# 6 - ECMT1020 Summer School 09 Multiple Regression(cont The...

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ECMT1020 Summer School 09 Multiple Regression (cont)

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The Multiple Regression Model Idea: Examine the linear relationship between i ) i ki k 2i 2 1i 1 0 i ε X β X β X β β Y + + + + + = Multiple Regression Model with k Independent Variables: Y-intercept Population slopes Random Error
Multiple Regression Equation The coefficients of the multiple regression model are estimated using sample data ki k 2i 2 1i 1 0 i X b X b X b b Y ˆ + + + + = Estimated (or predicted) value of Y Estimated slope coefficients Multiple regression equation with k independent variables: Estimated intercept In this chapter we will always use Excel to obtain the regression slope coefficients and other regression summary measures.

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Two variable model Y X 1 X 2 2 2 1 1 0 X b X b b Y ˆ + + = S l o p e f r v a i b X 1 lo fo r v ria le 2 Multiple Regression Equation (continued)
Multiple Regression With multiple regression selection of the variables becomes important. The analyst must rely on good theory, research, evidence and experience to ensure that all relevant variables are included. Failure to include all relevant variables may make the ensuing analysis invalid . Testing the variables and the model for relevance and testing of the assumptions behind the model is important. Violation of assumptions may make subsequent analysis invalid.

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Estimation of Regression Models Estimation of multiple regression models is similar to estimation with simple regression (not the chart option). EXCEL or Stat Tools will provide the estimation routine which is a generalisation of line of best fit (Least squares). The analyst needs to ensure as far as possible that all data for all variables are available in the desired format. Missing data for some observations of independent variables means that the entire observation is typically not used.
Testing Regression Models Significance tests: The estimation of the sample model needs to be tested for inferences about the true population model. The significance tests of regression models will be for the overall model and the individual variables. Overall model test: This is a test of whether the entire model selected is a relevant explanatory model for the Y variable. It is conducted using an F test with a statistic provided in estimation output.

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Is the Model Significant? F Test for Overall Significance of the Model Shows if there is a linear relationship between all of the X variables considered together and Y Use F-test statistic Hypotheses: H 0 : β 1 = β 2 = … = β k = 0 (no linear relationship) H 1 : at least one β i ≠ 0 (at least one independent variable affects Y)
F Test for Overall Significance Test statistic: where F has (numerator) = k and (denominator) = (n – k - 1) degrees of freedom 1 k n SSE k SSR MSE MSR F - - = =

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6 - ECMT1020 Summer School 09 Multiple Regression(cont The...

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