metadata: authors: 0: author: Dang,Hai-Anh H.
1: author: Lanjouw,Peter F.
count: World
docna: 0: docna: Measuring poverty dynamics with synthetic
            panels based on cross-sections
docty: Policy Research Working Paper
subtopic: Labor & Employment Law,Educational Sciences,Poverty Impact Evaluation,Poverty Monitoring & Analysis,Inequality,Poverty Assessment,Poverty Diagnostics,Poverty Lines,Small Area Estimation Poverty Mapping,Crime and Society
teratopic: Macroeconomics and Economic Growth,Science and Technology Development,Poverty Reduction,Education
abstracts: cdata!: Panel data conventionally underpin the
            analysis of poverty mobility over time. However, such data
            are not readily available for most developing countries. Far
            more common are the "snap-shots" of welfare
            captured by cross-section surveys. This paper proposes a
            method to construct synthetic panel data from cross sections
            which can provide point estimates of poverty mobility. In
            contrast to traditional pseudo-panel methods that require
            multiple rounds of cross-sectional data to study poverty at
            the cohort level, the proposed method can be applied to
            settings with as few as two survey rounds and also permits
            investigation at the more disaggregated household level. The
            procedure is implemented using cross-section survey data
            from several countries, spanning different income levels and
            geographical regions. Estimates fall within the 95 percent
            confidence interval -- or even one standard error in many
            cases -- of those based on actual panel data. The method is
            not only restricted to studying poverty mobility but can
            also accommodate investigation of other welfare outcome dynamics.
repnme: repnme: Measuring poverty dynamics with synthetic
            panels based on cross-sections
display_title: Measuring poverty dynamics with
            synthetic panels based on cross-sections
listing_relative_url: /research/2013/06/17919450/measuring-poverty-dynamics-synthetic-panels-based-cross-sections
topicv3: Poverty m1326687 695,Chronic Poverty m1326690 1155,Criminal Justice m1326993 1014,Education m1327048 470,Household Surveys m1504071 2761
publishtoextweb_dt: 2013-06-01T00:00:00Z
historic_topic: Macroeconomics and Economic Growth,Science and Technology Development,Poverty Reduction,Education
no_of_pages: 70
author_keys: 0: author_key: 747967
1: author_key: 95180
model_extractions: WPS6504 Policy Research Working Paper 6504 Measuring Poverty Dynamics with Synthetic Panels Based on Cross-Sections _Hai-Anh Dang Peter Lanjouw_ The World Bank Development Research Group Poverty and Inequality Team June 2013
Policy Research Working Paper 6504

# **Abstract**

Panel data conventionally underpin the analysis of poverty mobility over time. However, such data are not readily available for most developing countries. Far more common are the “snap-shots” of welfare captured by cross-section surveys. This paper proposes a method to construct synthetic panel data from cross sections which can provide point estimates of poverty mobility. In contrast to traditional pseudo-panel methods that require multiple rounds of cross-sectional data to study poverty at the cohort level, the proposed method can be applied to settings with as few as two survey rounds and also permits investigation at the more disaggregated household level. The procedure is implemented using cross-section survey data from several countries, spanning different income levels and geographical regions. Estimates fall within the 95 percent confidence interval— or even one standard error in many cases—of those based on actual panel data. The method is not only restricted to studying poverty mobility but can also accommodate investigation of other welfare outcome dynamics.

This paper is a product of the Poverty and Inequality Team, Development Research Group. It is part of a larger effort by the World Bank to provide open access to its research and make a contribution to development policy discussions around the world. Policy Research Working Papers are also posted on the Web at http://econ.worldbank.org. The authors may be contacted at hdang@worldbank.org and p.f.lanjouw@vu.nl.

_The Policy Research Working Paper Series disseminates the findings of work in progress to encourage the exchange of ideas about development issues. An objective of the series is to get the findings out quickly, even if the presentations are less than fully polished. The papers carry the names of the authors and should be cited accordingly. The findings, interpretations, and conclusions expressed in this paper are entirely those of the authors. They do not necessarily represent the views of the International Bank for Reconstruction and Development/World Bank and its affiliated organizations, or those of the Executive Directors of the World Bank or the governments they represent._ Produced by the Research Support Team
# **Measuring Poverty Dynamics with Synthetic Panels Based on Cross-Sections**

Hai-Anh H. Dang and Peter F. Lanjouw<sup>*</sup>

**Keywords:** transitory and chronic poverty, mobility, welfare, synthetic panels JEL Codes: C53, D31, I32, O15

> _*_ Dang <u>(hdang@worldbank.org; corresponding author) and Lanjouw (p.f.lanjouw@vu.nl) are</u> respectively economist and professor with the Poverty and Inequality Unit, Development Research Group, World Bank, and VU University Amsterdam. We thank  Francois Bourguignon, Alan Dorfman, Chris Elbers, Francisco Ferreira, Gary Fields, Paul Glewwe, Bill Greene, Bo Honore, Dean Jolliffe, Aart Kraay, Christoph Lakner, Yue Man Lee, Michael Lokshin, David McKenzie, Reema Nayar ,Tuoc Van Phan, Sergiy Radyakin, Carolina Sanchez-Paramo, Renos Vakis, Roy van der Weide, Nobuo Yoshida, and participants at meetings of the Econometric Society, International Association for Applied Econometrics, International Conference on Panel Data, and seminars at IFPRI, University of New South Wales, and World Bank for helpful discussions on earlier versions of this paper. We further thank Renos Vakis and Leonardo Lucchetti for their help with the Peruvian data.
# **I. Introduction**

To effectively reduce poverty, we want to understand the factors that help households escape poverty as well as those that induce them to remain in, or fall back into, poverty. It is commonly claimed that we need panel data to answer these questions, especially at the household or individual level. However, for a variety of reasons, cross-sectional data are far more common, in most developing countries, than panel data. Panel data collection can be very costly, for example, and can also pose a number of logistical and capacity-related challenges. For whatever reason, the scarcity of panel data has rendered the analysis of welfare dynamics difficult, if not impossible, in many developing country settings.

To overcome the non-availability of panel data, there have been a number of studies that develop pseudo-panels (or synthetic panels) out of multiple rounds of cross-sectional data. Following the seminal contributions of Deaton (1985), synthetic panels based on cohorts have been widely used to track income and consumption outcomes over time (e.g., Deaton and Paxson, 1994; Banks, Blundell, and Brugiavini, 2001; Pencavel, 2007). Other outcomes have also been analyzed with synthetic panels; these include, for example, labor responses to tax reforms (Blundell, Duncan, and Meghir, 1988), the returns to academic and vocational qualifications (Mcintosh, 2006), and household demand for private medical insurance (Propper, Rees, and Green, 2001). Notably, since cross-section samples are typically refreshed each time that the surveys are fielded, synthetic panels are possibly less exposed to the concerns surrounding attrition and measurement error that are often leveled at actual panel 2
data.<sup>1</sup> Thus, unsurprisingly, the econometrics of pseudo-panel data is a rapidly growing field of research.<sup>2</sup> Perhaps because of their emphasis on cohorts rather than the household or individual, pseudo panel methods have not been widely applied to the study of poverty dynamics. Two notable exceptions are Bourguignon, Goh and Kim (2004) and Güell and Hu (2006) who construct synthetic panels at the household level. However, these two approaches require certain assumptions that may not always be easily satisfied in available cross sections: the former requires at least three rounds of cross section data and assumes a first-order autoregression (AR (1)) process through which past household or individual incomes (earnings) can affect present outcomes; the latter is exclusively restricted to duration analysis.

Against this background, a recent paper by Dang, Lanjouw, Luoto, and McKenzie (2014) proposes both parametric and non-parametric approaches to construct synthetic panels at the household level from two rounds of cross sections with rather parsimonious assumptions.<sup>3</sup> These synthetic panels are then used to predict lower-bound and upper-bound estimates of household transitions into and out of poverty in Vietnam and Indonesia. Drawing on both cross sectional data and genuine panel data, the authors compare mobility estimates based on synthetic panels to those that would be obtained from actual panel data and find that the “true” estimate of mobility (as revealed by the actual panel data) is generally sandwiched between

> 1 See, for example, Glewwe and Jacoby (2000) and Kalton (2009) for recent overviews of the advantages and disadvantages of cross sections and panel data in both developing and richer country contexts.

