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Spatial disaggregation of poverty and disability: application to Tanzania

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Abstract

Estimating poverty measures for disabled people in developing countries is often difficult, partly because relevant data are not readily available. We extend the small-area estimation developed by Elbers, Lanjouw and Lanjouw (2002, 2003) to estimate poverty by the disability status of the household head, when the disability status is unavailable in the survey. We propose two alternative approaches to this extension: Aggregation and Instrumental Variables Approaches. We apply these approaches to data from Tanzania and show that both approaches work. Our estimation results show that disability is indeed positively associated with poverty in every region of mainland Tanzania.

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Notes

  1. See Elwan (1999); Haveman and Wolfe (2000), and Yeo and Moore (2003) and studies cited therein for earlier studies on this and related topics.

  2. The proportion of disabled people would also depend on the way it is defined and measured. See Online Appendix for a related discussion on this issue.

  3. See Mitra et al. (2017) for a review of literature on extra cost of disability (i.e., additional income that disabled people will need to achieve a given level of standards of living).

  4. For example, community-level indicators are often merged into the census and survey. Once covariates from auxiliary datasets are merged into the census and survey, we regard these covariates to be part of the census and survey datasets for our purpose.

  5. Both Mont and Cuong (2011) and Loyalka et al. (2014) provide poverty estimates by disability status only for urban and rural areas in Vietnam and China, respectively. Hoogeveen (2005) discussed below reports poverty rates by the disability status only for four regions. Mont and Nguyen (2018) provide poverty measures at the level of six regional estimates. We report poverty rates by the disability status of the household heads for 21 regions.

  6. Mont and Nguyen (2018) also apply the ELL method to derive poverty measures for people with and without disability at a spatially disaggregated level. However, it does not allow for the difference in model parameters between these two groups.

  7. Note that the regression in the second step is unweighted, since the accuracy of \({\hat{u}}^2_k\) as an estimate of \(\sigma ^2_{u,k}\) does not systematically improve as the number of observations in each cluster k grows. This is because the error term \(\eta \) does not go away by aggregation. Weighting can actually make the estimate of \(\sigma ^2_{\eta }\) less accurate.

  8. If either \({\tilde{\sigma }}^{2,(r)}_{\eta }\) or \({\tilde{\sigma }}^{2,(r)}_{\epsilon }\) is negative, we redraw until both are positive.

  9. Some regions were split after the survey was conducted. In 2002, the Manyara region broke out of the Arusha region and the census data allow us to distinguishes between Manyara and Arusha regions, but the survey data do not.

  10. See National Bureau of Statistics (2002) for the weights to derived the adult equivalence scale, which is similar to those cited in Collier et al. (1986). See World Bank (2015) for additional discussions on the adult equivalence scale used in Tanzania.

  11. Evidence from Malawi and Zambia indicates that self-reported disabled people, on average, experience much more activity limitations than self-reported non-disabled people (Loeb and Eide 2004, 2006).

  12. In Online Appendix Table A1, we report the distribution of types of disability.

  13. Online Appendix Table A2 also shows that the head’s economic disability status is more strongly correlated with poverty than the presence of an economically disabled member, showing the importance of household head’s disability status relative to other member’s disability status. Online Appendix also provides further discussions.

  14. In Online Appendix Table A4, we also report the summary statistics and correlations of \({\hat{u}}^2_k\), \({\bar{A}}_{k, \text {non-disabled}}\), and \({\bar{A}}_{k, \text {disabled}}\).

  15. Online Appendix Table A6 reports the point estimates and standard errors by disability status in each region.

  16. Note, however, that there is an ongoing effort to improve the data collection standards and data availability (Abualghaib et al. 2019; Groce and Mont 2017). The methodology proposed in this paper will remain relevant because it takes time to improve data availability and because it is still useful to see the change from the past situation.

  17. In addition, we also implemented a version of Hoogeveen (2005), where we use the same set of covariates as those in Table 2, except that we replace the individual disability status with the cluster-level prevalence of disability. The point estimate (standard error) of \(P_0\) for non-disabled households, disabled, and mainland Tanzania are 0.490 (0.009), 0.554 (0.010), and 0.491 (0.008), respectively. Hence, poverty appears to be overestimated when a comparable model is used. While this apparent overestimation may be driven by a particular specification we use, the standard errors also appear to be too low, particularly for non-disabled households. This is also the case when the Instrumental Variables Approach is used. This results is expected, since Hoogeveen (2005) does not take into consideration the fact that the cluster-level prevalence of disability is just a proxy for the individual-level disability status.

