Chapter 15: Advanced Panel Methods

Opening Question

When standard difference-in-differences won't work—because parallel trends is implausible or you have only one treated unit—what alternatives exist for causal inference with panel data?

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Chapter Overview

Chapter 13 established difference-in-differences as a workhorse for policy evaluation. But DiD requires parallel trends: the assumption that treated and control units would have evolved similarly absent treatment. This assumption often fails. Sometimes treated units are structurally different from available controls. Sometimes there is only one treated unit—a single country, state, or firm—making "parallel trends" conceptually awkward. And sometimes treatment effects interact with unobserved factors in ways that standard fixed effects cannot capture.

This chapter develops three methods that extend panel data causal inference beyond DiD:

1. Synthetic control: Construct a weighted combination of control units that matches the treated unit's pre-treatment trajectory, then use this "synthetic" unit as the counterfactual.

2. Interactive fixed effects: Model outcomes as depending on unit-specific responses to time-varying latent factors, relaxing the additive separability of standard two-way fixed effects.

3. Matrix completion: Treat the counterfactual outcomes as missing entries in a matrix and use low-rank assumptions to impute them.

These methods share a common insight: when a single set of control units doesn't provide a valid counterfactual, we can construct one by exploiting the panel structure.

What you will learn:

• How to construct and evaluate synthetic controls

• Inference methods for synthetic control (permutation, conformal)

• The interactive fixed effects model and its estimation

• When matrix completion methods provide an advantage

• How to choose among these methods in practice

Prerequisites: Chapter 13 (Difference-in-Differences), Chapter 3 (Statistical Foundations)

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15.1 The Synthetic Control Method

Motivation: The Single Treated Unit Problem

Consider evaluating German reunification's effect on West German GDP. There is one West Germany. Standard DiD would compare Germany to "control" countries, but which countries? France, UK, and the US have different industrial structures, growth trajectories, and institutional contexts. Any single country comparison is problematic; averaging across dissimilar countries doesn't obviously help.

The synthetic control insight: instead of choosing a single comparison or arbitrary average, construct a weighted combination of control units that best reproduces the treated unit's characteristics before treatment.

Setup and Notation

Consider $J+1$ units observed over $T$ periods, where unit 1 is treated at time $T_0 + 1$ and units $2, \ldots, J+1$ ("the donor pool") remain untreated throughout. Let:

• $Y_{it}$ = outcome for unit $i$ at time $t$

• $Y_{it}(0)$ = potential outcome without treatment

• $Y_{it}(1)$ = potential outcome with treatment

For $t > T_0$, the treatment effect on the treated unit is:

$\tau_{1t} = Y_{1t}(1) - Y_{1t}(0) = Y_{1t} - Y_{1t}(0)$

We observe $Y_{1t}$ but not $Y_{1t}(0)$. The challenge is constructing a credible estimate of $Y_{1t}(0)$.

The Synthetic Control Estimator

A synthetic control is a weighted average of control units:

$\hat{Y}_{1t}^{SC} = \sum_{j=2}^{J+1} w_j Y_{jt}$

where weights $w = (w_2, \ldots, w_{J+1})'$ satisfy $w_j \geq 0$ and $\sum_j w_j = 1$.

The treatment effect estimate is:

$\hat{\tau}_{1t} = Y_{1t} - \hat{Y}_{1t}^{SC}$

Definition 15.1 (Synthetic Control): A synthetic control for treated unit 1 is a weighted average of untreated donor units, with non-negative weights summing to one, chosen to match the treated unit's pre-treatment characteristics.

Intuition: If the synthetic control tracks the treated unit closely before treatment, any post-treatment divergence reflects the treatment effect. The synthetic control "is" what the treated unit would have looked like without treatment.

Choosing Weights

Abadie, Diamond, and Hainmueller (2010) propose choosing weights to minimize:

$\|X_1 - X_0 w\|_V = \sqrt{(X_1 - X_0 w)' V (X_1 - X_0 w)}$

where:

• $X_1$ = vector of pre-treatment characteristics for the treated unit

• $X_0$ = matrix of pre-treatment characteristics for donor units

• $V$ = diagonal matrix of variable importance weights

What goes into $X$?

