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1. Introduction to Bayesian Methods in Comparative Case Studies
Comparative case studies are increasingly common in social sciences, especially when using time-series cross-sectional (TSCS) data. These studies often deal with:
- Small samples of aggregate entities
- Non-random interventions on a few units
- Delayed treatment effects
These characteristics pose challenges in predicting counterfactual outcomes and making statistical inferences. The Synthetic Control Method (SCM) has become popular, but it has limitations in fully addressing inference and prediction challenges. Bayesian methods provide a compelling alternative.
Why is Bayesian Inference Necessary?
- Interpretability of Uncertainty: Bayesian measures are straightforward to understand.
- Heterogeneity Capture: Bayesian multilevel modeling effectively captures various sources of data heterogeneity and dynamics.
- Flexibility: It can accommodate flexible functional forms, reducing model dependency and incorporating modeling uncertainties.
This article explores a Bayesian causal inference framework to estimate treatment effects in comparative case studies, focusing on the posterior predictive distribution of treated counterfactuals. This approach uses a low-rank approximation of the observed untreated outcome matrix to predict treated counterfactuals, relying on the latent ignorability assumption.
2. Understanding Bayesian Causal Inference: Posterior Predictive Distributions
This section dives into the setup, notations, and causal quantities, developing posterior predictive distributions of counterfactuals based on key assumptions.
2.1 Setting Up the Framework and Defining Estimands
Notations and Definitions
- $i = 1, 2, …, N$ represents the unit of observation.
- $t = 1, 2, …, T$ represents the time period of observation.
- A binary treatment $w_{it}$ indicates whether a unit receives an intervention, which remains active once initiated (staggered adoption).
Key Concepts
- Timing of Adoption: Denoted as $a_i$, it signifies when unit i adopts the treatment. The set of possible adoption times is $mathbb{A} = {1,2,…, T, c}$, where $a_i = c > T$ means the unit is never treated.
- Treated vs. Control Units: Units adopting the treatment at any observed time ($a_i = 1, 2, …, T$) are treated. Units never adopting the treatment by period T ($a_i = c$) are controls.
- Pre-Treatment Periods: The number of pre-treatment periods for treated unit i is $T_{0,i} = a_i – 1$.
Assumptions
To ensure accurate causal inference, two critical assumptions are made:
Assumption 1: Cross-Sectional Stable Unit Treatment Value Assumption (SUTVA)
Potential outcomes of unit i depend only on the treatment status of unit i:
$textbf{y}{it} (textbf{W}) = textbf{y}{it} (textbf{w}_i), forall i, t$
This assumption rules out cross-sectional spillover effects, simplifying the analysis by focusing solely on the impact of treatment within each unit.
Assumption 2: No Anticipation
For all unit i and all time periods before adoption $t < a_i$:
$y_{it}(ai) = y{it}(c), mbox{ for } t < a_i$
This assumption asserts that potential outcomes before treatment are unaffected by future treatment adoption, ensuring no anticipatory effects influence the results.
Estimands: Quantifying Treatment Effects
Under Assumptions 1 and 2, the treatment effect for treated unit i at $t geq a_i$ is:
$delta{it} = y{it}(ai) – y{it}(c), mbox{ for } a_i leq t leq T$
This measures the difference between the observed post-treatment outcome and the counterfactual outcome had the unit never been treated.
The sample average treatment effect on the treated (ATT) for units under treatment for p periods is:
$delta{p} = frac{1}{N{tr,p}} sum_{i: T – p + 1 leq ai leq T} delta{i, a_i+p-1}$
where $N_{tr,p}$ is the number of treated units treated for p periods in the sample.
2.2 Delving into the Assignment Mechanism
Bayesian causal inference views causal inference as a missing data problem. The goal is to impute counterfactuals from their posterior predictive distribution $Pr (textbf{Y}(textbf{W})^{mis} |textbf{X}, textbf{Y}(textbf{W})^{obs}, textbf{W})$ using observed outcomes, covariates, and the assignment mechanism. For the average treatment effect on the treated (ATT), the primary objective is to predict the untreated outcomes of treated units, denoted as $textbf{Y}(textbf{0})^{mis}$.
