Decomposition of inter-regional poverty gap in India: a spatial approach

Abstract

This paper examines the impact of various socio-economic factors on consumption in rural West Bengal, an eastern state of India, using a regression-based technique reformulated in a spatial framework. The difference of incidences of poverty (head count ratios) in two parts (North and South) of rural West Bengal is then decomposed using the familiar Oaxaca decomposition methodology into an aggregate characteristics effect, which is interpreted as a resource effect and an aggregate coefficients effect, which is interpreted as an efficiency effect. An important observation from the present analysis is that the poorer North Bengal has a scarcity in terms of availability of characteristics (resources) compared to that in South Bengal and the resource scarcity in North Bengal is the dominant factor causing the poverty gap between the two parts of West Bengal. Thus, attention needs to be paid to North Bengal with respect to enhancement of important policy variables like education level, Government aid, and employment opportunities.

Appendices

Appendix 1

1.1 Finding the incidence of poverty

From (4)

$$\begin{aligned} \left( {\frac{y}{z}} \right) ^{*}=\text{ MX }\beta +M\varepsilon =X^{*}\beta +\varepsilon ^{*}, \end{aligned}$$

(21)

where $M=\left( {I-\rho W} \right) ^{-1}$ is the Leontief inverse, $X^{*}= \text{ MX }, \varepsilon ^{*}= M\varepsilon$.

Let $p_{_i}$ be the probability of the ith household being poor.

Then,

$$\begin{aligned}&p_{_i} =\text{ prob }\left( {\left( {\frac{y}{z}} \right) _{i }^*\prec 0} \right) = \text{ prob }\left( {X_i^*\beta +\varepsilon _i^*\prec 0} \right) \nonumber \\&\quad =\text{ prob }\left( {\varepsilon _i^*\prec -X_i^*\beta } \right) =\text{ prob }\left( {\frac{\varepsilon _i^*-E\left( {\varepsilon _i^*} \right) }{\sqrt{\mathrm{var}\left( {\varepsilon _i^*} \right) }}\prec \frac{-X_i^*\beta -E\left( {\varepsilon _i^*} \right) }{\sqrt{\mathrm{var}\left( {\varepsilon _i^*} \right) }}} \right) \nonumber \\&\quad =\varPhi \left( {\frac{-X_i^*\beta }{\sigma _i }} \right) ;\left[ {E\left( {\varepsilon _i^*=0} \right) ,\;\mathrm{var}\left( {\varepsilon _i^*} \right) =\sigma _i^2,say} \right] ; \nonumber \\&\quad =\varPhi \left( {X_i^{**} \beta ^{*}} \right) ; \left[ {\beta ^{*}=-\beta ,\; X_i^{**} =\frac{X_i^*}{\sigma _i }} \right] . \end{aligned}$$

(22)

The incidence of poverty for any region (R) will be asymptotically equal to the sample average of $p_{_i} {^{\prime }}$s.

Thus, $H_R =\frac{1}{n^{R}}\sum _{i=1}^{n^{R}} {\varPhi \left( {X_{i_{R}}^{**} \hat{{\beta }}_{R }^*} \right) }$; R = A, B, where $n^{R}$ is the number of households in region R.

Appendix 2

$$\begin{aligned} C^{1}\equiv C_\mathrm{con}^{1} +C_\mathrm{spat}^{1} \end{aligned}$$

Proof

$$\begin{aligned} X^{**}=\sum {^{-1}}\text{ MX }. \end{aligned}$$

Substituting $X^{**}=\sum {^{-1}}\text{ MX }$ from (21) and (22) with $\sum {{=}diag\left( {\sigma _1,\sigma _2 ,\ldots ,\sigma _n} \right) }$, where $\sigma _{i }^2 ={\text{ var }}\left( {\varepsilon _i^*} \right)$, $C^{1}$ can be written as

