Modeling Heterogeneity by Structural Varying Coeﬃcients Models in Presence of Endogeneity

Modeling Heterogeneity by Structural VaryingCoefficients Models in Presence of Endogeneity

Stefan Sperlich, Giacomo Benini, Raoul Theler, Virginie Trachsel

Universite de Geneve

Geneva School of Economics and Management

Stefan Sperlich (Uni Geneve) Vaying Coefficients 1 / 22

Preliminaries

Causality and Correlation

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Preliminaries

Statistical Data Analysis and Causality

consider Y = ϕ(D,X1,X2, ..., ε) to study the effect/impact of D on Y

Disentangling causality from correlation is one of the fundamentalproblems of data analysis. Every time the experimental methodology –typical in some hard sciences – is not applicable, it becomes almostimpossible to separate causality from observed correlations usingnon-simulated data.


Preliminaries

Statistical Data Analysis and Nonparametrics

Boons and Banes of Nonparametric Statistics

no functional form misspecification

’no need’ to any specification (?)

curse of dimensionality

slower convergence rates, smoothing parameter, numerics,...

problems of interpretation


Preliminaries

Proposition

both sides can gain from modeling

well known in econometrics: structural models

well known in nonparametrics: semiparametric methods

Comment:certainly, in (pure) econometric theory, nonparametric methods are’standard’, though, ...


Preliminaries

Preliminary considerations and confusions

For data as above Y ,D ∈ IR , X ∈ IRq

Of interest is E [Y |X = x ,D = d ] = m(x , d) modeled typically as

m(x , d) = dα + x ′β or y = dα + x ′β + ε

and want to study the impact of D, say α

For D continuous interpret α as ∂Y /∂d on average, for D binary

ATE ({D = 0} → {D = 1}) = E [Y 1 − Y 0]

The notion in average is enticing as people are tempted to think first’on average given (x , d)’ but more often ’on average over all’

Potential problem: endogeneity


Heterogeneity in regression

Heterogeneous Returns / Elasticities

Economies of Scale in agriculture: Severance-Lossin and Sperlich (1999)

analyzed Wisconsin farms. Found increasing returns to scale∑qj β′j > 1.

logYi = β0 +∑q

j=1βj(logXi ,j) + εi

Efficiency of labor offices: Profit and Sperlich (2004) studiedtime-space variation of Job-Matching and their sources Q

Flexible Engle curves and Slutsky (A)Symmetry: Pendakur and

Sperlich (2009) estimated consumer behavior in Canada on pricevariation controlling for real expenditures Q, β(·) vector, A(·) matrix

exp.shares = β(Q) + A(Q) prices , Q = $(x , p)

etc.



What do the standard regression methods estimate?

Having heterogeneous returns in mind, write (with D in X , α in β)

Yit = Ditαit + X ′itβ + εit = Ditα + X ′itβ + Dit(αit − α) + εit︸︷︷︸=:εit

need E [εit |Xit ,Dit ] = 0

where αit might be a function of vector Qit (may include Dit ,Xit)

Neglecting further interaction and other sources of endogeneity

functional misspecification or say Qit can generate endogeneity



Methods for VCM (implementation)

To estimate VCM, there exist quite a bit in R(though often only for very specific models)

For RCM or MEM anywaybut also deterministic ones:

package NP Hayfield and Racine (2008, 2012); kernels

SVCM Heim et al. (2007, 2012); space varying spline coefficients

BayesX (incl. Belitz, Brezger, Kneib, Lang, 20??); splines

GAMLSS Stasinopoulos (2005, 2012); ML and splines



Methods for VCM (theory)

See ISReview: Park, Mammen, Lee and Lee (2013)

Kernel local pol. smoothing (Fan and Zhang, 1999,2000,2008)

local maximum likelihood (e.g. Cai et al, 2000)

spline methods (e.g. Chiang et al., 2001)

smooth backfitting (Mammen and Nielsen, 2003; Roca-Pardinas and Sperlich,

2010)

Bayesian structured additive models (Fahrmeir et al., 2004)

Particularly large literature on αt , βt

less literature regarding VCM for panel data

Again, not mentioned: random coefficient and mixed effects models



Example: Econ. growth and inequality: Model∆Yit = ρ log(Yi ,t−1) +α1,it log(Kit) +α2,it log(Lit) +β1 log(Depit) +δi + εit

Fully Parametric Semiparametric VCM

FE RE middle-DV Gini-DV

log(Yt−1) −0.010∗∗∗ −0.006∗∗∗ −0.006∗∗∗ -0.005∗∗

(0.002) (0.001) (0.002) (0.002)log(Dept) −0.001 −0.002 −0.008∗∗∗ −0.008∗∗∗

(0.002) (0.002) (0.002) (0.002)log(Kt) 0.021∗∗∗ 0.023∗∗∗

(0.001) (0.001)log(Lt) −0.016∗∗∗ −0.014∗∗∗

(0.001) (0.001)

∆Yit = ρ log(Yi ,t−1) + g1(ineqi ,t−3)lKit + g2(ineqi ,t−3)lLit + β1lDepit +

h1(ineqi ,t−3, lKi ,t−3, v1,it) + h2(ineqi ,t−3, lLi ,t−3, v2,it) + δi + εit



Example: Econ. growth and inequality: functions

0.00

0.05

0.10

0.15

0.20

0.3 0.4 0.5Middel Class

g_1(

gini)_

hat

Returns to Physical Capital

0.00

0.05

0.10

0.15

0.3 0.4 0.5Middel Class

g_2(

gini)_

hat

Returns to Human Capital

0.05

0.10

0.2 0.3 0.4 0.5 0.6Gini Index

g_1(

gini)_

hat

Returns to Physical Capital

0.00

0.05

0.10

0.15

0.2 0.3 0.4 0.5 0.6Gini Index

g_2(

gini)_

hat

Returns to Human Capital


Heterogeneity and IV regression

The Deus ex Machina principle in economics

Murray (2006) Jo Economic Per-spectives called it (seriously) theeconomists’ long lever to move theworld (Archimedes)

