probability of purchase, amount of purchase, and the demographic incidence of the lottery tax

Journal of Public Economics 54 (1994) 121-143. North-Holland

Probability of purchase, amount of purchase, and the demographic incidence of the lottery tax

Frank Scott and John Garen*

Department of Economics, University of Kentucky, Lexington, KY 40506, USA

Received April 1992, final version received February 1993

When consumers’ indifference curves are discontinuous at zero, perh;_,- due to stigma or fixed costs of purchase, then factors influencing the amount purchased will have a different etfect on the probability of purchase. Estimation of the demand for a commodity like lottery tickets requires a procedure that accounts for this. Tobit estimation, commonly used in this literature, does not allow for this generalization. We investigate individual lottery ticket purchases using data from a survey of households. Our results are quite disparate from earlier studies regarding the effects of many demographic variables on probability of play, level of play, and expected lottery ticket purchases, and imply a different demographic incidence of the lottery tax than previous studies suggest.

1. Introduction

In response to the phenomenal growth in state lotteries, empirical investigations of lottery purchases have become fairly common. Much of the analysis has focused on determining the economic burden of the implicit lottery tax.’ Other studies have analyzed determinants of aggregate lottery ticket sales while still others have used micro data to examine individual purchases of lottery tickets.’ Typically, lottery purchases are regressed on

Correspondence to: F. Scott, Department of Economics, University of Kentucky, Lexington, KY 40506, USA.

*We wish to thank participants in a workshop at the University of Chicago and in the Applied Microeconomics Workshop at the University of Kentucky for comments. Also, discussions with Dan Black and Weiren Wang were very fruitful. Financial support was provided by the National Science Foundation (grant no. RII-8610671) and the Commonwealth of Kentucky through the EPSCoR program.

‘Clotfelter and Cook (1989, pp. 223-227) summarize economic studies of the incidence of the lottery tax. They conclude that, since expenditures seem to be flat across income classes, the lottery tax is decidedly regressive.

‘Recent analysis of aggregate sales include Stover (1987), who estimates potential revenues that states can expect from different types of lottery games, and Hersch and McDougall (1989), who compare county-level lottery sales with voting patterns in lottery adoption elections. In addition to Clotfelter and Cook (1989, pp. 10&104), two other recent lottery studies using individual data to analyze demographic aspects of the tax incidence are Livernois (1987) and Borg and Mason (1988).

0047-2727/94/$07.00 0 1994 Elsevier Science B.V. All rights reserved SSDI 0047-2727(93)07376-L

122 F. Scott and .I. Garen, Demographic incidence of the lottery tax

income and several demographic variables such as age, sex, race, and schooling. Essentially, these latter studies estimate the demand for lottery tickets. The results obviously indicate the nature of demand for the lottery product. However, they also reveal the ‘demographic’ burden of the implicit lottery tax, i.e. which groups bear more of the burden of the tax.

This paper presents estimates of the demand for lottery tickets using sample selection methods not previously utilized in this literature. Our results are quite disparate from earlier studies regarding the effects of many demographic variables on expected lottery ticket purchases. Our findings imply that several variables affect the probability of purchase differently than the level of purchases. Accordingly, this implies a different demographic incidence of the lottery tax than previous studies suggest.

We investigate individual lottery ticket purchases using data from a survey of Kentucky households. As with many goods, there are individuals who have zero purchases of lottery tickets. Implementing this model normally calls for use of the tobit [Tobin (1958)] technique since it accounts for the censoring of purchases at zero. This is the method commonly used in the empirical investigation of lottery demand.3

For some goods, however, the criteria that determine whether or not purchases are positive differ from those that determine the amount purchased. For example, Cragg (1971) argued that the purchase of some durable goods involves substantial search costs that partly determine whether or not to make a purchase. Cogan (1980, 1981) considered the case where there are fixed costs associated with supplying positive hours of work that affect the labor force participation decision. Moftitt (1983) developed a model where receipt of AFDC carries a stigma and thus influences the decision whether to apply for benefits. In each of these cases the decision to acquire a good is affected by additional factors that may not influence the amount purchased

in the same manner. State lottery tickets are an example of a commodity the consumption of

which may carry a stigma. Even though the gambling experience may provide a thrill, some consumers may suffer a loss of utility simply by participating in the lottery. For example, religious beliefs may cause disutility of lottery play and deter some individuals from participating in the lottery, but have no effect on the level of play among those who do purchase tickets. If that happens, then estimation of lottery demand using a technique that constrains religion to have the same impact on participation as it does on play will result in faulty conclusions. The tobit procedure implicitly imposes

3The tobit method has been generalized to deal with related models. Wales and Woodland (1983) developed a procedure for handling many goods where purchases of a subset of those goods may be censored at zero. Deaton and Irish (1984) treated the case where consumption of a good is not necessarily timed to coincide with expenditure, but the analyst only has data on expenditure.

F. Scott and J. Garen, Demographic incidence of the lottery tax 123

this restriction. Thus, tobit is a logical intitial choice for estimating lottery ticket demand because of the censoring problem, but whether it is an unduly restrictive specification is a testable hypothesis.

We estimate lottery demand using Heckman’s likelihood function, which does not impose this restriction, and compare the results to those from tobit estimation.4 Also, we estimate the lottery ticket demand function by a consistent two-step procedure and develop a Chow test to determine if the tobit restrictions are appropriate.5 We find that they are not and come to different conclusions about the structure of lottery demand than is implied by tobit.

2. Theoretical model

Consider the purchase of commodities the consumption of which may carry a stigma or the purchase of which involves some fixed cost.6 Define T as the number of lottery tickets purchased and m their unit price.’ Let X be a composite commodity of all other goods with price normalized to unity, and denote income as Y Suppose the individual’s utility function is given by*

v= UW,yT)-$, i

ifT>O, U( I: Oh if T=O.

