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Ch4 The Multiple Linear Regression Model

Ch4 The Multiple Linear Regression Model. The assumptions underlying the multiple linear regression model； The least-square estimates of parameters and tests; The implication of regression coefficients; Adjusted goodness-of-fit statistic； Standardized coefficients and elasticity.

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Ch4 The Multiple Linear Regression Model

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  1. Ch4 The Multiple Linear Regression Model • The assumptions underlying the multiple linear regression model； • The least-square estimates of parameters and tests; • The implication of regression coefficients; • Adjusted goodness-of-fit statistic； • Standardized coefficients and elasticity.

  2. 1. The Multiple Regression Model • General form of multiple regression model: • The simplest form: three-variable model:

  3. 1. Assumptions of Model • The relationship between Y and X is linear. • The X’s are nonstochastic variables. • No exact linear relationship exists between two or more independent variables. • The error has zero expected value: • The error term has constant variance for all observations, i.e. • Errors corresponding to different observations are statistically independent. Thus, • The error term is normally distributed.

  4. 2. Estimates of Parameters • The multiple linear regression model • Estimate Criterion:

  5. 3. Test of Parameters • Gauss-Markov theorem is still valid for multiple regression model given assumptions. • To test the null hypothesis:

  6. 4. Goodness-of-Fit (Review) TSS = RSS + ESS

  7. 5. Several Problems Using • is sensitive to the number of independent variables included in the regression model; • When the intercept is 0, the use and interpretation of could be difficult. • All statistical results are dependent on the implied assumption: the model is correct;

  8. 6. Adjusted • Adjusted account for the number of degrees of freedom. • The relationship between and:

  9. Properties of • If k=1, then . • If k is greater 1, then . • can be negative. • always increases when new variables are added to regression model, while is not.

  10. 7.Testing the Significance of • To test the null hypothesis:

  11. Note Even though none of the regression coefficients are found to be significant according to individual t tests, the F test of the significance of regression equation may allow for rejection of the null hypothesis.

  12. Regression statistics in the interpretation of the model • Standardized coefficients; • Elasticity.

  13. Standardized coefficients • Standardized coefficients describe the relative importance of the independent variables in a multiple regression model; • Standardized process: • Relationship between two coefficients:

  14. Elasticity • Elasticity measures the effect on the dependent variable of a 1 percent change in independent variable; • Calculation: • Elasticity is unit-free and its value may be positive or negative. Large elasticity implies that the dependent variable is very responsive to changes in the independent variables.

  15. Example: Interest Rate • The estimate equation is

  16. Example: US Cobb-Douglas Production Function • Y: US output • K: Capital stock • L: Employment

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