Christian Belzil

Testing the Specification of the Mincer Wage Equation

Keywords: Mincer Regressions, Heterogeneity, Random Coefficient Models, returns to schooling, returns to experience. JEL Classification: J2-J3. 1

Using panel data taken from the NLSY, I perform the joint estimation of i) a reduced-form dynamic model of the transition from one grade level to the next with observed and unobserved heterogeneity, and ii) a flexible version of the celebrated Mincerian wage equation with skill heterogeneity, non linearity in schooling, non-separability between the effects of schooling and experience and heteroskedasticity (after conditionning on unobserved skills). The model rejects all symplifying assumptions common in the empirical literature. In particular, the log wage regression is highly convex, even after conditionning on unobserved and observed skills. Skill heterogeneity is also found to be over-estimated when non-linearity is ignored. After conditioning on skill heterogeneity, schooling has a positive effect on wage growth.

Introduction and objectives

The Mincerian wage regression is one of the most widely used tools of empirical economics. The literature making use of empirical wage regressions is vast. Mincerian wage regressions have been applied to numerous areas of Labor Economics. The Mincer wage equation plays a central part in the literature devoted to the returns to education as well as in the literature on wage inequality. It is also used to investigate statistical discrimination, gender differences in wages and occupation (and sectoral choices) choices. [START_REF] Belzil | Unobserved Ability and the Return to Schooling[END_REF] This paper is driven by one major objective. It is to obtain estimates of all the components of the Mincerian wage regression within an econometric specification i) in which schooling is endogenous, ii) in which the wage regression is estimated as flexibly as in the structural estimation literature, (iii) in which the number of parametric/distributional assumptions is kept to a minimal level. The econometric model is based on two distinct components; a reduced-form dynamic model of schooling attainment based on the hazard specification of the transition from one grade level to the next with observed and unobserved heterogeneity and a non-linear Mincerian wage regression model with observed and unobserved skill heterogeneity.

To meet this objective, I perform four main tasks. First, I obtain panel estimates of all the key components of the Mincerian wage regression function in a context where i) skill heterogeneity affects the intercept term, the return to schooling and the return to experience, ii) the local return to schooling may v a r yw i t hg r a d el e v e l( t h er e t u r nt oc o l l e g em a ybed i fferent than the return to grade school or high school), and where iii) the returns to experience depend on accumulated schooling. Secondly, I perform statistical tests of these various hypotheses (skill heterogeneity, non-linearity, and separability) in order to shed light on the optimal specification of the celebrated Mincerian wage regression function. Thirdly, I perform some variance decompositions of the individual specific intercepts and slopes in order to assess the relative importance of parents background variables, pure individual heterogeneity and accumulated schooling (for the returns to experience) in explaining skill heterogeneity in the labor market. Finally, in order to evaluate the reliability of the most popular model specifications found in the literature (obtained when various dimensions of the most general model specification are removed), I compare the estimates of the first and second moments of the returns to schooling and experience obtained under various scenarios.

The main results are as follows. The model rejects all simplifying assumptions common in the empirical literature. I find that the degree of convexity of the wage regression, as measured by the difference in the local returns to schooling before and after high school graduation, is dependent on the allowance for skill heterogeneity. However, the log wage regression remains highly convex, even after conditioning on unobserved and observed skills. The convexity is acute and it is therefore not solely a reflection of omitted skill heterogeneity. Not surprisingly, skill heterogeneity is also found to be quite important, but I also find that ignoring non-linearity inflates the crosssectional variance in the returns to schooling. After conditioning on skill heterogeneity, there is a positive correlation between accumulated schooling and the individual specific returns to experience. This is consistent with the view that accumulated schooling may have a causal effect on wage growth.

The results reported here are in line with those found in the structural literature. The estimates of the returns to schooling, much lower than point estimates reported in the OLS/IV literature, seem to suggest that the discrepancy between structural estimates and OLS/IV estimates may well be explained by differences in the econometric specification of the wage regressions, but not by the parametric assumptions required to achieve structural estimation.

The paper is structured as follows. In Section 2, i discuss some background literature. The empirical mo d e li se x p o s e di nS e c t i o n3 . S e c t i o n 4 is devoted to the results of the statistical tests. The structural parameter estimates are discussed in Section 5 and the relative importance of skill heterogeneity and non-linearities is st u d i e di nS e c t i o n6 . I nS e c t i o n7 ,Ii nvestigate the importance of allowing for non-separability. The conclusion is in Section 8.

Background Literature

For a long time, empirical models have been based on the ad-hoc assumptions that individual differences in market skills can be captured in the intercept term of the wage regression function and that log wages vary linearly with schooling. The validity of these assumptions has however been seriously questioned in recent years and many economists have examined the stability of the stylized facts about age earnings profil e sr e p o r t e di nM i n c e r( 7 4 ) . [START_REF] Belzil | Structural Estimates of the Intergenerational Education Correlation[END_REF] Consequently, economists have started to pay particular attention to the introduction of heterogeneity in the slopes of the wage regression, to potential non-linearity (the convexity of the wage schooling relationship) and to the separability between education and experience.

With regards to skill heterogeneity, the random coefficient representation of the wage regression function has gained in popularity, along with the literature on estimating average treatment effects. [START_REF] Belzil | A Structural Analysis of the Correlated Random Coefficient Wage Regression Model with an Application to the OLS/IV Puzzle[END_REF] At the same time, others have paid a particular attention to potential non-linearities explained by differences in local returns to the schooling across grade levels (Belzil and Hansen, 2002). [START_REF] Belzil | Risk Aversion, Education and Earnings Dispersion[END_REF] Furthermore, the recognition that post-schooling human capital investments should be treated as endogenous is likely to translate into new waves of empirical work which, among other things, should question the validity of the separability assumption (Rosenzweig and Wolpin, 2000).

