Exceptional Argentina Di Tella, Glaeser and Llach - Thomas Piketty

2.2 Income distribution effects
Following Carvalho Filho y Chamon (2006) we explore also the possibility that the
amount of bias may change along the Engel curve thus allowing estimating different
mismeasurements in earnings growth for different income levels. Using a semiparametric
specification and assuming, as before, that the biases are the same for the
food and non food bundles, we have that:
wijt
[ ln( 1+ ∏
Ft
) − ln( + ∏
Nit
)]
t
[ Yit
− ln( 1+ ∏
Gt
) − ln( 1+
EGit
)] + θ
x
X
ijt
+
ijt
= φ + γ
1
+ f ln ∑ µ . (13)
x
The function f [ Y − ln( 1+ ∏ ) − ln( 1+
E )]
t
ln
it
Gt
Git
may be estimated non parametrically
using the differencing method of Yatchew (1997).
To apply this method we sort observations by income. The difference between two
observations can be written as:
φ
γ
{[ ln( 1+ ∏
Ft
) − ln( 1+ ∏
Nit
)] − [ ln( 1+ ∏
Ft
) − ln( + ∏
Ni−
t
)]}
[ lnY
− ln( 1+ ∏ ) − ln( 1+
E )] − f [ lnY
− ln( 1+ ∏ ) − ln( E )]
wijt − w i −1 jt
= +
1
1
+ f
t it
Gt
Git t i− 1t
Gt
1+
Gi−1t
+∑ θ
x
( X
ijt
− X
i−1 jt
) + µ
ijt
− µ
i−1
jt
. (14)
x
As we have sorted by incomes, incomes are pretty similar so
ln
( + ∏ ) − ln( 1+
E ) ≅ lnY
− ln( 1+ ∏ ) − ln( E )
Yit
ln 1
Gt
Git
i− 1t
Gt
1+
Gi−1t
− . (15)
Assuming that f
t
is a smooth function
[ lnY
ln( 1+ ∏ ) − ln( 1+
E )] ≅ f [ lnY
− ln( 1+ ∏ ) − ln( E )]
f
t it
Gt
Git t i− 1t
Gt
1+
Gi−1t
− . (16)
So equation (14) becomes:
{[ ln( 1+ ∏ ) − ln( 1+ ∏ )] − [ ln( 1+ ∏ ) − ln( + ∏ )]}
wijt − w i −1 jt
= + γ
Ft
Nit
Ft
1
Ni−1t
+∑ θ
x
( X
ijt
− X
i−1 jt
) + µ
ijt
− µ
i−1
jt
.
x
φ (17)
Note that equation (17) is a linear function (with coefficients identical to those of (13))
so that we can consistently estimate it by OLS, and construct the linear part of the
prediction of w , called ŵ , to arrive to:
ijt
ijt
t
ijt
ijt
[ lnYit
− ln( 1+ ∏Gt
) − ln( + EGit
)] +
ijt
w − wˆ = f
1 µ . (18)