# Interactions or multigroup SEM?

Let's suppose I have model with three latent variables (L1, L2, L3). These three latent variables are included as regressors in a path explaining Y (a measured variable), so the equation would be: Y =~ L1 + L2 + L3.

The survey was gathered in three different cities (City_A, City_B, and City_C) and I have theoretical reasons to believe that regression coefficiente for L3 should be different.

Now, in standard econometrics or statistics the intuition would be to add interactions. The Equation would be:

Y = L1 + L2 + L3 + City_B + City_C + L3City_B + L3City_C

Where City_A was used as the reference variable and the "effect" is just L3.

My first equation is if in SEM, that apporoach is still valid.

As I cannot add interactions with latents in lavaan and STATA is extremely slow, there are two alternatives I am considering. The first is to run a Multigroup SEM by city setting everything to be the same except for the path of L3 on Y. That way I will get three estimabes of the L3 coefficent: L3_A, L3_B, and L3_C.

My second question is if this approach is correct or even preferable than the interactions. Moreover, I am wondering if both models are the same. An additional issue I am considering is if I should evaluate invariance of L3 for the multigroup to make sense. The reason why I think the effect of L3 on Y vaies by city is because the construct is probably perceived different in every city.

The second alternative I am considering is to run a CFA with L1, L2 and L3, and take out the scores for the three variables. Then I can run a regression using L1, L2, and L3 with the city interactions. I have read some papers where authors follow the mentioned approach but using EFA or PCA, I prefer CFA.

My third question is if I should stick to the multigroup alternative or go with the CFA + linear regression.

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