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help for povdeco                               Stephen P. Jenkins (August 2006)
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Poverty indices, with optional decomposition by subgroup

povdeco varname [weights] [if exp] [in range] [, pline(#)
varpline(zvar) bygroup(groupvar) summarize]

fweights and aweights are allowed; see help weights.

Description

povdeco estimates three poverty indices from the Foster, Greer and
Thorbecke (1984) class, FGT(a). FGT(0) is the headcount ratio (the
proportion poor); FGT(1) is the average normalised poverty gap; FGT(2) is
the average squared normalised poverty gap. The larger a is, the greater
the degree of `poverty aversion' (sensitivity to large poverty gaps).
Optionally provided are statistics such as mean income amongst the poor
and also decompositions of these indices by population subgroup. Poverty
decompositions by subgroup are useful for providing poverty `profiles' at
a point in time, and for analyzing secular trends in poverty using
shift-share analysis. Unit record (`micro' level) data are required.

Typically one's data are in one of two forms. Either (1) the money
incomes for each unit i, x_i, are equivalised using an equivalence scale
factor, m_i, so that y_i = x_i / m_i, and the poverty line is a single
(common) value, in the same units as equivalised income, z. Or (2)
incomes are not equivalised, but there are different poverty lines
depending on (for example) household type. Suppose the poverty line for
unit i is z_i.  Observe that if z_i = z * m_i, FGT poverty index
calculations based on {y_i,z} give exactly the same answers as
calculations based on {x_i,z_i}, i=1,...,n.  For the common poverty line
case, use pline(#) to specify the poverty line. For the heterogeneous
poverty line case, use varpline(zvar) to specify the poverty lines.

For the characterization of the FGT(a) poverty index, see Foster et al.
(1984).  Seidl (1988) and Zheng (1997) survey the literature on aggregate
poverty indices.

groupvar must take non-negative integer values only. To create such a
variable from an existing variable, use the egen function group.  By
default, observations with missing values on groupvar are excluded from
calculations when the bygroup option is specified. If you wish to include
them, create a new variable with the egen function group and use its
missing option. The egen function group is also useful for multi-way
decompositions. E.g. for a decomposition by sex and region, create a new
groupvar defining sex-region combinations by specifying sex and region in
group(varlist).

Bootstrapped standard errors for the estimates of the indices can be
derived using bootstrap. Standard errors derived using linearization
methods can be calculated straightforwardly by first creating for each
FGT(a) index, a variable equal to I_i * [ ( z_i - y_i ) / z_i) ] ^ a,
where I_i = 1 if observation i is poor and 0 otherwise. Then estimate the
mean of each of these new variables using svy mean having first
appropriately svyset your data. See Jenkins (2006) for examples.

Technical details

Consider a population of persons (or households ...), i = 1,...,n, with
income y_i, and weight w_i. Let f_i = w_i / N, where N = SUM w_i.  (In
what follows all sums are over all values of whatever is subscripted.)
When the data are unweighted, w_i = 1 and N = n.  The poverty line is z_i
for each i, and the poverty gap for person i is max(0, z_i-y_i). For the
common poverty line case (1) above, z_i = z, all i.

Suppose there is an exhaustive partition of the population into
mutually-exclusive subgroups k = 1,...,K.

The FGT class of poverty indices is given by

FGT(a) = SUM f_i * I_i * [ ( z_i - y_i ) / z_i) ] ^ a, a >= 0,

where I_i = 1 if y_i < z_i and I_i = 0 otherwise.

Each FGT(a) index can be additively decomposed as

FGT(a) = SUM v_k * FGT_k(a)

where v_k = N_k / N is the weighted number of persons in subgroup k
divided by the weighted total number of persons (subgroup population
share), and FGT_k(a), poverty for subgroup k, is calculated as if each
subgroup were a separate population.

Also displayed when subgroup decompositions requested, for each k, are:

subgroup poverty 'share', S_k = v_k * FGT_k(a) / FGT(a), and

subgroup poverty 'risk', R_k = FGT_k(a) / FGT(a) = S_k / v_k.

