Christian Belzil

Jørgen Hansen

Nicolai Kristensen

ESTIMATING COMPLEMENTARITY BETWEEN EDUCATION AND TRAINING *

Keywords: Skill Complementarity, Dynamic Treatment Effects, Dynamic Programming, Random Coefficients. Classification JEL: I2, J2, J3

In this paper, we formulate and estimate a structural model of post-schooling training that explicitly allows for possible complementarity between initial schooling levels and returns to training. Precisely, the wage outcome equation depends on accumulated schooling and on the incidence of training. The effect of training on wage growth depends on individual permanent endowments as well as on education. We find evidence of statistically significant complementarity, i.e. the higher educated receive the highest return to the MBA-type training considered here.

Introduction

In most schooling systems, education is accumulated through successive steps conditional on completion of a lower schooling level. There is therefore rarely any di¤erence in educational attainment among a cross-section of individuals currently attending the same education (grade) level. However, this is not the case in post-schooling training participation, where educational background may di¤er widely across participants. For this reason, it is natural to investigate whether the incidence of training and its e¤ect on wages depends on schooling.

This question motivates the current paper. More precisely, we estimate the e¤ect of training on post-training wage outcomes in a context where individual returns depend on permanent individual-speci…c heterogeneity and on the level of schooling achieved by the individual.

The issue of possible correlation between individual returns to training and accumulated schooling is far from new. In a series of papers (i.a. Heckman 2000, Carneiro and Heckman 2003, Knudsen et al. 2006, and Cunha and Heckman 2007), Heckman and co-authors argue that complementarity in learning exists. Skill formation is a dynamic process with strong synergies that lead early investments to promote later investments: skill begets skill.

In a dynamic skill accumulation model, complementarity may take at least two distinct forms. First, accumulated skills may decrease the psychic cost of investing in further skill accumulation. In such a case, individual age earnings pro…les would tend to display positive correlation between schooling and post-schooling wage growth, even if the wage regression is separable in education and experience (or any post-schooling training activities). However, with data on earnings, schooling and training decisions, the econometrician would be able to estimate the returns to education and training within a classical (separable) Mincer wage equation.

A second type of dynamic skill accumulation model would arise if, for a given level of individual permanent endowments, the wage gain associated with training participation depends on accumulated schooling. In such a case, schooling would also magnify the e¤ect of post-schooling human capital investments. However, for a given level of skill endowment, the wage equation would no longer be linearly separable in schooling and training. As a result, the returns to additional training will be higher for individuals with a relatively high formal schooling, say a college degree, vis-à-vis individuals with a lower level of education, say high school.

In both cases, the policy implications of complementarity are important and far-reaching: Investment in early childhood +development is more cost-e¤ective all else equal, vis-à-vis seeking to upgrade quali…cations of unskilled adult workers. As there exists a widespread belief that OECD countries have experienced a rapid shift in demand toward more skilled workers, the implications of the studies sited above are very important for how to e¤ectively design policies that deal with this shift in demand of skilled labor. 1 At a pure econometric level, estimating the e¤ects of training on wages is relatively complicated. Because both training and schooling decisions are potentially a¤ected by individual permanent endowments, IV estimation, typically based on the availability of policy shocks (or natural experiments), is ill-equipped to tackle this issue. Indeed, IV estimation is usually associated to the notion of Local Average Treatment E¤ect (LATE), which was introduced by Imbens and Angrist (1994). If the dispersion of individual wage gains re ‡ects di¤erences in schooling as well as di¤erences in skills (which would be the case if there is complementarity), classical IV would hardly be able to disentangle the causal e¤ect of schooling from the heterogeneity component. Because the LATE is an average return for a sub-population a¤ected by a policy change (the instrument), it is not even clear that such an estimate would be of any help in determining the incidence (or the degree) of complementarity between schooling and training. 2 In this paper, we focus on dynamic skill accumulation that takes the form of a causal e¤ect of education level on individual private returns (wage gains) to training participation. We investigate the degree of complementarity between formal schooling level and MBA-type of training (described in detail in section 2.3). To this end, we use data from Danish administrative employer-employee records spanning from 1980 to 2003 and condition the sample on participants with highest completed formal schooling in one of two levels. Level 1 includes high school degree or some (partly) vocational education at an equivalent level (12-13 years) while Level 2 consists of people with at least a bachelor degree or its equivalence (15 years or more). This provides us with a unique opportunity to analyze heterogeneity and complementarity in returns to training between di¤erent skills groups. We do this by estimating the returns to training using a structural model of the 1 Heckman and co-authors are not the only scholars …nding results in support of complementarity, see e.g. Shaw (1989) and Meghir and Palme (2005). However, complementarity in learning has not gone unchallenged. For instance, Krueger (2003) argues for the opposite view that returns to training are at least as big at the bottom of the income distribution as at the top.

