Let 1 be the first column vector of the design matrix X. Show that H1=1 for the multiple linear regression case(p-1>1). (H is hat matrix, i.e., H=X(X'X)^-1X')
The followings are my reasoning so far.
Let H=[r1 r2 .. rn]', where rn is a row vector of H.
Then r1*1=1(scalr). (* inner product)
why r1*1 should be 1?
I think prooving H1=H is the same as answering the question above.
But I can not push the thing further.
What property of H should I use?
I know H is idempotent and symmetric.
Do I have to use any other property which I don't know?
Best Answer
$$H_{n\times k}= X\left(X'X\right)^{-1}X' \Rightarrow X'H = X'$$
The first row of $X'$ is a row of ones, so $\left[X'\right]_{1j}=1$ . Denoting $h_{ij}$ the typical element of $H$, the typical element of the first row of $X'H$ is
$$\left[X'H\right]_{1j} = \sum_{i=1}^n h_{ij} = \left[X'\right]_{1j}= 1 \;\;\forall j$$
But $\left[X'H\right]_{1j}$ is the sum of the elements of the $j$ column of $H$, i.e. it is the inner product of this column with the vector of ones. And this hold for all columns of $H$. $QED$.