
β δ β ! β
= − + +
∑
∑
∑= = =
1
1 1
1
(4.23)
This is the simulated maximum likelihood method. When H is fixed and tends to infinity, the estimator is biased and the asymptotic bias is of the order of 1
H . However, when H and n tend to infinity in such a way that n
H tends to zero, then the simulated maximum likelihood estimator is consistent and asymptotically efficient. Finally, the last three methods can be combined to form the pseudo-simulated maximum likelihood and the quasi-generalized pseudo-simulated maximum likelihood methods. In this study, the pseudo-simulated maximum likelihood was performed with H = 200. The asymptotic variance-covariance matrix of the estimators computed by this method is the same as those of the corresponding pseudo- maximum likelihood, again when H and n tend to infinity in such a way that n
H tends to zero.
4.2.4. CML Estimators
In the previous models, firm specific effects are introduced into the Poisson parameter λit in order to take into account the heterogeneity arising from the panel structure of data.
Assuming that these specific effects are random, the compound Poisson model leads to more general models such as the negative binomial and the QGPML ones. Nevertheless, as stressed by BGVR (1995), this way of introducing the heterogeneity relies on the strong assumption that the firms’ unobserved effects are independent of the explanatory variables. If this assumption is not satisfied, the previous estimators are not consistent and we know, from panel data analysis, that in this case the fixed effect specification has to be used. In order to allow for fixed effects to be correlated with regressors, two alternative econometric approaches are considered: the conditional maximum likelihood estimator developed by HHG (1984) and a non-linear GMM estimator proposed by Montalvo (1993) following Chamberlain (1992) and applied by BGVR (1995), BGW (1995) and Crépon and Duguet (1997). Both approaches rely on a fixed effect specification of the firm unobserved heterogeneity. Finally, the ‘robustness’ of the spillover specification is investigated by imposing stronger assumptions on the residuals of the previous GMM estimator and by considering an alternative one based on a linear feedback model.
HHG (1984) developed fixed-effects Poisson and negative binomial models based on the conditional maximum likelihood approach of Anderson (1970). The key point of this approach consists of conditioning on the sum over time of patents for a given firm. This allows removal of the firms’ specific effects from the distribution of the dependent patent variable conditionally to the sum of patents over the whole period. Moreover, the authors showed that the derivation of the Poisson fixed-effect model leads to a multinomial logit distribution while a negative multivariate hypergeometric distribution is obtained from the negative binomial fixed- effect model. It should be noted that although these CML estimators allow one to get around the problem of correlated firm specific effects, their consistency relies on the crucial assumption of strict exogeneity of explanatory variables (Montalvo, 1993; BGVR, 1995). This assumption is hard to justify in the patent-R&D relationship where the patenting of an innovation is likely to call for further R&D. For instance, activities such as developing, testing, or improving a new product are in many cases undertaken after the patent application itself.
Hence the technological determinants of the patenting process should be considered more as weakly exogenous or predetermined rather than strictly exogenous.
4.2.5. GMM Estimators
The next approach investigated in this chapter still allows for correlated fixed effects but relaxes the strict exogeneity assumption of regressors. This approach departs from the multiplicative fixed effects model (MFEM):
( )
Pit =exp xitβ +εi +uit (4.24)
From this expression, the generalized method of moments (GMM) framework of Hansen (1982) can be implemented by forming the following set of conditional mean restrictions:
( ) ( )
E P zit| is,εi =exp xitβ+εi ,∀ ≤s t (4.25) where zis represents any set of instruments such that equation 4.25 holds. This raises the question of the choice of the optimal instruments set. As in Montalvo (1993) and BGVR (1995), the following instruments are considered:
( )
zis = 1,k ,...,k ,s ,...,si1 is i1 is (4.26)
though efficiency gains of the GMM estimator could be achieved by considering additional
instruments62.
Conditions 4.25 can not be used directly because they depend on the unobserved fixed effects. However, these effects can be removed by the following quasi-differenced transformation proposed by Chamberlain (1992):
( )
[ ]
{ }
E Pit −Pit 1+ exp xit−xit 1+ β |zis = ∀ ≤0, s t (4.27)
Since the conditioning set is dated at period s, these orthogonality conditions remain valid under weak exogeneity of regressors. Moreover, if the instruments are strictly exogenous, then observations for all periods become valid instruments and this implies additional orthogonality conditions in equation 4.2763. Following Arellano and Bond (1991) and Mairesse and Hall (1996), the validity of these additional conditions can be tested by performing Sargan difference tests in a sequential way.
