# Synthetic Control and Inference

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## Abstract

**:**

## 1. Introduction

## 2. Placebo Test and Synthetic Control

## 3. Monte Carlo

- Vary the values of $\alpha $’s such that (a) none of the components of ${\omega}_{\ast}$ dominates; (b) only two of the elements are non-zero.
- Vary the values of $\gamma $’s such that the unbalanced unobservable factors C(i) (a) disappear; and (b) are present.
- Vary ${T}_{0}$ such that the estimation errors in the weights are (a) prominent; and (b) negligible.

## 4. Possible Alternatives to Placebo Tests

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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1. | Doudchenko and Imbens (2016) also consider a slightly more general requirement ${Y}_{0,t}\approx \alpha +{\sum}_{j=1}^{J}{w}_{j}{Y}_{j,t}$. This is a sensible way to enhance accuracy of synthetic control viewed as a point estimator. It also provides a link to the difference-in-differences estimator. Because our focus is on inferential aspects of the problem, we simplify notation and analysis by abstracting away from the intercept term. |

2. | Using the notation consistent with this paper, Equation (1) in Abadie et al. (2010) takes the form ${Y}_{j,t}\left(0\right)={\lambda}_{t}+{\left({\theta}_{t}-{\lambda}_{t}\right)}^{\prime}{Z}_{j}+{\gamma}_{j}^{\prime}{\delta}_{t}+{\u03f5}_{j,t}$, so the factor structure in Equation (3) of this paper is a special case of Equation (1) in Abadie et al. (2010), where ${Z}_{j}=1$, ${\delta}_{1t}=1$ and ${\gamma}_{1j}={\alpha}_{j}$, i.e., it is a special case where the ${Z}_{j}$ does not exist and the first element of ${\delta}_{t}$ is time invariant. |

3. | Conley and Taber (2011), who proposed a similar test, cite Bertrand et al. (2004) when they discuss placebo tests. Abadie et al. (2010) reference many other papers that precede Bertrand et al. (2004). |

4. | |

5. | We are using the fact that the symmetry implies the equality of marginal distributions, and therefore, the lack of equality of marginal distributions is a sufficient condition for violation of symmetry. |

6. | |

7. | |

8. | It is straightforward to prove that under stationarity assumption, the only model that allows the synthetic controls to trace the trajectory of the outcome for the treated (i.e., ${Y}_{0,t}={\sum}_{j=1}^{J}{\omega}_{j}{Y}_{j,t}$ for some $\omega $) is a linear factor model with $Var\left({\u03f5}_{j,t}\right)=0$. |

9. | See Ferman and Pinto (2017) for related discussion on the bias of the synthetic control estimator. |

10. | We set ${\theta}_{t}\sim N\left(0,1\right)$ in Table 2. We also considered the case where ${\theta}_{t}=0$. Although the results for this case are not reported here in the paper, they were qualitatively similar to the ${\theta}_{t}\sim N\left(0,1\right)$ case. They are available upon request. (When the adding-up constraint was imposed, the two cases gave the same results. Without the adding-up constraint, these two specifications give slightly different results.) |

11. | This can be done by assuming that ${\alpha}_{0}={\sum}_{j=1}^{J}{\omega}_{j}{\alpha}_{j}$ and ${\theta}_{t}=0$. |

12. | The results are available upon request. |

13. | If one were to assume that ${Y}_{0,t}\left(1\right)={Y}_{0,t}\left(0\right)+\beta $, the factor model in Equation (3) becomes
$$\begin{array}{cc}\hfill {Y}_{j,t}& ={\alpha}_{j}+{\theta}_{t}+{\gamma}_{j}^{\prime}{\delta}_{t}+{\u03f5}_{j,t},\phantom{\rule{1.em}{0ex}}t=1,\dots ,{T}_{0},\hfill \\ \hfill {Y}_{j,{T}_{0}+1}& ={\alpha}_{j}+{\theta}_{{T}_{0}+1}+{\gamma}_{j}^{\prime}{\delta}_{{T}_{0}+1}+{\u03f5}_{j,{T}_{0}+1},\phantom{\rule{1.em}{0ex}}j=1,\dots ,J,\hfill \\ \hfill {Y}_{0,{T}_{0}+1}& =\beta +{\alpha}_{0}+{\theta}_{{T}_{0}+1}+{\gamma}_{0}^{\prime}{\delta}_{{T}_{0}+1}+{\u03f5}_{0,{T}_{0}+1}.\hfill \end{array}$$
Using the pre-treatment data, one can consistently estimate $\left({\alpha}_{j},{\gamma}_{j}^{\prime}\right)$ ($j=0,1,\dots ,J$) and $\left({\theta}_{t},{\delta}_{t}^{\prime}\right)$ as long as $J,{T}_{0}\to \infty $. Using the control outcome for the period $t={T}_{0}+1$ along with $\left({\alpha}_{j},{\gamma}_{j}^{\prime}\right)$ consistently estimated, one can consistently estimate $\left({\theta}_{{T}_{0}+1},{\delta}_{{T}_{0}+1}^{\prime}\right)$, which is possible if $J\to \infty $. Combining $\left({\alpha}_{0},{\gamma}_{0}^{\prime}\right)$ as well as $\left({\theta}_{{T}_{0}+1},{\delta}_{{T}_{0}+1}^{\prime}\right)$, one can make an inference of $\beta $. |

