Margarita Genius

Prof Strazzera Elisabetta
email: strazzera@unica.it

Crenos Dres

Applying the Copula Approach to Sample Selection Modelling

Keywords: sample selection, bivariate distributions, FIML, copulas Sample selection, FIML, Bivariate Distributions, Copulas JEL: C34, C51, H41, Q26

The limited availability of tractable multivariate distributions undermines the validity of the standard parametric approach to sample selection modelling. Copula distributions can be very useful in situations where the applied researcher has a prior on the distributional form of the margins, since the modelling of the latter is separated from that of the dependence structure. The present paper first presents an application to female work data. Afterwards, the approach is analysed in an application to contingent valuation data on recreational values of forests. It is shown that the copula approach is especially beneficial in case of strong departures from the hypothesis of normality.

Introduction

Endogenous sampling is a pervasive problem in applied microeconometrics. In an extensive survey on the topic of sample selection modelling, [START_REF] Vella | Estimating Models with Sample Selection Bias: A Survey[END_REF] affirms that "the ability to estimate and test econometric models over nonrandomly chosen sub-samples is unquestionably one of the more significant innovations in microeconometrics". While progress in the econometric analysis and treatment of sample selection cannot be denied, the debate is still open on what is the best procedure to be followed to obtain robust estimates from sample selection models.

In general, Full Information Maximum Likelihood (FIML) estimates are recognized as the most efficient, as long as the underlying models are correctly specified. The proviso is important, since FIML sample selection models are typically based on the assumption of bivariate normality of the joint distribution, which implies that the marginals are themselves univariate normals.

Unfortunately, this assumption can often be seen as unduly restrictive: this is, in general, the case for the two fields of application chosen in the present paper to illustrate the copula approach to sample selection, i.e. models of labour supply, and models of contingent valuation.

Sample selection issues arise in the context of labour supply because not all individuals participate in the labour market. In this context, [START_REF] Heckman | Simple Estimators for Treatment Parameters in a Latent Variable Framework with an Application to Estimating the Returns to Schooling[END_REF] suggest that since the wage density tends to be fat tailed, "the family of Student-t distributions offers an attractive and potentially more appropriate class of models for the treatment parameters than those implied by the benchmark Normal model".

In contingent valuation studies selectivity may be induced by people refusing to state, or deliberately misrepresenting, their reservation price for the good under analysis. The estimates of Willingness To Pay (WTP) based on the truncated sample of valid responses may be biased. Sample selection models can be used to detect and correct selectivity bias generated by protest behaviour: see [START_REF] Donaldson | Limited Dependent Variables in Willingness to Pay Studies: Applications in Health Care[END_REF], [START_REF] Alvarez-Farizo | Estimating the Benefits of Agri-Environmental Policy: Econometric Issues in Open-Ended Contingent Valuation Studies[END_REF], Kontoleon and Swanson (2002) (2002,2003). The Heckman's sample selection model might not be considered suitable in many applications, as the WTP distribution is generally non-normal: skewed and platikurtic distributions often provide a better fit to the data.

In an effort to attain more flexibility in sample selection modelling, a conspicuous stream of research has focused on non-parametric or semi-parametric methods, which do not require stringent distributional assumptions. Unfortunately, semiparametric methods impose some costs: for example, the intercept in the outcome equation is not identified, and its estimation requires additional procedures (such as those proposed by Heckman, 1990, or Andrews andSchafgans, 1996). Estimation of the covariance matrix of the parameters is more demanding than in the parametric case (see Vella, 1998, pp. 143-44). Furthermore, the choice of the bandwidth can affect the resulting estimates: in particular, problems of overfitting have been reported when crossvalidation techniques are used in conjunction with kernel estimates [START_REF] Mroz | Overfitting and Biases in Nonparametric Kernel Regressions Using Cross Validated Bandwidths: a Cautionary Note[END_REF], and this is especially so in two-stage estimation problems. On the other hand, if no cross-validation or optimal criteria are used to select the bandwidth, then many estimation rounds using different bandwidths are needed to ensure the resulting estimates do not differ drastically across bandwidths.

Another path of research maintains the parametric structure of the standard Heckman's model, but allows for other distributional assumptions. Early works in this direction are [START_REF] Olsen | A Least Squares Correction for Selectivity Bias[END_REF] and [START_REF] Lee | Some Approaches to the Correction of Selectivity Bias[END_REF][START_REF] Lee | Generalized Econometric Models with Selectivity[END_REF], who propose two-step methods where the assumption of normality is relaxed into an assumption of linear relationship between the disturbances. Of particular interest for our paper is the FIML model suggested by Lee in the same papers cited above (1982,1983). It is shown that the FIML approach could be maintained even in presence of a non normal joint distribution: all that is needed is that the econometrician knows (or, has good priors on) the non normal distributions generating the errors. It is sufficient to apply the inverse standard normal distribution function on the non normal marginals to transform them into normal variates, so that the bivariate normal (BVN) distribution can be safely applied. This procedure is a particular case of the copula approach suggested by [START_REF] Smith | Modelling Sample Selection Using Archimedean Copulas[END_REF] to model sample selection using a FIML framework and relaxing the restrictive BVN distributional assumption of the standard Heckman's model.

Broadly speaking, a copula is a function that links separately specified marginals into a multivariate distribution on [0,1] n . The copula representation of the multivariate distribution allows different specifications for the marginals and greater flexibility in the specification of the dependence, therefore bypassing some of the limitations of bivariate normality mentioned above. As will be seen in the course of the paper, this is especially useful in situations where the researcher might have some prior knowledge of the marginal distributions and also when asymmetry and/or fat tails in the bivariate distribution are suspected.

A limitation of the aforementioned Lee's copula is that while allowing great flexibility in the specification of the marginals, it still restricts the type of dependence to linear correlation, and the resulting distribution may still be unsuitable to fit the data. This could prove to be crucial in cases where dependence other than linear correlation exists, since the lack of the latter could lead to the erroneous conclusion that no sample selection bias exists. Other copulas, allowing a wider range of dependency patterns, would be more appropriate in such cases. [START_REF] Smith | Modelling Sample Selection Using Archimedean Copulas[END_REF] indicates a special class of copulas, namely the Archimedean copulas, easy to implement and quite flexible to fit a variety of distributional shapes.

In this paper, we first show how the copula approach works when the assumption of normality of the joint distribution is patently violated. It is a labour supply application, based on Martins ' (2001) work on female labour participation and wages. The copula parametric approach is compared to the Heckman's FIML, and to the semiparametric 2-step method that Martins uses to correct selectivity bias in the wages estimates. Next, we examine a case where departure from normality is only slight. We use contingent valuation data on recreational values of forests, published by Strazzera et al. (2003). They estimated a sample selection model using both the Heckman's FIML and 2-step methods, which we now compare to the results obtained from the copula approach.

The paper is organized as follows: the next section describes the copula models and their application to the sample selection problem; section 3 shows how the copula approach works in comparison to the standard Heckman's FIML model, and the semiparametric method on female labour data. The fourth section is devoted to the application of the copula approach to contingent valuation data on the recreational value of forests, characterized by selectivity bias due to protest responses to the WTP question. Several models are estimated, allowing testing of different dependence structures and distributional assumptions for the marginals. Section 5 concludes the paper.

