BLS WORKING PAPERS

U.S. DEPARTMENT OF LABOR 

Bureau of Labor Statistics 

OFFICE OF EMPLOYMENT AND UNEMPLOYMENT 

STATISTICS 

 

Reexamining the Returns to Training: Functional Form, Magnitude, and Interpretation 

Harley Frazis, U.S. Bureau of Labor Statistics 

Mark A. Loewenstein, U.S. Bureau of Labor Statistics 

 Working Paper 367

 July 2003 

The views expressed are those of the authors and do not necessarily reflect the policies of the U.S. Bureau of Labor 

Statistics or the views of other staff members. 
Reexamining the Returns to Training: 

Functional Form, Magnitude, and Interpretation 

Harley Frazis Mark A. Loewenstein 

Bureau of Labor Statistics Bureau of Labor Statistics 

2 Massachusetts Ave. NE 2 Massachusetts Ave. NE 

Suite 4945 Suite 4130 

Washington D.C. 20212 Washington D.C. 20212 

Phone: 202-691-7394 Phone: 202-691-7385 

Frazis_H@bls.gov Loewenstein_M@bls.gov 

The authors thank Dan Black, Marilyn Manser, Terra McKinnish, and participants in a 

session of the 1998 Southern Economic Association meetings and at seminars at Virginia 

Commonwealth University and George Washington University for helpful comments. 

We thank Alexander Eidelman for assistance in constructing the NLSY dataset, Hector 

Rodriguez for help with figures, and James Spletzer both for help with the NLSY data 

and for comments. The views expressed here are those of the authors and do not 

necessarily reflect the views of the U.S. Department of Labor or the Bureau of Labor 

Statistics. 
Abstract 

 

This paper estimates the wage returns to training, paying careful attention to the 

choice of functional form. Both the National Longitudinal Survey of Youth (NLSY) and 

Employer Opportunity Pilot Project (EOPP) datasets indicate that the return to an extra 

hour of formal training diminishes sharply with the amount of training received. A cube 

root specification fits the data best, but the log specification also does well. The linear 

and quadratic specifications substantially understate the effect of training. 

 

If wages are not adjusted continuously, estimating the total effect of training 

requires that one include lagged and lead training as well as current training in the 

regression equation. Consequently, the NLSY is ideally suited to estimate the total return 

to training. We find very large returns to formal training. These returns are sharply 

reduced when one adjusts for heterogeneity in wage growth. Returns are reduced further 

when one takes into account the effect of promotions and the fact that direct costs are a 

substantial portion of the total cost of training. The mixed continuous-discrete nature of 

the training variable means that measurement error can cause estimates of the effects of 

short spells of training to be biased upward, but we demonstrate that the maximum 

upward bias in estimated returns at the geometric mean is relatively small. 

After correcting for confounding factors, we are left with a return to training that 

is several times the returns to schooling. Heterogeneity in returns explains how returns to 

formal training can be so high while most workers do not get formal training. In the 

EOPP data, the return to training is significantly higher in more complex jobs. With 

unobserved heterogeneity in returns, our estimates can be regarded as the return to 

training for the trained, but cannot be extrapolated to the untrained. 
1 

 I. Introduction 

In recent years, a substantial literature analyzing the extent and consequences of 

on-the-job training has emerged, taking advantage of new datasets with direct measures 

of training. Studies find support for the human capital model’s prediction that a worker’s 

wage is positively related to past investments in his training.1 Indeed, Brown (1989) 

reports that "within-firm wage growth is mainly determined by contemporaneous 

productivity growth". Similarly, Barron, Black, and Loewenstein (1989) note that 

“training is one of only a few variables affecting wage and productivity growth.” 

However, in many respects the literature on training lags behind the more 

developed literature on the returns to schooling. While studies of the rate of return to 

schooling are numerous, we are aware of few studies that attempt to estimate rates of 

return to training.2

 At a more basic empirical level, while the best simple functional form 

to characterize the earnings-schooling relationship has been settled since Heckman and 

Polachek (1974), researchers have paid little attention to the choice of the appropriate 

functional form for the earnings-training relationship. Differences in functional forms 

across studies makes comparisons difficult. This difficulty is compounded by the fact 

that researchers using different functional forms have tended to use different datasets: 

while users of the Employer Opportunity Pilot Project (EOPP) data and the closely 

 

1

 A non-exhaustive list of references here includes Altonji and Spletzer (1991), Barron, Berger, and Black 

(1999), Barron, Black and Loewenstein (1989), Bartel (1995), Brown (1989), Lengermann (1999), Lillard 

and Tan (1986), Loewenstein and Spletzer (1996, 1998, 1999a), Lynch (1992), Mincer (1988), Pischke 

(2001) and Veum (1995). 

2

 Mincer’s (1989) review article in Education Researcher calculates rates of return in the range of 32-48 

percent before depreciation. Bartel (1995), using a company dataset, estimates the rate of return to training 

at 58 percent before depreciation; her calculation includes direct costs of training. Allowing for 

depreciation substantially reduces these numbers—Mincer’s range after correction is from 4 to 26 percent, 

using Lillard and Tan’s (1986) estimated 15-20% depreciation rate; Bartel’s is 42 percent with 10 percent 
2 

related Small Business Administration (SBA) data have generally used log specifications 

(for example, see Barron, Black, and Loewenstein 1989 and Barron, Berger, and Black 

1997b), researchers using the National Longitudinal Study of Youth (NLSY) have used 

linear specifications (for example, Lynch 1992, Parent 1999) or specifications that 

estimate the return to a spell of training without making use of information on the 

duration of the spell (Loewenstein and Spletzer 1996, Lengermann 1999). 

This paper has three goals. First, we seek to perform a service similar to 

Heckman and Polachek (1974) by investigating the choice of the appropriate functional 

form for formal training in a wage equation. Second, we derive estimates of the rate of 

return to formal training. We use both our preferred functional form and, where possible, 

a semi-nonparametric estimator to derive rate-of-return estimates. Third, because 

estimated returns from our fixed-effect regressions are quite high--over 150 percent for 

typical values of training-- and incidence low (less than 25% of job spells in our NLSY 

sample have any formal training) we consider possible explanations for the seeming 

contradiction between high rates of return and low incidence of training. 

Most of our analysis relies on NLSY data. Of the various datasets with training 

information, the NLSY provides the best information on formal training at all levels of 

tenure. In our discussion of how to estimate rates of return to training, we demonstrate 

that under reasonable assumptions, wage regressions must include lagged and lead 

training as well as current training in order to produce a rate of return estimate. The 

NLSY, which is longitudinal, is thus ideally suited to estimate the return to training. We 

supplement our NLSY analysis with estimates based on EOPP data. EOPP is an 

 depreciation and 26 percent with 20 percent deprecation. Interestingly, Lengermann (1999) finds no 

evidence that the return to training depreciates with time. 
3 

employer survey and provides the best information on formal and informal training at the 

start of the job. The EOPP data enable us to check that our key NLSY results are not 

distorted by our inability to control for informal training or by the specific survey 

methods used in the NLSY. 

Our results from both the NLSY and EOPP indicate that the return to an extra 

hour of training diminishes sharply with the amount of training received. We find that a 

cube root specification fits the data best, but the log specification also does well. These 

specifications, along with our semi-nonparametric estimator, indicate that there are very 

substantial returns to the initial interval of formal training. In contrast, in both datasets 

the linear specification substantially understates the effect of training, and in the NLSY 

the quadratic also substantially understates the effect of training. 

In section III, we develop a simple theoretical framework that allows us to 

estimate the rate of return to training. Our estimate of the rate of return is very high. 

Rates of return at median positive training of 60 hours are in the 150-175 percent range 

for the better-fitting parsimonious specifications and over 180 percent for the Fourier 

series. These numbers are much higher than, for example, estimates of returns to 

schooling, and present a puzzle in view of the fact that only a minority of respondents in 

either the NLSY or EOPP have any formal training. 

In section IV, we discuss five potential explanations as to why estimated returns 

to training are so high: heterogeneity in wage growth, measurement error, promotions, 

direct costs of training, and heterogeneity in returns to training. Using the cube root 

specification as our base, we show how correcting for these various factors causes one to 

adjust the estimated return to training. 
4 

The estimated return to training is sharply reduced when one adjusts for 

heterogeneity in wage growth, a result similar to that obtained by Pischke (2001). 

Returns are reduced further when one takes into account the effect of promotions and the 

fact that direct costs are a substantial portion of the total cost of training. The mixed 

continuous-discrete nature of the training variable means that measurement error can 

cause estimates of the effects of short spells of training to be biased upward, but we 

demonstrate that the maximum upward bias in estimated returns at the geometric mean is 

relatively small. 

After correcting for confounding factors, we are still left with a return to training 

that is several times the returns to schooling. Heterogeneity in returns explains how 

returns to formal training can be so high while most workers do not get formal training. 

Both the EOPP data and the NLSY provide evidence of such heterogeneity: in EOPP, the 

return to training is significantly higher in more complex jobs, and in the NLSY 

managers and professionals have higher returns to training than do other occupations. If, 

as seems reasonable, part of the heterogeneity in returns is due to unobservable factors, 

our estimates can be regarded as the return to training for the trained, but cannot be 

extrapolated to the untrained. 

II. Functional Form of the Training-Wages Relationship 

In this section, we compare several different simple functional forms to determine 

which best describes the relationship between training and log wages. In view of the 

advances over the last several decades in the theory and computational feasibility of nonparametric and semi-parametric estimators, it might be questioned whether this is a useful 

task. Why not characterize the relationship between training and wages non-
5 

parametrically? We think that it is useful to recommend a particular parsimonious 

specification for applications where one might not have the luxury of estimating a more 

flexible functional form. For example, one might have a specification where training is 

interacted with other variables or contains lead and lag terms such that estimating a more 

flexible function form is not practical. (We present an example of this in Section 4.) 

Alternatively, training may not be the focus of the analysis, but one may want to control 

for the effect of training on wages when estimating other effects. For example, Altonji 

and Pierret (2001) seek to correct for the effect of training when examining the effect of 

cognitive skills on wage growth.

NLSY Data

The NLSY is a dataset of 12,686 individuals who were aged 14 to 21 in 1979. 

These youth were interviewed annually from 1979 to 1994, and every two years since 

then. Response rates were over 90 percent for each year until 1996, and as of 2000 were 

83 percent. We use data from the 1979 through 2000 surveys.3 

The training section of the survey begins with the question, “Since [the date of the 

last interview], did you attend any training program or any on-the-job training designed 

to help people find a job, improve job skills, or learn a new job?” Individuals who 

answer yes to this question are then asked a series of detailed questions about each of 

their different training spells. In 1988 and thereafter, individuals are asked about the 

duration of their various training spells in weeks and the average number of hours each 

week that were spent in training. For each training spell in a given year, we have 

 

3

 Individuals were not interviewed in 1995. From 1996 on, the survey is being conducted every other year. 
6 

calculated the number of hours spent in training as the product of the duration in weeks 

and the average number of hours spent in training during a week.4

 

The training questions were changed somewhat in 1988. From 1979-1986, 

detailed information was obtained only on training spells that lasted longer than one 

month.5

 We have used the information contained in the later surveys to impute hours 

spent in training for training spells in the early surveys that last less than one month. 

Besides conditioning on the fact that a spell lasts less than one month, our imputations 

also condition on an individual’s age.6

The focus of our analysis is training whose explicit cost is at least partly paid for 

by the employer.7

 Information on who paid for training is available only after 1987; prior 

to 1987, we include only company training and spells less than one month (The post 

1987 data indicate that company training was generally paid for by the employer. Prior 

 

4

 Individuals are not asked about the number of weeks spent in training if their training spell is in progress 

at the time of the interview, but instead this information is obtained in a subsequent interview after the 

training spell is completed. Thus, besides being asked about training spells that began since the last 

interview, individuals are also asked about spells that were still in progress at the time of the last interview. 

