Kernel and ridge matching

Each treated unit is compared with a kernel-weighted average of the controls of its stratum within one bandwidth of its propensity score. Ridge matching replaces the average with Frölich’s (2004) ridge-regularized local-linear fit. The default bandwidth ("pair_distance") is set from the 1:1 nearest-neighbor distances; CrossValidation(outcome=) chooses it by cross-validating an outcome. Standard errors are the influence function’s, with the weights held fixed.

import polars as pl

from match_stream import CrossValidation, match_kernel, simulate_matching

pl.Config.set_tbl_hide_dataframe_shape(True)

df = simulate_matching(100_000)
kw = dict(exact=["r", "i"], verbose=False)

m = match_kernel("d ~ x1 + x2 + x3", df, **kw)
m.summary()
m.effect("y").summary()

r = match_kernel("d ~ x1 + x2 + x3", df, kernel="epan", ridge=True, **kw)
print(f"\nridge, bandwidth {r.bandwidth:.5f}:")
r.effect("y").summary()

grid = [0.002, 0.005, 0.01, 0.02, 0.05, 0.1]
cv = match_kernel("d ~ x1 + x2 + x3", df, ridge=True,
                  bandwidth=CrossValidation(outcome="y", grid=grid), **kw)
print("\ncross-validation:")
print(cv.bandwidth_info.cv)
print(f"chosen: {cv.bandwidth}")
cv.effect("y").summary()

Output:

match-stream: kernel matching, epan kernel, estimand ATT
  distance: ps   bandwidth: 0.000417348 (pair_distance)
  exact: r, i (20 strata)
               available     matched   unmatched
  treated         25,497      23,716       1,781
  control         74,503      63,809      10,694
### y: kernel matching on ps
Estimand: ATT   Matched focal units: 23,716
Standard errors: influence function, weights fixed, clustered by unit (87,525)
┌─────────────┬──────────┬────────────┬───────────┬──────────┐
│ Coefficient ┆ Estimate ┆ Std. Error ┆ t value   ┆ Pr(>|t|) │
│ ---         ┆ ---      ┆ ---        ┆ ---       ┆ ---      │
│ str         ┆ f64      ┆ f64        ┆ f64       ┆ f64      │
╞═════════════╪══════════╪════════════╪═══════════╪══════════╡
│ ATT:y       ┆ 0.990778 ┆ 0.012987   ┆ 76.290996 ┆ 0.0      │
└─────────────┴──────────┴────────────┴───────────┴──────────┘

ridge, bandwidth 0.00042:
### y: ridge matching on ps
Estimand: ATT   Matched focal units: 23,716
Standard errors: influence function, weights fixed, clustered by unit (87,525)
┌─────────────┬──────────┬────────────┬───────────┬──────────┐
│ Coefficient ┆ Estimate ┆ Std. Error ┆ t value   ┆ Pr(>|t|) │
│ ---         ┆ ---      ┆ ---        ┆ ---       ┆ ---      │
│ str         ┆ f64      ┆ f64        ┆ f64       ┆ f64      │
╞═════════════╪══════════╪════════════╪═══════════╪══════════╡
│ ATT:y       ┆ 0.99224  ┆ 0.013223   ┆ 75.040366 ┆ 0.0      │
└─────────────┴──────────┴────────────┴───────────┴──────────┘

cross-validation:
┌───────────┬───────────┬─────────┐
│ bandwidth ┆ criterion ┆ weight  │
│ ---       ┆ ---       ┆ ---     │
│ f64       ┆ f64       ┆ f64     │
╞═══════════╪═══════════╪═════════╡
│ 0.002     ┆ 2.010846  ┆ 74203.0 │
│ 0.005     ┆ 1.950153  ┆ 74203.0 │
│ 0.01      ┆ 1.929642  ┆ 74203.0 │
│ 0.02      ┆ 1.919829  ┆ 74203.0 │
│ 0.05      ┆ 1.914433  ┆ 74203.0 │
│ 0.1       ┆ 1.912883  ┆ 74203.0 │
└───────────┴───────────┴─────────┘
chosen: 0.1
### y: ridge matching on ps
Estimand: ATT   Matched focal units: 25,494
Standard errors: influence function, weights fixed, clustered by unit (99,997)
┌─────────────┬──────────┬────────────┬───────────┬──────────┐
│ Coefficient ┆ Estimate ┆ Std. Error ┆ t value   ┆ Pr(>|t|) │
│ ---         ┆ ---      ┆ ---        ┆ ---       ┆ ---      │
│ str         ┆ f64      ┆ f64        ┆ f64       ┆ f64      │
╞═════════════╪══════════╪════════════╪═══════════╪══════════╡
│ ATT:y       ┆ 1.011057 ┆ 0.012162   ┆ 83.131974 ┆ 0.0      │
└─────────────┴──────────┴────────────┴───────────┴──────────┘

Notes.

• The default bandwidth ("pair_distance") is 1.5 times the 90th percentile of the nonzero 1:1 nearest-neighbor distances. It is narrow: here 1,781 treated units have no control within it and are unmatched.

• Cross-validation predicts each control’s outcome from the other controls (leave one out), for each bandwidth on the grid. It picks the bandwidth with the smallest mean squared error over a fixed set of controls.

  • Here the outcome is smooth in the score, so the widest bandwidth wins and nearly every treated unit is matched (25,494 of 25,497).

  • match-stream warns when the minimum is at the edge of the grid: widen CrossValidation(grid=) then.

• Computation. Kernel and ridge weights come from window sums, so no pairs are formed (see How it works).