Palgrave Handbook of Econometrics: Applied Econometrics

Andrew M. Jones 561

For example, in Aakviket al.(2005) the treatment is a Norwegian vocational
rehabilitation (VR) program and the outcome is a binary measure of employment.
Analysis is based on a 10% sample of all those who applied for VR in 1989. To define
the treatment effects of interest, Aakviket al., specify a discrete choice model with
a common factor structure. There is a switching regression for the binary indicator
of employment under the two treatment regimes:

y^1 i =f 1 (xi,u 1 i)= 1 (x′iβ 1 ≥u 1 i)

y^0 i =f 0 (xi,u 0 i)= 1 (x′iβ 0 ≥u 0 i),

(12.8)

along with a latent variable model for the assignment of treatment:

di= 1

if

di∗=z′iβd−udi>0.

(12.9)

The error terms are assumed to have a common factor structure:

udi=−ηi+εdi

u 1 i=−α 1 ηi+ε 1 i

u 0 i=−α 0 ηi+ε 0 i.

(12.10)

Estimation is by FIML, assuming that the error components are jointly normal.
Given this set-up, the treatment effects of interest can be defined as follows:

MTE(x,u)=E

(

|x,d∗= 0

)

=E

(

|x,ud=z′iβd

)

(12.11)

ATE(x)=

∫

u

MTE(x,u)dF(u)=E(|x) (12.12)

ATE=E(ATE(x))=

∫

x

E(|x)dF(x) (12.13)

ATT(x,u)=E

(

|x,d= 1

)

=E

(

|x,ud<z′iβd

)

, (12.14)

whereF(u)is the distribution ofuandF(x)is the distribution of thexs. Aakvik
et al.(2005) do not use the concept of the LATE in their study but, based on the
notation of their model, it could be expressed as:

LATE(x,z, ̃z)=E

(

|x,z′iβd<ud< ̃z′iβd

)

, (12.15)

where, for illustration, it is assumed that assignment to treatment is monotonically
related to a single instrument that takes two values z andz ̃, wherez′iβd<z ̃′iβd. The
LATE defines the treatment effect for all those individuals who are induced into
the treatment by the change in the instrument (see, e.g., Basuet al.,2007).
The nonlinear model is identified by functional form, but an exclusion restriction
is also imposed by including an instrument – the degree of rationing of VR places
in the individual’s locality – inz, but not inx. The apparent positive impact of