While "skill heterogeneity" and "non-linearity" are not mutually exclusive, they are rarely confronted. This oversight might be a serious drawback. If the individuals who have higher market ability also have a comparative advantage in schooling (experience higher returns to schooling) and acquire more schooling, the convexity of the wage regression function might only reflect dynamic self-selection (merely a composition effect). That is, as we move [START_REF] Belzil | Structural Estimates of the Intergenerational Education Correlation[END_REF] For more details, see Heckman, Lochner and Todd, 2005 and Lemieux, 2003. [START_REF] Belzil | A Structural Analysis of the Correlated Random Coefficient Wage Regression Model with an Application to the OLS/IV Puzzle[END_REF] The term "correlated random coefficient wage regression model" is often used to refer to the standard Mincerian wage regression model where all coefficients are individual specific. Recent papers devoted to specification and estimation issues surrounding a random coefficient model of the wage regression include Heckman and Vitlacyl (1998 2005), Woolridge (1998, 2000), Angrist and Imbens (1994) and Card (2000)). Belzil and Hansen (2006) present a structural analysis of the correlated random coefficient wage regression model and show that all treatment effect parameters may be obtained within a structural framework. [START_REF] Belzil | Risk Aversion, Education and Earnings Dispersion[END_REF] Belzil and Hansen (2002) used a structural dynamic programming model to obtain flexible estimates of the wage regression function from the National Longitudinal Survey of Youth (NLSY) and found that a model with constant local returns is strongly rejected in favor of a highly convex log wage regression function composed of 8 segments. The average return over the entire range (around 4% per year) is found to be much lower than what is usually reported in the literature. The degree of convexity of the wage regression (as well as its change over time) is also pointed in Mincer (1997), Lemieux (2003) and Deschênes (2001).

toward higher levels of schooling, the local returns to schooling may turn out to be estimated from an increasingly large proportion of high ability workers. If so, allowing for cross-sectional heterogeneity in the slope parameter (s) of the wage regression might obviate the need for a flexible (non-linear) specification of the wage regression function and facilitate estimation. Equally, if the wage regression is truly convex (the returns increase with grade level), estimates of the returns to schooling obtained in a standard linear random coefficient framework might over-estimate the importance of cross-sectional heterogeneity. [START_REF] Cameron | Life Cycle Schooling and Dynamic Selection Bias: Models and Evidence for Five Cohorts of American Males[END_REF] Knowing the relative importance of the non-linearity and the skill heterogeneity hypotheses is fundamental for those interested in estimating the returns to schooling. In the literature, it is customary to estimate the log wage regression function using Instrumental Variable (IV) techniques and interpret the estimates within a linear random coefficient framework. The linearity assumption is therefore crucial. [START_REF] Cameron | Estimation of Educational Borrowing Constraints[END_REF] However, if the linear wage regression is not supported by the data and the form of the wage regression function is unknown, the estimation method is more complicated. Currently, the relative merits of both model specifications are unknown. A casual review of the recent literature would reveal that labor economists tend to favor the skill heterogeneity hypothesis. This preference is the result of ad-hoc assumptions. It is not founded on any evidence.

Similarly, the independence between education and the return to experience, typically illustrated by the fact that age earnings profiles are approximately parallel across broad education groups, is also being questioned (Heckman, Lochner and Todd, 2005, and Lemieux, 2003). This suggests that log wages regression may not be separable in education and experience and, in particular, that the return to experience may be affected by schooling. Indeed, many economic models may be consistent with this. [START_REF] Card | The Causal Effect of Education on Earnings[END_REF] Finally, it should be noted that the literature is not only characterized by the diversity of applications and by differences at the level of the functional form and the stochastic specification, but also by a variety of estimation methods. While the vast majority of econometric estimates of the returns to schooling or experience are obtained in an OLS or an IV framework, estimates have also been obtained using structural dynamic programming techniques based on maximum likelihood methods (or their simulated counterparts). There is a surprising discrepancy between estimates obtained in a structural framework and those obtained in a standard OLS/IV framework. While OLS and IV estimates are typically high (estimates lying between 10% and 15% per year are often reported for the US), structural estimates (such as those reported in Keane and Wolpin, 1997 and Belzil and Hansen, 2002) are much lower. [START_REF] Card | Can Falling Supply Explain the rising Return to college for Younger men? A Cohort-Based Analysis[END_REF] T h e s er e s u l t sa r ed i fficult to reconcile, as each estimation method is based on a large number of assumptions. [START_REF] Carneiro | Estimating Distributions of Treatment Effects with an Application to the Returns to Schooling and measurement of the Effects of Uncertainty on College Choice[END_REF] The model is estimated using data from the National longitudinal survey of Youth (79-90). I restrict myself to this period because the resulting sample is virtually the same sample used by Belzil and Hansen (2002a) and Keane and Wolpin (1997). I can then compare returns to schooling and experience obtained from structural models with those obtained from a reduced-form approximation of the dynamic discrete choice. A brief description of the sample data is found in Appendix. The empirical likelihood function maximizes the joint probability of the observed schooling attainment and a particular wage history observed between 1979 and 1990. The estimation method is flexible. It is semi-parametric in spirit and allows for observed and unobserved heterogeneity in all dimensions. Each component of the wage regression (intercept term, returns to schooling and returns to experience) require 11 parameters (4 support points and 7 observable regressors). It also allows for a flexible estimation of the error shock of the grade transition model and the post schooling wage distribution by assuming that the errors are drawn from a mixture of 5 normal distributions. [START_REF] Chen | Does College Attendance Increase wage Volatility? Working Paper[END_REF] AsfarasIknow,thisisthemostgeneral Mincer wage regression ever estimated.