> 2 See, for example, Inoue (2008) for a recent development and a brief review of this literature.

> 3 This method focuses on temporal imputation or survey-to-survey imputation over time. For related studies that investigate spatial imputation or survey-to-census imputation, see, e.g., Elbers, Lanjouw, and Lanjouw (2003), some recent economic applications of which include Agostini and Brown (2010), Elbers et al. (2007), and Demombynes and Ozler (2005). More broadly, this method is related to the literature on identifying the bounds on the joint distribution for outcomes in different samples (see, e.g., Cross and Manski, 2002) and the statistical literature on imputing missing data (see, e.g., Little and Rubin, 2002). See also Ridder and Moffitt (2007) for a recent review on the econometrics of data combination.

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the upper bounds and lower bounds based on the synthetic panels.<sup>4</sup> In particular, the interval between the bound estimates can be narrowed if an appropriate range for the correlation between the error terms can be postulated. Dang et al. (2014) suggest that panel data, where available from other sources such as other countries with similar characteristics, might be scrutinized to identify such a narrower range. Despite its infancy, applications of Dang et al.’s method in various settings have been yielding encouraging results.<sup>5</sup> In this paper we generalize the method introduced by Dang et al. in several important aspects. First, by introducing a method to approximate the appropriate correlation term and its theoretical upper bound using each country’s _own_ cross sectional surveys, we overcome the limitations of having to draw on external estimates from actual panels in similar settings. Notably, such “similar” actual panels might neither be available, nor offer very convincing estimates (see further discussion in section III below). Thus with the more general framework offered here, we can apply our method to study poverty dynamics using _just only_ two rounds of cross section data that satisfy our (rather standard) assumptions. The approximation of point values for the correlation term allows us to move beyond the _bound_ estimates proposed by Dang et al. (2014) to actual _point_ estimates of poverty mobility. This represents a major advance in terms of interpretation, and potential application in practice. In particular, we can investigate different measures of poverty dynamics, such as the population shares in different poverty status categories in both survey periods considered together (i.e., unconditional or joint probabilities) or the population shares in different poverty status categories in one period

> 4 For presentational convenience, we generally refer to household movements between poverty or consumption categories as mobility.

> 5 For example, recent applications/ validations of Dang et al.’s method against true panel data include Bierbaum and Gassmann (2012) for the Kyrgyz Republic, Cruces et al., (forthcoming) for Chile, Nicaragua, and Peru, and Martinez et al. (2013) for the Philippines.

4
given their welfare status in the other period (i.e., conditional probabilities). The former measure provides one way to define chronic poverty,<sup>6</sup> and the latter permits the study of poverty transitions. We also provide the formulae for the standard errors on point estimates, which is not available with the bound estimates offered by Dang et al (2014).

Second, we generalize the construction of these synthetic panels to settings where more than two rounds of data are available. This offers useful results since few actual panel datasets in developing countries span more than two periods; and in those cases where they do, they may be suffereing heavily from attrition problems. By considering poverty mobility in more than two periods, we can investigate richer inter-temporal profiles of movement into and out of poverty.

Third, our proposed framework further extends Dang et al. (2014)’s investigation of household transitions into and out of poverty to a much more general setup of household movements among different consumption groups. A typical description of the former is usually shown in a 2x2 poverty transition matrix—where household movement is tracked between a poor category or non-poor category from one period to the next—while an example of the latter is a 5x5 consumption transition matrix, where household movement is tracked not just for two poverty categories but for five consumption quintiles.

We validate our estimates both theoretically with a Monte Carlo simulation exercise and empirically by using cross sectional and actual panel survey data from several high-income and developing countries including Bosnia-Herzegovina, Lao PDR, Peru, the United States, and Vietnam. We find that our synthetic panel estimates are close to—and mostly lying within

> 6 Also see Calvo and Dercon (2009) and Foster (2009) for more discussion on different definitions of chronic poverty. We restrict our discussion in this paper to a money-metric measure of poverty, for a multidimensional measure see Alkire and Foster (2011).

5
the 95 percent confidence intervals or even one standard error in many cases—those of actual panel data. Assuming our estimation model is valid, the standard errors on our model-based synthetic panel estimates are also found to be smaller than the sampling-based standard errors of actual panel data (an advantage derived due to the fact that sample sizes of cross-section survey data generally exceed those of the actual panel data).

This paper consists of seven sections. We start in the next section with a brief description of the method introduced in Dang et al., which is followed by our generalization of this method and estimation procedures in Section III; the Monte Carlo simulation exercise is provided in Section IV. We describe the data in Section V before applying our method to investigate poverty dynamics in Section VI, and Section VII concludes.

# **II. Bound Estimates on Poverty Mobility**

Let _xij_ be a vector of household characteristics observed in survey round _j, j_ = 1 or 2, that are also observed in the other survey round for household _i, i_ = 1,..., N. A key requirement is that these household characteristics are time variant. Subject to data availability, these household characteristics can include such clearly time-invariant variables as sex, ethnicity, religion, language, place of birth, and parental education as well as variables that can be converted into time invariant versions based, for example, on information about household heads’ age and education. The vector _xij_ can also include time-varying household characteristics if retrospective questions about the round-1 values of such characteristics are asked in the second round survey.

Let _yij_ then represent household consumption or income in survey round _j_ . The linear projection of household consumption (or income) on household characteristics for each survey round is given by 6
Let _zj_ be the poverty line in period _j_ . We are interested in knowing such quantities as which represents the percentage of households that are poor in the first period but nonpoor in the second period (considered together for two periods), or which represents the percentage of poor households in the first period that escape poverty in the second period.

If panel data are available, we can easily estimate the quantities in (3a) and (3b); otherwise, we can use synthetic panels for this purpose. Assume that the underlying population being sampled in survey rounds 1 and 2 are the same, or more specifically, ��� � ���, and ���|��� and ���|��� have identical distributions (Assumption 1 in Dang et al, 2014). We can rely on the time-invariant variables _xij_ that are collected in both survey rounds to predict the consumptions in period 1 for households interviewed in period 2, and vice versa.<sup>7</sup> In particular, assume that ��� and ��� have a bivariate normal distribution with correlation coefficient  and standard deviations ���and ��� respectively (Assumption 2’). If  is known, Dang et al. propose to estimate quantity (3a) by

> 7 In other words, this assumption implies that households in period 2 that have similar characteristics to those of households in period 1 would have achieved the same consumption levels in period 1 or vice versa.

7
where  2  . stands for the standard bivariate normal cumulative distribution function (cdf) ) (and  2  . stands for the standard bivariate normal probability density function (pdf)).

However, since  is usually unknown in most contexts, assume also that it is bounded by the interval [0, 1] (Assumption 2’’).<sup>8</sup> Since for any _x, y_ , and _ρ_ ,  <u>2 </u> _x_ <u>,</u> _<u>y</u>_ <u>, </u>   2  _x_ , _y_ ,   0  (Sungur, 1990), equation (4) indicates that a lower (higher) value of  means a higher (lower) probability of being poor in the first period but non-poor in the second period. Thus the lower bound and upper bound estimates of mobility can be established by identifying the appropriate range of values for the correlation term  . Absent any other information, Dang et al. (2014) suggest one can start by assuming that  is either 0 or 1. However, by examining empirical estimates from actual panel data for other countries, they propose a narrower range for Indonesia during 1997-2000 of [0.3, 0.7]. This in turn yields a lower bound and upper bound interval of [8.1, 11.8] for the proportion of households that were poor in 1997 but non-poor in 2000, which encompasses the “true” proportion rate of 10.1 percent based on actual panel data (Dang et al, 2014).

