Recovering Latent Regime Size from Second and Fourth Moments

A stylized one-period microstructure model writes the observed shock as $$Y = S a + \varepsilon,$$ where $S$ takes values $+1$ and $-1$ with equal probability, $a>0$ is an unknown regime magnitude, and $\varepsilon \sim N(0,\sigma^2)$ is independent noise.

From data, the empirical second moment is $m_2 = 5$ and the empirical fourth moment is $m_4 = 43$.

Use the method of moments to estimate $a$ and $\sigma^2$.