The Copula Approach to Sample Selection

The structure of the sample selection model (in its simplest parametric form) is a two-equation system: the first equation is the

Selection equation < + + = 0 0 0 1 1 i i i i i z if z if Y ' '
(1) which determines the observability or not for all the members in the sample of the second equation, the

Outcome equation

i i i u x Y + = ' 2 (2); F o r P e e r
( ) = = = + = 1 0 1 2 1 0 1 1 i i i i Y x y y i i Y i y y H y g I z F I l ' ) , ( ) ( ln ) ( ln ' (3)
where g is the pdf of u i , and F is the cdf of i . This model was originated in [START_REF] Gronau | Wage Comparisons -A Selectivity Bias[END_REF] and [START_REF] Heckman | Shadow Prices, Market Wages, and Labor Supply[END_REF], who specified H as a Bivariate Normal. This distributional assumption is still the paradigm in FIML sample selection modelling, due to ease of implementation and relative flexibility in modelling correlation 1 . Unfortunately, distributional misspecification will, in general, produce inconsistent estimates of the parameters: see [START_REF] Vella | Estimating Models with Sample Selection Bias: A Survey[END_REF] for a thorough discussion.

A recent trend is to relax the normality assumption by using semiparametric methods, which do not impose parametric forms on the error distribution. As explained in the introduction of this paper, this strategy imposes several costs. [START_REF] Lee | Some Approaches to the Correction of Selectivity Bias[END_REF][START_REF] Lee | Generalized Econometric Models with Selectivity[END_REF] suggests a different approach: even if the stochastic parts of the two equations are specified as non-normal, they can be transformed into random variables that are characterized by the bivariate normal distribution. This transform, which involves the use of the inverse standard normal distribution, is an example of a bivariate copula function, which is defined as follows:

Definition: A 2-dimensional copula is a function [0,1] [0,1] : C 2
, with the following properties:

1 As opposed, for example, to the bivariate logistic that restricts correlation to a narrow range:

2 2 3 , 3 . F o r P e e r R e v i e w
For every

[ ] 0 C(u,0) C(0,u) , u = = " 1 0 , ; For every [ ] u u C and u u C u = = " ) , ( ) , ( , , 1 1 1 0 ; For every [ ] [ ] 2 1 2 1 2 2 1 1 1 , 0 1 , 0 ) , ( ), , ( v v and u u with v u v u # # × " : 0 ) , ( ) , ( ) , ( ) , ( 1 1 2 1 1 2 2 2 + v u C v u C v u C v u C .
The last condition is the two-dimensional analogue of a nondecreasing one-dimensional function.

The theoretical basis of multivariate modeling by copulas is provided by a theorem due to Sklar (1959).

Sklar's Theorem

Let H be a joint distribution function with margins F 1 and F 2 , which are, respectively, the cumulative distribution functions of the random variables x 1 and x 2 . Then there exists a function

C such that )) ( ), ( ( ) , ( 2 2 1 1 2 1 x F x F C x x H = , for every R x x " 2 1 ,
, where R represents the extended real line. Conversely, if C is a copula and F 1 and F 2 are distribution functions, then the function H defined above is a joint distribution function with margins F 1 and F 2 .

Since the copula function "links a multidimensional distribution to its one-dimensional margins" [START_REF] Sklar | Random Variables, Distribution Functions, and Copulas -a Personal Look Backward and Forward[END_REF], the name "copula" (connection) is explained. The parametric copula approach ensures a high level of flexibility to the modeler, since the specification of the margins F 1 and F 2 can be separated from the specification of the dependence structure through the function C and an underlying parameter % , which governs the intensity of the dependence 2 .

The aforementioned Lee's inverse normal transformation corresponds to specifying a bivariate normal copula with non-normal margins. Although it is computationally straightforward, and flexible in the specification of the marginals, its use in empirical work has been relatively scant: the reason may be that the type of dependence allowed for by this copula is restricted to linear 2 The present work only deals with parametric copulas.

) v ( ) u ( )) v , u ( C ( & & & % + = .
The additive structure of copulas in this class makes estimation of the maximum likelihood, and calculation of the score function, relatively easy. Furthermore, the family is sufficiently large so as to allow a wide range of distributional shapes (right or left skewness, fat or thin tails, etc.).

Another characteristic of copulas that can be valuable to the applied researcher is the capability of accommodating both positive and negative dependence. Copulas ranging from the lower Fréchet bound (perfect negative dependence as ' *

) to the upper Fréchet bound (perfect positive dependence as ' *

) are said to be comprehensive. A measure of dependence commonly used in econometrics applications is linear correlation; however, this measure is valid only when dealing with elliptical copulas (such as the BVN). Alternative measures of dependence include Kendall's ( (K ( ) and Spearman's ) (S ) ), which are measures of concordance3 . The former is defined as follows:

( ) ( )

0 0 < > = ) )( ( ) )( ( Y Y X X P Y Y X X P K ( .
Another expression for K ( is in terms of copulas (see Nelsen,cit.,p. 129):

++ = 2 1 0 1 4 ] , [ ) , ( ) , ( v u dC v u C K ( , F o r P e e r R e v i e w
9 that is the expression we will use to compute it when a closed form expression is not available. The measure proposed by Spearman is given by ( ) ( ) ( )

0 0 3 < , > , = ) ) ( ( ) )( ( Y Y X X P Y Y X X P S ) where ( ) ' , ' ) , ( ), , ( Y X Y X Y X
and are three independent random vectors with a common distribution function H whose margins are F and G.

Also in this case we have a copula expression:

[ ] ++ = 2 1 0 3 12 , ) , ( v u uvdC S )
For continuous random variables the above measures are measures of concordance, which implies that they take values in [-1,1], taking the value zero when we have independence (see Nelsen,cit.,p. 136 for a definition of concordance measure). Spearman's ) can be interpreted as a correlation coefficient between the cdfs of the two variables. We recall that the linear (or Pearson) correlation is not a measure of dependence: for example, In some applications symmetry may be an undesirable feature, and asymmetric copulas may be preferred. The Clayton copula exhibits asymmetry in the sense that there is a clustering of values in the left tail of the joint distribution: exactly the opposite to the Joe copula, which exhibits a strong clustering of values in the right tail. The Gumbel copula is similar to the Joe, but with a thinner tail. Unfortunately, the last three copulas, just as the most part of Archimedean copulas (one exception is the Frank copula), are monotonic: they cannot accommodate negative dependence.

