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Chapter 12: Q143SE (page 807)

To model the relationship between y, a dependent variable, and x, an independent variable, a researcher has taken one measurement on y at each of three different x-values. Drawing on his mathematical expertise, the researcher realizes that he can fit the second-order model Ey=β0+β1x+β2x2 and it will pass exactly through all three points, yielding SSE = 0. The researcher, delighted with the excellent fit of the model, eagerly sets out to use it to make inferences. What problems will he encounter in attempting to make inferences?

Short Answer

Expert verified

The researcher has fit a second-order model to the data and taken one measurement on y at 3 different values for x. The SSE value is coming out to be zero indicating that the data points lie perfectly on the line. However, since the SSE value is zero, the researcher might face issues while testing the significance of beta parameters or the goodness of fit test for the model.

Step by step solution

01

Second-order model

The researcher has fit a second-order model to the data and taken one measurement on y at 3 different values for x. The SSE value is coming out to be zero indicating that the data points lie perfectly on the line

02

Problems with second-order model

However, since the SSE value is zero, the researcher might face issues while testing the significance of beta parameters or the goodness of fit test for 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 α = .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 α = .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 α = .05.

Question: Bordeaux wine sold at auction. The uncertainty of the weather during the growing season, the phenomenon that wine tastes better with age, and the fact that some vineyards produce better wines than others encourage speculation concerning the value of a case of wine produced by a certain vineyard during a certain year (or vintage). The publishers of a newsletter titled Liquid Assets: The International Guide to Fine Wine discussed a multiple regression approach to predicting the London auction price of red Bordeaux wine. The natural logarithm of the price y (in dollars) of a case containing a dozen bottles of red wine was modelled as a function of weather during growing season and age of vintage. Consider the multiple regression results for hypothetical data collected for 30 vintages (years) shown below.

  1. Conduct a t-test (atα=0.05 ) for each of the βparameters in the model. Interpret the results.
  2. When the natural log of y is used as a dependent variable, the antilogarithm of a b coefficient minus 1—that is ebi - 1—represents the percentage change in y for every 1-unit increase in the associated x-value. Use this information to interpret each of the b estimates.
  3. Interpret the values of R2and s. Do you recommend using the model for predicting Bordeaux wine prices? Explain

Question: Manipulating rates of return with stock splits. Some firms have been accused of using stock splits to manipulate their stock prices before being acquired by another firm. An article in Financial Management (Winter 2008) investigated the impact of stock splits on long-run stock performance for acquiring firms. A simplified version of the model fit by the researchers follows:

E(y)=β0+β1x1+β2x2+β3x1x2

where

y = Firm’s 3-year buy-and-hold return rate (%)

x1 = {1 if stock split prior to acquisition, 0 if not}

x2 = {1 if firm’s discretionary accrual is high, 0 if discretionary accrual is low}

a. In terms of the β’s in the model, what is the mean buy and- hold return rate (BAR) for a firm with no stock split and a high discretionary accrual (DA)?

b. In terms of the β’s in the model, what is the mean BAR for a firm with no stock split and a low DA?

c. For firms with no stock split, find the difference between the mean BAR for firms with high and low DA. (Hint: Use your answers to parts a and b.)

d. Repeat part c for firms with a stock split.

e. Note that the differences, parts c and d, are not the same. Explain why this illustrates the notion of interaction between x1 and x2.

f. A test for H0: β3 = 0 yielded a p-value of 0.027. Using α = .05, interpret this result.

g. The researchers reported that the estimated values of both β2 and β3 are negative. Consequently, they conclude that “high-DA acquirers perform worse compared with low-DA acquirers. Moreover, the underperformance is even greater if high-DA acquirers have a stock split before acquisition.” Do you agree?

Question: Failure times of silicon wafer microchips. Refer to the National Semiconductor study of manufactured silicon wafer integrated circuit chips, Exercise 12.63 (p. 749). Recall that the failure times of the microchips (in hours) was determined at different solder temperatures (degrees Celsius). The data are repeated in the table below.

  1. Fit the straight-line modelEy=β0+β1xto the data, where y = failure time and x = solder temperature.

  2. Compute the residual for a microchip manufactured at a temperature of 149°C.

  3. Plot the residuals against solder temperature (x). Do you detect a trend?

  4. In Exercise 12.63c, you determined that failure time (y) and solder temperature (x) were curvilinearly related. Does the residual plot, part c, support this conclusion?

Question: Failure times of silicon wafer microchips. Researchers at National Semiconductor experimented with tin-lead solder bumps used to manufacture silicon wafer integrated circuit chips (International Wafer-Level Packaging Conference, November 3–4, 2005). The failure times of the microchips (in hours) were determined at different solder temperatures (degrees Celsius). The data for one experiment are given in the table. The researchers want to predict failure time (y) based on solder temperature (x).

  1. Construct a scatterplot for the data. What type of relationship, linear or curvilinear, appears to exist between failure time and solder temperature?
  2. Fit the model,E(y)=β0+β1x+β2x2, to the data. Give the least-squares prediction equation.
  3. Conduct a test to determine if there is upward curvature in the relationship between failure time and solder temperature. (use.α=0.05)
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