Horace (today Heckman orDeaton), however, instructed poets(economists) that they must neverresort to it



Instrumental Variable Estimation (IV) when βi constantSimplified presentation merging D,X and α, β:

Y = X ′β + ε , 0 = E [ε] 6= E [ε|X ] say ’because of’ Xk

for whatever reason - but you have instruments W (include X−k) s.th.Cov(X ,W ) 6= 0 & E [ε|W ] = 0. Then

β = Cov(W ,X )−1Cov(W ,Y ) = Cov(W ,X )−1{Cov(W ,X )β+Cov(W , ε)}

Control function: We may write the ’selection equation’

Xk = g(W ,X−k) + v , 0 = E [v |W ,X−k ] ⇒ v

arising from idea that instruments are variables that induce variation in X

(β, h) = Cov(X , X )−1Cov(X ,Y ) with X = (X , v)

E [Y |X , v ] = X ′β + v ′h



Instrumental Variable Estimation - Varying Coefficients βi

β = (∑i

WiX′i )−1

∑i

WiYi = (∑i

WiX′i )−1

∑i

WiX′i βi + RTi

if Wi is mean-independent from ri = βi − β we get identification – howrealistic is it without being weak instrument?

List of assumptions increases significantly

But even then, what does it estimate? Consider D and W binary

αIV =E [Y |W = 1]− E [Y |W = 0]

E [D|W = 1]− E [D|W = 0]= LATE

extension to discrete and continuous D, W harder to understand

IV gives large variances (Jean-Marie Dufour)


Toward structural modeling: VCM with IVs

Toward structural modeling

As the existence of such an instrument is quite unlikely

usefulness, credibility, interpretability, but also est. quality increase bymodeling

Yi = β(Qi )′Xi +

εi︷︸︸︷e ′iXi + εi , βi = β(Qi ) + ei

IV conditions get more realistic

variation over LATE should get reduced

’usual’ advantages of non- and semiparametric data analysis apply

Not hard to extend existing methods for endo- and heterogeneity problemsto VCM (already done in paper with PhD students)



Example: Mincer’s wage equation: standard IV

log(wagei ) = β0 + αeduci + β1experi + β2exper2i + εi

educi = γ0 + γ1Wi + γ2experi + γ3exper2i + vi

as educ endogenous; typical IV Wi are parental educ

IV: feduc meduc feduc & meduc

educ 0.075∗∗∗ 0.043∗∗ 0.060∗∗∗

(0.015) (0.016) (0.014)exper 0.040∗∗∗ 0.038∗∗ 0.039∗∗∗

(0.005) (0.005) (0.005)exper2 −0.001∗∗ −0.001∗∗ −0.001∗∗

(0.000) (0.000) (0.000)Constant 1.486∗∗∗ 1.915∗∗∗ 1.678∗∗∗

(0.201) (0.208) (0.185)

Schultz (2003) argues also, that returns to educ varies with exper



Example: Mincer’s wage equation: VCM IV

log(wagei ) = β0+g(experi )educi+β1experi+β2exper2i +h(Zi , experi , vi )+εi

0.04

0.05

0.06

0.07

0.08

0 10 20 30Exper

g_1(

expe

r)_ha

t

Returns to EducationIV Father

0.04

0.05

0.06

0.07

0.08

0 10 20 30Exper

g_1(

expe

r)_ha

t

Returns to EducationIV Mother

The functions α(·) for IV feduc (left) and meduc (right)

Remark: Bands are constructed with a special wild bootstrap.



But still, αi is a function of Winteresting ways to look at LATE when W is continuous;maybe most popular one is the marginal treatment effect MTE

Let D be binary, D = 11{P(W ) ≥ v} (impose monotonicity)

Are interested in surplus regarding an incentive, say W given X

S [P(W ) = p] = E [Y 1 − Y 0|v ≤ p] p quantile of v

the definition of the marginal TE is simply

MTE (u) = E [Y 1 − Y 0|u = v ]⇒ S(p) =

∫ p

0MTE (u)du

a nonparametric estimate can be obtained by

∂S(p)/∂p = ∂E [Y |P(W ) = p]/∂p

under a certain set of conditions etc.

Remark: prescinding from X and Q in notation



Estimation and testing

Simplest implementation would be

Two-step estimation with

firstly, semiparametric probit or logit for P(D = d |Q) with splinesinside link

secondly, semiparametric VCM (or partial additive) estimation ofmain fctn

Have presently joint projects on

theory paper on inference in nonparametric structural equations (withE.Mammen),especially on testing separability and significance using smoothbackfitting

creating an R package for these methods (J. Roca-Pardinas) includingadaptive bandwidth choices



Example: Export Promotion

The standard model is

log(Yit) = α log(budgetit) + β log(popit) + δi + λt + εit

where

the EPA log(budgetit) could be endogenous.

α heterogeneous, i.e. be modeled as fctn of Q or/and W

these could be composition and sources of budget, etc.

other predictors Q are eg. the structure of the EPA

or the employment of budgets

The results α(Q) allowed us to give country (or EPA) specific results onreturns, efficiency etc., that is, to make policy relevant statements.

Remark: not presented because all excluded instruments had no significantimpact on α function, the others little or no impact on log(budgetit).


Modeling Heterogeneity by Structural Varying Coeﬃcients Models in Presence of Endogeneity

Economy & Finance

Transcript of Modeling Heterogeneity by Structural Varying Coeﬃcients Models in Presence of Endogeneity