Alfred DeMaris - Regression with Social Data, Modeling Continuous and Limited Response Variables

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excluded due to these sources of selectivity. In all likelihood, unobserved character-

istics of couples which inﬂuenced their refusal to answer items on violence, their

failure to remain together as a couple, or their inability to be resurveyed also aﬀect

their proclivity for violence. The authors therefore employed a sample-selection

model to adjust for this problem.

As another example of selection eﬀects: providing information on particular

items may be dependent on individuals’ choosing to engage in an activity, a choice

that can be considered the product of a selection propensity. For example, studies of

earnings necessarily include valid data from only those electing to participate in the

labor force. However, the response variable can be considered to be the hourly wage

oﬀer, which is relevant to all those who are of working age. For those who work, the

hourly wage oﬀer is the actual wage earned. For those who are not working, the

hourly wage oﬀer is unobserved (Wooldridge, 2000). Hence, estimates of regressor

eﬀects using only working people are likely to be biased. As Wooldridge (2000,

p. 558) explains: “Because working may be systematically correlated with unob-

servables that aﬀect the wage oﬀer, using only working people...might produce

biased estimators of the parameters in the wage function.” Researchers should be

judicious, however, in how they deﬁne the target population. Otherwise, virtually every

analysis would seem to need correction for sample selectivity. Excluding, for exam-

ple, single people from a regression analysis of the propensity to dissolve an exist-

ing marital union (e.g., Booth et al., 1984) doesn’t result in selection bias provided

that it is the population of currently married couples that is of interest. As Stolzenberg

and Relles (1990, p. 408) observe: “. . . the severity of censoring, and probably the

severity of censoring bias, is aﬀected by substantive decisions about the population

to which one wishes to draw inferences.”

Estimation

Although equation (9.30) posits a continuous selection propensity, in fact, all that is

observed is whether or not a case in selected into the current sample. Therefore the

model that is actually estimated employs a selection equation based on a binary indi-

cator, Z

⫽

冦

and the condition for observing the outcome of interest is

is missing if Z

⫽ 0,

⫽ Y

if Z

⫽ 1.

The selection equation then becomes (based on the reasoning articulated in

Chapter 7)

P(Z

⫽ 1 冟 w

) ⫽ Φ(w

⬘⬘

γγ

), (9.32)

if Z

ⱕ 0,

if Z

⬎ 0

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where, due to identiﬁability requirements, σ

is now assumed to equal 1. The model

can be estimated with maximum likelihood [see Breen (1996) for an expression for

the log-likelihood]. An alternative procedure that has also been employed exten-

sively is the Heckman two-step, or Heckit (Greene 2003, p. 784) estimator. It is con-

structed as follows. First, estimate (9.32) as a probit model, employing the full

sample of cases. Then using w

and γ

from that analysis, estimate λ(α

) as

(α

) ⫽

ᎏ

φ(

(

⬘

)

ᎏ

Finally, estimate equation (9.31) with OLS using only those with nonmissing scores

on Y

and adding λ

(α

) as a regressor in the equation. This results in a consistent esti-

mate of

ββ

. However, the estimates of σ

and coeﬃcient standard errors produced by

OLS software are not correct because the error term is heteroscedastic. Greene (2003)

provides expressions for σ

and for the correct asymptotic variance–covariance

matrix of parameter estimates and has incorporated them into the Heckit procedure in

LIMDEP. This procedure also generates an estimate of ρ equal to θ

/σ

, but as this is

not a sample correlation, the estimate can fall outside the range [⫺1, 1]. When it does,

LIMDEP reports it as 1 or ⫺1 (see the example below). It should be noted here that

sample-selection models are not limited to probit and linear regression for the selec-

tion and substantive equations, respectively. Programs such as LIMDEP oﬀer a vari-

ety of alternative speciﬁcations for both equations [see, e.g., Greene (1998) for

details].