Figures 1 and2 show the plots of some copulas (Clayton, Lee, Gumbel, Joe) based on standard Normal and Logistic marginals, and the BVN standard model. The results are summarized in table 2: the first two columns contain Martins' estimates of the parametric (FIML, BVN) model and of the 2-step semiparametric model, respectively in the first and in the second column. The standard errors reported in table 2 for the BVN model are calculated from the inverse of the computed Hessian, and differ slightly from those reported by Martins, apparently calculated from the cross product of the first derivatives. In the selection equation, the husband's wage seems to have no significant effect on the decision to participate in the labour market, while in the wage equation the only coefficient that is significant at the 5% level is the educational attainment. Martins shows that the HH test [START_REF] Horowitz | Testing a parametric model against a semiparametric alternative[END_REF] rejects the

Sample selection modelling on female labour supply data

Probit for the participation equation at the 5% level at bandwidth greater than 0.55, and argues that a semiparametric approach can be useful to overcome the misspecification problem. The estimates of the selection equation parameters in the semiparametric model can be obtained up to a factor of proportionality (i.e. one of the coefficients is normalized to one), so they are not directly comparable to the competing models; it can be noticed however that the coefficient of the husband wage becomes significant in the semiparametric model. Focusing on the wage equation, significant estimates are obtained for the educational level and the two variables related to potential experience, while the 5% level of significance is not attained for the two interaction terms between potential experience and children.

The 2-step semiparametric estimator is in general less efficient in comparison to the FIML estimator, provided the latter is correctly specified. We show now how the copula approach allows fairly easy estimations while relaxing the distributional assumptions imposed by the standard method, based on the BVN distribution. As a first step, the margins should be specified, based on some explorative analysis of the data, or theoretical priors. For the selection equation, applying the HH test to the Logit specification, we observe that it is not rejected at the 5% level up to bandwidth marginals. In the last column of table 4 we report the estimates obtained from the Joe copula model.

The parameter of the t distribution is estimated along with the other parameters. Its value, about 3, indicates very heavy tails in the distribution: we recall that for =1 the t distribution is a Cauchy, while for >30 it approximates a Normal. In the selection equation, the husband's wage is significant at the 5% level; in the wage equation the two interaction terms between potential experience and children are not statistically significant, while all the other estimates are significant at the 1% level. These results are close to those obtained with the 2-step semiparametric estimator, but they have been obtained with less computations than those required by the semiparametric approach, since the latter entails approaching the estimation as a two-step procedure and trying several bandwidths both for the first step estimates and for the constant term of the wage equation.

Furthermore, the copula approach allows estimation of the dependence structure, not estimated in the semiparametric model, which is important to analyse the statistical significance of the self selection effect (especially when the FIML BVN model does not produce a reliable estimate of the selection parameter). Also, observing the level and sign of the selection parameter may be useful for the interpretation of the self-selection process.

The approach using copulas can very easily be implemented using any software that allows for user specified likelihood functions such as GAUSS, LIMDEP, STATA, or EVIEWS. Model selection criteria such as Akaike or tests such as [START_REF] Vuong | Likelihood Ratio Tests for Model Selection and Non-Nested Hypothesis[END_REF] can be used as an aid in selecting between any two competing models. In the example above, the Akaike and Schwarz information criteria which use a penalization for the number of parameters in a model as well as the

Sample Selection Modelling on Contingent Valuation Data

In the following we present an application of the copula approach to the analysis of data on recreational benefits provided by forests and woodlands in Scotland. The study was conducted by the Queens University Belfast, and published by Strazzera, Genius, Hutchinson and Scarpa in Environmental and Resource Economics (2003).

The questionnaires were administered on-site in selected forest and woodlands sites used for recreation, through face-to-face interviews. Individuals were asked various questions aimed at conveying information about their demographic and socio-economic characteristics, interests and hobbies, previous excursions to forests, and details on the present visit. Afterwards, they were asked if they would be willing to pay a given entry fee (bid) to the forest, were this the only possibility to maintain public access to the forest. The fee was supposed to be paid by the respondent for each person in the party. The initial bid amounts t used were uniformly distributed across visitors, and were chosen on the basis of initial estimates of the WTP distribution obtained from extensive pilot studies. Next, individuals were asked the exact amount they would be willing to pay as an entry charge to the forest for each component of the party. We first estimate a standard FIML model, based on the assumption of bivariate normality of the joint distribution: column 1 of Table 4 reports the parameter estimates for the best fitting regressions for the two equations (participation and valuation), selected by means of likelihood ratio tests for nested specifications from more comprehensive models. The explanatory variables in the participation equation are: the amount the individual was asked to pay at the first stage of the elicitation process (i.e. the bid multiplied by the number of people in the party); the number of visits to the forest where the interview took place, or to other forest sites during the past year; time spent in the forest; parking expenditure; income (class 2); and a dummy variable indicating whether the individual was alone or in a party when visiting the forest. It can be observed that higher tendered bids induce a higher probability of a protest response. People who frequently visit forests are also more probably protesters, and this can be explained as a reaction to the reallocation of their property rights (in the Coasian sense). On the other hand, people who spent more time in the forest are less likely to protest, as well as people who paid a parking fee for the current visit, while the effect of income is not clear-cut.

The valuation equation specifies log WTP as the dependent variable. The results indicate a standard downward sloping demand curve (more frequent visitors to the forest are willing to pay less per visit). Time spent at the site and the appreciation of the recreational benefits given by the forest have, as expected, a positive effect. Also parking expenditures are positively correlated with stated WTP, and this can be easily explained by considering that the object of the elicitation question was a ticket inclusive of parking fees. Income has also the expected effect since the lower income categories are willing to pay less on average; males are willing to pay more than females.

The negative estimate for the coefficient of Children seems to indicate that respondents placed lower values for children in their party; but the effect must be somehow counter-balanced, since the coefficient estimate for party size close to one indicates that there is some proportionality between the total amount the respondent is willing to pay and the number of people in the pool.

Although this model does not show evident symptoms of misspecification (namely, instability of the coefficient estimates, and the correlation coefficient close to its boundary), we wish to investigate the tenability of the assumption of bivariate normality for the joint distribution. The following step involves the analysis of the distributional specification of the two margins. As in the previous case, both the [START_REF] Horowitz | Semiparametric estimation of a work-trip mode choice model[END_REF] and [START_REF] Horowitz | Testing a parametric model against a semiparametric alternative[END_REF] tests are applied to check the normality assumption for the selection equation. For the valuation equation we apply the Pagan-Vella test for normality. The results of the latter (F-statistic: 2.81) would lead to rejection of the hypothesis of normality for the valuation equation at a 1% level of significance, though not at a 5% level of significance. The HH test does not reject the probit model for the participation equation at all selected bandwidths; the Horowitz test at bandwidth h=1 rejects the Probit (Figure 1), while at the same bandwidth the Logit is not rejected (Figure 2).

After estimating the model under different distributional specifications (Normal, Logistic, Extreme Value) for either margin, we select the logistic-logistic specification as the one giving the best fit as measured by the Akaike and Schwarz criteria. The last columns of Table 4 report results

for the best fitting model, i.e. the Joe copula, which under all distributional assumptions performed better than the competing models. Its opposite, the Clayton copula, is also reported for demonstrative purposes. We also show results for the copula is in fact quite similar to the fitted BVN distribution, as the plots in Fig. 5 show.

Table 5 reports the estimates and confidence intervals for the measures of central tendency of WTP, obtained from the BVN and the Joe copula with Logistic marginals. Since the parameter estimates do not differ much across models, the mean and median values estimates obtained from them are also very close. The plots reported in Figure 5 are useful to explain this result: while the fitted Joe copula exhibits some skewness and fatter tails with respect to the fitted BVN, yet the divergence is not dramatic. Using the copula approach when there is a weak departure from normality of the joint distribution would not produce major changes in the estimates, but there is some gain in precision, resulting in narrower confidence intervals.

