Instrumental variables in Structural Equation Modeling (SEM)
I have estimated a full SEM model where I have the measurement equation and the structural equation using a cross-sectional dataset. I want to estimate the effect of a latent variable L on y1, y2, and y3. The command used is:
sem (d1 d2<-L) (y1 y2 y3<-L x1 x2), latent(L) cov(e.y1*e.L e.y2*e.L e.y3*e.L) nolog
d1 and d2 are observed indicators used to measure the latent variable L. x1 and x2 are exogenous explanatory variables. y1, y2, y3 are the main dependent variables. Now my thesis supervisor said that the indicator d2 is endogenous to the y2 so I should handle this. Now my questions are:
As the endogenous variable d2 does not appear directly in my equation y2,
1) how to test the endogeneity of d2 in the equation y2?
2) how do I address this issue if I suppose I manage to have an instrument Z? And how to test the validity of such an instrument?
Here is what I did, but I don't know which one is correct:
Option 1) sem (d1 d2<-L) (y1 y2 y3<-L x1 x2) (L<-Z), latent(L) cov(e.y2*e.L) nolog
Option 2) sem (d1 d2<-L) (y1 y2 y3<-L x1 x2) (d2<-Z), latent(L) cov(e.y2*e.d2) nolog
3) How to address the issue if I don't manage to get a valid instrument?
Thank you.
sem (d1 d2<-L) (y1 y2 y3<-L x1 x2), latent(L) cov(e.y1*e.L e.y2*e.L e.y3*e.L) nolog
d1 and d2 are observed indicators used to measure the latent variable L. x1 and x2 are exogenous explanatory variables. y1, y2, y3 are the main dependent variables. Now my thesis supervisor said that the indicator d2 is endogenous to the y2 so I should handle this. Now my questions are:
As the endogenous variable d2 does not appear directly in my equation y2,
1) how to test the endogeneity of d2 in the equation y2?
2) how do I address this issue if I suppose I manage to have an instrument Z? And how to test the validity of such an instrument?
Here is what I did, but I don't know which one is correct:
Option 1) sem (d1 d2<-L) (y1 y2 y3<-L x1 x2) (L<-Z), latent(L) cov(e.y2*e.L) nolog
Option 2) sem (d1 d2<-L) (y1 y2 y3<-L x1 x2) (d2<-Z), latent(L) cov(e.y2*e.d2) nolog
3) How to address the issue if I don't manage to get a valid instrument?
Thank you.
