OPTIMISING ANTI-POVERTY TRANSFERS WITH QUANTILE ... - Ivie

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Its solution is easy to calculate: ˆβ =(X 0 X) −1 X 0 Y ,whereX is the matrix
composed of the X i0 as vectors line, and Y is the vector of which coordinates are
the y i . One problem with the OLS method is that it is likely to predict well the
mean living standards (because E(ŷ|X) =E(X ˆβ|X) =ȳ, the population mean of
the living standards), but not necessarily the living standards of the poor, nor the
living standards of the households close to the poverty line.
Our approach is to use instead quantile regressions for generating the predictions
of the living standards. Such method, centered on a given quantile θ ∈]0, 1[
ensures that the conditional expectation of the θ th quantile of the predictor is Xβ
(E(q θ (ŷ|X)) = Xβ,whereq θ (.|X) is the conditional quantile function centered on
the θ th quantile. The predictions of living standards around the θ th quantile of
living standards should be better determined than with OLS. To be able to predict
well the living standards of the poor or the near poor, a good choice of parameter
θ seems therefore such that the θ th quantile of y is close to the poverty line. In
that way, the targeting scheme can be said to ‘focus on the poor’.
Quantile regression estimates can be obtained as solutions to the following
optimisation program.
min
β
n
X
i=1
¡
y i − X i0 β ¢£ θ − 1 [yi −X i0 β