SDID Estimator - Unit and Time Weights

Summary

SDID is computed in four steps (Algorithm 1): (1) set a ridge penalty $\zeta$ from the scale of one-period outcome changes; (2) solve a penalised, intercept-augmented synthetic-control problem for unit weights $\hat{\omega }$ on the simplex; (3) solve the transposed problem — regress each control unit’s post-period average on its pre-period outcomes — for time weights $\hat{\lambda }$ on the simplex; (4) run TWFE DiD weighted by ${\hat{\omega }}_i{\hat{\lambda }}_t$. Equivalently, ${\hat{\tau }}^{sdid}$ is a weighted double difference $\hat{\tau }(\omega ,\lambda )$ of the four blocks of $Y$. The intercepts are what distinguish SDID weights from SC weights: they only need to make trends parallel, not identical, because the fixed effects absorb level gaps.

Overview

Partition the outcome matrix by control/treated rows and pre/post columns:

$$Y=\left(\begin{array}{cc}{Y}_{co,pre}& Y_{co,post}\\ Y_{tr,pre}& Y_{tr,post}\end{array}\right)$$

SC uses $Y_{co,pre}$ to learn how to combine rows (units) to mimic the treated rows. SDID does that and uses $Y_{co,pre},Y_{co,post}$ to learn how to combine columns (pre-periods) to mimic the post-period columns. The symmetry is the key idea: unit weights are a “vertical regression”, time weights are a “horizontal regression”, and both are fit only on cells that are never treated.

Main Content

Unit weights (eq. 2.1) ^def-sdid-unit-weights

$$({\hat{\omega }}_0,{\hat{\omega }}^{sdid})=\arg \underset{{\omega }_0\in \mathbb{R},\omega \in \Omega }{\min }\sum _{t=1}^{T_{pre}}{\left({\omega }_0+\sum _{i=1}^{N_{co}}{\omega }_i{Y}_{it}-\frac{1}{N_{tr}}\sum _{i=N_{co}+1}^N{Y}_{it}\right)}^2+{\zeta }^2{T}_{pre}\Vert \omega {\Vert }_2^2$$ $$\Omega =\left\{\omega \in {\mathbb{R}}_{+}^N:\sum _{i=1}^{N_{co}}{\omega }_i=1,{\omega }_i=N_{tr}^{-1}\text{for }i>N_{co}\right\}$$

Two differences from Abadie–Diamond–Hainmueller weights: (a) the intercept ${\omega }_0$ — the weighted controls need only be parallel to the treated average, since ${\alpha }_i$ in the final regression absorbs constant gaps; (b) the ridge penalty, following Doudchenko & Imbens (2016), which disperses the weights and makes them unique. With ${\omega }_0=0$ and $\zeta =0$, (2.1) is exactly an ADH weight choice for $N_{tr}=1$.

Regularisation parameter (eq. 2.2) ^def-sdid-zeta

$$\zeta =(N_{tr}{T}_{post}{)}^{1/4}\hat{\sigma },{\hat{\sigma }}^2=\frac{1}{N_{co}(T_{pre}-1)}\sum _{i=1}^{N_{co}}\sum _{t=1}^{T_{pre}-1}{\left({\Delta }_{it}-\bar{\Delta }\right)}^2,{\Delta }_{it}=Y_{i(t+1)}-Y_{it}$$

$\hat{\sigma }$ is the standard deviation of first differences among controls in the pre-period — “the size of a typical one-period outcome change” — and $(N_{tr}{T}_{post}{)}^{1/4}$ is a theoretically motivated scaling (Theorem 1 requires $(N_{tr}{T}_{post}{)}^{1/2}\log N_{co}=o({\zeta }^2)$). No cross-validation is involved.

Time weights (eq. 2.3) ^def-sdid-time-weights

$$({\hat{\lambda }}_0,{\hat{\lambda }}^{sdid})=\arg \underset{{\lambda }_0\in \mathbb{R},\lambda \in \Lambda }{\min }\sum _{i=1}^{N_{co}}{\left({\lambda }_0+\sum _{t=1}^{T_{pre}}{\lambda }_t{Y}_{it}-\frac{1}{T_{post}}\sum _{t=T_{pre}+1}^T{Y}_{it}\right)}^2$$ $$\Lambda =\left\{\lambda \in {\mathbb{R}}_{+}^T:\sum _{t=1}^{T_{pre}}{\lambda }_t=1,{\lambda }_t=T_{post}^{-1}\text{for }t>T_{pre}\right\}$$

No ridge penalty here (only a numerical $\zeta =10^{-6}\hat{\sigma }$ for uniqueness, fn. 3). The asymmetry “reflects the fact we allow for correlated observations within time periods for the same unit, but not across units within a time period” beyond the factor structure (p. 7). Interpretation: the weighted average of a control unit’s history should predict its post-period average up to a constant ${\lambda }_0$.

