import numpy as np, pandas as pd, matplotlib.pyplot as plt, warnings
warnings.filterwarnings("ignore")
from scipy.stats import f as fdist
BLUE="#2b6cb0"; RED="#c53030"; GREEN="#2f855a"; ORANGE="#dd6b20"; GREY="#a0aec0"; PURP="#6b46c1"
def sim(seed,L,strength,n=500,beta=1.0,rho=2.0):
    r=np.random.default_rng(seed); Z=r.normal(0,1,(n,L)); u=r.normal(0,1,n)
    D=Z@np.full(L,strength)+u+r.normal(0,1,n); Y=beta*D+rho*u+r.normal(0,1,n); return Z,D,Y
def tsls(Z,D,Y):
    n=len(Y); Zc=np.column_stack([np.ones(n),Z]); Xc=np.column_stack([np.ones(n),D])
    PZ=Zc@np.linalg.solve(Zc.T@Zc,Zc.T@Xc); b=np.linalg.solve(PZ.T@PZ,PZ.T@Y); res=Y-Xc@b
    s2=res@res/(n-2); return b[1], np.sqrt((s2*np.linalg.inv(PZ.T@PZ))[1,1])
def AR_covers(Z,D,Y,b0,alpha=0.05):                       # F-test all instruments =0 in OLS(Y-b0 D ~ 1+Z)
    n,L=Z.shape; e=Y-b0*D; X1=np.column_stack([np.ones(n),Z]); X0=np.ones((n,1))
    r1=e-X1@np.linalg.lstsq(X1,e,rcond=None)[0]; r0=e-X0@np.linalg.lstsq(X0,e,rcond=None)[0]
    F=((r0@r0-r1@r1)/L)/(r1@r1/(n-L-1)); return F<fdist.ppf(1-alpha,L,n-L-1)
rows=[]
for L,st,lab in [(1,1.0,"strong (1 instrument)"),(30,0.03,"many weak (30, AK-style)")]:
    wc=[]; ac=[]; bs=[]; ols=[]
    for s in range(800):
        Z,D,Y=sim(s,L,st); b,se=tsls(Z,D,Y); wc.append(abs(b-1)<=1.96*se); ac.append(AR_covers(Z,D,Y,1.0)); bs.append(b); ols.append(np.polyfit(D.mean(1) if L>1 else D,Y,1)[0] if False else np.polyfit(D,Y,1)[0])
    rows.append((lab,np.mean(bs),np.mean(ols),100*np.mean(wc),100*np.mean(ac)))
    print(f"{lab:26s}: 2SLS={np.mean(bs):.2f} (true 1, OLS={np.mean(ols):.2f}) | Wald cov {100*np.mean(wc):.0f}% | AR cov {100*np.mean(ac):.0f}%")
fig,ax=plt.subplots(figsize=(8,4.2))
lab=[r[0] for r in rows]; x=np.arange(len(lab)); w=0.35
ax.bar(x-w/2,[r[3] for r in rows],w,color=RED,label="2SLS Wald")
ax.bar(x+w/2,[r[4] for r in rows],w,color=GREEN,label="Anderson-Rubin")
ax.axhline(95,color="k",ls="--",label="95% nominal"); ax.set_xticks(x); ax.set_xticklabels(lab,fontsize=9)
ax.set_ylabel("CI coverage of the true effect"); ax.set_title("Many weak instruments: 2SLS-Wald coverage collapses; AR holds")
for i,r in enumerate(rows): ax.text(i-w/2,r[3]+2,f"{r[3]:.0f}%",ha="center",fontsize=8); ax.text(i+w/2,r[4]+2,f"{r[4]:.0f}%",ha="center",fontsize=8)
ax.legend(); plt.tight_layout(); plt.show()
print("With 30 weak instruments the 2SLS estimate is dragged from the truth (1) toward OLS, and its Wald interval covers the")
print("truth almost never -- the many-weak-instruments trap behind the Bound-Jaeger-Baker critique. Anderson-Rubin stays valid.")

strong (1 instrument)     : 2SLS=1.00 (true 1, OLS=1.67) | Wald cov 95% | AR cov 95%

many weak (30, AK-style)  : 2SLS=1.84 (true 1, OLS=1.99) | Wald cov 4% | AR cov 94%

With 30 weak instruments the 2SLS estimate is dragged from the truth (1) toward OLS, and its Wald interval covers the
truth almost never -- the many-weak-instruments trap behind the Bound-Jaeger-Baker critique. Anderson-Rubin stays valid.