2 The empirical relevance of the LATE parameter is not unanymously recognized. For instance, several economists have claimed that the LATE parameter is not "policyrelevant" for a variety of reasons, see e.g. Heckman and Vytlacil (2007).

training decision, in which the wage equation is formulated in a correlated random coe¢ cient wage regression model (CRCWRM) in the vein of Belzil and Hansen (2007).

The main …nding of the paper is the existence of complementarity between schooling and training. More precisely, we …nd that individual returns (wage gains) to training are magni…ed by individual schooling level. The results therefore indicate a dynamic causal e¤ect of schooling and imply that the standard representation of the Mincer wage equation, which is built on an assumption of separability between schooling and training, may be erroneous. 3 While it is not clear that the result regarding complementarity may be extrapolated to all possible post-schooling investment decisions, we believe that one of the major contribution of this paper is the development of a methodology that address possible non-separability between the e¤ects of schooling and post-schooling human capital investments on wages.

The paper is organized as follows. In the next section, we present the Danish data that have been used to implement our model. The setup of the behavioral model is found in section 3. The estimation strategy is discussed in section 4 while the parameter estimates are presented in section 5. Section 6 provides a discussion of the main …ndings while section 7 concludes the paper.

Data

Danish Register Data

Data come from administrative records maintained by Statistics Denmark. The records constitute a longitudinal database that contains information about all individuals aged 15 to 74 (demographic characteristics, education, labor market experience, tenure and earnings) and employees in all workplaces in Denmark during the period 1980-2003. Persons and workplaces are matched at the end of November each year. Consequently, only changes of employment November-to-November are accounted for, not intervening changes.

The database has been merged with a course register database that contains information on all government co-sponsored formal training courses, vocational as well as classroom training. This register includes detailed information on the business training program described below.

The Training Program

We analyze the e¤ects of a government co-sponsored training program in business administration4 , which has a long tradition in Denmark dating back more than 50 years and until recently this training program was the only MBA-type training program available in Denmark. The curriculum consists of studies in business economics with classes taking place in the evening (or in a few instances on weekends). The designed course load is equivalent to two years'full-time studies, divided into two parts of equal size. It is possible to stop after Part I but a diploma is obtained after completion of the second part only5 . The program is considered as training although it clearly has some academic component to it and as such is close in nature to education and (most) participants work full-time both during Part I as well as Part II of the program. 6 In Part I of the program, students are introduced to fundamental theories and methods in social science. The minimum educational background required in order to be eligible to enter Part I is a high school degree.

Part II of the program includes more in-depth studies in some …eld of business economics. 7 Students who have completed Part I successfully and who have a minimum of two years (full-time equivalents) of relevant work experience are eligible for Part II.

In this paper, the sample is restricted to individuals who have completed Part I of the program and we model their decision to either stop after Part I or continue (perhaps after one or more sabatical years) with Part II of the program. As mentioned in the introduction, we select individuals who belong to one of two levels of education; High School or College and above.8

Descriptive Statistics

We limit our sample to males, born 1960-1970 who completed Part I of the training program before 1989. We follow these individuals from the time they complete Part I and as long as we observe them in the data.

By conditioning on cohorts born 1960 or later, keeping in mind that the register database starts in 1980, we ensure that we observe individuals in our sample from around the time their labor market experience potentially starts accumulating (around age 19-20).

This leaves us with a sample of about 3,500 individuals (54,000 observations) of which 2,654 individuals have no schooling beyond high school while 857 individuals have completed college or more, cf. Table 1. About 75 percent choose to continue with Part II.

[Table 1 about here] Looking at the raw wage di¤erences between individuals who select into Part II of the training versus those who only take Part I reveals that the median wage among those who complete Part II is markedly higher than those who do not, cf. Figure 1. Of course, this does not convey any information about whether this is an e¤ect of training or purely a sign of positive selection of the most able and motivated.

[Figure 1 about here]

In formulating our structural model, we need to know the approximate time between completion of Part I and start of Part II (if ever entering). Not surprisingly, Figure 2 shows that about 90 percent of those who continue do so immediately, whereas the remaining participants enroll within the following two years. Likewise, the model takes into account the fact that duration of Part II of the training program, although it typically takes two years, may stretch over 4-5 years, cf. Figure 3.9

[Figures 2 and 3 about here]

From the data we can observe that the reason for prolonged duration for many training participants is not that they fail to pass exams but rather that they deliberately choose a less intensive course-load and for this reason stretch the duration over more than two years. This observation is taken into account when we estimate the probability that it takes more than two years to complete Part II.

A Structural Dynamic Model

As mentioned above, we consider individuals who have completed Part I of the training program, and we model their decision to continue in the training program (to enter Part II) or not. Every decision is made at the beginning of each period.

The Model

Choice Set

Individuals are assumed to be rational and forward-looking. In every period, they make choices in order to maximize their expected net present value of utility until they retire.