(1)

This utility function was used by Moffttt (1983) to analyze AFDC participation. The parameters y and $ reflect the ‘stigma’ associated with lottery play. If $ > 0, the individual feels some disutility from positive purchases of lottery tickets. If y < 1, the individual feels disutility with additional lottery play.’

4As pointed out by an anonymous referee, the two-part model allows one to distinguish between the ‘extensive’ and ‘intensive’ margins of demand. Since some effects occur more through elTects on the probability of any demand, and other effects occur more through the quantity demanded if demanding any, tobit may not be the preferred approach.

‘Lin and Schmidt (1984) develop a Lagrange multiplier test for tobit versus the Cragg (1971) model. The latter allows for different effects on the probability and on the amount. However, the formulation here requires testing tobit against the Heckman model.

6This model draws on the work of Cogan (1980) and Moffttt (1983). ‘The price of a lottery ticket should be viewed as its purchase price net of expected winnings

from the purchase of the ticket. This is a slight variation of Clotfelter and Cook’s (1987) definition of price as the cost of a dollar’s worth of expected prize winnings.

8Ticket purchases, 7; enter the utility function directly, indicating that the act of playing the lottery provides utility. Thus, any assumption about attitudes toward risk is not critical as one can observe positive T even with risk aversion. For this reason, and because our emphasis is on the influence of stigma, we do not concentrate on modeling attitudes toward risk. One can view the utility function in (1) as implicitly containing risk attitudes. For a recent explanation of lottery play that is consistent with general risk aversion, see Quiggin (1991).

‘The parameters $ and 7 shift the utility function, thus we seem to be discussing taste factors. However, the analysis is perfectly consistent with treating (I/ and y as shifters of household production functions.

124 F. Scott and J. Garen, Demographic incidence of the lottery tax

OTHER GOODS

C

T

Fig. 1

An indifference curve for this utility function is depicted as the locus ABC in fig. 1. For positive purchases of lottery tickets, the indifference curve is as normally drawn. However, when ticket purchases are zero, the indifference curve has a ‘tail’ of AB. This vertical distance represents II/. It indicates that when T=O, the individual does not experience the disutility of lottery play and so is just as happy at lower income, OA, as with infinitesimally small but positive play and income OB. Note that when $ =O, the usual indifference curve arises, and so the standard analysis is a special case of this model. Note also that as y becomes smaller, the indifference curve rotates through B, flattening out.

The individual is assumed to maximize utility by choosing T and X subject to the budget constraint Y =X+m7: The following two steps characterize the solution. First, find the utility-maximizing values of T and X given that T > 0. This is equivalent to treating I) as fixed and ignoring it, and thus the choices of T and X are not affected by $. Second, determine if utility is higher with positive lottery purchases or zer0.l’

The first step involves maximizing U(X,yT) subject to the budget constraint, which entails the usual tangency of an indifference curve to the budget line:

“Treating 1(1 as a lixed cost of purchasing lottery tickets yields the same solution. The difference is that I) affects the budget constraint in this case. The ‘tail’ on the indifference curve becomes a ‘spike’ on the budget constraint and the solution to the maximum problem is identical.

F. Scott and J. Gwen, Demographic incidence of the lottery tax 125

yU,JU,=m. (2)

The solutions to this problem, T* and X*, are functions of I: m, and y as

X*=X(I:m,y). (3)

The effects of the exogenous variables Y, m, y, and $ on T* are the comparative statics dT*/dE; aT*pm, and aT*/dy, while dT*/dl//=O.

Usually the analysis ends here and ignores the possibility of a positive *. Eq. (3) summarizes the solution for the point of tangency of an indifference curve and the budget line. If T* is negative, then the purchase of lottery tickets is censored at zero, otherwise T* is the amount of purchases observed. Tobit estimation is appropriate.

With the possibility of a positive $, however, this is no longer the case. Even if T* > 0, the individual may refrain from purchasing lottery tickets if the disutility of positive play is large enough. The individual will play the lottery only if T* >O and if

U(X*,yT*)-1//-U(I:O)>O. (4)

To analyze this decision more carefully, let us follow Cogan (1980) and consider the following approach to determining whether lottery play is positive. Define m" as the reservation price of a lottery ticket, i.e. the highest price at which the individual will buy any tickets. It is given by the solution for m when

u(X( X m”, Y), T( I: m”, 9) - $ - VI: 0) = 0. (5)

When Y =OA in fig. 1, the value of m” is shown by the slope of the segment AR. Clearly, m” is a function of Y y, and tj.

The value of T that corresponds to ma is called the reservation purchases of lottery tickets. It is denoted To in fig. 1 and is given by To= T(xm', y). If the actual price of lottery tickets is less than the reservation price, i.e. m<mO, the individual can reach a higher level of utility by buying a positive amount of lottery tickets. Because demand curves slope downward, it will be the case that T* > To. If m > m", the individual would attain a lower level of utility by purchasing T*, so no tickets are bought. Here, T* < To.

In summary, if T* > To, then positive purchases of T* are observed. If T* < To, then zero purchases are observed. This generalizes the standard analysis of corner solutions. In the usual case, the reservation price is the slope of the indifference curve at T =0 and reservation purchases are, by definition, zero. With $ >O, To is positive.


To see why non-zero $ requires in any empirical analysis a separate consideration of the determinants of whether to play and the determinants of how many tickets to purchase if one plays, let us consider the effects of the exogenous variables Y m, y, and Ic, on participation. Participation occurs if T* - T”>O. If $ =0 and thus To =O, the effect of, say, a change in income on participation is simply the effect of the change in income on T*. If $>O and To > 0, then the effect of a change in income on participation is its effect on T*- To. This is not equal to dT*/dY because dT’/dY #O. Such reasoning holds for all the exogenous variables.”