A Reduced-form Dynamic Model of Schooling and Wages

The econometric strategy is based on two items; a hazard function of grade completion and a wage regression model flexibly specified.

Schooling attainments

The econometric model used to deal with the endogeneity of schooling attainment is a hazard function model of grade transition. I generate the hazard function from an individual/grade specifici n d e xγ * iS , is expressed as

γ * iS = γ 0S + X 0 i γ 1S + γ 2S • θ G i + ε S it (1)
where γ 0S ,γ 1S and γ 2S are vectors of grade specific intercept and slopes to be estimated. Without loss of generality, I define the index, γ * iS , as the difference between utility of leaving school after completing grade S minus the utility of continuing in school beyond grade level S. The decision to stop is recorded in a variable γ iS =1w he nγ * iS > 0a n dγ iS =0ifn ot . T hec o nditional probability of stopping school with grade level S (the hazard rate) is given by z S=s (γ * i,S )w h e r ez S=s (.) is a cumulative distribution function of ε S=s it .T h e r ea r ea sm a n yz S=s (.) 0 s as there are possible grade levels. The continuation probability is equal to one minus the hazard rate. The term θ G i represents an individual specific unobserved term affecting the propensity to acquire schooling. The vector X i is composed of observable family characteristics; father's education, mother's education, an interaction term between father's and mother's schooling, household income, Armed Forces qualification tests (AFQT) scores, number of siblings and an indicator equal to one if the individual has been raised by both biological parents and 0 if not. Yearly household income is reported as of 1978 and measured in units of $1,000. AFQT scores are corrected for the level of schooling at the time when the test was taken. [START_REF] Deschênes | unobserved Ability, Comparative Advantage and the Rising Return to Education in the United States: A Cohort Based Approach[END_REF] Note that θ G i is assumed to be orthogonal to X i . This approach amounts to the estimation of a vector of grade level specific intercept terms for each type, along with the restriction that the distance between each type specific intercept (at one particular grade level) is the same at all different grade levels.

The Mincerian Wage Regression with Heterogeneity, Non-linearity and Non-separability, and heteroskedasticity

The log wage received by individual i,a tt i m et,i sg i v e nb y log

w it = ϕ w i + ϕ i (S i ,Exper it )+ε w it (2)
where S i denotes schooling and Exper it is accumulated experience at date t. I use actual experience as opposed to potential experience and assume that it is exogenous (see Appendix 1). I assume that ε w it has density f w S=s (.). In order to estimate the model, I choose a tractable form for ϕ i (S i ,Exper it ), that is

ϕ i (S i ,Exper it )=ϕ S i (S i )+ϕ E i (S i ) • Exper it (3)
• ϕ S i (.)=ϕ i1 • S i + δ 2 • S ic -where ϕ i1 =exp(X 0 i β s + θ s i ) -where S ic = S i -12 if S i > 12 and S ic =0ifS i ≤ 12 • ϕ E i =exp(X 0 i β E + τ 1 • S i + τ 2 • S ic + θ E i ) • ϕ w i = X 0 i β w + θ w i • (θ G i ,θ s i ,θ E i ,θ w i
) are jointly distributed with CDF H(.). In order to approximate H(.) as accurately as possible, I assume that there are 4 types of individuals. Each type is therefore endowed with a vector

(θ G i ,θ s i ,θ E i ,θ w i )fork =1, 2...4.
Theprobabilit yofbelongingtot ypek, p k , are estimated using logistic transforms.

p k = exp(q 0 k ) P 4
j=1 exp(q 0 j ) and with the restriction that q 4 =0.

• z S=s (.) is approximated with a mean-mixture of 5 normal random variables; that is

z S=s (.)= M X m=1 P * m(s) (s) • Φ(µ m (s),σ m (s))
where Φ(µ m ,σ m ) denotes the normal cdf and where σ m (s)=1f o rm = 1, 2, ..M. Because I allow for 4 different (type specific) intercepts, I impose the following identification conditions; µ m (s) = 0 for one m, and . This is true for all possible s.

• f w S=s (.) is approximated with a mixture of 5 unrestricted normal densities; that is

f w S=s (ε w it )== M X m=1 P * m • φ(µ m ,σ m )
where φ(µ m ,σ m ) denoted the standard normal density. For identification purpouses, I also impose µ m (s)=0foronem.

Altogether, the definitions of ϕ S i (.),ϕ E i and ϕ w i allow for skill heterogeneity, non-linearities in the return to schooling (with two levels) and for a causal effect of accumulated schooling on the return to experience. [START_REF] Eckstein | Youth Employment and Academic Performance in High School[END_REF] The positivity of ϕ i1 and ϕ E i are imposed in order to eliminate the possibility of unrealistic values for predicted wages or for the returns to schooling and experience. Note that I focus on linear returns to experience because the model is fit on a sample of young workers and wages are observed over a period over which the concavity of earnings profile has most likely not set in yet. The allowance for a possible correlation between θ G i and labor market skill heterogeneity (θ s i ,θ E i ,θ w i ) will capture any endogeneity in schooling which may persists even after conditioning on X.