  18. In Tanzania, social exclusion of disabled people may not be among the most important issues facing those who are disabled (Kisanji 1995a, b; UNICEF 1999).

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Funding

An earlier version of this study was supported by SMU Research Grant (07-C208-SMU-007).

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  1. School of Economics, Singapore Management University, 90 Stamford Road, 178903, Singapore, Singapore

    Tomoki Fujii

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  1. Tomoki Fujii
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Fujii declares that he has no conflict of interest.

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All procedures performed in studies involving human participants were in accordance with the ethical standards of the Singapore Management University and with the 1964 Helsinki declaration and its later amendments or comparable ethical standards.

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This research grew out of my consulting work for the World Bank. I benefited from discussion with Chris Elbers, Hans Hoogeveen, Denis Leung, Wietze Lindeboom, Roy van der Weide, and an anonymous referee. An earlier version of this study was supported by SMU Research Grant (07-C208-SMU-007). Usual caveats apply.

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Appendices

Appendix

Appendix A: Comparison of regional poverty estimates

Table 5 provides the survey-only estimates of poverty rates at this level in column (1) and their corresponding SAE estimates in column (2) at the level of 20 survey regions. The survey-only and SAE estimates are generally close and have a strong positive correlation of 0.71. The null hypothesis for a two-sided z-test of equality of the survey-only and SAE estimates is not rejected at a five percent level of significance in all regions. The null hypothesis for the joint \(\chi ^2\)-test of the equality of the two estimates in all regions was also not rejected. Therefore, on balance, even though the difference between survey-only and SAE estimates can be large in absolute value for a few regions, there is no evidence that survey-only and SAE estimates are systematically different.

In column (3), we report the share of people living in disabled households. As the comparisons of column (3) with columns (1) and (2) show, there is a strong positive correlation between the poverty rate and share of people in disable households. The correlation coefficient between columns (1) and (3) [(2) and (3)] is 0.66 [0.68]. Because the calculation of the correlation between columns (1) and (3) does not involve any imputation, the positive correlation cannot be attributed to imputation.

Table 5 Comparison of regional poverty rates between survey-only and SAE estimates
Full size table

Appendix B: Proofs

Proof of Proposition 1

We first establish the consistency of the cluster-level OLS estimator \({\hat{\varvec{\beta }}}_{COLS}\equiv ({\bar{\varvec{X}}}^T_C{\bar{\varvec{X}}}_C)^{-1}{\bar{\varvec{X}}}_C{\bar{\varvec{Y}}}_S\). Letting \(j=0\) in Assumptions A3–A5 and using \(K_S\rightarrow \infty \) as \(K_C\rightarrow \infty \), we have:

$$\begin{aligned} {\hat{\varvec{\beta }}}_{COLS}-\varvec{\beta }= & {} (K_S^{-1}{\bar{\varvec{X}}}^T_C{\bar{\varvec{X}}}_C)^{-1}K_S^{-1}{\bar{\varvec{X}}}_C(\varvec{X}_S\varvec{\beta }+\varvec{U}_S)-\varvec{\beta }\\= & {} (K_S^{-1}{\bar{\varvec{X}}}^T_C{\bar{\varvec{X}}}_C)^{-1}K_S^{-1}{\bar{\varvec{X}}}_C({\bar{\varvec{X}}}_S-{\bar{\varvec{X}}}_C)\beta \\ {}{} & {} +(K_S^{-1}{\bar{\varvec{X}}}^T_C{\bar{\varvec{X}}}_C)^{-1}K_S^{-1}{\bar{\varvec{X}}}_C{\bar{\varvec{U}}}_S\xrightarrow {p}0_{L} \end{aligned}$$

