Variance estimation under monotone non-response for a panel survey
Section 7. Conclusion

In this paper, we considered variance estimation accounting for weighting adjustments in panel surveys. We proposed both an approximately unbiased variance estimator and a simplified variance estimator for estimators of totals, complex parameters and measures of change, which covers most cases that may be encountered in practice. Our simulation results indicate that the proposed variance estimator performs well in all cases considered. The simplified variance estimator tends to overestimate the variance of the expansion estimator for totals, and to overestimate the variance for calibrated estimators of totals when the calibration variables lack of explanatory power for the variable of interest. However, the simplified variance estimator performs well for the estimation of ratios and change in totals with calibrated weights, even if the calibration model is not appropriate for the study variable.

The assumption of independent response behaviour is usually not tenable for multi-stage surveys, since units within clusters tend to be correlated with respect to the response behaviour. In this context, estimation of response probabilities based upon conditional logistic regression in the context of correlated responses has been studied by Skinner and D’Arrigo (2011), see also Kim, Kwon and Park (2016). Extending the present work in the context of correlated response behaviour is a challenging problem for further research.

Acknowledgements

We thank the Editors, an Associate Editor and the referees for useful comments and suggestions which led to an improvement of the paper.

Appendix

Estimation of the variance due to non-response for Response Homogeneity Groups

We consider the model of Response Homogeneity Groups introduced in Section 2.5. Recall that this model may be summarized as follows: at each time δ = 1, , t , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacqaH0oazcaaI9aGaaGymaiaaiYcaca aMe8UaeSOjGSKaaGilaiaaysW7caWG0bGaaiilaaaa@3C44@ the sub-sample s δ 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGZbWaaSbaaSqaaiabes7aKjabgk HiTiaaigdaaeqaaaaa@363D@ is partitioned into C ( δ 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGdbWaaeWaaeaacqaH0oazcqGHsi slcaaIXaaacaGLOaGaayzkaaaaaa@376A@ groups s δ 1 c , c = 1, , C ( δ 1 ) . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGZbWaa0baaSqaaiabes7aKjabgk HiTiaaigdaaeaacaWGJbaaaOGaaGilaiaaysW7caWGJbGaaGypaiaa igdacaaISaGaaGjbVlablAciljaaiYcacaaMe8Uaam4qamaabmaaba GaeqiTdqMaeyOeI0IaaGymaaGaayjkaiaawMcaaiaac6caaaa@47D5@ The response probabilities are assumed to be constant within the groups.

This model is equivalent to the logistic regression model in (2.18), with

z i δ = [ 1 { i s δ 1 1 } , , 1 { i s δ 1 C ( δ 1 ) } ] . ( A .1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWG6bWaa0baaSqaaiaadMgaaeaacq aH0oazaaGccaaI9aWaamWaaeaacaaIXaWaaiWaaeaacaWGPbGaeyic I4Saam4CamaaDaaaleaacqaH0oazcqGHsislcaaIXaaabaGaaGymaa aaaOGaay5Eaiaaw2haaiaaiYcacaaMe8UaeSOjGSKaaGilaiaaysW7 caaIXaWaaiWaaeaacaWGPbGaeyicI4Saam4CamaaDaaaleaacqaH0o azcqGHsislcaaIXaaabaGaam4qamaabmaabaGaeqiTdqMaeyOeI0Ia aGymaaGaayjkaiaawMcaaaaaaOGaay5Eaiaaw2haaaGaay5waiaaw2 faamaaCaaaleqabaWefv3ySLgznfgDOfdaryqr1ngBPrginfgDObYt UvgaiqqacqWFKksLaaGccaaMb8UaaGOlaiaaywW7caaMf8UaaGzbVl aaywW7caaMf8UaaiikaiaabgeacaqGUaGaaGymaiaacMcaaaa@7134@