3. Econometric model

T* denotes the value of T at the tangency of an indifference curve and the budget line. It is given in eq. (3) as T* = T( Km, y). Let the vector X include income and factors that affect the stigma parameter y. The price, m, does not vary in our sample, and so is included in the constant term. Also, let u reflect unobservables that affect y.

Now, a first-order approximation to T* is

T*=Xfi+u, (6)

where p is the set of coefficients corresponding to X. We do not adopt a particular functional form for the utility function and solve explicitly for T*. The reason is that to solve explicitly for the entire model involves finding explicit solutions for T*, m”, and To. Even with particular functional forms for the utility function, we generally cannot find explicit solutions for the entire model. Thus, we rely on first-order approximations. Therefore, the estimated parameters cannot be interpreted as parameters of the utility function.

The reservation price, m”, can be written as m” =m”( Y y, (I/), implying that reservation purchases are To = T( rm”( x:y, II/), y), Thus, To is affected by income and anything that influences the stigma parameters y and $. This is the vector of variables X in eq. (6). l2 It is clear, however, that these variables

’ ‘The effects of the exogenous variables on the reservation purchases are given by

dT” ST’ aTo dm”

dY r?Y-+c?m dy’

lZIf some variables affect only the stigma or fixed cost of purchase, the matrix X in (7) need not be identical to that in eq. (6).


have a different effect on To than T*. Also, unobservables affect To differently than T". A first-order approximation to To can be written as

T'=Xcr+&, (7)

where the coefficient vector TV differs from that in (6) and E is a disturbance affecting To which differs from U.

Assume that u in eq. (6) and E in eq. (7) are normally distributed with zero mean and non-zero covariance.’ 3 Define the dummy variable d, where d = 1 if positive lottery purchases are made and d =0 otherwise. The probability that d=l is Pr{T*>T’} or

Pr{Xfi+u>Xa+s}=Pr{u-s>-X(fi-U)}=@(XS/g,), (8)

where 6 =(/% c(), q = u- E with variance a:, and @( .) is the cumulative standard normal.

Letting T be observed ticket purchases, eqs. (6), (7), and (8) define a model where

T=Xp+u, if T*>T',

and

T=O, if T*sT', (9)

and where Pr {T* > To} is given by (8). The parameters p and 6, along with the cross-equation covariance of the disturbances, can be estimated by maximum likelihood methods as in Heckman (1979). Identification is achieved through the non-linearity introduced by the assumption of normality and through exclusion restrictions. Below, we test for and do not reject the normality assumption.r4 Also, we experiment with various exclusion restrictions.

In the simpler model, $ = 0 and To = 0. Let the T* equation remain as in (6). Now, the probability that d = 1 is Pr {T* > 0} or

Pr {U > - xp} = @(xp/a,). (10)

The model is then

T=X/i’+u, if T*>O,

and

13The assumption of normality is tested and accepted in the empirical section of the paper. Thus, we proceed with the discussion of the econometric model with that assumption.

14An alternative approach is to not specify a distribution and utilize semiparametric methods as in Newey et al. (1990).


T=O, if T*sO, (11)

with Pr {T* > 0} as in (10). Tobit is the maximum likelihood estimator of this model.

Notice that in tobit, the single parameter vector p determines the effect of the exogenous variables on both the probability and the level of lottery play. The more general model allows for different effects on probability and level of purchase. We estimate both the tobit and the more general model and make various comparisons to determine which is appropriate.

We also develop a Chow test for the tobit restrictions using a consistent two-step estimator. The two-step procedure is as follows. The first step is to estimate a probit equation for lottery participation, i.e. whether d =0 or d= 1, which generates estimates of 0 =6/o,,. The second step involves estimation of ticket purchases via OLS for those who do participate. Consider

E(TId=l)=XB+E(tr(y> -X6)

= XB + Ccql~,l c4(x~l~,)l@~xcTJl~ (12)

where 4 is the standard normal probability density function and @ is the associated cumulative density function. We can replace S/c, with 6 from the probit equation and then estimate

T = X/3 + Ca,,,/cr,l MxbWH^)l + u, (13)

where v is a random disturbance. Estimation of (13) by OLS yields consistent parameter estimates.

In the tobit version of this problem recall that T = Xb + u if d = 1 and that Pr {d = l} = Pr {U > -X/3} = @(X/?/g,). This is where the restriction comes in. If one estimates a probit for participation, then the estimated coefficients, 8, should be equal to /?/a, if the tobit model is correct.

We can test the appropriateness of this restriction by estimating the probit and im,posing the restriction that a=&r,, in the second-stage OLS. To see this, with the tobit restriction we have

E(T/d=l)=Xg+E(uIu>-XII)=X/3+o,[~(XP/a,)/~(XP/o,)]. (14)

When we substitute 8 for b/o, and 00, for p we get

xea, + o,[$(xe)/@(xe)] = o,[XQ+ 4(X@/@(XQ]. (15)

We can now create a single variable, [Xe+ $(X&@(X@], and estimate

T = o,,[Xe+ @(X@/@(X@} + v (16)

by OLS. We have imposed the restriction that the first-stage coefhcients are

proportional to the second-stage coefficients. This restriction applies only to

F. Scott and J. Garen, Demographic incidence of’the lottery tax 129

estimation of the second-stage OLS equation. Thus, we can compare the sum of the squared residuals of the restricted and unrestricted regressions in a Chow test.

Our estimation procedure enables us to separately estimate the effects of exogenous variables on the probability of lottery play and the level of play and to test whether it is appropriate to restrict the effects to be proportional. In the data analyzed below, this restriction generates substantial differences in the estimates of the model. Thus, while we are not able to explicitly estimate the values of the stigma parameters of the utility function, if stigma is significant, then the proportionality restriction will fail.

