**Schema 2.2. Gross change parameter computation from household continuity rules for Figure 2.1**

|One-t<br>|o-one continuityrules*<br><br>|||One-to-manycontinu<br>|ityrule<br>||
|---|---|---|---|---|---|---|
|Household|t<sup>∗</sup><br>a<br>b|c<br>D|Household|t<sup>∗</sup>|a<br>b|c<br>d|
|_t_<br>Povertys|tatus atgiven time^<br>0<br>0|1<br>0|_t_<br>Povertys|tatus atgiven time^|0<br>0|1<br>0|
|A|0<br>1<br>0|0<br>0|A|0|1<br>0|1/3<br>1|
|B|1<br>0<br>1|0<br>0|B|1|0<br>2/3|1/3<br>0|
|C|1<br>0<br>0|1<br>0|C|1|0<br>1/3|1/3<br>0|
|Hhld<br>|||Hhld<br>||||
|ouseo<br>U2←1|Computation<br>Parameter|%|ouseo<br>U2←1|Computation|Parameter|%|
|Positive change event|1(Bb)<br>1|33.3|Positive change event|2/3(Bb)+1/3(Cb)|1|25.0|
|Negative change event|0<br>0|0.0|Negative change event|1/3(Ac)|1/3|8.3|
|Stationaryevent|1(Aa)+ 1(Cc)<br>2|66.7|Stationary event|1 (Aa)+1(Ad)+1/3 (Bc)+<br>1/3(Cc)|8/3|66.7|
|Total househ|olds<br>3|100|Total ho|useholds|4|100.0|
|*Continuity rule follows t<br>^Cross-sectional poverty s|he head of household<br>tatus=1, non-poverty status=0||^Cross-sectional poverty|status=1, non-poverty s|tatus=0||

# 3. SAMPLING SETTING AND ESTIMATOR

As anticipated in the introduction, we consider the case where longitudinal and cross-sectional computed using data from a longitudinal survey in which individuals were first observed at estimates referring to the current time (e.g., the year 2024) denoted as time tt are to be intermediate between (e.g., the year 2020) in a refresh of the initial survey (based on well-specified tracking and continuity rules), in which a new sample is also independently selected. This observational time tt<sup>∗</sup> (e.g., the year 2015) and subsequently observed again in year tt<sup>∗∗</sup> setting is very attractive as it summarizes most of the estimation and representativeness tt and tt<sup>∗</sup> problems that can arise in actual large-scale longitudinal studies. The following subsections formalize, using specific notation, the aspects related to sampling design, movers, and direct estimation.

# 3.1. Sampling design

We introduce below some symbols to better illustrate the operations just described. Initial time , from the cross-sectional ,l tt<sup>∗</sup> tt<sup>∗</sup> . We select a fixed sample size of households, SS= ∑ ,l ,l individuals of sample tt<sup>∗</sup> tt<sup>∗</sup> tt<sup>∗</sup> tt<sup>∗</sup> population HH , where household l∈HH enters the sample with inclusion probability ππ . SS tt<sup>∗</sup> tt<sup>∗</sup> l∈HH,l) member of household l is included in the sample tt∗ tt<sup>∗</sup> tt<sup>∗</sup> includes mm households, being mm ππ . We observe all NN tt<sup>∗</sup> = ∑l=1 ,l denote the household l. Thus, the jj-th (jj= 1, ... , NN in terms of individuals. mmtt∗ tt<sup>∗</sup> tt<sup>∗</sup> with the same probability of inclusion in their household. Let nn NN Intermediate time tt<sup>∗</sup> , of households from the cross-sectional realized sample size of SS tt<sup>∗∗</sup> tt<sup>∗∗</sup> . We select a new sample, SS . We tt<sup>∗∗</sup> tt<sup>∗∗</sup> tt<sup>∗</sup> tt<sup>∗∗</sup> population HH . SS is selected independently from SS and is a fixed-size sample of mm tt<sup>∗∗</sup> tt<sup>∗∗</sup> ,kk households. Household kk∈HH of individuals. enters in the sample with the inclusion probability ππ tt<sup>∗∗</sup> ,kk observe in the survey all NN individuals of sampled household kk. In this way, we observe a tt<sup>∗∗</sup> time, the enumerators track sample of nn people who they are able to track, as guided tt<sup>∗</sup> by specific Moreover, starting from sample SS tracking rules . Tracking rules are fundamental in deciding whether to follow people interviewed in the previous tt<sup>∗</sup> tt<sup>∗</sup> and collect the information of interest on all nn . For example, as mentioned in the who moved from where they were interviewed at time tt<sup>∗</sup> 8