• Pre-treatment outcomes (often several time periods)

• Covariates predicting outcomes (e.g., GDP per capita, industry composition)

• Functions of pre-treatment outcomes (means, growth rates)

Choosing $V$: The importance matrix $V$ determines which characteristics matter most for matching. Common approaches:

• Equal weighting

• Inverse variance weighting

• Cross-validation: choose $V$ to minimize prediction error in early pre-treatment periods

Example: German Reunification

Abadie, Diamond, and Hainmueller (2015) estimate reunification's effect on West German GDP per capita.

Setup:

• Treated unit: West Germany (reunification in 1990)

• Donor pool: 16 OECD countries

• Pre-treatment period: 1960-1989

• Matching variables: GDP per capita, trade openness, inflation, industry share, schooling

Synthetic Germany: The algorithm selects weights primarily on Austria (42%), US (22%), Japan (16%), Switzerland (11%), and Netherlands (9%). This combination matches West Germany's pre-reunification GDP trajectory closely.

Results: Pre-1990, real and synthetic Germany track closely. Post-1990, actual West German GDP falls increasingly below synthetic Germany. By 2003, the gap is approximately 1,600 USD per capita—a cumulative cost of reunification equivalent to about one year's economic growth.

Why synthetic control here? No single OECD country matches Germany's pre-1990 trajectory. But a weighted combination does. This combination provides a more credible counterfactual than any single comparator or unweighted average.

Practical Implementation

Step 1: Define the donor pool

Include units that:

• Are plausibly similar to the treated unit

• Were not affected by the treatment (no spillovers)

• Have data available for all periods

Exclude units that:

• Experienced their own major shocks during the study period

• Are fundamentally different (e.g., developing countries in a developed-country study)

Step 2: Select matching characteristics

Include:

• Pre-treatment outcome values (multiple time points)

• Predictors of the outcome

• Pre-treatment trends or growth rates

Avoid:

• Post-treatment variables

• Variables affected by anticipation of treatment

Step 3: Estimate weights and examine fit

• Run the optimization

• Check pre-treatment fit visually and via RMSPE (root mean squared prediction error)

• Examine weights: are they sensible? Concentrated on few units?

Step 4: Assess sensitivity

• Leave-one-out: Re-estimate dropping each donor unit

• Alternative specifications: Different $V$ matrices, different matching variables

• Backtesting: Apply method to earlier periods without treatment

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15.2 Inference for Synthetic Control

The Inference Challenge

Standard inference methods don't apply to synthetic control:

• We have one treated unit (no sampling variation in the traditional sense)

• Weights are estimated, creating model uncertainty

• Treatment effects may be heterogeneous across time

Permutation Inference

The key insight: if treatment had no effect, the treated unit's post-treatment deviation from its synthetic control should be no larger than deviations we'd observe for untreated units.

Placebo tests: Apply synthetic control to each donor unit, pretending it was treated at time $T_0$. Compute the post/pre RMSPE ratio:

$\text{Ratio}_j = \frac{RMSPE_j^{post}}{RMSPE_j^{pre}}$

If the treated unit's ratio is extreme relative to placebo ratios, the effect is unlikely due to chance.

p-value calculation: The p-value is the fraction of placebo ratios at least as large as the treated unit's ratio:

$p = \frac{\#\{j: \text{Ratio}_j \geq \text{Ratio}_1\}}{J+1}$

With $J+1$ units, the minimum p-value is $1/(J+1)$. This limits power with small donor pools.

Definition 15.2 (Permutation p-value): The permutation p-value for synthetic control equals the proportion of donor units whose post/pre RMSPE ratio is at least as large as the treated unit's ratio.

Conformal Inference

Chernozhukov et al. (2021) develop conformal inference for synthetic control, providing exact finite-sample confidence intervals under weaker assumptions than permutation tests.

Key idea: Under the null of no treatment effect, the treated unit's residuals should be exchangeable with residuals from donor units. This allows constructing confidence intervals by inverting hypothesis tests.

