Bivariate Probit Estimation for Panel Data: a two-ste Gauss-Hermite Quadrature Aroach with an alication to roduct and rocess innovations for France Benoît MULKAY Université de Montellier January 05 Preliminary, Do not quote! Abstract This aer describes two methods for comuting a bivariate robit model on anel data with correlated random e ects. A rst aroach using simulated maximum likelihood has been already resented in the literature. An alternative method based on a two-ste Gauss-Hermite quadrature in order to evaluate the likelihood function is roosed in this article. A simulation shows the imortance to estimate the correlation in random e ects and the correlation between both equations. Finally an alication is erformed to estimate the determinants of roduct or rocess innovations on a anel of French rms. It shows a very large correlation between individual e ects. JEL code : C33, C35, O3. Address : Faculté d Economie, Avenue Raymond Dugrand CS 79606, 34960 Montellier Cedex. E-mail : benoit.mulkay@univ-mont.fr. The author thanks Jacques Mairesse and Wladimir Raymond, as well as the articiants to the 0th anel data conference (Tokyo, July 04), for helful comments on this aer.
Introduction The estimation of a robit model on anel data is now usual. Many softwares roose such method of estimation which relies on individual random e ects because the xed e ects aroach is not valid due to the incidental arameters roblem in the non-linear anel data model. In a seminal aer, Butler and Mo tt (98) suggested to integrating the density conditional over the distribution of the individual e ects in order to eliminate them by taking an average density. They roosed to use a Gauss-Hermite Quadrature to comute this integral for each individual in the anel. On the other hand, many emirical roblems imly two binary variables. The classic bivariate robit model is now common for cross-section data, but no usual rocedure is available for anel data where there is an individual random e ect. In fact, it is interesting to estimate the correlation between the individual e ects in the two equations, because it shows how the unoberved heterogeneity of individuals is correlated accross equation, while there is still a correlation between the idiosyncratic error terms in the two equations. On the line of multivariate robit model, Lee and Oguzoglu (007) and Kano (008) have roosed a simulated maximum likelihood aroach where the individuals e ects are integrated out by comuting the double integral by simulation. But this rocedure could be very time-consuming even with fast modern comuter. In this article, an alternative aroach based on a two-ste Gauss-Hermite Quadrature is used in order to comute this double integral, which should be rather in the line of the Butler and Mo tt (98) aroach. Such a method has been already investigated in the context of a Heckman selection model on anel data by by Raymond et al (007, 00). I adat their method in the case of a bivariate anel data model in the section. A simulation analysis is done in Section 3 in order to show the imortance of taking account individual e ects in estimation of a robit model on anel data. The searated estimation of the two robit models shows clearly that they are consistent due to the fact that the model is correctly seci ed and that the correlations between the individual e ects or between the error terms are only of second order. In fact like in a seemingly unrelated regression equations model, there is only a gain in e ciency of taking account of the covariance structure of the error terms comosed of an individual e ect and a idiosyncratic error. However the estimation of the correlations is of interest in order to assess the e ects of unobserved heterogeneity on each equation. Finally in Section 4, we resent an alication of this rocedure in the case of the estimation of the determinants of roduct and rocess innovations on a anel of French rms during the eriod 000-007. The French data are annual, but they indicate only whether a rm, with R&D exenditures, introduces a roduct or a rocess innovation during the given year. The model exlaining the roduct or rocess innovations is simle because it deends only on the size of the rm and on the R&D intensity. There is a ositive e ect of the size of the rm of the same magnitude for both tyes of innovation, while there is also a ositive e ect of the R&D intensity, but this e ect is non-