Options

bygroup(groupvar) requests inequality decompositions by population
subgroup, with subgroup membership summarized by groupvar.

summarize requests presentation of the mean of varname, the mean among
the poor, and the mean poverty gap among the poor.

pline(#) is used to specify the poverty line # in the common poverty line
case.

varpline(zvar) is used to specify the variable zvar containing the values
of poverty line for each observation in the heterogeneous poverty
line case.

Saved results

r(fgt0), r(fgt1), r(fgt2)   FGT(a), for a = 0, 1, 2

r(mean)                     mean
r(meanpoor)                 mean among the poor
r(meangappoor)              mean poverty gap among the poor
r(N), r(sumw)               number of observations, sum of weights

If the bygroup option is specified, also saved are:

r(fgt0_k)                   FGT(a), for a = 0, 1, 2, and
r(fgt1_k)                   each subgroup k, where the values of k
r(fgt2_k)                   correspond to the values of groupvar
in the estimation sample. See r(levels) below.

r(mean_k)                   mean among subgroup k
r(meanpoor_k)               mean among the poor in subgroup k
r(meangappoor_k)            mean poverty gap among the poor in subgroup k
r(n_k), r(sumw_k)           number of subgroup observations, subgroup sum o
> f weights
r(v_k)                      subgroup population share, v_k
r(share0_k)                 FGT(0) poverty share among subgroup k
r(share1_k)                 FGT(1) poverty share among subgroup k
r(share2_k)                 FGT(2) poverty share among subgroup k
r(risk0_k)                  FGT(0) poverty risk among subgroup k
r(risk1_k)                  FGT(1) poverty risk among subgroup k
r(risk2_k)                  FGT(2) poverty risk among subgroup k

r(levels)                   macro containing the set of values of groupvar
(the number of unique values = K)

For the convenience of users of earlier versions of these programs, a
selected set of estimates is also saved in global macros, as follows.

S_FGT0, S_FGT1, S_FGT2      FGT(a), for a = 0, 1, 2

Examples

. povdeco x [aw = wgtvar], pline(100)

. povdeco x [aw = wgtvar], pline(100) by(famtype)

. povdeco x, varpline(z)

. povdeco x if sex==1, pl(100) summarize

. // bootstrapped standard errors for FGT(2) in Stata version 8

. preserve

. keep if x > 0 & x < .

. version 8: bootstrap "povdeco x, pline(100)" fgt2 = r(fgt2), reps(100)

. restore

. // bootstrapped standard errors for FGT(2) in Stata version 9

. preserve

. keep if x > 0 & x < .

. bootstrap fgt2 = r(fgt2), reps(100): povdeco x, pline(100)

. restore

. // multi-way decomposition

. egen sexXregion = group(sex region)

. povdeco x, pline(100) by(sexXregion)

Author

Stephen P. Jenkins <stephenj@essex.ac.uk>
Institute for Social and Economic Research
University of Essex, Colchester CO4 3SQ, U.K.

Acknowledgements

For comments and suggestions, I am grateful to Philippe Van Kerm and Nick
Cox.

References

Foster, J.E., Greer, J., and Thorbecke, E. 1984.  A class of decomposable
poverty indices.  Econometrica 52: 761-766.

Jenkins, S.P. 2006. Estimation and interpretation of measures of
inequality, poverty, and social welfare using Stata. Presentation at
North American Stata Users' Group Meetings 2006, Boston MA.
http://econpapers.repec.org/paper/bocasug06/16.htm.

Seidl, C. 1988. Poverty measurement: a survey. In: D. Bös, M. Rose and C.
Seidl (eds.), Welfare and Efficiency in Public Economics. Heidelberg:
Springer-Verlag.

Zheng, B. 1997. Aggregate poverty indices.  Journal of Economic Surveys
11: 123-162.

Also see

poverty if installed; ineqdeco if installed.

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