In our model, the choice set is very simple. Upon completion of Part I of the training program, individuals have to decide whether to continue with the second part of the program or not. All choices are made within the …rst three years after completion of Part I, but workers may be active in training up to and including period 5 in the model. Di¤erences in training duration mainly re ‡ect individual choices of various intensity levels (and not failure to pass exams).

Utility of Work

The instantaneous utility from work, denoted U W ( ), is set equal to the real gross log wages, ln w it . As the individuals in our sample experience only very low degrees of unemployment, we model their career paths as if they are full-time employed in every period. The log wage received by individual i in period t is given by

U W it ( ) = ln w it = ' 1i T R it + ' 2 Exper it + (1)
' 3 Exper 2 it + i I i;college + W i + " W it ;
where I college is an indicator which equals 1 if the individual has a college degree (including bachelors and masters) and zero otherwise, i is an individual-speci…c parameter, T R it is an indicator for having completed Part II of the training, ' 1i is the individual-speci…c wage return to training, and ' 2 and ' 3 measure the e¤ects of experience and experience squared. Unobserved heterogeneity is accommodated both through the individual speci…c intercept term, w i , and through the returns to education, training and education and training interacted (the complementarity term). Finally, " W it represents idiosyncratic stochastic wage shocks.

The individual-speci…c returns to training can be written as

' 1i = I i;college ' ci + ! 1i
where the term ' ci is an unrestricted type i speci…c parameter that represents the di¤erence in returns to training for those with a college degree relative to those with a high school diploma. A positive ' ci would imply that schooling has a causal e¤ect on the return to training and it could be interpreted as evidence in favor of complementarity in learning. At this stage, it is important to note that, in general, ! 1i is correlated with I i;college : Note also that because college attendance is a¤ected by individual heterogeneity, the di¤erence between actual training returns of high school graduates and college graduates would generally be di¤erent from ' ci :

Utility of Training Participation

Non-pecuniary utility is measured in monetary equivalents and enters the model as utility of learning and utility of leisure (disutility of loss of leisure), and these utility components are allowed to depend on a series of observable covariates which, following Belzil and Hansen (2002, 2003 and 2007), enter the model exogenously.

The per-period utility of selecting into the second part of the training is therefore given by

U T R it = X T R it + T R i + " T R it (2)
The matrix of covariates, X T R it , includes age and age squared, marital status, number of children and local labor market conditions, i.e. the local unemployment rate (by education group, gender, and age) across time. It also includes an indicator for whether college is completed or not and tuition fee. Lastly, the utility of training participation depends on unobserved individual heterogeneity, T R i , orthogonal to observed covariates while " T R it represents idiosyncratic stochastic shocks. The unobserved utility component, T R i , captures unobserved heterogeneity which arises if individuals are endowed with di¤erent unobserved pre-training skills or di¤er in terms of their intrinsic motivation. The term may also capture idiosyncratic knowledge about the time and e¤ort required for completion of the training. Each of these utility components can not be separately identi…ed so we think of T R i as the combined net outcome of unobserved factors.

The Stochastic Shocks

We assume that the random shocks (" T R it , " W it ) are normally distributed and independent of each other as well as independent of training, formal schooling and experience.

The Utility Function

The instantaneous utility is given as the sum of equation 1 and equation 2

U it = U W it + I d=1 U T R it ; (3)
where the components of U W ( ) and U T R ( ) are speci…ed above. The choice variable, d, equals 1 if the individual chooses to continue with Part II of the training program, and zero otherwise.

Solving the Problem

The Optimization Problem and Solution Method

For sake of clarity, the state space is partitioned into two components. The …rst one, s it , includes period-speci…c idiosyncratic random shocks to utility (" T R it ; " W it ), accumulated experience (Exper it ), and schooling (I college ) for individual i at time t. The second one, T R it , is equal to 1 if individual i has completed Part II in period t, and 0 if not.

The law of motion implies that the state variable, T R it ; equals 0 until the training course is actually completed.

Suppose that the decision to enter Part II is made at the beginning of period t: As individuals face two options, two value functions have to be speci…ed. First, the value function of choosing full-time work and foregoing the training opportunity at time t (denoted V W it (:)); is simply

V W it (:) = U W it (s it ; T R it = 0) + EV i;t+1 (s it+1 ; T R it+1 = 0)
where EV i;t+1 is the expected maximum utility, cf. below.

