A Second Course in Statistics Regression Analysis William Mendenhall Terry Sincich Seventh Edition......

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1 A Second Course in Statistics Regression Analysis William Mendenhall Terry Sincich Seventh Edition......

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3 and TPF&C uses multiple regression models to accomplish this salary evaluation. The Compensation Management Service, as TPF&C calls it, measures both the internal and external consistency of a company s pay policies to determine whether they reflect the management s intent. The dependent variable y used to measure executive compensation is annual salary. The independent variables used to explain salary structure include the variables listed in Table 5. The management at TPF&C has found that executive compensation models that use the natural logarithm of salary as the dependent variable are better predictors than models that use salary as the dependent variable. This is probably because salaries tend to be incremented in percentages rather than dollar values. Thus, the multiplicative model we propose (in its linear form) is ln(y) = β 0 + β 1 x 1 + β 2 x 2 + β 3 x 3 + β 4 x 4 + β 5 x 5 + β 6 x1 2 + β 7x 3 x 4 + ε We have included a second-order term, x1 2, to account for a possible curvilinear relationship between ln(salary) and years of experience, x 1. Also, the interaction term x 3 x 4 is included to account for the fact that the relationship between the number of employees supervised, x 4, and corporate salary may depend on gender, x 3. For example, as the number of supervised employees increases, a male s salary (with all other factors being equal) might rise more rapidly than a female s. (If this is found to be true, the firm will take steps to remove the apparent discrimination against female executives.) EXECSAL Table 5 List of independent variables for executive compensation example Independent Variable Description x 1 x 2 x 3 x 4 x 5 Years of experience Years of education 1 if male; 0 if female Number of employees supervised Corporate assets (millions of dollars) x 6 x 2 1 x 7 x 3 x 4 A sample of 100 executives is selected and the variables y and x 1,x 2,...,x 5 are recorded. (The data are saved in the CD file named EXECSAL.) The multiplicative model is fit to the data using MINITAB, with the results shown in the MINITAB printout, Figure 20. (a) Find the least squares prediction equation, and interpret the estimate of β 2. (b) Locate the estimate of s and interpret its value. (c) Locate Ra 2 and interpret its value. (d) Conduct a test of overall model utility using α =.05. (e) Is there evidence of gender discrimination at the firm? Test using α =.05. (f) Use the model to predict the salary of an executive with the characteristics shown in Table

4 Figure 20 MINITAB multiple regression output for Example 10 Table 6 Values of independent variables for a particular executive x 1 = 12 years of experience x 2 = 16 years of education x 3 = 0(female) x 4 = 400 employees supervised x 5 = $160 million (the firm s asset value) x1 2 = 144 x 3 x 4 = 0 Solution (a) The least squares model (highlighted on the MINITAB printout) is ln(y) = x x x x x x x 7 Because we are using the logarithm of salary as the dependent variable, the β estimates have different interpretations than previously discussed. In general, a parameter β in a multiplicative (log) model represents the percentage 221

5 increase (or decrease) in the dependent variable for a 1-unit increase in the corresponding independent variable. The percentage change is calculated by taking the antilogarithm of the β estimate and subtracting 1 (i.e., e ˆβ 1). For example, the percentage change in executive compensation associated with a 1-unit (i.e., 1-year) increase in years of education x 2 is (e ˆβ 2 1) = (e ) =.031. Thus, when all other independent variables are held constant, we estimate executive salary to increase 3.1% for each additional year of education. A Multiplicative (Log) Model Relating y to Several Independent Variables ln(y) = β 0 + β 1 x 1 + β 2 x 2 + +β k x k + ε where ln(y) = natural logarithm of y Interpretation of β s (e β i 1) 100% = Percentage change in y for every 1 unit increase in x i, holding all other x s fixed (b) The estimate of the standard deviation σ (shaded on the printout) is s =.066. Our interpretation is that most of the observed ln(y) values (logarithms of salaries) lie within 2s = 2(.066) =.132 of their least squares predicted values. A more practical interpretation (in terms of salaries) is obtained, however, if we take the antilog of this value and subtract 1, similar to the manipulation in part a. That is, we expect most of the observed executive salaries to lie within e 2s 1 = e =.141, or 14.1% of their respective least squares predicted values. (c) The adjusted R 2 value (highlighted on the printout) is Ra 2 =.936. This implies that, after taking into account sample size and the number of independent variables, almost 94% of the variation in the logarithm of salaries for these 100 sampled executives is accounted for by the model. (d) The test for overall model utility is conducted as follows: H 0 : β 1 = β 2 = =β 7 = 0 H a : At least one of the model coefficients is nonzero. Mean square for model Test statistic: F = = (shaded in Figure 20) MSE p-value =.000 (shaded in Figure 20) Since α =.05 exceeds the p-value of the test, we conclude that the model does contribute information for predicting executive salaries. It appears that at least one of the β parameters in the model differs from 0. (e) If the firm is (knowingly or unknowingly) discriminating against female executives, then the mean salary for females (denoted μ F ) will be less than the The result is derived by expressing the percentage change in salary y, as(y 1 y 0 )/y 0,wherey 1 = the value of y when, say, x = 1, and y 0 = the value of y when x = 0. Now let y = ln(y) and assume the log model is y = β 0 + β 1 x.then { y = e y = e β 0 e β1x e β 0 when x = 0 = e β 0 e β 1 when x = 1 Substituting, we have y 1 y 0 y 0 = eβ0 e β1 e β0 e β 0 = e β

