Dynamic Covariate Balancing DiD#
The diddynamic module implements the dynamic covariate balancing (DCB) estimator of
Viviano and Bradic (2026) for panel data where
treatments change over time. It handles settings where treatment assignments depend on
high-dimensional covariates, past outcomes, and past treatments, and where outcomes and
time-varying covariates may depend on the entire trajectory of past treatments.
Think about a country deciding whether to adopt democratic institutions. That decision is shaped by past economic performance, the political trajectories of neighbours, and dozens of other observable factors. Once the country transitions, its GDP, trade flows, and institutional quality all change, which in turn influence whether democracy persists in the next period. This kind of feedback loop between treatments and outcomes over time is precisely what makes dynamic treatment regimes so challenging to analyse. The standard DiD toolkit, built for settings where treatment is a one-time permanent event, simply cannot accommodate this back-and-forth.
Why standard approaches fall short#
Before introducing DCB, it helps to understand what goes wrong with the usual tools.
Standard DiD and event studies assume staggered adoption, where once a unit receives treatment it stays treated forever. When countries can switch in and out of democracy based on past economic outcomes, the parallel trends assumption breaks down (Ghanem et al., 2022; Marx et al., 2022). TWFE compounds the problem by collapsing different treatment sequences into a single coefficient, and negative weighting gets worse because the pool of “control” units shifts every period.
Standard local projections (Jordà, 2005) regress observed outcomes on current and lagged treatments plus covariates. The trouble is that this model is written in terms of observed rather than potential outcomes. With dynamic treatment selection, the resulting coefficient conflates the causal effect with the distribution of future treatment decisions, which means it depends on the propensity score. In the empirical application of Viviano and Bradic (2026), local projections substantially underestimate long-run treatment effects compared to DCB.
Inverse probability weighting (IPW) takes a different approach, reweighting each unit by the probability of the treatment sequence it actually experienced. For \(T\) periods, the weight for unit \(i\) is the product of \(T\) conditional probabilities,
and this product can blow up fast. If any single-period propensity score is close to zero, the whole product explodes. In many empirical settings the estimated probability of following a given treatment path for just two consecutive periods already drops below 0.1 for some units, making IPW weights for longer histories wildly variable. DCB sidesteps this entirely by constructing balancing weights through a quadratic program that never estimates the propensity score at all.
Setup and notation#
We observe a panel of \(n\) i.i.d. units over \(T\) periods. For unit \(i\) in period \(t\), let \(X_{i,t}\) denote time-varying covariates, \(D_{i,t} \in \{0,1\}\) the binary treatment, and \(Y_{i,t}\) the outcome. All the information available up to (but not including) the treatment decision at time \(t\) is collected in the history vector
which grows with \(t\) as each additional period contributes its own covariates, treatments, and outcomes. Since those covariates and outcomes may themselves depend on earlier treatments, we also define the potential history under treatment path \(d_{1:(t-1)}\),
capturing the covariates and outcomes that would have been observed had the unit followed treatment path \(d_{1:(t-1)}\).
What we are estimating#
The target is the average treatment effect of two treatment histories \(d_{1:T}\) and \(d'_{1:T}\),
where \(Y_T(d_{1:T})\) is the potential outcome at the final period under the full history \(d_{1:T}\). This captures the total effect, including both direct effects on the outcome and indirect effects that propagate through intermediate covariates and outcomes.
A few concrete examples make this more tangible.
\(\text{ATE}((1,1),(0,0))\) asks what happens when a unit is treated for two consecutive periods compared to untreated for two periods. This is the most common target in short-panel applications.
\(\text{ATE}((1,0),(0,0))\) isolates the direct effect of a single period of treatment that is then reversed. Comparing this to \(\text{ATE}((1,1),(0,0))\) reveals how much of the total effect comes from sustained versus one-time exposure.
With long panels, we often want to average over earlier treatment assignments and focus on the last \(h\) periods. The resulting estimand,
\[\mathbb{E}\bigl[Y_T(D_{1:(T-h)}, d_{(T-h+1):T})\bigr] - \mathbb{E}\bigl[Y_T(D_{1:(T-h)}, d'_{(T-h+1):T})\bigr],\]is what the
histories_lengthoption targets. Varying \(h\) traces out how the treatment effect evolves with exposure length.
Identifying assumptions#
Identification rests on three core assumptions that generalise the standard unconfoundedness framework to the dynamic setting. We develop these first for two periods, where the logic is easiest to follow, and then state the general versions.
The two-period case#
With two periods, we observe \((X_{i,1}, D_{i,1}, Y_{i,1}, X_{i,2}, D_{i,2}, Y_{i,2})\) for each unit and define \(H_{i,2} = [D_{i,1}, X_{i,1}, X_{i,2}, Y_{i,1}]\). The potential outcome \(Y_{i,2}(d_1, d_2)\) represents the outcome a unit would achieve if it received treatment \(d_1\) in period 1 and \(d_2\) in period 2.
