BLS WORKING PAPERS

U.S. DEPARTMENT OF LABOR

Bureau of Labor Statistics

 

OFFICE OF EMPLOYMENT RESEARCH

AND PROGRAM DEVELOPMENT

Reexamining the Returns to Training: Functional Form,

Magnitude, and Interpretation

Harley Frazis and Mark A. Loewenstein

Working Paper 325

August 1999

 

The views expressed are those of the author(s) and do not reflect the policies of the U.S. Bureau of Labor Statistics or the

views of other staff members. The authors thank Dan Black, Marilyn Manser, Terra McKinnish, and participants in a session of

the 1998 Southern Economic Association meetings 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.
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: 202-606-7394 Phone: 202-606-7385

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 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 examines the appropriate functional form and the size of the wage

returns to training. In both the National Longitudinal Survey of Youth (NLSY) and

Employer Opportunity Pilot Project (EOPP) datasets a log specification fits best. In the

NLSY, the full effect of training occurs with a lag as long as two years, training on

previous jobs is a substitute for training on the current job, and the return to training

declines with labor market experience. The EOPP data indicate that formal and informal

training are perfect substitutes; however, an hour of formal training has a much greater

effect on wages than does an hour of informal training.

We find very large returns to formal training in both the NLSY and EOPP. 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

minimal. Heterogeneity in returns is a more plausible explanation of the high estimated

return to 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.
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.”

While there is a widespread agreement that training is positively associated with

wage growth, relatively little attention has been paid to the size of the effect.2 Researchers

have paid even less attention to the choice of the appropriate functional form. Indeed, the

variation in functional forms across studies makes it difficult to compare estimated rates of

return. 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 related Small Business

 

1

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

(1999), Barron, Black and Loewenstein (1989, 1993), Bartel (1995), Brown (1989), Lengermann (1999),

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

Pischke (1999) and Veum (1995).

2

 We are aware of two attempts to calculate rates of return to on-the-job training. 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

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

evidence that the return to training depreciates with time.
2

Administration (SBA) data have generally used log specifications (for example, see

Barron, Black, and Loewenstein 1989 and Barron, Black, and Berger 1999), 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).

In this paper, we use the NLSY and EOPP datasets to narrow the range of

plausible estimates of the return to on-the-job training by carefully investigating the choice

of the appropriate functional form. The NLSY and EOPP datasets complement each other

very well. The NLSY is a longitudinal survey of workers and provides the best

information on formal training at all levels of tenure. EOPP is an employer survey and

provides the best information on formal and informal training at the start of the job. Thus,

for example, the NLSY allows one to examine the time pattern of the return to training

and the effect of training in previous jobs on the return to training in the current job, while

the EOPP data allow one to examine the extent to which informal training is a substitute

for formal training.

Our key findings from the NLSY and EOPP are very similar. The results from

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

the amount of training received. In fact, the pattern of returns is accounted for very well

by a log specification. Our estimates from both datasets imply there are very high returns

to the initial interval of formal training, especially for workers with low levels of tenure

and experience. For example, the NLSY data show a wage return to the first 40 hours of

formal training as large as 8 percent for persons with low levels of tenure and experience
3

(in the EOPP data, this return is 6 percent), which is roughly the same magnitude as

estimated returns to a year of schooling.

The large size of the estimated returns to formal training raises the question as to

exactly what these estimates represent. As explained below, because of the mixed

continuous-discrete nature of formal training, measurement error can potentially lead to

the overestimation of the returns to short spells. We model the effect of measurement

error and find that under reasonable assumptions, the maximum upward bias in estimated

returns at the geometric mean of training is minimal.

Heterogeneity is a more plausible explanation of the high estimated return to

training. Indeed, the EOPP dataset provides direct evidence of heterogeneity in returns.

As discussed below, with unobserved heterogeneity in returns, our estimates can be

regarded as the (average) return to training for the trained. But despite our control for

heterogeneity in wage levels by means of a fixed effect, this return cannot be extrapolated

to the untrained.

It is interesting to contrast our results with the literature on the returns to

schooling. Some have suggested that the effects of measurement error on the one hand,

and ability bias and heterogeneity in returns on the other, roughly cancel each other out,

leaving OLS estimates as a good guide to the return to schooling for the average member

of the population (Ashenfelter and Zimmerman 1997, Ashenfelter and Rouse 1998). This

is clearly not the case for formal training—taking heterogeneity into account appears

essential to making sense of estimated returns.
4

The remainder of the paper is organized as follows. We analyze the NLSY data in

section II and the EOPP data in section III. We then turn to questions of interpretation in

section IV, and conclude in section V.

II. Analysis of the NLSY

Data

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

These youth have been interviewed annually since 1979, and the response rate has been 90

percent or greater in each year. We use data from the 1979 through 1994 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 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

 

3

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

year.

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

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

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.7 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 classification.

The two key training variables used in the empirical work to follow are total

 

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

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

the individual’s main job.
6

accumulated completed training on the current job and total accumulated completed

training on past jobs. These variables are obtained by adding the training a worker has

completed in the current year to the training he has received in all previous years.

Basic Results

Our basic specification is:

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

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 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 εijt is a mean

zero error term, homoscedastic and uncorrelated with Xijt and across jobs, persons and

periods. All specifications are run as fixed-effect regressions within jobs. The vector X

includes experience and experience squared, tenure and tenure squared, age, a dummy for

ever married, years of education (which occasionally changes within a job), year dummies,

and interactions of tenure with the following: age, female, AFQT,8 years of education,

ever married, union, two dummies for initial occupation in the job, and missing AFQT and

missing union. As additional controls for training, we include a count of spells with

missing training duration (most of these occur before 1988) and a dummy for training

ongoing at the time of the interview.

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

union. We also exclude observations with real wages below $1 or above $100 in 1982-84

 

8

 Specifically, the residual from a regression of AFQT on dummies for year of birth.
7

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). We also exclude the

military subsample, active members of the armed forces, the self-employed, those in farm

occupations, and observations where the respondent was enrolled in school at any time

between interviews. The resulting sample has 61,033 observations from 15,876 jobs.

Descriptive statistics are shown in table 1.

The results for different functional forms for training are shown in table 2. The top

panel shows results using the entire sample. We include an “incidence” specification that

counts total number of training spells. The “log” specification is ln(T+1), where T is

number of hours of training.9 The table shows adjusted R2s (explained variance as a

proportion of within-job variance) and total effects of training at the median number of

hours of training for those with positive training. The differences in fit appear slight.

