Step 1. Matching Pre-Treatment Time Series

The DSC algorithm first estimates the speed relationship between the pre-treatment segments of the target time series $\mathbf {y}_{1}$ and the donor time series $\mathbf {y}_{j}$ . This is achieved by employing the DTW method to align the pre-treatment portion of $\mathbf {y}_{1}$ with $\mathbf {y}_{j}$ and storing the warping path matrix.

The warping path matrix stores the results of DTW alignments and is formally defined as follows:

$$ \begin{align*} \mathbf{P}_{j} = [p_{j,\upsilon,t}], \quad \upsilon,t\in [1,N],\\ p_{j,\upsilon,t} = \begin{cases} 1, \hfill & \text{ if } y_{j,\upsilon} \text{ matches } y_{1,t},\\ 0, \hfill & \text{otherwise}. \end{cases} \end{align*} $$

As illustrated in Figure 3, the points of $\mathbf {y}_{1}$ are matched to the points of $\mathbf {y}_{j}$ by DTW in a manner that minimizes the total distance between the matched points.Footnote ¹⁰ The warping path matrix $\mathbf {P}_{j}$ captures the speed difference between $\mathbf {y}_{1}$ and $\mathbf {y}_{j}$ .

Figure 3 Warping path. The left figure shows data points matched in DTW, connected by dashed lines. The right figure displays the corresponding warping path matrix, where only matched pairs (ones) are shown. The time series $\mathbf {y}_{j}$ initially progresses at a rate $2\times $ slower (indicated in red) than $\mathbf {y}_{1}$ but later becomes $2\times $ faster (in blue) than $\mathbf {y}_{1}$ .

We first match the pre-treatment target time series $\mathbf {y}_{1,1:T}$ to the donor time series $\mathbf {y}_{j}$ (see the first part of Figure 2). The point $y_{1,T}$ is matched to $y_{j,C}$ .Footnote ¹¹ Let $\mathbf {y}_{j,pre} = \mathbf {y}_{j,1:C}$ denote the pre-treatment part of $\mathbf {y}_{j}$ and $\mathbf {y}_{j,post} = \mathbf {y}_{j,C:N}$ the post-treatment portion. Similarly, we define $\mathbf {y}_{1,pre}=\mathbf {y}_{1,1:T}$ and $\mathbf {y}_{1,post}=\mathbf {y}_{1,T:N}$ . Using DTW, we obtain a warping path from $\mathbf {y}_{j,pre} \to \mathbf {y}_{1,pre}$ , which is stored in a $C\times T$ matrix $\mathbf {P}_{j,pre}$ .

The warping path $\mathbf {P}_{j,pre}$ is used in step three to warp the time axis of $\mathbf {y}_{j,pre}$ with the goal of minimizing the speed difference between $\mathbf {y}_{j,pre}$ and $\mathbf {y}_{1,pre}$ . Additionally, it is employed along with the warping path derived from step two to adjust the speed of $\mathbf {y}_{j,post}$ .

Step 2. Matching Pre- and Post-Treatment Donor Time Series

Learning the warping path between the post-treatment segments $\mathbf {y}_{j,post}$ and $\mathbf {y}_{1,post}$ is more challenging. We cannot directly align the two sequences, because their differences are due not only to their different speeds but also to the treatment effect. Aligning them would artificially remove that treatment effect. So the post-treatment warping path $\mathbf {P}_{j,post}$ , that is, the inherent differences in speed, must be learned from the pre-treatment path $\mathbf {P}_{j,pre}$ .

To extract similar short-term sequences in $\mathbf {y}_{j,post}$ and $\mathbf {y}_{j,pre}$ and therefore infer the warping path $\mathbf {P}_{j,post}$ from $\mathbf {P}_{j,pre}$ , we use a double-sliding window approach.Footnote ¹² In essence, the double-sliding window serves as a dynamic “lens”—the first window scans through the post-treatment time series to identify segments requiring alignment, while the second window sifts through the pre-treatment time series to find the most similar segments, which facilitates optimal matching and warping of patterns between the two time series. As illustrated in the second part of Figure 2, we slide a target window $Q_{u}$ in $\mathbf {y}_{j,post}$ and a reference window $R_{i}$ in $\mathbf {y}_{j,pre}$ . For each short-term sequence $Q_{u}$ in $\mathbf {y}_{j,post}$ , we find the best-matching sequence $R^{*}$ from $\mathbf {y}_{j,pre}$ , and record the warping path $Q_{u}\to R^{*}$ as $\mathbf {P}_{j,Q_{u}\to R^{*}}$ . We also extract the corresponding pre-treatment warping path $\mathbf {P}_{j,R^{*}}:R^{*}\to \mathbf {y}_{1}$ from $\mathbf {P}_{j,pre}$ . Next, $\mathbf {P}_{j,R^{*}}$ is adjusted based on $\mathbf {P}_{j,Q_{u}\to R^{*}}$ to account for the differences between $Q_{u}$ and $R^{*}$ . The resulting adjusted warping path is $\mathbf {P}_{j,Q_{u}}$ . Once the sliding target window $Q_{u}$ reaches the right boundary $y_{j,N}$ , we merge all the resulting warping paths $\mathbf {P}_{j,Q_{u}}$ to obtain the final warping path $\mathbf {P}_{j,post}$ for the post treatment donor time series.Footnote ¹³

