Graham, Imbens, and Ridder (2006)

Complementarity and Aggregate Implications of Assortative Matching

These slides are based on the working paper “Complementarity and Aggregate Implications of Assortative Matching” by Bryan S. Graham, Guido W. Imbens, and Geert Ridder, May 14, 2006.

Presentation by Jason Blevins, Duke Applied Microeconometrics Reading Group, March 6, 2007.

Basic Model








Identifying Assumption

That is, the average output we would see if all firms were assigned W=w equals the average output among firms that actually have W=w. The distribution of potential outcomes must be the same in the subpopulation of firms that were assigned W=w as that in the overall population. This is the analogous assumption to that of the binary treatment effect model of Rosenbaum and Rubin (1983).

Production Function

Unconfoundedness implies that among firms with identical X and Z, the (counterfactual) average output of firms if we assigned W=w to all firms is equal to the actual average output of firms that are in fact assigned W=w.

Quantities of Interest

Positive Matching

Graphical Example

Graphical example of Positive Assortative Matching


An Example

An Example

Negative Matching


Estimation of g

g^(w,x,z)= iY iK(vV ib) iK(vV ib)

Support problems: we are trying to learn about a counterfactual allocation that may involve areas of the support for which we have few observations to estimate g.

Estimation of CDFs

F^ X(x)=N 1 i1(X ix) F^ W(w)=N 1 i1(W iw)

F^ W 1(q)=inf w𝒲1{F^ W(w)q}

This is the inverse of the empirical CDF of W.


β^ pam=1N i=1 Ng^[F^ W 1(F^ X(X i)),X i,Z i]

β^ nam=1N i=1 Ng^[F^ W 1(1F^ X(X i)),X i,Z i]

Correlated Matching

Correlated Matching

Correlated Matching

Normal Copula

ϕ(x 1,x 2,ρ)=12π1ρ 2exp[12(1ρ 2)(x 1 22ρx 1x 2+x 2 2)],

ϕ c(x 1,x 2,ρ)=ϕ(x 1,x 2,ρ)Φ(c,c,ρ)Φ(c,c,ρ)[Φ(c,c,ρ)Φ(c,c,ρ)].

H W,X(w,x)=Φ c[Φ c 1(F W(w)),Φ c 1(F X(x));ρ].

Normal Copula

h W,X(w,z)=ϕ c[Φ c 1(F W(w)),Φ c 1(F X(x));ρ]f W(w)f W(x)ϕ c[Φ c 1(F W(w))]ϕ c[Φ c 1(F X(x))]

Correlated Matching

β cm(ρ,τ) =τE[Y] +(1τ)g(w,x,z)dΦ(Φ 1(F W|Z(w|z)),Φ 1(F X|Z(x|z));ρ)F Z(z).

β cm(ρ,0)=g(w,x,z)ϕ c[Φ c 1(F W(w)),Φ c 1(F X(x));ρ]ϕ c[Φ c 1(F W(w))]ϕ c[Φ c 1(F X(x))]f W(w)f X,Z(x,z)dwdxdz

β rm=β cm(0,0)=[g(w,z,z)dF W|Z(w|z)dF X|Z(x|z)]dF Z(z).


β^ cm(ρ,0)=1N 2 i=1 N j=1 Ng^(W i,X j,Z j)ϕ c[Φ c 1(F^ W(W i)),Φ c 1(F^ X(X j));ρ]ϕ c[Φ c 1(F^ W(W i))]ϕ c[Φ c 1(F^ X(X i))]

β^ cm(ρ,τ)=τβ^ sq+(1τ)β^ cm(ρ,0).


Summary Statistics

VariableMeanStd. dev.
Ed. child13.062.38
Ed. mother11.202.87
Ed. father11.203.64


VariableCoefficientStd. Err.
Ed. mother-0.04100.0360
Ed. father-0.07700.0290
Ed. mother20.01100.0023
Ed. father20.01100.0015
Ed. mother × Ed. father0.00140.0029


ρ β̂ cs std(β̂ cs)