5

 Training questions were not asked in 1987. 

6

 In the later surveys, individuals were explicitly asked about both the weekly duration of training and the 

year and month that a training spell began and ended. In the early surveys, individuals were asked about 

the year and month that a training spell began and ended, but were not explicitly asked about the number of 

weeks that a training spell lasted. Inspection of the post-1987 data reveals that 4 weeks is the best estimate 

for the weekly duration of training when a training spell ends in the subsequent month, 8 weeks is the best 

estimate for the weekly duration of training when a training spell ends in the second month after it began, 

and so on. Individuals are not asked about the starting year and month of a training spell that was in 

progress at the time of the last interview. For the pre-1987 data, we have obtained this information by 

carrying it forward from the year in which a training spell initially began. We have not had to do this for 

the post-1987 data because in the later years individuals were explicitly asked about the weekly duration of 

training spells that were in progress at the time of the last interview. 

7

Naturally, this cost can be passed on to the worker in the form of a lower wage. Eighty-fiver percent of 

the training spells in the NLS are at least partially paid for by the employer. We focus on this training 

because it would appear to correspond most closely to the on-the-job training concept referred to by Becker 

and subsequent human capital theorists. We use this framework when discussing rates of return to training 

in the next section. 
7 

to 1987, individuals with spells less than one month were not asked about the type of 

training they received; the post 1987 data indicate that short spells are generally 

employer-paid). Training not paid for by the employer (after 1987) and non-company 

training (prior to 1987) are accumulated in a separate variable. 

In investigating the effect of training on wages, it is important to distinguish 

between training that took place on the current job and training that took place on other 

jobs. By comparing the beginning and ending dates of a training spell with the date that 

the individual started working at his current job, we are able to classify a training spell as 

occurring on the current job or on a previous job.8

 When there is some ambiguity as to 

whether training occurred on the current job or in a previous job, we classify the training 

as occurring in the current job. Our results are not sensitive to this choice. 

The key training variable used in the empirical work to follow is total 

accumulated completed training on the current job. This variable is obtained by adding 

the training a worker has completed in the current year to the training he has received in 

all previous years on the current job. 

Basic Results

Our basic specification is: 

(1) ijt ijt T i ij t ijt W = X β + f β +α +θ +ω + ε 1 2 ln ( )

for person i in job j at time t, where W is the wage rate, X is a vector of time-varying 

control variables, T is hours of training on the current job, f(⋅) varies by specification, αi

and θij are permanent person and job-match specific error terms, ωt is a year effect, and 

 

8

 In cases where the individual holds more than one job simultaneously, we assume that training occurs on 

the individual’s main job. 
8 

εijt is a mean zero error term, uncorrelated with Xijt. All specifications are run as fixedeffect regressions within jobs. As the fixed effect will absorb both unchanging individual 

characteristics and job characteristics, the X vector is mostly comprised of functions of 

tenure and interactions of tenure with other variables. Specifically, the X vector consists 

of tenure, tenure squared, tenure cubed, and interactions of the three tenure terms with 

age at start of job, experience at start of job, AFQT,9

 years of education, ever married, 

part-time, union, two dummies for initial occupation in the job, Black, Hispanic, female, 

enrolled in school, and missing value indicators for AFQT, union, and part-time. Years 

of education (which occasionally changes within a job), non-employer-paid training,10

and dummies for ever married, part-time, enrolled in school, missing part-time, and year 

dummies are also included. As additional controls for training, we include a count of 

spells with missing training duration (most of these occur before 1988). 

We exclude observations with missing values on variables other than AFQT, 

union, and part-time. We also exclude observations with real wages below $1 or above 

$100 in 1982-84 dollars, or with log wages where the absolute value of the difference 

with the job mean is greater than 1.5 (which is a little more than 7.5 standard deviations). 

Finally, we exclude the military subsample, and jobs where for half or more observations 

on that job the respondent is an active member of the armed forces, self-employed, in a 

farm occupation, or enrolled in school at any time between interviews. The resulting 

sample has 75,698 observations from 17,809 jobs. 

 

9

 Specifically, the residual from a regression of AFQT on dummies for year of birth. 

10 A preliminary functional form analysis showed that a linear specification was best for non-employer-paid 

training. 
9 

Descriptive statistics are shown in Table 1. Table 2 gives more detail on the 

distribution of training by showing selected percentiles of the positive distribution of both 

the stock of training and of training during the previous year. Note that the distribution is 

quite skewed to the right. Log-normality appears to be a good approximation of the 

distribution. Using the Box-Cox transformation B(T;  λ λ (T −1) as a transformation 

of the distribution of T to normality (where λ=0 corresponds to a log-normal distribution 

and λ=1 to a normal distribution; see Greene 2000, p. 444-46), the estimated value of λ 

for the positive training sample is .03 for the stock of training and -.01 for training in the 

previous year. 

As an aid to determining the best simple functional form, we make a different use 

of the Box-Cox transformation by first estimating a model where f (T) = B(T;λ). 

	
 	

	rrespond to f (T) = T and f (T) = ln(T) , respectively. The 

estimation was done by non-
	
 

(bootstrap standard error .062), very close to the value of 1/3 corresponding to the cube 

root. 

The results for different functional forms for training are shown in Table 3.11 In 

addition to the cube root, we include linear, log, and quadratic specifications, a 

specification where the training variable is a dummy indicating whether any training has 

been completed on the current job, and a specification with both this dummy and a linear 

term. In the “log” specification, f(T) = ln(T+1), where T is number of hours of training. 

The table shows 2 R s (explained variance as a proportion of within-job variance) and the 

total effect at the median number of hours of training (60), where the median is calculated 

 

11 Estimating regressions for men and women separately gives results quite similar to Table 3. 
10 

across all observations with a positive stock of training on the current job. The 

differences in fit appear slight. However, the best-fitting specification—the cube root—

increases 2 R over twice as much as the worst-fitting specification—the linear—relative 

to the fit excluding training variables. The quadratic specification and the dummy 

specification are little improvement on the linear, while the log specification is close to 

the cube root. The dummy-plus-linear specification has a somewhat lower fit than the 

log and the cube root. The results indicate that the effect of training on wages is highly 

non-linear, with the effect declining more rapidly than implied by a quadratic 

specification. There is no evidence that the presence of an incidence effect explains the 

non-linearity.12

To contrast the effects of training on wages implied by the different functional 

forms, the last column in Table 3 shows the predicted effect of training at the median of 

the distribution of positive hours of training. The implied effect of the median hours of 

training differs by more than a factor of 12 between the different specifications. The log 

specification shows the largest effect, over 4 percent, with the cube root yielding a 

slightly smaller effect. The linear and quadratic specifications apparently greatly 

understate the impact of training on wages. 

One might suspect that the better fit of the log and cube root specifications simply 

reflects the fact that these functions’ compression of the right tail of the training 

distribution reduces the influence of outliers. To test for this, we omitted the top one 

percent of the distribution of positive training. The total effect of the median amount of 

positive training increases for the linear and quadratic specifications, but is still far below 

 

12 Adding a dummy term to the cube root specification yields a dummy coefficient that is negative, small in 

magnitude, and not significant. 
11 

the other specifications. The cube root specification is still the best fitting; excepting 

perhaps the quadratic, there is no marked improvement in the fit of the other 

specifications relative to the cube root. 

All of the above specifications are parsimonious, with the rate of decline 

determined by the functional form. To compare the patterns of returns implied by these 

specifications with those obtained from less restrictive specifications, we use a seminonparametric estimator: the Fourier series expansion (Gallant 1981). A Kth order 

Fourier series is a linear combination of cosine and sine terms, or 

f * (T) ( cos( jT) sin( jT)) 2 j

K

j 1

= ∑ α1 j +α =. A function’s Fourier expansion has the 

property that the differences between the value of a function f and the value of its Fourier 

expansion f* and between the derivatives of f and the derivatives of f* can be minimized 

to an arbitrary degree over the range of the function by choosing K to be sufficiently 

large. It thus provides a global approximation to the true function, rather than a local 

approximation (as in a Taylor series expansion).13 

In practice, linear and quadratic terms are usually added to the expansion. 

Moreover, for non-periodic functions the variable T needs to be transformed to a variable 

T* such that 0 < T* < 2π, after which the expansion can be implemented as: 

*( *) * * ( cos( *) sin( *)) 2

1

1

2

1 2 f T T T jT j jT

K

j

= δ +δ +∑ α j +α =. 

In our case, due to the essentially log-normal distribution of training, it is 

computationally convenient to work with the log of training as the basis for the Fourier 

 

13 Other semi-parametric estimation methods are harder to adapt to the fixed-effect setup. Li and Stengos 

(1996) consider fixed-effect estimation of β1, but it is not possible to estimate f(T) directly using their 

method. 
12 

expansion. We thus adopt the transformation T*=0.001+c ln(T+1), with c chosen so that 

	
		 		 	



prediction errors = ∑ − −

i

i XiK iK CV y

2

) ˆ ( β , where XiK is the complete vector of 

regressors for the Kth order expansion and β −iK ˆ is the corresponding coefficient vector 

from a regression omitting observation i. Andrews (1991) shows this criterion is 

asymptotically optimal in the sense that the probability of choosing the K that minimizes 

the expected sum of squared errors converges to 1 as the sample size increases, even in 

the presence of heteroscedasticity.14 We searched all orders of the expansion from K=1 

to 14. The order K was 13 for both the complete and outlier-omitted sample. 

We calculate the statistic Q2 ≡ 1- 2

2

( * ( *) 0)

( ( ) * ( *))

∑

∑

−

−

f T

f T f T

 to obtain a convenient 

summary measure of the closeness of fit between an arbitrary specification f(T) and the 

estimated Fourier series f*(T*).15 Analogous to the traditional R2, which measures the 

percentage reduction in the sum of the squared distance between the dependent variable 

and the predicted value relative to a model with only a constant, Q2

 measures the 

percentage reduction in the squared distance between the Fourier series and f(T) relative 

to a specification which omits training. As can be seen in the third column of Table 3, 

the cube root specification is closest to the Fourier series, and the linear specification is 

the furthest. Indeed, the cube root specification explains over 80 percent of the squared 

distance between the Fourier series and a specification without training, while the linear 

specification explains only 33-48 percent depending on the sample. 

 

14 However, asymptotic optimality is not proven if observations are not independent. We are unaware of 

equivalent results for this case. 
13 

Figure 1 plots the effect of training estimated in the sample without outliers for all 

specifications except for the dummy specification. The effect is plotted against log 

training since a linear scale would overly compress the range where the data are 

concentrated. The range of the figure is further restricted to the 5th through 95th

percentile of the positive training distribution. The volatile nature of the Fourier series 

apparent in the figure suggests that much of the variation in the Fourier function 

unexplained by the better-fitting functional forms is spurious. Consistent with the Q2

statistics, the figure shows that the linear and quadratic specifications fit the basic pattern 

of returns in the Fourier series expansion worse than the other functional forms over most 

of the range of the data, especially between the 25th and 75th percentiles. The dummylinear specification is also somewhat below the Fourier series for most of the range 

between the 25th and 75th percentiles. 

Why do the linear and quadratic functional forms track the Fourier series so 

poorly, especially in the middle of the positive training distribution? In our fixed-effect 

regressions, observations with large deviations of training from average training will have 

a disproportionately large effect on the training coefficient. (Indeed, the justification for 

discarding training outliers stems from the fact that erroneous observations in the tails 

will have particularly damaging effects.) Specifications such as the linear should tend to 

predict better in the right tail of the distribution and worse in the middle of the training 

distribution than specifications like the log that compress the training distribution. The 

linear function’s tendency to fit the right tail will lead to an especially poor fit in the 

middle of the training distribution when linearity is a misspecification. 