Advantages over permutation:

• Valid with fewer donor units

• Allows for serial correlation in outcomes

• Provides confidence intervals, not just p-values

Interpreting Results

Strong evidence requires:

1. Good pre-treatment fit (low pre-RMSPE)

2. Large post-treatment gap

3. Extreme rank in placebo distribution

Weak evidence occurs when:

• Pre-treatment fit is poor (synthetic control doesn't track treated unit)

• Many placebo units show similar or larger gaps

• Gap emerges gradually or only in later periods

Example: Tobacco Control in California

Abadie, Diamond, and Hainmueller (2010) evaluate California's 1988 Proposition 99, which increased cigarette taxes and funded anti-smoking programs.

Placebo analysis: Running synthetic control for all 38 control states, California's post-1988 per-capita cigarette consumption decline is the most extreme—larger than any placebo. The p-value is approximately 0.026 (1/38).

This demonstrates that California's smoking reduction was unlikely to have occurred by chance given the variation observed across other states.

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15.3 Extensions and Variations

Multiple Treated Units

When multiple units receive treatment (possibly at different times), several approaches exist:

Pooled synthetic control: Construct a synthetic control for each treated unit separately, then average the treatment effects.

Synthetic DiD (Section 15.4): Combines synthetic control with difference-in-differences, exploiting both cross-sectional and time-series variation in constructing counterfactuals.

Convexity Constraints

Standard synthetic control restricts weights to be non-negative and sum to one. This ensures the synthetic unit interpolates within the donor pool, avoiding extrapolation.

Advantages of convexity:

• Transparent: easy to see which units contribute

• Conservative: synthetic control is bounded by observed data

• Interpretable: weights have clear meaning

Limitations:

• May not achieve good fit if treated unit is outside the donor pool's convex hull

• Cannot give negative weight to units that should be "unlike" the treated unit

Box: The Convex Hull Problem—When Synthetic Control Fails

The convex hull is the set of all weighted averages of the donor units (with non-negative weights summing to one). Geometrically, imagine stretching a rubber band around the donor units in pre-treatment characteristic space—everything inside is achievable.

The problem: If the treated unit lies outside this hull, no synthetic control can match it well. The algorithm will place all weight on the closest boundary point, but a gap remains.

Visual intuition: Imagine two characteristics (GDP per capita and industrialization). If your treated unit has both the highest GDP and highest industrialization, no weighted average of lower-on-both donors can match it—you'd need to extrapolate beyond what any control achieved.

Figure 15.1: The convex hull problem. The treated unit (red diamond) lies outside the convex hull of control units (blue shaded region). The best synthetic control (green square) can only reach the hull boundary, leaving an extrapolation gap.

Warning signs:

• Large pre-treatment RMSPE despite optimization

• Weights concentrated on one or two extreme donors

• Treated unit's characteristics exceed all donors on multiple dimensions

• Algorithm puts 100% weight on a single donor

What to do:

1. Expand the donor pool if plausible units were excluded

2. Use fewer matching characteristics—focus on the most important

3. Consider augmented SC (Ben-Michael et al. 2021) which combines regression adjustment with SC

4. Use SDiD which handles extrapolation more gracefully

5. Be honest: If no good synthetic match exists, report this as a limitation rather than forcing a poor fit

The deeper lesson: Synthetic control is fundamentally an interpolation method. It excels when the treated unit is "typical" relative to the donor pool. It struggles with extreme or unusual treated units—precisely those that may be most policy-relevant.

Alternatives: Some researchers allow negative weights (relaxing convexity) or penalized regression methods (e.g., LASSO) for weight selection.

Covariate Adjustment

The original synthetic control matches on pre-treatment outcomes and covariates. Recent extensions:

Bias correction: Adjust for remaining covariate imbalance using regression:

$\hat{\tau}_{1t}^{BC} = Y_{1t} - \hat{Y}_{1t}^{SC} - \hat{\beta}'(X_1 - X_0 w)$

Augmented synthetic control: Ben-Michael, Feller, and Rothstein (2021) combine synthetic control with outcome modeling, achieving doubly-robust properties.

