Question

2. Let Yi-Au + β124 + εί, (jz 1,2, . . . ,n), where the εί are independent N(0, σ2). ow that the correlation coefficient ofBo and βί is-n /(nDx (b) Derive an F-statistic for testing H : β-0.

Answer 1

2. Here are the five steps of the overall F-test for regression

1. State the null and alternative hypotheses:
   
   • H₀: β₁ = β₂ = ... = β_p-1 = 0
     
     H₁: β_j ≠ 0, for at least one value of j
2. Compute the test statistic assuming that the null hypothesis is true:
   
   • F = MSM / MSE = (explained variance) / (unexplained variance)
3. Find a (1 - α)100% confidence interval I for (DFM, DFE) degrees of freedom using an F-table or statistical software.
   
4. Accept the null hypothesis if F ∈ I; reject it if F ∉ I.
   
5. Use statistical software to determine the p-value
6. For multiple regression models, we have this remarkable property: SSM + SSE = SST.
   
7. Corrected Degrees of Freedom for Model: DFM = p - 1
   
8. Degrees of Freedom for Error: DFE = n - p
   
9. Corrected Degrees of Freedom Total: DFT = n - 1
       Subtract 1 from n for the corrected degrees of freedom.
       Horizontal line regression is the null hypothesis model.
   
10. For multiple regression models with intercept, DFM + DFE = DFT.
    
11. Mean of Squares for Model: MSM = SSM / DFM
    
12. Mean of Squares for Error: MSE = SSE / DFE
        The sample variance of the residuals.
    
13. Mean of Squares Total: MST = SST / DFT
        The sample variance of the y-variable.
    
14. In general, a researcher wants the variation due to the model (MSM) to be large with respect to the variation due to the residuals (MSE).