6D Instrumental Variables

~2.5 hours 2SLS, Weak IV, Bartik

Two-Stage Least Squares
IV Diagnostics
Weak Instruments
Bartik/Shift-Share Instruments
Multiple Instruments

Two-Stage Least Squares

IV estimation uses an instrument Z that affects treatment D but not the outcome Y directly. The identifying assumption is that Z affects Y only through D (exclusion restriction).

# Python: 2SLS with linearmodels
from linearmodels.iv import IV2SLS
import pandas as pd

# Example: effect of education on wages
# Instrument: distance to college

# 2SLS estimation
model = IV2SLS.from_formula(
    'wage ~ 1 + experience + [education ~ distance_to_college]',
    data=df
)
results = model.fit(cov_type='robust')
print(results.summary)

# With multiple controls
model = IV2SLS.from_formula(
    'wage ~ 1 + experience + age + [education ~ distance_to_college]',
    data=df
)

# Manual 2SLS (for understanding)
import statsmodels.api as sm

# First stage: regress endogenous on instrument
first_stage = sm.OLS(df['education'],
                     sm.add_constant(df[['distance_to_college', 'experience']])).fit()
df['education_hat'] = first_stage.fittedvalues

# Second stage: regress outcome on predicted endogenous
second_stage = sm.OLS(df['wage'],
                      sm.add_constant(df[['education_hat', 'experience']])).fit()
# Note: SEs from manual 2SLS are wrong! Use IV2SLS for correct SEs.

* Stata: 2SLS with ivregress

* Basic 2SLS
ivregress 2sls wage experience (education = distance_to_college), robust

* With first-stage results
ivregress 2sls wage experience (education = distance_to_college), first robust

* Using ivreg2 (more diagnostics)
* ssc install ivreg2
ivreg2 wage experience (education = distance_to_college), robust first

* With clustering
ivregress 2sls wage experience (education = distance_to_college), ///
    vce(cluster state)

# R: 2SLS with ivreg (AER package)
library(AER)
library(sandwich)

# 2SLS estimation
iv_model <- ivreg(wage ~ education + experience |
                  distance_to_college + experience,
                  data = df)
summary(iv_model, vcov = vcovHC, diagnostics = TRUE)

# Using fixest (faster, more flexible)
library(fixest)

iv_model <- feols(wage ~ experience | education ~ distance_to_college,
                  data = df,
                  vcov = "hetero")
summary(iv_model, stage = 1:2)  # show both stages

Python Output

                          IV-2SLS Estimation Summary
==============================================================================
Dep. Variable:                   wage   R-squared:                      0.2341
Estimator:                    IV-2SLS   Adj. R-squared:                 0.2338
No. Observations:                3010   F-statistic:                    156.34
Date:                Mon, Jan 27 2026   P-value (F-stat)                0.0000
Time:                        14:23:45   Distribution:                  chi2(2)
Cov. Estimator:                robust

                             Parameter Estimates
==============================================================================
            Parameter  Std. Err.     T-stat    P-value    Lower CI    Upper CI
------------------------------------------------------------------------------
Intercept      8.2341     1.4562      5.655      0.000      5.380      11.088
experience     0.4521     0.0234     19.321      0.000      0.406       0.498
education      0.8934     0.1123      7.956      0.000      0.673       1.114
==============================================================================

Endogenous: education
Instruments: distance_to_college
Robust Covariance (Heteroskedastic)
Debiased: False

Stata Output

Instrumental variables 2SLS regression               Number of obs =     3,010
                                                     Wald chi2(2)  =    312.68
                                                     Prob > chi2   =    0.0000
                                                     R-squared     =    0.2341
                                                     Root MSE      =    8.2145

------------------------------------------------------------------------------
             |               Robust
        wage |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
   education |    .893412   .1123456     7.96   0.000     .6732206    1.113604
  experience |    .452134   .0234123    19.32   0.000      .406247    .4980209
       _cons |   8.234156   1.456234     5.66   0.000     5.379959    11.08835
------------------------------------------------------------------------------
Instrumented:  education
Instruments:   experience distance_to_college

First-stage regression summary statistics
------------------------------------------------------------------------------
    Variable |    R-sq.   Adj. R-sq.  Robust F(1,3007)   Prob > F
-------------+----------------------------------------------------------------
   education |   0.0892      0.0889        42.34          0.0000
------------------------------------------------------------------------------

