# Synthetic Control Methods for Comparative Case Studies: Estimating the Effect of California s Tobacco Control Program

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3 Abadie, Diamond, and Hainmueller: Synthetic Control Methods for Comparative Case Studies 495 unit. In Section 3 we discuss this assumption in the context of our empirical investigation. Let α it = Y I it YN it be the effect of the intervention for unit i at time t, and let D it be an indicator that takes value one if unit i is exposed to the intervention at time t, and value zero otherwise. The observed outcome for unit i at time t is Y it = Y N it + α it D it. Because only the first region (region one is exposed to the intervention and only after period T 0 (with 1 T 0 < T, we have that { 1 ifi = 1 and t > T0, D it = 0 otherwise. We aim to estimate (α 1T0 +1,...,α 1T.Fort > T 0, α 1t = Y I 1t YN 1t = Y 1t Y N 1t. Because Y I 1t is observed, to estimate α 1t we just need to estimate Y N 1t. Suppose that YN it is given by a factor model Y N it = δ t + θ t Z i + λ t μ i + ε 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 (not affected 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 i 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. Consider a (J 1 vector of weights W = (w 2,...,w such that w j 0forj = 2,...,J + 1 and w 2 + +w = 1. Each particular value of the vector W represents a potential synthetic control, that is, a particular weighted average of control regions. The value of the outcome variable for each synthetic control indexed by W is w j Y jt = δ t + θ t w j Z j + λ t w j μ j + w j ε jt. Suppose that there are (w 2,...,w such that w j Y j1 = Y 11, w j Y jt 0 = Y 1T0, w j Y j2 = Y 12,..., and w j Z j = Z 1. In Appendix B, we prove that if T 0 t=1 λ t λ t is nonsingular, then, ( Y1t N w j Y T 0 T0 1 jt = w j λ t λ n λ n λ s (ε js ε 1s s=1 n=1 (2 w j (ε jt ε 1t. (3 In Appendix B we prove also that, under standard conditions, the mean of the right-hand side of Equation (3 will be close to zero if the number of preintervention periods is large relative to the scale of the transitory shocks. This suggests using α 1t = Y 1t w j Y jt for t {T 0 + 1,...,T} as an estimator of α 1t. Equation (2 can hold exactly only if (Y 11,...,Y 1T0, Z 1 belongs to the convex hull of {(Y 21,...,Y 2T0, Z 2,..., (Y 1,...,Y T0, Z }. In practice, it is often the case that no set of weights exists such that Equation (2 holds exactly in the data. Then, the synthetic control region is selected so that Equation (2 holds approximately. In some cases, it may not even be possible to obtain a weighted combination of untreated units such that Equation (3 holds approximately. This would be the case if (Y 11,...,Y 1T0, Z 1 falls far from the convex hull of {(Y 21,...,Y 2T0, Z 2,...,(Y 1,...,Y T0, Z }. Notice, however, that the magnitude of such discrepancy can be calculated for each particular application. So for each particular application, the analyst can decide if the characteristics of the treated unit are sufficiently matched by the synthetic control. In some instances, the fit may be poor and then we would not recommend using a synthetic control. Even if there is a synthetic control that provides a good fit for the treated units, interpolation biases may be large if the simple linear model presented in this section does not hold over the entire set of regions in any particular sample. Researchers trying to minimize biases caused by interpolating across regions with very different characteristics may restrict the donor pool to regions with similar characteristics to the region exposed to the event or intervention of interest. Notice that, even if taken at face value, Equation (1 generalizes the usual difference-in-differences (fixed-effects model that is often applied in empirical studies in the social sciences. The traditional difference-in-differences (fixed-effects model can be obtained if we impose that λ t in Equation (1 is constant for all t. That is, the difference-in-differences model allows for the presence of unobserved confounders but restricts the effect of those confounders to be constant in time, so they can be eliminated by taking time differences. In contrast, the factor model presented in this section allows the effects of confounding unobserved characteristics to vary with time. Under this model, taking time differences does not eliminate the unobserved confounders, μ j. However, a synthetic control such that w j Z j = Z 1 and w j μ j = μ 1, (4 would provide an unbiased estimator of Y N 1t. Choosing a synthetic control in this manner is, of course, not feasible because μ 1,...,μ are not observed. However, under fairly standard conditions (see Appendix B, the factor model in Equation (1 implies that a synthetic control can fit Z 1 and a long set of preintervention outcomes, Y 11,...,Y 1T0, only as long as it fits Z 1 and μ 1, so Equation (4 holds approximately. Synthetic controls can provide useful estimates in more general contexts than the factor model considered thus far. Consider, for example, the following autoregressive model with

