# SCM Using Time Series

## Introduction

This vignette illustrates the syntax of SCMT models. For a more general introduction to package `MSCMT` see its main vignette.

Although SCM models are usually based on time series data of predictor variables, standard SCM estimation does not exploit this particular characteristic. Instead, time series data of predictors are either aggregated, mostly by calculating (a bunch of) means, or every instant of time is considered as a separate input variable with individual predictor weight. With package `MSCMT`, a time series of a predictor variable can be considered as single input variable without the need of aggregation, an extension of SCM called SCMT, see Klößner and Pfeifer (2015).

This vignette illustrates the syntax of SCMT models and how SCMT models may lead to more meaningful predictor weights without drawbacks concerning the model fit.

## Definition of the Standard Model

We use the `basque` dataset in package `Synth` as an example and replicate the preparation of the data from the main vignette of this package:

    library(Synth)
    data(basque)
    library(MSCMT)
    Basque <- listFromLong(basque, unit.variable="regionno", time.variable="year", unit.names.variable="regionname")
    school.sum <- with(Basque,colSums(school.illit + school.prim + school.med + school.high  + school.post.high))
    Basque$school.higher <- Basque$school.high + Basque$school.post.high
    for (item in c("school.illit", "school.prim", "school.med", "school.higher"))
      Basque[[item]] <- 6 * 100 * t(t(Basque[[item]]) / school.sum)

We also replicate model specification of the main vignette which reproduces the model in Abadie and Gardeazabal (2003):

    treatment.identifier <- "Basque Country (Pais Vasco)"
    controls.identifier  <- setdiff(colnames(Basque[[1]]),
                                    c(treatment.identifier, "Spain (Espana)"))
    times.dep  <- cbind("gdpcap"                = c(1960,1969))
    times.pred <- cbind("school.illit"          = c(1964,1969),
                        "school.prim"           = c(1964,1969),
                        "school.med"            = c(1964,1969),
                        "school.higher"         = c(1964,1969),
                        "invest"                = c(1964,1969),
                        "gdpcap"                = c(1960,1969),
                        "sec.agriculture"       = c(1961,1969),
                        "sec.energy"            = c(1961,1969),
                        "sec.industry"          = c(1961,1969),
                        "sec.construction"      = c(1961,1969),
                        "sec.services.venta"    = c(1961,1969),
                        "sec.services.nonventa" = c(1961,1969),
                        "popdens"               = c(1969,1969))
    agg.fns <- rep("mean", ncol(times.pred))

Estimation of the model gives:

    res <- mscmt(Basque, treatment.identifier, controls.identifier, times.dep, times.pred, agg.fns, seed=1, single.v=TRUE, verbose=FALSE)
    res

    ## Specification:
    ## --------------
    ##
    ## Model type:     SCM
    ## Treated unit:   Basque Country (Pais Vasco)
    ## Control units:  Andalucia, Aragon, Principado De Asturias,
    ##                 Baleares (Islas), Canarias, Cantabria, Castilla Y Leon,
    ##                 Castilla-La Mancha, Cataluna, Comunidad Valenciana,
    ##                 Extremadura, Galicia, Madrid (Comunidad De),
    ##                 Murcia (Region de), Navarra (Comunidad Foral De),
    ##                 Rioja (La)
    ## Dependent(s):   gdpcap with optimization period from 1960 to 1969
    ## Predictors:
    ##                 school.illit          from 1964 to 1969, aggregated via 'mean',
    ##                 school.prim           from 1964 to 1969, aggregated via 'mean',
    ##                 school.med            from 1964 to 1969, aggregated via 'mean',
    ##                 school.higher         from 1964 to 1969, aggregated via 'mean',
    ##                 invest                from 1964 to 1969, aggregated via 'mean',
    ##                 gdpcap                from 1960 to 1969, aggregated via 'mean',
    ##                 sec.agriculture       from 1961 to 1969, aggregated via 'mean',
    ##                 sec.energy            from 1961 to 1969, aggregated via 'mean',
    ##                 sec.industry          from 1961 to 1969, aggregated via 'mean',
    ##                 sec.construction      from 1961 to 1969, aggregated via 'mean',
    ##                 sec.services.venta    from 1961 to 1969, aggregated via 'mean',
    ##                 sec.services.nonventa from 1961 to 1969, aggregated via 'mean',
    ##                 popdens               from 1969 to 1969, aggregated via 'mean'
    ##
    ##
    ## Results:
    ## --------
    ##
    ## Result type:    Ordinary solution, ie. no perfect preditor fit possible
    ##                 and the predictors impose some restrictions on the outer
    ##                 optimization.
    ## Optimal W:      Baleares (Islas)     : 21.92728%,
    ##                 Cataluna             : 63.27857%,
    ##                 Madrid (Comunidad De): 14.79414%
    ## Dependent loss: MSPE ('loss V'): 0.004286071,
    ##                 RMSPE          : 0.065468095
    ## (Optimal) V:    Single predictor weights V requested. The optimal weight
    ##                 vector V is:
    ##                                                         max.order
    ##                 school.illit.mean.1964.1969          1.578398e-05
    ##                 school.prim.mean.1964.1969           1.578398e-05
    ##                 school.med.mean.1964.1969            1.578398e-05
    ##                 school.higher.mean.1964.1969         2.903475e-04
    ##                 invest.mean.1964.1969                2.990163e-04
    ##                 gdpcap.mean.1960.1969                9.992528e-01
    ##                 sec.agriculture.mean.1961.1969       1.578398e-05
    ##                 sec.energy.mean.1961.1969            1.578398e-05
    ##                 sec.industry.mean.1961.1969          1.578398e-05
    ##                 sec.construction.mean.1961.1969      1.578398e-05
    ##                 sec.services.venta.mean.1961.1969    1.578398e-05
    ##                 sec.services.nonventa.mean.1961.1969 1.578398e-05
    ##                 popdens.mean.1969.1969               1.578398e-05
    ##                 ----------
    ##                 pred. loss                           3.374961e-04
    ##                 (Predictor weights V are standardized by sum(V)=1)
    ##