$$\begin{aligned} C^{1}&= \left[ \left\{ \overline{\sum {_\mathrm{A}^{-1} \left( {I+\rho _\mathrm{A} W_\mathrm{A} +\rho _\mathrm{A}^2 W_\mathrm{A}^2 +\ldots } \right) X_\mathrm{A}}}\right. \right. \\&\left. \left. -\overline{\sum {_\mathrm{B}^{-1} \left( {I+\rho _\mathrm{B} W_\mathrm{B} +\rho _\mathrm{B}^2 W_\mathrm{B}^2 +\ldots } \right) X_\mathrm{B} } } \right\} \hat{{\beta }}_\mathrm{A}^*\right] \\&\quad \times \phi \left( {\overline{X_\mathrm{A}^{**} } \beta _\mathrm{A}^*}\right) ; \quad \left[ \because {X^{*}=\left( {I-\rho W} \right) ^{-1}X} \right] \\&= \left[ \left( {\overline{\sum \nolimits _\mathrm{A}^{-1} {X_\mathrm{A} } } -\overline{\sum \nolimits _\mathrm{B}^{-1} {X_\mathrm{B} } } } \right) \hat{{\beta }}_\mathrm{A}^*+\left( {\rho _\mathrm{A}\overline{\sum \nolimits _\mathrm{A}^{-1} {W_\mathrm{A} X_\mathrm{A}}} -\rho _\mathrm{B}\overline{\sum \nolimits _\mathrm{B}^{-1} {W_\mathrm{B} X_\mathrm{B} } } } \right) \hat{{\beta }}_\mathrm{A}^*\right. \\&\quad \left. +\left( {\rho _\mathrm{A}^2 \overline{\sum \nolimits _\mathrm{A}^{-1} {W_\mathrm{A}^2 X_\mathrm{A} } } -\rho _\mathrm{B}^2 \overline{\sum \nolimits _\mathrm{B}^{-1} {W_\mathrm{B}^2 X_\mathrm{B} } } } \right) \hat{{\beta }}_\mathrm{A}^*\right] \times \phi \left( {\overline{X_\mathrm{A}^{**} } \hat{{\beta }}_\mathrm{A}^*} \right) \end{aligned}$$

[Higher ordered terms involving $\rho W$ are ignored in empirical exercises (Lesage and Charles 2008)]

$$\begin{aligned}&= \left\{ {\left( {\overline{\sum \nolimits _\mathrm{A}^{-1} {X_\mathrm{A}}} -\overline{\sum \nolimits _\mathrm{B}^{-1} {X_\mathrm{B} } } } \right) \hat{{\beta }}_\mathrm{A}^*} \right\} \quad \times \phi \left( {\overline{X_\mathrm{A}^{**} } \hat{{\beta }}_\mathrm{A}^*} \right) \\&\quad + \left\{ \left( {\rho _\mathrm{A}\overline{\sum \nolimits _\mathrm{A}^{-1} {W_\mathrm{A}X_\mathrm{A} } } -\rho _\mathrm{B} \overline{\sum \nolimits _\mathrm{B}^{-1} {W_\mathrm{B} X_\mathrm{B} } } } \right) \hat{{\beta }}_\mathrm{A}^*\right. \\&\quad \left. +\left( {\rho _\mathrm{A}^2 \overline{\sum \nolimits _\mathrm{A}^{-1} {W_\mathrm{A}^2 X_\mathrm{A} } } -\rho _\mathrm{B}^2 \overline{\sum \nolimits _\mathrm{B}^{-1} {W_\mathrm{B}^2 X_\mathrm{B} } } } \right) \hat{{\beta }}_\mathrm{A}^*\right\} \, \times \,\phi \left( {\overline{X_\mathrm{A}^{**} } \hat{{\beta }}_\mathrm{A}^*} \right) . \\&D^{1}\equiv D_\mathrm{con}^{1} +D_\mathrm{spat}^{1}. \end{aligned}$$

$\square$

Proof

Writing $X^{**}=\sum {^{-1}} \text{ MX },\, D^{1}$ can be written as

$$\begin{aligned} D^{1}&= \overline{\sum \nolimits _\mathrm{B}^{-1} {\left( {I+\rho _\mathrm{B} W_\mathrm{B} +\rho _\mathrm{B}^2 W_\mathrm{B}^2 +\cdots } \right) } X_\mathrm{B} } \left( {\hat{{\beta }}_\mathrm{A}^*-\hat{{\beta }}_\mathrm{B}^*} \right) \phi \left( {\overline{X_\mathrm{B}^{**} } \hat{{\beta }}_\mathrm{B}^*} \right) \\&= \left\{ {\overline{\sum \nolimits _\mathrm{B}^{-1} {X_\mathrm{B} } } \left( {\hat{{\beta }}_\mathrm{A}^*-\hat{{\beta }}_\mathrm{B}^*} \right) } \right. +\left. \rho _\mathrm{B} \overline{\sum \nolimits _\mathrm{B}^{-1} {W_\mathrm{B} X_\mathrm{B} } } \left( {\hat{{\beta }}_\mathrm{A}^*-\hat{{\beta }}_\mathrm{B}^*} \right) \right. \\&\left. +\rho _\mathrm{B }^2 \overline{\sum \nolimits _\mathrm{B}^{-1} {W_\mathrm{B }^2 X_\mathrm{B} } } \left( {\hat{{\beta }}_\mathrm{A}^*-\hat{{\beta }}_\mathrm{B}^*} \right) \right\} \phi \left( {\overline{X_\mathrm{B}^{**} } \hat{{\beta }}_\mathrm{B}^*} \right) \end{aligned}$$