Consistent with the data, assume that the second part of the training period always last at least two periods. In order to capture the intrinsic randomness in the time needed to complete the program, we introduce a continuation probability, ; which represents the probability that the training program will have a duration beyond two years. This probability is parameterized simply as

(d t = 1) or not, denoted V T R
it (s it ); can be expressed by the Bellman equation

V T R it (s it ) = U T R it (s it ; T R it = 0) + EfV T R i;t+1 (s it+1 ; T R it+1 = 0)g (4)
where

E(V T R it+1 j s it+1 ; T R it+1 = 0) = U T R it+1 (s it+1 ; T R it+1 = 0) + f[(1 ) EV W it+2 (s it+2 ; T R it+2 = 1) + EV T R it+2 (s it+2 ; T R it+2 = 0)]g
With probability (1 ) he completes Part II in period t + 2, while with probability the training lasts an additional period. 10 The optimal value function, at date t; is EV it (:) = EM AXfV T R it (:); V W it (:)g The optimal choice is found using backward induction. Once an individual enters into Part II of the training program, he will complete the training in either 2 or 3 periods. In this sense, his outcome is deterministic and the problem can be seen as an optimal stopping problem: once a person enters the training program, there are no more choices to be made. Alternatively, if he does not enter in period t, he has the option of entering in the next period. In Appendix A, we include a more formal explanation of how to compute the EM AX used in the backward recursions.

Estimation

The Initial Condition

Although we are not modeling schooling explicitly, it is still important to allow for the distribution of unobserved heterogeneity to be arbitrarily correlated with college completion. This is because individual permanent endowments that may help explain training participation are also likely to have played a role in schooling choices. As a consequence, we formulate the schooling decision as a binary probit equation which also depends on unobserved heterogeneity. That is, the probability of initially having completed college, is simply

(X S i + S i )
where (:) is the standard normal cumulative distribution function, X S i includes mothers and fathers highest completed education and S i is unobserved heterogeneity correlated with other components ( i ; ' ci ; ! 1i ; T R i ; W i ; dur i ):

Unobserved Heterogeneity

As is customary in this type of models, we introduce unobserved heterogeneity in the form of …nite mixtures (Heckman and Singer, 1984). In particular, we assume that there are K types of individuals, each endowed with a value of ( T R k ; W k ; k ; ' ck ; ! 1k ; S k ); k = 1; 2; :::; K. 11

Identi…cation

As is the case for most structural models set in an intertemporal framework, the estimation of our model requires some explicit functional forms. 12 For instance, we make explicit assumptions concerning the form of the utility function and the distribution of random shocks. Based on these assumptions, we use panel data to identify the probability of actual choices and outcomes. Exogenous variation is obtained from time-varying tuition costs, changes in 1 1 The probability of a given type can then be estimated as p k = exp(q k ) P K j=1 exp(q k ) ;where one of the q 0 k s is normalized to 0. We employ K = 4: 1 2 Although IV models are typically presented without explicit functional form restrictions, their interpretation also require implicit restrictions, which may take the form of very speci…c functional form assumptions, cf. Keane (2006).

the distance to nearest institution and local unemployment rates (by gender, region, age and education). 13 Tuition fees were introduced starting from 1990 (prior to this year the training was free of charge) and have been rising (in real 1980 prices) since then (Figure 5). 14 [Figures 4 and5 about here] This introduces time and spatial variation, which helps in the identi…cation of the non-pecuniary part of the model.

The Likelihood Function

For a given set of parameter values, one can solve the "inner" optimization problem using backward recursions. Then, given a set of actual observed choices, the likelihood of these choices is computed (in an "outer"algorithm).

The likelihood consists of four parts: 15 (a) The probability of observing a particular sequence of choices. This sequence can be expressed as products of marginal probabilities, conditional on the unobserved heterogeneity components, T R k and W k , and is denoted L 1 (k). (c) The joint densities of observed wages from = 1 until nper i (the number of periods in which individual i is observed), at observed wage w , which can be factored as the product of a normal conditional probability multiplied by the marginal wage density f ( ), denoted L 3 (k).

1 3 In particular note that only about half of the institutions that o¤er Part I of the training program also o¤er Part II (Figure 4). This change in distance to Part II institutions provides variation, which should be helpful for identi…cation. However, empirically it turned out not to be signi…cant and it was therefore excluded from the estimation.

1 4 Note that prices di¤er between institutions. This could raise a concern for quality di¤erences in the training also. However, a recent report from the Ministry of Education concludes that dispite the substantial di¤erences in tuition costs there appears to be no quality di¤erence across institutions (MoE, 2000).

1 5 Details of how the model is solved and how the likelihood is constructed can be found in Appendix A.

(d) The probability of having a college degree (i.e. endogenizing the initial conditions), denoted L 4 (k).

The k-index represents the discrete type mixture for the unobserved heterogeneity components. Hence, the log-likelihood to be maximized is the product over types

Ln L = ln 4 X k=1 p k L 1 (k) L 2 (k) L 3 (k) L 4 (k):
(5)

Estimates of the Structural Model

The presentation of the structural parameter estimates is split into 4 di¤erent sets; those pertaining to the non-pecuniary costs of training (Table 2), the wage equation (Table 3), the continuation probability and initial conditions (Table 4) and the distribution of unobserved heterogeneity components (Table 5).

