First Course Ding: Chapter 23

An econometric perspective on instrumental variables

Chapter 23 is where the design-based IV story becomes an econometrics workflow: controls, overidentification, and alternative representations of the same parameter.

from pathlib import Path

import numpy as np
import pandas as pd

import crabbymetrics as cm

np.set_printoptions(precision=4, suppress=True)


def repo_root():
    for candidate in [Path.cwd().resolve(), *Path.cwd().resolve().parents]:
        if (candidate / "ding_w_source").exists():
            return candidate
    raise FileNotFoundError("could not locate ding_w_source from the current working directory")

def iv_moments(theta, data):
    resid = data["y"] - data["x"] @ theta
    return data["z"] * resid[:, None]


def iv_jacobian(theta, data):
    del theta
    return -(data["z"].T @ data["x"]) / data["x"].shape[0]

1 Card’s Returns-To-Schooling Data

card = pd.read_csv(repo_root() / "ding_w_source" / "card1995.csv")
y = card["lwage"].to_numpy(dtype=float)
d = card["educ"].to_numpy(dtype=float)[:, None]

controls = [
    "exper",
    "expersq",
    "black",
    "south",
    "smsa",
    "reg661",
    "reg662",
    "reg663",
    "reg664",
    "reg665",
    "reg666",
    "reg667",
    "reg668",
    "smsa66",
]
x = card[controls].to_numpy(dtype=float)
z_excl = card[["nearc2", "nearc4"]].to_numpy(dtype=float)

keep = np.isfinite(y) & np.isfinite(d[:, 0]) & np.isfinite(x).all(axis=1) & np.isfinite(z_excl).all(axis=1)
y = y[keep]
d = d[keep]
x = x[keep]
z_excl = z_excl[keep]

2 Closed-Form TwoSLS And Overidentified GMM

ols = cm.OLS()
ols.fit(np.column_stack([d[:, 0], x]), y)
ols_hc1 = ols.summary(vcov="hc1")

tsls = cm.TwoSLS()
tsls.fit(d, x, z_excl, y)
tsls_hc1 = tsls.summary(vcov="hc1")

X = np.column_stack([np.ones(len(y)), d[:, 0], x])
Z = np.column_stack([np.ones(len(y)), x, z_excl])

gmm = cm.GMM(iv_moments, jacobian_fn=iv_jacobian, max_iterations=200, tolerance=1e-9)
gmm.fit({"x": X, "y": y, "z": Z}, np.zeros(X.shape[1]))
gmm_summary = gmm.summary(vcov="sandwich", omega="iid")

print("OLS return to education:", round(float(ols_hc1["coef"][0]), 4))
print("OLS HC1 SE:", round(float(ols_hc1["coef_se"][0]), 4))
print("TwoSLS return to education:", round(float(tsls_hc1["coef"][0]), 4))
print("TwoSLS HC1 SE:", round(float(tsls_hc1["coef_se"][0]), 4))
print("Two-step GMM return to education:", round(float(gmm_summary["coef"][1]), 4))
print("Two-step GMM SE:", round(float(gmm_summary["se"][1]), 4))
print("GMM J-statistic:", round(float(gmm_summary["j_stat"]), 4), "with df =", gmm_summary["j_df"])

OLS return to education: 0.0747
OLS HC1 SE: 0.0036
TwoSLS return to education: 0.1571
TwoSLS HC1 SE: 0.0526
Two-step GMM return to education: 0.1551
Two-step GMM SE: 0.0522
GMM J-statistic: 1.3567 with df = 1

3 Control-Function View

first_stage = cm.OLS()
first_stage.fit(np.column_stack([x, z_excl]), d[:, 0])
pi = first_stage.summary()
d_hat = first_stage.predict(np.column_stack([x, z_excl]))
resid = d[:, 0] - d_hat

control_function = cm.OLS()
control_function.fit(np.column_stack([d[:, 0], resid, x]), y)
cf_hc1 = control_function.summary(vcov="hc1")

print("TwoSLS coefficient:", round(float(tsls_hc1["coef"][0]), 4))
print("Control-function coefficient on education:", round(float(cf_hc1["coef"][0]), 4))

TwoSLS coefficient: 0.1571
Control-function coefficient on education: 0.1571

By Chapter 23, the IV problem has several equivalent front ends:

• closed-form TwoSLS
• overidentified linear IV as two-step GMM
• a control-function representation

In crabbymetrics, those are now all available without leaving the same Rust-backed core.