Working Paper: Gender Differences in Charitable Giving

Evidence from the COPPS 2007 Wave

Una OsiliEmail: uosili@iupui.edu

Department of Economics

Indiana University Purdue University Indianapolis

Director of Research

Center on Philanthropy at Indiana University

J. Elliott MillerEmail: jeemille@.iupui.edu

Center on Philanthropy at Indiana University

Debra MeschEmail: dmesch@iupui.edu

Director, Women’s Philanthropy Institute

School of Public and Environmental Affairs

Indiana University Purdue University Indianapolis

Abstract

Philanthropic researchers and policy makers studying charitable bequests face a number of complexissues in building models which can inform policy decisions and help those seeking donations. Borrowingmethods from health economics, biostatistics, and the actuarial risk literature for the analysis of costs,we utilize One-Part and Two-Part OLS and GLM models and extensions on the 2007 wave of the CenterOn Philanthropy’s Panel Study, the charitable giving section of the Panel Study of Income Dynamics.Findings show that single women are both more likely to give, and to give more after controlling fordifferences in income, wealth, and demographics.

Keywords: Box-Cox Transformation; Retransformation Problem; Generalized Linear Models; Quasi-likelihoodGeneralized Linear Models; Two-Part (Hurdle) Models; Duan’s Smearing Factor; Extended EstimatingEquations.

1

1 Previous Literature

The economics literature found that men were more generous when it was less costly to give while women

were more generous when it was more expensive to give, highlighting the difference in ”demand curves

for altruism” between males and females [Anderoni and Vesterlund, 2001]. Other empirical studies from

economics have found that charitable giving between the sexes can differ by factors such as age [List, 2004],

as a result of cognitive ability and personality [Ben-Ner, Kong, and Putterman, 2004], by gender pairings or

same-sex groups [Kamas, Preston, and Baum, 2008], or due to the proportion of females in the household

[Pharoah and Tanner, 1997].

In addition, men and women do exhibit different charity choices and patterns of donating money.

Males tend to concentrate their giving among a few charities, whereas females are more likely to spread

the amounts they give across a wide range of charities e.g., [Andreoni, Brown, and Rischall, 2003]; [Piper

and Schnepf, 2008]. These studies found that women were more likely to donate to human service, health,

and education while men were more likely to donate to adult recreation and sports. In a study by [Eckel

and Grossman, 2003], men and women exhibited a high degree of similarity in their choice for charities,

but women were more generous compared to men in six of the ten cases. Previous research also indicates

that women tend to give to organizations that have had an impact on them or someone they know person-

ally [Parsons, 2004, Burgoyne, Young, and Walker, 2005]. Subsequently, much of the empirical research

indicates that men and women exhibit different charity choices and patterns of donating money. This

paper seeks to clarify the role that gender plays in determining the amount donated to charitable causes,

especially by comparing the different amounts between female and male headed households. Further, using

the same dataset, we examine the differences between men and women’s giving by charitable area.

2 Models and Methods

2.1 Modeling Giving Incidence

(HAVE IDEA TO REWORK THIS SECTION AND HOW TO MEASURE THE GENDER EFFECT)

START WITH HOW TO MEASURE GENDER EFFECT, DIFFERENT FOR DISCRETE AND CTS

COVARIATES¿ DIFFERENT. One method evaluete at sample means othe raverage over the distribution.

We show both these approaches answer research questions.

Since their introduction in the 1930’s and 1940’s Binary Response Models such as probit and logit have

permeated all disciplines of applied statistical research [Long, 1997, Agresti, 2002]. We develop these models

in this paper because they are in a sense the simplest specification of giving, and extensively examine the

2

marginal and incremental effects of common explanatory variables of giving found in the literature. These

models also are examples of generalized linear models, a framework we exploit for the modeling of amount

given. Binary response models also function as the first part of the two-part model, a technique for dealing

with excess ”zeros” in a giving amount models discussed below.

We [Allison, 1999] argues that we are often interested in comparing how the effects of variables differ

across groups (e.g. is the effect of education on income greater for men than it is for women?). HOWEVER,

when doing logistic regression, there is a potential pitfall in cross-group comparisons that, Allison claims,

has largely gone unnoticed. Unlike linear regression coefficients, coefficients in binary regression models

are confounded with residual variation (unobserved heterogeneity). Differences in the degree of residual

variation across groups can produce apparent differences in slope coefficients that are not indicative of true

differences.

[Allison, 1999] proposes a test that removes the effect of residual variation by assuming that the co-

efficients for at least one independent variable are the same in both groups. Unfortunately, a researcher

might lack sufficient theoretical or empirical information to justify such an assumption. Making an ad hoc

decision that some regression coefficients a

Tests of predicted probabilities provide an alternative approach for comparing groups that is unaf-

fected by group differences in residual variation and does not require assumptions about the equality of

regression coefficients for some variables. Group comparisons are made by testing the equality of predicted

probabilities at different values of the independent variables.

Mandalla, Long and Allison. Long mentions that even though the coefficients cannot be compared the

predicted probabilities are invariant to the problem.

This method requires thinking differently about apparent differences.. No longer a single test for

indicating statistically significant differences but allows for differences at different levels of variables.

Group comparisons in logit and probit using predicted probabilities1 J. Scott Long Indiana University

June 25, 2009

2.2 Modeling the Amount Given to Charity

Much like the analysis of health cost and expenditure data, the philanthropic researcher studying charitable

bequests faces a number of econometric issues which have to be addressed in order to draw valid inferences.

First; by definition, charitable giving data is non-negative. This OLS violation must be accommodated by

any model of donation amount and yields the two alterative approaches to modeling such data found in

the health literature; one based on transformation of the outcome, and the other based on use of a suitable

link function in the generalized linear model approach. Second; a significant fraction of observations have

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zero (without any) giving. This ”spike” at zero becomes much more substantial especially when analysis

is performed on specific giving sectors. Solutions to this problem are based on modeling conditional

distribution (givers) and full distribution (givers and non-givers) separately, or finding a suitable model that

accommodates both. The analysis of cost literature has identified two alterative approaches to modeling

such data; one based on transformation of the outcome, and the other based on generalized linear models.

Also many studies in the literature examine conditional and unconditional giving separately, and the two-

part model represents a compromise between the two. Third; the distribution of gifts for those that give

are heavily right-skewed and typically exhibit leptokurtic or ”heavy” upper tails1. Although there are

many observations of small donations, there are a substantial number of mega-gifts creating a leptokurtic

or ”heavy-tailed distribution. Also note that skewness does not imply a heavy upper tail. Fourth; the

response of amount given to any given covariate may be nonlinear. This is typically accommodated by

including nonlinear specifications of the suspect covariates in the regression or through the use of non-

linear model. Finally, Giving response may change both by the method of gift (private and corporate

foundational gifts inherently different from individual gifts) and or by level of gift.

The degree and severity of each of these issues differ by data set, and in turn, the degree to which they

must be accommodated depends on the specific research questions being addressed. This implies that no

one model that works for any given situation. We will also see that in fixing one problem, one can often

create new challenges.

.

Transformation of the outcome (Y ) using OLS regression typically on ln(Y ) or MLE for Box-Cox trans-

formations usually overcomes the problem of skewness and may reduce the problems of heteroscedasticity

and kurtosis [Box and Cox, 1964], but does not result in a model of the mean µ(x) on the original dollar

scale, but on log dollar scale. To draw consistent inferences about the mean µ(x) or any fun thereof (BASU

2009)

Basically it boils down to choose between a potentially asymptotically biased (inconsistent) model

based on transformation of Y or the consistent but much less efficient method based on GLM approaches

[Manning and Mullahy, 2001]

Equally important is to build models scientific, policy, and decision making questions. Our research

focus is on pragmatic rather than explanatory exploration.

Two

OUTLINE: (1) Discuss how effects are measured, introduce recycled predictions, explain econometric

1Another potential complication is the distribution of gift amounts may be bimodal, see [Deb and Trivedi, 1997, Deb andHolmes, 2000] for a discussion and solution using finite mixture models.

4

problems with measuring charitable bequests, Introduce models from health literature of cost modeling,

and explain in general the strengths and weaknesses of each.

2.3 How to measure the Gender Effect?

The primary goal of generosity modeling is to draw inferences based on mean gifts or incidence to determine

how different covariates affect mean donations. This is accomplished by the use regression models where

response variable Y is modeled as a function of covariate vector X = (X1, X2, . . . , Xp)T via a mean

function µ(x) ≡ E(Y | X = x). In general, interest lies in estimating the effect of one or more of the

covariates Xj on Y , and the appropriate interpretation depends on weather Xj is continuous or discrete.

