Notes on treatment-effect高级计量Notes(美国A&M大学,甘犁教授)

Average Treatment Effect

Li Gan

Nov, 2007

1. The Regression Method:

We are interested in average changes in outcome y . Denote 1 if with treatment, and 0 without treatment. Average Treatment Effect is defined as:

ATE = E(y 1 – y 0) (1)

The difficulty in estimating is that we observe y1 or y0, not both, for each person. More precisely, let w = 1 if treatment. The observed outcome y can be written as:

y = (1-w ) y 0 + w y 1. (2)

If w is independent of y , then:

E(y 1-y 0) = E(y 1-y 0 |w ) = E(y 1|w =1) – E(y 0|w =0)

In fact, we only need the weak assumption (rather than independence): mean independence: E(y 0|w ) = E(y 0), E(y 1|w ) = E(y 1).

Now let:

()()0

,0,11110000=+==+=v E v y v E v y μμ

Therefore, (2) can be written as:

(3) ()(01001010)1(v v w v w wy y w y ?++?+=+?=μμμ)

First, assume conditional mean independence:

Assumption 1 (ATE 1): (a) E(y 0|w,x ) = E(y 0|x ), and (b) E(y 1|w,x ) = E(y 1|x )

Intuition: even though y 1 and y 0 may be correlated with w , they are uncorrelated with w if we partial out x .

Taking expectation of (3) (and with ATE 1):

E(y|w,x ) = μ0 + αw + g 0(x ) + w (g 1(x )- g 0(x )), (4)

where α=μ1- μ0 is the Average Treatment Effect (ATE), and g i (x )=E (v i |x ).

Linearization of g i (x ):

E(y|w,x ) = μ0 + αw + x β0 + w (x-ψ)δ,

where ψ=E(x). The last term is to ensure that g 1(x )- g 0(x )=0. So the regression to estimate ATE α is:

y i on 1, w i , x i , w i (x i – x )

Here the control functions involve not just xi, but also interactions of the covariates with the treatment variable.

We can estimate treatment effect conditional on x:

()()δα???x x x E T A ?+=

2. Propensity Score:

Let p(x ) = Pr(w =1|x ).

(w – p(x )) y = (w – p(x ))(wy 1 + (1-w ) y 0)

= wy 1 – p(x ) (1-w ) y 0 – p(x )wy 1

Take conditional expectation with respect to y:

E y [(w – p(x ))y|w,x ]= wm 1(x )– p(x ) (1-w ) m 0(x )– p(x )wm 1(x ),

where E(y j |w,x )= E(y j |x )=m j (x ). Taking expectation with respect to w:

E w {E y [(w – p(x ))y|w,x ]|x }

= E w [wm 1(x )– p(x ) (1-w ) m 0(x )– p(x )wm 1(x )]

= p(x )m 1(x )– p(x ) (1- p(x )) m 0(x )– p(x ) p(x )m 1(x )

= m 1(x )p(x )(1-p(x ))- m 0(x )p(x )(1- p(x ))

=(m 1(x )-m 0(x ))p(x )(1-p(x ))

Therefore,

()()

()()))

(1)(()(01x p x p y x p w E x m x m ATE ??=?= A simple and popular estimator in program evaluation is obtained from OLS regression:

y i on 1, w i , ()i x p

?

where coefficient for w i is the estimate of the treatment effect. In other words, the estimated propensity score plays the role of the control function.

3. Dummy Endogenous Variables

Consider the model again:

E(y|w,x ) = μ0 + αw + x β0 + u 0, (4)

w is endogenous. Again, w = 1 if treated, and 0 otherwise.

Assume that Pr(w=1|x,z ) = G(x, z; γ)

Procedure 1:

(1) Estimate the binary response model Pr(w i =1|x i ,z i ) = G(x i ,z i ;γ), and obtain the

fitted values . i

G ?(2) Estimate (4) using instruments 1, and x i . i

G ?

Procedure 1 has important robustness property:

(a) Because we use as an IV, the model Pr(w i =1|x i ,z i ) = G(x i ,z i ;γ) does not have to be correctly specified.

i

G ?(b) Technically, α and β are identified even if we do not have extra variables

excluded from x. But can rarely justify the estimator in this case.

Suppose that w given x follows a probit model (no z). Because G(x, γ)

= Φ(γ0 +x γ1), is a nonlinear function of x, it is not perfectly correlated

with x, so it ca nbe used as IV for w.