Due to staggered adoption, where the treatment assignment matrix $textbf{W}$ is determined by the adoption time vector $mathcal{A}$, the posterior predictive distribution of $textbf{Y}(textbf{0})^{mis}$ can be written as:
$Pr (textbf{Y}(textbf{0})^{mis}|textbf{X}, textbf{Y}(textbf{0})^{obs}, mathcal{A}) propto Pr(textbf{X}, textbf{Y}(textbf{0})) Pr(mathcal{A}|textbf{X}, textbf{Y}(textbf{0}))$
The underlying “science” $Pr (textbf{X}, textbf{Y}(textbf{0}))$ and the treatment assignment mechanism $Pr (mathcal{A}|textbf{X}, textbf{Y}(textbf{0}))$ help predict the counterfactuals.
Assumption 3: Individualistic Assignment and Positivity
- Individualistic Assignment: The adoption time of unit i does not depend on the covariates or potential outcomes of other units, given its covariates $textbf{X}_i$ and potential outcomes $textbf{Y}_i(textbf{0})$.
- Positivity: $0 < Pr(a_i|textbf{X}_i, textbf{Y}_i(textbf{0})) < 1$ for all unit i. Each unit has a non-zero chance of getting treated.
The assignment mechanism may correlate with $textbf{Y}_i(textbf{0})^{mis}$, requiring further restrictions to avoid confounding.
Assumption 4: Latent Ignorability
Given observed pre-treatment covariates $textbf{X}_i$ and latent variables $textbf{U}i = (u{i1}, u{i2}, …, u{iT})$, the assignment mechanism is independent of missing or observed untreated outcomes:
$Pr(a_i|textbf{X}_i, textbf{Y}_i(textbf{0}), textbf{U}_i) = Pr(a_i|textbf{X}_i, textbf{Y}_i(textbf{0})^{mis}, textbf{Y}_i(textbf{0})^{obs}, textbf{U}_i) = Pr(a_i|textbf{X}_i, textbf{U}_i)$
$textbf{U}_i$ captures unit-level heterogeneity and unit-specific time trends. This assumption extends strict exogeneity, ruling out dynamic feedback from past outcomes on current and future treatment assignments, conditional on $textbf{U}_i$.
Assumption 5: Feasible Data Extraction
For each unit i, a covariate vector $textbf{U}_i$ exists, such that the matrix $textbf{U} = (textbf{U}_1, …, textbf{U}_N)$ can be approximated by lower-rank matrices ($r ll min {N, T}$):
$textbf{U} = boldsymbol{Gamma}^{prime}textbf{F}$
where $textbf{F} = (textbf{f}_1, …, textbf{f}_T)$ is a factor matrix, and $boldsymbol{Gamma} = (boldsymbol{gamma}_1, …, boldsymbol{gamma}_N)$ is a matrix of factor loadings.
This assumption, consistent with factor-augmented approaches, allows for the decomposition of unit-specific time trends into common trends with heterogeneous impacts.
2.3 Performing Posterior Predictive Inference
By integrating the latent ignorability assumption (Assumption 4), $textbf{U}$ is considered part of the covariates, leading to:
$Pr (textbf{Y}(textbf{0})^{mis}|textbf{X}^{prime}, textbf{Y}(textbf{0})^{obs}, mathcal{A}) propto Pr(textbf{X}^{prime}, textbf{Y}(textbf{0}))$
This simplifies counterfactual prediction by making the treatment assignment mechanism ignorable, reducing the task to modeling the underlying science $Pr (textbf{X}^{prime}, textbf{Y}(textbf{0}))$.
Assumption 6: Exchangeability
When $textbf{U}$ is known, the sequence ${(textbf{X}{it}^{prime}, y{it}(c))}_{i = 1,…,N; t = 1,…, T}$ is exchangeable, meaning their joint distribution is invariant to permutations in the index it.
