generate synthetic instrumental variables (IV) for all endogenous regressors. These instruments are some linear combination of exogenous or control variables. In the second stage, the model is estimated by substituting instrumental variables for the endogenous regressors. This approach can be defined as follows. We estimate the following three-equation model: (4) (5) (6) where and is the number of years of education for the father and mother of the child. are a ith set of control variables as listed in Section 3. Lewbel (2012) shows that in the presence of two endogenous variables, and , their regression coefficients and can be consistently identified if there exist some exogenous variables in , satisfying the following three conditions: (i) E() = 0, E(, E(; (ii) Cov(,Cov; and (iii) Cov ( and Cov (, where can be a subset of the exogenous variables or itself. Under the above assumptions, ( and ( can be used as the excluded instrument vector for and . The strength of the instruments corresponds to the degree of heteroscedasticity of and with respect to 5. Results Table 2 reports the regression coefficients and the correlation coefficients for father-son pairs. The dependent variable is the years of education attained by the son and the independent variable is the number of years of schooling of the father. With the exception of the youngest cohorts, the average years of schooling of sons and their fathers has seen an upward trend. All regression and correlation coefficients are positive and statistically significant; however, there is a steady decline in their magnitude across birth cohorts. This implies that parental education still
generate synthetic instrumental variables (IV) for all endogenous regressors. These instruments are some linear combination of exogenous or control variables. In the second stage, the model is estimated by substituting instrumental variables for the endogenous regressors. This approach can be defined as follows. We estimate the following three-equation model: (4) (5) (6) where and is the number of years of education for the father and mother of the ith child. are a set of control variables as listed in Section 3. Lewbel (2012) shows that in the presence of two endogenous variables, and , their regression coefficients and can be consistently identified if there exist some exogenous variables in , satisfying the following three conditions: (i) E() = 0, E(, E(; (ii) Cov(,Cov; and (iii) Cov ( and Cov (, where can be a subset of the exogenous variables or itself. Under the above assumptions, ( and ( can be used as the excluded instrument vector for and . The strength of the instruments corresponds to the degree of heteroscedasticity of and with respect to 5. Results Table 2 reports the regression coefficients and the correlation coefficients for father-son pairs. The dependent variable is the years of education attained by the son and the independent variable is the number of years of schooling of the father. With the exception of the youngest cohorts, the average years of schooling of sons and their fathers has seen an upward trend. All regression and correlation coefficients are positive and statistically significant; however, there is a steady decline in their magnitude across birth cohorts. This implies that parental education still