Mathematics · Linear Regression and Hypothesis Testing
Consider the multiple linear regression model Y = Xβ + ε, where Y is an n × 1 observed data vector with n > 5; X is an n × 5 matrix of known constants with rank(X) = 5; β = (β0, β1, β2, β3, β4)T and ε = (ε1, ..., εn)T, where εi for i = 1, ..., n are independent and identically distributed N(0, 1) random variables. Consider testing the linear hypothesis H0: β1 = β2 = β3 = β4 = c (a known constant) against the alternative H1: H0 is not true. Which of the following statements is true?
Verified Solution4 OptionsUpdated: October 2026👁️1 View
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- 1The sum of squares of residuals under H0 follows a central χ2 distribution with (n - 5) degrees of freedom.
- 2The sum of squares of residuals under H0 follows a central χ2 distribution with (n - 1) degrees of freedom.
- 3The test statistic follows a central F-distribution with (5, n - 1) degrees of freedom.
- 4The test statistic follows a central F-distribution with (4, n - 5) degrees of f reedom.✓
Correct answer
The test statistic follows a central F-distribution with (4, n - 5) degrees of f reedom.
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