1. An Empirical Investigation of the Quantity-Quality Model in Mexico
Emla Fitzsimons
Bansi Malde
Institute for Fiscal Studies, London
30th August 2008

2. Introduction
Objective: to estimate the causal effect of family size on
children’s education
Quantity-quality model (Becker,1960; Becker and
Lewis,1973; Becker and Tomes,1976):
As quantity (number of children) rises, total cost of quality (investment into children) also rises, thus decreasing the demand for quality
 So frequently observed negative correlation b/w family size and child quality consistent with theory
1st instrument is based on parental preferences for having at least one boy. Mention that we experimented with bb but not powerful in first stage

3. Introduction But fertility is endogenous
to empirically test whether family size has a causal effect on quality requires exogenous changes in fertility that are uncorrelated with preferences/budget constraints
We use two natural experiments as instrumental
variables for family size
(1) having successive births of females
(2) birth of twins
1st instrument is based on parental preferences for having at least one boy. Mention that we experimented with bb but not powerful in first stage

4. Related Literature Similar instruments have been used in previous
Twins - Rosenzweig and Wolpin,1980 (-);
Rosenzweig and Zhang,2006 (-); Li, Zhang and
Zu, 2007 (-); Ponczek and Souza,2007 (-); Qian,
2006 (-,+); Cáceres,2006 (-)
Sex composition - Angrist, Lavy and
Schlosser,2006 (0); Lee, 2004 (-); Baez,2007 (-)
Note here I have omitted refs that do not look at children’s educ as outcome

5. Methodology Estimate the model Y=b0+b1X+b2F+u
OLS  b2 biased if omitted variables affect both Y and F
So estimate using IV methodology
Mention: where
Y: 0-1 indicator of child participation in school;
years of schooling
X: child characteristics such as age, sex, birth order;
parental characteristics such as age, education
F: family size
u: unobserved factors that affect Y and that may be correlated with F
Mention: Coefficient of interest is b2
Parent’s preferences an e.g. of omitted vbles

6. F = a0 + a1g2 + a2g3 + a3g1g2g3 +a5t3 + a6X + u
Methodology
We consider first- and second-born children separately;
also separately by sex
Instruments for family size:
First-born girls
Exogenous increase in family size from 2 to 3:
- 1st 2 births are females; twins at 2nd birth
F = a0 + a1g1g2 + a2t2 + a3X + x
First- and second-born girls
Exogenous increase in family size from 3 to 4:
- 1st 3 births are females; twins at 3rd birth
F = a0 + a1g2 + a2g3 + a3g1g2g3 +a5t3 + a6X + u
Parent’s preferences an e.g. of omitted vbles
The 2nd 1st stage eqn is for first-borns: for second-borns, practically the same but control for g1 instead of g3. I wouldn’t even go into this unless someone is confused and asks!
F = a0 + a1g1 + a2g3 + a3g1g2g3 +a5t3 + a6X + u
Mention: the gg instrument assumes parents want to have at least one boy, which is valid in our context
Mention that in this way we can test sensitivity of 1st born girl results to different instruments
F = a0 + a1g1g2 + a2t2 + a3X + x
F = a0 + a1g2 + a2g3 + a3g1g2g3 + a5t3 + a6X + u

7. Methodology First-born boys
Exogenous increase in family size from 2 to 3:
- twins at 2nd birth
First- and second-born boys
Exogenous increase in family size from 3 to 4:
- twins at 3rd birth
Parent’s preferences an e.g. of omitted vbles
Mention: the gg instrument assumes parents want to have at least one boy, which is valid in our context
F = a0 + a1g1g2 + a2t2 + a3X + x
F = a0 + a1g2 + a2g3 + a3g1g2g3 + a5t3 + a6X + u

8. Data Cross-sectional socio-economic Census data
collected across households in marginalised
rural areas throughout 31 states in Mexico b/w
1996 and 1999 (ENCASEH)
School participation and years of schooling of year olds
Sample of 617,807 households across 42,186 villages
Large samples contained in these data are extremely advantageous and provide us with sufficient variation in these 2 random events
Note sample selection: drop households in which both parents are not married, households in which the eldest child is 18 or above, households that reported more than one household head and some suspect observations.

