Consider a linear model $X\beta + \epsilon$, where $E(\epsilon) = 0$ and a fixed deterministic $n\times p$ design matrix $X$. $\beta = (\beta_1, ..., \beta_p)^T$,
$rank(X) = p$
Explain wheter the claim is true or not:
If all explanatory variables are orthogonal (uncorrelated), dropping one explanatory variable will
not change the remaining p-values
The solution in our exercise says the claim is true. Can anyone give insights why this might be true? (if it is)
Your solutions manual is wrong. For orthogonal design, the values of $\beta_i$ will not change after dropping a variable. However, your estimate for $\sigma$, namely $\hat{\sigma}$, will, as will the number of degrees of freedom of the $t$-distributions you use to test your coefficients. This will change the $p$-values you observe, as @daniel shows in his simulation.