linear because it is decreasing u to a R&D intensity which is roughly 50 % of the total turnover of the rm. Finally the unobserved heterogeneity a ects both equation with a very high ositive correlation of 90 %, even though the estimated correlation between the idiosyncratic errors terms is about 50 %. In consequence the individual characteristics have the same imact on both tye of innovations. The random e ect bivariate robit. The bivariate robit model Here we resent brie y the bivariate robit model. Thismodel is comosed by latent variables y and y which are exlained by exogenous variables x and x and by ossibly correlated error terms " and ", normally distributed with unit variances. and correlation coe cient : y x 0 + " y x 0 + " where " " " i:i:d:n 0 0 ; If the data are observed on several individuals only, we obtain the classical bivariate robit model when the observed variables y and y are de ned as : y (y > 0) y (y > 0) where (:::) is the indicator function with value one if the exression in arenthesis is true, and zero otherwise. The maximum likelihood estimator is then simly obtained with the classical transformation : q j y j such that the robability of a given choice between the 4 ossible con gurations of choice is : Pr (Y y ; Y y j x ; x ; ; ; ) [q (x 0 ) ; q (x 0 ) ; q q ] with () is the cumulative density fonction of the bivariate standard normal distribution : [u ; u ; ] Z u Z u Z u Z u (z ; z ; ) dz dz z ex + z z z ( ) dz dz See for examle : Greene (008, Section XXI.6). The classical normalization of variances to unity is done here because only the signs of the latent variables are observed. Therefore the scale does not matter. 3
where () is its robability density function of a bivariate standard normal variable with correlation. As the N observations of the samle are indeendent, the log-likelihood function is given by : ln $ nx ln q;i x 0 ;i ; q ;i x 0 ;i ; q ;i q ;i i which should be maximized to obtain the maximum likelihood estimator of the bivariate robit model 3. Greene (008) gives the analytic rst and second order conditions of the estimation roblem. When the observations come from a anel of individuals observed during a given time eriod (suosed here for simlicity to be the same for all individuals such that the anel is balanced), there is often an individual e ect to take account of the unobserved heterogenity of the individuals. However with a nonlinear model, like the robit model, if the individual e ects are treated as xed or correlated with the exlanatory variables, there is an incidental arameters roblem (Neyman and Scott, 948; Lancaster, 000; or Cameron and Trivedi; 006). Thus we need to assume that individual e ects are not correlated with the exlanatory variables, and we use a random e ects model with a seci ed distribution. These random e ects are then eliminated by integrating over the distribution. The univariate robit case has been rst studied by Butler and Mo tt (98) and Skrondal and Rabe-Hasketh (004). We generalize this univariate random e ect robit model to the case of two latent variables for i ; :::; N individuals and t ; :::; T time eriods : y ;it x 0 ;it + ;i + " ;it y ;it x0 ;it + ;i + " ;it for i ; :::; N and t ; :::; T. where : 8 >< >: " it i ";it 0 i:i:d:n ; " ;it 0 0 i:i:d:n ; ;i 0 ;i Here we assume imlicitly that the observations are indeendant over time and accross indivisuals. The exlanatory variables are exogenous with resect to the error terms and with the individual random e ects. This last hyothesis could be relaxed by introducing the average values of the regressors along the lines roosed by Mundlak (978) if the individual e ect can be decomosed on a linear combination of the averaged regressors lus an uncorrelated e ects. The observed model is: y;it y;it > 0 y ;it y ;it > 0 3 See for examle the estimation rocedure birobit in Stata (Hardin, 996). 4
Let us de ne the classical transformation of the observed variables : q;it y ;it q ;it y ;it. The individual joint density function Because of the indeendance of observations over time, the conditional joint density for the T observations of the i th individual is: f i (y i j X i ; i ; ; ) TY f it (y it j X it ; i ; ; ) t As we have assumed a normal distribution for the error terms in the latent model, the density for an observation is given as in the bivariate robit model above: Pr (Y y ; Y y j x ; x ; i ; ; ) (q (x 0 + ) ; q (x 0 + ) ; q q ) where the individual random e ects are added u to the conventional observable arts of the latent functions. The joint density for an individual, conditional to the vector of the individual random e ects i ( ;i ; ;i ), is then: f i (y i j X i ; i ; ; ) TY q ;it x 0 ;it + ;i ; q;it x 0 ;it + ;i ; q;it q ;it t Assuming a normal disribution for these individual random e ects with variances and resectively and a correlation coe cient, the density function for the individual e ects is given by : g i i j ; ; ( ex ( ( ) ) " ;i ;i ;i + ;i #) This density function does not deend on observables but on the three arameters which should be estimated. The unconditional (to the individual random e ects) joint density for the i th individual is obtained by averaging over the 5
