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## An exercise…

If you’re eager to take a pen…

Suppose that

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

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

Let the sample size be $n$. 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)

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