ch06 - 6-1 Introduction To Empirical Models6-1 Introduction...

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Unformatted text preview: 6-1 Introduction To Empirical Models6-1 Introduction To Empirical Models6-1 Introduction To Empirical Models6-1 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 straight-line 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 xis6-1 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 Y-values at each xand that the variance of this distribution is the same at each x.6-1 Introduction To Empirical Models6-1 Introduction To Empirical Models6-1 Introduction To Empirical ModelsA Multiple Regression Model:6-1 Introduction To Empirical Models6-1 Introduction To Empirical Models6-2 Simple Linear Regression6-2.1 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 model6-2 Simple Linear Regression6-2.1 Least Squares Estimation Suppose that we have n pairs of observations (x1, y1), (x2, y2), , (xn, yn).6-2 Simple Linear Regression6-2.1 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 6-6.6-2 Simple Linear Regression6-2.1 Least Squares EstimationUsing Equation 6-8,the nobservations in the sample can be expressed asThe sum of the squares of the deviations of the observations from the true regression line is6-2 Simple Linear Regression6-2.1 Least Squares Estimation6-2 Simple Linear Regression6-2.1 Least Squares Estimation6-2....
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ch06 - 6-1 Introduction To Empirical Models6-1 Introduction...

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