distribution of these individual e ects : y i j X i ; ; ; ; ; `i Z + Z + Z + Z + f i (y i j X i ; i ; ; ) g i i j ; ; d ;i d ;i " Y T q ;it x 0 ;it + ;i ; q;it x 0 ;it + ;i ; q;it q ;it # t ( ) () ( ex ( ) " ;i ;i ;i + ;i #) d ;i d ;i.3 Decomosition of the double integral The evaluation of the individual likelihood function () requires the comutation of a double integral. Lee and Oguzoglu (007) and Kano (008) have roosed a method of comutation by simulation where ;i and ;i are randomly drawn in the bivariate normal distribution 4. The individual joint density (unconditional to the individual random e ects is aroximated by : y i j X i ; ; ; ; ; ' where `i Z + Z + R a (r) ;i r t f i (y i j X i ; i ; ; ) g i j ; ; d ;i d ;i " RX Y T q ;it and a (r) ;i a (r) ;i a (r) ;i x 0 ;it + a (r) ;i ; q ;it x 0 ;it + a (r) ;i # ; q ;it q ;it () are R random draws in a bivariate normal distribution:! 0 i:i:d:n 0 ; However the comutation should be very time-consuming and imrecise even though we use modern simulator like GHK or Halton simulators, because we need to comute R cumulative density function with a large value of R in order to obtain su cient recision in the log-likelihood function. Instead we use the two-ste Gauss-Hermite quadrature technique originally roosed in a coule of aers by Raymond et al. (007, 00) in the case of an 4 Miranda (00) suggests the same rocedure in an unublished aer resented at the Mexican Stata Conference in 00. 6
Heckman samle selection model on anel data. This method relies on a decomosition of the two-dimensional normal distribution for the individual e ects into a one-dimensional marginal distribution and a one-dimensional conditional distribution. The unconditional joint density for the i th individual is rewritten as: y i j X i ; ; ; ; ; `i Z + Z + f i (y i j X i ; i ; ; ) ( " ( ex ( + ) Z + Z + ex ( ( ) f i (y i j X i ; i ; ; ) " ;i which can be in turn rewritten as: with `i y i j X i ; ; ; ; Z + H i ( ;i ) ( ) ex ( ;i ( H i ( ;i ) ex Z + ( ) ) #) ) #) " ;i ex " f i (y i j X i ; i ; ; ) " ;i d d # ( ;i / ) d ;i (3) ;i # ( ;i / ) d ;i d ;i #) ;i d ;i Let us evaluate this last function by using a gauss-hermite Quadrature by doing a change in variable such that ( / ) z ( ) with d ( )dz such that 5 : Z + H ( ) ( ) ( ) ( z ex Z + ) `i y i j X i ; z ( ex ( ) z ); ; ; ( ) f i y i j X i ; z ( ); ; ; ( ) ex z ex z dz : 5 We dro the individual index i for the clarity of the exosition. dz 7
This is a Gaussian integral which can be aroximated by a Gauss-Hermite quadrature with weights! m and abscissas a m for M integration oints (m ; :::; M) 6 : Z + MX f (z) e z dz '! m f (a m ) m Thus the H ( ) function is aroximated by : H i ( ;i ) ' MX m "! m f i y i j X i ; a m ( ); ;i; ; ex Now the second ste of the rocedure is to introduce this function in the individual joint density `i y i j X i ; ; ; ; ; above (3) with a second change in variables ( / ) z ( ) with d ( )dz to obtain: `i y i j X i ; ; ; ; ; Z " # + ( / ) H ( ) ex d ;i Z + ( ) Z + MX m! m f i y i j X i ; a m ( ); ; ; " ex Z + MX m a m # ex " # ( / ) d ;i! m f i y i j X i ; a m ( ); z ( ); ; " # " ex z ( )a m ex MX m! m`i z # dz y i j X i ; a m ( ); z ( ); ; ex [a m z ] ex z dz A second Gauss-Hermite quadrature can be used to comute this Gaussian integral. For P integration oints ( ; :::; P ), we have the weights! and the abscissas a. Finally the individual joint density unconditional to the individual e ects can be aroximated by : 6 The more the number of oints, the more recise is the aroximation. Genrally the number of oints is set to 8; or 6 (see Cameron and Trivedi, 005, Section XII.3.). The values of weights! m and abscissas a m can be found in mathematical textbooks. a m # : 8
' `i y i j X i ; ; ; ; ; (4)! ( ) PX MX TY!! m ex [a m a ] (q u ;m ; q u ; ; q q ) m t where the arguments of the bivariate cumulative density function are : u ;m x 0 + a m ( ) u ; x 0 + a ( ) Finally as the individuals are indeendent, the log-likelihood function should be exressed as: ln $ NX i ln `i y i j X i ; ; ; ; ; N ln () + N ln +! NX PX MX TY ln!! m ex [a m a ] (q ;i u ;m;i ; q ;i u ;m;i ; q ;i q ;i ) (5) i m In order to maximize this log-likelihood function, we can use the usual transformations for the correlation coe cients: 8 < a tanh ln + : a tanh ln + t or ( ex( ) ex( )+ ex( ) ex( )+ At each evaluation of the likelihood function, it is necessary to comute N M P cumulative density functions of the bivariate normal variables with this two-ste quadrature, which seems much more reasonable relative to the comutation of N R cumulative density functions for the simulated method. In fact we should have a su ciently good aroximation with M P oints in the Gauss-Hermite quadrature, even though we shoud take at least R 00 oints for the comutation by simulation with less recision. The two rocedures of estimation of the bivariate robit model by maximum likelihood have been written in a Stata rogram either with the simulated maximum likelihood or with the Gauss-Hermite quadrature 7. 7 These rograms uses the maximum likelihood rocedures in Stata by Gould et al. (00). 9
