General Studies · Regression Analysis
An analyst fits a multiple linear regression model Yi = β0 + β1X1i + ... + β4X4i + εi, i = 1, 2, ..., n, using the method of least squares. However, the coordinator insists that the intercept and two regressors Z1 = X1 + X3 and Z2 = X2 - X4 are enough to represent the model. Suppose the coordinator's claim can be tested in the form of a general linear hypothesis, viz., H0: LTβ = 0 against HA: H0 is not true, where βT = (β0, β1, ..., β4). Suppose we have n observations on the response Y and each regressor. Further, assume that the errors with or without restrictions are independent N(0, σ2) variables. Then, which of the following statements is true?
Verified Solution4 OptionsUpdated: October 2026👁️1 View
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- 1A possible choice of LT is 0 1 0 -1 00 0 1 0 1 matrix transposition representation equivalent to testing β1 = β3 and β2 = -β4.✓
- 2The test statistic for testing H0 against HA follows an F-distribution with (2, n - 4) degrees of freedom under H0.
- 3Sum of squares residuals under the restrictions LTβ = 0 follows σ2χ2n-3 distribution.
- 4Sum of squares residuals without the restrictions LTβ = 0 follows a non-central σ2χ2n-4 distribution.
Correct answer
A possible choice of LT is 0 1 0 -1 00 0 1 0 1 matrix transposition representation equivalent to testing β1 = β3 and β2 = -β4.
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