A Two‐level GREG Estimator for Consistent Estimation in Household Surveys

Konrad, Anne; Burgard, Jan Pablo; Münnich, Ralf Article — Published Version A Two‐level GREG Estimator for Consistent Estimation in Household Surveys International Statistical Review Provided in Cooperation with: John Wiley & Sons Suggested Citation: Konrad, Anne; Burgard, Jan Pablo; Münnich, Ralf (2021) : A Two‐level GREG Estimator for Consistent Estimation in Household Surveys, International Statistical Review, ISSN 1751-5823, Wiley, Hoboken, NJ, Vol. 89, Iss. 3, pp. 635-656, https://doi.org/10.1111/insr.12460 This Version is available at: https://hdl.handle.net/10419/284842 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. http://creativecommons.org/licenses/by-nc/4.0/ A Two-level GREG Estimator for Consistent Estimation in Household Surveys Anne Konrad , Jan Pablo Burgard and Ralf Münnich Economic and Social Statistics Department, University of Trier, Trier, Germany E-mail: [email protected] Summary Household surveys provide information on both person-level and household-level characteristics. To ensure consistent estimates between both levels, statistical offices often use integrated weights that are equal for all persons within a household and the household itself. However, these integrated weights ignore the individual patterns of the persons, and the heterogeneity within a household is no longer reflected. As an alternative to integrated weighting, we propose a two-level generalised regression estimator that is capable of both ensuring consistent person and household estimates and allowing for different weights for persons within a household. A Monte Carlo simulation supports the superiority of our two-level generalised regression estimator compared with integrated weighting. Key words: Integrated weighting; household surveys; generalised regression estimator; calibration; consistent estimation. 1 Introduction Household surveys are an important source of socio-economic and demographic data for scientific researchers as well as for political decision makers. The provided data allow for the estimation of both person and household characteristics. Estimation is characterised by assigning weights to the observed data to infer from the sample to the population (cf. Deville & Särndal, 1992, p. 376). These weights are frequently composed of the design weights and multiple adjustments. The design weight, as the inverse of the inclusion probability, accounts for the sampling design. To improve the efficiency of the resulting estimators, the design weights are often adjusted such that the final estimates of the auxiliary variables agree with their known population totals (cf. Särndal, 2007). A separate weight adjustment at the person and household levels does not necessarily ensure that the estimates of variable that are common to person and household data set coincide between both levels. However, survey users strive to obtain consistent estimates for common variables. Consistent estimates in this context are defined as obtaining the same estimates independent of using the person or the household data set. The vital role of consistency between person-level and household-level estimates is further emphasised as one principle in the European Statistics Code of Practice (cf. Eurostat, 2011, Principle 14). International Statistical Review (2021), 89, 3, 635–656 doi: 10.1111/insr.12460 © 2021 The Authors. International Statistical Review published by John Wiley & Sons Ltd on behalf of International Statistical Institute. This is an open access article under the terms of the Creative Commons Attribution-NonCommercial License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited and is not used for commercial purposes. The current practice to ensure consistent estimates in household surveys is integrated weighting originated by Lemaître & Dufour (1987). The method of integrated weighting produces one single weight for all persons within the same household. This single integrated person weight is then assigned one to one to the household to which the person belongs. Consistency is thereby ensured by using the same weights to estimate person and household characteristics. The strict requirement of equal weights does no longer reflect the heterogeneity of the individual persons within a household, and the individual patterns of the persons are lost. It is intuitive that for very volatile variables, such as income, the resulting estimates might be significantly influenced when the same weights are assigned to all persons within a household, independently of whether they are top earners, children or inactive persons. As a remedy, we propose a two-level generalised regression (GREG) estimator that ensures consistent estimates while overcoming the strict requirement of equal weights for all persons within the same household and the household itself. The underlying idea is to constrain the consistency requirements to variables that are common to the person and household data set. For this purpose, we adopt the idea of incorporating the common variables as additional auxiliaries known from the literature on multiple independent surveys. In contrast to integrated weighting, our proposed two-level GREG estimator uses the original auxiliary information and herein allows for different weights for the persons within a certain household. As a result, consistency is ensured directly and solely for the relevant variables instead of indirectly by aggregating the individual information per household. The paper is organised as follows: Section 2 briefly outlines the basic framework of cluster sampling and GREG estimators. Section 3 introduces integrated weighting and discusses consequences of the strict requirement of equal weights. In Section 4, we propose the two-level GREG estimator as alternative to integrated weighting. As a benchmark to our proposed estimator, we discuss in Section 5 the generalised least squares (GLS) adjustment algorithm, which can also be adopted to ensure consistent estimates. A Monte Carlo (MC) simulation study (Section 6) compares our proposed two-level GREG estimator with integrated weighting and the GLS adjustment algorithm. Section 7 contains concluding remarks. 