In linear models, the appropriate effect is always given by the regression coefficients βj , regardless if Xj is

continuous or discrete, but in non-linear (generalized linear models) (more general models) it is quantified

via more general functions of µ(x). If Xj is continuous the appropriate functional in economics is the

partial derivative of µ(x) with respect to covariate xj in vector x = (x1, x2, . . . , xp)T which I will denote as

Dj(µ;x) ≡∂µ(x)

∂xj(1)

which is the (instantaneous/ infinitesimal) rate of change in µ(X) with respect to Xj evaluated at X = x.

When Xj is an indicator variable (like gender) we define Dj(µ;x−j) to be the discrete difference in µ(x)

at two different levels of Xj , i.e.:

Dj(µ;x−j) ≡ µ(xj = 1, x−j)− µ(xj = 0, x−j) (2)

where x−j is the vector x without xj . In most applications, any particular combination of values, such as

X = x represents only a negligible fraction of the population and what the researcher really wants to assess

is the effect of Xj in the whole population. This measure in statistics is the expected value of Dj(µ;X)

taken over the population distribution of X. This parameter is termed the marginal effect of the covariates

in the sense used by econometricians [Greene, 1997, pg. 824] and is given by:

ξj ≡ EX [Dj(µ;X)] (3)

and is the population average rate of change in µ(X) with respect to Xj , controlling for other factors X−j .

When Xj is binary, the parameter of interest is the incremental effect given by:

πj ≡ EX−j [Dj(µ;X−j)] (4)

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where the expectation is taken overX−j , marginally with respect toXj . The parameter πj is the population

average contrast in the mean of Y for Xj = 1 and Xj = 0. Again, the expectation is taken over X, but since

Xj is fixed at 0 or 1 in Dj(µ;X−j), πj only involves the marginal distribution of X−j . The interpretation

of both ξj and πj are as effects of Xj on the mean of Y , adjusting for all other covariates in the model,

where this adjustment is to the population distribution of X [Basu, 2005].

To evaluate the “average” or “overall” marginal effect, two approaches are frequently used. One ap-

proach is to compute the marginal effect at the level of the sample means, [see Long, 1997, pg. 71-79]

for a detailed discussion of this approach with respect to binary response models. The other approach is

to compute marginal effects for each observation and then calculate the sample average of these effects

to obtain the overall marginal effect. This method known in the literature as the method of recycled pre-

dictions [StataCorp, 2005], also referred to as averaging the individual marginal effects [Greene, 1997],

and predictive margins [Graubard, Edward, and Korn, 1999]. Both the approaches yield similar results for

large samples. For smaller samples, however, averaging the individual marginal effects is preferred [Greene,

1997, pg. 339350]. [Pull Green 1997(Greene, 1997, p. )., . Dont matchup

SHORTEN TO JUST A PARAGRAPH

2.4 OLS Regression on ln(Y ) or√Y and the Retransformation Problem

Setting the ”zeros” problem aside, consider the traditional econometric approach of dealing with skewed

outcomes: transforming outcome Y usually by either a logarithmic or Box-Cox transformation, followed by

regression of transformed Y using OLS. Consider the semi-log equation of transformed response Y defined

only for y > 0 :

ln(y) = x′iβ + εi with E[εi | x] = 0 (5)

Estimation yields estimates for E[ln(y) | x] not ln(E[y | x]). 2 From a policy perspective, OLS

regression on log transformed Y yields estimates in terms of log-dollars, a scale which is inherently hard

to interpret. It is possible to go from E[ln(y) | x] to E[y | x] by retransformation [Duan, 1983, Manning,

1998], however the retransformation problem is that in many cases the transformation back to the dollar

scale can cause bias in the estimates of the mean response E[y | x], especially if the error ε is heteroscedastic

in x.

To find the conditional mean of y, as opposed to ln(y), taking anti-logs on both sides and then expec-

tations:

E[y | y > 0,x] = exp(x′β)E[exp(ε|x)] (6)

2For yi > 0∀i, (∏n

i=1 yi)1n = exp

∑ni=1

1nln yi i.e. exponentiated arithmetic mean of logged data equals the geometric mean

of untransformed data.

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If ε is normal and homoscedastic with log-scale variance σ2 , then E[exp(ε|x)] = σ2/2. and the expected

value of a conditionally log-normal variable y is:

E[y | x] = exp

(x′β +

σ2

2

)(7)

Duan [1983] notes that if the error term is not normally distributed re-transformation results in bias in the

expectation. Duan’s solution to consistently estimate the expectation is a nonparametric re-scaling of the

re-transformed predictor known as a smearing factor where E[exp(ε|x)] is estimated using the average of

the exponentiated residuals from OLS regression on ln(y) :

S =1

N

N∑i=1

exp(εi), where εi = ln(yi)− x′iβ (8)

This correction is aimed at deriving consistent estimates in the overall mean prediction in the case of non-

normal differences in variances, skewness and kurtosis assuming only assuming the errors are independent

and identically distributed. If the error ε is heteroscedastic in x i.e. S depends on x because distributional

form or scale of residuals related to x [Manning, 1998] argues the retransformation should model the

heteroscedasticity. For example if the errors are different by gender (or any other indicator) one can employ

separate smearing factors for men and women. However, if the errors are heteroscedastic in a continuous

variable (like income) it would be necessary to estimate smearing factors which vary continuously with the

covariate. Note that the use of separate smearing factors for subgroup eliminates the efficiency gain from

log transformation because one cannot assume that the standard errors (p-values) derived for the log of

gift amount applies to the arithmetic mean of untransformed cost. To evaluate the statistical significance

of covariates one will need to bootstrap the standard errors.

Another problem with log-transformed model is that y = 0 cannot be logged. If modeling the en-

tire distribution of gifts is of interest, one can substitute a small value (typically 1) but the logarithm

transformation sensitive to small values 3.

An analogous formula exists for the square root transformation: this time instead of being multiplicative

the smearing factor is additive

√y = x′β + ε (9)

So E[y|x] = φ+ (x′β)2

φ =1

N

N∑i=1

εi2 (10)

3A change of giving from ln($1) to ln($2), an insignificant change, has the same effect as ln($10, 000) to ln($20, 000), a verysignificant change.

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2.5 Quasi-likelihood Generalized Linear Models (GLMs)

Second class of models relevant for modeling bequest amount have received considerable usage in the bio-

statistics and health economics literature for modeling health care costs [Mullahy, 1998, Blough, Madden,

and Hornbrook, 1999, Manning and Mullahy, 2001]. GLM models have a number of advantages over re-

gression of transformed outcome Y ; First, since model predictions are based on transforming µ(x) instead

of Y they eliminate the retransformation problem. Second, GLMs can be fit to the entire sample since

“zeros” in the domain of Y are accommodated, and if “excess zeros” are an issue, they can be utilized as

the second part of a two-part model. Third, the quasi-likelihood formulation [Wedderburn, 1974] separates

the specification of the mean function µ = g−1(.) along with the linear predictor from the variance function

h(.). This separation, has the advantage that if the mean function (link and linear predictor) are correctly

specified, then the estimates will be unbiased, and specification of the variance function in affects only the

efficiency (precision) of the estimates 4.

As the name implies, Generalized Linear Models (GLMs) are an extension of the classical linear model

and were first proposed by [McCullagh and Nelder, 1989]. To formulate a GLM, one must specify:

1. The linear predictor (which is defined as in the traditional linear model):

ηi = xTi β (11)

where xi is the [p × 1] vector of covariates for observation i and β is the [1 × p] vector of unknown

regression parameters.

2. A strictly monotone, differentiable link function, g(.), describes how the expected value of response

yi is related to the linear predictor:

g(µi) = ηi = xTi β (12)

where µi = E[yi | xi]. 5

3. The response variables y1, y2, . . . are assumed i.i.d. from a distribution of the Real Exponential Class

(Family) 6. Note this assumption implies the existence of a variance function relating the variance

4Choice between link and variance choice can be made canonical inks that re appropriate.5Monotonicity assures that the link function is invertible, and the inverse of the link function yields the mean function

g−1(ηi) = µi. To see this simply apply the inverse to both sides of g(µi) = ηi; g−1(g(µi)) = g−1(ηi) = g−1(xT

i β) ⇒ µi =g−1(xT

i β). The inverse link function ensures that xTi β when we insert the estimates β maintains the Gauss-Markov assumptions

for linear models and all of the results follow even though the outcome variable is non-normal.6The (Real) Exponential family, classified first by (Fisher 1934) developed the idea that most of the applied probability

mass and density functions such as the Gaussian, Binomial, Poisson, Exponential, Weibull, Gamma, and many others can bethought of as special cases of a more general classification.

