You can allow a covariance/correlation between error terms, but it is rather uncommon (though not totally implausible or impossible) to allow the error term for one variable to influence another observed variable (e.g., if two observed variables share the same wording or other method effect). In that situation, it is more common (and more useful) to introduce an additional factor (i.e., a method or indicator-specific factor) as a residual (specific) factor for the items in question. That way, you can separate random measurement error variance from systematic (i.e., reliable) but specific variance that is shared between specific items. For an example, see Figure 2 in

Article The Relationship between Spatial Abilities and Representatio...

The "Rater 2" factor for FR2 and TC2 in the path diagram is a residual factor that reflects common method (rater-specific) variance in their model.

Hi ,

In my opinion we should not do the same till the time you have very strong theoretical background , generally you will not have any case/theory or history of such scenario hence its not recommended.

Regards,

Dr. Uday Bhale

Structural equation modeling (SEM) is a method that allows researchers to test complex hypotheses about the relationships among observed and latent variables (Kline, 2015). Observed variables are those that can be measured directly, such as test scores or survey responses. Latent variables are those that cannot be measured directly, such as intelligence or academic achievement. SEM involves specifying a model that represents the causal structure of the variables and estimating the parameters of the model using data. One of the advantages of SEM is that it can provide information about the fit of the model to the data, as well as the modification indices that suggest how the model can be improved (Byrne, 2016).

One of the questions that may arise when using SEM is whether it is appropriate to connect an observed variable and an error variable if the output suggests it in modification indices. An error variable is a latent variable that represents the measurement error or unexplained variance of an observed variable. Connecting an observed variable and an error variable means adding a path between them in the model, which implies that there is a direct effect of one variable on another. This may not make sense theoretically or conceptually, as error variables are usually assumed to be independent of other variables in the model. Therefore, connecting an observed variable and an error variable may not be a good practice in SEM, unless there is a strong justification for doing so based on substantive knowledge or previous research (Brown, 2015).