2 Basic Framework Household surveys are often realised by means of single-stage cluster sampling, characterised by sampling all units within a selected cluster. The sampling process of single-stage cluster sampling consists of two stages. At the first stage, from a finite population of households Uh¼f1;…;g;…;Mg, a sample s h is selected according to the sampling design p(·), where p(s h ) is the probability of selecting s h . The size of the sample s h is denoted by m. Subscript hrefers to the household level. Let U g be the population of persons within household gof size N g . The probability sampling design generates for every household a known inclusion probability πg¼Prðg∈shÞ¼∑sh:g∈shpðshÞwith π g >0. At the second stage, all persons within a selected household are sampled. The finite population and the sample of persons are denoted by Up¼∪g∈UhUg¼f1;…;i;…;Ngand sp¼∪g∈shUg, respectively. Subscript prefers to the person level. The sample size is given by n¼∑g∈shNg. Because all persons within a selected household are sampled, it follows that πi¼Prði∈spÞ¼ Prðg∈shÞ¼πg. Let y i be a variable of interest of person i. For simplicity, we assume full response. The objective is to estimate the unknown population total Ty;p¼∑i∈Upyiat the person level. A basic design unbiased estimator for the population total T y,p is the Horvitz–Thompson (HT) estimator given by ^ THT y;p¼∑i∈spπ1 iyi(cf. Horvitz & Thompson, 1952). 636 KONRAD ET AL. International Statistical Review (2021), 89, 3, 635–656 © 2021 The Authors. International Statistical Review published by John Wiley & Sons Ltd on behalf of International Statistical Institute. A widely used model-assisted estimator incorporating available auxiliary variables is the GREG estimator (cf. Cassel et al., 1997; Särndal, 1980; Isaki & Fuller, 1982, Wright, 1983). The vector containing the auxiliary variables of person iis defined as xi¼ ðxi1;…;xiq;…;xiQÞTwhere xi1¼1 determines the intercept. The corresponding total vector Tx¼ðTx1;…;Txq;…;TxQÞTof dimension Qis known from censuses, registers or other reliable sources. The GREG estimator relies on a linear regression model that specifies the relationship between the variable of interest and the auxiliaries. The person-level regression model ξis given by yi¼xT iβþϵifor alli∈Up(1) with EξðϵiÞ¼0, VξðϵiÞ¼viσ2and EξðϵiϵjÞ¼0 for all i≠j.E ξ and V ξ denote the expectation and the variance with respect to the model ξ. The scale factor v i , with v i >0, has to be known and describes the residual pattern. Based on the assisting model (1), the linear GREG estimator for the unknown population total Typ¼∑i∈Upyiis obtained from ^ TGREG y;p¼ ^ THT y;pþ ^ BTðTx ^ THT xÞ;(2) where ^ B¼∑i∈spπ1 ixixT i  1∑i∈spπ1 ixiyiis an HT-type least squares estimate for β.Itisassumed that the matrix ∑i∈spπ1 ixixT i  1is nonsingular. According to expression (2), the GREG estimator can be interpreted as an HT estimator plus an adjustment term. This adjustment term is composed by the difference of the known and the estimated totals of the auxiliaries weighted by the magnitude of the relationship between the variable of interest and the auxiliary variables. Alternatively to expression (2), the GREG estimator can be written in linearly weighted form ^ TGREG y;p¼P i∈sp wiyi with wGREG i¼π1 iþπ1 iv1 ixiP i∈sp π1 iv1 ixixT i  1 Tx ^ THT x  :(3) An important property of the GREG estimator is that the sums of the auxiliary variables weighted by (3) are consistent with the known totals, that is, ∑i∈spwGREG ixi¼Tx. Based on the first-order Taylor linear approximation, the GREG estimator is approximately design unbiased under mild design conditions on the assisting model and on the sampling design (cf. Särndal, 2007, p. 103). The design unbiasedness of the GREG estimator does not depend on whether the population is really generated by the model ξ. The efficiency is, indeed, influenced by the predictive power of the model (cf. Särndal et al., 1992, p. 227, p. 239). The variance estimator of the GREG estimator under single-stage cluster sampling approximated by Taylor linearisation is given in terms of the residuals (cf. Särndal et al., 1992, p. 129, p. 235). 637Consistent Estimation in Household Surveys International Statistical Review (2021), 89, 3, 635–656 © 2021 The Authors. International Statistical Review published by John Wiley & Sons Ltd on behalf of International Statistical Institute. ^ Vð ^ TGREG y;pÞ¼ X g∈shX k∈sh △gk πgk X i∈spX j∈sp wGREG iriwGREG jrj(4) with residuals ri¼yixT i ^ B,wGREG idefined in (3) and △gk :¼πgk πgπk. Further correction methods can be applied in order to stabilise and improve the variance estimates (cf. D’Arrigo & Skinner, 2010; Kott, 2009). The GREG estimator can be seen as a special case of a broader class of calibration estimators where the calibrated weights are chosen as close as possible to the original design weights (cf. Deville & Särndal, 1992; Deville et al., 1993). Closeness between both weights is measured via a pre-specified distance function. For the requirements on the distance, see, for example, in Haziza and Beaumont (2017, p. 213). The choice of the chi-squared distance function leads to the GREG weights defined in (3). Further distance functions are discussed in Deville & Särndal (1992), Deville et al. (1993), Huang & Fuller (1978), Alexander (1987), Singh & Mohl (1996) and Stukel et al. (1996). The calibration estimator generated by different distance functions asymptotically equals the GREG estimator (Deville & Särndal, 1992). Thus, for large sample sizes, the choice of the distance function has only a minor impact on the properties of the calibration estimator. Singh & Mohl (1996) and Stukel et al. (1996) extended this finding to modest sample sizes. A current review of calibration methods and distance functions can be found in Devaud & Tillé (2019). 