# The synthetic control method compared to difference in diff

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1 The synthetic control method compared to difference in differences: discussion April 2014

2 Outline of discussion 1 Very brief overview of synthetic control methods 2 Synthetic control methods for disaggregated data 3 Matching vs/and synthetic control methods 4 Indirect effects

3 Synthetic control methods: overview Goal: to evaluate the impact of a treatment implemented at the aggregate level in one (or very few) unit using a small number of controls to build the counterfactual Synthetic control methods use (long) longitudinal data to build the weighted average of non-treated units that best reproduces characteristics of the treated unit over time, prior to treatment this is the synthetic cohort impact of treatment is quantified by a simple difference after treatment: treated vs synthetic cohort

4 Synthetic control methods: overview Goal: to evaluate the impact of a treatment implemented at the aggregate level in one (or very few) unit using a small number of controls to build the counterfactual Synthetic control methods use (long) longitudinal data to build the weighted average of non-treated units that best reproduces characteristics of the treated unit over time, prior to treatment this is the synthetic cohort impact of treatment is quantified by a simple difference after treatment: treated vs synthetic cohort

5 Synthetic control methods: formalisation Units: j = 0,1,..., where j = 0 is the treated and j = 1,..., are controls Time frame: t = 1,...,T 1 split in two periods - before treatment t = 1,...,T 0 and after treatment t = T 0 + 1,...,T 1 Potential and observed outcomes for the treated unit are ( Y0t 0,Y ) 0t 1 and { Y0t 0 for t = 1,...,T 0 Y 0t = Y0t 1 for t = T 0 + 1,...,T 1 Aim is to estimate α 0t = Y 1 0t Y 0 0t for t = T 0 + 1,...,T 1

6 Synthetic control methods: formalisation Model of untreated outcomes for unit j = 0,..., and time t = 1,...,T 1 Y 0 jt = δ t + θ t Z j + λ t µ j + ε jt Z j are the observed, pre-treatment covariates µ j are permanent unobserved variables δ t are common time effects ε jt are unobserved transitory shocks at the unit level with zero mean

7 Synthetic control method: formalisation Choose W = ( w 1,...,w ) [0,1], adding to 1, to minimise distance in pre-treatment characteristics between treated and weighted average of controls Treatment effect estimated by the simple difference ˆα 0t = Y 0t w j Y jt for t = T 0 + 1,...,T 1 Ideally, one would want to select W such that w j Z j = Z 0 and w j µ j = µ 0 so ˆα 0t is unbiased (ε is mean-independent of (Z, µ) and independent across units and over time) Not feasible since µ is unobserved

8 Synthetic control methods: formalisation Solution: choose W satisfying w j Z j = Z 0, w j Y j1 = Y 01,..., Bias can be bounded under mild conditions ( ) 1 p E(ˆα 0t α 0t ) < β p 1 m p max T p 1, 0 w j Y jt 0 = Y 0T0 σ T0

9 Synthetic control methods: bias ( ) 1 p E(ˆα 0t α 0t ) < β p 1 m p max T p 1, 0 Bias is small when T 0 is large relative to scale of ε σ T0 Intuition: a synthetic cohort can fit ( Z 0,Y 01,...,Y 0T0 ) for a large T 0 only if it fits (Z 0, µ 0 ) But a large does not help reducing the bias once these conditions are met w j Z j = Z 0, w j Y j1 = Y 01,..., w j Y jt 0 = Y 0T0

10 Synthetic control methods for disaggregated data Some issues 1 Many controls not necessarily beneficial - although this depends on how small is and whether the aggregate treated unit lies outside the domain of the controls 2 Scale of the transitory shock can be larger at the disaggregated level if the aggregate outcome is the average of outcomes for smaller units in which case need more time periods to keep bias down 3 Possibly more serious interpolation bias

11 Synthetic control methods for disaggregated data Some issues 1 Many controls not necessarily beneficial - although this depends on how small is and whether the aggregate treated unit lies outside the domain of the controls 2 Scale of the transitory shock can be larger at the disaggregated level if the aggregate outcome is the average of outcomes for smaller units in which case need more time periods to keep bias down 3 Possibly more serious interpolation bias

12 Synthetic control methods for disaggregated data Some issues 1 Many controls not necessarily beneficial - although this depends on how small is and whether the aggregate treated unit lies outside the domain of the controls 2 Scale of the transitory shock can be larger at the disaggregated level if the aggregate outcome is the average of outcomes for smaller units in which case need more time periods to keep bias down 3 Possibly more serious interpolation bias

13 Synthetic control methods for disaggregated data Some issues 1 Many controls not necessarily beneficial - although this depends on how small is and whether the aggregate treated unit lies outside the domain of the controls 2 Scale of the transitory shock can be larger at the disaggregated level if the aggregate outcome is the average of outcomes for smaller units in which case need more time periods to keep bias down 3 Possibly more serious interpolation bias

14 Synthetic control methods for disaggregated data Interpolation bias The synthetic control method relies on the linearity of the model of untreated outcomes Y 0 jt = δ t + θ t Z j + λ t µ j + ε jt Even if linearity is violated, the model can be a good local approximation But bias can be large if the characteristics of control units are far from those of the treated In aggregate studies: pick and choose the control units that more closely resemble the treated But this is difficult to implement with more and smaller units Or when treated and control units are of different nature

15 Matching and synthetic control methods DID has been used with matching to relax strict functional form assumptions Matching selects controls that have, each of them, characteristics ( Zj,Y j1,...,y jt0 ) close to those of the treated Matching is a local estimator hence it is less sensitive to interpolation bias and allows for a more general specification of the model of untreated outcomes

16 Matching and synthetic control methods A more general model, under the assumption that ε is balanced conditional on ( Z j,y j1,...,y jt0 ) : Y 0 jt = f t ( Zj, µ j ) + εjt Matching is unbiased if it removes unobserved permanent differences between treated and control units But it may be impossible to match closely on the many characteristics ( Zj,Y j1,...,y jt0 ) Combine matching with synthetic cohorts for more disaggregated data? if f is smooth, a local polynomial approximation of f is accurate matching prior to applying a synthetic control method could help ensuring the comparability of admissable controls

17 Indirect effects At the aggregate level, it is quite plausible that what happens in one unit affects other units Abadie s and co-authors applications California s tobacco control programme may have influenced behaviour and legislation in other states The unification of Germany (and indeed aggregate shocks to its economy) may affect the economic outcomes of closely connected countries More similar control units may be more exposed to indirect effects Trade-off between interpolation bias and indirect effects

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