5 Conclusions and future research

Iván A. Carrillo and Alan F. Karr

Previous

We have proposed a novel approach to combining different cohorts of a longitudinal survey. The major requirement of our method is that there is a cross-sectional survey weight for each wave, or that one can be built from available information. This weight should allow for statistical inference to the population of interest at the corresponding wave. In that case, our method should perform better than usual estimation procedures (where the auto-correlation is not incorporated) in many practical situations, in particular when there is a high auto-correlation among responses from the same subject.

In general, survey practitioners avoid as much as possible the use of multiple survey weights. However, in the case of rotating panels this is an appealing approach for at least two reasons. On the one hand, it allows for the use of all the available data in a clear and cohesive way in a single analysis procedure. On the other hand, we have shown how readily available cross-sectional survey weights can be directly used for longitudinal analysis, without the need to develop, store, and distribute an additional longitudinal weight or weights.

Our method is directly applicable to any kind of longitudinal survey as long as there are cross-sectional survey weights available (or these can be created) at each wave, and these weights represent the population of interest at the particular wave.

For the theory that we developed about the variance of the estimator proposed, we utilized the (cross-sectional) design weights $w_{ij},$ which are the inverse of the inclusion probabilities. Yet for the application in our model for salary in the SDR we used the final (cross-sectional) survey weights, which are not the original design weights, but adjusted (in the usual way) weights. This mismatch requires further exploration.

Similarly, in our derivations of the variance, we assumed that the cohorts were independent. However, the SDR does not totally satisfy this assumption for two reasons. Firstly, at any particular wave, the selection of the sample from the old cohorts is not performed independently across cohorts. In order to reduce the number of strata, since 1991 the NSF has collapsed strata over year of degree receipt for the old cohorts. Additionally, the post-stratification adjustments made to the design weights do not condition over cohort either, and as a result, weights are shared across cohorts. This sampling selection scheme and weighting adjustment procedure violate the independence across cohorts. Some additional calculations (included in the Appendix) have shown that the independence among cohort is not such a crucial requirement for our variance estimation method to produce good approximations, as explained in Section 3.3.1. In future research we plan to evaluate in more detail the impact of this issue.

Acknowledgements

This research was supported by NSF grant SRS-1019244 to the National Institute of Statistical Sciences (NISS). Any opinions, findings, and conclusions or recommendations expressed in this publication are those of the authors and do not necessarily reflect the views of the National Science Foundation. The authors thank Paul Biemer of RTI International, Stephen Cohen and Nirmala Kannankutty of the National Center for Science and Engineering Statistics at NSF, and Criselda Toto, formerly of NISS, for numerous insightful discussions during the research. We are also grateful to the Associate Editor and two referees for their useful suggestions.

Appendix - Proofs

To develop an expression for $C_{\xi },$ we first simplify ${\Psi }_s(\beta ){{\Psi }^{\prime }}_U(\beta ).$ Let $F_{i(k)}=B_i{\text{I}}_i(\text{U})e_{i(k\dots 3)}$ for $k=1,2,3,$ then we have:

$N^2{\Psi }_s\left(B\right)\Psi {\prime }_U\left(B\right)={\displaystyle \sum _{i\in s}}{B}_i{W}_i{e}_i{\displaystyle \sum _{i\in U}e{\prime }_i{\text{I}}_i\left(\text{U}\right)B{\prime }_i}=\left[{\displaystyle \sum _{i\in s}{B}_i{W}_i{e}_i}\right]\left[{\displaystyle \sum _{i\in s}{F^{\prime }}_{i\left(1\right)}}+{\displaystyle \sum _{i\notin s}{F^{\prime }}_{i\left(1\right)}}\right]$
$={\displaystyle \sum _{i\in s}}{B}_i{W}_i{e}_i{\displaystyle \sum _{i\in s}{F^{\prime }}_{i\left(1\right)}}+{\displaystyle \sum _{i\in s}{B}_i{W}_i{e}_i}{\displaystyle \sum _{i\notin s}{F^{\prime }}_{i\left(1\right)}}$
$={\displaystyle \sum _{i\in s}}{B}_i{W}_i{e}_i{e^{\prime }}_k{B^{\prime }}_i+{\displaystyle \sum _{i\in s}}{\displaystyle \sum _{\begin{array}{c}k\in s\\ k\ne i\end{array}}{B}_i{W}_i{e}_i{e^{\prime }}_k{\text{I}}_k\left(\text{U}\right){B^{\prime }}_k}+\text{A}\mathrm{,}$

