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Use census  years (1982 and 1990) OR (1982 and 2000) for your analysis.

Estimate an OLS model with the following variables:

Dependent (LHS) variable:  Fertility outcomes -  i) Did you have another child (if you want, you may limit your analysis to only the sample with one daughter)? ii) Among those with another child was the next birth male? (again, you may stay restricted to the sample with one daughter)

Key explanatory variable: Create a binary variable labelled  policy (=0) if mothers were not exposed to one child policy and faced a zero-fine regime (1982 census parents); 1 if they were subjected to the one-child policy  (1990/and 2000 census parents).  

Control variables: following OP, use age and years of education of both parents; province fixed effects, a fixed-effect for the year of  the previous birth.

Create a table similar to  OP's Table 6. Do not forget the sample restrictions (refer to Table 6): 

· parents who have not yet had a son 

· mothers aged 35–40 

· also, the regressions are estimated using the mothers in this group and their births in the 1982,1990, and 2000 China census samples between 1964 and 2000.

An alternative idea using fines (a bit more complicated than the previous one):  Examine the impact of being located in a high-fine settingvs low-fine setting on fertility outcomes. In this case, you will replace policy by another binary variable finerate (=0 if mother resides in a low-fine area; 1 if high-fine area).

OP offers Stata replication package to create the fine rates and, subsequently impute the fine rates to census parents using fines and policy rules.  Use it to create the variable finerate.