Assumption 1 (No Anticipation)
For \(d_1 \in \{0,1\}\), let \(Y_{i,1}(d_1, 1) = Y_{i,1}(d_1, 0)\) and \(X_{i,2}(d_1, 1) = X_{i,2}(d_1, 0)\).
Intermediate outcomes and covariates depend only on past treatments, not on future ones. Treatment at \(t = 2\) has no contemporaneous effect on covariates.
This is a standard restriction in the causal inference literature. It allows forward-looking behaviour (a unit may choose treatment in anticipation of future benefits) but rules out effects from treatment realisations that haven’t happened yet.
Assumption 2 (Sequential Ignorability)
For all \((d_1, d_2) \in \{0,1\}^2\),
\(Y_{i,2}(d_1, d_2) \perp D_{i,2} \mid D_{i,1}, X_{i,1}, X_{i,2}, Y_{i,1}\),
\((Y_{i,2}(d_1, d_2), H_{i,2}(d_1)) \perp D_{i,1} \mid X_{i,1}\).
Part (A) says that, once we condition on everything observed through period 1, the second-period treatment is as good as random. Part (B) says the same for the first-period treatment conditional on baseline covariates. Together, these allow treatment decisions to depend on all observed history (including past outcomes and treatments) as long as there are no unobserved confounders after conditioning.
Assumption 3 (Potential Local Projections)
For some \(\beta_{d_1,d_2}^{(1)} \in \mathbb{R}^{p_1}\) and \(\beta_{d_1,d_2}^{(2)} \in \mathbb{R}^{p_2}\),
This is where DCB departs from both standard local projections and from IPW. Following the spirit of Jordà (2005), Assumption 3 imposes linearity, but on expected potential outcomes rather than observed ones. This distinction matters greatly. A model on realised outcomes would tie the estimated treatment effect to the propensity score, whereas a potential outcome model does not. Coefficients can differ across treatment histories \((d_1, d_2)\), and the dimensions \(p_1, p_2\) are allowed to grow with \(n\), so the model can accommodate large numbers of covariates and their transformations. In high dimensions, the linearity can be relaxed to an approximation accurate to \(o(n^{-1/2})\), which is enough for valid inference.
General case#
Moving from two to \(T\) periods is conceptually straightforward. The key change is that sequential ignorability now conditions on the full history \(H_{i,t}\) at each period rather than just baseline covariates or a single lag.
Assumption 1’ (No Anticipation, General)
For any \(d_{1:T} \in \{0,1\}^T\) and \(t \leq T\), the potential history \(H_{i,t}(d_{1:T})\) is constant in \(d_{t:T}\).
Assumption 2’ (Sequential Ignorability, General)
For all \(d_{1:T} \in \{0,1\}^T\) and each \(t \leq T\),
Assumption 3’ (Potential Local Projections, General)
For some \(\beta_{d_{1:T}}^{(t)} \in \mathbb{R}^{p_t}\),
Assumption 4 (Overlap)
\(P(D_{i,t} = d_t \mid H_{i,t}) \in (\delta, 1 - \delta)\) for some \(\delta \in (0,1)\) and each \(t \in \{1, \ldots, T\}\).
Overlap ensures that the dynamic balance constraints introduced below are feasible. Intuitively, every unit must have a positive probability of following the target treatment path at each step. The true IPW weights turn out to be one feasible set of weights, but DCB will find better ones.
From assumptions to estimable quantities#
The assumptions above connect the potential outcome model to things we can actually measure. In the two-period case, the identification result is especially clean.
Lemma (Identification, Two Periods). Under Assumptions 1–3, for any \((d_1, d_2) \in \{0,1\}^2\),
and, iterating the conditional expectation,
Read from top to bottom, the logic is recursive. The first line uses sequential ignorability (Assumption 2A) to swap the observed outcome for the potential outcome, then applies the linear model (Assumption 3) to express the conditional expectation in terms of \(H_{i,2}(d_1)\). The second line iterates backward, using Assumption 2B and the first-period projection to push the conditioning down to baseline covariates \(X_{i,1}\).
This two-step recursion is the key insight of the paper. It connects the marginal structural models literature (Robins et al., 2000) to local projections in economics, and it motivates the backward estimation strategy at the heart of DCB.
For \(T\) periods, the same recursion applies at each step. For every \(t\),
and projecting backward,
These relationships tell us how to estimate the coefficients from data (regress backward through time) and how to recover the potential outcome \(\mu_T(d_{1:T})\) by combining those estimates with balancing weights.