3 A simulation A simulation of the rocedures for the estimation of the bivariate robit model has been erformed in order to assess the e ect of neglecting the correlation between the two equations, and between the unobserved heterogeneity in each equation. A set of observations for N individuals during T eriods has been generated for a bivariate latent rocess: y ;0 + ; x + ; x + + " where " y ;0 + ; x + ; x + + " " 0 i:i:d:n ; " 0 where the exogenous variables x and x have been drawn indeendently for each observations in a standard normal distribution. The individual e ects and have been also drawn into a bivariate normal distribution with correlation : 0 i:i:d:n 0 ; : Then the observable deendent variables are constructed on the basis of the sign of the corresonding latent variables: y (y > 0) y (y > 0) In the following simulations, the number of individuals has been set to 000 with 0 eriods for each individuals, such that there are 0 000 observations in the anel data set which corresonds to the usual size of such data. The true structural arameters in the model are the following : ;0 0:50; ; :00; ; 0:00 and ;0 0:50; ; 0:50; ; :00. Therefore the second exlanatory variable aears only in the second equation. The correlation coe cient of the error terms has been set to 0:50, the same value has the correlation coe cient between the individual random e ects : 0:50, while the standard deviation of these individual e ects are the same: :00. The observed atterns of resonse in this simulated model is the shown in the Table.This simulated data set exhibits an association between both deendent variable with a Kendall s-t b measure of association of 0. with a standard error of 0.00, as well a Pearson Chi-squared of 49.55 showing clearly a ositive signi cant association between the two observed deendent variables. Moreover the tetrachoric correlation is 0.349 with a standard error 0.05 which is less than the assumed correlation between the error terms in the latent model. 0
y 0 Total y 0 9.7 %.4 % 4.0 % 8. % 9.9 % 58.0% Total 57.7 % 4.3 % 00 % Table : Contingency table of the binary variables in the simualted model. This simulated data set exhibits an association between both deendent variable with a Kendall s-t b measure of association of 0. with a standard error of 0.00, as well a Pearson Chi-squared of 49.55 showing clearly a ositive signi cant association between the two observed deendent variables. Moreover the tetrachoric correlation is 0.349 with a standard error 0.05 which is less than the assumed correlation between the error terms in the latent model. The model is estimated by a ooled bivariate robit method where there are no individual e ects as a benchmark for estimations. Then it is estimated using the Gauss-Hermite Quadrature (with oints) allowing individual e ects. We roceed to four estimations : the rst one (estimation ) with the individual random e ects but with a zero correlation between error terms ( 0) and a zero correlation between the individual e ects ( 0), the second estimation () allows for an estimated correlation between the error terms (), wile the third estimation (3) allows only a correlation between the individual e ects (). Finally the last estimation (4) is the comlete model where both correlations must be estimated. The standard likelihood ratio tests are erformed in order to verify the assumtion about the individual e ects and the correlations in the model. The Gauss-Hermite Quadrature rocedure with integration oints is here faster by 40 % than the simulated maximum likelihhod rocedures erformed on the same dataset and on the same comuter. Even though the convergence is quite fast in three or four iterations starting with the initial values from the two univariate anel robit estimations, it takes hovever between 7 minutes (for the rst estimation) to 4 minutes (for the last estimation) to erform such a regression 8 on 0 000 observations for a model with only 3 arameters in each equation! The benchmark estimation is clearly biased for the structural arameters of each equation because there is no individual e ects. Only a correlation between the idiosyncratic error terms is estimated with an estimated value (0.5) close to the theoretical correlation (0.50). Let us remark that the arameter estimates are less than the half of their theoretical values. A likelihood ratio test rejects clearly this hyothesis of no individual random e ects. Introducing individual random e ects in the estimation but with no correlation is equivalent to two distinct estimation of a random e ect robit model for each equation. The structural arameter estimates are now close to their theoretical value, taking 8 The estimation are erformed on a Dell OtiPlex 900 with a i7 Intel rocessor running at 3.4 Ghz. The rocedures are written in a standard code for maximum likelihhod estimation with Stata software.