8

of the outcome variable to it’s expectation:

Vi ≡ V ar(Yi|Xi) = h(µi, θ) (13)

GLMs modern extended form which uses quasi-likelihood estimation based on the iteratively reweighted

least squares (IRLS) algorithm [Wedderburn, 1974] relaxes this last requirement of specifying the distri-

bution of the errors, and only requires specifying a proportional functional relationship between mean and

variances usually taken to be of the Power Variance family V ar[y | x] ∝ (E[y | x])θ. Taking θ2 ∈ 0, 1, 2, 3

corresponds to the named distributions

and correspond to the named distributions ”” Specifying (θ = 0) corresponds to constant variance,

(θ = 1) variance is proportional to the mean or “Poisson like” while (θ = 2) or (θ = 3), the variance is

specified to be proportional to mean squared “Gamma- like model” or mean cubed “Inverse-Gaussian” In

general, there is no theoretical need for scale parameter to be integer and models of the form h() cane

implemented n STATA. Also, the (IRLS) algorithm requires using robust variance estimates [Huber, 1967,

White, 1980] for consistent estimates of variances.

2.6 Extended Estimating Equations

[Basu and Rathouz, 2005, Basu, 2005] developed an extension of the traditional quasi-likelihood GLM called

Extended Estimating Equations (EEE) where they use a Box-Cox style power link function along with a

variance structure, that are directly estimated with the traditional coefficients from the data allowing for

a more flexible functional form for the mean and variance estimators. The model estimates the mean and

variance parameters jointly from the data

Let Y denote the Amount given to charity. Note by definition Y ≥ 0. Let X = (X0, X1, . . . , Xp)′ be

the vector of covariates in the model where X0 is a vector of ones.

Letting µi = µ(Xi) a Generalized Linear Model (GLM) McCullagh and Nelder [1989] is specified by

g(µi) = ηi, ηi = XTi β where β is the vector of regression coefficients. The link function g(.) relating µi

to the linear predictor is taken to be strictly monotone and differentiable. Let variance of the variable be

given by Vi = V ar(Yi|Xi).

Following [Box and Cox, 1964] Basu and Rathouz defined the parametric family of variance functions

ηi = g(µi;λ) =

µλi −1λ , if λ = 0

ln(µi), if λ = 0

(14)

Generalized Quasi-Likelihood Linear Models additionally specify the functional form of the variance

9

function Define the variance function V (yi) = h(µi; θ1, θ2)

First, following Hardin and Hilbe [2001] the Power Variance as h(µi; θ1, θ2) = θ1µθ2i . A second para-

metric family Quadratic Variance is defined by h(µi; θ1, θ2) = θ1µi + θ2µ2i . 1 shows hoe the common

quasi-likelihood GLM variance families covered by their model.

Table 1: Distributional Cases Covered under Power Variance and Quadratic Variance in EEE

Power Quadratic Distribution

θ1 θ2 θ1 θ2

1 1 1 0 Poisson> 0 2 0 > 0 Gamma> 0 3 + + Inverse Gaussian+ + 1 > 0 Negative Binomial

Duplicated from [Basu, 2005]+ Indicates no reduction to GLM.

Since IRLS algorithm (maximum quasi-likelihood) optimization required for consistent estimates of

variance function [Huber, 1967, White, 1980].

2.7 Two-Part Models

A common feature in many charitable giving studies is a large number of people who do not donate in the

period of interest while the remainder give a varying amount of charity. In the sample of single headed

households analyzed here in the COPPS, (Diff answers with weights ???) 49.1 % did not give to charity

in 2006. Among those that do give, there are typically a substantial number of outliers giving many time

the average gift. In such studies, to understand the effect of a predictor on charitable amounts one cannot

ignore either the fact that many people do not give and among those that do give the distribution is highly

skewed. The two-part regression technique (cite here)[11] explicitly accounts for such data structures by

separately modeling the decision to give and the amount given conditional; on giving. The Two-part

model is a mixture distribution model with a point mass component at zero. The zero/nonzero outcome is

modeled by a binary choice model, typically logit or probit, and then the magnitude of non-zero responses

will be modeled conditionally by a continuous distribution technique, usually ordinary least squares on a

monotone transformation (usually log transformation) of the outcome to account for the skewness or an

appropriate form of a quasi-likelihood generalized linear model.

ADD STANDARD DISCUSSION OF TPM HERE.

SHOW MIXTURE DENSITY, and EXPECTATION (Shows Why Expectation works) Let Y denote

the charitable donation in dollars for a single head of household. Let x be vector of covariates. and θ a

10

vector of parameters. Given X = x, Let

I =

1, if Y > 0

0, otherwise.

(15)

fY (y; θ | x) =

Pr(I = 0 | x) if y = 0;

Pr(I = 1 | x)fY (y; θ | I = 1, x), if y > 0

0, if y < 0

(16)

For non-negative response Y the expectation for individual i can always be decomposed as

E[yi | xi] = Pr(yi > 0 | xi)E[yi | xi, yi > 0] (17)

Here, Part #1 specifies the probability of giving, specified either by a probit model

Pr(yi > 0) = Φ(x′iβ) (18)

where Φ is the CDF of the standard normal, or a logit

Pr(yi > 0) =ex

′iβ

1 + ex′iβ

(19)

INCREMENTAL EFFECTS ARE HARD IN TPM: REQUIRES DOUBLE DIFFERENCE To measure

the incremental effect of gender in the combined two-part model: let xd be the female indicator. Then

E[y | xd = 1]− E[y | xd = 0]

=(Pr(y > 0 | xd = 1)− Pr(y > 0 | xd = 0)

)E[y | y > 0, xd = 1]

− Pr(y > 0 | xd = 0)(E[y | y > 0, xd = 1]− E[y | y > 0, xd = 0]

) (20)

Calculating marginal and incremental effects in Two-part models becomes much more complicated if

interaction terms between covariates Xj and Xk are included as regressors in the non-linear first part [Ai

and Norton, 2003]. The calculation requires taking the full derivative or double difference. Further compli-

cations arise if there is heteroscedasticity in the second part which affects retransformation. Alternatives

to the scalar smearing factor include employing multiple smearing factors by group [Manning, 1998], using

an appropriate GLM in the second part [Blough, Madden, and Hornbrook, 1999] or explicitly modeling

the heteroscedasticity [Ai and Norton, 2000, 2008]

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3 Data Description

The data used in this study was drawn from the 2007 wave of the Center on Philanthropy Panel Study

(COPPS), a module of the Panel Study of Income Dynamics (PSID). The PSID is a longitudinal study that

surveys the same 8,000 households since 1968. Giving and volunteering questions were subsequently added

into the PSID in 2001 by the Center on Philanthropy at Indiana University and the Center has conducted

waves for 2001, 2003, 2005, 2007, and 2009 (forthcoming). The COPPS dataset represents the premier

source for examining giving and volunteering patterns by the same households over time with regards to

economic and demographic variables.

We restricted our analysis to single-headed households in the year 2007 (i.e., male and female-headed

households) because the COPPS data is collected at the house-hold level, preventing the separation and

analysis of giving patterns between males and females within married households. Previous research, finds

that married households give more and are more likely to give than singles. What about our sample? We

further restricted the analysis to those who had complete records for both the giving portion of the giving

and volunteering questionnaire and exogenous control variables. The final sample drawn from the COPPS

had 2408 individuals. The reader can refer to 3 in the Appendix for the definitions of all variables used in

the analysis.

Tables 4.2 and 4.2 show summary and descriptive statistics of total amount given to charity decomposed

by charitable sub-sector for the entire sample (unconditional) and for those who gave a donation in 2006

(conditional) respectively. Examination of Table 4.2 shows 41.55 % of single households did not donate to

charity with the percentage of zeros increasing dramatically when examining gifts by sub-sector. Given the

”spike of zeros”, measures of central tendency such as the mean and median for the entire sample is are

hard to interpret. Furthermore, amount of charitable donation are highly-right skewed with leptokurtic

right tails, violating the Gauss-Markov assumptions required of standard estimation techniques.

Table 4.2 shows sampling weight adjusted summary statistics for decomposed by gender. Of the

N = 2408 single household heads about two-thirds are female NF = 1573 and NM = 835. The fact that

there are roughly twice as many single women in the sample than single men is due to the the fact that

the PSID consists of a low income over-sample and restricting our analysis to singles yield proportionably

more people in the over-sample. Since single women are proportional more in the over-sample. 62.8% of

single females gave to some charity as compared to 51.5% of single males. Single women on average gave

18.86$ more than single men in (2006). However, men’s total amount given is much more (almost twice as)

variable than their single women counterparts SDM ≈ $4000 while SDF ≈ $2000 Single men, on average,

are more likely to never have married, make $17,297 more, and are $21,258 wealthier than their single

12

female counterparts. Single Women in the sample are on average 6yrs older, 17% more likely to regularly

attend church and to to be widowed, but the two groups are quite similar in all other aspects.

Univariate tests of mean differences were performed 7.

