BLS WORKING PAPERS

U.S. DEPARTMENT OF LABOR

Bureau of Labor Statistics

 

OFFICE OF COMPENSATION AND

AND WORKING CONDITIONS

Belated Training: The Relationship Between Training, Tenure, and Wages

Mark A. Loewenstein and James R. Spletzer

Working Paper 296

September 1996

 

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. We thank Harley Frazis and seminar participants at Rutgers University, the Census Bureau's Center for

Economic Studies, and the NLS Summer Workshop for helpful comments.
Belated Training:

The Relationship Between Training, Tenure, and Wages

September 13, 1996

Mark A. Loewenstein James R. Spletzer

Bureau of Labor Statistics Bureau of Labor Statistics

2 Massachusetts Ave NE, Room 4130 2 Massachusetts Ave NE, Room 4945

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

Phone: 202-606-7385 Phone: 202-606-7393

E-Mail: Loewenstein_M@bls.gov E-Mail: Spletzer_J@bls.gov

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. We thank Harley Frazis and seminar

participants at Rutgers University, the Census Bureau's Center for Economic Studies, and the NLS

Summer Workshop for helpful comments.
Abstract

 The increasing and concave age earnings profile is one of the most commonly accepted facts in

economics. The human capital model attributes rising wage-tenure and wage-experience profiles to

investments in worker productivity. Therefore, an implicit assumption in the human capital model

is that training must be occurring throughout a worker's stay at an employer. However, the

empirical training literature has thus far either implicitly or explicitly assumed that training is

concentrated at the beginning of the employment relationship. Using data from the National

Longitudinal Survey of Youth, we find that training in the later years of tenure is the norm rather

than the exception. To eliminate the bias from belated information, we estimate a fixed effects

wage equation with a two stage procedure that uses previous training and fixed individual and

match-specific variables as instruments. While instrumental variables estimation is the standard

method of correcting for measurement error, instrumental variables estimates are not consistent

when the explanatory variable measured with error (in our case, training) can only take on a

limited range of values. Nevertheless, the relative magnitudes of the OLS and the IV training

coefficients provide information that enables us to correct for measurement error bias without any

need for external information on misclassification rates. Our results provide strong support for the

human capital model in that most wage growth beyond the first year of tenure is due to training.
I. Introduction

 Age-earnings profiles are virtually always increasing and concave. The most commonly

accepted explanation, first offered by Becker (1962) and Mincer (1962), stresses workers' human

capital investments while on the job. More recently, some economists have argued that the wage

returns to tenure and experience may reflect other factors. For example, a positive return to

experience can be explained by the fact that individuals who have been in the labor force longer are

more likely to have located good job matches.1 And Lazear (1981) has argued that a positive

return to tenure may reflect the fact that employers defer wage compensation in order to offer

workers incentives not to shirk.

 Only in recent years has information on explicit measures of on-the-job training been collected,

making possible direct tests of the human capital hypothesis. Studies using these data find support

for the human capital model's prediction that a worker's wage is positively related to past

investments in his training.2 Indeed, Brown (1989) reports that "within-firm wage growth is

mainly determined by contemporaneous productivity growth."

 If the human capital explanation of the positive wage-tenure relationship is correct, then

training should persist well after the employment relationship has begun. However, the literature

as a whole pays little attention to the actual timing of training investments. In fact, largely because

of data limitations, most studies seem to assume that training is concentrated at the start of the

employment relationship. One exception is a recent paper by Bartel (1995), who analyzes the

 

1

 Similarly, in a cross section, positively sloped wage tenure profiles can be explained by the fact

that good matches are more likely to survive. In fact, Abraham and Farber (1987) and Altonji and

Shakotko (1987) assert that once one controls for the ultimate duration of a job, tenure has no

additional effect on wages. However, Topel (1991) argues that these estimates are biased

downward and that tenure is positively related to wages even after one controls for the pure job

matching effect.

2

 These studies include Altonji and Spletzer (1991), Barron, Berger, and Black (1994a), Barron,

Black, and Loewenstein, (1989, 1993), Bartel (1995), Brown (1989), Duncan and Hoffman

(1978), Lillard and Tan (1992), Loewenstein and Spletzer (1994, 1996b), Lynch (1992), Mincer

(1988), Parsons (1989), Pergamit and Shack-Marquez (1987), and Veum (1995).
2

1986-90 personnel records of a large manufacturing company. Bartel finds that “although the

probability of receiving training and the amount of training are highest for the newly hired

employees, an important observation is that experienced employees at this company continue to

receive formal training. For example, in 1990, 47% of the individuals who were hired before 1980

received some formal training. This is a significant finding since it indicates that training is not

confined to new hires but is an ongoing process at this company.”

 There are in fact strong theoretical reasons why one might expect to see significant amounts of

training beyond the first year of employment. In a simple stationary world with linear costs, the

human capital model predicts that training will be concentrated at the start of the employment

relationship. Under these conditions, there is no reason for an employer to delay a training

investment beyond a worker's initial period of employment because this merely reduces the time

period over which returns will be realized. Of course, as pointed out some time ago by Ben-Porath

(1967) and Blinder and Weiss (1976), a convex cost function for training can cause training to be

spread out over time.

 Perhaps more importantly, the simple argument above implicitly presumes that the training cost

function is invariant over time and that the employer and worker both have full information about

the quality of their match at the beginning of the employment relationship. For example, it seems

quite likely that a worker may be better able to absorb expensive training after an initial period of

"learning by doing" in which he has been acclimatized to his job and work environment. If the cost

of training the worker is sufficiently lower after he has had some experience with the employer,

then it may pay the employer to defer some training until it is cheaper, even if this means foregoing

the returns to training during the early part of the employment relationship.

 Furthermore, as pointed out some time ago by Johnson (1978) and Jovanovic (1979),

information about the quality of a firm-worker match is often revealed only after a period of time.

An employer who delays training can target his training investments on his most promising
3

workers, perhaps concentrating on training only those workers who are targeted for promotion. In

addition, by deferring training, an employer and worker can significantly lower the probability of

making a costly investment in a bad match that is unlikely to last very long (either because the

worker turns out to have a low match-specific productivity or because he discovers that he does not

like the non-pecuniary attributes of the employer's job).

 In this paper, we use recent data from the National Longitudinal Surveys of Youth in order to

thoroughly analyze the relationship among training, tenure, and wages. Our incidence analysis in

section II indicates that a significant amount of recurring and delayed training occurs well beyond

the start of the employment relationship. In section III, we turn to the motivating question of the

human capital model's prediction that wage growth is caused by productivity increases and the

implication that wage growth in the later years of tenure is at least partly caused by belated spells

of training. In order to eliminate potential biases from individual fixed effects, match specific fixed

effects, and belated information, we estimate fixed effects wage equations using a two stage

procedure that corrects for the endogeneity of training in the wage growth equation. One of our

key results is that the bias due to measurement error in training incidence is quite large.

 As discussed in section IV, instrumental variable estimation in the presence of measurement

error is complicated by the fact that the classical errors in variable model is not applicable when

the explanatory variable that is measured with error can only take on a limited range of values. We

show that the relative magnitudes of the IV and the OLS training coefficients provide information

about the amount of measurement error. This enables us to correct for measurement error bias

without relying on external information on misclassification rates such as might be provided by a

validation study. After correcting for measurement error bias, we find that the returns to training

are quite large and much larger than the tenure coefficients that measure wage growth in the

absence of training. Similar to Brown (1989), these results provide strong support for the human

capital model.
4

II. The Relationship Between Training and Tenure

 In this section, we examine the relationship between training and tenure using data from the

National Longitudinal Survey of Youth (NLSY).3 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. The NLSY data contain detailed

information on wages, tenure, and training. We utilize data from the 1988 through 1991 NLSY

surveys because these data provide information on all training spells regardless of their duration.4

To ensure that we observe all training spells at a given employer, we restrict our sample to

individuals who start a new job within one year of the 1988 survey or later. This restriction leaves

us with between one and four years of data for individuals aged 23-34. Because we omit persons

with more than one year of tenure in 1988 who do not change jobs between 1988 and 1991, our

sample likely has an above average proportion of high turnover individuals. After eliminating

observations where an individual is not employed or where there are missing training data, we end

up with a sample made up of 6,145 individuals who in total contribute 15,743 person-year

observations. The NLSY training question is "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?" In our sample of 15,743 person-year observations, the average annual

training incidence is 12.18 percent.

 

3

 Lynch (1991, 1992) characterizes training incidence and determines the effects of training on

wages and turnover using the 1979-1983 NLSY data. Loewenstein and Spletzer (1996b) use the

1988-1991 NLSY data to analyze the division of the costs and returns to training. Veum (1995)

also analyzes the 1988-1991 NLSY data.

4

 The training section of the NLSY questionnaire was re-designed in 1988. Prior to 1988,

information was only obtained on training spells that lasted longer than one month. Data from

1988-1991 indicate that 64.5% of training spells are less than four weeks in duration. There were

no training questions in the 1987 survey. Data from 1992 and 1993 were not yet available at the

time we started this project.
5

 Table 1 provides information on the relationship between a worker's tenure and the probability

that he receives training in a given year. The integer value of tenure=1 in column 1 applies to those

persons with tenure between 1 week and 52 weeks, tenure=2 applies to those persons with tenure

between 53 and 104 weeks, and so forth. The statistics in column 2 are the sample sizes associated

with each year of tenure, and the statistics in column 3 are the probabilities that individuals with

the specified tenure received training on the current job in the previous year.