3 Integrated Weighting In the integrated weighting approach, introduced by Lemaître & Dufour (1987), consistency between person-level and household-level estimates is ensured by calculating weights at one level and then assigning these weights one to one to the respective other level. As a consequence thereof, it is not necessarily guaranteed that the weights at the person level sum up to the number of persons in the population and that simultaneously the weights at the household level sum up to the total number of households in the population. As it is an important property, we call this compliance at both levels of the sum of the weights with the population values as the integrated property. To ensure the integrated property, an additional variable has to be incorporated into the auxiliary variables xi. Therefore, we define at the person level x∘ i¼ðxi0;xi1;xi2;…;xiQÞT¼ðN1 g;1;xi2;…;xiQÞT¼ðN1 g;xiÞT as the integrated auxiliary vector of dimension (Q+ 1), which sums up within each household gto x∘ g¼ðxg0;xg1;xg2;…;xgQÞT¼ð1;Ng;xg2;…;xgQÞT: Superscript ∘indicates the integrated property. The corresponding known and estimated total vectors of dimension (Q+ 1) are denoted by Tx∘¼ðM;TxÞTand ^ THT x°¼ð ^ THT x0; ^ THT xÞT, respectively. 3.1 Point and Variance Estimator To produce the same weights for all persons within the same household, the individual auxiliary variables are replaced by the household mean value. The household mean value is determined by  x∘ g¼N1 g∑i∈Ugx∘ i, which is assigned to all persons within the household. At the 638 KONRAD ET AL. International Statistical Review (2021), 89, 3, 635–656 © 2021 The Authors. International Statistical Review published by John Wiley & Sons Ltd on behalf of International Statistical Institute. person level, the household mean value is denoted as  x∘ i. By the substitution of  x∘ ias auxiliary variables in the assisting model (1), the integrated GREG estimator results expressed by ^ TINT y;p¼ ^ THT y;pþ ^ B∘TðTx∘ ^ THT x°Þ(5) with ^ B∘¼∑i∈spπ1 iv1 i  x∘ i  x∘ i T  1∑i∈spπ1 iv1 i  x∘ iyi. The corresponding integrated person weights are given by wINT i¼1 πi þ  x∘T i πiviX i∈sp  x∘ i  x∘T i πgvg ! 1 ðTx∘ ^ THT x°Þ;(6) which are equivalent to the household weights ¼1 πg þ  x∘T i πgvgX g∈sh  x∘ i  x∘T i πgvg ! 1 ðTx∘ ^ THT x°Þ ¼wINT g: Note that vg¼∑i∈Ugvi. Inserting vi¼1 into (6) results in the integrated GREG estimator introduced by Lemaître & Dufour (1987). Inserting vi¼N1 g, in turn, gives the integrated GREG estimator proposed by Nieuwenbroek (1993). The choice vi¼N1 gcan be interpreted that the variance of the variable of interest decreases with the household size (Nieuwenbroek, 1993, p. 9). As the integrated GREG estimator is a special case of (2), it is also design unbiased. Its variance estimator is approximated by Taylor linearisation that is given by ^ Vð ^ TINT y;pÞ¼ X g∈shX k∈sh △gk πgk wINT grINT gwINT krINT k(7) with residuals rINT g¼ygx∘ g T^ B∘. As an alternative to integrated weighting, Zieschang (1986), Luery (1986) and Alexander (1987) discussed a calibration estimator using the chi-squared distance function to produce household weights. When these weights are applied for both person and household characteristics, the calibration weights are asymptotically equivalent to the integrated weights defined in (6) using vi¼N1 g. Verma & Clémenceau (1996) proposed to extend the household-level auxiliary variables in the calibration estimator by the person-level information. For this purpose, the sample distribution of the person auxiliaries is inflated by the household size. Branson & Wittenberg (2014) suggested a minimum cross-entropy approach to produce integrated weights. Wittenberg (2010) showed that the minimum cross-entropy approach is equivalent to the raking estimator introduced by Deming & Stephan (1940) and to the calibration estimator with a multiplicative distance function. Compared with the integrated weights defined in (6), the minimum cross-entropy weights are prevented from being negative. Isaki et al. (2004) used quadratic programming to produce household weights suitable for the estimation of person and household characteristics. Park & Fuller (2005, p. 8) showed that quadratic programming is equal to the calibration estimator with a truncated linear distance function. When dropping the bounds in the distance function, quadratic programming generates weights that are asymptotically equivalent to the integrated weights defined in (6) with vi¼N1 g. Neethling & Galpin (2006) empirically compared a calibration estimator with integrated weighting using a linear and a 639Consistent Estimation in Household Surveys International Statistical Review (2021), 89, 3, 635–656 © 2021 The Authors. International Statistical Review published by John Wiley & Sons Ltd on behalf of International Statistical Institute. multiplicative distance function. Boonstra et al. (2003) and Houbiers (2004) introduced repeated weighting to enforce consistency among contingency tables of survey estimates. We do not pursue repeated weighting because we focus on producing global weights that are suitable for all survey variables. 