Chapter 24 Building Regression Models Copyright 2011 Pearson

24. 1 Identifying Explanatory Variables What explanatory variables belong in a regression model for stock returns? § Initial model motivated by theory such as CAPM § Seek additional variables that improve fit and produce better predictions § The process is typically complicated by correlated explanatory variables (i. e. , collinearity) 3 of 46 Copyright © 2011 Pearson Education, Inc.

24. 1 Identifying Explanatory Variables The Initial Model § Build a model that describes returns on Sony stock § CAPM provides a theoretical starting point: use % change for the whole stock market as an explanatory variable 4 of 46 Copyright © 2011 Pearson Education, Inc.

24. 1 Identifying Explanatory Variables The Initial Model – Scatterplot Association appears linear, two

24. 1 Identifying Explanatory Variables The Initial Model – Timeplot of Residuals Locates outliers in time (Dec. 1999 and Apr. 2003). No evidence of dependence. 6 of 46 Copyright © 2011 Pearson Education, Inc.

24. 1 Identifying Explanatory Variables The Initial Model – Regression Results 7 of 46

24. 1 Identifying Explanatory Variables The Initial Model – Residual Plot Aside from the

24. 1 Identifying Explanatory Variables The Initial Model – Check Normality Aside from the

24. 1 Identifying Explanatory Variables The Initial Model – Proceed to Inference § Estimates are consistent with CAPM. § The estimated intercept is not significantly different from zero with a p-value of 0. 6964. § The estimated slope is highly significant with a pvalue less than 0. 0001. 10 of 46 Copyright © 2011 Pearson Education, Inc.

24. 1 Identifying Explanatory Variables Identifying Other Variables § Research in finance suggests other variables, should be added to the initial model. § Three of these variables are: percentage change in the DJIA (Dow % Change) and differences in performance between small and large companies (Small-Big) and between growth and value stocks (High-Low). 11 of 46 Copyright © 2011 Pearson Education, Inc.

24. 1 Identifying Explanatory Variables Correlation Matrix 12 of 46 Copyright © 2011 Pearson

24. 1 Identifying Explanatory Variables Scatterplot Matrix 13 of 46 Copyright © 2011 Pearson

24. 1 Identifying Explanatory Variables Identifying Other Variables § The correlation matrix indicates that percentage changes in the DJIA and in the whole market index are highly correlated. § The scatterplot matrix indicates that the association between the response and these variables appear linear. 14 of 46 Copyright © 2011 Pearson Education, Inc.

24. 1 Identifying Explanatory Variables Adding Explanatory Variables § The data consist of 168 observations with four candidate explanatory variables. § Begin model building by including all four variables in the multiple regression model. 15 of 46 Copyright © 2011 Pearson Education, Inc.

24. 1 Identifying Explanatory Variables MRM with All Four Explanatory Variables 16 of 46

24. 1 Identifying Explanatory Variables Residual Plot: Residuals vs. Fitted Values Outliers are still present; however, this and other residual plots show the conditions for MRM are satisfied. 17 of 46 Copyright © 2011 Pearson Education, Inc.

24. 1 Identifying Explanatory Variables MRM with All Four Explanatory Variables § The F-statistic is 21. 59 with p-value of 0. 0001; this multiple regression equation explains statistically significant variation in percentage changes in the value of Sony stock. § Based on the t-statistics, only the variable Small. Big improves a regression that contains all of the other explanatory variables. 18 of 46 Copyright © 2011 Pearson Education, Inc.

24. 1 Identifying Explanatory Variables MRM with All Four Explanatory Variables § Adding other explanatory variables to the initial model alters the slope for Market % Change. § This once important variable is no longer statistically significant in explaining percentage changes in the value of Sony stock. 19 of 46 Copyright © 2011 Pearson Education, Inc.

24. 2 Collinearity Marginal and Partial Slopes § There is a high correlation between Market % Change and Dow % Change (r = 0. 89). § This collinearity produces imprecise estimates of the partial slopes. § It explains the difference between the marginal and partial slopes for Market % Change. 20 of 46 Copyright © 2011 Pearson Education, Inc.

24. 2 Collinearity Variance Inflation Factor (VIF) § Variance inflation factor: quantifies the amount of unique variation in each explanatory variable and measures the effect of collinearity. § The VIF for is 21 of 46 Copyright © 2011 Pearson Education, Inc.

24. 2 Collinearity Results for Sony Stock Value Example 22 of 46 Copyright ©

24. 2 Collinearity Results for Sony Stock Value Example § Is High-Low not statistically significant because it is redundant or simply unrelated to the response? § Because it has a VIF near 1, collinearity has little effect on this variable (not redundant). § Generally, VIF > 5 or 10 suggests redundancy. 23 of 46 Copyright © 2011 Pearson Education, Inc.

24. 2 Collinearity Signs of Collinearity § R 2 increases less than we’d expect. § Slopes of correlated explanatory variables in the model change dramatically. § The F-statistic is more impressive than individual t-statistics. 24 of 46 Copyright © 2011 Pearson Education, Inc.