However, the best-fitting specification—the log—increases adjusted R2 several times as

much as the worst-fitting specification—the linear—relative to the fit excluding training

variables. The quadratic specification and the incidence specification are little

improvement on the linear, while the square root specification is close to the log. The

quadratic specification has the advantage of allowing one to check for decreasing returns

since the estimated coefficient on the quadratic term provides a test of the null hypothesis

 

9

 Note that this is a special case of the more general specification f(T)=ln(sT+1). Non-linear least squares

yields an estimated value of s that equals .18, but is not significantly different from one. Note also that the

model E(y| X, T) = a0+Xβ + γf(T), f(T)= ln(sT+c), where c is a parameter to be estimated, is not identified,

so setting c equal to 1 is without loss of generality.
8

of constant or increasing returns to training. The null is rejected at the 5 percent level

(t=−1.95).

The implied effect of the median hours of training (or the median number of spells)

differs by more than a factor of 30 between the different specifications, with the log

specification showing the largest effect and the linear and quadratic specifications

apparently greatly understating the effect. The implied annualized rate of return of

training, assuming a work-year of 2000 hours, ranges from 3 percent to 92 percent.

One might suspect that the better fit of the log and square 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, and also the top one percent of the

distribution of number of spells. The total effect of the median amount of positive training

is similar in the sample without outliers to the total sample for all specifications with the

possible exception of the quadratic. The log specification is still the best fitting; there is

no marked improvement in the fit of the other specifications relative to the log. The t

statistic on the quadratic term increases in magnitude to –3.79, providing a stronger

rejection of the hypothesis of constant returns.

Our best fitting specification, the log, implies large initial returns to training that

decline steeply. 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 estimate two

more general functional forms. The first of these is a spline specification with transition

points at the 25th, 50th, and 75th percentiles of the positive training distribution.
9

Spline results for the entire sample are shown in table 3. The return to an hour of

training declines steeply from .00078 between the 0th and 25th percentiles to .00022

between the 50th and 75th percentiles to .00001 between the 75th and 100th percentiles.

The negative return between the 25th and 50th percentile is not statistically significant and

most likely due to sampling error. The results for the sample with outliers omitted are

similar, although in this case the return between the 75th and 100th percentiles is negative

but not statistically significant.

As a more flexible functional form, we estimate a Fourier series expansion

(Gallant, 1981). A Fourier series expansion of K terms of a function f(T) is:

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

K

j 1

= ∑ α1j + α

=

.

In practice, linear and quadratic terms are usually added. Moreover, for non-periodic

functions the variable T needs to be transformed to a variable T* such that 0 < T* < 2π.

The expansion is then implemented as:

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

K

j

j 2

1

1

2

= δ1 +δ2 +∑ α + α

=

.

The Fourier expansion has the property that the differences between both the level of the

true function and its derivatives, and the level of the Fourier expansion and its derivatives,

can be minimized to an arbitrary degree over the range of the function by choosing K to be
10

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

a local approximation (as in a Taylor series expansion).10

In our case, due to the essentially log-normal distribution of training,11 it is more

convenient to work with the log of training as a basis for the Fourier expansion rather than

training itself. Our transformation is T*=0.001+ln(T+1)/2, which has a range of .001 to

4.24 in our outlier omitted data. We added terms to the expansion until 2 R decreases; this

turned out to be K=4.

We use the statistic Q2≡ 1- 2

2

(f *(T*) 0)

(f(T) f *(T*))

∑

∑

−

−

 as a convenient summary

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

Fourier series f*(T*).12 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 2, the log

specification is closest to the Fourier series, and the linear specification is the furthest.

Indeed, the log specification explains about 90 percent of the squared distance between

 

10 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.

11 We estimated the Box-Cox transformation to normality − λ

λ

(T 1) , where λ=0 corresponds to a lognormal distribution and λ=1 to a normal distribution. For the positive training sample, λ is estimated to

be -.03.

12 We are grateful to Dan Black for suggesting this type of statistic.
11

the Fourier series and zero, while the linear specification only explains about 30 percent.

Figure 1 plots the estimated effect of training for the linear, log, quadratic, square

root, and Fourier series specifications in the sample without outliers. The effect is plotted

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

are concentrated. As indicated by the Q2 statistic, the figure shows that the log

specification fits the basic pattern of returns in the Fourier series expansion better than the

other functional forms over most of the range of the data.13 (Not surprisingly, the Fourier

series is fairly erratic, especially in ranges of the data near the maximum or minimum.)

One can see that the log specification is closer to the Fourier series than the other

specifications in an interval starting below the 25th and ending above the 75th percentile.

Why do the functional forms other than the log track the Fourier series so poorly,

even 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.14 (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

 

13 As indicated in table 2, the relative performance of the log specification is even better when we do not

drop outliers from the sample.

14 Specifically, the coefficient on training is given by 

∑

∑

−

− − ω

β =

2

2

f (T )) ˆ( f (T )

f (T ))(lnW ˆ )

ˆ

( f (T )

ˆ

, where

f(T)

ˆ

 and ωˆ denote the predicted values of f(T) and ln W from regressions of f(T) and ln W on X and

the fixed effects. Note that 2

ˆb is a weighted sum of the (lnW − ωˆ ) observations, with the absolute

value of the weights proportional to the absolute value of f (T )

ˆ

f (T )− .
12

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.

As a final specification test, we added squared and cubed terms (i.e., ln(T)2 and

ln(T)3) to the log specification. These terms were not significant (p=.35). We accordingly

adopt the log specification for the remainder of this section.

The Timing of the Returns to Training

The above specifications implicitly assume that all of the wage returns to training

occur shortly after training is completed. We now examine specifications with lagged

training to determine the time pattern of the return to training. Table 4 shows returns to

log training for training lagged up to three years, with lagged training coded as zero if the

respondent was not with the current employer during the relevant period. As can be seen,

lagged training has sizable and statistically significant effects on wages, but the lag does

not extend beyond two years. Failure to take into account lagged effects results in a

sizable understatement of the total returns to training. Summing the training coefficients

for the specification with training lagged two years (or three years), the total return to a

log hour of training is .012, compared to .008 without lags. At the median hours of

training, this corresponds to an annualized return of 137 percent.

Substitutability with Other Forms of On-the-Job Training.

One might expect training with a previous employer to be a substitute for training
13

with the current employer. More broadly, one might expect tenure on the current job, or

labor market experience in general, to be a substitute for formal training. Here we

examine interactions of log training on the current job with other forms of on-the-job

human capital accumulation.

In column 1 of table 5, we interact (log) training on the current job (lagged twice)

with log training on previous jobs.15 We sum coefficients over all lags to obtain both the

total wage effect of training on the current job and the interaction effect of training on

previous jobs. We find a small but statistically significant negative interaction, implying

that previous job training and current job training are substitutes. The effect on the

coefficients for current job training is relatively slight. The implied effect of training for a

respondent with no previous job training is about 10 percent higher than was found in

table 4.

We interact training on the current job with tenure and tenure squared in column 2.

