Question

Consider a regression analysis with three independent variables \(x_{1}, x_{2}\), and \(x_{3}\). Give the equation for the following regression models: a. The model that includes as predictors all independent variables but no quadratic or interaction terms b. The model that includes as predictors all independent variables and all quadratic terms c. All models that include as predictors all independent variables, no quadratic terms, and exactly one interaction term d. The model that includes as predictors all independent variables, all quadratic terms, and all interaction terms (the full quadratic model)

Short answer

a) \(y = \beta_{0} + \beta_{1}x_{1} + \beta_{2}x_{2} + \beta_{3}x_{3}\) \nb) \(y = \beta_{0} + \beta_{1}x_{1} + \beta_{2}x_{2} + \beta_{3}x_{3} + \beta_{4}x_{1}^{2} + \beta_{5}x_{2}^{2} + \beta_{6}x_{3}^{2}\) \nc) Three alternatives: \(y = \beta_{0} + \beta_{1}x_{1} + \beta_{2}x_{2} + \beta_{3}x_{3} + \beta_{4}x_{1}x_{2}\), \(y = \beta_{0} + \beta_{1}x_{1} + \beta_{2}x_{2} + \beta_{3}x_{3} + \beta_{5}x_{1}x_{3}\), \(y = \beta_{0} + \beta_{1}x_{1} + \beta_{2}x_{2} + \beta_{3}x_{3} + \beta_{6}x_{2}x_{3}\) \nd) \(y = \beta_{0} + \beta_{1}x_{1} + \beta_{2}x_{2} + \beta_{3}x_{3} + \beta_{4}x_{1}^{2} + \beta_{5}x_{2}^{2} + \beta_{6}x_{3}^{2} + \beta_{7}x_{1}x_{2} + \beta_{8}x_{1}x_{3} + \beta_{9}x_{2}x_{3}\)

Step-by-step solution

Part (a): No quadratic or Interaction terms

Since no quadratic or interaction terms are needed, the equation would just include the predictors. The model would be: \(y = \beta_{0} + \beta_{1}x_{1} + \beta_{2}x_{2} + \beta_{3}x_{3}\)

Part (b): Quadratic terms

Now quadratic terms need to be included. A quadratic term is a variable squared. This gives the model: \(y = \beta_{0} + \beta_{1}x_{1} + \beta_{2}x_{2} + \beta_{3}x_{3} + \beta_{4}x_{1}^{2} + \beta_{5}x_{2}^{2} + \beta_{6}x_{3}^{2}\)

Part (c): One Interaction term

Interaction terms are terms where variables multiply each other. Since only one interaction term should be included, three models need to be created for each possible pair. The models are: \(y = \beta_{0} + \beta_{1}x_{1} + \beta_{2}x_{2} + \beta_{3}x_{3} + \beta_{4}x_{1}x_{2}\), \(y = \beta_{0} + \beta_{1}x_{1} + \beta_{2}x_{2} + \beta_{3}x_{3} + \beta_{5}x_{1}x_{3}\), \(y = \beta_{0} + \beta_{1}x_{1} + \beta_{2}x_{2} + \beta_{3}x_{3} + \beta_{6}x_{2}x_{3}\)

Part (d): Full Quadratic Model

A full quadratic model includes all interaction and quadratic terms. This yields the model: \(y = \beta_{0} + \beta_{1}x_{1} + \beta_{2}x_{2} + \beta_{3}x_{3} + \beta_{4}x_{1}^{2} + \beta_{5}x_{2}^{2} + \beta_{6}x_{3}^{2} + \beta_{7}x_{1}x_{2} + \beta_{8}x_{1}x_{3} + \beta_{9}x_{2}x_{3}\)