 The training incidence statistics appearing in table 1 indicate quite clearly that a substantial

amount of training occurs after the first year of employment. As can be seen in column 3, annual

training incidence nearly doubles from 9.89 percent in the first year of tenure to 17.21 percent in

the second year of tenure, and is then roughly constant around 18 percent in the third and fourth

years of tenure.5

 A question that comes immediately to mind after looking at table 1 is the extent to which the

training spells occurring after the first year of tenure are first time delayed training spells and the

extent to which they constitute recurring training spells by individuals who have already received

training in the past. Altonji and Spletzer (1991) find that previous training is associated with a

higher probability of present training, but their data does not enable them to determine whether or

not this previous training occurred in the current job. Lynch (1992) and Mincer (1988) show that

individuals who have received training in past jobs are more likely to receive training in their

present job, but they do not look at the relationship between the likelihood of current training and

previous training in the current job. To determine the empirical relationship between current

 

5

 Recall that we are classifying a person as having one year of tenure if he has been on his current

job between 1 and 52 weeks at the time of his interview. Our estimate of first year training may

therefore understate the total amount of training that takes place in the first 52 weeks of

employment. Although this alone might lead us to overstate the amount of training provided in the

second year of tenure, we are likewise misclassifying training in the latter part of the second year

of tenure as occurring in the third year, and so forth. Note too that while our estimate of first year

training may understate the total amount of training that takes place in the first 52 weeks of

employment, it is an overestimate of training that takes place at the very start of the match.
6

training and previous training in both the current job and in past jobs, we decompose the belated

training spells in our sample into those that are first time delayed spells and those that are recurring

spells.

 Before turning to the data, we should note that the relationship between the receipt of previous

training and the probability of current training is theoretically ambiguous, as there are two

conflicting effects. Diminishing returns to training by themselves would lead to an inverse relation

because they would imply that, other things the same, the expected gain to providing current

training is lower for individuals who have had previous training than for individuals who have not

had previous training. Of course, other things may not be the same. There is considerable

heterogeneity among workers. Some can absorb training more easily and cheaply than others. The

existence or lack of previous training provides a signal about the individual. To the extent that

previous training signals a comparative advantage in the receipt of training, we would expect

individuals who have received previous training to be more likely to receive current training.

 The preceding argument focuses on characteristics that are specific to individuals and thus

applies to both previous training in the current job and training in a past job. However, job

matching considerations suggest that the probability of current training may be more closely

related to previous training in the current job than to previous training in past jobs. Since it only

pays to incur the costs of specific training if the match between a worker and an employer is a

good one that is likely to last a long time, the occurrence of training indicates that a match is high

quality. If training has a larger return in high quality matches, individuals who have received

previous training in the current job may be more likely to receive a recurring spell of training in the

current year.

 Let us now estimate the effect of previous training on the probability of current training. To

help fix ideas, let Tτ be an indicator variable equal to one if a worker in his τ

th year of tenure
7

receives training during year τ, and let PCJτ be the number of previous spells of training that a

worker in his τ

th year of tenure has received in his current job (only counting one spell per

year). Τhe probability of receiving training during the τ

th year of tenure can be written as

 (1) Pr( ) Pr( ) Pr( ) τ τ τ τ

τ

T T PCJ k PCJ k

k

= = ∑ = = =

=

−

1 1

0

1

,

where Pr(Tτ=1|PCJτ=k) is the worker's probability of receiving training in his τ

th year of tenure

conditional on his having received k previous spells of training in his current job. Equation (1)

indicates that the probability of receiving training during the τ

th year of tenure is merely the

weighted sum of the conditional probabilities Pr(Tτ=1|PCJτ=k), where the weights are the

probabilities of having received k spells of previous training in the current job.

 In the absence of explanatory variables, the maximum likelihood estimator of the conditional

probability Pr(Tτ=1|PCJτ=k) is ρτ,k, where ρτ,k denotes the proportion of individuals with tenure τ

and k spells of previous training in the current job who receive training in the current year. This

estimator is asymptotically normally distributed with mean Pr(Tτ=1|PCJτ=k) and variance

ρτ,k(1-ρτ,k)/Nτ,k, where Nτ,k denotes the number of sample observations where a worker is in his

τ

th year of tenure and has received k spells of previous training in the current job.

 The maximum likelihood estimates of the conditional probabilities Pr(Tτ=1|PCJτ=k) appear in

column 1 of table 2. The number of sample observations Nτ,k are also listed in table 2 in the

column labeled Cell Size.6 The results in column 1 indicate quite strongly that individuals with

 

6

 A careful examination of table 2 reveals that the cell sizes sum to less than 15743. To ensure

that we do not double count a single training spell that is mentioned at two consecutive interviews,

we distinguish between training spells that were ongoing at the time of last year's interview and

those that were completed when estimating Pr(Tτ=1|PCJτ=k>0). As expected, the estimated
8

previous training in their current job are more likely to receive current training. For example, the

probability of receiving training in the second year of tenure is 13.44 percent if an individual did

not receive training during his first year of tenure, but is 30.61 percent if he did receive training

during his first year of tenure. The difference between these two probabilities is statistically

different from zero.7 Similarly, individuals are significantly more likely to receive training in their

third and fourth year of tenure if they have received previous training at the same job.

Furthermore, there is some evidence that the probability of receiving training is positively related to

the number of previous training spells. For example, 28.07 percent of individuals in their third

year of tenure who have received training in one of the last two years receive training, whereas

45.10 percent of individuals in their third year of tenure who have received training in each of the

last two years receive training. This finding that individuals who have received previous training in

their current job are significantly more likely to receive training in the current year suggests that

there may be substantial individual and/or match fixed effects in training incidence.

 While individuals with previous training in their current job are more likely to receive current

training than those who have not, the incidence of first time (delayed) training occurring in the later

years of tenure is not insignificant. As noted above, the probability of receiving training in the

second year of tenure is 13.44 percent if an individual did not receive prior current job training. In

years three and four, this probability is still 11 to 12 percent. One possible explanation for this

delayed training is belated information about match quality: a worker and his employer may want

to defer making significant investments in training until they have had sufficient time to ascertain

the quality of their match. A second explanation is that a worker may be better able to absorb

 

probabilities for the former are near one. The estimated probabilities in table 2 are for individuals

who do not have an ongoing training spell at the time of the previous interview.

7

 Under the null hypothesis that Pr(Tτ=1|PCJτ=k)=Pr(Tτ=1|PCJτ=k-1), the test statistic

(ρτ,k-ρτ,k-1)/[ρτ,k(1-ρτ,k)/Nτ,k+ρτ,k-1(1-ρτ,k-1)/Nτ,k-1]

1/2 has a standard normal distribution. For

τ=2 and k=1, this test statistic is 6.68.
9

expensive training after an initial period of learning by doing in which he has become acclimatized

to his job and work environment.8

 We can further decompose the sample mean of training for each year of tenure conditional on

the number of previous spells of training in the same job into the means for distinct subgroups

defined by whether or not a worker has received previous training in a past job. That is, let POJτ

be an indicator variable equal to 1 if a worker in his τ

th year of tenure received previous training in

other jobs.9 Let Pr(Tτ=1|PCJτ=k,POJτ>0) denote a worker's probability of receiving training in

his τ

th year of tenure conditional on his having received k previous spells of training in the current

job and at least one spell of training in past jobs, and let Pr(Tτ=1|PCJτ=k,POJτ=0) denote a

worker's probability of receiving training in his τ

th year of tenure conditional on his having

received k previous spells of training in his current job but no spells of training in past jobs. The

probability of receiving training during the τ

th year of tenure can be written as

 (2) Pr( ) Pr( ) Pr( ) τ τ τ τ τ τ

τ

T T PSJ k, POJ PSJ k, POJ

k

= = ∑ = = = = =

=

−

1 1 0 0

0

1

+ ∑ = = > = >

=

−

Pr( ) Pr( ) τ τ τ τ τ

τ

T PSJ k, POJ PSJ k, POJ

k

1 0 0

0

1

.

 

8

 Technological change is another possible reason for delayed training. Indeed, Bartel and

Sicherman (1995) find that technological change increases the incidence of training for individuals

who did not receive training in the prior year. For a thorough analysis of delayed training using

several different data sets, see Loewenstein and Spletzer (1996a).

9

 Note that we are using an indicator variable rather than the number of previous spells of training

in past jobs. Our original specifications controlled for both the number of previous training spells

in past jobs and the number of years observed in past jobs. The number of years observed in past

jobs controls for the left censoring of our panel. We are able to reject the hypothesis that the

training probabilities differ by the number of years observed in past jobs. Although we can not

reject the hypothesis that the training probabilities differ by the number of spells observed in past

jobs, this appears to be the result of sample size rather than meaningful coefficient differences.

Therefore, in order to minimize the number of reported coefficients in table 2, we aggregate the
10

 Maximum likelihood estimates of the conditional probabilities in equation (2) are reported in

column 2 of table 2.10 For individuals in their first year of tenure, previous training in other jobs is

associated with a significantly higher probability of training. Specifically, while the probability of

receiving training is 8.94 percent for individuals who have not had previous training in past jobs, it

is 28.76 percent for individuals who have had such training. However, for other years of tenure,

the relationship between previous training in past jobs and current training is not so clear-cut. In

fact, while previous training in other jobs tends to raise the probability of training if an individual

has had previous training in the current job, it appears to slightly lower the probability of training

if the individual has not had previous training in the current job.