3.2 Consequences of Integrated Weighting The requirement of equal weights for all person within and the household itself causes various consequences. First, due to the aggregation of the individual auxiliary variables to constructed household mean values, the outcome values of the original auxiliary variables are redistributed within a household. In explanation, suppose the original auxiliary x i indicates whether the person is male or female. Whereas the original auxiliary has only two possible outcome values, 0 or 1, the number of possible values in the integrated approach increases with the number of household members: for a single-person household, it is 0 or 1; for a two-person household, 0, 1/2 or 1; for a three-person household, 0, 1/3, 2/3 or 1; for a four-person household, 0, 1/4, 2/4 or 3/4; and so on. The increased number of outcome values in the integrated weighting approach may enlarge the variation of the integrated weights. Second, through the substitution of the original auxiliary variables by the household mean values, the variances and covariances of the integrated auxiliaries only capture the variances and covariances between the households. Ignoring the within variance and within covariance implies that the heterogeneity of the persons within a household is not taken into account. Thus, the integrated weighting approach does not exploit all available sample information when estimating the coefficient vector in the estimator. The lower the variation of the auxiliaries, the less stable the projection onto the space spanned by the auxiliaries and the higher the variance of the integrated coefficients. Therefore, we expect that the integrated coefficients vary more compared with the coefficients resulting from an ordinary GREG estimator defined in (2), which, in turn, decreases the efficiency of the final estimates. This argumentation contradicts Steel & Clark (2007) who stated that the correlation within households is not relevant to estimate population totals in single-stage cluster sampling because sampled and non-sampled persons live in distinct households. In the simulation study, we examine which argument prevails. Third, the one-to-one weight assignment in the integrated approach tacitly assumes that the strength of the relationship between the variable of interest and the auxiliary variables is identical at both levels. However, Robinson (1950) showed that the correlations for the same variables can be different at the individual level than at the aggregated level. This phenomenon is known as ecological fallacy. In case of ecological fallacy, the coefficients at both levels differ, and in the extreme case, also their signs differ. The difference between the coefficients at both levels affects the efficiency of the integrated estimates because the GREG estimator is model assisted. Thus, even if its approximately design unbiasedness does not depend on the correctness of the model, its efficiency, in turn, relies on the strength of the relationship between the variable of interest and the auxiliaries. Fourth, the integrated GREG estimator requires x i0 as additional auxiliary variable to ensure the integrated property. The simulation study in Section 6 validates whether these consequences of integrated weighting affect the quality of the final estimates. 4 Proposed Two-level Generalised Regression Estimator As an alternative to integrated weighting, we propose a two-level GREG estimator that is capable of both ensuring consistent person and household estimates and allowing for different weights for persons within a household. 640 KONRAD ET AL. International Statistical Review (2021), 89, 3, 635–656 © 2021 The Authors. International Statistical Review published by John Wiley & Sons Ltd on behalf of International Statistical Institute. 4.1 Core Idea The idea underlying our proposed two-level GREG estimator is to constrain the consistency requirements to variables that are common to the person and household data set. By incorporating the common variables as additional auxiliaries, our two-level GREG estimator produces consistent estimates of these variables. Thus, consistency is ensured more directly and only for the relevant variables, instead of indirectly by aggregating the individual information per household as it is done in the integrated weighting approach. To implement the two-level GREG estimator, we adapt the method given by Renssen & Nieuwenbroek (1997), known from the literature on combining information from multiple independent surveys. However, there are considerable differences between multiple independent surveys and household surveys in terms of the definition of common variables, the dependence of the surveys and differing target populations. The main advantage of our two-level GREG estimator compared with integrated weighting is that the original individual rather than the constructed aggregated auxiliaries are utilised. Therefore, the individual pattern of the persons is retained. Furthermore, our two-level GREG estimator consists of separate person-level and household-level estimators, providing two further advantages. First, the different calculation levels of person and household characteristics, which prevent the problems caused by ecological fallacy, are considered. Second, the variable selection process is more flexible because different auxiliary variables can be incorporated in the person-level estimator than in the household-level estimator. Finally, no additional auxiliary variable is required to enforce the integrated property. We propose two different two-level GREG estimators: a naïve and an extended two-level GREG estimator. The difference is given by the estimator for the unknown common variable totals. Whereas the naïve approach is easier to implement, because only the household-level estimator has to be adjusted by the common variables, the extended two-level GREG estimator enables to insert the best available estimate of the unknown common variable totals. 