Linear Regression Analysis

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Linear Regression Analysis

• Paskorn _at_ DssG

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Regression Analysis

• Regression analysis is a statistical tool for the
  investigation of relationships between variables.
• E.g, the effect of a price increase to demand
• The relationship between the mean value of a
  random variable (e.g., product sales) and the
  corresponding values of one or more independent
  variables (e.g. promotion expenditures).

sales
Promotion expenditure
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Regression Analysis Techniques

• There are several regression analysis models.
• simple linear regression model
• 1 random variable.
• 1 independent variable.
• multiple linear regression model
• 1 random variable (e.g., product sales).
• Many independent variables (promotion
  expenditure, product color)
• Nonlinear model
• The relationship of a random variable and
  independent variables is not linear.

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Simple Linear Regression Model

• The most common form of estimation equation in
  regression analysis is a linear relationship
  between a random variable and an independent
  variable.

Y a ßX
Y dependent (random) variable a interception
on Y-axis ß slope
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Assumptions

• Assumption 1 The value of the dependent variable
  Y is claimed to be a random variable, which is
  dependent on fixed (non random) values of the
  independent variable X
• Assumption 2 A theoretical straight-line
  relationship exists between X and the expected
  value of Y ( E(Y) )for each of the possible
  values of X.

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yi yi ei
yi theoretical value
yi a ßxi ei
ei error
E(e) 0
a
yi observed value
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Estimating the Population Regression Coefficients

• The population regression coefficients a and ß
  are estimated by using n pairs (population) of
  observations (x1,y1),(x2,y2) (xn,yn)
• The following process find a sample regression
  line that best fits the sample of observations
  (the number of observations is less than the
  number of population)
• The sample estimates of a and ß can be
  represented as a and b respectively.

yi a bxi
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Finding a and b

• The process to find a and b is called ordinary
  least square (OLS) process method.
• This OLS minimize the sum of the squares of
  errors.

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Example
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