Tobacco and Alcohol: Complements or Substitutes? - A Statistical Guinea Pig Approach

have not properly identified γc , estimates for this parameter cannot contribute to
our research question which, in turn, exclusively has to be answered on basis of the
estimate for γ_a . This problem is not solved by imposing cross-equation restrictions.
Yet, even worse, imposing such restrictions might render ^a biased, too.²¹

6 Model Extensions

6.1 Non-Linear Specifications

In the model presented so far the degree of complementarity in the consumption of
alcohol and tobacco is captured by the fixed coefficients γ . Yet, the assumption that
the degree of complementarity in consumption does not depend on consumption levels
seems to be quite strong. One way to address this problem is specifying γ as a
function of endogenous variables, i.e. γa = γa(c*_t), γ_c analogously. In this case, even
if a simple linear relationship is assumed, a reduced form model representation does
not longer exist in closed form. Nevertheless, a linear instrumental variable estimator
that deals with non-linear functions of endogenous regressors as if they were additional
explanatory variables still can be applied in order to obtain estimates for the structural
parameters, see Wooldridge (2002: 232). For this approach additional instruments are
required. Interaction terms of the original instruments z_it or, alternatively, squares and
higher order powers of ^*_it and ^*_it seem to be the most obvious choices, see Wooldridge
(2002: 237). Rather than a linear specification we chose a quadratic one

( ⅛ )²

Ya = Ya 1 + Ya ∙2----

1000

Yc = Yc 1 + Yc 2““—,

1000

which corresponds to cubic structural equations. This choice is for technical reasons.
Fitted values for (α*_it)³ and (c*_it)³ still can be obtained from simple Tobit regressions
with (a_it)³ and (c_it)³ serving as left hand side variables. In contrast - due to non-
negativeness - squares do not allow for this approach. Several versions of non-linear
model specifications are estimated, using (i) (a*_it)³ and (^'*_it)³, (ii) (^'*_it)², (^'*_it)², (^'*_it)³,
and (^'*_it)³ and, (iii) interaction terms of the elements of z_ait and, respectively, z_cit as

²¹ Since the likelihood function of the Tobit model is globally concave, binding inequality constraints
will always result in corner solutions. In our case [γ_a ^_c] = [—0 — 0.224] maximizes the restricted
likelihood function. I.e. the supposably biased estimator for γ_c prevails over the consistent one for γ_a
if the restriction sign(γ_a) = sign(γ_c) is imposed.

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