TABLE 1.

Summary of Parameter Constraints and Assumptions of Various Product Indicator Approaches (Sources: Jöreskog & Yang, 1996; Kelava et al., 2011; Ma, 2010; Wall & Amemiya, 2001)

Model Specification	CPI	GAPI	UPI
1. Factor loadings of the product indicators on ξ1ξ2. For example, λX1X4 = λX1λX4	Yes	Yes	No
2. E(ξ1ξ2) = Cov(ξ1, ξ2)	Yes	Yes	Yes
3. Var(ξ1ξ2) = Var(ξ1)Var(ξ2) + Cov2 (ξ1, ξ2)	Yes	No	No
4. Cov(ξ1,ξ1ξ2) = Cov(ξ2, ξ1ξ2) = 0	Yes	No	No
5. Variances of the unique factors of the product indicators. For example, Var(δX1X4) = λX1 Var(ξ1)Var(δX4) + λX4 Var(ξ2)Var (δX1) + Var(δX1)Var(δX4)	Yes	Yes	No
6. Zero covariances between the unique factors of exogenous indicators and those of product indicators (assuming zero covariances among the unique factors of exogenous indicators)	Yes	Yes	Yes
7. Covariances between unique factors of product indicators that share the same exogenous indicators (assuming zero covariances among the unique factors of exogenous indicators). For example, Cov(δX1X4, δX1X5) = λX4λX5Var(ξ2)Var(δX1)	Yes	Yes	No
8. Normality assumptions of exogenous indicators from model specification	Yes	More Liberal	Most Liberal

Note. CPI means constrained product indicator approach. GAPI means generalized appended product indicator approach. UPI means unconstrained product indicator approach. As noted in 8, model constraints 3, 4, 5, 6, and 7 assume multivariate normality for the CPI approach. Constraint 7 is not a concern if the two exogenous latent variables have equal numbers of indicators. The distributional assumptions are relaxed for the GAPI and UPI approaches.