Sum of subpopulation estimates not equal to estimate of total. Is this correct?

I am studying a population that consists of several sub-populations. I need models for the entire population and each of the sub-populations. Strangely, the total for the estimated variable of interest across the sub-populations differs from the estimated variable of interest for the population as a whole.

Some details. The variable of interest (VOI) is the number of people who have done something. The observations are for several years and for several countries -- this is panel data.

I defined the dependent variable (DV) as the logit of the ratio (VOI / population of the country). This transformation makes the most sense to me and produces the best fit for VOI.

I calculate the estimated VOI by estimating DV using the model and then undoing the transformation. For each sub-population, I found the models using stepwise regression with fixed effects.

I need to predict what would happen to VOI if the independent variables had certain values. The predicted VOI is the estimated VOI with the independent variables set to those values.

The problem is with the predicted VOI. Logically, the sum of VOI across the sub-populations should equal to the VOI for the population as a whole, since these are numbers of people. In practice, the sum of the predicted VOI for the sub-populations is very different from the overall predicted VOI.

I don't think that this is wrong per se. In each case, I am getting the best prediction conditional on my knowledge (or lack of it) of the sub-population. Is this correct?

But even if it's not wrong, it looks bad. People who are reading the paper will complain.

Ideas?

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"Fitted values"? "The models"? Could you please tell us something about what these are, how they are fitted, and what kind of data you have? Otherwise, it's impossible to tell what you're doing. –  whuber Nov 1 '11 at 18:27
See, statisticians are complaining about it already, as well ;). –  StasK Nov 2 '11 at 2:40
@whuber I've just edited the post to include more details. I am just trying to be concise, but I realize details are important too. Thanks. –  Jessica Nov 2 '11 at 14:50
Quite helpful, thanks: those details were needed. One more thing: how are the independent variables among the subpopulations related to those within the populations? Could you give some examples of the IVs you are using? –  whuber Nov 2 '11 at 14:54
I have a small number of potential IVs. They are exactly the same for all the sub-populations and the full population. The IVs indicate the presence / absence of particular laws in each country / year. For each model, the stepwise regression selects a few of these. I could throw in the others, but they would have large p-values. –  Jessica Nov 2 '11 at 14:59