ABSTRACT
One commonly used statistical approach to assessing whether β2 is a zero vector, found in most text books on multiple linear regression (see e.g., Kleinbaum et al., 1998, Draper and Smith, 1998, and Dielman, 2001), is to test the hypotheses
H0 : β2 = 0 against Ha : β2 = 0. (4.3)
Applying the partial F test (1.7), H0 is rejected if and only if
(SSR of model (4.1)−SSR of model (4.2))/k MS residual of model (4.1)
> f αk,ν (4.4)
where f αk,ν is the upper α point of the F distribution with k and ν = n− (p+ 1) degrees of freedom. Equivalently, from (1.11), H0 is rejected if and only if
(Aβˆ)′(A(X′X)−1A′)−1(Aβˆ)/k MS residual of model (4.1)
where A is a k× (p+1) matrix given by A = (0,Ik). In fact, we have
SSR of model (4.1)−SSR of model (4.2)
= (Aβˆ)′(A(X′X)−1A′)−1(Aβˆ)
from Theorems 1.3 and 1.5. The inferences that can be drawn from this partial F test are that if H0 is
rejected then β2 is deemed to be a non-zero vector and so at least some of the covariates xp−k+1, · · · ,xp affect the response variable Y , and that if H0 is not rejected then there is not enough statistical evidence to conclude that β2 is not equal to zero. Unfortunately, this latter case is often falsely interpreted as β2 is equal to zero and so model (4.2) is accepted as more appropriate than model (4.1).Whether β′2x2 can be deleted from model (4.1) is better judged based on the magnitude of β′2x2. The partial F test above provides no information on the magnitude of β′2x2, irrespective whether H0 is rejected or not. On the other hand, a confidence band on β′2x2 provides useful information on the magnitude of β
4.2 Hyperbolic confidence bands
The next theorem provides a hyperbolic confidence band for β′2x2 over the whole space of x2 ∈ k. Let βˆ2 = Aβˆ denote the estimator of β2, which is made of the last k components of βˆ. Clearly βˆ2 has the distributionNk(β2,σ2V), where V is the k× k partition matrix formed from the last k rows and the last k columns of (X′X)−1, i.e. V = A(X′X)−1A′.
