TESTING U- SHAPED, INVERTED U- SHAPED OR …

1 WIEN| 28 MAY 2015 TESTING U- SHAPED , INVERTED U- SHAPED OR other nonlinear RELATIONSHIPS IN business - S nke Albers Professor of Marketing and Innovation K hne Logistics University, Hamburg AGENDA 2 Rationale for U-shape or INVERTED U-shape TESTING for U-shapes Alternative regressions (Weighting inverse to frequency) Alternative functional forms Nonparametric methods: Trend lines in Excel Spline Regressions Kernel regression Local polynomial smoothing AGENDA 3 Rationale for U-shape or INVERTED U-shape TESTING for U-shapes Alternative regressions (Weighting inverse to frequency) Alternative functional forms Nonparametric methods: Trend lines in Excel Spline Regressions Kernel regression Local polynomial smoothing RATIONALE FOR U- SHAPE 4 Operations Cost curves are typically assumed u- SHAPED that have somewhere a minimum.

2 CSR Barnett and Salomon (SMJ 2012) found a u- SHAPED relationship between corporate financial performance (ROA) and corporate social performance. Happiness Happiness is u- SHAPED with respect to age (Frijters and Beatton, 2012). Barnett Michael L. and Robert M. Salomon (2012): Does it pay to be really good? Addressing the shape of the relationship between social and financial performance, Strategic Management Journal, 33 (11), 1304-1320. Frijters, Paul and Tony Beatton (2012): The mystery of the U- SHAPED relationship between happiness and age, Journal of Economic Behavior & Organization, 82 (2/3), 525-542. RATIONAL FOR INVERTED U- SHAPE 5 Profit Financial Performance depends on sales and costs and thus must exhibit a maximum. Sales Manchanda and Chintagunta (MarkLett 2004) argue that a sales response function might exhibit a super-saturation where sales declines after a certain point because of too much selling effort.

3 Manchanda, Puneet and Pradeep K. Chintagunta (2004): Responsiveness of Physician Prescription Behavior to Salesforce Effort: An Individual Level Analysis, Marketing Letters, 15 (2-3), 129-145. SATISFACTION LOYALTY FUNCTIONAL FORMS AND MODERATING FACTORS 6 Functional Form Characteristics Existence Moderating factors Moderating factors Product category and market characteristics Customer economic and demographic variables 1 Linear Ngobo (1999) Anderson (1994) Homburg and Giering (2001) Streukens and Ruyter (2004) Seiders et al. (2005) Magi (2003) Seiders et al. (2005) 2 Concave Jones and Sasser (1995) None None Ngobo (1999) 3 Convex Jones and Sasser (1995) None None Keiningham et al. (2003) 4 S- SHAPED Ngobo (1999) None None 5 Inverse S- SHAPED Homburg et al. (2005) None None Keiningham et al. (2003) nonlinear (undefined) Mittal and Kamakura (2001) None Mittal and Kamakura (2001) Dong, Songting, Min Ding, Rajdeep Grewal, and Ping Zhao (2011): Functional forms of the satisfaction loyalty relationship , International Journal of Research in Marketing, 28 (1), 38 50 AGENDA 7 Rationale for U-shape or INVERTED U-shape TESTING for U-shapes Alternative regressions (Weighting inverse to frequency) Alternative functional forms Nonparametric methods: Trend lines in Excel Spline Regressions Kernel regression Local polynomial smoothing TYPICAL TEST FOR U- SHAPE 8 In most empirical work trying to identify U shapes, the researcher includes a nonlinear (usually quadratic) term in an otherwise standard regression model.

4 If this term is significant and, in addition, the estimated extremum is within the data range, it is common to conclude that there is a U- SHAPED relationship . Jo Thori Lind and Halvor Mehlum (2010): With or Without U? The Appropriate Test for a U- SHAPED relationship , OXFORD BULLETIN OF ECONOMICS AND STATISTICS, 72 (1), 109-118 QUADRATIC REGRESSION LEADS TO SIGNIFICANT relationship BUT DOES NOT REPRESENT SHAPE 9 y = 72,798x0,3664 R = 0,2345 y = 80,032ln(x) + 11,537 R = 0,2062 y = -0,011x2 + 3,2774x + 183,76 R = 0,1973 0,00100,00200,00300,00400,00500,00600,00 700,00020406080100120140160180200 Sales QUADRATIC REGRESSION FOR EXAMPLE DATA 10 Regression Statistics Multiple R 0,44417807 R Square 0,19729416 Adjusted R Square 0,19568392 Standard Error 95,7654713 Observations 1000 ANOVA df SS MS F Significance F Regression 2 2247350,79 1123675,4 122,524509 2,6386E-48 Residual 997 9143512,42 9171,0255 Total 999 11390863,2 Coefficients Standard Error t Stat P-value Optimum Intercept 183,760289 10,1702193 18,0684686 2,4001E-63 148,99 Budget 3,27737825 0,32226047 10,1699667 3,4904E-23 Budget2 -0,01099834 0,00235221 -4,67575754 3.

5 3313E-06 TESTING OF SIGNIFICANCE OF QUADRATIC TERM 11 TESTING the significance of the quadratic term might be misleading because the quadratic and linear term are highly correlated and therefore we get non-essential correlation which leads to inflated standard errors. A better test is whether the additional variance of the introduction of a quadratic term is significant. other ways of avoiding are transforming the polynomial terms. TRANSFORMING QUADRATIC TERMS 12 It is recommended to mean-center the original and quadratic variables which removes non-essential correlation and thus leads to meaningful standard errors. However, the interpretation of the coefficients is changing. Another alternative is to orthogonalize the polynomial variables. In case of a quadratic term it can be done by regressing the linear term on the quadratic term and using only the residuals which are orthogonal to the linear term.

