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Unformatted text preview: 61 Introduction To Empirical Models61 Introduction To Empirical Models61 Introduction To Empirical Models61 Introduction To Empirical ModelsBased on the scatter diagram, it is probably reasonable to assume that the mean of the random variable Y is related to x by the following straightline relationship:where the slope and intercept of the line are called regression coefficients.The simple linear regression modelis given bywhere is the random error term.We think of the regression model as an empirical model.Suppose that the mean and variance of are 0 and 2, respectively, thenThe variance of Ygiven xis61 Introduction To Empirical ModelsThe true regression model is a line of mean values:where 1can be interpreted as the change in the mean of Yfor a unit change in x.Also, the variability of Y at a particular value of xis determined by the error variance, 2.This implies there is a distribution of Yvalues at each xand that the variance of this distribution is the same at each x.61 Introduction To Empirical Models61 Introduction To Empirical Models61 Introduction To Empirical ModelsA Multiple Regression Model:61 Introduction To Empirical Models61 Introduction To Empirical Models62 Simple Linear Regression62.1 Least Squares EstimationThe case of simple linear regression considers a single regressoror predictorx and a dependentor response variableY.The expected value of Yat each level of xis a random variable:We assume that each observation, Y, can be described by the model62 Simple Linear Regression62.1 Least Squares Estimation Suppose that we have n pairs of observations (x1, y1), (x2, y2), , (xn, yn).62 Simple Linear Regression62.1 Least Squares EstimationThe method of least squaresis used to estimate the parameters, and 1by minimizing the sum of the squares of the vertical deviations in Figure 66.62 Simple Linear Regression62.1 Least Squares EstimationUsing Equation 68,the nobservations in the sample can be expressed asThe sum of the squares of the deviations of the observations from the true regression line is62 Simple Linear Regression62.1 Least Squares Estimation62 Simple Linear Regression62.1 Least Squares Estimation62....
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 Fall '09
 Kamman

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