The equation (2.2) leads to the estimated response probabilities

p ^ i δ = i s δ 1 c k i δ r i δ i s δ 1 c k i δ for i s δ 1 c . ( A .2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaceWGWbGbaKaadaqhaaWcbaGaamyAaa qaaiabes7aKbaakiaai2dadaWcaaqaamaaqababaGaam4AamaaDaaa leaacaWGPbaabaGaeqiTdqgaaOGaamOCamaaDaaaleaacaWGPbaaba GaeqiTdqgaaaqaaiaadMgacqGHiiIZcaWGZbWaa0baaWqaaiabes7a KjabgkHiTiaaigdaaeaacaWGJbaaaaWcbeqdcqGHris5aaGcbaWaaa beaeaacaWGRbWaa0baaSqaaiaadMgaaeaacqaH0oazaaaabaGaamyA aiabgIGiolaadohadaqhaaadbaGaeqiTdqMaeyOeI0IaaGymaaqaai aadogaaaaaleqaniabggHiLdaaaOGaaGzbVlaaywW7caqGMbGaae4B aiaabkhacaaMf8UaaGzbVlaadMgacqGHiiIZcaWGZbWaa0baaSqaai abes7aKjabgkHiTiaaigdaaeaacaWGJbaaaOGaaGOlaiaaywW7caaM f8UaaGzbVlaaywW7caaMf8UaaiikaiaabgeacaqGUaGaaGOmaiaacM caaaa@71EC@

We first consider the case when the reweighted estimator is computed at time t = 1. MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWG0bGaaGypaiaaigdacaGGUaaaaa@34F9@ In the estimator of the variance due to non-response given in (2.21), the vector γ ^ 1 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacuaHZoWzgaqcamaaDaaaleaacaaIXa aabaGaaGymaaaaaaa@3526@ simplifies as

γ ^ 1 1 = ( i s 1 s 0 1 y i 1 π i p ^ 1 1 i s 1 s 0 1 k i 1 , , i s 1 s 0 C ( 0 ) y i 1 π i p ^ C ( 0 ) 1 i s 1 s 0 C ( 0 ) k i 1 ) . ( A .3 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacuaHZoWzgaqcamaaDaaaleaacaaIXa aabaGaaGymaaaakiaai2dadaqadaqaamaalaaabaWaaabeaeaadaWc baWcbaGaamyEamaaBaaameaacaWGPbGaaGymaaqabaaaleaacqaHap aCdaWgaaadbaGaamyAaaqabaaaaaWcbaGaamyAaiabgIGiolaadoha daWgaaadbaGaaGymaaqabaWccqGHPiYXcaWGZbWaa0baaWqaaiaaic daaeaacaaIXaaaaaWcbeqdcqGHris5aaGcbaGabmiCayaajaWaa0ba aSqaaiaaigdaaeaacaaIXaaaaOWaaabeaeaacaWGRbWaa0baaSqaai aadMgaaeaacaaIXaaaaaqaaiaadMgacqGHiiIZcaWGZbWaaSbaaWqa aiaaigdaaeqaaSGaeyykICSaam4CamaaDaaameaacaaIWaaabaGaaG ymaaaaaSqab0GaeyyeIuoaaaGccaaISaGaaGjbVlablAciljaaiYca caaMe8+aaSaaaeaadaaeqaqaamaaleaaleaacaWG5bWaaSbaaWqaai aadMgacaaIXaaabeaaaSqaaiabec8aWnaaBaaameaacaWGPbaabeaa aaaaleaacaWGPbGaeyicI4Saam4CamaaBaaameaacaaIXaaabeaali abgMIihlaadohadaqhaaadbaGaaGimaaqaaiaadoeadaqadaqaaiaa icdaaiaawIcacaGLPaaaaaaaleqaniabggHiLdaakeaaceWGWbGbaK aadaqhaaWcbaGaam4qamaabmaabaGaaGimaaGaayjkaiaawMcaaaqa aiaaigdaaaGcdaaeqaqaaiaadUgadaqhaaWcbaGaamyAaaqaaiaaig daaaaabaGaamyAaiabgIGiolaadohadaWgaaadbaGaaGymaaqabaWc cqGHPiYXcaWGZbWaa0baaWqaaiaaicdaaeaacaWGdbWaaeWaaeaaca aIWaaacaGLOaGaayzkaaaaaaWcbeqdcqGHris5aaaaaOGaayjkaiaa wMcaamaaCaaaleqabaWefv3ySLgznfgDOfdaryqr1ngBPrginfgDOb YtUvgaiqqacqWFKksLaaGccaaIUaGaaGzbVlaaywW7caaMf8UaaGzb VlaaywW7caGGOaGaaeyqaiaab6cacaaIZaGaaiykaaaa@9C99@