6 In case of a polynomial of degree larger than two the Orthpoly command in Stata can be used in order to get a set of orthogonal variables. RESULTS OF DIFFERENT TRANSFORMATIONS OF THE QUADRATIC TERM 13 Regression Statistics df SS MS Multiple R 0,44417807 Regression 2 2247350,79 1123675,395 R Square 0,19729416 Residual 997 9143512,42 9171,025498 Adjusted R Square 0,19568392 Total 999 11390863,2 Standard Error 95,7654713 F Significance F Observations 1000 122,5245089 2,6386E-48 Coefficients Standard Error t Stat P-value Intercept 183,760289 10,1702193 18,0684686 0,0000 Budget 3,27737825 0,32226047 10,1699667 0,0000 Budget2 -0,01099834 0,002352205 -4,67575754 0,0000 Coefficients Standard Error t Stat P-value Intercept 317,169735 3,313744506 95,7133944 0,0000 Budget centered 2,2072784 0,143486002 15,3832316 0,0000 Budget2 centered -0,01099834 0,002352205 -4,67575754 0.

7 0000 Coefficients Standard Error t Stat P-value Intercept 218,84814 6,864418523 31,8815263 0,0000 Budget 1,89176875 0,126629319 14,9394213 0,0000 Budget2 orthogonal -0,01099834 0,002352205 -4,67575754 0,0000 QUADRATIC relationship LEADS TO WRONG CONCLUSION 14 y = 0,0101x3 - 0,5157x2 + 45,799x + 21,503 y = 0,1528x2 + 34,691x + 57,834 0100020003000400050006000700080000102030 40506070 Sales per Physician Frequency Moving Average Albers, S nke (2012): Optimizable and Implementable Aggregate Response Modeling for Marketing Decision Support, International Journal of Research in Marketing, 29 (2), 111-122, p. 112 SUFFICIENT TEST FOR U- OR INVERTED U- SHAPE 15 The typical test with a quadratic test only provides a necessary condition for a u-shape but not a sufficient one. Sufficient tests are given when A visual inspection suggests such a relationship (Albers 2012) When the data of the smaller (larger) than optimal values exhibit a negative (positive) slope in the regression (Kostyshak 2015) testable with a utest in Stata Albers, S nke (2012): Optimizable and Implementable Aggregate Response Modeling for Marketing Decision Support, International Journal of Research in Marketing, 29 (2), 111-122 Kostyshak, Scott (2015): Non-parametric TESTING of U- SHAPED Relationships, Working Paper, Princeton U- TEST IN STATA FOR U- OR INVERTED U- SHAPE 16.

8 Utest Budget Budget2 Specification: f(x)=x^2 Extreme point: Te s t : H1: Inverse U shape vs. H0: Monotone or U shape Lower bound Upper bound Interval Slope t-value P>t .0535067 Overall test of presence of a Inverse U shape: t-value = P>t = .0535 QUADRATIC REGRESSION FOR EXAMPLE DATA 17 Regression Statistics Multiple R 0,44417807 R Square 0,19729416 Adjusted R Square 0,19568392 Standard Error 95,7654713 Observations 1000 ANOVA df SS MS F Significance F Regression 2 2247350,79 1123675,4 122,524509 2,6386E-48 Residual 997 9143512,42 9171,0255 Total 999 11390863,2 Coefficients Standard Error t Stat P-value Optimum Intercept 183,760289 10,1702193 18,0684686 2,4001E-63 148,99 Budget 3,27737825 0,32226047 10,1699667 3,4904E-23 Budget2 -0,01099834 0,00235221 -4,67575754 3,3313E-06 y = 4,2087x - 321,88 R = 0,4342 0,00100,00200,00300,00400,00500,00600,00 170,00175,00180,00185,00190,00195,00200.

9 00 AGENDA 18 Rationale for U-shape or INVERTED U-shape TESTING for U-shapes Alternative regressions (Weighting inverse to frequency) Alternative functional forms Nonparametric methods: Trend lines in Excel Spline Regressions Kernel regression Local polynomial smoothing ALTERNATIVE REGRESSION FOR DETECTING SHAPES 19 Although OLS has good properties for predicting the response variable given a random draw of independent variables, these properties do not directly translate to estimating the global shape of a regression function when the distribution of the independent variable is not uniform (Albers 2012, Kostyshak 2015). => A solution is using an inverse frequency weighting with GLS (Albers 2012, Albers 2015, Kostyshak 2015) Albers, S nke (2012): Optimizable and Implementable Aggregate Response Modeling for Marketing Decision Support, International Journal of Research in Marketing, 29 (2), 111-122 Albers, S nke (2015): What Drives Publication Productivity at German Universities?

10 , Schmalenbach business Review, 67 (1, 2015), 6-33 Kostyshak, Scott (2015): Non-parametric TESTING of U- SHAPED Relationships, Working Paper, Princeton INVERSE FREQUENCY WEIGHTED REGRESSION 20 data interval AGENDA 21 Rationale for U-shape or INVERTED U-shape TESTING for U-shapes Alternative regressions (Weighting inverse to frequency) Alternative functional forms Nonparametric methods: Trend lines in Excel Spline Regressions Kernel regression Local polynomial smoothing ALTERNATIVE FUNCTIONAL FORMS 22 Name Function Elasticity Constant Elasticity Semi-logarithmic Diminishing Elasticity Modified Exponential Log-Reciprocal Albers, S nke (2012): Optimizable and Implementable Aggregate Response Modeling for Marketing Decision Support, International Journal of Research in Marketing, 29 (2), 111-12, p.