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15.4 Synthetic Difference-in-Differences

Arkhangelsky et al. (2021) combine synthetic control and DiD, addressing limitations of each:

• DiD's problem: Parallel trends may not hold unconditionally

• SC's problem: Poor fit when no convex combination matches the treated unit

The SDiD Estimator

SDiD reweights both units and time periods:

$\hat{\tau}^{SDiD} = \sum_{i: D_i=1} \sum_{t: Post_t=1} \omega_i \lambda_t (Y_{it} - \mu)$

where $\omega_i$ are unit weights (like SC) and $\lambda_t$ are time weights (emphasizing pre-treatment periods similar to post-treatment periods).

Intuition

Unit reweighting: Like synthetic control, upweight control units that track treated units pre-treatment.

Time reweighting: Unlike standard DiD which equally weights pre-periods, emphasize pre-treatment periods where control units' time series patterns resemble the treatment period.

The result: a DiD estimator that doesn't require unconditional parallel trends—only parallel trends for the reweighted comparison.

Figure 15.2: How different methods weight control units over time. DiD weights all controls equally across all periods. Synthetic control concentrates unit weights but treats pre-periods equally. SDiD reweights both dimensions—more weight on similar units AND on pre-periods that resemble post-periods.

When to Use SDiD vs. SC vs. DiD

Setting	Recommended Method
Single treated unit, good SC fit	Synthetic control
Multiple treated units, staggered timing	Modern DiD (Ch. 13)
Multiple treated units, PT questionable	SDiD
Single treated unit, poor SC fit	SDiD or interactive FE

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15.5 Interactive Fixed Effects

Beyond Additive Separability

Standard two-way fixed effects assumes:

$Y_{it} = \alpha_i + \gamma_t + \tau D_{it} + \varepsilon_{it}$

This is additively separable: unit effects and time effects contribute independently. But many outcomes reflect interactions between unit characteristics and time-varying factors.

Example: Economic growth depends on both country characteristics (institutions, human capital) and global conditions (commodity prices, technology shocks). Countries respond differently to the same global shock—export-oriented economies boom when world demand rises; commodity exporters benefit from price spikes.

The interactive fixed effects model captures this:

$Y_{it} = \alpha_i + \gamma_t + \lambda_i' F_t + \tau D_{it} + \varepsilon_{it}$

where:

• $\lambda_i$ = unit-specific factor loadings ($r \times 1$ vector)

• $F_t$ = time-varying common factors ($r \times 1$ vector)

• $\lambda_i' F_t$ = unit-specific responses to common shocks

Definition 15.3 (Interactive Fixed Effects): An interactive fixed effects model allows unit-specific responses to time-varying latent factors, capturing heterogeneous trends that additive two-way fixed effects cannot accommodate.

Intuition: Units travel along different trend trajectories determined by their factor loadings. Treatment effects are identified by deviations from these unit-specific trajectories.

Estimation

Bai (2009): Proposes iterative estimation:

1. Given $F_t$, estimate $(\alpha_i, \lambda_i, \tau)$ by cross-sectional regression

2. Given $(\lambda_i)$, estimate $F_t$ by principal components

3. Iterate until convergence

Number of factors: Use information criteria (Bai-Ng) or cross-validation to select $r$.

Inference: With large $N$ and $T$, standard errors can be computed; bootstrap is often used in practice.

When Interactive FE Helps

Interactive fixed effects addresses violations of parallel trends that arise from:

• Heterogeneous time trends: Units with different underlying growth rates

• Differential responses to shocks: Some units more sensitive to business cycles

• Omitted trending confounders: Variables that trend differently across units

Example: Evaluating Economic Policies

Consider evaluating a state-level policy (tax incentive, regulation) adopted by some states. Standard DiD assumes treated and control states would have followed parallel paths. But states differ in industrial composition, meaning they respond differently to national business cycles.

Interactive FE approach:

1. Extract common factors from economic variables (employment, output)

2. Estimate state-specific loadings

3. Treatment effect is deviation from state's factor-implied trajectory

Gobillon and Magnac (2016) show this can substantially change policy effect estimates compared to standard DiD.