Conclusions

The copula representation of the bivariate distribution underlying the sample selection model allows different specifications for the marginals and great flexibility in the specification of the dependence. In a recent paper, [START_REF] Smith | Modelling Sample Selection Using Archimedean Copulas[END_REF] suggests the use of copula functions, and in particular Archimedean copulas, to correct selectivity bias in data affected by endogenous sampling. In this

  copula functionals allow greater flexibility in the dependence structure. In consideration of their simple mathematical structure,[START_REF] Smith | Modelling Sample Selection Using Archimedean Copulas[END_REF] advocates use of Archimedean copulas for application to selectivity models.Archimedean copulas are functions generated by an additive continuous, convex decreasing function &, with &(1)=0. If, in addition, &(0)=', the generator is strict. In general, Archimedean copulas have the following form:

  In a study published by the Journal of Applied Econometrics (2001) Martins applies both parametric and semiparametric methods to the estimation of the participation and wage equations for married women in Portugal. The author shows that the 2-step semiparametric estimator is more efficient than the parametric ML estimator. The parametric model is based on a wrong assumption of bivariate normality for the joint distribution function: testing for normality of residuals in the participation equation leads to rejection of the hypothesis. Estimation of a 2-step semiparametric model is shown to produce more efficient estimates. In the following we show how the copula approach works in this context. The data set is a sample from the Portuguese Employment Survey, interview year 1991. The sample used in the analysis consists of 2339 observations on married women, 1400 of whom were employed. Martins estimates a participation equation, regressing the dependent variable (which takes a value 1 if the woman participates in the labour force, and zero otherwise) on the following regressors: AGE (age in years), AGE2 (age squared), EDU (years of education), CHILD (the number of children under 18 in the household), YCHILD (number of children under the age of 3) LHUSWG (log of husband's wage). The outcome equation regresses the log of wages on the PEXP (potential experience years, calculated as age-edu-6), PEXP2 (PEXP squared), PEXPCHD (PEXP multiplied by CHILD), PEXPCHD2 (PEXP2 multiplied by CHILD).

  and is not rejected at 10% level for bandwidth h=1: the Logistic could be a candidate for the error distribution in the participation model. For the wage equation, a[START_REF] Pagan | Diagnostic Tests for Models Based on Individual Data: A Survey[END_REF] test indicates a strong departure from normality, and, following[START_REF] Heckman | Simple Estimators for Treatment Parameters in a Latent Variable Framework with an Application to Estimating the Returns to Schooling[END_REF] line of argument, a better choice would be a Student-t distribution. Thus, we estimate different copula models based on the Logistic (for the participation equation) and Student-t (for the wage equation)

Figure 1 .

 1 Figure 1. Plots of BVN, Gumbel, Joe and Clayton Copulas: Normal marginals.

  Vuong test favor the Joe

	copula with logistic and t marginals over the standard bivariate normal model (Vuong's statistic is
	8.7 and the test is asymptotically normal).
	F o
	r
	P e
	e
	r
	R e
	v i
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	13

Table 3

 3 gives summary statistics for the data used in this analysis: mean and standard deviation of the covariates for the full sample, and for the sub-sample of non protesters. Full descriptions of these variables are given in an Appendix. It can be seen that there are 535 protest responses, which amounts to 18% of the sample.

	The models are estimated using different covariate specifications related to the effect of socio-
	economic or personal characteristics, such as income, education, age, sex; or features of the visit,
	such as the number and age of components of the party, expenses for parking or food, activities
	engaged in during the visit, previous visit experiences.
	F o
	r
	P e
	e
	r
	R e
	v i
	e w
	14

  Lee copula, since it is fairly well known in the econometrics literature: recent applications include Von Ophem (2000) and Heckman et al. Parameter estimates do not change dramatically across copulas, but it can be observed that for most parameters the Joe and the Clayton copulas show departures in opposite directions from the benchmark estimates. The estimate of % in the Clayton copula, and its associated standard error, would indicate lack of dependence; however, this is due to the fact that the type of left tail clustering assumed by this copula is not compatible with our data, and the value of the loglikelihood confirms the relatively bad fit. The parameter % is not directly comparable across copulas, but Kendall's ( and Spearman's ) are. The Akaike and Schwarz criteria indicate the Joe copula, which exhibits the highest degree of dependence, as the best fitting model. However, unlike the first application, in this case the Vuong test fails to reject the BVN model: the fitted Joe

	(2001).
	F o
	r
	P e
	e
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	v i
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	16

Table : 3. Means and standard deviations (in parenthesis) by groups of respondents

 : that copula models are flexible and easy tools to deal with sample selection while improving efficiency. First, we examined a case where the data show strong departures from the hypothesis of normality. Using data published by[START_REF] Martins Fraga | Parametric and Semiparametric Estimation of Sample Selection Models: An Empirical Application to the Female Labour Force in Portugal[END_REF], we could see that the copula approach entail efficiency gains of the parameter estimates in a full information setting, and the improvement of the overall goodness of fit with respect to the BVN model is confirmed by the Vuong's test for model selection. We then applied the copula approach to WTP data collected to assess the use value of forests for recreation. This data had been modelled in a paper byStrazzera et al. (2003) by means of standard parametric sample selection models. Here, the tenability of the assumption of bivariate normality implicit in the standard Heckman's model is checked, and it is found that the hypothesis of normality for the joint distribution of errors in the outcome equation could be rejected, even though the departure from normality is not strong. The copula approach is applied to analyse different hypotheses on both the dependence structure and the distributional shape of the margins. Several copula models were estimated, and the best fitting model was a Joe copula, i.e. a model suitable for asymmetric, right-tailed joint distributions, linking two logistic distributions. However, the Joe copula and the BVN model are nearly equivalent, as indicated by the result of the Vuong selection test. As it perhaps could be expected, in this case the advantage of using the copula approach instead of the standard Heckman's model is not as important as in the case of strong departures from normality, even though it allows some gain in the precision of the estimates.

