DISCUSSION PAPER SERIES

IZA DP No. 2870

Do You Receive a Lighter Prison Sentence
Because You Are a Woman? An Economic Analysis
of Federal Criminal Sentencing Guidelines
Supriya Sarnikar
Todd Sorensen
Ronald L. Oaxaca
June 2007

Forschungsinstitut
zur Zukunft der Arbeit
Institute for the Study
of Labor

Do You Receive a Lighter Prison
Sentence Because You Are a Woman?
An Economic Analysis of Federal
Criminal Sentencing Guidelines
Supriya Sarnikar
Westfield State College

Todd Sorensen
University of Arizona
and IZA

Ronald L. Oaxaca
University of Arizona
and IZA

Discussion Paper No. 2870
June 2007

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IZA Discussion Paper No. 2870
June 2007

ABSTRACT
Do You Receive a Lighter Prison Sentence
Because You Are a Woman? An Economic Analysis of
Federal Criminal Sentencing Guidelines*
The Federal criminal sentencing guidelines struck down by the U.S. Supreme Court in 2005
required that males and females who commit the same crime and have the same prior
criminal record be sentenced equally. Using data obtained from the United States Sentencing
Commission’s records, we examine whether there exists any gender-based bias in criminal
sentencing decisions. We treat months in prison as a censored variable in order to account
for the frequent outcome of no prison time. Additionally, we control for the self-selection of
the defendant into guilty pleas through use of an endogenous switching regression model. A
new decomposition methodology is employed. Our results indicate that women receive more
lenient sentences even after controlling for circumstances such as the severity of the offense
and past criminal history.

JEL Classification:
Keywords:

J78, K14, K42

discrimination, criminal justice, decomposition analysis,
limited dependent variable analysis

Corresponding author:
Ronald L. Oaxaca
Department of Economics
University of Arizona
McClelland Hall, Room 401QQ
Tucson, AZ 85721-0108
USA
E-Mail: rlo@email.arizona.edu

*

The authors gratefully acknowledge the helpful comments of participants at the 2006 Midwestern
Economics Association meetings, the 2006 Western Economics Association meetings, and the 2006
WPEG conference at the University of Kent and of seminar participants at IZA.

Gender equity has been one of the major global social issues to emerge out of the
20th century. A major focus of economists in this regard is on disparate labor market
outcomes for men and women. Emphasis is placed on human capital explanations for
gender wage gaps though there is some scope for other explanations, such as Becker
taste-driven discrimination, statistical discrimination, and market power. Labor market outcomes potentially a¤ect future criminal activities, and criminal activities can
a¤ect labor market outcomes. This paper examines the gender equity issue in the
criminal justice arena and notes that labor market outcomes and criminal justice
outcomes can be jointly determined. A popular perception in the criminal justice
system is that female criminal behavior is a less serious problem than male criminal
behavior. Detailed statistics compiled by the Bureau of Justice Statistics show that
women commit fewer o¤enses than men and substantially di¤erent types of o¤enses
than men. However, the statistics also reveal a rising trend in o¤enses committed
by females and an increase in the incarceration of females in recent years. Beyond
the labor market implications of gender equity in the criminal justice system there is
also a concern for allocative e¢ ciency regarding resources devoted to deterrence and
incarceration.
The Federal Sentencing Guidelines that arose out of the Sentencing Reform Act
of 1984 and that were subsequently struck down by the U.S. Supreme Court in 2005
(consolidated cases of United States v. Booker, No. 04-104, and United States v.
Fanfan, No. 04-105) required that males and females who commit the same crime and
have the same prior criminal record receive equal sentences. Critics of the sentencing
guidelines argue that women should be accorded separate treatment because females
who are caught in the criminal justice system “enter it due to circumstances that are
distinctly di¤erent from those of men”1 .
1

Others argue that gender is not a factor

“Research on Women and Girls in the Justice System.” National Institute of Justice Report

(September 2000) at page iii. Available at http://www.ncjrs.gov/pd¢ les1/nij/180973.pdf.

2

that should enter into the sentencing decision. The Supreme Court in its split 5
to 4 decision argued that the mandatory guidelines violated the rights of criminal
defendants to have a jury rather than a judge decide if defendants had committed all
elements of a given crime. Consequently, the guidelines are only advisory to judges
who may increase the length of sentences if they determine that the circumstances
based on jury determination or admission of the defendant merit a longer prison
sentence (Chicago Daily Law Bulletin 27 December 2005). The 2005 decision created
some ambiguity in how far judges could stray from the now "advisory" sentencing
guidelines. Currently, the Supreme Court is considering a case in which a sentencing
judge departed from the guidelines, only to have an appellate court overrule the
judge’s sentencing decision. The decision the Supreme Court hands down in this case
may have as large an impact on the way sentences are given as the 2005 case2 (New
York Times Feb 20, 2007)
Whether the circumstances in which a crime is committed should be a consideration
in criminal justice is not a question that we propose to answer here. Rather, we
address the question of whether or not women do indeed receive more lenient sentences
despite the sentencing guidelines. The answer to this question is important to both
sides of the debate. Those in the justice system who favor equal treatment but
believe that women are let o¤ too lightly may be especially harsh when judging a
female accused of crime, while those who favor separate treatment of women but
believe that they are treated equally may be less stringent. Thus, perceptions of
unequal treatment, when they are not based on systematic study and sound facts,
may lead to actual inequality in the justice system. A systematic study of whether
bias actually exists is therefore not only necessary but timely given the rising trend
in o¤enses committed by women and the increase in female incarceration rates as
2

The pending cases are Clairborne v. United States, No. 06-5618 and Rita v. United States, No.

06-5754.

3

evidenced by the data compiled by the Bureau of Justice Statistics. Further work
can begin to better tie the relationship between gender equity in the criminal justice
system with gender equity in the labor market.
An unpublished paper by Oaxaca and Sarnikar (2005) [henceforth OS] uses a rich
data set on sentencing outcomes from the United State Sentencing Commission to estimate separate logistic regressions for men and women, where the dependent variable
is a binary variable measuring whether or not convicted individuals received federal
prison time. While summary statistics from their data set show that females are less
likely to receive prison time than males, more sophisticated analysis can take account
of covariates that can explain some or all of the gender sentencing di¤erential.
In this paper, we consider outcomes from the sentencing process more broadly for a
sample of whites who were convicted while the mandatory sentencing guidelines were
still in e¤ect. Speci…cally, we look beyond the binary Prison/No-Prison outcome to a
continuous measure of prison sentence. Ideally, one would want to take into account
the fact that defendants must choose whether or not to plea bargain or to take their
chances in a trial. Given that we work with a sample of convicted individuals (we
do not have data on acquittals), we model the probability of whether the conviction
was the result of a trial versus a plea bargain. We treat sentences handed down due
to guilty pleas as outcomes from one regime, while sentences given to defendants
convicted in a trial are treated as outcomes from a separate regime. This approach
allows characteristics to be weighted di¤erently depending on the path to conviction.
One would expect that the average sentences of identical defendants facing identical
charges should be lower in the plea regime.3

In our data set, around 25% of all

criminal sentences involve no prison time. Because of this considerable mass point at
3

If the sentences were lighter in the trial regime it would be di¢ cult to believe that defendants

would ever do anything but plead not guilty, as this would generate a positive probability of facing
no sentence at all.

4

zero, it may be inappropriate to consider the distribution of sentencing outcomes to
be continuous. Also, the plea vs. trial regime is a choice variable for the defendant
so that we must account for self selection in our model. Accordingly, we treat the
outcome variable as a mixed discrete continuous variable. Therefore, our econometric
model is a censoring (Tobit) switching regression with endogenous switching, which
we estimate by full information maximum likelihood (FIML).
To measure how much of the male/female sentencing di¤erential can be attributed
to di¤erences in the characteristics of men and women, compared to how much of the
di¤erential can be explained by di¤erences in the weights applied to these characteristics by judges, we develop a new decomposition. This decomposition builds upon
Neuman and Oaxaca (2004), which addresses the issue of selectivity in the context of
a Heckman sample selection model. We expand this analysis to decompose di¤erentials in the switching regression model with censoring. Our approach takes account
of the fact that predicted outcome means will not generally match sample outcome
means because of the highly non-linear nature of the model.
Within our data set, the scarcity of observations on females and the preponderance
of observations in the plea regime conspire to leave us with an insu¢ cient sample
size to properly apply FIML to estimate the sentencing determination model separately for females. In the decomposition developed here, we exploit an insight from
Oaxaca and Ransom (1994) that allows us to decompose the male-female regime and
sentencing di¤erentials without actually estimating the model for females. Rather
than comparing weights from a male only and female only model, we instead are
able to compare the estimated parameters from the model for males with parameters
estimated in a pooled model for males and females.

