A Structural Analysis of the Correlated Random Coe¢ cient Wage Regression Model1

We estimate a …nite mixture dynamic programming model of schooling decisions in which the log wage regression function is set in a random coe¢ cient framework. We also analyze the determinants of 3 counterfactual experiments (a college attendance subsidy, a high school graduation subsidy and an overall decrease in the rate of time preference) and examine a proposition often claimed in the “Average Treatment E¤ects” literature; that the discrepancy between OLS and IV estimates of the returns to schooling may be explained by the relatively higher returns experienced by those a¤ected by exogenous policy changes. We …nd that the average return to experience upon entering the labor market (0.0863) exceeds the average return to schooling (0.0576) and we …nd more cross-sectional variability in the returns to experience than in the returns to schooling. Labor market skills (as opposed to taste for schooling) appear to be the prime factor explaining schooling attainments. We …nd little evidence in favor of a positive correlation between reactions induced by an exogenous experiment and the individual speci…c returns to schooling. Key Words: Random Coe¢ cient, Returns to Schooling, Comparative Advantages, Dynamic Programming, Dynamic Self-Selection. JEL Classi…cation: J2-J3. ha ls hs -0 02 01 35 0, v er si on 1 22 J ul 2 00 9 1 Introduction and Objectives In this paper, we investigate the empirical properties of the correlated random coe¢ cient wage regression model (CRCWRM) using a structural dynamic programming model.1 The term “correlated random coe¢ cient wage regression model”refers to the standard Mincerian log wage regression function in which the coe¢ cients may be arbitrarily correlated with the regressors (education and experience). While the comparative advantages representation of the labor market is far from being new (Roy, 1951, Becker and Chiswick, 1966 and Willis and Rosen, 1979), economists have only recently paid particular attention to the speci…cation and the estimation of linear wage regression models set in a random coe¢ cient framework (Heckman and Vitlacyl (1998, 2000), Wooldridge (1997, 2000), Angrist and Imbens (1994), Card (2000) and Meghir and Palme (2001)). In this recent branch of the literature, it is customary to estimate the log wage regression function using Instrumental Variable (IV) techniques and interpret the estimates in a framework where the returns to schooling are individual speci…c. This surge of new research is understandable. In a context where schooling is understood as the outcome of individual decision making within a dynamic framework, rational individuals base their schooling decisions partly on absolute and comparative advantages in the labor market and partly on their taste for schooling. As a consequence, the random coe¢ cients (the returns to schooling and experience), as opposed to only the individual speci…c intercept terms, will normally be correlated with individual schooling attainments. In a linear wage regression, individual di¤erences in the intercept term represent a measure of absolute advantage in the labor market while di¤erences in slopes re‡ect individual comparative advantages in human capital acquisition via schooling and experience. While it might be tempting to focus solely on heterogeneity in the returns to schooling (and assume homogeneous returns to experience), this approach is likely to be unsatisfactory. If the returns to schooling and experience are truly correlated, ignoring individual di¤erences in the return to labor market experience is likely to a¤ect the estimates of the returns to schooling as well as the causal link between labor market ability and schooling (dynamic self-selection). Modeling wage regressions in a random coe¢ cient framework therefore requires the allowance for 1The term “correlated random coe¢ cient wage regression model”is also used in Heckman and Vitlacyl (1998). 1 ha ls hs -0 02 01 35 0, v er si on 1 22 J ul 2 00 9 heterogeneity in the returns to experience.2 As it stands, very little is known about the empirical properties of the CRCWRM. For instance, those interested in estimating the returns to schooling by IV techniques usually ignore higher moments such as the variance of the returns to schooling and experience, or use a reduced-form framework which cannot disclose the covariances between realized schooling and the individual speci…c coe¢ cients. However, these quantities are important. They may shed light on the importance of comparative advantages in the labor market and help comprehend the determinants of individual schooling attainments. Finally, they may help quantify the “Ability Bias”(OLS bias) arising in estimating the returns to schooling using regression techniques. Obviously, a random coe¢ cient regression model provides a realistic framework to evaluate the relative importance of labor markets skills and taste for schooling in explaining cross-sectional di¤erences in schooling attainments. Virtually all recent work on empirical earnings functions is directly or indirectly based on a random coe¢ cient framework. For this reason, it deserves some attention.3 Our main objective is to investigate the empirical properties of the CRCWRM. These include the population average returns to schooling and experience, the relative dispersions in the returns to schooling and experience, and the relative importance of labor market skills and individual speci…c taste for schooling in explaining cross-sectional di¤erences in schooling attainments. We estimate a …nite mixture structural dynamic programming model of schooling decisions with 8 unknown types of individuals, where each type is characterized by a speci…c log wage regression function (linear) as well as a speci…c utility of attending school.4 The estimation of a mixed 2Individual di¤erences in the return to experience may be explained by comparative advantages in on-the-job training, learning on the job, job search or any other type of postschooling activities enhancing market wages. Allowing for heterogeneity in the returns to experience is especially important if individual post-schooling human capital investments are unobserved (which is the case in most data sets). 