Title
synth -- Synthetic control methods for comparative case studies
Syntax
synth depvar predictorvars , trunit(#) trperiod(#) [ counit(numlist) xperiod(numlist) mspeperiod() resultsperiod() nested allopt unitnames(varname) figure keep(file) customV(numlist) optsettings ]
Dataset must be declared as a (balanced) panel dataset using tsset panelvar timevar; see tsset. Variables specified in depvar and predictorvars must be numeric variables; abbreviations are not allowed.
Description
synth implements the synthetic control method for causal inference in comparative case studies. synth estimates the effect of an intervention of interest by comparing the evolution of an aggregate outcome depvar for a unit affected by the intervention to the evolution of the same aggregate outcome for a synthetic control group. synth constructs this synthetic control group by searching for a weighted combination of control units chosen to approximate the unit affected by the intervention in terms of the outcome predictors. The evolution of the outcome for the resulting synthetic control group is an estimate of the counterfactual of what would have been observed for the affected unit in the absence of the intervention. synth can also be used to conduct a variety of placebo and permutation tests that produce informative inference regardless of the number of available comparison units and the number of available time-periods. See Abadie and Gardeazabal (2003) and Abadie, Diamond, and Hainmueller (2010) for details.
Required Settings
predictorvars the list of predictor variables. By default, all predictor variables are averaged over the entire pre-intervention period, which ranges from the earliest time period available in the panel time variable specified in tsset timevar to the period immediately prior to the intervention specified in trperiod. Missing values are ignored in the averages. The user has two options to flexibly specify the time periods over which predictors are averaged:
(1) xperiod(numlist) allows to specify a common period over which all predictors should be averaged; see below for details.
(2) For each particular predictor the user can specify the period over which the variable will be averaged. For this, synth uses a specialized syntax. The time period is specified in parenthesis directly following the variable name, e.g. varname(period) with no blanks between the variable name and its period. period can contain either a single period, a numlist of periods, or several periods concatenated by a "&". The periods refer to the panel time variable specified in tsset timevar. For example, assume the time periods are given in years, and there are four predictors X1, X2, X3, and X4 then:
. synth Y X1(1980) X2(1982&1986&1988) X3(1980(1)1990) X4
indicates that: X1(1980): the value of the variable X1 in the year 1980 is entered as a predictor.
X2(1982&1986&1988): the value of the variable X2 averaged over the years 1982, 1986, and 1988 is entered as a predictor.
X3(1980(1)1990): the value of the variable X3 averaged over the years 1980,1981,...,1990 is entered as a predictor.
X4: since no variable specific period is provided, the value of the variable X4 is averaged either over the entire pretreatment period (default) or the period specified in xperiod(numlist) and then entered as a predictor.
trunit(#) the unit number of the unit affected by the intervention as given in the panel id variable specified in tsset panelvar; see tsset. Notice that only a single unit number can be specified. If the intervention of interest affected several units the user may chose to combine these units first and then treat them as a single unit affected by the intervention.
trperiod(#) the time period when the intervention occurred. The time period refers to the panel time variable specified in tsset timevar; see tsset. Only a single number can be specified.
Options
counit(numlist) a list of unit numbers for the control units as given in the panel id variable specified in tsset panelvar; see tsset. counit() should be specified as a list of integer numbers (see numlist) and contain at least two control units. The list of control units specified constitute what is called the `donor pool', the set of potential control units out of which the synthetic control unit is constructed. Notice that counit is optional, if no counit is specified, the donor pool defaults to all units available in the panel id variable specified in tsset, excluding the unit affected by the intervention specified in trunit.