9. Is sex composition random?
Comparison of mean characteristics by sex composition of 2 first-borns, for hhs
with female first-borns
Female,
male=1
female=1
Difference
Father’s age
40.979
40.976
-0.002
Mother’s age
36.878
36.86
-0.018
Mother’s age at first birth
21.805
21.777
-0.028
Father’s years of education
3.691
3.687
-0.003
Mother’s years of education
3.326
3.32
-0.006
Birth spacing b/w 1st and 2nd children
2.831
2.833
0.002
Family size
4.186
4.249
0.063*
Explain what family size is; here we do really well despite v large sample sizes
Put * when diffs signif

10. Difference after matching
Are twin births random?
Comparison of mean characteristics by twins at 2nd birth
Twins at second
birth=0
birth=1
Difference
Difference after matching
Father’s age
41.036
42.215
1.179*
0.106
Mother’s age
36.926
37.939
1.013*
0.880*
Mother’s age at first birth
22.272
23.236
0.963*
0.015
Father’s years of education
3.68
3.736
0.056
0.032
Mother’s years of education
3.315
3.488
0.174*
0.079
Birth spacing b/w 1st and 2nd children
2.825
3.346
0.521*
0.492*
Family size
4.242
4.583
0.341*
0.367*
Explain how we do the matching
Mention that we get a similar picture when we look at ggg, twins at 3rd birth etc
For father’s age and mother’s age in final column, we control for mother’s age at first birth

11. Results - School participation
FEMALE FIRST-BORNS
12-14
15-17
LPM
IV1
IV2
Family size
-0.025**
(0.001)
-0.001
(0.016)
-0.031
(0.030)
-0.023**
(0.025)
(0.033) N
88,649
78,701
95,850
87,273
Over-identification test -
0.588
1.223
0.043
0.810
First stage
2nd birth girl
0.149**
(0.009)
0.107**
(0.012)
2nd birth twins
0.694**
(0.061)
0.459**
(0.066)
2nd and 3rd births girls
0.083**
(0.017)
0.087**
(0.020)
3rd birth twins
0.646**
(0.057)
0.603**
(0.069)
F test
192.09
69.58
61.92
48.65
- We are showing trimmed results throughout
Estimate using linear IV - results robust to non-linear IV
Std errors clustered at village level
Stress that we have v strong 1st stages throughout; also we can do overid tests for females

12. Results - School participation
MALE FIRST-BORNS
12-14
15-17
LPM
IV1
IV2
Family size
-0.021**
(0.001)
-0.025
(0.043)
0.021
(0.027)
-0.015
(0.041)
0.048
(0.040) N
93,593
82,119
112,929
102,254
First stage
2nd birth twins -
0.450**
(0.065)
0.440**
(0.057)
3rd birth twins
0.581**
(0.061)
0.496**
(0.054)
F test
47.51
91.34
58.72
83.31

13. Results - School participation
SECOND-BORNS
FEMALES 12-14
MALES 12-14
LPM IV
Family size
-0.027**
(0.001)
-0.039
(0.027)
-0.021**
-0.038
(0.039) N
83,151
86,039
Over-identification test -
0.217
First stage
3rd birth girl
0.105**
(0.019)
3rd birth twins
0.640**
(0.069)
0.467**
(0.061)
F test
52.89
59.62
Mention that not robust - weak 1st stage - so we don’t consider them

