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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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