Co-variance between exogenous and endogenous latents

[katja.si...@gmail.com]

Hi,

Attached is the model I working on. The double errors represent co-variances between the endogenous and exogenous latent variables. I am trying to use these covariances  to show that there is some sort of a feedback loop. 

However, I have several doubts about my model. I read on some web sites that modelling the co-variance between exogenous and endogenous variables is not correct and should throw an error during the execution. Thus, I wanted to check what does lavaan assume about this co-variance?

Also this is my code. This is my first project using SEM. I apologize if I am not making much sense with this question.

model_d3.1<-'

ITMC =~ q20a+q20b+q22a

BMC=~q20i+q20j+q20k

TR =~ q24a+q24b+q24c

SHU=~q25a+q25b+q25c

ITSC=~q14b+q14c +q14d +q14f +q14g+q14j

TR ~ITMC +q11_dic+q9a+q8 +LNFTE +ProfitNo

SHU~ BMC +TR +q8+q11_dic+q9a +LNFTE +ProfitNo

ITSC~SHU  +q11_dic+q9a+q8 +LNFTE +ProfitNo

ITSC~~ITMC

ITSC~~BMC

'

fit_controls <- sem(model_d3.1, data=data_cio,estimator="DWLS",

                    ordered=c("q11_dic","q9a","ProfitNo","q20a","q20b","q22a",

                              "q20i","q20j","q20k",

                              "q24b","q24c","q24a",

                              "q25c","q25b","q25a","q25d",

                              "q14b","q14c","q14d","q14f","q14g","q14j"))

fitMeasures(fit_controls,c("chisq","df","pvalue","CFI","TLI","RMSEA","GFI","NFI","SRMR","RMR","SMC"))

[Edward Rigdon]

     To make sense of this, imagine a model without the residual covariances between ITSC and ITMC / BMC. This model does not fully account for the covariance between these variables. But there are many alternative models that could account for it. One is your model with residual covariances--which could suggest a feedback relation as you describe, or it could mean excluded third variables affecting ITSC, ITMC, BMC. If "excluded third variables" is the reason, then ITMC and BMC are not really exogenous at all, and that makes your model misspecified, even if it fits well. Other alternative models might involve more paths from ITMC and BMC through the chain of mediators. (It could require additional variables, or research steps, to distinguish the empirical evidence for one alternative model over another. Short of that, you may be left with no way to tell.)

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