First review the results of the model including both main and interaction effects:

reg price c.weight##c.mpg

The coefficients on weight and mpg are both significant and positive, but the coefficient on their product is significant and negative.

To run the model without main effects, replace ## with #:

reg price c.weight#c.mpg

First note that the R-squared of this model is essentially zero, as is its F-statistic: this model can't predict its way out of a paper bag. In addition, the coefficient on the product is now zero—and the standard error is quite small. What happened?

Since c.weight#c.mpg is made up of weight and mpg, you can't change it without changing one or both of its components. The first model suggests this is a complicated affair: increasing weight and/or mpg increases the expected price through its main effect, but decreases the expected price through the product. The second model, since it does not include the main effects, has to attribute the entire change to the effect of the product, and the combined effect turns out to be zero.

Having main and interaction effects cancel each other out so precisely is unusual. But it's even more unusual for main effects not to matter at all. If you leave them out, the model will have to attribute their effects to the interaction term. This might increase its coefficient or decrease it, but the important thing is that it will be wrong. The bottom line is that you should always include the main effects whenever you include an interaction.

Last Revised: 1/19/2011