Interpreting main effects model with omitted interaction terms

[Alice Campbell]

Suppose I have two regressors, task availability (Xa) and task participation (Xp), and a DV Y. One can only participate in a task if it is available, but one can choose not to participate even if a task is available. Baseline model Y ~ Xa, should be the total effect of task availability. Y ~ Xa + Xa * Xp, adds the effect of participation. Now, if the interaction model is the true model, would Y ~ Xa be biased, because of omitted variable problem? That is, the total effect cannot be estimated using the baseline model?

[Phil Bromiley]

You didn't get a quick answer. You'll increase your chances of a useful answer by following the FAQ on asking questions - provide Stata code in code delimiters, readable Stata output, and sample data using dataex.

Your explanation is hard to follow. If the correct model includes the interaction, a model without the interaction is sort of giving you the average of the main and interaction averaged over the actual frequency of observations in the data. This is problematic since it depends on frequency of sampling, not on covariances.

No, the baseline model is not the total effect. Personally, I am strongly in favor of using the full model and interpreting it using margins.

[Alice Campbell]

Note that Y has a valid value, with or without task participation.