An inspection of equation ( 3) reveals that, in this particular framework, the returns to schooling vary with experience (education causes wage growth).

For a given number of years of experience, the marginal effect of a year of schooling is given by

δ log w it δS i = δϕ S i (.) δS i + δϕ E i (.) δS i • Exper it (4)
Focussing on the marginal effect of post-high school training, we get that

δ log w it δS i = ϕ S i (.)+δ 2 +[ϕ E i (S i ) • (τ 1 + τ 2 ) • Exper it ]( 5 )
In the literature, it is customary to assume that (τ 1 = τ 2 =0 , so that there is no distinction between the returns to schooling as measured at entrance in the labor market and the returns measured several years beyond school completion. In the present model, ϕ S i (S i )+δ 2 is a measure of the marginal effect of schooling on wages only at entrance in the labor market (when Exper it = 0). The growing pattern of the returns to schooling will be illustrated in Section 6.

The Likelihood Function

The likelihood function is the joint probability of observing a level of schooling attainment, S i , and a particular wage history (w i1 ...w i1990 ). Given type k, it has three components; the probability of having continued in school until S years of schooling is achieved (L 1k ), the joint probability of stopping school with S y e a r so fs c h o o l i n ga n de n t e r i n gt h el a b o rm a r k e ta to b s e r v e d wage w i1 (denoted L 2k ) and the density of observed wages until 1990 (denoted L 3k ). The entry wage (w i1 ) and the decision to stop school must be treated as dependent, even after conditioning on unobserved heterogeneity, because the schooling decision is most likely based, among other things, on starting wages. [START_REF] Geweke | Mixture of Normals Probit Models[END_REF] Given a distributional assumption for the hazard (and therefore the school continuation probabilities),L 1k is simply Π S-1 j=1 (1z j (.)) while L 2k would typically require the evaluation of the conditional probability of stopping school (conditional on the observed entry wage) times the marginal distribution of the entry wage. Finally L 3k is just the product of wages densities (w i2 , ...w i1990 ). For a given type k, the likelihood is therefore

L k = L 1k •L 2k •L 3k
and the log likelihood function to be maximized is

log L =log K X k=1 p k • L k (6)
where each p k represents the population proportion of type k.

Identification

In order to understand identification of the model, it is informative to consider the literature on estimating hazard functions, as well as from results on estimating discrete choices with mixtures of normals. To estimate the model, I choose to approximate the error shock by a mixture of normal distributions.

As shown in Geweke and Keane (1997), a mixture of normals is able to approximate a wide range of possible distribution in the context of a binary discrete choice model (provided some scaling conditions). It significantly outperform standard probit or logit models. At the same time, and consistent with panel data models, the repeated observations on labor market wages allow me to identify the person specific intercept and slopes. Again, using mixture of normals (with no restrictions) allows me to approximate the error shock as flexibly as possible. Because both the distribution of wage shocks and error shocks generating the sequence of discrete choices are estimated flexibly, and because I interpret the model as a flexible approximation to a dynamic discrete choice model, I estimate the joint likelihood without any exclusion restriction (a feature of most structural models).

Note that the grade transition model set in (1) is a special case of the reduced-form discrete choice model analyzed in Heckman and Navarro (2006). They prove non-parametric identification of several classes of dynamic discrete choice models (including discrete hazard functions) to which they append outcome equations. Unlike Heckman and Navarro (2006), I do not impose a curvature condition on the latent utility equation. Instead, I rely solely on the flexibility of the normal mixture specification to obtain parametric identification. Intuitively, identification may be more easily understood by noting that the joint likelihood uses not only the firstmomentofthewage equation, but also higher moments.

Searching for the Best Specification

As a first step, I estimated the most general model specification and reestimated several restricted versions that allowed me to perform likelihood ratio tests. In order to implement the model, I have initially investigated a version with observed heterogeneity and gradually included unobserved types. Various experiments have indicated that it is not necessary to go beyond 4 types. [START_REF] Gosling | The Changing Distribution of Male Wages in the UK[END_REF] It should be noted that the 7 regressors representing family background are highly correlated. As a consequence, I treated the set of regressors as a single block which can proxy skill heterogeneity and chose not to remove the variables that may turn out to be insignificant in one of the components of the wage regression.

There are 3 natural hypotheses of interest. The first one is that the effect of schooling on log wages is linear (δ 2 =0 ) . The second hypothesis is that, conditional on unobserved heterogeneity, the returns to experience are unaffected by accumulated schooling (τ 1 = τ 2 = 0). The third hypothesis is that skill heterogeneity is accounted for in the wage intercept and that a random coefficient specification is not required. This boils down to imposing β s = β E =0,θ s 1 = θ s 2 = ....θ s k , and θ E 1 = θ E 2 = ....θ E k (the "classical representation" of the Mincer wage regression). This test hinges on the assumption that there is a fixed (known) number of types. [START_REF] Heckman | Earnings Functions, Rates of Return, and Treatment Effects: The Mincer Equation and Beyond[END_REF] A summary of the likelihood ratio tests is found in Table 1 below. As is clear from the test statistics reported in Table 1, all three hypotheses are strongly rejected at the 1% level and, as a consequence, the optimal specification requires non-linearities in schooling, dependence between the returns to experience and accumulated schooling, and skill heterogeneity in the slopes. The evidence is overwhelming and does not require further discussion. This specification is now the baseline model which can be used to investigate several issues, which are addressed below. [START_REF] Gosling | The Changing Distribution of Male Wages in the UK[END_REF] Indeed, I tried with 6 types but it turned out that the parameter estimates and the correlation estimates were practically not affected by the decision to go to 4 types. This is most likely explained by the relatively large number of observed regressors already included. [START_REF] Heckman | Earnings Functions, Rates of Return, and Treatment Effects: The Mincer Equation and Beyond[END_REF] Considering the number of types as fixed is relatively standard in the empirical literature where the estimation method consists of a relatively complicated mixed likelihood function. Aside from the case of a single spell duration model, non-parametric estimation of K is rarely achieved in the empirical literature (see Heckman and Singer, 1984).

