Estimation of Disaggregate-Level Poverty Incidence in Odisha Under Area-Level Hierarchical Bayes Small Area Model

Abstract

Sustainable development goal-1 of the United Nations is to end poverty in all its forms everywhere. The estimates of poverty related parameters obtained from large scale sample survey are often available at large domain level (e.g. state level). But, poverty rates are not uniformly distributed across the regions. The regional variations are masked in such large domain level estimates. However, for monitoring the progress of poverty alleviation programmes aimed at reduction of poverty often require micro or disaggregate level estimates. The traditional survey estimation approaches are not suitable for generating the reliable estimates at this level because of sample size problem. It is the main endeavor of Small Area Estimation (SAE) approach to produce micro level statistics with acceptable precision without incurring any extra cost and utilizing existing survey data. In this study, the Hierarchical Bayes approach of SAE has been applied to generate reliable and representative district level poverty incidence for the State of Odisha in India using the Household Consumer Expenditure Survey 2011–2012 data of National Sample Survey Office and linked with Population Census 2011. The results show the precise performance of model based estimates generated by SAE method to a greater extent than the direct survey estimates. A poverty map has also been produced to observe the spatial inequality in poverty distribution.

Appendix

Full conditional distributions for the Gibbs sampler under four HB models are presented below. Let, $\tilde{p}\, = \,\left( {p_{1w} , \ldots ,p_{mw} } \right)^{T}$, ${\mathbf{P}}\, = \,\left( {P_{1} , \ldots ,P_{m} } \right)^{T}$, ${\mathbf{X}}\, = \,\left( {{\mathbf{x}}_{1}^{T} , \ldots ,{\mathbf{x}}_{m}^{T} } \right)^{T}$, ${\mathbf{x}}_{i}^{T} = \,\left( {x_{i1} , \ldots ,x_{ik} } \right)$, ${\varvec{\upbeta}}\,{ = }\,\left( {\beta_{1} , \ldots ,\beta_{k} } \right)^{T}$ and k represents number of auxiliary variates and usually $x_{i1}$ is taken to be 1 $\forall \,\,{\text{i = }}\,1, \ldots ,{\text{m}}$.

The full conditional distributions for the M1 are given as,

• (1)
  $$P_{i} \,|\,{\varvec{\upbeta}},\,\sigma_{v}^{2} ,\,\tilde{p}\sim\,N\,\left( {\frac{{\sigma_{v}^{2} }}{{\sigma_{v}^{2} + \,\sigma_{ei}^{2} }}\,p_{iw} \, + \,\frac{{\sigma_{ei}^{2} }}{{\sigma_{v}^{2} + \,\sigma_{ei}^{2} }}{\mathbf{x}}_{i}^{T} {\varvec{\upbeta}},\,\frac{{\sigma_{ei}^{2} \sigma_{v}^{2} }}{{\sigma_{v}^{2} + \,\sigma_{ei}^{2} }}} \right)\,;$$
• (2)
  $${\varvec{\upbeta}}\, |\,P_{i} ,\,\sigma_{v}^{2} ,\,{\tilde{\text{p}}}\,\sim\,{\text{N}}\,\,\left( {\left( {{\mathbf{X}}^{T} {\mathbf{X}}} \right)^{ - 1} {\mathbf{X}}^{T} {\mathbf{P}} ,\,\sigma_{\text{v}}^{ 2} \left( {{\mathbf{X}}^{T} {\mathbf{X}}} \right)^{ - 1} } \right)\,;$$
• (3)
  $$\sigma_{v}^{2} \,|\,{\varvec{\upbeta}},\,P_{i} ,\,\tilde{p}\,\sim\,IG\,\left( {a + \frac{m}{2},\,\,b + \,\frac{{\sum\limits_{i = 1}^{m} {\left( {P_{i} \, - \,{\mathbf{x}}_{i}^{T} {\varvec{\upbeta}}} \right)^{2} } }}{2}} \right).$$

The full conditional distributions for the M2 are given as follows,

• (1)
  $$P_{i} \,|\,{\varvec{\upbeta}},\,\sigma_{v}^{2} ,\,\tilde{p}\, \propto \,\frac{1}{{P_{i} \,\left( {1 - \,P_{i} } \right)\sqrt {\sigma_{ei}^{2} \sigma_{v}^{2} } \,}}\,\exp \left( { - \,\frac{{\left( {p_{iw} \, - P_{i} \,\,} \right)^{2} }}{{2\sigma_{ei}^{2} }}\, - \,\frac{{\left( {\log it\left( {P_{i} } \right)\, - {\mathbf{x}}_{i}^{T} {\varvec{\upbeta}}\,\,} \right)^{2} }}{{2\sigma_{v}^{2} }}} \right)\,;$$
• (2)
  $${\varvec{\upbeta}}\, |\,{\text{P}}_{\text{i}} ,\,\sigma_{\text{v}}^{ 2} ,\,{\tilde{\text{p}}}\sim\,{\text{N}}\,\,\left( {\left( {{\mathbf{X}}^{T} {\mathbf{X}}} \right)^{ - 1} {\mathbf{X}}^{T} {\text{logit}}({\mathbf{P}}) ,\,\sigma_{\text{v}}^{ 2} \left( {{\mathbf{X}}^{T} {\mathbf{X}}} \right)^{ - 1} } \right)\,;$$
• (3)
  $$\sigma_{v}^{2} \,|\,{\varvec{\upbeta}},\,P_{i} ,\,\tilde{p}\,\sim\,IG\,\left( {a + \frac{m}{2},\,\,b + \,\frac{{\sum\limits_{i = 1}^{m} {\left( {\log it\left( {P_{i} } \right) - \,{\mathbf{x}}_{i}^{T} {\varvec{\upbeta}}} \right)^{2} } }}{2}} \right).$$

