# Determining the numerical value of the parameters of a Moving Average process

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: $$$$y_{t} = \epsilon_{t}+\alpha\epsilon_{t-1}$$$$ with $$\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 $$\alpha$$?

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