4 496 Journal of the American Statistical Association, June 2010 time-varying coefficients: Y N it+1 = α ty N it + β t+1 Z it+1 + u it+1, Z it+1 = γ t Y N it + t Z it + v it+1, where u it+1 and v it+1 have mean zero conditional on F t = {Y js, Z js } 1 j N,s t. Suppose that we can choose {w j } 2 j N such that w j Y jt 0 = Y 1T0 and (5 w j Z jt 0 = Z 1T0. (6 Then, the synthetic control estimator is unbiased even if data for only a single pretreatment period are available. See Appendix B for details. 2.3 Implementation Let W be a (J 1 vector of positive weights that sum to one. That is, W = (w 2,...,w with w j 0forj = 2,...,J + 1 and w w = 1. Each value of W represents a weighted average of the available control regions and, therefore, a synthetic control. Notice that, although we define our synthetic controls as convex combinations of unexposed units, negative weights or weights larger than one can be used at the cost of allowing extrapolation. The outcome variable of interest is observed for T periods, t = 1,...,T, for the region affected by the intervention, Y 1t, and the unaffected regions, Y jt, where j = 2,...,J + 1. Let the (T 0 1 vector K = (k 1,...,k T0 define a linear combination of preintervention outcomes: Ȳi K = T 0 s=1 k sy is.forexample, if k 1 = k 2 = = k T0 1 = 0 and k T0 = 1, then Ȳ K = Y it0, the value of the outcome variable in the period immediately prior to the intervention. If k 1 = k 2 = =k T0 = 1/T 0, then Ȳi K = T 1 T0 0 s=1 Y is, the simple average of the outcome variable for the preintervention periods. Consider M of such linear combinations defined by the vectors K 1,...,K M.Let X 1 = (Z 1, ȲK 1 1,...,ȲK M 1 be a (k 1 vector of preintervention characteristics for the exposed region, with k = r + M. Similarly, X 0 is a (k J matrix that contains the same variables for the unaffected regions. That is, the jth column of X 0 is (Z j, ȲK 1 j,...,ȳ K M j. The vector W is chosen to minimize some distance, X 1 X 0 W, between X 1 and X 0 W, subject to w 2 0,...,w 0, w 2 + +w = 1. One obvious choice for Ȳ K 1 i,...,ȳ K M i is Ȳ K 1 i = Y i1,...,ȳ K T 0 i = Y it0, that is, the values of the outcome variable for all the available preintervention periods. In practice, however, the computation of the weights w 2,...,w can be simplified by considering only a few linear combination of preintervention outcomes and checking whether Equation (2 holds approximately for the resulting weights. To measure the discrepancy between X 1 and X 0 W, we will employ X 1 X 0 W V = (X 1 X 0 W V(X 1 X 0 W, where V is some (k k symmetric and positive semidefinite matrix, although other choices are also possible. If the relationship between the outcome variable and the explanatory variables in X 1 and X 0 is highly nonlinear and the support of the explanatory variables is large, interpolation biases may be severe. (For instance, an equally weighted combination of a 65%-White 35%-Nonwhite state and a 95%-White 5%-Nonwhite state will approximate the outcome of a 80%-White 20%-Nonwhite state if that outcome is approximately linear in the racial composition of the states. However, if the outcome is highly nonlinear in racial composition, the quality of the approximation may be poor. In that case, W can be chosen to minimize X 1 X 0 W plus a set of penalty terms specified as increasing functions of the distances between X 1 and the corresponding values for the control units with positive weights in W. Alternatively, as mentioned in Section 2.2, interpolation biases can be reduced by restricting the comparison group to units that are similar to the exposed units in term of the values of X 1. Although our inferential procedures are valid for any choice of V, the choice of V influences the mean square error of the estimator. An optimal choice of V assigns weights to linear combinations of the variables in X 0 and X 1 to minimize the mean square error of the synthetic control estimator. Sometimes this choice can be based on subjective assessments of the predictive power of the variables in X 1 and X 0. The choice of V can also be data-driven. One possibility is to choose V such that the resulting synthetic control region approximates the trajectory of the outcome variable of the affected region in the preintervention periods. Following Abadie and Gardeazabal (2003, in the empirical section of this article we choose V among positive definite and diagonal matrices such that the mean squared prediction error of the outcome variable is minimized for the preintervention periods (see Abadie and Gardeazabal 2003, appendix B, for details. Alternatively, if the number of available preintervention periods in the sample is large enough, researchers may divide them into an initial training period and a subsequent validation period. Given a V, W (V can be computed using data from the training period. Then, the matrix V can be chosen to minimize the mean squared prediction error produced by the weights W (V during the validation period. 2.4 Inference The standard errors commonly reported in regression-based comparative case studies measure uncertainty about aggregate data. For example, Card (1990 uses data from the U.S. Current Population Survey to estimate native employment rates in Miami and a set of comparison cities around the time of the Mariel Boatlift. Card and Krueger (1994 use data on a sample of fast food restaurants in New Jersey and Pennsylvania to estimate the average number of employees in fast food restaurants in these two states around the time when the minimum wage was increased in New Jersey. The standard errors reported in these studies reflect only the unavailability of aggregate data on employment (for native workers in Miami and other cities, and in fast food restaurants in New Jersey and Pennsylvania, respectively. This mode of inference would logically produce zero standard errors if aggregate data were used for estimation. However, perfect knowledge of aggregate data does not eliminate all uncertainty about the parameters of interest. That is, even if aggregate data are used for estimation, in most cases researchers would not believe that there is no remaining uncertainty about the value of the parameters of interest. The reason is that not all uncertainty about the value of the estimated parameters comes from lack of knowledge of aggregate data. In comparative case studies, an additional source of uncertainty derives from ignorance about the ability of the control group to

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