45 Panel Matching for Time-Series Cross-Sectional Data
The matching methods covered earlier in this part treat each unit as a single cross-sectional observation. Time-series cross-sectional (TSCS) data, where the same units are observed repeatedly over time and treatment can switch on and off, calls for a different matching logic: instead of matching a treated unit to similar control units at a single point in time, we match a treated unit-period to control unit-periods that share its recent treatment history. Imai, Kim, and Wang (2023) formalize this as panel matching, implemented in the PanelMatch package, and show that a properly weighted matching estimator is what a dynamic two-way fixed-effects regression is implicitly trying, and often failing, to approximate under treatment effect heterogeneity (Imai, Kim, et al. 2021).
This section develops the panel matching framework, replicates the package’s canonical democratization-and-growth application, and works through the diagnostics that distinguish a credible panel matching estimate from an unweighted two-way fixed-effects regression that happens to produce a similar point estimate.
45.1 Why Two-Way Fixed Effects Is Not Enough
A common dynamic linear specification for TSCS data with a policy indicator \(X_{it}\) regresses the outcome on current and lagged treatment, unit and time fixed effects, and lagged outcomes to soak up serial correlation: \[ Y_{it} = \alpha_i + \gamma_t + \beta X_{it} + \sum_{\ell=1}^{L}\left(\rho_\ell Y_{i,t-\ell} + \zeta_\ell' Z_{i,t-\ell}\right) + \varepsilon_{it}. \] Imai and Kim (2019, 2021) show that this specification, and the simpler static two-way fixed-effects model without lagged outcomes, is approximately equivalent to a variance-weighted average of all pairwise comparisons between units and between time periods, and that these weights can be negative whenever treatment effects vary over time or across units (Imai and Kim 2019, 2021). When that happens, the sign of the estimated \(\hat\beta\) can differ from the sign of every unit-level effect actually in the data, the same “forbidden comparison” problem that motivates the Goodman-Bacon decomposition for two-period-style difference-in-differences. Panel matching addresses the same underlying problem from a design-based rather than a regression-based angle.
45.2 The Panel Matching Framework
Let \(X_{it} \in \{0,1\}\) denote the treatment, \(D_{it} = X_{it}(1-X_{i,t-1})\) the event of treatment onset, and \(L\) the number of pre-treatment periods used to define comparability.
45.2.1 Matched Sets
For every treated onset \((i,t)\) with \(D_{it}=1\), the matched set collects control units that share the treated unit’s treatment history over the preceding \(L\) periods and were themselves untreated at \(t\): \[ \mathcal{M}_{it} = \{\, i' : i' \neq i,\ X_{i't}=0,\ X_{i't'} = X_{it'} \text{ for } t'=t-1,\dots,t-L \,\}. \] This is the panel analogue of a risk set in survival analysis, restricted to comparisons with an identical recent treatment trajectory rather than merely a shared time index.
45.2.2 Refinement
Raw matched sets can be large and heterogeneous on other pre-treatment characteristics, so they are refined either by Mahalanobis distance, \[ S_{it}(i') = \frac{1}{L}\sum_{\ell=1}^{L} \sqrt{(V_{i,t-\ell}-V_{i',t-\ell})' \Sigma_{i,t-\ell}^{-1} (V_{i,t-\ell}-V_{i',t-\ell})}, \] or by a propensity score for treatment onset conditional on the same lagged covariate history, \(e_{it}(\{V_{i,t-\ell}\}_{\ell=1}^L) = \Pr(X_{it}=1 \mid \{V_{i,t-\ell}\}_{\ell=1}^L)\), used either to keep the \(J\) closest controls or to weight all controls in the set.