 

 15 We are grateful to Dan Black for suggesting this type of statistic. 
14 

EOPP

The NLSY provides strong evidence that returns to formal training decline greatly 

with the quantity of such training. As a check on these results, we now look at the 

evidence provided by EOPP. Unlike the NLSY, EOPP is not a longitudinal survey, and it 

only contains information on training at the start of the job, which causes difficulties for 

the rate of return analysis below. But EOPP does provide good measures of both formal 

and informal training. It also provides a measure of the number of weeks it takes a new 

employee to become fully qualified if he or she has the necessary school provided 

training but no experience in the job, which we refer to as "job complexity" as suggested 

by Barron, Berger, and Black (1999).16 We will make use of this measure later on in our 

analysis. 

EOPP’s information on formal training comes from employers’ reports about the 

number of hours specially trained personnel spent giving formal training to the most 

recently hired worker during his first three months of employment. We obtain a measure 

of informal training by summing (1) the number of hours that line supervisors and 

management personnel spent giving the most recently hired worker informal 

individualized training and extra supervision, (2) the number of hours that co-workers 

spent away from other tasks in providing the most recently hired worker with informal 

individualized training, and (3) the number of hours that a new worker typically spends 

watching others do the job rather than do it himself. 

Employers in EOPP provide information about the average wage paid to a worker 

who has been in the most recently filled position for two years, allowing one to estimate a 

 

16 For more information about the survey and the training questions, see Barron, Black, and Loewenstein 

(1989). 
15 

pseudo fixed-effect equation. In the estimations that follow, the dependent variable is the 

difference between the logarithm of the wage after two years and the logarithm of the 

starting wage paid to the most recently hired worker. Besides the training variables, we 

include the following explanatory variables in all of our estimated equations: the most 

recently hired worker’s age, gender, years of education, tenure, and dummy variables 

indicating whether the worked had received any vocational training or belonged to a 

union. In addition, we include the logarithm of the number of employees at the 

establishment, dummies indicating whether the most recently filled position was parttime or seasonal, two occupational dummies, and dummy variables for missing 

education, tenure, and union.17 Finally, we also include as controls several variables that 

are less commonly found in other datasets–the most recently hired worker’s relevant 

employment experience in jobs having some application to the position for which he was 

hired, relevant experience squared, and the logarithm of the job complexity measure 

described above. 

We exclude observations with missing values for any variables other than tenure, 

union, or years of education. We also exclude observations where wage growth is more 

than seven deviations above or below the sample mean. Finally, we exclude farm and 

government jobs. The resulting sample has 1,715 observations. 

Sample means are reported in Table 4. Note that the bulk of training is informal. 

Ninety-five percent of workers receive informal training during the first three months of 

employment, but similar to the NLSY only 13 percent of workers receive formal training. 

And while mean informal training for those with any informal training is 132 hours, 

 

17 Employers are (implicitly) asked about the starting wage paid to the most recently hired worker at the 

time he was hired, but about the average wage currently paid to workers with two years experience in the 
16 

mean formal training for those with any formal training is only 72 hours. In a 

preliminary analysis we found that the log is the best fitting simple specification for 

estimating the wage effect of informal training. Consequently, in our analysis of formal 

training, we include the log of informal training as one of our control variables. Our 

analysis of formal training is not sensitive to our treatment of informal training. 

We again begin our analysis of the effect of formal training by estimating a Box-

!		

	
 "#$%		
	

the NLSY. (Not surprisingly, since there are only 219 observations with positive formal 

training, the standard error for the EOPP estimate is quite high (.291)). 

The results for the various functional forms for training are shown in Table 5. 

The EOPP results are in general agreement with those from the NLSY. Once again, the 

linear specification performs the worst: like the NLSY data, the EOPP data indicate quite 

clearly that there are diminishing returns to training. The cube root specification 

performs the best in the sample without outliers and second best in the complete sample. 

Furthermore, when one uses the cube root specification, the estimated effect of training in 

the EOPP sample is similar to the estimated effect in the NLSY. 

The quadratic specification fits best in the complete sample and comes closest to 

the estimated Fourier function, but in light of the fact that we only have 219 observations 

with positive formal training, we would not place too much weight on the Fourier results. 

Indeed, the simple cube root and quadratic specifications both have a higher 2 R than the 

Fourier series. The volatility of the results of the quadratic specification between samples 

makes us reluctant to recommend it as an alternative to the cube root. In addition to the 

 

job. Since wages increase over time, tenure is positively correlated with wage growth. 
17 

sensitivity to outliers in the NLSY, the predicted effect of training at the EOPP median 

positive training of 38 hours using a quadratic specification is three times higher in EOPP 

than in the NLSY if one omits outliers (and nine times higher using the complete 

sample). For the log and cube root specifications the effects estimated using EOPP are 

only about 50 percent higher than those from the NLSY.18

III. Rates of Return to Training 

Our best-fitting specifications in the NLSY indicate that 60 hours of formal 

training, the median positive amount of training, increases wages by 3-4 percent. The 

estimated effects of training in EOPP are even larger, as high as 5 percent for the median 

positive training of 38 hours. Relatively short training spells thus have substantial effects 

on wages. For comparison, current estimates of the effect of a year of school on wages 

are about 10 percent for the U.S. (see Jaeger 2003 for example). Here we examine how 

one can obtain estimates of the rate of return to the training investment from the 

coefficients in a wage regression. We take as our starting point a simple model in which 

a worker’s wage always reflects his productivity. We then modify this model to take into 

account frictions in the wage setting process. 

 Rate of Return Calculations Using Coefficients on Lagged, Current, and Lead Training

Consider a worker whose value of marginal product is given by q = g(T), where T

denotes training and where g’ 0. We allow workers with no training to have positive 

productivity -- that is, g(0) > 0. But we assume that while receiving training, a worker 

 

18 As shown below, one year may not be sufficient to capture the full effect of training in the NLSY. When 

one allows one lead and one lag, the estimated return to training in the NLSY is substantially closer to the 

EOPP estimated return, which is itself estimated over a two year period . Another possible explanation for 

the higher estimated return in EOPP is that workers receiving greater amounts of training during the first 

three months may also receive more training during the rest of the two year period. Consistent with this, 

Loewenstein and Spletzer (1996) find that within the NLSY, training incidence is highly correlated within 

jobs. 
18 

produces nothing. For the time being, we neglect the direct cost of training. Assuming 

for expositional convenience that the job match is infinitely-lived, the present value of the 

stream of output of an employee who receives T years of training from time t to t + T is 

∫

∞

+

− = − − =

T t r

g T rT V T g T r t d ( ) exp( ) ( ) ( ) exp( (τ )) τ , where r is the discount rate. The 

internal rate of return IRR for a training investment of T years is defined as the value of r 

such that V(T) = V(0). Simple algebra establishes that T

g T g IRR ln( ( )) − ln( (0)) = . 

It is easy to establish that the present value of output is maximized at a value of T

where MR(T) = r and 0 ( ) <

dT

dMR T , where ( )ln( ( )) ’( ) ( )

g T

g T

dT

d g T MR T ≡ ≡ . Given 

diminishing returns, infra-marginal returns will be greater than marginal returns and 

IRR ( MR(u)du)(1/T) MR(T)

T

o

= > ∫ . Thus, given rapidly diminishing returns to training, 

a high IRR clearly does not imply sub-optimal investment in training. In contrast, 

assuming that our fixed-effect estimates of returns can be interpreted structurally, high 

marginal returns would imply sub-optimal investment in training. We return to this issue 

below. 

Note also that high observed average IRR’s do not imply the existence of 

economic rents. If jobs with good training opportunities did offer economic rents, 

workers would enter these jobs, driving down output prices and wages. In equilibrium, 

the wage profile for a given job would depend upon the training it offers, but, other things 

the same, the present value of the wage stream would be equalized across all jobs. 

How will the effect of training on productivity be reflected in wage growth? 

Suppose for the moment that the worker bears all the cost and realize all the gain to 
19 

training -- because, say, training is general (see Becker 1975). Let wt denote the worker’s 

wage at time t and let Tt denote his accumulated training at time t. If wages were 

adjusted continuously, then we would have wt = g(Tt). That is, the wage at any moment 

in time is determined solely by the contemporaneous stock of completed training; lagged 

and lead values of training do not affect the wage. 

In reality, frictions in wage setting prevent wages from being adjusted 

continuously. Consider an example where the worker is hired at time 0 and wages are 

adjusted once every year. Suppose that a training spell of length T starts and ends 

between t=1 and t =2. That is, letting τ1 be the date that training starts and τ2 be the date 

that training ends, 1 < τ1 < τ2 < 2. Then the worker’s wage is g(0) during the first year 

of employment (0 &t < 1) and g(T) during the third year (2 &t < 3). During the second 

year (1 &t < 2), the wage is given by: ) (0) ( wt = π 1 g + π 2 g T , where π 1 = τ1 – 1 is the 

fraction of the second year that the worker works before receiving training and π 2 = 2 

- 2 is the remaining time after the receipt of training. Wages during the period that 

training takes place are thus a weighted average of pre- and post-training productivity, 

with the weights adding up to less than one because there is no production during training 

itself. Taking first differences one obtains ln(w2 ) − ln(w1 ) = 

ln( (0) ( )) ln( (0)) π 1 g + π 2 g T − g and )) ln( ) ln( ) ln( ( )) ln( (0) ( w3 − w2 = g T − π 1 g + π 2 g T . 

Note that the effect of training will be spread over two periods - the period of training 

itself and the period after training has been completed. Note also that if 2 is sufficiently 

small relative to the time spent in training, ln(w2) - ln(w1) will be negative. 

Now consider a regression of wage observations on the stock of (completed) 

training accumulated on the job. Observations are recorded by a survey at times t1, t2, t3, 
20 

with k - 1 < tk < k. How should training enter the regression? To answer this question, 

let us return to our example. Note that two cases are possible. If training is completed 

before the survey date t2, then T1 = 0 and T2 = T, so that )) ln( ) ln( (0) ( wt 2 = π 1 g + π 2 g T2

and )) ln( ) ln( ( wt3 = g T2 . That is, the training T enters the wage equation at time t2 as 

current training and at time t3 as lagged training. On the other hand, if training is 

completed after the survey date t2, then T2 = T1 = 0 and T3 = T, so that 

ln( ) ln( (0) ( )) wt 2 = π 1 g + π 2 g T3 and )) ln( ) ln( ( wt3 = g T3 . In this case, the training T enters 

the wage equation at time t2 as lead training and at time t3 as current training. 

If the sample is a mixture of the two cases, then current, lagged, and lead training 

all belong in the wage equation. In our example, if the proportion p of individuals 

complete training T before the interview date t2, the observed effect of lagged training is 

[ln( ( )) ln( (0) ( ))] p g T − π 1 g +π 2 g T , the observed effect of current training is 

(1 )[ln( ( )) ln( (0) ( ))] [ln( (0) ( )) ln( (0))] − p g T − π 1 g + π 2 g T + p π 1 g + π 2 g T − g , and the 

observed effect of lead training is (1 )[ln( (0) ( )) ln( (0))] − p π 1 g + π 2 g T − g . The total 

effect, ln(g(T)) - ln(g(0)), is the sum of these three effects. Accordingly, to estimate the 

IRR of training it is necessary to include one lead and one lag term.19 

The foregoing has assumed that the worker bears all the costs (in terms of 

foregone production) and obtains all the returns to training. If the training is to some 

extent firm-specific, or if there are frictions in the labor market that cause the firm to 

share in the cost of general training [Loewenstein and Spletzer 1998, Acemoglu and 

 

19 In our estimations, we set lagged training to zero the first period a worker is in a job. Lead training for 

the last period is set to the worker’s final training in the job. If a worker leaves a job at time τ after survey 

date tN but before survey date tN+1, final training is obtained by adding training between tN and τ to training 

at time tN. 
21 

Pischke 2001], then the wage effect will underestimate the return to training in terms of 

productivity.20 The observed wage effect is thus a lower bound (subject to caveats 

explained later). 