account their standard errors. This is rather the case for the second equation, while the rst one resents estimates a litle bit smaller than theitr theoretical values. However the estimated standard deviations of the individual e ects are lower than exected for both equations. The likelihood ratio tests of the correlations between individual e ects and/or between the error terms in the model clearly accet the resence of such correlations in the estimations. Moreover these estimated correlations have a very small estimated standard error, even though they are non-linear transformations of the estimated arameters in constructing interval con dence foor these correlations. Benchmark () () (3) (4) Equation ;0 0: 0:49 0:440 0:436 0:446 [ 0:50] (0 :03 ) (0 :047 ) (0 :047 ) (0 :05 ) (0 :05 ) ; 0:408 0:98 0:97 0:94 0:939 [ :00] (0 :04 ) (0 :07 ) (0 :06 ) (0 :07 ) (0 :07 ) ; 0:005 0:005 0:004 0:004 0:003 [ 0:00] (0 :03 ) (0 :00 ) (0 :00 ) (0 :00 ) (0 :00 ) Equation ;0 0:8 0:53 0:507 0:545 0:5 [ 0:50] (0 :03 ) (0 :067 ) (0 :06 ) (0 :067 ) (0 :065 ) ; 0:0 0:489 0:493 0:495 0:497 [ 0:50] (0 :04 ) (0 :0 ) (0 :0 ) (0 :03 ) (0 :0 ) ; 0:465 :06 :008 :09 :0 [ :00] (0 :04 ) (0 :08 ) (0 :08 ) (0 :08 ) (0 :08 ) Standard Error of Individual E ects 0 :777 :737 :00 :995 [ :00] (0 :098 ) (0 :095 ) (0 :8 ) (0 :7 ) 0 :668 :637 :90 :895 [ :00] (0 :09 ) (0 :090 ) (0 :4 ) (0 : ) Correlations 0:5 0 0:534 0 0:476 [ 0:50] (0 :04 ) (0 :046 ) (0 :036 ) 0 0 0 0:550 0:536 [ 0:50] (0 :07 ) (0 :07 ) Log Likelihood 97:8 7896:7 786:4 776:9 7688:3 Standard errors of estimates in arenthesis. True value of arameters in squared brackets in rst column. Table : Simulation Results If a correlation between the error terms in both equations is allowed ( 6 0), the estimated results are closer from the theoretical values, while the standard deviation of the individual e ect are again under-estimated. In the oosite if
only a correlation between individual random e ects is allowed in the estimation ( 6 0), there are small changes in the structural arameters estimates, even though the estimated standard deviations of the individual e ects are now close from their theoretical values. The same conclusions are obtained in the full model where both correlations are estimated. All estimated arameters are now very close from their theoretical values, and the hyothesis of no correlations between individual e ects and between error terms is clearly rejected by the likelihood ratio tests. 4 An alication to roduct and rocess innovations In this section, we investigate the behavior of roduct and rocess innovations on a anel of French rms on the eriod 999-007. The data comes from the annual R&D surveys conllected each year by the Ministry of Research. The 999 reform of the R&D surveys in France introduced two new question s about the roduct or the rocess innovations. These question are stated as : "During the year, did your enterrise or your grou introduce new or signi cantly imroved goods coming from the R&D activity of your rm?" (Yes or No) "During the year, did your enterrise or your grou introduce new or signi cantly imroved methods of manufacturing or roducing goods or services coming from the R&D activity of your rm?" (Yes or No) These questions are slightly di erent from the usual Community Innovation Survey (CIS) questionnaire because in the latter the time eriod is rolonged over 3 years. For examles in the CIS 004 questions, the rst words are relaced by During the three years 00 to 004,.... Moreover in the French R&D surveys, only innovations coming from the R&D done by the rm are considered. That excludes the innovations which were introduced without any R&D e ort. On the other hand, the roduct or rocess innovations can be done by another rm in the grou. This is why the answers to the CIS surveys and the R&D surveys are not directly comarable. But the most imortant di erence is that in CIS surveys, the innovations are accounted for on the three years eriod. A second roblem arises from the fact that rms has many di culties to disentangle roduct or rocess innovations, even though the de nitions from the Oslo manual are quite recise (see the discussion in Mairesse and Mohnen, 00). When a rm introduces a new roduct on the market, it changes and imroves also the methods of roduction. Therefore, the roduct and rocess innovations is linked at the rm level. Even though this roblem of measurement is a serious one, we will consider both tyes of innovations in the following. 3
