Consider the Laplace intercept-only example at the end of Section 6.3.6, so y = + u

Consider the Laplace intercept-only example at the end of Section 6.3.6, so $y=\mu +u$. Then GMM estimation is based on $\mathrm{E}[\mathbf{h}(\mu )]=\mathbf{0}$, where $\mathbf{h}(\mu )=[(y-$ ${\mu ),(y-\mu {)}^3]}^{\mathrm{\prime }}$.

(a) Using knowledge of the central moments of $y$ given in Section 6.3.6, show that ${\mathbf{G}}_0=E[\mathrm{\partial }\mathbf{h}/\mathrm{\partial }\mu ]=[-1,-6{]}^{\mathrm{\prime }}$ and that ${\mathbf{S}}_0=E\left[{\mathbf{h}\mathbf{h}}^{\mathrm{\prime }}\right]$ has diagonal entries 2 and 720 and off-diagonal entries 24.

(b) Hence show that ${\mathbf{G}}_0^{\mathrm{\prime }}{\mathbf{S}}_0^{-1}{\mathbf{G}}_0=252/432$.

(c) Hence show that ${\hat{\mu }}_{\text{OGmm }}$ has asymptotic variance $1.7143/\mathrm{N}$.

(d) Show that the GMM estimator of $\mu$ with $\mathbf{W}={\mathbf{I}}_2$ has asymptotic variance $19.14/N$.