Choosing instruments

Download the script: instruments.py

The risk of many instruments. Every lag of every instrumenting variable, in every period, is a column. With 10 periods, gmm(y, 2:.) alone is 36 columns. Too many instruments overfit the endogenous variables, and the Hansen test loses its power to reject: its p-values drift toward 1 (Roodman 2009b).

What to compare. The usual remedies are to:

  • limit the lags;

  • collapse the instruments (one per lag);

  • replace them by their principal components.

Then look at what changes: the instrument count, the estimates, the Hansen and AR(2) tests, and the model and moment selection criteria (MMSC; lower is better).

base = "y ~ l(y, 1) + x + w | year"
specs = {
    "all lags": "gmm(y, 2:.) + gmm(x, 1:.) + iv(w)",
    "lags 2-3": "gmm(y, 2:3) + gmm(x, 1:2) + iv(w)",
    "collapsed": "gmm(y, 2:., collapse=True) + gmm(x, 1:., collapse=True) + iv(w)",
}
for name, inst in specs.items():
    r = dpd_stream(f"{base} | {inst}", df, panel=("id", "year"))
    ...

┌───────────┬──────────┬──────────┬─────────────┬──────────┬──────────┬─────────────┐
│ spec      ┆ rho      ┆ se       ┆ instruments ┆ hansen_p ┆ ar2_p    ┆ mmsc_bic    │
╞═══════════╪══════════╪══════════╪═════════════╪══════════╪══════════╪═════════════╡
│ all lags  ┆ 0.595532 ┆ 0.004602 ┆ 89          ┆ 0.486069 ┆ 0.540762 ┆ -748.768287 │
│ lags 2-3  ┆ 0.593107 ┆ 0.005802 ┆ 40          ┆ 0.743126 ┆ 0.546    ┆ -283.593002 │
│ collapsed ┆ 0.591788 ┆ 0.005324 ┆ 26          ┆ 0.96834  ┆ 0.5477   ┆ -152.372356 │
└───────────┴──────────┴──────────┴─────────────┴──────────┴──────────┴─────────────┘

(The true ρ is 0.6.)

Forward orthogonal deviations

The problem with gaps. First differences lose two observations at every gap in a panel: the one missing, and the next, which has no difference. Forward orthogonal deviations (transform="fod", xtabond2’s orthogonal) subtract the mean of all later observations, so they lose only one.

fd: 29,519 observations, rho = 0.6017 (0.0081)
fod: 31,902 observations, rho = 0.6013 (0.0058)

Principal components

pca=True replaces the GMM-style instruments by the principal components of their correlation matrix. It keeps those with eigenvalues of 1 or more, as xtabond2 does; components= fixes the number instead. The Kaiser–Meyer–Olkin statistic measures how well the instruments suit the reduction.

pca: 11 principal components (KMO 0.904); rho = 0.5752 (0.0198)

Instruments as objects

The formula’s gmm() and iv() terms can also be passed as Gmm and Iv objects. That is convenient in a loop:

from dpd_stream import Gmm, Iv

for b in (3, 4, 5):
    r = dpd_stream("y ~ l(y, 1) + x + w | year", df, panel=("id", "year"),
                   gmm=[Gmm("y", lags=(2, b)), Gmm("x", lags=(1, b - 1))], iv=[Iv("w")])

lags 2-3: rho = 0.5931, Hansen p = 0.743, difference-in-Hansen p (x's instruments) = 0.818
lags 2-4: rho = 0.5934, Hansen p = 0.355, difference-in-Hansen p (x's instruments) = 0.689
lags 2-5: rho = 0.5931, Hansen p = 0.410, difference-in-Hansen p (x's instruments) = 0.581

The suboptions

Suboption

Effect

gmm(x, a:b)

lags a to b; . for no upper limit; 1:. by default, as xtabond2

collapse=True

one instrument per lag and variable (or collapse=True on the call, for every group)

eq='diff' / 'level' / 'both'

which equation the instruments are for, in system GMM; 'level' gives differenced instruments for the levels equation

split=True

system GMM: the levels instruments get their own difference-in-Hansen test

passthru=True

levels-equation instruments are not differenced

orthogonal=True

backward orthogonal deviations of the variable, to pair with transform="fod"

iv(x, eq=..., passthru=True, mz=True)

IV-style: one column per variable, transformed for the transformed equation unless passthru; mz: missing becomes 0 instead of dropping the observation
The script
"""Choosing instruments: lag limits, collapsing, forward orthogonal
deviations, principal components, and the tests that guide the choice.

    python examples/instruments.py

Too many instruments overfit the endogenous variables and weaken the
Hansen test (Roodman 2009b); this compares instrument sets on one panel.
"""

import polars as pl

from dpd_stream import Gmm, Iv, dpd_stream
from dpd_stream.simulate import simulate_dynamic

df = simulate_dynamic(5_000, 10, rho=(0.6,), seed=2)
P = ("id", "year")
base = "y ~ l(y, 1) + x + w | year"

specs = {
    "all lags": "gmm(y, 2:.) + gmm(x, 1:.) + iv(w)",
    "lags 2-3": "gmm(y, 2:3) + gmm(x, 1:2) + iv(w)",
    "collapsed": "gmm(y, 2:., collapse=True) + gmm(x, 1:., collapse=True) + iv(w)",
}
rows = []
for name, inst in specs.items():
    r = dpd_stream(f"{base} | {inst}", df, panel=P, verbose=False)
    rows.append(dict(spec=name, rho=r.beta[0], se=r.se[0], instruments=r.n_instruments,
                     hansen_p=r.hansen["p"], ar2_p=r.ar_test(2)["p"],
                     mmsc_bic=r.mmsc["bic"]))
print(pl.DataFrame(rows))

# ---- forward orthogonal deviations keep more observations with gaps
gappy = simulate_dynamic(5_000, 10, rho=(0.6,), p_gap=0.1, seed=3)
for tr in ("fd", "fod"):
    r = dpd_stream(f"{base} | gmm(y, 2:4) + gmm(x, 1:3) + iv(w)", gappy, panel=P,
                   transform=tr, verbose=False)
    print(f"{tr}: {r.n_obs:,} observations, rho = {r.beta[0]:.4f} ({r.se[0]:.4f})")

# ---- principal components of the GMM-style instruments (xtabond2's pca)
r = dpd_stream(f"{base} | gmm(y, 2:.) + gmm(x, 1:.) + iv(w)", df, panel=P, pca=True,
               verbose=False)
print(f"pca: {r.pca['components']} principal components (KMO {r.pca['kmo']:.3f}); "
      f"rho = {r.beta[0]:.4f} ({r.se[0]:.4f})")

# ---- the same specification as objects, e.g. built in a loop
for b in (3, 4, 5):
    r = dpd_stream(base, df, panel=P, gmm=[Gmm("y", lags=(2, b)), Gmm("x", lags=(1, b - 1))],
                   iv=[Iv("w")], verbose=False)
    print(f"lags 2-{b}: rho = {r.beta[0]:.4f}, Hansen p = {r.hansen['p']:.3f}, "
          f"difference-in-Hansen p (x's instruments) = "
          f"{[d for d in r.diff_hansen if 'x' in d['group']][0]['difference_p']:.3f}")