After some algebra, the variance estimator in (2.21) may be rewritten as

V ^ 1 nr ( Y ^ 1 ) = c = 1 C ( 0 ) ( 1 p ^ c 1 ) ( p ^ c 1 ) 2 i s 1 s 0 c ( y i 1 π i k i 1 j s 1 s 0 c y j 1 π j j s 1 s 0 c k j 1 ) 2 . ( A .4 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaceWGwbGbaKaadaqhaaWcbaGaaGymaa qaaiaab6gacaqGYbaaaOWaaeWaaeaaceWGzbGbaKaadaWgaaWcbaGa aGymaaqabaaakiaawIcacaGLPaaacaaI9aWaaabCaeqaleaacaWGJb GaaGypaiaaigdaaeaacaWGdbWaaeWaaeaacaaIWaaacaGLOaGaayzk aaaaniabggHiLdGcdaWcaaqaamaabmaabaGaaGymaiabgkHiTiqadc hagaqcamaaDaaaleaacaWGJbaabaGaaGymaaaaaOGaayjkaiaawMca aaqaamaabmaabaGabmiCayaajaWaa0baaSqaaiaadogaaeaacaaIXa aaaaGccaGLOaGaayzkaaWaaWbaaSqabeaacaaIYaaaaaaakmaaqafa beWcbaGaamyAaiabgIGiolaadohadaWgaaadbaGaaGymaaqabaWccq GHPiYXcaWGZbWaa0baaWqaaiaaicdaaeaacaWGJbaaaaWcbeqdcqGH ris5aOWaaeWaaeaadaWcaaqaaiaadMhadaWgaaWcbaGaamyAaiaaig daaeqaaaGcbaGaeqiWda3aaSbaaSqaaiaadMgaaeqaaaaakiabgkHi TiaadUgadaqhaaWcbaGaamyAaaqaaiaaigdaaaGcdaWcaaqaamaaqa babaWaaSqaaSqaaiaadMhadaWgaaadbaGaamOAaiaaigdaaeqaaaWc baGaeqiWda3aaSbaaWqaaiaadQgaaeqaaaaaaSqaaiaadQgacqGHii IZcaWGZbWaaSbaaWqaaiaaigdaaeqaaSGaeyykICSaam4CamaaDaaa meaacaaIWaaabaGaam4yaaaaaSqab0GaeyyeIuoaaOqaamaaqababa Gaam4AamaaDaaaleaacaWGQbaabaGaaGymaaaaaeaacaWGQbGaeyic I4Saam4CamaaBaaameaacaaIXaaabeaaliabgMIihlaadohadaqhaa adbaGaaGimaaqaaiaadogaaaaaleqaniabggHiLdaaaaGccaGLOaGa ayzkaaWaaWbaaSqabeaacaaIYaaaaOGaaGOlaiaaywW7caaMf8UaaG zbVlaaywW7caaMf8UaaiikaiaabgeacaqGUaGaaGinaiaacMcaaaa@8E20@

We now consider the case when the reweighted estimator is computed at time t = 2. MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWG0bGaaGypaiaaikdacaGGUaaaaa@34FA@ We focus on the simpler case when the same system of RHGs is kept over time. In the estimator of the variance due to non-response given in (2.22), the vectors γ ^ 2 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacuaHZoWzgaqcamaaDaaaleaacaaIYa aabaGaaGymaaaaaaa@3527@ and γ ^ 2 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacuaHZoWzgaqcamaaDaaaleaacaaIYa aabaGaaGOmaaaaaaa@3528@ simplify as