Limitations

Data requirements: Needs substantial pre-treatment periods to estimate factors and loadings.

Factor selection: Results can depend on number of factors chosen.

Weak factors: If factor structure is weak, standard DiD may be preferred.

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15.6 Matrix Completion Methods

Framing the Problem

Think of the outcome data as a matrix $Y$ where rows are units and columns are time periods. We observe:

• All entries for control units

• Pre-treatment entries for treated units

• Post-treatment entries for treated units (observed with treatment effect)

The counterfactual $Y_{it}(0)$ for treated units post-treatment is a missing entry we want to impute.

The Netflix Analogy: Why Matrix Completion Works

Netflix faces a similar problem: predicting which movies you'll enjoy when you've only rated a tiny fraction of their catalog. They have a massive user-movie matrix with mostly missing entries.

The key insight is that this matrix has low rank—it can be well-approximated by a small number of underlying factors. Why? Because users have "types" (action fans, rom-com enthusiasts, documentary watchers) and movies have "types" (genre, pacing, visual style). A user's rating is largely determined by how their type matches the movie's type.

In panel data, the logic is identical. Units (states, firms, individuals) have underlying "types" that determine their outcomes, and time periods have "shocks" that affect different types differently. If we know a unit's type from its observed outcomes, we can predict its missing counterfactual outcomes—just as Netflix predicts your rating for an unwatched movie.

This is why the method is called "matrix completion": we're filling in the missing entries of a matrix using the structure revealed by the observed entries.

Low-Rank Assumptions

Matrix completion assumes the counterfactual matrix $Y^{(0)}$ (outcomes without treatment) has low rank:

$Y_{it}(0) = \sum_{k=1}^{r} \lambda_{ik} f_{kt}$

This is equivalent to an interactive fixed effects structure. The insight: low-rank matrices can be recovered from partial observations under certain conditions.

Athey et al. (2021): Matrix Completion for Causal Inference

Procedure:

1. Estimate $\hat{Y}^{(0)}$ using nuclear norm regularization (encourages low rank)

2. Treatment effect: $\hat{\tau}_{it} = Y_{it} - \hat{Y}_{it}^{(0)}$ for treated observations

Advantages:

• Unified framework covering SC and interactive FE as special cases

• Allows for missing data in control units

• Regularization handles high-dimensional settings

Inference: Provide confidence intervals based on approximate normality of the regularized estimator.

Connection to Synthetic Control and IFE

Method	What it's doing
Synthetic control	Imputing missing entries by weighted average of rows
Interactive FE	Imputing via explicit factor model
Matrix completion	Imputing via low-rank approximation (agnostic about direction)

Matrix completion is the most general: it nests the others as special cases when the same assumptions hold.

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Practical Guidance

When to Use Each Method

Situation	Recommended Method
Single treated unit, long pre-period	Synthetic control
Single treated unit, short pre-period	Interactive FE or matrix completion
Few treated units, good donor pool	Synthetic control (pooled)
Many treated units, staggered timing	Modern DiD or SDiD
Heterogeneous trends across units	Interactive FE
Many missing values in panel	Matrix completion
Parallel trends plausible	Standard DiD (simpler, more transparent)

Common Pitfalls

Pitfall 1: Overfitting the pre-treatment period Matching on many pre-treatment outcomes can achieve perfect pre-fit while missing the underlying data-generating process.

How to avoid: Use cross-validation; leave out some pre-treatment periods for validation. Match on meaningful predictors, not just raw outcomes.

Pitfall 2: Extrapolation in synthetic control If the treated unit lies outside the donor pool's convex hull (on important dimensions), no non-negative weight combination will match well.

How to avoid: Check that treated unit's characteristics are within the range of the donor pool. Consider augmented synthetic control or SDiD if extrapolation is needed.

Pitfall 3: Ignoring inference uncertainty Synthetic control point estimates look precise, but inference is challenging with few units.

How to avoid: Always report placebo tests and p-values. Use conformal inference for sharper bounds. Be honest about limited power.