This means that we can obtain consistent estimates of \(\sigma ^2_\eta \) and \(\sigma ^2_{\epsilon ,g}\) by running a regression of the squared OLS residual \({\hat{u}}^2_k\) on a constant and \({\bar{A}}_{k,1},\cdots ,{\bar{A}}_{k,G}\). Hence, we can obtain a consistent estimate \({\hat{\varvec{\Omega }}}_A\) of \(\varvec{\Omega }_A\) by replacing \(\sigma ^2_\eta \) and \(\sigma ^2_{\epsilon ,g}\) with their consistent estimates in the definition of \(\varvec{\Omega }_A\). Then, letting \({\hat{\varvec{J}}}_1\equiv {\hat{\varvec{\Omega }}}_A\varvec{W}_S{\hat{\varvec{\Omega }}}_A\), we have \({\hat{\varvec{J}}}_1\xrightarrow {p}\varvec{J}_1\) as \(K_C\rightarrow \infty \). From this and assumptions A1–A5 and using \(K_S\rightarrow \infty \) as \(K_C\rightarrow \infty \), we have the following results as \(K_C\rightarrow \infty \):

$$\begin{aligned} {\hat{\varvec{\beta }}}_{AGG}-\varvec{\beta }{} & {} = ({\bar{\varvec{X}}}^T_C{\hat{\varvec{\Omega }}}_{A}^{-T/2}{\bar{\varvec{W}}}_S\nonumber \\{} & {} \quad {\hat{\varvec{\Omega }}}_{A}^{-1/2}{\bar{\varvec{X}}}_C)^{-1}{\bar{\varvec{X}}}^T_C{\hat{\varvec{\Omega }}}_{A}^{-T/2}{\bar{\varvec{W}}}_S {\hat{\varvec{\Omega }}}_{A}^{-1/2}({\bar{\varvec{X}}}_S\varvec{\beta }+{\bar{\varvec{U}}}_S)-\varvec{\beta }\nonumber \\{} & {} = ({\bar{\varvec{X}}}^T_C {\hat{\varvec{J}}}_1{\bar{\varvec{X}}}_C)^{-1}{\bar{\varvec{X}}}^T_C{\hat{\varvec{J}}}_1 ({\bar{\varvec{X}}}_S\varvec{\beta }+{\bar{\varvec{U}}}_S)-\varvec{\beta }\nonumber \\{} & {} = (K^{-1}_S{\bar{\varvec{X}}}^T_C{\hat{\varvec{J}}}_1{\bar{\varvec{X}}}_C)^{-1}K^{-1}_S {\bar{\varvec{X}}}^T_C {\hat{\varvec{J}}}_1({\bar{\varvec{X}}}_S-{\bar{\varvec{X}}}_C)\varvec{\beta }\nonumber \\{} & {} \quad +(K_S^{-1}{\bar{\varvec{X}}}^T_C{\hat{\varvec{J}}}_1{\bar{\varvec{X}}}_C)^{-1} K_S^{-1}{\bar{\varvec{X}}}^T_C{\hat{\varvec{J}}}_1{\bar{\varvec{U}}}_S\nonumber \\{} & {} \xrightarrow {p} 0_L, \end{aligned}$$
(9)
$$\begin{aligned} \sqrt{K_S}({\hat{\varvec{\beta }}}_{AGG}-\varvec{\beta }){} & {} = (K^{-1}_S{\bar{\varvec{X}}}^T_C{\hat{\varvec{J}}}_1 {\bar{\varvec{X}}}_C)^{-1}K^{-1/2}_S {\bar{\varvec{X}}}^T_C{\hat{\varvec{J}}}_1({\bar{\varvec{X}}}_S-{\bar{\varvec{X}}}_C)\varvec{\beta }\nonumber \\{} & {} \quad +(K_S^{-1}{\bar{\varvec{X}}}^T_C{\hat{\varvec{J}}}_1{\bar{\varvec{X}}}_C)^{-1}K_S^{-1/2} {\bar{\varvec{X}}}^T_C{\hat{\varvec{J}}}_1{\bar{\varvec{U}}}_S \nonumber \\{} & {} \xrightarrow {d} {\mathcal {N}}(0,\Xi ^{-1}_1(\Xi _0+\Xi _2)\Xi ^{-1}_1), \end{aligned}$$
(10)

where \(\Xi _0\equiv \lim _{K_S\rightarrow \infty } K^{-1/2}_S {\bar{\varvec{X}}}^T_C{\hat{\varvec{J}}}_1({\bar{\varvec{X}}}_S-{\bar{\varvec{X}}}_C) \varvec{\beta }\varvec{\beta }^T({\bar{\varvec{X}}}_S-{\bar{\varvec{X}}}_C)^T{\hat{\varvec{J}}}_1{\bar{\varvec{X}}}_C\). Notice that the between-cluster variations in the regressors are already differenced out in \(({\bar{\varvec{X}}}_S-{\bar{\varvec{X}}}_C)\). Therefore, \(({\bar{\varvec{X}}}_S-{\bar{\varvec{X}}}_C)\) is influenced by the within-cluster variation in the regressor. With a sufficiently large fraction of households sampled in each cluster, the contribution of \(\Xi ^{-1}_1\Xi _0\Xi ^{-1}_1\) to the asymptotic variance of \({\hat{\varvec{\beta }}}_{AGG}\) is small relative to \(\Xi ^{-1}_1\Xi _2\Xi ^{-1}_1\). Hence, the asymptotic variance of \({\hat{\beta }}_{AGG}\) is approximately equal to \(\Xi ^{-1}_1\Xi _2\Xi ^{-1}_1\). \(\square \)