	Page 23 of 28	Submitted Manuscript
	F o FULL SAMPLE Mean WTP (£) Median WTP … … Children 0.88 (1.08) Alone 0.07 (0.26) Time 4.71 (0.75) Parking 0.23 (0.48) Past 1.51 (1.35) paper we show F o F o F o F o F o Other 1.40 (1.26) Improved 0.92 (0.27) F o	NON PROTESTERS 4.23(3.6) 3 0.88 (1.076) 0.06 (0.23) 4.77 (0.73) 0.26 (0.51) 1.39 (1.23) 1.35 (1.22) 0.92 (0.26)
	Income	r r r r r r r
	1: <16000 2: 16000-30000 Male Sample size	P e e r P e e r P e e r P e e r P e e r P e e r 0.32 (0.47) 0.47 (0.50) 0.31 (0.46) 0.49 (0.50) 0.65 (0.48) P e 0.65 (0.48) 2964 e 2429 r
			R e R e R e R e R e R e R e
			v i v i v i v i v i v i v i
			e w e w e w e w e w e w e w
			17 22

Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK

Table 4 . Estimates of BVN, 2-Step parametric and Copula Models for CV Data

 4

	Variables	BVN	2-step param.	Joe-NN	Lee-LL	Clayton-LL	Joe-LL
	Constant	0.743	0.728	0.695	1.213	1.194	1.156
		(0.201) *	(0.200)	(0.205)	(0.355)	(0.353)	(0.361)
	Bid1	-0.354	-0.329	-0.358	-0.629	-0.595	-0.637
		(0.036)	(0.036)	(0.035)	(0.064)	(0.066)	(0.063)
	Alone	-0.636	-0.642	-0.597	-1.086	-1.106	-1.049
		(0.107)	(0.107)	(0.105)	(0.182)	(0.183)	(0.179)
	Time	0.193	0.191	0.201	0.345	0.341	0.354
		(0.039)	(0.039)	(0.040)	(0.070)	(0.069)	(0.071)
	Park	0.584	0.579	0.583	1.227	1.208	1.231
		(0.094)	(0.092)	(0.093)	(0.209)	(0.207)	(0.207)
	Past	-0.134	-0.132	-0.133	-0.237	-0.231	-0.240
		(0.021)	(0.021)	(0.021)	(0.037)	(0.037)	(0.037)
	Other	-0.070	-0.075	-0.061	-0.116	-0.126	-0.104
		(0.021)	(0.022)	(0.021)	(0.038)	(0.038)	(0.038)
	Inc2	0.168	0.167	0.162	0.282	0.284	0.278
		(0.057)	(0.057)	(0.057)	(0.102)	(0.102)	(0.101)
	Constant	-0.666	-1.010	-0.717	-0.632	-0.543	-0.647
		(0.113)	(0.129)	(0.114)	(0.113)	(0.113)	(0.112)
	Children	-0.074	-0.086	-0.078	-0.077	-0.074	-0.080
		(0.018)	(0.018)	(0.018)	(0.018)	(0.018)	(0.018)
	Time	0.184	0.222	0.194	0.181	0.171	0.187
		(0.019)	(0.020)	(0.019)	(0.019)	(0.019)	(0.019)
	Park	0.267	0.336	0.273	0.283	0.265	0.282
		(0.028)	(0.030)	(0.028)	(0.026)	(0.026)	(0.025)
	Past	-0.115	-0.147	-0.121	-0.121	-0.111	-0.124
		(0.012)	(0.013)	(0.012)	(0.012)	(0.012)	(0.012)
	Male	0.067	0.069	0.068	0.078	0.078	0.080
		(0.028)	(0.028)	(0.027)	(0.027)	(0.027)	(0.027)
	Party	0.937	0.943	0.938	0.938	0.940	0.940
		(0.046)	(0.045)	(0.047)	(0.045)	(0.045)	(0.045)
	Improved	0.190	0.194	0.186	0.166	0.160	0.161
		(0.050)	(0.050)	(0.050)	(0.052)	(0.052)	(0.052)
	Inc1	-0.181	-0.175	-0.181	-0.183	-0.185	-0.183
		(0.037)	(0.037)	(0.037)	(0.037)	(0.037)	(0.037)
	Inc2	-0.142	-0.104	-0.137	-0.140	-0.152	-0.140
		(0.035)	(0.035)	(0.035)	(0.034)	(0.034)	(0.034)
		0.649		0.639	0.367	0.364	0.356
		(0.011)		(0.010)	(0.007)	(0.008)	(0.006)
	%	0.287 (0.074)		1.954 (0.308)	0.337 (0.078)	0.115 (0.109)	1.760 (0.193)
	K (	0.185		0.345	0.219	0.054	0.297
	S )	0.275		0.491	0.323	0.081	0.428
	Log-lik	-3606		-3600	-3590	-3596	-3584

Table 5 : Means and Standard Deviations from BVN and Joe-LL Copula Model

 5

		BVN	Joe-LL
	Mean
	WTP	3.518	3.550
	C.I. Mean
	>	3.392	3.433
	<	3.645	3.667
	Median
	WTP	2.851	2.855
	C.I. Med.
	>	2.739	2.762
	<	2.962	2.949

Other measures of dependence rely on the criterion of dependence between random variables: for a definition, see[START_REF] Nelsen | An Introduction to Copulas[END_REF] p. 170.

Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UKSubmitted Manuscript

 ,(( (log of) time passed in the forest (minutes) Parking:

(log of) cost of parking (£) Past:

(log of) number of visits to the forest in the past year Others:

(log of) number of visits to other forests in the past year Improved: the forest has improved recreation: 1-yes; 0-no Income: Household income (£) 1 <15999 2 16000<30000 3 30000 and above


Rafael L Myro

Alberto Colino

Patricio Pérez

Growht

ECONOMIC GROWTH IN A WORLD OF IDEAS: THE US AND THE LEADING EUROPEAN COUNTRIES

d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

With the introduction of an ideas function, Paul Romer started an interesting new line of research in economic growth models [START_REF] Romer | Endogenous Technological Change[END_REF]. After some useful changes in Romer's original specification to make it more generally applicable, Charles Jones performed a growth accounting exercise for the US economy along the period that contained at least two innovative contributions [START_REF] Jones | Sources of U.S. Economic Growth in a World of Ideas[END_REF]. The first was to relate Solow's residual path in the US to the number of scientists and technicians in the five countries concentrating the main weight of world tech´s innovation (US, UK, Germany, France and Japan, a group defined by the above-mentioned author as G-5).

The second was to distinguish between the part of the residual that can be attributed to the increase of population in a stationary state (and consequently the number of scientists and technicians, which is assumed to increase proportionally) and the part that can be assigned to the transition towards such a state.

Considering this line of research to be of great interest, we extend here the application of Romer´s and Jones´ ideas functions specifications to the three European Countries integrating the G-5 Group, Germany, France and the United Kingdom, trying to offer a explanation of the evolution of the aggregate multifactor productivity in them.

However, as these economies are technologically dependent to a certain extent, at least from the US, our purpose require mainly to focus in the mechanisms that capture the international dissemination of scientific knowledge as much as this proposed by Jones seems too simple even for the technologically more independent economies as it is that of the US. In fact, we show that the introduction of a catching-up process to a technological frontier improves the model for the European economies. The convenience of such an approach was already suggested years ago by [START_REF] Nelson | Investments in Humans, Technological Diffusion, and Economic Growth[END_REF] and more recently by [START_REF] Barro | Technological Diffusion, Convergence and Growth[END_REF] and either have implicit or explicitly been assumed in the empirical works by [START_REF] Coe | International R&D Spillovers[END_REF]Helpman (1995), De La Fuente (1995), Engelbrecht (1997) or [START_REF] Bils | Does Schooling Cause Growth ?[END_REF]. Jones´ generalization of the ideas function originally proposed by Romer adopts the following form in its discrete time version [START_REF] Jones | R&D Based Models of Economic Growth[END_REF][START_REF] Jones | Sources of U.S. Economic Growth in a World of Ideas[END_REF]:

t At t A H A = + 1 [1] A 0 > 0; > 0, 0 < 1; < 1,
Where A is the ideas stock, H A is the world research effort in the production of A, and is a parameter to measure the degree of dissemination. [START_REF] Jones | Sources of U.S. Economic Growth in a World of Ideas[END_REF] approaches the value A t through the observed multi-factor productivity (TFP) 1 and H A as the sum of the number of researchers in the G-5 countries, assuming that the human capital they embody is the same in all those countries.

