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**Unformatted text preview: **What is regression? Regression : modelling expectation (or mean or average) Assumes expectation of a single response (or dependent) variable is a function of one or more other variables Important: intent of the regression analysis determines the content of the regression analysis Lecture 2 – p. 1/32 Purposes of regression Prediction Covariate selection (variable screening) Model specification Parameter estimation Lect u Simple linear regression Basic, review, but very important Building block for future concepts Can visualize the complete regression problem easily Lecture 2 – p. 3/32 The problem Data comes from Example 2.1 in the text Explanatory variable (or covariate of interest) is the accounting rate on a stock (represents input into investment) Dependent variable (or response variable) is the marke t rate (or return) Lect u Accounting data-10 10 20 30 40 5 10 15 20 ACCOUNTINGRATE MARKETRATE Lecture 2 – p. 5/32 Accounting data Goal: Build a model to predict the market rate using the accounting rate and to make inference about the quantitative nature of their relationship How do we do this? Important: State the model and the assumptions Lect u SLR model Let y be the variable representing the response, x be the variable representing the covariate of interest Simple linear regression assumes y = β + β 1 x + ǫ where β is the intercept, β 1 is the slope and ǫ is the model error Usually index by i = 1 , ..., n , where y i = β + β 1 x i + ǫ i Lecture 2 – p. 7/32 Basic assumptions One conditions on the x i values (i.e. we treat x as constant, rather than as a random variable) and assumes the error around the measurement of x is negligible ǫ i are uncorrelated random variables with mean E ( ǫ i ) = 0 and variance V ar ( ǫ i ) = σ 2 Note: variance does not depend on i or X ! y i are random variables such that: E ( y i | x i ) = β + β 1 x i V ar ( y i | x i ) = σ 2 Lect u Centered model Computationally speaking, sometimes useful to center covariates Can write: y i = β * + β 1 ( x i- ¯ x ) + ǫ i where β * = β + β 1 ¯ x Lecture 2 – p. 9/32Lecture 2 – p....

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