45.2.3 Estimator
The treatment effect at horizon \(F\) periods after onset is the weighted difference-in-differences, \[ \widehat{\text{ATT}}(F) = \frac{1}{\sum_i\sum_t D_{it}} \sum_i \sum_{t} D_{it} \left[ (Y_{i,t+F}-Y_{i,t-1}) - \sum_{i' \in \mathcal{M}_{it}} w_{it}^{i'} (Y_{i',t+F}-Y_{i',t-1}) \right], \] which Imai, Kim, and Wang show is numerically equivalent to a properly weighted two-way fixed-effects regression, so panel matching does not discard the fixed-effects machinery, it corrects the implicit weights that machinery uses (Imai, Kim, et al. 2021).
45.2.4 Identifying Assumptions
Three assumptions justify the estimator: no interference between units (a unit’s outcome does not depend on other units’ treatment status), limited treatment carryover (the effect of treatment onset is fully realized within the \(L\)-period matching window plus \(F\)-period lead), and parallel trends conditional on matched treatment history, meaning treated and matched control units would have followed the same outcome trajectory absent the treated unit’s onset, given that both share the same treatment, outcome, and covariate history over the preceding \(L\) periods.
45.3 Replication: Democratization and Economic Growth
We replicate the package’s canonical application, based on Acemoglu, Naidu, Restrepo, and Robinson’s data on democratic transitions and growth, bundled directly with the package.
# install.packages("PanelMatch")
library(PanelMatch)
library(tidyverse)
data(dem)
head(dem[, c("wbcode2", "year", "dem", "y", "tradewb")])
#> wbcode2 year dem y tradewb
#> 1 2 1960 0 NA NA
#> 2 2 1961 0 NA 11.47824
#> 3 2 1962 0 NA 12.97522
#> 4 2 1963 0 NA 18.52119
#> 5 2 1964 0 NA 25.75280
#> 6 2 1965 0 NA 29.31384dem is binary (democratization), y is log GDP per capita, and tradewb is a time-varying covariate (trade openness) used for refinement. Since version 3 of the package, the panel structure is declared once in a PanelData object, which every downstream function then consumes.
dem_panel <- PanelData(
panel.data = dem,
unit.id = "wbcode2",
time.id = "year",
treatment = "dem",
outcome = "y"
)
DisplayTreatment(
dem_panel,
xlab = "year",
ylab = "country code",
legend.position = "none"
)
45.3.1 Constructing and Refining Matched Sets
pm_obj <- PanelMatch(
lag = 4,
panel.data = dem_panel,
refinement.method = "mahalanobis",
covs.formula = ~ tradewb,
size.match = 5,
qoi = "att",
lead = 0:4,
match.missing = TRUE,
forbid.treatment.reversal = FALSE,
use.diagonal.variance.matrix = TRUE
)
summary(pm_obj$att)
#> $overview
#> wbcode2 year matched.set.size
#> 1 4 1992 74
#> 2 4 1997 2
#> 3 6 1973 63
#> 4 6 1983 73
#> 5 7 1991 81
#> 6 7 1998 1
#> 7 12 1992 74
#> 8 13 2003 58
#> 9 15 1991 81
#> 10 16 1977 63
#> 11 17 1991 81
#> 12 18 1991 81
#> 13 22 1991 81
#> 14 25 1982 72
#> 15 26 1985 76
#> 16 31 1993 65
#> 17 34 1990 79
#> 18 36 2000 57
#> 19 38 1992 74
#> 20 40 1990 79
#> 21 40 1996 1
#> 22 40 2002 1
#> 23 41 1991 81
#> 24 45 1993 65
#> 25 47 1999 59
#> 26 50 1978 63
#> 27 52 1979 60
#> 28 55 1978 63
#> 29 56 1992 74
#> 30 57 1995 57
#> 31 59 1990 1
#> 32 64 1995 57
#> 33 65 1970 58
#> 34 65 1979 60
#> 35 65 1996 57
#> 36 70 1994 58
#> 37 70 2005 1
#> 38 72 1975 61
#> 39 73 1984 74
#> 40 75 1966 46