5

LITERATURE
Since the seminal work of Becker (1968), there has been a signi…cant amount of
research aimed at understanding the economics of crime. In the basic economic model
of crime, a rational individual decides whether or not to allocate his/her time to
criminal activity by comparing the expected net return from criminal activity to the
expected returns from legitimate activity. The expected net return to criminal activity
consists of the potential …nancial and psychic bene…ts (B) of committing the crime
minus the cost (C) of committing the crime. The cost to the individual of committing
the crime is determined as the product of the probability (p) of being caught by law
enforcement and the severity of the punishment (S). If the returns to legitimate labor
market activity is the wage (W), then a rational, risk-neutral individual will engage
in criminal activity only if B-pS > W. This static model therefore predicts that
criminal activity can be deterred by either increasing the probability of detection(p),
the severity of punishment(S) and/or the wage rate (W) in the labor market.
Economists have since subjected these theoretical predictions to empirical testing
using econometric models of varying degrees of sophistication. Ehrlich (1973) and
Levitt (1997) estimate the impact of increased law enforcement presence on crime
and …nd that increasing law enforcement e¤orts have the desired e¤ect of lowering
the incidence of crime. Ehrlich (1977) estimated the deterrence e¤ect of capital punishment on crime. Witte (1980) …nds that the deterrence e¤ect of higher legal wages
was small compared to the deterrence e¤ects of the severity and certainty of state
imposed penalties. Johnson, Kantor and Fishback (2007) study the e¤ect of social
insurance on crime rates. Block and Gerety (1995) reports on laboratory experiments
that examine di¤erences between the criminal population and the general population
in the relative responsiveness to the deterrence e¤ects of severity of punishment versus
the deterrence e¤ects of the certainty of punishment. The results showed that convicts

6

were more deterred by increases in the certainty of punishment whereas the student
subjects were more deterred by increases in the severity of punishment. Kuziemko
(2006) uses New York State’s reinstatement of the death penalty to identify the effect of capital punishment on plea bargaining outcomes. Freeman (1999), Grogger
(1998), and Gould, Weinberg and Mustard (2002) …nd that falling real wages were a
signi…cant determinant of increasing crime rates during the decades of the 1970s and
1980s.
The link between deterrence e¤orts and crime rates is an endogenous one. Decisions to increase law enforcement e¤orts are often made in response to increasing crime
rates. Similarly, di¢ culty in …nding legitimate labor market employment might push
some individuals into criminal activity but the fact that an individual has engaged in
criminal activity also would lower that individual’s probability of …nding legitimate
employment. Myers (1983) investigates whether poor labor market prospects postrelease a¤ect the re-integration of ex-convicts into the mainstream. Using di¤erent
data sets, Myers …nds that better wages post-release signi…cantly reduced recidivism.
Witte and Reid (1980) also …nd that receiving a high wage on the …rst job after being
released from prison decreases recidivism and that the wage rate received by a prison
‘releasee’ depends mostly on the demand side characteristics such as the industry
and occupation rather than on the accumulated human capital of the ‘releasee’. Imai
and Krishna (2004) estimate a dynamic model of criminal behavior and show that
expected future adverse consequences in the labor market prove to be an e¤ective
deterrent to crime. Waldfogel (1994) estimated the e¤ects of conviction and imprisonment on post-conviction income and employment probabilities and found that the
state-imposed sanctions were much smaller in comparison to the “market sanction”,
which he estimated as the income lost due to conviction and imprisonment. Also, the
“market sanction” was signi…cant only for those o¤enders who worked at jobs that
required much trust. Grogger (1995) used longitudinal data and concluded that the
7

strong negative correlation between arrests and subsequent labor market sanctions
that was found in earlier cross-sectional studies was largely due to unobserved characteristics that in‡uence both criminal and labor market behavior. Grogger (1995)
however does …nd that there are signi…cant negative consequences of arrests in the
labor market but that they are short-lived.
Consistent with the predictions of the economic model of crime, the Sentencing
Reform Act of 1984 (SRA 1984) increased the length of punishment for almost all
crimes, eliminated probation and reduced the possibility of parole for good behavior.
Kling (2006) estimates the e¤ect of this increased severity of punishment on labor
market prospects of criminals post-release. Kling …nds that there is no signi…cant
adverse e¤ect on employment or earnings of criminals due to longer incarceration
lengths and concludes that this may be because prison rehabilitation programs may
be o¤setting the loss of potential work experience and human capital depreciation
while in prison.
The sentencing guidelines formulated pursuant to the SRA 1984 aimed to provide
uniform sanctions for the same crime by eliminating gender, age, or racial disparities
in sentencing. While economists have studied the deterrence e¤ect of severity of punishment quite extensively, relatively little literature exists on the optimality and desirability of uniform sentencing. Lott (1992) argues against uniform sentencing based
on the …nding that market sanctions in the form of lost incomes, opportunity costs
of imprisonment and the adverse impact of incarceration on labor market prospects
are disproportionately higher for individuals with higher incomes. Since the expected
total monetary penalty includes the reduction in legitimate earnings capability post
release, Lott argues that the state-imposed punishments should be proportionately
adjusted. Moreover, since mere conviction can restrict the post-conviction opportunities for higher income individuals more severely than for lower skilled people, Lott
argues that in order to equalize the severity of punishment, higher income individuals
8

would have to be convicted much less frequently than low-income criminals. One
could argue that such equalization could be addressed by di¤erential sentence length.
However, the sentencing guidelines explicitly prohibited sentencing judges from considering factors such as the defendant’s socioeconomic status, race, sex, age, and
religion. The punishment was to be proportional to the severity of the crime and the
defendant’s criminal history alone. Judicial discretion to change the sentence based
on characteristics of the defendant was thus severely restricted under the guidelines.
Several studies in the criminology literature have examined gender and racial disparities in sentencing prior to the formulation of sentencing guidelines. See Tonry
(1996) for a survey of these studies. Whether the guidelines have been successful in
reducing the disparity has also been studied extensively both in the criminology and
the law and economics literature. Anderson, Kling and Stith (1999), Kempf-Leonard
and Sample (2001) study sentencing disparities before and after the federal sentencing
guidelines. Mustard (2001) looks at racial and gender disparities in sentencing under
the federal guidelines and …nds that observed disparities in sentencing are mainly due
to the special circumstances when judges are allowed to depart from the guidelines
and not due to discriminatory tastes of judges. Schanzenbach (2005) estimates the
e¤ect of judicial demographics on sentencing outcomes and …nds that increasing the
proportion of female judges increases the gender disparity in sentencing and interprets this as evidence that male judges are paternalistic and therefore lenient towards
female o¤enders.
Almost all of the studies mentioned infer gender-based discrimination in sentencing
from the statistically signi…cant coe¢ cient on a dummy variable indicating the gender of the criminal o¤ender. Yet, sentencing discrepancies may be observed merely
because a judge takes into account extralegal circumstances of the defendant. If the
circumstances of male and female criminal defendants are substantially di¤erent, as
claimed by several authors, then the consideration of circumstances by judges may
9

appear as gender-based bias even when the judge exhibits no such discriminatory
tastes. Verdier and Zenou (2004) show that when there is statistical discrimination
in the labor market and everyone believes that blacks, for example, are more likely to
engage in criminal activity, then such beliefs lead to lower wages for blacks. When the
opportunity cost of a crime is thus lowered, such beliefs become self-ful…lling and lead
to higher crime rates among blacks. It is therefore important to thoroughly investigate whether any bias actually exists in the criminal justice system since perceived
bias may itself lead to actual bias. Given the adverse labor market consequences of
incarceration, unequal treatment of men and women in the criminal justice system
may lead to unequal prospects for men and women in the labor market as well.
Our research design separates the e¤ect of di¤erences in circumstances from the
e¤ect of di¤erences in weights attached to circumstances by judges. If a judge attaches
di¤erent weights to the same circumstances of a male and a female o¤ender, then we
may attribute that to a gender-based bias. But if a judge attaches the same weights to
circumstances but on average awards di¤erent sentences to male and female o¤enders
then that di¤erence in sentencing might be due to di¤erences in circumstances of the
two defendants. Oaxaca and Sarnikar (2005) use decomposition analysis to investigate
whether there exists any leniency towards women in the binary decision of whether or
not to imprison a convicted person. The results of this decomposition show that the
di¤erences in characteristics explain more than 100% of the gender sentencing gap.
If, when determining whether or not to sentence a woman to prison, judges applied
the same weights on characteristics as they use for men, women would actually be
slightly less likely to face prison.
DATA
The data used in this study are obtained from the United States Sentencing Commission’s data collection e¤orts and pertain to cases that terminated in convictions
10