There is a substantial interaction effect. The effect of a one-unit increase in log training

declines at the rate of .002 log points per year of tenure at zero tenure (calculated as the

sum of the coefficients on the interactions of linear tenure with log training on the current

job); a one-unit increase in log training now increases log wages by .022 at zero tenure

and zero training on previous jobs. In column 3, we further interact training on the

current job with experience and experience squared. We find that training is a substitute

for early career experience. The interaction with (linear) tenure is cut in half and is no

longer significant, while the effect of a one-unit increase in log training declines at the rate
14

of .002 log points per year of experience at zero experience.16 The increase in log wages

from a one-unit increase in log training with zero experience (and hence zero tenure) and

no previous job training is estimated as .029, more than twice the return to log training

estimated in table 4, although it should be noted that there are few observations with zero

experience.17,18

To make clearer the effect of the various forms of human capital acquisition on the

return to training, table 6 shows estimated returns to 40 hours of training for 12

hypothetical respondents with various values of the relevant characteristics. We assign

either zero or the median positive value (78.9) of training on the current job, and similarly

for training on previous jobs (here the median positive value is 200 hours). We assign the

25th, 50th, and 75th percentile values of tenure and experience, varying both together (that

is, we assign a respondent with the 25th percentile value of experience the 25th percentile

value of tenure).

As one might expect, training on the current job is the primary determinant of the

return to further training. The return to training declines by a factor of nine as one moves

 

15 Note that the noninteracted duration of training at previous employers is absorbed into the fixed effect.

Work by Loewenstein and Spletzer (1998,1999b) and Lengermann (1999) indicates that training received

at a previous employer results in a higher wage at the current employer.

16 The declines in the return to training with tenure and experience are non-linear. In results not shown,

interacting a linear tenure term with training without also interacting tenure squared shows little effect of

tenure on the return to training, and similarly for experience.

17 While observations where the respondent is enrolled in school are excluded from the wage regression,

we do not exclude experience gained while enrolled in school from our experience measure.

18 The change in the NLSY training sequence does not appear to have influenced our estimation results.

In several of the above specifications, including the specification in column 3 of table 5, we have

interacted the training variables with an indicator for year before 1987. These interactions are never

either singly or jointly statistically significant at conventional levels.
15

from zero to the median positive value; the annualized rates of return to 40 hours of

training for those with no training on the current job are in the 200-400 percent range,

compared to 20 to 45 percent for those with the median positive value. The effects of

training in previous jobs, tenure, and experience on the return to further training are

smaller, but still substantial. As one simultaneously goes from the 25th to the 75th

percentile of tenure and experience and from zero to the median positive value of training

on previous jobs, the return to training is cut by more than half.

III. Analysis of the EOPP

Data

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

the quantity of such training. While the NLSY has the important advantage of being a

longitudinal survey, making it possible to keep track of the training that individuals receive

over their working life, the NLSY does not contain information on informal training that is

useful for our present purposes.19 To look at the nature of the return to informal training,

we now turn to the Employer Opportunity Pilot Project (EOPP) survey. Unlike the

NLSY, EOPP is not a longitudinal survey, and it only contains information on

 

19 The NLSY began asking detailed questions about informal on-the-job training in the 1993 survey.

After completing the formal training questions, an individual in the NLSY is asked whether he had to

learn new job skills in the past 12 months because of some change at work. The ensuing sequence of

"informal training" questions in the 1993 NLSY is designed to measure training that was not already

recorded in the preceding sequence of formal training questions. However, as noted by Loewenstein and

Spletzer (1999a), the routing patterns in the informal training questions limit their use as a source of

information regarding on-the-job skill acquisition: individuals who did not experience changes at work

within the past 12 months are not routed into the detailed training questions. The informal training

questions have been revamped in the most recent survey.
16

training at the start of the job. However, EOPP has the advantage of providing good

measures of both formal and informal training.20 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. Our previous analysis of the

NLSY indicates that the pattern of wage returns to training is accounted for very well by a

log specification. We would like to know whether the same is true for informal training.

An additional question of interest is the precise nature of the substitution between informal

and formal training.

EOPP is not a longitudinal study, so it is not possible to estimate a true fixed-effect

wage equation. However, 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 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.

 

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

(1989).
17

Besides the training variables, we include the following explanatory variables in all of our

estimated equations: the most recently hired worker’s age, number of years of education,

gender, a dummy variable indicating whether or not the most recently hired worker

belonged to a union, the logarithm of the number of employees at the establishment, and

two occupational dummies. Also included as control variables are 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 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.

We exclude observations with missing values for any of the variables. The

resulting sample has 1,550 observations. Sample means are reported in table 7. Clearly,

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 134 hours, mean formal training for those with any formal training is

only 75 hours. Thus, not only are workers much more likely to receive informal training

than formal training, but informal spells last longer.

Results

Following the previous literature (Barron, Black, and Loewenstein 1989, for

example), we first look at specifications in which formal and informal training are simply

summed to obtain total training. Note that this specification makes two implicit

assumptions. First, it presumes that formal and informal training are perfect substitutes.
18

Second, it presumes that the rate of substitution is one-to-one since an hour of informal

training has exactly the same effect on wages as does an hour of formal training. An

equivalent way of expressing the latter idea is that formal and informal training have equal

weights in constructing the aggregate “total training”. In our first set of equations, log

wage growth is regressed against total training and other control variables. The results for

different functional forms are shown in table 8. Table 8 mirrors table 2 and shows

adjusted R2s and total effects of training at the median number of hours of training. The

top panel shows results using the entire sample, while the bottom panel reports the

estimation results when one omits observations in the top 1 percent of the training

distribution. As with the NLSY equations, the differences in fit are slight since all

specifications capture the tendency for wages to increase with the stock of training.

Nevertheless, the EOPP results, like the NLSY results, provide strong indication of

diminishing returns to training; the linear specification is clearly the worst-fitting.

The estimated return to training in EOPP is remarkably close to the estimated

return in the NLSY. Recall (from column 2 of table 5) that the total return (allowing for

lags as long as two years) to a log hour of formal training when tenure is zero is .022 in

the NLSY. If one does not control for informal training in EOPP, one obtains a

coefficient of .018 on log hours of formal training.21 (As reported in column 2 of table 9,

this coefficient falls to .016 when one adds log hours of informal training to the equation.)

Unlike the case with the NLSY, the choice of the best-fitting functional form is not

so clear in EOPP: the square root specification yields a higher adjusted R2 than the log

 

21 Loewenstein and Spletzer (1999a), using a slightly different EOPP sample, report a coefficient of .021

on log hours of formal training when informal training is not controlled for.
19

specification, which in turn yields a higher adjusted R2 than the quadratic. This order is

reversed when one omits training outliers. Before attempting to pass judgment on the

appropriate functional form, we examine more carefully the assumptions that formal and

informal training are perfect substitutes and that an hour of formal training is equivalent to

an hour of informal training, assumptions that are implicit in all of the specifications in

table 8. The most natural place to begin investigating this question is with the log

specification.

