Path Analysis assumptions: endogenous variables cannot share error covariance?

In this website: http://mb3is.megx.net/gustame/constrained-analyses/path-analysis

"If variables are believed to share extraneous variables, they should be considered as exogenous variables rather than endogenous variables."

Does path analysis really have such assumption? If so, why?

I have this question because I would like to predict two outcomes (math at age 5, math at age 6) by four predictors in my path model, and I believe the two outcomes have shared residual covariance.

I think the author of that link is being a bit severe. The default, in structural equation modeling, is for residual variances to be uncorrelated, but you can posit correlations between them if you think it makes sense to do so.

However, residual correlations will weaken the wow factor of your model. Presumably, you are saying that math at age 6 is related to math at age 5 and 4 predictors ... let's say gender, grade 1 teacher, parental homework involvement, hours of computer games. What would it mean if the residual variances are dependent? How could that happen? One way is if there is another variable kicking around like, say, attendance at a tutorial program, which some kids have and others don't - and which influences math at age 5 and 6, but which you have not included in the model. This is the "extraneous variable" the author is talking about.

So your research hypothesis is now: math at age 6 depends on math at age 5 plus 4 variables I have measured, plus a bunch of stuff I didn't think about. This is probably true in most educational research, but not the stuff of riveting publications.

You didn't specify exactly what your model is going to look like. Do the "four predictors" just feed into "math at age 5"? Or do they feed in to both time points? If they represent realities that persist from age 5 to 6, you might want to feed them in to both outcomes. For example, perhaps the grade 1 teacher is good because the school is good and the kindergarten teacher is also good, meaning that a school effect should feed into outcome at age 5 and outcome at age 6. If you just feed the 4 predictors into one of the math outcomes, you will get a non-zero residual covariance from the predictor effect that was not properly included in the model specification.

• Thanks @Placidia, the predictors are teacher at age 5 and homework at age 5 and teacher at age 6 homework at age 6 (with homework at age 5 predicts age 6 too). i believe there are residual variance of math at age 5 and 6 (like writing skills, time management skills) but I do not care about those in this model. That's why I add a residual variance between math at age 5 and 6.... Sep 8, 2015 at 1:45
• I have add my model in the question. (Teacher 5 doesn't correlate sig. with math at 6 and hence no path was drawn between them).... Sep 8, 2015 at 1:56
• @ceoec A lot of path analysis software will include correlations between teacher5 and teacher6 automatically. You have to suppress them if you don't want them. I would still put a regression arrow from math5 to math6, given that math is cumulative and it makes sense. Then you might find that you don't need the correlation between the residual errors. If you omit the arrow from math5 to math6, but have significant residual correlation, that almost undoubtedly means that the regression relationship was needed. Sep 8, 2015 at 12:48
• you spot the key difficulties i am facing -- i do not want to regress math6 on math5 -- the reason i didn't put regression arrow from math5 to math6 is I do not want to use math5 to predict math6, i only would like to see how teacher and homework play an unique role in math6... math5 and math6 do have very significant residual correlation... Sep 8, 2015 at 12:55
• Once I use math5 to predict math6, the role of teacher and homework at 5 and 6 disappeared. Sep 8, 2015 at 12:57