5T h e P a r a m e t e r E s t i m a t e s

In what follows, I discuss the parameter estimates and, in particular those pertaining to the wage returns to schooling and experience. The entire set of parameter estimates for the most general model specification is found in a sequence of tables ranging from Table 2A to Table 2F. Because there is a very large number of parameters required to approximate the distribution of the random shocks, and because these same parameters are relatively difficult to interpret, I report the resulting means, variances, and skewness coefficients for both the grade transition (Table 2G) and the wage distribution (Table 2H). The first and second moments of returns to schooling and experience are found in table 3. A Variance decomposition is documented in Table 4.

The Effects of Parents Background Variables

Parents background variables affect grade transition as well as wages. As mentioned earlier, these variables are highly collinear. The objective is to treat them as a single block of variables that are used as proxies for individual endowments. Indeed, their relative explanatory power with respect to grade transition and to wages will be discussed below (in the variance decomposition section).

Parents Background Variables and Schooling Attainments

The estimates of the effects of parents background variables on the hazard rate are found in the first column of Table 2A. After taking into account the interaction term between mother's and father's schooling, the estimates indicate that the school continuation probability increases with parents' schooling. [START_REF] Heckman | A Method for Minimizing the impact of distributional Assumptions in Econometric models for Duration Data[END_REF] The parameter estimates also imply that schooling attainments will increase with household income, AFQT scores and decrease with the number of siblings. Those raised with both biological parents also tend to leave school later. These results are consistent with what has been reported in Belzil and Hansen (2002), Eckstein and Wolpin (1999) and Cameron and Heckman (1998 and 2000). Similar results are also present in numerous other studies. They do not require more discussion.

Parents Background Variables and Wages

Overall, the level of significance of the family background variables in the wage regression (reported in Table 2B) is somewhat lower than what was found in the grade transition equation. Notwithstanding this, there is evidence that most variables associated with higher schooling attainments (lower hazards) are also associated with higher returns to schooling, higher returns to experience and higher wage intercepts. This conclusion is reached after the examination of the effects of parents background variables and after taking into account the interaction terms. In particular, the returns to schooling andexperienceincreasewithfather'sschooling and mother's schooling. Both the intercept term and the returns to experience also increase with family income and AFQT scores. However, the returns to schooling appear to be decreasing with both parents income and AFQT scores, although the effects are relatively small.

Non-Linearity

The results found in Table 2C show strong evidence in favor of the convexity of the wage-schooling relationship. Without loss of generality, these estimates of the returns to education are measured upon entrance in the labor market (when Exper it =0). The parameter estimate for δ 2 (equal to 0.0406), along with the estimates for β s (Table 2B), imply an average return to schooling equal to 0.0403 per year of schooling prior to high school graduation and 0.0804 in college (Table 3A). This is consistent with evidence presented in Belzil and Hansen (2002) and seems to indicate that the non-linear (convex) shape of the wage schooling relationship is acute and, furthermore, not a reflection of omitted skill heterogeneity. This issue will be addressed in Section 6. [START_REF] Heckman | Structural Equations, Treatment Effects and Econometric Policy Evaluations[END_REF] Aside from convexity, it should also be noted that, when compared to the estimates of the returns to schooling reported in the IV literature, these estimates are small. [START_REF] Heckman | Instrumental Variables Methods for the Correlated Random Coefficient Model[END_REF] However, they are comparable with the relatively lower estimates obtained in the structural literature. [START_REF] Hsiao | The Analysis of Panel data[END_REF]

Non-Separability: The Effect of Schooling on Wage Growth

After conditioning on skill heterogeneity, and taking into account the endogeneity of schooling, the estimate for τ 1 and τ 2 (found in Table 2C) indicate that there is a positive correlation between accumulated schooling and the individual specific returns to experience. However, this positive correlation is mostly explained by schooling acquired beyond high school graduation. This is illustrated by the relatively small value of the estimate for τ 1 (0.0033) and the much larger value for τ 2 (0.0573). This is consistent with the view that accumulated schooling may have a causal effect on wage growth.

Unobserved Heterogeneity

An inspection of the unobserved heterogeneity support points, along with the type probabilities (Table 2D and Table 2E), reveals that those types who will experience higher schooling attainments (lower hazard rates) will also experience higher returns to schooling and experience. More precisely, and after conditioning on observed attributes, type 3 individuals (representing 52% of the population) will experience the highest level of schooling (the lowest hazard) and will be endowed with the second highest returns to schooling and experience. The highest return to schooling and experience is achieved by type 2 individuals while type 4 individuals will obtain the lowest returns.