Proof of Proposition 2

First, note that \(N_D\) is approximately proportionate to \(K_D\) and that \(N_D\rightarrow \infty \) as \(K_D\rightarrow \infty \) for \(D\in \{C,S\}\). The consistency and asymptotic normality of the TS2SLS estimator can be derived from a variant of the standard argument (White 1984, Chap. 5). Under assumptions A1-A2 and A6-A9 and using \(K_S\rightarrow \infty \) as \(K_C\rightarrow \infty \), we have the following results as \(K_S\rightarrow \infty \):

$$\begin{aligned}{} & {} {\hat{\varvec{\beta }}}_{TS2SLS}-\varvec{\beta }\\{} & {} \quad =(N_C^{-1}\varvec{Z}^T_C\varvec{W}_C \varvec{X}_C)^{-1}(N_C^{-1}\varvec{Z}^T_C\varvec{W}_C \varvec{Z}_C)(N_S^{-1}\varvec{Z}^T_S\varvec{W}_S \varvec{Z}_S)^{-1}N_S^{-1}\varvec{Z}^T_S\varvec{W}_S \varvec{X}_S\varvec{\beta }\\{} & {} \qquad +(N_C^{-1}\varvec{Z}^T_C\varvec{W}_C \varvec{X}_C)^{-1}(N_C^{-1}\varvec{Z}^T_C\varvec{W}_C \varvec{Z}_C)(N_S^{-1}\varvec{Z}^T_S\varvec{W}_S \varvec{Z}_s)(N_S^{-1}\varvec{Z}^T_S\varvec{W}_S \varvec{U}_s)-\varvec{\beta }\\{} & {} \quad \xrightarrow {p}(\varvec{Q}_0^{-1}\varvec{Q}_1\varvec{Q}_1^{-1}\varvec{Q}_0-I)\varvec{\beta }=0,\\{} & {} \sqrt{K_C}(\varvec{\beta }_{TS2SLS}-\varvec{\beta })\\{} & {} \quad =\sqrt{K_C}\left[ (N_C^{-1}\varvec{Z}^T_C\varvec{W}_C \varvec{X}_C)^{-1}(N_C^{-1}\varvec{Z}^T_C\varvec{W}_C \varvec{Z}_C)(N_S^{-1}\varvec{Z}^T_S\varvec{W}_S \varvec{Z}_S)^{-1}N_S^{-1}\varvec{Z}^T_S\varvec{W}_S \varvec{X}_S-I\right] \varvec{\beta }\\{} & {} \qquad +\sqrt{K_C}\left[ (N_C^{-1}\varvec{Z}^T_C\varvec{W}_C \varvec{X}_C)^{-1}(N_C^{-1}\varvec{Z}^T_C\varvec{W}_C \varvec{Z}_C)(N_S^{-1}\varvec{Z}^T_S\varvec{W}_S \varvec{Z}_S)(N_S^{-1}\varvec{Z}^T_S\varvec{W}_S \varvec{U}_S)\right] \\{} & {} \quad \xrightarrow {d}{\mathcal {N}}(0,\varvec{Q}_0^{-1}\varvec{Q}_2 \varvec{Q}_0^{-1}). \end{aligned}$$

Notice that \(\varvec{Q}_1\) does not show up in the final expression because \((N_C^{-1}\varvec{Z}^T_C\varvec{W}_C \varvec{Z}_C)(N_S^{-1}\varvec{Z}^T_S\varvec{W}_S \varvec{Z}_S)^{-1}\xrightarrow {p} \varvec{Q}_1\varvec{Q}_1^{-1}=I\) as \(K_C\rightarrow \infty \). \(\square \)

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Fujii, T. Spatial disaggregation of poverty and disability: application to Tanzania. Empir Econ 66, 705–734 (2024). https://doi.org/10.1007/s00181-023-02478-8

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