xperiod(numlist) a list of time periods over which the predictor variables specified in predictorvars are averaged. The list of time periods refers to the panel time variable specified in tsset timevar. For example, if the specified panel time variable is given in years, xperiod(1980(1)1988) indicates that the predictor variables are averaged over all years from 1980, 1981,...,1988. See numlist on how to specify lists of numbers. If no xperiod is specified, xperiod defaults to the entire pre-intervention period, which by default ranges from the earliest time period available in the panel time variable to the period immediately prior to the intervention. Notice that the period of the intervention itself is excluded from the average and missing entries are ignored. Also notice that variable-specific time periods always take precedence over xperiod. Usually, xperiod is specified to contain a number of pre-intervention periods, although post-intervention time periods could be included if the predictors are not affected by the intervention.
mspeperiod(numlist) a list of pre-intervention time periods over which the mean squared prediction error (MSPE) should be minimized. The list of time periods refers to the panel time variable specified in tsset timevar; see tsset. The MSPE refers to the squared deviations between the outcome for the treated unit and the synthetic control unit summed over all pre-intervention periods specified in mspeperiod(numlist). See numlist on how to specify lists of numbers. If no mspeperiod() is specified, mspeperiod() defaults to the entire pre-intervention period ranging from the earliest time period available in the panel time variable to the period immediately prior to the intervention. Notice that the period of the intervention itself is excluded from mspeperiod(). Usually, the mspeperiod() is specified to cover the whole pre-treatment period up to the time of the intervention, but other choices are possible.
resultsperiod(numlist) a list of time periods over which the results of synth should be obtained in the optional figure (see figure), the optional results dataset (see keep), and the return matrices (see ereturn results)). The list of time periods refers to the panel time variable specified in tsset timevar. If no resultsperiod is specified, resultsperiod defaults to the entire period, which by default ranges from the earliest to the latest time period available in the panel time variable.
nested by default synth uses a data-driven regression based method to obtain the variable weights contained in the V-matrix. This method relies on a constrained quadratic programming routine, that finds the best fitting W-weights conditional on the regression based V-matrix. This procedure is fast and often yields satisfactory results in terms of minimizing the MSPE. Specifying nested will lead to better performance, however, at the expense of additional computing time. If nested is specified synth embarks on an fully nested optimization procedure that searches among all (diagonal) positive semidefinite V-matrices and sets of W-weights for the best fitting convex combination of the control units. The fully nested optimization contains the regression based V as a starting point, but often produces convex combinations that achieve even lower MSPE. If customV is specified and nested is specified, the user supplied V-matrix form the starting point for the nested optimization. All parameters of both optimizers can be tuned by the user depending on his application (see optimset).
allopt if nested is specified (see nested) the user can also specify allopt if she is willing to trade-off even more computing time in order to gain fully robust results. Sometimes the search space may contain local minima such that the nested optimization procedure starting from the regression based V-matrix may not find the global minimum in the parameter space. allopt provides a robustness check by running the nested optimization three times using three different starting points (the regression based V, equal V-weights, and a third procedure that uses Stata's ml search procedure to find good starting values. synth returns the best result of all three attempts. This option usually will take three times the amount of computing time compared to the nested option. Often allopt will lead to no improvement over just the nested method, but sometime allopt produces even better results.
unitnames(varname) a string variable that contains unit names. The unit names refer to the unit numbers in the panel id variable specified in tsset pvar; see tsset. If unitnames is provided the results will be displayed with unit numbers labeled by their respective unit names. So for example, if the user has two variables in his dataset called country_numbers (numeric) and country_names (string), unitnames(country_names) could be specified to display the results using country names instead of numbers. Alternatively, if the user does not specify unitnames, but his panel id variable is labeled, the labels from the latter will be used.
figure if specified synth produces a line plot with outcome trends for the treated unit and the synthetic control unit for the years specified in resultsperiod().
keep(filename) saves a dataset with the results in the file filename.dta. This dataset can be used to further process the results.
If keep(filename) is specified, filename.dta will hold the following variables:
_time: A variable that contains the respective time period (from the tsset panel time variable (timevar)) for all periods specified in resultsperiod().
_Y_treated: The observed outcome depvar for the treated unit specified in tr() for each time period specified in resultsperiod().