(c) In principle, it important to recognize that Procedure 1 is not the same as

using G as a regressor in place of w .

y i on 1, and x i . i

G ?

Consistency of the OLS estimators from the regression:

(5) i

i i i u x G y +++=00?βαδ would rely on G( ) to be correctly specified. Note that (5) also has problems with standard errors that need to be corrected.

Allow interact term:

()i i i i i i e x x w x w y +?+++=δβαδ00 (6)

Procedure 2:

(a) Estimate Pr(w i =1|x i ,z i ) = G(x i ,z i ;γ)

(b) Use 1, and x i , and i G ?()x x G i

i ?? as IVs. Discussions are the same as before.

4. Regression discontinuity

It is useful to distinguish between two general settings, the Sharp and the Fuzzy Regression Discontinuity designs. In the sharp design, the assignment w i is a

deterministic function of one of the covariates, the forcing (or treatment-determining) variable x :

Sharp design:

w i = 1(x i > x 0)

All units with x i > x 0 are assigned to the treatment group (and participation is mandatory for these individuals), and all units with x i ≤ x 0 are assigned to the control group. In this sharp design, we look at the discontinuity in the conditional expectation of the outcome given the covariates to uncover the ATE :

)|(]|[lim ]|[lim 0010

0x x y y E x y E x y E ATE x x x x =?=?=?+→→

Fuzzy design:

E(w i |x i = x ) = Pr(w i = 1|x ) is discontinuous at known value x 0.

The sharp and fuzzy designs differ in that in the sharp design the treatment

assignment is deterministic given x , while the fuzzy design the treatment assignment may depend on additional factors unobserved by econometrician. In both designs, the discontinuity point x 0 is known.

Assumption (RD):

(i) ()x w E w x x |lim 0+→+= and ()x w E w x x |lim 0

?→?= exist.

(ii) w + ≠ w - In Angrist and Lavy (1999), an identifying assumption would be that the class size for a student in a school with a number of pupils approaching (for example) 800 above differs from that of a student in a school with a number of pupils approaching 800 from below.

Assumption: E(y 1i – y 0i |x i = x ) is continuous in x at x 0.

This assumption is valid where we have reason to believe that person close to threshold c are similar and thus would experience similar outcome absent treatment.

Theorem: ATE, denoted as α:

?

+?

+??=w w y y α Proof:

Let Δ to be a small positive number.

()()

()()()()

()()()()()(()()()()()()()

Δ??Δ++Δ??Δ+=Δ??Δ++Δ???Δ+?=Δ?+??Δ++?=)Δ??Δ+00000000000010010001000100||||||||||||x y E x y E x w E x w E x y E x y E x w y y E x w y y E x y w y y E x y w y y E x y E x y E α

As Δ ?0, we have:

()?+?+?=?w w y y α

Here we use the fact (assumption) that E(y 0) is continuous at x 0 without treatment.

The conclusion follows.

Given this theorem, we can obtain an estimate of α by estimating y +, y -, w +, and w -. There are several ways to estimate this. The most popular way is to do it non-parametrically.

In practice,

()()()()∑∑∑∑<

h x x x h x x x y y

i i i i i i

Note for a sharp design RD, w + - w - = 1. For a fuzzy design RD,

()()()()∑∑∑∑<

00

0000011?11?x x h x x x h x w w h x x x h x x x w w

i i i i i i

where h is the bandwidth. An interesting note is that this is numerically equivalent to an IV estimator for the regression of y i on w i for people in the subsample

using ()h x x h x i +<

Practically, for a sharp design,

1. Graph the data by computing the average value of the outcome variable over a set of bins. The bandwidth has to be large enough to have sufficient amount of

precision so that the plots look smooth on either side of the cutoff value, but at the same time small enough to make the jump around the cutoff value clear.

2. Estimate the treatment effect by running linear regression on both sides of the cutoff point. Since we propose to use a rectangular kernel, these are just standard

regression estimated within a bin of width h on both sides of cutoff point. Note that:

i.Standard errors can be computed using standard leas square methods

(robust standard errors)

ii.The optimal bandwidth can be chosen using cross-validation methods. Fuzzy design:

1.Graph the average outcome over a set of bins as in the case of SRD, but also

graph the probability of treatment.

2.Estimating the treatment effect using TSLS.