Nuances

Some elements of the sample-selection model are characterized by nuances that need

further explanation. First, notice that selection bias comes about because of a

nonzero correlation, ρ, between ε and u, since the impact of the omitted term in

equation (9.31) is θ ⫽ ρσ

, and σ

would never be zero. What does this correlation

mean? As in other situations involving correlated errors (e.g., seemingly unrelated

regression equations or factor-analysis models) ρ represents a residual correlation

between two outcomes that remains after all observable eﬀects have been accounted

for. In this particular case, ρ is the correlation between Z* and Y* that is not

accounted for by their mutual dependence on a set of observable explanatory vari-

ables. As an example, in the simulated data above, Corr(Z*,Y*) ⫽ .852, while

ρ ⫽ .707. Thus, only 1 ⫺ (.707/.852), or 17% of the Z*–Y* correlation, is due to the

explanatory variable, X, while the remainder is due to the correlation between the

two disturbance terms. If the correlation between Z* and Y* were accounted for

entirely by the explanatory variables in both equations, ρ would be zero and selec-

tion bias would not be a problem. This is tantamount to the situation of exogenous

selection (Wooldridge, 2000), or selection based on the independent variables, which

causes no problems of bias (Wooldridge, 2000). On the other hand, ρ is nonzero to

the extent that latent characteristics of observations aﬀect both individuals’ propen-

sity to respond and their score on the substantive outcome [see also Berk (1983)] and

this is what induces selection bias.

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Second, w and x can be the same set of regressors, but it is better if we have at

least one factor in w that is not in x. Otherwise, the coeﬃcients in equation (9.31)

are identiﬁed only because λ

(α

) is a nonlinear translation of the model’s regres-

sors. Even so, having w and x be the same regressors causes substantial collinear-

ity problems, since x and λ

(α

) will then tend to be highly correlated. Ideally, we

want to choose the unique element of w to be a factor that is unrelated to the out-

come.

Third, how is λ(α

) to be interpreted? This term can be considered the hazard of

exclusion (Berk, 1983). The higher its value, the more a given case possesses char-

acteristics associated with exclusion from the sample. Why? Once again, regard

Figure 9.3. Remember that as z ⫽ w

⬘γ becomes increasingly positive, equation

(9.32) tells us that Φ(w

⬘γ), the probability of being selected into the sample, also

increases, and φ(w

⬘γ) decreases to zero, which means that λ(α

) shrinks toward

zero. As w

⬘γ becomes increasingly negative, Φ(w

⬘γ), the probability of being

selected, shrinks toward zero, while φ(w

⬘γ) also decreases to zero but at a slower

rate, which implies that λ(α

) becomes large. Therefore, larger values of λ(α

)

reﬂect a lower probability of inclusion into the sample.

Fourth, θ, the “eﬀect” of the hazard of exclusion, can be misleading, since it is

opposite to intuition. For one thing, it should probably be thought of only as an

association parameter, since the hazard of exclusion does not actually “cause” Y*.

Additionally, the sign of this eﬀect is the same as the sign of ρ, since σ

is always

positive. If ρ is positive, for instance, whatever unobserved factors raise the prob-

ability of selection also elevate the outcome. A positive “eﬀect” of the hazard of

exclusion in this case indicates that the tendency to be included—not excluded—

is associated with a higher mean outcome. This is a subtle point that can easily

cause confusion. As an example, Berk’s (1983) exposition of sample selection

eﬀects considered potential bias in the regression model for satisfaction with jury

duty brought about by using only the sample that responded to a mail survey. The

eﬀect of λ

(α

) in the model for satisfaction with jury duty was seen to be nega-

tive, implying a negative value for ρ. The temptation is to conclude that exclusion

was associated with less satisfaction, or that the dissatisﬁed were less likely to

respond. Yet a negative ρ means that, net of observed covariates, the tendency to

respond was associated with less satisfaction; in other words, the dissatisﬁed were

more likely to respond.