While there are some rms which innovates only in roduct or in rocess, the statistical di erence between both tyes of innovations are thin. There are also cross relationshis between roduct and rocess innovations. The samle of the French R&D surveys covers a 8 years eriod : from 000 to 007. Only rms with at least 4 consecutive years of data are retained in the samle. There are 74 rms, corresonding to 506 observations in the unbalanced samle. 6.5 % of rms reort an innovation in a new roduct during the year, while there are 60.3 % of rms indicating a rocess innovation. But large rms are more innovative than smaller rms. When the share of innovators are weighted by emloyment, the rate of innovation rises to 8 % for both roduct and rocess innovations. In fact about 60 % of small and medium-sized rms reort an innovation, either in roduct or in rocess, while 75 % of large rms (more than 000 emloyees) introduce an innovation during a given year. Process Innovation NO YES TOTAL Product NO 6.7 %.8 % 38.5 % Innovation YES 3.0 % 48.5 % 6.5 % TOTAL 39.7 % 60.3 % 00 % 506 observations, 74 rms, 000-007. Table 3 : Share of Product and Process Innovators in France There is a ositive and large association between roduct and rocess association. Nearly half of the observations in the samle lead to both tyes of innovation, while routhly a quarter of the samle reorts no innovation at all, neither in roduct nor in rocess, even though the rms are doing R&D during the year. Finally % of observations show only a roduct innovation, while 3 % only a rocess innovation. The Kendall s- B measure of association is 0.479 with an asymtotic standard error of 0.007 showing a large and ositive association between both tyes of innovations. Finally the tetrachoric correlation is 0.689 with a standard error 0.008. This clearly demonstrates the link between both tye of innovations at the rm level. But this high correlation can be due to the unobserved characteristics of the rm, or rather to an idiosyncratic shock a ecting both innovations at each eriod. We will estimate a simle bivariate robit model determining each tye of innovations at the rm level to illustrate which correlations are the most imortant at the rm level. In this simle model, the roduct or the rocess innovations are determined by the size of the rm, measured by the log of its total emloyment: log (L), and by the R&D intensity: RY, i.e. the total R&D exenditure divided by the total turnover of the rms. The squared value of the R&D intensity (RY ) is also introduced in the model in order to cature a non linear e ect of the R&D intensity 9. A full set of time dummies is also considered in the estimation. They 9 A non-linear e ect of the size of the rm have been tested with the squared log of total emloyment. But its arameter estimates is never signi cantly di erent from zero. 4
are always jointly signi cant in all the estimations below. Table 4 shows the main result of the analysis for the classical Probit model without any individual e ect and for the Panel Probit model with individual random e ects. The usual univariate estimations are done searately for each equations, while the bivariate estimations are resented with both correlations between individual e ects and between the error terms. This last estimations is done by using our double Gauss-Hermite Quadrature rocedures using integration oints for both dimensions. It takes more than 40 minutes to obtain the convergence to a maximum for the log-likelihood function but after only 5 iterations. PROBIT PANEL PROBIT UNIVARIATE BIVARIATE UNIVARIATE BIVARIATE PRODUCT INNOVATION log(l) 0:084 0:088 0:4 0:4 (0 :007 ) (0 :007 ) (0 :05 ) (0 :04 ) (RY ) :585 :630 :835 :658 (0 :80 ) (0 :79 ) (0 :37 ) (0 :307 ) (RY ) :550 :583 :749 :554 (0 :80 ) (0 :80 ) (0 :30 ) (0 :300 ) Const: 0:579 0:596 0:780 0:758 (0 :059 ) (0 :060 ) (0 :00 ) (0 :098 ) PROCESS INNOVATION log(l) 0:085 0:090 0:4 0:7 (0 :007 ) (0 :007 ) (0 :05 ) (0 :05 ) (RY ) :484 :53 :59 :443 (0 :8 ) (0 :80 ) (0 :30 ) (0 :3 ) (RY ) :560 :598 :57 :493 (0 :8 ) (0 :8 ) (0 :33 ) (0 :303 ) Const: 0:8 0:86 :075 :094 (0 :060 ) (0 :06 ) (0 :0 ) (0 :00 ) STANDARD DEVIATIONS AND CORRELATIONS 0 0 0:990 0:508 (0 :08 ) (0 :08 ) 0 0 :00 0:56 (0 :09 ) (0 :09 ) 0 0:74 0 0:464 (0 :009 ) (0 :06 ) 0 0 0 0:899 (0 :005 ) Log:Likelihood 63:68 4699:08 4308:4 3087:34 506 Observations, 74 Firms, 000-007 Standard errors in arenthesis. Full set of time dummies not reorted here Table 4 : Parameter Estimates 5