γ ^ 2 1 = ( i s 2 s 1 1 y i 2 π i p ^ 1 1 i s 2 s 1 1 k i 1 , , i s 2 s 1 C ( 0 ) y i 2 π i p ^ C ( 0 ) 1 i s 2 s 1 C ( 0 ) k i 1 ) , ( A .5 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacuaHZoWzgaqcamaaDaaaleaacaaIYa aabaGaaGymaaaakiaai2dadaqadaqaamaalaaabaWaaabeaeaadaWc baWcbaGaamyEamaaBaaameaacaWGPbGaaGOmaaqabaaaleaacqaHap aCdaWgaaadbaGaamyAaaqabaaaaaWcbaGaamyAaiabgIGiolaadoha daWgaaadbaGaaGOmaaqabaWccqGHPiYXcaWGZbWaa0baaWqaaiaaig daaeaacaaIXaaaaaWcbeqdcqGHris5aaGcbaGabmiCayaajaWaa0ba aSqaaiaaigdaaeaacaaIXaaaaOWaaabeaeaacaWGRbWaa0baaSqaai aadMgaaeaacaaIXaaaaaqaaiaadMgacqGHiiIZcaWGZbWaaSbaaWqa aiaaikdaaeqaaSGaeyykICSaam4CamaaDaaameaacaaIXaaabaGaaG ymaaaaaSqab0GaeyyeIuoaaaGccaaISaGaaGjbVlablAciljaaiYca caaMe8+aaSaaaeaadaaeqaqaamaaleaaleaacaWG5bWaaSbaaWqaai aadMgacaaIYaaabeaaaSqaaiabec8aWnaaBaaameaacaWGPbaabeaa aaaaleaacaWGPbGaeyicI4Saam4CamaaBaaameaacaaIYaaabeaali abgMIihlaadohadaqhaaadbaGaaGymaaqaaiaadoeadaqadaqaaiaa icdaaiaawIcacaGLPaaaaaaaleqaniabggHiLdaakeaaceWGWbGbaK aadaqhaaWcbaGaam4qamaabmaabaGaaGimaaGaayjkaiaawMcaaaqa aiaaigdaaaGcdaaeqaqaaiaadUgadaqhaaWcbaGaamyAaaqaaiaaig daaaaabaGaamyAaiabgIGiolaadohadaWgaaadbaGaaGOmaaqabaWc cqGHPiYXcaWGZbWaa0baaWqaaiaaigdaaeaacaWGdbWaaeWaaeaaca aIWaaacaGLOaGaayzkaaaaaaWcbeqdcqGHris5aaaaaOGaayjkaiaa wMcaamaaCaaaleqabaWefv3ySLgznfgDOfdaryqr1ngBPrginfgDOb YtUvgaiqqacqWFKksLaaGccaaMb8UaaGilaiaaywW7caaMf8UaaGzb VlaaywW7caaMf8UaaiikaiaabgeacaqGUaGaaGynaiaacMcaaaa@9E2E@