 The estimates in columns 1 and 2 of table 2 provide some evidence of individual fixed effects

and substantial evidence of match fixed effects in training. To determine whether these effects can

be explained by observable heterogeneity, we estimate an ordinary least squares equation that

includes various individual and job characteristics in addition to variables indicating the presence

or absence of prior training. The estimated equation is reported in column 3 of table 2.11

Although not listed, the coefficients on education and the armed force qualifying test are positive,

consistent with both the existing literature and our priors that more able individuals are more likely

to receive training. As is apparent from column 3, much of the fixed effect in training cannot be

explained by observables. The point estimates of the conditional probabilities fall by

 

number of training spells in past other jobs into an incidence measure. This aggregation does not

affect the main conclusions cited in the text.

10 A careful examination of the cell sizes in table 2 reveals that many individuals in our sample are

not observed in past jobs. For example, no past job is observed for 71.4 percent of the

observations with tenure=1. We have estimated the probability of receiving training during the τ

th

year of tenure for those individuals with no observed past jobs, but we do not report these

coefficients in table 2.

11 As is well-known, OLS is not entirely appropriate for estimating a conditional probability

model such as this. We present OLS estimates in column 3 because they facilitate comparison

with the results in column 2 (in the absence of explanatory variables, the point estimates of the

coefficients would be identical). Our conclusions do not change when we estimate the equation in

column 3 using a probit model.
11

approximately 20 to 25 percent when we include the explanatory variables, but most remain

statistically different from zero. Furthermore, the pattern that the probability of training increases

with the number of previous training spells in the current job still holds when we control for

observable heterogeneity.

III. Training and Wage Growth

 Our results in the previous section indicate that a substantial amount of recurring and delayed

training occurs well beyond the start of the employment relationship. In this section, we investigate

the extent to which this belated training can explain wage growth. A finding that training is

positively associated with wage growth would be strong evidence in favor of the human capital

model. However, some care needs to be taken to ensure that our training coefficients are not

biased. Our results in the previous section indicate that there are significant individual and match

specific fixed effects in training incidence. Furthermore, it appears that the match specific effect in

training incidence may evolve over time as belated information becomes available. If there are

similar effects in the wage equation, then simple OLS coefficients will be biased.

 To fix ideas, suppose the log wage of a worker i who is in his tth year of tenure at employer j is

given by

 (3) w T T c X c X c X u v ijt t ij

t

ik

k

j

= + ∑ + ∑ + i + ij + ijt + i + ijt + ijt

= =

−

γ β δ ε τ τ

τ 1 1

1

1 2 3

~

,

where γt is a tenure specific effect on wages common to all individuals, Tijτ is a variable indicating

whether individual i received any training in his τ

th year of tenure at his current job j, ~

Tik ≡ ∑ Tikτ

τ

is

the total number of times that individual i received training at his kth employer, Xi is a vector
12

consisting of individual-specific variables (such as ability, race, or gender), Xij is a vector of

match-specific variables (such as employer size), and Xijt is a vector of observable variables that

can vary within a match over time (such as the individual's marital status). Using a standard

decomposition of the error term, ui is an individual fixed effect that captures the net wage effect of

unobserved person-specific variables, vijt is an individual-job match effect, and εijt is a transitory

mean zero error component that is uncorrelated with both the explanatory variables and the fixed

effects. Note that equation (3) allows training in different years of tenure to have different wage

effects. It also allows for the possibility that training in the current and previous jobs may have

different wage effects.12

 As noted above, we are concerned with the possibility that the individual-job match effect varies

over time due to the arrival of belated information. If we assume that the belated information is

additive, the individual-job match fixed effect can be notationally written as

vijt = vijt-1 + θijt = vij + ij

t

τ

τ

θ

=

∑

1

, where vij is the net wage effect of unobserved job match variables at

the start of the job and θijτ is the net wage effect of belated information that arrives during the τ

th

year of tenure. Equation (3) can thus be rewritten as

 (4) w T T c X c X c X u v ijt t ij ik i ij ijt i ij ij ijt

t

k

t j

= + ∑ + ∑ + + + + + + ∑ +

= =

−

=

γ β δ θ ε τ τ τ

τ τ

~

1 2 3

1 1

1

1

.

 Consistent with the findings of others, our results in the preceding section indicate that more

educated workers and more able workers are more likely to receive training. Since our AFQT

measure is certainly not a perfect measure of ability and since we have no measure of the quality of

the job match, there are likely to be significant individual and match specific fixed effects in our

 

12 For a detailed analysis of the differential wage effects of training in the current and previous
13

wage equation. If training incidence is positively correlated with these effects (that is, if E[Tijt⋅

(ui+vij)]>0), OLS estimation of (4) will result in training coefficients that are biased upward. We

can eliminate this bias by estimating a wage growth equation for individuals who do not change

jobs. Specifically, if we take first differences of equation (4) for persons who have been at their

current employer for more than one year, we obtain

 (5) wijt - wijt-1 = (γt - γt-1) + β θ ε ε t Tijt ijt ijt ijt ijt ijt + c3 X − X −1 + + − −1

( ) .

 The positive correlation between Tijt and (ui+vij) does not cause any problems for OLS

estimation of equation (5) because the error components ui and vij are first differenced away.13

However, while first differencing eliminates the bias caused by the error components ui and vij,

there still remains the potential bias caused by the belated information effect θijt. A positive

correlation between training Tijt and belated information θijt will cause an upward bias in the

training coefficient βt. As is well known, we can eliminate this bias if we replace the training

variable Tijt by an appropriate instrumented value that is correlated with Tijt but not with the

belated information θijt. While it is often difficult to find suitable instruments, this is not a problem

in the current case. Note that previous training does not belong in the fixed effects wage growth

equation (5). Since the results in the previous section indicate that previous training is correlated

with current training, previous training can be used in the construction of an instrument for current

 

job, see Loewenstein and Spletzer (1996b).

13 Note that first differencing the wages of an individual who switches jobs eliminates individual

fixed effects but not match specific fixed effects. It is for this reason that we focus on the wage

growth of job stayers.
14

training. The variables Xi and Xij also do not belong in the fixed effects wage growth equation and

thus provide another source of exogenous variation in our training instrument.14

 The instrumental variable technique used above to eliminate the belated information bias is also

the standard method of correcting for measurement error. Recent research by Ashenfelter and

Krueger (1994) and Card (1995) suggests that measurement error in education may cause the

estimated returns to education to be biased downward by as much as 15%. One suspects that, if

anything, training is more difficult to measure than education. Indeed, using a unique survey of

matched employer-employee responses to the same training questions, Barron, Berger, and Black

(1994b) find that the correlation between worker reported and employer reported incidence of onsite (off-site) formal training is only .318 (.377). 15 While a positive correlation between training

and belated information would cause an upward bias in the training coefficient in equation (5),

measurement error in the training variable would lead to a downward bias. We do not have an apriori prediction regarding the net effect of these two biases; it is an empirical question whether the

instrumented training coefficient will be higher or lower than the OLS coefficient.

 Let us now turn to the actual estimation results. Table 3 presents the results of estimating a

wage level equation. To facilitate comparison with the results to follow we only include job stayers

in the estimation (that is, we only include those observations where the length of tenure is greater

than one year). As seen in column 2, when no other explanatory variables are included in the

 

14 Theoretically, it does not matter if we obtain predicted training incidence from OLS or probit

estimation because in either case the predicted training measure will be correlated with the actual

measure of training and uncorrelated with the error; see, for example, Kelejian (1971). (The

preceding statement assumes that training is measured without error. The case of measurement

error is analyzed in detail below.) As is well known, when predicted training is obtained via OLS,

two stage least squares and instrumental variable estimation are equivalent. However, as noted by

Heckman (1978), the instrumental variable estimator is not equivalent to the two stage estimator

when the first stage estimation is probit because residuals from the prediction of training will not

generally be orthogonal to the explanatory variables in the wage growth equation. Heckman

recommends two stage least squares on the grounds that it is easier to use. Consequently, we

choose to report results based on the linear estimator.
15

equation, the estimated returns to training are quite high - the coefficients on training in the current

job range from .1246 to .1993, and the coefficient on training in prior jobs is .1203. As noted

above, there is considerable evidence that more able workers are more likely to receive training.

To the extent that explanatory variables such as education and AFQT score control for

productivity differences among workers we should expect their inclusion in the wage equation to

significantly reduce the estimated return to training. This is indeed the case. However, as seen in

column 3 of table 3, even when we include a rich set of additional explanatory variables, the

estimated returns to training are still substantial, as the coefficients on training in the current job

range from .0554 to .0773, and the coefficient on training in prior jobs is .0576.16

 As discussed above, training coefficients in a simple OLS wage level equation suffer from at

least two distinct sources of bias. The fact that high ability workers in high quality job matches are

more likely to receive training causes the OLS training coefficients to be biased upward, and the

effects of belated information relating to the match specific effect also suggests an upward bias.

We can eliminate the bias caused by unobserved fixed individual and match differences by

estimating a (fixed effects) wage growth equation. Estimated wage growth equations are reported

in table 4. Column 1 of table 4 indicates that when we regress wage growth against tenure

indicators alone, all three of the tenure coefficients are positive and the first two are statistically

different from zero. The coefficient of .0491 on the indicator variable for the second year of tenure

tells us that between the first and second year of tenure workers' wages on average rise by about

five percent. Similarly, the coefficient of .0259 on the indicator variable for the third year of

 

15 In contrast, Ashenfelter and Krueger (1994) report that the correlation for self reported years of

schooling between identical twins is .658.

