Missing panel outcomes

Masked matrix completion, available-case SDID and overlap-specific ridge

Missing outcomes are not zeros. Pass an N\times T outcome matrix containing NaN, or supply fit(Y, W, observed=mask) to ignore selected entries. The three estimators use the same observation mask but different fitting rules.

Estimator What a missing outcome changes
Matrix completion Removes that cell from the low-rank fitting loss.
Synthetic DID Removes its contribution from both the numerator and denominator of a weighted average.
Horizontal ridge Selects donors available at prediction time and fits on their jointly observed pre-periods.

W is a known treatment schedule, separate from outcome availability. When adoption dates are known, construct W[i, t] = (t >= adoption[i]), even where Y[i, t] is missing. Unknown treatment is not automatically coded as untreated.

A panel with holes

This example has six donors, two treated units and a constant treatment effect of 1.25. It removes donor and treated outcomes both before and after adoption.

Code 
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
from matplotlib.colors import ListedColormap
from IPython.display import display
import crabbymetrics as cm

rng = np.random.default_rng(941)
time = np.arange(14)
donors = rng.normal(size=(6, 1)) + rng.normal(size=(6, 2)) @ np.vstack(
    [np.sin(time / 3), time / 8]
)
baseline = 0.3 * donors[0] + 0.7 * donors[3] + 0.4
Y = np.vstack([donors, baseline - 0.15, baseline + 0.15])
W = np.zeros_like(Y)
W[6:, 8:] = 1
Y += 1.25 * W
observed = np.ones(Y.shape, dtype=bool)
for i, t in [(0, 1), (1, 3), (2, 10), (3, 12), (6, 2), (7, 9)]:
    observed[i, t] = False
Code 
state = np.where(observed, W, 2)
fig, ax = plt.subplots(figsize=(9, 3.2), layout="constrained")
im = ax.imshow(state, aspect="auto", vmin=0, vmax=2,
               cmap=ListedColormap(["#487ba0", "#df8f3e", "#dddddd"]))
ax.set(xlabel="Period", ylabel="Unit", title="Observed untreated, observed treated, missing")
bar = fig.colorbar(im, ax=ax, ticks=[0, 1, 2])
bar.ax.set_yticklabels(["Untreated", "Treated", "Missing outcome"])
plt.show()

Fit all three estimators

The explicit mask ignores the hidden values in Y. Replacing those values with NaN and omitting observed gives the same fits.

Code 
models = {
    "Matrix completion": cm.MatrixCompletion(lambda_fraction=0.03,
                                             max_iterations=3000, tolerance=1e-9),
    "Horizontal ridge": cm.HorizontalPanelRidge(penalty=0.2),
    "Synthetic DID": cm.SyntheticDID(zeta_omega=0.3, zeta_lambda=0.3,
                                     max_iterations=20000),
}
rows = []
for name, model in models.items():
    model.fit(Y, W, observed=observed)
    s = model.summary()
    rows.append({"Estimator": name, "ATT": s["att"],
                 "Observed treated cells": s["n_treated_observed"],
                 "Missing outcomes": s["n_missing"]})
display(pd.DataFrame(rows).round(4))
Estimator ATT Observed treated cells Missing outcomes
0 Matrix completion 0.9614 11 6
1 Horizontal ridge 1.2536 11 6
2 Synthetic DID 0.8099 11 6

All three ATTs target the observed treated cells. Matrix completion and ridge average their observed post-treatment prediction errors. SDID also subtracts its time-weighted pre-treatment gap. SDID’s displayed counterfactual uses the unit-fit intercept, so averaging its diagnostic effect matrix is not the SDID ATT. Missing treated outcomes have no reported effect, even when a counterfactual can be predicted there.

Matrix completion: mask the loss

Let R_{it} indicate an observed outcome and define \Omega=\{(i,t):R_{it}=1,\ W_{it}=0\}. The objective is

\min_{L,\alpha,\gamma} \frac{1}{|\Omega|}\sum_{(i,t)\in\Omega} (Y_{it}-L_{it}-\alpha_i-\gamma_t)^2+\lambda\lVert L\rVert_*.

The same mask is used in fixed-effect updates, the singular-value-thresholding step, penalty scaling and convergence diagnostics. Neither missing entries nor treated outcomes enter the loss. Low-rank predictions can cover both.

Each unit and period must have an observed untreated outcome. With both fixed effects enabled, the observed-untreated unit–period graph must be connected. These are support checks, not a guarantee that a particular low-rank model fits.

SDID: normalize over available observations

For donor weights \omega, the synthetic outcome at period t is

s_t(\omega)= \frac{\sum_{i\in C}R_{it}\omega_iY_{it}} {\sum_{i\in C}R_{it}\omega_i}.

