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Chapter 12: Q121E (page 802)

Consider fitting the multiple regression model

E(y)= β0+β1x1+ β2x2+β3x3+ β4x4 +β5x5

A matrix of correlations for all pairs of independent variables is given below. Do you detect a multicollinearity problem? Explain


Short Answer

Expert verified

In this question, x4 and x2 has a correlation of 0.93 and x4 and x5 has a correlation of 0.86. These correlation numbers are very high indicating a strong positive relationship between x4 and x2 and x4 and x5 respectively. Thus, the problem of multicollinearity exists in the model.

Step by step solution

01

Multicollinearity check

Multicollinearity is checked by checking the correlation amongst the independent variables. If there is high correlation amongst any two independent variables, it is said that the problem of multicollinearity exists in the model.

02

Application of multicollinearity check

In this question, x4 and x2 has a correlation of 0.93 and x4 and x5 has a correlation of 0.86. These correlation numbers are very high indicating a strong positive relationship between x4 and x2 and x4 and x5 respectively. Thus, the problem of multicollinearity exists in the model.

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Most popular questions from this chapter

Question: The Excel printout below resulted from fitting the following model to n = 15 data points: y=β0+β1x1+β2x2+ε

Where,

x1=(1iflevel20ifnot)x2=(1iflevel30ifnot)

Question: Orange juice demand study. A chilled orange juice warehousing operation in New York City was experiencing too many out-of-stock situations with its 96-ounce containers. To better understand current and future demand for this product, the company examined the last 40 days of sales, which are shown in the table below. One of the company’s objectives is to model demand, y, as a function of sale day, x (where x = 1, 2, 3, c, 40).

  1. Construct a scatterplot for these data.
  2. Does it appear that a second-order model might better explain the variation in demand than a first-order model? Explain.
  3. Fit a first-order model to these data.
  4. Fit a second-order model to these data.
  5. Compare the results in parts c and d and decide which model better explains variation in demand. Justify your choice.

Write a model that relates E(y) to two independent variables—one quantitative and one qualitative at four levels. Construct a model that allows the associated response curves to be second-order but does not allow for interaction between the two independent variables.

Question: Company donations to charity. The amount a company donates to a charitable organization is often restricted by financial inflexibility at the firm. One measure of financial inflexibility is the ratio of restricted assets to total firm assets. A study published in the Journal of Management Accounting Research (Vol. 27, 2015) investigated the link between donation amount and this ratio. Data were collected on donations to 115,333 charities over a recent 10-year period, resulting in a sample of 419,225 firm-years. The researchers fit the quadratic model,E(y)=β0+β1x+β2x2, where y = natural logarithm of total donations to charity by a firm in a year and x = ratio of restricted assets to the firm’s total assets in the previous year. [Note: This model is a simplified version of the actual model fit by the researchers.]

  1. The researchers’ theory is that as a firm’s restricted assets increase, donations will initially increase. However, there is a point at which donations will not only diminish, but also decline as restricted assets increase. How should the researchers use the model to test this theory?
  2. The results of the multiple regression are shown in the table below. Use this information to test the researchers’ theory at. What do you conclude?

Going for it on fourth down in the NFL. Refer to the Chance (Winter 2009) study of fourth-down decisions by coaches in the National Football League (NFL), Exercise 11.69 (p. 679). Recall that statisticians at California State University, Northridge, fit a straight-line model for predicting the number of points scored (y) by a team that has a first-down with a given number of yards (x) from the opposing goal line. A second model fit to data collected on five NFL teams from a recent season was the quadratic regression model, E(y)=β0+β1x+β2x2.The regression yielded the following results: y=6.13+0.141x-0.0009x2,R2=0.226.

a) If possible, give a practical interpretation of each of the b estimates in the model.

b) Give a practical interpretation of the coefficient of determination,R2.

c) In Exercise 11.63, the coefficient of correlation for the straight-line model was reported asR2=0.18. Does this statistic alone indicate that the quadratic model is a better fit than the straight-line model? Explain.

d) What test of hypothesis would you conduct to determine if the quadratic model is a better fit than the straight-line model?
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