where $\text{A}=({\displaystyle {\sum }_{i\in s}{B}_i{W}_i{e}_i})({\displaystyle {\sum }_{i\notin s}{F^{\prime }}_{i(1)}}),$ and let $\text{B}={\displaystyle {\sum }_{i\in s}}{\displaystyle {\sum }_{\begin{array}{c}k\in s\\ k\ne i\end{array}}{B}_i{W}_i{e}_i{e^{\prime }}_k{\text{I}}_k(\text{U}){B^{\prime }}_k.}$ The two sums in $\text{A}$ are model-independent, $e_i$ and ${e^{\prime }}_k$ (in $\text{B)}$ are two model-independent terms, and A and B both have model-expectation zero; therefore, $E_{\xi }\left[{\Psi }_s(\beta ){{\Psi }^{\prime }}_U(\beta )\right]=N^{-2}{\displaystyle {\sum }_{i\in s}}{B}_i{W}_i{E}_{\xi }\left[e_i{e^{\prime }}_i\right]{B^{\prime }}_i=N^{-2}{\displaystyle {\sum }_{i\in s}}{B}_i{W}_i{\Sigma }_i{B^{\prime }}_i=N^{-1}{\widehat{H}}_{\Sigma V}(\beta );$ equation (3.9) follows.

We now develop the expression for${\text{Var}}_p\left[{\Psi }_s({\beta }_N)\right],$ the design variance of the estimating function; we redefine $B_i={(\partial {{\mu }^{\prime }}_i/\partial \beta )|}_{\beta ={\beta }_N}{V}_i^{-1}$ and $e_i=y_i-{\mu }_i({\beta }_N);$ then

${\text{Var}}_p\left[{\Psi }_p\left({\beta }_N\right)\right]{\text{=Var}}_p\left(\frac{1}{N}{\displaystyle \sum _{i\in s}{B}_i{W}_i{e}_i}\right)$
$=\frac{1}{N^2}{\text{Var}}_p\left({\displaystyle \sum _{i\in s_{1\left(1\right)}}}{B}_i{W}_i{e}_i+{\displaystyle \sum _{i\in s_{2\left(2\right)}}}{B}_i{W}_i{e}_i+{\displaystyle \sum _{i\in s_{3\left(3\right)}}}{B}_i{W}_i{e}_i\right)$
$=\frac{1}{N^2}{\text{Var}}_p\left({\displaystyle \sum _{i\in s_{1\left(1\right)}}}{B}_i{W}_i{e}_i\right)+\frac{1}{N^2}{\text{Var}}_p\left({\displaystyle \sum _{i\in s_{2\left(2\right)}}}{B}_i{W}_i{e}_i\right)$
$+\frac{1}{N^2}{\text{Var}}_p({\displaystyle \sum _{i\in s_{3(3)}}}{B}_i{W}_i{e}_i)=D_{(1)}+D_{(2)}+D_{(3)}\mathrm{,}$

where, for line (A.1), we assume that the (three) cohorts are design-independent. Now, $N^2{D}_{(1)}={\text{Var}}_p\left[{\displaystyle {\sum }_{i\in U_{1(1)}}}{B}_i{W}_i\text{Diag}\left\{I_{i(1)}\right\}{e}_i\right]={\text{Var}}_p\left[{\displaystyle {\sum }_{i\in U_{1(1)}}}{B}_i{W}_i\text{Diag}\left\{e_i\right\}{I}_{i(1)}\right],$ where $\text{Diag}\left\{e\right\}$ is, for a column vector $e,$ a diagonal matrix with diagonal entries being the elements of $e,$ and $I_{i(1)}=(I_i(s_{1(1)}),I_i(s_{2(1)})I_i(s_{1(1)})\mathrm{,}{I_i(s_{3(1)})I_i(s_{2(1)})I_i(s_{1(1)}))}^{\prime }.$