It is remarkable that the mean of the lagged dependent variable `gdpcap.mean.1960.1969` is by far the most important predictor with a weight of 0.9992528, all other predictors are only marginally relevant due to their tiny (at most 0.0002990163) weights.1

## Removing the Lagged Dependent Variable

Omitting the lagged dependent variable `gdpcap.mean.1960.1969` from the model definition, however, leads to a significant increase of the dependent loss:

    times.pred <- times.pred[,-6]
    agg.fns <- rep("mean", ncol(times.pred))
    res2 <- mscmt(Basque, treatment.identifier, controls.identifier, times.dep, times.pred, agg.fns, seed=1, single.v=TRUE, verbose=FALSE)
    res2

    ## Specification:
    ## --------------
    ##
    ## Model type:     SCM
    ## Treated unit:   Basque Country (Pais Vasco)
    ## Control units:  Andalucia, Aragon, Principado De Asturias,
    ##                 Baleares (Islas), Canarias, Cantabria, Castilla Y Leon,
    ##                 Castilla-La Mancha, Cataluna, Comunidad Valenciana,
    ##                 Extremadura, Galicia, Madrid (Comunidad De),
    ##                 Murcia (Region de), Navarra (Comunidad Foral De),
    ##                 Rioja (La)
    ## Dependent(s):   gdpcap with optimization period from 1960 to 1969
    ## Predictors:
    ##                 school.illit          from 1964 to 1969, aggregated via 'mean',
    ##                 school.prim           from 1964 to 1969, aggregated via 'mean',
    ##                 school.med            from 1964 to 1969, aggregated via 'mean',
    ##                 school.higher         from 1964 to 1969, aggregated via 'mean',
    ##                 invest                from 1964 to 1969, aggregated via 'mean',
    ##                 sec.agriculture       from 1961 to 1969, aggregated via 'mean',
    ##                 sec.energy            from 1961 to 1969, aggregated via 'mean',
    ##                 sec.industry          from 1961 to 1969, aggregated via 'mean',
    ##                 sec.construction      from 1961 to 1969, aggregated via 'mean',
    ##                 sec.services.venta    from 1961 to 1969, aggregated via 'mean',
    ##                 sec.services.nonventa from 1961 to 1969, aggregated via 'mean',
    ##                 popdens               from 1969 to 1969, aggregated via 'mean'
    ##
    ##
    ## Results:
    ## --------
    ##
    ## Result type:    Ordinary solution, ie. no perfect preditor fit possible
    ##                 and the predictors impose some restrictions on the outer
    ##                 optimization.
    ## Optimal W:      Cataluna             : 85.0814%,
    ##                 Madrid (Comunidad De): 14.9186%
    ## Dependent loss: MSPE ('loss V'): 0.008864545,
    ##                 RMSPE          : 0.094151712
    ## (Optimal) V:    Single predictor weights V requested. The optimal weight
    ##                 vector V is:
    ##                                                       max.order
    ##                 school.illit.mean.1964.1969          0.02710923
    ##                 school.prim.mean.1964.1969           0.02710923
    ##                 school.med.mean.1964.1969            0.09108599
    ##                 school.higher.mean.1964.1969         0.23068005
    ##                 invest.mean.1964.1969                0.02710923
    ##                 sec.agriculture.mean.1961.1969       0.02710923
    ##                 sec.energy.mean.1961.1969            0.02710923
    ##                 sec.industry.mean.1961.1969          0.23068005
    ##                 sec.construction.mean.1961.1969      0.02710923
    ##                 sec.services.venta.mean.1961.1969    0.02710923
    ##                 sec.services.nonventa.mean.1961.1969 0.02710923
    ##                 popdens.mean.1969.1969               0.23068005
    ##                 ----------
    ##                 pred. loss                           0.31473799
    ##                 (Predictor weights V are standardized by sum(V)=1)
    ##

The dependent loss (MSPE) increased considerably from 0.0042861 to 0.0088645. Trying to give more meaning to the economic predictors in this way obviously has the drawback of worsening the fit of the dependent variable.

