Chapter 24 Building Regression Models Copyright © 2011 Pearson Education, Inc 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) 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 of 46 Copyright © 2011 Pearson Education, Inc 24.1 Identifying Explanatory Variables The Initial Model – Scatterplot Association appears linear, two outliers identified of 46 Copyright © 2011 Pearson Education, Inc 24.1 Identifying Explanatory Variables The Initial Model – Timeplot of Residuals Locates outliers in time (Dec 1999 and Apr 2003) No evidence of dependence of 46 Copyright © 2011 Pearson Education, Inc 24.1 Identifying Explanatory Variables The Initial Model – Regression Results of 46 Copyright © 2011 Pearson Education, Inc 24.1 Identifying Explanatory Variables The Initial Model – Residual Plot Aside from the two outliers, residuals have similar variances of 46 Copyright © 2011 Pearson Education, Inc 24.1 Identifying Explanatory Variables The Initial Model – Check Normality Aside from the two outliers, residuals are nearly normal of 46 Copyright © 2011 Pearson Education, Inc 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 4M Example 24.1: MARKET SEGMENTATION Mechanics – Estimation Results 32 of 46 Copyright © 2011 Pearson Education, Inc 4M Example 24.1: MARKET SEGMENTATION Mechanics – Examine Plots MRM conditions are satisfied 33 of 46 Copyright © 2011 Pearson Education, Inc 4M 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 4M 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 to 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 4M 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 4M 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 4M Example 24.2: RETAIL PROFITS Method Several high correlations are present (shaded in table) and indicate the presence of collinearity 38 of 46 Copyright © 2011 Pearson Education, Inc 4M 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 4M Example 24.2: RETAIL PROFITS Mechanics – Estimation Results 40 of 46 Copyright © 2011 Pearson Education, Inc 4M Example 24.2: RETAIL PROFITS Mechanics – Examine Plots These and other plots (not shown here) indicate that all MRM conditions are satisfied 41 of 46 Copyright © 2011 Pearson Education, Inc 4M 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 4M 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 4M 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 ... 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... 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... explanatory variable and measures the effect of collinearity  The VIF for x j is VIF ( x j )  1 Rj 21 of 46 Copyright © 2011 Pearson Education, Inc 24. 2 Collinearity Results for Sony Stock Value