I am learning about dummy coding and effect coding. Intuitively, they make sense. I understand the interpretations of the intercepts in both examples: mean of the reference condition in dummy coding and the grand mean in effect coding.
I'm struggling to understand what it is in the maths, however, that makes centering the coding on 0 lead to an intercept that reflects the grand mean.
In dummy coding, it's clear the intercept is the mean of the reference condition:
\begin{align} \hat{Y}_{group1,i} = \beta_0 + \beta_1 \times 0 \\ = \beta_0 \end{align}
and the slope the difference between this reference condition and the other predictor (in a 2 condition regression)
\begin{align}\hat{\beta}_1 = \overline{Y}_\text{group2} - \beta_0\\ = \overline{Y}_\text{group2} - \overline{Y}_\text{group1}\end{align}
But how does centering the predictors around 0, e.g. (-0.5,0.5) affect the GLM equation such that the intercept now represents the grand mean?
I've looked up loads of sites and they all just say that the intercept is the grand mean when using effects coding, but im yet to see why that's the case mathematically.