[Tables 2,3, 4 and 5 about here]:

The Cost of Training

Given the additive form of the Utility function, negative parameter estimates for the non-pecuniary utility components, equation 2, indicate that an increase in a speci…c variable tends to decrease the utility of training (increase the psychic cost). We note that the results indicate that the incidence of training increases with the local unemployment rate and decreases with tuition. Other things being equal, those who were older when they terminated Part I are less likely to enrol in Part II. Interestingly, the e¤ect of accumulated education (the college indicator) on the net utility of training is signi…cantly negative for alle four types (the weighted average is equal to -0.81). This does not necessarily imply that education increase the psychic cost of training. To see the argument, one has to re-interpret the negative e¤ect of education as a composite e¤ect on distinct components of the per-period utility of training. For instance, the utility of training may incorporate (i) a pure consumption value of the training cost (prestige associated with training, etc.), (ii) a component that re ‡ects psychic learning costs, or (iii) a component that measure the e¤ect of education on time availability for training (for instance, reduction in leisure may be higher for college educated).

Finally, the utility (or the cost) of training decreases (increases) with the number of young children.

The Wage Equation

Estimates of the wage equation are found in Table 3. The individual-speci…c return to college completion (ranging from 0.01 to 0.17) indicates a large premium to college completion. The population average return is equal to 0.15. When reported on a per-annum basis, these estimates imply wage returns to education (ignoring the e¤ect of education on the return to training) ranging roughly between 4 percent and 5 percent per year.

Not surprisingly, training completion is also associated with a relatively high wage return. For those who have completed high school, the returns range between 0.09 and 0.28. As Part I does not result in a diploma part of these estimated returns to Part II (vis-à-vis Part I) may also be returns to credentials.

For those who have completed college, the return to training is magni…ed by 0.10 (weighted average across types), after conditioning on unobserved heterogeneity. The population average of the individual-speci…c return to training is 0.14. Evidently, individual di¤erences in returns to training are dominated by individual permanent endowments.

Returning to the issue of complementarity, our estimates indicate a statistically signi…cant degree of complementarity between schooling and training. Interestingly, complementarity di¤ers markedly in magnitude across the four types ranging from zero (type 3) to 022 (type 2).

The Continuation Probability and Initial Conditions

The estimates in Table 4 show that there is considerable heterogeneity in the time required to complete Part II of the training program. As expected, older and more experienced workers appear to complete the program faster than younger and less experienced workers. Moreover, the duration of Part II is longer for those who took longer time to complete the …rst part of the training program.

The initial condition equation indicates, as is known in most countries, that the probability of graduating from college is highly correlated with parents'education. Unobserved heterogeneity is also found important.

The Heterogeneity Distribution

A thorough review of the main heterogeneity components (utility of training, utility from college and training, return to college, return to training and complementarity in training) and their standard deviations reveals a fair degree of cross-sectional dispersion in individual endowments. This is particularly true for the individual returns to training (the ! 1i ) with a standard deviation equal to 0.09 and the complementarity parameter (' ci ) with a standard deviation equal to 0.12.

Individuals of types 1 and 4 are endowed with the highest utility of training (lower learning cost) as well as the highest returns to college (these types also constitute the majority of the population, about 85%). This may be interpreted as evidence that both the wage return to college and the psychic costs of training are a¤ected by skills that are academic in nature. However, ignoring the e¤ect that schooling may have on returns to training, the wage return to training appears to be higher for those who are endowed with lower utility of training (types 2 and 3). This result is not surprising within a coherent behavioral model. A high positive correlation between utilities and returns (to training) would most likely imply a very large incidence of post-schooling training. Obviously, the frequency of post-schooling training predicted by the model is naturally limited by the observed frequencies in the data, since structural parameters must maximize the likelihood of observing the data (including training decisions frequencies).

Discussion

We now discuss a few issues that are naturally raised in the present context. In particular, we consider (i) the degree of heterogeneity bias that would prevail if we estimated the wage equation by OLS, (ii) the dynamic e¤ects of schooling on post-schooling wage growth (the return to training), and (iii) the e¤ect of an increase in training costs (tuition).

Heterogeneity Bias

One of the advantages of estimating a structural dynamic model with unobserved heterogeneity is that the estimated parameters of the Mincer wage regression are interpreted as causal e¤ects net of heterogeneity bias. For instance, in the context of a model with endogenous schooling decisions and exogenous post-schooling wage growth, a structural model o¤ers an explicit opportunity to quantify the classical "ability bias".

In the present model, the OLS bias is a complicated quantity. The di¤erence between OLS estimates and their true population counterpart depends on multi-variate heterogeneity (individual wage intercept, individual returns to college, and individual returns to training). It is therefore interesting to compare our estimates of the return to training and education (and possibly interactions) with parameters from an OLS regression.

Overall, the degree of heterogeneity bias appears to be important. The OLS estimate of the e¤ect of college on earnings -equal to 0.91 -is about 6 times as high as the average type-speci…c estimates. On the other hand, the OLS estimate of the return to training, around 0.13, appears to be only slightly smaller than the true average return to training (about 0.14).

