6. Extensions

Paul Knottnerus

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In this section we briefly discuss a number of extensions of the AC estimates estimator described in the preceding section. Firstly, we pay attention to the situation whereby regression estimators, say ${\widehat{\overline{Y}}}_{REG,k}$ and ${\widehat{\overline{X}}}_{REG,k},$ are used instead of SRS estimators $\left(k=2,12\text{ and }23\right).$ To avoid a notational burden, we look at the situation with one explanatory variable, say $z;$ a generalization for more auxiliaries is straightforward. Furthermore, for simplicity’s sake, we assume that the estimated regression coefficients, denoted by $b_{yz2}$ and $b_{xz2},$ stem from $s_2.$ In order to derive the aligned composite estimators in this situation, we only need to evaluate (co)variance terms of the form $\mathrm{cov}\left({\widehat{\overline{Y}}}_{REG,k},{\widehat{\overline{X}}}_{REG,l}\right)$ in the different formulas $\left(k,l=2,12\text{ and }23\right).$ This evaluation can be done as follows. Replace the $Y_i$ and $X_i$ in the formulas by the corresponding (estimated) residuals from a regression on $Z_i$ and a constant. That is,

$$\mathrm{cov}\left({\widehat{\overline{Y}}}_{REG,k},{\widehat{\overline{X}}}_{REG,l}\right)=\mathrm{cov}\left({\overline{y}}_k^{*},{\overline{x}}_l^{*}\right),(6.1)$$

where the (estimated) residual variables $Y_i^{*}$ and $X_i^{*}$ are defined by

$$\begin{array}{l}{Y}_i^{*}=Y_i-{\overline{y}}_k-b_{yz2}\left(Z_i-{\overline{z}}_k\right)=Y_i-b_{yz2}{Z}_i+const.\\ X_i^{*}=X_i-{\overline{x}}_l-b_{xz2}\left(Z_i-{\overline{z}}_l\right)=X_i-b_{xz2}{Z}_i+const.\end{array}$$

The term $\mathrm{cov}\left({\overline{y}}_k^{*},{\overline{x}}_l^{*}\right)$ on the right-hand side of (6.1) can be calculated in the same manner as $\mathrm{cov}\left({\overline{y}}_k^{},{\overline{x}}_l^{}\right),$ discussed in preceding sections; see also formula (A.8) in Appendix A.3 and recall $\mathrm{var}\left({\widehat{\overline{Y}}}_{REG,k}\right)=\mathrm{cov}\left({\widehat{\overline{Y}}}_{REG,k},{\widehat{\overline{Y}}}_{REG,k}\right).$ In addition, the same approach can be applied when use is made of ratio estimators such as ${\widehat{\overline{Y}}}_{R,k}={\overline{y}}_k\overline{Z}/{\overline{z}}_k$ and ${\widehat{\overline{X}}}_{R,l}={\overline{x}}_l\overline{Z}/{\overline{z}}_l.$ That is, the residual variables $Y_i^{*}$ and $X_i^{*}$ are now to be read as

$$Y_i^{*}=Y_i-\frac{{\overline{y}}_2}{{\overline{z}}_2}{Z}_i\text{ and }{X}_i^{*}=X_i-\frac{{\overline{x}}_2}{{\overline{z}}_2}{Z}_i.$$

An alternative option for taking an auxiliary variable into account is to extend both the parameter vector $\theta$ and the set of prior restrictions. For instance, in Example 5.1 the parameter $\theta$ was implicitly defined by $\theta ={\left(\overline{D},\overline{Y},\overline{X}\right)}^{\prime }.$ When the variable $z$ is observed in samples 12 and 23, the new, extended ${\widehat{\theta }}_0$ is given by

$${\widehat{\theta }}_0={\left({\widehat{\overline{D}}}_{OLP},{\overline{y}}_{23},{\overline{x}}_{12},{\overline{z}}_{23},{\overline{z}}_{12},{\overline{z}}_2\right)}^{\prime }$$

and the extended set of prior restrictions is

$$\begin{array}{l}{\theta }_2-{\theta }_1-{\theta }_3=0;\\ {\theta }_4-{\theta }_5=0;\\ {\theta }_4-{\theta }_6=0;\\ {\theta }_4=\overline{Z}.\end{array}$$

Hence, the new $c$ is $c={\left(0,0,0,\overline{Z}\right)}^{\prime }.$ In this way the efficiency of ${\widehat{\theta }}_0$ can be further improved.

Secondly, another extension regards births and deaths. With respect to deaths, the population in period $t-12$ can be divided into two (post)strata: one consisting of the deaths in period $t$ and one consisting of the enterprises existing in periods $t-12$ and $t.$ Using such a poststratification still leads to an asymptotically unbiased estimator for the population mean at period $t,$ provided there are no births. In order to take births into account, one should draw an appropriate sample from this substratum of births especially when the number of births is substantial, and when there are no realistic assumptions with respect to the total turnover in this substratum in month $t.$

Finally, we examine the situation whereby a combination of quarterly and semesterly data is to be analysed. Suppose that in quarters 2, 4 and 6 semesterly samples are drawn which need not be the same as the quarterly samples in those quarters. In order to explain the AC estimates estimator in this situation, consider six consecutive quarterly SRS estimates for the quarterly means of the turnover, say ${\overline{y}}_1,{\overline{y}}_2,{\overline{y}}_3,{\overline{y}}_4,{\overline{y}}_5,{\overline{y}}_6,$ and three semesterly SRS estimates for the semesterly means of turnover, say ${\overline{x}}_2,{\overline{x}}_4$ and ${\overline{x}}_6;$ note that the subscript refers to the quarter of observation and not to a sample set as before. Furthermore, suppose that the following growth ratios are to be estimated: $G_{62}=Y_6/Y_2,H_{62}=X_6/X_2$ and $H_{64}=X_6/X_4$ as well as the corresponding quarterly and semesterly totals. In order to obtain a consistent set of estimators for totals (means) and growth rates, define in analogy with the approach in Section 5

$${\widehat{\theta }}_0={\left({\widehat{G}}_{62,OLP},{\widehat{H}}_{62,OLP},{\widehat{H}}_{64,OLP},{\overline{y}}_1,{\overline{y}}_2,{\overline{y}}_3,{\overline{y}}_4,{\overline{y}}_5,{\overline{y}}_6,{\overline{x}}_2,{\overline{x}}_4,{\overline{x}}_6\right)}^{\prime }.$$

The corresponding set of restrictions is

$$\begin{array}{cc}\hfill {\theta }_9-{\theta }_1{\theta }_5& ={\overline{Y}}_6-G_{62}{\overline{Y}}_2=0\hfill \\ \hfill {\theta }_{12}-{\theta }_2{\theta }_{10}& ={\overline{X}}_6-H_{62}{\overline{X}}_2=0\hfill \\ \hfill {\theta }_{12}-{\theta }_3{\theta }_{11}& ={\overline{X}}_6-H_{64}{\overline{X}}_4=0\hfill \\ \hfill {\theta }_4+{\theta }_5-{\theta }_{10}& ={\overline{Y}}_1+{\overline{Y}}_2-{\overline{X}}_2=0\hfill \\ \hfill {\theta }_6+{\theta }_7-{\theta }_{11}& ={\overline{Y}}_3+{\overline{Y}}_4-{\overline{X}}_4=0\hfill \\ \hfill {\theta }_8+{\theta }_9-{\theta }_{12}& ={\overline{Y}}_5+{\overline{Y}}_6-{\overline{X}}_6=0.\hfill \end{array}$$

The matrix $V_0$ can be estimated in a similar manner as described in Sections 2 and 4.

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