def AR_pval(Z,D,Y,b0):
    n=len(Y); Z=Z.reshape(n,-1); L=Z.shape[1]; e=Y-b0*D
    X1=np.column_stack([np.ones(n),Z]); r1=e-X1@np.linalg.lstsq(X1,e,rcond=None)[0]
    r0=e-np.mean(e); F=((r0@r0-r1@r1)/L)/(r1@r1/(n-L-1)); return 1-fdist.cdf(F,L,n-L-1)
grid=np.linspace(-4,6,1201)
# a strong and a weak single-instrument sample
Zs,Ds,Ys=sim(0,1,1.0); Zw,Dw,Yw=sim(0,1,0.06)
def ar_set(Z,D,Y): inc=grid[[AR_pval(Z,D,Y,g)>0.05 for g in grid]]; return inc
def wald(Z,D,Y): b,se=tsls(Z,D,Y); return b-1.96*se,b+1.96*se
fig,ax=plt.subplots(1,2,figsize=(13,4.4))
for a,(Z,D,Y),lab in zip(ax,[(Zs,Ds,Ys),(Zw,Dw,Yw)],["strong instrument","weak instrument"]):
    pv=np.array([AR_pval(Z,D,Y,g) for g in grid]); a.plot(grid,pv,color=BLUE,lw=2)
    a.axhline(0.05,color=RED,ls="--"); a.axvline(1.0,color=GREEN,ls=":",label="true β=1")
    lo,hi=wald(Z,D,Y); a.axvspan(lo,hi,color=ORANGE,alpha=.2,label=f"2SLS Wald CI")
    inc=ar_set(Z,D,Y); a.axvspan(inc.min(),inc.max(),color=BLUE,alpha=.12,label="AR set")
    a.set_xlabel("β₀"); a.set_ylabel("AR p-value"); a.set_title(f"{lab}"); a.legend(fontsize=7)
plt.tight_layout(); plt.show()
lo,hi=wald(Zw,Dw,Yw); inc=ar_set(Zw,Dw,Yw)
print(f"weak sample: 2SLS-Wald CI [{lo:.2f}, {hi:.2f}]  vs  AR set [{inc.min():.2f}, {inc.max():.2f}]"+(" (touches grid edge = effectively unbounded)" if inc.min()<=grid[1] or inc.max()>=grid[-2] else ""))
print("The AR set is the range of effect values the instruments cannot rule out. Under a weak instrument it is wide/unbounded")
print("-- correctly reporting little information -- whereas the Wald CI stays narrow and can exclude the truth.")

weak sample: 2SLS-Wald CI [-1.28, 3.12]  vs  AR set [-4.00, 6.00] (touches grid edge = effectively unbounded)
The AR set is the range of effect values the instruments cannot rule out. Under a weak instrument it is wide/unbounded
-- correctly reporting little information -- whereas the Wald CI stays narrow and can exclude the truth.

# the memorable thresholds from Lee, McCrary, Moon & Weidner (2022)
print("Lee-McCrary-Moon-Weidner (2022) tF, 5% level:")
print("  first-stage F > 104.7  -> the usual t > 1.96 is valid (c_F = 1.96)")
print("  first-stage F  ~ 10    -> c_F is much larger than 1.96; the conventional CI badly understates uncertainty")
print("  the classic 'F > 10' rule of thumb is NOT enough for valid conventional inference.")
# Card first-stage F for context (computed from scratch)
d=pd.read_csv("card.csv"); ctrl=["exper","expersq","black","south","smsa","reg661","reg662","reg663","reg664","reg665","reg666","reg667","reg668","smsa66"]
d=d.dropna(subset=["lwage","educ","nearc4"]+ctrl).reset_index(drop=True); n=len(d)
C=np.column_stack([np.ones(n)]+[d[c].values for c in ctrl]); Z=d["nearc4"].values; D=d["educ"].values
Xf=np.column_stack([Z,C]); rf=D-Xf@np.linalg.lstsq(Xf,D,rcond=None)[0]; r0=D-C@np.linalg.lstsq(C,D,rcond=None)[0]
F_card=((r0@r0-rf@rf)/1)/(rf@rf/(n-Xf.shape[1]))
print(f"\nCard first-stage F = {F_card:.1f}  -> ABOVE the old 'F>10' rule, but FAR BELOW 104.7:")
print(f"   the conventional 2SLS CI is too narrow; valid inference (AR / tF) must be wider. (ivDiag: c_F ~ 2.93.)")

