I am referring to a confirmatory factor analysis output. I am curious to know how do we interpret the "Variance Parameters" Output containing the "error variance estimates" if the predictors are significant (p < 0.05). Do we essentially require the error variance estimates to be significant? What does that essentially mean? (2) How do we interpret the value of LM stat in the "Rank Order of the 10 largest LM Stat forPath Relations?"
It doesn't mean much for the error variance to be significant.
It's a variance. It has to be greater than zero (unless it's exactly zero, which is very unlikely), so a significance test that tells you it's significantly greater than zero (or not significantly greater than zero) doesn't tell you much interesting.
Therefore we don't care if it's significant, and we don't require it to be significant.
I'm not familiar with the Rank Order of the LM Stat. I guess that's the lagrange multiplier, but questions about interpreting results from specific software are usually considered to be off topic on CrossValidated.