14. | Indeed, Abadie et al. (2010) (Section 2.2) consider some other model (in addition to the factor model) for motivation of the synthetic control. |

DGP No. | $\mathit{\alpha}$’s | $\mathit{\gamma}$’s | Variations |
---|---|---|---|

1 | ${\alpha}_{0}=(2J+1)/3,{\alpha}_{1}=1,\cdots ,{\alpha}_{J}=J$ | ${\gamma}_{0}={\gamma}_{1}=\cdots ={\gamma}_{J}=0$ | 1(a), 2(a) |

2 | ${\alpha}_{0}=(2J+1)/3,{\alpha}_{1}=1,\cdots ,{\alpha}_{J}=J$ | ${\gamma}_{0}=2,{\gamma}_{1}=\cdots ={\gamma}_{J}=1$ | 1(a), 2(b) |

3 | ${\alpha}_{0}=5/3,{\alpha}_{1}=1,{\alpha}_{2}=2,{\alpha}_{3}=\cdots ={\alpha}_{J}=0$ | ${\gamma}_{0}={\gamma}_{1}=\cdots ={\gamma}_{J}=0$ | 1(b), 2(a) |

4 | ${\alpha}_{0}=5/3,{\alpha}_{1}=1,{\alpha}_{2}=2,{\alpha}_{3}=\cdots ={\alpha}_{J}=0$ | ${\gamma}_{0}=2,{\gamma}_{1}=\cdots ={\gamma}_{J}=1$ | 1(b), 2(b) |

DGP | J | ${\mathit{\sigma}}_{\mathit{\delta}}^{2}$ | Estimated Weights | True Weights | ||||||
---|---|---|---|---|---|---|---|---|---|---|

${\mathit{T}}_{0}=40$ | ${\mathit{T}}_{0}=80$ | ${\mathit{T}}_{0}=400$ | ${\mathit{T}}_{0}=800$ | ${\mathit{T}}_{0}=40$ | ${\mathit{T}}_{0}=80$ | ${\mathit{T}}_{0}=400$ | ${\mathit{T}}_{0}=800$ | |||