over the period 1996-2002. The data set is available from the Federal Justice Resource
Statistics Center. In order to abstract from sentencing issues associated with race and
ethnicity, we have con…ned our attention to convicted white males and white females.
There were a total of 45,060 sentencing cases in our sample (37,104 cases for males
and 7,956 cases for women).
Table 1 presents a summary of the share of sentences involving no prison time.
Overall, a higher percentage of females receive no prison time upon conviction. This is
true for both the trial and guilty plea regimes. For both males and females, conviction
by a guilty plea is associated with a larger percentage of sentences involving no prison
time.
The variables reported in Table 2 are the ones we have constructed for use in our
sentence determination model. Both the measure of …nal o¤ense level and the criminal
history variable are set according to a …xed formula. To calculate the o¤ense level,
the case is assigned a base level for o¤ense and then adjusted for various aggravating
circumstances such as the use of a …rearm in the crime or obstruction of justice, or for
mitigating circumstances such as acceptance of responsibility. The criminal history
measure is a function of both the length of prior imprisonments and how recently these
sentences were given.4 . While men on average are awarded longer prison sentences
(42 months) than women (17 months), the severity of their o¤enses as measured by
the …nal o¤ense level scores are greater on average than those of women. Also, men
on average have a higher past criminal history score than women. Convicted men are
on average two years older than convicted women and are more likely to have private
counsel. A higher percentage of men are college graduates (13% vs. 7%).
4

For details on their construction of these variables, please see the following documents on the

USSC’s website:
http://www.ussc.gov/training/sent_ex_rob.pdf
http://www.ussc.gov/training/material.htm

11

In Table 3 we present summary statistics pertaining to the average length of sentences imposed on both men and women in each of our sample years. Note that in
each year the average male sentence is more than twice that of the average female
sentence. If one were to only consider these summary statistics and no covariates, it
would appear that women receive considerably lighter sentences than do males, and
that this di¤erence is considerably greater in the trial regime. Overall and in the
trial regime, average male sentences generally declined over the sample period while
average female sentences actually rose. Average sentences in the plea regime tended
to rise for both males and females.
ECONOMETRIC MODEL
Below we describe the econometric methods used to estimate the necessary parameters to decompose the sentence di¤erentials. First, we describe the model we
use to decompose the sentence di¤erence into an explained portion (di¤erences in
characteristics) and an unexplained portion (di¤erences in weights).
Sentencing
In our data set, we observe the sentencing outcomes for defendants whose cases
reach the sentencing phase. Recall that there are two ways in which a defendant’s
case can reach the sentencing phase. While a signi…cant number of defendants faced
sentencing after being convicted by a jury, the most frequent way a defendant reached
the sentencing phase was by pleading guilty. Plea bargains reached with a prosecutor
are often the reason for this guilty plea; these defendants are sentenced under what
we call the plea regime. When a defendant pleads not guilty, but is convicted in a
trial, they are sentenced under the trial regime. We de…ne y as the months in prison
the defendant is sentenced to, X as the vector of the individual’s characteristics, and

12

as the vector of weights on the defendant’s characteristics in the respective regimes.
Equation (1) represents sentencing outcomes when an individual pleads guilty or is
convicted by trial:
yi =

XP i
XT i

P
T

+ "Pi if defendant is in plea regime
+ "Ti if defendant is in trial regime.

(1)

Although the formal model permits di¤erences in the covariates appearing in each
sentencing regime, the empirical speci…cation actually used in this paper restricts
covariates to be identical in both sentencing regimes.
The very nature of a plea bargain suggests that the process determining the sentence
of the defendant will not be the same in the two regimes. We would then expect the
sentences received by two otherwise identical defendants to depend upon the way in
which they reached the sentencing phase. Put another way, the weights applied to
an individual’s characteristics will be di¤erent depending on which sentencing regime
the defendant is facing. Accordingly, it may be inappropriate to pool observations
from individuals in these two regimes into a single sentencing equation. If individuals
were exogenously selected into one of the two regimes, we could simply estimate the
two models separately.
In order to more formally take account of the regime outcome conditional upon
conviction, let

P

represent the probability of a guilty plea,

ability of going to trial and being convicted, and

T &A

T &C

represent the prob-

represent the probability of

going to trial and being acquitted. Conditional upon prosecution, these probabilities sum to 1. Because we do not have observations on those who went to trial
and were acquitted, we can only estimate the following conditional probabilities:
P

=

and

T &C

; which sum to 1 and where P C is the
+ T &C
P + T &C
probability that one’s conviction was from a guilty plea and T C is the probability
PC

TC

=

P

that one’s conviction was by trial. Let the variable s represent the conditional latent
variable corresponding to a defendant’s conviction by trial. The variable s takes on
13

a value of 1 if the defendant’s conviction is by trial, and a value of 0 if the defendant
enters a guilty plea. The vector index variable Zi is a set of variables a¤ecting this
probability. Accordingly, the binary regime determination model may be expressed
as

si = Zi + ui
1 if si > 0
si =
0 if si 0:

(2)
(3)

Correlation between unobservables in the plea decision stage and unobservables in the sentencing stage will create non random selection that will prevent us
from obtaining consistent estimates of the parameters if they are estimated by OLS
or Tobit.

To account for this self-selection, we model the sentence determination

process using a switching regression model with endogenous switching. We assume
that the error term from each regime’s sentence determination equation follows a
bivariate normal distribution with the error term from the selection equation. The
nature of this model requires that an explicit distributional assumption be made.
The structure of the error terms is given in the following variance-covariance matrix,
where T denotes the trial regime, P denotes the plea regime, and s denotes the binary
selection equation (the variance of which is normalized to 1)5 :
0
1
1
Ps
Ts
B
C
B
C
2
V = B Ps
PT C
P
@
A
Ts

5

PT

(4)

2
T

The errors in the two sentencing regimes could be correlated; however the model neither requires

nor provides identi…cation of this parameter.

14

The likelihood function of the model is then:
N
Y
1
yi XT i T
L =
Pr(ui >
T

i=1

si

Zi j"T i )

T

yi

1

XP i

P

P

1 si

Zi j"P i )

Pr(ui

P

(5)

This expression is simpli…ed once we take account of the conditional distribution
of u on " :

L =

N
Y
i=1

(

(

1
P

1

yi

XT i

T

Zi +

T

Ts
T

1

T

yi

XP i

(yi

Zi

P

Ps
P

1

P

XT i

T)

Ts

(yi

XP i

P)

Ps

!)si

!)1

si

(6)

One additional econometric problem we face is the non-continuous distribution of
the dependent variable. Because sentence length cannot be negative, and nearly 25%
of our sample receives no prison time, it may be necessary to account for this mass
point at 0 in order to obtain consistent estimates.6 In the context of our switching
regression model, we treat the dependent variable as a mixed discrete continuous
variable, with limit observations at 0. The sentence outcome is now represented as

yP i = XP i P + "Pi if defendant is in plea regime
yP i if yP i > 0 and si = 0
yP i =
0 if yP i 0 and si = 0
yT i = XT i T + "Ti if defendant is in trial regime
yT i if yT i > 0 and si = 1
yT i =
0 if yT i 0 and si = 1
6

(7)
(8)
(9)
(10)

We also estimate the model without accounting for censoring; the log-likelihood obtained is

signi…cantly lower than that obtained in the model where we account for the censoring.

15

The likelihood for the switching regression with endogenous switching and censoring
allows four di¤erent types of entries to the likelihood function: limit and non-limit
observations in both of the regimes. The likelihood function is

L =

N
Y

2

XTi

(

; Zi ;

T

i=1

(

s i li
T

1

yT i

T

1
P

XT i

T

2

Ts

XP i

Zi +

Ts
T

(yT i

1
Zi

P

2

XT i

; Zi ;

T)

Ts
Ps
P

1

P

where l = 1 for limit observations and

(1 si )li
P

P

T

yP i

XP i

(yP i
Ps

XP i

Ps

!)si (1
P)

!)(1

li )

si )(1 li )

(11)

represents the cumulative bivariate normal

distribution.
DECOMPOSING SENTENCING DIFFERENTIALS
To examine how much of the gender di¤erence in sentences is due to leniency toward
one sex or the other, we apply empirical methods developed in the labor economics
literature to estimate gender bias in criminal sentencing outcomes.

