To simplify notation letµ(θ) = S1 PS
s=1ki(u s
i;θ, ωis|usi ∈ECLSB) be the vector of simulated moments given the parameter vector θ. Taking the derivative of the objective function with respect to θ yields the following first order conditions:
∂µ ∂θ 0 ˆ θN W−1[ 1 N N X i=1 [Ki −µ(θN)] = 0
A Taylor expansion around µ(θ0) gives:
µ(θN) = µ(θ0) + ∂µ ∂θ θ∗ (θN −θ0)
for someθ∗betweenθ0 andθN. Plugging the Taylor expansion into the first order condition, premultiplying and rearranging gives:
√ N(θN −θ0) = ∂µ ∂θ 0 ˆ θN W−1∂µ ∂θ θ∗ −1 ∂µ ∂θ 0 ˆ θN W−1 1 √ N N X i=1 [Ki−µ(θ0)
Applying a Central Limit Theorem gives the following variance-covariance matrix for the limiting distribution:
∂µ ∂θ 0 ˆ θN W−1∂µ ∂θ θ ∗ −1 ∂µ ∂θ 0 ˆ θN W−1V[K−µ(θ0)]W−1 ∂µ ∂θ θˆN ∂µ ∂θ 0 ˆ θN W−1∂µ ∂θ θ ∗ −1 I then approximate ∂µ∂θ ˆ θN
using the matrix of numerical partial derivatives cal- culated at the optimal parameter value andV[K−µ(θ0)] is approximated by the variance-covariance of the moments at the optimal parameter value.
Moments
I use 622 moments in the estimation. The moments are as follows (number of moments in parentheses):
• Average child cognitive skills 1. by mother’s marital status (2) 2. by race (2)
3. by mother’s education (4) 4. by father’s education (4) 5. by father’s experience (6)
6. by mother’s age at first birth (4) 7. by number younger siblings (2)
8. by lagged cognitive achievement quartile (4)
9. by lagged hours and quartiles of home and child care quality () 10. by number older siblings (3)
11. by age of focal child (7) 12. by age of mother (12)
13. by lagged home inputs, child care inputs and hours of care (36) 14. lagged cognitive achievement (4)
15. variance by lagged cognitive achievement (4) 16. standard deviation (1)
1. by parents’ marital status (2) 2. by race (2)
3. by mother’s education (4) 4. by father’s education (4) 5. by father’s experience (6)
6. by mother’s age at first birth (4) 7. by number younger siblings (2) 8. by number older siblings (3) 9. by age of focal child (7) 10. by age of mother (12) 11. standard deviation (1)
• Percent in part-time child care 1. by parents’ marital status (2) 2. by race (2)
3. by mother’s education (4) 4. by father’s education (4) 5. by father’s experience (6)
6. by mother’s age at first birth (4) 7. by number younger siblings (2) 8. by number older siblings (3) 9. by age of focal child (7)
10. by age of mother (12) 11. standard deviation (1)
• Average child care quality
1. by parents’ marital status (2) 2. by race (2)
3. by mother’s education (4) 4. by father’s education (4) 5. by father’s experience (6)
6. by mother’s age at first birth (4) 7. by number younger kids (2) 8. by number older kids (3) 9. by age of focal child (5) 10. by age of mother (11) 11. standard deviation (1)
12. autocorrelation between rounds (1)
• Average child care price 1. by marital status (2) 2. by race (2)
3. by mother’s education (4) 4. by father’s education (4) 5. by father’s experience (6)
6. by mother’s age at first birth (4) 7. by number younger kids (2) 8. by number older kids (3) 9. by age of focal child (7) 10. by age of mother (12) 11. standard deviation (1)
12. autocorrelation between rounds (1)
• Percent in Head Start
1. by parents’ marital status (2) 2. by race (2)
3. by mother’s education (4) 4. by father’s education (4) 5. by father’s experience (5)
6. by mother’s age at first birth (4) 7. by number younger kids (2) 8. by number older kids (3)
9. average Head Start child care quality (1)
10. standard deviation Head Start child care quality (1)
• Average home quality 1. by marital status (2) 2. by race (2)
3. by mother’s education (4) 4. by father’s education (4) 5. by father’s experience (6)
6. by mother’s age at first birth (4) 7. by number younger kids (2) 8. by number older kids (3) 9. by age of focal child (7) 10. standard deviation (1)
11. autocorrelation between rounds (1)
• Percent in full-time labor force participation 1. by parents’ marital status (2)
2. by race (2)
3. by mother’s education (4) 4. by father’s education (4) 5. by father’s experience (6)
6. by mother’s age at first birth (4) 7. by number younger siblings (2) 8. by number older siblings (3) 9. by age of focal child (7) 10. by age of mother (12) 11. standard deviation (1)
• Percent in part-time labor force participation 1. by parents’ marital status (2)
2. by race (2)
3. by mother’s education (4) 4. by father’s education (4) 5. by father’s experience (6)
6. by mother’s age at first birth (4) 7. by number younger siblings (2) 8. by number older siblings (3) 9. by age of focal child (7) 10. by age of mother (12) 11. standard deviation (1)
• Joint distribution of labor supply and child care hours (8)
• Child care transition probabilities between rounds (8)
• Labor supply transition probabilities between rounds (8)
• Average mother’s wage: 1. by marital status (2) 2. by race (2)
3. by mother’s education (4) 4. by father’s education (4) 5. by father’s experience (6)
6. by mother’s age at first birth (4) 7. by number younger kids (2) 8. by number older kids (3) 9. by age of focal child (7) 10. by age of mother (12) 11. standard deviation (1)
12. autocorrelation between rounds (1)
• Average husband’s income: 1. by race (2)
2. by mother’s education (4) 3. by father’s education (4) 4. by father’s experience (6)
5. by mother’s age at first birth (4) 6. by number younger kids (2) 7. by number older kids (3) 8. by age of focal child (8) 9. standard deviation (1)
10. autocorrelation between rounds (1)
• Percent married: 1. by race (2)
3. by mother’s age at first birth (4) 4. by number younger kids (2) 5. by number older kids (3) 6. by age of focal child (7) 7. by age of mother (12) 8. standard deviation (1)
• Number of children less than 5: 1. by marital status (2) 2. by race (2)
3. by mother’s education (4) 4. by father’s education (4) 5. by father’s experience (6)
6. by mother’s age at first birth (4) 7. by age of mother (12)
8. standard deviation (1)
• Number of children older than 5: 1. by marital status (2)
2. by race (2)
3. by mother’s education (4) 4. by father’s education (4) 5. by father’s experience (6)
6. by mother’s age at first birth (4) 7. by age of mother (10)