Marijuana medicalization and motor vehicle fatalities: a synthetic control group approach

Abstract

Objectives

This paper reports a quasi-experimental evaluation of California’s 1996 medical marijuana law (MML), known as Proposition 215, on statewide motor vehicle fatalities between 1996 and 2015.

Methods

To infer the causal impact of California’s MML enactment on statewide motor vehicle fatalities, we construct a synthetic control group for California (i.e., California had it NOT enacted MMLs) as a weighted sum of annual traffic fatality time series from a donor pool of untreated (no MML) states. The post-MML difference between California and its constructed counterfactual reflects the net effect of MMLs on statewide traffic fatalities. The synthetic control group design avoids the problematic homogeneity assumptions intrinsic to panel regression models, which have been employed in prominent studies of this topic.

Results

California’s 1996 MML appears to have produced a large, sustained decrease in statewide motor vehicle fatalities amounting to an annual reduction between 588 and 900 vehicle fatalities. This finding is consistent across a wide range of model specifications and donor pool restrictions. In-sample placebo test results suggest that the estimated intervention effect is unlikely to be a spurious artifact and the “leave-one-out” sensitivity analysis demonstrates that the effect is not being driven by an individual or ensemble of influential donor pool states.

Conclusions

Our focus on California as a case study limits our ability to generalize our estimate of the MML intervention on motor vehicle fatalities in California to other MML states; however, state-level MML interventions have major differences in their policy dimensions that seem unlikely to “average out” through aggregation.

Appendices

Appendix 1. Synthetic control group algebra

Suppose that ${Y}_{it}^M$ is given by a factor model,

$$ {Y}_{it}^M={\delta}_t+{0}_t{Z}_{i+}{\lambda}_t{\mu}_t+{\varepsilon}_{it} $$

(1)

where δ_t is an unknown common factor with constant factor loadings across units, Z_i is a (r × 1) vector of observed covariates that are unaffected by the intervention, θ_t is a (1 × r) vector of unknown parameters, λ_t is a (1 × F) vector of unobserved common factors, μ_t is an (F × 1) vector of unknown factor loadings, and the error terms ε_it are unobserved transitory shocks at the region level with zero mean.

Let W = (w₂, …, w_J + 1) be a (J × 1) vector of donor pool unit weights, selected via a constrained quadratic programming optimization routine, to minimize (X₁-X₀W)′V(X₁-X₀W). Where X₁ is a (K × 1) vector of pre-intervention values of K predictive factors of outcome Y for the treated unit, X₀ is a (K × J) matrix of the same K predictive factors for the untreated units, and V is a (K × K) positive definite matrix whose diagonal elements are weights reflecting the relative importance of the factors in X₁ and X₀. Each value of the vector W represents a potential synthetic control (i.e., a weighted combination of control units). The value of the outcome variable Y for each synthetic control indexed by W is

$$ \sum \limits_{j=2}^{J+1}{w}_j{Y}_{jt}={\delta}_t+{\uptheta}_t\sum \limits_{j=2}^{J+1}{w}_j{Z}_j+{\lambda}_t\sum \limits_{j=2}^{J+1}{w}_j{\mu}_j+\sum \limits_{j=2}^{J+1}{w}_j{\upvarepsilon}_{jt} $$

Therefore,

$$ {Y}_{it}^N-\sum \limits_{j=2}^{J+1}{w}_j{Y}_{jt}^N={\uptheta}_t\left({Z}_{1^{-}}\sum \limits_{j=2}^{J+1}{w}_j{Z}_j\right)+{\lambda}_t\left(\mu -\sum \limits_{j=2}^{J+1}{w}_j{\mu}_j\right)+\sum \limits_{j=2}^{J+1}{w}_j\left({\varepsilon}_{1t}-{\varepsilon}_{jt}\right) $$

(2)

As Abadie, Diamond, and Hainmueller (2010) note, we assume that the error terms ε_it are independent across units and time. However, the unobserved residual u_it = λ_tμ_t + ε_it may be correlated across units and time regardless of the assumed independence of ε_it because of the presence of the term λ_tμ_t. If the term λ_t is assumed to be constant for all times t, then Eq. (1) produces the traditional fixed-effects difference-in-differences model. Since the model presented in Abadie, Diamond, and Hainmueller (2010) allows the effect of unobserved confounders to vary over time, taking time differences does not eliminate the potential bias of unobserved confounders μ_t. Suppose that there are $\left({w}_2^{\ast },\dots, {w}_{J+1}^{\ast}\right)$ such that

$$ {\displaystyle \begin{array}{cc}\sum \limits_{j=2}^{J+1}{w}_j^{\ast }{Y}_{j1}={Y}_{11},& \sum \limits_{j=2}^{J+1}{w}_j^{\ast }{Y}_{j2}={Y}_{12},\dots, \\ {}\sum \limits_{j=2}^{J+1}{w}_j^{\ast }{Y}_{j{T}_0=}{Y}_{1{T}_0},\mathrm{and}& \sum \limits_{j=2}^{J+1}{w}_j^{\ast }{Z}_j={Z}_1\end{array}} $$

(3)

then the effect of the intervention can be estimated as the difference between the observed outcome variable ${Y}_{it}^M$and the estimated outcome for the synthetic control unit:

$$ {\widehat{\alpha}}_{1t}={Y}_{1t}+\sum \limits_{j=2}^{J+1}\left({w}_j^{\ast }{Y}_{jt}\right) $$

(4)

Equation (3) can hold exactly only if (Y₁₁,...,Y_1T0,${\boldsymbol{Z}}_1^{\acute{\mkern6mu}}$) belongs to the convex hull of {(Y₂₁,...,Y_2T0, ${\boldsymbol{Z}}_2^{\acute{\mkern6mu}}$), ..., (Y_J + 11,...,Y_J + 1T0,${\boldsymbol{Z}}_{J+1}^{\acute{\mkern6mu}}$)}. Often, no set of weights exists that would allow for Eq. (3) to hold exactly. When that is the case, weights are selected so that Eq. (3) holds approximately.

Appendix 2.

Table 3 Synthetic control group composition

Table 4 Leave-one-out sensitivity analysis composition comparison