Bayesian DiD

DiD

• 이중차분법(Difference-in-Differences; DiD): 평행 추세 가정 하에 정책 시행에 따른 인과적 효과를 반사실과의 추세 차이로써 추정하는 준실험설계법(Quasi-Experimental)

• 평행 추세 가정(Parallel Trends Assumption): 관측 불가능한 반사실(counterfactual), 즉 실험군이 실제로 정책이 적용되지 않은 상태를 대리하는 논리적 장치로서, 정책이 없었더라면 실험군의 추세는 통제군과 동일했을 것이라는 가정

  \[\begin{aligned} \underbrace{\mathbb{E}\left[Y_{i,t}(0)-Y_{i,s}(0) \mid D_{i}=1\right]}_{\text{experimental group}} &\approx \underbrace{\mathbb{E}\left[Y_{i,t}(0)-Y_{i,s}(0) \mid D_{i}=0\right]}_{\text{control group}}\\ \underbrace{\mathbb{E}\left[Y_{i,t}(0) \mid D_{i}=1\right]}_{\text{counterfactual}} - \underbrace{\mathbb{E}\left[Y_{i,s}(0) \mid D_{i}=1\right]}_{\text{factual}} &\approx \underbrace{\mathbb{E}\left[Y_{i,t}(0) \mid D_{i}=0\right] - \mathbb{E}\left[Y_{i,s}(0) \mid D_{i}=0\right]}_{\text{control group}}\\ \therefore \underbrace{\mathbb{E}\left[Y_{i,t}(0) \mid D_{i}=1\right]}_{\text{counterfactual}} &\approx \underbrace{\mathbb{E}\left[Y_{i,s}(0) \mid D_{i}=1\right]}_{\text{factual}}\\ &\quad + \underbrace{\mathbb{E}\left[Y_{i,t}(0) \mid D_{i}=0\right] - \mathbb{E}\left[Y_{i,s}(0) \mid D_{i}=0\right]}_{\text{control group}} \end{aligned}\]

• 정책의 인과적 효과(Average Treatment effect on the Treated; ATT): 정책 시행에 따른 실제 결과와 정책 미시행에 따른 반사실적 결과 간 차이

  \[\begin{aligned} \delta &=\mathbb{E}\left[Y_{i,t}(1)-Y_{i,t}(0) \mid D_{i}=1\right]\\ &=\underbrace{\mathbb{E}\left[Y_{i,t}(1)\mid D_{i}=1\right]}_{\text{factual}} - \underbrace{\mathbb{E}\left[Y_{i,t}(0)\mid D_{i}=1\right]}_{\text{counterfactual}}\\ &\approx \underbrace{\mathbb{E}\left[Y_{i,t}(1)\mid D_{i}=1\right] - \mathbb{E}\left[Y_{i,s}(0) \mid D_{i}=1\right]}_{\text{experimental group}}\\ &\quad+ \underbrace{\mathbb{E}\left[Y_{i,t}(0) \mid D_{i}=0\right] - \mathbb{E}\left[Y_{i,s}(0) \mid D_{i}=0\right]}_{\text{control group}} \end{aligned}\]

Model

• 2시점 2단위 모형

  \[\begin{aligned} Y_{i,t}(D_{i} \times T_{t}) &=\alpha + \beta D_{i} + \gamma T_{t} + \delta(D_{i} \times T_{t}) + \epsilon_{i,t} \end{aligned}\]
  • $t \in {0,1}$: 2시점(정책 시행 전과 후)
  • $i \in {0,1}$: 2단위(실험군과 통제군)
  • $Y_{i,t}$: 단위 $i$ 의 시점 $t$ 에서 결과
  • $\alpha$: 통제군의 사전 평균
  • $\beta$: 실험군과 통제군 간 고정 수준 차이
  • $D_{i} \in {0,1}$: 단위 $i$ 의 처리군 여부
  • $\gamma$: 사전-사후 공통 추세 변화
  • $T_{t} \in {0,1}$: 시점 $t$ 의 사후 여부
  • $\delta$: 정책의 인과적 효과(ATT)