4 Estimation Results

To address our research questions we focus on measuring the gender effect after controlling for observable

differences between men and women, and determining the effect of income in both the probability of giving

and in the amount given.

4.1 Giving Incidence Models

To determine the gender effect in the incidence of giving, we fit three versions of binary response models

commonly used in the literature; linear probability model (LPM), logit, and probit. ?? shows the estimated

coefficients and the estimated incremental effect of gender computed by the method of recycled predictions

8 All three models show highly significant effect of gender, with estimated incremental effects indicating

that single women are about 9% more likely to give than their single male counterparts. The estimated

logit coefficients are about 1.72 larger than those of the probit, a result which is due to the different iden-

tifying assumptions of the two models, however the yield nearly identical predictions (Pearson Correlation

Coefficient of (0.9984) between the two) [See Long, 1997, pg. 47-49.] 9 Several of the model assumptions of

the LPM are violated including heteroscedasticity, non-normality of the errors, and predicted probabilities

less than zero or exceeding one, and the linear functional form violates many empirical economic findings,

all reasons for the development of logit and probit models in the first place. Given the similarity in the fit

of the logit and probit models and the fact that the logit model is proportional in the odds 10, we choose

to examine effects of the covariates in the logit model in detail. Given the logit model is non-linear, no sin-

gle approach to interpretation can describe the relationship between an exogenous variable and predicted

probability of giving, so we examine the effects using odds ratios effect of the predicted probabilities in

tables 4.2 4.2 respectively.

4.2 displays the factor change coefficients and 95% confidence intervals for giving incidence for the logit

model. Being female increases the odds of giving by a factor of 1.6979 holding all other variables constant.

7Difference in means were tested after adjusting for the non-random sampling weights used in the PSID. Bartletts test ofequal variances was performed, and Swaiter approximation was used in these cases. adjusted methods weight

8The standard errors for the incremental effect were estimated via bootstrap methods (TO Be DONE).9The logit model assumes the identification assumption V ar(ε | x) = π2/3 while the probit assumes V ar(ε | x) = 1 so

V ar(εL | x) = (π2/3)V ar(εP | x).10For the Logit Model, the effect of variable xk in the odds is independent of the level of xk or the level of any other variable

in the model, which aids in interpretation.

13

Other notable positive effects were found for being retired (as opposed to being unemployed or disabled),

tax itemization, religions attendance, and for attained education. Being retired, itemizing federal income

taxes, and monthly religious attendance have percentage changes in the odds of giving of 43.5%, 458.7%,

and 233.1% respectively. [Think Education Table OF Predicted Prob mentioned in next paragraph Might

be Useful] Since the odd ratio is a multiplicative coefficient the magnitudes of positive and negative effects

should be compared by taking the inverse of the negative effects, and are presented in the last column of

4.2.

A constant factor change in the odds does not correspond to a constant factor change in the probability

[See Long, 1997, pg.82]. Furthermore, the effect on the probability depends on the level of all the variables

in the model. Recall the S-shape of the logistic pdf, when the range of predicted probabilities is between

0.3 and 0.7, the relationship between the P r(yi = 1 | x′iβ) and xk is nearly linear and one can use the

estimated coefficients to summarize the effect. Also if the range of the predicted probability is small the

effect will also be approximately linear. 4.2 shows that the marginal and incremental effects holding all

other variables constant at the mean and median, and all effects are nearly linear in this range. (NOTE

THE SMALL EFFECT OF INCOME AND THE STAT INSIGNIFICANT ONE OF WEALTH, NEED

TO ADD PERHAPS A SQUARE BUT WHAT ABOUT NEG VALUES?)

The next step to understanding the effects of the independent variables is to determine. The range

of the predicted probabilities. Little can be learned by analyzing variables whose range of probabilities is

small, such as the indicator for working. For variables with large ranges, the endpoints dictate

Next thought was that this specification may be to restrictive, what if single men and women not only

differ in their levels of the covariates but respond differently to these effects? This suggests running separate

regressions for men and women 11. Unlike least squares, the coefficients from nonlinear GLMs cannot be

directly compared, the differences in estimates are confounded with unobserved residual variation. Using

Long’s suggestion that the predicted probabilities are invariant to the identification problem we examine

the effect of gender and income in this setting. 4.2 shows the predicted probability for men and women

over the distribution of income holding all other variables at the sample mean. This plot shows that

women and men respond differently at different levels of income. 4.2 plots the difference in predicted

probabilities from 4.2 with a corresponding 95% confidence interval calculated via the delta-method. When

the confidence interval crosses the zero-line the difference is statistically insignificant. This plot shows that

gender differences in giving incidence are insignificant at lowest and highest levels of income, but women

and more likely to give in the income range $28,000 to $275,000, a range including more than half the

sample (%53 pull exact numbers). Furthermore the apparent higher male incidence at the lowest end is

11equivalently excluding the intercept and interacting gender with all other variables in the model

14

insignificant, and there is virtually no difference at levels of income exceeding $275,000.

4.1.1 Interpreting the Income Plots OUTLINE

OUTLINE OF INCOME STORY: ADDED BONUS WE FIND GENDER DIFFERENCES TOO!

Plot [1] without square shows nonlinear pattern to response of giving incidence to income and income.

Also shows apparent gender differences in predicted giving incidence over lower levels of income but no

difference in incidence at higher levels. This model doesn’t allow for separate effects of variables by

gender. Plot [2] shows that predicted incidence for men and women when allowing them to have difference

responses to income. See women are much more responsive to income then men. Plot [3] shows that this

difference is statistically significant over the range of income 35, 000−280,000, which constitutes most of the

support (and distribution) of income. Plot show two more things, (1) (Tail of Plot: ‘No Difference is stat

significant?’) (2)increasing decreasing pattern of differences(Possibility we are not properly capturing the

effect of income before comparing the gender difference. We test this hypothesis that the giving incidence

is a nonlinearly increasing or inversely U-shaped function of household income. Mention must use centered

income and wealth squared to get ME. Examining Table X (Logit with Square) show that the the coefficients

for income (+) and income squared (-) [(and wealth (+), wealth squared (-))] support the hypothesis that

the propensity of giving depends nonlinearly on income. The coefficients for wealth and wealth squared are

not individually significant at the conventional 5-percent level they are jointly significant.(Cannot reject

null hypothesis Likelihood Ratio Test of βwealth = βwealth2 = 0 χ2[2] = 1.97 p-value= 0.3743.

This sign pattern yield two possibilities for the response of incidence to income the effect can be strictly

increasing or increasing over some range to some maximum, and then decreasing past this point (parabola

down). The effect is parabola down if the value of income that maximizes the linear prediction.falls within

the range of income. Invoke the invariance property of maximum likelihood estimators that maximum

value of income that also maximizes linear prediction of two and is given by by estimate argmaxincome =

− βinc

βinc2= 532.5695 which is in the support of income, so we parabola down. Similarly for wealth − βwealth

βwealth2=

48541.526 which falls outside the range of wealth so wealth has a strictly positive effect on incidence.

(CHANGE INC2WEALTH SQUARED IN SUMMARY STATS) Capture total effect of income as weighted

linear combination of main effect income (income centered square=0) and ( effect of everything held at

mean)

Missing CI PLOT OF GENDER DIFFERENCES WITH INC INC SQUARED.

15

4.2 Model Predictions and Diagnostics: Amount Given Models

Primary diagnostic concern is having a systematic bias as a function of covariates, Secondary concern is

efficiency (precision). GLMs unbiased as link as mean function (linear predictor and link) are correctly

specified.,

First question in modeling amount of donations either with transformed OLS or GLMs is to determine

the the relationship between the linear index x′β and E[y | x], to determine either a transformation to

near normality or to determine the functional form of the link function. The health economics literature

offers little guidance for the applied researcher on how to identify the correct link function. In fact, most

health care studies of costs simply assume the log link. No single diagnostic test can that identifies the

appropriate link, and the literature has generalized (modified)a multitude of tests designed originally for

OLS to models with a linear index x′β.

The Box-Cox estimate of the link parameter of λ = 0.0976 95% CI: (0.0690, 0.1261). This is ”close”

but statistically significant difference. The Box-Cox test is a relatively weak test and is known to be

biased in the face of heteroscedasticity (FIND CITATION), but can function as a guide to model building.

Other within sample diagnostic tests include the Pregibon Link Test [Pregibon, 1980] and the Ramsey

RESET test [Ramsey, 1969, Thursby and Schmidt, 1977] which check the linearity of the response on

scale of estimation, and the Pearson Correlation Test [McCullagh and Nelder, 1989] (Basu pg512) and

Modified Hosmer-Lemeshow Tests [Hosmer and Lemeshow, 2000] which check for systematic bias estimate

of prediction of E[Y | X] on the $ scale. All these tests are non-constructive,i.e they tell you might have

a problem but do not recommend a fix.