γ ^ 2 2 = ( i s 2 s 1 1 y i 2 π i p ^ 1 1 p ^ 1 2 i s 2 s 1 1 k i 2 , , i s 2 s 1 C ( 0 ) y i 2 π i p ^ C ( 0 ) 1 p ^ C ( 0 ) 2 i s 2 s 1 C ( 0 ) k i 2 ) . ( A .6 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacuaHZoWzgaqcamaaDaaaleaacaaIYa aabaGaaGOmaaaakiaai2dadaqadaqaamaalaaabaWaaabeaeaadaWc baWcbaGaamyEamaaBaaameaacaWGPbGaaGOmaaqabaaaleaacqaHap aCdaWgaaadbaGaamyAaaqabaaaaaWcbaGaamyAaiabgIGiolaadoha daWgaaadbaGaaGOmaaqabaWccqGHPiYXcaWGZbWaa0baaWqaaiaaig daaeaacaaIXaaaaaWcbeqdcqGHris5aaGcbaGabmiCayaajaWaa0ba aSqaaiaaigdaaeaacaaIXaaaaOGabmiCayaajaWaa0baaSqaaiaaig daaeaacaaIYaaaaOWaaabeaeaacaWGRbWaa0baaSqaaiaadMgaaeaa caaIYaaaaaqaaiaadMgacqGHiiIZcaWGZbWaaSbaaWqaaiaaikdaae qaaSGaeyykICSaam4CamaaDaaameaacaaIXaaabaGaaGymaaaaaSqa b0GaeyyeIuoaaaGccaaISaGaaGjbVlablAciljaaiYcacaaMe8+aaS aaaeaadaaeqaqaamaaleaaleaacaWG5bWaaSbaaWqaaiaadMgacaaI YaaabeaaaSqaaiabec8aWnaaBaaameaacaWGPbaabeaaaaaaleaaca WGPbGaeyicI4Saam4CamaaBaaameaacaaIYaaabeaaliabgMIihlaa dohadaqhaaadbaGaaGymaaqaaiaadoeadaqadaqaaiaaicdaaiaawI cacaGLPaaaaaaaleqaniabggHiLdaakeaaceWGWbGbaKaadaqhaaWc baGaam4qamaabmaabaGaaGimaaGaayjkaiaawMcaaaqaaiaaigdaaa GcceWGWbGbaKaadaqhaaWcbaGaam4qamaabmaabaGaaGimaaGaayjk aiaawMcaaaqaaiaaikdaaaGcdaaeqaqaaiaadUgadaqhaaWcbaGaam yAaaqaaiaaikdaaaaabaGaamyAaiabgIGiolaadohadaWgaaadbaGa aGOmaaqabaWccqGHPiYXcaWGZbWaa0baaWqaaiaaigdaaeaacaWGdb WaaeWaaeaacaaIWaaacaGLOaGaayzkaaaaaaWcbeqdcqGHris5aaaa aOGaayjkaiaawMcaamaaCaaaleqabaWefv3ySLgznfgDOfdaryqr1n gBPrginfgDObYtUvgaiqqacqWFKksLaaGccaaIUaGaaGzbVlaaywW7 caaMf8UaaGzbVlaaywW7caGGOaGaaeyqaiaab6cacaaI2aGaaiykaa aa@A460@

