Modeling Heterogeneity by Structural Varying Coeﬃcients Models in Presence of Endogeneity

Modeling Heterogeneity by Structural Varying
Coefficients Models in Presence of Endogeneity
Stefan Sperlich, Giacomo Benini, Raoul Theler, Virginie Trachsel
Université de Genève
Geneva School of Economics and Management
Stefan Sperlich (Uni Genève) 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 fundamental
problems of data analysis. Every time the experimental methodology –
typical in some hard sciences – is not applicable, it becomes almost
impossible to separate causality from observed correlations using
non-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
q
j βj > 1.
logYi = β0 +
q
j=1
βj (logXi,j ) + εi
Efficiency of labor offices: Profit and Sperlich (2004) studied
time-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 price
variation 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 + Xitβ + εit = Ditα + Xitβ + Dit(αit − α) + εit
=: it
need E[ it|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 anyway
but 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-Pardiñas 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 + it
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, ˆv1,it) + h2(ineqi,t−3, lLi,t−3, ˆv2,it) + δi + εit

Example: Econ. growth and inequality: functions
0.00
0.05
0.10
0.15
0.20
0.3 0.4 0.5
Middel Class
g_1(gini)_hat
Returns to Physical Capital
0.00
0.05
0.10
0.15
0.3 0.4 0.5
Middel Class
g_2(gini)_hat
Returns to Human Capital
0.05
0.10
0.2 0.3 0.4 0.5 0.6
Gini 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 0.6
Gini Index
g_2(gini)_hat
Returns to Human Capital

Heterogeneity and IV regression
The Deus ex Machina principle in economicsHorace (today Heckman or
Deaton), however, instructed poets
(economists) that they must never
resort to it

Instrumental Variable Estimation (IV) when βi constant
Simpliﬁed presentation merging D, X and α, β:
Y = X β + , 0 = E[ ] = E[ |X] say ’because of’ Xk
for whatever reason - but you have instruments W (include X−k) s.th.
Cov(X, W ) = 0 & E[ |W ] = 0. Then
β = Cov(W , X)−1
Cov(W , Y ) = Cov(W , X)−1
{Cov(W , X)β+Cov(W , )}
Control function: We may write the ’selection equation’
Xk = g(W , X−k) + v , 0 = E[v|W , X−k] ⇒ ˆv
arising from idea that instruments are variables that induce variation in X
(β, h) = Cov( ˜X, ˜X)−1
Cov( ˜X, Y ) with ˜X = (X, ˆv)
E[Y |X, ˆv] = X β + ˆv h

Instrumental Variable Estimation - Varying Coefficients βi
ˆβ = (
i
Wi Xi )−1
i
Wi Yi = (
i
Wi Xi )−1
i
Wi Xi βi + RTi
if Wi is mean-independent from ri = βi − β we get identification – how
realistic 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 by
modeling
Yi = β(Qi ) Xi +
i
ei Xi + 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 problems
to VCM (already done in paper with PhD students)

Example: Mincer’s wage equation: standard IV
log(wagei ) = β0 + αeduci + β1experi + β2exper2
i + i
educi = γ0 + γ1Wi + γ2experi + γ3exper2
i + 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 +β2exper2
i +h(Zi , experi , ˆvi )+εi
0.04
0.05
0.06
0.07
0.08
0 10 20 30
Exper
g_1(exper)_hat
Returns to Education
IV Father
0.04
0.05
0.06
0.07
0.08
0 10 20 30
Exper
g_1(exper)_hat
Returns to Education
IV 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 W
interesting 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
0
MTE(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 splines
inside link
secondly, semiparametric VCM (or partial additive) estimation of
main fctn
Have presently joint projects on
theory paper on inference in nonparametric structural equations (with
E.Mammen),
especially on testing separability and significance using smooth
backfitting
creating an R package for these methods (J. Roca-Pardiñas) including
adaptive bandwidth choices

Example: Export Promotion
The standard model is
log(Yit) = α log(budgetit) + β log(popit) + δi + λt + it
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 on
returns, efficiency etc., that is, to make policy relevant statements.
Remark: not presented because all excluded instruments had no significant
impact on α function, the others little or no impact on log(budgetit).

Modeling Heterogeneity by Structural Varying Coeﬃcients Models in Presence of Endogeneity

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