Finite Feedback Cycling in Structural Equation Models

Abstract

In models containing reciprocal effects, or longer causal loops, the usual effect estimates assume that any effect touching a loop initiates an infinite cycling of effects around that loop. The real world, in contrast, might permit only finite feedback cycles. I use a simple hypothetical model to demonstrate that if the world permits only a few effect cycles, many coefficient estimates are substantially biased. If the world permits additional partial-cycle use in addition to full cyclings around the causal loop, some of the effect estimates are proper, and a full set of proper effect estimates can be recovered by hand calculations involving the model total effects. If the world permits no additional partial-cycle use, it might not be possible to recover proper estimates from the usual output.

It is not the equations representing the causal model, but rather the calculations of the covariance implications of the model, that change with limited cycling possibilities. Unfortunately, the features required to permit direct estimation of limited-cycle effects are not under user control in common structural equation programs, so estimation and detailed investigation of models with finite cycling of effects around feedback loops awaits new programming. To obtain unbiased estimates with limited causal cyclings, the researcher must continue to strive to specify the proper effect locations but must also attend to the number of full and partial causal cyclings permitted by the world. Determining the appropriate number of cycles is not a matter to be delegated to a statistician; it is something the researcher must attend to as a matter of substantive theory, methodology, and model interpretation.

Notes

¹For example, in LISREL, the Model syntax line might include a statement like BP = 4 (for Beta Powers = 4) to signal that only the first four powers of the B matrix were to be used. The powers of the B matrices might be designated as B0, B1, B2, B3, B4, and so on. A LISREL command like “Value 0.0 B4(1,1)” would specify that the row 1, column 1 entry in the fourth power matrix be set to “0.0” (as in Equation 13), and a command like “Free B4(3,2)” would free the only nonzero element in the fourth power matrix (as in Equation 13). Preferably, the initial specification of, and final numerical values in, the B power matrices should be printed so the model specification can be checked. (It would be helpful, but might be asking too much, to expect that the symbolic form of the matrices as in Equation 8 could be printed.) The other programming requirements should be minimal if a BP (Beta Power) specification on LISREL's Model line switched to invoking a subroutine summing the relevant B power matrices under the usual program name assigned the calculation of (I — B)⁻¹. The other program calculations that ordinarily call (I — B)⁻¹ (such as the derivatives in the minimization function) would thereby automatically call the appropriate replacement matrix.