GuidesSynthetic control and interrupted time series

Synthetic control and interrupted time series

Estimate the effect of a policy on one treated unit by building a weighted combination of untreated units that tracks it before the policy. Then compare with an interrupted time series, which uses the treated unit alone.

Run the program#

The bundled smoking panel records cigarette sales per person in 39 US states from 1970 to 2000 (Abadie, Diamond and Hainmueller, 2010). California's Proposition 99, a tobacco tax, took effect in 1989; the other 38 states are the donors.

from ergodic import interrupted_time_series, synthetic_control, timeseries
from ergodic.datasets import load_smoking

# 1. Load the panel: packs per person, by state and year.
smoking = load_smoking()
print(smoking.data.n_entities, "states,", smoking.data.periods[0], "to", smoking.data.periods[-1])

# 2. Fit a synthetic California from the other states, before 1989.
fit = synthetic_control(smoking.data, "cigsale", treated="California", at=1989)
for state, weight in sorted(fit.weights.items(), key=lambda item: -item[1])[:5]:
    print(f"  {state:12} {weight:.2f}")
print(f"pre-1989 tracking error: {fit.pre_rmspe:.1f} packs")

# 3. Read the effect: the mean gap after 1989, with placebo inference.
low, high = fit.interval()
print(f"effect {fit.value:.1f} packs, placebo band [{low:.1f}, {high:.1f}]")
print(f"placebo p-value {fit.p_value:.3f}, gap in {fit.periods[-1]}: {fit.gaps[-1]:.1f}")

# 4. Fit an interrupted time series on California alone.
rows = smoking.data.to_pandas()
california = rows[rows["state"] == "California"][["year", "cigsale"]]
its = interrupted_time_series(timeseries(california, time="year"), "cigsale", at=1989)
low, high = its.interval()
print(f"interrupted time series: {its.value:.1f} packs, 95% interval [{low:.1f}, {high:.1f}]")

39 states, 1970 to 2000
  Utah         0.21
  Nevada       0.14
  New Mexico   0.13
  Idaho        0.12
  Montana      0.11
pre-1989 tracking error: 2.3 packs
effect -21.0 packs, placebo band [-39.3, -2.7]
placebo p-value 0.051, gap in 2000: -27.9
interrupted time series: -28.3 packs, 95% interval [-40.8, -15.8]

• Step 2. Five states carry most of the weight. The weights are nonnegative and sum to one. Before 1989 the synthetic California misses the real one by about 2 packs per person a year.
• Step 3. After 1989, California sold about 21 fewer packs per person a year than its synthetic copy. By 2000 the gap was about 28 packs.
• Placebo inference. The fit is rerun with each donor state as if it were treated. Only one placebo has a post-to-pre error ratio as large as California's, so the p-value is 2/39. The band adds and subtracts the 95th percentile of the placebo states' mean gaps, in absolute value.
• Step 4. The interrupted time series projects California's own pre-1989 trend forward. Its estimate is larger, because it attributes any departure from California's own linear pre-1989 trend to the law, including a change that would have happened without it.

Read a synthetic control#

synthetic_control returns a SyntheticControlEffect. value is the mean of observed minus synthetic over the periods from at on. observed, synthetic and gaps run over periods, ready to plot.

The weights fit the outcome path only. They do not match covariates such as income or retail price, so they can differ from published synthetic controls that do. There is no sampling standard error for one treated unit: std_error is None, and interval() is a placebo band, not a confidence interval.

Name the treated unit in one of two ways: treated= and at=, or treatment=, a 0/1 column that marks exactly one unit. donors= restricts the donor pool. Donors with a missing period are dropped with a warning; at least 2 must remain.

When to use an interrupted time series#

interrupted_time_series(data, outcome, *, at, seasonality=None, hac_lags=None) needs no donors. It fits a level, a trend, a jump at at and a change in slope, and reports the mean gap between the fit and the projected pre-trend. Its standard errors are Newey-West, which allow for autocorrelated residuals.

• level_change and slope_change hold the jump and the slope change, each with a standard error.
• seasonality=12 adds Fourier terms for a monthly cycle.
• It assumes the pre-period trend would have continued and that nothing else changed at at. Here the other 38 states also fell, on average, about 20 packs below their own linear pre-1989 trend, which breaks that assumption. Prefer synthetic control when donors exist.

Next steps#