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I'm trying to work on #1 from Chapter 15 of Greene's Econometric Analysis and I'm confused about how to use the Delta method. The problem statement is:

For the normal distribution $μ_{2k} = \frac{\sigma^{2k}(2k)!}{k!2^k}$ and $μ_{2k+1} = 0, k = 0, 1,\ldots$. Use this result to analyze the two estimators $b_1= \frac{m_3}{m_2^{3/2}}$ and $b_2=\frac{m_4}{m_2^2}$. where $$ m_k = \frac{1}{n} \sum_{i=1}^n(x_i-\bar{x})^k $$ The following result will be useful: $$ \text{Asy.Cov}[\sqrt{n}m_j,\sqrt{n}m_k]=\mu{j+k}−\mu_j\mu_k+ jk\mu_2\mu_{j−1}\mu_{k−1} − jmu_{j−1}\mu_{k+1} −kmu_{k−1}\mu_{j+1}. $$ Use the delta method to obtain the asymptotic variances and covariance of these two functions, assuming the data are drawn from a normal distribution with mean $\mu$ and variance $\sigma^2$. (Hint: Under the assumptions, the sample mean is a consistent estimator of μ, so for purposes of deriving asymptotic results, the difference between x and μ may be ignored. As such, no generality is lost by assuming the mean is zero, and proceeding from there.) Obtain V, the 3 × 3 covariance matrix for the three moments, then use the delta method to show that the covariance matrix for the two estimators is:

$$ \mathbf{JVJ'}= \begin{bmatrix} 6/n & 0 \\ 0 & 24/n \end{bmatrix} $$

My question is how to properly apply the Delta method. The statement from the book is: If $\mathbf{z_n}$ is a $K\times 1$ sequence of vector-valued random variables such that $\sqrt{n}(\mathbf{z_n}-\boldsymbol{\mu}) \xrightarrow{d} N(\mathbf{0},\boldsymbol\Sigma)$ and if $\mathbf{c}(\mathbf{z_n})$ is a set of $J$ continuous functions of $\mathbf{z_n}$ not involving $n$, then $$\sqrt{n}(\mathbf{c}(\mathbf{z_n})-\mathbf{c}(\boldsymbol{\mu}) \xrightarrow{d} N(\mathbf{0},\mathbf{C}(\boldsymbol\mu)\boldsymbol\Sigma\mathbf{C}(\boldsymbol\mu)') $$, where $\mathbf{C}(\boldsymbol\mu)$ is the $J\times K$ matrix of partial derivatives.

Here, are my $\mathbf{z_n}$ the moments $m_2$, $m_3$, and $m_4$?

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  • $\begingroup$ Yes, they are. And the vector $\mathbf{c}$ collects the ratios $b_1$ and $b_2$. $\endgroup$ – Christoph Hanck Jun 14 '17 at 8:24
  • $\begingroup$ My problem is that the answer key says: The elements of $J$ are $$ \begin{bmatrix} \frac{\partial \sqrt{b_1}}{\partial m_2} & \frac{\partial \sqrt{b_1}}{\partial m_3} & \frac{\partial \sqrt{b_1}}{\partial m_4} \\ \frac{\partial b_2}{\partial m_2} & \frac{\partial b_2}{\partial m_3} & \frac{\partial b_2}{\partial m_4} \\ \end{bmatrix} $$ Wouldn't that be from taking the derivative of $c(\mathbf{z_n})$ rather than the derivative of $c(\mathbf{m})$? $\endgroup$ – user21359 Jun 14 '17 at 15:34
  • $\begingroup$ $J$ is the number of rows, that cannot be. Also, please add the self-study tag. $\endgroup$ – Christoph Hanck Jun 14 '17 at 15:36
  • $\begingroup$ $\mathbf{J}$ is the matrix noted above, the $\mathbf{C}(\mathbf{\boldsymbol\mu})$ $\endgroup$ – user21359 Jun 14 '17 at 15:37
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    $\begingroup$ This thread may be helpful: stats.stackexchange.com/questions/164804/… $\endgroup$ – Christoph Hanck Jun 14 '17 at 15:49

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