To be able to carry out the OLS estimation, expressing A in a more handled way, and obviating as far as possible the fact that the time series of both H At and A t are non stationary clearly exhibiting increasing trends, he divides [1] by A t , which allows him to express the first term in annual change rates, in the following way:

= + / 1 1 t At t t A H A A [2]
1 Following Jones, A t is a latent unobservable variable defining the technological frontier. The total factor productivity (TFP) of the US economy is suppose to fit this frontier with some measurement error, i.e. making . Then he log-linearizes the resulting equation [2] around a path where both the value of A t measured through the TFP, and that of H At , grow at constant rates, obtaining a standard error-correction model with the following form:

1 2 0 1
) log (log log

+ + + + t t At A t A H g A [3] Were )) / log( 1 ( 0 A A g g is a constant, A g is the rate of growth of A and t A t t g + + +
1 1 is a error term. 2 is equal to 1/ and the parameter is obtained from the linear tendency in the logarithm of H At and A t , working then as an integration parameter which could also be approximated by the ratio between the rates of growth of the two mentioned variables2 .

From the three coefficients of the log-linearized equation, 0 , and 2 , the implicit values of and are derived, so we have all the relevant coefficients of the original ideas function.

Table 1 shows the results of the estimates of [3] for Germany, France and the United Kingdom, for the period 1950-1999, which also includes Jones' results for the US referred to the period 1950-1993, without imposing any value for . Our own estimation for the US is also shown, taking in the longer period between 1950 and 2001. Appendix 1 details the sources of the data and of the methods employed in their processing.

[Table1]

The results obtained for the three G-5 countries are in line with those for the US. The values of are somewhat lower for the European countries, but clearly higher than 1, contrary to what was expected, which seems to reveal the problems of non stationarity and endogenity signaled by Jones. ´s values are always negative, but not greatly so in the cases of Germany and France, where the R 2 s are higher. It is worth pointing out as well that the parameter values obtained for the UK have more in common with their US equivalents than with the values for the other European countries. Finally, the inclusion in the estimation for the US of the years corresponding to the productivity boom does not significantly alter the value of the coefficients obtained for that country, as might have been supposed given the short length of the time period added. It seems that this boom will only prove to be important in the long term if it shows continuity over the coming years.

Whatever the case, these results tend to underline the goodness of fit of Romer's model and of the specification adopted by Jones. Of course, the points made above regarding the robustness of and should not be ignored. Nevertheless here we are more interested in checking out the elementary mechanism of the international dissemination of knowledge implicit in the model (the scientists and technicians in the remaining G-5 countries), because, from the results obtained, it could be easily proved that the European countries would never attain technological convergence with the US, even if a similar proportion of the population were employed in research activities, now standing at around 1 per cent. This result might be partially due to the scale effect on H that is implicit in the ideas function, or to limits on the international dissemination of knowledge. However, it could also be a consequence of the model's failure to capture the influence of ideas from abroad, given the absence of explicit variables to measure the relative technological backwardness of the European countries. It is advisable, then, to introduce a technological catch-up term in order to show convergence towards the technological frontier, as suggested by several authors already quoted.

To explore this last possibility, we tested the real contribution of researchers working in different countries, after which we concluded that, in the estimations carried out, only those scientists and technicians working in each country are significant. The introduction of scientists working in the other G-5 countries does not alter the results obtained with this crucial variable. If this is true for Germany, France and Great Britain 3 , it is even more so in the case of the US, the leading generator of ideas within 3 The detailed results are available from the authors upon request. the G-5. Thus, as it can be observed in Table 2, if, in the estimation of Jones´ original ideas function for the US (expression [1]), the number of scientists in the remaining G-5 countries is introduced gradually and separately, the coefficients obtained for them all are statistically insignificant.

[Table 2]

The conclusion seems clear: the mechanism for the international dissemination of ideas considered by the reference model is not suitable even for the leading country.

Therefore, if this deficiency was overcome the model would be capable of providing better results, above all when technologically dependent European countries are concerned. We try to do this in the following section.

But, before coming to that, we feel it would be helpful to illustrate the technological backwardness of European countries through Figure 1, which registers "research intensity" in each country, defined as the proportion of scientists and technicians in relation to total employment. Only recently have Germany and France approached the levels reached by the ratio quoted for the US, to a large degree as a consequence of the latter's very appreciable slowdown in growth from the mid-1960s onwards.

Nevertheless, in the 1990s the United States once again leapt ahead of the technological leaders of Europe in this area. But even if the ratios displayed by the US and Europe had been closer, it seems undeniable that, just as the models by Romer and Jones suppose, the volume of new ideas depends on the number of scientists, not merely on their proportion of the working population.

[Figure 1]

It is also interesting to point out that the variable employed here to measure the R&D activity, the number of researchers, H At , seems to be very close in its time evolution to another more commonly chosen, the stock of technological capital 4 . To verify this, we took the calculations of this last mentioned variable by [START_REF] Martín | Serie de Indicadores de Convergencia Real para España, el Resto[END_REF], arrived at using real R&D expenditure and the permanent inventory method, assuming a given rate of depreciation. Then we calculated the technological capital indicator per 4 Bönte (2003) shows that a long run relation between productivity and R&D capital stock exists in the US economy. Nevertheless, through an application to the Italian economy, [START_REF] Atella | Do R&D Expenditures Really Matter for TFP?[END_REF] show that this relationship depend on the measures of Solow´s residual and on the aggregation level of the analysis. [Figure 2]

What proves surprising about the indicator constructed for technological capital per worker is that it registers lower values for the US and UK than for Germany and France, perhaps due to differences in the relative wages of researchers measured in common currency. Actually, such result seems to encourage the use of the number of researchers as indicator of the dimension of R&D activities, particularly in international comparisons.

A technological frontier convergence model for the leading European countries

On the basis of the aforementioned research works on technological international spillovers, a more general model may be formulated for any technologically dependent country, taking one country or a group of leading countries as the technological frontier, l.

) ( 1 µ µ + = t lt At t A A H A [4]
where the increase of ideas stock for the technologically dependent country during the period t+1 depends on its value in t and on the value in that same year of the ideas stock of the country considered to be on the technological frontier, l, as well as the number of researchers H A in the country in time t. Following Jones´ intuition, in expression [4] we do not impose constant returns to scale in the ideas stocks. To do that we would have to make =1. In our estimates, we explore the importance of such restriction.

If the country being studied is also the leader in technology, i.e. if it is the l country, the previous expression is simplified and becomes transformed as follows:

F o r P e e r R e v i e w t At t A H A = +1
which is the formulation suggested by Jones, a particular case of the more general model posed here. When =1 we have the more restricted model by Romer . In terms of rates of variation, the above expression adopts this form:

1 1 + = t At t t A H A A
In the Romer case, as =1, a direct relationship is established between the growth rates of A and H, where the scale effect on H is more evident.