These meth-

ods have the advantage of decomposing gender di¤erences in sentencing outcomes
into two di¤erent components –one due to di¤erences in observable circumstances of
males and females convicted by the criminal justice system and another due to differences in unobserved circumstances or attitudes of judges towards the sexes. Such
decomposition is achieved by a three-step analysis.
The …rst step typically involves estimation of our empirical model for males and
females where the dependent variable is the length of the prison sentence. Here,
instead of estimating the empirical model separately for both males and females, we
estimate the model for males only. This approach is consistent with viewing the
unexplained gap as a residual. It is also necessary in our case, as the relatively small
16

number of female observations in the trial regime means that we are unable to identify
a number of parameters in an estimation of the model for females only. This approach
allows us to decompose the di¤erential without estimating the female weights, thus
circumventing the problem.
Our analysis departs from previous studies in the second step and adds greater
insight into the decision-making process that might lead to gender-based di¤erences
in criminal sentencing. In the second step, we predict the average sentence length
for females if they faced the male weights. In the third and …nal step, we use results
from the …rst two steps and decompose the di¤erences in length of sentences for
males and females into two components: one attributable to male-female di¤erences
in circumstances and a second attributable to unobserved di¤erences in attitudes of
judges towards the sexes and unobserved di¤erences in circumstances.
Decomposition methods such as the one described above were …rst developed in
labor market studies of gender and racial wage di¤erences [(Oaxaca 1973)] but have
not been used in studies of gender or racial bias in criminal sentencing decisions. Such
a method of estimating bias is valuable since it not only estimates any gender-based
di¤erences in sentencing outcomes but it also identi…es whether the observed bias
is due to gender di¤erences in circumstances or due to gender-based di¤erences in
weights attached to circumstances by judges.
In addition to the problems with identifying the female weights, we face two additional challenges which force us to expand beyond the Oaxaca (1973) decomposition.
The issue of selection bias in decompositions is addressed by Neuman and Oaxaca
(2004) in the context of a Heckit model. We are able to build o¤ of this work in the
decomposition we develop, as the Heckit is essentially a special case of an endogenous
switching regression model. Finally, we must account for the existence of the limit
observations in our data set.

17

Decomposing Sentencing Outcomes by Regime

First, consider the sentence determination equation for the trial regime:

yT i = XT i T + "T i if defendant is in the trial regime
yT i if yT i > 0; si = 1
yT i =
0 if yT i 0; si = 1

(12)
(13)

The expected value of a sentence in the trial regime is derived in Appendix 1.
De…ne the sample average sentence in the trial regime as yT m for males and yT f for
females.

The sample is composed of NT m men and NT f women.

The average

predicted value of sentences for males is de…ned as:

ybT m

NT m
1 X
=
ybT mi ;
NT m i=1

(14)

where ybT mi is the predicted sentence for the ith male in the trial regime. However, in

a …nite sample the predicted mean and the sample mean terms will not necessarily
be equal, i.e.

ybT m

NT m
NT m
1 X
1 X
=
ybT mi 6= yT m =
yT mi in general.
NT m i=1
NT m i=1

Assuming that the underlying model can be consistently estimated, we would have

plim(b
yT m

yT m) = 0

(15)

plim(b
yT f

y T f ) = 0:

(16)

When the predicted mean outcome does not match the sample mean outcome, we
have sample mean prediction error. The proportionate sample mean prediction errors
for males and females can be expressed as
18

It follows from consistency that

bT m = y T m
ybT m
bT f = y T f :
ybT f

y
plim(b) = plim
yb

(17)
(18)

= 1:

Appendix 2 contains a more detailed discussion of the use of sample mean error
predictions in the nonlinear decompositions adopted in this paper.
The average value of sentences for females in the trial regime using male weights is
de…ned as:

y^T0 f =

Nf
X

y^T0 f i

i=1

(19)

NT f

where y^T0 f i is a …tted value of the ith female sentence had they faced the male weights.
We decompose the di¤erence in average sentences in the trial regime as follows:

yT m

y T f = bT m (b
yT m

ybT0 f ) + (bT m

bT f )b
yT0 f + bT f ybT0 f

ybT f :

(20)

The …rst term in eq (20 ) measures the explained sentencing gap while the unexplained
gap is the sum of the last two terms. Note that the second term measures the
contribution of gender di¤erences in the sample mean prediction error while the last
term measures the contribution of gender di¤erences in the estimated parameters of
the model.7 It is therefore possible to separate out the e¤ect of gender di¤erences
7

Of course there are many instances in which there is no discrepancy between sample means and

predicted sample means, e.g. the linear regression model with a constant term, the logit model with
a constant term, and the second stage regression of a heckit sample selection model.

19

in ^T if the econometrician estimates both bT m and bT f . While we are able to
decompose the di¤erence in outcomes into the portion caused by di¤erences in weights

and di¤erences in characteristics, we will be unable to isolate the di¤erence caused
by weights into a portion caused by di¤erent ^T terms. However, if it is the case
that bT m

bT m yb0
Tf

bT f

0, the unexplained gap is totally captured by bT f ybT0 f

ybT f

ybT f . Under these circumstances one could identify the predicted mean
i
1 h
0
0
b
outcome for females as ybT f ybT f
yT m yT f
bT m
ybT f :
Tm y
bT m
The decomposition of sentences in the plea regime follows closely that of the trial

regime. Now using male weights from the plea regime, the …tted value of the length
of sentence in the regime becomes y^P , which di¤ers slightly in form from y^T .8
Decomposing Regime Choice

Now consider a decomposition of regime choice. Consider the regime determination
model given in (2) and (3) where a positive outcome indicates conviction by trial. The
observed proportion of females and males going to trial are, respectively

pT f =

pT m =

Nf
X

sf i

i=1

(21)

Nf
N
m
X
smi
i=1

(22)

Nm

We de…ne the di¤erence in outcomes for males and females as the observed di¤erences
in proportions of males and females in the trial regime, pT m

pT f .

Recall that we do not estimate the model separately for females.
8

However, we

The …tted value is now for individuals who are "selected in" in the plea equation, rather than the

"selected out" observations in the conviction by trial equation. The form of the selectivity term will
di¤er slightly. See Appendix 1 for the expressions governing the calculations of the mean outcomes.

20

are still able to decompose the di¤erence in male and female outcomes into the portion caused by di¤erences in characteristics and the portion caused by di¤erences in
weights. We go about these single model decompositions by decomposing di¤erentials
using only the estimated weights for males.
Here, we decompose the di¤erence in the propensity of males and females to be
convicted by trial regime using only male weights. Consider the regime determination
model estimated for males:

smi = Zmi m + ui
1 if smi > 0
smi =
0 if smi 0

(23)
(24)

The estimated weights in this model allow us to obtain a predicted probability of
conviction by trial for each individual in the sample:

p^T mi = (Zmi ^ m )

(25)

We compute the average predicted probability by averaging the individual predicted
probabilities:
p^T m

Nm
X
(Zmi ^ m )
=
Nm
i=1

(26)

Note that in the probit model, unlike the logit model, the average predicted probability of entering the trial regime will not necessarily equal the proportion of the
sample who do in fact enter the regime (for further work on the decomposition of
di¤erentials in the context of a probit model, see Fairly (2005) and Yun (1999)).
In practice the di¤erence is typically negligible. However, the selection probability
parameters in our model are obtained from FIML applied to the joint estimation of
the selection probability and sentencing equations. Hence, there is a need to scale the
mean predicted probabilities when conducting a decomposition of gender di¤erences
21

in the propensity to be convicted via the trial regime. As above for the sentencing
outcomes, the sample mean (probability) prediction errors for males can be expressed
as follows:
bsm = pT m
p^T m

(27)

The same consistency argument applies here as in the case of sentencing outcomes.
We estimate the average predicted probability of females being in the trial regime
had they faced the same weights as the males:

p^0T f

Nf
Nf
X
X
p^0T f i
(Zf i ^ m )
=
=
Nf
Nf
i=1
i=1

(28)

The di¤erence in the average probability of conviction via the trial regime can then
be decomposed as follows:

pT m

pT f = (pT m

bsm p^0 ) + (bsm p^0
Tf
Tf

pT f )

(29)

where the …rst term on the right hand side represents the di¤erence in probabilities
that can be attributed to di¤erences in characteristics, and the second term represents
the part of the di¤erence that can be attributed to di¤erences in weights.
Total Decomposition
Consider an algebraic decomposition of sentencing di¤erences by regime. De…ne
ym as the average sentence for males in our sample, and yf as the average sentence
for females. Each gender’s average sentence will be a weighted average of the average
sentence in the two regimes:

ym = yT m pT m + yP m (1
yf = yT f pT f + yP f (1
22

pT m )
pT f )

(30)
(31)

The di¤erence in average sentences can then be expressed as

ym

yf = yT m pT m + yP m (1

pT m )

yT f pT f

Adding and subtracting the terms yT f pT m and yP f (1

yP f (1

pT f )

pT m ), and collecting terms

appropriately yields

ym

yf = (yT m
+(yT f

yT f ) pT m + (yP m
yP f ) (pT m

yP f ) (1

pT m )

pT f ):

(32)

The …rst two terms in (32) can be interpreted as a weighted average of the di¤erences
in mean sentence outcomes for men and women (weighted by the probability of being
in each of the two regimes). The …nal term can be interpreted as the di¤erence in
mean sentence outcomes that can be attributed to gender di¤erences in the propensities of being in the trial regime (weighted by the di¤erences in mean outcomes among
females in the two regimes).
Recall how we decomposed each of the single decomposition terms.