4.2 Naïve Two-level Generalised Regression Estimator In the naïve approach, consistency is ensured by incorporating the common variables in the household-level estimator. The estimator at the person level, in turn, remains unaffected by the consistency requirements between the levels. Let ci¼ðci1;…;cil;…;ciLÞTbe the person-level common variable vector of dimension L, which sums up per household to ∑i∈Ugci¼cg¼ ðcg1;…;cgl;…;cgLÞT. The totals of the common variables are unknown and have to be estimated by ~ Tc. The subscript indicating the level of estimation is skipped, because due to the consistency requirements, ~ Tcmust be equal at the person and household levels. At the person level, the naïve two-level GREG estimator (abbreviated with TL1) is given by an ordinary GREG estimator as defined in (2); that is, ^ TTL1 y;p¼ ^ TGREG y;p¼ ^ THT y;pþ ^ BTðTx ^ THT xÞ:(8) Accordingly, ^ TTL1 y;pcontains only the auxiliary variables xi, but not the common variables. The corresponding weights wTL1 i¼wGREG iare obtained from (3). For simplicity, we assume vi¼1. At the household level, a separate estimator is implemented. To emphasise that different auxiliary variables can be included at both levels, we denote ag¼ðag1;…;agk;…;agKÞTas the K dimensional auxiliary vector of household g. The known vector of the totals is given by Ta.To ensure consistency between person-level and household-level estimates, the common variables 641Consistent Estimation in Household Surveys International Statistical Review (2021), 89, 3, 635–656 © 2021 The Authors. International Statistical Review published by John Wiley & Sons Ltd on behalf of International Statistical Institute. have to be included into the estimator as additional auxiliaries. To estimate the unknown vector of the common variable totals, we suggest the proposed person-level estimator (8) applied to the common variables, because the common variables are originally person characteristics. Thus, ~ Tc¼ ^ TTL1 c;p¼ ^ TGREG c;p¼X i∈sp wGREG ici:(9) Given (9), the proposed two-level GREG estimator at the household level is obtained from ^ TTL1 y;h¼ ^ THT y;hþÊT aTa ^ THT a  þÊT c ^ TGREG c;p ^ THT c;p  ;(10) where the coefficients Êaand Êcare simultaneously estimated by Êa Êc ! ¼X g∈sh π1 g ag cg !ag cg ! T "# 1X g∈sh π1 g ag cg ! yg:(11) It is assumed that the matrix ∑g∈shπ1 g ag cg !ag cg ! T is of full rank K+L. To quantify the impact of ensuring consistency on our proposed estimator at the household level, we decompose (10) into an ordinary GREG estimator, which does not ensure consistent estimates, and an adjustment term capturing the effect caused by including the common variables as additional auxiliaries. For the decomposition, an orthogonal decomposition of the coefficients (cf. Seber, 1977) is applied, given by Êa¼ ^ Ba ^ FaÊc;(12) where ^ Baarises from ^ TGREG y;h¼ ^ THT y;hþ ^ BT aðTa ^ THT aÞ(13) as an ordinary GREG estimator solely containing agas auxiliaries. Coefficient matrix ^ Fais obtained from ^ TGREG c;h¼ ^ THT c;hþ ^ FT aTa ^ THT a  . Coefficient vector Êcis defined in (11). Hence, the product of ^ Faand Êccaptures the effects of the common variables on the variable of interest neglected by ^ Ba. Inserting the orthogonal decomposition (12) into (10), we obtain ^ TTL1 y;h¼ ^ TGREG y;hþÊT c ^ TGREG c;p ^ TGREG c;h  :(14) According to (14), the adjustment term capturing the impact induced by the consistency requirements depends on Êcand the difference between the person-level and household-level estimates for the common variable totals. The greater the difference between the two estimates, the greater the adjustment term. The partial coefficient Êccan, alternatively to (11), be expressed in terms of residuals as Êc¼X g∈sh rFa grFa g T ! 1P g∈sh rFa grBa g(15) with rBa g¼yg ^ BT aagand rFa g¼cg ^ FT aagresulting from regressing the variable of interest 642 KONRAD ET AL. International Statistical Review (2021), 89, 3, 635–656 © 2021 The Authors. International Statistical Review published by John Wiley & Sons Ltd on behalf of International Statistical Institute. Merkouris (2004) differ from the original GLS adjustment algorithm (27) and (28) only with respect to weighting factor q. Guandalini & Tillé (2017) showed that (36) and (37) are the optimal coefficients if the samples were independent and the sample size is very large. The variance estimators arise considering the weighting factor qin (32) and (33), as done in (36) and (37). 5.3 Comparison of the Two-level Generalised Regression Estimator and the Generalised Least Squares Adjustment Algorithm The differences between our two-level GREG estimator and the GLS adjustment algorithm arise from how consistency is conceptually ensured and the estimation of the common variable totals. Our two-level GREG estimator uses the same estimated common variable totals in the person-level and household-level estimators. As estimator for the unknown common variable totals, we propose a person-level estimator, because in household surveys, it is more prevalent that the common variables are originally person characteristics that are assigned in aggregated form to household-level data set. In contrast, the GLS adjustment algorithm enforces consistent person-level and household-level estimates more indirectly through the pooling of the auxiliary information. The final estimates of the unknown common variable totals are determined by a weighted average of the single person-level and household-level estimates. Therefore, the same common variable information