∗ ∗∗ (tt←) tt←tt tt←tt (3.12) SS = SS ∪SS be the longitudinal sample-union at current time tt, obtained by joining together the two individuals. tt←tt<sup>∗</sup> tt←tt<sup>∗∗</sup> (tt←) (tt←) tt←tt<sup>∗</sup> tt←tt<sup>∗∗</sup> longitudinal samples SS and SS . The sample SS has mm = mm + mm (tt←) tt←tt<sup>∗</sup> tt←tt<sup>∗∗</sup> households, and nn = nn + nn in the standard form as a weighted sum of the data in the sample union as: FFtt←tt∗∗ We may express the estimator YY<sup>�</sup> , = � �NNtt,kk FFtt←tt∗∗ tt,kkii SS(tt←),kk (3.13) YY<sup>�</sup> kk∈ SS(tt←) ii=1 yy dd . With straightforward algebraic manipulation, starting from the above formula (3.6), ... , (3.11) we SS(tt←),kk (tt←) where dd is the direct sample weight of household kk of sample SS have ∗∗ ∗ = 1 SStt←tt∗,kk tt←tt<sup>∗</sup> , ααdd if kk∈SS SS(tt←),kk = � SStt←tt∗∗,kk tt←tt = 0 kk,tt←tt (3.14) dd (1 −αα)dd if kk∈SS ∧ FF SStt←tt∗∗,kk tt←tt<sup>∗∗</sup> kk,tt←tt<sup>∗</sup> dd if kk∈SS ∧ FF are given in formula (3.8) and (3.11) as the base weights of the sample-union. SStt←tt∗,kk SStt←tt∗∗,kk where the expressions for dd and dd SS(tt←),kk respectively. In the following, we denote dd As of the definition of the αα value, Singh and Mecatti (2011) provide an in-depth illustration of the different approaches proposed in literature for fixing the optimal value of αα in the context of multiple frame surveys. Hartley (1962, 1974) proposes choosing αα to minimize the variance (1974) suggests opting for a simpler alternative expression which is always positive, even if it FFtt←tt∗∗ of YY<sup>�</sup> . Unfortunately, the solution depends on the variable yy and αα may be negative. Hartley drawback and closely approximates the Hartley’s solution depends on yy. We suggest here an even simpler solution which does not suffer from the above ~~.~~ ∗ ∗∗ tt←tt<sup>∗</sup> mm (3.15) αα= tt←tt tt←tt mm + mm is known as factor of multiplicity. In the case illustrated in Section 5 below, this factor equals 2. tt←tt<sup>∗</sup> tt←tt<sup>∗∗</sup> If mm ≅mm , as in the case illustrated in Section 5 below, then αα= 0.5. The reciprocal of αα is known as factor of multiplicity. In the case illustrated in Section 5 below, this factor equals 2. 3.3.2. Calibration estimator may not be FFtt←tt∗∗ does not represent SS(tt←),kk The direct estimates YY<sup>�</sup> (see 3.13) obtained with the direct final weights dd E and the estimates could under-cover the actual population for that FFtt←tt∗∗ accurate at least for three reasons, thus requiring adjustment. First, YY<sup>�</sup> segment. Second, non-response may introduce bias in the estimates, as the nonresponding tt←tt∗∗ the aggregate YY population may have different characteristics from those who agree to participate in the survey. Third, there may be differences in the sample estimates with the known totals of some auxiliary . High values of the differences are often a sign of inaccuracy, resulting from either the abovementioned under-coverage problem or if tt variables available on the cross-sectional population HHsome subpopulations exhibited higher non-response rates than others.

The direct estimates can be adjusted jointly for the three issues above using the calibration estimator (Deville and Särndal 1992; Singh and Mohl 1996). This estimator defines calibrated 13