Two-Step Poverty Estimation

Pudney’s Estimator

Many problems outlined in the previous section arise from the attempt to infer the conditional mean of the “artificial” poverty variable directly. An alternative is to conduct the analysis in two steps: fir s t one estimates the conditional income distribution, and then computes the conditional mean of poverty from there. The effect of the covariates on poverty can be analysed by discretely varying X and comparing the predicted conditional poverty values.

The most appealing way to estimate a conditional income distribution may be a kernel regression of X on y, as a kernel does not place restrictions on the shape of the density function. In practice, however, this is infeasible when one wants to include a larger number of covariates. A feasible option is to fit the Gallant and Nychka (1987) semi- nonparametric conditional density, an approach that has been advocated by Pudney (1999). In this case, the density is specified as a normal multiplied by a suitably chosen polynomial:

(3 .5) = where h{s ) = ^ a ^ s^ (j){s)

(T(%) _t=o

H(X) and o(X) are the first two moments o f the distribution that determine its location and scale, and that may vary with the covariates. Implementing (3.5) requires specifying parametric forms for //(Zj and g(X) (usually linear or log-linear), and estimating //, a, and the cr-coefficients from the data. The order o f the polynomial k is chosen by

selection techniques such as goodness-of-fit statistics or asymptotic likelihood tests. Note that the higher moments of the distribution are assumed to be independent of X.

An Alternative Technique

The biggest disadvantage of Pudney’s technique is the considerable computational effort needed to implement it. It goes far beyond the requirements for the tobit-type estimators that have been discussed in the previous section.

This chapter’s main objective is to show that these complications can be avoided. It will outline a consistent estimator of similar generality, which, however, is comparatively simple, as it is based only on standard linear regression techniques and a summing-up procedure, exploiting some properties o f ordinary least squares (OLS). The technique should therefore be well within reach o f practitioners who do not want to spend too much time on technical hitches.

The Basic Model

Starting point is a standard log-income regression

(3.6) logy. = P ' X . +u. ,

where Z, is a vector of individual or household characteristics, and

(3.7) M. ~ iid .

(3.6) and (3.7) form a simple model o f the conditional log-income distribution:^^ the distribution’s first moment is a linear function of the covariates, and all higher moments are independent from X. Because o f (3.7), OLS estimates the coefficient vector ^ consistently. Moreover, as demonstrated in Theil (1971, p. 378 ), it is true that for each OLS disturbance

Income, earnings, and consumption are often found to be roughly log-normally distributed (see above), suggesting a log-normal specification. Furthermore, as will be shown in the applied section, the conditional poverty estimator takes a particularly simple form if a log-linear specification is combined with the Watts poverty index. However, the following could o f course be generalised to continuous income transformations other than the log.

(3.8) w. = lo g } ;.-y § 'X ,,

(3.9) p \ïm { û -- U - ) = 0 V i=l,...,A ^as N —> <50 .

Thus, if the true disturbances m. have a limiting distribution, the û. also have a limiting distribution, and both limiting distributions are identical (for a proof, see, e.g., Rao (1965, p. 101); formally,

(3.10) w , u. V /= l,...,iV a s AT->oo,

where d denotes “convergence in distribution” .

Next, consider the log-income distribution conditional on a specific characteristics vector X j. Under assumptions (3.6) and (3.7), its mean is consistently predicted by

p 'X^. As the disturbances are independent from the covariates by (3.7) and the covariate vector X, is fixed by construction, it follows from (3.10) and the Slutzky theorem that

(3.11) y^'Xj-HWy—-— SiS N — ><50.

The limiting distribution of y^’X, +u- , however, is, under the above assumptions, the distribution of log-income conditional on X , . A consistent estimate of poverty conditional on X, is obtained by plugging (3.11) into the poverty index (3.1):

(3.12) ZI , ) = ^ X F, (exp(;â' Z , +«,-), z)* 7(exp(y8' Z , + «, ) < z ) .

The effect of specific covariates on poverty can be analysed by suitably varying the set of characteristics from X, to and comparing f ( y ; z l X J with ^ (y iz lX ^ ) .

Figure (3.1) on page 114 illustrates how equation (3.12) works. (3.6) predicts the mean of log-income conditional on X , . Adding the regression residuals provides an estimate of the conditional density. Consequently, the area that is both underneath the

conditional density and left of the poverty line estimates the headcount ratio conditional on X j. More elaborate poverty measures require weighing the density with the deprivation function F. Changing the characteristics vector from X, to X^ shifts the conditional mean and, with it, the entire conditional distribution. As a consequence, the area underneath the poverty line changes, and with it predicted conditional poverty.^^

Heteroscedasticity

The above model is rather restrictive, as the iid assumption imposes that only the mean of the log-income distribution depends on X. Other aspects of the distribution may vary with the covariates too, however. As an example, consider the case where (log) income

dispersion is positively correlated with a dummy covariate D. Switching D on (setting it

from zero to one) has, therefore, two effects. First, it shifts the conditional mean o f log income as in figure 3.1. But in addition, it increases the dispersion of the distribution, which may be interpreted as an increase of income risk. As illustrated in figure 3.2, this in turn increases the fraction of the distribution below the poverty line, and therefore poverty. Neglecting this effect will underestimate the poverty risk associated with D= 1.

The iid assumption can be relaxed, however, by employing a standard control for heteroscedasticity. Suppose

(3.13) Var(Uj) = a f { X j ) = <7^*co{X.), i= l,...,N \ ^ c o - = N .

It is well known that OLS estimates the p coefficient vector consistently also under (3.13). Moreover, (3.10) holds under heteroscedasticity, too (see Harvey (1976)). Adding the OLS residuals on to the conditional mean as before, however, will not consistently estimate the conditional log-income distribution in this setting, as this ignores that the disturbances’ expected deviation from the mean changes with the

63

Inserting (3.8) gives P'X^ +w, = log y, P \ X - - X , ) . Hence, (3.12) can be rewritten as

1 ^

(3.12)’ F ( y ; z l X , ) = -^5^F(exp(log>', + F ’(X. -X ,)),z )* /(e x p (lo g y ^ + P \ X . - X , ) ) < z ) ,

N i=j

which gets rid of the regression residuals. Thus, the estimated conditional log-income distribution - and

therefore predicted poverty - is based on an N-time average o f a transformation of the observed log-

income distribution, where incomes are manipulated according to the weighed distance between the observed characteristics X- and the fixed characteristics X ,. The OLS coefficients serve as weights. 1 am grateful to Stephen Jenkins and John Ermisch for this observation.

covariates. In other words, the disturbances’ expected deviation conditional on may deviate from the unconditional standard deviation. Taking this into account requires adjusting the residual distribution as

Adding (3.14) to the conditional mean and inserting (3.13) gives

(3.15) f i ’X ^ + U i * ^ û ) ( X , ) , 1=1...N,

whose limiting distribution is the log-income distribution conditional on Xj (homoscedasticity is, obviously, the special case where û)=\).

Hence, in addition to P and the w., predicting the conditional income distribution requires a consistent estimate o f Cû{X), the dispersion factor. It expands the residual distribution if ûJ(X, ) > 1 (as in figure 3.2), and contracts it if ûJ(X, ) < 1.

Cù{X) can be obtained from the OLS residuals if one is prepared to assume that the

disturbances’ variance relates to the covariates as

(3.16) <rf(X ,) = A 'X,.

Then, forming the regression equation

(3.17) wf = erf + f . with £-~iid,

and replacing u f by its estimate u f gives

I Q Û

(log) P overty Line

o: D en sity conditional on X +: D en sity conditional on X;

0 0

Log Income