For convenience, the specification using the logarithm of total training is reported

again in column 1 of table 9. Column 2 of table 9 shows what happens when one includes

the logarithm of formal training and informal training as separate arguments. Note that

the second estimation yields a better fit, as measured by the adjusted R2. As can be seen in

column 3, adding an interaction term between formal and informal training does not yield

a further improvement in fit.

The equations in columns 1 and 2 can be viewed as special cases of a more general

specification of the form,

(2a) ln(Wit) – ln(Wi0) = ln(ψ(Tformal,Tinformal)) + Xiβ + εit,

(2b) ψ(Tformal,Tinformal) = A(b(Tformal+1)-ρ+ (1-b) (Tinformal+1)-ρ)

-k/ρ

,

where Wi0 and Wit are worker i’s starting wage and post-training wage, respectively, and

Tformal and Tinformal are the quantities of formal and informal training. Note that the function

ψ(⋅,⋅) is a generalized constant elasticity of substitution production function.22 The

 

22As with the simple log specification, we have modified the standard “constant elasticity of substitution”

specification to ensure that the wage function is defined when formal and informal training are both zero.

In the standard constant returns to scale specification, k = 1. Our more general specification allows for
20

parameter ρ is closely related to the elasticity of substitution, σ: σ = 1/(1+ρ). When ρ is

positive and large, σ is small and informal training cannot be easily substituted for formal

training. In the limit when ρ → 0, the production function becomes generalized CobbDouglas, or ψ(Tformal,Tinformal) = A((Tformal)

b

(Tinformal)

1-b)k

. Finally, when ρ → -1, isoquants

become linear. In this case, formal and informal training are perfect substitutes and can be

combined to form a single aggregate; the parameter b indicates the relative weights that

should be placed on formal and informal training when they are aggregated into total

training. The two types of training are weighted equally only when b = .5.

We estimated equation (2) with nonlinear least squares. The estimation results are

summarized in column 4 of table 9. If we do not restrict the parameter ρ to the

economically sensible range, the unconstrained nonlinear least squares estimate of ρ is –

2.97; this parameter is estimated very imprecisely, with a standard error of 9.38. A value

of ρ less than – 1 can be ruled out on economic grounds since it implies that isoquants are

not convex to the origin. When one imposes the corner restriction that ρ = – 1, the

adjusted R2 increases to .0952: a test of the restriction based on the difference in the sum

of squared residuals yields a chi-squared statistic with a p-value of .541.23 The estimate of

b in the restricted specification is .95. The EOPP data thus indicate that formal and

informal training are very good substitutes--indeed, our constrained point estimate is that

they are perfect substitutes--and that an hour of formal training should be counted as being

 

increasing (k > 1) and decreasing returns (k < 1). As is true with the NLSY data, the EOPP data strongly

imply diminishing returns.
21

worth 19 hours of informal training.24 These results are not driven by outliers. When one

omits observations in the top 1 percent of the training distribution, the estimated values of

ρ and b are both unchanged.

We estimated linear, quadratic, and square root specifications treating formal and

informal training as perfect substitutes and compared these specifications with ones

analogous to columns 2 and 3 of table 9. For each functional form, the best fitting

specification is that in which formal and informal training are treated as perfect substitutes.

As reported in table 10, the estimated weight on an hour of formal training is always much

higher than that on an hour of informal training. As was the case with the NLSY, the log

specification does somewhat better than the square root specification, which has an

adjusted R2 of 0.0930. The log specification also does significantly better than the

quadratic.

Unlike the NLSY, the estimated return to training in EOPP does not appear to be

affected by experience. When one interacts relevant experience and relevant experience

squared with aggregate training, there is little change in the coefficient on aggregate

training and the interaction terms are insignificant.

The various specifications are graphed in figures 2 and 3 along with the estimated

fourth order Fourier series (as with the NLSY, the length of the expansion that maximizes

adjusted R2; interestingly, when one does not omit outliers, the log specification actually

 

23 Imposing the restriction that ρ = -1, 2b) reduces to ψ(Tformal, Tinformal)=A (bTformal + (1-b)Tinformal +1)k

.

This is a special case of the more general specification ψ(Tformal, Tinformal)=A (s (bTformal + (1-b)Tinformal)

+1)k. The non-linear least squares point estimate of s equals 1.28, but is very imprecise.

24 Note, however, that the estimated wage elasticity with respect to formal training (.26) is only a little

higher than the estimated wage elasticity with respect to informal training (.15).
22

has a higher adjusted R2 than the Fourier series). The weights assigned to formal and

informal training are allowed to vary among all the functional forms. Figure 2 portrays the

estimated return to formal training when informal training is assigned its median value for

the subsample that receives positive formal training. Similarly, figure 3 depicts the

estimated return to informal training for those with the median value of formal training

(zero). When one confines one’s attention to the ranges of the graphs where most of the

data are located, the linear specification is again clearly the worst-fitting while the log

appears to be the best-fitting. The square root and quadratic specifications perform

similarly for formal training. In the case of informal training, the square root does about

as well as the log and considerably better than the quadratic.

In summary, the EOPP data, like the NLSY data, indicate that the return to

training diminishes sharply with training. However, a comparison of figures 1, 2 and 3

reveals that the differences in fit between the linear and diminishing return specifications

are not as great as in the NLSY. The same conclusion emerges when one examines the

increase in adjusted R2 relative to the fit excluding training variables and when one looks

at the proportion of the squared distance between the Fourier series and zero explained by

the various specifications (reported in column 3 of table 10). In the EOPP data, the effect

of the median hours of formal training on wages is higher by a factor of three for the log

specification than for the linear. This is a substantial difference, but smaller than the factor

of 30 that we obtained from the NLSY data.

 
23

As noted above, the EOPP data indicate that formal and informal training are

sufficiently close substitutes that they can meaningfully be summed to form a measure of

aggregate training. In forming this sum, formal training receives a much higher weight

than informal training, implying that an hour of formal training has a significantly greater

wage effect than an hour of informal training. When one calculates the estimated effect of

median training on a worker’s wage, one finds that the lower marginal effect of informal

training is in large part offset by the fact that workers receive much more informal training

than formal training. Specifically, adopting the log specification, the median hours of

formal training increase a worker’s wage by 6.2 percent, while the median hours of

informal training raise the wage by 3.8 percent. The implied annualized rates of return of

training are 308 percent and 103 percent, respectively.