 Similarly we can get $N^2{D}_{(2)}={\text{Var}}_p\left[{\displaystyle {\sum }_{i\in U_{2(2)}}}{B}_i{W}_i\text{Diag}\left\{e_i\right\}{I}_{i(2)}\right]$ and $N^2{D}_{(3)}={\text{Var}}_p\left[{\displaystyle {\sum }_{i\in U_{3(3)}}}{B}_i{W}_i\text{Diag}\left\{e_i\right\}{I}_{i(3)}\right],$ where $I_{i(2)}={(\mathrm{0,}{I}_i(s_{2(2)}),I_i(s_{3(2)})I_i(s_{2(2)}))}^{\prime },$ and $I_{i(3)}={(0,0,I_i(s_{3(3)}))}^{\prime }.$

 Now, let us concentrate on $D_{(1)};$ letting $C_i=B_i{W}_i\text{Diag}\left\{e_i\right\},$ we have:

$N^2{D}_{\left(1\right)}{\text{=Var}}_p\left({\displaystyle \sum _{i\in U_{1\left(1\right)}}}{C}_i{I}_{i\left(1\right)}\right)=\text{Var}\left\{E\left[{\displaystyle \sum _{i\in U_{1\left(1\right)}}}{C}_i{I}_{i\left(1\right)}|s_{1\left(1\right)}\right]\right\}$
$+E\left\{\text{Var}\left[{\displaystyle \sum _{i\in U_{1\left(1\right)}}}{C}_i{I}_{i\left(1\right)}|s_{1\left(1\right)}\right]\right\}$
$\text{=Var}\left\{E\left[E\left({\displaystyle \sum _{i\in U_{1\left(1\right)}}}{C}_i{I}_{i\left(1\right)}|s_{2\left(1\right)},s_{1\left(1\right)}\right)|s_{1\left(1\right)}\right]\right\}$
$+E\{\text{Var}\left[E\left({\displaystyle \sum _{i\in U_{1\left(1\right)}}}{C}_i{I}_{i\left(1\right)}|s_{2\left(1\right)},s_{1\left(1\right)}\right)|s_{1\left(1\right)}\right]\left(A.2\right)$
$+E\left[\text{Var}\left({\displaystyle \sum _{i\in U_{1\left(1\right)}}}{C}_i{I}_{i\left(1\right)}|s_{2\left(1\right)},s_{1\left(1\right)}\right)|s_{1\left(1\right)}\right]\}$
$=N^2{D}_{\left(1\right)1}+N^2{D}_{\left(1\right)2}+N^2{D}_{\left(1\right)3}.$

Let us do each of the terms in (A.2) in turn, beginning with $N^2{D}_{1(1)},$ we have:

$E({\displaystyle \sum _{i\in U_{1(1)}}}{B}_i{W}_i\text{Diag}\left\{e_i\right\}{I}_{i(1)}|s_{2(1)},s_{1(1)})={\displaystyle \sum _{i\in U_{1(1)}}}{B}_i{W}_i\text{Diag}\left\{e_i\right\}{I}_{i(1)}^{(1)},$

where $I_{i(1)}^{(1)}={(I_i(s_{1(1)}),I_i(s_{2(1)})I_i(s_{1(1)})\mathrm{,}{\pi }_{i3|s_{2(1)}}{I}_i(s_{2(1)})I_i(s_{1(1)}))}^{\prime },$ then

$E\left[E({\displaystyle \sum _{i\in U_{1(1)}}}{C}_i{I}_{i(1)}|s_{2(1)},s_{1(1)})|s_{1(1)}\right]={\displaystyle \sum _{i\in U_{1(1)}}}{C}_i{I}_{i(1)}^{(2)}={\displaystyle \sum _{i\in U_{1(1)}}}{B}_i{\text{I}}_i^{(1)}(\text{U})\text{Diag}\left\{I_{i(1)}^{(2)}\right\}{e}_i$

$$={\displaystyle \sum _{i\in U_{1(1)}}}{F}_i{\left[\frac{I_i(s_{1(1)})}{{\pi }_{i1}},\frac{I_i(s_{1(1)})}{{\pi }_{i1}},\frac{I_i(s_{1(1)})}{{\pi }_{i1}}\right]}^{\prime }$$

$$={\displaystyle \sum _{i\in U_{1(1)}}}{F}_i{1}_3\frac{I_i(s_{1(1)})}{{\pi }_{i1}}={\displaystyle \sum _{i\in U_{1(1)}}}{w}_{i1(1)}{F}_{i(1)}{I}_i(s_{1(1)})\mathrm{,}$$

