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I have to find MLE estimators of $\theta$ and $\sigma^{2}$. However, I could not solve these two equations. Do you have any idea about how to proceed?

$$ \frac{-\theta}{1-\theta^{2}}+\frac{\theta}{\sigma^{2}}y_{1}^{2} + \frac{y_{1}}{\sigma^{2}}(y_{2}-\theta y_{1})=0 \\ \frac{-1}{\sigma^{2}}+\frac{1}{2\sigma^{4}}(1-\theta^2)y_{1}^2 + \frac{1}{2\sigma^4}(y_{2}-\theta y_{1})^2=0 $$

Only $\theta$ and $\sigma^2$ are variables. The rest is constant.

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Work the first equation to give you a value for $\sigma^2$ depending on $\theta$. Substitute that in the second equation to get some polynomial equation for $\theta$. Then either (1) the equation factors nicely and you can get an explicit expression for $\theta$, or (2) it doesn't, in which case you can always solve it numerically.

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