Understanding Marginal vs. Partial Coefficients: A Case Study of House Prices

The marginal coefficient for the number of bathrooms represents the change in the predicted price of the house for every additional bathroom, holding all other variables constant. The partial coefficient for the number of bathrooms represents the change in the predicted price of the house for every additional bathroom, adjusting for the effects of the other variables in the model.

It is possible that the marginal coefficient for the number of bathrooms is larger than the partial coefficient because the number of bathrooms is highly correlated with other variables in the model, such as the number of square feet or lot size. In this case, the marginal coefficient may be capturing the effect of these other variables as well, leading to an inflated estimate of the effect of the number of bathrooms alone.

On the other hand, it is also possible that the partial coefficient for the number of bathrooms is larger than the marginal coefficient because the effect of the number of bathrooms is being masked by the other variables in the model. In this case, the partial coefficient is a more accurate estimate of the effect of the number of bathrooms alone, after adjusting for the effects of the other variables.

Ultimately, the difference between the marginal and partial coefficients for the number of bathrooms depends on the nature of the correlations between the explanatory variables in the model and the specific effects that each variable has on the response variable.