Lecture12-1

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Click to edit Master subtitle style BUAD 310 Applied Business Statistics 11/10/09

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2 A Quick Review The simple linear regression model assumes that the mean of the response variable y depends on an explanatory variable x according to a linear equation For any fixed value of x , response variable y varies normally around this mean and has a SD & that is the same for all values of x x x y 1 0 | β μ + =
3 Multiple Regression Multiple regression : the response variable y depends on k explanatory variables (e.g., three variables) x 1, x 2, x 3, …, xk. The mean response is a function of these variables The observed values of y vary about their means. We can think of subpopulations of responses, each corresponding to a particular set of values for all the explanatory variables x 1, x 2, x 3, …, xk. In each subpopulation y varies normally with a mean given by the population regression equation (*). The SD is the same in all subpopulations. (*) ... 2 2 1 1 0 k k y x x x β μ + + + + =

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4 Simple vs. Multiple Regression Simple regression Data: ( x 1, y 1) ( x 2, y 2) ( xn , yn ) Assumed model: yi = β0 + β1 x i + ε i ε i ~ N (0, σ) Parameters: β0, β1, σ Multiple regression Data: ( x 11, x 12, …, x 1 k , y 1) ( x 21, x 22, …, x 2 k , y 2) ( xn 1, xn 2, …, xnk , yn ) Assumed model: yi = β0 + β1 xi 1 + β2 xi 2 + … + β k xik + ε i ε i ~ N (0, σ) Parameters: β0, β1, β2, …, β k , σ
5 Multiple Linear Regression Model The model is y = µ y | x 1, x 2,…, xk + ε = β0 + β1 x 1 + β2 x 2 + … + β k xk + ε Assumptions for multiple regression are stated about the model error terms, “ ’s

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6 The Multiple Regression Model Idea: Examine the linear relationship between  ε x β x β x β β y k k 2 2 1 1 0 + + + + + = k k 2 2 1 1 0 x b x b x b b y ˆ + + + + = Population model: Y-intercept Population slopes Random Error Estimated  (or predicted)  value of y Estimated slope coefficients Estimated multiple regression model: Estimated intercept
7 Regression Model Assumptions Assumptions about the model error terms, I ’s Mean of Zero Assumption The mean of the error terms is equal to 0 Constant Variance Assumption The variance of the error terms σ2 is the same for every combination of values of x 1, x 2, …, xk Normality Assumption The error terms follow a normal distribution for every combination of values of x 1, x 2, …, xk ±ndependence Assumption The values of the error terms are statistically independent of each other

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Estimating the Coefficients Estimate the model coefficients β0, β1, β2, …, β k from the data by b 0, b 1, b 2, …, bk . The method of least squares chooses the values of the bi ’s that make the sum of squares of the residuals as small as possible, i.e., that minimize ( 29 = - - - - - n i ik k i i i x b x b x b b y 1 2 2 2 1 1 0 ... Note
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