is used twice, once in its initial form at the person level and once in aggregated form at the household level. However, it is questionable to what extent the aggregated household-level information, supplementary to the person-level information, helps to predict the common variable totals. Furthermore, when comparing the coefficients in the two-level GREG estimators (15) and (22) with the coefficient in the GLS adjustment algorithm (29), (30), (36) and (37), it becomes evident that the latter coefficients simultaneously use person-level and household-level information by the term ∑i∈sprFx irFxT iþ∑g∈shrFa grFaT g  . However, as mentioned in the previous paragraph, in the context of household surveys, it seems questionable to what extent the household-level auxiliary information helps to predict the person-level variables. Finally, the variance estimators of the common variables differ. In the two-level GREG estimator, the variance of the common variable totals depends solely on the person-level variance estimator. In contrast, in the GLS adjustment algorithm, the variance estimation for the common variable totals (31) is more demanding, because the variances at both levels and their covariances are required. 6 Simulation Study An MC simulation study compares the performance of the naïve (TL1) and extended two-level GREG estimators (TL2) with the integrated GREG estimator with scale factors vi¼ 1 (INT1) and vi¼N1 g(INT2) and with the GLS weighting algorithm according to Zieschang (1990) (ZIE) and according to Merkouris (2004) (MER). The simulation study is based on the synthetic and open accessible data set AMELIA, which is derived from EU-SILC (cf. Burgard et al., 2017). To reduce computational burden, we use the data of only one out of four regions. The population consists of approximately 2.6 million persons and 0.9 million households. We draw R¼1000 MC samples via simple random sampling. As sampling size, we choose m¼200. The MC average sample size of persons is ∑1000 r¼1nr¼577. The auxiliaries consist of 18 indicator variables and an intercept: two sex categories, four age categories and four marital status categories plus the cross-classification of sex by age (four 649Consistent Estimation in Household Surveys International Statistical Review (2021), 89, 3, 635–656 © 2021 The Authors. International Statistical Review published by John Wiley & Sons Ltd on behalf of International Statistical Institute. categories) and sex by marital status (four categories). Within the integrated GREG estimators, we also include the additional auxiliary, N1 g, required to ensure the integrated property. The choice of solely categorical variables induces that the GREG estimator is equivalent to a post-stratification estimator (cf. Zhang, 2000). For a fair comparison, we also incorporate the household size as further auxiliary variable into the two-level GREG estimator and into the GLS adjustment algorithm. We chose three types of variables of interest at the person level. Firstly, classical person-level characteristics: personal income inc and social income soc. Secondly, a variable related with household size: personal income by household size inc_hs.In practice, estimates for subgroups or domains are often of as much interest as population totals. Thus, thirdly, we analyse some cross-classifications of age by sex by marital status: age2_sex1_ms4(age class 20–39 years, female, widowed) and age4_sex0_ms2(male, age class 60 years and older, married) as well as the cross-classifications of employment by sex: act1_sex0 (at work, male) and act1_sex1(at work, female). Note that age, sex and marital status are used as auxiliary variables in the assisting model. The variables of interest at the household level are regular taxes on wealth taxes, total disposable household income disp_inc and regular inter-household cash transfer received cash_trans. We compare two different sets of common variables. In case 1 (L¼2), the common variables are inc and soc. In case 2, there are L¼10 common variables including inc,soc,dou as degree of urbanisation with three categories (densely populated area, intermediate area and thinly populated area), basic activity status act with three categories (at work, unemployed and inactive persons) and four different income components. In order to explore the quality of the point estimates, we compare the empirical mean squared error (RRMSE). Suppose ^ Tyras the resulting total estimate for the r-th MC replication with r¼ 1;…;R.Define EMCð ^ TÞ¼R1∑R r¼1 ^ Tr, where the quantity EMCð ^ TÞdenotes the empirical expectation of the estimator ^ T. Let Tbe the true value. Then RRMSE is defined as RRMSEð ^ TÞ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 R∑ R r¼1 ^ TrT  2 T2 s: To examine the spread of the weights, Table 1 outlines the summary statistics for the weights of the competing methods divided by the design weight for all R= 1 000 MC samples (about 577 Table 1. Summary statistics for the weights. L¼2common variables L ¼10 common variables Mean SD Min Max Range Mean SD Min Max Range Person level INT1 1.00 0.28 1.30 8.24 9.54 1.00 0.28 1.30 8.24 9.54 INT2 1.00 0.32 0.85 5.32 6.17 1.00 0.32 0.85 5.32 6.17 TL1 1.00 0.22 0.01 5.09 5.07 1.00 0.22 0.01 5.09 5.07 TL2 1.00 0.23 0.22 5.12 5.34 1.00 0.24 1.95 5.11 7.05 ZIE 1.00 0.22 0.03 5.11 5.08 1.00 0.73 1.15 5.74 6.88 MER 1.00 0.22 0.03 5.11 5.08 1.00 0.73 1.15 5.74 6.88 Household level INT1 1.00 0.33 1.30 8.24 9.54 1.00 0.33 1.30 8.24 9.54 INT2 1.00 0.29 0.85 5.32 6.17 1.00 0.29 0.85 5.32 6.17 TL1 1.00 0.32 1.16 5.19 6.36 1.00 2.79 8.38 8.92 17.31 TL2 1.00 0.33 1.20 5.21 6.41 1.00 2.79 8.66 9.23 17.89 ZIE 1.00 0.32 1.16 5.18 6.35 1.00 0.75 1.94 6.07 8.02 MER 1.00 0.32 1.16 5.18 6.35 1.00 0.75 1.94 6.07 8.02 650 KONRAD ET AL. International Statistical Review (2021), 89, 3, 635–656 © 2021 The Authors. International Statistical Review published by John Wiley & Sons Ltd on behalf of International Statistical Institute. 