All the arameters estimates are highly signi cant because there is a lot of observations ( 506) in the samle, while there are u to 6 arameters to estimate in the full bivariate model. Moreover the likelihood ratio tests reject clearly the assumtions of the absence of individual e ects, and the zero correlations between these individual e ects or between the equations. The introduction of individual e ects rises the e ect of the size on the innovations in roduct or in rocess. The ect of the R&D intensity is ositive but decreasing for all the estimations because the arameter of the level is ositive, even though the arameter of its squared value is negative. In consequence the e ect of R&D intensity on innovations increases u to a maximum which is routhly 50 % of the total turnover of the rm. The change of the arameter estimates for the R&D intensity is mixed according to the methods of estimation. The bivariate robit model are similar for the roduct innovation, while its e ect seems to be a lit bit smaller for the rocess innovations with the anel bivariate robit estimation. The estimated standard deviations of the individual e ects are smaller when we take account of the correlation between these e ects. They are roughly divided by, even though the correlation between the individual e ects is very large with an estimates of b 0:90. Therefore the unobserved individual characteristics of the rm seems to a ect in the same way the robability to innovate in roduct and in rocess. Finally the bivariate anel robit estimation allows to disentangle the correlation between the individual e ects from the correclation in the idiosyncratic error terms between the two equations, which is recisely estimated with b 0:46. The Table 5 resents the comutation of the marginal e ect comuted at the mean value in the samle. The e ect of the log emloyment is small but if the size of a rm is twice the size of another rm (then log (L) increases of 0:69), the robability to innovate in roduct or in rocess will be higher by.6 %. PROBIT PANEL PROBIT UNIVARIATE BIVARIATE UNIVARIATE BIVARIATE PRODUCT INNOVATION log(l) 0:09 0:030 0:037 0:037 (RY ) 0:548 0:563 0:60 0:543 (RY ) 0:536 0:547 0:574 0:509 PROCESS INNOVATION log(l) 0:09 0:03 0:037 0:038 (RY ) 0:50 0:56 0:494 0:467 (RY ) 0:536 0:549 0:5 0:484 506 Observations, 74 Firms, 000-007 Table 5 : Marginal E ects at the Mean Using the bivariate anel robit estimation, the e ect of the R&D intensity is also ositive and reach a maximum at 54 % for the roduct innovation and 6
48 % for the rocess innovation. Comaring the case where the rm does not erform R&D with these maximum oints, the robability of an innovation in roduct increases by 4.5 %, while the robability of a rocess innovation will be higuer by.3 %. These gures seems to be reasonable because about 60 % of rms in the samle innovates in roduct or in rocess. 5 Conclusions In this article, an alternative metod of estimation of a bivariate robit model on anel data is resented. In such bivariate robit model on anel data, the likelihood function imlies to integrate the density conditional over the distribution of the individual random e ects in order to eliminate them by taking an average density. In the literature, some aers use simulations in order to comute the double integral. The alernative method relies on a double Gauss-Hermite quadrature rocedure in order to evaluate the double integral. This aer develos the log-likelihood function in this case and a rogram is written in Stata to estimate such model. This rogram should be otimized in the future in order to reduce estimation time, may be by using an adatative Gauss-Hermite rocedure. On anel data, it is imortant to introduce individial seci c e ects in order to avoid the omitted variable bias. This is shown in a simulation exercise where the ooled bivariate robit model is clearly rejected when there is