Ordered logit, demand for sex

*n n n ; n 1,2, , , N(1) y x

Dependent variable

nj

1 if client n belongs to category j; j 1,2,3,4(2) y

0 otherwise

Ordered structure

*nn1 1

*nn2 1 2*nn3 2 3*nn4 3

y 1 if y

y 1 if y(3)

y 1 if y

y 1 if y

Choice probability

*n n n nnj j 1 j j 1 j(4) P(y 1) P( y ) P( x x )

Probability distribution, Logistic

u1(5) F(u)

1 e

Ordered Logit

n nnj j j 1(6) P(y 1) F( x ) F( x )

n1 1 n

n2 2 n 1 n

n3 3 n 2 n

3

n4 nj 3 nj 1

P(y 1) F( x )

P(y 1) F( x ) F( x )

P(y 1) F( x ) F( x )

P(y 1) 1 P(y 1) 1 F( x )

Thus

4

nj n4 3 nj 1

[P(y 1)] 1so that P(y 1) 1 F( x )

Likelihood

nj

0 n

4 n

yN 4

n nj j-1n=1 j=1

where

F( - x ) 0

F( - x ) 1

(7) L( , ) = F( - x ) -F( - x )

Marginal effects

n nnj j 1 j

n n n

P(y 1) F( x ) F( x );for j 1,2,3,4

x x x

Marginal effects

1 n

1 n

1 n

1 n

1 n

1 n

n1 1 n

n n

( x )1 n 1 n( x )

( x )1 n( x )

1 n n

( x )n1 1 n

1 n ( x )n n

n1

P(y 1) F( x )?

x x

1F( x ) ln F( x ) ln(1 e )

1 eF( x )1 1

eF( x ) x 1 e

P(y 1) F( x ) eF( x )

x x 1 e

P(y 1)

1 n 1 nn

F( x )(1 F( x ))x

etc

Marginal effects

n11 n 1 n

n

n21 n 1 n 2 n 2 n

n

n32 n 2 n 3 n 3 n

n

n43 n 3 n

n

P(y 1)F( x )[1 F( x )]

x

P(y 1){F( x )[1 F( x )] F( x )[1 F( x )]}

x

P(y 1){F( x )[1 F( x )] F( x )[1 F( x )]}

x

P(y 1){F( x )[1 F( x )]}

x

Utility function, use of condoms

jnnj nj j 0,1; n 1, 2, , , N(10) U x ;

Probability of using condoms

n1

K K

1k nk k nkk 0 k 0

n1 n0 K K K

0k nk 1k nk k nkk 0 k 0 k 0

n0k 1k 0k

exp( x ) exp( x )(11) P(U U )

exp( x ) exp( x ) 1 exp( x )

where

1 condom0 no comdom

, and x 1.

Log likelihood

n1 n1N K K

y 1 yn nn1 n1k k

k 0 k 0n 1L( ) [ ( x )] [1 ( x )]

What money buys: clients of street sex workers in the US

• Maria Laura Di Tommaso, Marina della Guista, Isilda Shima and Steinar Strøm

Table A1. Dependent variable for the ordered logit

Frequency of sex with sex worker during last year . No of Obs 582Frequency per cent

=1 never 25.4

=2 once 27.0

=3 more than 1 but less than once per month 35.0

=4 1 to 3 times per month 12.5

VariablesOrdered Logit Logit: Probability of

being a “regular” client Logit: Probability of using condom

Education =1 college or more; =0 otherwise

0.160(0.194)

0.067(0.243)

0.067(0.474)

Work status =1 Full time; =0 otherwise 0.655**(0.281)

0.656*(0.347)

0.476(0.564)

Race =1 if non white; =0 white 0.491***(0.186)

0.201(0.226)

1.121**(0.576)

Job =1executives/business managers;=0 otherwise

-0.125(0.170)

-0.151(0.209)

-0.023(0.415)

Marriage =1 married; =0 otherwise -0.312*(0.173)

-0.118(0.213)