_Y_synthetic: The estimated outcome depvar for the synthetic control unit estimated using the convex combination of the control units specified in co() for each time period specified in resultsperiod().
_Co_Number: A variable that contains the unit number (from the tsset panel unit variable (panelvar) for each control unit specified in co(). If unit names are supplied via conames() the unit numbers will be labeled accordingly (each control unit number is labeled with its respective name from conames().
_W_weight: A variable that contains the estimated unit weight for each control units specified in co().
replace replaces the dataset specified in keep(filename) if it already exists.
customV(numlist) by default synth uses a data-driven regression based method to obtain the variable weights contained in the V-matrix. customV() gives the user the option to supply custom V-Weights instead. Notice that the V-weights determine the predictive power of the respective variable for the outcome of interest over the pre-intervention period. Highly predictive variables should be given a high weight, so that the unit affected by the intervention and the synthetic control unit match strongly on this predictor. Weights are specified as a list with weights appearing in the same order as the predictors listed in predictorvars. One weight must be supplied for each predictor. See the papers in the references for details. For now, only one weight per variable is allowed (the V matrix is diagonal), but future releases will allow non-diagonal V matrices to be supplied.
Optimization Settings: ----------------------------------------------------------------------------- Control parameters for the constrained quadratic optimization routine:
The constrained quadratic optimization routine is based on an algorithm that uses the interior point method to solve the constrained quadratic programming problem (see Vanderbei 1999 for more details on the interior point method). It is implemented via a C++ plugin and has the following tuning parameters:
margin(real) Margin for constraint violation tolerance. Default is 5 percent (ie. 0.05). maxiter(#) Maximum number of iterations. Default is 1000. sigf(#) Precision (no of significant figures). Default is 7. bound(#) Clipping bound for the variables. Default is 10.
Additional control parameters for the nested optimization routine:
If nested is specified, a nested optimization will be performed using the constrained quadratic programming routine and Stata's ml optimizer. By default, synth uses the maximize default settings. The user may tune the maximize settings depending on his application (e.g. like synth ... , iterate(20) ). -----------------------------------------------------------------------------
Saved Results
By default, synth ereturns the following matrices, which can be displayed by typing ereturn list after synth is finished (also see ereturn).
e(V_matrix): A diagonal matrix that contains the normalized variable weights in the diagonal.
e(X_balance) : A matrix that juxtaposes the predictor values for the unit affected by the intervention and the synthetic control unit. The matrix has two columns (treated and synthetic) and as many rows as predictors.
e(W_weights) : A matrix that contains the unit numbers and unit weights, ie. the relative contribution of each control unit to the synthetic control unit. The matrix has two column and as many rows as control units.
e(Y_treated) : A matrix that contains the values of the response variable for the treated unit for each time period. The matrix has one column and as many rows as time periods specified in resultsperiod().
e(Y_synthetic) : A matrix that contains the values of the response variable for the synthetic control unit for each time period. The matrix has one column and as many rows as time periods specified in {cmd:resultsperiod().
e(RMSPE) : A one by one matrix that contains the Root Mean Squared Prediction Error (RMSPE)}.
Examples
Load Example Data: This panel dataset contains information for 39 US States for the years 1970-2000 (see Abadie, Diamond, and Hainmueller (2010) for details). . sysuse smoking
Declare the dataset as panel: . tsset state year
Example 1 - Construct synthetic control group: synth cigsale beer(1984(1)1988) lnincome retprice age15to24 cigsale(1988) cigsale(1980) cigsale(1975), trunit(3) trperiod(1989)
In this example, the unit affected by the intervention is unit no 3 (California) in the year 1989. The donor pool (since no counit() is specified) defaults to the control units 1,2,4,5,...,39 ( ie. the other 38 states in the dataset). Since no xperiod() is provided, the predictor variables for which no variable specific time period is specified (retprice, lnincome, and age15to24) are averaged over the entire pre-intervention period up to the year of the intervention (1970,1981,...,1988). The beer variable has the time period (1984(1)1988) specified, meaning that it is averaged for the periods 1984,1985,...,1988. The variable cigsale will be used three times as a predictor using the values from periods 1988, 1980, and 1975 respectively. The MSPE is minimized over the entire pretreatment period, because mspeperiod() is not provided. By default, results are displayed for the period from 1970,1971,...,2000 period (the earliest and latest year in the dataset).