After determining an appropriate link, the researcher wishing to employ a quasi-likelihood GLM must

determine the relationship between the mean and the variance functions on the raw scale, [Manning and

Mullahy, 2001] suggested using the Modified Park Test [Park, 1966]. This procedure, regresses the squared

residuals (on the raw scale) from a provisional model on the predictions:

ln((yi − yi)2) = γ0 + γ1 ln(yi) + ϵi (21)

The coefficient γ1 in 21 indicates which variance function is appropriate (γ = 0) Gaussian NLLS, (γ = 1)

variance proportional to mean, (γ = 2) variance exceeds the mean, (γ = 3) Wald or Inverse Gaussian 12.

For example when this procedure is applied to the provisional 1-Pt Gamma GLM model on the COPPS

data yielded the estimate γ1 = (S.E. = x.xxx) and there was little sensitivity with respect to the choice of

the provisional model. This suggests that the best model falls between variance proportional to the mean

12Literature recommends using a GLM model to perform this test since the log-transformed squared residual raises the issueof retransformation bias

16

or standard deviation proportional to mean γ1 =

The following one and two-part models for amount given were fit and the parameter estimates are

presented in 4.2: (1) Tobit model on transformed outcome ln(Y + 1) which has been used by (cite?) 13.

(2)Also fit were one-part log-link Poisson and Gamma Families, the Extended Estimating Equation, and

a power-link power family suggested by the results of the extended estimating equation model.

Four versions of Two-Part models; each using the logit model as the first part ”hurdle” component,

with ln(y) normal-theory, common smearing factor (S = 1.9048), and separate smearing factors by gender

(SF = 1.9410, SM = 1.8265). log-link gamma family GLM.

All models preform surprising well given the 48.2% zeros in the COPPS. 4.2 show the average predictions

by decile plotted against the average gift by decile.

010

0020

0030

0040

0050

00P

redi

cted

and

Act

ual A

mou

nt

0 500 1000 1500 2000 2500 3000 3500Mean Prediction by Decile

Actual Amount1Pt−GLM Polsson1Pt−GLM Gamma1Pt−EEE2Pt− Lognormal2Pt−OLS 1−Smear2Pt−OLS 2−Smears2Pt−GLM Gamma

Figure 1: average predictions by decile plotted against the average gift by decile

All versions of the one part models show a statistically significant gender effect except the 1-part

Log-link Poisson family. To asses the magnitude of the gender and income effects on the dollar scale we

estimated the incremental effect (IE) for Female and the marginal effect for income via the method of

recycled predictions,

The Extended Estimation Equations estimate for the link parameter λ = 0.2508 95% CI: (0.1372,

0.3643) indicates the optimal link is neither the identity nor log but the fourth-root link (RIGHT?). The

estimated Power Variance parameters θ1 = 3.4188 95% CI:(2.5302, 4.3074), θ2 = 1.5307 (95% CI: 1.4003,

1.6612) show the optimal variance is somewhere between the Poisson and Gamma types. These result

shows the advantage of estimating the link and variance within the model.

The Two-Part Models13The log transformation was taken to make deal with the non-normality (skewness and kurtosis) while the constant was

added to accommodate the ”zeros” since ln(0) = −∞

17

Table 2: Comparison of Mean Predictions and Estimates of I.E. Gender M.E. Income

Amount Model Mean $ Std. Dev. Min. Max. I.E. Female M.E. Income

Actual Amount 865.7184 2791.150 0.00 70000.001Pt-GLM Poisson 865.7184 2149.733 23.24 58595.76 141.94 (105.80)1Pt-GLM Gamma 1087.302 3008.146 15.26 43020.87 329.06 (137.52)1Pt-EEE 921.6954 1745.698 4.72 30795.71 174.62 (126.43) 8.94 (3.159)2Pt-OLS Log Normal 999.7866 1909.499 3.50 32369.022Pt-OLS Common Smear 946.3597 1807.459 3.31 30639.272Pt-OLS 2 Smears 943.5065 1785.344 3.18 29380.772Pt-GLM Gamma 925.8159 2259.940 5.26 51257.93 143.26

N = 2408, Sampling Weight Adjusted (delta method s.e. in brackets () and bootstrap s.e. in braces[] )

0.2

5.5

.75

1P

r(P

ositi

ve G

ivin

g)

0 100 200 300 400 500 600Income ($ 1000’s)

Female Male

Effect of Income on Pr(Positive Giving)

Figure 2: Predicted Probability of Giving by Gender over Distribution of Income Logit Model

18

0.2

5.5

.75

1P

r(P

ositi

ve G

ivin

g)

0 200 400 600Income

Female Male

wg03_logitgroupcompare.do 2011−05−26 −jem

Effect of Income on Pr(Positive Giving)

Figure 3: Predicted Probability of Giving by Gender over the Distribution of Income Separate EffectsLogit Model

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Source: wg03_logitgroupcompare_F3.eps from wg03_logitgroupcompare.do jem 2011−06−20

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Figure 5: Difference In Predicted Probabilities with 95% C.I.

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Figure 6: Predicted Probability of Giving by Gender over Distribution of Income (Income2) Separate Effects LogitModel

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Actual Amount1Pt−GLM Polsson1Pt−GLM Gamma1Pt−EEE2Pt− Lognormal2Pt−OLS 1−Smear2Pt−OLS 2−Smears2Pt−GLM Gamma

Figure 7: average predictions by decile plotted against the average gift by decile

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23

Table 3: Code Book of Variables

Response Variables

total Total amount donated to charities in 2007.gtotal Indicator Variable =1 if Household donated to charity in 2007.

Sub-sector

Religion Giving to religious organizations.Combined Combination of purposes. United Way, The United Jewish Appeal, Catholic Charities, or

local community foundations.Needy Organizations which help people in need of food, shelter, or other basic necessities.Health Health care organizations such as hospitals, nursing homes, mental health facilities, cancer or

heart lung associations, or health telethons.Education Colleges and Universities, Parent Teacher Associations, libraries, or scholarship funds.Youth Youth or family services such as scouting, boys’ and girls’ clubs, sport leagues, Big Brothers-

Sisters, foster care, or family counseling.Cultural Organizations that support or promote the arts, culture, or ethnic awareness. Examples:

museums, theaters, orchestras, public broadcasting, or ethnic cultural awareness programs.Community Organizations that improve neighborhoods and communities. Examples: community associ-

ations or service clubs.Environment Organizations that preserve the environment; such as organizations for conservation efforts,

animal protection, or parks.International International or peace organizations that provide aid or promote world peace. Examples:

international children’s funds, disaster relief, or human rights.Other Catch all for organizations which do not fit into the above categories.

Control Variables

African American = 1 if head is African-American; = 0 otherwise.Female =1 if Head of Household is Female; =0 if male.Hispanic = 1 if head is Hispanic; = 0 otherwise. Note ”White” and ”Other” are the reference category.Healthy = 1 if household head was in good or excellent health; = 0 otherwise.Age Age of the household head in years.nkids5 Number of children present in the household age 5 or less.nkids618 Number of children present in the household age 6 to 17.Attend Church =1 if head attended religious services 1 or more times a month;= 0 otherwise.Less Than High School = 1 if head did not graduate from high school; = 0 otherwise. (Reference category)High School = 1 if head received highs school diploma or GED.; = 0 otherwise.Some College = 1 if head received associates degree or attended four year college program but did not

receive Bachelor’s degree; = 0 otherwise.College = 1 if heads received B.A. or B.S degree; = 0 otherwise.Graduate School = 1 if head received M.A., M.S., or Ph.D.; = 0 otherwise.Working = 1 if head was employed in 2007; = 0 otherwise.Retired = 1 if head was retired in 2007; = 0 otherwise.Unemployed = 1 if head was unemployed in 2007; = 0 otherwise.Disabled = 1 if head was disabled and unable to work in 2007; = 0 otherwise. Note that unemployed

and disabled are the reference category.Wealth Total value of household wealth (in $1000’s) excluding home equity. Note: negative values of

Wealth were recoded to zero to accommodate a potential non-linear wealth effect by includingWealth2 as the covariate specification.

Wealth House Household wealth (in $1000’s) including home equity.Income Total value of household income in $1000’s in 2006. Note: one reported negative value of

income was recoded to zero to accommodate a potential non-linear income effect by includingincome2 as an additional covariate.