Lee-McCrary-Moon-Weidner (2022) tF, 5% level:
  first-stage F > 104.7  -> the usual t > 1.96 is valid (c_F = 1.96)
  first-stage F  ~ 10    -> c_F is much larger than 1.96; the conventional CI badly understates uncertainty
  the classic 'F > 10' rule of thumb is NOT enough for valid conventional inference.

Card first-stage F = 13.3  -> ABOVE the old 'F>10' rule, but FAR BELOW 104.7:
   the conventional 2SLS CI is too narrow; valid inference (AR / tF) must be wider. (ivDiag: c_F ~ 2.93.)

# 2SLS Wald and AR set on Card (from scratch)
Zmat=np.column_stack([Z,C]); Xmat=np.column_stack([D,C]); Y=d["lwage"].values
PZ=Zmat@np.linalg.solve(Zmat.T@Zmat,Zmat.T@Xmat); b2=np.linalg.solve(PZ.T@PZ,PZ.T@Y); iv=b2[0]
res=Y-Xmat@b2; s2=res@res/(n-Xmat.shape[1]); se=np.sqrt((s2*np.linalg.inv(PZ.T@PZ))[0,0])
def AR_p_ctrl(b0):
    e=Y-b0*D; X1=np.column_stack([Z,C]); r1=e-X1@np.linalg.lstsq(X1,e,rcond=None)[0]; r0=e-C@np.linalg.lstsq(C,e,rcond=None)[0]
    F=((r0@r0-r1@r1)/1)/(r1@r1/(n-X1.shape[1])); return 1-fdist.cdf(F,1,n-X1.shape[1])
g=np.linspace(-0.1,0.5,3001); inc=g[[AR_p_ctrl(x)>0.05 for x in g]]
wald_ci=(iv-1.96*se,iv+1.96*se); ar_ci=(inc.min(),inc.max()); cF=2.934; tf_ci=(iv-cF*se,iv+cF*se)
print(f"2SLS estimate: educ = {iv:.4f} (SE {se:.4f})")
print(f"  2SLS Wald 95% CI : [{wald_ci[0]:.4f}, {wald_ci[1]:.4f}]  (excludes 0 -- but F<<104.7, so invalid/too narrow)")
print(f"  Anderson-Rubin   : [{ar_ci[0]:.4f}, {ar_ci[1]:.4f}]  (robust; wider; just excludes 0)")
print(f"  tF (c_F={cF})    : [{tf_ci[0]:.4f}, {tf_ci[1]:.4f}]  (robust & conservative; INCLUDES 0)")
fig,ax=plt.subplots(figsize=(9,3.6))
for i,(nm,ci,c) in enumerate([("2SLS Wald",wald_ci,ORANGE),("Anderson-Rubin",ar_ci,BLUE),("tF (Lee et al)",tf_ci,PURP)]):
    ax.plot(ci,[i,i],color=c,lw=4,solid_capstyle="round"); ax.plot([iv],[i],"o",color="k",ms=6)
    ax.text(ci[1]+0.005,i,nm,va="center",fontsize=9)
ax.axvline(0,color=RED,ls="--",label="no effect"); ax.axvline(iv,color=GREY,ls=":",label=f"2SLS point {iv:.2f}")
ax.set_yticks([]); ax.set_ylim(-0.5,2.8); ax.set_xlabel("return to a year of schooling (log points)"); ax.set_title("Card: valid weak-IV inference is wider — and borderline")
ax.legend(loc="upper left",fontsize=8); plt.tight_layout(); plt.show()
print("The naive Wald interval's neat significance is an artifact of treating F=13 as 'strong'. Robust inference widens the")
print("interval; the effect is real by AR but not by the conservative tF -- honestly borderline, as the IV-schooling debate has long held.")