After some algebra, the variance estimator in (2.22) may be rewritten as

V ^ 2 nr ( Y ^ 2 ) = c = 1 C ( 0 ) ( 1 p ^ c 1 ) p ^ c 2 i s 2 s 1 c ( y i 2 π i p ^ c 1 k i 1 j s 2 s 1 c y j 2 π j j s 2 s 1 c k j 1 ) 2 + c = 1 C ( 0 ) ( 1 p ^ c 2 ) i s 2 s 1 c ( y i 2 π i p ^ c 1 p ^ c 2 k i 2 j s 2 s 1 c y j 2 π j j s 2 s 1 c k j 2 ) 2 . ( A .7 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaafaqaaeGacaaabaGabmOvayaajaWaa0 baaSqaaiaaikdaaeaacaqGUbGaaeOCaaaakmaabmaabaGabmywayaa jaWaaSbaaSqaaiaaikdaaeqaaaGccaGLOaGaayzkaaaabaGaaGypam aaqahabeWcbaGaam4yaiaai2dacaaIXaaabaGaam4qamaabmaabaGa aGimaaGaayjkaiaawMcaaaqdcqGHris5aOWaaSaaaeaadaqadaqaai aaigdacqGHsislceWGWbGbaKaadaqhaaWcbaGaam4yaaqaaiaaigda aaaakiaawIcacaGLPaaaaeaaceWGWbGbaKaadaqhaaWcbaGaam4yaa qaaiaaikdaaaaaaOWaaabuaeqaleaacaWGPbGaeyicI4Saam4Camaa BaaameaacaaIYaaabeaaliabgMIihlaadohadaqhaaadbaGaaGymaa qaaiaadogaaaaaleqaniabggHiLdGcdaqadaqaamaalaaabaGaamyE amaaBaaaleaacaWGPbGaaGOmaaqabaaakeaacqaHapaCdaWgaaWcba GaamyAaaqabaGcceWGWbGbaKaadaqhaaWcbaGaam4yaaqaaiaaigda aaaaaOGaeyOeI0Iaam4AamaaDaaaleaacaWGPbaabaGaaGymaaaakm aalaaabaWaaabeaeaadaWcbaWcbaGaamyEamaaBaaameaacaWGQbGa aGOmaaqabaaaleaacqaHapaCdaWgaaadbaGaamOAaaqabaaaaaWcba GaamOAaiabgIGiolaadohadaWgaaadbaGaaGOmaaqabaWccqGHPiYX caWGZbWaa0baaWqaaiaaigdaaeaacaWGJbaaaaWcbeqdcqGHris5aa GcbaWaaabeaeaacaWGRbWaa0baaSqaaiaadQgaaeaacaaIXaaaaaqa aiaadQgacqGHiiIZcaWGZbWaaSbaaWqaaiaaikdaaeqaaSGaeyykIC Saam4CamaaDaaameaacaaIXaaabaGaam4yaaaaaSqab0GaeyyeIuoa aaaakiaawIcacaGLPaaadaahaaWcbeqaaiaaikdaaaaakeaaaeaacq GHRaWkdaaeWbqabSqaaiaadogacaaI9aGaaGymaaqaaiaadoeadaqa daqaaiaaicdaaiaawIcacaGLPaaaa0GaeyyeIuoakmaabmaabaGaaG ymaiabgkHiTiqadchagaqcamaaDaaaleaacaWGJbaabaGaaGOmaaaa aOGaayjkaiaawMcaamaaqafabeWcbaGaamyAaiabgIGiolaadohada WgaaadbaGaaGOmaaqabaWccqGHPiYXcaWGZbWaa0baaWqaaiaaigda aeaacaWGJbaaaaWcbeqdcqGHris5aOWaaeWaaeaadaWcaaqaaiaadM hadaWgaaWcbaGaamyAaiaaikdaaeqaaaGcbaGaeqiWda3aaSbaaSqa aiaadMgaaeqaaOGabmiCayaajaWaa0baaSqaaiaadogaaeaacaaIXa aaaOGabmiCayaajaWaa0baaSqaaiaadogaaeaacaaIYaaaaaaakiab gkHiTiaadUgadaqhaaWcbaGaamyAaaqaaiaaikdaaaGcdaWcaaqaam aaqababaWaaSqaaSqaaiaadMhadaWgaaadbaGaamOAaiaaikdaaeqa aaWcbaGaeqiWda3aaSbaaWqaaiaadQgaaeqaaaaaaSqaaiaadQgacq GHiiIZcaWGZbWaaSbaaWqaaiaaikdaaeqaaSGaeyykICSaam4Camaa DaaameaacaaIXaaabaGaam4yaaaaaSqab0GaeyyeIuoaaOqaamaaqa babaGaam4AamaaDaaaleaacaWGQbaabaGaaGOmaaaaaeaacaWGQbGa eyicI4Saam4CamaaBaaameaacaaIYaaabeaaliabgMIihlaadohada qhaaadbaGaaGymaaqaaiaadogaaaaaleqaniabggHiLdaaaaGccaGL OaGaayzkaaWaaWbaaSqabeaacaaIYaaaaOGaaGzaVlaai6cacaaMf8 UaaGzbVlaaywW7caaMf8UaaGzbVlaacIcacaqGbbGaaeOlaiaaiEda caGGPaaaaaaa@D9C4@

If we further assume that k i δ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGRbWaa0baaSqaaiaadMgaaeaacq aH0oazaaaaaa@357C@ is constant over times δ = 1, 2 , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacqaH0oazcaaI9aGaaGymaiaaiYcaca aMe8UaaGOmaiaacYcaaaa@38A2@ and may thus be rewritten as k i , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGRbWaaSbaaSqaaiaadMgaaeqaaO Gaaiilaaaa@3490@ the expression in (A.7) simplifies as