Denote the

portion of the di¤erence attributed to di¤erences in characteristics (the explained
portion) as E. The portion of the di¤erence attributed to gender di¤erences in the
parameters (the unexplained portion) is denoted as U . Each portion also contains a
subscript denoting the part of the estimation from which it originates:

23

yT m

yP m

yT f =

yP f

i h
ybT0 f ) + (bT m

h
bT m (b
yT m

= ET + UT
h
= bP m (b
yP m

i h
ybP0 f ) + (bP m

= EP + UP

pT m

bsm p^0 ) + (bsm p^0
Tf
Tf

pT f = (pT m

bT f )b
yT0 f + bT f ybT0 f

ybT f

bP f )b
yP0 f + bP f ybP0 f

ybP f

i
i

(33)

(34)

pT f )
(35)

= Es + Us

The decomposition of the overall gender sentencing gap can then be expressed as
ym

yf = [(ET + UT ) pT m + (EP + UP ) (1
+(yT f

pT m )]
(36)

yP f ) (Es + Us )

= ET pT m + EP (1
|

+UT pT m + UP (1
|

pT m ) + Es (yT f
{z
E

pT m ) + Us (yT f
{z
U

yP f )
}

yP f ) ;
}

where E is the total amount of the overall gender sentencing gap that is explained
by di¤erences in characteristics, and U is the total unexplained gap associated with
di¤erences in weights.
We note that a more straight forward total decomposition of the mean sentencing
di¤erences between men and women can be calculated as
ym

yf = ym

where
y^f0

=

P

i

^m y^0 + (^m y^0
f
f

p^0T f i y^T0 f i + 1
Nf

(37)

yf )

p^0T f i y^P0 f i

and
^m =

P

i

[^
pT mi y^T mi + (1
Nm
24

p^T mi ) y^P mi ]

1
ym

:

In this decomposition y^f0 is the mean …tted overall sentence for females using the
male weights. Empirically, it turns out that both (36 ) and (37) yield virtually
identical values of the total explained and unexplained portions of the overall gender
sentencing gap. However, a shortcoming of the decomposition given by (37) is that
it obscures the sources of the overall gender sentencing gap revealed by the more
detailed decomposition given in (36).
RESULTS
Formal theory does not o¤er very much guidance on the actual speci…cation of the
regime selection and sentencing equations. The sentencing guidelines largely con…ned
federal court judges to considering only current o¤ense level and criminal history
when passing sentence. Speci…cally, the guidelines exclude race, sex, national origin,
creed, religion, and socioeconomic status. Furthermore, employment and family ties
and responsibilities are also not to be considered in awarding criminal sentences.
With only limited exception, age and education are not supposed to be relevant
for sentencing decisions. Judges are permitted to award lighter prison sentences to
elderly defendants. Since we have data on these various potential factors, we are able
to empirically determine the extent to which they turn out to in‡uence sentences
because of, or despite, the guidelines. The variables that appear jointly in the regime
selection and sentencing equations are indicators for females (in the pooled sample),
education, marital status, the circuit court district, and year while the continuous
variables appearing jointly pertain to prior criminal history, number of dependents,
and age. An indicator for U.S. citizenship appears in the regime selection equation
but not in the sentencing equations. While judges should not take into account the
nationality of a defendant when determining her sentence, citizenship should serve
as a proxy for this defendant’s knowledge of and experience with the U.S. criminal
justice system; we would expect risk averse individuals with less knowledge of how
25

this process works to be less likely to take their chances in a trial rather than striking
a plea bargain deal. An indicator for a defendant’s …ne being waived appears in the
sentencing equation but not in the regime selection equation. This variable serves
as a crude proxy for income. Also, a cubic polynomial function of the severity of
the …nal o¤ense level appears in the sentencing equations but are excluded from the
regime selection equation. Both the …ne variable and the …nal criminal o¤ense level
variables are not determined at the time that the individual makes the decision about
going to trial. Given that defendants do not have perfect foresight, these variables
should determine the …nal sentence given but not a¤ect the plea decision.
Although our data span both cases and years, it is not treated as a panel. The data
are available as separate cross-sections by case for each year. Each case corresponds to
all prosecutions ending in convictions of an individual in the given year and the total
prison time awarded. While it is theoretically possible for an individual to appear in
more than one year’s cross-section, we suspect that this is not very common. Among
males the average prison sentence is 3.5 years over a period of 7 years. This does
not leave much time for multiple year convictions unless o¤enses are committed while
the individual is in prison. In the case of females the average prison sentence is 1.4
years over the period of our study. This would allow for multiple year convictions
except that the crime rate is still much lower for females. Female cases account for
just under 18% of the total number of cases in our data set.
To get a sense of whether or not there may be favoritism towards women, we …rst
estimate our model on a pooled sample of males and females, including an indicator
variable for whether the observation is that of a female o¤ender. In Table 4 we present
parameter estimates from this pooled sample of males and females The estimated
coe¢ cient on the female indicator variable is negative and signi…cant in the selection
equation, indicating that women are less likely to obtain their convictions via the trial
regime, where average sentences are higher. More educated and married individuals
26

are more likely to obtain their convictions through trial rather than through guilty
pleas. Being a U.S. citizen is associated with a lower probability of obtaining one’s
conviction via trial as opposed to a guilty plea. The chances that one would obtain
their conviction via trial rather than via a guilty plea rise with age until around 73
years after which the trial regime probability declines. The circuit court district in
which the conviction took place does a¤ect the probability of conviction via trial vs.
guilty plea. The year indicators (where 2002 is the omitted reference group) suggest
that the probability of obtaining conviction via trial relative to guilty plea steadily
declined over time. A more extensive past criminal history was positively associated
with conviction by trial vs. a guilty plea. Having a private defense counsel has a
statistically signi…cant negative impact on the probability of conviction by trial.
The estimated coe¢ cients on the female gender indicator are negative and statistically signi…cant in both sentencing regimes, but they are of a greater magnitude (in
absolute value) in the trial regime. Even before we allow all weights to di¤er by gender, this indicates that women may receive lighter sentences than men. This would
seemingly violate the sentencing guidelines. Contrary to the guidelines, marital status and number of dependents do a¤ect prison sentences, but only in the plea regime.
Married defendants receive shorter sentences in the plea regime. Having more dependents leads to shorter sentences in the plea regime. Age and education exhibit some
e¤ect on sentences though ordinarily these are not considered relevant by the guidelines. Sentence length rises with age and peaks at 69 years if one is convicted in the
trial regime and peaks at 29 years in the plea regime. Although the guidelines permit
lighter sentences for the elderly, a peak of 29 years in the plea regime and the strong
signi…cance of the age terms in the trial regime would not seem to be entirely consistent with the guidelines. Education appears to lower sentences in the plea regime
and raise them in the trial regime. Those who have been convicted and had …nes
waived receive longer sentences in the plea regime. If this variable adequately proxies
27

incomes of the defendants, then it would seem that poorer defendants receive longer
sentences in the plea regime. As expected the extent of a defendant’s criminal history
and severity of current …nal criminal o¤ense contribute to longer prison sentences in
both regimes. The signs and magnitudes of the linear, quadratic, and cubic terms
jointly imply that, for all relevant values of the variable, as the severity of the crime
for which one is convicted increases, sentence length increases at an increasing rate.
Having a private defense counsel lowered prison sentences in both conviction regimes.9
Similar to the case with conviction regime selection, the circuit court district in which
the conviction took place does a¤ect sentence lengths. The estimated coe¢ cients on
the time indicator variables reveal that, ceteris paribus, sentence length had been
declining over time in the trial regime while rising in the plea regime. Estimates of
the correlations between the conviction regime error and the sentencing regime errors
suggest that unobservables in the selection equation are negatively correlated with
unobservables in the trial sentencing equation and positively correlated with unobservables in the plea regime. Roughly speaking, this means that those who are more
likely to select into the conviction by trial regime can expect shorter sentences in the
trial regime and longer sentences in the plea regime. While this is a sensible result,
one potential problem is that the estimated correlation coe¢ cient between the regime
selection equation error term and the plea regime sentencing error term is close to
the boundary value of 1. It is probably the case that this extreme estimate of the
correlation coe¢ cient is caused by the fact that ninety …ve percent of the sample
9

If the choice of defense counsel and the conviction regime are jointly determined, then the choice

of defense counsel would be endogenous in the model. Accordingly, we estimate a model to determine
if the decision to be represented by a private attorney is jointly determined with regime choice. By
estimating the model with a bivariate probit, we can test for this possible correlation in the two
error terms related to these decisions. Our estimation …nds the error term correlation coe¢ cient to
be insigni…cant, suggesting that the coe¢ cient on the defense counsel variable in the main model is
consistently estimated.