2SLS estimate: educ = 0.1315 (SE 0.0550)
  2SLS Wald 95% CI : [0.0238, 0.2392]  (excludes 0 -- but F<<104.7, so invalid/too narrow)
  Anderson-Rubin   : [0.0250, 0.2848]  (robust; wider; just excludes 0)
  tF (c_F=2.934)    : [-0.0298, 0.2928]  (robust & conservative; INCLUDES 0)

The naive Wald interval's neat significance is an artifact of treating F=13 as 'strong'. Robust inference widens the
interval; the effect is real by AR but not by the conservative tF -- honestly borderline, as the IV-schooling debate has long held.

import zipfile, scipy.sparse as sp
ak = pd.read_csv("qob.csv.gz")
n = len(ak); y = ak.lwklywge.values; educ = ak.educ.values.astype(float)
print(f"Angrist & Krueger (1991): {n:,} men, born 19{ak.yob.min()}-19{ak.yob.max()}, "
      f"{ak.pob.nunique()} states of birth")
print(f"  mean log weekly wage {y.mean():.4f};  mean schooling {educ.mean():.3f} years")
print()

# quarter of birth really does move schooling -- the first stage is visible in the raw means
qm = ak.groupby("qob").educ.mean()
print("mean years of schooling by quarter of birth:")
for q, v in qm.items(): print(f"   Q{q}  {v:.4f}")
print(f"  Q4 - Q1 = {qm[4]-qm[1]:+.4f} years -- small, real, and the whole basis of the design")
print()

def dummies(codes, drop_first=True):
    u = np.unique(codes); u = u[1:] if drop_first else u
    idx = {v: j for j, v in enumerate(u)}
    r, c = [], []
    for i, v in enumerate(codes):
        if v in idx: r.append(i); c.append(idx[v])
    return sp.csc_matrix((np.ones(len(r)), (r, c)), shape=(n, len(u)))

ones = sp.csc_matrix(np.ones((n, 1)))
D_yob, D_pob = dummies(ak.yob.values), dummies(ak.pob.values)
W = sp.hstack([ones, D_yob, D_pob]).tocsc()           # exogenous controls: YOB and POB main effects

def interacted(qob):                                   # QOB x YOB and QOB x POB -- the 180-instrument set
    blocks = []
    for q in (2, 3, 4):
        M = sp.diags((qob == q).astype(float))
        blocks += [M @ sp.hstack([ones, D_yob]).tocsc(), M @ D_pob]
    return sp.hstack(blocks).tocsc()

def qob_only(qob):                                     # the simple set: three quarter dummies
    return sp.hstack([sp.csc_matrix((qob == q).astype(float).reshape(-1, 1)) for q in (2, 3, 4)]).tocsc()

def ak_tsls(Zx):
    """2SLS of log wage on schooling, instrumented by Zx, controlling for W."""
    A = sp.hstack([W, Zx]).tocsc(); X = sp.hstack([sp.csc_matrix(educ.reshape(-1, 1)), W]).tocsc()
    AA = (A.T @ A).toarray(); AX = np.asarray((A.T @ X).todense()); Ay = np.asarray(A.T @ y).ravel()
    pi = np.linalg.lstsq(AA, AX, rcond=None)[0]
    XPX = AX.T @ pi; b = np.linalg.lstsq(XPX, pi.T @ Ay, rcond=None)[0]
    res = y - np.asarray(X @ b).ravel(); k = np.linalg.matrix_rank(XPX)
    se = np.sqrt((res @ res / (n - k)) * np.linalg.pinv(XPX)[0, 0])
    pe = np.linalg.lstsq(AA, np.asarray(A.T @ educ).ravel(), rcond=None)[0]
    rss1 = ((educ - np.asarray(A @ pe).ravel()) ** 2).sum()
    WW = (W.T @ W).toarray(); pw = np.linalg.lstsq(WW, np.asarray(W.T @ educ).ravel(), rcond=None)[0]
    rss0 = ((educ - np.asarray(W @ pw).ravel()) ** 2).sum()
    q = np.linalg.matrix_rank(AA) - np.linalg.matrix_rank(WW)
    F = ((rss0 - rss1) / q) / (rss1 / (n - np.linalg.matrix_rank(AA)))
    return b[0], se, F, q