IV. Further Discussion and Interpretation of the Key Findings

Under the best fitting specification, the estimated returns to formal training in the

NLSY are very large. The results in table 6 show returns to the first 40 hours of formal

training as large as 8 percent for persons with low levels of tenure and experience, roughly

the same magnitude as estimated returns to a year of schooling. Marginal returns at the

median amount of formal training are an order of magnitude smaller but still quite large;

the smallest number in table 6 still implies an annualized rate of return that exceeds 20

percent. Similarly in the EOPP data, the estimated high weight of formal training in our

training aggregate implies a very high return to formal training. The first 40 hours of

formal training, for a worker with the median hours of informal training, increase wages by

6.2 percent.
24

Only a minority of respondents in both the NLSY and EOPP have any formal

training. This is puzzling in view of the high estimated returns. Estimated wage gains

should be, if anything, less than gains in productivity.25 Taking the results literally, it

would appear that potentially profitable investments in training are not being made. One

possibility is that the high returns for short spells are an artifact of measurement error;

another is that they reflect heterogeneity in returns to training. We discuss each in turn.

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.

This mixed continuous-discrete structure implies that, as explained below, estimates of the

effect of short spells of training may well 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

 

25 In the classic theory of human capital (Becker 1975), workers incur all the costs and realize all the

returns to general human capital, but share the costs and returns to specific human capital with employers.

Wage growth should therefore equal productivity growth when training is general but be less than

productivity growth when training is specific. Loewenstein and Spletzer (1998) argue and provide

evidence that contract enforcement considerations can lead to employers sharing the returns and costs to

even purely general training. Also see Acemoglu and Pischke (1999). Using subjective measures of

productivity in the EOPP and SBA data, Barron, Berger, and Black (1997b) find that the productivity

growth associated with training is several times the wage growth.
25

g¢¢ ≤ 0. Since we do not observe true training, our reduced form estimation does not yield

the function g, but instead yields 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; we do not consider distortions caused by

incorrect functional forms.)

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

training and error in the duration of spells that are classified correctly. Misclassification 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 4. 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.
26

To formalize this intuition and to bound the effects of measurement error, we need

to put further structure on the problem. For convenience, we disregard classification

error. We model duration error x as multiplicative, so that T=T*x for T* > 0. Let

u = ln(x) = ln(T) − ln(T*) denote the measurement error in (non-misclassified) log

training. We assume that u is distributed independently of ln(T*) and that E(u) = 0.

Letting n(⋅) denote the density for ln(T*) and f(⋅) denote the density for log measurement

error u, the conditional density for ln(T*) given observed (non-misclassified) training T=

T0 is given by

(3) η( ln(T*),T0) = 

∫ −

−

∞

0

0

0

(ln(T ) ) ( )

(ln( ) ln( *)) (ln( *))

x x dx

T T T

f n

f n

 .

The densities n(⋅) and f(⋅) are assumed to be unimodal and symmetric. We should note

that our measurement error assumptions are consistent with our data in that the training

distributions in both the NLSY and EOPP are both approximately log-normal.26 In

addition, 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.

Let h(⋅) be the function implicitly defined by h(ln(T)) = g(T). Taking a secondorder Taylor expansion of h(⋅) around ln(T0), the expected return to training for an

 

26 Estimating the Box-Cox transformation to normality − λ

λ

(T 1) 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 −.03 in the NLSY. Quantile plots also show that log-normality is a good approximation

in both datasets.
27

individual with observed non-misclassified training T = T0 can be expressed as

 (4) E(ln W|T= T0) = g(T0) + h'(ln(T0))E(ln(T*|T= T0)- ln(T0))

 + (1/2) h''(ln(T0))E((ln(T*)- ln(T0))2|T= T0) + R(T0),

where R(T0) is the error in the approximation.27

Note that when observed training T0 is low, measurement error is more likely to be

negative than positive, which means that the expected value of true training will exceed T0.

Conversely, the expected value of true training will be less than observed training if

observed training is high. In fact, let m* ≡ exp(E(ln(T*))) denote the geometric mean of

the training distribution. Then under our assumption that the densities n(⋅) and f(⋅) are

unimodal and symmetric, one can show that

(5) E(ln(T*)|T= T0) 

>

=

<

 ln(T0) as T0 

<

=

>

m* .

Consequently, for observed training at the geometric mean, (4) reduces to

 (6) E(ln W|T= m*) = g(m*) + (1/2) h''(ln(m*)) 2sln(T*)|T =m* + R(m*),

where 2sln(T*)|T =m* is the conditional variance of the log of true training when T = m*.

 

27 Note that

(a) E(ln W|T= T0) = h(ln(x) (ln(x),T )dx 0

0

∫ h

∞

.

Taking a second-order Taylor expansion around ln(T0), one can write

(b) h(ln(x)) = h(ln(T0)) + h'(ln(T0))(ln(x) - ln(T0)) + (1/2)h''(ln(T0))( ln(x) - ln(T0))2+ r(x),

where the term r(x) denotes an error term. Substituting (b) into (a) yields (4), where R(T0) =E(r(x)|T=T0)

≡ r( x ) (ln( x ),T )dx.

0

0

η ∫

∞
28

 If the return to training g(⋅) is linear in log training, then the function h(⋅) is linear

and h''(T0) = R(T0) = 0. It thus follows immediately from (6) that E(ln W|T = m*) =

g(m*), i.e., the estimated return to training at the geometric mean should equal the true

return. In the NLSY data, the implied annualized rate of return at the geometric mean of

88 hours is 85 percent for our simplest log specification, and 126 percent when one

allows for lags. In the EOPP data, the annualized rate of return at the geometric mean of

37 hours is an even higher 330 percent. (The higher rate of return in EOPP can be

attributed to the fact that the workers in the survey are just starting their current

employment relationship and consequently have lower accumulated training on the

current job.28) Clearly, measurement error cannot explain the high estimated return to

training at the geometric mean if the return to training is log-linear.

If the true return to training declines at a greater than logarithmic rate, then h(⋅) is

concave and (1/2) h''(m*)

2

sln(T*)|T =m*+ R(m*) < 0. In this case, it follows from (6) that

E(ln W|T= m*) < g(m*): the estimated return to observed training at (or not too far below)

the geometric mean will be less than the true return. Thus, measurement error can also

not explain the high estimated return to training if the true return to training declines at a

greater than logarithmic rate.

Finally, if the true return to training declines at a slower than logarithmic rate, then

h(⋅) is convex and the estimated return to non-misclassified observed training at (or not

too far above) the geometric mean will exceed the true return. To estimate the potential

 

28 Decreasing returns implies that the rate of return is inversely related to spell length. As discussed

above, the return to training is inversely related to job tenure.
29

upward bias, let the return to true training be given by g(T*) = c(T*)a, 0< a < 1, which

means that h(x) = c(exp(x))a and

(7) h''(m*) = α

2

h(m*).

In addition, suppose that measurement error and true training are both distributed

lognormally. This implies that observed training is also distributed lognormally, which as

noted above is a good approximation with our data. Given lognormality, one can show

that29

(8) 2sln(T*)|T =m* ≤

2

ln( )

(1/ 4)s T.