4. Data

Information on lottery participation and expenditures was gathered as part of a semiannual survey conducted by the University of Kentucky’s Survey Research Center. Six hundred and seventy-six Kentucky households were randomly polled by telephone and asked over 90 questions on a variety of topics. Specific questions on lottery participation, frequency of play, and average amount bet were asked. The period covered in the survey was the first six months of the Kentucky lottery, when only an instant game was available. Among respondents the mean weekly expenditure on lottery tickets is $3.00, which matches up closely with the $2.63 actually spent on average by adult Kentuckians during the survey period.

In addition to questions on lottery participation and play, questions were asked about household income, place of residence, and receipt of public assistance in the forms of food stamps, Medicaid, public housing, and AFDC. The respondent was also asked about years of formal education, age, sex, race, marital status, employment status, religion, and previous gambling on thoroughbred and harness racing (both of which are legal in Kentucky). Table 1 contains formal definitions of these variables along with summary statistics.

The variables representing income are household income, the receipt of public assistance variables, and employment status. The price of a lottery ticket in this sample does not vary. The variables hypothesized to shift $ and y are education, age, sex, race, marital status, place of residence, religion, and previous gambling. This data set allows a fuller analysis of individual demand for lotteries than has been possible so far.”

15Compared with Livernois (1987) and Clotfelter and Cook (1987, 1989, pp. 102-103), these data include several new variables as well as other variables that are more complete. The effects of previous gambling behavior, marital status, religion, and welfare recipiency on lottery participation and play have not been previously analyzed. Clotfelter and Cook (1989, Appendix table A.6, pp. 258-259) do include a number of variables of this sort in an analysis of individual participation in commercial gambling (which does not include lottery play). Their empirical results are quite similar to our results in column B of table 2.

Tab

le

1

Var

iabl

e de

fini

tions

an

d su

mm

ary

stat

istic

s.

Ent

ire

sam

ple

Lot

tery

pl

ayer

s n=

582

n=24

8

Var

iabl

e ~_

~ ~

LO

TT

ER

Y

EX

PE

ND

ITU

RE

S

BE

TT

OR

RU

RA

L

SMA

LL

TO

W

N

MA

RR

IED

MA

LE

WH

ITE

UN

EM

PL

OY

ED

FO

OD

STA

MP

S

AF

DC

Def

initi

on

~~

___

.__

Ave

rage

do

llar

expe

nditu

res

per

mon

th

on

the

Ken

tuck

y lo

ttery

= 1

if

the

indi

vidu

al

has

prev

ious

ly

gam

bled

on

ho

rser

acin

g;

= 0

othe

rwis

e

= 1

if

the

hous

ehol

d liv

es

in

a ru

ral

area

; =0

ot

herw

ise

= 1

if

the

hous

ehol

d liv

es

in

a sm

all

tow

n;

=0

othe

rwis

e

= 1

if

the

indi

vidu

al

is

curr

ently

m

arri

ed;

=0

othe

rwis

e

= 1

if

the

indi

vidu

al

is

mal

e;

=0

othe

rwis

e

= 1

if

the

indi

vidu

al

is

whi

te;

=0

othe

rwis

e

= 1

if

the

indi

vidu

al

is

unem

ploy

ed;

=0

othe

rwis

e

= 1

if

the

hous

ehol

d re

ceiv

es

food

st

amps

; =0

ot

herw

ise

= 1

if

the

hous

ehol

d re

ceiv

es

AFD

C;

=0

othe

rwis

e

X

S.D

.

13.0

0 50

.7

0.25

0.

43

0.31

0.

46

0.35

0.

48

0.67

0.

47

0.45

0.

49

0.93

0.

24

0.03

0.

17

0.06

0.

24

0.04

0.

19

s S.

D.

29.5

6 73

.2

0.38

0.

49

0.27

0.

45

0.35

0.

48

0.64

0.

48

0.48

0.

50

0.95

0.

23

0.05

0.

22

0.05

0.

22

0.03

0.

16

ME

DlC

AID

PU

BL

IC

HO

USI

NG

RE

L-

RE

FOR

MA

TIO

N

RE

L-P

IET

IST

IC

RE

L-

FUN

DA

ME

NT

AL

IST

RE

L-B

AP

TIS

T

RE

L-R

OM

AN

C

AT

HO

LIC

INC

OM

E

AG

E

ED

UC

AT

ION

= 1

if t

he

hous

ehol

d re

ceiv

es

Med

icai

d;

=0

othe

rwis

e 0.

03

0.16

= 1

if t

he

hous

ehol

d re

ceiv

es

publ

ic

hous

ing

assi

stan

ce;

= 0

othe

rwis

e

= 1

for

Pres

byte

rian

, L

uthe

ran,

E

pisc

opal

ian,

an

d ot

her

Ref

orm

atio

n E

ra

relig

ions

; =0

ot

herw

ise

= 1

for

Met

hodi

st,

A.M

.E.,

Dis

cipl

es

of

Chr

ist,

and

othe

r Pi

etis

tic

relig

ions

; =0

ot

herw

ise

= 1

for

Chu

rch

of

God

, U

nite

d M

issi

onar

y,

Naz

aren

e,

Pent

ecos

tal,

Chu

rch

of

Chr

ist,

Prim

itive

B

aptis

t, an

d ot

her

neo-

Fund

amen

talis

t re

ligio

ns;

= 0

othe

rwis

e

0.04

0.

21

0.03

0.

16

0.06

0.

24

0.14

0.

35

0.01

0.07

0.

25

0.14

0.06

0.