By de Finetti’s theorem, this allows for writing the posterior predictive distribution of $textbf{Y}(textbf{0})^{mis}$ as:
$Pr (textbf{Y}(textbf{0})^{mis}|textbf{X}^{prime}, textbf{Y}(textbf{0})^{obs}, mathcal{A}) propto int underbrace{left(prod_{it in S1} f(y{it}(c)^{mis}|textbf{X}{it}, boldsymbol{theta}^{prime}) right)}{text{posterior predictive distribution}} underbrace{left(prod_{it in S0} f(y{it}(c)^{obs}|textbf{X}{it}, boldsymbol{theta}^{prime})right)}{text{likelihood}} pi(boldsymbol{theta}) d boldsymbol{theta}$
where $boldsymbol{theta}$ are the parameters governing the DGP of $y{it}(c)$ given $textbf{X}^{prime}{it}$.
This framework reduces to building a parametric model to estimate parameters based on the likelihood of observed outcomes and predicting missing potential outcomes using the posterior predictive distribution.
3. Modeling and Implementation Strategies
3.1 Developing a Multilevel Model with Dynamic Factors
Assumption 7: Defining the Functional Form
The untreated potential outcomes for unit $i = 1, …, N$ at $t = 1, …, T$ are specified as:
$y{it} (c) = textbf{X}{it}^{prime}boldsymbol{beta}{it} + boldsymbol{gamma}{i}^{prime}textbf{f}{t} + epsilon{it}$
where:
$boldsymbol{beta}{it} = boldsymbol{beta} + boldsymbol{alpha}{i} +boldsymbol{xi}_{t}$
and
$boldsymbol{xi}{t} = Phi{xi} boldsymbol{xi}{t – 1} + boldsymbol{e}{t}, textbf{f}{t} = Phi{f} textbf{f}{t – 1} + boldsymbol{nu}{t}$
Components of the Model
- $textbf{X}{it} boldsymbol{beta}{it}$: Captures the relationships between observed covariates and the outcome. The coefficients $boldsymbol{beta}_{it}$ vary across units and over time.
- $boldsymbol{beta}$: Mean of $boldsymbol{beta}_{it}$, shared by all observations.
- $boldsymbol{alpha}{i}$ and $boldsymbol{xi}{t}$: Zero-mean unit- and time-specific “residuals” of $boldsymbol{beta}_{it}$, respectively.
- $boldsymbol{gamma}{i}^{prime}textbf{f}{t}$: Multifactor term, where $textbf{f}{t}$ and $boldsymbol{gamma}{i}$ are factor and factor loading vectors, approximating $textbf{U}_{i}$.
- $epsilon_{it}$: i.i.d. idiosyncratic errors.
Model Flexibility
The Dynamic Multilevel Latent Factor Model (DM-LFM) allows the slope coefficient of each covariate to vary by unit, time, or both. This is illustrated by rewriting the model in a reduced matrix format:
$boldsymbol{y}{i}(c) = boldsymbol{X}{i}boldsymbol{beta} + boldsymbol{Z}{i}boldsymbol{alpha}{i} + boldsymbol{A}{i} boldsymbol{xi} + textbf{F}boldsymbol{gamma}{i} + boldsymbol{epsilon}_{i}$
Where:
- $textbf{F}$: Factor matrix.
- $boldsymbol{Z}_{i}$: Covariates with unit-specific slopes.
- $boldsymbol{A}_{i}$: Covariates with time-specific slopes.
The model’s systemic part is $mathbb{E}[textbf{y}{i}(c)]=textbf{X}{i}boldsymbol{beta}$, with other components defining the variance of composite errors.
3.2 Bayesian Stochastic Model Specification Search
To address model selection challenges, Bayesian stochastic model searching reduces mis-specification risks and incorporates model uncertainty using shrinkage priors.
Bayesian Lasso Shrinkage
Applied to $boldsymbol{beta}$ using a hierarchical setting:
$beta{k} | tau{beta{k}}^{2} sim mathcal{N}(0, tau{beta{k}}^{2}), tau{beta{k}}^{2} | lambda{beta} sim Exp(frac{lambda{beta}^{2}}{2}), lambda{beta}^{2} sim mathcal{G}(a{1}, a{2}), k=1,…, p_{1}$
The tuning parameter $lambda$ controls sparsity and shrinkage, analogous to a regularization penalty in frequentist Lasso regression.