16 We have reestimated all of our wage equations with the duration of training included as an

additional explanatory variable. Once one controls for training incidence, the duration coefficients

are small in magnitude and statistically insignificant. This may be due to the fact that duration is

usually relatively short (as detailed by Loewenstein and Spletzer (1994), the mean duration of

completed formal training spells is 111 hours, but yet the median duration of completed formal

training spells is 32 hours) and is likely measured with considerable error.
16

tenure indicates that between the second and third year of tenure wages grow by over two and a

half percent, while the coefficient of .0204 on the indicator variable for the fourth year of tenure

indicates that wages grow by about two percent between the third and fourth years of tenure. As

seen in column 2, the tenure coefficients fall substantially when training indicators are added to the

wage growth equation. In column 3, the addition of (first-differenced) explanatory variables

further reduces the tenure coefficients, but has little effect on the training coefficients. In accord

with our theoretical discussion, the fixed effects training coefficients are smaller than the training

coefficients in the wage level equation.17 Nevertheless, all the training coefficients in the fixed

effects equation are positive and the estimated returns to second and third year training are 4.8 and

6 percent, respectively. The coefficient on third year training is statistically different from zero at

the 5 percent level.

 Estimating a wage growth equation eliminates the biases that are due to unobserved fixed

effects in the wage level equation. However, as discussed above, belated information about match

quality could still lead to upward biased training coefficients in a wage growth equation while

measurement error could lead to coefficients that are biased downward. To examine these

questions further, we estimate the wage growth equation using two stage least squares. Besides

previous training in the current job and previous training in other jobs, the explanatory variables in

the first stage OLS equation also include fixed demographic and job characteristics. The estimated

second stage wage growth equation is reported in column 4 of table 4.18 The coefficient estimates

on second, third, and fourth year training are .063, .1285, and .0814 respectively. These estimated

returns to training are markedly higher than those in column 3, suggesting that the downward bias

 

17

 Furthermore, when one uses job movers to estimates the returns to first year training, one finds

that the estimated returns to training in the first year are considerably higher than the estimated

returns to training in later years. This is consistent with our hypothesis that the OLS training

coefficients are partly picking up the returns to good job matches - returns which can be

differenced out for job stayers, but not for job movers.
17

in the OLS training coefficients due to measurement error swamps any possible upward bias

stemming from belated information about match quality. Still, it seems hard to believe that the

measurement error effect can be nearly as large as our estimated equations suggest. We address

this question in the next section.

IV. The Effects of Measurement Error on the Returns to Training

 The potential upward bias in simple OLS training coefficients stemming from the endogeneity

of training has been the object of a substantial amount of attention, but the downward bias from

measurement error has received little, if any, discussion. A comparison of columns 3 and 4 in table

4 reveals that the instrumental variable coefficients on second and third year training are

approximately three times higher than the corresponding OLS estimates. And the estimated return

to fourth year training is 23 percent higher using the instrumental variable estimation.

 Is it reasonable that measurement error can have such a large impact on our estimation results?

Exploration of this question requires some additional notation. Let Tit* be an indicator variable

equal to 1 if individual i receives training during his tth year of tenure (for the rest of this section,

we drop the "j" subscript since our wage growth equation is estimated only for job stayers). The

econometrician observes reported training Tit rather than actual training Tit*, where the difference

is the measurement error ωit=Tit-Tit*. Taking this measurement error into account, the wage

growth regression becomes:

 (6) Δwit = (γt - γt-1) + βtTit + c3ΔXit + φit ,

where φit = θit + Δεit - βtωit

 

18

 The first stage equations for second, third, and fourth year training have R2s of .16, .19, and .28,
18

 To simplify the exposition, we will assume for now that training is the only explanatory

variable affecting wage growth. We will show later that the analysis can be generalized to include

other explanatory variables. When c3 = 0, the wage growth equation is given by

 (6’) Δwit = (γt - γt-1) + βtTit + φit.

In this case, it is straightforward to show that the estimated OLS coefficient on year t training will

have the probability limit

 (7) plim $

cov( , )

var( )

cov( , )

var( )

β β β

ω θ

t

OLS

t t

it it

it

it it

it

T

T

T

T

= − + .

In accordance with the discussion in Section III, equation (7) suggests two possible sources of bias

in the OLS training coefficients in column 3 of table 4: a non-zero correlation between training Tt

and the measurement error ωi

t

, and a non-zero correlation between training Tit and the belated

information θit. While the bias due to belated information, cov(Tit,θit)/var(Tit), is presumably

positive, we show below that the bias due to measurement error,

-βtcov(Tit,ωit)/var(Tit), is unambiguously negative.19 A priori, the overall bias in the estimated OLS

coefficient is thus of indeterminate sign. The OLS coefficient is biased downward (upward) if the

bias due to measurement error is larger (smaller) in magnitude than the bias due to the belated

information about match quality.

 For notational convenience, define ηt≡Pr(Tit*=1|Tit=0) as the probability that an individual who

reports not receiving training actually receives training, and define νt≡Pr(Tit*=0|Tit=1) as the

 

so the correlations between our training measures and training instruments are relatively high.
19

probability that an individual who reports receiving training does not actually receive training. In

the discussion that follows, it will also be helpful to define the conditional probabilities α0t≡

Pr(Tit=1|Tit*=0) and α1t≡Pr(Tit=0|Tit*=1). Note that α0t is the probability that an individual who

does not receive training incorrectly reports that he receives training while α1t is the probability

that an individual who receives training incorrectly reports that he does not receive training.

Furthermore, define pt≡Pr(Tit=1) as the probability of a reported training spell and define pt*≡

Pr(Tit*=1) as the probability of a true training spell. The joint distribution of Tit and Tit* is

summarized by the following frequency table:

Row %

Column %

Tit*=0 Tit*=1

Tit=0 (1-ηt

)(1-pt)

(1-α0t)(1-pt*)

ηt(1-pt)

α1tpt*

Tit=1 νt

pt

α0t(1-pt*)

(1-νt)pt

(1-α1t)pt*

In interpreting the above frequency table, note that the probability that an individual neither

receives nor reports training can be expressed as either (1-ηt)(1-pt) or (1-α0t)(1-pt*). A similar

statement applies with respect to the other joint probabilities.

 Since both training Tit and the measurement error ωit are discrete variables, equation (7) can be

simplified as:

(8) plim $ Pr( , ) Pr( ){Pr( , ) Pr( , )}

Pr( ){ Pr( )}

* * *

βt β

OLS it it it it it it it

it it

t

T T T T T T T

T T

= −

= = − = = = − = =

= − =





























1

1 0 1 1 0 0 1

1 1 1

 

19

 More precisely, measurement error lowers the absolute value of the estimated coefficient. The

statement in the text presumes that βt > 0.
20

+













cov( , )

var( )

T

T

it it

it

θ

=[1− (νt + ηt) + bt]βt,

where b

T

T

t

it it

it t

≡

























cov( , )

var( )

θ

β

1

 will be referred to as the proportional belated information bias.

The proportionate bias of -(νt+ηt) due to measurement error in a bivariate variable such as training

incidence has been derived previously by Aigner (1973) and Freeman (1984). This bias is

unambiguously negative in sign; measurement error in the training variable causes the OLS

training coefficient to be biased downward.20

 Now let us turn to the instrumental variable estimation. The training instrument Zt has been

constructed so as to be uncorrelated with the belated information θt. We have not yet made any

assumptions regarding the correlation between the instrument Zt and the measurement error ωt. In

the absence of explanatory variables, the estimated instrumental variables (IV) coefficient on

training in the tth year of tenure in equation (6’) has probability limit:

 

20 Using the above frequency table, we can rewrite the OLS bias attributable to measurement error

in terms of {α0,α1,p*} rather than in terms of {η,ν}. Holding constant α0 and α1, one can easily

show that the OLS bias η+ν is a U-shaped function of p* that equals 1 when p* is either 0 or 1.

This implies that the OLS bias is especially large when p* is either very small or very large. If p*

is close to zero, the signal to noise ratio in reported training is quite small for given levels of α0 and 

α1, which merely says that a large proportion of those reporting training will not actually have

received training. Although η will be close to zero, ν will be close to one and the OLS coefficient

will be near zero. (Similarly, if p* is close to one, then η will be close to one and ν will be close to

zero.) As can be seen from Table 1, only about 18 percent of the individuals in our sample report

training in their second, third, and four years of tenure. As a consequence, even if the probabilities

of correct responses from individuals who receive and do not receive training are as high as 95

percent (that is, if α0 = α1 = .05), the OLS training coefficient is biased downward by 25 percent.

Furthermore, it is easy to show that the OLS bias η+ν rises with an increase in the probability of

an incorrect response. For instance, if the probability that individuals respond incorrectly increases
21

 (9) plim $

cov( , )

cov( , )

β

ω

t β

IV it it

it it

t

Z

Z T

= −























1  .

 Equation (9) indicates that the IV training coefficient is consistent if and only if the instrument

Zt is uncorrelated with the measurement error ωt. To determine possible causes of correlation, note

that we can always write the predicted training measure as

 (10) Zit=κt+Tit*δt+eit,

where κt=E(Zit|Tit*=0) and δt=E(Zit|Tit*=1)-E(Zit|Tit*=0) can be interpreted as the intercept and

slope from a regression of the predicted training measure on the true unobserved training measure.

Substituting (10) into equation (9) and using the fact that Tit=Tit*+ωit, we obtain:

 (11) plim $

cov( *, ) cov( , )

var( *) cov( *, ) cov( , )

β

δ ω ω

δ δ ω ω

t β

IV t it it it it

t it t it it it it

t

T e

T T e

= −

+

+ +























1  .

