Hi Judit,

you as a theorist have to think about whether such a "feedback loop" is something which you would expect and thus, hypothesize, and estimate. If you do, there are some difficulties because in order to isolate and estimate both parts of the loop, you need "instrumental variables". That is, for each involved variable, you need one unique predictor which is only related to the respective predictor but not to the other predictor. The latter is called the exclusion restriction/assumption. If you have at least two instruments per predictor, this restriction could be tested by the model chisquare test.

If you are rather interested in the X-->Y effect but you are skeptical that some simultaneity may exist, you need just one or better, several instruments only for X. Both are part of the so-called endogeneity problem with which especially observed data analyses are plagued.

Below please find some background.

Best

Holger

Angrist, J. D., & Krueger, A. B. (2001). Instrumental variables and the search for identification: From supply and demand to natural experiments. Journal of Economic Perspectives, 15(4), 69-85. doi:10.1257/jep.15.4.69

Antonakis, J., Bendahan, S., Jacquart, P., & Lalive, R. (2010). On making causal claims: A review and recommendations. The Leadership Quarterly, 21, 1086-1120. doi:10.1016/j.leaqua.2010.10.0

Bascle, G. (2008). Controlling for endogeneity with instrumental variables in strategic management research. Strategic Organization, 6, 285-327. doi:10.1177/1476127008094339

Baiocchi, M., Cheng, J., & Small, D. S. (2014). Instrumental variable methods for causal inference. Statistics in Medicine, 33(13), 2297-2340.

Bollen, K. A. (2012). Instrumental variables in sociology and the social sciences. Annual Review of Sociology, 38, 37-72. doi:10.1146/annurev-soc-081309-150141

Maydeu-Olivares, A., Shi, D., & Rosseel, Y. (2019). Instrumental variables two-stage least squares (2SLS) vs. maximum likelihood structural equation modeling of causal effects in linear regression models. Structural Equation Modeling: A Multidisciplinary Journal, 26(6), 876-892.

Paxton, P. M., Hipp, J. R., & Marquart-Pyatt, S. (2011). Nonrecursive models. Endogeneity, reciprocal relationships, and feedback loops. Thousand Oaks: Sage.

Wong, C.-S., & Law, K. S. (1999). Testing reciprocal relations by nonrecursive structural equation models using cross-sectional data. Organizational Research Methods, 2(1), 69-78.

It is very true what Holger says, I have seen those models in instrumental variables that evaluate dyads, parent-child relationships, family-work and person-environment, that explain phenomena based on a feedback between two dependent factors one of the other

Hello Judit,

Holger Steinmetz 's response is good, because he's pointing to the need to consider whether bi-directional influence makes sense in the context of theory surrounding the variables and population with which you're working.

To cut to the chase, though, path analysis (as explicated by Sewall Wright) does not allow for bi-directional influence...that invokes a more complex SEM framework (and a different computational approach than good old OLS regression).

Good luck with your work.

Wright, S. (1920). The relative importance of heredity and environment in determining the piebald pattern of guinea-pigs. Proceedings of the National Academy of Sciences, 6, 320-332. doi:10.1073/pnas.6.6.320

Wright, S. (1934). The method of path coefficients. Annals of Mathematical Statistics, 5, 161-215. doi:10.1214/aoms/1177732676

This monograph is often considered the official explication of path analysis.

Hi David Morse , I am a bit confused about your statement that path analysis does not allow to model loops. The whole book by Pamela Paxton is on how to do it. Or did you mean something specific? As it turns out, the inventor of instrumental variable was even Sewall Wright's father, Phillip Wright (see the nice paper by James Stock):

https://scholar.harvard.edu/files/stock/files/who_invented_iv_regression.pdf

Further, there is a huge literature on "nonrecursive path analysis". But as I said, perhaps you mean something what I do not get...

Best,

Holger

Hi there, thanks a lot for all your replies. I attached a sketch of the model that I intend to test, this is solely based on theory. The penciled arrow from L to C would make sense, based on empirical (qualitative...) studies, but I'm afraid I wouldn't have the "instrumental variable" Holger Steinmetz suggested. thx...

Meanwhile I start reading the recommended literature :)

I mistakenly attached an earlier version of the model, this one is the right one, sry.

Hi Judit,

oh my, this is a quite complex model with tons of assumptions which will certainly fail and lead to a misfit (at least one). Hence, at any rate, I would try to concentrate on some portion.

One portion could indeed be the effect of C on L. To test this effect and to estimate it consistently even if there is a reverse effect due to L (the grey one), you can use T as an instrument. Furthermore, I can be a conditional instrument, that is, it is no instrument per se (as it correlates directly with L via their joint dependence of M, but if you control for M, this connection is blocked and I becomes conditionally independent, see for further background, see

Van Der Zander, B., Textor, J., & Lis̈kiewicz, M. (2015). Efficiently finding conditional instruments for causal inference. IJCAI International Joint Conference on Artificial Intelligence(Ijcai), 3243-3249.

That is, to test the effect of C on L, you could to the following

a) You conduct a two-stage-least squares regression and use T and I, controlling for M. The Sagan-test will inform you if your assumptions that T and I are uncorrelated with L (the relevant exclusion restriction in order to support the viability of the instruments ).

b) You could to a path analysis with SEM in which T, I and M have an effect on C and C has an effect on L. The fit of this model will tell you 1) if there is some residual correlation between C and L which either is due to reverse causation or unobserved confounding) and 2) whether the exclusion restrictions hold.

In this model, beware of interpreting the effects of I and M on C. As you see from your model, they have no meaning besides their instrumental role.

If this test holds (rejecting the feedback loop), you could go forward towards your larger model.

Beyond all of that, there is a chance to fest the feedback loop although such a test would stand on shaky ground. As you see from your path diagram, M could serve as an instrument for L. So, you could set up a small model in which M is a predictor of L and T is a predictor for C. So you would have only one instrument for each variable which would require both a strong theoretical argument (especially as you cannot test the exclusion restriction) as well as strong effects of the instruments. When you have several instruments (as in the prior procedure), the joint relationship counts--but with only one, you have to have a strong effect.

In this model, and assumed the strong effects, you could test both effects directions. Additionally, it could be necessary to estimate the error covariance between L and C when there is further unobserved confounding.

HTH

Holger

Wow, Holger Steinmetz , quite an answer, thanks for your time. I wonder if you read the edits I made, first I mistakenly attached an earlier version of the model, but your answer seems valid for the second version as well. Thanks, really.

Hello all,

My comment about S. Wright and the formulation of path models (and the relative simplicity of OLS regression methods) was based on the family of recursive models. In that context, the restriction on bi-directional influence is essential. Ditto the absence of latent variables, though you may certainly find examples in the literature wherein investigation of relationships among latent variables are referred to as "path models" or "paths." The semantic distinctions have been somewhat ignored as the analytic tools and software packages for handling all types of structural models have become more plentiful. One case in point would be the occasional use of the term, "causal models," for situations in which no direct manipulation of variables in a hypothesized model ever takes place and data are collected at one point in time.

Folks who have published studies of reciprocal / bi-directional influence tend to rely on different data collection schemes (e.g., cross-lagged panel models); parameter estimation via methods other than OLS regression; latent variables; or some combination of these. That takes you out of the arena of pure path models, as explicated by Wright.

Good luck to Judit with her research work.