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Chapter 14: Problem 11

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

Expert verified
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

01

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: \(y = \beta_{0} + \beta_{1}x_{1} + \beta_{2}x_{2} + \beta_{3}x_{3}\)
02

Part (b): Quadratic terms

Now quadratic terms need to be included. A quadratic term is a variable squared. This gives the model: \(y = \beta_{0} + \beta_{1}x_{1} + \beta_{2}x_{2} + \beta_{3}x_{3} + \beta_{4}x_{1}^{2} + \beta_{5}x_{2}^{2} + \beta_{6}x_{3}^{2}\)
03

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: \(y = \beta_{0} + \beta_{1}x_{1} + \beta_{2}x_{2} + \beta_{3}x_{3} + \beta_{4}x_{1}x_{2}\), \(y = \beta_{0} + \beta_{1}x_{1} + \beta_{2}x_{2} + \beta_{3}x_{3} + \beta_{5}x_{1}x_{3}\), \(y = \beta_{0} + \beta_{1}x_{1} + \beta_{2}x_{2} + \beta_{3}x_{3} + \beta_{6}x_{2}x_{3}\)
04

Part (d): Full Quadratic Model

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

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Most popular questions from this chapter

The article "The Caseload Controversy and the Study of Criminal Courts" (Journal of Criminal Law and Criminology [1979]: 89-101) used a multiple regression analysis to help assess the impact of judicial caseload on the processing of criminal court cases. Data were collected in the Chicago criminal courts on the following variables: $$ \begin{aligned} y &=\text { number of indictments } \\ x_{1} &=\text { number of cases on the docket } \end{aligned} $$ \(x_{2}=\) number of cases pending in criminal court trial system The estimated regression equation (based on \(n=367\) observations) was $$ \hat{y}=28-.05 x_{1}-.003 x_{2}+.00002 x_{3} $$ where \(x_{3}=x_{1} x_{2}\) a. The reported value of \(R^{2}\) was . 16. Conduct the model utility test. Use a \(.05\) significance level. b. Given the results of the test in Part (a), does it surprise you that the \(R^{2}\) value is so low? Can you think of a possible explanation for this? c. How does adjusted \(R^{2}\) compare to \(R^{2}\) ?

The article "Effect of Manual Defoliation on Pole Bean Yield" (Journal of Economic Entomology [1984]: \(1019-1023\) ) used a quadratic regression model to describe the relationship between \(y=\) yield \((\mathrm{kg} /\) plot \()\) and \(x=\mathrm{de}-\) foliation level (a proportion between 0 and 1 ). The estimated regression equation based on \(n=24\) was \(\hat{y}=\) \(12.39+6.67 x_{1}-15.25 x_{2}\) where \(x_{1}=x\) and \(x_{2}=x^{2} .\) The article also reported that \(R^{2}\) for this model was .902. Does the quadratic model specify a useful relationship between \(y\) and \(x ?\) Carry out the appropriate test using a \(.01\) level of significance.

The article "The Influence of Temperature and Sunshine on the Alpha-Acid Contents of Hops" (Agricultural Meteorology [1974]: \(375-382\) ) used a multiple regression model to relate \(y=\) yield of hops to \(x_{1}=\) mean temperature \(\left({ }^{\circ} \mathrm{C}\right)\) between date of coming into hop and date of picking and \(x_{2}=\) mean percentage of sunshine during the same period. The model equation proposed is $$ y=415.11-6060 x_{1}-4.50 x_{2}+e $$ a. Suppose that this equation does indeed describe the true relationship. What mean yield corresponds to a temperature of 20 and a sunshine percentage of \(40 ?\) b. What is the mean yield when the mean temperature and percentage of sunshine are \(18.9\) and 43, respectively? c. Interpret the values of the population regression coefficients.

Suppose that the variables \(y, x_{1}\), and \(x_{2}\) are related by the regression model $$ y=1.8+.1 x_{1}+.8 x_{2}+e $$ a. Construct a graph (similar to that of Figure \(14.5)\) showing the relationship between mean \(y\) and \(x_{2}\) for fixed values 10,20 , and 30 of \(x_{1}\). b. Construct a graph depicting the relationship between mean \(y\) and \(x_{1}\) for fixed values 50,55, and 60 of \(x_{2}\). c. What aspect of the graphs in Parts (a) and (b) can be attributed to the lack of an interaction between \(x_{1}\) and \(x_{2}\) ? d. Suppose the interaction term \(.03 x_{3}\) where \(x_{3}=x_{1} x_{2}\) is added to the regression model equation. Using this new model, construct the graphs described in Parts (a) and (b). How do they differ from those obtained in Parts (a) and (b)?

Explain the difference between a deterministic and a probabilistic model. Give an example of a dependent variable \(y\) and two or more independent variables that might be related to \(y\) deterministically. Give an example of a dependent variable \(y\) and two or more independent variables that might be related to \(y\) in a probabilistic fashion.
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