Step 3. Warping Donor Time Series

In the third step, the DSC algorithm uses the previously estimated warping path $\mathbf {P}_{j,pre}$ to warp the pre-treatment donor series $\mathbf {y}_{j,pre}$ . Similarly, it uses $\mathbf {P}_{j,post}$ from the second step to warp $\mathbf {y}_{j,post}$ .Footnote ¹⁴ The resulting time series are combined into a single warped donor time series $\mathbf {y}_{j}^{w}$ :

$$ \begin{align*} \mathbf{y}_{j}^{w} &= [\mathbf{y}^{w}_{j,pre}, \mathbf{y}^{w}_{j,post}]\nonumber \\ &=[\mathbf{P}_{j,pre}(\mathbf{y}_{j,pre}), \mathbf{P}_{j,post}(\mathbf{y}_{j,post})]. \end{align*} $$

After being warped, the inherent speed differences between $\mathbf {y}_{j}^{w}$ and $\mathbf {y}_{1}$ are minimized, while the differences caused by the intervention are unchanged. Finally, using the warped donor time series , we apply the SC method to construct a counterfactual of $\mathbf {y}_{1}$ to minimize the impact of the speed problem.Footnote ¹⁵

All methods discussed in this paper have been implemented in an accompanying R package.Footnote ¹⁶

4.1 Monte Carlo simulation

We first use a Monte Carlo simulation to replicate the types of empirical challenges faced with real data. We generate hundreds of artificial panel datasets, each containing time series for ten units observed over 100 periods. In each sample of 10 series, one time series is designated as the “treated” unit, while the remaining nine serve as the donor pool to construct the SC (i.e., $J=9$ , $N=100$ ).Footnote ¹⁷

All 10 time series follow a common autoregressive integrated moving average (ARIMA) process but exhibit differing speeds. A time series’ speed is either random or a function of the time series direction (increasing or decreasing). The idea behind it is to capture the possibility that speed may vary as a function of the direction of the underlying series. Economic crashes (i.e., a decreasing series), for example, may unfold faster than recoveries (Reinhart and Rogoff Reference Reinhart and Rogoff2014). A dataset-specific parameter $\psi \in (0,1)$ determines the extent to which this occurs. For instance, $\psi =0$ implies that the speed will be entirely governed by a random normal noise. Conversely, if $\psi =1$ , the speed will completely depend on the direction of the time series.

For each dataset, we implement DSC and standard SC to construct counterfactuals for the treated unit. A key advantage of the Monte Carlo simulation, as opposed to real observational data, is that we know the true treatment effect and thus the true counterfactual. Consequently, we can assess how well counterfactuals formed using different methods approximate the post-treatment series. In accordance with common practice in the literature, we evaluate the performance of each method over the 10 periods following the treatment, that is, from $t=61$ to $t=70$ .

To evaluate the quality of the SC generated by each method, we compute the 10-period post-treatment mean squared error (MSE) for the estimated treatment effects. Specifically, the MSE for dataset d is defined as: $\textrm {MSE}_{d}=\frac {1}{10}\sum _{t=61}^{70}(\widehat {\tau }_{t,d}-\tau _{t})^{2}$ , where $\widehat {\tau }_{t,d}$ denotes the estimated treatment effects using one of the two SC methods. A small MSE indicates that the estimated treatment effect closely fits the true effect imposed during the data generation process. For each simulation, we calculate the MSE for the standard SC approach— $\textrm {MSE}_{SC}$ —and the MSE for our DSC approach— $\textrm {MSE}_{DSC}$ . The log ratio of these two values yields a measure r representing the relative performance of each method, that is, $r = \log \left (\textrm {MSE}_{DSC}/\textrm {MSE}_{SC}\right )$ . Negative log ratios suggest superior performance of our method via lower MSE. We compute the log ratio r for each simulated dataset and conduct a t-test on the resulting $100$ log ratios to assess whether the ratio is significantly different from 0 ( $p<0.01$ ).

The estimated treatment effects for all datasets are illustrated in Figure 4. We see that, in the 10 periods following the treatment, the average estimated treatment effects from the two methods are close to the true effects. However, the new method, DSC, produces a significantly smaller error area than the standard method. Specifically, the average logged MSE for the standard SC method ( $\overline {log(\textrm {MSE}_{SC})}$ ) is $0.90$ , while the value for the DSC method ( $\overline {log(\textrm {MSE}_{DSC})}$ ) is $-0.50$ .Footnote ¹⁸ Moreover, our approach outperforms the standard method 77% of the time (Wilcox test $< 0.001$ ). These results suggest a strong expected benefit to using this method and are in line with our theory: Section 2 shows that the estimated treatment effect from SC is potentially biased and has a larger variance. In the Monte Carlo study, we do not directly observe bias because the biases around the true values tend to average out to zero. But it is clear in the Monte Carlo results that the SC estimator has a higher variance, and hence that the DSC estimator of the treatment effect is more efficient.

