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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 $m_4$?

user21359
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  • Yes, they are. And the vector $\mathbf{c}$ collects the ratios $b_1$ and $b_2$. – Christoph Hanck Jun 14 '17 at 08:24
  • 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 $c(\mathbf{m})$? – user21359 Jun 14 '17 at 15:34
  • $J$ is the number of rows, that cannot be. Also, please add the self-study tag. – Christoph Hanck Jun 14 '17 at 15:36
  • $\mathbf{J}$ is the matrix noted above, the $\mathbf{C}(\mathbf{\boldsymbol\mu})$ – user21359 Jun 14 '17 at 15:37
  • I am not sure I get your question. If they are the same, it does not matter w.r.t. what you take the derivative. Either way, for the delta method, you evaluate the derivatives in the matrix $\mathbf{C}$ at the population moments, not the sample expressions. – Christoph Hanck Jun 14 '17 at 15:43
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    This thread may be helpful: https://stats.stackexchange.com/questions/164804/testing-for-normality-in-non-normal-distributions-with-zero-skewness-and-zero-ex – Christoph Hanck Jun 14 '17 at 15:49

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