I put this question here as it has a pure maths element to it, even though it has a statistical twist. Basically, I have been given the table of data:
$$\begin{matrix} i & \mathrm{Response} \, y_i & \mathrm{Covariate} \, x_i \\ 1 & -1 & -0.5 \\ 2 & -0.25 &-0.25 \\ 3 & 1 & 0 \\ 4 & 1.2 & 0.25 \\ 5 & 2.6 & 0.5 \end{matrix}$$
And for here, I need to create an orthogonal regression model. In other words, I need to come up with a model in the form
$$Y_i = \alpha_0 + \alpha_1(ax_i + b) + \alpha_2(dx_i^2 + ex_i + f)$$
Where
$$\sum_i^n ax_i + b = dx_i^2 + ex_i + f = 0$$
and $\alpha_{0,1,2}$ are just constants.
From looking at it, I can see that the linear term can just be $x_i$ as $\sum_i^n x_i = 0$, but I'm stuck on how to solve for the quadratic bit.
How can I do this?