1 | 0.1 | 0.076 | 0.067 | 0.082 | 0.090 | 0.074 | 0.070 | 0.079 | 0.091 | |

$J=20$ | 1 | 0.076 | 0.067 | 0.082 | 0.090 | 0.074 | 0.070 | 0.079 | 0.091 | |

10 | 0.076 | 0.067 | 0.082 | 0.090 | 0.074 | 0.070 | 0.079 | 0.091 | ||

0.1 | 0.079 | 0.080 | 0.080 | 0.093 | 0.079 | 0.090 | 0.084 | 0.096 | ||

$J=40$ | 1 | 0.079 | 0.080 | 0.080 | 0.093 | 0.079 | 0.090 | 0.084 | 0.096 | |

10 | 0.079 | 0.080 | 0.080 | 0.093 | 0.079 | 0.090 | 0.084 | 0.096 | ||

0.1 | 0.100 | 0.087 | 0.113 | 0.083 | 0.094 | 0.087 | 0.114 | 0.085 | ||

$J=80$ | 1 | 0.100 | 0.087 | 0.113 | 0.083 | 0.094 | 0.087 | 0.114 | 0.085 | |

10 | 0.100 | 0.087 | 0.113 | 0.083 | 0.094 | 0.087 | 0.114 | 0.085 | ||

2 | 0.1 | 0.193 | 0.178 | 0.175 | 0.190 | 0.174 | 0.176 | 0.174 | 0.190 | |

$J=20$ | 1 | 0.568 | 0.530 | 0.559 | 0.529 | 0.557 | 0.518 | 0.561 | 0.531 | |

10 | 0.838 | 0.843 | 0.837 | 0.837 | 0.839 | 0.828 | 0.838 | 0.838 | ||

0.1 | 0.193 | 0.219 | 0.198 | 0.208 | 0.200 | 0.213 | 0.193 | 0.207 | ||

$J=40$ | 1 | 0.561 | 0.623 | 0.586 | 0.589 | 0.556 | 0.593 | 0.588 | 0.583 | |

10 | 0.809 | 0.853 | 0.863 | 0.844 | 0.832 | 0.858 | 0.854 | 0.846 | ||

0.1 | 0.238 | 0.253 | 0.245 | 0.230 | 0.239 | 0.245 | 0.241 | 0.233 | ||

$J=80$ | 1 | 0.617 | 0.623 | 0.631 | 0.619 | 0.610 | 0.629 | 0.624 | 0.622 | |

10 | 0.865 | 0.870 | 0.870 | 0.866 | 0.861 | 0.894 | 0.867 | 0.868 | ||

3 | 0.1 | 0.115 | 0.123 | 0.110 | 0.125 | 0.114 | 0.113 | 0.109 | 0.123 | |

$J=20$ | 1 | 0.115 | 0.123 | 0.110 | 0.125 | 0.114 | 0.113 | 0.109 | 0.123 | |

10 | 0.115 | 0.123 | 0.110 | 0.125 | 0.114 | 0.113 | 0.109 | 0.123 | ||

0.1 | 0.152 | 0.151 | 0.149 | 0.153 | 0.153 | 0.154 | 0.146 | 0.154 | ||

$J=40$ | 1 | 0.152 | 0.151 | 0.149 | 0.153 | 0.153 | 0.154 | 0.146 | 0.154 | |

10 | 0.152 | 0.151 | 0.149 | 0.153 | 0.153 | 0.154 | 0.146 | 0.154 | ||

0.1 | 0.180 | 0.168 | 0.173 | 0.153 | 0.178 | 0.164 | 0.166 | 0.153 | ||

$J=80$ | 1 | 0.180 | 0.168 | 0.173 | 0.153 | 0.178 | 0.164 | 0.166 | 0.153 | |

10 | 0.180 | 0.168 | 0.173 | 0.153 | 0.178 | 0.164 | 0.166 | 0.153 | ||

4 | 0.1 | 0.201 | 0.219 | 0.193 | 0.214 | 0.207 | 0.213 | 0.195 | 0.209 | |

$J=20$ | 1 | 0.536 | 0.518 | 0.525 | 0.522 | 0.533 | 0.520 | 0.525 | 0.520 | |

10 | 0.837 | 0.827 | 0.817 | 0.814 | 0.841 | 0.809 | 0.824 | 0.812 | ||

0.1 | 0.238 | 0.253 | 0.249 | 0.253 | 0.241 | 0.268 | 0.239 | 0.249 | ||

$J=40$ | 1 | 0.549 | 0.597 | 0.583 | 0.579 | 0.552 | 0.576 | 0.583 | 0.582 | |

10 | 0.809 | 0.842 | 0.856 | 0.852 | 0.828 | 0.843 | 0.859 | 0.848 | ||

0.1 | 0.287 | 0.293 | 0.300 | 0.275 | 0.279 | 0.301 | 0.297 | 0.278 | ||

$J=80$ | 1 | 0.610 | 0.623 | 0.655 | 0.629 | 0.608 | 0.625 | 0.653 | 0.634 | |

10 | 0.861 | 0.871 | 0.874 | 0.860 | 0.866 | 0.880 | 0.865 | 0.866 |

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**MDPI and ACS Style**

Hahn, J.; Shi, R.
Synthetic Control and Inference. *Econometrics* **2017**, *5*, 52.
https://doi.org/10.3390/econometrics5040052

**AMA Style**

Hahn J, Shi R.
Synthetic Control and Inference. *Econometrics*. 2017; 5(4):52.
https://doi.org/10.3390/econometrics5040052

**Chicago/Turabian Style**

Hahn, Jinyong, and Ruoyao Shi.
2017. "Synthetic Control and Inference" *Econometrics* 5, no. 4: 52.
https://doi.org/10.3390/econometrics5040052