Lasso-Like Hierarchical Shrinkage
Imposed on $boldsymbol{alpha}{i}$, $boldsymbol{xi}{t}$, and $boldsymbol{gamma}_{i}$ to determine covariate inclusion. Variance parameters $omega^{2}$ are allowed to take the value zero, re-parameterizing the model as:
$boldsymbol{alpha}{i}=boldsymbol{omega}{boldsymbol{alpha}} cdot tilde{boldsymbol{alpha}}{i}, boldsymbol{xi}{t} =boldsymbol{omega}{boldsymbol{xi}} cdot tilde{boldsymbol{xi}}{t}, boldsymbol{gamma}{i}=boldsymbol{omega}{boldsymbol{gamma}} cdot tilde{boldsymbol{gamma}}_{i}$
With Lasso priors assigned to each $omega{alpha}$, $omega{xi}$, and $omega_{gamma_j}$, the algorithm decides whether a covariate is included, whether its coefficient varies by time or across units, and the number of latent factors. This process searches and averages models, incorporating model uncertainty.
3.3 Implementation: A Step-by-Step Guide to DM-LFM
Implementing a DM-LFM involves three main steps:
- Model Searching and Parameter Estimation: Specify and estimate the DM-LFM model with Bayesian shrinkage to sample parameters from their posterior distributions, $boldsymbol{theta}{it}^{(g)} sim pi (boldsymbol{theta}{it}| mathfrak{D})$, where $mathfrak{D}$ is the set of untreated observations.
- Prediction and Integration: Conduct Bayesian prediction by generating counterfactuals $y{it}(c)^{mis}$ for each treated unit from its posterior predictive distribution:
$f(y{it}(c)^{mis}|textbf{X}, textbf{Y}(textbf{0})^{obs}) propto int f(y{it}(c)^{mis}|textbf{X}{it}, boldsymbol{theta}{it}) pi (boldsymbol{theta}{it}|mathfrak{D})dboldsymbol{theta}_{it}$. - Inference and Diagnostics: Make inferences about the causal effect $delta{it}$ for each treated unit i, by summarizing the empirical posterior distribution of $delta{it}$ formed by $delta^{(g)}{it} = y{it}(a{i}) – y^{(g)}{it}(c), g=1,…, G$.
4. Simulation Studies: Validating the Bayesian DM-LFM
In this section, the DM-LFM is validated through simulated examples and comparative analyses, demonstrating its effectiveness and advantages over existing methods.
4.1 A Simulated Example: Deconstructing the DM-LFM
To illustrate the DM-LFM’s functionality, a panel dataset with 50 units and 30 time periods is simulated using the following DGP:
$y{it} = delta{it}w{it} + textbf{X}{it}^{prime}boldsymbol{beta}{it} + {boldsymbol{gamma}}{i}^{prime} textbf{f}{t} + epsilon{it} = delta{it}w{it} + textbf{X}{it}^{prime}(boldsymbol{beta}+boldsymbol{alpha}{i}+boldsymbol{xi}{t}) + {boldsymbol{gamma}}{i}^{prime} textbf{f}{t} + epsilon{it}$
Here:
- $w{it}$ is the treatment indicator, and $delta{it}$ is the treatment effect.
- $textbf{X}_{it}$ is a vector of 10 covariates, including an intercept and nine time-varying variables with non-zero, unit- and time-varying coefficients.
- Seven units receive treatment starting from Period 21 until Period 30, while the remaining 23 units are never treated.
- The heterogeneous treatment effects are defined as $delta{it,t>20}= t -20+ tau{it}$, where $tau_{it} stackrel{i.i.d.}{sim} N(0,1)$.
- The factor vector $textbf{f}_{t}$ is two-dimensional, with both factors following an AR(1) process.
- The probability of treatment is correlated with the sum of the unit’s factor loadings $tilde{boldsymbol{gamma}}_{i}$, which are i.i.d. $N(0,1)$.
This setup introduces biases in causal estimates if factors and covariates are not properly accounted for.
