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Chapter 12: 121E (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 x2has 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 x2and 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 x2has 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 x2and x4 and x5 respectively. Thus, the problem of multicollinearity exists in the model.

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Most popular questions from this chapter

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?

Question:If the analysis of variance F-test leads to the conclusion that at least one of the model parameters is nonzero, can you conclude that the model is the best predictor for the dependent variable ? Can you conclude that all of the terms in the model are important for predicting ? What is the appropriate conclusion?

Suppose the mean value E(y) of a response y is related to the quantitative independent variables x1and x2

E(y)=2+x1-3x2-x1x2

a) Identify and interpret the slope forx2

b) Plot the linear relationship between E(y) andx2for role="math" localid="1649796003444" x1=0,1,2, whererole="math" localid="1649796025582" 1≤x2≤3

c) How would you interpret the estimated slopes?

d) Use the lines you plotted in part b to determine the changes in E(y) for eachrole="math" localid="1649796051071" x1=0,1,2.

e) Use your graph from part b to determine how much E(y) changes whenrole="math" localid="1649796075921" 3≤x1≤5androle="math" localid="1649796084395" 1≤x2≤3.

Suppose you fit the quadratic model E(y)=β0+β1x+β2x2to a set of n = 20 data points and found R2=0.91, SSyy=29.94, and SSE = 2.63.

a. Is there sufficient evidence to indicate that the model contributes information for predicting y? Test using a = .05.

b. What null and alternative hypotheses would you test to determine whether upward curvature exists?

c. What null and alternative hypotheses would you test to determine whether downward curvature exists?

Suppose you used Minitab to fit the model y=β0+β1x1+β2x2+ε

to n = 15 data points and obtained the printout shown below.

  1. What is the least squares prediction equation?

  2. Find R2and interpret its value.

  3. Is there sufficient evidence to indicate that the model is useful for predicting y? Conduct an F-test using α = .05.

  4. Test the null hypothesis H0: β1= 0 against the alternative hypothesis Ha: β1≠ 0. Test using α = .05. Draw the appropriate conclusions.

  5. Find the standard deviation of the regression model and interpret it.
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