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Is it possible to determine the numerical value of the parameters of a moving average process just by looking at the correlogram (Autocorrelation function and Partial Autocorrelation function) of the model?

Suppose we have a simple $MA(1)$ process: \begin{equation} y_{t} = \epsilon_{t}+\alpha\epsilon_{t-1} \end{equation} with $\epsilon_{t} \sim WN(0, \sigma^{2}_{\epsilon})$.

Just by looking at the Autocorrelation function and Partial Autocorrelation function of the process without estimating a proper regression, can we derive the value of $\alpha$?

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I computed the correlation between $y_{t}$ and $y_{t-1}$ as follows: \begin{equation} Cov(y_{t}, y_{t-1})=E[y_{t}y_{t-1}]-E[y_{t}]E[y_{t-1}]=E[y_{t}y_{t-1}] = E[(\epsilon_{t}+\alpha \epsilon_{t-1})(\epsilon_{t-1}+\alpha \epsilon_{t-2})] = E[\alpha \epsilon_{t-1}^{2}] = \alpha \sigma^{2}_{\epsilon} \end{equation} \begin{equation} Var(y_{t})=Var(y_{t-1})=\sigma^{2}_{\epsilon}+\alpha^{2}\sigma^{2}_{\epsilon}=\sigma^{2}_{\epsilon}(1-\alpha^2) \end{equation} Therefore: \begin{equation} Corr(y_{t}, y_{t-1})=\frac{Cov(y_{t}, y_{t-1})}{Var(y_{t})} = \frac{\alpha}{1+\alpha^2} \end{equation} From wich I can get the value of alpha.

Thank you for the intuition!

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