Testing Gateway Theory: do cigarette prices affect illicit drug use?

682 M. Beenstock, G. Rahav / Journal of Health Economics 21 (2002) 679–698
The solution to this problem is to instrument C t−1 in Eq. (1). No doubt Z and X largely
overlap. An element of Z is the price of cigarettes (and other variables discussed in Section 4)
that prevailed when individual n was growing up (P n(t−1) ), which are hypothesized to
determine the demand for cigarettes but not the demand for cannabis. We expect C t−1 to
vary inversely with P t−1 . There is no reason to suspect that P t−1 and v t−1 are correlated
(nor P t−1 and u t ). If S t does not depend upon P t−1 , then P t−1 identifies the effect of C t−1
upon S t in Eq. (1).
The next step in the gateway chain concerns the relationship between cannabis and
hard drugs. The hypothesis of interest here is does exposure to cannabis induce a higher
probability of subsequently using hard drugs? We denote H n = 1 if individual n used hard
drugs after using cannabis and 0 otherwise. We express this as follows:
H n(t+1) = λY n(t+1) + δS nt + ρ y D y + w n(t+1) (3)
where Y is a vector of controls for hard drug use. Gateway Theory suggests δ>0. Since
we suspect that E(wu) >0 unobserved heterogeneity is likely to bias upwards estimates
of δ. Ideally, we seek exclusion restrictions for Y and X such as drug price data, which are
not available in Israel. Instead, we suggest using as an instrument for S t in Eq. (3) its fitted
value from IV estimation of Eq. (1). We refer to this as “domino” identification because
the natural experiment is indirect. Clearly, such indirect identification weakens the power
of tests on the value of δ. Random exposure to cigarettes at time t − 1 induces cannabis
consumption at time t, which in turn, induces hard drug use at time t + 1. If β and δ are
properly identified, and there is a causal gateway effect, we can expect a chain reaction
where raising the price of cigarettes will reduce cigarette smoking, which will reduce the
subsequent use of cannabis, which, in turn, will reduce the subsequent use of hard drugs.
Note that Eqs. (1)–(3) control for cohort effects. This is possible because we use several
surveys in Section 4 (y = year of survey − age). Had there been only one survey it would
not have been possible to identify the separate effects of P t−1 and birth cohort upon drug
consumption.
2.2. Recursive bivariate probit and two-stage procedures
If u and v in Eqs. (1) and (2) happen to be bivariate normal with cov(u, v) = ρ then
the model may be estimated by maximum likelihood as a recursive bivariate probit (RBP)
model (Maddala, 1983, pp. 122–123, Greene, 2000, pp. 852–825). Note that in this case C in
Eq. (1) does not have to be instrumented because, in contrast to the linear probability model,
it is not estimated by least squares. However, Maddala points out that identification requires
that X omit covariates in Z.IfX = Z the model is not identified, even parametrically.
Maddala suggests that an alternative procedure is to specify prob (C ∗ > 0) in Eq. (1)
instead of C, where C ∗ denotes the underlying latent variable that measures the propensity
to smoke cigarettes. In this case, it may be shown that a two-stage procedure provides
consistent estimates of the parameters in Eqs. (1) and (2). In the first stage, Eq. (2) is
estimated by probit or logit, and in the second stage the predicted probability of C obtained
from the first stage is used to replace C in Eq. (1).
There are advantages and disadvantages to both procedures. The main disadvantage of
RBP is that it may be sensitive to parametric assumptions about the unobserved heterogeneity.