Estimation#
With the identification result in hand, estimation proceeds in two stages.
Recursive coefficient estimation#
The coefficients are estimated by working backward from the final period, with penalised regression at each step. Two model specifications are available, trading off flexibility against sample efficiency.
The fully interacted model starts by regressing \(Y_{i,T}\) onto \(H_{i,T}\) using only units with \(D_{i,1:T} = d_{1:T}\). It then regresses the fitted values \(H_{i,t+1}\hat{\beta}_{d_{1:T}}^{(t+1)}\) onto \(H_{i,t}\) for units with \(D_{i,1:t} = d_{1:t}\), working backward to \(t = 1\). This allows completely heterogeneous treatment effects but limits the sample at each step to units on the target path.
The linear model instead regresses \(Y_{i,T}\) onto \((H_{i,T}, D_{i,1:T})\) using all units, leaving treatment indicators unpenalised. It then plugs in the target history for the treatment indicators and proceeds backward. This pools information across treatment paths, improving precision with long histories at the cost of assuming treatment effects are additive and linear.
Both specifications use LASSO with cross-validated penalty, keeping treatment indicators
unpenalised to avoid shrinking the treatment effect toward zero. An alternative
lasso_subsample strategy partitions the data into separate fitting and evaluation sets,
which can improve stability when the sample is small relative to the number of covariates.
Sequential balancing weights#
Here is where DCB diverges most sharply from existing methods. Rather than estimating propensity scores and inverting them (as IPW does), DCB finds weights that directly balance the covariate distributions across treatment groups. The balancing is done sequentially, one period at a time, through a series of quadratic programs.
To see why balancing matters, consider the two-period estimator,
where \(\bar{X}_1\) is the sample mean. In low dimensions, the last term alone would give a consistent estimator. But with many covariates, the LASSO estimates \(\hat{\beta}\) have non-negligible bias, and the first two terms are the correction. They reweight the period-specific residuals so that the high-dimensional estimation error washes out.
When we decompose the estimation error, the key term is
This has a beautiful product-of-rates structure. Each factor is the product of a coefficient estimation error and a covariate imbalance term. To make the bias vanish at the \(o(n^{-1/2})\) rate needed for valid inference, we need the weighted covariates under \(\hat{\gamma}_t\) to be close to those under \(\hat{\gamma}_{t-1}\).
The first imbalance term, \(\|\bar{X}_1 - \hat{\gamma}_1^\top X_1\|_\infty\), is the same static balancing condition that appears in cross-sectional studies (Athey et al., 2018).
The second, \(\|\hat{\gamma}_2^\top H_2 - \hat{\gamma}_1^\top H_2\|_\infty\), is new. It requires that the second-period histories be balanced after reweighting by the first-period weights. This is the dynamic balancing condition that gives the method its name.
For \(T\) periods, the estimator generalises to
and the estimation error decomposes into three terms. Define the residuals and prediction gaps as
Then
The bias \((I_1)\) is the product of coefficient errors and imbalances, exactly as in the two-period case. The remaining terms \((I_2)\) and \((I_3)\) are mean-zero as long as the weights satisfy two natural conditions: (i) \(\hat{\gamma}_t\) depends only on \((H_{i,t}, D_{i,t})\) and not on the outcome \(Y_{i,T}\), and (ii) \(\hat{\gamma}_{i,t} = 0\) whenever the unit’s treatment path doesn’t match the target. Both are built into the quadratic program by construction.
The DCB algorithm#
With the motivation in place, the algorithm itself is simple. Initialise \(\hat{\gamma}_{i,0} = 1/n\) and for each \(t \in \{1, \ldots, T\}\), solve
In words, at each period we find the weights with the smallest \(\ell_2\) norm (which maximises the effective sample size) subject to the dynamic balance constraint, a summing-to- one normalisation, non-negativity, an upper bound on individual weights, and the requirement that only units on the target treatment path receive positive weight.
The tuning parameters are set by the theory. The balance tolerance scales as
and a data-driven grid search selects the smallest \(K_{1,t}\) for which the quadratic
program has a feasible solution. This minimises bias first, then variance. When
adaptive_balancing=True, the algorithm tightens the constraints on covariates with large
estimated coefficients, further improving balance where it matters most.
Computationally, this is a sequence of \(T\) quadratic programs with linear constraints, so the cost scales polynomially in \(n\) and \(p\).
Why DCB beats IPW#
A natural question is why not just use IPW. The answer comes from a clean theoretical result. The normalised IPW weights,
are themselves a feasible solution to the DCB quadratic program. Since DCB minimises the \(\ell_2\) norm over a constraint set that includes IPW, the DCB weights are guaranteed to have variance no larger than IPW. And in practice the improvement is often dramatic. The effective sample size diagnostic \(1/(n\|\hat{\gamma}_t\|^2)\) typically shows DCB retaining several times more effective observations than IPW at the same horizon.