 Equation (11) indicates two potential sources of bias in the instrumental variables training

coefficient. The first of these is a possible correlation between et and ωt. Recall that previous

training, fixed individual characteristics, and fixed job match characteristics were the identifying

variables used in forming the predicted training variables used to estimate the second stage wage

growth equation in column 4 of table 4. Our motivation for using an individual's past training in

the construction of an instrument for current training is our earlier finding that individuals who

report training in the past are more likely to report training today. Of course, the implicit

 

from 5 to 10 percent, the bias in the OLS training coefficient escalates from 25 percent to 51

percent.
22

assumption here is that the positive correlation between past and present training arises at least in

part because true training is correlated over time. However, the positive correlation between past

and present training could occur if individuals tend to consistently misreport training over time. If

individuals who falsely report (not) receiving training in the past are more likely to falsely report

(not) receiving training today, then et and ωt will be positively correlated. As seen from equation

(11), this positive correlation will lead to a downward bias in the instrumental variable estimates of

the training coefficients in column 4 of table 4.

 We can eliminate the downward bias due to serial correlation in the misreporting of training

over time simply by dropping past training from the first stage training incidence equation.

Column 5 of table 4 reports the estimated wage growth equation when past training is omitted from

the first stage training incidence equation. Comparing columns 4 and 5, we see that, as expected,

dropping past training from the first stage training incidence equation leads to a moderate increase

in all of the estimated training coefficients in the second stage wage growth equation. Henceforth,

our discussion will refer to the estimates in column 5, so that we may assume that cov(eit,ωit) = 0.

 Unfortunately, the second source of bias in the instrumental variable training coefficient is not

so readily eliminated. This bias stems from a non-zero correlation between the true value of

training Tit* and the measurement error ωit. In the case of classical measurement error,

cov(Tit*,ωit)=0 and the instrumental variable coefficient is unbiased. However, with a discrete

variable such as training incidence, the assumption that Tit* and ωit are uncorrelated cannot hold.21

To see this, note that if an individual receives training so that Tit*=1, then ωit can only take on the

values {0,-1}. And if an individual does not receives training so that Tit*=0, then ωit can only take

on the values {0,1}. It follows immediately that Tt* and ωt must be negatively correlated. Thus,

while measurement error leads to a downward bias in the OLS training coefficients, it leads to an
23

upward bias in the IV training coefficients. To show this formally, we need merely simplify

equation (11) as:

 (12) plim $

cov( *, )

var( *) cov( * )

β

ω

ω

t β

IV it it

it it it

t

T

T T

= −

+























1 

 = 1

1 0 1 0 1 1 0

1 1 1

−

= = + = = = − = =

= = − =

























Pr( * , )) Pr( * ){Pr( * , } Pr( * , )

Pr(( * ) Pr( ) Pr(, )

T T T T T T T

T T T

it it it it it it it

it it it

βt

 =

1

1− 0 1













t t

t

α α

β

-

.

Assuming that α0t + α1t < 1, it follows immediately from (12) that the instrumental variable

estimates of the training coefficients are upward biased in the presence of measurement error.22

 From equation (12), we see that the proportional upward bias in the instrumental variable

estimate is 1/(1-α0t-α1t). If we had consistent estimates for α0t and α1t, say α$

0t andα$1t , we could

obtain a consistent estimate of βt simply by multiplying the instrumental variable estimate by (1-

α$

0t - α$1t ). But if we do not observe true training, how can we estimate α0t and α1t?

 One possible approach would be to use external information on misclassification rates, such as

might be provided by a validation study. Indeed, Card (1996) takes this approach in his recent

analysis of the wage effects of unions.23 An alternative approach is to note that the greater are the

 

21 We are indebted to Harley Frazis for initially pointing this out and for helping us to simplify the

derivation of the IV bias.

22

 If α0t + α1t > 1, then the measurement error in reported training is so severe that either a)

someone who reports receiving training is more likely not to have received training than to have

received training or b) someone who reports not having received training is more likely to have

received training than not to have received training. Not surprisingly, in this case, the estimated IV

training coefficient will not even have the right sign.

23

 Interestingly, Card finds that “union status misclassification errors lead to a 50-75 percent

attenuation in the average wage changes of observed union joiners and leavers, relative to the true

wage changes of actual joiners and leavers.”
24

mismeasurement probabilities α0t and α1t, the more severe is the downward bias in the OLS

training coefficient and the upward bias in the instrumental variable coefficient. Thus, the ratio of

the OLS and the IV training coefficients contains information about the size of α0t and α1t. Note

that since νt and ηt can be expressed in terms of pt, α0t and α1t, both the OLS coefficient in

equation (8) and the IV coefficient in equation (12) are functions of βt, pt, α0t and α1t, and the

proportional belated information bias bt. If one assumes that measurement error has an expected

value of 0, then one can solve for α0t and α1t as functions of the training probability pt, the

proportional belated information bias bt, and the expected values of the training coefficients $βt

OLS

and $βt

IV . After the appropriate calculations, the measurement error parameters {α ,α } 0t 1t

 can be

written as:

 (13) {α ,α } ,( ) 0 1

2 2

1

2 4

1

1 1

2 4

1

t t t

t t

t

t

t t

t

p

b b

r

p

b b

r

= + − +

















− + − +





































,

where rt ≡ plim 

$

$

β

β

t

IV

t

OLS .

24

 

24

 If we do not assume something about the expected value of measurement error or the value of

either α0t or α1t, we do not have enough information to solve for the key parameters of interest.

Specifically, our model reduces to the following two equations: E(ω) = [α0(1-p)-α1p]/(1-α0-α1)

and (1 ) 1 /

1 1

1

1

0 1 1

1 0

0 1

0 0

0 1

− − −

−

−

− −











 − −

−

− −











 +











α α  =

α α

α α

α α

p α α

p

p

p

b r. Assuming that E(ω)=0

allows us to solve for α0and α1 as functions of r and b. In a similar vein, Card (1996) closes his

measurement error model by assuming that α0=α1 (in Card’s notation, q0=1-q1). Our empirical

results for the returns to training turn out not to be very sensitive to the assumption that the

expected value of measurement error is zero. Assuming that the two misclassification parameters

are identical yields similar estimates of βt, as do other plausible assumptions. We choose to report

estimates based on the assumption that E(ωt) = 0 since this is equivalent to assuming that pt=pt*,

whereas the alternative assumption that α0t=α1t results in pt* estimates of .08 and .10 in the second
25

 Substituting (13) into (12) yields:

 (14) plim $βt β

IV

t

t

t

t

b

r

b

=

+ −





























1

4

1

2

2

.

Let $

$

$

rt

t

IV

t

OLS =

β

β

 denote the ratio of the estimated instrumental variable and OLS training coefficients

and let

 (15) $

$

$ βt β

t

t

t

t

b IV

r

b

≡ + −

















2

4

1

2

 .

Conditional on the belated information bias bt, 

$βt

 is clearly a consistent estimator of βt. It is

straightforward to show that 

t

IV

$β is asymptotically normally distributed with variance

 (16) Vt = 

1

1

2 n

Var Z Z

p

Z Z T T

n

t

it t it

it t it t

i

n

t

t

{( ) }

lim

( )( )

−

− −





















=

∑

φ

 = 

1

2 n

Var Z Z

t Z T

it t it

it it

{( ) }

(cov( , ))

− φ

,

 

and third years of tenure. Since these latter estimates are roughly half the reported probability of

training, we regard them as implausibly low.
26

where nt is the number of individuals in the sample with t years of tenure, Zt Zit

i

nt

≡ ∑

=1

, and

Tt Tit

i

nt

≡ ∑

=1

.

25 Thus, $βt

 is asymptotically normal distributed with mean βt and variance

Vt

b

r

t b

t

t

2

2

4

1

2

+ −















 $

.

 For expositional convenience, we have thus far assumed that training is the only explanatory

variable in the wage growth equation. It is straightforward to extend the analysis so as to allow

additional explanatory variables. Specifically, suppose that instead of (6’), the wage growth

equation is the more general equation (6). Adopting a similar argument to that in Card (1996), one

can then show that

 (8’) plim $

[ ( )]

β

ν η β

α α

t

OLS

t t t

t

t

t

R

R

= b

− + −

− −

+

1

1

2

1 0

2

,

where Rt

2

denotes the theoretical R-squared coefficient from a linear regression of Tit against the

other explanatory variables in the wage growth equation, and the belated information bias is now

given by bt = 

cov( , | )

var( | )

T X

T X

it it it

it it t

θ

β

Δ

Δ

























1

.

26 The appendix demonstrates that the insrumental variable

 

25 Note that in the standard instrumental variable formulation, we would have

Vt = 

1

2

n

Var Z Z

t Z T

it t it

it it

{( ) }

(cov( , ))

− φ

= 

1

2

n

Var Z Z

t Z T

it t it

it it

[ ( )]var( )}

(cov( , ))

− φ

. In our current formulation, we cannot

carry out the last step because Zit and φit are not independent.

26

 The derivation of (8’) assumes Tit is the only explanatory variable in the wage growth equation

that is measured with error. The derivation also assumes that cov(ΔXit,ωit| Tit*) = 0. Our

derivation differs from Aigner’s in that Aigner assumes that cov(ΔXit, ωit) = 0 even though ΔXit

may be correlated with Tit and Tit*.
27

bias does not change when additional explanatory variables are included in the equation. Using

(8’) and (12) to solve for α0t and α1t, one obtains

 (13’) α0

2

2

2 2 2 2

1

2

1

4

1

1

t t 1

t

t

t

t

t

p t t

b

R

b

R

r

= + − − − + − R + R

















( ) ( ) ( ) ,

 α1

2

2

2 2 2 2

1 1

2

1

4

1

1

t t 1

t

t

t

t

t

p t t

b

R

b

R

r

= − + − − − + − R + R

















( ) ( ) ( ) ( ) .

