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Unformatted text preview: Econ 508: Multiple Regression Juan Fung MSPE March 4, 2011 Juan Fung (MSPE) Econ 508: Multiple Regression March 4, 2011 1 / 20 The multiple regression model Multiple influences We can extend the simple regression model Y i = β 1 + β 2 X i + i , to capture other influences on the behavior of Y , Y i = β 1 + β 2 X i + β 3 Z i + i . Y is the dependent variable; X , Z are the independent variables; is a random error; and i is the index for the i th observation. Juan Fung (MSPE) Econ 508: Multiple Regression March 4, 2011 2 / 20 The multiple regression model Multiple influences The model, Y i = β 1 + β 2 X i + β 3 Z i + i , hypothesizes a linear relationship between the dependent variable, Y , and the independent variables, X , Z . As in the single regression model, the objective is to explain the mean behavior of Y in terms of 1. A deterministic component: the conditional mean, E [ Y i  X i , Z i ] = β 1 + β 2 X i + β 3 Z i , and 2. A stochastic component: the error, i . This is familiar, but interpretations differ. For example, the equation is no longer a line but a plane. What does a slope coefficient mean in this case? Juan Fung (MSPE) Econ 508: Multiple Regression March 4, 2011 3 / 20 The multiple regression model More generally... You may have k independent variables, X 1 , .. . , X k , plus intercept, Y i = β + β 1 X 1 , i + ··· + β k X k , i + i , where X 1 , i is the i th observation of the variable X 1 , etc. Matrix notation The model can be written more compactly using matrices, Y = X β + , where Y = Y 1 . . . Y n , = 1 . . . n are n × 1 vectors; β = β . . . β k is a ( k + 1) × 1 vector; and X is a n × ( k + 1) matrix. Juan Fung (MSPE) Econ 508: Multiple Regression March 4, 2011 4 / 20 The multiple regression model Y = X β + Thus, in matrix notation the regression model is Y 1 . . . Y n = X 10 ··· X 1 k . . . . . . . . . X n ··· X nk β . . . β k + 1 . . . n , where X ik is the i th observation of the k th regressor, and X i = 1 , ∀ i , represents the constant term....
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 Spring '08
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 Regression Analysis, multiple regression model, Juan Fung

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