Statistics with Economics and Business Applications.ppt

Statistics with Economics and Business Applications Chapter 12 Multiple Regression Analysis A brief exposition Introduction We can use the same basic ideas in simple linear regression to analyze relationships between a dependent variable and several independent variables Multiple regression is an extension of the simple linear regression for investigating how a response y is affected by several independent variables, x1, x2, x3,…, xk. Our objective are find relationships between y and x1, x2, x3,…, xk predict y using x1, x2, x3,…, xk Example Fatness (y) may depend on x1 = age x2 = sex x3 = body type Monthly sales (y) of the retail store may depend on x1 = advertising expenditure x2 = time of year x3 = state of economy x4 = size of inventory Some Questions Which of the independent variables are useful and which are not? How could we create a prediction equation to allow us to predict y using knowledge of x1, x2, x3 etc? How strong is the relationship between y and the independent variables? How good is this prediction? The General Linear Model y = b0+ b1x1 + b2x2 +…+ bkxk + e y is the dependent variable. b0, b1, b2,..., bk are unknown parameters x1, x2,..., xk are independent predictor variables The deterministic part of the model, E(y) = b0+ b1x1 + b2x2 +…+ bkxk , describes average value of y for any fixed values of x1, x2,..., xk . The observation y deviates from the deterministic model by an amount e. e is random error. We assume random errors are independent normal random variables with mean zero and a constant variance s2 The Method of Least Squares Data: n observations on the response y and the independent variables, x1, x2, x3, …xk. The best-fitting prediction equation is We choose our estimates to minimize The computation is usually done by a computer Steps in Regression Analysis Example Minitab Output Minitab Output Minitab Output Minitab Output Historical Note Where does the name “regression”