Rate of Return Results.

Table 6 shows results for the specifications considered in Table 3, with terms for 

lagged and lead training added. (Wage observations for the year 2000 were omitted, as 

lead training is not observed.) The functional form comparisons match those of Table 3, 

with observed wage effects about 25-35 percent higher. The order for K in the Fourier 

series expansion is 13 for the complete sample and 2 for the outlier-omitted sample; 

evidently the additional terms are needed only to track the behavior of the function for 

training outliers. 

Setting a year equal to 2000 hours, we compute rates of return for T hours of 

training as T

f T

IRR T

∑ t

− =

1

1

2000 ( )

( )

β

, where β-1 is lagged, β0 is current, and β1 is lead 

training. Rates of return at median positive training of 60 hours are in the 150-175 

percent range for the better-fitting parsimonious specifications and over 180 percent for 

the Fourier series. Because series estimates potentially pick up local features of the 

wage-training function, the estimated return at a specific point may not be representative 

of returns over larger intervals and is likely to have a high standard error. Accordingly, 

for the Fourier series estimates, we calculate mean returns for the 25th through 75th 

percentiles of the distribution of positive training (to correspond to median training) --

hereafter referred to as the mid-range return. (For the parametric estimators, the 

 

20 Other contracting situations are plausible. For example, the costs of training may be shared by workers 
22 

estimated mid-range returns are similar to the estimates at the median.) The estimated 

mid-range returns, shown in the first two rows of Table 7, are in the neighborhood of 180 

percent, comparable to that for the log specification. 

IV. Further Discussion and Interpretation of the Key Findings 

Under the best fitting specifications, the effect of formal training on wages is 

quite large. Rates of return for formal training estimated from the NLSY are in the 150-

180 percent range for the median positive hours of training. The effect of training on 

wages in EOPP is of a comparable order of magnitude. These numbers are much higher 

than, for example, estimates of returns to schooling. The numbers also present a puzzle 

in view of the fact that only a minority of jobs in both the NLSY and EOPP have any 

formal training--13% in EOPP in the first three months, and about 25 percent in the 

NLSY as of the last observation on the job. Taking the results literally, it would appear 

that potentially profitable investments in training are not being made. In this section we 

discuss five potential explanations as to why estimated returns to training are so high: 

heterogeneity in wage growth, measurement error, promotions, direct costs of training, 

and heterogeneity in returns to training. 

Heterogeneity in Wage Growth 

Our fixed-effect regressions control for all factors whose effect on wages remains 

unchanged during a job match. However, unobserved factors that affect both wage 

growth and training will bias fixed-effect estimates of the return to training. To test 

whether individuals who receive more training tend to have higher wage growth even in 

the absence of training, we add interactions of tenure, tenure squared, and tenure cubed 

with the cube root of an individual’s final observed training in the current job to the 

 who do not receive training. We leave these considerations for another paper. 
23 

NLSY wage equation.21 (A preliminary analysis showed that the cube root of final 

training fit best.) If workers with higher wage growth self-select into training, then the 

estimated effect of final training on wage growth should be positive and the coefficients 

on lead, current and lagged training should fall. 

This is in fact what we observe. The third and fourth rows of Table 7 shows midrange rates of return for the Fourier series.22 The rate of return to training falls by 55 

percentage points to about 125 percent, and the final training interactions are jointly 

significant at the 1 percent level. The interaction coefficients imply that respondents who 

end up with 60 hours of training average about .8 percent per year more rapid wage 

growth initially and about .6 per year after 2.5 years. 

When the (cube root of) final training interactions are added, the differences in fit 

among the various functional forms become smaller, as shown in Table 8. The Q2

statistic favors the log and cube root specifications. The rate of return for the Fourier 

series is higher than that for the parametric specification both at the median itself and 

between the 25th and 75th percentiles. Table 9 shows the coefficients for the leads and 

lags in the cube root specification, with and without final training interactions. Note that 

the lag and lead coefficients decline greatly in magnitude. 

The identification of the final-training/tenure interaction coefficients merits closer 

examination. These coefficients are identified because interactions of stocks of (current, 

 

21 Pischke (2001) controls for unobserved wage growth heterogeneity by including an interaction between 

tenure and an individual fixed effect in a wage regression that already includes a (noninteracted) fixedeffect term. Our approach is more flexible in that it allows for a more flexible tenure interaction – for 

example, individuals who acquire more training may tend to have higher wage growth in the first few years 

of tenure but not later. Pischke finds that adding the fixed-effect tenure interaction sharply reduces the 

coefficient on a training dummy, but not on training duration, suggesting that high growth individuals 

select into short duration training spells. 

22 As for the specification without final training interactions, K equals 13 for the complete sample and 2 for 

the outlier-omitted sample. 
24 

lead, and lagged) training with tenure are excluded from our regression, as are leads and 

lags beyond one period. In particular, training led M-t periods, where M is the year an 

individual leaves his current job, is precisely final training, so that interactions of training 

led M-t periods with tenure will be equal to final training times tenure. 

Excluding long leads of the stock of training is in our judgment clearly 

appropriate. This still raises the question of whether interactions of the stock of training 

with tenure should enter the wage equation. One might expect that job tenure and 

training would to some extent be substitutes, so that training that occurs after long tenure 

with the employer would have lower returns. We use the cube root specification to 

examine this question, as Fourier series estimates are quite imprecise. Results are shown 

in Table 10 for the outlier-omitted sample (results from the full sample are similar). For 

comparison purposes, column 1 reports the results of the cube root specification with the 

final training interactions (i.e., the specification reported in column 4 of Table 9). 

Column 2 shows what happens when one adds interactions of tenure with current, 

lead, and lagged training. The point estimates indicate that returns to training decline 

with tenure, especially in the first few years, as one would expect if job training were a 

substitute for on-the-job learning. However, the interactions are imprecisely estimated; 

the joint p value of the interactions is 0.57. In the bottom half of Table 10 we show the 

derivative of the effect of training (at the median of 60 hours) with respect to tenure at 

various points in the distribution of tenure. None of the derivatives examined are 

statistically significant at conventional levels. 

Attempting to simultaneously estimate leads and lags for training, final 

training/tenure interactions, and interactions of lead and lagged training with tenure may 
25 

well be demanding more from our data than they can reasonably be expected to show. 

The small size of the lead and lagged coefficients in Table 9 is another indication that it 

may be overly ambitious to attempt to estimate interactions of lead and lagged training 

with tenure. Accordingly, Column 3 of Table 10 shows the results of interacting current 

training with tenure, but not lead or lagged training. Here there is stronger, though not 

overwhelming, evidence that the effect of training declines with tenure. The p value of 

the interactions declines to 0.12, and the slope of the effect of training with respect to 

tenure at the 25th percentile of tenure is substantial and significant at the 5 percent level. 

The results suggest a decline in the returns to training with increased tenure in the first 

few years on the job. 

While the estimated return for the median tenure is similar whether or not tenure 

interactions are included, the inclusion of interactions with tenure increases the estimated 

return for low values of tenure. Using the cube root specification, the return at the 25th

percentile of tenure and 60 hours of training is 123 percent. This compares to a 95 

percent return when the tenure interaction is omitted . Both of these estimates are 

somewhat conservative given that the Fourier series returns are higher.23 

In summary, heterogeneity in wage growth is responsible for a significant part of 

the apparent high returns to training.24 Furthermore, it is difficult to estimate both final 

training effects and tenure interactions with the stock of training, but there is evidence 

that the returns to training decline with tenure. After correcting for wage growth 

 

23 The inclusion of interactions of tenure with current training also increases the effect of the final-training 

interactions: aside from the increase in wages immediately after being trained, respondents who end up 

with 60 hours of training are estimated to have 1.6 percent higher wage growth at the beginning of the job 

and 0.8 percent higher wage growth after 2.5 years. 
26 

heterogeneity, estimated returns to training for workers with the median positive value of 

training are in the neighborhood of 125 percent; returns for workers with low tenure are 

likely higher. 

Measurement Error

Substantial measurement error in training has been reported by Barron, Berger, 

and Black (1997a). In the standard analysis, measurement error results in estimates that 

are biased downward. However, the case of formal training is more complicated because 

of its mixed continuous-discrete nature: a majority of our sample report receiving no 

formal training, and those who report positive formal training report varying amounts. 

As explained below, this mixed continuous-discrete structure implies that estimates of the 

effect of short spells of training may be biased upward. 

To determine the likely effects of measurement error on our OLS results, let T* 

denote true training and T denote observed training. In addition, let g(T0) denote the 

return to training for those whose true training is T0. Abstracting from other covariates 

for convenience, g(T0) = E(ln W|T*=T0) − E(ln W|T*=0), where presumably g′ > 0 and g′′

≤ 0. Since we do not observe true training, the data do not reveal the function g, but 

instead reveal f, where f(T0) = E(ln W|T=T0) − E(ln W|T=0) is the expected return to 

training for an individual whose observed training is T0. (We assume throughout that we 

consistently estimate f.) 

One can distinguish between two types of measurement error: misclassification of 

training and error in the duration of spells that are classified correctly. Misclassification 

 24 A previous draft of this paper found no effect of the final training-tenure interactions on returns to 
27 

in turn can be subdivided into forgotten training, where T = 0 but T* > 0, and false 

training, where T*=0 but T > 0. Provided that both types of misclassification error are 

independent of the residual ε in equation (1), misclassification unambiguously reduces 

the observed return to training, f(T0). To see this, note that if there is any forgotten 

training, E(ln W|T=0) > E(ln W|T*=0). And if there is any false training of length T0, 

then E(ln W|T= T0) < E(ln W|T= T0, T* > 0). The greater is either type of 

misclassification error, the smaller is the observed return to training, f(T0). 

To gain intuition on the effects of duration error, consider figure 2. For ease of 

exposition, the figure assumes away misclassification. Line G in the figure represents the 

true function g(T), which goes through the origin. Under standard conditions, 

measurement error in the positive training sample will flatten the observed function, as 

shown in line F in the figure. If there is no misclassification error, E(ln W|T=0) = E(ln 

W|T*=0), so earnings of those with no training will be consistently estimated. However, 

for any level of training 0 < T0 < M, E(ln W|T= T0) > E(ln W|T*= T0), implying that the 

returns to training in this range will be overestimated. 

As positive training is approximately log-normal, the simplest assumption is that 

duration error is log-normal. We show in the Appendix, that if the return to training is a 

linear function of ln(T*), then the effects of duration error will cancel out at the geometric 

mean. If the return to training declines at a slower (faster) than logarithmic rate, then 

duration error will cause the return to training to be biased upward (downward). Using 

parameters from the NLSY and assuming a 2 period structure to the data, we show in the 

Appendix that under reasonable assumptions, the maximum proportional bias in the 

 training. The difference in results is due to the addition of tenure cubed to the specification, and improved 

measurement of final training. 
28 

estimated return to geometric mean training from duration error is less than 20 percent. 

Monte Carlo evidence establishes that any positive bias is likely to be smaller in longer 

panels. Taking into account plausible magnitudes of classification error (also discussed 

in the Appendix), downward bias from classification error should offset any upward bias 

from duration error. 

Promotions

While general heterogeneity in wage growth does not completely explain the 

large estimated returns to training, it is possible that employees are offered training after 

increases in their job responsibilities. This might cause us to falsely attribute wage 

increases to training that are in fact due to promotions. Both the NLSY and EOPP 

contain data on promotions, so we can estimate the extent that correcting for promotions 

reduces our estimates of the effect of training. 