Fifth, an examination of the nature of the bias associated with omitting λ(α

)

reveals the conditions under which sample selectivity does, and does not, create

problems. For simplicity of exposition, let’s assume that there is only one regressor,

although the principles generalize to any number of regessors. The substantive equa-

tion is

Y*⫽ β

⫹ β

X ⫹ ε,

where ε is normal with zero mean and variance σ

. The selection equation is

Z*⫽ γX ⫹ u,

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where u is normal with zero mean and variance 1. We deﬁne Z ⫽ 1 if Z*⬎ 0,

and Z ⫽ 0 otherwise. Furthermore, Y ⫽ Y* is observed only when Z ⫽ 1. The mean

of Y is then

E(Y 冟X, Z ⫽ 1) ⫽ β

⫹ β

X ⫹ ρσ

λ,

where λ ⫽ φ(γX )/Φ(γX). The equation for Y is then

Y ⫽ β

⫹ β

X ⫹ ρσ

λ ⫹ υ,

where Cov(X,υ) ⫽ 0. If we estimate Y using OLS and omit λ, what is b

consistent

for? We have

⫽

ᎏ

cov

,Y )

ᎏ

plim b

⫽

ᎏ

Cov

(

X,Y)

ᎏ

⫽

⫽⫽

⫹

(9.33)

The rightmost expression in equation (9.33) suggests that bias is a function of ρ and

the association between X and λ. Most important, both must be nonzero for bias to be

a problem. This means that for any particular focus variable, X

, selection bias is pres-

ent whenever ρ is nonzero and X

is signiﬁcantly correlated with λ—demonstrated by

having a signiﬁcant eﬀect in equation (9.32). If one or the other of these conditions

fails to hold, selection bias is not a problem for one’s analysis, at least in regard to the

X in question.

Simulation

Recall the incidentally truncated simulation data discussed above. As noted there, the

underlying substantive model is Y*⫽⫺2 ⫹ 1.5X ⫹ ε, and 60% of the cases were trun-

cated incidentally. In this particular simple example, w ⫽ x, since they are both the

same regressor, X. Also, as ρ⫽ .707 and σ

⫽ 2, θ⫽ ρσ

⫽ .707(2) ⫽ 1.414. Table 9.5

shows the true values of the parameters, as well as the estimated values based on OLS,

the Heckit procedure, and ML estimation of the sample-selection model for the 399

cases with no missing Y values. Only the substantive estimates, not the selection equa-

tion estimates, are shown for the sample-selection models. Also, notice the absence of

lambda for the ML results. The direct incorporation of ρ and w

⬘γ into the likelihood

function obviates the need to include lambda as a separate regressor. In that there was

a positive correlation between ε and u, and X had a positive eﬀect on both selection and

outcome, what do we expect the nature of the bias of the OLS slope to be? This is

ρσ

Cov(X,λ)

ᎏᎏ

⫹ ρσ

Cov(X,λ)

ᎏᎏᎏ

Cov(X, β

⫹ β

X ⫹ ρσ

λ ⫹ υ)

ᎏᎏᎏ

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SAMPLE-SELECTION MODELS 339

a little tricky. We have to remember that X’s eﬀect on the hazard of exclusion is oppo-

site its eﬀect on the probability of inclusion, which in this case implies a negative

Cov(X,λ). Hence, by expression (9.33), the OLS b

should be an underestimate of β

This is, in fact, what Table 9.5 shows, as the OLS estimate is .825, whereas the Heckit

estimate of 1.644 and the MLE of 1.37 are both much closer to the true value of 1.5.

Although the Heckit and ML estimates of the slope are about equally distant from the

true parameter value, the ML estimates for β

, σ, and ρ are considerably better than the

Heckit estimates. When possible, ML is to be preferred over the Heckman two-step

procedure. However, whereas the Heckit model can always be estimated, the ML tech-

nique may at times fail to converge (see the applications below).

Applications of the Sample-Selection Model

I illustrate two applications in this section. In the ﬁrst, I examine selection eﬀects in

a regression of academic self-esteem at time 2 (T2) on measures of prior (T1) aca-

demic self-esteem, academic achievement, and controls for a sample of 423 students

in introductory sociology at Bowling Green State University (Bradley, 2000). The

data were collected during the course of a semester, with the outcome variable meas-

ured several weeks after the explanatory variables. A total of 649 students were ini-

tially enrolled in the study, but only 423 provided nonmissing data on the response.

Hence, 35% of the data were incidentally truncated. For this example, I expect that

ρ will be positive: unmeasured correlates of inclusion in the sample (e.g., diligence

in attending class and in responding to the survey) should be associated with higher

academic self-esteem.