Thus, letting R$

t

2

denote the estimated R-squared coefficient from a linear regression of observed

training against the other explanatory variables in the wage growth equation,

 (15’) $(

$

(

$

) )

$

(

$

)

$

(

$

)

$ βt β

t

t t

t

t t

t

t t

b IV R R

r

R R

b

≡ − + + − + − − R

















2

2 2 2 2 2 2

4

1 2

1

1

2

1

is a consistent estimator of βt.

 The estimated returns to training that result from the calculations described above are presented

in Table 5.27 In interpreting this table, recall that these estimates depend on the magnitude of the

belated information bias, bt. Column 1 indicates the parameter estimates when there is no belated

information. Columns 2 and 3 indicate the parameter estimates that result when bt is .05 and .1,

respectively.

 

27

 As noted if footnote 26, the derivations of (8’) and (15’) assume that Tit is the only variable in

the wage growth equation that is measured with error. The estimates in Table 5 are therefore

obtained from single year OLS and IV equations instead of the pooled equations presented in Table

4. The pooled and single year estimates are very close.
28

 When there is no belated information bias, the estimated returns to training in column 1 are

.038 in the second year of tenure, .093 in the third year of tenure, and .069 in the fourth year of

tenure. An increase in the belated information bias results in a relatively small drop in the

estimated returns to training. For example, an increase in bt from 0 to 10 percent causes the

estimated returns for second and third year training to fall by about 10 percent and the estimated

return for fourth year training to fall by about 6 percent.

 The corrected wage coefficients in Table 5 are considerably lower than the IV coefficients in

column 5 of table 4. Nevertheless, the corrected coefficients still indicate that the effect of training

on wages is quite substantial. Most notably, training in the third or fourth year of tenure increases

wage growth during that year by somewhere between 6 and 9 percent. The effect of training

becomes even more striking when one considers our finding in the previous section that training

incidence is correlated over time for a given individual. If an individual receiving training in the

third year also received training in the second year of tenure, his total wage growth from year 1 to

year 3 is 12 to 13 percent higher than an individual who did not receive training in either year.

 Besides presenting the estimated returns to training, we also present our estimates of the

measurement error parameters α0t and α1t in table 5.28 Note that these estimates, like the estimated

returns to training, depend on the magnitude of the belated information bias. When there is no

belated information bias and a full set of explanatory variables are included in the wage growth

equation, α0t≡Pr(Tit=1|Tit*=0) is estimated to be .08 for t=2, .08 for t=3, and .02 for t=4.

Similarly, α1t≡Pr(Tit=0|Tit*=1) is estimated to be .377 for t=2, .358 for t=3, and .09 for t=4. In

other words, during the second and third years of tenure, eight percent of persons who did not

 

28 Let p$

t denote the proportion of individuals in the sample with tenure t who report receiving

training. Substituting p$

t

 and $

$

$

r

t

t

IV

t

OLS =

β

β

for pt and rt in equation (13’) yields consistent estimates

α$

0t

and α$

1t

of the measurement error parameters α0t and α1t.
29

receive training report that they did receive training, and roughly a third of those who received

training reported that they did not receive training. These proportions of misreporting fall sharply

in the fourth year of tenure.

 Our estimates thus indicate that there is a sizable amount of misreporting in the NLSY training

data. One may ask whether these measurement error estimates are reasonable. Barron, Berger,

and Black (1994b) is the only study we know of that has sought to quantify the measurement error

in training. Although their survey is different from the NLSY data used here, it is still instructive

to compare their results with ours.29 In the Barron-Berger-Black data, employers and employees

report similar amounts of training: thirty percent of workers and thirty-three percent of employers

report formal training. However, while the means are similar, the correlation between worker

reported training and employer reported training is only .369. If the employer's response were

always correct but the worker sometimes misreported, then it can be shown that the Barron-BergerBlack data imply that α0

 = Pr(Worker reports training|Employer does not report training) = .183

and α1 = Pr(Worker does not report training|Employer reports training) = .457. These

measurement error estimates are even larger than ours, although it should be kept in mind that the

reported probability of training is almost twice as high in the Barron-Berger-Black data as in the

NLSY data. Alternatively, if one assumes that workers and employers are equally likely to

misreport training, then it can be shown that the Barron-Berger-Black data imply that α0 = .125

and α1 = .27. These estimates are similar to those obtained here; at the very least, our estimates do

not look unreasonable.

 

29 Barron, Berger, and Black ask both an employee and his employer about on-site formal

training, off-site formal training, and various types of informal training during the first two weeks

of employment. See Barron, Berger, and Black for a more complete description and see

Loewenstein and Spletzer (1994) for a further comparison of "EOPP-like" surveys and the NLSY.

We are grateful to Dan Black for providing us the figures for aggregate "formal training."
30

 Finally, any conclusions with respect to the human capital model require unbiased tenure

coefficients. Of course, a bias in the estimated training coefficients will also lead to a bias in all of

the other variables in the wage growth regression, including the tenure-specific intercepts. Having

obtained a consistent estimate, $βt, of the training coefficient, we can obtain consistent estimates of

the coefficients on the remaining variables by estimating the equation

 (17) Δwt - 

$βt Tit = (γt

 - γt-1) + c3ΔXit + ψt,

where ψt ≡ θit + Δεit - βtωit + (

$ β β )

t − t Tit . The resulting tenure coefficients are presented in the

lower panel of table 5. As is evident from a comparison of tables 4 and 5, the consistently

estimated tenure coefficients are not substantially different from the instrumental variable tenure

coefficients.

 Using these unbiased estimates of the training coefficients and the tenure coefficients, the fact

that the training coefficients are substantially larger than the corresponding tenure coefficients in

the later years of tenure provide strong support for the human capital model. Between the first and

second year of tenure, the wages of workers who do not receive training grow by approximately

three and one-half percent, but the wages of workers who receive training grow by approximately

seven percent. Individuals who do not receive training between their second and third years of

tenure experience no wage growth, whereas workers who receive training in their third year of

tenure experience wage growth between eight and nine percent. Similarly, individuals who do not

receive training in their fourth year of tenure receive essentially no wage growth, whereas workers

who receive training in their fourth year of tenure experience wage growth of approximately six to

seven percent. Interestingly, unexplained wage growth is greatest between the first and second year

of tenure. We suspect that this may reflect the fact that the NLSY training questions likely miss
31

most informal training, training which is likely to be most important early in the employment

relationship.30

VI. Conclusions and Discussion

 The human capital model attributes rising wage-tenure and wage-experience profiles to

investments in worker productivity. Therefore, an implicit assumption in the human capital model

is that training must be occurring throughout a worker's stay at an employer. There are at least

two factors that may lead to significant amounts of belated training. First, workers are often better

able to absorb expensive training after an initial period of "learning by doing" in which they have

been acclimatized to their job and work environment. Second, information about the quality of a

firm-worker match is often revealed only after a period of time. An employer who delays training

can lower the probability of investing in bad matches.

 The data presented in section II indicate quite clearly that a substantial amount of training

occurs after the first year of employment. Annual training incidence nearly doubles from 9.89

percent in the first year of tenure to 17.21 percent in the second year of tenure, and is then roughly

constant around 18 percent in the third and fourth years of tenure. The data also indicate that there

are significant match specific fixed effects in training: individuals who have received previous

training at their current employer are more likely to receive current training.

 Section III of the paper examined the relationship between training and wages. Taking first

differences for job stayers eliminates potential biases stemming from unobserved fixed individual

and match effects, but does not eliminate the bias from belated information. Specifically, if an

 

30

 Since there is a positive correlation between the receipt of formal training and the receipt of

informal training (see Loewenstein and Spletzer (1994) for evidence on this score), the coefficient

on formal training should partly pick up the returns to informal training. If informal training is

more important early in the employment relationship, then this by itself should cause the coefficient

on second year training to exceed those on third and fourth year training. Our results indicate that

this effect is more than offset by higher returns to belated training.
32

employer belatedly offers a worker a raise and invests in his training upon discovering that his

match-specific productivity is high, then the training coefficient in the fixed effects wage equation

will still be biased upward. To eliminate the bias from belated information, we have estimated the

fixed effects equation with a two stage procedure that uses previous training and fixed individual

and match-specific variables as instruments.

 The instrumental variable technique is also the standard method of correcting for measurement

error. While the potential upward bias in the OLS training coefficient caused by a likely positive

correlation between unobserved worker ability and training has been the object of a fair amount of

discussion, the downward bias in the OLS training coefficient caused by measurement error has

received little, if any, attention.31 Recent research suggests that measurement error in education

may cause the estimated returns to education to be biased downward by as much as 15%. One

suspects that, if anything, training is more difficult to measure than education.

 Indeed, our instrumental variables estimation yields substantially higher estimated returns to

training than does our OLS estimation. However, instrumental variables estimation does not yield

consistent estimates when the explanatory variable measured with error is dichotomous, or more

generally, when the explanatory variable measured with error can take on only a limited range of

values. The difficulty stems from the fact that the measurement error in such a variable cannot be

classical, but must be negatively correlated with the variable’s true value. In our application, the

fact that the instrumented training incidence measure is negatively correlated with the measurement

 

31 A notable exception here is Brown (1989), who uses training data from the Panel Study of

Income Dynamics. This survey asks a worker how long it would take the average person to

become fully trained and qualified in a job like his, and Brown assumes an individual to be

receiving training if his job tenure is less than the time it takes to become fully trained and

qualified. Brown points out that the length of a training spell may be measured with error because

an individual's initial set of skills may differ from that of the "average person" and because it is not

clear whether a respondent interprets "job" as referring to the current position [as Brown assumes],

the current firm, or possibly the current occupation. Interestingly, Brown (1994) demonstrates that

measurement error may actually be inversely correlated with observed training, causing the

estimated returns to training to be biased upward.
33

error leads to an upward biased estimate of the returns to training. The relative magnitudes of the

IV and OLS training coefficients provide information about the size of measurement error bias.