On top of lower variance, DCB has a major theoretical advantage. AIPW in high dimensions requires consistent estimation of both the propensity score and the outcome model, each at rate \(o_p(n^{-1/4})\). DCB requires only the outcome model condition, with no assumption on the propensity score at all. The balancing weights absorb the role of the propensity score through the quadratic program, and the product-of-rates structure
replaces the usual product of propensity score and outcome model errors with a product of outcome model error and balance tolerance, which is controlled mechanically rather than estimated.
Weight stability across periods. A further reassurance comes from the following result. The DCB weights satisfy
for a finite constant \(c_t\). The first bound says DCB beats IPW at every period. The second says the weights’ norm is controlled from one period to the next, preventing the explosive growth that plagues IPW in long panels.
Inference#
Convergence and variance#
Under the identifying assumptions and the coefficient consistency condition
the DCB estimator converges at the parametric \(n^{-1/2}\) rate, even with high-dimensional covariates, as long as
This condition is satisfied by LASSO under standard sparsity and restricted eigenvalue assumptions, either with sub-Gaussian covariates (\(\max_t \|\hat{\beta}^{(t)} - \beta^{(t)}\|_1 = O_p(n^{-1/4})\)) or uniformly bounded covariates (\(\max_t \|\hat{\beta}^{(t)} - \beta^{(t)}\|_1 = o_p(1/\log n)\)).
A natural worry with sequential estimation is that errors compound over time. They do, but in a controlled way. The total bias is bounded by
with the balance constants growing at rate \(K_{1,t} = \log^{1/2}(t)\). In practice, the effective sample size shrinks with longer histories, which is why reporting effects at multiple history lengths is the recommended diagnostic.
The analytical variance estimator is
with three terms reflecting three sources of uncertainty. The first captures the final-period residual variance (weighted by the squared final-period weights), the second captures the between-period prediction gaps (weighted by the squared intermediate weights), and the third captures the baseline covariate variation. The normalised estimator is asymptotically standard normal,
For the ATE comparing two histories \(d_{1:T}\) and \(d'_{1:T}\) with \(d_1 \neq d'_1\), the two estimators use disjoint sets of units, so
When conditioning on baseline covariates \(X_1\), the baseline variation term drops out,
and the ATE variance is the sum of these conditional variances.
Critical values and clustering#
Two flavours of critical values are available. Gaussian quantiles use the standard normal and give tighter intervals when the balancing weights are well-behaved. Robust quantiles use a chi-squared distribution that accounts for weight estimation error, providing valid coverage under weaker conditions. Robust quantiles are the default and are generally recommended.
When within-cluster correlation is a concern (countries in the same region, patients in the same hospital), the analytical variance is replaced by a cluster-robust sandwich,
where \(\psi_i\) is the influence function and \(\mathcal{C}_c\) is the set of units in cluster \(c\). This is valid under arbitrary within-cluster dependence.
Practical extensions#
Pooled regression#
By default, the estimator targets the outcome in the final period. With pooled=True,
the regression pools all periods into a single model with time fixed effects,
which increases the effective sample size at the cost of assuming the treatment effect is stable across periods. Standard errors are automatically clustered at the unit level to account for serial correlation, unless a larger clustering variable is specified.
Treatment history length#
With long panels, using the full treatment history thins the sample because only units
on the exact target path get positive weight. The histories_length option lets you
estimate effects at multiple horizons \(h \in \{h_1, \ldots, h_K\}\), each using the
last \(h\) elements of the treatment sequences. The resulting estimand,
averages over prior assignments and isolates the effect of the last \(h\) periods. Reporting a range of \(h\) values (say 1 through 10 in a long panel) traces out how the effect builds with exposure and reveals the precision trade-off at each horizon.
Impulse response#
Setting impulse_response=True flips the treatment sequences for each \(h\) to
\(d_{1:h} = (1, 0, \ldots, 0)\) versus \(d'_{1:h} = (0, \ldots, 0)\). This
measures the effect of a one-period treatment shock at increasing horizons, much like an
impulse response function in time series. It is particularly useful for studying how a
transient policy intervention propagates through the system over time.
Heterogeneous effects across periods#
The final_periods option estimates treatment effects at multiple final time points,
holding the treatment histories fixed. This reveals whether the same treatment sequence
produces different outcomes at different calendar times, which could reflect time-varying
confounders, cohort effects, or secular trends.
Note
For formal proofs, the connection to marginal structural models, detailed comparisons with local projections and DiD, and extensive Monte Carlo evidence, see the full paper by Viviano and Bradic (2026).