Question

Question: Determine which pairs of the following models are “nested” models. For each pair of nested models, identify the complete and reduced model.

$$\begin{gathered} \mathit{a}\mathbf{.}\mathbf{}\mathit{E}\mathbf{\left(}\mathit{y}\mathbf{\right)}\mathbf{=}{\mathit{\beta }}_{\mathbf{0}}\mathbf{+}{\mathit{\beta }}_{\mathbf{1}}{\mathit{x}}_{\mathbf{1}}\mathbf{+}{\mathit{\beta }}_{\mathbf{2}}{\mathit{x}}_{\mathbf{2}} \\ \mathit{b}\mathbf{.}\mathbf{}\mathit{E}\mathbf{\left(}\mathit{y}\mathbf{\right)}\mathbf{=}{\mathit{\beta }}_{\mathbf{0}}\mathbf{+}{\mathit{\beta }}_{\mathbf{1}}{\mathit{x}}_{\mathbf{1}} \\ \mathit{c}\mathbf{.}\mathit{E}\mathbf{\left(}\mathit{y}\mathbf{\right)}\mathbf{=}{\mathit{\beta }}_{\mathbf{0}}\mathbf{+}{\mathit{\beta }}_{\mathbf{1}}{\mathit{x}}_{\mathbf{1}}\mathbf{+}{\mathit{\beta }}_{\mathbf{2}}{{\mathit{x}}_{\mathbf{1}}}^{\mathbf{2}} \\ \mathit{d}\mathbf{.}\mathit{E}\mathbf{\left(}\mathit{y}\mathbf{\right)}\mathbf{=}{\mathit{\beta }}_{\mathbf{0}}\mathbf{+}{\mathit{\beta }}_{\mathbf{1}}{\mathit{x}}_{\mathbf{1}}\mathbf{+}{\mathit{\beta }}_{\mathbf{2}}{\mathit{x}}_{\mathbf{2}}\mathbf{+}{\mathit{\beta }}_{\mathbf{3}}{\mathit{x}}_{\mathbf{1}}{\mathit{x}}_{\mathbf{2}} \\ \mathit{e}\mathbf{.}\mathbf{}\mathit{E}\mathbf{\left(}\mathit{y}\mathbf{\right)}\mathbf{=}{\mathit{\beta }}_{\mathbf{0}}\mathbf{+}{\mathit{\beta }}_{\mathbf{1}}{\mathit{x}}_{\mathbf{1}}\mathbf{+}{\mathit{\beta }}_{\mathbf{2}}{\mathit{x}}_{\mathbf{2}}\mathbf{+}{\mathit{\beta }}_{\mathbf{3}}{\mathit{x}}_{\mathbf{1}}{\mathit{x}}_{\mathbf{2}}\mathbf{+}{\mathit{\beta }}_{\mathbf{4}}{{\mathbf{x}}^{\mathbf{2}}}_{\mathbf{1}}\mathbf{+}{\mathit{\beta }}_{\mathbf{5}}{{\mathbf{x}}_{\mathbf{2}}}^{\mathbf{2}} \end{gathered}$$

Short answer

Answer

a. Option a is a nested model in two variables for option d and option e.

b. Option b is a nested model for all of the options.

c. Option c is a nested model for option e and a complete model for option b.

d. Option d is a nested model for option e and a complete model for option a.

e. Option e is a complete model for all of the options.

Step-by-step solution

Nested models 

Option a is a nested model in two variables for option d and option e.

Fixed Representation 

Option b is a nested model for all of the options.

Enclosed reproduction

Option c is a nested model for option e and a complete model for option b.

Ingrained Imitation

Option d is a nested model for option e and a complete model for option a.

Impacted duplicate

Option e is a complete model for all of the options.