Our results indicate that this bias is substantial: if there is no belated information bias, the IV

training coefficients are biased upward by as much as 75%.

 After correcting for measurement error bias, our estimated wage growth equation provides

strong support for the human capital model. The training coefficients are positive and are

substantially larger than the corresponding tenure coefficients, particularly, for the third and fourth

years of tenure. Between the first and second year of tenure, the wages of workers who do not

receive training grow by approximately three and one-half percent, but the wages of workers who

receive training grow by approximately seven percent. And while individuals who do not receive

training experience basically no wage growth between the second and fourth years of tenure,

workers who receive training in either the third or fourth years of tenure experience wage growth of

approximately six to nine percent.

 Finally, we may note that it is quite possible that the unexplained wage growth between the first

and second years of tenure is itself due to human capital accumulation. The training question in

the 1988-1991 NLSY surveys appears to primarily measure formal training. The NLSY did not

begin explicitly asking about informal training until 1993. Loewenstein and Spletzer (1994)

provide evidence that the training measure in the earlier surveys misses most informal training,

Thus, our present finding that unexplained wage growth is greatest between the first and second

year of tenure may simply reflect the fact that informal training is most important early in the

employment relationship. A thorough investigation of this hypothesis must be deferred until future

NLSY surveys using the new informal training questions become available.
34

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Table 1: Training-Tenure Relationship, 1988-1991 NLSY



Tenure Sample Size Percent Trained

 1 11014 9.89%

 2 3179 17.21%

 3 1162 18.16%

 4 388 18.30%


2

Table 2: Training Incidence

 | Cell |

Probability | Size (1) (2) (3) |

Pr(T1=1|PCJ1=0) | 11014 .0989 * |

 | (.0028) |

Pr(T1=1|PCJ1=0,POJ1=0) | 2763 .0894 * .0689 |

 | (.0054) (.0406) |

Pr(T1=1|PCJ1=0,POJ1>0) | 386 .2876 * .2362 * |

 | (.0230) (.0433) |

 | |

Pr(T2=1|PCJ2=0) | 2753 .1344 * |

 | (.0065) |

Pr(T2=1|PCJ2=0,POJ2=0) | 655 .1267 * .0941 * |

 | (.0130) (.0419) |

Pr(T2=1|PCJ2=0,POJ2>0) | 62 .0968 * .0589 |

 | (.0376) (.0566) |

Pr(T2=1|PCJ2=1) | 343 .3061 * |

 | (.0249) |

Pr(T2=1|PCJ2=1,POJ2=0) | 80 .2500 * .1757 * |

 | (.0484) (.0532) |

Pr(T2=1|PCJ2=1,POJ2>0) | 30 .3333 * .2732 * |

 | (.0861) (.0706) |

 | |

Pr(T3=1|PCJ3=0) | 850 .1176 * |

 | (.0110) |

Pr(T3=1|PCJ3=0,POJ3=0) | 159 .0881 * .0625 |

 | (.0225) (.0474) |

Pr(T3=1|PCJ3=0,POJ3>0) | 6 .0000 -.0523 |

 | (.0000) (.1333) |

Pr(T3=1|PCJ3=1) | 228 .2807 * |

 | (.0298) |

Pr(T3=1|PCJ3=1,POJ3=0) | 42 .2381 * .1713 * |

 | (.0657) (.0630) |

Pr(T3=1|PCJ3=1,POJ3>0) | 5 .2000 .1901 |

 | (.1789) (.1603) |

Pr(T3=1|PCJ3=2) | 51 .4510 * |

 | (.0697) |

Pr(T3=1|PCJ3=2,POJ3=0) | 6 .3333 * .2353 |

 | (.1924) (.1338) |

Pr(T3=1|PCJ3=2,POJ3>0) | 2 1.0000 * .9294 * |

 | (.0000) (.2235) |

 | |

Pr(T4=1|PCJ4=0) | 251 .1116 * .1116 * .0761 |

 | (.0199) (.0199) (.0452) |

Pr(T4=1|PCJ4=1) | 84 .2143 * .2143 * .1288 * |

 | (.0448) (.0448) (.0526) |

Pr(T4=1|PCJ4=2) | 30 .4000 * .4000 * .2979 * |

 | (.0894) (.0894) (.0699) |

Pr(T4=1|PCJ4=3) | 10 .3000 * .3000 * .2154 * |

 | (.1449) (.1449) (.1069) |

 | |

Explanatory Variables | No No Yes |

1988-1991 NLSY. Sample size = 15743.

Columns 1-2: Maximum likelihood estimates. Column 3: OLS coefficients. Standard

 errors in parentheses. * implies statistically different from zero at the 5%

 level of significance (one tailed test).

The statistics reported above are only for those individuals who did not have an

 ongoing training spell (PCJ>0) at the time of the previous interview. Statistics

 are not reported for individuals who had a training spell ongoing at the date of

 the last interview. Statistics are not reported in columns 2-3 for individuals
3

 who did not have a Prior Other Job observed.

Explanatory variables are year dummies, AFQT, race, gender, age, marital status,

 urban residence, local area unemployment rate, SMSA, union, experience and

 experience squared, firm size, education, school attendance in the previous

 year, multiple site firm, number of previous jobs, government employment,

 part-time employment, number of children, and industry and occupation.
4

Table 3: Wage Level Regressions (Job Stayers)

 | Mean (1) (2) (3) |

Tenure=2 | .6701 1.9936 * 1.9223 * 1.7761 * |

 | (.0085) (.0158) (.0883) |

Tenure=3 | .2477 2.0503 * 1.9378 * 1.7449 * |

 | (.0140) (.0205) (.0906) |

Tenure=4 | .0821 2.1313 * 1.9730 * 1.7443 * |

 | (.0243) (.0328) (.0990) |

 | |

Tenure=1 x Training | .1329 .1607 * .0554 * |

 | (.0209) (.0169) |

Tenure=2 x Training | .1789 .1246 * .0628 * |

 | (.0196) (.0158) |

Tenure=3 x Training | .0630 .1993 * .0773 * |

 | (.0325) (.0259) |

Tenure=4 x Training | .0151 .1535 * .0739 |

 | (.0672) (.0533) |

 | |

# Training Spells, | .0302 .1203 * .0576 |

 Previous Jobs | (.0383) (.0307) |

 | |

Explanatory Vars. | No No Yes |

1988-1991 NLSY. OLS Regression Coefficients. Standard errors in parentheses.

 Dependent variable is the log real wage. Mean (Standard deviation) of dependent

 variable is 2.0190 (.4659). Sample Size = 4432 (Job Stayers, Tenure>2).

* implies statistically different from zero at the 5% level of significance (two

 tailed test).

Equations 2-3 include controls for whether the training in the current year is

 ongoing at the date of the interview, and controls for whether no training is

 observed in previous jobs because of left censored data.

Explanatory variables are year dummies, AFQT, race, gender, age, marital status,

 urban residence, local area unemployment rate, SMSA, union, experience and

 experience squared, firm size, education, school attendance in the previous

 year, multiple site firm, number of previous jobs, government employment,

 part-time employment, number of children, and industry and occupation.
5

Table 4: Within Job Wage Growth Regressions

 | Mean (1) (2) (3) (4) (5) |

Tenure=2 | .6701 .0491 * .0458 * .0379 * .0297 .0277 |

 | (.0052) (.0057) (.0127) (.0156) (.0164) |

Tenure=3 | .2477 .0259 * .0187 * .0068 -.0084 -.0128 |

 | (.0085) (.0094) (.0144) (.0172) (.0187) |

Tenure=4 | .0821 .0204 .0090 .0110 .0054 .0029 |

 | (.0147) (.0163) (.0191) (.0229) (.0243) |

 | |

Tenure=2 x Training | .1171 .0171 .0190 .0630 .0666 |

 | (.0152) (.0152) (.0515) (.0537) |

Tenure=3 x Training | .0454 .0520 * .0482 * .1285 * .1412 * |

 | (.0240) (.0239) (.0610) (.0692) |

Tenure=4 x Training | .0151 .0575 .0600 .0814 .0837 |

 | (.0418) (.0417) (.0851) (.0894) |

 | |

Predicted (OLS) | |

 Tenure x Training | No No No Yes Yes |

1st Stage Variables | Tt-τ,X X |

Explanatory Vars. | No No Yes Yes Yes |

1988-1991 NLSY. Dependent variable is log real wage growth. Mean (Standard deviation) of

 dependent variable is 0.0410 (.2809). Sample Size = 4432. OLS Regression Coefficients.

 Columns 1-3: OLS Standard errors in parentheses. Columns 4-5: 2SLS Standard errors in

 parentheses. * implies statistically different from zero at the 5% level of significance

 (two tailed test).

Equations 2-5 include controls for whether the training in the current year is ongoing at the

 date of the interview and whether training was ongoing at the date of the previous interview.