where $I_{i(1)}^{(2)}={(I_i(s_{1(1)})\mathrm{,}{\pi }_{i2|s_{1(1)}}{I}_i(s_{1(1)})\mathrm{,}{\pi }_{i3|s_{2(1)}}{\pi }_{i2|s_{1(1)}}{I}_i(s_{1(1)}))}^{\prime },$ 
${\text{I}}_i^{(1)}(\text{U})=\text{diag}\left[I_i(U_1)/{\pi }_{i1},I_i(U_2)/({\pi }_{i1}{\pi }_{i2|s_{1(1)}}),I_i(U_3)/({\pi }_{i1}{\pi }_{i2|s_{1(1)}}{\pi }_{i3|s_{2(1)}})\right],$
 $F_i=B_i{\text{I}}_i(\text{U})\text{Diag}\left\{e_i\right\},$ and $1_3={(\mathrm{1,1,1})}^{\prime };$ this implies that $N^2{D}_{(1)1}=\text{Var}\left[{\displaystyle {\sum }_{i\in U_{1(1)}}}{w}_{i1}{B}_i{\text{I}}_i(\text{U})e_i{I}_i(s_{1(1)})\right]=\text{Var}\left[{\displaystyle {\sum }_{i\in s_{1(1)}}}{w}_{i1}{F}_{i(1)}\right].$

For $N^2{D}_{(1)2},$ we have:

$$\begin{array}{l}E({\displaystyle \sum _{i\in U_{1(1)}}}{C}_i{I}_{i(1)}|s_{2(1)},s_{1(1)})\\ ={\displaystyle \sum _{i\in U_{1(1)}}}{B}_i{W}_i\text{Diag}\left\{e_i\right\}{I}_{i(1)}^{(1)}={\displaystyle \sum _{i\in U_{1(1)}}}{B}_i{\text{I}}_i^{(1)}(\text{U})\text{Diag}\left\{I_{i(1)}^{(3)}\right\}{e}_i\\ ={\displaystyle \sum _{i\in U_{1(1)}}}{B}_i{\text{I}}_i(\text{U})\text{Diag}\left\{e_i\right\}[\frac{I_i(s_{1(1)})}{{\pi }_{i1}}\mathrm{,}\frac{I_i(s_{2(1)})I_i(s_{1(1)})}{{\pi }_{i1}{\pi }_{i2|s_{1(1)}}}\mathrm{,}{\frac{I_i(s_{2(1)})I_i(s_{1(1)})}{{\pi }_{i1}{\pi }_{i2|s_{1(1)}}}]}^{\prime }\\ ={\displaystyle \sum _{i\in s_{1(1)}}}{w}_{i2}{B}_i{\text{I}}_i(\text{U})\text{Diag}\left\{e_i\right\}{\left[{\pi }_{i2|s_{1(1)}}\mathrm{,}{I}_i(s_{2(1)})\mathrm{,}{I}_i(s_{2(1)})\right]}^{\prime }\mathrm{,}\end{array}$$

where $I_{i(1)}^{(3)}={(I_i(s_{1(1)})\mathrm{,}{I}_i(s_{2(1)})I_i(s_{1(1)})\mathrm{,}{\pi }_{i3|s_{2(1)}}{I}_i(s_{2(1)})I_i(s_{1(1)}))}^{\prime };$ then,

$\begin{array}{l}\text{Var}\left[E({\displaystyle \sum _{i\in U_{1(1)}}}{C}_i{I}_{i(1)}|s_{2(1)}\mathrm{,}{s}_{1(1)})|s_{1(1)}\right]\\ =\text{Var}\left[{\displaystyle \sum _{i\in s_{1(1)}}}{w}_{i2}{B}_i{\text{I}}_i(\text{U})\text{Diag}\left\{e_i\right\}{I}_{i(1)}^{(4)}|s_{1(1)}\right]\\ =\text{Var}\left[{\displaystyle \sum _{i\in s_{1(1)}}}{w}_{i2}{B}_i{\text{I}}_i(\text{U})\text{Diag}\left\{e_i\right\}{\left[\mathrm{0,}{I}_i(s_{2(1)})\mathrm{,}{I}_i(s_{2(1)})\right]}^{\prime }|s_{1(1)}\right]\\ =\text{Var}\left[{\displaystyle \sum _{i\in s_{1(1)}}}{w}_{i2}{B}_i{\text{I}}_i(\text{U})\text{Diag}\left\{e_i\right\}{I}_i(s_{2(1)})1_{02}|s_{1(1)}\right]\\ =\text{Var}\left[{\displaystyle \sum _{i\in s_{2(1)}}}{w}_{i2}{B}_i{\text{I}}_i(\text{U})e_{i(2\dots 3)}|s_{1(1)}\right]\mathrm{,}\text{ (A.3)}\end{array}$

