---
title: "SCM Using Time Series"
vignette: >
  %\VignetteIndexEntry{SCM Using Time Series}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
output: 
  html_vignette:
    toc: true
bibliography: ../inst/REFERENCES.bib
---

```{r, echo = FALSE}
knitr::opts_chunk$set(
  fig.width  = 7,
  fig.height = 4,
  fig.align  = "center",
#  cache      = TRUE,
  autodep    = TRUE
)
```

## Introduction

This vignette illustrates the syntax of SCM**T** models. For a more general introduction to package `MSCMT` see its [main vignette](WorkingWithMSCMT.html).

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 SCM**T**, see @KP16.

This vignette illustrates the syntax of SCM**T** models and how SCM**T** 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](WorkingWithMSCMT.html) of this package:

```{r}
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](WorkingWithMSCMT.html) which reproduces the model in @Abadie2003:

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

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 `r res$v[6,"max.order"]`, 
all other predictors are only marginally relevant due to their tiny 
(at most `r format(max(res$v[-6,"max.order"]),scientific=FALSE)`) 
weights.^[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).]

## 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:
```{r}
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
```

The dependent loss (MSPE) increased considerably from `r res$loss.v` to `r res2$loss.v`. 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:
```{r}
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
```

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 (`r res3$loss.v`) is even smaller than that of the original model (`r res$loss.v`) which used the dependent variable's mean as an extra economic predictor. 
``r rownames(res3$v)[which.max(res3$v[,"max.order"])]`` has now become the most important predictor with weight `r max(res3$v[,"max.order"])`, 
all other predictor weights are at least `r format(min(res3$v[,"max.order"]),scientific=FALSE)`.

## 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