Furthermore, calculations reveal that the proportional error term, R(m*)/ g(m*) is

negligible. It therefore follows from (6) and (8) that the maximum possible proportional

upward bias in the estimated return to training at the geometric mean is

2

ln( )

(1/ 8)s T h''(ln(m*))/h(ln(m*)) = 2

ln( )

(1/ 8)s T a

2

. The variance of observed log training in

the NLSY is 2.41. In EOPP, the variance of observed log formal training is 1.54. Thus, if

a = 1/2, the proportional bias in the estimated return to geometric mean training is less

than 7.5% in the NLSY and 4.9% in EOPP. Even if the return to training is linear in

 

29 Letting 2

s ln(T*) , 

2

sln(T ), and 2 s u denote the unconditional variances of ln(T*), ln(T), and u, the

coefficient of linear correlation between ln(T*) and ln(T) is given by

r = 

2 2

2 2

ln(T*) u

ln(T*)

ln(T*) ln(T) ln(T*) ln(T) ln(T*) ln(T)

cov(ln(T*),ln(T)) E(ln(T*)ln(T)) E(ln(T*))E(ln(T)) E(ln(T*) ) E(ln(T*))

s s

s

s s s s s s +

=

−

=

−

= .

If ln(T*) and ln(T) are normal, the conditional variance of ln(T*) is given by

(a) var(ln(T*)|ln(T0)) = (1-r

2

)

2

ln(T*) s = 

2

ln( )

2 2

ln( )

2

2

ln( *)

2

2

ln( *)

2

( )

T

u T u

u T

u T

s

s s s

s s

s s −

=

+

.

Partially differentiating (a) with respect to 2s u, one finds that holding 2

sln(T ) constant, var(ln(T*)|ln(T0)
30

training, i.e., a = 1, the proportional bias in the estimated return to geometric mean

training is about 30% in the NLSY and 20% in EOPP. As discussed above,

misclassification error will reduce these amounts. Thus, even if the true return to training

declines at a slower than logarithmic rate, measurement error does not appear to explain

the high estimated returns to training

If one drops that part of the sample that reports not having received training and

estimates the returns to training only for the subsample reporting positive training, our

measurement error problem reduces to the standard one and the estimated return to

training should be biased downward. When we estimate returns to training in the NLSY

using only the subsample reporting positive training, the log functional form is still the best

fitting (but the square root is close) and the returns to training increase slightly. Similar

results obtain in EOPP. These results need to be interpreted cautiously because the

parameter estimates are not very precise, but they suggest that either misclassification is

important in tempering the upward bias for short spells in the full sample or that

measurement error may not be such a serious problem after all.30

 

is maximized when 2s u = (1/2) 2

sln(T ). Substituting into (a), one obtains the result that var(ln(T*)|ln(T0))

≤

2

ln( )

(1/ 4)s T.

30 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 would imply,

among other things, 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 activities are misclassified as training, described above as “false training”.

Accounting for this would complicate our analysis considerably. The EOPP, with a single formal training

item, is not subject to this problem. We conclude from the similarity of the results between the two

datasets that this complication is probably not a major concern.
31

In conclusion, even if one allows for the maximum possible measurement error

bias, the estimated return to training is still very high. Either the required rate of return to

training investments is very high, or our results cannot be taken as a guide to the return to

formal training for the untrained.

Heterogeneity

There are reasons why investments in training may have a high required rate of

return. If the skills involved are firm-specific, the time horizon over which the returns to

training are realized is the worker’s tenure with the firm. This may be much shorter than

the working life over which investments in schooling are realized.31 Formal training may

also have a high direct cost, especially if specialized skills are involved. However, we

think it improbable that these factors completely explain our results.

If a week of formal training raises productivity by 8 percent, a worker need only be

with a firm for 13 weeks for a training investment to break even. We would expect direct

costs to be a major reason for a low training incidence in small firms, which cannot take

advantage of economies of scale in training. However, while the incidence of training

increases with establishment size, in our NLSY data slightly less than 50 percent of

respondents from establishments with over 1,000 employees ever receive any formal

training during the time they are in our wage sample, and similarly for respondents from

establishments with over 2,000 or over 5,000 employees.

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
32

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

differ across jobs. The EOPP dataset provides direct evidence of heterogeneity in returns.

Recall that one of the control variables in our wage growth regression is 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. Barron, Berger, and Black (1999) suggest that this variable

provides a measure of job complexity. Consistent with this interpretation, the log number

of weeks until fully qualified is positively related to wage growth. The last column in table

9 reports the effects of interacting hours of aggregate training with the number of weeks it

takes a new employee without previous experience to become fully qualified. The

coefficient on the interacted variable is positive and quite large. And adding the interacted

variable to the wage growth equation reduces the coefficients on non-interacted aggregate

hours of training and non-interacted weeks until fully qualified (not reported in the table)

by factors of three and nine, respectively; in fact, the coefficients on both non-interacted

variables are no longer significantly different from zero. These results provide strong

evidence that the return to training varies greatly across jobs.

In addition to heterogeneity in returns to training, heterogeneity in wage growth

may also affect our results. 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

have modified the NLSY wage equation to include interactions of tenure and tenure

 

31 Interestingly, Loewenstein and Spletzer (1999b) provide evidence that a substantial portion of on-thejob training is general. Nevertheless, Royalty (1996) does find empirical support for the proposition that
33

squared with an individual’s final observed training in the current job. If workers with

higher wage growth self-select into training, then the coefficient on the tenure - terminal

training interaction term should be positive and the coefficients on current and lagged

training should fall.32 However, this turns out not to be the case. When added to the

specification in column 3 of table 4, the tenure- terminal training interaction terms are not

significant and have little effect on the key training coefficients. We conclude that for our

data, heterogeneity in wage growth has little effect on the estimated return to training;

heterogeneity in returns to training appears to be far more important.

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 wage levels by means

of the fixed effect. To see this, consider the follow simplified wage model,

(9) ln Wit = ai + βig(Tit) + eit ,

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

we now disregard measurement error and again abstract from other covariates in the wage

equation.

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

indicates that training is higher for more productive workers,33 presumably because their

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

 

training is less frequent on high-turnover jobs.

32 Note the similarity between our use of terminal training and Abraham and Farber’s (1987) use of

completed tenure to control for unobserved heterogeneity in estimating the effect of tenure on wages.

33 For example, see Barron, Berger, and Black (1999).
34

lower for more able individuals in more productive jobs, that is, if cov(αi,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. We noted above that the return to

training is higher for individuals who are in jobs that require more time to be fully

qualified. Loewenstein and Spletzer (1999) demonstrate that hours of aggregate training

are strongly positively correlated with the number of weeks it takes a worker without

experience to be fully qualified.

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

(first differences) is given by:

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

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

 = E(βi| Ti2= T0)g(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 b g(T0), but, as is clear from (10), consistently
35

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

34

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(bi|T=T0)g¢(T0) + 

T

i T T

∂

∂E( | = ) b 0

g(T0),

and which will exceed E(bi|T=T0)g¢(T0) if 

T

i T T

∂

∂E( | = ) b 0

 > 0: estimation of g¢ is

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

training can be expected to have a higher return.