3Heterogenity in realized returns to schooling may also arise if the local returns change with the level to schooling. In a recent paper, Belzil and Hansen (2002a) used a structural dynamic programming model to obtain ‡exible estimates of the wage regression function from the National Longitudinal Survey of Youth (NLSY). They found that the log wage regression is highly convex and found returns to schooling much lower than what is usually reported in the existing literature although the local returns may ‡uctuate between 1% (or less) and 13% per year. 4In this paper, we disregard using available measures like AFQT scores. We do so for two main reasons. First, AFQT scores may often re‡ect di¤erences in schooling (as 2 ha ls hs -0 02 01 35 0, v er si on 1 22 J ul 2 00 9 likelihood function has two main advantages. It can capture any arbitrary correlation between any of the heterogeneity components and it obviates the need to incorporate all parents’background variables in each single heterogeneity component or to select, somewhat arbitrarily, which heterogeneity components are correlated with household background variables and which ones are not. A second objective is to illustrate the importance of population heterogeneity and, more speci…cally, to analyze the characteristics of the subpopulation (s) most a¤ected by a counterfactual experiment. This is an important issue. In the literature, estimates of the returns to schooling obtained using instrumental variable techniques are often higher than OLS estimates.5 It is often postulated that these results are explained by the fact that those individuals more likely to react to an exogenous policy change must have higher returns to schooling than average. As far as we know, this claim has neither been proved nor veri…ed empirically in any direct fashion. To illustrate the importance of heterogeneity in the reactions to treatment, we conducted 3 counterfactual experiments. We …rst increased the level of the utility of attending school so to mimic the e¤ect of a college subsidy. As this experiment is targeted at speci…c school levels, we also simulated a decrease in the rate of time preference. This experiment is more likely to a¤ect school attendance at all levels, although it is perhaps more di¢ cult to associate an empirical counterpart to it.6 Finally, we simulated a high school graduation subsidy which, for instance, might be targeted at reducing high school drop out behavior. In all cases, the experiment represents a “truly exogenous”change which may serve as a basis for a natural experiment. A third and …nal objective is to investigate the notion of “Ability Bias”in a context where the notion is much deeper than the usual correlation between tests are not taken at the same age by all young individuals). Second, and perhaps more importantly, we are interested in developing a methodology which may be applicable to a wide range of panel data sets. As far as we know, most panel data sets of labor market histories do not contain observed measures of skill heterogeneity such as AFQT scores and the like. 5The validity of very high returns to schooling, reported in a simple regression framework have been seriously questioned (see Manski and Pepper, 2000 and Belzil and Hansen, 2002a). It is also interesting to note that empirical evidence also suggests that standard wage regressions augmented with observable measures of ability (such as test scores and the like) lead to a decrease in the estimated returns to schooling. 6In the literature, it is sometimes argued that di¤erences in credit constraints may be captured in the discount rate (see Cameron and Taber, for a recent example). 3 ha ls hs -0 02 01 35 0, v er si on 1 22 J ul 2 00 9 the individual speci…c intercept terms of the wage regression and realized schooling attainments. As market ability heterogeneity is multi-dimensional in our model, our estimate of the Ability Bias (OLS bias) is not only explained by the correlation between the individual speci…c intercept term and realized schooling but also by the simultaneous correlations between schooling and experience and the individual speci…c deviations from population average returns to schooling and experience. The model is implemented on a panel of white males taken from the National Longitudinal Survey of Youth (NLSY). The panel covers a period going from 1979 until 1990. The main results are as follows. We …nd population average returns to schooling which are much below those reported in the existing literature. Our estimates are also much lower than those obtained using standard OLS techniques. The average return to experience upon entering the labor market (0.0863) exceeds the average return to schooling (0.0576) and we …nd more cross-sectional variability in the returns to experience than in the returns to schooling. The returns to schooling and experience are found to be positively correlated. Not surprisingly, the correlated random coe¢ cient wage regression model …ts wage data very well. It can explain as much as 78.5% of the variation in realized wages. Overall, the dynamic programming model indicates that labor market skills are the prime factor explaining schooling attainments as 82% of the explained variation is indeed explained by individual comparative and absolute advantages in the labor market and only 18% is explained by individual di¤erences in taste for schooling. Moreover, realized schooling attainments