• 다시점 다단위 모형(processing time per unit is the same)

  \[\begin{aligned} Y_{i,t}(D_{i} \times T_{t}) &=\alpha_{i} + \gamma_{t} + \delta_{i}(D_{i} \times T_{t}) + \epsilon_{i,t} \end{aligned}\]
  • $t=1,2,\cdots,K$: 다시점
  • $i=1,2,\cdots,N$: 다단위
  • $Y_{i,t}$: 단위 $i$ 의 시점 $t$ 에서 결과
  • $\alpha_{i}$: 단위 $i$ 의 고정 효과
  • $\gamma_{t}$: 시점 $t$ 의 고정 효과
  • $D_{i} \in {0,1}$: 단위 $i$ 의 처리군 여부
  • $T_{t} \in {0,1}$: 시점 $t$ 의 사후 여부
  • $\delta_{i}$: 단위 $i$ 에 대하여 정책의 인과적 효과(ATT)

Bayesian Method

flat bayesian

• frequentist ols:

  \[Y_{i,t} =\alpha_{i}+\gamma_{t}+\delta_{i}(D_{i} \times T_{t})+\epsilon_{i,t}\]

• bayesian likelihood:

  \[\begin{aligned} Y_{i,t} \mid \alpha_{i},\gamma_{t},\delta_{i},\sigma &\sim \mathcal{N}(\mu_{i,t},\sigma^{2})\\ \mu_{i,t} &= \alpha_{i} + \gamma_{t} + \delta_{i}(D_{i} \times T_{t}) \end{aligned}\]

• prior of parameters:

  \[\begin{aligned} \alpha_{i} &\overset{\mathrm{i.i.d}}{\sim} \mathcal{N}(0,10^{2}) \quad i=1,\cdots,N\\ \gamma_{t} &\overset{\mathrm{i.i.d}}{\sim} \mathcal{N}(0,10^{2}) \quad t=1,\cdots,T\\ \delta_{i} &\overset{\mathrm{i.i.d}}{\sim} \mathcal{N}(0,10^{2}) \quad i=1,\cdots,N\\ \sigma &\sim \mathrm{Half-cauchy}(2) \end{aligned}\]

• posterior estimation:

  \[\begin{aligned} \underbrace{p(\alpha_{i},\gamma_{t},\delta_{i},\sigma \mid Y_{i,t})}_{\text{posterior}} &\propto \underbrace{p(Y_{i,t} \mid \alpha_{i},\gamma_{t},\delta_{i},\sigma)}_{\text{likelihood}}\\ &\quad\times\underbrace{p(\alpha_{i})p(\gamma_{t})p(\delta_{i})p(\sigma)}_{\text{prior}} \quad \mathrm{s.t.} \quad \alpha_{i}\perp\gamma_{t}\perp\delta_{i}\perp\sigma \end{aligned}\]

• refer. cauchy distribution

  • $X \sim \mathrm{Cauchy}(x_{0},\gamma)$:

    \[f(x) = \frac{1}{\pi \gamma \left[1 + \left(\frac{x - x_{0}}{\gamma}\right)^{2} \right]}\]

  • $X \sim \mathrm{Half-Cauchy}(\gamma)$: absolute value distribution of cauchy

    \[f(x) = \frac{2}{\pi \gamma \left[1 + \left(x/\gamma\right)^{2} \right]}, \quad x>0\]

hierarchical bayesian

• level 1:

  \[Y_{i,t} \mid \alpha_{i},\gamma_{t},\delta_{i},\sigma \sim \mathcal{N}(\mu_{i,t},\sigma^{2})\]

• level 2:

  \[\begin{aligned} \alpha_{i} \mid \mu_{\alpha},\tau_{\alpha} &\sim \mathcal{N}(\mu_{\alpha},\tau_{\alpha}^{2}) \quad i=1,\cdots,N\\ \gamma_{t} \mid \mu_{\gamma},\tau_{\gamma} &\sim \mathcal{N}(\mu_{\gamma},\tau_{\gamma}^{2}) \quad t=1,\cdots,T\\ \delta_{i} \mid \mu_{\delta},\tau_{\delta} &\sim \mathcal{N}(\mu_{\delta},\tau_{\delta}^{2}) \quad i=1,\cdots,N \end{aligned}\]