Algorithm 1 — SDID ^alg-sdid

Input: $Y$, $W$. Output: ${\hat{\tau }}^{sdid}$.

1. Compute $\zeta$ via (2.2).
2. Compute unit weights ${\hat{\omega }}^{sdid}$ via (2.1).
3. Compute time weights ${\hat{\lambda }}^{sdid}$ via (2.3).
4. Solve the weighted TWFE regression

$$({\hat{\tau }}^{sdid},\hat{\mu },\hat{\alpha },\hat{\beta })=\arg \min \sum _{i,t}(Y_{it}-\mu -{\alpha }_i-{\beta }_t-W_{it}\tau {)}^2{\hat{\omega }}_i^{sdid}{\hat{\lambda }}_t^{sdid}$$

Covariates (fn. 4): apply SDID to residuals $Y_{it}^{res}=Y_{it}-X_{it}\hat{\beta }$ from a regression of $Y_{it}$ on time-varying exogenous $X_{it}$.

Weighted double-differencing representation (eqs. 2.4-2.5, 4.3) ^thm-sdid-double-diff

For any $\omega \in \Omega$, $\lambda \in \Lambda$ define

$$\hat{\tau }(\omega ,\lambda )={\omega }_{tr}^{\top }{Y}_{tr,post}{\lambda }_{post}-{\omega }_{co}^{\top }{Y}_{co,post}{\lambda }_{post}-{\omega }_{tr}^{\top }{Y}_{tr,pre}{\lambda }_{pre}+{\omega }_{co}^{\top }{Y}_{co,pre}{\lambda }_{pre}$$

Then ${\hat{\tau }}^{did}=\hat{\tau }(\text{uniform},\text{uniform})$ and ${\hat{\tau }}^{sdid}=\hat{\tau }({\hat{\omega }}^{sdid},{\hat{\lambda }}^{sdid})$. Equivalently, every estimator in the family is

$$\hat{\tau }={\hat{\delta }}_{tr}-\sum _{i=1}^{N_{co}}{\hat{\omega }}_i{\hat{\delta }}_i,{\hat{\delta }}_{tr}=\frac{1}{N_{tr}}\sum _{i>N_{co}}{\hat{\delta }}_i$$

with unit-level “adjusted outcomes”

$${\hat{\delta }}_i^{sc}=\frac{1}{T_{post}}\sum _{t>T_{pre}}{Y}_{it},{\hat{\delta }}_i^{did}=\frac{1}{T_{post}}\sum _{t>T_{pre}}{Y}_{it}-\frac{1}{T_{pre}}\sum _{t\le T_{pre}}{Y}_{it},{\hat{\delta }}_i^{sdid}=\frac{1}{T_{post}}\sum _{t>T_{pre}}{Y}_{it}-\sum _{t\le T_{pre}}{\hat{\lambda }}_t^{sdid}{Y}_{it}$$

So the final regression never has to be run as a regression: SDID is four weighted block means.

What the intercepts and fixed effects buy: invariance

Because of ${\omega }_0$, ${\lambda }_0$ and the two-way fixed effects, ${\hat{\tau }}^{sdid}$ is unchanged if $L_{it}\leftarrow L_{it}+{\alpha }_i+{\beta }_t$ for any ${\alpha }_i,{\beta }_t$ (§4, p. 20). DiD shares this invariance; SC is invariant to ${\beta }_t$ shifts only (fn. 8). This is why SDID does not require the treated unit to lie in the convex hull of control levels — a classic SC failure mode discussed in Synthetic Control Requirements.

What the time weights do

Contrast three conventions for the pre-period baseline (p. 9):

• DiD: all pre-periods equally, ${\lambda }_t=1/T_{pre}$.
• Event studies: only the last pre-period (${\lambda }_{-1}=1$), since coefficients are normalised to $\ell =-1$ — see Event Study Designs and Dynamic Treatment Effects.
• SDID: data-driven. Periods are chosen so that “the weighted average of historical outcomes predict average treatment period outcomes for control units, up to a constant.”