28

represent convictions via guilty pleas.
In Table 5 we report the FIML estimates based on just the male sample. Since
the results for males are qualitatively the same as those for the pooled sample, we
do not separately discuss these estimates. The major purpose behind estimating the
model separately for males is to provide us with the necessary parameter estimates
to compute the decomposition of gender di¤erences in prison sentences.
Decomposition results are reported in Tables 6 through 8. We begin with Table 6
which presents mean sentencing outcomes by regime and regime selection di¤erences
as well as predicted outcomes using estimated male weights. On average men are
awarded nearly 25 more months of prison than women. This varies by sentencing
regime. For those convicted by trial, men received an average of 69 more months of
prison than women. Among those who plead guilty, men received an average of almost
22 more months of prison time than women. A higher percentage of men than women
received their convictions via trial vs. a guilty plea, 5.5% vs. 3.5%. From the …tted
(predicted mean) sentences for males, we are able to calculate the proportionate mean
sample prediction errors. The most accurate prediction corresponds to the plea regime
which is the one into which the vast majority of the cases fall. The last column of
Table 6 reports the predicted outcomes for females using the FIML estimated weights
for men and are comparable to the calculated …tted values for men reported in the
next to the last column in Table 6. For the actual decompositions, the proportionate
mean prediction errors for men are applied to the predicted outcomes for women
obtained using the estimated male weights. The …gures in Table 6 clearly imply that
if females had faced the same sentence determination process as men, they would
have experienced longer prison sentences in each regime, though still less than those
of men, and would have had a higher propensity to have received their convictions
from the trial regime as opposed to the plea regime.
Our decompositions of gender sentencing di¤erences in each regime and gender
29

di¤erences in conviction regime probabilities are reported in Table 7. Di¤erences in
the female mean characteristics explain 46% of the gender sentencing di¤erential in
the trial regime and 66% of the sentencing di¤erential in the plea regime. We observe
that of the 69 month sentencing gap that favors women in the trial regime, nearly 38
months of the gap cannot be accounted for by gender di¤erences in circumstances.
Of the 22 month sentencing gap that favors women in the plea regime, 7 months of
the gap cannot be accounted for by gender di¤erences in circumstances. Only about
21% of the 2.1 percentage point gender gap in the propensity to obtain conviction in
the trial regime can be explained by gender di¤erences in characteristics. Females are
also less likely to be sentenced in the trial regime, though their characteristics suggest
they would actually be more likely to be sentenced in this regime if they were to face
the male weights (though still less likely than males).
In Table 8 we parse out the components that add to the overall gender sentencing
di¤erence across both conviction regimes. These components weight the explained
and unexplained portions of the sentencing gaps in each regime by the probabilities
of being in each regime and gender di¤erences in these probabilities. Of the nearly
25 month overall gender sentencing gap favoring women, 3.8 months (15.4%) arises
from gender sentencing di¤erences in the trial regime. Gender sentencing di¤erences
in the plea regime account for a little over 20 months (81.6%) of the overall gap. The
remainder of less than one month (3.0%) is accounted for by gender di¤erences in
conviction regime probabilities. Overall, the explained portion of the gap accounts for
about 15.4 months (62.7%) of the total gender sentencing di¤erence. This leaves about
9.5 months (38.3%) that cannot be explained by gender di¤erences in circumstances.
Table 8 disaggregates the explained and unexplained portions of the overall sentencing
gap by contributions from each sentencing regime and sentencing regime probabilities.
The plea regime accounts for the largest contribution to the overall explained gap (13.5
months or 87.6%) and to the overall unexplained gap (6.8 months or 72.0%). In fact
30

the largest single component of the constituent parts of the overall gender sentencing
gap is the 13.5 month explained gap from the plea regime which accounts for 54.0%
of the overall advantage of women in awarded sentences.
CONCLUSION
Unlike any studies in the literature so far, our study separates observed gender differences in sentencing into two di¤erent components –one attributable to di¤erences
in circumstances of male and female criminal defendants, and the second attributable
to di¤erences in attitudes of sentencing judges towards male and female defendants
and the di¤erences due to unobservable characteristics of the male and female defendants. Our model takes account of the joint determination of sentences by regime
and conviction regime selection as well as censoring occasioned by sentences that do
not involve prison time. We are able to determine the role of gender di¤erences in
selection regime probabilities. Such decomposition provides a better insight into the
decision-making process of sentencing judges. Knowing whether judges consider extralegal circumstances in their decision making is important, but knowing how they
consider extralegal circumstances is useful to policy makers in deciding how to reform sentencing guidelines to ensure equal treatment. This study not only examines
whether judges consider extralegal circumstances but if they do, it asks whether they
attach the same weight to circumstances of males and females. Even in light of the
Supreme Court’s decision in 2005 to strike down the Federal Sentencing Guidelines,
our results may o¤er some guidance as to what to expect now that judges are less
constrained in imposing sentences.
We …nd that women receive prison sentences that average a little over 2 years
less than those awarded to men. Even after controlling for circumstances such as
the severity of the o¤ense and past criminal history, women receive more lenient
sentences. Approximately 9.5 months of the female advantage cannot be explained
31

by gender di¤erences in individual circumstances. In other words if women faced the
same sentencing structure as men, women would on average receive 15.4 months less
prison time than men rather than 24.9 months less prison time. Most of the gender
gap arises from convictions via guilty pleas, which account for the vast majority of
the convictions observed in our data. Besides gender, we …nd evidence that judges
took into account factors such as family circumstances which are expressly prohibited
from consideration when awarding sentences.
One should bear in mind that our data permit us to examine only the end stage of
the criminal justice system. A more comprehensive treatment would take account of
the fact that before arriving at the judge for sentencing, a defendant must also pass
through a jury or possible plea bargain with a prosecutor. Before reaching this stage,
other groups, such as the police and the prosecution, have the potential to create bias
in the criminal justice system. Future work will focus on separating out di¤erential
outcomes layer by layer, as well as making explicit the impact of gender bias in the
criminal justice system on gender di¤erences in labor market outcomes.

32

Appendix 1: Expected Value of Dependent Variable with Censoring
The expected value of a censored dependent variable is simply the product of the
probability of observing a non-limit observation and the expected value of the dependent variable given that it is a non-limit observation, plus the probability of observing
a limit observation times the expected value of the dependent variable given that it
is a limit observation. Because the censoring point is at zero, the expected value of
limit observations is 0, causing the second term to drop from the expression. We …rst
consider the trial regime:

E[yT i jsi = 1] = Pr(yT i > 0jsi = 1) E[yT i jyT i > 0 \ si = 1]
+ Pr(yT i

0jsi = 1) E[yT i jyT i

0 \ si = 1]

= Pr(yT i > 0jsi = 1) E[yT i jyT i > 0 \ si = 1]
+ Pr(yT i

0jsi = 1) 0

= Pr(yT i > 0jsi = 1) E[yT i jyT i > 0 \ si = 1]

Consider each of the two right hand size terms separately.

(1)

First, consider the

probability of observing a non-limit observation, conditional upon selection. From
our speci…cation of the data generating process for y and s, we can express this as
the function of two random variables, " and u.