• level 3:

  \[\begin{aligned} \mu_{\alpha} &\sim \mathcal{N}(0,10^{2}), \quad \tau_{\alpha} \sim \mathrm{Half-cauchy}(2)\\ \mu_{\gamma} &\sim \mathcal{N}(0,10^{2}), \quad \tau_{\gamma} \sim \mathrm{Half-cauchy}(2)\\ \mu_{\delta} &\sim \mathcal{N}(0,10^{2}), \quad \tau_{\delta} \sim \mathrm{Half-cauchy}(2) \end{aligned}\]

• posterior estimation:

  \[\begin{aligned} \underbrace{p(\alpha_{i},\gamma_{t},\delta_{i},\sigma \mid Y_{i,t})}_{\text{posterior}} &\propto \underbrace{p(Y_{i,t} \mid \alpha_{i},\gamma_{t},\delta_{i},\sigma)}_{\text{likelihood}}\\ &\quad\times \underbrace{p(\alpha_{i} \mid \mu_{\alpha},\tau_{\alpha})p(\mu_{\alpha})p(\tau_{\alpha})}_{\propto p(\mu_{\alpha},\tau_{\alpha} \mid \alpha_{i})} \quad &\mathrm{s.t.} \quad \mu_{\alpha} \perp \tau_{\alpha}\\ &\quad\times \underbrace{p(\gamma_{t} \mid \mu_{\gamma},\tau_{\gamma})p(\mu_{\gamma})p(\tau_{\gamma})}_{\propto p(\mu_{\gamma},\tau_{\gamma} \mid \gamma_{t})} \quad &\mathrm{s.t.} \quad \mu_{\gamma} \perp \tau_{\gamma}\\ &\quad\times \underbrace{p(\delta_{i} \mid \mu_{\delta},\tau_{\delta})p(\mu_{\delta})p(\tau_{\delta})}_{\propto p(\mu_{\delta},\tau_{\delta} \mid \delta_{i})} \quad &\mathrm{s.t.} \quad \mu_{\delta} \perp \tau_{\delta}\\ &\quad\times p(\sigma) \end{aligned}\]

uncertainty in treatment assignment and timing

• policy exposure uncertainty:

  \[\begin{aligned} D_{i} \mid \pi_{i} &\sim \mathrm{Bernoulli}(\pi_{i})\\ \pi_{i} &\overset{\mathrm{i.i.d}}{\sim} \mathrm{Beta}(1,1) \end{aligned}\]

• policy activation uncertainty (processing time per unit is the same):

  \[\begin{aligned} T_{t} \mid \phi_{t} &\sim \mathrm{Bernoulli}(\phi_{t})\\ \phi_{t} &\overset{\mathrm{i.i.d}}{\sim} \mathrm{Beta}(1,1) \end{aligned}\]

• parallel trends assumption:

  \[\begin{gathered} \mathbb{E}\left[Y_{i,t}(0)-Y_{i,s}(0) \mid \pi_{i}\right] = \mathbb{E}\left[Y_{i,t}(0)-Y_{i,s}(0)\right]\\ \Updownarrow\\ (Y_{i,t}(0)-Y_{i,s}(0)) \perp D_{i} \mid \pi_{i} \end{gathered}\]

• posterior estimation:

  \[\begin{aligned} &p(\alpha_{i},\gamma_{t},\delta_{i},\sigma,\pi_{i},\phi_{t} \mid Y_{i,t}, D_{i}, T_{t})\\ &\propto \underbrace{p(Y_{i,t} \mid \alpha_{i},\gamma_{t},\delta_{i},\sigma) \cdot p(\alpha_{i})p(\gamma_{t})p(\delta_{i})p(\sigma)}_{\propto p(\alpha_{i},\gamma_{t},\delta_{i},\sigma \mid Y_{i,t})}\\ &\quad\times \underbrace{p(D_{i}\mid\pi_{i})\cdot p(\pi_{i})}_{\propto p(\pi_{i}\mid D_{i})}\\ &\quad\times \underbrace{p(T_{t}\mid\phi_{t})\cdot p(\phi_{t})}_{\propto p(\phi_{t}\mid T_{t})} \end{aligned}\]