synthReturn

The synthReturn R package implements the synthetic matching method originally suggested by Acemoglu et al. (2016) and modified by Kreitmeir et al. (2023) to estimate the average treatment effect $\widehat{\phi}$ of an event on the stock return of all treated firms. The new version of the synthetic matching method by Kreitmeir et al. (2023) accommodates (i) multiple event dates and (ii) deals with missing values directly instead of relying on the assumption that they are equal to 0.

For more details on the empirical framework, please see the Empirical Framework Section below.

If you end up using this package, please cite our paper:

• Kreitmeir, D., Lane, N., and Raschky, P. A. (2023). The value of names - Civil society, information, and governing multinationals. SSRN WP 3751162, https://ssrn.com/abstract=3751162

Installation

To install the most recent version of the synthReturn package from GitHub:

# install.packages("devtools")
devtools::install_github("davidkreitmeir/synthReturn")

Short examples

The following is an illustration of the method for a simulated dataset with (i) two event-dates and (ii) no missing values. For details on the generation of the simulated dataset(s), please see data-raw/sim_ret.R.

Let's first get the data ready:

library(synthReturn)
# Load data in that comes in the synthReturn package
data(ret_two_evdates)

Now, to estimate the average treatment effect $\widehat{\phi}$ with the synthetic matching method, we can use the synthReturn function:

results <- synthReturn(
  data = ret_two_evdates,
    tidname = "treatid",
    cidname = "controlid",
    rname = "ret",
    dname = "date",
    edname = "eventdate",
    estwind = c(-100,-1),
    eventwind = c(0,5),
    estobs_min = 1,
    eventobs_min = 1,
    placebo = TRUE,
    ngroup = 2,
    ndraws = 10
)

results$ate

For the case that returns are missing for firms either in the treatment or control group, you can set a threshold for the minimum of non-missing trading days during both the estimation (estobs_min) and event window (eventobs_min). In this example, I require each firm to have non-missing returns for at least 90% of trading days during both, the estimation and event window (Note that the default is no missing returns, i.e. 100%).

# Load data in that comes in the synthReturn package
data(ret_two_evdates_na) # 5% of all returns missing

results <- synthReturn(
  data = ret_two_evdates,
    tidname = "treatid",
    cidname = "controlid",
    rname = "ret",
    dname = "date",
    edname = "eventdate",
    estwind = c(-100,-1),
    eventwind = c(0,5),
    estobs_min = 0.9,
    eventobs_min = 0.9,
    placebo = TRUE,
    ngroup = 2,
    ndraws = 10
)

results$ate

Empirical Framework

A synthetic match for each company $i$ in the treatment group is found by solving the following optimization problem:

$$\underset{\{ w_{j}^{i}\}_{j \in \text{Control group}}}{\arg\min} \underset{t \in \text{Estimation Window}}{\sum \left[ R_{it} - \underset{j \in \text{Control group}}{\sum w_{j}^{i}R_{jt}} \right]^{2}} \quad \text{s.t.} \quad \underset{j \in \text{Control group}}{ \sum w_{j}^{i} = 1} \quad \text{and} \quad w_{j}^{i} \geq 0$$

where $R_{it}$ and $R_{jt}$ is the daily return on date $t$ for the treatment, respectively control company and ${w_{j}^{i*}}$ is the weight for control firm $j$ in the optimal weighting for firm $i$.

The optimization problem above boils down to a quadratic programming problem, as the objective function is quadratic and the two constraints are linear. I.e. the problem can be written as:

$$\underset{\mathbf{w} \in \mathbb{R}^{J}}{\arg\min} f \left( \mathbf{w} \right) = \frac{1}{2} \mathbf{w}^\intercal \mathbf{D} \mathbf{w} - \mathbf{w}^\intercal \mathbf{b} \quad \text{s.t.} \quad \mathbf{A}_{1}\mathbf{w} = 1 \quad \text{and} \quad \mathbf{A}_{2}\mathbf{w} \leq \mathbf{0}$$

where $\mathbf{w} \in \mathbb{R}^{J}$ is a vector containing the optimal weights for each of the $j = 1,...,J$ companies; $\mathbf{D} = \in \mathbb{R}^{J \times J}$ is symmetric and equal to $\mathbf{R}^\intercal \times \mathbf{R}$ with matrix $\mathbf{R} \in \mathbb{R}^{T \times J}$ containing the returns during the estimation window of length $T$ for all control companies $J$; $b \in \mathbb{R}^{J}$ and equal to $\mathbf{R}^\intercal \times \mathbf{r}$ with $\mathbf{r} \in \mathbb{R}^{T}$ comprising the returns of the treated firm over the estimation window; $A_{1} \in \mathbb{R}^{T \times J}$ and $A_{2} \in \mathbb{R}^{J \times J}$ are identity matrices and $\mathbf{0} \in \mathbb{R}^{J}$ a vector of zeros.

Reformulating the optimization problem allows the use the dual method of Goldfarb and Idnani (1982, 1983) for solving quadratic programming problem implemented in quadprog::solve.QP.

After finding the optimal weights $w_{j}^{i*}$, the abnormal return of the treated firm $i$ is given by the difference between its return $R_{it}$ and the return for the synthetic firm $w_{j}^{i*}R_{jt}$:

$$\widehat{AR}_{it} = R_{it} - \underset{j \in \text{Control group}}{\sum w_{j}^{i*}R_{jt}}$$

The cumulative abnormal return for the period $0$ to $k$ is adjusted for the "goodness" of the synthetic match for all firms in the treatment group:

$$\widehat{\phi}\left(0,k\right) = \frac{\underset{i \in \text{Treatment group}}{\sum \frac{\sum_{\tau_{1}=0}^{\tau_{2}} \widehat{AR}_{it}}{\widehat{\sigma_{i}}}}}{\underset{i \in \text{Treatment group}}{\sum \frac{1}{\widehat{\sigma_{i}}}}} \quad \text{where} \quad \widehat{\sigma_{i}} = \sqrt{\frac{\underset{t \in \text{Estimation Window}}{\sum \left(\widehat{AR}_{it}\right)^{2}}}{T}},$$

where $\widehat{\phi}\left(0,k\right)$ is the cumulative effect over the period $\tau_{1}=0$ to $\tau_{2}$ in the event window. The treatment effect is, hence, a weighted average of each event-firm specific effect, with greater weight given to the estimated effects for which the synthetic firm tracks the return of the treated company more closely during the estimation window.

To draw inference, confidence intervals are constructed by randomly drawing placebo treatment groups, as suggested by Acemoglu et al. (2016). To accommodate multiple event dates synthRetrun draws $K \times E$ placebo treatment groups of size $N$, where $K$ is the number of random draws at each event date $e$, with the number of (unique) event dates equaling $E$. The cumulative abnormal return effect is significant at the 10%, 5%, or 1% level if the actual estimated treatment effect $\widehat{\phi}$ lies outside of the interval that contains the $\left[5,95\right]$, $\left[2.5,97.5\right]$, or $\left[0.5,99.5\right]$ percentiles of the placebo treatment effects $\widehat{\phi}_{\text{placebo}}$.