Pr(yT i > 0jsi = 1) = Pr("T i < XT i

T jui

< Zi )

(2)

By Bayes’rule we can express this as the joint probability that a non-limit observation
is selected into the trial regime, divided by the probability of that observation being

33

in the trial regime. This term can then be expressed using values from the cumulative
normal and cumulative bivariate normal distributions.
Pr( "TTi <

Pr(yT i > 0jsi = 1) =
2(

=

XT i

T

T

XT i

T

T

\ ui < Z i )

Pr(ui < Zi )

; Zi ;

sT )

(3)

:

(Zi )

Finally, we must consider the expected value of the dependent variable, given that
it is a non-limit observation in the trial regime. Recall that non-limit observations
take on the value

E[yT i jyT i > 0 \ si = 1] = E[yT i jyT i > 0 \ si = 1]
= E[yT i jyT i > 0 \ si > 0]
XT i T
"T i
<
\ ui < Z i ] :
= E[yT i j
T

(4)

T

This expected value appears similar to the expected value of the dependent variable
in the main equation of the Heckit model: it is truncated by the draw for the error
term in the selection equation. It also appears similar to the expected value of the
dependent variable in the Tobit model: it is truncated by the draw for the error term
in the main equation. This incidence of "double truncation" however, is substantially
more complex than the single truncation in either the Tobit or the Heckit. We derive
it for our model based on page 72 of Johnson and Kotz (1972):
XT i

E[yT i jyT i > 0 \ si = 1] =
(

+

Tf

sT

(

2(

XT i
T

T

XT i
T

T

T

; Zi ;
1

) (p

1

1

( Zi ) ( p

1

34

sT )

2
sT

2
sT

[ Zi

[ XT i

XT i
T

T

T

])

)

( Zi )])

(5)

The resulting expected value of the length of sentence in the trial regime is:

E[yT i jsi = 1] =

2(

XT i
T

T

; Zi ;

(Zi )

sT )

E[yT i ]

We can then de…ne the y^T (X; Z; ^m ) = E[yT i ] as given above.

35

(6)

Appendix 2: A Note on Sample Mean Prediction Error in
Decompositions
In decomposition analysis, the standard term to decompose is the di¤erence between the sample mean of the dependent variable for two groups. De…ne the sample
mean values for groups m and f as y m and y f , where each group has Nm and Nf
members, respectively. After estimating an econometric equation for both of the
groups, we can then calculate …tted values ybmi and ybf i for each individual in groups
m and f , respectively. The average …tted value for members of these groups is:
ybm
ybf

Nm
1 X
=
ybmi
Nm i=1

(1)

Nf
1 X
ybf i
=
Nf i=1

(2)

De…ne ybfo i as the …tted value of an observation in group f , had that individual faced

the group m estimated parameters. The mean of this variable for group f is then:

ybf0

Nf
1 X 0
=
yb
Nf i=1 f i

(3)

By adding and subtracting the ybf0 term, the decomposition is then expressed as:
ym

ybf0 ) + (b
yf0

y f = (y m

yf )

(4)

where the …rst term expresses the di¤erence in the left hand side variable which can
be attributed to di¤erences in the characteristics of the two groups, and the second
term expresses the di¤erence caused by di¤erences in the parameters the two groups
face.
Assuming that the underlying model can be consistently estimated, we would have

36

plim(b
ym

ym) = 0

(5)

plim(b
yf

yf ) = 0

(6)

However, in a …nite sample, the yb and y terms will not necessarily be equal. We can

express the sample mean prediction error in the model as follows:

y m = bm ybm
It follows from consistency that

y f = bf ybf

plim(b) = plim

The decomposition can now be expressed as:

ym

(7)

y f = (bm ybm

(8)

y
=1
yb

ybfo ) + (b
yfo

bf ybf )

(9)

The impact of the estimation error becomes more clear if, instead of adding and
subtracting ybfo , we instead add and subtract
ym

y f = (bm ybm

= (bm ybm
= (y m

bfo
my

bm yb0 ) + (bm yb0
f

f

bm yb0 ) + bm
f

bm yb0 ) + (bm yb0
f

f

bf ybf )

bf yb0 + bf yb0
f
f

yf )

ybf

(10)
(11)

Thus, the b terms contribute to both the explained and unexplained portions of

the mean decomposition.

In principle it is possible to separate out the e¤ect of gender di¤erences in the b

parameter from the e¤ect of di¤erences in other parameters eq (10). However, this is
37

only feasible if the econometrician estimates both the bm and bf terms. In our case,

we lack su¢ cient data to identify the weights in the model for females. Consequently,
we only are able to decompose the di¤erence in mean outcomes into the portion caused
by di¤erences in weights and di¤erences in characteristics according to eq (11).

38

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42

Table 1
Percentage of Sentences Involving No Prison Time

Year
1996
1997
1998
1999
2000
2001
2002

Total (%)
25.26
25.25
21.63
21.98
23.21
21.67
22.42

Males
Trial (%)
6.45
4.83
4.56
9.76
4.78
9.90
5.63

Plea (%)
27.00
26.99
22.94
22.74
24.13
22.10
22.86

Total (%)
44.41
41.85
37.67
39.97
38.03
35.84
42.57

Females
Trial (%)
12.73
21.74
18.60
34.88
10.26
14.81
17.39

Plea (%)
46.34
42.80
38.42
40.15
39.00
36.32
43.04

Table 2
Variable Definitions and Summary Statistics

Variable
TOTALMONTHS
REGIME
FEMALE
FINEWAIV
HISCHOOL
GED
SOMECOLL
COLLGRAD
NUMDEPEN
MARRD
CITIZN
DEFENSEP
XCRHISSR
CRIMHIS1
CRIMHIS2
CRIMHIS3
CRIMHIS4
CRIMHIS5
CRIMHIS6
XFOLSOR
XFOLSOR2
XFOLSOR3
AGE
AGE2
CIRC1
CIRC2
CIRC3
CIRC4
CIRC5
CIRC6
CIRC7
CIRC8
CIRC9
CIRC10
CIRC11

Description
Length of prison sentence in months
Indicator for trial regime
Indicator for female
Indicator of fine being waived
Indicator for high school education
Indicator for general equivalency diploma
Indicator for some college attended
Indicator for a college degree or higher
Number of dependents
Indicator for married or cohabiting
Indicator for US citizen
Indicator for private counsel
Final criminal history category
Final criminal history category =1
Final criminal history category =2
Final criminal history category=3
Final criminal history category =4
Final criminal history category =5
Final criminal history category =6
Final offense level
Final offense level squared
Final offense level cubed
Age of defendant
Age of defendant squared
Circuit indicators
Circuit indicators
Circuit indicators
Circuit indicators
Circuit indicators
Circuit indicators
Circuit indicators
Circuit indicators
Circuit indicators
Circuit indicators
Circuit indicators

Overall
Mean
Std. Dev
37.27
73.12
0.05
0.22
0.18
0.38
0.84
0.37
0.24
0.43
0.13
0.34
0.26
0.44
0.12
0.32
1.13
1.41
0.26
0.44
0.95
0.21
0.36
0.48
2.11
1.60
0.58
0.49
0.12
0.32
0.13
0.33
0.06
0.24
0.04
0.19
0.08
0.27
16.80
8.30
351.02
325.02
8507.77
11499.56
37.74
11.14
1548.66
888.58
0.03
0.16
0.11
0.31
0.04
0.21
0.05
0.21
0.10
0.30
0.08
0.28
0.04
0.20
0.08
0.27
0.27
0.44
0.05
0.23
0.14
0.35

Males
Mean
Std. Dev
41.67
78.03
0.06
0.23
0.00
0.00
0.83
0.38
0.23
0.42
0.14
0.34
0.25
0.43
0.13
0.33
1.13
1.44
0.26
0.44
0.95
0.22
0.37
0.48
2.23
1.66
0.54
0.50
0.12
0.33
0.13
0.34
0.07
0.25
0.04
0.20
0.09
0.29
17.37
8.35
371.49
332.24
9140.23
11899.77
38.08
11.22
1576.29
900.04
0.03
0.17
0.12
0.32
0.05
0.21
0.05
0.21
0.10
0.30
0.08
0.27
0.04
0.20
0.08
0.27
0.27
0.44
0.05
0.22
0.14
0.35

Females
Mean
Std. Dev
16.76
37.07
0.03
0.18
1.00
0.00
0.87
0.34
0.28
0.45
0.11
0.32
0.29
0.45
0.07
0.26
1.09
1.29
0.25
0.43
0.97
0.16
0.30
0.46
1.57
1.16
0.74
0.44
0.09
0.29
0.10
0.29
0.03
0.17
0.02
0.13
0.03
0.16
14.11
7.51
255.55
269.16
5558.17
8832.43
36.16
10.59
1419.79
820.88
0.02
0.14
0.08
0.27
0.04
0.19
0.05
0.21
0.12
0.32
0.09
0.29
0.04
0.20
0.09
0.29
0.28
0.45
0.06
0.24
0.13
0.34

1996
1997
1998
1999
2000
2001
2002

Year indicators
Year indicators
Year indicators
Year indicators
Year indicators
Year indicators
Year indicators