000 values). It becomes apparent that at the person level, both integrated GREG estimators INT1 and INT2 have the highest ranges. Note that INT1 and INT2 show the same minimum and maximum at the person and household levels because of the one-to-one weight assignment between the levels. Moreover, both integrated approaches remain unaffected by the increase of the number of common variables. The same is true for TL1, which is at the person level equal to an ordinary GREG estimator. For L¼2, the weights of both proposed two-level GREG estimators TL1 and TL2 have a considerably smaller range than the integrated weights. For TL1 as ordinary GREG estimator, this observation confirms with the results found by Lemaître & Dufour (1987) and Rottach & Hall (2005). Moreover, the standard deviation is lower for TL1 and TL2 than for both integrated approaches. The higher variation of the integrated weights is caused by the increased number of outcome values of the auxiliary variables as described in Section 7. However, even if the weights of TL2 vary less, their range increases with the number of common variables. This means that in some MC samples, extreme weights emerge. The household weights vary more than the person weights except for INT2, which weights are modelled at the household level. At the household level, TL1 and TL2 suffer from the increase of the common variables. As mentioned in Section 4.4, there is a considerable literature on methods to reduce the range of the weights. Accordingly, all estimators under consideration can be adjusted using other calibration distance functions than the chi-squared distance function. However, because negative weights do not affect the statistical properties of the resulting estimators, we refrain focusing on these methods in this paper. Figures 1 and 2 plot the ratios of the MSEs of integrated weighting relative to our two-level GREG estimators and relative to the GLS adjustment algorithm. Both plots give almost the Figure 1. Ratios of INT1 relative to the two-level generalised regression estimator and the generalised least squares adjustment algorithm. 651Consistent Estimation in Household Surveys International Statistical Review (2021), 89, 3, 635–656 © 2021 The Authors. International Statistical Review published by John Wiley & Sons Ltd on behalf of International Statistical Institute. similar pictures on the efficiency of the estimators relative to INT1 and INT2. With respect to the person level, it becomes evident that all squares and circles indicating TL1 and TL2 lie nearby or to the right of the red line. Therefore, TL1 and TL2 perform at least as well as integrated weighting independent from the number of common variables. The greatest gains in precision are realised for TL2. Here, all variables benefit from inserting improved estimates for the common variable totals estimated by a specialised auxiliary variable set for each common variable. Actually, for the common variable inc, the variance of integrated weighting exceeds the variance of the two-level GREG estimator up to 73%. There are also significant efficiency gains realised for the subdomains of inc and hs, because of the correlation structure with the common variables in the assisting model. Interestingly, these precision gains are less pronounced for smaller and larger households. This observation confirms with the different explanatory power of the assisting models, which ranges from 0.19 to 0.22 for smaller and larger households and from 0.39 to 0.44 for the medium-size households. The high efficiency gains with respect to income and income-related variables are a hint that the proposed two-level GREG estimators better capture the heterogeneity within a household for variables that are differently distributed within a household. The efficiency improvements of the two-level GREG estimator compared with integrated weighting are a hint that the within variance of the households should not be ignored. Because the person-level TL1 estimator is equivalent with an ordinary GREG estimator, the ratios of INT1/TL1 or INT2/TL1 are comparable with the results given in the literature on integrated weighting. Except for the cross-classification of inc and hs, our results are in line with the statement given by Lemaître & Dufour (1987), Steel & Clark (2007) and van den Figure 2. Ratios of INT2 relative to the two-level generalised regression estimator and the generalised least squares adjustment algorithm. 652 KONRAD ET AL. International Statistical Review (2021), 89, 3, 635–656 © 2021 The Authors. International Statistical Review published by John Wiley & Sons Ltd on behalf of International Statistical Institute. Brakel (2013,2016) that the inefficiencies of integrated weighting would be limited compared with an ordinary GREG estimator. MER and ZIE perform in all scenarios nearly identical. Even if the common variables are included as additional auxiliaries, no considerable efficiency gains are realised with respect to inc and soc. For the cross-classifications of inc and hs, ZIE and MER perform more efficient than integrated weighting. Therefore, the explanatory power of household size outweighs the explanatory power of income with the variable of interest in the assisting model. However, the efficiency of ZIE and MER suffers with increasing number of common variables. At the household level and for L¼2, our two-level GREG estimators outperform integrated weighting; however, there are considerably efficiency losses for L¼10 and taxes,disp_inc and cash_trans. Therefore, in order to avoid an efficiency loss for the estimation of household characteristics, the number of common variables should be chosen with caution. Nevertheless, the efficiency gains at the person level of our proposed two-level GREG estimator are convincing, in particular for variables related to the common variables. 