X0 = sp.hstack([sp.csc_matrix(educ.reshape(-1, 1)), W]).tocsc()
XX = (X0.T @ X0).toarray()
b_ols = np.linalg.lstsq(XX, np.asarray(X0.T @ y).ravel(), rcond=None)[0]
r0 = y - np.asarray(X0 @ b_ols).ravel()
se_ols = np.sqrt((r0 @ r0 / (n - np.linalg.matrix_rank(XX))) * np.linalg.pinv(XX)[0, 0])
b3, se3, F3, q3 = ak_tsls(qob_only(ak.qob.values))
b180, se180, F180, q180 = ak_tsls(interacted(ak.qob.values))

print(f"  {'specification':32} {'return to schooling':>20} {'SE':>8} {'t':>7} {'first-stage F':>15}")
print(f"  {'OLS':32} {b_ols[0]:>20.4f} {se_ols:>8.4f} {b_ols[0]/se_ols:>7.1f} {'--':>15}")
print(f"  {'2SLS, 3 quarter dummies':32} {b3:>20.4f} {se3:>8.4f} {b3/se3:>7.2f} {F3:>15.2f}")
print(f"  {'2SLS, ' + str(q180) + ' interactions':32} {b180:>20.4f} {se180:>8.4f} {b180/se180:>7.2f} {F180:>15.2f}")
print()
print(f"Adding {q180-q3} instruments cut the standard error from {se3:.4f} to {se180:.4f} -- a {100*(1-se180/se3):.0f}% gain in")
print(f"apparent precision -- while the first-stage F fell from {F3:.1f} to {F180:.2f}. Section 3's threshold for")
print(f"valid conventional inference is 104.7. The celebrated specification is short of it by a factor of {104.7/F180:.0f}.")
print()
print("Note also where the estimate went. The simple specification says", f"{b3:.4f};", "OLS says", f"{b_ols[0]:.4f}.")
frac = (b180 - b_ols[0]) / (b3 - b_ols[0])
print(f"The {q180}-instrument version lands at {b180:.4f}, {100*(1-frac):.0f}% of the way back toward OLS -- the direction")
print("many weak instruments always push, because a first stage that is mostly noise reproduces the")
print("endogenous variation it was supposed to purge.")
print()

print("THE PLACEBO. Quarter of birth replaced by a uniform random draw, same specification, 200 times.")
print()
rng = np.random.default_rng(20260813)
pl = np.array([ak_tsls(interacted(rng.integers(1, 5, size=n)))[:3] for _ in range(200)])
tpl = pl[:, 0] / pl[:, 1]
print(f"  estimates      mean {pl[:,0].mean():.4f}   range [{pl[:,0].min():.4f}, {pl[:,0].max():.4f}]")
print(f"  standard errors mean {pl[:,1].mean():.4f}   (the real specification reports {se180:.4f})")
print(f"  first-stage F  mean {pl[:,2].mean():.3f}   max {pl[:,2].max():.3f}   (pure noise gives about 1)")
print()
print(f"  significant at 5%:                  {(np.abs(tpl)>1.96).sum()} of {len(pl)}")
print(f"  |t| range {np.abs(tpl).min():.2f} to {np.abs(tpl).max():.2f}, median {np.median(np.abs(tpl)):.2f}")
print(f"  larger than the real 2SLS estimate: {(pl[:,0]>b180).sum()} of {len(pl)}")
print(f"  larger than OLS:                    {(pl[:,0]>b_ols[0]).sum()} of {len(pl)}")
print()
print(f"Random numbers, run through the published specification, return {pl[:,0].mean():.4f} with a t-statistic")
print(f"whose median is {np.median(np.abs(tpl)):.1f}. Nearly every draw clears the 5% bar. {(pl[:,0]>b180).sum()} of {len(pl)} produce an estimate")
print("LARGER than the real one. Nothing in the output announces that the instruments are noise.")
print()
print(f"And the standard errors are not lying. The spread of the placebo estimates is {pl[:,0].std(ddof=1):.4f}, against a")
print(f"mean reported standard error of {pl[:,1].mean():.4f} -- calibrated almost exactly. The estimator has an")
print("accurate view of its own sampling variability and no view whatsoever of the fact that it is")
print("estimating nothing. That is the failure to internalise: a standard error answers 'how much would")
print("this move across samples', never 'is this quantity identified'.")
print()
print(f"The one honest signal is the first stage. The placebo F never exceeds {pl[:,2].max():.2f}; the real one is {F180:.2f}.")
print("So the real instruments do carry information -- genuinely more than nothing, and nowhere near")
print("enough. Which is the whole subject of this page: F = 2.58 against a requirement of 104.7.")