The GMM method based on equation 4.27 has several advantages with respect to the full parametric Poisson fixed-effect model. First, it does not impose the equality of the first two conditional moments. Second, it allows for heteroskedasticity and any serial correlation pattern of the error terms. Finally, it relaxes the strict exogeneity assumption of the regressors. This greater flexibility comes at the price of less efficient estimators in general. However, following Crépon and Duguet (1997), more structure can be put on the GMM estimator, in particular in terms of restricted serial correlation, by imposing additional conditions in equation 4.27 whose validity can in turn be tested using Sargan difference tests64.
Besides the question of efficiency gains, Crépon and Duguet put forwards an economic motivation for testing restrictions on residual correlation as well. On the one hand, once the fixed effects are accounted for, the estimates of current and past values of R&D and spillovers could reflect the existence of correlated random shocks in the knowledge production function
62See Crépon and Duguet (1997) for an application.
63Given equations 4.1, 4.26 and 4.27, there are 2T + 1 instruments (including the constant term) and (T - 5) quasi-differences, where T = 9. Two cases have to be distinguished. First, in case of weak exogeneity of the zis, the number of orthogonality conditions as in equation 4.27 is (T - 5)[2(T - 4)/2 + 2(T - 4 - τ) + 1], where τ represents the extent to which the zis are weakly exogenous, i.e. τ = 1 (τ = 2,...) means that s = t (t - 1,...) in equation 4.27. Second, if the zis are strictly exogenous, than equation 4.27 implies (T - 5)(2T + 1) orthogonality conditions, i.e. (T - 5)[(2T + 1) - 2(T - 4)/2 + 2(T - 4 - τ) + 1] additional conditions.
64 In order to restrict serial correlation, the authors suggest two methods, one of which consists of adding past values of the dependent variable in the set of instruments:
( )
zis* = 1,x ,...,x ,i1 is Pi1,...,Pis−1 . (4.28) This implies (T - 5)[(T - 4)/2 + (T - 5 - τ)] additional orthogonality conditions in case of weakly exogenous zis.
due to the presence of serially correlated residuals. On the other hand, if the hypothesis of no serial correlation can not be rejected, then this function should rather be viewed as a steady process. This argument is actually more crucial than it may appear at first because of the presence of current and lagged values of technological spillovers in the knowledge-production function. Indeed, these variables are characterized by outside R&D, and hence, they might reflect neglected serial correlation to the extent that the total amount of R&D performed outside a given firm at different time periods picks up such random innovation shocks.
4.2.6. Dynamic specification
Finally, the presence of serial correlation may be viewed as an issue of dynamic misspecification as well. For instance, if the past patenting activity is an important determinant of current outcomes, then the omission of this determinant may be reflected in correlated residuals. Here also, if no restriction is imposed on serial correlation, these effects may again be picked up by the spillover variables. In order to investigate this last issue, an alternative dynamic specification of the knowledge-production function has been estimated in line with BGVR (1995). The authors propose a linear feedback model (LFM) which leads to the following quasi-differenced orthogonality conditions:
( ) ( ) ([ ) ]
E Pit P P P x x z s t
it-1
*
it +1 it
*
it**
it+1
**
is
− − − − *
= ∀ ≤
ρ ρ exp β* | 0, (4.29)
where Pit* (P Pit it Pit )
, ,
= −1 −2 , ρ′ =(ρ ρ ρ1, 2, 3),xit** (kit it)
= ,s and β′ =* (β βk, s)
The specification on which equation 4.29 is based is characterized by the presence of lagged values of the patent variable among the regressors. In this particular case, we know from the literature of panel data65 that if the fixed effects are removed by first (or quasi) differencing and if t - 2 lagged and higher values of Pit are used as instruments for ∆Pit-1, then consistent estimates can be obtained as long as the residuals are not serially correlated. Using the same GMM framework as before, it is possible to test nested hypotheses regarding serial correlation by performing Sargan difference test statistics. Finally performing a non nested J test à la Davidson-MacKinnon, it is possible to compare this last model with the previous one.
65 For a discussion, see Anderson and Hsiao (1981), Arellano (1989), Arellano and Bond (1991), Arellano and Bover (1995) or Ahn and Schmidt (1995).