The 1988-90 NLSY surveys asked respondents whether their job responsibilities 

had been increased since the last interview. Respondents also were asked whether they 

had received a promotion and, if promoted, whether responsibilities had increased as a 

result of the promotion. In 1996-2000, respondents were asked separate questions about 

changes in job responsibilities and promotions. We focus on changes in job 

responsibilities because a "promotion" after training may merely be a recognition of the 

worker’s increased productivity. Using promotion variables produces similar results. 

We total changes in responsibilities within each job separately over the years 

1987-1990 and 1994-2000 and estimate a wage equation over both sub-periods (where 
29 

the job is unchanged, separate fixed-effects are estimated in each sub-period). 25 We find 

that adding the change in responsibilities variable to a Fourier series specification that 

includes final training interactions reduces the mid-range rate of return by 48 percentage 

points in the outlier-omitted sample. As reported in Table 7, one is thus left with a rate of 

return of 75 percent. Similarly, in the cube root specification, adding the change in 

responsibilities variable reduces the sum of the training coefficients by .0030, implying a 

reduction in the estimated rate of return of about 39 percentage points. Applying this to 

the results in Table 8 produces a rate of return of 56 percent in the outlier-omitted 

sample. 

 It is very likely that there is mutual causation between training and promotions. 

For example, in the SEPT95 sample of employees (Frazis et al. 1997), of those who 

received formal training from their current employer, 14 percent reported receiving a 

promotion when training was satisfactorily completed and 40 percent reported that 

training was necessary for future advancement (categories are not mutually exclusive). 

Thus, not surprisingly, training helps workers get subsequent promotions. We clearly 

have an identification problem; while giving an able worker more responsibilities may 

increase productivity in the absence of training, a worker’s improved ability to carry out 

more advanced job duties should properly be considered to be part of the return to the 

training investment. The above specification attributes all promotion-induced wage 

growth to promotions per se as opposed to the training that may have made the 

promotions possible. The estimated 40-50 percentage point reduction in the estimated 

effect of training is clearly too large. 

 

25 To create a uniform variable about changes in job responsibilities, in the 1988-90 period we count an 

individual as experiencing a change in responsibilities if a) he answers affirmatively to the change in 
30 

A more reasonable way of accounting for promotions when estimating the return 

to training is to control for promotions when estimating the effect of the current stock of 

training, but not when estimating the effect of lagged training-that is, we calculate the 

return to training as βLt + βLt+1 + βSt-1, where βLt is the training coefficient for period t in 

the long regression including promotions and βSt is the coefficient for period t in the short 

regression omitting promotions. This procedure in effect attributes promotion-induced 

wage growth to the promotion if the promotion occurs roughly concurrently with or some 

time before training, and to training if the promotion is realized some time after training. 

This approach, which is still probably too conservative, yields a reduction in the 

estimated rate of return from promotions of 34 (30) percentage points using the Fourier 

series estimates (cube root specification), resulting in a rate of return of 89 (64) percent. 

In contrast to the NLSY, the EOPP data provide no indication that the estimated 

return to training is partly due to the effect of promotions. Employers in EOPP are asked 

whether the last worker hired has received a promotion and, if so, how many months after 

being hired. We have added an indicator variable that takes on a value of 1 when a 

worker has received a promotion within two years of being hired as an additional 

explanatory variable. While the coefficient on the promotion dummy is positive and 

significant, there is virtually no effect on the formal training coefficient. 

Direct Costs of Training

The 1995 Survey of Employer Provided Training (SEPT95) estimated that, in its 

sampling frame of firms with 50 or more employees, wages and salaries of trainers, 

payments to outside trainers, tuition reimbursements, and contributions to training funds 

 

responsibilities question or b) indicates that his responsibilities have changed as a result of a promotion. 
31 

totaled $300 per employee in 1994. The survey also estimated that wages and salaries 

paid to employees while in formal training totaled $224 over the period May-October 

1995 (Frazis et al. 1997). Pro-rating the wage and salary cost of employees to a full year, 

the wages paid to workers receiving training appear to account for only about 60 percent 

of the total costs of training; other direct costs account for the remaining 40 percent. 

Applying this to our previous results, we obtain an estimated rate of return of about 40 -

50 percent. 

Heterogeneity in Returns to Training

One strongly suspects that our estimated returns are greater than could be realized 

by workers without formal training were they to get such training. Since the skills 

required for different jobs are heterogeneous, it makes sense that the returns to training 

differ across jobs. Both the NLSY and the EOPP data provide direct evidence of 

heterogeneity in returns. 

We interact the cube root of the current stock of training with job characteristics 

in the NLSY: the two occupational dummies and the part-time indicator. (The 

specification also includes final training interactions.) Results are shown in Table 11. 

Both the managerial and professional dummy and the part-time dummy have strong 

positive effects on the returns to training, with managerial and professional jobs having 

an 80 percentage points greater rate of return to training of 60 hours than do blue collar 

jobs (the difference is strongly significant). Managerial and professional employees are 

more likely to receive formal training, which is consistent with their higher returns. The 

positive effect of part-time is harder to interpret, since part-time status is negatively 
32 

associated with training. One possibility is that it reflects higher required effects of 

training on productivity to make investing in a part-time employee worthwhile.26

EOPP contains a variable that may more directly reflect training requirements for 

a job. Recall that one of the control variables in our wage growth regression is the 

logarithm of the number of weeks it takes a new employee in the most recently filled 

position to become fully trained and qualified if he or she has the necessary school 

provided training but no experience in the job, which we refer to as "job complexity". 

Consistent with this interpretation, "job complexity" is positively related to wage growth, 

as can be seen in column 1 of Table 12, which reports the key coefficients on training and 

job complexity in our preferred EOPP specification (i.e., the cube root specification in 

Table 5). 

Column 2 shows the effect of interacting formal training and job complexity and 

column 3 shows the effect of also including an informal training interaction. The 

noninteracted training coefficients fall when the job complexity-training interactions are 

included, and the job complexity – training interactions are both positive. But the 

standard errors are very high. Apparently, attempting to estimate the separate effects on 

wage growth of formal training, informal training, job complexity, and job complexitytraining interactions places too great demands on the EOPP data. We therefore look for a 

more parsimonious specification. 

We begin by considering a specification of the form: 

E(∆wi |X, Tformal, Tinformal) = ln(ψ(Tformal,Tinformal)) + Xiβ , 

 

26 In spite of the huge effect of part-time training on wages (60 hours of part-time training raises the wage 

by 6.7 percent), the rate of return to training part-time blue collar workers is slightly less than the return to 

training full-time managers and professionals if one calculates the rate of return using 1000 hours for a 

work year instead of 2000 hours. 
33 

ψ(Tformal,Tinformal) = A(b(Tformal+1)-ρ

+ (1-b) (Tinformal+1)-ρ)

-k/ρ

. 

The perfect substitute specification corresponds to = -1, and the Cobb-Douglas 

specification corresponds to the limit of ψ as ρ → 0. The perfect substitute specification 

turns out to be best fitting, although the parameter is very imprecisely estimated. 

Having determined that the perfect-substitute specification is reasonable, we next 

estimate a Box-Cox model of the form 

E(∆wi |X, Tformal, Tinformal) =(1/λ)(b(Tformal+1) +(1-b)(Tformal + 1))λ

-1 + Xiβ 

The estimated value of λ is .02, indicating that the log is a good choice for 

functional form when one aggregates formal and informal training. The results for the 

log – perfect substitute specification are reported in column 4 of Table 12.27 Finally, 

column 5 shows the results of interacting aggregate training with job complexity. The 

coefficient on the interacted variable is positive and quite large. The return to aggregate 

training increases by 50 percent going from the 25th to the 75th percentile of weeks until 

fully qualified. In the presence of measurement error in both variables, this is likely to be 

a severe understatement of the effect of job complexity on returns to training.28 

The EOPP and NLSY results provide evidence that the return to training varies 

greatly across jobs. If some of the heterogeneity in returns is unobservable, as seems 

likely, then our results do not reflect the returns to training that could be obtained by the 

average member of the population. This is in spite of our control for heterogeneity in 

 

27 Note that in obtaining aggregate training, the estimated weight on formal training is .935 and the weight 

on informal training is only .055. 

28 Barron, Berger, and Black (1997a) find substantial discrepancies between employer and employee 

reports of weeks until fully qualified. Griliches and Ringstadt (1970) demonstrate that measurement error 

is likely to more severely bias downward the magnitude of the coefficient of a quadratic term than a linear 

term where the true model is quadratic. For similar reasons, measurement error in both hours of training 
34 

wage levels by means of the fixed effect. To see this, consider the following simplified 

wage model that abstracts from covariates other than training: 

(9) ln Wit = αi + βi (Tit) + eit , 

where E(eit) = E(αi) = 0, E(βi) = β , and eit is independent of α and β. 

Both α and β are potentially correlated with T. There is ample evidence that 

training is higher for more productive workers,29 presumably because their cost of 

training is lower and/or their return to training is higher. If the cost of training is lower 

for more able individuals in more productive jobs, that is, if cov(α,T) > 0, then OLS 

estimates of the return to training will be biased upward. 

Fixed-effect estimation eliminates any potential bias stemming from a positive 

correlation between unmeasured ability α and training. However, fixed-effect estimates 

of the return to training do not purge the effect of a correlation between β and T. The 

EOPP data provide evidence of just such a correlation. Using this data, Loewenstein and 

Spletzer (1999b) demonstrate that hours of aggregate training are strongly positively 

correlated with job complexity. And, as noted above, the return to training is higher for 

individuals who are in more complex jobs. 

To analyze the bias in fixed-effect estimation, consider a situation where we have 

two periods of data, with training always equal to 0 when t=1 and varying across the 

sample when t=2. The expected value of the return to training estimated by fixed effects 

(which, in this case, is equivalent to first differences) is given by: 

 and number of weeks until fully qualified severely downward biases the estimated magnitude of their 

interaction. 

29 For example, see Barron, Berger, and Black (1999). 
35 

(10) f(T0) = E(ln Wi2|Ti2=T0) − E(ln Wi1|Ti2=T0) 

 = E(αi|Ti2= T0 ) − E(αi|Ti2= T0 ) + E(βi (T0)| Ti2= T0) 

 = E(βi| Ti2= T0) (T0). 

One can distinguish between the return to training for the average member of the 

population and the return to training for the trained (see, for example, Heckman and Robb 

1985 and Heckman 1997). Fixed-effect regressions do not estimate the return to training 

for the average member of the population β (T0), but, as is clear from (10), consistently 

estimate the effect of a given amount of training for those with that amount of training.

30 

In particular, our high estimated returns to short spells of training are not overestimates of 

the return to training for those with such spells. However, this does not mean that one 

would expect individuals who do not receive formal training to have realized such returns 

had they been trained. Indeed, any reasonable model would predict that E(βi|T=T0) > 

E(βi|T=0): individuals with training should tend to have a higher return than those with no 

training. 

Without the appropriate structural restrictions, it is not possible to estimate the 

expected return to training of workers who do not receive training. Similar comments 

apply to estimates of the marginal return to training, which will be estimated as 

(11) f′(T0) = E(βi|T=T0) ′(T0) + T

i T T

∂

∂E( | = ) β 0 (T0), 

 

30 Note that the example given, with zero training in the first period followed by varying amounts in the 

second period, is exactly the situation in EOPP. As with measurement error, the situation is more 

complicated in the multiperiod NLSY dataset, where the estimated return g(T0) will partly reflect average 

returns and partly reflect marginal returns. When we omit observations with (within-job) accumulated 

training greater than zero but less than final observed training--thus bringing the situation closer to that in 

EOPP--the results are virtually identical to those in table 2. 
36 

and which will exceed E(βi|T=T0) ′(T0) if T

i T T

∂

∂E( | = ) β 0 > 0: estimation of is 

confounded by a composition effect stemming from the fact that individuals with more 

training can be expected to have a higher return. 