Results and Observations
The estimated Average Treatment Effect on the Treated (ATT) with 95% credibility intervals is compared using a Bayesian DiD model and a DM-LFM model. The DiD model assumes constant coefficients and no factors, while the DM-LFM incorporates both.
- DiD Model: Multiple estimates in the pre-treatment periods are away from zero, indicating significant biases in the post-treatment ATT estimates.
- DM-LFM Model: Performs significantly better, showing no pre-trend with ATT estimates close to the true values, covered by the 95% credibility intervals.
Furthermore, the DM-LFM accurately selects non-zero components of the covariate coefficients and identifies the number of factors to be two, consistent with the DGP.
4.2 Monte Carlo Evidence: Assessing Model Performance
Several Monte Carlo exercises are conducted to study the properties of the DM-LFM and compare it with existing methods.
Exercise 1: Component-Wise Analysis and Sample Size Effects
Samples are simulated using the same DGP, varying the sample size (both the number of units N and pre-treatment periods $T_{0}$). The full DM-LFM model is compared with three simpler variants:
- Model with covariates but without factors (analogous to DiD).
- Model without covariates but with 10 latent factors (analogous to Gsynth without time-varying covariates).
- Model with factors and covariates with fixed coefficients.
The results show that the DM-LFM outperforms the other three models in terms of bias, standard deviation, root mean squared errors (RMSE), and coverage. Each key component of the model contributes to improved causal effect estimation, with the factor term having the most impact, and covariates with varying coefficients improving precision and coverage.
Exercise 2: DM-LFM vs. SCM and Gsynth
The performance of the DM-LFM is compared with the Synthetic Control Method (SCM) and Gsynth, demonstrating its superiority in RMSE. The DM-LFM outperforms Gsynth when the true number of factors is unknown or when factors are numerous with weak signals.
5. Empirical Applications: Real-World Scenarios
Here, the Bayesian DM-LFM is applied to two real-world examples, showcasing its practical utility and advantages in handling complex data structures.
5.1 Economic Impact of German Reunification
The DM-LFM is applied to analyze the economic impact of German reunification, incorporating pre-treatment time-invariant covariates, including averages of trade openness, inflation rate, industry share, schooling, and investment rate, consistent with prior research. The initial model specified is:
$y{it}(c) = textbf{X}{i}(boldsymbol{beta} + boldsymbol{xi}{t}) + textbf{f}{t}boldsymbol{gamma}{i} + epsilon{it}$
This model excludes unit fixed effects and unit-varying coefficients, maintaining consistency with the SCM while including time-varying coefficients.
Model Selection and Factor Analysis
The results indicate the presence of four to six significant factors, supported by the bimodal posteriors of their corresponding $omega$ values. The model effectively captures the essential underlying structures influencing the economic outcomes.
Goodness-of-Fit and Counterfactual Predictions
The counterfactual predictions of the DM-LFM are compared with those of the SCM. Both methods yield similar results, indicating that the GDP per capita of the counterfactual West Germany would have been higher than that of the actual West Germany during most of the post-reunification period, excluding the initial years after reunification.
Treatment Effects and Placebo Testing
Estimated effects of reunification on West Germany using both the SCM and the Bayesian DM-LFM are analyzed. To ensure the robustness of causal estimates, a placebo test is conducted by artificially setting the period 1987–1989 (three years before reunification) as the treatment period. The estimated effects during this placebo period are close to zero, bolstering confidence in the identification assumptions and the reliability of the causal estimates.
5.2 Election Day Registration and Voter Turnout: A Case Study
The DM-LFM is applied to analyze the effect of Election Day Registration (EDR) on voter turnout in the United States. A full DM-LFM model is specified, including time-varying covariates (universal mail-in registration and motor voter registration) and 10 factors.
Model Specification and Covariates
Given the limited number of covariates, shrinkage is not imposed on their $boldsymbol{beta}$ values. Instead, shrinkage priors are assigned to $boldsymbol{alpha}{i}$, $boldsymbol{xi}{t}$, and $boldsymbol{gamma}_{i}$ to better account for individual and temporal variations.