Itemize = 1 if head filed itemized deduction on their federal income taxes in 2007; = 0 otherwise.Marginal Tax Rate = effective tax rate (Osili should define)North East = 1 if Household located in North-East” Census Region; = 0 otherwise.North Central = 1 if Household located in North-Central Census Region; = 0 otherwise.South = 1 if Household located in North-East” Census Region; = 0 otherwise.West - Other = 1 if Household located in West Census Region, Hawaii, or Alaska, or listed a foreign country

as primary residence; = 0 otherwise. (Reference Category)Never Married = 1 if head has never been married in 2007; = 0 otherwise.Separated = 1 if head is currently separated (living separately) from spouse in 2007; = 0 otherwise.Widowed = 1 if head is widowed; = 0 otherwise.Divorced = 1 if head is divorced; = 0 otherwise.

Table 4: Summary Statistics: Amount Given To Charity

Amount $ Mean Std. Dev. Median Skewness Kurtosis Maximum % 10×Mean % Positive

Total 865.7184 2791.150 150 16.304 375.493 70000 0.83 58.45Religion 494.1459 2166.027 0 20.659 550.578 60000 1.79 36.10

Combined 89.0329 631.369 0 27.612 1014.465 25000 1.74 20.89Needy 102.6763 386.539 0 8.563 95.815 5000 1.29 27.15Health 56.1568 376.692 0 15.768 295.365 8000 0.50 19.79

Education 38.4836 381.723 0 18.756 398.895 10000 1.45 10.54Youth 12.7579 86.070 0 13.692 241.864 2000 1.95 9.54

Culture 13.9666 78.262 0 10.789 178.414 2000 1.91 7.67Community 9.3686 105.482 0 19.965 459.301 3000 1.33 4.79

Environment 16.7032 155.853 0 24.336 797.378 6000 1.29 8.75International 11.8000 108.196 0 16.654 333.455 2500 1.16 5.75

Other 20.6266 176.970 0 20.018 568.690 8000 1.54 6.61

N = 2408Sampling Weight AdjustedSource: wg00 SummaryStats2.do

Table 5: Summary Statistics: Amount Given To Charity Conditional on Giving

Amount $ Obs. Mean Std. Dev. Median Maximum Skewness Kurtosis

Total 1225 1481.2168 3524.521 640 70000 13.3434 243.3744Religion 791 1368.7338 3436.264 600 60000 13.5752 226.9506

Combined 452 426.1656 1329.480 200 25000 13.4172 233.8609Needy 533 378.2182 668.408 200 5000 4.8156 30.5895Health 356 283.7171 808.623 100 8000 7.216 61.9959

Education 225 365.1235 1126.180 100 10000 6.0756 42.7821Youth 187 133.6615 248.498 50 2000 4.4265 26.7094

Culture 124 182.1337 222.735 100 2000 3.4673 21.7196Community 89 195.3894 444.782 50 3000 4.3093 22.6247

Environment 133 190.9986 496.186 50 6000 7.6153 78.5253International 103 205.2555 406.746 50 2500 3.9869 20.5435

Other 121 312.0734 621.190 130 8000 5.5689 45.7022

Source: wg00 SummaryStats2.do, Sampling Weight Adjusted

25

Tab

le6:Summary

Statistics:

Men

VersusWomen

Males

Females

Variable

Mean

Std

.Dev.

Min

Max

Mean

Std

.Dev.

Min

Max

XF−

XM

Resp

onse

Variables

GaveToCharity

0.5150

0.500

01

0.6275

0.484

01

0.1125∗∗∗

TotalAmtGiven

854.0758

3785.375

070000

872.9402

1934.516

035500

18.8644

TotalAmtGiven

ifGiving>

01658.3608

5150.147

470000

1391.0398

2290.252

135500

−267.3210

ControlVariables

Demogra

phic

Variables

AfricanAmerican

0.1535

0.361

01

0.2333

0.423

01

0.0798∗∗∗

Hispanic

0.0621

0.241

01

0.0603

0.238

01

−0.0017

healthy

0.8017

0.399

01

0.7687

0.422

01

−0.0330∗

age

50.9470

15.803

21

98

57.2672

16.887

23

101

6.3202

#kids<

60.0644

0.330

04

0.0799

0.329

03

0.0155

#kids≥

6and<

18

0.1508

0.536

07

0.3322

0.778

07

0.1814

Atten

dChurch

0.3307

0.471

01

0.5040

0.500

01

0.1733∗∗∗

Highest

Education

Attained

Education(Y

ears)

13.1701

2.533

017

12.8002

2.586

017

−0.3699

LessThanHighSchool

0.1540

0.361

01

0.1940

0.396

01

0.0401∗∗∗

HighSchool(D

iploma/GED)

0.3612

0.481

01

0.3727

0.484

01

0.0115

SomeCollegeorAss.

0.3544

0.479

01

0.3476

0.476

01

−0.0068

College(B

A/BS)

0.1202

0.325

01

0.1156

0.320

01

−0.0046∗

Graduate

School

0.1304

0.337

01

0.0856

0.280

01

−0.0447

EmploymentSta

tus

working

0.6441

0.479

01

0.5620

0.496

01

−0.0821∗∗∗

retired

0.1939

0.396

01

0.2651

0.442

01

−0.0672∗∗∗

unem

ployed

0.0972

0.296

01

0.1114

0.315

01

0.0143

disabled

0.0648

0.246

01

0.0615

0.240

01

−0.0033

Income-W

ealth

-Tax

Wealth($1000’s)

207.6267

871.985

013128.03

186.7967

1926.982

048389.21

20.83

WealthHouse

($1000’s)

277.6462

933.665

−281.9261

13614.11

269.1545

1965.931

−243.0397

49069.72

−8.4917

Income($1000’s)

54.9276

66.785

0600

37.6437

38.615

0632.75

17.2840

Item

ize

0.2370

0.426

01

0.2730

0.446

01

0.0360∗

MarginalTaxRate

0.8497

0.109

0.63

10.8853

0.115

0.54

10.0356

Censu

sRegion

NorthEast

0.1726

0.378

01

0.1763

0.381

01

0.0037

NorthCen

tral

0.2585

0.438

01

0.2779

0.448

01

0.0193

South

0.3469

0.476

01

0.3624

0.481

01

0.0155

West-Other

0.2220

0.416

01

0.1834

0.387

01

−0.0386∗∗

Marita

lSta

tus

Never

Married

0.3775

0.485

01

0.2680

0.443

01

−0.1095∗∗∗

Sep

arated

0.0718

0.258

01

0.0663

0.249

01

−0.0055

Widow

ed0.1258

0.332

01

0.3080

0.462

01

0.1822∗∗∗

Divorced

0.4249

0.495

01

0.3577

0.479

01

−0.0672∗∗∗

N=

835

1573

Sum

ofW

gt=

22552.402

36357.672

Statisticsare

samplingweightadjusted

,Standard

errors

inparentheses

*p<

0.10,**p<

0.05,***p<

0.01

26

Tab

le7:

GivingIncidence:Parameter

Estim

ates

FullSample

Sep

arate

Effects

LPM

Probit

Logit

Logit

(Males)

Logit

(Fem

ales)

female

0.0994∗∗∗

0.328∗∗∗

0.563∗∗∗

(0.0186)

(0.0653)

(0.111)

africanamerican

−0.0778∗∗∗

−0.262∗∗∗

−0.439∗∗∗

−0.232

−0.482∗∗∗

(0.0222)

(0.0762)

(0.130)

(0.241)

(0.157)

hispanic

−0.0404

−0.138

−0.177

−0.611

0.0578

(0.0366)

(0.126)

(0.216)

(0.415)

(0.263)

hea

lthy

0.0311

0.0640

0.126

−0.163

0.280∗

(0.0217)

(0.0739)

(0.125)

(0.221)

(0.155)

age

0.00336∗∗∗

0.00998∗∗∗

0.0171∗∗∗

0.0214∗∗∗

0.0129∗∗

(0.000763)

(0.00269)

(0.00458)

(0.00811)

(0.00576)

retired

0.0767∗∗

0.215∗∗

0.343∗

0.805∗∗

0.116

(0.0314)

(0.109)

(0.185)

(0.361)

(0.219)

working

0.0110

0.0203

0.00333

0.494∗

−0.372∗

(0.0253)

(0.0877)

(0.148)

(0.262)

(0.191)

#kids<

6−0.0434

−0.132

−0.249

−0.515∗

−0.0877

(0.0271)

(0.0902)

(0.159)

(0.310)

(0.195)

#kidS6to18

−0.0168

−0.0557

−0.0909

0.156

−0.230∗∗

(0.0135)

(0.0470)

(0.0806)

(0.167)

(0.0952)

atten

dch

urch

0.215∗∗∗

0.733∗∗∗

1.220∗∗∗

1.240∗∗∗

1.254∗∗∗

(0.0181)

(0.0640)

(0.109)

(0.193)

(0.135)

highschool

0.0926∗∗∗

0.254∗∗∗

0.425∗∗∗

0.218

0.550∗∗∗

(0.0271)

(0.0921)