no individual e ects in estimation. The searated estimation of the two robit models is clearly consistent due to the fact that the model is correctly seci ed and that the correlations between the individual e ects or between the error terms are only of second order. However a bivariate robit model allows also to estimate consistently the correlation between the individual random e ect and between the idiosyncratic error terms in the equations model. But the rocedure should be long even thouh the number of iterations is reduced. This rocedure is alied in the case of the estimation of the determinants of roduct and rocess innovations on a anel of French rms during the eriod 000-007. Here the model exlaining the roduct or rocess innovations is simle because it deends only on the size of the rm and ositively on the R&D intensity. There is a ositive e ect of the size of the rm of the same magnitude for both tyes of innovation, while there is also a ositive e ect of the R&D intensity, but this e ect is non-linear because it is decreasing u to a R&D intensity which is roughly 50 % of the total turnover of the rm. Finally the estimated correlation between the idiosyncratic errors terms is about 50 % indicating that a shock a ect in the same sense with a high magnitude both tyes of innovations. The unobserved heterogeneity also a ects both roduct and rocess innovations with a very high ositive correlation of 90 %, whicjh can be due to the fact that our model is very simle. The rm s unobserved characteristics may 7
lead to a rm s innovative behaviour for both innovations, but these characteristics should come from the internal organization of the rm or from the market on which it oerates. A further investigation of these determinants should be on the next agenda of research. The large correlated e ects could be also the sign of a high ersistence of innovative behaviour at the rm s level. The rm s characteristics can also a et ersistenly the roduct or rocess innovations. We should investigate the ersistence of this innovative behaviour in a following aer. 8
References BUTLER, J. S. and Robert MOFFITT (98) : "A comutationnaly Ef- cient Quadrature Procedure for the One-Factor Multinomial Probit Model", Econometrica, 50(3), May 98, 76-764. CAMERON Colin A. and Pravin K. TRIVEDI (005) : Microeconometrics: Methods and Alications, Cambridge University Press. GOULD, William W., Je rey S. PITBLADO, and Brian P. POI (00) : Maximum Likelihood Estimation with Stata (4th edition), College Station, TX: Stata Press. GREENE, William (003) : Econometric Analysis (5th edition), Prentice- Hall - Pearson Education. HARDIN, James W. (996) : "sg6: Bivariate robit models", Stata Technical Bulletin Rerints, vol. 6,. 5 58. KANO Shigeki (008) : "Like Husband, Like Wife: A Bivariate Dynamic Probit Analysis of Sousal Obesities", mimeo, Osaka Prefecture University. LANCASTER, Tony (000) : "The Incidental Parameter Problem since 948", Journal of Econometrics, 95, 39-43. LEE, Wang-Sheng and Umut OGUZOGLU (007) : "Well-Being and Ill- Being: A Bivariate Panel Data Analysis", IZA Discussion Paer Series No. 308, October 007. MAIRESSE, Jacques and Pierre MOHNEN (00) : Using innovation Surveys for Econometric Analysis, NBER Working aer No. 5857. MIRANDA, Alfonso (00) : "Dynamic Bivariate Probit Models for Panel Data", aer resented at the Mexican Users Grou Meeting, available at htt://fmwww.bc.edu/reec/msug00/mex0sug_miranda.df MUNDLAK, Yairn (978) : "On the Pooling of Time Series and Cross Section Data", Econometrica, 46, 69-85. NEYMAN, J. and Elizabeth SCOTT (948) : "Consistent Estimates Based on Partially Consistent Observations", Econometrica, 6(), January 948, -3. RAYMOND Wladimir, Pierre MOHNEN, Franz PALM, and Sybrand SCHIM VAN DER LOEFF (007) : "The Behavior of the Maximum Likelihood Estimator of Dynamic Panel Data Samle Selection Models", CESIfo Working aer No. 99. RAYMOND Wladimir, Pierre MOHNEN, Franz PALM, and Sybrand SCHIM VAN DER LOEFF (00) : "Persistence of Innovation in Dutch Manufacturing: Is it Surious?", Review of Economics and Statistics, 9(3), August 00, 495-504. SKRONDAL, Anders and Sohia RABE-HASKETH (004) : Generalized Latent Variable Modelling: Multilevel, Longitudinal and Structural Equation Models, Boca Raton, FL, Chaman and Hall. 9