With serially correlated noise, the oracle time weights shrink not toward zero but toward the autoregression vector $\psi ={\Sigma }_{pre,pre}^{-1}{\Sigma }_{pre,post}{\lambda }_{post}$ (eq. 4.9) — the population coefficient from regressing the post-period average error on pre-period errors. This is why SDID can be more precise than DiD even when TWFE is correctly specified: it differences out the predictable part of the post-period noise.

Staggered adoption (Appendix §8)

“With staggered adoption the weighted DID regression approach in SDID does not work directly.” The paper’s fix: for each adoption date $a$, form a block-assignment sub-panel of the never-treated units plus the cohort adopting at $a$; run SDID on each; average the estimates with weights equal to each sub-panel’s share of treated unit-period cells. (Alternatively split by time periods.) This mirrors the cohort-by-cohort logic of Group-Time Average Treatment Effects.

Examples

The California weights (Appendix §7.2). Time weights put all mass on the last three pre-years:

Year	DID ${\lambda }_t$	SDID ${\hat{\lambda }}_t$
1988	0.053	0.427
1987	0.053	0.206
1986	0.053	0.366
1970–1985	0.053 each	0.000

Unit weights: SC is sparse — Utah 0.396, Montana 0.232, Nevada 0.204, Connecticut 0.104, New Hampshire 0.045, Colorado 0.013, Delaware 0.004, everything else 0. SDID spreads mass across roughly 30 of 38 states (largest: Nevada 0.124, New Hampshire 0.105, Connecticut 0.078, Delaware 0.070), a consequence of the ridge penalty and the intercept. Figure 1 shows that under DiD and SC a single state (New Hampshire) has very high influence ${\hat{\omega }}_i({\hat{\delta }}_{tr}-{\hat{\delta }}_i)$, whereas “SDID does not give any state particularly high influence.”

Code sketch (NumPy/CVXPY; the double-difference form avoids fitting the regression):

import numpy as np, cvxpy as cp
 
def sdid(Y, N0, T0):
    N, T = Y.shape; N1, T1 = N - N0, T - T0
    d = np.diff(Y[:N0, :T0], axis=1)
    sigma = d.std()                               # eq. (2.2): divisor N_co (T_pre - 1)
    zeta = (N1 * T1) ** 0.25 * sigma
 
    # unit weights: rows = pre-periods, cols = control units
    w, w0 = cp.Variable(N0, nonneg=True), cp.Variable()
    target = Y[N0:, :T0].mean(axis=0)
    cp.Problem(cp.Minimize(cp.sum_squares(w0 + Y[:N0, :T0].T @ w - target)
                           + zeta**2 * T0 * cp.sum_squares(w)),
               [cp.sum(w) == 1]).solve()
 
    # time weights: rows = control units, cols = pre-periods
    l, l0 = cp.Variable(T0, nonneg=True), cp.Variable()
    target_t = Y[:N0, T0:].mean(axis=1)
    cp.Problem(cp.Minimize(cp.sum_squares(l0 + Y[:N0, :T0] @ l - target_t)
                           + (1e-6 * sigma)**2 * N0 * cp.sum_squares(l)),
               [cp.sum(l) == 1]).solve()
 
    omega = np.r_[-w.value, np.ones(N1) / N1]     # signed unit contrast
    lam   = np.r_[-l.value, np.ones(T1) / T1]     # signed time contrast
    return omega @ Y @ lam, w.value, l.value      # eq. (4.3)

Connections

• Synthetic Control — the ${\omega }_0=0,\zeta =0$ special case of the unit-weight problem.
• Synthetic Control Requirements — convex-hull and fit requirements that the intercept relaxes.
• Synthetic Control Extensions — penalised SC and augmented SC; with a linear outcome model $\hat{m}(Y_{i,pre})={\hat{\lambda }}_0+Y_{i,pre}{\hat{\lambda }}_{pre}$ constrained to the simplex, augmented SC equals SDID-without-intercept (§6, eq. 6.1).
• Fixed-Effects Model — step 4 is a weighted TWFE regression.
• Covariate Balance and Matching Diagnostics — SDID weights are balancing weights, but for unobserved factors $\Gamma ,{\rm Y}$ rather than observed covariates (§6, p. 34).

See Also

• SDID vs DiD vs Synthetic Control
• SDID Inference - Bootstrap, Jackknife and Placebo
• Generalized Synthetic Control Method
• Differences-in-Differences