(0.156)

(0.303)

(0.184)

someco

lleg

e0.161∗∗∗

0.441∗∗∗

0.760∗∗∗

0.590∗

0.805∗∗∗

(0.0285)

(0.0964)

(0.164)

(0.306)

(0.198)

colleg

e0.230∗∗∗

0.763∗∗∗

1.258∗∗∗

1.010∗∗∗

1.419∗∗∗

(0.0341)

(0.123)

(0.209)

(0.365)

(0.271)

gradschool

0.133∗∗∗

0.392∗∗

0.596∗∗

0.145

0.838∗∗

(0.0449)

(0.172)

(0.294)

(0.491)

(0.385)

item

ize

0.239∗∗∗

0.920∗∗∗

1.663∗∗∗

2.125∗∗∗

1.274∗∗∗

(0.0223)

(0.0895)

(0.168)

(0.299)

(0.208)

inco

me

0.00367∗∗∗

0.0137∗∗∗

0.0231∗∗∗

0.0159∗∗∗

0.0343∗∗∗

(nohouse)

(0.000389)

(0.00152)

(0.00268)

(0.00369)

(0.00436)

inco

me2

−0.00000603∗∗∗

−0.0000222∗∗∗

−0.0000371∗∗∗

−0.0000260∗∗∗

−0.0000495∗∗∗

(0.000000842)

(0.00000311)

(0.00000550)

(0.00000769)

(0.0000117)

wea

lth

0.00000491

0.000127

0.000303

0.000331

0.000363

(0.0000179)

(0.000120)

(0.000238)

(0.000434)

(0.000440)

wea

lth2

−2.84e−

10

−2.81e−

09

−6.19e−

09

−4.22e−

09

−8.08e−

09

(3.90e−

10)

(3.64e−

09)

(7.93e−

09)

(0.000000104)

(1.08e−

08)

intercep

t−0.0969∗

−1.864∗∗∗

−3.156∗∗∗

−3.178∗∗∗

−2.549∗∗∗

(0.0532)

(0.189)

(0.326)

(0.560)

(0.417)

IEFem

ale

0.0994

0.0929

0.0938

N2408

2408

2408

835

1573

ll−1272.1

−1190.8

−1189.2

−410.4

−758.5

aic

2582.2

2421.6

2418.4

858.7

1555.1

bic

2692.1

2537.3

2534.1

948.5

1656.9

Sta

ndard

errors

inpare

nth

ese

s*

p<

0.10,**

p<

0.05,***

p<

0.01

Sampling

WeightAdju

sted

27

Table 8: SAME PORTRAIT ORIENTATION Giving Incidence: Parameter Estimates

Full Sample Separate EffectsLPM Probit Logit Logit (Males) Logit (Females)

female 0.0994 ∗ ∗∗ 0.328 ∗ ∗∗ 0.563 ∗ ∗∗(0.0186) (0.0653) (0.111)

african american −0.0778 ∗ ∗∗ −0.262 ∗ ∗∗ −0.439 ∗ ∗∗ −0.232 −0.482 ∗ ∗∗(0.0222) (0.0762) (0.130) (0.241) (0.157)

hispanic −0.0404 −0.138 −0.177 −0.611 0.0578(0.0366) (0.126) (0.216) (0.415) (0.263)

healthy 0.0311 0.0640 0.126 −0.163 0.280∗(0.0217) (0.0739) (0.125) (0.221) (0.155)

age 0.00336 ∗ ∗∗ 0.00998 ∗ ∗∗ 0.0171 ∗ ∗∗ 0.0214 ∗ ∗∗ 0.0129 ∗ ∗(0.000763) (0.00269) (0.00458) (0.00811) (0.00576)

retired 0.0767 ∗ ∗ 0.215 ∗ ∗ 0.343∗ 0.805 ∗ ∗ 0.116(0.0314) (0.109) (0.185) (0.361) (0.219)

working 0.0110 0.0203 0.00333 0.494∗ −0.372∗(0.0253) (0.0877) (0.148) (0.262) (0.191)

# kids < 6 −0.0434 −0.132 −0.249 −0.515∗ −0.0877(0.0271) (0.0902) (0.159) (0.310) (0.195)

# kids −0.0168 −0.0557 −0.0909 0.156 −0.230 ∗ ∗(0.0135) (0.0470) (0.0806) (0.167) (0.0952)

attend church 0.215 ∗ ∗∗ 0.733 ∗ ∗∗ 1.220 ∗ ∗∗ 1.240 ∗ ∗∗ 1.254 ∗ ∗∗(0.0181) (0.0640) (0.109) (0.193) (0.135)

high school 0.0926 ∗ ∗∗ 0.254 ∗ ∗∗ 0.425 ∗ ∗∗ 0.218 0.550 ∗ ∗∗(0.0271) (0.0921) (0.156) (0.303) (0.184)

some college 0.161 ∗ ∗∗ 0.441 ∗ ∗∗ 0.760 ∗ ∗∗ 0.590∗ 0.805 ∗ ∗∗(0.0285) (0.0964) (0.164) (0.306) (0.198)

college 0.230 ∗ ∗∗ 0.763 ∗ ∗∗ 1.258 ∗ ∗∗ 1.010 ∗ ∗∗ 1.419 ∗ ∗∗(0.0341) (0.123) (0.209) (0.365) (0.271)

grad school 0.133 ∗ ∗∗ 0.392 ∗ ∗ 0.596 ∗ ∗ 0.145 0.838 ∗ ∗(0.0449) (0.172) (0.294) (0.491) (0.385)

itemize 0.239 ∗ ∗∗ 0.920 ∗ ∗∗ 1.663 ∗ ∗∗ 2.125 ∗ ∗∗ 1.274 ∗ ∗∗(0.0223) (0.0895) (0.168) (0.299) (0.208)

income 0.00367 ∗ ∗∗ 0.0137 ∗ ∗∗ 0.0231 ∗ ∗∗ 0.0159 ∗ ∗∗ 0.0343 ∗ ∗∗(no house) (0.000389) (0.00152) (0.00268) (0.00369) (0.00436)

income2 −0.00000603 ∗ ∗∗ −0.0000222 ∗ ∗∗ −0.0000371 ∗ ∗∗ −0.0000260 ∗ ∗∗ −0.0000495 ∗ ∗∗(0.000000842) (0.00000311) (0.00000550) (0.00000769) (0.0000117)

wealth 0.00000491 0.000127 0.000303 0.000331 0.000363(0.0000179) (0.000120) (0.000238) (0.000434) (0.000440)

wealth2 −2.84e− 10 −2.81e− 09 −6.19e− 09 −4.22e− 09 −8.08e− 09(3.90e− 10) (3.64e− 09) (7.93e− 09) (0.000000104) (1.08e− 08)

intercept −0.0969∗ −1.864 ∗ ∗∗ −3.156 ∗ ∗∗ −3.178 ∗ ∗∗ −2.549 ∗ ∗∗(0.0532) (0.189) (0.326) (0.560) (0.417)

IE Female 0.0994 0.0929 0.0938

N 2408 2408 2408 835 1573ll −1272.1 −1190.8 −1189.2 −410.4 −758.5aic 2582.2 2421.6 2418.4 858.7 1555.1bic 2692.1 2537.3 2534.1 948.5 1656.9

Standard errors in parentheses * p < 0.10, ** p < 0.05, *** p < 0.01

Sampling Weight Adjusted

28

Tab

le9:AmountGiven:Parameter

Estim

ates

1-P

art

Models

2-P

art

Models

Tobit

1Pt-P

oisson

1Pt-G

am

ma

1Pt-E

EE

Logit

[1]

Log

OLS

[2A

]2Pt-G

am

ma

[2B]

female

1.056∗∗∗

0.164

0.303∗∗∗

0.228∗∗

0.563∗∗∗

0.0756

0.0803

(0.211)

(0.121)

(0.117

(0.0973)

(0.146)

(0.0915)

(0.0881)

african

american

−1.778∗∗∗

0.256∗

0.0215

0.0299

−0.439∗∗∗

0.0871

0.291∗∗∗

(0.263)

(0.139)

(0.132)

(0.116)

(0.162)

(0.117)

(0.107)

hispanic

−0.261

−0.0783

0.0530

0.00400

−0.177

0.0431

0.151

(0.434)

(0.247)

(0.275)

(0.245)

(0.281)

(0.198)

(0.231)

healthy

0.218

0.119

0.100

0.204

0.126

0.00999

−0.00575

(0.251)

(0.168)

(0.134)

(0.131)

(0.170)

(0.135)

(0.106)

age

0.0414∗∗∗

0.00585

0.0133∗∗∗

0.00884∗∗

0.0171∗∗∗

0.00374

0.00423

(0.00860)

(0.00513)

(0.00464)