Example 2 - Construct synthetic control group: synth cigsale beer lnincome(1980&1985) retprice cigsale(1988) cigsale(1980) cigsale(1975), trunit(3) trperiod(1989) fig
This example is similar to example 1, but now beer is averaged over the entire pretreatment period while lnincome is only averaged over the periods 1980 and 1985. Since no data is available for beer prior to 1984, synth will inform the user that there is missing data for this variable and that the missing values are ignored in the averaging. A results figure is also requested using the fig option.
Example 3 - Construct synthetic control group: synth cigsale retprice cigsale(1970) cigsale(1979) , trunit(33) counit(1(1)20) trperiod(1980) fig resultsperiod(1970(1)1990)
In this example, the unit affected by the intervention is state no 33, and the donor pool of potential control units is restricted to states no 1,2,...,20. The intervention occurs in 1980, and results are obtained for the 1970,1971,...,1990 period.
Example 4 - Construct synthetic control group: synth cigsale retprice cigsale(1970) cigsale(1979) , trunit(33) counit(1(1)20) trperiod(1980) resultsperiod(1970(1)1990) keep(resout)
This example is similar to example 2 but keep(resout) is specified and thus synth will save a dataset named resout.dta in the current Stata working directory (type pwd to see the path of your working directory). This dataset contains the result from the current fit and can be used for further processing. Also to easily access results recall that synth routinely returns all result matrices. These can be displayed by typing ereturn list after synth has terminated.
Example 5 - Construct synthetic control group: synth cigsale beer lnincome retprice age15to24 cigsale(1988) cigsale(1980) cigsale(1975) , trunit(3) trperiod(1989) xperiod(1980(1)1988) nested
This is again example 2, but the nested option is specified, which may produce lower loss at the expense of additional computing time. Also, xperiod() is specified indicating that predictors are averaged for the 1980,1981,...,1988 period.
Example 5 – Run placebo in space: . tempname resmat forvalues i = 1/4 { synth cigsale retprice cigsale(1988) cigsale(1980) cigsale(1975) , trunit(`i') trperiod(1989) xperiod(1980(1)1988) matrix `resmat' = nullmat(`resmat') \ e(RMSPE) local names `"`names' `"`i'"'"' } mat colnames `resmat' = "RMSPE" mat rownames `resmat' = `names' matlist `resmat' , row("Treated Unit")
This is a code example to run placebo studies by iteratively reassigning the intervention in space to the first four states. To do so, we simply run a four loop each where the trunit() setting is incremented in each iteration. Thus, in the first run of synth state number one is assigned to the intervention, in the second run state number two, etc, etc. In each run we store the RMSPE and display it in a matrix at the end.
References
Abadie, A., Diamond, A., and J. Hainmueller. 2010. Synthetic Control Methods for Comparative Case Studies: Estimating the Effect of California's Tobacco Control Program. Journal of the American Statistical Association 105(490): 493-505.
Abadie, A. and Gardeazabal, J. 2003. Economic Costs of Conflict: A Case Study of the Basque Country. American Economic Review 93(1): 113-132.
Vanderbei, R.J. 1999. LOQO: An interior point code for quadratic programming. Optimization Methods and Software 11: 451-484.
Authors
Jens Hainmueller, jhainm@mit.edu MIT
Alberto Abadie, alberto_abadie@harvard.edu Harvard University
Alexis Diamond, adiamond@fas.harvard.edu IFC