Ch06 - -1 Introduction To Empirical Models-1 Introduction To Empirical Models-1 Introduction To Empirical Models-1 Introduction To Empirical Models

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Unformatted text preview: -1 Introduction To Empirical Models-1 Introduction To Empirical Models-1 Introduction To Empirical Models-1 Introduction To Empirical Models Based 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 model is given by where ε is the random error term. We think of the regression model as an empirical model. Suppose that the mean and variance of ε are 0 and σ 2 , respectively, then The variance of Y given x is-1 Introduction To Empirical Models • The true regression model is a line of mean values: where β 1 can be interpreted as the change in the mean of Y for a unit change in x . • Also, the variability of Y at a particular value of x is determined by the error variance, σ 2 . • This implies there is a distribution of Y-values at each x and that the variance of this distribution is the same at each x .-1 Introduction To Empirical Models-1 Introduction To Empirical Models-1 Introduction To Empirical Models A Multiple Regression Model:-1 Introduction To Empirical Models-1 Introduction To Empirical Models-2 Simple Linear Regression 6-2.1 Least Squares Estimation • The case of simple linear regression considers a single regressor or predictor x and a dependent or response variable Y . • The expected value of Y at each level of x is a random variable: • We assume that each observation, Y , can be described by the model-2 Simple Linear Regression 6-2.1 Least Squares Estimation • Suppose that we have n pairs of observations ( x 1 , y 1 ), (x 2 , y 2 ), …, ( x n , y n ). • The method of least squares is used to estimate the parameters, β 0 and β 1 by minimizing the sum of the squares of the vertical deviations in Figure 6-6.-2 Simple Linear Regression 6-2.1 Least Squares Estimation • Using Equation 6-8, t he n observations in the sample can be expressed as • The sum of the squares of the deviations of the observations from the true regression line is-2 Simple Linear Regression 6-2.1 Least Squares Estimation-2 Simple Linear Regression 6-2.1 Least Squares Estimation-2 Simple Linear Regression 6-2.1 Least Squares Estimation-2 Simple Linear Regression 6-2.1 Least Squares Estimation-2 Simple Linear Regression 6-2.1 Least Squares Estimation6-2....
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This note was uploaded on 12/25/2010 for the course ALL 0204 taught by Professor 79979 during the Spring '10 term at National Chiao Tung University.

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Ch06 - -1 Introduction To Empirical Models-1 Introduction To Empirical Models-1 Introduction To Empirical Models-1 Introduction To Empirical Models

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