The second example, based on the NVAWS, is a regression of posttraumatic

stress disorder symptoms (PTSD) for a sample of 331 women who had been victim-

ized by their current intimate partner. This is a subset of a larger sample of 1829

women who have ever been victimized by an intimate partner; however, only those

currently partnered were asked about PTSD. Because the response is relevant to all

women victimized by intimate partners, and because fully 82% of the sample is inci-

dentally truncated, selection eﬀects pose a serious threat to inference. Predictors

employed in this analysis are largely measures of the severity of victimization

Table 9.5 Regression with Simulated Data, Showing Eﬀects

of Incidental Truncation on Parameter Estimates

True

Regressor Parameters Values OLS Heckit MLE

Intercept β

⫺2.000 1.623 ⫺3.032 ⫺1.488

X β

1.500 .825 1.644 1.370

Lambda θ 1.414 2.093

σ 2.000 1.769 2.333 2.044

.496 .203 .213

ρ .707 .897 .694

Note: n ⫽ 399.

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340 TRUNCATED AND CENSORED REGRESSION MODELS

perpetrated by any adult, including the current partner. Again, I expect a positive ρ,

since unmeasured factors associated with being in a currently abusive relationship

(e.g., growing up in a violent or unstable family) should also be predictive of higher

levels of PTSD.

Academic Self-Esteem. Table 9.6 presents the regression models for T2 academic

self-esteem (interval variable with higher scores reﬂecting greater academic self-

esteem), showing estimates from the Heckman two-step procedure as well as ML,

both estimated using LIMDEP, with uncorrected OLS (or, simply, “OLS”) for com-

parison. Predictors include high school GPA, college GPA, high school grades

(coded in half-point increments from 0 ⫽ “mostly F’s” to 4 ⫽ “mostly A’s”), male

(dummy for being male), T1 academic self-esteem (interval variable with higher

scores reﬂecting greater academic self-esteem), and T1 test anxiety (interval variable

with higher scores reﬂecting greater test anxiety). As a unique predictor for the

selection models, I chose an indicator of whether the student pays his or her own

tuition as a variable aﬀecting selection, but not self-esteem. The reasoning is that stu-

dents who pay their own tuition are much more likely to attend every class. Since

missing data on the response is due primarily to absenteeism, paying one’s own

tuition should be a predictor of inclusion in the valid sample. There is no reason, on

the other hand, why paying one’s own tuition would per se boost self-esteem.

The only factor signiﬁcantly related to selection is college GPA, with higher

GPAs presaging a greater probability of inclusion. Lambda, the hazard of exclusion

in the Heckit model, has a positive slope, as expected. Similarly, the ML estimate of

Table 9.6 Unstandardized OLS and ML Estimates of Sample-Selection Models of T2

Academic Self-Esteem

Heckman Two-Step

Uncorrected

Regressor OLS

Selection Response Selection Response

Intercept 10.518*** ⫺.991 ⫺8.002 ⫺1.003 9.473

Pays own tuition ⫺.080 ⫺.105

High school GPA .851 .140 1.830 .141 .907

High school grades ⫺.306 .061 .235 .060 ⫺.275

College GPA .873** .237** 2.644 .240** .972

Male .678 .006 .694 .006 .679

T1 academic self-esteem .591*** ⫺.0003 .585*** .0001 .590***

T1 test anxiety ⫺.091*** .002 ⫺.074 .002 ⫺.090***

Lambda 14.086

3.343 10.799 3.366

.573 .576

1.000 .237

n ⫽ 423.

n ⫽ 649.

* p ⬍ .05. ** p ⬍ .01. *** p ⬍ .001.

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SAMPLE-SELECTION MODELS 341

ρ is .237. However, neither θ

nor ρ

(reported as 1.000 in Heckit) is signiﬁcant, sug-

gesting that there may not be much of a selection problem in these data. This is also

evident in the relative lack of diﬀerences between OLS and ML estimates of the

parameters (the Heckit estimates are notoriously poor if selectivity is minimal; more

about this below). Nonetheless, for instructive purposes, consider the change in the

estimate for college GPA from the OLS to the ML estimates. Panel (a) of Figure 9.5

(a)

College

GPA

Hazard of

Exclusion

Academic

Self-Esteem

−

(b)

Minor

Violence

Hazard of

Exclusion

PTSD

−

Severe

Violence

Hazard of

Exclusion

PTSD

Child

Abuse

Hazard of

Exclusion

PTSD

−

Figure 9.5 Sample-selection bias in the analyses of academic self-esteem and PTSD.

