# Prove: Least Squares Prediction Equation Contains Mean Point


Note: This is Exercise 11.81 in Mathematical Statistics with Applications, 5th Ed., by Wackerly, Mendenhall, and Scheaffer.

My Work So Far: What we need to do is show that $$\overline{y}=\hat\beta_0+\hat\beta_1\overline{x}_1+\hat\beta_2\overline{x}_2 +\cdots+\hat\beta_k\overline{x}_k.$$ Now if we set up the matrices \begin{align*} \mathbf{x}&=\szdb{ \begin{matrix} 1&x_{11}&x_{21}&\cdots&x_{k1}\\ 1&x_{12}&x_{22}&\cdots&x_{k2}\\ \vdots &\vdots &\vdots &\ddots &\vdots\\ 1&x_{1n}&x_{2n}&\cdots&x_{kn} \end{matrix}}\\ \mathbf{a}&=\szdb{ \begin{matrix} 1\\ \overline{x}_1\\ \overline{x}_2\\ \vdots \\ \overline{x}_k \end{matrix} }\\ \mathbf{y}&=\szdb{ \begin{matrix} y_1\\y_2\\ \vdots \\ y_n \end{matrix}}, \end{align*} then we know $${\hat\beta}=(\mathbf{x}^T\mathbf{x})^{-1}\mathbf{x}^T\mathbf{y},$$ and we are asked to prove $$\overline{y}=\mathbf{a}^T\hat\beta.$$ Here $$n$$ is the number of data points, and $$k$$ is the number of features.

My Questions: I have no idea where to go from here, or even whether this is the right approach to begin. What's a hint?

One place to start would be to replace your $$a^T$$ vector with $$\frac1n$$ times a row vector of $$n$$ 1's times $$x$$. This is just the matrix version of finding the means ($$\bar{x}$$) of the columns. When you substitute that (and $$\hat{\beta}$$) into your last equation you will have a piece that computes means times the hat matrix times the y vector. Look up the properties of the hat matrix and/or remember that $$y_i = \hat{y_i} + \hat\epsilon_i$$ and remember or look up what the mean of the observed residuals is to continue.
• Thanks for your answer! Just to make sure I'm understanding, you're recommending $$\mathbf{a}^T=\frac1n[\underbrace{\begin{matrix}1&1&\dots&1\end{matrix}}_{n\times}]?$$ Oct 1, 2021 at 16:38
• Or do you mean $$\mathbf{a}^T=\frac1n\left[ \underbrace{\begin{matrix} 1 &1 &1 &\dots &1 \end{matrix}}_{n\times} \right]\mathbf{x}?$$ Oct 1, 2021 at 17:48
• I can see that you meant the latter expression, with $\mathbf{x}$ in it. Now I have a new question: I can see that your suggestion allows me to get to $$\mathbf{a}'\hat\beta =\frac1n\left[ \underbrace{\begin{matrix} 1 &1 &1 &\dots &1 \end{matrix}}_{n\times} \right]Py.$$ But why should $\overline{y}=\overline{\hat{y}}?$ Oh, your last comment. Gotta follow it out! Oct 1, 2021 at 18:06
1. If the mean of all the $$x$$s was zero, the mean point would be $$(0,\dots,0,\hat\beta_0)$$. The matrix $$X^TX$$ would be block diagonal, with a block for the intercept and a block for everything else, which will imply $$\hat\beta_0=\bar y$$. So the fitted (hyper)plane passes through the mean point.
2. Reparametrising $$x_i\mapsto x_i-\bar x_i$$ relabels the x axes but it doesn't change whether the fitted (hyper)plane goes through a point. So if the result holds when the $$x$$s have mean zero, it holds in general