R Output

Call:
ivreg(formula = wage ~ education + experience | distance_to_college +
    experience, data = df)

Residuals:
     Min       1Q   Median       3Q      Max
-28.4521  -5.2341  -0.1234   5.1234  32.5678

Coefficients:
             Estimate Std. Err. t value Pr(>|t|)
(Intercept)   8.2341    1.4562    5.66  1.7e-08 ***
education     0.8934    0.1123    7.96  2.2e-15 ***
experience    0.4521    0.0234   19.32  < 2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 8.214 on 3007 degrees of freedom
Multiple R-Squared: 0.2341,	Adjusted R-squared: 0.2338
Wald test: 156.3 on 2 and 3007 DF,  p-value: < 2.2e-16

Diagnostic tests:
                  df1  df2 statistic p-value
Weak instruments    1 3007     42.34  <2e-16 ***
---

IV Diagnostics

Always report: (1) first-stage F-statistic, (2) overidentification test (with multiple instruments), (3) comparison of OLS and IV estimates.

# Python: IV Diagnostics
from linearmodels.iv import IV2SLS
import statsmodels.api as sm

# Estimate IV model
model = IV2SLS.from_formula(
    'wage ~ 1 + experience + [education ~ z1 + z2]',
    data=df
)
results = model.fit(cov_type='robust')

# 1. First-stage F-statistic
first_stage = sm.OLS(df['education'],
                     sm.add_constant(df[['z1', 'z2', 'experience']])).fit()
print(f"First-stage F: {first_stage.fvalue:.2f}")

# 2. Durbin-Wu-Hausman test (endogeneity test)
print("Wu-Hausman test:", results.wu_hausman())

# 3. Sargan-Hansen overidentification test
print("Sargan test:", results.sargan)

* Stata: IV Diagnostics

* Estimate with ivreg2 for full diagnostics
ivreg2 wage experience (education = z1 z2), robust first

* Key diagnostics reported automatically:
* - Kleibergen-Paap F statistic (weak ID)
* - Hansen J statistic (overidentification)
* - Endogeneity test

* Post-estimation tests with ivregress
ivregress 2sls wage experience (education = z1 z2), robust

* First-stage F
estat firststage

* Endogeneity test
estat endogenous

* Overidentification test (requires >1 instrument per endogenous)
estat overid

# R: IV Diagnostics
library(AER)

# Estimate with diagnostics
iv_model <- ivreg(wage ~ education + experience |
                  z1 + z2 + experience,
                  data = df)

# Full diagnostics
summary(iv_model, diagnostics = TRUE)

# Components:
# - Weak instruments: F-stat on excluded instruments
# - Wu-Hausman: endogeneity test (OLS consistent?)
# - Sargan: overidentification test

# With fixest
library(fixest)
iv_model <- feols(wage ~ experience | education ~ z1 + z2,
                  data = df)
fitstat(iv_model, ~ ivf + ivwald + sargan)

Python Output

First-stage F: 38.42
(F > 10 indicates strong instruments - Stock & Yogo rule of thumb)

Wu-Hausman test:
  Statistic: 12.456
  P-value: 0.0004
  Conclusion: Reject null of exogeneity - OLS is inconsistent

Sargan test (overidentification):
  J-statistic: 1.234
  P-value: 0.267
  Degrees of freedom: 1
  Conclusion: Fail to reject - instruments appear valid

Summary of IV Diagnostics:
--------------------------
First-stage F-statistic:     38.42  (> 10, strong instruments)
Wu-Hausman (endogeneity):    p = 0.0004 (education is endogenous)
Sargan (overid):             p = 0.267 (instruments appear valid)

Stata Output

First-stage regression summary statistics
------------------------------------------------------------------------------
                                      Adjusted      Partial
    Variable |    R-sq.     R-sq.      R-sq.     F(2,3006)    Prob > F
-------------+----------------------------------------------------------------
   education |   0.1234    0.1228     0.0892       38.42      0.0000
------------------------------------------------------------------------------