where $I_{i(1)}^{(4)}={\left[{\pi }_{i2|s_{1(1)}}\mathrm{,}{I}_i(s_{2(1)})\mathrm{,}{I}_i(s_{2(1)})\right]}^{\prime },1_{02}={(\mathrm{0,1,1})}^{\prime },$ and line (A.3) is because, conditional on $s_{1(1)},{\pi }_{i2|s_{1(1)}}$ is constant and therefore the variance of that component is zero. This means that:

$N^2{D}_{(1)2}=E\left\{\text{Var}\left[E({\displaystyle \sum _{i\in U_{1(1)}}}{C}_i{I}_{i(1)}|s_{2(1)}\mathrm{,}{s}_{1(1)})|s_{1(1)}\right]\right\}$
$=E\left\{\text{Var}\left[{\displaystyle \sum _{i\in s_{2(1)}}}{w}_{i2}{F}_{i(2)}|s_{1(1)}\right]\right\}$
$=\text{Var}\left[{\displaystyle \sum _{i\in s_{2(1)}}}{w}_{i2}{F}_{i(2)}\right]-\text{Var}\left\{E\left[{\displaystyle \sum _{i\in s_{2(1)}}}{w}_{i2}{F}_{i(2)}|s_{1(1)}\right]\right\}$
$=\text{Var}\left[{\displaystyle \sum _{i\in s_{2(1)}}}{w}_{i2}{F}_{i(2)}\right]-\text{Var}\left\{E\left[{\displaystyle \sum _{i\in s_{2(1)}}}{w}_{i2|s_{1(1)}}{w}_{i1}{F}_{i(2)}|s_{1(1)}\right]\right\}$
$=\text{Var}\left[{\displaystyle \sum _{i\in s_{2(1)}}}{w}_{i2}{F}_{i(2)}\right]-\text{Var}\left\{{\displaystyle \sum _{i\in s_{1(1)}}}{w}_{i1}{F}_i\right\}\mathrm{.}$

We can, similarly, show that:

$N^2{D}_{(1)3}=E\left\{E\left[\text{Var}({\displaystyle \sum _{i\in U_{1(1)}}}{B}_i{W}_i\text{Diag}\left\{e_i\right\}{I}_{i(1)}|s_{2(1)}\mathrm{,}{s}_{1(1)})|s_{1(1)}\right]\right\}$
$=E\left\{E\left[\text{Var}({\displaystyle \sum _{i\in s_{3(1)}}}{w}_{i3}{I}_i(s_{2(1)})I_i(s_{1(1)})F_{i(3)}|s_{2(1)},s_{1(1)})|s_{1(1)}\right]\right\}$
$=E\{\text{Var}\left[{\displaystyle \sum _{i\in s_{3(1)}}}{w}_{i3}{I}_i(s_{2(1)})F_{i(3)}|s_{1(1)}\right]$
$-\text{Var}\left[E({\displaystyle \sum _{i\in s_{3(1)}}}{w}_{i3}{I}_i(s_{2(1)})F_{i(3)}|s_{2(1)}\mathrm{,}{s}_{1(1)})|s_{1(1)}\right]\}$
$=E\left\{\text{Var}\left[{\displaystyle \sum _{i\in s_{3(1)}}}{w}_{i3}{F}_{i(3)}|s_{1(1)}\right]-\text{Var}\left[{\displaystyle \sum _{i\in s_{2(1)}}}{w}_{i2}{F}_{i(3)}|s_{1(1)}\right]\right\}$
$=\text{Var}\left[{\displaystyle \sum _{i\in s_{3(1)}}}{w}_{i3}{F}_{i(3)}\right]-\text{Var}\left[E({\displaystyle \sum _{i\in s_{3(1)}}}{w}_{i3}{F}_{i(3)}|s_{1(1)})\right]$
$-\text{Var}\left[{\displaystyle \sum _{i\in s_{2(1)}}}{w}_{i2}{F}_{i(3)}\right]+\text{Var}\left[E({\displaystyle \sum _{i\in s_{2(1)}}}{w}_{i2}{F}_{i(3)}|s_{1(1)})\right]$
$=\text{Var}\left[{\displaystyle \sum _{i\in s_{3(1)}}}{w}_{i3}{F}_{i(3)}\right]-\text{Var}\left[{\displaystyle \sum _{i\in s_{1(1)}}}{w}_{i1}{F}_{i(3)}\right]$
$-\text{Var}\left[{\displaystyle \sum _{i\in s_{2(1)}}}{w}_{i2}{F}_{i(3)}\right]+\text{Var}\left[{\displaystyle \sum _{i\in s_{1(1)}}}{w}_{i1}{F}_{i(3)}\right]\mathrm{.}$