0.090(0.412)

Control dislike 0.276***(0.096)

0.220*(0.118)

-0.062(0.234)

Age 0.017*(0.009)

0.030***(0.011)

-0.031(0.020)

Factor1 'againstg ender violence' 0.181*(0.108)

0.274**(0.136)

0.464*(0.259)

Factor2 'against prostitution'

-0.159*(0.094)

-0.199*(0.112)

-0.400*(0.222)

Factor3 'sex workers not different and dislike their job'

0.198**(0.101)

0.200*(0.124)

-0.102(0.242)

Factor4 'like relationships' -0.536***(0.112)

-0.641***(0.137)

-0.351(0.266)

Factor5 'variety dislike' -0.968***(0.121)

-1.031***(0.151)

0.692***(0.281)

Factor6 'relationship troubles ' -0.026(0.109)

0.006(0.137)

0.482*(0.293)

Threshold 1 0.788(0.550)

Threshold 2 2.233***(0.559)

Threshold 3 4.452***(0.580)

Constant -2.501***(0.692)

3.643***(1.339)

# of observationsMcfaddens rho

5820.14

5820.18

5700.71

Table 7: Marginal effects in the ordered logit

VariablesNever with sex workers

Once with sex workers

More than 1 time but less then once per month

1 to 3 times per month

Education =1 college or more;=0 otherwise

-0.0269(0.033)

-0.012(0.014)

0.027(0.033)

0.012(0.014)

Work status =1 Full time; =0 otherwise

-0.123**(0.059)

-0.033***(0.008)

0.113**(0.048)

0.0429***(0.015)

Race =1 if non white;=0 white -0.077***(0.028)

-0.044**(0.018)

0.079***(0.029)

0.0425**(0.017)

Job =1executives/business managers=0 otherwise

0.02(0.028)

0.01(0.014)

-0.02(0.028)

-0.010(0.013)

Marriage =1 married; 0 otherwise 0.051*(0.0287)

0.026*(0.015)

-0.052*(0.029)

-0.025*(0.014)

Control Dislike -0.045***(0.016)

-0.023***(0.008)

0.046***(0.017)

0.022***(0.008)

Age -0.002**(0.002)

-0.001*(0.0008)

0.002*(0.0015)

0.001*(0.0007)

Factor1 'Against gender violence' -0.029*(0.018)

-0.015*(0.0094)

0.030*(0.018)

0.014*(0.0088)

Factor2 'Against prostitution' 0.026*(0.015)

0.013*(0.0083)

-0.026*(0.015)

-0.012*(0.0077)

Factor3 'Sex workers not different and dislike their job'

-0.032**(0.016)

-0.016*(0.009 )

0.033**(0.0172)

0.016*(0.0083)

Factor4 'Like Relationships' 0.088***(0.0186)

0.045***(0.011)

-0.09***(0.020)

-0.043***(0.009)

Factor5 'Variety dislike' 0.159***(0.02)

0.085***(0.015)

-0.162***(0.024)

-0.078***(0.012)

Factor6 'Relationship troubles' 0.004(0.017)

0.002(0.009)

-0.004(0.018)

-0.002(0.008)

Other examples

• Tax evasion and detection probabilities

• What is the chance for being detected when evading taxes?

4

*n n n

nj

*nn1 1

*nn2 1 2*nn3 2 3

*nn4 3

*nn5 4

; n 1,2, , , N

q

(1) q x

1 if individual n 'sanswer belongs to category j; j 1,2,3,4,5(2) q

0 otherwise

1 if q

q 1 if q

(3) q 1 if q

q 1 if q

q 1 if q

n nnj j j 1

5

nn5 4njj 1

(4) P(q 1) F( x ) F( x )

(5) [P(q 1)] 1so that P(q 1) 1 F( x )

The questions

What is the chance of being detected:

1. Will certainly be detected

2. Will almost certainly be detected

3. Will perhaps be detected

4. Will almost certainly not be detected

5. Will certainly not be detected