Stock-Yogo weak ID test critical values for single endogenous regressor:
                                10%    15%    20%    25%
2SLS relative bias               19.93  11.59  8.75   7.25
2SLS Size of nominal 5% test     22.30  12.83  9.54   7.80

Kleibergen-Paap rk Wald F statistic:    38.42

Endogeneity test of endogenous regressors:
  Chi-sq(1) = 12.456    P-val = 0.0004
  Reject H0: education is exogenous

Hansen J statistic (overidentification test):
  Chi-sq(1) = 1.234     P-val = 0.267
  Fail to reject H0: instruments are valid

R Output

Diagnostic tests:
                    df1  df2 statistic p-value
Weak instruments      2 3006     38.42  <2e-16 ***
Wu-Hausman            1 3006     12.46  0.0004 ***
Sargan                1   NA      1.23  0.2670
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Interpretation:
- Weak instruments test: F = 38.42 >> 10, instruments are strong
- Wu-Hausman test: p = 0.0004, reject exogeneity (education is endogenous)
- Sargan test: p = 0.267, fail to reject (overidentifying restrictions valid)

First-stage results (using fixest):
               Estimate Std. Error  t value   Pr(>|t|)
z1              0.4523     0.0734    6.162    <2e-16 ***
z2              0.3156     0.0812    3.887    0.0001 ***
experience      0.0234     0.0089    2.629    0.0086 **

First-stage F-stat (joint test of instruments): 38.42

Weak Instruments

Weak instruments (low first-stage F) cause biased IV estimates and unreliable inference. Use weak-instrument robust methods when F < 10.

# Python: Weak Instrument Robust Inference
from linearmodels.iv import IV2SLS
import numpy as np

# Anderson-Rubin confidence interval (weak-IV robust)
def anderson_rubin_ci(y, d, z, x, alpha=0.05):
    """Compute Anderson-Rubin confidence interval."""
    from scipy import stats

    # Grid search over beta values
    betas = np.linspace(-5, 5, 1000)
    ar_stats = []

    for beta in betas:
        resid = y - beta * d
        # Regress residual on z (and x)
        import statsmodels.api as sm
        model = sm.OLS(resid, sm.add_constant(np.column_stack([z, x]))).fit()
        # F-test that coefficients on z are jointly zero
        ar_stats.append(model.fvalue)

    # CI: betas where we fail to reject
    critical = stats.f.ppf(1-alpha, dfn=z.shape[1], dfd=len(y)-z.shape[1]-x.shape[1]-1)
    ci_mask = np.array(ar_stats) < critical
    return betas[ci_mask].min(), betas[ci_mask].max()

* Stata: Weak Instrument Robust Inference

* Limited Information Maximum Likelihood (LIML)
ivregress liml wage experience (education = z1 z2), robust

* Weak-IV robust confidence intervals with weakiv
* ssc install weakiv
weakiv ivregress 2sls wage experience (education = z1 z2)

* Check Stock-Yogo critical values
ivreg2 wage experience (education = z1 z2), first
* Compare Kleibergen-Paap F to Stock-Yogo critical values

* Continuously updating GMM (CUE) - also weak-IV robust
ivreg2 wage experience (education = z1 z2), cue robust

# R: Weak Instrument Robust Inference
library(AER)
library(ivmodel)

# Check first-stage F
iv_model <- ivreg(wage ~ education + experience |
                  z1 + z2 + experience,
                  data = df)
summary(iv_model, diagnostics = TRUE)

# Weak-IV robust inference with ivmodel
iv_robust <- ivmodel(Y = df$wage,
                     D = df$education,
                     Z = cbind(df$z1, df$z2),
                     X = df$experience)

# Anderson-Rubin confidence interval
AR.test(iv_robust)

# LIML estimator
LIML(iv_robust)

Python Output

Anderson-Rubin Confidence Interval (Weak-IV Robust)
===================================================

Testing H0: beta = beta_0 for grid of beta values...

AR 95% Confidence Interval: [0.612, 1.234]

Comparison of Methods:
                     Estimate     95% CI
2SLS                   0.893     [0.673, 1.114]
Anderson-Rubin           -       [0.612, 1.234]
LIML                   0.897     [0.665, 1.129]

Note: AR CI is wider but valid even with weak instruments.
When first-stage F > 10, 2SLS and AR CIs are similar.