# **III. Point Estimates on Poverty Mobility in a Generalized Framework**

Despite its relevance for identifying bound estimates on poverty dynamics at the household level, the method introduced in Dang et al. (2014) suffers from several drawbacks. First, unless one is content to work with the extreme case of  in the [0, 1] range, a group of countries with actual panel data that is comparable to the country under investigation must be found so that a more reasonable empirical range of values for  can be identified. This task

> 8 Dang et al. (2014) provide several reasons why this assumption can be expected to hold and show this is the case using household survey data from several countries. Also see Dang et al. for more discussion on the implications for these assumptions and proofs for the bound estimates.

8
would require a certain degree of homogeneity for these countries, since  may vary depending on a host of factors caused by any difference ranging from economic structures to modeling methods and survey designs. Complicating this issue even further, for the same country,  is likely to be different for different household welfare outcomes; thus an appropriate range of correlation term for, say, household food consumption must be estimated separately from that for household non-food consumption. Over time,  might also change.

Second, this method provides at best bound estimates, rather than point estimates. While bound estimates are certainly more useful than no estimates at all in the absence of true panel data, they also leave room for greater precision. There always exists a tradeoff between precision and encompassment with the bound approach: a larger bound interval is more likely to encompass the true rates but will be less precise, and vice versa.

In this section we generalize the method introduced in Dang et al. (2014) by presenting a method to estimate  based only on a country’s own cross sectional surveys. We discuss the asymptotic properties of this new point estimator. We then extend the method to settings with three survey rounds or more. We maintain the Dang et al. (2014) framework described in the previous section.

# **III.1. Theoretical Estimates for ρ**

We offer the following proposition to obtain the simple correlation coefficient between household consumption in two survey rounds  _yi_ 1<sup>_y_</sup> _i_ 2 , which is closely related to  .

**Proposition 1- Approximate estimation of**  _yi_ 1<sup>_y_</sup> _i_ 2

_Assume household consumption follows a simple linear dynamic data-generating process_ ' _given by yi_ 2     _yi_ 1  _i_ 2 _(*), where_<sup></sup> _i_ 2<sup>_is the random error term. Also assume that the_</sup> _sample size of each household survey round is large enough (or N_   ), _the number of cohorts (C) constructed from the survey data is fixed, and the cohort dummy variables satisfy_ 9
_the relevance and exogeneity criteria for instrumental variables for yi1 in (*). The simple correlation coefficient_  _yi_ 1<sup>_y_</sup> _i_ 2 _can then be approximated with the synthetic panel cohort-level simple correlation coefficient_  _yc_ 1<sup>_y_</sup> _c_ 2 _, where c indexes the cohorts constructed from the household survey data._

# **Proof**

See Appendix 1.

In the absence of true panel data, we do not observe yi1 for the same household with household consumption in period 2, but we can predict it by projecting household consumption in period 1 on the cohort dummy variables; this process is equivalent to an instrumental variables (IV) estimation where the instrumental variables are the cohort dummy variables, and practically results in taking period-by-period sample averages within the head of household i’s cohort (Verbeek, 2008). Thus a consistent estimate for  _yi_ 1<sup>_y_</sup> _i_ 2 (and  ) requires that these instruments are relevant (i.e., statistically significant in the regression of household consumption on themselves) and exogenous (i.e., being uncorrelated with the error term<sup></sup> _i_ 2<sup>).</sup> While the former assumption can be easily checked given the available data, the latter needs to be assumed and depends on our assumptions or on prior knowledge about the specific data under consideration. For example, while we expect the latter assumption holds in most contexts, it would be violated if there are cohort effects in the random error term<sup></sup> _i_ 2<sup>(see, for</sup> example, Moffitt (1993) or McKenzie (2004)).<sup>9</sup> Furthermore, in addition to the relevance and exogeneity conditions, good instruments also need to be strong for unbiased estimates (Stock and Yogo (2005)), which effectively requires cohort dummy variables to be strongly

> 9 On a related note, surveys that focus on a particular age group of the population (e.g., youth surveys) or with particular designs (e.g., oversampling certain age cohorts) are unlikely to provide consistent estimates for the whole population.

10
correlated with household consumption; fortunately, as with the relevance condition, this additional condition can be easily checked using the cross sections.<sup>10</sup> Cohorts can be constructed from age or some combination of age and other time-invariant characteristics such as gender or ethnicity, as long as the cell size for each cohort is large enough. The assumptions stated in Proposition 1 on large cohort sizes are standard in the traditional pseudo-panel literature (e.g., type 1 asymptotics in Verbeek (2008)) and helps preclude measurement errors with cohort means. While it appears reasonable to assume that N tending to infinity given the usually large number of households interviewed with current household surveys, there seems to be no current consensus in the literature on how large nc should be.<sup>11</sup> Thus Proposition 1 provides an approximation of  _yi_ 1<sup>_y_</sup> _i_ 2 based on asymptotic theory, and how well this approximation turns out to be in practice is an empirical issue. We will see later in our empirical estimates using household surveys that estimation results are rather encouraging. Furthermore Corollary 2.1, to be discussed below, will provide a lower value for  _yi_ 1<sup>_y_</sup> _i_ 2 as a check on our cohort estimate.

Armed with an estimate for  _yi_ 1<sup>_y_</sup> _i_ 2 , we can then proceed to propose an estimate for the partial correlation coefficient  , which in turn helps provide the point estimate for poverty

> 10 Cohorts with a weaker correlation with household consumption would capture less variation in the latter. Clearly, the implicit assumption underlying traditional pseudo-panel analysis is that cohort dummy variables have a strong relationship with household consumption. In the extreme case, consumption (or poverty) mobility can happen entirely within cohorts, but we would expect this phenomenon to be rare in practice. Furthermore, this case would be easily detected since it will result in the synthetic panel cohort-level simple correlation coefficient  _yc_ 1<sup>_y_</sup> _c_ 2 being equal to 0 (i.e., since cohort means remain unchanged over time). We would like to thank David McKenzie for suggesting this point to us.

> 11 Monte Carlo simulations by Verbeek and Nijman (1992) suggests that cohort sizes of 100 to 200 are sufficient, (but also see, e.g., Devereux (2007) for a related simulation). Given a typical sample size of 5000 for most household surveys, if the household head’s age is used as the cohort variables, we can have approximately between 25 and 50 such cohorts of this sample size (depending on their age distribution).

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mobility. We also provide an upper value on  as a robustness check on our estimates for this parameter.

# **Proposition 2- Point estimate of** 

_Let R_ 2 _j , for j= 1, 2, respectively represent the coefficients of determination obtained from estimating equations (1) and (2), and xi represent the vector of household time-invariant characteristics. The partial correlation coefficient_  _can be estimated by_

# **Corollary 2.1- Another approximation of** 

_If_  1  2 _, the partial correlation coefficient_  _can also be estimated by_

# **Corollary 2.2- Upper value of** 

_Assume that the error terms_  _ij in equations (1) and (2) follows the traditional household random effects model and can be broken down as_  _ij_  _ui_  _vij where conditional on the observed household characteristics, the unobserved household effects ui has a normal distribution with mean 0 and variance_  _u_ 2 _, the idiosyncratic error terms_<sup>_v_</sup> _i_ 1<sup>_andv_</sup> _i_ 2<sup>_both have_</sup> _a normal distribution with means 0’s and variance_  _v_ 2 _, and the covariance between_<sup>_v_</sup> _i_ 1<sup>_andv_</sup> _i_ 2<sup>_is 0. An upper value for the partial correlation coefficient_</sup><sup>_is given by the simple_</sup> _correlation coefficient_  _yi_ 1<sup>_y_</sup> _i_ 2 _._

**Corollary 2.3- Lower value of**  _yi_ 1<sup>_y_</sup> _i_ 2 _The simple correlation coefficient_  _yi_ 1<sup>_y_</sup> _i_ 2 _for household consumption between the two survey rounds is greater than or equal to its lower value_

# **Proof**

See Appendix 1.

