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Suppose that

$$ {{y}_{i}}=\alpha +\beta x_{i}^{2}+{{\varepsilon }_{i}}$$,

where $$ {{{\varepsilon }_{i}}}$ are iid with $latex E({{\varepsilon }_{i}})=0$$, $$ E(\varepsilon _{i}^{2})=\sigma _{{}}^{2}$$, $$ E(\varepsilon _{i}^{3})=\tau$$, while the regressor $${{x}_{i}}$$ is deterministic: $$ {{x}_{i}}=\gamma^{i}$$, $$ \gamma \in \left( 0,1 \right)$$.

Let the sample size be $latex n$. Discuss as fully as you can the asymptotic behavior of the least squares estimates $$ (\hat{\alpha },\hat{\beta },\hat{\sigma }_{{}}^{2})$$ of $$(\alpha ,\beta ,\sigma _{{}}^{2})$$ as $$ n\to \infty$$. (by Stanislav Anatolyev)