24

= 1

for

Sout

hern

an

d ei

ghte

en

othe

r ca

tego

ries

of

B

aptis

t re

ligio

ns;

= 0

othe

rwis

e

0.12

0.

33

0.40

0.

49

0.38

= 1

for

Rom

an

Cat

holic

; =0

ot

herw

ise

Tot

al

annu

al

fam

ily

pre-

tax

inco

me

in

thou

sand

s of

do

llars

A

ge

in

year

s

0.12

0.

32

25.1

03

15.5

92

45.4

17

.5

0.18

27.4

90

40.5

Yea

rs

of

form

al

educ

atio

n 12

.3

3.1

12.6

0.f

1

0.34

0.49

0.39

15.3

65

15.1

2.4

Not

e:

The

se

lect

ion

of

cate

gori

es

and

the

grou

ping

of

re

ligio

ns

into

th

ose

cate

gori

es

was

do

ne

by

the

Uni

vers

ity

of

Ken

tuck

y Su

rvey

R

esea

rch

Cen

ter.


5. Empirical results

The version of the model where stigma or fixed costs of purchase are restricted to zero calls for estimation of lottery demand with tobit. The empirical model is T* =Xp+u, with T= T* if T* >0 and T =0 otherwise, where T is observed purchases and X is a vector of explanatory variables described in table 1. These results are contained in column A of table 2.

The tobit estimates indicate that those who have previously gambled on thoroughbred or harness horseracing are likely to spend more money on the lottery. Being married significantly reduces lottery expenditures. Other things the same, those classified as neo-fundamentalist Protestants spend less, while Roman Catholics tend to spend more. While coefficients on the following variables are not significant at usual levels, being unemployed has a positive effect on expenditures, non-whites have higher expenditures, and more formal education has a negative effect on expenditures. While none of the coefi- cients on the welfare variables is significant, three of the four have negative signs. The quadratic specification for age implies that expenditures reach a maximum at 28 years and decline after that. Neither coefficient on the quadratic specification of income is significant, indicating no significant effect of income on lottery purchases.’ ‘j

These results are broadly consistent with Livernois (1987) and Clotfelter and Cook (1987, 1989), who also used tobit to analyze lottery demand. Both found income to have no discernible effect on lottery expenditures, and education to have a negative effect. Clotfelter and Cook found blacks to spend more than whites, males to spend more than females, those aged 25-54 to spend more than the very young or very old, and no difference between urban and rural locations.17

The more general model of lottery demand uses maximum likelihood estimation as developed by Heckman (1979). The results of estimating the probability of participating in the lottery are contained in column B of table 2. Prior gambling activity makes it more likely that an individual will be a lottery player. Married persons are less likely to participate. Being unemployed increases the likelihood of participation. Neo-fundamentalist Prot- estants are less likely, and Roman Catholics are more likely to be lottery

161n different specifications we tried income in linear, quadratic, and categorical forms. We report the quadratic specification. In linear form income is not signiiicant in either the tobit or the maximum likelihood models, nor is it significant when dummy variables for different income categories are used.

“It is natural to ask how comparable these results for Kentucky, where less than 50 percent of residents live in metropolitan areas, are to less rural states. Clotfelter and Cook (1989, pp. 102-103) used tobit to analyze lottery expenditures in 1984 by Maryland residents, where more than 90 percent of residents live in metropolitan areas. To facilitate the comparison, we replicated Clotfelter and Cook’s tobit specification using our data. The results were generally consistent, with coefficients being of the same sign and order of magnitude.


players. The rate of participation in the lottery declines with more formal education. Participation in the lottery increases and then decreases with age and with income, reaching maximums at age 25 and $30,000, respectively.

Receipt of public assistance does not have any apparent impact on lottery participation. Religious affiliations other than Roman Catholicism and neo- fundamentalist Protestant have no discernible effect on whether or not an individual purchases lottery tickets. Race, sex, and place of residence also do not significantly affect lottery participation.

The lottery participation decision is influenced by most of the variables that turn up significant in the tobit equation on lottery expenditures. It remains to be seen whether these variables still have some effect on how much people actually spend on the lottery, given that they participate. The results of this estimation are contained in column C of table 2. Previous betting behavior, marital status, age, income, education, and religion all have no apparent impact on how many tickets lottery players buy each month. While race has no effect on the participation decision, among those who do play, non-whites spend significantly more than whites. Food stamp and AFDC recipients who play the lottery buy fewer tickets and Medicaid recipients buy more tickets; however, the sample size of welfare recipients who play the lottery is too small for much to be made of that result.‘*

Recall that the coefficients for the determinants of participation are (p - a)/~,, while those determining the level are /?. It seems clear from table 2 that the two sets of coefficients are not proportional, thus CI is not zero. This can be illustrated by considering the effect of variables whose coefficients are statistically significant in the tobit model. For example, the tobit estimates imply that previous gambling behavior increases ticket purchases, both by increasing the probability of ticket purchase and the level of purchases. The more general estimates indicate, however, that only the probability of lottery play increases by a statistically significant amount.

Similarly, the tobit estimates suggest that married individuals, those with more education, and those with fundamentalist religious affiliations spend less on the lottery, while the unemployed and Catholics spend more. However, the estimates of columns B and C reveal that these effects operate only on the probability of lottery play and have no significant effect on the amount of play, given participation. Also, the tobit estimates indicate that whites and those on AFDC spend less on the lottery. Columns B and C show that this operates only by affecting the amount of purchases, given participation, but without affecting the probability of lottery play.

In the estimation of columns B and C, there are no exclusion restrictions and identification is accomplished by the normality assumption. It seems

‘*Of the 248 individuals who bought lottery tickets, 13 received food stamps, 7 received AFDC, and 7 received Medicaid.