are more strongly correlated with individual di¤erences in returns to experience than in returns to schooling. The importance of individual heterogeneity in the level of reactions to policy changes is well illustrated by the 3 counterfactual experiments that we conducted. In all cases (the college attendance subsidy, the high school graduation subsidy and the overall decrease in the rate of time preference), the reactions induced by the arti…cial instrument are a complicated (non-linear) function of individual speci…c skills and, moreover, the determinants of the reactions will di¤er according to the nature of the experiment. More speci…cally, it appears that the determinants of the reactions are very much a¤ected by whether or not the experiment is targeted at a particular school level. However, homogeneity is strongly rejected. Unlike what is often claimed in the average treatment e¤ects literature, the discrepancy between IV and OLS is not necessarily explained by a positive correlation between individual 4 ha ls hs -0 02 01 35 0, v er si on 1 22 J ul 2 00 9 speci…c schooling reactions and the returns to schooling. While it is possible to reconcile our results with conventional wisdom about IV estimates of the returns to schooling in one of our experiments (the general decrease in the discount rate), a positive correlation between individual speci…c reactions and the returns to schooling is not a “general consequence”of any exogenous change in the determinant of school attendance. The paper is structured as follows. The empirical dynamic programming model is exposed in Section 2. The goodness of …t is evaluated in Section 3. A discussion of the estimates of the return to schooling and experience are found in Section 4. In Section 5, we illustrate the links between labor market skills and dynamic self-selection. In Section 6, we analyze the determinants of the individual speci…c reactions to 3 counterfactual experiments and examine a proposition often claimed in the “Average Treatment E¤ects” literature; that the discrepancy between OLS and IV estimates of the returns to schooling may be explained by the relatively higher returns experienced by those a¤ected by exogenous policy changes. In Section 7, we discuss the links between our estimates and those reported in the literature and reexamine the notion of Ability Bias in a context where the regression function allows for a rich speci…cation of absolute and comparative advantages. The conclusion is in Section 8. 2 An Empirical Dynamic ProgrammingModel of Schooling Decisions with Comparative Advantages In this section, we introduce the empirical dynamic programming model. While the theoretical structure of the problem solved by a speci…c agent is similar to the model found in Belzil and Hansen (2002a), the di¤erent stochastic speci…cation and, especially, the allowance for a rich speci…cation of absolute and comparative advantages requires a full presentation. Young individuals decide sequentially whether it is optimal or not to enter the labor market or continue accumulate human capital. Individuals maximize discounted expected lifetime utility over a …nite horizon T and have identical preferences. Both the instantaneous utility of being in school and the utility of work are logarithmic. The control variable, dit; summarizes the stopping rule. When dit = 1; an individual invests in an additional year 5 ha ls hs -0 02 01 35 0, v er si on 1 22 J ul 2 00 9 of schooling at the beginning of period t. When dit = 0, an individual leaves school at the beginning of period t (to enter the labor market). Every decision is made at the beginning the period and the amount of schooling acquired by the beginning of date t is denoted Sit: 2.1 The Utility of Attending School The instantaneous utility of attending school, U (:); is formulated as the following equation7 U (:) = (Sit) + i + " it (1) in which " it i:i:d N(0; 2 ) represents a stochastic utility shock, the term i represents individual heterogeneity (ability) a¤ecting the utility of attending school and (:) captures the co-movement between the utility of attending school and grade level. We assume that individuals interrupt schooling with exogenous probability and, as a consequence, the possibility to take a decision depends on a state variable Iit: When Iit = 1; the decision problem is frozen for one period. If Iit = 0; the decision can be made. When an interruption occurs, the stock of human capital remains constant over the period.9 2.2 The Utility of Work Once the individual has entered the labor market, he receives monetary income ~ wt; which is the product of the yearly employment rate, et; and the wage rate, wt: The instantaneous utility of work, U(:) U(:) = log( ~ wt) = log(et wt) 7The utiliy of school could be interpreted as the monetary equivalent (on a per hour basis) of attending school. 8The interruption state is meant to capture events such as illness, injury, travel, temporary work, incarceration or academic failure. 9The NLSY does not contain data on parental transfers and, in particular, does not allow a distinction in income received according to the interruption status. As a consequence, we ignore the distinction between income support while in school and income support when school is interrupted. In the NLSY, we …nd that more than 85% of the sample has never experienced school interruption. 6 ha ls hs -0 02 01 35 0, v er si on 1 22 J ul 2 00 9 2.3 The Correlated Random Coe¢ cient Wage Regression Model The log wage received by individual i, at time t, is given by logwit = '1i Sit + i ('2 Experit + '3 Exper it) + i + "it (2) where '1i is the individual speci…c wage return to schooling and i is an individual speci…c factor multiplying the e¤ect of experience ('2) and the e¤ect of experience squared ('3). The term w i represents an individual speci…c intercept term. We assume that