With similar calculations, we obtain the corresponding expressions for $N^2{D}_{(2)},N^2{D}_{(2)2},N^2{D}_{(2)3},$ and $N^2{D}_{(3)}=N^2{D}_{(3)3}.$

Finally, we sketch the development of an expression for ${\text{Var}}_p\left[{\Psi }_s({\beta }_N)\right]$ without assuming independence among cohorts. First, notice that ${\Psi }_s({\beta }_N)$ can be written as:

$\begin{array}{l}{\displaystyle \sum _{i\in s_1}{B}_i{\text{I}}_i(\text{U})}\left[\begin{array}{ccc}{w}_{i1}& & \text{O}\\ & w_{i2}& \\ \text{O}& & w_{i3}\end{array}\right]\left[\begin{array}{c}{e}_{i1}\\ e_{i2}\\ e_{i3}\end{array}\right]+{\displaystyle \sum _{i\in s_2}{B}_i{\text{I}}_i(\text{U})}\left[\begin{array}{ccc}0& & \text{O}\\ & w_{i2}& \\ \text{O}& & w_{i3}\end{array}\right]\left[\begin{array}{c}0\\ e_{i2}\\ e_{i3}\end{array}\right]\\ -{\displaystyle \sum _{i\in s_1}{B}_i{\text{I}}_i(\text{U})}\left[\begin{array}{ccc}0& & \text{O}\\ & w_{i2}& \\ \text{O}& & w_{i3}\end{array}\right]\left[\begin{array}{c}0\\ e_{i2}\\ e_{i3}\end{array}\right]\\ +{\displaystyle \sum _{i\in s_3}{B}_i{\text{I}}_i(\text{U})}\left[\begin{array}{ccc}0& & \text{O}\\ & 0& \\ \text{O}& & w_{i3}\end{array}\right]\left[\begin{array}{c}0\\ 0\\ e_{i3}\end{array}\right]-{\displaystyle \sum _{i\in s_2}{B}_i{\text{I}}_i(\text{U})}\left[\begin{array}{ccc}0& & \text{O}\\ & 0& \\ \text{O}& & w_{i3}\end{array}\right]\left[\begin{array}{c}0\\ 0\\ e_{i3}\end{array}\right]\\ ={\displaystyle \sum _{i\in s_1}{w}_{i1}{B}_i{\text{I}}_i(\text{U})e_i}-{\displaystyle \sum _{i\in s_1}{w}_{i1}{B}_i{\text{I}}_i(\text{U})}\left[\begin{array}{c}0\\ e_{i2}\\ e_{i3}\end{array}\right]+{\displaystyle \sum _{i\in s_2}{w}_{i2}{B}_i{\text{I}}_i(\text{U})}\left[\begin{array}{c}0\\ e_{i2}\\ e_{i3}\end{array}\right]-{\displaystyle \sum _{i\in s_2}{w}_{i2}{B}_i{\text{I}}_i(\text{U})}\left[\begin{array}{c}0\\ 0\\ e_{i3}\end{array}\right]\\ +{\displaystyle \sum _{i\in s_3}{w}_{i3}{B}_i{\text{I}}_i(\text{U})}\left[\begin{array}{c}0\\ 0\\ e_{i3}\end{array}\right];\end{array}$

letting $z_i=B_i{\text{I}}_i(\text{U})e_i,z_{i(2\dots 3)}=B_i{\text{I}}_i(\text{U}){\left[\mathrm{0,}{e}_{i2}\mathrm{,}{e}_{i3}\right]}^{\prime },$ and $z_{i(3\dots 3)}=B_i{\text{I}}_i(\text{U}){\left[\mathrm{0,0,}{e}_{i3}\right]}^{\prime },$ ${\text{Var}}_p\left[{\Psi }_s({\beta }_N)\right]$ can be expanded as:

$\begin{array}{l}{\text{Var}}_p\left[{\displaystyle \sum _{i\in s_1}{w}_{i1}{z}_i}\right]+{\text{Var}}_p\left[{\displaystyle \sum _{i\in s_1}{w}_{i1}{z}_{i(2\cdots 3)}}\right]\\ +{\text{Var}}_p\left[{\displaystyle \sum _{i\in s_2}{w}_{i2}{z}_{i(2\cdots 3)}}\right]+{\text{Var}}_p\left[{\displaystyle \sum _{i\in s_2}{w}_{i2}{z}_{i(3\cdots 3)}}\right]\\ +{\text{Var}}_p\left[{\displaystyle \sum _{i\in s_3}{w}_{i3}{z}_{i(3\cdots 3)}}\right]-2{\text{Cov}}_p\left[{\displaystyle \sum _{i\in s_1}{w}_{i1}{z}_i\mathrm{,}}{\displaystyle \sum _{i\in s_1}{w}_{i1}{z}_{i(2\cdots 3)}}\right]\\ +2{\text{Cov}}_p\left[{\displaystyle \sum _{i\in s_1}{w}_{i1}{z}_i\mathrm{,}}{\displaystyle \sum _{i\in s_2}{w}_{i2}{z}_{i(2\cdots 3)}}\right]\\ -2{\text{Cov}}_p\left[{\displaystyle \sum _{i\in s_1}{w}_{i1}{z}_i\mathrm{,}}{\displaystyle \sum _{i\in s_2}{w}_{i2}{z}_{i(3\cdots 3)}}\right]+2{\text{Cov}}_p\left[{\displaystyle \sum _{i\in s_1}{w}_{i1}{z}_i\mathrm{,}}{\displaystyle \sum _{i\in s_3}{w}_{i3}{z}_{i(3\cdots 3)}}\right]\\ -2{\text{Cov}}_p\left[{\displaystyle \sum _{i\in s_1}{w}_{i1}{z}_{i(2\cdots 3)}\mathrm{,}}{\displaystyle \sum _{i\in s_2}{w}_{i2}{z}_{i(2\cdots 3)}}\right]+2{\text{Cov}}_p\left[{\displaystyle \sum _{i\in s_1}{w}_{i1}{z}_{i(2\cdots 3)}\mathrm{,}}{\displaystyle \sum _{i\in s_2}{w}_{i2}{z}_{i(3\cdots 3)}}\right]\\ -2{\text{Cov}}_p\left[{\displaystyle \sum _{i\in s_1}{w}_{i1}{z}_{i(2\cdots 3)}\mathrm{,}}{\displaystyle \sum _{i\in s_3}{w}_{i3}{z}_{i(3\cdots 3)}}\right]-2{\text{Cov}}_p\left[{\displaystyle \sum _{i\in s_2}{w}_{i2}{z}_{i(2\cdots 3)}\mathrm{,}}{\displaystyle \sum _{i\in s_2}{w}_{i2}{z}_{i(3\cdots 3)}}\right]\\ +2{\text{Cov}}_p\left[{\displaystyle \sum _{i\in s_2}{w}_{i2}{z}_{i(2\cdots 3)}\mathrm{,}}{\displaystyle \sum _{i\in s_3}{w}_{i3}{z}_{i(3\cdots 3)}}\right]-2{\text{Cov}}_p\left[{\displaystyle \sum _{i\in s_2}{w}_{i2}{z}_{i(3\cdots 3)}\mathrm{,}}{\displaystyle \sum _{i\in s_3}{w}_{i3}{z}_{i(3\cdots 3)}}\right]\\ ={\text{Var}}_p\left[{\displaystyle \sum _{i\in s_1}{w}_{i1}{z}_i}\right]+{\text{Var}}_p\left[{\displaystyle \sum _{i\in s_2}{w}_{i2}{z}_{i(2\cdots 3)}}\right]-{\text{Var}}_p\left[{\displaystyle \sum _{i\in s_1}{w}_{i1}{z}_{i(2\cdots 3)}}\right]\\ +{\text{Var}}_p\left[{\displaystyle \sum _{i\in s_3}{w}_{i3}{z}_{i(3\cdots 3)}}\right]-{\text{Var}}_p\left[{\displaystyle \sum _{i\in s_2}{w}_{i2}{z}_{i(3\cdots 3)}}\right]\\ +2{\text{Cov}}_p\left[{\displaystyle \sum _{i\in s_1}{w}_{i1}{z}_{i(1\cdots 1)}\mathrm{,}}{\displaystyle \sum _{i\in s_2}{w}_{i2}{z}_{i(2\cdots 3)}}\right]-2{\text{Cov}}_p\left[{\displaystyle \sum _{i\in s_1}{w}_{i1}{z}_{i(1\cdots 1)}\mathrm{,}}{\displaystyle \sum _{i\in s_1}{w}_{i1}{z}_{i(2\cdots 3)}}\right]\\ +2{\text{Cov}}_p\left[{\displaystyle \sum _{i\in s_1}{w}_{i1}{z}_{i(1\cdots 2)}\mathrm{,}}{\displaystyle \sum _{i\in s_3}{w}_{i3}{z}_{i(3\cdots 3)}}\right]-2{\text{Cov}}_p\left[{\displaystyle \sum _{i\in s_1}{w}_{i1}{z}_{i(1\cdots 2)}\mathrm{,}}{\displaystyle \sum _{i\in s_2}{w}_{i2}{z}_{i(3\cdots 3)}}\right]\\ +2{\text{Cov}}_p\left[{\displaystyle \sum _{i\in s_1}{w}_{i1}{z}_{i(2\cdots 2)}\mathrm{,}}{\displaystyle \sum _{i\in s_2}{w}_{i2}{z}_{i(3\cdots 3)}}\right]-2{\text{Cov}}_p\left[{\displaystyle \sum _{i\in s_1}{w}_{i1}{z}_{i(2\cdots 2)}\mathrm{,}}{\displaystyle \sum _{i\in s_3}{w}_{i3}{z}_{i(3\cdots 3)}}\right]\text{ (A.4)}\\ +2{\text{Cov}}_p\left[{\displaystyle \sum _{i\in s_2}{w}_{i2}{z}_{i(2\cdots 2)}\mathrm{,}}{\displaystyle \sum _{i\in s_3}{w}_{i3}{z}_{i(3\cdots 3)}}\right]-2{\text{Cov}}_p\left[{\displaystyle \sum _{i\in s_2}{w}_{i2}{z}_{i(2\cdots 2)}\mathrm{,}}{\displaystyle \sum _{i\in s_2}{w}_{i2}{z}_{i(3\cdots 3)}}\right]\mathrm{.}\end{array}$