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342 TRUNCATED AND CENSORED REGRESSION MODELS

illustrates the eﬀect of having omitted the correction for selectivity in the model. As

college GPA has a positive eﬀect on academic self-esteem and a negative eﬀect on

the hazard of exclusion (since it has a positive eﬀect on the probability of inclusion),

and as the hazard of exclusion has a positive “eﬀect” on academic self-esteem, the

impact of college GPA is underestimated without correcting for selectivity. The

eﬀect of college GPA therefore goes from .873 in OLS to 2.644 in Heckit and .972

in ML, although neither of the latter estimates is signiﬁcant. At any rate, as selectiv-

ity does not seem to be a problem in this analysis; the OLS estimates would be pre-

ferred over the others.

PTSD. Table 9.7 shows OLS and Heckit estimates for the regression of PTSD.

Predictors include dummies for having experienced minor physical assault,

severe physical assault, having had to take time oﬀ due to the physical assault,

and having experienced rape with penetration, as well as the interval variables

annual income and the child abuse index, as described above. In this particular

case, the ML procedure did not converge, a pitfall of this technique, as noted

above. The unique predictor of selection chosen here was an indicator of being

married to the partner. As marriage implies more barriers to the termination of

relationships, I would expect those who are married to the current partner to be

more likely to be found in currently abusive partnerships. On the other hand, there

is no particular reason why marital status per se should aﬀect PTSD. The selec-

tion model results reveal that being married to the partner is, indeed, predictive

of inclusion in the current sample. So are having experienced minor physical

Table 9.7 Unstandardized OLS Estimates of Sample-Selection Models

of PTSD

Heckman Two-Step

Uncorrected

Regressor OLS

Selection Response

Intercept 31.727*** ⫺1.582*** 9.329

Married to partner .491***

Minor physical assault ⫺1.464 .727*** 6.793

Severe physical assault 5.657*** ⫺.350*** 1.937

Took time oﬀ due to assault 6.590*** ⫺.002 5.929**

Child abuse index 1.092** .060*** 1.631***

Rape with penetration 3.453* ⫺.123 1.312

Income ⫺.963** ⫺.027 ⫺1.206***

Lambda 12.741**

13.994 17.631

.209 .228

.723

n ⫽ 331.

n ⫽ 1829.

* p ⬍ .05. ** p ⬍ .01. *** p ⬍ .001.

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assault and a greater severity of child abuse. Interestingly, having experienced

severe physical assault lowers the probability of inclusion. This is reasonable,

since those experiencing severe assault, which was most likely at the hands of an

intimate partner, might be especially likely to avoid ending up in another (i.e., a

current) abusive liaison.

In this example we ﬁnd that lambda is also quite signiﬁcant, and, as expected, ρ

is positive with a value of .723. These ﬁndings suggest that selectivity bias could be

a problem in this analysis. In particular, the coeﬃcients for minor assault, severe

assault, and child abuse should all be aﬀected. Panel (b) in Figure 9.5 illustrates the

nature of the biases. We should ﬁnd the eﬀects of minor physical assault and child

abuse to be suppressed by omission of lambda, while the eﬀects of severe violence

should be confounded with the hazard. A comparison of OLS coeﬃcients with those

of the Heckit procedure show that these expectations are borne out. The coeﬃcient

for minor physical assault increases from ⫺1.464 in OLS to 6.793 in Heckit, although

neither coeﬃcient is signiﬁcant. The coeﬃcient for child abuse increases from 1.092

in OLS to 1.631 in Heckit, with both estimates being signiﬁcant. The coeﬃcient for

severe physical assault, on the other hand, is reduced to insigniﬁcance after control-

ling for the hazard of exclusion, going from 5.657 in OLS to 1.937 in Heckit.

Apparently, the impact of severe physical assault is overestimated in OLS when fail-

ing to correct for the fact that severely victimized women are more likely to be

excluded from the sample, and the hazard of exclusion is associated with a greater

mean PTSD.