[Higher ordered terms involving $\rho W$ are ignored in empirical exercises (Lesage and Charles 2008)]

$$\begin{aligned}&= \left\{ {\overline{\sum \nolimits _\mathrm{B}^{-1} {X_\mathrm{B} } } \left( {\beta _\mathrm{A}^*-\beta _\mathrm{B}^*} \right) } \right\} \phi \left( {\overline{X_\mathrm{B}^{**} } \hat{{\beta }}_\mathrm{B}^*} \right) \\&\quad +\left\{ {\rho _\mathrm{B} \overline{\sum \nolimits _\mathrm{B}^{-1} {W_\mathrm{B} X_\mathrm{B} } } \left( {\beta _\mathrm{A}^*-\beta _\mathrm{B}^*} \right) +\rho _\mathrm{B }^2 \overline{\sum \nolimits _\mathrm{B}^{-1} {W_\mathrm{B }^2 X_\mathrm{B} } } \left( {\beta _\mathrm{A}^*-\beta _\mathrm{B}^*} \right) } \right\} \phi \left( {\overline{X_\mathrm{B}^{**} } \hat{{\beta }}_\mathrm{B}^*}\right) ; \\&\equiv D_\mathrm{con}^{1} +D_\mathrm{spat}^{1}. \end{aligned}$$

$\square$

Appendix 3

1.1 Statistical precision of $C, D, C_k$, and $D_k$

For finding the precision of aggregate (C and D) as well as individual $(C_k\,\text{ and }\,D_k)$ effects, asymptotic variances of C, D, $C_k$, and $D_k$ are found following Yun (2005) and Chattopadhyay (2012) as.

1.2 Asymptotic variance of $C, D, C_k$, and $D_k$

The aggregate characteristics effect, C is a function of the estimated coefficients of region A and the autoregressive coefficient of region B i.e.,

$C=f\left( {\hat{{\rho }}_\mathrm{A},\hat{{\beta }}_\mathrm{A}^*,\hat{{\rho }}_\mathrm{B}} \right) ;$ using (9) and (10).

Using the delta method, the asymptotic variance of C $\left( {{=}\sigma _\mathrm{C}^2 } \right)$ can thus be obtained³⁴ as

$$\begin{aligned} \sigma _C^2 =\left( {\frac{\partial C}{\partial \hat{{\rho }}_\mathrm{A}} \frac{\partial C}{\partial \hat{{\beta }}_\mathrm{A}^*}}\right) \sum {\left( {\rho _\mathrm{A},\beta _\mathrm{A }^*} \right) } \left( {\frac{\partial C}{\partial \hat{{\rho }}_\mathrm{A} } \frac{\partial C}{\partial \hat{{\beta }}_\mathrm{A}^*}} \right) ^{T} \quad +\left( {\frac{\partial C}{\partial \hat{{\rho }}_\mathrm{B}}} \right) \sum {\left( {\rho _\mathrm{B}}\right) }\left( {\frac{\partial C}{\partial \hat{{\rho }}_\mathrm{B} }} \right) ^{T}; \end{aligned}$$

where $\sum {\left( {\rho _\mathrm{A}, \beta _\mathrm{A }^*} \right) }=$ asymptotic variance–covariance matrix of $\left( {\rho _\mathrm{A}, \beta _\mathrm{A }^*} \right) , \sum {\rho _\mathrm{B}} =$ asymptotic variance of $\rho _\mathrm{B}$.

Again the aggregate coefficients effect, D is a function of the coefficients $\left( {\beta ^{*}}\right)$ of both the regions A and B and of the autoregressive coefficient of region B, i.e., $D=f\left( {\hat{{\beta }}_A^*;\hat{{\rho }}_\mathrm{B},\beta _\mathrm{B}^*} \right)$; using (12) and (13).