(0.00427)

(0.00592)

(0.00374)

(0.00337)

retire

d0.966∗∗∗

−0.0873

0.297

0.274

0.343

−0.135

−0.0595

(0.356)

(0.178)

(0.191)

(0.170)

(0.243)

(0.152)

(0.145)

work

ing

0.207

−0.526∗∗

−0.171

−0.132

0.00333

−0.372∗∗∗

−0.376∗∗∗

(0. 302)

(0.220)

(0. 160)

(0.154)

(0. 177)

(0.133)

(0. 130)

nkid

s5−1.111∗∗∗

−0.409∗∗

−0.265∗

−0.110

−0.249

−0.229∗∗

−0.269∗∗

(0.342)

(0.201)

(0.140)

(0.0946)

(0.178)

(0.113)

(0.117)

kid

s618

−0.538∗∗∗

−0.0974

−0.102

−0.00411

−0.0909

−0.0899

−0.0284

(0.165)

(0.106)

(0.0835)

(0.0660)

(0.0865)

(0.0641)

(0.0626)

high

school

1.452∗∗∗

0.771∗∗∗

0.518∗∗

0.325∗

0.425∗∗

0.433∗∗

0.359∗∗

(0.331)

(0.192)

(0.219)

(0.174)

(0.200)

(0.183)

(0.176)

somecollege

2.396∗∗∗

0.941∗∗∗

0.691∗∗∗

0.613∗∗∗

0.760∗∗∗

0.572∗∗∗

0.330∗

(0.343)

(0.186)

(0.214)

(0.172)

(0.215)

(0.181)

(0.171)

college

3.034∗∗∗

1.311∗∗∗

0.963∗∗∗

0.961∗∗∗

1.258∗∗∗

0.661∗∗∗

0.557∗∗∗

(0.396)

(0.210)

(0.242)

(0.194)

(0.268)

(0.196)

(0.188)

gra

dsc

hool

2.213∗∗∗

1.561∗∗∗

1.113∗∗∗

1.166∗∗∗

0.596

0.827∗∗∗

0.867∗∗∗

(0.504)

(0.298)

(0.351)

(0.342)

(0.383)

(0.256)

(0.253)

wealth

0.000207

0.000283∗∗∗

0.000267∗∗∗

0.000435∗∗∗

0.000303

0.000230∗∗∗

0.000273∗∗∗

(0.000187)

(0.0000269)

(0.0000710)

(0.000137)

(0.000408)

(0.0000890)

(0.0000560)

wealth2

−5.41e−

09

−5.47e−

09∗∗∗

−5.74e−

09∗∗∗

−8.53e−

09∗∗∗

−6.19e−

09

−4.17e−

09∗∗

−5.30e−

09∗∗∗

(4.09e−

09)

(5.78e−

10)

(1.49e−

09)

(2.73e−

09)

(8.55e−

09)

(1.83e−

09)

(1.18e−

09)

income

0.0420∗∗∗

0.00908∗∗∗

0.0191∗∗∗

0.0146∗∗∗

0.0231∗∗∗

0.00836∗∗∗

0.00926∗∗∗

(0.00428)

(0.00198)

(0.00217)

(0.00202)

(0.00361)

(0.00180)

(0.00153)

income2

−0.0000666∗∗∗

−0.0000143∗∗∗

−0.0000267∗∗∗

−0.0000206∗∗∗

−0.0000371∗∗∗

−0.0000102∗∗∗

−0.0000118∗∗∗

(0.00000910)

(0.00000444)

(0.00000365)

(0.00000426)

(0.00000651)

(0.00000327)

(0.00000270)

itemize

2.441∗∗∗

0.731∗∗∗

0.740∗∗∗

0.846∗∗∗

1.663∗∗∗

0.528∗∗∗

0.400∗∗∗

(0.238)

(0.119)

(0.103)

(0.102)

(0.227)

(0.0908)

(0.0882)

goto

church

2.646∗∗∗

1.131∗∗∗

1.340∗∗∗

1.047∗∗∗

1.220∗∗∗

0.902∗∗∗

0.863∗∗∗

(0.203)

(0.112)

(0.102)

(0.101)

(0.140)

(0.0891)

(0.0853)

inte

rcept

−5.655∗∗∗

4.052∗∗∗

2.949∗∗∗

−3.091∗∗∗

−3.156∗∗∗

4.746∗∗∗

5.370∗∗∗

(0.646)

(0.364)

(0.385)

(0.326)

(0.421)

(0.310)

(0.285)

se4.284∗∗∗

(0.0964)

λ0.251∗∗∗

0.0579)

θ1

3.419∗∗∗

(0.453)

θ2

1.531∗∗∗

(0.0665)

I.E.Female

139.72

322.21

174.62

143.26

(126.43)

N2408

2408

2408

2408

2408

1225

1225

ll−4566.6

−1357948.3

−16861.1

−1189.2

−1933.6

−9777.1

aic

9175.2

2715934.7

33760.1

2418.4

3905.2

19592.1

bic

9296.7

2716044.6

33870.0

2534.1

4002.3

19689.2

Robust

Sta

ndard

errors

inpare

nth

ese

s*

p<

0.10,**

p<

0.05,***

p<

0.01

I.E.calculate

dvia

recycled

pre

dictions,

s.e.ofi.e.forEEE

calculate

dvia

delta-m

eth

od.

29

Table 10: Odds Ratio Interpretation of Regressors: Logit Giving Incidence

Factor Change in Factor Change in % ChangeVariable Coefficient Odds (0 → 1) 95% C.I. Odds (1 → 0) in Odds

female 0.5633 1.7565 1.412 2.185 0.5693 75.6african american −0.4390 0.6447 0.500 0.831 1.5511 −35.5hispanic −0.1771 0.8377 0.549 1.279 1.1937 −16.2healthy 0.1256 1.1338 0.887 1.449 0.8820 13.4age 0.0171 1.0172 1.008 1.026 0.9831 1.7retired 0.3429 1.4090 0.981 2.023 0.7097 40.9working 0.0033 1.0033 0.750 1.342 0.9967 0.3nkids5 −0.2491 0.7795 0.571 1.064 1.2828 −22.0nkids618 −0.0910 0.9131 0.780 1.069 1.0952 −8.7attend church 1.2198 3.3866 2.735 4.193 0.2953 238.7high school 0.4251 1.5297 1.126 2.077 0.6537 53.0some college 0.7598 2.1379 1.550 2.949 0.4677 113.8college 1.2579 3.5178 2.337 5.296 0.2843 251.8grad school 0.5957 1.8143 1.019 3.229 0.5512 81.4itemize 1.6630 5.2750 3.797 7.328 0.1896 427.5income/income2

wealth/wealth2

Source: wg02 incidence1.do

Table 11: Probabilities of Charitable Giving Over the Range of Each Independent Variable for the LogitModel

M.E. (I.E.) M.E. (I.E.)

Variable At Max At Min Range of P r at Mean s.e. at Median s.e.

female 0.6969 0.5669 0.1300 0.1300 (0.026) 0.1028 (0.022)african american 0.5663 0.6695 −0.1032 −0.1032 (0.031) −0.0827 (0.024)hispanic 0.6107 0.6519 −0.0412 −0.0412 (0.051) −0.0355 (0.042)healthy 0.6557 0.6268 0.0289 0.0289 (0.029) 0.0255 (0.025)age 0.8028 0.5098 0.2930 0.0039 (0.001) 0.0036 (0.001)retired 0.7064 0.6307 0.0757 0.0757 (0.039) 0.0760 (0.043)working 0.6498 0.6490 0.0008 0.0008 (0.034) 0.0007 (0.031)nkids5 0.4107 0.6537 −0.2429 −0.0567 (0.036) −0.0519 (0.033)nkids618 0.5010 0.6549 −0.1539 −0.0207 (0.018) −0.0189 (0.017)attend church 0.7863 0.5207 0.2656 0.2656 (0.022) 0.2914 (0.026)high school 0.7095 0.6149 0.0946 0.0946 (0.034) 0.0954 (0.034)some college 0.7593 0.5960 0.1633 0.1633 (0.033) 0.1773 (0.036)college 0.8445 0.6069 0.2376 0.2376 (0.031) 0.3006 (0.047)grad school 0.7645 0.6415 0.1230 0.1230 (0.054) 0.1367 (0.070)itemize 0.8640 0.5463 0.3177 0.3177 (0.024) 0.3932 (0.036)income/income2 0.5498 0.0039 (0.0004)wealth/wealth2

Notes: N = 2408, Marginal effects (M.E) for continuous variables and Incremental effects (I.E.) for dichotomous variableswere calculated holding all other variables at the sampling weight adjusted means or medians.

Range = P r(y = 1 | x,maxxk)− P r(y = 1 | x,minxk)

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