Statistics for business decision making and analysis robert stine and foster chapter 24

24.1 Identifying Explanatory VariablesWhat explanatory variables belong in a regression model for stock returns?. 24.1 Identifying Explanatory VariablesThe Initial Model  Build a model

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)

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

24.1 Identifying Explanatory Variables

The Initial Model – Scatterplot

Association appears linear, two outliers identified

24.1 Identifying Explanatory Variables

The Initial Model – Timeplot of Residuals

Locates outliers in time (Dec 1999 and Apr 2003)

No evidence of dependence

24.1 Identifying Explanatory Variables

The Initial Model – Regression Results

24.1 Identifying Explanatory Variables

The Initial Model – Residual Plot

Aside from the two outliers, residuals have similar

24.1 Identifying Explanatory Variables

The Initial Model – Check Normality

Aside from the two outliers, residuals are nearly

normal.

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

p-value less than 0.0001

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)

24.1 Identifying Explanatory Variables

Correlation Matrix

24.1 Identifying Explanatory Variables

Scatterplot Matrix

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

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

24.1 Identifying Explanatory Variables

MRM with All Four Explanatory Variables

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

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

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

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

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

2

1

1 )

24.2 Collinearity

Results for Sony Stock Value Example

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

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.

24.2 Collinearity

Signs of Collinearity (Continued)

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

 Variance inflation factors increase

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

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

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.

4M 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.

4M Example 24.1:

MARKET SEGMENTATION

Mechanics – Estimation Results

4M Example 24.1:

MARKET SEGMENTATION

Mechanics – Examine Plots

MRM conditions are satisfied.

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

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

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.

4M Example 24.2: RETAIL PROFITS

Method

Use multiple regression The response

variable is profit Examine the correlation

matrix and the scatterplot matrix.

4M Example 24.2: RETAIL PROFITS

Method

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

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.

4M Example 24.2: RETAIL PROFITS

Mechanics – Estimation Results

4M Example 24.2: RETAIL PROFITS

Mechanics – Examine Plots

These and other plots (not shown here) indicate

that all MRM conditions are satisfied

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

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.

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.

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