fig, ax = plt.subplots(1, 2, figsize=(12, 4.2))
ax[0].hist(pl[:, 0], bins=28, color=GREY, edgecolor="white")
for v, c, lab in [(b_ols[0], BLUE, f"OLS {b_ols[0]:.4f}"), (b180, RED, f"real 2SLS {b180:.4f}"),
                  (b3, GREEN, f"3 instruments {b3:.4f}")]:
    ax[0].axvline(v, color=c, lw=2.2, label=lab)
ax[0].set_xlabel("estimated return to schooling"); ax[0].set_ylabel("placebo runs")
ax[0].set_title("200 runs with randomly generated quarters of birth"); ax[0].legend(fontsize=8)

ax[1].hist(np.abs(tpl), bins=28, color=GREY, edgecolor="white")
ax[1].axvline(1.96, color=RED, lw=2.2, ls="--", label="|t| = 1.96")
ax[1].axvline(abs(b180 / se180), color=BLUE, lw=2.2, label=f"real specification {abs(b180/se180):.1f}")
ax[1].set_xlabel("|t| on the placebo estimate"); ax[1].set_ylabel("placebo runs")
ax[1].set_title(f"{(np.abs(tpl)>1.96).sum()} of {len(pl)} random-instrument runs are 'significant'")
ax[1].legend(fontsize=8)
plt.tight_layout(); plt.show()
print("Left: the placebo estimates straddle OLS, which is where irrelevant instruments send 2SLS.")
print("Right: the significance the specification manufactures out of random numbers.")

Angrist & Krueger (1991): 329,509 men, born 1930-1939, 51 states of birth
  mean log weekly wage 5.8999;  mean schooling 12.770 years

mean years of schooling by quarter of birth:
   Q1  12.6881
   Q2  12.7447
   Q3  12.8054
   Q4  12.8394
  Q4 - Q1 = +0.1514 years -- small, real, and the whole basis of the design

  specification                     return to schooling       SE       t   first-stage F
  OLS                                            0.0673   0.0003   194.4              --
  2SLS, 3 quarter dummies                        0.1077   0.0195    5.52           36.04
  2SLS, 180 interactions                         0.0928   0.0093    9.98            2.58

Adding 177 instruments cut the standard error from 0.0195 to 0.0093 -- a 52% gain in
apparent precision -- while the first-stage F fell from 36.0 to 2.58. Section 3's threshold for
valid conventional inference is 104.7. The celebrated specification is short of it by a factor of 41.

Note also where the estimate went. The simple specification says 0.1077; OLS says 0.0673.
The 180-instrument version lands at 0.0928, 37% of the way back toward OLS -- the direction
many weak instruments always push, because a first stage that is mostly noise reproduces the
endogenous variation it was supposed to purge.

THE PLACEBO. Quarter of birth replaced by a uniform random draw, same specification, 200 times.

  estimates      mean 0.0651   range [0.0250, 0.1129]
  standard errors mean 0.0150   (the real specification reports 0.0093)
  first-stage F  mean 0.994   max 1.284   (pure noise gives about 1)

  significant at 5%:                  197 of 200
  |t| range 1.52 to 7.27, median 4.44
  larger than the real 2SLS estimate: 8 of 200
  larger than OLS:                    87 of 200

Random numbers, run through the published specification, return 0.0651 with a t-statistic
whose median is 4.4. Nearly every draw clears the 5% bar. 8 of 200 produce an estimate
LARGER than the real one. Nothing in the output announces that the instruments are noise.

And the standard errors are not lying. The spread of the placebo estimates is 0.0150, against a
mean reported standard error of 0.0150 -- calibrated almost exactly. The estimator has an
accurate view of its own sampling variability and no view whatsoever of the fact that it is
estimating nothing. That is the failure to internalise: a standard error answers 'how much would
this move across samples', never 'is this quantity identified'.

The one honest signal is the first stage. The placebo F never exceeds 1.28; the real one is 2.58.
So the real instruments do carry information -- genuinely more than nothing, and nowhere near
enough. Which is the whole subject of this page: F = 2.58 against a requirement of 104.7.

Left: the placebo estimates straddle OLS, which is where irrelevant instruments send 2SLS.
Right: the significance the specification manufactures out of random numbers.

Causal Inference III(b) — Weak-Instrument-Robust Inference¶

When the first-stage F is not enough: Anderson-Rubin confidence sets and the tF procedure¶

1. The coverage collapse — why the 2SLS Wald CI fails¶

2. The Anderson-Rubin confidence set — from scratch¶

3. The tF procedure — recalibrating "F > 10" to "F > 104.7"¶

4. Card returns-to-schooling under valid inference¶

5. The instruments that were never there — Angrist & Krueger on the real data¶

6. Summary¶