CJT 765: Structural Equation Modeling

1. CJT 765: Structural Equation Modeling Class 10: Non-recursive Models

2. Outline of Class • Non-recursive models • Equilibrium • Panel Designs • Identification Issues: Order and Rank Conditions again and more this time!

3. What is a non-recursive model? • Model with direct feedback loops (causal paths) • Model with correlated disturbances which have causal paths between the endogenous variables with correlated disturbances • Model with indirect feedback loops (Y1--> Y2---> Y3--> Y1)

4. Why do we need non-recursive models? • One-way relationship does not reflect what we believe social reality to be • To provide jobs for people with strong statistical and matrix algebra training • Correlated disturbances reflects assumption that corresponding endogenous variables share at least one common omitted cause

5. Other Peculiarities of Non-recursive Models • Variables in feedback loops have indirect effects on themselves! • Total effect of a variable on itself is an estimate of sum of all possible cycles through the other variable, i.e., an infinite series. • Multiple R2 may be inappropriate for endogenous variables involved in feedback loops.

6. The Equilibrium Assumption • Any changes in the system have already manifested their effects and the system is in a steady state. • That is, particular estimates of reciprocal causal effects do not depend on the particular time point of data collection. • The causal process has basically dampened out and is not just beginning.

7. Panel Design • Do they solve the non-recursive problem? • They are one possible solution • Not necessarily recursive depending on disturbance correlations

8. Necessary but not Sufficient Conditions for Identification: Counting Rule • Counting rule: Number of estimated parameters cannot be greater than the number of sample variances and covariances. Where the number of observed variables = p, this is given by [p x (p+1)] / 2

9. Necessary but not Sufficient Conditions for Identification: Order Condition • If m = # of endogenous variables in the model and k = # of exogenous variables in the model, and ke = # exogenous variables in the model excluded from the structural equation model being tested and mi = number of endogenous variables in the model included in the equation being tested (including the one being explained on the left-hand side), the following requirement must be satisfied: ke > mi-1

10. Necessary but not Sufficient Conditions for Identification: Rank Condition • For nonrecursive models, each variable in a feedback loop must have a unique pattern of direct effects on it from variables outside the loop. • Specifically, the rank condition is met for the equation of an endogenous variable is the rank of the reduced matrix is > the total number of endogenous variables minus 1.

11. Berry’s Algorithm for the Rank Condition • Create a system matrix • Create a reduced matrix • If rank of reduced matrix for each endogenous variable > m-1, the rank condition is met.

12. What to do about an under-identified model? • Add equality or proportionality constraints (equality makes the reciprocal causation not very interesting, proportionality requires prior knowledge) • Add exogenous variables such that: • Number of additional observations > number of new parameters added • Numbers of excluded variables for endogenous variables are each > 1 • Respecified model meets the rank condition