7 Conclusion Consistent person-level and household-level estimates are a major concern of statistical offices. Integrated weighting produces consistent estimates by using the same weights for all persons within a household and the household itself. The consequences of equal weights are an increased number of factor values of the auxiliaries and the ignorance of heterogeneity of the persons within households. As remedy, we proposed a two-level GREG estimator to achieve consistency without the strict requirement of equal weights for all household members. To ensure consistent estimates, we restrain the consistency requirements to the variables common to the person-level and household-level data sets. By using the same estimated total for the common variables at both levels, consistency is ensured more directly and only for the relevant variables. In many household surveys, only few variables are provided on both the person and household levels such that the number of additional auxiliary variables is limited. Our MC simulation study supports the superiority of our proposed two-level GREG estimator compared with integrated weighting in particular with respect to the common variables and variables correlated with them. However, the number of variables for which consistency is required should be chosen carefully in order to avoid inefficiencies for some household estimates. The choice between the naïve and extended two-level GREG estimators is characterised by a trade-off between of the implementation expenses and quality of the final estimates. In case of the extended two-level GREG estimator, specialised variables have to be specified and the variance estimation is more demanding, but there are significant efficiency gains possible. These efficiency gains depend on the strength of the relation between the common variables and variables of interest. With this two-level GREG estimator, we contradict the assumption prevailing in the literature that equal weights of persons within the same household and the household itself are indispensable to ensure consistent person and household estimates (cf. Lavallée, 1995, p. 27 Estevao & Särndal, 2006, p. 139; Steel & Clark, 2007, p. 51; Branson & Wittenberg, 2014, p. 20; van den Brakel, 2016, p. 149). Another application field for our proposed two-level GREG estimator is integrated surveys where several smaller surveys are integrated in a core sample. In this context, consistent estimates between the core sample and the subsample are desired. However, in integrated surveys, the number of common variables might be large. This enlarges the range of the weights and results in less stable estimates. Therefore, the two-level GREG estimator has to be extended to handle with a variety of auxiliary variables. The combination with generalised calibration using soft constraints may additionally be considered in this case (Burgard et al., 2019). 653Consistent Estimation in Household Surveys International Statistical Review (2021), 89, 3, 635–656 © 2021 The Authors. International Statistical Review published by John Wiley & Sons Ltd on behalf of International Statistical Institute. For most survey practitioner, negative weights may be undesirable. Therefore, further research should focus on methods that prevent negative weights within the two-level GREG estimator framework. A possible starting point would be the substitution of the chi-squared distance function by a linear truncated or multiplicative distance function. However, an extension to various other calibration functions seems straightforward but urges using adequate numerical iteration methods. Further future research should address the effects of consistency requirements on non-response adjustment. In general, methods to prevent a non-response bias proceed at the person level. Hence, the adjusted person weights are no longer necessarily equal within a household. In order to still guarantee consistency, Eurostat (cf. European Commission, 2014, p. 40) recommends averaging the adjusted person weights within a household and assigning this average weight to all household members. In contrast, our two-level GREG estimator allows a non-response adjustment at the person level without the need for a subsequent averaging process of the resulting weights. The incorporation of the common variables guarantees consistency even in the case of non-response adjustment. Therefore, individual response patterns are retained. This flexibility reinforces the superiority of our two-level GREG estimator compared with integrated weighting. ACKNOWLEDGEMENTS The research was conducted within the RIFOSS research project, which is funded by the German Federal Statistical Office. We profoundly thank the editor and the anonymous reviewers for the valuable, detailed and helpful reviews. References Alexander, C.H. 1987. A class of methods for using person controls in household weighting. Surv. Methodol.,13(2), 183–198. Boonstra, H.J.H., van den Brakel, J.A., Knotterus, P., Nieuwenbroek, N.J. & Renssen, R.H. 2003. A strategy to obtain consistency among tables of survey estimates. 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[Received May 2019; accepted May 2021] 656 KONRAD ET AL. International Statistical Review (2021), 89, 3, 635–656 © 2021 The Authors. International Statistical Review published by John Wiley & Sons Ltd on behalf of International Statistical Institute.