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Several remarks are in order here. First, while the first way of estimating  given in (5) naturally follows from our framework provided by equations (1) and (2) and provides more accurate results, the second way of estimating  in (6) is somewhat neater and perhaps more amenable to interpretation. It suggests that, given the estimated parameters in equations (1) and (2) are close to each other, the partial correlation coefficient for household consumption can be interpreted as the simple correlation coefficient purged of (the geometric mean of) its multiple correlation with household (time-invariant) characteristics in the two survey rounds, and then reweighted by (the geometric mean of) the shares of the unexplained predicted errors. In our validation exercise to be discussed below, these two formulae give very similar estimates for  .<sup>12</sup> Second, the variance-covariance matrix of the time-invariant household characteristics var( _xi_ ) in expression (5) is the same for each round of true panel data, but can vary for the cross sectional surveys. It may thus be useful to check one’s data to make sure these matrices are similar; otherwise, one may separately try the variance from each survey round to see if there is any difference in poverty estimates. In our empirical estimates discussed below, these variance-covariance matrices are very similar between survey rounds and make almost no difference to our estimate whether we use the one from the first survey round or from the second survey round.<sup>13</sup>

> 12 Strictly speaking, we only require  1  2'   2  1' instead of assuming  1  2 for Corollary 2.1, but we make the above assumption for convenience. Note that for the three-variable case, the two formulae in (5) and (6) are identical. Also note that another way, still, to estimate  is using the recursion formula for partial correlation coefficients provided by Anderson (2003, p.41); however, this formula requires much more calculations than the given formulae above, thus we do not discuss it further.

> 13 We abuse the notations var(x) and var(y) to refer to both the population true quantities and their sample estimates to keep the expressions simpler. Similarly, we subsequently use N to refer to both the total population and the sample survey.

13
Third, the assumption on the traditional household random effects in Corollary 2.2 is rather standard and should be satisfied as long as the variances of  _ij_ are similar. As long as there are unobserved household characteristics that are not controlled for in the regression, 2  _u_ will be positive. However, the more prediction power the model has due to inclusion of previously unobserved household time-invariant characteristics (e.g., through better data collection), the less variance these unobserved household characteristics have or the smaller 2  _u_ is.

Fourth, we do not estimate  the same way we do with  _yi_ 1<sup>_y_</sup> _i_ 2 as in Proposition 1, but need go through one more step with Proposition 2. The reason is straightforward once we recall that the cohort aggregation method in Proposition 1 is akin to an instrumental variable method where the cohort dummy (or age) variables work as the instruments (Moffitt, 1993). Thus, since the predicted error terms obtained from equations (1) and (2) are netted of age (and other time-invariant characteristics), when these error terms are aggregated by cohorts again, they would tend to zero.<sup>14</sup> On the other hand, we do not estimate  using the same procedures in Proposition 1, by, say, leaving out the age variables in estimating equations (1) and (2) since  obtained this way is different by construction from (and will overestimate) the partial correlation coefficient we are interested in.

Finally, it is rather straightforward to see that, given our assumption that  is non-negative, the numerators in equations (5) and (6) are also non-negative, thus leading to Corollary 2.3.<sup>15</sup>

> 14 In fact, an informal check on a lower value for the partial correlation coefficient can be done by just implementing the same procedures in Proposition 1, where the predicted error terms are obtained from estimating equations (1) and (2) (including the age variable). However, as discussed above, while this estimated partial correlation can provide some value for checking purposes, it is likely to be not statistically different from 0. 15 Note that the seemingly intuitive result that the partial correlation coefficient should be less than or equal to the simple correlation coefficient (e.g., since the former results when all possible correlation with other household characteristics are removed from the latter) stated in Corollary 2.2 may only hold in our context of

14
Since the lower values of  _yi_ 1<sup>_y_</sup> _i_ 2 provided by Corollary 2.3 are derived in a different way, these represent a robustness check on our estimate based on the cohort analysis in Proposition 1 above. While these “tests” may not be powerful in the sense that they can give a tight estimate just under  _yi_ 1<sup>_y_</sup> _i_ 2 , at least they can provide some assurance that our cohort estimate of  _yi_ 1<sup>_y_</sup> _i_ 2 should satisfy a lower bound estimate and provide a positive estimate for  . Also a practical use of expression (8) is, given the R<sup>2</sup> ‘s from two cross sections, we can use their geometric mean as a lower value to quickly gauge the strength of the simple correlation between household consumption in the two periods. A similar use of Corollary 2.1 when we know  _yi_ 1<sup>_y_</sup> _i_ 2 provides a shorthand calculation for the partial correlation  .

# **III.2. Further Discussion on Point Estimates and Poverty Dynamics**

To fully characterize the distribution for the point estimates in (4), we provide below its asymptotics.

# **Proposition 3- Asymptotic results for point estimates for two periods**

_Assuming that household consumption can be explained by household characteristics as stated in equations (1) and (2) and all the standard regularity conditions are satisfied for each equation (i.e., X_ '  / _N_ <sup>_p_</sup> 0 _and X_ ' _X_ / _N_  _p M finite and positive definite)._<sup>_16_</sup> _Let P represent household i’s (i=1,..., N) quantity of poverty dynamics (e.g., P_  _P_ ( _yi_ 1  _z_ 1 _and yi_ 2  _z_ 2) _), dj an indicator function that equals 1 if the household is poor and equals -1 if the household is non-poor in period j, j= 1, 2,_  _d_<sup></sup> _d_ 1<sup>_d_</sup> 2  , and  _yi_ 1 _yi_ 2 , _d_  _d_ 1 _d_ 2  _yi_ 1 _yi_ 2 , _our point estimates are distributed as_ the multiple correlation between household consumption and household characteristics, but not in general. For example, where the R<sup>2</sup> ’s in expression (6) are not multiple correlation coefficients but just bivariate correlation coefficients, they can take on negative values that will invalidate this equality. This is the well-known suppression problem in the statistics literature; see, e.g., Friedman and Wall (2005) for a recent discussion. 16 As is the usual practice, vectors of time-invariant characteristics _xi_ ’s (kx1) are transposed into row vectors and stacked on top of each other to form the matrix _X_ (nxk), and the vectors of error terms _ε_ (nx1) are formed similarly from the scalars _εi_ ’s.

15
_The covariance-variance matrix V can be decomposed into two components, one due to sampling errors and the other due to model errors assuming these two errors are uncorrelated such that V_   _s_   _m ._ ˆ ˆ _with n_  3  _m_ , _V_ (  1) _and V_ (  2 ) _being respectively the estimated asymptotic covariancevariance matrix for the estimated coefficients obtained from equations (1) and (2), V_ (  ˆ  _m_ ) _being approximated by_ (8( _N_ 4 _N_  7)3  )ˆ2  2 _m , and V_ (  ˆ _yi_ 1 _yi_ 2 , _d_ ) _the estimated asymptotic variance obtained from Proposition 1._

# **Proof**

See Appendix 1.

16
A couple of remarks are in order. First, the model variance will be smaller the better fit we have for our regressions in equations (1) and (2). And the larger the sample sizes used for prediction, the further the sampling variance can be reduced; thus, this points to the advantages of cross sections over panel data when the former has much larger sample sizes than the latter. A natural extension of this would be to pool estimates from the two cross sections for a larger sample size to reduce the sampling errors even more.<sup>17</sup> Whether the model variance or the sampling variance is the dominant component would depend on the dynamics of the underlying regression relationship and the overall precision of our theoretical models. As shown later, our estimation results also indicate that the sampling variance is significantly larger than the model variance, which is consistent with findings in the small area estimation literature (see, e.g., Rao (2003, p. 35)). But more importantly, when the postulated estimation model is correct, it follows that the model-based variances (synthetic panel estimates in our case) are usually smaller than the design-based variances (weighted estimates based on actual panel data) (Matloff, 1981; Binder and Roberts, 2009).<sup>18</sup> Second, the formulae in Proposition 3 are general and can be used to obtain our estimates using data either from the first or the second survey round, where the subscript _j_ in<sup>_x_</sup> _ij_<sup>should</sup> be adjusted accordingly to indicate data for the corresponding survey round. Estimates using either survey rounds are theoretically equivalent since the following identity always holds

> 17 Since we are mostly interested in estimating the means, assuming that the sample sizes of the cross sections are similar, we can simply use the corresponding population weight for each cross section. For estimates of other quantities such as totals, the population weights when pooling two cross sections can be adjusted by dividing by half; see, for example, Botman and Jack (1995) for a related discussion with the National Health Interview Surveys, and Kish (1999, 2002) for overviews on combining surveys.