To summarize the effects of measurement error and heterogeneity in returns to

training on estimated returns: Under reasonable assumptions, average returns at or above

the geometric mean of training are underestimated or at worst slightly overestimated due

 

34 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 (see fn. 30), 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

to measurement error. Heterogeneity in returns does not affect this conclusion as long as

estimated returns to a given amount of training are interpreted as the average return for

those with that amount of training. Measurement error in duration will cause marginal

returns to training to be underestimated,35 while heterogeneity in returns will likely cause

marginal returns to be overestimated (a situation similar to that found in the literature on

returns to schooling; see, for example, Ashenfelter and Rouse 1998). We conclude that

those with training around the geometric mean of the NLSY (EOPP) do have annualized

returns to training of at least 100 (300) percent, but that these returns cannot be

extrapolated to the untrained.

V. Conclusion

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

wage returns to training and their magnitude. We find that the logarithmic form appears

to fit best. In the EOPP dataset, which contains extensive data on both formal and

informal training, we find that formal and informal training are perfect substitutes, but an

hour of formal training has a much larger effect on wages than an hour of informal

training.

In both the NLSY and EOPP datasets, we find very large returns to formal training

when we use the best-fitting functional form. These rates of return are an order of

magnitude higher than those implied by the frequently-used linear specification,

 

35 False training may cause marginal returns to be overestimated in ranges where the frequency of false

training is declining. We regard this as likely to be of small importance in practice.
37

showing that choice of functional form has a large effect. Given returns of this magnitude,

it is puzzling that only a minority of each sample receives formal training. It is very

unlikely that the high return to training can be due to measurement error. Heterogeneity

of returns is a more compelling explanation of the puzzle. We find direct evidence in the

EOPP dataset that returns to training increase with job complexity. With heterogeneity in

returns, our results cannot be considered structural estimates in the sense of showing the

return to training for an average member of the population. Neither 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

estimates can be interpreted as the average 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 suggest 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).36 We leave more complete analysis of heterogeneity in returns to training as a

topic for future research.

 

36 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.
38

While structural interpretation is difficult due to heterogeneity and measurement

error, the size of the returns combined with the short length of most training spells creates

a strong presumption that returns must be steeply declining. The returns of 6-8 percent

for an initial week of training early in the job match that we find are almost certainly much

less in succeeding weeks. One might speculate that the brief but powerful training spells

we observe reflect training that is oriented toward teaching how to perform specific jobrelated tasks rather than broader human capital development. For example, teaching a

computer programmer a new programming language may greatly increase his or her

productivity in a given job, but training much beyond the basics of the language may have

quite small effects. Broader development of programming skills may be left to prior

schooling or to job experience. Stepping back from structural interpretation, the

logarithmic specification for the returns to formal training stands in stark contrast to

specifications compatible with the returns to schooling, which are broadly linear and

exhibit increasing returns—“sheepskin effects”--at levels associated with degrees.
39

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42

Table 1

Descriptive Statistics, NLSY

Variable Mean Std. Dev. Min. Max.

Ln Wage 1.85 0.47 0.00 4.53

# train. spells, current job 0.60 1.33 0.00 21.00

Training Hours, Training > 0 351.59 1070.96 0.50 19200.00

Ln (Training + 1), Training > 0 4.51 1.55 0.41 9.86

Year=1980 0.02 0.14 0.00 1.00

Year=1981 0.03 0.17 0.00 1.00

Year=1982 0.04 0.21 0.00 1.00

Year=1983 0.05 0.22 0.00 1.00

Year=1984 0.06 0.24 0.00 1.00

Year=1985 0.07 0.25 0.00 1.00

Year=1986 0.07 0.26 0.00 1.00

Year=1987 0.08 0.26 0.00 1.00

Year=1988 0.08 0.27 0.00 1.00

Year=1989 0.09 0.28 0.00 1.00

Year=1990 0.09 0.28 0.00 1.00

Year=1991 0.08 0.27 0.00 1.00

Year=1992 0.08 0.27 0.00 1.00

Year=1993 0.08 0.27 0.00 1.00

Year=1994 0.06 0.24 0.00 1.00

Black 0.25 0.43 0.00 1.00

Hispanic 0.18 0.38 0.00 1.00

Age 27.62 4.19 16.00 37.83

Female 0.50 0.50 0.00 1.00

AFQT (residual) -0.06 20.16 -65.48 45.94

Years education 12.62 2.21 0.00 20.00

Ever married 0.61 0.49 0.00 1.00

Union 0.19 0.39 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) 0.29 0.46 0.00 1.00

Missing AFQT 0.06 0.23 0.00 1.00

Missing Union 0.06 0.24 0.00 1.00

Any ongoing training 0.03 0.18 0.00 1.00

# spells missing hrs, current job 0.17 0.62 0.00 9.00

n 15,876

Obs. 61,033
43

Table 2

Returns to Training for Different Functional Forms, NLSY

Specification 2 R Fraction

Fourier

Series

Explained

Total

Effect at

Median

Complete Sample

No Training Vars. 0.1366 -- --

Linear 0.1371 .281 0.001

Quadratic 0.1371 .347 0.002

Square root 0.1378 .729 0.016

Log 0.1381 .906 0.036

# spells 0.1375 .559 0.009

Fourier series 0.1383 -- 0.031

n 15,876

Obs 61,033

Training Outliers Omitted*

No Training Vars. 0.1349 -- --

Linear 0.1354 .313 0.002

Quadratic 0.1356 .459 0.006

Square root 0.1361 .721 0.019

Log 0.1364 .896 0.037

# spells 0.1361 .610 0.012

Fourier series 0.1367 -- 0.032

n 15,847

Obs. 60,646

*Top 1% of training duration, number of spells omitted.
44

Table 3

Returns to Training in Spline Specification, NLSY

Interval Slope

(x 100)

Difference

with

previous

segment

Complete Sample

T≤ 40 0.078

(0.017)

40≤T≤78.9 -0.009 -0.086

(0.024) (0.037)

78.9≤T≤219.2 0.022 0.030

(0.007) (0.029)

219.2≤T 0.001 -0.021

(0.000) (0.007)

n 15,876

Obs 61,033

2 R 0.1383

Training Outliers Omitted#

T≤ 40 0.078

(0.017)

40≤T≤78.9 -0.011 -0.089

(0.024) (0.037)

78.9≤T≤219.2 0.026 0.037

(0.008) (0.029)

219.2≤T -0.000 -0.026

(0.001) (0.008)

n 15,847

Obs. 60,646

2 R 0.1366

#

Top 1% of training duration, number of spells omitted.
45

Table 4

Lagged Effects of Training on Log Wages, NLSY

1) 2) 3) 4)

