Question

(2) Suppose the original regression is given by y = β0 + β1x1 + β2x2 + β3x3 + u. You want to test for heteroscedasticity using F test. What auxiliary regression should you run? What is the

null hypothesis you need to test?

Answer 1

Given the regression function Y = β0 + β1X1 + β2X2, ..., +βkXk we wish to test the hypothesis: H0 : (βk−q+1 = 0), ..., (βk = 0) Against the alternative H1 : at least one of the restricted parameters are different from zero. 1. Regress y on the restricted set of independent variables and save the residuals u˜ 2. Regress u˜ on all of the independent variables and obtain the R2 , name it R2 u , using the underscore u to distinguish this from the R2 from the population regression function this is known as the auxiliary regression 3. Compute LM = nR2 u , that is the sample size multiplied by the R2 from the regression of the residuals on all of the independent variables 4. Compare the LM to the appropriate critical values, c, in a χ 2 q distribution Rejection Rule: If LM > c we are unable to reject the alternative hypothesisGiven the regression function Y = β0 + β1X1 + β2X2, ..., +βkXk we wish to test the hypothesis: H0 : (βk−q+1 = 0), ..., (βk = 0) Against the alternative H1 : at least one of the restricted parameters are different from zero. 1. Regress y on the restricted set of independent variables and save the residuals u˜ 2. Regress u˜ on all of the independent variables and obtain the R2 , name it R2 u , using the underscore u to distinguish this from the R2 from the population regression function this is known as the auxiliary regression 3. Compute LM = nR2 u , that is the sample size multiplied by the R2 from the regression of the residuals on all of the independent variables 4. Compare the LM to the appropriate critical values, c, in a χ 2 q distribution Rejection Rule: If LM > c we are unable to reject the alternative hypothesis