coefficients for debt: in the proportion of patients seen with Medicare regression for IMPeds doctors, and in the malpractice claim regression for surgeons. Both are
precisely measured zeroes.
22
VI. Cohort Wage Growth Restrictions
While I am interested in lifetime specialty specific wage paths, my data set consists of a single cross-section of doctors with fewer than ten years prac-
tice experience. Let observed wage growth rates in each specialty for the first, second, third, and fourth decades of doctors’ careers growth rates be
, ,
s s
s 1
2
l l
l , and
s3
l respectively where s indexes over the five specialties. Using only the information in
the YPS, only
s0
c —the growth rate in wages over the first decade post-residency—is identified by
s0
l . Any identification of the growth parameters after the first decade of practice
, ,
s s
s 1
2 3
c c
c come from the smoothing in the overlap polynomial since there are no doctors old enough in the YPS to provide information on
,
s s
1 2
l l , and
s3
l . In fact ,
,
s s
s 1
2 3
c c
c are not identified at all when σ the smoothing parameter approaches zero. Clearly, a cross-sectional identification strategy alone will not
work—I need other information about specialty-specific differential wage growth in the out decades.
The 1986 and 1996 SCMP report the average wage by specialty in four different age categories—younger than 36 years old, between 36 and 45 years old, between 46 and
55 years old, and between 56 and 65 years old. Thus, I observe wages for the 1985 cohort of physicians both in 1985 and in 1995 after the cohort has aged ten years. The
bottom rows of Table 1 report these cohort growth rates. To see how these numbers are constructed, consider a specific example—FP doctors between 36 and 46 years old in
1985. The 1986 SCMP reports the average wage of this cohort of doctors in 1985, while the 1996 SCMP reports their average wage in 1995, after they have aged 10 years. Let
waget,a is the average wage in year t for age cohort a. From these two numbers a sim- ple formula yields the average yearly wage growth rate, κ, over the decade:
, ,
wage a
wage a
10 1995
10 1
1985
10
+ =
+ l
This calculation implicitly assumes that the physicians in my 1990 data set will experience the same wage growth rates as the physicians in each age category did
between 1985 and 1995. For example, I assume that when the physicians in my data set are 45 years old, they will experience the same wage growth as 45-year-old physi-
cians did between 1985 and 1995. This assumption is false if the wage growth of physicians is subject to an idiosyncratic shock at some point in time in the future. In
defense of this assumption, however, I submit that any projection of future wages would be subject to this error. In fact, this assumption is standard in other contexts—
for example, life expectancy is often calculated using this assumption—and is an improvement over cross-sectionally based projections.
23
Bhattacharya 127
22. The mean probability of ever having a malpractice claim filed among surgeons is 28 percent, while the mean proportion of patients seen with Medicare by IMPeds doctors is 21 percent.
23. Cross-sectionally based projections assume that current 45 year old physicians will have the same wages 10 years from now as current 55 year old physicians have now.
Incorporating this cohort-based, specialty-specific wage growth information requires deriving a set of restrictions on the wage growth parameters in the model.
Let w
s
t,a be the average wage of physicians in specialty s at time t for the cohort aged a years. Since there is only a single cross-section of physicians in my data, the
age of the physicians and the time in the future are not separately identified. Doctors’ years of experience, t, can be related to their current age, years of residency training
in specialty s train
s
, and the age they received their MD age
MD
by the following identity:
t a
age train
11
MD s
= -
- Because of this linear relationship, the age argument in w
s
t,a can be suppressed without lost generality. The growth rate in wages from year t to t+1 is given by:
w t w t
w t 12
1
s s
s
+ -
However, the SCMP wage growth data give average growth over a decade rather than over a single year, and they average over age ranges rather than over calendar
time. The average yearly wage growth rate for physicians over the decade starting from age to age + 10 is given by:
10 age
冱
w a age
train w a
age train
w a age
train 13
10 1
1
s MD
s s
MD s
s MD
s
- -
- -
+ -
- -
+ =
age a
where identity Equation 11 is substituted for t. Equation 1 of the empirical model can be used to predict w
s
t over the course of a doctor’s career. I derive cohort wage restrictions by setting the empirical model’s
predictions about the average wage growth equal to the actual wage growth observed in the SCMP data. First, I set
w t
s
equal to w
st
from Equation 1 and then plug these wages into Equation 13. All the time invariant terms in Equation 1 cancel out of both
the numerator and denominator, leaving a set of three since there are three age cohorts that I follow in the SCMP data—I do not follow the 1985 cohort of 56 to 65
year olds restrictions on gt;
s
c , the time varying determinant of log wage in Equation 1:
24 k
10 k
10 1
+
; ;
; ;
冱 冱
exp exp
exp N
g t g t
g t for s
k 14
1 10
1 1
1 5
1 3
N s
s s
ks 36
f f
+ -
= =
= c
c c
l
= -
i 45
= age
For ease of exposition, Equation 11 is not plugged in. The final parameter estimates maximize the likelihood function Equation 8 subject to Equation 14.
25
The Journal of Human Resources 128
24. It is useful to think of the YPS micro level data as identifying the physician wage level in each specialty and its growth over the first ten years of practice. The wage growth rates over the final 30 years of practice
are effectively identified by the macro data from the SCMP. 25. There are several sets of additional parameter restrictions required to identify the model. First, α
s
must be set to zero for one of the specialties, just as would be the case in a multinomial logit model. I arbitrarily
set the parameters in the Radiology branch to zero. Second, since there are variables in the logit part of the likelihood function that take on different values for the different specialties, the parameters associated with
them must be common among all four specialty branches, just as would be the case in a conditional logit
VII. Wage and Selection Equation Estimates