Interestingly, the OLS estimate of the e¤ect of schooling on the return to training is negative and therefore does not suggest any complementarity. The estimate, which is equal to -0.33, when compared to the average type-speci…c structural estimate (0.10), may be interpreted as evidence in favor of important heterogeneity bias.

The Explanatory Power of Schooling on Individual Returns to Training

As noted earlier, di¤erences between individual returns to training of college graduates and high school graduates should be a¤ected by the causal e¤ect of schooling on training returns as well as a composition e¤ect. To verify this claim and in order to quantify the importance of complementarity, we regress individual-speci…c returns to training on a college completion indicator. To compute individual returns, we needed to simulate schooling outcomes by type so as to recover the relationship between individual endowments and schooling. Perhaps surprisingly, this regression indicates that the di¤erence in returns between college graduates and high school graduates is around -0.24. In other words, high school graduates have higher returns to training, even though the true causal e¤ect is positive 0.10 (on average). This indicates the importance of self-selection. If many individuals endowed with a high level of labor market skills (including a high return to business training) stop at high school, the results may display a large di¤erential between college and high school graduates in terms of returns to business training. Yet, this di¤erential is not inconsistent with a positive causal e¤ect of education and training.

This regression also allows us to decompose the variance of individual returns to training into two components; unobserved skills and accumulated schooling. The resulting R 2 ; equal to 0.26, indicates that di¤erences in schooling explains one fourth of the distribution of individual wage returns to training. The rest is explained by individual speci…c permanent endowments. This appears to be true whether or not we condition on individual endowments.

This result is consistent with the importance of persistent unobserved heterogeneity displayed in the distribution of individual endowments. As the degree of cross-sectional dispersion in individual skills and tastes is relatively high, increasing the level of schooling for a randomly selected individual will only have a limited e¤ect on the return to training. This type of result is common in structural models that allow for individual heterogeneity. 16

Increasing Training Tuition

In order to illustrate the role of heterogeneity in training enrolment, we simulated an increase of DKK 10,000 in training tuition. 17 The reactions to such a hypothetical policy depend on a number of individual-speci…c factors, such as non-pecuniary utility from attending training and the …nancial bene…ts of such activities. On average, the fee increase resulted in a drop in (predicted) enrolment by about 25 percent. The decrease in enrolment varies considerably across individual types, with Type 3 showing the largest drop (26.1 percent) and Type 1 individuals showing the smallest drop (23.3 percent). From the structural estimates, we know that Type 2 individuals receive the least non-pecuniary utility from attending Part II of the training program while Type 4 individuals receive the most utility. At the same time, the wage gain from training is lowest for Type 4 individuals and sig-ni…cantly higher for Type 2 individuals. Hence, in this particular example, the di¤erence in reactions to a fee increase is mainly driven by di¤erences in non-monetary utilty from attending training.

Obviously, these results are highly dependent on the partial equilibrium nature of our model. In a general equilibrium framework, increasing access to post-schooling training would not only a¤ect the composition of the population of participants, but would also a¤ect wage and training policies adopted by …rms. As a result, these e¤ects should really be understood as upper bounds.

Conclusion

In this paper, we have investigated the causal e¤ect that schooling may have on subsequent wage gains following training. Speci…cally, we investigated the complementarity hypothesis and found that training seems to be slightly more bene…cial to college graduates than to high school graduates. The result of signi…cant complementarity found here does not apply to all other types of training and as such the results cannot be interpreted as evidence of complementarity in general. The key contribution of this paper is the proposed model, which yields consistent estimation of returns to both education and training without imposing separability between schooling and training.

Obviously, the results reported herein apply to a context where the posteducation training course itself has an academic component. In the present context, education and training seem to represent activities that are close in nature.

At this stage, several questions remain unanswered. While education has a small causal e¤ect on the wage gain associated to Part II of the program, it may still be true that it has major e¤ect on the probability of entering, or completing, Part I. If so, it appears important to model training decisions from the beginning of the program.

It should be understood that we have investigated only one type of endogenous post-schooling activities (namely training) by which education can a¤ect post-schooling wage gains. However, within a dynamic skill accumulation framework, accumulated education can a¤ect (positively or negatively) the psychic cost of a large number of other post-schooling activities, or increase its associated wage gain. As the range of post-schooling investment opportunities is wide, it is unlikely that the econometrician can access data on all relevant investment outcomes, cf. Rosenzweig and Wolpin (2000). This conjecture presents major challenges to applied econometricians. In the presence of any type of post-schooling dynamics, the logic behind IV strategies collapse, and structural modeling becomes more complicated. 18 Our plan is to address this fundamental issue in the near future.

A Appendix: The Solution Algorithm and the Loglikelihood

A.1 The Solution Algorithm

The value function of going into training can be written as (omitting individual index i)

V T R t (s t ) = U T R t + E[V t+1 (s t+1 ) j s t ; d = 1] + " W t + " T R t ; (A1)
and the value function of not going into training is given by

V W t (s t ) = U W t + E[V t+1 (s t+1 ) j s t ; d = 0] + " W t ; (A2)
where U T R signi…es the deterministic part of the utility function.