> 18 Notably, this decomposition of the error terms is similar in spirit to the decomposition in Elbers, Lanjouw, and Lanjouw (ELL) (2003), which is related to the familiar formula in sampling statistics that the mean squared error of the estimate is composed of the variance of the estimate plus its bias squared. However, the key difference between that study and ours is that ELL impute data from a survey into a census, but we are imputing data from a survey to a survey. Thus while ELL do not need to take into account the sampling errors, we have to model these sampling errors explicitly. An alternative to this analytical error would be the bootstrap error.

17
_P_ ( _yi_ 1  _z_ 1 _and yi_ 2  _z_ 2)  _P_ ( _yi_ 2  _z_ 2 _and yi_ 1  _z_ 1) . Also note that while the number of households is the same for actual panel data across survey rounds, it can vary for cross sectional data, thus the number of observations (N) in the variance approximation for _V_ (  ˆ  _m_ ) should be adjusted accordingly.

Effective poverty reduction strategies require a good understanding of the proportions of the population that remains in certain poverty statuses in both periods, as well as the proportions of the population that move into or out of poverty in one period given their poverty status in the other period. Roughly speaking, the former outcomes include absolute numbers of poverty proportions (or joint probabilities) such as chronic poverty rates, while the latter outcomes include relative numbers of poverty proportions (or conditional probabilities) such as the proportion of the poor in the first period that exit poverty in the second period. The former outcomes thus provide a simultaneous view of poverty dynamics over time, but the latter outcomes emphasize its sequential nature, and both measures combined would provide a rich picture of poverty dynamics. We thus provide the following Corollary to Proposition 3 to provide the asymptotic results for such cases.

# **Corollary 3.1- Asymptotic results for point estimates of relative quantities of poverty dynamics for two periods**

_Given the same assumptions in Proposition 3, let Pi1 and Pi,12 respectively represent household i’s (i=1,..., N) quantities of poverty dynamics in period j (j= 1, 2) and both periods (e.g., Pij_  _P_ ( _yij_  _z j_ ) _and Pi_ ,12  _P_ ( _yi_ 1  _z_ 1 _and yi_ 2  _z_ 2 ) _), dj an indicator function that equals 1 if the household is poor and equals -1 if the household is non-poor in period j, and_  _d_<sup></sup> _d_ 1<sup>_d_</sup> 2  . _And let the sampled averaged estimated quantities of poverty dynamics_ 18
_where the covariance-variance matrix Vr can be estimated as_

**Proof** See Appendix 1.

Note that we estimate  (.) using equations (1) or (2) and its variance as discussed above, but do not use the corresponding sample-based statistics (i.e., poverty headcount ratio) to be consistent with the way we estimate  2 (.) . If the model has good fit,  (.) would be very similar to the sample-based poverty headcount ratio but has much smaller variance.<sup>19</sup> However, since we have to estimate both the numerators and denominators in the ratios (and their standard errors), this would reduce the accuracy of our estimates compared to those for the absolute quantities of poverty dynamics provided in Proposition 3.

# **III.3. Mobility for Three Periods or More**

We now generalize this method to the general setting where there are three or even more rounds of survey data. More generally, assume that there are _k_ rounds of survey data and

> ~~ˆ ˆ~~

> 19 Another practical implication is that if we divide  2 (.) by the sample poverty rate instead of  (.) , this ratio can be larger than 100 percent when we consider estimates for certain subpopulation groups.

19
household consumption levels can be explained by household characteristics for survey round _j_<sup>_th_</sup> in the following equations where j= 1,..., k. We are interested in knowing such quantities as where zj is the poverty line in period j and the relation sign (~ ) indicates either the larger sign (>) or smaller sign (<).

It is rather straightforward to see that the formula to calculate such quantities in (16) on data from the j<sup>th</sup> survey round is the generalized version of (4) as follows where  _k_  . stands for the k-variate normal cumulative distribution function, and _d j_ , with _j=_ _1,...., k_ an indicator variable that equals 1 when<sup>_y_</sup> _ij_<sup>is smaller than the corresponding poverty</sup> line in the same period (i.e., household _i_ is poor in period _j_ ), and equals -1 otherwise. Note that the matrix  of partial correlation coefficient is symmetric and is represented by rounds under consideration and all the elements on the diagonal are 1s.  _djl_ stands for the 20
correlation coefficient between equation _j_ and _l_ and equals _d j_ * _dl_ *  _jl_ ; there are such _k_ <u>(</u> _k_  1) 2 correlation coefficients in the correlation coefficient matrix  .

However, compared to the previous case of two periods, the computation now becomes more involved since the number of integral dimensions corresponds to the number of survey rounds. Estimates will be likely to be less accurate for three periods, and longer periods in general, than those for two periods due to increased layers of (modeling and sampling) errors.

We can then generalize Proposition 3 to any setting with two periods or more. Estimates on the relative quantity of poverty dynamics can be obtained by extending the result in Corollary 3.1 to more periods, but again, note that estimates will be likely to be less accurate the more periods we consider.

# **Proposition 4- Asymptotic results for point estimates for k periods**

_Assuming that household consumption can be explained by household characteristics as stated in the following equations_ ' _yij_   _j xij_   _ij_ (18) _, j= 1,..., k and all the standard regularity conditions are satisfied for each equation (i.e., X_ '  / _N_ <sup>_p_</sup> 0 _and X_ ' _X_ / _N_  _p M finite and positive definite). Let P represent household i’s (i=1,..., N) quantity of poverty dynamics (e.g., P_ ( _yi_ 1 ~ _z_ 1 , _yi_ 2 ~ _z_ 2, _yi_ 3 ~ _z_ 3,...., _yik_ ~ _zk_ ) _), dj an indicator function that equals 1 if the household is poor and equals -1 if the household is non-poor in period j, and d jl_  _d jd l_ , _our point estimates are distributed as_ _The covariance-variance matrix V can be decomposed into two components, one due to sampling errors and the other due to model errors assuming these two errors are uncorrelated such that V_   _s_   _m ._ _The first component_  _s is due to the sampling errors and can be estimated using the bootstrap method._ _To make notations less cluttered, let_  ( _jxl_ ) _represent the matrix of estimated coefficients obtained from (18),_  _k_ (.) _the standard k-variate normal probability, and_ 21
ˆ _similarly, let_  (  _m_ ,  _n_ ) _be the (k-2)x(k-2) partial correlation matrix given_ <sup>ˆ</sup> _m and_ <sup>ˆ</sup> _n with the_ ˆ ˆ ˆ _off-diagonal entries_  ˆ _dst_ . _mn_  <u></u> 1  _<u>dst</u>_  <u>.</u> _<u>m</u>_ ˆ _dsn_ 2  . <u></u> _m_ _<u>dsn</u>_ 1. _<u>m</u>_  _<u>dtn</u>_ ˆ _dtn_ 2. _<u>m</u>_ . _m for s, t= 1,..., k and s_ , _t_  _m_ , _n . The_ _with V_ (  ˆ _m_ ) _being the asymptotic covariance-variance matrix for the estimated coefficients obtained from the corresponding equation in (18) and V_ (  ˆ  _m_ ) _being approximated by_ (8 _N_  7)  ˆ  2 _m_ (4 _N_  3)2 _._

# **III.4. Mobility between Different Consumption Groups**

The results in Proposition 3 can be straightforwardly extended to settings with more than two consumption groups as follows.