Heteroskedasticity and Skewness

Because I assume a separate distribution for the random shocks, for each grade level, the results are therefore obtained under arbitrary form of heteroskedasticity, skewness or kurtosis. Because of the relative complexity of [START_REF] Heckman | Instrumental Variables Methods for the Correlated Random Coefficient Model[END_REF] The OLS estimate fluctuates depending on which year of the panel is chosen. However, for those periods considered, it averages around 10% per year. Indeed, the pooled OLS estimate is equal to 9.9%. [START_REF] Hsiao | The Analysis of Panel data[END_REF] As pointed out in Belzil (2006), IV estimates are typically obtained in a framework where accumulated experience is ignored and implicitly included in the error term.

the formulas that link each parameter (the P * 0 m s,theµ m s and the σ 0 m s)tothe resulting moments (variance, skewness and kurtosis), it is difficult to perform a formal test of equality of these moments across schooling levels.

In view of the recent literature aimed at distinguishing ex-ante risk from unobserved heterogeneity, it may be particularly interesting to investigate how the variance of log wages behave across grade levels. The sequence of grade level specific variances, found in Table 2H (column 2), does not provide strong evidence in favor of (or against) a generally increasing pattern. The variance of the stochastic component of log wages fluctuates around 0.30, but it there is no clear pattern that allows to link the variance with schooling. Heteroskedasticity does not appear to be a serious issue for log wages. Similarly, the degree of skewness resulting from the distributions appears quite mild. Negative skewness appears as frequent as positive skewness, but furthermore, the degree of skewness rarely exceeds 0 by a large amount.

Variance Decompositions

Some variance decompositions are found in Table 4. These may be used to infer the relative importance of parents background variables, unobserved skills and schooling (for the returns to experience) in explaining skill heterogeneity. The main findings seem to indicate that, while modeling wage regressions in ac o n t e x tw h e r et h ec o e fficients are allowed to be correlated with observed characteristics is important, skill heterogeneity is captured mostly through unobserved skills. More precisely,

• Only 9% of the cross sectional variations in returns to schooling is explained by parents background variables while 91 % is explained by unobserved skills

• 24% of the cross-sectional variations in the returns to experience are explained by parents background variables while 60% are explained by unobserved skills. Interestingly, accumulated schooling explains 18% of the returns to experience.

• 38% of the cross sectional variations in the wage intercept are explained by parents background variables while 62% are explained by unobserved skills.

6 Assessing the Relative Importance of Heterogeneity and Non-linearity.

At this stage, it is natural to investigate the consequence of ignoring either skill heterogeneity or non-linearity on the accuracy of the estimates of the returns to schooling. After all, most estimates published in the literature (based on IV methods) are based on cross-section data and on model specifications where the wage regressions are assumed to be linear in schooling.

While determining the degree of convexity might appear as a pure statistical issue at first glance, it is not really so. As discussed in Belzil 2006, the college/high-school wage premium may be easily explained in presence of high psychic costs. Determining the importance of non-linearities may therefore be a key step in evaluating the importance of psychic costs in college attendance decisions. [START_REF] Imbens | Identification and Estimation of Local Average Treatment Effects[END_REF] The reliability of various model specifications may be investigated by comparing estimates of the returns to schooling upon labor market entrance obtained when various dimensions of the most general model specification are removed. In Table 5, I perform such comparisons. I also report estimates of the returns to schooling and experience in the case where skill heterogeneity and non-linearity are omitted (in column 4) and compare them to the estimates already reported. For the specifications with skill heterogeneity, I report the mean return to schooling as well as the standard deviation.

The results indicate that, to a certain extent, the degree of convexity of the wage schooling relationship is affected by the omission of skill heterogeneity. The difference between the returns in high school and in college, of the order of 4 percentage points in a flexible model (column 3), which allows for both skill heterogeneity and non-linearities, is now increased to 7 percentage point when skill heterogeneity is not controlled for (column 2). This result is not surprising and illustrates the importance of dynamic self-selection.

However, as indicated by the rejection of the linear model, a fair degree of convexity persists. At the same time, the consequences of ignoring nonlinearity are also quite spectacular. The estimate for the population average return to schooling in a linear model, which is around 6% per year (column 1), seriously over estimates the return to high school training (averaging 0.0376) and under estimates the return to post high school training (0.0864).

Ignoring both nonlinearity and skill heterogeneity (including in the intercept term), raises the return to schooling to 9% (column 4) and creates the statistical illusion that log wages are increased by more than 9% per year, regardless of the level of schooling.

Finally, another consequence of ignoring non-linearity is the exaggeration of the importance of skill heterogeneity. This is illustrated by the increase in the standard deviation of the returns to schooling from 0.0266 (when nonlinearity is accounted for) to 0.0359 (when it is ignored).

In short, the results indicate that both non-linearity and heterogeneity are important and, perhaps more importantly, that ignoring either of those aspects may have serious consequences.

7 The Importance of Non-Separability Among all particular dimensions that Ih a v ee x a m i n e d ,t h ei s s u eo fs e p a r ability of log wages in education and the returns to experience may be the most interesting from an economic standpoint. While non-linearity and heterogeneity may be seen as "statistical" issues, the absence of separability suggests the relevance of modeling wage growth. As stated earlier, wage growth may not only be seen from a human capital perspective but also from a pure incentive angle. Indeed, in the literature on labor market incentives and personnel economics, wage growth is also related to firm payment mechanisms (promotions, tournaments and various delayed payment schemes).