In this last expression, the first thing we notice is that all the diagonal elements in all the covariance terms are exactly equal to zero; this means that whether or not the cohorts are independent of one another, expression (3.13) is exact for the variance terms.
To analyze the importance of the covariance terms, we concentrate on the term in line (A.4); the conclusion for the other terms is the same; note that this term can be written as:

$2{\text{Cov}}_p\left[{\displaystyle \sum _{i\in s_1}{w}_{i1}}\frac{\partial {{\mu }^{\prime }}_i}{\partial \beta }{V}_i^{-1}\left(\begin{array}{c}{e}_{i1}\\ e_{i2}\\ 0\end{array}\right)\mathrm{,}{\displaystyle \sum _{i\in s_3}{w}_{i3}}\frac{\partial {{\mu }^{\prime }}_i}{\partial \beta }{V}_i^{-1}\left(\begin{array}{c}0\\ 0\\ e_{i3}\end{array}\right)-{\displaystyle \sum _{i\in s_2}{w}_{i2}}\frac{\partial {{\mu }^{\prime }}_i}{\partial \beta }{V}_i^{-1}\left(\begin{array}{c}0\\ 0\\ e_{i3}\end{array}\right)\right];$

Property 3.1 states that if the cohorts are design-independent, all the covariance terms are exactly equal to zero. In addition to that, from this last expression we conclude, trivially, that if the waves are design-independent, all the covariance terms are equal to zero too. This formula for the term in line (A.4) also implies that if the individual weights do not vary greatly between consecutive waves, and there is a high overlap between consecutive waves, the covariance terms are not too large. Finally, if the overlap is small, it is reasonable to assume design-independence between the waves, and then the covariance terms can be safely approximated by zero.

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