14. Results - Years of schooling
FEMALE FIRST-BORNS
12-14
15-17
OLS
IV1
IV2
Family size
-0.104** (0.004)
0.081 (0.084)
(0.110)
-0.166** (0.006)
(0.128)
(0.152) N
91,429
91,888
99,885
99,993
Over-identification test -
0.955
0.917
0.077
0.196
First stage
2nd birth girl
0.143** (0.009)
0.105** (0.011)
2nd birth twins
0.675** (0.060)
0.450** (0.065)
2nd and 3rd births girls
0.072** (0.016)
0.075** (0.018)
3rd birth twins
0.623** (0.050)
0.581** (0.063)
F test
191.02
82.80
61.54
48.59
I would go v quickly through these results

15. Results - Years of schooling
MALE FIRST-BORNS
12-14
15-17
OLS
IV1
IV2
Family size
-0.106**
(0.004)
0.347*
(0.159)
(0.104)
-0.159** (0.005)
0.061
(0.187)
0.162
(0.199) N
95,961
95,269
117,506
117,118
First stage
2nd birth twins -
0.443
(0.068)**
0.441**(0.056)
3rd birth twins
0.597**
(0.054)
0.499**
(0.048)
F test
42.42
122.47
62.01
110.21

16. Results - Years of schooling
SECOND-BORNS
FEMALES 12-14
MALES 12-14
OLS IV
Family size
-0.114**
(0.004)
0.154
(0.099)
-0.112**
(0.005)
-0.107
(0.143) N
85,803
88,272
Over-identification test -
0.552
First stage
3rd birth girl
0.112** (0.019)
3rd birth twins
0.626**
(0.067)
0.473
(0.061)**
F test
55.46
59.98

17. Identification assumption
Sex composition of first n births / twin births have no direct
effect on children’s education. Only affect it through effect
on family size
Concern – economies of scale from same-sex children? If
so, this may affect household resources and thereby
education (Rosenzweig and Wolpin, 2000)
May bias upwards the effects of family size on education
Mention Rosenzweig and Zhang,2006 if someone mentions lower birthweight

18. Validity of Instruments
Any evidence of economies of scale, for example via sharing
of clothing?
Can look at this using Mexican data set PROGRESA, where
we observe expenditures on children’s clothing and shoes,
separately by sex
2 (reassuring) things to note:
 only 2.2% (2.1%) of total household expenditure is on children’s clothing (shoes)
 in our sample, amongst
gg=1 households: ave # girls=3.1, ave # boys=1.2
gb=1 households: ave # girls=2.1, ave # boys=2.1
- large family sizes in both - so similar opportunities for economies
of scale in both?
Progresa much smaller sample but can think of it as representative of ours
Mention we’re considering v poor rural hhs, so this argument less valid
Stress that hhs are really spending v little on these goods to begin with so alleviates concerns about extent of importance of ec of scale

19. Validity of Instruments
Compare monthly expenditures across households with gg=1
and gb=1
Notes: tobit estimation; control for total household expenditure,
family size
Expenditure on
Food
Non food
Children’s
clothing
shoes gg
-9.197
(9.122)
7.573
(8.660)
0.844
(1.997)
-0.563
(1.30) N
1,977
1,970
1,966
Detail:
gg=1: ave # girls=3.3, ave # boys=1.3
gb=1: ave # girls=2.2, ave # boys=2.2
Mention: observe no statistical differences in children’s clothing or shoes (or in food, which we wouldn’t expect to have observed – though maybe boys consume more calories etc so not so clear to me)

20. Ongoing Work
1. Obtain bounds on the IV parameter estimates, for different correlations b/w the instruments and model errors
- do this using method of Ashley (2007): provide ‘guess’ as to sign and size of correlation b/w endog vble and error term
2. Additional instruments (1st 4 births female; twins at 4th birth)
3. Child labour as an outcome (Deb and Rosati,
2004; Ponzcek and Souza, 2007)
Apart from deeper investigation of economies of scale work, here is what we’re planning to do
Note as far as I can see Ashley is unpublished – I saw ages ago that it’s forthcoming in J of Applied Econometrics, but I can’t find that anymore! If someone asks, it’s definitely an 07 working paper
Child labour: mention that only these 2 papers take into a/c endogeneity of family size and consider effects on child labour