At this stage, two issues naturally arise. First, if schooling affects wage growth (given unobserved skills), the returns to schooling must be redefined so to incorporate the fact that schooling facilitates access to high wage growth. The returns to schooling, defined for the early years of labor market experience, are found in Table 6. Despite the seemingly small estimate for τ 2 , it is clear that the return to post high-school training rises significantly. It goes from 0.08 (at entrance in the labor market) to 0.11 after 8 years of experience.

A second issue relates to the effect of assuming separability at the estimation level. To illustrate this, I re-estimated a conventional form of the regression (setting τ 1 and τ 2 to 0), and re-evaluated the returns to schooling upon entrance in the labor market and experience. The results are in Table 7.

Contrary to intuition, I find that imposing separability does not affect much the return to post high-school training. In this restricted version, the population average return to college training is 0.0876. However, the return to high school training is largely inflated by imposing separability. Its average, now equal to 0.0498, is almost 50% higher than in the non-separable model (0.0498/0.0376). This is a severe over-estimation which, as far as I know, is practically never discussed in the literature. Although non-linearity and skill heterogeneity have been investigated before and will likely be investigated by researchers in the future, the cause of non-separability deserves some more attention. Modeling the channels by which schooling affects wage growth (training opportunities, promotions,..etc.) appears to be most appropriate.

8C o n c l u s i o n

In this paper, I present econometric estimates of the celebrated Mincer wage regression obtained with a degree of flexibility which, as far as I know, has not been achieved before. The econometric model is novel. It is based on a reduced-form dynamic model of schooling attainment based on the hazard specification of the transition from one grade level to the next with observed and unobserved heterogeneity and a wage regression with the following attributes; skill heterogeneity, non linearity in schooling, non-separability between schooling and experience and heteroskedasticity. The model is estimated from a panel of young males taken from the National longitudinal survey of Youth (79-90). The empirical likelihood function maximizes the joint probability of the schooling attainment and the wage histories observed between 1979 and 1990.

The data reject all simplifying assumptions common in the empirical literature. I find that the degree of convexity of the wage regression, as measured by the difference in the local returns to schooling before and after high school graduation, is dependent on the allowance for skill heterogeneity. However, the log wage regression remains highly convex, even after conditioning on unobserved and observed skills. The convexity is acute and is therefore not solely a reflection of omitted skill heterogeneity. Not surprisingly, skill heterogeneity is also found to be quite important. After conditioning on skill heterogeneity, there is a positive correlation between accumulated schooling and the individual specific returns to experience. This is consistent with the view that accumulated schooling may have a causal effect on wage growth. Standard models based on the separability assumption have two major defects. First, they ignore the positive benefit of education on future wage growth. Secondly, they appear to over-estimate the returns to high-school education by a significant margin (as much as 15%).

Overall, the results presented therein are much more in line with those reported in the structural literature than in the OLS/IV literature. For instance, the population average return to college education upon entrance in the labor market, around 8% per year, is much inferior to IV estimates often exceeding 15%. The huge discrepancy between structural estimates and OLS/IV estimates does not seem to be attributable to the strong parametric assumptions required in structural models. After all, the estimation strategy proposed in this paper is based on a wage regression more generally specified than in the literature and on a relatively flexible way to treat the endogeneity of schooling decisions. The discrepancy between structural estimates and reduced-form estimates has been noticed relatively recently. As of now, it is certainly not well understood. The results reported here seem to suggest that it may well be explained by differences in the econometric specification of the wage regressions but only further investigation will enable applied econometricians to fully understand these marked differences. started to work full-time by 1990 (94% of our sample), will never return to school beyond 1990. Finally, one notes that the number of interruptions is relatively small. Unlike many reduced-form studies which use proxies for post-schooling labor market experience (see Rosenzweig and Wolpin), we use actual labor market experience. Actual experience accumulated is computed using the fraction of the year worked by a given individual. The availability of data on actual employment rates allows use to estimate the employment security return to schooling.

The average schooling completed (by 1990) is 12.8 years. As described in Belzil and Hansen (2000), it is clear that the distribution of schooling attainments is bimodal. There is a large fraction of young individuals who terminate school after 12 years (high school graduation). The next largest frequency is at 16 years and corresponds to college graduation. Altogether, more than half of the sample has obtained either 12 or 16 years of schooling. As a consequence, one might expect that either the wage return to schooling or the parental transfers vary substantially with grade level.

Table 1

 1

	Testing for skill heterogeneity, non-linearities
	and heteroskedasticity
		Likelihood Ratio # of restrictions critical value
		statistics		at 1% level
	Null Hypothesis
	Linear returns	15.7	1	6.6
	to schooling
	Effect of schooling	11.0	2	9.2
	on return to experience
	Homogenous returns	260.3	14	29.1
	schooling/experience

Table 3

 3 Skill heterogeneity: first and second moments of the returns to schooling and experience Note: The total shares for the returns to experience do not add to 100 because schooling is endogenous and therefore not orthogonal to background variables and skills.

	Table 2C
	Non-Linearity and
	Non-Separability parameters
	(with asymptotic t-ratios)
	Non-linearity	--
		(δ 2 )0 . 0 4 0 6
		(6.73)
	Non-Separability
		(τ 1 )0 . 0 0 3 5
		(1.62)
		(τ 2 )0 . 0 4 3 7
		(5.01)

Table A1 -

 A1 Descriptive Statistics Family income and hourly wages are reported in 1990 dollars. Family income is measured as of May 1978. The increasing number of wage observations is explained by the increase in participation rates.