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**Proposition 5- Asymptotic results for point estimates for mobility between different groups for two periods**

_Assuming that household consumption can be explained by household characteristics as stated in equations (1) and (2) and all the standard regularity conditions are satisfied for each equation (i.e., X_ '  / _N_ <sup>_p_</sup> 0 _and X_ ' _X_ / _N_  _p M finite and positive definite). Let P_<sup>_lm_</sup> _represent household i’s (i=1,..., N) probability of moving from consumption group l in period lm l_  1 _l m_  1 _m 1 to consumption group m in period 2, that is P_  _P_ ( _z_ 1  _yi_ 1  _z_ 1 _and z_ 2  _yi_ 2  _z_ 2 ) , _where l, m= 1,...,k, and the zj are the thresholds that separate the different consumption_ 0 _k groups, with_<sup>_z_</sup> _j_   _and_<sup>_z_</sup> _j_   _, for period j, j= 1, 2._ D _efining F_<sup>_l,m_</sup> _as_ Given _k_ consumption groups in each period, there are _kxk_ transitions in total. The formulae for the standard errors for the general case can be much more than those for mobility for three periods or more (and are at least as complicated). Thus we suggest estimation of the standard errors by the bootstrap method.<sup>20</sup>

# **III.5. Estimation Procedures**

A practical concern not yet discussed is whether or not equations (1) and (2) should be estimated with household weights. There appear to be both advantages and disadvantages with both approaches. Weighted regressions are especially relevant when the provided household weights were constructed to account for non-response or attrition bias or specifically based on the dependent variables (informative sampling); on the other hand, unweighted regressions are most relevant when the proposed super-population (i.e., equations (1) and (2)) model is correct and can provide some causal interpretation. Estimation without

> 20 Also note that our empirical estimates, discussed later, point to little, if any difference between the standard errors estimated using the analytical formulae offered in Proposition 3 and those using the bootstrap approach.

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weights in the former case results in biased estimates, while estimation with weights in the latter case yields inefficiency (i.e., larger standard errors).<sup>21</sup> Thus it seems advisable to estimate models both with and without weights and compare results, particularly where there is limited information on how the weights have been constructed.

Given the framework discussed above, we propose the following steps to obtain poverty mobility for two periods: _Step 1_ : Using the data in survey round 1, estimate equation (1) and obtain the predicted<sup>ˆ</sup> ' ˆ coefficients  1 , and the predicted standard error <sup></sup> _i_ 1 for the error term<sup></sup> _i_ 1<sup>. Using the data in</sup> survey round 2, estimate equation (2) and obtain similar parameters  ˆ2 ' and  ˆ<sup></sup> _i_ 2 .

_Step 2_ : Aggregate data in both survey rounds 1 and 2 by cohorts and obtain the estimated cohort-level simple correlation coefficient  ˆ _yi_ 1<sup>_y_</sup> _i_ 2 . Calculate  ˆ using Proposition 2, and check _Step 3_ : For each household in survey round j, calculate absolute quantities of poverty mobility household is poor and equals -1 if the household is non-poor in period j, j= 1, 2, and  ˆ _d_<sup></sup> _d_ 1<sup>_d_</sup> 2  ˆ . Calculate the standard errors using Proposition 3. Make the appropriate adjustments to obtain population-level numbers.

_Step 4_ : (If relevant) Calculate the population-level relative quantities of poverty mobility for 1 if the household is poor and equals -1 if the household is non-poor in period j, j= 1, 2, and  ˆ _d_<sup></sup> _d_ 1<sup>_d_</sup> 2  ˆ . Calculate the standard errors using Corollary 3.1.

> 21 See also Deaton (1997), Lorh (2010), and Pfeffermann (2011) for further discussion on this topic.

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The estimation procedures for three periods or more are similar, with poverty mobility rates and standard errors estimated in Steps 3 and 4 using Proposition 4 instead of Proposition 3.<sup>22</sup>

# **IV. Monte Carlo Simulation**

We validate our method based on simulated data in this section. Assuming household i’s consumption in periods 1 and 2 can be generated using the following model _yi_ 1   1   11 _x_ 1   12 _x_ 2   13 _x_ 3   14 _x_ 4   15 _x_ 5   16 _x_ 6   17 _x_ 7   18 _x_ 8  _vi_ 1 (24) _yi_ 2   2   21 _x_ 1   22 _x_ 2   23 _x_ 3   24 _x_ 4   25 _x_ 5   26 _x_ 6   27 _x_ 7   28 _x_ 8  _vi_ 2 (25) where the _x_ ’s are household head’s time-invariant characteristics, and _v_ ’s the random error terms. Since we may not be able to fully observe all household time-invariant characteristics in a survey, this model allows us to simulate situations ranging from one extreme with minimal data collected on the household (say, we only observe _x1_ ) to the other extreme with fully observed household characteristics (i.e., we can observe all the _x_ ’s). When the _x_ ’s are unobserved, they would, together with the random error term _vj_ , be absorbed in the (total) random error term _εj_ .

We then assume the following parameter values

> 22 We are working on a Stata program to calculate poverty mobility based on synthetic panels and will make it publicly available soon.

25
where N(0, c) stands for the normal distribution with mean 0 and variance c; BVN(., .) similarly represents the bivariate normal distribution with the vector of mean 0 and the given variance-covariance matrix.

Given these parameter values, we can calculate that var( _yi1_ )= 27, var( _yi2_ )= 38.6,  _yi_ 1<sup>_y_</sup> _i_ 2  .089 as well as a range of values for the partial correlation coefficient  corresponding to the number of time-variant characteristics we can observe in the model. Table 1 provides the values for  for different data situations. These range from very limited information on the time-invariant variables where we only observe _x1_ (i.e., _ρ_ = 0.88 and is almost identical to  _yi_ 1<sup>_y_</sup> _i_ 2 ) to a typical setting with just a few such variables (i.e., _ρ_ =0.58), and to an unusual setting where we fully observe all the _x_ ’s (i.e., _ρ_ =0.15). Note that without loss of generality, we assume a certain degree of correlation over time for the error terms _v_ ’s ( _ρ_ =0.15) which may represent the correlation due to factors other than household timeinvariant characteristics such as time-varying unexpected shocks. In data situations where we only observe some but not all of the time-invariant _x_ ’s, the unobserved _x_ ’s end up in the random error terms and contribute to the correlation  of these error terms over the two periods. In such situations, the contribution of the time-invariant component is indistinguishable from that of the time-varying component, and it is the sum total of these two components that matters for our simulation purposes (as well as in most practical settings).

To save space, we provide simulation results only for Models 1, 5, and 8 at three different sample sizes N= 1000, 4000, and 10000, with 1000 simulations for each model run. These sample sizes range from a small panel dataset to a decent-sized dataset and an unusually large one. We fix the poverty line in period 2 at the 30<sup>th</sup> percentile, and then graph in Figure 1 the true percentage of households that are poor in both periods (solid line) and its 95 percent 26
confidence intervals (shaded bands) and the estimated percentage using simulated data (dashed line) against the whole spectrum of poverty rates in the first period.

Figure 1 shows that estimated poverty rates using simulated data closely track the true rates and fall within their 95 percent confidence intervals, but the more time-invariant variables that are available, the better prediction we have. In particular, the dashed lines are almost indistinguishable from the solid line for the graphs where  = 0.15 or even where  = 0.58. When there is very limited information on these time-invariant variables ( _ρ_ = 0.88) and the sample sizes for the actual panel data are mid-sized or unusually large (N= 4000 or 10000), estimates (partially) fall outside the 95 percent confidence interval for the middle part of the distribution. But note that even in this case, if the sample size is not large enough (N= 1000), the synthetic panel estimates still compares favorably well to true poverty rates (which are less accurately estimated given the small sample size). Varying the model parameters or the poverty lines gives us similar results (not shown).

Overall, these simulation results are very encouraging and indicate that our method performs rather well under the theoretical setting where our assumption about the bivariate normality distribution of the error terms holds. We will examine in the next sections how our method performs in the real-life settings where this assumption may be violated.

# **V. Data**

To validate our method with real survey data, we analyze household panel survey data from Bosnia-Herzegovina (Bosnia-Herzegovina Living Standards Measurement Survey, BLSMS), Lao PDR (Expenditure and Consumption Survey, LECS), the United States (Panel Study of Income Dynamics, PSID), Peru (Peruvian National Household Survey, ENAHO), and Vietnam (Vietnam Household Living Standards Survey, VHLSS). We use two rounds 27
from the first two surveys and three rounds from the last three surveys, with data from the BLSMS in 2001-2004,<sup>23</sup> the LECS in 2002/03-2007/08, the PSIDs in 2005, 2007, and 2009, the ENAHOs in 2004, 2005, and 2006, and the VHLSSs in 2004, 2006, and 2008. The number of households hovers around 2,376 households for Bosnia-Herzegovina, 6,500 households for the LECS, 9,189 households for each round of the VHLSSs, more than 5,000 households for the PSIDs,<sup>24</sup> and almost 20,000 households for the ENAHOs.

