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I have read the previous discussions on interaction effects and main effects, and I have a question on the subject.

I am running a destination choice model (multinomial logit), and I have one continuous variable (distance in meters) and several dummy variables accounting for given behavior (main activity at destination) which I am hypothesizing as having an interaction effect.

In this particular model, the effect of Distance is as expected significant by itself, and also are significant the interactions between Distance and the dummy variables, however, when I try to include the dummy variables alone to get their main effects, the model cannot be estimated even after including only one dummy variable. I illustrate this below:

  1. Distance+Distance*Dummy2+Distance*Dummy3 = Significant main and interaction effects (s.e. of interaction not estimated)
  2. Distance*Dummy2+Distance*Dummy3 = Significant interaction effects
  3. Distance+Dummy1+Dummy2 = Model cant be estimated
  4. Distance+Dummy1+Dummy2+Distance*Dummy2+Distance*Dummy3 = Model cant be estimated

So my question would be:
1. if it is possible to account only for the interaction terms without the main effects given what I explained before?

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    $\begingroup$ First, better to try to figure out why the models 3 and 4 can't be estimated. It is possible to account for interaction effects without the main effects, but it's rarely a good idea. $\endgroup$ – Peter Flom Jun 30 '12 at 11:52
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    $\begingroup$ You should read the discussion on this CV question: including-the-interaction-but-not-the-main-effects-in-a-model, as it will help you to think about these topics. $\endgroup$ – gung - Reinstate Monica Jun 30 '12 at 13:50
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    $\begingroup$ Moreover, I can't quite follow how you are thinking about (understanding or interpreting) interaction effects. Here is another question you should read through: stats.stackexchange.com/questions/27724/…. $\endgroup$ – gung - Reinstate Monica Jun 30 '12 at 14:41
  • $\begingroup$ Thank your for your prompt replies. Regarding Peter Floms comment, i am trying to find out why this is happening but so far the dataset seem to contain no data input errors. (still working though). As for gungs comments, i have read the threads, and that was in part what prompted me to ask the question. I guess my main concern is whether or not, not including the main effects still has scientific validity or not. I understand the ideally one should include both the main effects and the interactions but in the case this cannot be done, is there still a point in estimating the model? $\endgroup$ – errorterm Jul 2 '12 at 5:39
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In any discrete choice model, there are two types of variables: those that vary across the alternatives, and those that are constant for each decision maker. The first become generic coefficients, and the second are alternative-specific. There is one generic $\beta$ that is the same for all alternatives, but each alternative has its own alternative-specific $\beta$. So for $J$ alternatives, each alternative-specific variable creates $J-1$ parameters.

In a typical discrete choice model, this isn't a problem. But a residential location choice model usually has $J$ somewhere near 1000. It is impractical to consider 999 different parameters for a single variable. This is why variables that do not vary across alternatives (like a person's income) are usually interacted with variables that do (like the average cost of a home in each zone). In standard econometrics, we typically include the non-interacted components to separate correlation between income and home price as much as possible. But in your problem, you don't really have a choice.

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