Summary 

To summarize our discussion: heterogeneity in wage growth, promotions, and 

direct costs are all partial explanations for the high estimated rates of return to training 

appearing in Tables 5 and 6. After correcting for these factors, we are left with returns in 

the neighborhood of 40-50 percent at the median positive level of training.31 

Measurement error likely leads to either minimal overestimates or to underestimates at 

this level of training. These returns are several times the returns to schooling and are 

very likely an underestimate in that they do not reflect cost-sharing with the employer. 

Returns appear to be higher for those with low tenure. Heterogeneity in returns is a 

potential explanation as to how returns to formal training can be so high while most 

workers do not get formal training. While those with formal training of 60 hours do have 

annualized returns to training of at least 40-50 percent, these returns cannot be 

extrapolated to the untrained. 

V. Conclusion 

This paper has investigated the related questions of the functional form and 

magnitude of the wage returns to formal training. Our results from both the NLSY and 

EOPP indicate that the return to an extra hour of training diminishes sharply with the 

amount of training received. A cube root specification generally fits the data best, but the 

 

31 This estimate turns out to be similar to those obtained by Bartel and Mincer, but this is by coincidence as 

our estimate is obtained very differently. Unlike Bartel and Mincer, we use a nonlinear specification, allow 
37 

log specification also does well. The linear specification always fits the data poorly and 

substantially understates the effect of training, and the quadratic specification is quite 

volatile. 

Our best fitting specifications indicate that there are very substantial returns to the 

initial interval of formal training. One explanation is heterogeneity in wage growth. 

Fixed-effect regressions control for all factors whose effect on wages remains unchanged 

during a job match. However, unobserved factors that affect both wage growth and 

training will bias fixed-effect estimates of the return to training. Controlling for 

heterogeneity in wage growth by adding interactions of tenure and an individual’s final 

observed training in the current job to the wage equation has the effect of sharply 

reducing the estimated return to training. 

Returns are reduced further when one takes into account the effect of promotions 

and the fact that direct costs are a substantial portion of the total cost of training. After 

correcting for confounding factors, we are still left with a rate of return in the 

neighborhood of 40-50 percent at the median positive level of training. This estimated 

return, which is several times that associated with schooling, is an underestimate since it 

does not take into account cost-sharing with the employer. 

Heterogeneity in returns explains how returns to formal training can be so high 

while most workers do not get formal training. Both NLSY and EOPP data provide 

evidence of such heterogeneity: the return to training is significantly higher for managers 

and professionals in the NLSY and in more complex jobs in EOPP. With heterogeneity 

in returns, our results cannot be considered structural estimates in the sense of showing 

 for one period lead and lagged effects as implied by theory, and control for heterogeneity in wage growth, 

promotions, and the direct cost of training. 
38 

the return to training for an average member of the population, nor can estimated 

marginal returns be interpreted as the marginal returns to any member of the population. 

However, under reasonable assumptions our fixed-effect method ensures that the 

estimated average return can be interpreted as the return to a given amount of training for 

those with that amount of training. 

Structural estimation of returns to training when there is heterogeneity presents 

challenges. While a fair amount of research on the econometrics of heterogeneous 

returns has recently been published (for example, Angrist, Imbens and Rubin 1996, 

Heckman 1997, Heckman and Vytacil 1998), there are two problems with applying this 

research to training. First, it is difficult to find a plausible instrument. Second, as with 

measurement error, the mixed continuous-discrete structure complicates the problem. 

The only paper that we are aware of that deals with a problem of this type is Kenney et al. 

(1979).32 We leave a more complete analysis of heterogeneity in returns to training as a 

topic for future research. 

 

32 Kenney et al. (1979) obtain structural estimates of the return to college education in a model where there 

is a mass point at zero years of college. In their model, the returns to entering college are heterogeneous, 

though the returns to years of college conditional on entering are not. 
39 

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42 

Table 1 

Descriptive Statistics, NLSY 

Variable Mean Std. Dev. Min. Max. 

Ln Wage 1.88 0.49 0.00 4.53

# train. spells, current job 0.54 1.34 0.00 21.00

Training Hours 40.60 209.14 0.00 9260.00

Ln (Training + 1) 1.02 1.90 0.00 9.13

Training Hours, Training > 0 164.60 396.11 0.50 9260.00

Ln (Training + 1), Training > 0 4.12 1.39 0.41 9.13

Years Tenure 3.80 3.68 0.00 22.77

Year=1980 0.02 0.13 0.00 1.00

Year=1981 0.03 0.16 0.00 1.00

Year=1982 0.04 0.19 0.00 1.00

Year=1983 0.04 0.20 0.00 1.00

Year=1984 0.05 0.22 0.00 1.00

Year=1985 0.06 0.23 0.00 1.00

Year=1986 0.06 0.24 0.00 1.00

Year=1987 0.06 0.24 0.00 1.00

Year=1988 0.07 0.25 0.00 1.00

Year=1989 0.07 0.26 0.00 1.00

Year=1990 0.07 0.26 0.00 1.00

Year=1991 0.07 0.25 0.00 1.00

Year=1992 0.07 0.25 0.00 1.00

Year=1993 0.07 0.25 0.00 1.00

Year=1994 0.06 0.24 0.00 1.00

Year=1996 0.06 0.24 0.00 1.00

Year=1998 0.06 0.23 0.00 1.00

Year=2000 0.05 0.21 0.00 1.00

Black 0.26 0.44 0.00 1.00

Hispanic 0.18 0.38 0.00 1.00

Age at start of job 25.28 4.78 14.90 41.19

Years experience at start of job 5.88 4.07 0.00 23.47

Female 0.50 0.50 0.00 1.00

AFQT (residual) 0.37 20.17 -65.48 45.94

Years education 12.77 2.27 0.00 20.00

Ever married 0.63 0.48 0.00 1.00

Union 0.20 0.40 0.00 1.00

Managerial/prof. (1st yr. in job) 0.17 0.38 0.00 1.00

Other white-collar (1st yr. in job) 0.34 0.47 0.00 1.00

Part-time 0.12 0.32 0.00 1.00

[Continued] 
43 

Table 1, continued 

Variable Mean Std. Dev. Min. Max.

Hours outside training on current 

job 18.47 165.59 0.00 5440.00

Missing AFQT 0.06 0.23 0.00 1.00

Missing Union 0.05 0.21 0.00 1.00

Missing Part-time 0.00 0.03 0.00 1.00

# spells missing tr. hrs, current job 0.01 0.12 0.00 4.00

Ever training spell, current employer 0.25 0.43 0.00 1.00

Any training spell, current period 0.10 0.30 0.00 1.00

 

N 17,809 

Obs. 75,698 
44 

Table 2 

Percentiles of Distribution of Hours of Training, Training > 0 

Percentile Cumulative 

Stock of 

Training 

Hours of 

Training During 

Previous Year 

10 8 4 

25 26 10 

50 60 40 

75 146 64 

90 360 160 

# > 0 18,673 7,589 
45 

Table 3 

Returns to Training for Different Functional Forms, NLSY 

Specification 2 R Fraction 

Fourier 

Series 

Explained 

Total 

Effect at 

Median 

Complete Sample 

No Training Vars. 0.2033 -- -- 

Dummy 0.2042 .624 0.031 

Linear 0.2040 .332 0.003 

Quadratic 0.2042 .461 0.005 

Cube root 0.2050 .842 0.036 

Log 0.2049 .822 0.041 

Dummy + Linear 0.2047 .732 0.029 

Fourier series 0.2057 -- 0.039 

 

N 17,809 

Obs 75,698 

 

Training Outliers Omitted* 

 

No Training Vars. 0.2023 -- -- 

Dummy 0.2032 .630 0.031 

Linear 0.2031 .481 0.007 

Quadratic 0.2035 .716 0.014 

Cube root 0.2040 .842 0.037 

Log 0.2038 .823 0.041 

Dummy + Linear 0.2037 .762 0.029 

Fourier series 0.2047 -- 0.034 

 

N 17,788 

Obs. 75,497 

*Top 1% of training duration omitted. 
46 

Table 4 

Descriptive Statistics, EOPP 

Variable Mean Std. Dev. Min. Max. 

Ln Wage Growth 0.19 0.20 -.56 1.51

Formal training indicator 0.13 0.33 0.00 1.00

Informal training indicator 0.95 0.22 0.00 1.00

Hrs. formal tr., formal tr. > 0 72.27 101.14 1.00 640.00

Hrs. informal tr., informal tr. > 0 131.72 175.03 1.00 2070.0

Ln (formal tr. + 1), formal tr. > 0 3.57 1.23 0.69 6.46

Ln (informal tr. + 1), inf. tr. > 0 4.23 1.21 0.69 7.64

Ln # weeks until fully trained 2.21 1.24 0.00 6.033

Years relevant experience 2.38 4.49 0.00 40.00

Rel. experience squared 25.76 108.31 0.00 1600.00

Age 26.89 9.10 16.00 64.00

Years education 12.47 1.65 2.0 24.00

Vocational schooling 0.28 0.45 0.00 1.00

Temporary or seasonal job 0.15 0.36 0.00 1.00

Part-time job 0.21 0.41 0.00 1.00

Union 0.11 0.28 0.00 1.00

Ln establishment size 2.87 1.51 0.00 8.60

Female 0.45 0.50 0.00 1.00

Managerial/professional 0.11 0.31 0.00 1.00

Tenure 1.32 1.59 0.00 29.92

Other white-collar 0.57 0.50 0.00 1.00

Missing Union 0.13 0.11 0.00 1.00

Missing Years education 0.03 0.18 0.00 1.00

Missing Tenure 0.03 0.18 0.00 1.00

Obs. 1,715
47 

Table 5

Returns to Training for Different Functional Forms, EOPP 

Specification 2 R Fraction 

Fourier 

Series 

Explained

Total 

Effect at 

Median 

Complete Sample 

No Formal Training 

Vars. 

0.1756 -- -- 

Dummy 0.1807 .587 0.044 

Linear 0.1813 .660 0.014 

Quadratic 0.1837 .951 0.031 

Cube root 0.1834 .873 0.047 

Log 0.1830 .840 0.052 

Dummy + Linear 0.1822 .790 0.036 

Fourier series 0.1833 0.041 

 

Obs 1,715 

 

Training Outliers Omitted* 

 

No Formal Training 

Vars. 

