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Chapter 12: Q154SE (page 810)

Question: Factors identifying urban counties. The Professional Geographer (February 2000) published a study of urban and rural counties in the western United States. Six independent variables鈥攖otal county population (x1), population density (x2), population concentration (x3), population growth(x4), proportion of county land in farms (x5,) and 5-year change in agricultural land base (x6)鈥攚ere used to model the urban/rural rating (y) of a county, where rating was recorded on a scale of 1 (most rural) to 10 (most urban). Prior to running the multiple regression analysis, the researchers were concerned about possible multicollinearity in the data. Below is a correlation matrix for data collected oncounties.

a. Based on the correlation matrix, is there any evidence of extreme multicollinearity?

b. The first-order model with all six independent variables was fit, and the results are shown in the next table. Based on the reported tests, is there any evidence of extreme multicollinearity?

Short Answer

Expert verified

Answer

a. From the correlation matrix, it can be seen that all the values are below 0.20. the negative sign just indicates the inverse relationship between the variables. But the correlation value for x1 and x2 is 0.45 and for x3 and x2 is 0.43 which is not very alarming but still indicate moderate multicollinearity.

b. The overall model p-value is less than 0.001 indicating the overall adequacy of the model however, the p-values of some coefficients like x2 x4 ,x6 and are very insignificant at 0.230, 0.860, and 0.580 which might indicate that the variables might be correlated and that there might be multicollinearity in the model.

Step by step solution

01

Given Information

There are six independent variables county population (x1), population density (x2), population concentration (x3), population growth(x4), farm land (x5), and agricultural change (x6).

02

Interpretation of correlation matrix 

From the correlation matrix, it can be seen that all the values are below 0.20. the negative sign just indicates the inverse relationship between the variables. But the correlation value for x1 and x2 is 0.45 and for x3 and x2 is 0.43 which is not very alarming but still indicate moderate multicollinearity.

03

Evidence for multicollinearity

The overall model p-value is less than 0.001 indicating the overall adequacy of the model however, the p-values of some coefficients like x2 x4 ,x6 and are very insignificant at 0.230, 0.860, and 0.580 which might indicate that the variables might be correlated and that there might be multicollinearity in the model.

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Most popular questions from this chapter

Question: Predicting elements in aluminum alloys. Aluminum scraps that are recycled into alloys are classified into three categories: soft-drink cans, pots and pans, and automobile crank chambers. A study of how these three materials affect the metal elements present in aluminum alloys was published in Advances in Applied Physics (Vol. 1, 2013). Data on 126 production runs at an aluminum plant were used to model the percentage (y) of various elements (e.g., silver, boron, iron) that make up the aluminum alloy. Three independent variables were used in the model: x1 = proportion of aluminum scraps from cans, x2 = proportion of aluminum scraps from pots/pans, and x3 = proportion of aluminum scraps from crank chambers. The first-order model, , was fit to the data for several elements. The estimates of the model parameters (p-values in parentheses) for silver and iron are shown in the accompanying table.

(A) Is the overall model statistically useful (at 伪 = .05) for predicting the percentage of silver in the alloy? If so, give a practical interpretation of R2.

(b)Is the overall model statistically useful (at a = .05) for predicting the percentage of iron in the alloy? If so, give a practical interpretation of R2.

(c)Based on the parameter estimates, sketch the relationship between percentage of silver (y) and proportion of aluminum scraps from cans (x1). Conduct a test to determine if this relationship is statistically significant at 伪 = .05.

(d)Based on the parameter estimates, sketch the relationship between percentage of iron (y) and proportion of aluminum scraps from cans (x1). Conduct a test to determine if this relationship is statistically significant at 伪 = .05.

Impact of race on football card values. University of Colorado sociologists investigated the impact of race on the value of professional football players鈥� 鈥渞ookie鈥� cards (Electronic Journal of Sociology, 2007). The sample consisted of 148 rookie cards of National Football League (NFL) players who were inducted into the Football Hall of Fame. The price of the card (in dollars) was modeled as a function of several qualitative independent variables: race of player (black or white), card availability (high or low), and player position (quarterback, running back, wide receiver, tight end, defensive lineman, linebacker, defensive back, or offensive lineman).

  1. Create the appropriate dummy variables for each of the qualitative independent variables.
  2. Write a model for price (y) as a function of race. Interpret the尾鈥檚 in the model.
  3. Write a model for price (y) as a function of card availability. Interpret the尾鈥檚 in the model.
  4. Write a model for price (y) as a function of position. Interpret the尾鈥檚 in the model.

Question: Risk management performance. An article in the International Journal of Production Economics (Vol. 171, 2016) investigated the factors associated with a firm鈥檚 supply chain risk management performance (y). Five potential independent variables (all measured quantitatively) were considered: (1) firm size, (2) supplier orientation, (3) supplier dependency, (4) customer orientation, and (5) systemic purchasing. Consider running a stepwise regression to find the best subset of predictors for risk management performance.

a. How many 1-variable models are fit in step 1 of the stepwise regression?

b. Assume supplier orientation is selected in step 1. How many 2-variable models are fit in step 2 of the stepwise regression?

c. Assume systemic purchasing is selected in step 2. How many 3-variable models are fit in step 3 of the stepwise regression?

d. Assume customer orientation is selected in step 3. How many 4-variable models are fit in step 4 of the stepwise regression?

e. Through the first 4 steps of the stepwise regression, determine the total number of t-tests performed. Assuming each test uses an a = .05 level of significance, give an estimate of the probability of at least one Type I error in the stepwise regression.

Question: Suppose you fit the regression modelE(y)=尾0+尾1x1+尾2x2+尾3x1+尾4x12+尾5x22to n = 30 data points and wish to test H0: 尾3 = 尾4 = 尾5 = 0

a. State the alternative hypothesis Ha.

b. Give the reduced model appropriate for conducting the test.

c. What are the numerator and denominator degrees of freedom associated with the F-statistic?

d. Suppose the SSE鈥檚 for the reduced and complete models are SSER = 1,250.2 and SSEC = 1,125.2. Conduct the hypothesis test and interpret the results of your test. Test using 伪 = .05.

Buy-side vs. sell-side analysts鈥� earnings forecasts. Refer to the Financial Analysts Journal (July/August 2008) comparison of earnings forecasts of buy-side and sell-side analysts, Exercise 2.86 (p. 112). The Harvard Business School professors used regression to model the relative optimism (y) of the analysts鈥� 3-month horizon forecasts. One of the independent variables used to model forecast optimism was the dummy variable x = {1 if the analyst worked for a buy-side firm, 0 if the analyst worked for a sell-side firm}.

a) Write the equation of the model for E(y) as a function of type of firm.

b) Interpret the value of尾0in the model, part a.

c) The professors write that the value of尾1in the model, part a, 鈥渞epresents the mean difference in relative forecast optimism between buy-side and sell-side analysts.鈥� Do you agree?

d) The professors also argue that 鈥渋f buy-side analysts make less optimistic forecasts than their sell-side counterparts, the [estimated value of尾1] will be negative.鈥� Do you agree?
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