ECON 3300 unit 3

ECON 3300 unit 3 - 1 Unit Three: Linear Regression Analysis...

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Unit Three: Linear Regression Analysis Linear regression analysis is a widely-used technique for describing the relationship between a quantitative variable Y (the dependent variable) and one or more other variables X 1 , X 2 , …, X p (the independent or predictor variables), at least one of which is quantitative. In a simple linear regression analysis, there is exactly one independent variable; in a multiple linear regression analysis, there are two or more independent variables. Linear regression analysis has various purposes, including: (1) to make predictions (example: A large retail department chain wants to assess whether it could predict, with reasonable certainty and precision, the first-year sales (Y) of a proposed retail store based on the proposed location (X 1 = 1 if mall, 0 if shopping center) and measures of population (X 2 ), per capita income (X 3 ), and competition (X 4 ) within the service area.) (2) to estimate impacts (example: A manufacturer of ceramic vases wants to estimate the impact of increasing the temperature (X 1 ) within the kiln, or increasing the pressure (X 2 ) applied when clay is injected into the mold, on the breaking strength (Y) of its ceramic vases.) (3) to test hypotheses (example: A researcher wants to test the hypothesis that the end-of-year price (Y) of the S&P500 would be positively related to the mid-year percentage of employed persons in the 45-to-64 year old group (X 1 ) after controlling for bond yields (X 2 ), GDP (X 3 4 ), and the maximum capital gains tax rate (X 5 ).) (4) to set standards (example: The manager of a landscaping company wants to estimate the relationship between the time (Y) it would take an employee to perform a landscaping job and the number of years of experience (X 1 ) of the employee as well as various characteristics of the job or site, including the number of trees to be planted (X 2 ), the number of bushes to be planted (X 3 ), and the condition of the terrain (X 4 = 1 if rocky, 0 otherwise) at the site.) Simple Linear Regression The (classical normal) simple linear regression model is expressed in the form Y = ß 0 + ß 1 X + ε , where: Y and X are quantitative variables ß 0 and ß 1 are real numbers. ε is called the error (or stochastic or random error) term. It represents the combined influence on Y of all factors other than X. Note : Examples of quantitative variables include such “plain” variables as Sales and Age and such transformed variables as ln(Assets) and Income . Assumptions of the model include (with x denoting any value in the presumed domain of X): Across all potential entities* with X = x : 1. E( ε ) = 0 ( E(Y) = ß 0 + ß 1 x). We will call this the linearity assumption . It follows from this assumption that an entity with
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ECON 3300 unit 3 - 1 Unit Three: Linear Regression Analysis...

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