Caveats Regarding Heckman’s Two-Step Procedure

Stolzenberg and Relles’s (1990, 1997) simulations compare the Heckman procedure

to uncorrected OLS under a range of conditions. They ﬁnd that, on average, the

Heckman procedure performs no better than uncorrected OLS. However, Heckit

appears to reduce bias consistently when two conditions are simultaneously met: (1)

ρ is very high and (2) x and w are very highly correlated (Stolzenberg and Relles,

1990). They note further that substantive equations with high R

’s can tolerate con-

siderable sample selectivity without showing much bias; and if the selection model

itself exhibits poor discriminatory power, bias is also likely to be minimal. In sum,

they suggest that if bias is very severe and the sample is large, Heckit probably

improves the estimates. But if bias is only moderate, or if samples have “only a few

hundred cases,” there is substantial risk that the Heckman procedure will make the

estimates worse (Stolzenberg and Relles, 1997, p. 503). One way to approach sus-

pected selection bias is to estimate the sample-selection model with ML if possible,

or with the Heckit procedure if not. If one’s focus variables are signiﬁcant predic-

tors of inclusion in the sample, and if θ

(Heckit) or ρ

(ML) are signiﬁcant, sample-

selection corrections are appropriate. If these conditions do not obtain, correction for

selectivity may not be warranted. Moreover, when using the Heckit approach, the

test of θ

should employ the asymptotically correct standard error found in programs

such as LIMDEP or STATA rather than relying on the t test in standard OLS

software.

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EXERCISES

9.1 Suppose that Y is normally distributed with mean 12 and standard deviation

4. If the distribution is truncated at y ⫽ 8, what are the mean and standard

deviation of the truncated distribution?

9.2. Suppose that Y is normally distributed with mean 12 and standard deviation

4. If the distribution is censored at y ⫽ 8, what are the mean and standard

deviation of the censored variable?

9.3 Suppose that y and z have a bivariate normal distribution with µ

⫽ 12, µ

⫽ 8,

⫽ 4, σ

⫽ 2, and ρ ⫽ .5. Suppose further that y is observed only when z ⬎ 5;

that is, the distribution of y is incidentally truncated at z ⫽ 5. What are the

mean and standard deviation for the incidentally truncated distribution of y?

9.4 Note that the inverse Mills ratio is expressed as λ(z) ⫽ φ(z)/[1 ⫺ Φ(z)].

However, for ⫺z we have the IMR as

λ(⫺z) ⫽

ᎏ

1 ⫺

φ(

⫺

(

⫺

)

ᎏ

⫽

ᎏ

φ(

(

)

ᎏ

⫽ λ(α),

where α ⫽⫺z, which is the form it takes in the censored (from below) and

sample-selected regression models discussed in this chapter. [Note that φ(z)/

Φ(z) in certain contexts is also denoted simply as “λ”.] Illustrate the practical

range of the IMR (in this latter form) by evaluating it at z equal to ⫺4, 0, and 4.

9.5 Using equation (9.16) and the ML estimates in Table 9.2 for the truncated

regression model of exam 1 score, give the predicted mean exam 1 score for

a student in the population of students scoring at least 70 on exam 1, with a

college GPA of 3.2, a math diagnostic score of 43, and an attitude score of 10.

9.6 Using equation (9.16) and the ML estimates in Table 9.2 for the truncated

regression model of exam 1 score, give the predicted mean exam 1 score for a

student, in the population of students scoring at least 70 on exam 1, with a col-

lege GPA of 2.5, a math diagnostic score of 37, and an attitude score of ⫺2.

9.7 Based on equation (9.23) and the estimates in model 1 of Table 9.3, give the

predicted observed CESD score for a woman 1 standard deviation below the

mean on open disagreement, in an unstable relationship, with a relationship

duration of 12 years.

9.8 Based on equation (9.23) and the tobit estimates in Table 9.4, give the pre-

dicted observed physical assault severity score for a 35-year-old woman with

child abuse index ⫽ 4, who has an education score of 5, is married and white,

and whose income score is 2.

9.9 Means for the regressors used in Table 9.3 are open disagreement ⫽ 0 (cen-

tered variable), female ⫽ .553, unstable relationship ⫽ .361, and relationship

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