Model identified at solution but not identified at start values

Let's be careful about separating model identification from model estimation. Model estimation is an iterative process of trying to find the "best" parameter values. Model identification -- in the SEM and statistics context -- is a single-step process of determining whether a unique set of parameter values can be found. The terminology gets muddled by the use of "system identification" (and sometimes "model identification") in time series and engineering to refer to the same thing as "model estimation" in SEM.

In SEM, we say a model is identified if there is no other set of parameter values that can produce the same results. The check that we use in mxCheckIdentification() is local. That is, you can and sometimes do get different answers about whether or not your model is identified depending on the parameter values. The same model might be locally identified for one set of free parameter values, but not for another set of free parameter values. This phenomena does not imply the model is "bad". To quote from the help page for mxCheckIdentification(): "You might get different answers about model identification depending on the free parameter values."

A typical situation where a model is locally identified by not globally identified is a factor model where all the loadings could change sign. One solution has all the factor loadings positive; the other solution has all the factor loadings negative. A local identification check with the factor loadings at zero will say the model is not locally identified. However, a local identification check at factor loadings not all near zero will say the model is locally identified. This is not a cause for concern for factor models, or any other model for that matter.

In many cases, starting values of all zeros produce locally not identified models. Shifting the starting values off zero will often produce locally identified models in these cases.