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Degrees of Freedom and Information Criteria for the Synthetic Control Method

Guillaume Allaire Pouliot, Zhen Xie, Ziyi Liu

Abstract

We provide an analytical characterization of the model flexibility of the synthetic control method (SCM) in the familiar form of degrees of freedom. We obtain estimable information criteria, which may be used to circumvent cross-validation when selecting either the tuning parameter in penalized variants of SCM or the weighting matrix in the SCM with covariates. We assess the impact of car license rationing in Tianjin; while a natural match is available, both it and other donors are noisy, inviting the use of SCM to average over approximately matching donors. The very large number of candidate donors calls for penalized variants of SCM and we observe that model selection using information criteria outperforms that based on cross-validation.

Degrees of Freedom and Information Criteria for the Synthetic Control Method

Abstract

We provide an analytical characterization of the model flexibility of the synthetic control method (SCM) in the familiar form of degrees of freedom. We obtain estimable information criteria, which may be used to circumvent cross-validation when selecting either the tuning parameter in penalized variants of SCM or the weighting matrix in the SCM with covariates. We assess the impact of car license rationing in Tianjin; while a natural match is available, both it and other donors are noisy, inviting the use of SCM to average over approximately matching donors. The very large number of candidate donors calls for penalized variants of SCM and we observe that model selection using information criteria outperforms that based on cross-validation.
Paper Structure (46 sections, 24 theorems, 107 equations, 13 figures, 2 tables)

This paper contains 46 sections, 24 theorems, 107 equations, 13 figures, 2 tables.

Table of Contents

  1. Introduction
  2. The Synthetic Control Method
  3. The Synthetic Control Method Without Covariates
  4. The Synthetic Control Method With Covariates
  5. The Penalized Synthetic Control Method
  6. Constrained Ridge SCM
  7. Elastic Net SCM
  8. Information Criteria and Degrees of Freedom
  9. The Degrees of Freedom of the Synthetic Control Methods
  10. Information Criteria for the Synthetic Control Methods
  11. Heteroskedasticity-Robust Information Criteria
  12. Discussion of Gaussian Assumption
  13. Forecasting Counterfactual Car Sales Under Rationing in Tianjin
  14. Empirical Approach
  15. Simulation
  16. ...and 31 more sections

Key Result

Proposition 3.1

Let $\tilde{\mathbf{X}}=(\mathbf{X}^\top,\mathbf{1}_p)^\top\in\mathbb{R}^{(n+1)\times p}$, an augmented donor matrix. Suppose that $\mathbf{Y}|\mathbf{X}$ follows the probability law stipulated in (GaussianAssumption) and (eq:Ahomoskedastic). Then, where $\mathcal{A}=\mathcal{A}\left(\mathbf{Y}\right):=\{ j : \hat{\beta}_{\mathrm{sc},j}(\mathbf{Y}) > 0 \}$ is the active support corresponding to a

Figures (13)

  • Figure 1: Synthetic control output of California Proposition 99 investigation. The degrees of freedom estimate is 5. The left-hand side and right-hand side panels are, respectively, our replications of Table 2 and Figure 2 of abadie2010synthetic.
  • Figure 2: Degrees of freedom of the synthetic control method without covariates (blue) and of the lasso (green) on the left-hand side, and of the best subset selection regression on the right-hand side.
  • Figure 3: Assessment of robustness to heteroskedasticity
  • Figure 4: Prediction MSE and SURE Estimates for Different $\lambda$ Values in PSCM (Shijiazhuang)
  • Figure 5: Assessment of robustness to Gaussian assumption
  • ...and 8 more figures

Theorems & Definitions (44)

  • Proposition 3.1: Degrees of Freedom of SCM Without Covariates
  • Corollary 3.1
  • Proposition 3.2: Degrees of Freedom of Penalized Synthetic Controls
  • Proposition 3.3: Degrees of Freedom of Constrained Ridge SCM
  • Remark 3.1
  • Remark 3.2
  • Proposition 3.4: Degrees of Freedom of Elastic Net SCM, tibshirani2012degrees
  • Remark 3.3
  • Proposition 3.5: Degrees of Freedom of SCM With Covariates
  • Corollary 3.2
  • ...and 34 more