24. 2 Collinearity Signs of Collinearity (Continued) § Standard errors for partial slopes are larger than those for marginal slopes. § Variance inflation factors increase. 25 of 46 Copyright © 2011 Pearson Education, Inc.

24. 2 Collinearity Remedies for Collinearity § Remove redundant explanatory variables. § Re-express explanatory variables (e. g. , use the average of Market % Change and Dow % Change as an explanatory variable). § Do nothing if the explanatory variables are significant with sensible estimates. 26 of 46 Copyright © 2011 Pearson Education, Inc.

24. 3 Removing Explanatory Variables Issues § After adding several explanatory variables to a model, some of those added and some of those originally present may not be statistically significant. § Remove those variables for which both statistics and substance indicate removal (e. g. , remove Dow % Change rather than Market % Change). 27 of 46 Copyright © 2011 Pearson Education, Inc.

4 M Example 24. 1: MARKET SEGMENTATION Motivation Within which magazine should a manufacturer of a new mobile phone advertise? One has an older audience. They collect consumer ratings on the new phone design along with consumers’ ages and reported incomes. 28 of 46 Copyright © 2011 Pearson Education, Inc.

4 M Example 24. 1: MARKET SEGMENTATION Method Use multiple regression with ratings as the response and age and income as the explanatory variables. Examine the correlation matrix and scatterplot matrix. 29 of 46 Copyright © 2011 Pearson Education, Inc.

4 M Example 24. 1: MARKET SEGMENTATION Method There is a high correlation between

4 M Example 24. 1: MARKET SEGMENTATION Method Association is linear with no outliers.

4 M Example 24. 1: MARKET SEGMENTATION Mechanics – Estimation Results 32 of 46

4 M Example 24. 1: MARKET SEGMENTATION Mechanics – Examine Plots MRM conditions are

4 M Example 24. 1: MARKET SEGMENTATION Mechanics The F-statistic has a p-value of < 0. 0001. The model explains statistically significant variation in the ratings. Although collinear, both predictors (age and income) are statistically significant. 34 of 46 Copyright © 2011 Pearson Education, Inc.

4 M Example 24. 1: MARKET SEGMENTATION Message The manufacturer should advertise in the magazine with younger subscribers. Based on the 95% confidence interval for the slope of Age, an affluent audience that is younger by 20 years assigns, on average, ratings that are 1 to 2 points higher than the older, affluent audience. Age changes sign when adjusted for differences in income. Substantively, this makes sense because younger customers with money find the new design attractive. 35 of 46 Copyright © 2011 Pearson Education, Inc.

4 M Example 24. 2: RETAIL PROFITS Motivation A chain of pharmacies is looking to expand into a new community. It has data for 110 cities on the following variables: income, disposable income, birth rate, social security recipients, cardiovascular deaths and percentage of local population aged 65 or more. 36 of 46 Copyright © 2011 Pearson Education, Inc.

4 M Example 24. 2: RETAIL PROFITS Method Use multiple regression. The response variable is profit. Examine the correlation matrix and the scatterplot matrix. 37 of 46 Copyright © 2011 Pearson Education, Inc.

4 M Example 24. 2: RETAIL PROFITS Method Several high correlations are present (shaded

4 M Example 24. 2: RETAIL PROFITS Method This partial scatterplot matrix identifies communities that are distinct from others. Linearity and no lurking variables conditions are met. 39 of 46 Copyright © 2011 Pearson Education, Inc.

4 M Example 24. 2: RETAIL PROFITS Mechanics – Estimation Results 40 of 46

4 M Example 24. 2: RETAIL PROFITS Mechanics – Examine Plots These and other

4 M Example 24. 2: RETAIL PROFITS Mechanics The F-statistic indicates that this collection of explanatory variables explains statistically significant variation in profits. The VIF’s indicate some explanatory variables are redundant and should be removed (one at a time) from the model. 42 of 46 Copyright © 2011 Pearson Education, Inc.

4 M Example 24. 2: RETAIL PROFITS Mechanics – Simplified Model This multiple regression separates the effects of birth rates from age (and income). It reveals that cities with higher birth rates produce higher profits when compared to cities with lower birth rates but comparable income and local population above 65. 43 of 46 Copyright © 2011 Pearson Education, Inc.

4 M Example 24. 2: RETAIL PROFITS Message Three characteristics of the local community affect estimated profits: disposable income, age and birth rates. Increases in each of these lead to higher profits. The data show that the pharmacy chain will have to trade off these characteristics in selecting a site for expansion. 44 of 46 Copyright © 2011 Pearson Education, Inc.

Best Practices § Begin a regression analysis by looking at plots. § Use the F-statistic for the overall model and a tstatistic for each explanatory variable. § Learn to recognize the presence of collinearity. § Don’t fear collinearity – understand it. 45 of 46 Copyright © 2011 Pearson Education, Inc.

Pitfalls § Do not remove explanatory variables at the first sign of collinearity. § Don’t remove several explanatory variables from your model at once. 46 of 46 Copyright © 2011 Pearson Education, Inc.