0.1749 -- 

Dummy 0.1801 .574 0.045 

Linear 0.1829 .837 0.018 

Quadratic 0.1831 .963 0.028 

Cube root 0.1834 .899 0.050 

Log 0.1827 .847 0.053 

Dummy + Linear 0.1832 .902 0.035 

Fourier series 0.1830 0.041 

 

Obs. 1,713 

*Top 1% of formal training duration observations omitted. 
48 

Table 6 

Rates of Return to Training for Different Functional Forms with Lagged and Lead 

Training, NLSY 

Specification 2 R Fraction 

Fourier 

Series 

Explained 

Total 

Effect at 

Median 

Implied 

Rate of 

Return at 

Median 

(%) 

Complete Sample 

No Training Vars. 0.1949 -- -- -- 

Dummy 0.1961 .581 .042 140 

Linear 0.1957 .299 .004 12 

Quadratic 0.1959 .378 .007 23 

Cube root 0.1969 .772 .045 149 

Log 0.1968 .759 .053 175 

Dummy + Linear 0.1965 .683 .040 132 

Fourier series 0.1970 -- .059 197 

 

 16,534 

Obs 69,800 

 

Training Outliers Omitted* 

 

No Training Vars. 0.1939 -- -- 

Dummy 0.1951 .734 .042 140 

Linear 0.1951 .590 .008 28 

Quadratic 0.1956 .790 .018 59 

Cube root 0.1959 .978 .048 159 

Log 0.1958 .949 .053 178 

Dummy + Linear 0.1957 .914 .040 134 

Fourier series 0.1960 -- .056 186 

 

n 16,502 

Obs. 69,573 

*Top 1% of training duration omitted. 
49 

Table 7 

Fourier Series Estimates of Mean Rates of Return for 25th - 75th Percentiles of Positive 

Training Distribution 

Without correction for heterogeneity in 

growth rates: 

 

Complete 

sample 

183 

 (33) 

 

Outlier-omitted 

sample 

178 

 (33) 

 

Corrected for heterogeneity in growth rates: 

 

Complete 

sample 

128 

 (36) 

 

Outlier-omitted 

sample 

124 

 (36) 

 

Corrected for heterogeneity in growth rates 

and promotions: 

 

Complete 

sample 

79 

(32) 

 

Outlier-omitted 

sample 

75 

(31) 

Corrected for heterogeneity in growth rates 

and promotions’ effect on lead and current 

training coefficients: 

 

Complete 

Sample 

 88 

 (33) 

 

Outlier-omitted 

sample 

 89 

 (30) 

 

*Standard errors are in parentheses 
50 

Table 8 

Rates of Return to Training for Different Functional Forms with Lagged and Lead 

Training and Final Training Interactions, NLSY 

Specification 2 R Fraction 

Fourier 

Series 

Explained 

Total 

Effect at 

Median 

Implied 

Rate of 

Return at 

Median 

(%) 

Complete Sample 

No Training Vars. 0.1970 -- -- -- 

Dummy 0.1973 .411 0.021 69 

Linear 0.1970 .063 0.001 2 

Quadratic 0.1971 .099 0.001 5 

Cube root 0.1974 .518 0.025 82 

Log 0.1974 .539 0.029 95 

Dummy + Linear 0.1973 .449 0.022 72 

Fourier series 0.1986 -- 0.043 144

 

n 16,534 

Obs 69,800 

 

Training Outliers Omitted* 

 

No Training Vars. 0.1959 -- -- 

Dummy 0.1962 .681 0.021 71 

Linear 0.1960 .316 0.003 10 

Quadratic 0.1963 .634 0.009 30 

Cube root 0.1963 .892 0.029 95 

Log 0.1963 .895 0.030 102 

Dummy + Linear 0.1963 .832 0.024 81 

Fourier series 0.1966 -- 0.040 133 

 

n 16,502 

Obs. 69,573 
51 

Table 9 

Selected Coefficients and Rates of Return, Cube Root Specification 

 Full Sample Outliers Omitted 

Lead Training1/3 0.0024 

(0.0014) 

0.0006 

(0.0014) 

0.0027 

(0.0018) 

0.0010 

(0.0018) 

Current Training1/3 0.0051 

(0.0012) 

0.0043 

(0.0012) 

0.0050 

(0.0015) 

0.0044 

(0.0013) 

Lagged Training1/3 0.0040 

(0.0010) 

0.0013 

(0.0011) 

0.0045 

(0.0012) 

0.0019 

(0.0015) 

Final Training1/3 x Tenure 0.0021 

(0.0008) 

0.0020 

(0.0009) 

Final Training1/3 x 

Tenure2

 -0.0001 

 (0.0001) 

 -0.0001 

 (0.0001) 

Final Training1/3 x 

Tenure3

/100 

 0.0001 

(0.0005) 

 0.0001 

(0.0005) 

Effect of Training at 

Median Positive Hours 

0.0448 

(0.0070) 

0.0246 

(0.0084) 

0.0476 

(0.0075) 

0.0285 

(0.0088) 

Rate of Return to 

Training at Median 

Positive Hours 

149 82 159 95 

 
52 

Table 10 

Effect of Tenure on Returns to Training, Outlier-omitted Sample 

Interactions of Tenure, Tenure2, 

Tenure3 with: 

Final 

Training 

Final 

Training, 

Current, 

Lead 

and 

Lagged 

Training 

Final 

Training, 

Current 

Training 

Effect of Median Positive Training at: 

25th Percentile of Tenure (1.1 years) 0.0285 

(0.0088)

 0.0365 

(0.0175) 

 0.0369 

(0.0120) 

50th Percentile of Tenure (2.5 years) 0.0285 

(0.0088)

 0.0283 

(0.0127) 

 0.0266 

(0.0101) 

75th Percentile of Tenure (5.0 years) 0.0285 

(0.0088)

 0.0226 

(0.0096) 

 0.0222 

(0.0089) 

 

Slope of Training Effect with Respect 

to Tenure at: 

 

25th Percentile of Tenure -0.0087 

 (0.0053) 

-0.0099 

(0.0042) 

50th Percentile of Tenure -0.0046 

 (0.0035) 

-0.0051 

(0.0027) 

75th Percentile of Tenure 0.0007 

 (0.0027) 

 0.0011 

(0.0022) 

Effect of Median Positive Training on 

Wage Growth at Median Tenure (Final 

Training Interactions) 

 0.006 0.008 0.008 

 

p value, Tenure-Training Interactions* 0.57 0.12 

* Current, lead and lagged training in column (2); current training in column (3). 
53 

Table 11 

Selected Coefficients and Rates of Return, Cube Root Specification with Job 

Characteristics Interactions, NLSY, Outlier-omitted sample. 

Coefficient 

Lead Training1/3 0.0010 

(0.0018) 

Lagged Training1/3 0.0016 

(0.0013) 

Current Training1/3 0.0012 

(0.0017) 

Initial Occ. Managerial/Professional 

x Current Training1/3 0.0063 

(0.0013) 

Initial Occ. Other White Collar 

x Current Training1/3 0.0015 

(0.0011) 

Part-Time x Current Training1/3 0.0133 

(0.0049) 

 

Effect at 60 

Hours 

Rate of 

Return at 

60 Hours 

 

Blue Collar 0.0152 

(0.0091) 

 51 

 (30) 

Managerial/Professional 0.0397 

(0.0099) 

133 

 (33) 

Other White-Collar 0.0212 

(0.0094) 

 71 

 (31) 

Part-Time Blue Collar 0.0672 

(0.0205) 

 112* 

 (34) 

*Calculated at work-year of 1000 hours.
54 

Table 12 

Interactions of Training and Job Complexity, EOPP 

Cube Root 

Formal 

Training, 

 Log Informal

Training

Cube Root 

 Formal 

Training,

Log Informal 

Training, 

Formal 

Training 

Interaction 

Cube Root 

Formal 

Training, 

Log Informal 

Training , 

Training 

Interactions 

Perfect 

Substitutes 

Perfect 

Substitutes, 

Job 

Complexity 

Interaction 

Coefficient 

 

Cube Root Formal 

Training 

 

 0.014 

 (0.004) 

 0.006 

(0.010) 

 0.008 

(0.010) 

 

Log Informal 

Training 

 

 0.015 

 (0.003) 

 0.015 

 (0.003) 

 0.008 

 (0.006) 

 

Log Number of 

Weeks Until 

Qualified 

 0.012 

 (0.004) 

 0.011 

 (0.004) 

 -0.000 

 (0.009) 

0.015 

 (0.003) 

 Cube Root Formal 

x Log Number of 

Weeks 

 

 0.003 

(0.004) 

 0.002 

 (0.003) 

 -0.001 

 (0.003) 

 

Log Informal x Log 

Number of Weeks 

 0.003 

 (0.002) 

 

Log Weighted 

Aggregate Training

 

0.026 

(0.004) 

0.013 

(0.008) 

Log Weighted 

Aggregate Training 

x Job Complexity 

 

0.0055 

(0.0028) 

Weight on Formal 

Training 

 

0.935 

(0.034) 

0.928 

(0.038) 

Obs. 1,715 1,715 1,715 1,715 1,715 

2 R 0.1834 0.1834 0.1841 0.1844 0.1857 
55 

Figure 1

Predicted effect of training on ln wages, various specifications

Fourier

Log

Cube Root

Dum.+Linear

Quad.

Linear

-0.02-0.04

0

0.080.060.040.02

0.1

0.12

1.5 2.5 3.5 4.5 5.5 6.5

Ln(Hrs. training+1)

Effect on ln wage

Fourier Log Cube Root Dum.+Linear Quad. Linear

25th percentile 75th percentile
56 

Figure 2 

M 

G 

F 

Hours of Training 

Ln Wages 
57 

Appendix. 

In this appendix, we give the details of our calculation of the maximum effect of 

duration error and of the potential effects of misclassification error. Let η = ln(T) −

ln(T*) denote the measurement error in (non-misclassified) log training. We assume that 

η is independent of ln(T*), and is distributed N(0, 2 σ η ) and that ln(T*) is distributed 

N(μ, 2

*σ ). These measurement error assumptions are consistent with our data in that the 

training distributions in the NLSY and EOPP are both approximately log-normal.33 In 

fact, note that in the NLSY reported training hours for each spell are the product of 

reported hours per week and reported spell duration in weeks, strongly implying a 

multiplicative element to the measurement error. 

Under our assumptions, the distribution of ln(T*) conditional on ln(T) is normal 

with mean (ln( ) ) * (ln( *) | ln( )) μ

σ

σ

E T T = μ + T − 2

T

2

 and variance 

(A1) 2 σ ln(T*)|ln(T ) = (1-ρ

2

) 2

* σ , 

where 

σ T

σ

ρ

* = denotes the correlation coefficient between ln(T*) and ln(T), and 

2 2 2

* σ T ≡ σ +σ η is the variance of observed log training. Consequently, if the return to 

training is a linear function of ln(T*), say g(T*) = βln(T*), then 

 (A2) E(ln W|ln(T)= μ) = β E(ln(T*)| ln(T)= μ) = g(exp(μ)) , 

so that the return to training is consistently estimated at the geometric mean of training. 

 

33 Estimating the Box-Cox transformation to normality (T - yields an estimate of λ of .03 for the EOPP 

positive formal training sample. Recall that λ=0 corresponds to log-normality, and that our estimate of λ is 

−.01 in the NLSY. Quantile plots also show that log-normality is a good approximation in both datasets. 
58 

If the return to training is not a linear function of ln(T*), then the effects of 

measurement error will not cancel out at the geometric mean, so that E(ln W|ln(T)= μ) ≠

g(exp(μ)). To see this, let h(⋅) be the function implicitly defined by h(ln(x)) = g(x), and 

note that E(ln W|ln(T)= ) = E(h(ln(T*))|ln(T)= ) = h(T*) (ln(T*))d ln(T*) ∫

∞

−∞

γ , where 

γ(⋅) is the (normal) density function for ln(T*) conditional on ln(T) = μ. Taking a 

second-order Taylor expansion of h(⋅) around , the expected return to training for an 

individual with observed non-misclassified training ln(T) = can be expressed as 

(A3) E(ln W|ln(T)= ) = h( ) + h’( ) *) (ln(T*) ) (ln(T*))d ln(T ∫

∞

−∞

− μ γ

 + (1/2) ’’( (ln( *)))(ln( *) ) (ln( *)) ln( *) 2 h T T T d T ∫

∞

−∞

τ − μ γ , 

 = g(exp(μ)) + (1/2) ’’( (ln( *)))(ln( *) ) (ln( *)) ln( *), 2 h T T T d T ∫

∞

−∞

τ − μ γ

where τ(ln(T*)) is some value between and ln(T*), and the second term on the right 

hand side of the first equation is 0 by virtue of the fact that E(ln(T*)|ln(T)= . From (A3), 

we see that the nature of the bias at μ is determined by the sign of h’’. If the true return to 

training declines at a slower than logarithmic rate so that h’’ > 0, the estimated return to 

non-misclassified observed training at the geometric mean will exceed the true return. 

 To estimate the potential upward bias if h’’ > 0, let the return to true training be given 

by g(T*) = c(T*)δ , 0< δ < 1, which means that h(x) = c(exp(x))δ . Further expanding the 

Taylor series in (A3), using the fact that h(n)

(x) = δ

n

 h(x), and rearranging terms, the 

proportional upward bias in the estimated return to training at the geometric mean is 
59 

((ln( *) ) | ln( ) ) ( ) (2 )!