Pitfall 4: Too many factors in interactive FE Overfitting the factor structure absorbs treatment effects.

How to avoid: Use information criteria to select number of factors. Check sensitivity to factor choice.

Implementation Checklist

• Clearly define the treated unit(s) and donor pool

• Justify why standard DiD parallel trends is questionable

• For SC: Check pre-treatment fit (RMSPE) and examine weights

• For SC: Conduct placebo tests on all donor units

• For SC: Report leave-one-out sensitivity

• For IFE: Select number of factors via information criteria

• Compare results across methods (SC, IFE, SDiD) for robustness

• Discuss remaining threats (spillovers, anticipation, structural breaks)

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Qualitative Bridge

Complementing Quantitative Counterfactuals

Synthetic control and related methods construct statistical counterfactuals. But they cannot tell us:

• Why the treatment occurred when and where it did

• What mechanisms drove the observed effects

• Whether the synthetic counterfactual captures relevant dimensions of similarity

Qualitative research addresses these gaps.

When to Combine

Justifying the donor pool: Historical and institutional analysis can justify which units belong in the donor pool. For German reunification, qualitative understanding of post-war European economic history supports including other OECD economies while excluding Eastern Bloc countries.

Understanding treatment assignment: Process tracing how and why a policy was adopted helps assess whether selection threatens identification. If California adopted Proposition 99 due to unique political factors unrelated to smoking trends, the synthetic control is more credible.

Interpreting results: What does it mean that West German GDP fell below synthetic Germany post-reunification? Economic historians provide mechanisms: absorption of East German workers, capital flows eastward, institutional integration costs.

Example: China's Special Economic Zones

Quantitative analysis can estimate SEZ effects on local GDP using synthetic control (constructing a "synthetic Shenzhen" from non-SEZ Chinese cities). But understanding why SEZs worked requires qualitative investigation:

• Institutional analysis: How did SEZ governance differ from standard Chinese administration?

• Case studies: What specific firms and industries drove growth?

• Historical context: How did proximity to Hong Kong matter for Shenzhen specifically?

The quantitative effect estimate gains meaning from qualitative mechanism analysis.

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Integration Note

Connections to Other Methods

Method	Relationship	See Chapter
Difference-in-Differences	SC/IFE relax DiD's parallel trends	Ch. 13
Factor Models	IFE uses factor structure; connection to forecasting	Ch. 7
Instrumental Variables	Can combine with IFE for endogenous treatment	Ch. 12
Bounds	When SC fit is poor, partial identification may be appropriate	Ch. 17

Triangulation Strategies

Results from advanced panel methods gain credibility when:

1. Multiple methods agree: SC, IFE, and SDiD yield similar estimates

2. Leave-one-out stability: Dropping any donor unit doesn't change conclusions

3. Backtesting: Applying the method to earlier "placebo" periods finds null effects

4. Qualitative corroboration: Mechanisms suggested by effect timing align with institutional knowledge

5. External comparisons: Other studies of similar policies find consistent effects

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Running Example: China and Synthetic Control

Constructing a Synthetic China

What would China's GDP trajectory have looked like without the post-1978 reforms? This is the "n=1" problem par excellence—there is only one China.

Synthetic control offers one approach. Hsiao, Ching, and Wan (2012) construct a synthetic China from other developing economies:

Donor pool: Large developing countries (India, Brazil, Mexico, Indonesia, etc.) and other Asian economies

Matching: Pre-1978 GDP per capita levels and trends

Challenge: No combination of other countries well-matches China's pre-1978 trajectory. China in 1978 had unusual characteristics:

• Large population

• Decades of central planning

• Recent disruption from Cultural Revolution

• Low base but substantial human capital

Results: The synthetic China substantially underperforms actual China post-1978. The gap—attributable to reforms—grows over time.

Limitations: Pre-fit is imperfect, raising concerns about counterfactual validity. Alternative donor pools yield different estimates. This illustrates both the promise (providing some counterfactual) and limitations (imprecision with poor fit) of synthetic control for transformative policy changes.