Except for the PSID that is implemented by the University of Michigan, all the other surveys are nationally representative surveys implemented by each country’s statistical agencies, with previous or current technical assistance from international organizations (the World Bank with Peru and Vietnam), leading universities (University of Essex with BosniaHerzegovina) or statistical agencies in richer countries (Statistics Sweden with Lao PDR). Also except for the PSID, all the other surveys are similar to the LSMS-type (Living Standards Measurement Survey) surveys supported by the World Bank in a number of developing countries and provide detailed information on household consumption and demographics, as well as schooling, health, employment, migration, and housing. The PSID has a more complex structure and provides similarly detailed, if not richer, information. All these surveys are widely used in academic studies (especially the PSID) as well as poverty assessment by the government and the donor community. We use the official poverty lines for Lao PDR, Peru, and Vietnam; for the USA, we use the poverty lines provided in the PSID data (which adjust for family size and demographics); for Bosnia-Herzegovina we use the 20<sup>th</sup> percentile of the consumption distribution in 2001 as the poverty line.

> 23 We build this dataset up on the basis of data from Demirguc-Kunt, Klapper and Panos (2011). 24 We only consider the sample persons in the PSID with non-zero longitudinal weights.

28
One particular feature the LECSs, VHLSSs and ENAHOs share is a rotating panel design, which collects panel data for a subset of each survey round between two adjacent years. Around one third and one half of the households in the first round are repeated in the next round for the LECs and VHLSSs respectively, and the corresponding repetition ratio for the ENAHOs is around one quarter. This combination of both cross-sectional data and panel data in one survey provides an appropriate setting for us to implement our procedures on the cross section components, and then validate our estimates against the true rates from the panel components for each country. For the BLSMSs and the PSIDs, there is no rotating panel design and so we use the panel halves, pretending that these are cross sectional data. To ensure comparability between estimates based on the panel and cross section components, we use household weights with our estimates for the ENAHOs and population weights for the remaining surveys.<sup>25</sup> We use income and household consumption as a household welfare measure respectively for the US and all the other countries.<sup>26</sup> Appendix 3 provides a more detailed description of these surveys and other data quality checks.

> 25 There can be both pros and cons with using true panel data versus a survey with a rotating panel design for validation purposes. On one hand, a rotating panel design may be more suitable since actual panel surveys usually have a smaller sample size than those of cross sections, and a reasonably large sample size is required to obtain accurate estimates for the cohort-level simple correlation coefficient as well as for our asymptotic results. On the other hand, an important requirement for using rotating panel surveys is that the data from the cross section component be similar to those from the panel component. For Peru, the household-weighted headcount poverty rates based on the actual panel component are around 5 percent lower than those based on the cross section component, and the population-weighted estimates are even more different. Thus while the Peruvian data are not perfect for validation purposes, we believe it is still useful to show estimates for this country using household weights. But note that this difference will introduce more noise into our estimates.

26 The PSID also has some information on household consumption but this measure is not commonly used to measure poverty and is much less comprehensive than those for other surveys.

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# **VI. Estimates on Poverty Dynamics**

# **VI.1. Approximation for ρ**

Consistent with the literature on pseudo-panel data, we restrict household heads’ age range to 25-55 for the first survey round and adjust this appropriately for later survey rounds (e.g., looking at the age cohort 27-57 if the next survey round is two years later). While this age range can be extended to include older people, it may be ill-advised to include those who are younger, at least since most household heads tend to be older than 25 in all the countries we look at.

After obtaining an estimate for  _yi_ 1<sup>_y_</sup> _i_ 2 from the synthetic panels (which are all highly statistically significant with p-values less than 0.01) based on age cohorts, we calculate  in two different ways using Proposition 2 and Corollary 2.1. Estimation results are very similar with the differences being at most 0.01; thus, we only show the estimates based on Proposition 2 in Table 2. Comparing estimates using the actual panels and the synthetic panels, the absolute difference for  ˆ ranges from 0.01 (Bosnia-Herzegovina) to 0.18 (the US during 20052007), which corresponds to a range of relative differences of 2 to 28 percent.

Nevertheless, it is reassuring to see that estimates for  are always less than those for  _yi_ 1<sup>_y_</sup> _i_ 2 , which is consistent with the hypothesis in Corollary 2.2; similarly, estimates for  _yi_ 1<sup>_y_</sup> _i_ 2 are also larger than its lower value estimates as can be estimated in Corollary 2.3(ii) (not shown). These results are very encouraging and suggest that our framework could well be applied to these cross sections to obtain an approximation for  in the absence of actual panel data.

# **VI.2. Overall Poverty Mobility**

To save space, we show estimates for poverty dynamics using the latest survey rounds available, that is for Bosnia-Herzegovina during 2001-2004, Lao PDR during 2002/0330
2007/08, Peru during 2005- 2006, the US during 2007-2009, and Vietnam during 2006- 2008 in Table 3.<sup>27</sup> Estimates for earlier survey rounds for the last three countries provided in Appendix 2, Table 2.2 offer qualitatively similar results.

We provide two ways to evaluate the goodness-of-fit for estimation results. First, we consider the precision of the synthetic panel point estimates by enumerating the number of times they fall within the 95 percent confidence interval around those from the actual panels. A more demanding test is to consider a similar statistics on a narrower band of one standard error around the actual panel estimates. Second, we consider the efficiency of the synthetic panel estimates by looking at the share of the overlap between the 95 percent confidence intervals of the synthetic panel estimates and the true estimates over the former’s 95 percent confidence interval. This test focuses on the proportional contribution of its variance in the total mean squared error and scores the accuracy of each estimate on a 100 percent scale.<sup>28</sup> Our estimation results using actual panel data and synthetic panel data are respectively displayed under the columns “Actual Panel” and “Synthetic Panel” in Table 3, and the goodness-of-fit results are shown in Table 4.<sup>29</sup> Results appear very encouraging with the synthetic panel point estimates being close to the true point estimates and lying within the 95 percent confidence intervals around the true estimates for almost all the cases (i.e., 19 out of 20). In fact, more than half of the synthetic panel point estimates fall within one standard error of the actual panel estimates (i.e., 11 out of 20; not shown). Furthermore, the difference between the true rates and our estimates seems negligible for certain mobility categories; for

> 27 Unless otherwise noted, we use data in the second survey round (xi2)—as the base year—for predictions. Estimates corresponding to those in Table 2 but use data in the first survey round are provided in Appendix 2, Table 2.3. Weighted regressions provide qualitatively similar but somewhat less accurate results, thus we use unweighted regressions.

> 28 See also Dorfman (2011) for a related test for the bias of the point estimate in the context of small area estimation.

> 29 Estimated parameters for equations (1) and (2) are provided in Appendix 2, Table 2.1.

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example, the percentage of those remaining poor in both periods for Peru during 2005-2006 were 29.9 percent and 30.9 percent respectively for the true rates and our estimates.

For the second test, Table 4 shows that all the synthetic panel estimates have half or more of their 95 percent confidence intervals being contained inside those around the true estimates. If we only consider the cases where the former should be completely contained inside the latter, more than four fifths (i.e., 17 out of 20) of the synthetic panel estimates pass this higher bar of 100 percent inclusion.