The unit-weight fit compares this with the observed treated-cohort pre-period mean, using a free intercept and the existing squared-weight penalty. Time weights use the analogous available-period average for each donor. Their target is that donor’s observed post-period mean, weighted by the number of observed treated outcomes at each post-period and renormalized over the donor’s observed periods.

Both weight problems have the form

\min_{a,\,b\in\Delta} \frac{1}{|\mathcal J|}\sum_{j\in\mathcal J} \left[m_j-a- \frac{\sum_k R_{jk}b_kX_{jk}}{\sum_k R_{jk}b_k}\right]^2 +\zeta^2\lVert b\rVert_2^2.

Rows without an observed target and columns without training observations are excluded. Missing cells enter neither sum. A zero available-weight denominator is an error, not a zero synthetic outcome.

This is an available-case extension of SDID. Unlike the complete-panel quadratic problem, the denominators depend on the weights, so the objective can be nonconvex. The implementation profiles out the intercept and uses simplex-projected gradient descent with backtracking, initialized uniformly. It reports its objective and projected-gradient residual and fails if the iteration budget expires without stationarity. Complete panels retain the existing active-set quadratic solver.

Within a cohort, the ATT averages observed post-treatment gaps using the number of observed treated cells per period, then subtracts the fitted time-weighted pre-gap. Cohorts are combined using their observed treated post-cell counts. Default penalties use adjacent observed donor pre-period differences; gaps are not bridged when estimating the noise scale.

Code 
diagnostics = models["Synthetic DID"].summary()["missing_diagnostics"]
rows = []
for d in diagnostics:
    for kind in ["unit", "time"]:
        rows.append({"Cohort": d["cohort"], "Weights": kind,
                     "Iterations": d[f"{kind}_iterations"],
                     "Objective": d[f"{kind}_objective"],
                     "Gradient residual": d[f"{kind}_projected_gradient"]})
display(pd.DataFrame(rows))
Cohort Weights Iterations Objective Gradient residual
0 8 unit 67 0.034660 7.976549e-09
1 8 time 7 0.999626 0.000000e+00

Ridge: missing outcomes can also be missing predictors

Horizontal ridge uses donor outcomes as regressors. Dropping a missing response does not resolve a missing donor regressor. For each cohort and prediction period:

  1. Select donors observed at that period with some pre-period overlap.
  2. Select pre-periods with an observed treated-cohort mean and observations for every selected donor.
  3. Fit ordinary ridge on that observed design, with an unpenalized intercept, and predict from the selected donors.

Periods with identical donor availability reuse the same fitted model. No donor feature is filled with zero or its mean. If an observed treated post-period has no donor/training overlap, the fit raises an explicit error. This rule can still run out of common pre-periods in a very sparse panel; it does not manufacture a full design by filling missing regressors.

summary()["overlap_models"] identifies the donors, training periods and coefficients used for each prediction. When a cohort needs multiple models, its legacy single cohort_coef/cohort_intercepts row is NaN rather than displaying one model as if it applied to every period.

Code 
overlap = models["Horizontal ridge"].summary()["overlap_models"]
display(pd.DataFrame([{key: value for key, value in model.items()
                       if key != "coef"} for model in overlap]))
cohort periods donors training_periods intercept
0 8 [0, 2, 4, 5, 6, 7, 8, 9, 11, 13] [0, 1, 2, 3, 4, 5] [0, 2, 4, 5, 6, 7] 1.058420
1 8 [1] [1, 2, 3, 4, 5] [0, 1, 2, 4, 5, 6, 7] 0.986261
2 8 [3] [0, 2, 3, 4, 5] [0, 2, 3, 4, 5, 6, 7] 0.993076
3 8 [10] [0, 1, 3, 4, 5] [0, 2, 4, 5, 6, 7] 0.878841
4 8 [12] [0, 1, 2, 4, 5] [0, 2, 4, 5, 6, 7] 1.119325

What the checks establish

The test suite compares masked MC against observed-only two-way FE least squares, ridge against independently constructed NumPy regressions, and SDID against SciPy optimization of the same normalized objectives. It also checks low-rank recovery, complete-panel parity, missing-value placeholder invariance, and that changing treated post outcomes cannot change the fitted counterfactual.

Handling missing entries removes a computational obstacle; it does not identify effects for selectively unobserved outcomes. Interpreting the observed-cell ATT still requires the estimator’s counterfactual assumptions and a suitable observation process.

API references

MatrixCompletion · SyntheticDID · HorizontalPanelRidge