The full conditional distributions for the M3 are given as below,

• (1)
  $$P_{i} \,|\,{\varvec{\upbeta}},\,\sigma_{v}^{2} ,\,\tilde{p}\, \propto \,\frac{1}{{P_{i} \,\left( {1 - \,P_{i} } \right)\sqrt {\,\,\xi_{i} \sigma_{v}^{2} } \,}}\,\exp \left( { - \,\frac{{\left( {p_{iw} \, - P_{i} } \right)^{2} }}{{2\,\xi_{i} }}\, - \,\frac{{\left( {\log it\left( {P_{i} } \right)\, - {\mathbf{x}}_{i}^{T} {\varvec{\upbeta}}\,\,} \right)^{2} }}{{2\sigma_{v}^{2} }}} \right)\,;$$
• (2)
  $${\varvec{\upbeta}}\, |\,P_{i} ,\,\sigma_{\text{v}}^{ 2} ,\,{\tilde{\text{p}}}\,\sim\,{\text{N}}\,\,\left( {\left( {{\mathbf{X}}^{T} {\mathbf{X}}} \right)^{ - 1} {\mathbf{X}}^{T} {\text{logit}}\left( {\mathbf{P}} \right) ,\,\sigma_{\text{v}}^{ 2} \left( {{\mathbf{X}}^{T} {\mathbf{X}}} \right)^{ - 1} } \right)\,;$$
• (3)
  $$\sigma_{v}^{2} \,|\,{\varvec{\upbeta}},\,P_{i} ,\,\tilde{p}\,\sim\,IG\,\left( {a\, + \frac{m}{2},\,\,b\,\, + \,\,\frac{{\sum\limits_{i = 1}^{m} {\left( {\log it\,\left( {P_{i} } \right)\, - \,{\mathbf{x}}_{i}^{T} {\varvec{\upbeta}}} \right)^{2} } }}{2}} \right).$$

Let, $\delta_{i} \, = \,\frac{{n_{i} }}{{deff_{i} }} - 1$.

Then the full conditional distributions for the M4 are given as follows,

• (1)
  $$P_{i} \,|\,{\varvec{\upbeta}},\,\sigma_{v}^{2} ,\,\tilde{p}\, \propto \,\frac{1}{{P_{i} \,\left( {1 - \,P_{i} } \right)\sqrt {\,\,\sigma_{v}^{2} } \,}}\,\frac{{p_{iw}^{{P_{i} \delta_{i} \, - \,1}} \,\left( {1 - \,p_{iw} } \right)^{{(1 - \,P_{i} )\delta_{i} \, - \,1}} }}{{\varGamma \,\left( {P_{i} \delta_{i} } \right)\,\varGamma \left( {\left( {1 - \,P_{i} } \right)\delta_{i} } \right)\,}}\exp \left( { - \,\,\frac{{\left( {\log it\left( {P_{i} } \right)\, - {\mathbf{x}}_{i}^{T} {\varvec{\upbeta}}\,\,} \right)^{2} }}{{2\sigma_{v}^{2} }}} \right)\,;$$
• (2)
  $${\varvec{\upbeta}}\, |\,P_{i} ,\,\sigma_{\text{v}}^{ 2} ,\,{\tilde{\text{p}}}\,\sim\,{\text{N}}\,\,\left( {\left( {{\mathbf{X}}^{T} {\mathbf{X}}} \right)^{ - 1} {\mathbf{X}}^{T} {\text{logit}}\left( {\mathbf{P}} \right) ,\,\sigma_{\text{v}}^{ 2} \left( {{\mathbf{X}}^{T} {\mathbf{X}}} \right)^{ - 1} } \right)\,;$$
• (3)
  $$\sigma_{v}^{2} \,|\,{\varvec{\upbeta}},\,P_{i} ,\,\tilde{p}\,\sim\,IG\,\left( {a\, + \frac{m}{2},\,\,b\,\, + \,\,\frac{{\sum\limits_{i = 1}^{m} {\left( {\log it\,\left( {P_{i} } \right)\, - \,{\mathbf{x}}_{i}^{T} {\varvec{\upbeta}}} \right)^{2} } }}{2}} \right).$$