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

2. Outline of Class Non-recursive models Non-recursive models Equilibrium Equilibrium Panel Designs Panel Designs Identification Issues: Order and Rank Conditions again and more this time! 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 direct feedback loops (causal paths) Model with correlated disturbances which have causal paths between the endogenous variables with correlated disturbances Model with correlated disturbances which have causal paths between the endogenous variables with correlated disturbances Model with indirect feedback loops (Y 1 --> Y 2- --> Y 3 --> Y 1) Model with indirect feedback loops (Y 1 --> Y 2- --> Y 3 --> Y 1)

4. Why do we need non-recursive models? One-way relationship does not reflect what we believe social reality to be 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 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 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! 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. 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 R 2 may be inappropriate for endogenous variables involved in feedback loops. Multiple R 2 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. 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. 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. The causal process has basically dampened out and is not just beginning.

7. Panel Design Do they solve the non-recursive problem? Do they solve the non-recursive problem? They are one possible solution They are one possible solution Not necessarily recursive depending on disturbance correlations 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 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 [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 k e = # exogenous variables in the model excluded from the structural equation model being tested and m i = 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: k e > m i -1 If m = # of endogenous variables in the model and k = # of exogenous variables in the model, and k e = # exogenous variables in the model excluded from the structural equation model being tested and m i = 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: k e > m i -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. 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. 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 system matrix Create a reduced matrix Create a reduced matrix If rank of reduced matrix for each endogenous variable > m-1, the rank condition is met. 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 equality or proportionality constraints (equality makes the reciprocal causation not very interesting, proportionality requires prior knowledge) Add exogenous variables such that: Add exogenous variables such that: Number of additional observations > number of new parameters addedNumber of additional observations > number of new parameters added Numbers of excluded variables for endogenous variables are each > 1Numbers of excluded variables for endogenous variables are each > 1 Respecified model meets the rank conditionRespecified model meets the rank condition