## SCMT without the Lagged Dependent Variable

Leaving the lagged dependent variable `gdpcap.mean.1960.1969` aside, but considering all other predictor variables as time series instead of aggregating their values leads to the following results:

    agg.fns <- rep("id", ncol(times.pred))   # Omitting agg.fns has the same effect (as "id" is the default)
    res3 <- mscmt(Basque, treatment.identifier, controls.identifier, times.dep, times.pred, agg.fns, seed=1, single.v=TRUE, verbose=FALSE)
    res3

    ## Specification:
    ## --------------
    ##
    ## Model type:     SCMT
    ## Treated unit:   Basque Country (Pais Vasco)
    ## Control units:  Andalucia, Aragon, Principado De Asturias,
    ##                 Baleares (Islas), Canarias, Cantabria, Castilla Y Leon,
    ##                 Castilla-La Mancha, Cataluna, Comunidad Valenciana,
    ##                 Extremadura, Galicia, Madrid (Comunidad De),
    ##                 Murcia (Region de), Navarra (Comunidad Foral De),
    ##                 Rioja (La)
    ## Dependent(s):   gdpcap with optimization period from 1960 to 1969
    ## Predictors:     school.illit          from 1964 to 1969,
    ##                 school.prim           from 1964 to 1969,
    ##                 school.med            from 1964 to 1969,
    ##                 school.higher         from 1964 to 1969,
    ##                 invest                from 1964 to 1969,
    ##                 sec.agriculture       from 1961 to 1969,
    ##                 sec.energy            from 1961 to 1969,
    ##                 sec.industry          from 1961 to 1969,
    ##                 sec.construction      from 1961 to 1969,
    ##                 sec.services.venta    from 1961 to 1969,
    ##                 sec.services.nonventa from 1961 to 1969,
    ##                 popdens               from 1969 to 1969
    ##
    ##
    ## Results:
    ## --------
    ##
    ## Result type:    Ordinary solution, ie. no perfect preditor fit possible
    ##                 and the predictors impose some restrictions on the outer
    ##                 optimization.
    ## Optimal W:      Baleares (Islas)            : 30.61618%,
    ##                 Cataluna                    : 25.64227%,
    ##                 Madrid (Comunidad De)       : 31.31920%,
    ##                 Navarra (Comunidad Foral De): 12.42236%
    ## Dependent loss: MSPE ('loss V'): 0.004212379,
    ##                 RMSPE          : 0.064902846
    ## (Optimal) V:    Single predictor weights V requested. The optimal weight
    ##                 vector V is:
    ##                                         max.order
    ##                 school.illit          0.001191182
    ##                 school.prim           0.002720510
    ##                 school.med            0.986287184
    ##                 school.higher         0.000127276
    ##                 invest                0.000127276
    ##                 sec.agriculture       0.008782917
    ##                 sec.energy            0.000127276
    ##                 sec.industry          0.000127276
    ##                 sec.construction      0.000127276
    ##                 sec.services.venta    0.000127276
    ##                 sec.services.nonventa 0.000127276
    ##                 popdens               0.000127276
    ##                 ----------
    ##                 pred. loss            0.013824839
    ##                 (Predictor weights V are standardized by sum(V)=1)
    ##

Notice that this specification’s model type is ‘SCMT’, in contrast to the previous models which were ‘SCM’ models. By using the ‘SCMT’ model, the dependent loss (0.0042124) is even smaller than that of the original model (0.0042861) which used the dependent variable’s mean as an extra economic predictor. `school.med` has now become the most important predictor with weight 0.9862872, all other predictor weights are at least 0.000127276.

## Summary

This vignette illustrated that considering predictors as true time series (without intermediate aggregation) may have various benefits. In this example, by excluding the mean of the lagged dependent variable from the set of economic predictors and considering all other predictors as time series, more meaningful predictor weights could be obtained and the dependent variable’s fit could be slightly improved, too.

## References

Abadie, Alberto, and Javier Gardeazabal. 2003. “The Economic Costs of Conflict: A Case Study of the Basque Country.” The American Economic Review 93 (1): 113–32. http://dx.doi.org/10.1257/000282803321455188.

Klößner, Stefan, and Gregor Pfeifer. 2015. “Synthesizing Cash for Clunkers: Stabilizing the Car Market, Hurting the Environment.” Annual Conference 2015 (Münster): Economic Development - Theory and Policy. Verein für Socialpolitik/German Economic Association. https://ideas.repec.org/p/zbw/vfsc15/113207.html.

1. Notice that the weight vector `v` is obtained by maximizing the order statistics of `v` (while fixing the sum of `v` to 1). This choice of ‘v’ attributes weights as large as possible to even the least relevant predictor(s).↩