# Should coefficient on interaction term be positive or negative?

I have the following model for housing prices

price = $$\beta_0$$ + $$\beta_1$$ sqrft + $$\beta_2$$ bedrooms + $$\beta_3$$ sqrft $$\times$$ bedrooms + $$\beta_4$$ bathroom,

where sqrft is square feet.

I am confused regarding the interaction term. Should $$\beta_3$$ be positive or negative? I think that the effect of additional bedroom on house price is higher for larger houses. Is this correct?

• A title emphasizing interaction effects would be more appropriate – emeryville May 4 '19 at 20:51

I think that the effect of additional bedroom on house price is higher for larger houses. Is this correct?

As @emeryville said, if your question is about what the result should look like, different people may have different theories. I think one can make a reasonable argument that going from 1 bedroom (small house) to 2 bedrooms should have a larger impact on the price than going from say 5 bedrooms (large house) to 6 bedrooms because the marginal value of adding an extra bedroom is larger in that case.

This is maybe outside the scope of your question, but based on the wording of your question, it sounds like you have a hypothesis which may not be supported by the results of your model. So I just want to add that this could mean that your theory was wrong, but it could also mean other things, like that there is some outlier in the dataset that greatly influences the results, or that the model was misspecified in that it does not accurately reflect the true relationship between your variables. If your estimated result was a negative $$\beta_{3}$$, but the true of value of that parameter is positive, that could mean that you have some omitted variables that resulted in a biased coefficient. For example, perhaps the neighborhood location is associated with whether the effect of adding a bedroom is higher or lower for larger houses (and location is also correlated with price). In a downtown location, both price and size could depend on different factors than a house in a suburb and the value of additional bedroom could be different. So you would need to add location to the regression equation to obtain an unbiased $$\beta_{3}$$ coefficient.

• Thank you for your reply. The original question was why it is reasonable to include the interaction term between bedroom and house size( as measured by sqft). Is there any theory which posits that the effect of bedrooms on price depends on house size? You make a valid point that for small houses the change in price will be higher when additional bedroom is added compared to a big house (b3 would be negative).But one may also say that the larger the house the greater effect on bedrooms on prices (b3 would be positive i). So why it is reasnoable to include the interaction term? – Ronaldo777 May 5 '19 at 10:14
• Deciding which variables, including interaction terms, to include in regression is typically based on domain knowledge and your research question, meaning that your model specification should lead to unbiased estimates of the relationship you are exploring. Interaction terms are also one way to model non-linear relationships, so you could first plot variables against each other and if you see that the relationship between two of the variables depends on the value of a third variable, that would be a good indication that there is an interaction effect that should be included in your model. – AlexK May 5 '19 at 19:26

The coefficient of the interaction effect $$\beta_3$$ could be either positive or negative depending on your theory. A positive value for the effect of the interaction term would indeed imply that the larger the house, the greater (more positive) the effect of bedrooms on prices is.

Here is a nice reference on Interaction effects between continuous variables.

• Thank you for your reply. The original question was why it is reasonable to include the interaction term between bedroom and house size( as measured by sqft). Is there any theory which posits that the effect of bedrooms on price depends on house size? So why it is reasonable to include the interact term? – Ronaldo777 May 5 '19 at 10:15