V ^ 2 nr ( Y ^ 2 ) = c = 1 C ( 0 ) ( 1 p ^ c 1 2 ) ( p ^ c 1 2 ) 2 i s 2 s 1 c ( y i 2 π i k i j s 2 s 1 c y j 2 π j j s 2 s 1 c k j ) 2 . ( A .8 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaceWGwbGbaKaadaqhaaWcbaGaaGOmaa qaaiaab6gacaqGYbaaaOWaaeWaaeaaceWGzbGbaKaadaWgaaWcbaGa aGOmaaqabaaakiaawIcacaGLPaaacaaI9aWaaabCaeqaleaacaWGJb GaaGypaiaaigdaaeaacaWGdbWaaeWaaeaacaaIWaaacaGLOaGaayzk aaaaniabggHiLdGcdaWcaaqaamaabmaabaGaaGymaiabgkHiTiqadc hagaqcamaaDaaaleaacaWGJbaabaGaaGymaiaaykW7cqGHsgIRcaaM c8UaaGOmaaaaaOGaayjkaiaawMcaaaqaamaabmaabaGabmiCayaaja Waa0baaSqaaiaadogaaeaacaaIXaGaaGPaVlabgkziUkaaykW7caaI YaaaaaGccaGLOaGaayzkaaWaaWbaaSqabeaacaaIYaaaaaaakmaaqa fabeWcbaGaamyAaiabgIGiolaadohadaWgaaadbaGaaGOmaaqabaWc cqGHPiYXcaWGZbWaa0baaWqaaiaaigdaaeaacaWGJbaaaaWcbeqdcq GHris5aOWaaeWaaeaadaWcaaqaaiaadMhadaWgaaWcbaGaamyAaiaa ikdaaeqaaaGcbaGaeqiWda3aaSbaaSqaaiaadMgaaeqaaaaakiabgk HiTiaadUgadaWgaaWcbaGaamyAaaqabaGcdaWcaaqaamaaqababaWa aSqaaSqaaiaadMhadaWgaaadbaGaamOAaiaaikdaaeqaaaWcbaGaeq iWda3aaSbaaWqaaiaadQgaaeqaaaaaaSqaaiaadQgacqGHiiIZcaWG ZbWaaSbaaWqaaiaaikdaaeqaaSGaeyykICSaam4CamaaDaaameaaca aIXaaabaGaam4yaaaaaSqab0GaeyyeIuoaaOqaamaaqababaGaam4A amaaBaaaleaacaWGQbaabeaaaeaacaWGQbGaeyicI4Saam4CamaaBa aameaacaaIYaaabeaaliabgMIihlaadohadaqhaaadbaGaaGymaaqa aiaadogaaaaaleqaniabggHiLdaaaaGccaGLOaGaayzkaaWaaWbaaS qabeaacaaIYaaaaOGaaGOlaiaaywW7caaMf8UaaGzbVlaaywW7caaM f8UaaiikaiaabgeacaqGUaGaaGioaiaacMcaaaa@9834@

with p ^ c 1 2 = δ = 1 2 p ^ c δ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaceWGWbGbaKaadaqhaaWcbaGaam4yaa qaaiaaigdacaaMc8UaeyOKH4QaaGPaVlaaikdaaaGccaaI9aWaaebm aeqaleaacqaH0oazcaaI9aGaaGymaaqaaiaaikdaa0Gaey4dIunaki aaykW7ceWGWbGbaKaadaqhaaWcbaGaam4yaaqaaiabes7aKbaaaaa@465A@ for c = 1, , C ( 0 ) . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGJbGaaGypaiaaigdacaaISaGaaG jbVlablAciljaaiYcacaaMe8Uaam4qamaabmaabaGaaGimaaGaayjk aiaawMcaaiaac6caaaa@3D9B@ This simplification of the variance estimator can be extended to the reweighted estimator at time t . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWG0bGaaiOlaaaa@3377@ Assuming that the RHGs are kept over time, and that k i δ = k i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGRbWaa0baaSqaaiaadMgaaeaacq aH0oazaaGccaaI9aGaam4AamaaBaaaleaacaWGPbaabeaaaaa@3857@ for any δ = 1, , t , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacqaH0oazcaaI9aGaaGymaiaaiYcaca aMe8UaeSOjGSKaaGilaiaaysW7caWG0bGaaiilaaaa@3C44@ the variance estimator in (2.12) may be written as