Coefficient

Current Log Training 0.0083 0.0057 0.0058 0.0058

(0.0010) (0.0011) (0.0011) (0.0011)

Log Training Lagged 1 Year 0.0051 0.0036 0.0036

(0.0009) (0.0010) (0.0010)

Log Training Lagged 2 Years 0.0029 0.0031

(0.0010) (0.0012)

Log Training Lagged 3 Years -0.0003

(0.0012)

Total Effect of Training 0.0083 0.0109 0.0123 0.0122

(0.0010) (0.0011) (0.0012) (0.0013)

Table 5

Interactions of Training on Current Job with Other Variables, NLSY

1) 2) 3)

Coefficient

Sum* of Log Training Coefficients 0.0136 0.0221 0.0294

(0.0013)

 (0.0029) (0.0042)

Sum of Interactions with Previous Training -0.0007 -0.0007 -0.0008

(0.0003)

 (0.0003) (0.0004)

Sum of Interactions with Tenure at 0 Tenure -0.0021 -0.0010

 (Quadratic Interaction with Tenure) (0.0007) (0.0009)

Sum of Interactions with Exper. at 0 Exper. -0.0019

 (Quadratic Interaction with Experience) (0.0011)

*Over lags.
46

Table 6

Increase in Log Wages from 40 Hours of Training for Selected Values of

Characteristics, NLSY

Initial

Hours of

Training,

Current

Job

Hours of

Training

on

Previous

Jobs

Years

Tenure

Years

Experience

Increase in

Log Wages

from 40 Hours

of Training

1.19 4.54 0.0801

 0 2.60 7.44 0.0660

0 5.00 10.65 0.0521

1.19 4.54 0.0651

200 2.60 7.44 0.0510

5.00 10.65 0.0372

1.19 4.54 0.0088

 0 2.60 7.44 0.0072

5.00 10.65 0.0057

78.9

1.19 4.54 0.0071

200 2.60 7.44 0.0056

5.00 10.65 0.0041
47

Table 7

Descriptive Statistics, EOPP

Variable Mean Std. Dev. Min. Max.

Ln Wage Growth 0.19 0.21 -1.99 1.95

Formal training indicator 0.13 0.34 0.00 1.00

Informal training indicator 0.95 0.21 0.00 1.00

Hrs. formal tr., formal tr. > 0 75.14 103.76 1.00 640.00

Hrs. informal tr., informal tr. > 0 134.17 177.70 1.00 2070.0

Ln (formal tr. + 1), formal tr. > 0 3.61 1.24 0.69 6.46

Ln (informal tr. + 1), inf. tr. > 0 4.25 1.21 0.69 7.64

Ln # weeks until fully trained 2.23 1.24 0.00 6.033

Years relevant experience 2.33 4.45 0.00 40.00

Rel. experience squared 25.24 110.42 0.00 1600.00

Age 26.57 8.84 16.00 64.00

Years education 12.47 1.66 2.00 24.00

Union 0.11 0.31 0.00 1.00

Ln establishment size 2.89 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

Other white-collar 0.56 0.50 0.00 1.00

Obs. 1,550
48

Table 8

Returns to Aggregate Training for Different Functional Forms, EOPP

Specification 2 R Total

Effect at

Median

Complete Sample

No Training Vars. 0.0714 --

Linear 0.0833 0.009

Quadratic 0.0838 0.014

Square root 0.0877 0.039

Log 0.0864 0.081

Obs. 1,550

Training Outliers Omitted*

No Training Vars. 0.0694 --

Linear 0.0753 0.009

Quadratic 0.0830 0.031

Square root 0.0803 0.035

Log 0.0813 0.074

Obs. 1,533

*Top 1% of training duration omitted.
49

Table 9

Returns to Formal and Informal Training for Various Log Specifications, EOPP

Unweighted Sum CobbDouglasCobb-Douglas

Plus FormalInformal

Interaction

 CES CES Plus

Aggregate

Training -

Number of

Weeks until

Qualified

Interaction

Coefficient

Log Unweighted

Aggregate Training

 0.019

(0.004)

Log Formal Training 0.016

 (0.004)

0.021

 (0.017)

Log Informal Training 0.015

(0.004)

 0.015

 (0.004)

Log Formal Training x Log

Informal Training

-0.001

 (0.003)

Log Weighted Aggregate

Training

0.029

( (0.005)

0.010

(0.009)

Log Weighted Aggregate

Training x Log Number

Weeks Until Fully

Qualified

0.007

(0.003)

Weight on Formal

Training

 0.950

 (0.028)

0.938

(0.035)

Obs. 1,550 1,550 1,550 1,550 1,550

2 R 0.0864 0.0922 0.0917 0.0952 0.0969
50

Table 10

Returns to Formal and Informal Training for Different Functional Forms,

EOPP

Specification 2 R Fraction

Fourier

Series

Explained

Weight on

Formal

Training

Total Effect

of Formal

Training at

Median

Positive

Value

Total Effect

of Informal

Training at

Median

Positive

Value

Complete Sample

Linear 0.0861 0.597 0.80 0.015 0.007

Quadratic 0.0882 0.699 0.87 0.031 0.009

Square root 0.0930 0.905 0.89 0.039 0.031

Log 0.0952 0.896 0.95 0.062 0.038

Fourier Series 0.0945 -- 0.90 0.052 0.052

Obs. 1,550

Training Outliers Omitted*

Linear 0.0785 0.0576 0.84 0.014 0.006

Quadratic 0.0824 0.0710 0.89 0.039 0.014

Square root 0.0852 0.0851 0.91 0.034 0.028

Log 0.0883 0.0861 0.95 0.051 0.037

Fourier Series 0.0890 -- 0.91 0.055 0.050

Obs. 1,533

*Top 1% of (unweighted) training duration omitted.
51

Figure 1. Estimated Effect of Hours of Training on Ln Wages for Different Specifications--

NLSY

-0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

0 1 2 3 4 5 6 7

Ln(Training Hours+1)

Change in Ln Wages

Log Square Rt. Quadratic Linear Fourier Series

25th percentile 75th percentile
52

Figure 2. Estimated Effect of Formal Hours of Training on Ln Wages for Different 

Specifications--EOPP

-0.05

0

0.05

0.1

0.15

0.2

0 1 2 3 4 5

Ln(Formal Hours+1)

Change in Ln Wages

Log Square Rt Quadratic Linear fourier series

25th percentile 75th percentile
53

Figure 3. Estimated Effect of Informal Hours of Training on Ln Wages for Different 

Specifications--EOPP

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0 1 2 3 4 5 6

Ln(Informal Hours+1)

Change in Ln Wages

Log Square Rt Quadratic Linear fourier series

25th percentile 75th percentile
54

Figure 4

M

G

F

Hours of Training

Ln Wages