If the country under study does not happen to be the leader, the corresponding expression is:

1 1 + ! " = µ t t lt At t t A A A H A A
in which the growth rate of A depends directly on H and on the technological distance from the leader, captured by the ratio A t,l / A t , where µ becomes a coefficient to measure the speed of convergence towards the technological frontier. It also depends on the stock of knowledge at the technological frontier, which is wider than the domestic stock.

To make the estimation procedure easier, equation [4] may be expressed in absolute values of A:

) ( 1 µ µ + = t lt At t A A H A . [5]
As De la Fuente points out (2002), this expression has similar implications to that by

+ + + = + µ [6]
The values of elasticities and no longer coincide with those in the original model of Jones, where the dependent variable is the variation of A. But it can be easily proved that the value of in Jones´ model is equal to that obtained in [6] multiplied by

! " + g 1 1 ,
where g is the rate of variation of A (De la Fuente, 2002) 5 .

In estimating this model, we discovered a high negative correlation between A t and t tl A A log log , generating problems of multi-collinearity and hence difficulting calculation of parameters´ true values. We decided, therefore, to carry out the estimation in two stages. First, we imposed the restriction of constant returns to scale ( 1)=0, just to estimate the values of and µ; then we used the residuals obtained in the estimation to calculate the value of the coefficient , regressing them on the log values of home ideas stock.

5 Similarly in Jones´ model correspond in [6] to ( 1)/g. When the parameter equality restrictions are removed, the results are worse for France, since µ does not prove significant, and for the United Kingdom, which obtains a negative value of . However, in this scenario, the values of this coefficient grow for Germany and France. When the equality restriction is imposed only on the values of , the results of the estimation improve for France but not for the UK.

However, the estimates forecast higher TFP than that registered for recent years, leading us to think that perhaps the model performs better avoiding the restriction of constant returns of scale, following Jones´ intuition. Nevertheless, when we approach the value of 1, using the residuals of the estimations already made, we get a negative and significant value for it only in the case of Germany, as shown in table 4. The estimate for that country is also robust to autocorrelation, following the standard Rao test for an error component model [START_REF] Bera | Test for the Error Component Model in the Presence of Local Misspecification[END_REF]. Anyway, the value obtained for Germany is very small, possibly indicating that could be close to 1.

[ to do so. This might, however, be because barriers to the entry of foreign technologies have tended to be on the rise in both countries, as Parente andPrescott (1993, 2000) suggest, even more than in the United Kingdom where less barriers would have been kept permanently.

[Figure 3]

The conclusion is that the ideas model can be improved in this direction, at least for the technologically dependent countries, or, to put it another way, for all but the US6 .

Nonetheless, it is undeniable that the United States also benefits from ideas created elsewhere, although it is difficult to capture this effect at an empirical level, since this demands the calculation of a more complex technological frontier, a combination of various countries. In fact, this might be the reason for the limited explanatory capacity of the model, highlighted in the obtained R 2 , though that in itself does not detract from the positive features of this specification, about which we offer more arguments below.

Conclusions

In this paper we have discussed some of the results that emerge from the ideas production model that was originally formulated by [START_REF] Romer | Endogenous Technological Change[END_REF], then generalized by [START_REF] Jones | R&D Based Models of Economic Growth[END_REF] and later used by the same author to explain the behavior of TFP in the US [START_REF] Jones | Sources of U.S. Economic Growth in a World of Ideas[END_REF]. We have analyzed the application of the model to the countries at the forefront of technological research in Europe, i.e. Germany, France and the UK, as well as to the US for a longer time period than that considered by Jones. We point out that Jones' empirical approach to capturing the international dissemination of ideas can be improved, as the TFP is completely dependent on each country's own scientists, both in Europe and in the US. Explicit consideration of the international circulation of ideas must involve the introduction of catch-up mechanisms in relation to a technological frontier. When this is done, the model seems to perform better and we observe that there is a process of convergence towards the technological frontier. In the explanation of the TFP evolution of some leading European countries, we have approached the distinction between the domestic creation of ideas from the capture of foreign ones. We have also made a first attempt to isolate the negative effect of the increased stock of ideas on the ability to discover new ones, adding new evidence to

Jones' intuitive instinct.

To advance further in this area of research would require, in addition to more depurated • Labour force. The starting point is the total employment in 1960, obtained from OECD Labour Force Statistics. The series for the following years was obtained by applying to that number the rates of variation provided by Eurostat, in European Economy. In contrast, the series for the preceding years, 1950-1960, is the result of deducting the annual variations provided by Maddison (1995b) from the number of people employed in 1960.

• Physical Capital. Fixed capital stock was calculated by means of the perpetual inventory method, obtaining the initial value of capital stock following the approach of [START_REF] Harberger | Private and Social Rates of Return to Capital in Uruguay[END_REF]. The depreciation rate used was 4 per cent. For the years between 1950 and 1960, the annual variation rates provided by Maddison (1995a) were applied to the value estimated for 1960. 1 9 5 0 1 9 5 3 1 9 5 6 1 9 5 9 1 9 6 2 1 9 6 5 1 9 6 8 1 9 7 1 1 9 7 4 1 9 7 7 1 9 8 0 1 9 8 3 1 9 8 6 1 9 8 9 1 9 9 2 1 9 9 5 1 9 9 8 2 0 0 1 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 1 9 5 1 1 9 5 3 1 9 5 5 1 9 5 7 1 9 5 9 1 9 6 1 1 9 6 3 1 9 6 5 1 9 6 7 1 9 6 9 1 9 7 1 1 9 7 3 1 9 7 5 1 9 7 7 1 9 7 9 1 9 8 1 1 9 8 3 1 9 8 5 1 9 8 7 1 9 8 9 1 9 9 1 1 9 9 3 1 9 9 5 1 9 9 7 1 9 9 9 1 9 5 1 1 9 5 3 1 9 5 5 1 9 5 7 1 9 5 9 1 9 6 1 1 9 6 3 1 9 6 5 1 9 6 7 1 9 6 9 1 9 7 1 1 9 7 3 1 9 7 5 1 9 7 7 1 9 7 9 1 9 8 1 1 9 8 3 1 9 8 5 1 9 8 7 1 9 8 9 1 9 9 1 1 9 9 3 1 9 9 5 1 9 9 7 1 9 9 9

YEARS

Germany

REAL ESTIMATE

C. UNITED KINGDOM

-0,06 -0,04 -0,02 0 0,02 0,04 0,06 0,08 1 9 5 1 1 9 5 3 1 9 5 5 1 9 5 7 1 9 5 9 1 9 6 1 1 9 6 3 1 9 6 5 1 9 6 7 1 9 6 9 1 9 7 1 1 9 7 3 1 9 7 5 1 9 7 7 1 9 7 9 1 9 8 1 1 9 8 3 1 9 8 5 1 9 8 7 1 9 8 9 1 9 9 1 1 9 9 3 1 9 9 5 1 9 9 7 1 9 9 9

YEARS

  the following pages we first expose and apply to Germany, France and the United Kigdom Romer´s and Jones´ ideas function specifications to afterward propose and estimate a new one which introduce a catching up process in technological knowledge. 1. Technical progress and scientific knowledge in the leading European countries compared to the US.

  and technicians) that show to be largely constant in the long run (Figure2), indicating parallel evolutions of capital and people involved in research activities.