V ^ t nr ( Y ^ t ) = c = 1 C ( 0 ) ( 1 p ^ c 1 t ) ( p ^ c 1 t ) 2 i s t s t 1 c ( y i t π i k i j s t s t 1 c y j t π j j s t s t 1 c k j ) 2 ( A .9 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaceWGwbGbaKaadaqhaaWcbaGaamiDaa qaaiaab6gacaqGYbaaaOWaaeWaaeaaceWGzbGbaKaadaWgaaWcbaGa amiDaaqabaaakiaawIcacaGLPaaacaaI9aWaaabCaeqaleaacaWGJb GaaGypaiaaigdaaeaacaWGdbWaaeWaaeaacaaIWaaacaGLOaGaayzk aaaaniabggHiLdGcdaWcaaqaamaabmaabaGaaGymaiabgkHiTiqadc hagaqcamaaDaaaleaacaWGJbaabaGaaGymaiaaykW7cqGHsgIRcaaM c8UaamiDaaaaaOGaayjkaiaawMcaaaqaamaabmaabaGabmiCayaaja Waa0baaSqaaiaadogaaeaacaaIXaGaaGPaVlabgkziUkaaykW7caWG 0baaaaGccaGLOaGaayzkaaWaaWbaaSqabeaacaaIYaaaaaaakmaaqa fabeWcbaGaamyAaiabgIGiolaadohadaWgaaadbaGaamiDaaqabaWc cqGHPiYXcaWGZbWaa0baaWqaaiaadshacqGHsislcaaIXaaabaGaam 4yaaaaaSqab0GaeyyeIuoakmaabmaabaWaaSaaaeaacaWG5bWaaSba aSqaaiaadMgacaWG0baabeaaaOqaaiabec8aWnaaBaaaleaacaWGPb aabeaaaaGccqGHsislcaWGRbWaaSbaaSqaaiaadMgaaeqaaOWaaSaa aeaadaaeqaqaamaaleaaleaacaWG5bWaaSbaaWqaaiaadQgacaWG0b aabeaaaSqaaiabec8aWnaaBaaameaacaWGQbaabeaaaaaaleaacaWG QbGaeyicI4Saam4CamaaBaaameaacaWG0baabeaaliabgMIihlaado hadaqhaaadbaGaamiDaiabgkHiTiaaigdaaeaacaWGJbaaaaWcbeqd cqGHris5aaGcbaWaaabeaeaacaWGRbWaaSbaaSqaaiaadQgaaeqaaa qaaiaadQgacqGHiiIZcaWGZbWaaSbaaWqaaiaadshaaeqaaSGaeyyk ICSaam4CamaaDaaameaacaWG0bGaeyOeI0IaaGymaaqaaiaadogaaa aaleqaniabggHiLdaaaaGccaGLOaGaayzkaaWaaWbaaSqabeaacaaI YaaaaOGaaGzbVlaaywW7caaMf8UaaGzbVlaaywW7caGGOaGaaeyqai aab6cacaaI5aGaaiykaaaa@9F54@

with p ^ c 1 t = δ = 1 t p ^ c δ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaceWGWbGbaKaadaqhaaWcbaGaam4yaa qaaiaaigdacaaMc8UaeyOKH4QaaGPaVlaadshaaaGccaaI9aWaaebm aeqaleaacqaH0oazcaaI9aGaaGymaaqaaiaadshaa0Gaey4dIunaki aaykW7ceWGWbGbaKaadaqhaaWcbaGaam4yaaqaaiabes7aKbaaaaa@46D4@ for c = 1, , C ( 0 ) . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGJbGaaGypaiaaigdacaaISaGaaG jbVlablAciljaaiYcacaaMe8Uaam4qamaabmaabaGaaGimaaGaayjk aiaawMcaaiaac6caaaa@3D9B@

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