Figure 4 Results from the simulation study with $95\%$ confidence intervals. The main graph showcases results drawn from Monte Carlo simulations where $\psi = 1$ . The gray-shaded region delineates the period over which performance is estimated. Red and blue lines represent the distribution of estimated treatment effects for the dynamic synthetic control and synthetic control methods, respectively. The true treatment effect is in black. An inset in the top-left corner demonstrates that larger $\psi $ values lead to improved performance—as evidenced by more negative t-values.

Finally, we also show that the results hold for all values of $\psi $ —the parameter which controls the extent to which speed varies as a function of the shape of the series (see the subfigure of Figure 4), although larger $\psi $ s lead to more significance (more negative t value). When $\psi =1$ , the average log ratio of MSEs is $\bar {r} = -1.39$ . This value indicates that, on average, the DSC method reduces the MSE of the estimated treatment effect by $75.19\%$ when compared to the standard SC method ( $1-e^{-1.39} \approx 75.19\%$ ).

4.2 Re-Evaluating Empirical Findings

We now show that these results are not limited to artificial data but extend to real-world empirical data. Specifically, we apply our method to three seminal articles in the SC literature: Abadie and Gardeazabal’s (Reference Abadie and Gardeazabal2003) analysis of the economic costs of terrorism in the Basque country; Abadie et al.’s (Reference Abadie, Diamond and Hainmueller2010) assessment of the effect of Proposition 99—a large-scale tobacco control program implemented in California in 1988; and Abadie et al.’s (Reference Abadie, Diamond and Hainmueller2015) evaluation of the economic impact of the 1990 German reunification on West Germany.Footnote ¹⁹

One challenge, of course, is that we do not know the true treatment effect for the main unit of interest. This effect must be estimated and, unlike in the Monte Carlo study, there is no way to argue that a particular result is more accurate or less biased than another. However, we do know that there is no treatment effect in the other (non-treated) units. Thus, we can assess the performance of the estimators by comparing how well their respective synthetic controls approximate the true post-treatment period for non-treated units. For example, we cannot know the true effect of the German reunification on West Germany’s GDP, but we do know that there was no treatment in Canada, and can thus attempt to match Canada’s post-1990 trajectory—as well as each of the untreated units.Footnote ²⁰ Due to the limited set of untreated units, however, this is not enough to yield a sufficiently large set of estimates for comparison. We therefore use jacknife resampling to generate more datasets in which we randomly remove one of the countries.

We now review each of the three datasets and the results obtained using each method.

4.2.1 Terrorism and GDP per Capita in the Basque Country

Abadie and Gardeazabal (Reference Abadie and Gardeazabal2003) find that the outbreak of terrorism in the late 1960s significantly affected per capita GDP in the Basque Country. When compared to an SC region without terrorism, the Basque Country’s GDP declined by about 10 percentage points.

Our estimate of the effect on the Basque country is similar to the standard SC estimate. However, when we build SCs for the untreated units themselves, we find that they are closer to the true trajectory than is the standard SC. Figure 5 (top) displays the distribution of our estimated treatment effects for all untreated units. Since these units did not receive a treatment, our SC should ideally be as close as possible to the post-treatment values of the time series. In other words, the average difference between the post-treatment series and the SC should deviate as little as possible from zero. Visually, we observe that the band for our approach is narrower than the one for standard SC.

Figure 5 Placebo tests, real data. We revisited the placebo tests reported in Abadie and Gardeazabal (Reference Abadie and Gardeazabal2003), Abadie et al. (Reference Abadie, Diamond and Hainmueller2010), and Abadie et al. (Reference Abadie, Diamond and Hainmueller2015). The plots report the placebo tests for each of these studies, using standard synthetic controls (SCs) (blue) and dynamic synthetic control (red). In addition, the estimated treatment effects for the treated units—Basque Country, California State, and West Germany—are shown as thick, brighter lines. For each study, find that our placebo estimates exhibit smaller variance than those using standard SCs, which do not account for variations in speed.

However, this plot does not capture the full extent of the true improvement, as we should be comparing the pairwise performance of each algorithm (instead of the pooled comparison shown here). To more formally demonstrate the improvement, we calculate the MSE for each SC compared to the unit of interest. We then calculate a t-test of $\log (\textrm {MSE}_{DSC}/\textrm {MSE}_{SC})$ . A negative value indicates that our MSE is smaller than the one obtained with the standard approach. We find that this is indeed the case. The log ratio of the MSEs is significantly less than zero ( $t=-7.91, p<0.0001$ ).Footnote ²¹

In terms of our ability to make inferences, the observed reduction in MSE when employing the DSC method suggests its superior efficiency compared to traditional SC methods. In short, our DSC approach more closely approximates the true treatment effects and therefore enhances the robustness and reliability of our causal inference.