Sample Selection

Walter Sosa-Escudero

Econ 507. Econometric Analysis. Spring 2012

April 23, 2012

Walter Sosa-Escudero Sample Selection

Preliminaries 1) Truncated normal distribution

X f(x), X|X < a: X truncated in a. Then

f(x|X < a) = f(x)Pr(X < a)

If X N(, 2), and recalling that

Pr(X < a) = Pr(

1/(X ) < 1/(a ))

= Pr(z < ),

(a )/, z (x )/.

f(x|X < a) =1

12pi

exp[12(x

)2]()

=(1/)(z)

()

Walter Sosa-Escudero Sample Selection

Result (no proof): if X N(, 2), then:

E(X|X < a) = ()()

Truncated to the right: expected value moves to the left(general).

How much? Depends on and 2

(z) (z)/(z) is known as the inverse Mills ratio

Walter Sosa-Escudero Sample Selection

Preliminaries 2) Latent variables and probits

Recall the probit model:

Pr(y = 1|x) = (x)

can be consistently estimated by MLE based on a randomsample (yi, xi), i = 1, . . . , n.

Consider the regression model

y = x + u, u N(0, 2)y not directly observable (a latent variable) but, instead, weobserve y = 1[y > 0].

Which parameters of the regression models can be estimatedconsistently with this information?

Walter Sosa-Escudero Sample Selection

Note that:

P (y = 1|x) = P (y > 0|x)= P (u > x|x)= P (u < x|x)= P (u/ < x/ | x)= (x)

This is a probit model with /.

Based on (yi, xi), we can estimate consistenly by MLE, evenwhen we cannot estimate and 2 separately.

Walter Sosa-Escudero Sample Selection

2 and are not identified with the sample (yi, xi). Forexample = 10 and 2 = 2 are observationally equivalent tothe case = 5 y 2 = 1. / is identified.

Walter Sosa-Escudero Sample Selection

Sample selectivity

Consider the regression model

yi = xi + ui

si is a selectivity variable: si = 1 observed, 0 if not.

We can think that there is a super sample of size N ofyi , xi, si and that we observe the sub sample y

i , xi, only

when si = 1.

Example: female labor productivity

Walter Sosa-Escudero Sample Selection

OLS under selectivity

With a random sample (yi , xi), consistency relies on:

E(ui|xi) = 0

which implies E(yi|xi) = xi.

Now we do not have a random sample, but one conditioned onsi = 1. Taking conditional expectations:

E(yi|xi, si = 1) = xi + E(ui|xi, si = 1)OLS based on the selected sample will be inconsistent, unlessE(ui|xi, si = 1) = 0.

Walter Sosa-Escudero Sample Selection

Not every selectivity mechanism makes OLS inconsistent.

If u is independent of x, OLS still consistent (why?).

If s = g(x), OLS still consistent.

Examples: wages and education. Males with odd SSN. Maleswith formal education?

Walter Sosa-Escudero Sample Selection

An estimable model under selectivity

Consider the following equations:{y1i = x

1i 1 + u1i (regression)

y2i = x2i 2 + u2i (selectivity)

Let y2i = 1[y2i > 0].

Example: y1i = wage, regression equantion determines wages based onpersons characteristics (x1i). y2i = net utility of work, x2i are observeddeterminantes of utility.

Walter Sosa-Escudero Sample Selection

Assumptions:

1 (y2i, x2i) observed for everyone.

2 (y1i, x1i) observed only if y2i = 1 (selected sample).

3 (u1i, u2i) are independent of x2i and have zero mean.

4 u2i N(0, 22).5 E(u1i|u2i) = u2. No-observables can be related.

Walter Sosa-Escudero Sample Selection

Selectivity bias

Here si y2i.

E(y1i|x1i, y2i = 1) = x1i1 + E(u1i | x1i, y2i = 1)= x1i1 + E

[E(u1i|u2i) | x1i, y2i = 1

]= x1i1 + E

[u2i | x1i, y2i = 1

]= x1i1 + E

[u2i | x1i, y2i > 0

]= x1i1 + E

[u2i | x1i , u2i < x2i2

]= x1i1 2 (x2i2/2)= x1i1 2 zi 6= x1i1

with zi (x2i2/2). OLS with the selected sample isinconsistent.

Walter Sosa-Escudero Sample Selection

E(y1i|x1i, y2i = 1) = x1i1 2 zi 6= x1i1

Inconsistency: omission of zi. Heckman (1979): selectivitybias as misspecification.

Inconsistency due to the correlation between u1i y u2i, , thatis 6= 0.

Walter Sosa-Escudero Sample Selection

Heckmans two-step estimator

Define u1i y1i x1i1 zi, 2. Solving:

y1i = x1i +

z + u1i

where, by construction E(u1i|x1i, y2i = 1) = 0.x1i, zi observable when y2i = 1: OLS of y1i on x1i and ziusing the selected sample gives consistent esimates of 1 and.Problem: zi (x2i2/2) is NOT observable, it depends on2 and 2.

Walter Sosa-Escudero Sample Selection

Note that u2i N(0, 22), hence:

P (y2i = 1) = P (y2i > 0) = P (u2i/2 < x

2i2/2) = (x

2i)

P (y2i = 1) is a probit model with unknown coefficients .

x2i and y2i are observed for everybody: can be estimatedusing probit.

Important: we cannot identify 2i and 2 separately but = 2i/2i.

Walter Sosa-Escudero Sample Selection

This suggests the following two-stage method:

Stage 1: Obtain estimates based on the probit modelP (y2i = 1) = (x

2i) using the full sample. Predict zi using

zi = (x2i).

Stage 2: Regress y2i on x1i and zi using the selected sample.This produces consistent estimates of 1 and

.

Walter Sosa-Escudero Sample Selection

Practical remarks

The method is consistent and asymptotically normal (methodof moments estimator). Standard inference works fine.

Careful with asympototic variance. The second stage isheteroskedastic. Requires correction. See Greene (Ch. 20).

A test of H0 : = 0 may be used to check sample selectivity.Under H0, the regression model with the selected sample ishomoskedastic, test can be based in a model without takingcare of heteroskedasticity.

Classic issue: low power when x1 is very similar to x2.

MLE? Requires bivariate normality. Complicated likelihood,rarely used.

Walter Sosa-Escudero Sample Selection