Skip to content
Part CCSIR NET June 2025the-intercept-is-a-parameter-so-five-of-them-leave-n-minus-5-not-n-minus-4

The intercept is a parameter so five of them leave n minus 5 not n minus 4

An analyst fits a multiple linear regression model Yi=β0+β1X1i+⋯+β4X4i+εi,i=1,2,…,nY_{i} = \beta_{0} + \beta_{1}X_{1i} + \dots + \beta_{4}X_{4i} + \varepsilon_{i}, i = 1, 2, \dots, n, using the method of least squares. However, his coordinator insists that the intercept and two regressors Z1=X1+X3Z_{1} = X_{1} + X_{3} and Z2=X2−X4Z_{2} = X_{2} - X_{4} 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=0H_{0} : L^{T}\beta = 0 against HA:H0H_A : H_{0} is not true, where βT=(β0,β1,…,β4)\beta^{T} = (\beta_{0}, \beta_{1}, \dots, \beta_{4}). 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)N(0, \sigma^{2}) variables. Then, which of the following statements are true?

  1. A.A possible choice of L is the 5 × 2 matrix with rows (0, 0), (1, 0), (0, 1), (−1, 0), (0, 1)✓ Correct
  2. B.The test statistic for testing H0H_{0} against H_A follows an F-distribution with (2, n − 4) degrees of freedom under H0H_{0}
  3. C.Sum of squares residuals under the restrictions LTβ=0L^{T}\beta = 0 follows σ2χn−32\sigma^{2}\chi^{2}_{n-3} distribution✓ Correct
  4. D.Sum of squares residuals without the restrictions LTβ=0L^{T}\beta = 0 follows a non-central σ2χn−42\sigma^{2}\chi^{2}_{n-4} distribution

You have the answer. Trap Analysis is why the other three were written.

Not a worked solution repeated four times — the specific reasoning error each wrong option was built to reward, and why this one tests execution slip.

See pricing

50 are analysed free — try those first.

The trap it tests

Execution slip

The idea was right. The computation was not.

Drill statements like this

Related counterexample: OLS is the BLUE in every linear model

More on this topic

The chapter behind this: Linear models, Gauss–Markov and ANOVA — free to read

From Linear Models and Multivariate › Gauss–Markov, regression, ANOVA basics

Last revised . Found a mistake? Tell us — corrections are the fastest thing we act on.

ShareWhatsAppTelegram