The probability that an agent enters into the second part of the training program in period t equals the probability that

V T R t (s t ) > V W t (s t ) or equivalently that " T R t > V W t (s t ) V T R t (s t ), where V T R denotes the deter- ministic part of V T R , i.e. V T R t (s t ) = U T R t (s t ) + E[V t+1 (s t+1 ) j s t ; d = 1],
and similarly for V In order to evaluate the expected value function, we need to integrate out the period-speci…c shocks (" W t ; " T R t ). As the shocks are assumed to be iid normally distributed, we can use the law of iterated expectations and write (omitting time index t) E(V

T R ) = Pr[V T R ( ) V W ( )] E[V T R j V T R > V W ]:
These two terms can in turn be written as:

Pr[V T R V W ] = Pr " " T R T R > V W V T R T R # = " V T R V W T R # (A3)
and

E(V T R j " T R > V W V T R ) = V T R + 2 4 V W V T R T R 1 V W V T R T R 3 5 T R : (A4)
see e.g. Wooldridge (2002).

Similar derivations for

E(V W ) = Pr[V T R ( ) < V W ( )] E[V W j V T R < V W ] yields Pr[V T R ( ) < V W ( )] = 1 " V T R V W T R # (A5)
and

E(V W j " T R > V W V T R ) = V W + 2 4 V T R V W T R 1 V T R V W T R 3 5 T R (A6)
Collecting A3-A6 we can compute the EM AX[ ] given by

EM AX[V T R ( ); V W ( )] = Pr[V T R ( ) V W ( )] E[V T R jV T R > V W ] + Pr[V T R ( ) < V W ( )] E[V W jV T R < V W ]: (A7)
A.2 The Log-Likelihood

As outlined in section 4:2, the log-likelihood can be written as

Ln L = ln X 4 k=1 p k L 1 (k) L 2 (k) L 3 (k) L 4 (k): (
L 2 (k) = I (enter part II) ( [ 3 ] p 3 (1 [ 3 ]) (1 p 3 ) )
, where [ 3 ] denotes the probability that training lasts 3 periods (conditional on ever entering), and where p 3 denotes an indicator=1 if actual duration was 3 or more periods and zero otherwise. The probability is parameterized as 3 = 30 + 31 (Duration of P art I) + 32 (Experience j ).

(c) L 3 (k) is the joint probability of being in the labor market in year + 1, at observed wage w +1 . Hence L

3 (k) = f (w +1 ) f (w +2 ) f (w +3 ) f (w +4 ) ... f (w nper i );
where nper i denotes the number of periods in which individual i is observed.

(d) L 4 (k) is the probability of entering with a college degree as initial value. This part of the likelihood can be written as ( college college (1 college ) (1 college) ), where college denotes the probability of having a college degree upon entering. This probability is parameterized as col lege = c0 + c1 (mothers education) + c2 (f athers education):

Note that when wage densities are simulated, we use all available wage information, i.e. taking into account that the panel is unbalanced. 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007

B Appendix : Figures and Tables

Slagelse

Copenhagen Odense Aalborg Herning Aarhus

Notes: Prior to 1990 the training was fully financed by the public. To get 2007 prices, multiply with a factor 2.7, i.e. the tuition fee in 2007 was about USD 7,000-10,000 in real 2007 prices.

  where (:) is the standard normal cumulative distribution function, X dur it includes experience and duration of Part I and dur i is unobserved heterogeneity correlated with other components ( i ; ' ci ; ! 1i ; T R i ; W i ): The value function associated with the decision to work full-time and continue with Part II of the training program

  (b) The probability of completing training in 2 or 3 periods conditional on entering into training, denoted L 2 (k).

  a) L 1 (k) is the probability of observing the actual choice path for a given individual, conditional on the unobserved heterogeneity components. L 1 (k) = Pr[(d 1 (k) = 1); (d 2 (k) = 1); (d 3 (k) = 1); (d 4 (k) = 1)].(b) L 2 (k) is the probability of completing training in 2 or 3 periods conditional on entering into training. We may write this part of the likelihood as

Figure

  Figure 1 Gross Median Wages 1982-2003, by Education and Training

Figure 4

 4 Figure 4 Location of Institutions Offering Part I and Part II of the Training Program, 1980-2004

Figure

  Figure 5 Tuition Fee for Part II 1990-2007, by main institution (USD 1980 prices)

Table 1 Number of Individuals, by Education and Training

 1 : Conditional on experience=7 years. Altogether there are 3,527 individuals in the sample, and about 54,000 observations. 29

			Part II Training
		No	Yes	Total
	High School	565	2,089	2,654
	College	222	635	857
		787	2,724	3,511

Note

Table 2 Non-pecuniary Utility of Attending Part II of the Training

 2 Notes: copenhagen and single are indicator variables. duration of Part I training is measured in months and tuition is measured in 1,000 DKK (1980 prices).