#> 41 75 1986 77
#> 42 78 1992 74
#> 43 80 1982 72
#> 44 81 2000 57
#> 45 82 2006 49
#> 46 83 1990 79
#> 47 84 1999 59
#> 48 96 2002 58
#> 49 97 2005 54
#> 50 101 1988 80
#> 51 104 2005 54
#> 52 105 2004 57
#> 53 109 1993 65
#> 54 110 1993 65
#> 55 112 1993 65
#> 56 115 1994 58
#> 57 116 1993 65
#> 58 118 1997 58
#> 59 119 1991 81
#> 60 120 1992 74
#> 61 123 1993 65
#> 62 124 1994 58
#> 63 128 1994 58
#> 64 133 1991 81
#> 65 134 1979 60
#> 66 134 1999 59
#> 67 135 1990 79
#> 68 138 1991 81
#> 69 138 2006 49
#> 70 141 1972 63
#> 71 141 1988 80
#> 72 142 1994 58
#> 73 143 1980 61
#> 74 144 1987 78
#> 75 149 1976 63
#> 76 150 1993 65
#> 77 154 1990 79
#> 78 155 1993 65
#> 79 158 1965 40
#> 80 158 1986 77
#> 81 159 2000 57
#> 82 162 2004 57
#> 83 163 1996 57
#> 84 163 2001 59
#> 85 164 1982 72
#> 86 167 1988 80
#> 87 168 1993 65
#> 88 169 1992 74
#> 89 177 1974 62
#> 90 177 1978 1
#> 91 183 1983 2
#> 92 187 1994 58
#> 93 188 1985 76
#> 94 199 1994 58
#> 95 201 1991 81
#> 96 202 1978 63
#> 97 17 2009 0
#> 98 70 1999 0
#> 99 82 1994 0
#> 100 109 1999 0
#> 101 133 1999 0
#> 102 143 1993 0
#> 103 167 1991 0
#> 104 177 1992 0
#> 105 183 1973 0
#>
#> $set.size.summary
#> Min. 1st Qu. Median Mean 3rd Qu. Max.
#> 0.00 57.00 62.00 55.75 74.00 81.00
#>
#> $number.of.treated.units
#> [1] 105
#>
#> $num.units.empty.set
#> [1] 9
#>
#> $lag
#> [1] 4lag = 4 sets \(L=4\): only control units with the same democratization status over the preceding four years enter a treated unit’s matched set. size.match = 5 keeps the five Mahalanobis-nearest controls per matched set.
45.3.2 Estimating the Treatment Effect
pe_obj <- PanelEstimate(
sets = pm_obj,
panel.data = dem_panel,
se.method = "bootstrap",
number.iterations = 1000
)
summary(pe_obj)
#> estimate std.error 2.5% 97.5%
#> t+0 -0.74479358 0.9007656 -2.610771 0.8962208
#> t+1 -0.49706679 1.6310276 -3.633504 2.7689336
#> t+2 -0.06272069 2.1380191 -4.219380 4.2171092
#> t+3 0.95377551 2.5300820 -3.786903 5.7865203
#> t+4 0.91032290 2.8315244 -4.766821 6.2680290
plot(pe_obj, ylab = "Estimated ATT of democratization on log GDP", main = "")
The estimated effects at each lead \(F=0,\dots,4\) trace out the dynamic path of democratization’s effect on income, directly comparable to an event-study plot but built from matched-set weighted differences rather than TWFE regression coefficients.
45.4 Show and Tell: Covariate Balance and Comparison to TWFE
45.4.1 Balance Diagnostics
Refinement should bring the treated and matched-control groups close on pre-treatment covariate trajectories; this is checked directly rather than assumed.
balance_check <- get_covariate_balance(
pm_obj,
panel.data = dem_panel,
covariates = c("tradewb")
)
summary(balance_check)
#> $pm_obj
#> tradewb_unrefined tradewb
#> t_4 -0.07245466 0.20189953
#> t_3 -0.20930129 0.06080927
#> t_2 -0.24425207 0.02641510
#> t_1 -0.10806125 0.08635853
#> t_0 -0.09493854 0.05376340Standardized differences well outside \(\pm 0.1\) at any lag indicate the refinement is not achieving comparability on that covariate, in which case the matching formula, the number of matches retained (size.match), or the lag window (lag) should be revisited before the effect estimate is treated as credible.