Thus using the delta method, the asymptotic variance of coefficients effect, $D \left( {{=}\sigma _\mathrm{D}^2 } \right)$ can be derived as

$$\begin{aligned} \sigma _\mathrm{D}^2 =\left( {\frac{\partial D}{\partial \hat{{\beta }}_\mathrm{A}^*}} \right) \sum {\left( {\beta _\mathrm{A}^*} \right) } \left( {\frac{\partial D}{\partial \hat{{\beta }}_\mathrm{A}^*}} \right) ^{T}+\left( {\frac{\partial D}{\partial \hat{{\rho }}_\mathrm{B} } \frac{ \partial D}{\partial \hat{{\beta }}_\mathrm{B}^*}} \right) \sum {\left( {\rho _\mathrm{B}, \beta _\mathrm{B}^*}\right) } \left( {\frac{\partial D}{\partial \hat{{\rho }}_\mathrm{B}} \frac{\partial D}{\partial \hat{{\beta }}_\mathrm{B}^*}} \right) ^{T}, \end{aligned}$$

where $\sum {\left( {\beta _\mathrm{A}^*} \right) }=$ asymptotic variance– covariance matrix of $\beta _\mathrm{B}^*$ and $\sum {\left( {\rho _\mathrm{B}, \beta _\mathrm{B}^*} \right) }$ = asymptotic variance–covariance matrix of $\left( {\rho _\mathrm{B}, \beta _\mathrm{B}^*} \right)$.

Now $\sum {\left( {\rho _\mathrm{A},\beta _\mathrm{A }^*}\right) }, \sum {\left( {\beta _\mathrm{A}^*}\right) }$ and $\sum {\left( {\rho _\mathrm{B} ,\beta _\mathrm{B}^*}\right) }$ can, respectively, be obtained from $\sum {\left( {\rho _\mathrm{A}, \beta _\mathrm{A}}\right) }, \sum {\left( {\beta _\mathrm{A} } \right) }$ and $\sum {\left( {\rho _\mathrm{B}, \beta _\mathrm{B} } \right) }$ by using the delta method, where

$$\begin{aligned} \sum {\left( . \right) } =\text{ asymptotic }\, \text{ variance-- }\text{ covariance }\, \text{ matrix }\, \text{ of } \left( .\right) . \end{aligned}$$

Using the same methodology as in above, the asymptotic variance of

$C_k \left( {=\sigma _{C_k }^2 } \right)$ and the asymptotic variance of $D_k \left( {{=}\sigma _{D_k }^2 } \right)$ can be obtained as

$$\begin{aligned} \sigma _{C_k }^2 =\left( {\frac{\partial C_k }{\partial \hat{{\rho }}_\mathrm{A} } \frac{\partial C_k }{\partial \hat{{\beta }}_\mathrm{A}^*}} \right) \sum {\left( {\rho _\mathrm{A},\beta _\mathrm{A }^*} \right) } \left( {\frac{\partial C_k }{\partial \hat{{\rho }}_\mathrm{A}} \frac{\partial C_k }{\partial \hat{{\beta }}_\mathrm{A}^*}} \right) ^{T}+ \left( {\frac{\partial C_k }{ \partial \hat{{\rho }}_\mathrm{B} }} \right) \sum {\left( {\rho _\mathrm{B} } \right) } \left( {\frac{\partial C_k }{\partial \hat{{\rho }}_\mathrm{B} }} \right) ^{T} \end{aligned}$$

and³⁵

$$\begin{aligned} \sigma _{D_k }^2 = \left( {\frac{\partial D_k }{\partial \hat{{\beta }}_\mathrm{A}^*}} \right) \sum {\left( {\beta _\mathrm{A}^*} \right) } \left( {\frac{\partial D_k }{\partial \hat{{\beta }}_\mathrm{A}^*}} \right) ^{T}+\left( {\frac{\partial D_k }{\partial \hat{{\rho }}_\mathrm{B}} \frac{\partial D_k }{ \partial \hat{{\beta }}_\mathrm{B}^*}} \right) \sum {\left( {\rho _\mathrm{B}, \beta _\mathrm{B}^*}\right) } \left( {\frac{\partial D_k }{\partial \hat{{\rho }}_\mathrm{B }} \frac{\partial D_k }{\partial \hat{{\beta }}_\mathrm{B }^*}} \right) ^{T}. \end{aligned}$$