0.13
0.13
0.13
0.14
0.14
0.15
0.17

0.33
0.34
0.34
0.35
0.35
0.36
0.37

0.13
0.14
0.13
0.14
0.14
0.15
0.17

0.33
0.34
0.34
0.34
0.35
0.36
0.38

0.12
0.13
0.14
0.15
0.15
0.15
0.16

0.33
0.34
0.35
0.36
0.35
0.36
0.36

Table 3
Mean Sentences in Months

Year
1996
1997
1998
1999
2000
2001
2002

Total
43.26
43.81
42.00
42.15
40.12
41.20
39.87

Males
Trial
133.44
136.47
124.29
112.62
108.01
94.50
111.85

Plea
34.92
35.93
35.66
37.79
36.75
39.24
38.00

Total
15.00
19.27
16.07
15.08
15.93
18.27
17.58

Females
Trial
44.37
94.50
36.17
30.66
44.85
59.19
55.26

Plea
13.21
15.73
15.28
14.51
14.92
17.34
16.88

Table 4
Censored Switching Regression with Endogenous Switching: Pooled Sample

Variable
Constant
FEMALE
FINEWAIV
HISCHOOL
GED
SOMECOLL
COLLGRAD
CITIZN
MARRD
NUMDEPEN
DEFENSEP
CRIMHIS2
CRIMHIS3
CRIMHIS4
CRIMHIS5
CRIMHIS6
XFOLSOR
XFOLSOR2
XFOLSOR3
AGE
AGE2x10-2
CIRC2
CIRC3
CIRC4
CIRC5
CIRC6
CIRC7
CIRC8
CIRC9
CIRC10
CIRC11
1996
1997
1998
1999
2000
2001
Sigma 0
Rho 0u
Sigma 1
Rho 1u
N
Log-Likelihood

Regime Selection
Parameter
Asmp Z
-2.685
-17.17
-0.198
-5.90
0.001
0.020
0.017
0.186
-0.043
0.053
-0.008
-0.039
0.038
0.030
0.019
0.107
0.077

0.03
0.57
0.55
4.88
-1.65
1.67
-0.97
-1.72
1.12
0.87
0.39
1.93
2.19

0.035
-0.024
-0.385
-0.252
-0.160
-0.266
-0.039
-0.023
-0.217
-0.242
-0.184
0.003
0.535
0.506
0.464
0.348
0.258
0.128

4.91
-2.79
-5.92
-3.37
-2.14
-3.98
-0.59
-0.31
-3.30
-4.04
-2.58
0.05
12.04
11.45
10.28
8.05
5.80
2.82

47.109
-0.663
215.842
0.994
45060
-189907.6

958.51
-67.95
102.38
1692.69

Trial Regime
Parameter
Asmp Z
-768.983
-17.66
-48.700
-5.67
3.944
0.80
-3.030
-0.41
4.163
0.49
0.882
0.12
30.596
3.31
10.319
-2.081
-9.220
22.950
29.251
39.424
76.201
74.597
13.632
-0.480
0.009
8.396
-6.123
-98.366
-50.159
-30.082
-60.598
-3.004
6.603
-36.302
-53.839
-34.254
7.136
123.182
115.919
104.472
71.825
56.937
25.239

2333

1.38
-1.12
-1.66
2.82
3.50
3.44
5.84
9.53
5.16
-4.73
7.43
4.93
-2.99
-6.30
-2.75
-1.70
-3.79
-0.19
0.36
-2.29
-3.68
-2.02
0.49
11.53
11.12
9.68
7.04
5.38
2.29

Plea Regime
Parameter
Asmp Z
-128.508
-30.91
-9.223
-10.29
4.695
5.73
-1.105
-1.50
1.192
1.53
-1.519
-2.20
-1.836
-1.73
-2.715
-0.873
-5.525
8.531
18.795
30.676
38.444
53.103
15.799
-0.723
0.014
0.249
-0.430
-6.391
1.783
12.412
10.880
8.319
17.144
-1.485
2.105
3.417
10.045
-0.603
-0.030
0.529
1.511
1.800
1.759

42727

-4.02
-4.55
-9.48
9.04
24.54
29.92
27.52
69.13
51.80
-53.66
79.57
1.45
-2.01
-3.24
0.79
5.70
5.49
4.00
7.76
-0.76
1.13
1.57
5.30
-0.56
-0.03
0.47
1.49
1.78
1.80

Table 5
Censored Switching Regression with Endogenous Switching: Males

Variable
Constant
FINEWAIV
HISCHOOL
GED
SOMECOLL
COLLGRAD
CITIZN
MARRD
NUMDEPEN
DEFENSEP
CRIMHIS2
CRIMHIS3
CRIMHIS4
CRIMHIS5
CRIMHIS6
XFOLSOR
XFOLSOR2
XFOLSOR3
AGE
AGE2x10-2
CIRC2
CIRC3
CIRC4
CIRC5
CIRC6
CIRC7
CIRC8
CIRC9
CIRC10
CIRC11
1996
1997
1998
1999
2000
2001
Sigma 0
Rho 0u
Sigma 1
Rho 1u
N
Log-Likelihood

Regime Selection
Parameter
Asmp Z
-2.638
-15.64
0.017
0.036
0.025
0.171
-0.042
0.055
-0.006
-0.049
0.033
0.041
0.035
0.113
0.077

0.51
0.94
0.77
4.18
-1.52
1.60
-0.70
-1.96
0.89
1.13
0.71
1.94
2.11

0.034
-0.024
-0.397
-0.258
-0.166
-0.278
-0.074
-0.034
-0.246
-0.259
-0.176
-0.013
0.538
0.514
0.481
0.357
0.257
0.128

4.46
-2.64
-5.80
-3.26
-2.10
-3.89
-1.06
-0.43
-3.52
-4.08
-2.33
-0.21
11.14
10.70
9.72
7.65
5.34
2.62

49.543
-0.692
224.546
0.994
37104
-163499.9

858.88
-72.20
93.54
1535.52

Trial Regime
Parameter
Asmp Z
-767.458
-15.67
3.340
0.60
-1.423
-0.17
7.070
0.75
1.778
0.21
26.557
2.58
11.390
-1.706
-11.229
22.167
32.000
43.492
78.039
76.024
11.693
-0.404
0.008
8.507
-6.336
-103.768
-52.720
-33.349
-61.521
-9.518
4.737
-42.872
-57.704
-30.395
4.682
126.383
120.490
111.016
75.419
55.756
24.521

2057

1.35
-0.83
-1.80
2.42
3.49
3.50
5.39
9.01
3.94
-3.54
5.97
4.43
-2.76
-6.08
-2.63
-1.71
-3.49
-0.53
0.24
-2.44
-3.59
-1.63
0.29
10.40
10.17
8.99
6.58
4.69
1.99

Plea Regime
Parameter
Asmp Z
-128.041
-26.57
5.149
5.52
-1.611
-1.88
1.172
1.32
-1.885
-2.37
-1.521
-1.28
-3.354
-0.948
-6.532
8.394
19.239
31.494
38.609
53.490
15.581
-0.699
0.014
0.185
-0.343
-6.478
1.005
12.327
9.425
7.088
15.882
-1.560
1.188
2.539
9.672
-1.064
-0.825
-0.272
1.409
1.362
1.539

35047

-4.30
-4.33
-9.74
7.75
21.88
27.17
24.59
61.44
43.19
-44.14
65.83
0.94
-1.40
-2.89
0.39
4.94
4.14
2.98
6.31
-0.69
0.56
1.01
4.48
-0.85
-0.68
-0.21
1.19
1.16
1.36

Table 6
Mean Sentences and Conviction-by-Trial Probabilities
Variable

ȳ
ȳ T
ȳ P
p̄ T
̂ T
̂ P
̂ s
̂

Males
41.673
120.845
37.027
0.055
0.852
0.992
0.894
0.962

Females
16.757
51.736
15.500
0.035

Difference
24.916
69.109
21.526
0.021

Male Fitted
43.320
141.770
37.320
0.062

Females Fitted
(Male Weights)
25.737
89.281
22.776
0.051

Table 7
Decomposition by Part
Variable

ȳ Tm − ȳ Tf
ȳ Pm − ȳ Pf
p̄ Tm − p̄ Tf

Explained

Unexplained

Total Gap

31.564

37.545

69.109

14.251

7.275

21.527

0.004

0.016

0.021

Table 8
Contribution to Total
Explained

E T p̄ Tm
E P 1 − p̄ Tm 
E s ȳ Tf − ȳ Pf 
E

Unexplained
1.750
13.461
0.159
15.370

U T p̄ Tm
U P 1 − p̄ Tm 
U s ȳ Tf − ȳ Pf 
U

Total Gap
2.081
6.872
0.593
9.546

E T p̄ Tm  U T p̄ Tm
E P 1 − p̄ Tm   U P 1 − p̄ Tm 
E s ȳ Tf − ȳ Pf   U s ȳ Tf − ȳ Pf 
EU

3.831
20.333
0.752
24.916