# Register-based sampling for household panels 6. Variance estimation

Parameters of the RIS are estimated as the ratio of two population totals

$$\widehat{R}=\frac{{\widehat{t}}_{y}}{{\widehat{t}}_{z}},\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}(6.1)$$

where ${\widehat{t}}_{y}$ and ${\widehat{t}}_{z}$ are GREG estimators defined by (5.1) or (5.2) in the case of person-based or household-based target variables, respectively. The variance of (6.1) under a sample design where core persons are drawn by means of stratified simple random sampling, and all household members of these core persons are included in the sample can be approximated by

$$V\left(\widehat{R}\right)=\frac{1}{{t}_{z}^{2}}{\displaystyle \sum _{h=1}^{H}\frac{{N}_{h}^{2}\left(1-{f}_{h}\right)}{{n}_{h}}\frac{1}{{N}_{h}-1}}\text{\hspace{0.17em}}{{\displaystyle \sum _{k=1}^{{N}_{h}}\left(\frac{{e}_{kh}}{{g}_{k}}-\frac{1}{{N}_{h}}{\displaystyle \sum _{{k}^{\prime}=1}^{{N}_{h}}\frac{{e}_{{k}^{\prime}h}}{{g}_{{k}^{\prime}}}}\right)}}^{2},\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}(6.2)$$

where ${f}_{h}={n}_{h}/{N}_{h},$ ${e}_{kh}=\left({y}_{kh}-{x}_{kh}^{t}{b}_{y}\right)-R\left({z}_{kh}-{x}_{kh}^{t}{b}_{z}\right),$ and ${b}_{y}$ and ${b}_{z}$ are the finite population regression coefficients of the regression of ${y}_{kh}$ and ${z}_{kh}$ respectively on ${x}_{kh}.$ An estimator for the variance specified in (6.2) is given by

$$\widehat{V}\left(\widehat{R}\right)=\frac{1}{{\widehat{t}}_{z}^{2}}{\displaystyle \sum _{h=1}^{H}\left(1-{f}_{h}\right)\frac{{n}_{h}}{{n}_{h}-1}}\text{\hspace{0.17em}}{{\displaystyle \sum _{k=1}^{{n}_{h}}\left({w}_{k}{\widehat{e}}_{k}-\frac{1}{{n}_{h}}{\displaystyle \sum _{{k}^{\prime}=1}^{{n}_{h}}{w}_{{k}^{\prime}}{\widehat{e}}_{{k}^{\prime}h}}\right)}}^{2},\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}(6.3)$$

where ${\widehat{e}}_{kh}=\left({y}_{kh}-{x}_{kh}^{t}{\widehat{b}}_{y}\right)-\widehat{R}\left({z}_{kh}-{x}_{kh}^{t}{\widehat{b}}_{z}\right)$ and ${\widehat{b}}_{y}$ and ${\widehat{b}}_{z}$ are the HT type estimators for ${b}_{y}$ and ${b}_{z}.$ These results follow directly from inserting first and second order inclusion expectations specified in (3.3) through (3.6) in the general approximation for the variance of the ratio of two GREG estimators and its estimator (Särndal et al. 1992, Section 7.13).

The same expressions for the variance can be derived from the variance expressions proposed for the Generalized Weight Share method in the case of indirect sampling. In Lavallée (1995), variance expressions for the HT estimator are based on the sampling design used to select the sample ${s}_{}^{A}$ of $n$ units from population ${U}_{}^{A}$ with transformed target variables, say ${z}_{i}.$ In this application each unit in ${U}^{A}$ has exactly one link with a unit in ${U}^{B}.$ As a result ${z}_{i}$ in Lavallée (1995) is in this case defined as the sum over the target variables of all elements in cluster $k,$ divided by the number of units in cluster $k$ with a link to population ${U}^{A},$ i.e., ${z}_{i}={y}_{k}/{g}_{k}$ for all $i\in {U}^{A}$ that have a link with cluster $k\in {U}^{B}.$ Inserting the first and second order inclusion probabilities for stratified simple random sampling without replacement and the transformed variables ${z}_{i}$ (where the target variable ${y}_{k}$ is preplaced by the residual of the regression on the cluster totals ${e}_{k})$ in the variance formula for a ratio gives (6.2). Result (6.3) follows in a similar way.

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