		Mean St dev. # of individuals
	Family Income/1000	36,904 27.61	1710
	father's educ	11.69 3.47	1710
	mother's educ	11.67 2.46	1710
	# of siblings	3.18	2.13	1710
	prop. raised in urban areas	0.73	-	1710
	prop. raised in south	0.27	-	1710
	prop in nuclear family	0.79	-	1710
	AFQT/10	49.50 28.47	1710
	Schooling completed (1990)	12.81 2.58	1710
	# of interruptions	0.06	0.51	1710
	duration of interruptions (year) 0.43	1.39	1710
	wage 1979 (hour)	7.36	2.43	217
	wage 1980 (hour)	7.17	2.74	422
	wage 1981 (hour)	7.18	2.75	598
	wage 1982 (hour)	7.43	3.17	819
	wage 1983 (hour)	7.35	3.21	947
	wage 1984 (hour)	7.66	3.60	1071
	wage 1985 (hour)	8.08	3.54	1060
	wage 1986 (hour)	8.75	3.87	1097
	wage 1987 (hour)	9.64	4.44	1147
	wage 1988 (hour)	10.32 4.89	1215
	wage 1989 (hour)	10.47 4.97	1232
	wage 1990 (hour)	10.99 5.23	1230
	Experience 1990 (years)	8.05	11.55	1230
	Note:

For an historical perspective on the Mincer equation, see[START_REF] Heckman | Earnings Functions, Rates of Return, and Treatment Effects: The Mincer Equation and Beyond[END_REF].

[START_REF] Carneiro | Estimating Distributions of Treatment Effects with an Application to the Returns to Schooling and measurement of the Effects of Uncertainty on College Choice[END_REF] adress this issue within a factor structure.

In general, the use of IV techniques requires separability between the instruments and the error term in the treatment equation and it also imposes monotonicity.

These include models of endogenous post-schooling human capital investments as well as various lifecycle incentive models where wages are upward sloping.

The reader will note that, strictly speaking, there is no such thing as "structural estimates of the returns to schooling". Structural estimation does not identify new parameters of the Mincerian wage regression. However, I use the term "structural estimates" to refer to economic models where endogenous schooling is modeled through the solution of an intertemporal model, in which the return to schooling plays a central part.

[START_REF] Carneiro | Estimating Distributions of Treatment Effects with an Application to the Returns to Schooling and measurement of the Effects of Uncertainty on College Choice[END_REF] The difference between IV and structural estimates is surveyed inBelzil (2006

).[START_REF] Chen | Does College Attendance Increase wage Volatility? Working Paper[END_REF] For the error distribution of the grade transition model, the 5 normals are subject to a scaling restriction (seeGeweke and Keane, 2000).

To do so, I regressed AFQT scores on schooling and kept the residual.

Another type of non-separability (ignored in this paper) could arise if the returns to schooling decline with experience (or age) because of depreciation.

This type of interdependence is more clearly defined within a structural model.

This may be seen after noting that, given the range of father's and mother's schooling, thenegativeeffect of the interaction term (found at each possible grade level) will dominate when evaluating the marginal effects for both mother's schooling and father's schooling.

The increasing degree of convexity of the wage regression is also pointed in[START_REF] Mincer | Changes in Wage Inequality[END_REF],Lemieux (2003) andDeschênes (2001).

This is issue is at the center of most structural models of schooling decisions. SeeBelzil (2006), for a survey.

This feature of the NLSY implies that there is a relatively low level of measurement error in the education variable.

Overall, interruptions tend to be quite short. Almost half of the individuals (45 %) who experienced an interruption, returned to school within one year while 73% returned within 3 years.

Appendix 1-The Data

The sample used in the analysis is extracted from the 1979 youth cohort of the T he N ational Longitudinal Survey of Y outh (NLSY). The NLSY is a nationally representative sample of 12,686 Americans who were 14-21 years old as of January 1, 1979. After the initial survey, re-interviews have been c o n d u c t e di ne a c hs u b s e q u e n ty e a ru n t i l1 9 9 6 . I nt h i sp a pe r ,w er e s t r i c to u r sample to white males who were age 20 or less as of January 1, 1979. We record information on education, wages and on employment rates for each individual from the time the individual is age 16 up to December 31, 1990.

The original sample contained 3,790 white males. However, we lacked information on family background variables (such as family income as of 1978 and parents' education). We lost about 17% of the sample due to missing information regarding family income and about 6% due to missing information regarding parents' education. The age limit and missing information regarding actual work experience further reduced the sample to 1,710.

Descriptive statistics for the sampleu s e di nt h ee s t i m a t i o nc a nbef o u n d in Table 1. The education length variable is the reported highest grade completed as of May 1 of the survey year and individuals are also asked if they are currently enrolled in school or not. [START_REF] Keane | The Career Decisions of Young Men[END_REF] This question allows us to identify those individuals who are still acquiring schooling and therefore to take into account that education length is right-censored for some individuals. It also helps us to identify those individuals who have interrupted schooling. Overall, the majority of young individuals acquire education without interruption. The low incidence of interruptions (Table 1) explains the low average number of interruptions per individual (0.22) and the very low average interruption duration (0.43 year) . In our sample, only 306 individuals have experienced at least one interruption. This represents only 18% of our sample and it is along the lines of results reported in Keane and Wolpin (1997). [START_REF] Lazear | Personnel Economics: Past Lessons and Future Directions[END_REF] Given the age of the individuals in our sample, we assume that those who have already