1st stage variables "Tt-τ" are indicators of training in previous years of tenure, number of

 previous training spells in other jobs, controls for whether the training in the previous

 year was ongoing at the date of the interview, and controls for whether no training is

 observed in previous jobs because of left censored data. 1st stage variables "X" are an

 intercept, AFQT, race, gender, age, marital status, urban residence, local area unemployment

 rate, SMSA, union, experience and experience squared, firm size, education, school attendance

 in the previous year, multiple site firm, number of previous jobs, government employment,

 part-time employment, number of children, and industry and occupation.

Explanatory variables are (in first differences) year dummies, marital status, local area

 unemployment rate, experience squared, education, part-time employment, school attendance

 in the previous year, and number of children.
6

Table 5: Derivation of Unbiased Coefficients

 (1) (2) (3)

b2 Belated Info.Bias, t=2 0 .05 .1

b3 Belated Info.Bias, t=3 0 .05 .1

b4 Belated Info.Bias, t=4 0 .05 .1

{α02,α12} {.080,.377} {.084,.397} {.088,.416}

{α03,α13} {.080,.358} {.084,.378} {.089,.397}

{α04,α14} {.020,.090} {.025,.11} {.029,.129}

Tenure=2 x Training (β2) 0.038

 (0.032)

 0.036

 (0.031)

 0.034

 (0.030)

Tenure=3 x Training (β3) 0.093 *

 (0.031)

 0.089 *

 (0.030)

 0.084 *

 (0.028)

Tenure=4 x Training (β4) 0.069

 (0.050)

 0.067

 (0.049)

 0.065

 (0.047)

Tenure=2 (γ2 - γ1) 0.033 *

 (0.012)

 0.034 *

 (0.012)

 0.034 *

 (0.012)

Tenure=3 (γ3 - γ2) -0.004

 (0.014)

 -0.003

 (0.014)

 -0.002

 (0.014)

Tenure=4 (γ4 - γ3) 0.006

 (0.018)

 0.007

 (0.018)

 0.007

 (0.018)

* implies statistically different from zero at the 5% level of significance (two tailed test).
Appendix for “Belated Training: The Relationship Between Training, Tenure, and Wages”

 This appendix shows how to generalize the analysis of measurement error in Section 4 so as

to include additional explanatory variables in the wage growth equation. To make the analysis tractable,

we assume that only one variable is measured with measurement error. Thus, instead of considering an

equation that pools all of the observations, we consider separate equations for each year of tenure. These

are the equations that are used to obtain the coefficients reported in Table 5.

 When training is the only explanatory variable in the wage growth equation,

 (A1) plim $

cov( , )

var( )

cov( , )

var( )

β β

ω θ

t

OLS

t

it it

it

it it

it

T

T

T

T

= −











1  +

 = β

θ

t

it it

it

it it

it

T T

T

T

T

cov( *, )

var( )

cov( , )

var( )











 + .

When the vector ΔXt of additional variables is included in the wage growth equation along with training,

the expected value of the OLS training coefficient is given by

 (A2) p

T T X

T X

T X

T X

t

OLS

t

it it it

it it

it it it

it it

lim $

cov( *, | )

var( | )

cov( , | )

var( | )

β β

θ

=











 +

Δ

Δ

Δ

Δ

Card (1996) shows that

cov( *, | )

var( | )

cov( *, )

T T X var( )

T X

T T

T

R

R

it it it

it it

it it

it

t

t

Δ

Δ

=

−

− −

−

2

1 0

2

1

1

α α

,
2

where Rt

2

denotes the theoretical R-squared coefficient from a linear regression of Tit against he other

explanatory variables in the wage growth equation. Thus, (A2) can be rewritten as

 (A3) p

T T

T

R

R

T X

T X

t

OLS

t

it it

it

t

t

it it it

it it

lim $

cov( *, )

var( ) cov( , | )

var( | )

β β

α α θ

=

−

− −

−

























+

2

1 0

2

1

1

Δ

Δ

 = β

η ν

α α

t

t t

t

t

t

R

R

b

1

1

1

2

1 0

2

− − −

− −

−

+

























.

 When training is the only explanatory variable in the wage growth equation, we have

 (A4) p

T

T T

t

IV

t

it it

it it it

lim $

cov( *, )

var( *) cov( *, ))

β β

ω

ω

= −

+











1 

 = β

ω

t

it

it it it

T

T T

var( *)

var( *) + cov( *, ))













 = βt

it

it it

T

T T

var( *)

cov( *, )











 ,

where we have used the fact that ωit = Tit - Tit*. When the vector ΔXt of additional variables is included in

the wage growth equation along with training, the expected value of the IV training coefficient is given by

 (A5) p

T X

T T X

t

IV

t

it it

it it it

lim $

var( *| )

cov( *, | )

β = β













Δ

Δ

.

Let
3

 (A6) Tit* = pt* + ( ΔXit − ΔXt)c + vit

denote the linear projection of Tt* on ΔXit , where ΔXt denotes the mean of ΔXit and E(vit) = E(vitΔXit) = 0.

The implied linear projection of Tit on ΔXit is

 (A7) Tit* = pt + (1-α1 - α0)( ΔXit − ΔXt)c + ξit.

It is straightforward to show that ξit is orthogonal to ΔXit . Combining (A5), (A6), and (A7), one obtains

 (A8) p

T p c X X

T p c X X T p c X X

t

IV

t

it t it t

it t it t it t it t

lim $

var( *) * ( )

cov( * * ( ), ( ) ( ))

β β

α α

=

− − −

− − − − − − − −













Δ Δ

Δ Δ 1 1 0 Δ Δ

 β

α α

t

it X X

it it X X

T c V c

T T c V c

t t

t t

var( *) '

cov( *, ) ( ) '

−

− − −

















Δ Δ

1 1 0 Δ Δ

,

where VΔXtΔXtis the population variance-covariance matrix of ΔXit. When E(ωit) = 0, it is straightforward

to show that var(Tit*) = pt(1- pt), cov(Tit*, Tit) = pt(1- α1 - pt), and

1- α1 - α0 = (1- α1 - pt)/( 1- pt). Thus, (A8) can be rewritten as

 (A9) p

p p c V c

p p c V c

p

p

t

IV

t

t t X X

t t X X

t

t

t t

t t

lim $

( ) '

( ) '

β β

α

α

=

− −

− − −

− −

−









































1

1

1

1

1

1

Δ Δ

Δ Δ

 = β

α

t

t

t

p

p

(1 )

1 1

−

− −













 = β

α α

t

1

1− 1 − 0











 .
4

 Now let us find the variance of $βt. As is well known, the instrumental and two stage least squares

estimators are equivalent. It is straightforward to show that

 (A20) (A20) 

$

$

$

.

$

.

( ) .

.

β β φ

φ

φ

t

TSLS

TSLS

TSLS

m

TSLS

t

m

t

t

n t

c

c

c

c

c

c

t

31

32

3

31

32

3

1

1

2

































=

































+

































−

z ' z z '

t t t

 = 

β

φ

φ

φ

t

m

t it t it

i

n

t i t it

i

n

t i t it

i

n

c

c

c

n Z Z

n d

n d

t

t

t

31

32

3

1

1

1

1

1

1

1

1

1

.

( )

( / ) ( )

( / )

.

.

( / )

































+

−

















































−

=

=

=

∑

∑

∑

z ' z t t

nt

,

where

zt ≡

Z d d d

Z d d d

Z d d d

t t t mt

t t t mt

n t n t n t n mt t t t t

1 11 12 1

2 21 22 2

1 2

...

...

. . . ...

. . . ...

...

































.

and dijt denotes the jth element in the vector ΔXit − ΔXt

.
5

n

n Z Z

n d

n d

t

t it t it

i

n

t i t it

i

n

t imt it

i

n

t

t

t

( / ) ( )

( / )

.

.

( / )

1

1

1

1

1

1

1

−

















































=

=

=

∑

∑

∑

φ

φ

φ

 is normal with mean 

cov( , )

cov( , )

.

.

cov( , )

Z

d

d

it it

i t it

imt it

φ

φ

φ

1

































and variance covariance matrix

At ≡

var(( ) ) cov(( ) , ) ... cov(( ) , )

cov(( ) , )

. lim

.

cov(( ) , )

Z Z Z Z d Z Z d

Z Z d

p

n

Z Z d

it t it it t it i t it it t it imt it

it t it i t it

it t it imt it

− − −

−

−





































φ φ φ φ φ

φ φ

σ

φ φ

φ

1

1

2 dt

'dt

.

Thus, 

$

$

$

.

$

βt

TSLS

TSLS

TSLS

m

TSLS

c

c

c

31

32

3

































is normal with variance covariance matrix Vt = (1/nt)plim( )

z ' z t t

n

−1 At plim( )

z ' z t t

n

−1

.

Letting Vt ≡ [vijt] , the variance of $βt

TSLS is simply [v ]

t

11 . Since

$

(

$

(

$

) )

$

(

$

)

$

(

$

)

$ βt β

t

t

t

t

b IV R R

r

R R

b

≡ − + + − + − − R

















2

2 2 2 2 2 2

4

1 2

1

1

2

1 and

b

R R

r

R R

b

R

t

t

t

2

2 2 2 2 2 2

4

1 2

1

1

2

( 1

$

(

$

) )

$

(

$

)

$

(

$ − + + − + − − )

















 converges in probability to

b

R R

r

b

R

t

t

t

2

2 2 2 2

4

1 2

1

2

( − + ( ) + − (1− ) , 

$βt

is asymptotically normal with variance

[v ]

b

R R

r

b

t R

t

t

t

11

2

2 2 2 2

2

4

1 2

1

2

( − + ( ) + − (1− )

















.