		Parameter Estimate	Std. Err.	t-value
	constant type 1	18.509	0.055	336.356
	constant type 2	16.638	0.039	422.603
	constant type 3	17.378	0.097	179.569
	constant type 4	18.689	0.085	221.249
	local unemployment rate	0.021	0.006	3.516
	cohort	0.034	0.018	1.945
	copenhagen (0/1)	-0.074	0.017	-4.362
	age when completing Part I	-0.198	0.004	-46.063
	single	-0.054	0.026	-2.037
	number of children	-0.186	0.040	-4.659
	duration of Part I training	-0.011	0.002	-6.375
	tuition (1,000 DKK)	-0.151	0.020	-7.572
	college (0/1), type 1	-0.818	0.138	-5.944
	college (0/1), type 2	-1.385	0.198	-7.001
	college (0/1), type 3	-0.199	0.010	-20.131
	college (0/1), type 4	-0.794	0.137	-5.811

Table 3 Wage Equation Parameters

 3

		Parameter Estimate	Std. Err.	t-value
	wage constant, type 1	11.556	0.010	1157.045
	wage constant, type 2	10.830	0.013	856.604
	wage constant, type 3	11.930	0.016	746.161
	wage constant, type 4	11.275	0.008	1334.856
	college, type 1	0.174	0.026	6.617
	college, type 2	0.005	0.030	0.182
	college, type 3	0.099	0.027	3.643
	college, type 4	0.168	0.019	9.094
	experience	0.083	0.002	52.452
	experience-squared	-0.002	0.000	-19.945
	training Part II, type 1	0.156	0.010	15.895
	training Part II, type 2	0.243	0.014	17.804
	training Part II, type 3	0.282	0.014	20.510
	training Part II, type 4	0.091	0.009	10.480
	complementarity term
	training Part II x college, type 1	0.087	0.023	3.796
	training Part II x college, type 2	0.222	0.027	8.154
	training Part II x college, type 3	-0.066	0.071	-0.922
	training Part II x college, type 4	0.119	0.016	7.388

Table 4 Remaining Parameter Estimates

 4

		Parameter Estimate	Std. Err.	t-value
	σ-ξ	0.907	0.020	44.581
	σ-w	0.425	0.001	294.388
	prob(duration > 2 years)
	duration constant, type 1	-1.676	0.060	-28.007
	duration constant, type 2	-1.850	0.009	-216.172
	duration constant, type 3	-1.303	0.069	-18.870
	duration constant, type 4	-1.543	0.064	-24.028
	experience	-0.189	0.018	-10.463
	duration of Part I training	0.009	0.003	3.322
	prob(college degree)
	college constant, type 1	-1.837	0.043	-43.118
	college constant, type 2	-1.563	0.058	-26.968
	college constant, type 3	-1.382	0.066	-20.912
	college constant, type 4	-1.770	0.043	-41.371
	mothers education	0.120	0.018	6.879
	fathers education	0.063	0.014	4.446

Table 5 Type Probabilities

 5

	q k	Std. Err.	t-value	p k

The misspeci…cation of the Mincer equation has already been noted in[START_REF] Heckman | Fifty Years of Mincer Earnings Regressions[END_REF], as well as in[START_REF] Belzil | Testing the Speci…cation of the Mincer Wage Equation[END_REF]

In Danish called "Handelsdiploma" or simply "HD".

So the Handelsdiploma (HD) requires Completion of both Part I and Part II.

In the nomenclature applied by Statistics Denmark this course is de…ned as postschooling training. Furthermore, it takes place in the evening time and is attended by full-time workers as many other types of training.

7 Possible …elds include: Finance, Organization, Accounting, Logistics & Supply Chain Management, Financial Counselling and Retail & Key Account

Management.[START_REF] Belzil | Testing the Speci…cation of the Mincer Wage Equation[END_REF] As the focus here is on complementarity it is not necessary to model selection into Part I as well. Such an extension would not be trivial since the duration of part I also can be of various length.

As can be seen from Figures2 and 3there is virtually no di¤erence between the two education groups in their "time to enrol" and "time to complete".

0 From data we observe that Part II sometimes lasts more than three periods. Any duration beyond two periods is lumped together here.

6 For instance,[START_REF] Keane | The Career Decisions og Young Men[END_REF] document that increasing college attendance subsidies would be relatively ine¤ective for the US, precisely because school attendance is primarily explained by individual unobserved factors that are already determined by age 16.1 7 In

prices.

[START_REF] Belzil | Testing the Speci…cation of the Mincer Wage Equation[END_REF] See[START_REF] Belzil | Calibration and IV estimation of a Wage Outcome Equation within a Dynamic Environment[END_REF].

C

Appendix: Model Validation