Complementary Approaches

Given synthetic control's limitations for China, triangulation is essential:

• Growth accounting (Ch. 6): Decompose growth into capital, labor, and TFP contributions

• DiD on SEZs (Ch. 13): Estimate effects of specific policies using within-China variation

• Time series methods (Ch. 16): Identify structural breaks in trend growth

• Qualitative analysis (Ch. 23): Institutional and historical explanation

No single method answers "what caused China's growth." But combining methods builds cumulative evidence.

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Summary

Key takeaways:

1. Synthetic control constructs counterfactuals for single treated units by weighting control units to match pre-treatment characteristics. It's most useful when no single control provides a valid comparison.

2. Inference is challenging with few units. Permutation tests compare the treated unit's pre/post deviation to placebo deviations; conformal inference provides sharper bounds.

3. Interactive fixed effects relaxes DiD's additive structure, allowing unit-specific responses to common time-varying factors. This captures heterogeneous trends that violate parallel trends.

4. Matrix completion frames counterfactual imputation as a missing data problem, using low-rank assumptions to fill in unobserved potential outcomes.

5. Method choice depends on the setting: number of treated units, quality of donor pool, plausibility of parallel trends, and data availability.

Returning to the opening question: When standard DiD fails—because parallel trends is implausible or there's only one treated unit—synthetic control, interactive fixed effects, and matrix completion provide alternatives. Each constructs a counterfactual differently: by explicitly weighting control units, by modeling factor structures, or by exploiting low-rank matrix assumptions. The best practice is often to apply multiple methods and assess whether conclusions are robust.

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Further Reading

Essential

• Abadie (2021), "Using Synthetic Controls: Feasibility, Data Requirements, and Methodological Aspects" - Comprehensive practical guide

• Arkhangelsky et al. (2021), "Synthetic Difference-in-Differences" - Combining SC with DiD

For Deeper Understanding

• Abadie, Diamond, and Hainmueller (2010), "Synthetic Control Methods for Comparative Case Studies" - Original methodological paper

• Abadie, Diamond, and Hainmueller (2015), "Comparative Politics and the Synthetic Control Method" - German reunification application

• Bai (2009), "Panel Data Models with Interactive Fixed Effects" - Key IFE paper

Advanced/Specialized

• Chernozhukov et al. (2021), "An Exact and Robust Conformal Inference Method for Counterfactual and Synthetic Controls" - Conformal inference

• Athey et al. (2021), "Matrix Completion Methods for Causal Panel Data Models" - Unifying framework

• Ben-Michael, Feller, and Rothstein (2021), "The Augmented Synthetic Control Method" - Bias correction

Applications

• Abadie and Gardeazabal (2003), "The Economic Costs of Conflict: A Case Study of the Basque Country" - Original SC application

• Donohue and Levitt (2001) / Foote and Goetz (2008) - Abortion and crime (SC reanalysis)

• Gobillon and Magnac (2016), "Regional Policy Evaluation: Interactive Fixed Effects and Synthetic Controls" - IFE application

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Exercises

Conceptual

1. Explain why synthetic control requires the treated unit to lie within the convex hull of the donor pool. What happens when this condition fails? What alternatives exist?

2. In what sense does interactive fixed effects "nest" standard two-way fixed effects? When does the distinction matter for treatment effect estimation?

3. Compare permutation inference and conformal inference for synthetic control. What are the key assumptions of each? When would you prefer one over the other?

Applied

1. Download state-level smoking data and replicate the California Proposition 99 analysis:

   • Construct a synthetic California from the donor pool

   • Produce placebo tests for all donor states

   • Calculate the permutation p-value

   • Assess sensitivity to donor pool composition

2. Using cross-country GDP data, construct a synthetic control for a country that underwent a major policy change (e.g., trade liberalization, EU accession). Discuss challenges in achieving good pre-treatment fit.

Discussion

1. A critic argues: "Synthetic control is just fancy curve fitting. If you try enough weight combinations, you can match any pre-treatment path, but that doesn't mean the post-treatment comparison is causal." How would you respond? What distinguishes valid from invalid synthetic control applications?