(ln | ln( ) ) ( ) 2

1

2

μ μ

δ

μ

μ μ = − = = − ∑

∞

=

E T T

h n

E W T h n

n

n

. 

Using (A1) and maximizing 2 σ ln(T *)|ln(T )=μ with respect to 2

* σ while holding 2 σT

constant, one can show that 2 σ ln(T *)|ln(T )=μ ≤ 2 (1/ 4)σ T . The variance of observed log 

training in the NLSY is 1.94. In EOPP, the variance of observed log formal training is 

1.27. Numerical calculations thus show that if δ = .33, the maximum proportional bias in 

the estimated return to geometric mean training is only about 3 percent in the NLSY and 

2 percent in EOPP. If δ = .75, the maximum proportional bias in the estimated return to 

geometric mean training is about 18 percent in the NLSY and 11 percent in EOPP. 

Classification error will cause underestimation of the returns at the geometric 

mean (and all other points). We can make a plausible estimate of the extent of 

classification error by using data from a 1993 survey matching workers’ and firms’ reports 

of training sponsored by the Upjohn Institute, similar in design to EOPP (Barron, Berger, 

and Black 1997).34 The Upjohn survey covers formal (and informal) training on the first 

four weeks on the job. 

We first deal with forgotten training, assuming for the time being that there is no 

false training. Recall that training effects are g(T) 'g(0), but we estimate f(T) 'f(0). 

With forgotten training, f(0) ()W|T=0) = E(g(T*)|T = 0). Let 

(1 ) ( ( *) | 0) ( ( *) | 0)

(1 ) ( ( *) | 0)

− = + >

− = ≡ p E g T T pE g T Tp E g T T

r denote the proportion of the total g(T*) 

that is forgotten. Disregarding duration error, ) ( ) ( T0 g T0 f = and 

 

34 We thank Dan Black for supplying us with the data from this survey. 
60 

(1 )(1 )

( ( ) | 0)

(1 )(1 )

( ( *) | 0) ( ( *) | 0)

p r

prE g T T

p r

prE g T T E g T T − −> = − −> = = . Normalizing g(0) to zero, 

the proportionate bias in the estimated return to training is thus given by 

(1 )(1 ) ( )

( ( ) | 0)

( ) (0)

( ) (0) ( ( ) (0))

0 0

0 0

p r g T

prE g T T

g T g

f T f g T g

− −

> = − −

− − − . 

We assume that reports of formal training by the employer that are not mentioned 

by the employee are forgotten training. Taking g(*	++		$,-.	


indicate that about 33 percent of the cube root of formal training is forgotten. In the 

NLSY, p = .1 and E(g(T) | T > 0) is 4.36. Assuming that the proportion r of the total 

g(T*) that is forgotten is the same in Upjohn as in the NLSY, forgotten training depresses 

the estimated rate of return by 6%. 

Now consider the effect of false training. Consider an example where the 

probability of forgetting a spell of training is .35 (compatible with Upjohn data) and the 

probability of reporting a spell where none exists is .01, where the observed probability 

of training is .1 as in the NLSY, and where the distribution of false training is the same as 

the distribution of true observed training. Note that 1 1 *) 0 p = p * (1−α ) + ( − p α , where 

p* is the true probability of training, α0 is the probability of false training (conditional on 

no true training), and α1 is the probability of forgetting training conditional on receiving 

training. The percentage of reported training that is false is (1− p*)α 0 / p , which solving 

for p* from the parameter values just given is 8.6%. 

In summary, even small amounts of false training cause substantial bias when the 

probability of training is relatively low. In our example, a false training probability of 

1% leads to an 8.6% downward bias in the estimated return to training. Combining this 
61 

bias with the bias from forgotten training, one obtains a number of the same order of 

magnitude as the maximum positive bias from duration error, so on net it seems unlikely 

that there is significant positive bias to estimated returns at the geometric mean. 

Our discussion neglects one potential complication in the NLSY: the fact that our 

measure of training stocks is not derived from a single questionnaire item, but is the sum 

of training flows accumulated across periods, each component of which is subject to 

misclassification and duration error. This greatly complicates the analysis. The EOPP, 

with a single formal training item, is not subject to this problem. 

One factor that might work in the direction of overestimating the return to training 

is that duration error in the stocks of training in the positive training sample would 

include the effects of misclassification error in the flows, as within the same job some 

spells of training are forgotten and some false training is reported. If, as indicated by 

Loewenstein and Spletzer (1996), training is highly correlated within jobs and if 

respondents sometimes forget to report training, the amount of training that respondents 

with positive training receive on average could be underestimated, resulting in 

overestimation of returns. On the other hand, the sum of measurement error across 

multiple spells will tend to become less similar to the log-normal distribution and more 

similar to the normal, which would tend to push down estimated returns at the geometric 

mean for any given functional form. 

We conducted a small Monte Carlo exercise in order to judge the probable effect 

of the panel nature of our data on the direction of the measurement error effect. The setup is as follows. Workers are observed for four periods t={0,1,2,3}. Time 0 corresponds 

to the start of the job, before training is observed. Training spell duration is positive in 
62 

periods 1-3 if the latent variable T** > 0. it i T = K − t + t + ρλ 2 ** .021 .0006 it ρ ε 2 + 1− , 

where λi is a person-specific fixed effect and it ε is a residual uncorrelated across 

periods, both distributed normally with unit variance, and where -that 

varies across specifications. The coefficients on tenure, here and below, are taken from 

the NLSY data. Training is forgotten with a constant probability α. The parameter K is a 

function of α such that the observed probability of training over the four periods is .10, 

corresponding to the data. 

Training duration for positive training observations is distributed log-normally, 

with T T t t e it > = − + +2 ln( * | ** 0) 3.84 .091 .0052 , where e is distributed normally with 

mean zero
	
% 
	-	
/-
	%	

duration error, e has standard deviation 1.27. In specifications with duration error, e has 

standard deviation 1.27 / 2 and a normal mean zero duration error variable with the 

same standard deviation (and independent of the other variables) is added to produce 

observed log training ln(Tit) (when training is not forgotten). Geometric mean training 

for an individual spell is about 40 hours with this specification, corresponding to the data. 

Wages are generated by the following process: 

i it

t

Wit = + g ∑Ti + t + t + + u ≤β κ τ

τ

2 ln( , where ) 1.92 ( *) .095 .0021 is a normally-distributed 

person-specific effect with mean zero and standard deviation 0.5 and u is a normallydistributed residual uncorrelated across periods with standard deviation 0.2. The function 

g varies across specifications, with the parameter β varying such that 60 hours of training 

increases log wages by .04. 
63 

For computational simplicity, we simulate estimation by splines, taking quartiles 

of the positive training distribution as our knot-points. We take as our bias measure the 

percentage bias of the predicted effect of training at the median of positive training for 

each simulated sample. For comparison purposes, we also simulate comparable 

specifications where we observe only periods t={0,1}. We computed 1,000 simulations 

per specification. 

Results are shown in Appendix Tables 1 and 2. Table 1 shows the results for the 

four-period setup. There appears to be a downward bias in each specification due to the 

spline functional form, in most cases small. (The small reported bias in the linear 

specification with no duration error and zero probability of forgetting is due to sampling 

error.) Measurement error in most cases appears to increase the downward bias. The 

possibility of overestimation due to respondents with positive training sometimes failing 

to report training appears to be a factor only for extreme specifications--specifically, the 

linear specification with perfect correlation of the propensity to train across time periods, 

moderate values of forgotten training, and no duration error. Duration error decreases 

underestimation in some specifications, but typically in specifications with a great deal of 

forgotten training, and not enough to lead to overestimation of returns. 

Appendix Table 2 shows results from the 2-period setup. These are in line with 

our theoretical analysis above: for some linear and T.75 specifications with no or moderate 

amounts of forgotten training duration error leads to overestimation of returns. In 

summary, our Monte Carlo exercise indicates that the fact that the NLSY has more than 

two periods of data makes it less likely that we are overestimating returns to median 

hours of training. 
64 

Appendix Table 1 

Percentage Bias at Median Hours of Training, Four-period Simulations 

True % bias at median 

functional 

form 

 Probability 

of 

forgetting No duration 

error 

Duration error 

present 

Ln 0.4 0 -4.0% -5.1%

 0.25 -11.9% -13.7%

 0.50 -29.6% -28.6%

 0.75 -66.7% -65.2%

 0.6 0 -4.0% -4.9%

 0.25 -12.0% -12.3%

 0.50 -27.5% -26.8%

 0.75 -62.9% -62.8%

 0.8 0 -3.2% -5.3%

 0.25 -12.0% -12.8%

 0.50 -25.0% -25.0%

 0.75 -59.8% -59.7%

 1 0 -7.9% -10.1%

 0.25 -12.3% -14.8%

 0.50 -25.5% -25.8%

 0.75 -52.0% -50.0%

 

Cube 0.4 0 -5.6% -9.1%

root 0.25 -14.2% -15.3%

 0.50 -32.2% -33.1%

 0.75 -70.4% -68.5%

 0.6 0 -4.5% -10.4%

 0.25 -16.4% -20.0%

 0.50 -35.2% -36.4%

 0.75 -71.1% -69.8%

 0.8 0 -4.7% -11.1%

 0.25 -20.1% -25.9%

 0.50 -39.8% -42.8%

 0.75 -70.5% -73.0%

 1 0 -34.2% -53.7%

 0.25 -52.9% -63.7%

 0.50 -63.9% -68.8%

 0.75 -80.7% -81.2%
65 

Appendix Table 1, continued 

True % bias at median 

functional 

form 

 Probability 

of 

forgetting No duration 

error 

Duration error 

present 

T.75 0.4 0 -4.4% -5.0%

 0.25 -14.5% -11.6%

 0.50 -28.6% -22.9%

 0.75 -69.4% -59.1%

 0.6 0 -5.2% -6.3%

 0.25 -11.4% -13.7%

 0.50 -27.9% -22.6%

 0.75 -69.1% -55.6%

 0.8 0 -3.9% -10.8%

 0.25 -9.4% -17.0%

 0.50 -24.2% -25.2%

 0.75 -64.0% -55.5%

 1 0 -8.6% -21.2%

 0.25 -10.8% -25.9%

 0.50 -21.7% -26.7%

 0.75 -56.9% -50.3%

 

Linear 0.4 0 -1.0% -2.1%

 0.25 -6.2% -6.4%

 0.50 -22.7% -14.6%

 0.75 -75.2% -47.0%

 0.6 0 2.1% -7.0%

 0.25 -5.5% -6.3%

 0.50 -14.2% -13.3%

 0.75 -67.5% -43.3%

 0.8 0 -1.1% -12.1%

 0.25 0.4% -10.4%

 0.50 -8.8% -10.4%

 0.75 -59.4% -40.2%

 1 0 -0.3% -12.0%

 0.25 14.2% -3.6%

 0.50 18.3% 1.5%

 0.75 -31.4% -22.9%

 
66 

Appendix Table 2 

Percentage Bias at Median Hours of Training, Two-period Simulations 

True functional form Probability % bias at median 

of 

forgetting No 

duration 

error 

Duration 

error 

present 

Ln 0 -3.8% -5.0%

 0.25 -6.4% -9.1%

 0.50 -15.2% -18.2%

 0.75 -44.6% -45.4%

 

Cube root 0 -4.3% -3.2%

 0.25 -9.2% -7.5%

 0.50 -18.9% -17.0%

 0.75 -50.0% -45.3%

 

T.75 0 -7.6% 5.7%

 0.25 -15.7% 0.2%

 0.50 -28.9% -10.8%

 0.75 -72.0% -46.0%

 

Linear 0 -1.5% 14.2%

 0.25 -11.3% 8.5%

 0.50 -30.6% -5.2%

 0.75 -91.3% -45.2%