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Suppose we were trying to estimate the impact of $Size$ of a house on its price while controlling for whether or not the house is located by the water. Suppose we constructed a dummy variable for houses by the water ($DW=1$ for houses by the water and $DW=0$ for houses not by the water) and were looking at the regression model $$Price = \beta_0 + \beta_1 Size+ \beta_2 DW + \mu$$

My Question: Would it be correct to interpret $\beta_1$ as: $\beta_1$ is the effect of size of price, regardless of whether or not a house is located by the water, and assuming that the effect of size on houses by the water is similar to the effect of size on houses not by the water?

I say “regardless of whether or not a house is located by the water” because I think I should account for both the cases when $DW=0$ and $DW=1$. Also, I say “assuming that the effect of size on houses by the water is similar to the effect of size on houses not by the water” due to the lack of the interaction term in the regression model.

Is my interpretation of $\beta_1$ correct? If not, how can it be improvised?

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$\beta_1$ is the relationship between Size and Price and the model inherently requires that this relationship is the same for all houses, regardless of the value of $DW$ (that is, regardless of whether the house is next to water). Another option, as you mentioned, if you believe the relationship between $Size$ and $Price$ could be different for houses next to or not next to water, would be to add an interaction term: $\beta_3 * Size * DW$.

In addition, rather than a dichotomous $DW$, you could turn this into a continuous variable by using distance from water as a regression variable, which would allow for differences in predicted $Price$ for houses of a given size but different distances from water. But maybe there's a big premium for houses right next to the water, so you might want an even more flexible model that allows $Price$ to change non-linearly with distance from the water.

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This answer to this question, and indeed to all "how do I interpret a linear regression model", is to remember two things:

  1. Regression is to estimate a function using data
  2. In order to calculate the effect on the outcome of the function resulting from a change in one inputs, we can simply calculate the derivative of the function w.r.t. the input i.e. $\frac{\partial Price }{\partial X_i}$

The interpretation of $\frac{\partial Price }{\partial X_i}$ really is: What happens with price when I change only $X_i$ and nothing else. From your model it's clear that $\frac{\partial Price }{\partial Size} = \beta_1$. As you say, the implicit assumption is that the marginal effect of size on price is constant no matter where the house is located - this may, or may not, be a reasonable assumption. However by interacting DW and Size, you can allow the model to capture that Size might be moderated by location.

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