Peng-Kai -

Someone on Stack Exchange pointed out the attached by Andrew Gelman:

 http://andrewgelman.com/2005/01/25/why_i_dont_use/

"Why I don’t use the term 'fixed and random effects.'"

Apparently he gives five definitions for fixed vs random.

I don't really know what your question is as I don't know what you mean by fixed effects, except that you do have this:

 yhat = fixed effect + x*betahat

Is that a common intercept across different groups? 

You also have this:

 mean(yi)-mean(xi)*betahat

I don't understand that.  What is that for?  In some specific cases, say the classical ratio estimator (a WLS estimator to the origin), or OLS, it should be exactly the estimated intercept, i think, and in other model unbiased cases the sum of residuals should be only asymptotically zero, leaving 'approximately' the 'estimate' of your intercept in your case.  ???   Are you looking at different ways to estimate the intercept? 

If you have different levels, perhaps your software is using different data in your two ways of estimating the intercept, if that is what you are doing.  (Don't forget to look at standard errors too.)  Perhaps R is grouping data differently and effectively weighting it differently in your two methods???

I don't know R, but it depends on the software as to exactly how it handles the data, and what estimators it uses, so you may need to consult any documentation you can find to see if R is really doing what you think it is doing.

Anyway, those are my guesses as to what you could investigate. 

Best wishes - Jim

without entering in the detail of your data, if you are a R user you should use the function(s) _predict()_ of R. The software automatically will select the method best suited to your data - lm, glm, lmer, nlme, ....