Tab

le

2

Est

imat

ion

of

part

icip

atio

n an

d le

vel

of

play

m

odel

s.

___~

In

depe

nden

t A

: T

obit

on

lotte

ry

-___

__

B:

Prob

abili

ty

of

lotte

ry

C:

Det

erm

inan

ts

of

lotte

ry

vari

able

ex

pend

iture

s pa

rtic

ipat

ion

expe

nditu

res

_.~_

~

INT

ER

CE

PT

__

___

4.55

7-

11

0.10

(0

.11)

(0

.02)

(0

.96)

BE

TT

OR

25

.15

0.69

-

10.2

5 (2

.65)

(4

.62)

(0

.22)

RU

RA

L

-4.8

6 0.

02

- 11

.24

(0.4

5)

(0.1

0)

(0.5

2)

SMA

LL

T

OW

N

3.34

0.

10

1.10

(0

.33)

(0

.69)

(0

.06)

MA

RR

IED

-

26.9

9 -0

.38

- 14

.56

(2.8

1)

(2.5

3)

(0.4

6)

MA

LE

8.

55

0.04

6.

07

(1.0

3)

(0.3

2)

(0.4

5)

WH

ITE

~

30.

22

0.08

-

64.5

6 (1

.69)

(0

.28)

(3

.76)

UN

EM

PL

OY

ED

44

.54

1.18

-

18.4

6 (1

.83)

(2

.63)

(0

.21)

FOO

D

STA

MPS

-

19.9

4 0.

16

- 83

.65

(0.8

5)

(0.4

5)

(1.7

9)

AF

DC

-

52.8

7 -0

.42

- 10

4.00

(1

.67)

(0

.84)

(1

.99)

ME

DIC

AID

41

.98

-0.2

3 18

2.24

(1

.77)

(0

.39)

(4

.95)

PU

BL

IC

HO

USI

NG

RE

L-R

EFO

RM

AT

ION

RE

L-P

IET

IST

IC

RE

L-F

UN

DA

ME

NT

AL

IST

RE

L-B

AP

TIS

T

RE

L-C

AT

HO

LIC

INC

OM

E

INC

OM

E

SQ

UA

RE

D

AG

E

AG

E-S

QU

AR

ED

ED

UC

AT

lON

n Rh

o

- 46

.43

(1.2

9)

2.98

(0

.15)

- 5.

50

(0.3

6)

- 37

.25

(2.1

2)

9.90

(0

.82)

30.8

0 (2

.09)

1.29

(1

.03)

-0.0

18

(0.9

5)

2.04

( 1

.44)

- 0.

036

(2.3

5)

- 2.

70

(1.5

8)

582

-0.6

0 (1

.21)

0.02

(0

.06)

-0.1

5 (0

.73)

-0.7

1 (2

.91)

-0.1

6 (0

.94)

0.49

(2

.23)

0.03

(1

.87)

-0.0

005

(1.9

1)

0.02

(1

.17)

-0.0

004

(2.2

5)

-0.0

4 (1

.78)

582 _

- 19

.68

(0.2

7)

4.84

(0

.03)

5.49

(0

.12)

14.5

9 (0

.22)

38.2

7 (1

.34)

16.4

6 (0

.40)

-0.9

4 (0

.26)

0.01

5 (0

.28)

0.81

(0

.22)

- 0.

009

(0.1

8)

- 1.

85

(0.3

3)

582

- 0.

06

(0.0

3)

____

_ N

ote:

f-

stat

istic

s ar

e in

pa

rent

hese

s.

E


worthwhile to examine that assumption. Bera et al. (1984) provide Lagrange multiplier tests of normality for limited dependent variable models. We apply their test in two steps. The first step involves estimating the probability of participation as in (8) and testing for the normality of the disturbance, q. If that normality is not rejected, the next step is to test for the normality of the error v in regression equation (13). In neither case is normality rejected, thus we have reasonable confidence in the estimation procedure and in identification.”

To determine whether our results are dependent on our specification without exclusion restrictions, we have also estimated models, with over- identifying exclusion restrictions. If there are variables that affect only the fixed component of stigma, $, or which are associated with fixed costs of purchasing lottery tickets, then they should be included in the participation equation but not included in the amount of purchase equation. While we can think of no overwhelmingly convincing exclusion restrictions, two possibili- ties are religion and location. It seems plausible that to many of the laity, if playing the lottery is viewed as a sin, it is as sinful to buy one ticket per month as it is to buy ten. If that is so, then the religion variables should be excluded from the second equation. Likewise, if one lives some distance from the nearest lottery vendor, then the transportation cost of purchase is the same regardless of the number of tickets purchased. To the extent that residents of small towns and rural locations face significant fixed costs of purchase, the location dummy variables should be excluded from the second equation.

The results of estimating the model with these exclusion restrictions are contained in table 3. Columns A and B contain the estimates when religion categories are excluded from the second stage. Columns C and D contain the estimates when location categories are excluded. A comparison of these results with those in table 2 indicates that no appreciable changes occur.2o

The proposition that the coefficients determining the probability of participation are proportional to those determining the level of play can be tested formally with a Chow test using the two-step estimation procedure that was outlined above. The sum of squared residuals from the regression that imposes the tobit restriction is 323.830. The unrestricted sum of squares is 246.609. There are 248 observations, 23 parameters, and one restriction. The F-statistic is 3.207 and exceeds 1.97, which is the l-percent critical value of the F-distribution with 22 and 225 degrees of freedom. Thus the tobit

“See the appendix for a complete discussion of the test. 2”We tried excluding both religion and location variables from the second stage, as well as

various combinations of the demographic and welfare variables. None of these made any appreciable difference in the results.


restriction is clearly rejected.21 While the Chow test indicates rejection of the tobit model in favor of the

Heckman maximum likelihood model, it is worthwhile to make less formal comparisons between the two models. We first compare the abilities of the tobit and Heckman maximum likelihood models to predict whether or not an individual participates in the lottery. We then compare the models’ abilities to predict the level of play among those who do choose to purchase a positive number of tickets. Cogan (1981) also uses these procedures.

