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Peter van Rijn edited this page Mar 15, 2021 · 6 revisions

The unit used for a list which, for each integer b from 1 to B includes the name of parameter , the estimated parameter , the estimated asymptotic standard deviation of derived from the square root of row b and column b of the estimated asymptotic covariance matrix

\begin{equation*} \widehat{\Cov}(\hat{\boldsymbol{\beta}})= \mathbf{T}[-\nabla^2\ell_V(\hat{\boldsymbol{\gamma}})]^{-1}[-\nabla^2\ell(\hat{\boldsymbol{\gamma}})] [-\nabla^2\ell_V(\hat{\boldsymbol{\gamma}})]^{-1}\mathbf{T}' \end{equation*}

of , the estimated asymptotic standard deviation of derived from the square root of row b and column b of the Louis (Louis, 1982) estimate

\begin{equation*} \widehat{\Cov}_L(\hat{\boldsymbol{\beta}})= \mathbf{T}[\boldsymbol{\Phi}_V(\hat{\boldsymbol{\gamma}})]^{-1}\boldsymbol{\Phi}(\hat{\boldsymbol{\gamma}}) [\boldsymbol{\Phi}_V(\hat{\boldsymbol{\gamma}})]^{-1}\mathbf{T}' \end{equation*}

of the asymptotic covariance matrix of , and the estimated asymptotic standard deviation of derived from the square root of row b and column b of the sandwich estimated asymptotic covariance matrix

\begin{equation*} \widehat{\Cov}_S(\hat{\boldsymbol{\beta}})= \mathbf{T}[-\nabla^2\ell_V(\hat{\boldsymbol{\gamma}})]^{-1}\boldsymbol{\Phi}(\hat{\boldsymbol{\gamma}}) [-\nabla^2\ell_V(\hat{\boldsymbol{\gamma}})]^{-1}\mathbf{T}' \end{equation*}

obtained without the assumption that the model holds (Huber, 1967; White, 1980; Haberman, 1989). If the Louis approximation is used for the negative Hessian matrix in implementation of the Newton-Raphson algorithm, then these three estimated asymptotic standard deviations are all the same. If complex sampling is used, then the estimated asymptotic standard deviation of derived from the square root of row b and column b of the estimated asymptotic covariance matrix

\begin{equation*} \widehat{\Cov}_S(\hat{\boldsymbol{\beta}})= \mathbf{T}[-\nabla^2\ell_V(\hat{\boldsymbol{\gamma}})]^{-1}\widehat{\Cov}(\nabla\ell(\boldsymbol{\gamma})) [-\nabla^2\ell_V(\hat{\boldsymbol{\gamma}})]^{-1}\mathbf{T}'. \end{equation*}

In listening9.txt, unitbeta is 11. An example of output appears in listeningbeta.csv. In this example, all expressions for standard errors yield rather similar results. The third column is , the fourth column is , and the fifth column is . In listeningp2.txt, an artificial example of complex sampling is provided in which primary sampling units are defined in terms of a sequence number. In this case, unitbeta is 11. The corresponding file is listeningpbeta.csv. Because the example is artificial, the added column for is quite similar to the other columns.

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