6 mean salary for males (denoted μ M ) with the same qualifications (e.g., years of experience, years of education, etc.) From our previous discussion of dummy variables, this difference will be represented by β 3,theβ coefficient multiplied by x 3 if we set number of employees supervised, x 4,equalto0.Sincex 3 = 1 if male, 0 if female, then β 3 = (μ M μ F ) for fixed values of x 1,x 2 and x 5,and x 4 = 0. Consequently, a test of H 0 : β 3 = 0 versus H a : β 3 > 0 is one way to test the discrimination hypothesis. The p-value for this onetailed test is one-half the p-value shown on the MINITAB printout (i.e.,.002/2 =.001). With such a small p-value, there is strong evidence to reject H 0 and claim that some form of gender discrimination exists at the firm. A test for discrimination could also include testing for the interaction term, β 7 x 3 x 4. If, as the number of employees supervised (x 4 ) increases, the rate of increase in ln(salary) for males exceeds the rate for females, then β 7 > 0. To see this, hold x 1,x 2,andx 5 constant in the model (e.g., x 1 = 10 years of experience, x 2 = 15 years of education, and x 5 = $120 million in corporate assets), then substitute x 3 = 0 into the equation to obtain: x 3 = 0 (Females): E{ln(y)} =β 0 + β 1 (10) + β 2 (15) + β 5 (120) + β 6 (15 2 ) + β 3 (0) + β 4 (x 4 ) + β 7 (0)(x 4 ) = β 0 + β 1 (10)+ β 2 (15)+ β 5 (120)+ β 6 (15 2 ) + β }{{} 4 (x }{{} 4 ) y-intercept (constant) slope Similarly, substituting x 3 = 1 into the equation we obtain: x 3 = 1 (Males): E{ln(y)} =β 0 + β 1 (10) + β 2 (15) + β 5 (120) + β 6 (15 2 ) + β 3 (1) + β 4 (x 4 ) + β 7 (1)(x 4 ) = β 0 + β 1 (10) + β 2 (15) + β 5 (120) + β 6 (15 2 ) + β }{{} 3 y-intercept (constant) + (β 4 + β 7 ) x }{{} 4 slope Thus, the slope for males (β 4 + β 7 ) exceeds the slope for females (β 4 ) only if β 7 > 0. The one-tailed p-value for testing H 0 : β 7 = 0 against H a : β 7 > 0(highlighted on the printout) is.002/2 =.001. Consequently, we reject H 0 and find further evidence of gender discrimination at the firm. (f) The least squares model can be used to obtain a predicted value for the logarithm of salary. Substituting the values of the x s shown in Table 6, we obtain ln(y) = ˆβ 0 + ˆβ 1 (12) + ˆβ 2 (16) + ˆβ 3 (0) + ˆβ 4 (400) + ˆβ 5 (160x) + ˆβ 6 (144) + ˆβ 7 (0) This predicted value is given at the bottom of the MINITAB printout, Figure 20, ln(y) = The 95% prediction interval, from to , is also highlighted on the printout. To predict the salary of an executive with these characteristics, we take the antilog of these values. That is, the predicted salary is e = $80, 992 (rounded to the nearest dollar) and the 95% prediction interval is from e to e (or from $70,834 to 223

7 $92,596). Thus, an executive with the characteristics in Table 6 should be paid between $70,834 and $92,596 to be consistent with the sample data. Warning To decide whether a log transformation on the dependent variable is necessary, naive researchers sometimes compare the R 2 values for the two models y = β 0 + β 1 x 1 + +β k x k + ε and ln(y) = β 0 + β 1 x 1 + +β k x k + ε and choose the model with the larger R 2.ButtheseR 2 values are not comparable since the dependent variables are not the same! One way to generate comparable R 2 values is to calculate the predicted values, ln(y), for the log model and then compute the corresponding ŷ values using the inverse transformation ŷ = êln(y). A pseudo-r 2 for the log model can then be calculated in the usual way: Rln(y) 2 = 1 (y i ŷ i ) 2 (y i ȳ i ) 2 R 2 ln(y) is now comparable to the R2 for the untransformed model. See Maddala (1988) for a discussion of more formal methods for comparing the two models. 12 Exercises 43 First-order model. Write a first-order linear model relating the mean value of y, E(y), to (a) two quantitative independent variables (b) four quantitative independent variables 44 Second-order model. Write a complete secondorder linear model relating the mean value of y, E(y), to (a) two quantitative independent variables (b) three quantitative independent variables 45 Qualitative predictors. Write a model relating E(y) to a qualitative independent variable with (a) two levels, A and B (b) four levels, A, B, C, and D Interpret the β parameters in each case. 46 Graphing a first-order model. Consider the firstorder equation y = 1 + 2x 1 + x 2 (a) Graph the relationship between y and x 1 for x 2 = 0, 1, and 2. (b) Are the graphed curves in part a first-order or second-order? (c) How do the graphed curves in part a relate to each other? (d) If a linear model is first-order in two independent variables, what type of geometric relationship will you obtain when E(y) is graphed as a function of one of the independent variables for various values of the other independent variable? 47 Graphing a first-order model. Consider the firstorder equation y = 1 + 2x 1 + x 2 3x 3 (a) Graph the relationship between y and x 1 for x 2 = 1andx 3 = 3. (b) Repeat part a for x 2 = 1andx 3 = 1. (c) If a linear model is first-order in three independent variables, what type of geometric relationship will you obtain when E(y) is graphed as a function of one of the independent variables for various values of the other independent variables? 48 Graphing a second-order model. Consider the second-order model y = 1 + x 1 x 2 + 2x1 2 + x2 2 (a) Graph the relationship between y and x 1 for x 2 = 0, 1, and 2. (b) Are the graphed curves in part a first-order or second-order? (c) How do the graphed curves in part a relate to each other? (d) Do the independent variables x 1 and x 2 interact? Explain. 224