45.4.2 Comparison to a Naive Dynamic TWFE Regression
library(fixest)
twfe_fit <- feols(
y ~ dem | wbcode2 + year,
data = dem,
cluster = ~wbcode2
)
modelsummary::modelsummary(
list("Static TWFE" = twfe_fit),
stars = TRUE,
gof_omit = "IC|Log|Adj"
)| Static TWFE | |
|---|---|
| + p < 0.1, * p < 0.05, ** p < 0.01, *** p < 0.001 | |
| dem | -10.112* |
| (4.316) | |
| Num.Obs. | 6934 |
| R2 | 0.970 |
| R2 Within | 0.011 |
| RMSE | 26.76 |
| Std.Errors | by: wbcode2 |
| FE: wbcode2 | X |
| FE: year | X |
The static TWFE coefficient on dem collapses every lead and every matched-set weight into a single number; comparing it to the leadwise panel matching estimates from pe_obj shows directly whether the TWFE estimate is an artifact of its implicit weighting or a reasonable summary of the panel matching dynamic path. A substantial divergence, particularly a sign flip or an estimate outside the panel matching confidence bands, is the signature of the negative-weighting problem discussed above.
45.5 Extension: Refinement Choice and Robustness
Two robustness checks are standard before reporting a panel matching estimate.
-
Refinement method. Reestimate with
refinement.method = "ps.match"or"ps.weight"in place of"mahalanobis"and confirm the ATT path is qualitatively stable; propensity-score refinement can behave differently than distance matching when the covariate history is high-dimensional relative to the number of treated onsets. -
Lag window. Reestimate with
lag = 2andlag = 6in place oflag = 4. A short lag window risks matching on too little treatment history to satisfy parallel trends; an overly long window shrinks the usable sample of matched sets and can induce more missingness in the refinement covariates.
pm_obj_ps <- PanelMatch(
lag = 4,
panel.data = dem_panel,
refinement.method = "ps.weight",
covs.formula = ~ tradewb,
qoi = "att",
lead = 0:4,
match.missing = TRUE,
forbid.treatment.reversal = FALSE
)
pe_obj_ps <- PanelEstimate(sets = pm_obj_ps, panel.data = dem_panel,
se.method = "bootstrap", number.iterations = 1000)
summary(pe_obj_ps)
#> estimate std.error 2.5% 97.5%
#> t+0 -1.2468200 0.8195036 -2.850091 0.2764356
#> t+1 -1.4896010 1.4830262 -4.588859 1.2605824
#> t+2 -1.2236333 1.9786283 -5.421453 2.3639696
#> t+3 -0.7999444 2.3439938 -5.721563 3.5926832
#> t+4 -1.0413993 2.6802145 -6.611552 3.854951445.6 Applied Relevance
Panel matching is the natural tool whenever a policy or business intervention rolls out to different units at different times and the analyst wants to avoid the implicit, potentially negative weighting a dynamic fixed-effects regression can silently impose. Typical applications include state-level policy adoption studies, staggered product or feature launches across markets, and firm-level technology adoption, all settings already discussed from the difference-in-differences side in Staggered Difference-in-Differences. Panel matching is a complementary, not competing, estimator in these settings: it targets the same group-time treatment effects as Callaway and Sant’Anna’s estimator, but does so by explicit unit matching on treatment history rather than by group-time aggregation, and comparing the two is itself a useful robustness check when both are available.
Key Takeaway: with repeated observations on the same units and treatment that turns on at different times, match unit-periods on their recent treatment history before comparing outcomes, rather than relying on a two-way fixed-effects regression to do that weighting implicitly. The two approaches coincide when treatment effects are homogeneous and diverge, sometimes in sign, when they are not.