Quantitative Business Analysis for Decision Making

1
Quantitative Business Analysis for Decision Making

• Multiple Linear
• Regression
• Analysis

2
Outlines

• Multiple Regression Model
• Estimation
• Testing Significance of Predictors
• Multicollinearity
• Selection of Predictors
• Diagnostic Plots

3
Multiple Regression Model

• Multiple linear regression model
• are slope coefficients of
• X1, X2 , ,Xk.
• quantifies the amount of change in
• response Y for a unit change in Xi when
• all other predictors are held fixed.

4
Multiple Regression Model (cont)

• In the model,
• is the mean of Y.
• Contributes to the variation in Y values from
  their mean , and
• is assumed normally distributed with mean 0
  and standard deviation

5
Sampling

• A random sample of n units is taken. Then for
• each unit k1 measurements are made
• Y, X1 , X2 , ., Xk

6
Estimated Model

• Estimated multiple regression model is
• Expressions for bi are cumbersome to
• write. is an estimate of

7
Standard Error
Sample standard deviation around the mean
(estimated regression model) is It is an
estimate of Standard error of (for
specified values of predictors) is denoted by
8
Testing Significance of a Predictor

• For comparing with a reference ,test
• statistic is
• and for estimating by a confidence
• interval,
• compute

9
Coefficient of Determination

• Coefficient of determination R2 quantifies the
  of
• variation in the Y-distribution that is accounted
  by the
• predictors in the model. If
• R2 80, then 20 variation in the
  Y-distribution is due to factors other than those
  in the model.
• R2 increases as predictors are added in the model
  but at the cost of complicating it.

10
Testing the Model for Significance

• Null hypothesis predictors in the relationship
  have no predictive power to explain the variation
  in Y-distribution
• Test statistic F . It
  has
• F- distribution with k and (n-k-1) degrees of
• freedoms for the numerator and denominator.

11
Multicollinearity and Selection of Predictors

• Multicollinearity - occurs when predictors are
  highly
• correlated among themselves. In its presence R2
  may be high,
• but individual coefficients are less reliable.
• Screening process (e.g. stepwise regression) can
  eliminate
• multicollinearity by selecting only those
  predictors that are not
• strongly correlated among themselves.

12
Diagnostic Plots

• Residuals are used to diagnose
  the validity of the model assumptions.
• A scatter plot of the residuals against the
  predicted values can serve as a diagnostic tool.
• A diagnostic plot can identify outliers, unequal
• variability, and need for transformation to
  achieve
• homogeneity etc.

13
Indicator Variables

• Indicator variables (also called dummy variables)
  are
• numerical codes that are used to represent
  qualitative
• variables.
• For example, 0 for men and 1 for women.
• For a qualitative variable with c categories,
  (c-1) indicator variables need to be defined.