In predicting the probability of participation, we classify a prediction as correct if the predicted probability exceeds one-half and d = 1 or if the predicted probability is less than one-half and d =O. Otherwise it is an incorrect prediction. The tobit model generated correct participation predictions 63.9 percent of the time. The Heckman maximum likelihood model generated correct participation predictions 69.9 percent of the time.

To compare the ability of each model to predict the level of play among participants, we computed the mean square of actual minus predicted purchases. The tobit model produced a mean squared error of 8281.9. The Heckman maximum likelihood model generated much more accurate predictions, with a mean squared error of 4252.9.

It seems clear that the tobit restriction that each X affects the probability and the level of play proportionally is not appropriate. It is natural, however, to ask if relaxing this assumption makes much difference in assessing the effects of the exogenous variables on expected lottery ticket purchases. Expected ticket purchases, E(T), are given by

E(T)=Pr {T>O} E(T/ T>O),

and it follows that

(17)

aE( T)/dX = [a Pr (T > O}/aX] [E( T 1 T > 0)]

+[aE(T) T>O)/dX][Pr {T>O)].

These can be computed for both the tobit and Heckman models with the estimated parameters from each model and values for X.‘*

The effect of a variable on expected ticket purchases depends on its effect on the probability of lottery play and on the level of play. While tobit may

21The disturbance term u in the OLS equations is heteroskedastic, and F-tests assume > > homoskedasticity. As Maddala (1983, p. 225) suggests, we correct for the heteroskedasticity using weighted least squares. The F-test is then applied to the sums of squared errors of the weighted regressions. These are what are reported in the text. Unweighted OLS estimates also reject the restriction.

2ZFor X we use the means for the whole sample. For Pr{T>O} we use the percent who purchased lottery tickets. For E(T ( T>O) we use the mean ticket purchases among those who bought positive amounts.


E: n

PU

BL

IC

HO

USl

NG

RE

L-R

EFO

RM

AT

ION

RE

L-P

IET

IST

IC

RE

L-F

UN

DA

ME

NT

AL

IST

RE

L-B

AP

TIS

T

RE

L-C

AT

HO

LIC

1NC

OM

E

INC

OM

E

SQU

AR

ED

AG

E

AG

E

SQ

UA

RE

D

ED

UC

AT

ION

n Rh

o

- 0.

60

(1.1

8)

0.02

(0

.06)

-0.1

5 (0

.74)

-0.7

1 (2

.91)

-0.1

6 (0

.96)

0.49

(2

.23)

0.03

(1

.88)

-0.0

005

(1.9

1)

0.02

(1

.20)

- 0.

0004

(2

.31)

-0.0

4 (1

.78)

582

-0.8

5 (0

.30)

0.01

(0

.32)

1.36

(0

.40)

- 0.

02

(0.4

4)

- 3.

09

(0.7

4)

248 0.

115

(0.1

8)

-0.6

0 (1

.20)

0.02

(0

.07)

-0.1

5 (0

.73)

-0.7

1 (2

.92)

-0.1

6 (0

.95)

0.49

(2

.26)

0.03

(1

.87)

-0.o

oO5

(1.9

2)

0.02

(1

.18)

- 0.

0004

(2

.27)

-0.0

5 (1

.79)

582

-20.

21

(0.2

8)

6.34

(0

.04)

5.83

(0

.21)

13.2

3 (0

.25)

36.6

8 (1

.32)

17.9

9 (0

.46)

-0.9

7 (0

.29)

0.02

(0

.34)

0.63

(0

.17)

-0.0

1 (0

.16)

- 2.

03

(0.4

1)

248 0.

115

(0.1

8)


Table 4

Decomposition of effects on expected ticket purchases.

Variable

EPr{T>O} A: aX- mE(Tj T>O) B: ~- dX - W->O)Pr~T>O~ C: aEcT,

3X

Tobit Heckman Tobit Heckman Tobit Heckman

BETTOR 3.69 7.95 3.73 -3.63 7.41 4.32

MARRIED - 3.86 -4.31 -3.91 - 6.60 - 1.77 - 10.91

WHITE -4.33 0.97 -4.37 -27.42 - 8.70 - 26.44

UNEMPLOYED 6.38 13.51 6.44 -6.61 12.82 6.90

EDUCATION -0.39 -0.51 -0.39 -0.84 -0.78 - 1.35

estimate these components incorrectly, the appropriately weighted sum of these components may be correct. Table 4 reveals that this is not generally true, however. This table shows the two components of LIE(T)/aX as in (18) and their sum for selected variables, using the parameter estimates from tobit and the more general model.

As column C of table 4 indicates, for several of the independent variables the total effect, aE(T)/BX, differs substantially between tobit and Heckman maximum likelihood. For example, tobit estimates the magnitude of the effect of race on expected ticket purchases to be three times smaller than that estimated by the Heckman maximum likelihood. Unemployment and education are also estimated by tobit and Heckman maximum likelihood to have appreciably different effects on expected ticket purchases.

These results imply that the demographic burden of the implicit lottery tax is quite different than tobit indicates. For example, holding other things constant, both non-whites and those with less schooling are estimated to have significantly greater expected expenditures than tobit indicates. Accord- ingly, these groups bear a larger burden of the implicit tax than tobit implies.