Factors Influencing Turnout
The model suggests that at least six factors significantly influence voter turnout, contrasting with the Gsynth model, which includes only two factors selected via cross-validation.
Covariate Effects and Parameter Estimates
The analysis reveals that while the intercept varies in both time and space dimensions, the varying parts of the slopes of the covariates are effectively shrunk to zero. The results are consistent with prior findings, indicating that the chosen covariates do not significantly explain the variation in voter turnout.
Posterior Distributions and Treatment Effects
Posterior distributions of counterfactual outcomes are generated for the treated states during post-treatment years to estimate the effect of EDR on voter turnout. The Average Treatment Effect on the Treated (ATT) is reported for the same duration of adoption, pooling posterior draws of individual treatment effects for each treated state in year p after adoption.
Comparative Analysis and Model Performance
A comparative analysis between the Gsynth and DM-LFM models reveals that the Bayesian 95% credibility intervals are considerably narrower than the 95% confidence intervals from Gsynth. This suggests that the Bayesian approach provides more precise estimates and better predictive performance of individual counterfactuals.
6. Discussion: Evaluating the Bayesian DM-LFM
The Bayesian DM-LFM distinguishes itself through its transparent assumptions, flexibility, and robust model validation, positioning it as a valuable tool for comparative case studies. Its ability to provide interpretable uncertainty estimates, manage complex relationships, and accommodate varied model specifications makes it particularly useful when the conventional “parallel trends” assumption is questionable.
This comparative analysis highlights the specific strengths of the DM-LFM, reinforcing its potential to advance quantitative social science research by offering deeper insights and more reliable causal inferences. By offering more interpretable results, and by being more flexible in the scenarios it can handle, the DM-LFM approach offers a potent Bayesian alternative to the synthetic control method for comparative case studies.
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FAQ: Bayesian Alternative to Synthetic Control
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What is the main advantage of using a Bayesian approach over traditional methods like Synthetic Control?
The Bayesian approach offers more interpretable uncertainty measures and greater flexibility in capturing data heterogeneity, leading to more robust inferences in comparative case studies.
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How does the latent ignorability assumption improve causal inference?
The latent ignorability assumption breaks the link between treatment assignment and control outcomes, allowing for more accurate estimation of causal effects by conditioning on both observed covariates and unobserved latent variables.
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Can the Dynamic Multilevel Latent Factor Model (DM-LFM) handle complex data structures?
Yes, the DM-LFM is specifically designed to handle complex data structures by allowing the slope coefficient of each covariate to vary by unit, time, or both, and by incorporating multiple latent factors.
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What is Bayesian stochastic model searching, and how does it reduce model mis-specification risks?
Bayesian stochastic model searching uses shrinkage priors to choose the number of latent factors and decide whether and how to include a covariate, reducing model dependency and incorporating modeling uncertainties.
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How does the DM-LFM compare to existing methods like DiD, SCM, and Gsynth?
The DM-LFM is more flexible, allowing for both time-invariant and time-varying covariates and coefficients that vary by unit and time, while DiD and SCM have limitations in covariate types and coefficient variability.
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Is the DM-LFM computationally expensive?
Yes, the Bayesian approach can be computationally more expensive than frequentist methods like Gsynth, especially when the sample size is large.
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When is the DM-LFM most applicable?
The DM-LFM is especially well suited for comparative case studies when the conventional “parallel trends” assumption is unlikely to hold, when there are multiple treated units, and when researchers need interpretable uncertainty estimates.
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What kind of data is suitable for the DM-LFM approach?
Time-series cross-sectional (TSCS) data, or long panel data, with a small number of aggregate entities, non-random interventions, and delayed treatment effects are suitable for the DM-LFM approach.
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Can the DM-LFM be used for policy evaluation?
Yes, the DM-LFM provides reliable causal estimates and uncertainty measures, making it suitable for policy evaluation where understanding the impact of interventions is crucial.
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Where can I find more information about the DM-LFM and its applications?
Visit compare.edu.vn to explore a wide range of comparative studies and access additional resources related to the DM-LFM and its applications.
