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 γa . 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), γc 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 zit 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
(5)
(6)
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 (ait)3 and (cit)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 zait and, respectively, zcit 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 [γa ^c] = [—0 — 0.224] maximizes the restricted
likelihood function. I.e. the supposably biased estimator for γc prevails over the consistent one for γa
if the restriction sign(γa) = sign(γc) is imposed.
19