As discussed above, the standard errors for the synthetic panel estimates consist of two components, the model errors and the sampling errors, with the latter’s variance expected to be larger than the former’s variance when the regressions have good fits.<sup>30</sup> This is indeed the case where (results not shown) the variances of the sampling errors are significantly larger than those for the model errors. Thus, since the sampling errors account for most of the errors with the synthetic estimates and the cross sections used for the synthetic estimates have larger sample sizes than panel data, the synthetic estimates unsurprisingly have smaller standard errors than those based on actual panel data. Table 3 shows that, for the US and BosniaHerzegovina where the sample size is the same for both actual panel and synthetic panel estimates, the standard errors for the latter are smaller than those for the former.<sup>31</sup> Table 5 provides the proportions of the population that exit or fall into poverty in the second period given their poverty status in the first period, using the results derived in Corollary 3.1. The goodness-fit-of tests are now shown at the bottom of this table to save

> 30 We also calculate the bootstrap standard errors by bootstrapping (yij, **x** ij) from its empirical distribution function (1,000 times) and applying the estimated parameters for equations (1) and (2) from the original samples. Estimation results are very similar to the analytical standard errors.

> 31 More generally, this is consistent with the well-known result in survey sampling that the model-based variances (synthetic panel estimates in our case) are usually smaller than the design-based variances (weighted estimates based on true panel data). See, e.g, Binder and Roberts (2009) for a recent review on this topic.

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space. Estimation results are, not surprisingly, slightly less accurate than those in Table 3 since both the numerators and denominators in the ratios in Corollary 3.1 are estimated. Due to one additional layer of estimates, now just more than four fifths (i.e., 18 out of 20) and one half (i.e., 10 out of 20) of the synthetic estimates are respectively within the 95 percent confidence intervals and one standard error of the true rates. Interestingly, all the estimates for Lao PDR fall within one standard error and the 95 percent confidence intervals of the true rates.

Similarly with the efficiency test, now almost all (i.e., 19 out of 20) and four fifths (i.e., 16 out of 20) of the synthetic panel estimates pass the 50 percent mark and 100 percent mark respectively.

# **VI.3. Poverty Mobility for Population Sub-Groups**

It is important to investigate poverty dynamics for population sub-groups for at least two reasons. First, policy makers are usually interested in focusing on smaller population groups rather than the whole population in designing social safety net programs; and second, synthetic panels usually have larger sample sizes than actual panel data, and thus the larger sample sizes the former has, the more accurate estimates it can bring.

We estimate and plot the estimated rates with their 95 percent confidence intervals for the absolute and relative measures of poverty dynamics against the true rates for the population categorized by ethnicity (i.e., ethnic minority groups), gender of household heads (i.e., female-headed households), education achievement (i.e., primary education or higher, lower secondary education or higher), and residence areas (i.e., urban households or regions the household live in) respectively for Peru in Figures 2 to 5 and Vietnam in Figures 6 to 9.<sup>32</sup>

> 32 We do not show all eight graphs for one country or for other countries to save space, but results are qualitatively similar.

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Clearly, these categorizations can overlap but they can provide a first cut at profiling poverty mobility for different groups, and we would expect an overlap between the synthetic panel estimates and the true rates for the groups whose heterogeneity mimics that of the whole country.

Not surprisingly, the 95 percent confidence intervals for synthetic panels estimates for both Peru and Vietnam are much smaller than those for the true rates with the gaps between the standard errors amplified roughly twice (i.e., multiplied by 1.96). Our estimates appear to be reasonably good, especially for those who are poor in either period or both periods. Except for a few cases (e.g., households where heads only achieve primary education or living in urban Selva in Figure 2, or households living in urban areas in Figure 7), there is much overlap between the true rates and our estimated rates.

# **VI.4. Poverty Mobility in Three Periods**

We turn next in Table 6 to examining our estimates on poverty mobility for three periods using data from all three survey rounds for the US in 2005-2007, and 2009, Vietnam in 2004, 2006, and 2008, and Peru in 2004, 2005, and 2006, where there are 8 possible poverty categories that each household can fall in in these three periods (for the unconditional probabilities).<sup>33</sup> As discussed above, we should expect estimates to be less accurate than those for two periods; however, our proposed method turns out to work quite well with more than two thirds (i.e., 17 out of 24) of all the point estimates being contained in the 95 percent confidence intervals around the true rates; the corresponding figure for the stricter test of one standard error is just one half (i.e., 12 out of 24). The efficiency test points to somewhat better

> 33 Estimation results for the trivariate normal probabilities in this table are calculated using the Stata algorithm by Cappellari and Jenkins (2006) with 100 Halton draws.

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results with more than two thirds (i.e., 17 out of 24) and almost two thirds (i.e., 15 out of 24) of the point estimates respectively pass the 50 percent mark and 100 percent mark.

# **VI.5. Consumption Dynamics between Different Groups**

A common way to study consumption dynamics is to divide the consumption distribution into five groups (quintiles), and look at the 5x5 transition matrix for mobility between two periods. We provide such estimates using data from Vietnam in 2006-2008 in Table 7, where the synthetic panel estimates are shown in panel B and the actual panel estimates shown in panel A. While estimates are off with some of the row and column totals, it is perhaps more useful to focus on the inner transitions since the former do not offer as much insight into mobility as the latter.<sup>34</sup> Estimation results are, again, rather encouraging with the majority of the different consumption categories (i.e., four-fifths of the inner transitions) falling within the 95 percent CIs of the true estimates, which are presented in bold. These estimates also pass the 100 percent mark of the coverage test.<sup>35</sup> Furthermore, some of the remaining estimates that fall just outside these 95 percent CIs around the true estimates appear practically close to the latter (e.g., the transition from quintile 3 to quintile 4 or from the richest quintile to quintile 2).

> 34  The row or column totals should sum up to 20 percent by definition and serve mostly as an indicator of prediction accuracy for these totals only. In addition, it may be useful to highlight the fact that our validation is predicated on the assumption that the true panel data for Vietnam have good quality. If the mobility in the true panel data is partly caused by spurious changes due to measurement errors (or attrition bias) in household consumption, our estimates based on the synthetic panel data would be more accurate since cross sections are free of such data issues. See also Dang and Lanjouw (2014) for a related study of mobility between three consumption groups: poor, vulnerable, and middle class.

> 35 Other useful statistics that can be calculated from Table 7, panel B, include the percentages of the population that have seen either an improvement or a decline or remained in the same quintile over time, which are respectively 24.7 percent, 27.3 percent, and 48 percent. These estimates are within the 95 CIs around those based on the actual panels.

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# **VII. Conclusion**

For effective poverty reduction policies to take place, we need to understand well poverty mobility over time. In the absence of panel data, it has historically been difficult to study poverty mobility in developing countries. However, there are by now at least two rounds of cross-sectional household survey data available in a large majority of developing countries. Our proposed method, which generalizes that initiated by Dang et al. (2014), offers a means to convert these cross-sectional survey data into synthetic panel data.

In particular, by moving away from the bound estimates in Dang et al. (2014) to point estimates, we show that our estimates are quite accurate and do not depend on additional information from ancillary data. Our method would thus seem applicable in most settings where two cross sections are available. We find that estimation results are good not only for the general population but for smaller population groups as well, and are associated with much tighter confidence intervals than even direct, panel-data based estimates in those settings where the sample sizes for the cross sections are large enough. Our estimates are validated against both simulated data and actual panel data spanning different income levels and geographical regions. In addition, our method can be readily extended to settings with three survey rounds or more, although predictions are more accurate for shorter periods.

It should be noted that our method needs not be restricted only to the analysis of poverty mobility. It can in principle also be employed to study dynamics more generally. Possible application can involve dynamics in the areas of labor (e.g., what is the percentage of those who move from the rural farm sector to the non-farm sector?), health (e.g., what is the percentage of those who have medical insurance in the first period but do not in the second period?) or finance (e.g., what is the percentage of borrowers who do not default on loans in the first period but do in the second period?).

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We come away with a growing sense that this basic methodology offers significant potential towards a better understanding of poverty dynamics in settings where panel data are absent, and can serve as a rather promising avenue for further research. For example, since measurement errors may be larger with richer households’ consumption, future research may investigate the heterogeneity of the error terms for different types of households. Another potential direction is to help better target social transfers by identifying a “vulnerability line” that is higher than the poverty line but households with a consumption level lower than this line are still subject to a considerable probability of falling back into poverty in the next period (Dang and Lanjouw, 2014). Another is to examine the extent that the estimates offered by our synthetic panels can improve on and correct for “bad” panel data estimates resulting from serious attrition problems.

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