Figure 3

 3 Figure 3 depict the best estimates for the three countries which fit well the true TFP values. The estimates are particularly good for Germany. An explanation for the decrease in TFP over recent years in Germany and France would seem to rely in the reduction of the external source of ideas, perhaps due to the fact that the European countries have gradually drawn closer to the US ideas stock and have slowly continued

  data and better estimates, the introduction of variables to explain what determines an increases the speed of the catch-up process, paying attention to commercial flows and foreign investment, as vehicles of transmission, in line with suggestions contained in certain recent studies.

FIGURE

  FIGURE 1.-RESEARCH INTENSITY (per cent of scientists and technicians in total employment)

  FIGURE 2.-TECHNOLOGICAL CAPITAL STOCK PER RESEARCHER Millions of 1995 euros per researcher

  FIGURE 3. TOTAL FACTOR PRODUCTIVITY IN THE EUROPEAN COUNTRIES: REAL AND ESTIMATES ANNUAL RATESA. GERMANY

Table 3 ]

 3 Table3offers the first set of estimations for Germany, France and the United Kingdom, assuming that the US defines the technological frontier they tend towards. The estimation is made jointly for them all, using the SUR method, testing the introduction of different restrictions on the parameters. Constant returns to scale in the foreign stock of ideas are assumed at this first stage.[The first estimation, in which a common is imposed and for the catch-up coefficient, µ, offers positive results. The value of R 2 is high and the values of and µ are significant, except in the case of the UK. The values for prove to be lower than in Jones' original model, but are not comparable to them as was explained above. The comparable values for Germany, France and the United Kingdom would be,

respectively, 0.56, 0.65 and 0.65, a result very much in line with what would be expected from the theoretical model. The speed of convergence towards the frontier is always in excess of 8 per cent, and frequently reaches levels around 15 per cent, guaranteeing convergence with the US in the stationary state, particularly for Germany.

Table 4 ]

 4

Table 1 .

 1 Estimation of the ideas function for 1950-1999 Estimates of the equation [3]. Log-linearized modelThe dependent variable is logA t+1 West standard errors, robust to heterokedasticity and autocorrelation, in brackets. The first estimation for the US is that carried out by Jones.

	• Investment. Gross Capital Investment at 1990s constant prices was calculated
	from Eurostat (European Economy Appendix). The values for 1950-1960 were
	calculated using the variation rates provided by Madison (1995a).

Table 3 .

 3 -Frontier convergence estimates for Germany, France and the United Kingdom.

		Estimates of the equation [6]. SUR method
			The dependent variable is log A t+1
		Germany	France United Kingdom	Germany France United Kingdom
		-0.118	Common	Common	-0.171	-0.208	-0.103
		(.029)			(.055)	(.050)	(.030)
		0.020	0.018	0.007	0.028	0.035	-0.002
		(.005)	(.006)	(.005)	(.010)	(.009)	(.006)
	µ	0.132	Common	Common	0.169	Common Common
		(.014)			(.025)
	R 2	0.52			0.54
	DW	1.73			1.79
		Standard error values in brackets. is not estimated here implicitly assuming it is equal to 1

Table 3 (

 3 Cont). -Frontier convergence estimates

		Estimates of the equation [6]. SUR method
			The dependent variable is log A t+1
	Germany	France United Kingdom		Germany France United Kingdom
		-0.125 In common In common	-0.252	-0.047	-0.093
		(.034)			(.071)	(.095)	(.043)
		0.019	0.021	-0.002	0.041	0.007	-0.002
		(.006)	(.006)	(.007)	(.012)	(.017)	(.007)
	µ	0.149	0.123	0.195	0.208	0.083	0.151
		(.016)	(.019)	(.053)	(.033)	(.050)	(.062)
	R 2	0.54			0.55
	DW	1.79			1.87
	Standard error values in brackets. is no estimated here implicitly assuming it is equal to 1

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Table 4 .

 4 -Estimates of the impact of Log At The dependent variable is the residual of estimates in Table3

	___________________________________________________________________________________
		Germany		France	United Kingdom
	____________________________________________________________________________________
		1	2
	_____________________________________________________________________________________
	constant	0.0171	0.0213	0.0100	-0.0096
		(.008)	(.009)	(.009)	(.022)
	( 1)	-0.0205	-0.0275	-0.0098	0.0143
		(.006)	(.006)	(.006)	(.017)
	R 2	0.17	0.26	0.03	0.002
	DW	1.79	1.63	1.69	1.64
	______________________________________________________________________
	Standard error values brackets

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This parameter E is supposed to be more faithful than . In fact,[START_REF] Jones | Sources of U.S. Economic Growth in a World of Ideas[END_REF] obtains implausible values for in his estimates, much higher than 1, as these are not driven by time trends and so the OLS estimators are subject to bias because of measurement error and endogeneity. So he opts for imposing values to along a justifiable range, from 0.25 to 1, deriving from them a spectrum of estimates for that presumably must include its true value.

Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript

Naturally, a great deal of work remains to be done in various areas, ranging from improving data and extending their reach to other countries, to the specification of the model and its application to different time periods, taking into consideration the possibility of structural breaks in R&D process in the mid 60s, as[START_REF] Pérez | Trend Breaks in the Research and Development Process[END_REF] show. .

, O40 -General < O4 -Economic Growth and Aggregate Productivity < O -Economic Development, Technological Change, and Growth, O33 -Technological Change: Choices and Consequences|Diffusion Processes < O3 -Technological Change|Research and Development < O -Economic Development, Technological Change, and Growth, O30 -General < O3 -Technological Change|Research and Development < O -Economic Development, Technological Change, and , TOTAL FACTOR PRODUCTIVITY, TECHNICAL PROGRESS,

APPENDIX 1. DATA SOURCES

• GDP per Hour. The data for GDP at constant prices for the 1990s were calculated using Eurostat (Statistical appendix to European Economy). The values corresponding to the period 1950-1960 are based on the GDP Movement series provided by Maddison (1995b). Weekly working hours in non-agricultural activities were obtained from the Work Statistics Directories, published by the International Labor Organization (ILO), whilst it was necessary to use various issues of the OECD Labour Force Statistics in order to estimate some of the values for the United Kingdom.

• Human Capital. The data for average years of educational training for the population sector over 25 years of age come from De la Fuente and Doménech ( 2001) and (in the case of Germany) from [START_REF] Barro | International Data on Educational Attainment: Updates and Implications[END_REF].

• Engineers and Scientists assigned to R+D activities. The source is the same as in [START_REF] Jones | Sources of U.S. Economic Growth in a World of Ideas[END_REF], though it was necessary to estimate for 1994-1999 using the exponential smoothing method. For the years prior to 1960, it was assumed that the ratio of "research intensity" for each of the three European countries in relation to the US was the same in 1950 as in 1960. This ratio was interpolated for the intermediate years. To obtain the number of scientists and technicians a multiplication was made based on employment.