Copyright © 2011 Pearson Education, Inc. Building Regression Models Chapter 24

Copyright © 2011 Pearson Education, Inc.

Building Regression Models

Chapter 24

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)


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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


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The Initial Model – Scatterplot

Association appears linear, two outliers identified.


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The Initial Model – Timeplot of Residuals

Locates outliers in time (Dec. 1999 and Apr. 2003).No evidence of dependence.


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The Initial Model – Regression Results


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The Initial Model – Residual Plot

Aside from the two outliers, residuals have similar variances.


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The Initial Model – Check Normality

Aside from the two outliers, residuals are nearly normal.


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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 p-value less than 0.0001.


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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).


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Correlation Matrix


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Scatterplot Matrix


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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.


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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.


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MRM with All Four Explanatory Variables


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Residual Plot: Residuals vs. Fitted Values

Outliers are still present; however, this and other residual plots show the conditions for MRM are satisfied.


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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.


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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.


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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.


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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


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24.2 Collinearity

Results for Sony Stock Value Example


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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.


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24.2 Collinearity

Signs of Collinearity

R2 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.


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24.2 Collinearity

Signs of Collinearity (Continued)

Standard errors for partial slopes are larger than those for marginal slopes.

Variance inflation factors increase.


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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.


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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).


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4M 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.


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Method

Use multiple regression with ratings as the response and age and income as the explanatory variables. Examine the correlation matrix and scatterplot matrix.


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Method

There is a high correlation between age and income that implies collinearity.


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Method

Association is linear with no outliers.


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Mechanics – Estimation Results


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Mechanics – Examine Plots

MRM conditions are satisfied.


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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.


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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.


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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.


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Method

Use multiple regression. The response variable is profit. Examine the correlation matrix and the scatterplot matrix.


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Method

Several high correlations are present (shaded in table) and indicate the presence of collinearity.


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Method

This partial scatterplotmatrix identifies communities that aredistinct from others.

Linearity and no lurking variables conditions are met.


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Mechanics – Estimation Results


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Mechanics – Examine Plots

These and other plots (not shown here) indicate that all MRM conditions are satisfied.


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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.


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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.


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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.


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Best Practices

Begin a regression analysis by looking at plots.

Use the F-statistic for the overall model and a t-statistic for each explanatory variable.

Learn to recognize the presence of collinearity.

Don’t fear collinearity – understand it.


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Pitfalls

Do not remove explanatory variables at the first sign of collinearity.

Don’t remove several explanatory variables from your model at once.


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Documents

Copyright © 2011 Pearson Education, Inc. Building Regression Models Chapter 24