Another interesting feature of table 4 is the sizable differences in the effects of several variables in columns A and B for the Heckman procedure. Column A contains the impact of a change in X on the probability of playing, given the expected level of ticket purchases among those who play. Column B contains the impact of a change in X on the expected level of ticket purchases among those who do play, given the probability of participation. Thus, prior gambling activity, race, and being unemployed exhibit sharp differences in their effects on the probability versus the level of play, while marital status and education exhibit similar component effects.

6. Conclusions

Tobin (1958) initiated the empirical analysis of the demand for goods when


some consumers refrain from purchasing. His estimation technique imposes specific constraints on the consumer’s utility function. In some cases such constraints may not be appropriate. This paper has utilized a more general approach to estimating censored regression models of consumer demand, drawing from the work of Heckman (1979) and others.

We have shown that when consumers’ indifference curves are discontinuous at zero, perhaps due to a stigma or fixed costs of purchase, then use of tobit results in mis-specification. Estimation of the demand for a commodity like lottery tickets requires that a more general maximum likelihood procedure be used. Use of tobit leads one to assume that determinants of lottery participation and level of play have proportional effects when, in fact, this is not the case. This leads to error in estimating the effect of the exogenous variables on expected lottery purchases.

Many empirical studies of demand are done with observations where zero is purchased. Analysts typically do not separate the probability of buying any of the commodity from the quantity demanded of those who buy positive amounts. Some factors may exert their influence more through effects on the probability of any demand, while others may work more through the quantity demanded if demanding any. Our results suggest that researchers should proceed with caution when estimating demand for goods with non-purchasers in the sample.

Appendix

The procedures of Bera et al. (1984) provide Lagrange multiplier tests of the null hypothesis of normality against the alternative of the Pearson distribution. The normal is a special case of the Pearson. Their tests are only for univariate distributions, but the model of this paper utilizes the bivariate normal. We know of no derivation of a test against the bivariate Pearson. Thus, the approach of testing for normality in two steps is adopted.

The first step is to estimate the probability of participation as in (8) and test for the normality of the disturbance q. Assuming normality is not rejected, then the next step is to test for the normality of the error u in regression equation (13).

The Pearson distribution for any random variable e is given by

d ln g(e) c, -e

de c,-cc,+c,e2’ (A.11

where g(e) is the density function. If c1=O=c2, then the distribution is normal. Thus, the test is if these two parameters are different from zero. Letting L be the log-likelihood function of the model being estimated and 8 the maximum likelihood estimates under the null hypothesis of normality, the Lagrange multiplier statistic is


which is asymptotically distributed as chi-square with two degrees of freedom. Bera et al. (1984) show how this statistic is computed from the probit estimates of a binary dependent variable model.

This statistic is computed from the probit estimates of eq. (8). Its value is 0.1113. The 5 percent critical value for a chi-square variable with two degrees of freedom is 5.99. Normality is not rejected.

Given this, and under the null hypothesis of normality of u in (9), regression (13) and the normality of the disturbance v follows. A similar Lagrange multiplier test of the normality of v is easily constructed based on the results in Bera et al. (1984). It is computed from least squares estimates of (13). However, v is heteroskedastic, so the data are weighted appropriately to transform v to homoskedastic. The test statistic is then computed from the results of estimation of (13) with the weighted data. For the model without exclusion restrictions, its value is 4.97. For the model with exclusion of the religious variables from this equation, the value of the test statistic is 5.18. Normality is not rejected in either case.

References

Bera, A., C. Jarque and L.F. Lee, 1984, Testing the normality assumption in limited dependent variable models, International Economic Review 25, 563-578.

Borg, M. and P. Mason, 1988, The budgetary incidence of a lottery to support education, National Tax Journal 41, 75-85.

Clotfelter, C. and P. Cook, 1987, Implicit taxation in lottery finance, National Tax Journal 40, 533-546.

Clotfelter. C. and P. Cook. 1989. Selline hone: State lotteries in America (Harvard University Press,‘Cambridge, MA).’ - I

Cogan, J., 1980, Labor supply with costs of labor market entry, in: J. Smith, ed., Female labor SUDD~V: Theorv and estimation (Princeton University Press, Princeton, NJ) 327-364.

Coga&*J, 1981, Fixed costs and labor supply, Economktrica 49, 945-963. Cragg, J., 1971, Some statistical models for limited dependent variables with application to the

demand for durable goods, Econometrica 39,829~844. Deaton, A. and M. Irish, 1984, Statistical models for zero expenditures in household budgets,

Journal of Public Economics 23, 59-80. Heckman, J., 1979, Sample selection bias as a specification error, Econometrica 42, 153-161. Hersch, P. and G. McDougall, 1989, Do people put their money where their votes are? The case

of lottery tickets, Southern Economic Journal 56, 32-38. Lin, T. and-P. Schmidt, 1984, A test of the tobit specification against an alternative suggested by

Crapg. Review of Economics and Statistics 66, 174177. Livernois, J., 1987, The redistributive effect of lotteries: Evidence from Canada, Public Finance

Quarterly 15, 339-351. Maddala, G., 1983, Limited dependent and qualitative variables in econometrics (Cambridge

University Press, Cambridge). Moflitt, R., 1983, An economic model of welfare stigma, American Economic Review 73,

1023-1035.


Newey, W., J. Powell and J. Walker, 1990, Semiparametric estimation of selection models: Some empirical results, American Economic Review 80, 324328.

Quiggin, J., 1991, On the optimal design of lotteries, Economica 58, 1-16. Stover, M., 1987, The revenue potential of state lotteries, Public Finance Quarterly 15, 428440. Tobin, J., 1958, Estimation of relationships for limited dependent variables, Econometrica 26,

2436. Wales, T. and A. Woodland, 1983, Estimation of consumer demand systems with binding non-

negativity constraints, Journal of Econometrics 21, 263-285.

probability of purchase, amount of purchase, and the demographic incidence of the lottery tax

Documents