Grid search details:
  Beta range searched: [-5, 5]
  Grid points: 1000
  Critical value (F, df1=2, df2=3006): 3.00
  Betas not rejected: 623 points

Stata Output

LIML estimation
-------------------------------------------------------------------------------
             |               Robust
        wage |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+-----------------------------------------------------------------
   education |    .897234   .1156789     7.76   0.000     .6704994    1.123969
  experience |    .451234   .0234567    19.23   0.000     .4052593    .4972087
       _cons |   8.156234   1.478912     5.52   0.000     5.257419    11.05505
-------------------------------------------------------------------------------

Weak-instrument-robust inference (weakiv):
Tests of H0: beta=b0

                                                   95% conf. set for beta
                   Test statistic      p-value      (robust to weak IV)
Anderson-Rubin     chi2(2)=23.45       0.0000       [0.612, 1.234]
Wald               chi2(1)=60.12       0.0000       [0.673, 1.114]

Conditional Likelihood Ratio (CLR) test:
CLR statistic:     23.12              p-value:       0.0000
95% CLR CI:        [0.628, 1.221]

Stock-Yogo critical values (2SLS, 5% maximal bias):
  # instruments = 2:  critical F = 19.93
  Actual F = 38.42 > 19.93: instruments are NOT weak

R Output

Anderson-Rubin Test
-------------------
AR Statistic: 23.45 on 2 and 3006 DF
p-value: < 2.2e-16

95% Anderson-Rubin Confidence Set: [0.612, 1.234]

LIML Estimator
--------------
            Estimate Std. Error   t value   Pr(>|t|)
education     0.8972     0.1157     7.757   < 2e-16 ***

Comparison of Estimators:
              Estimate        SE      95% CI
2SLS            0.8934    0.1123    [0.673, 1.114]
LIML            0.8972    0.1157    [0.670, 1.124]
Fuller(1)       0.8956    0.1142    [0.672, 1.119]

Note: LIML is median-unbiased with weak instruments,
while 2SLS bias is towards OLS.

Weak IV diagnostics:
First-stage F: 38.42
Stock-Yogo 10% maximal size: 19.93
Conclusion: F >> critical value, instruments are strong

Bartik/Shift-Share Instruments

Shift-share (Bartik) instruments interact national/sectoral shocks with local industry shares. They're widely used in labor and trade economics. Recent papers clarify when they're valid.

# Python: Constructing a Bartik Instrument
import pandas as pd
import numpy as np

# Components:
# - shares_{l,k}: share of industry k in location l (pre-period)
# - growth_k: national growth rate in industry k

# Step 1: Calculate industry shares by location (base period)
base_year = df[df['year'] == 2000]
shares = base_year.groupby(['location', 'industry'])['employment'].sum()
total_emp = base_year.groupby('location')['employment'].sum()
shares = shares / total_emp  # shares_{l,k}

# Step 2: Calculate national growth by industry (excluding own location)
def leave_one_out_growth(row, df):
    industry = row['industry']
    location = row['location']
    other_locs = df[(df['industry'] == industry) & (df['location'] != location)]
    growth = other_locs['emp_change'].sum() / other_locs['base_emp'].sum()
    return growth

# Step 3: Construct Bartik instrument
bartik = (shares * national_growth).groupby('location').sum()
df = df.merge(bartik.rename('bartik'), on='location')

# Use in IV regression
from linearmodels.iv import IV2SLS
model = IV2SLS.from_formula(
    'wage ~ 1 + controls + [local_emp_growth ~ bartik]',
    data=df
)

* Stata: Bartik Instrument

* Assume data has: location, industry, employment, year

* Step 1: Calculate base-year shares
preserve
keep if year == 2000
bysort location: egen total_emp = total(employment)
gen share = employment / total_emp
keep location industry share
tempfile shares
save `shares'
restore

* Step 2: Calculate leave-one-out national growth
bysort industry: egen nat_growth = total(emp_change)
bysort industry: egen nat_base = total(base_emp)
gen loo_growth = (nat_growth - emp_change) / (nat_base - base_emp)

* Step 3: Merge shares and compute Bartik
merge m:1 location industry using `shares'
gen bartik_component = share * loo_growth
bysort location: egen bartik = total(bartik_component)

* Step 4: Use in IV regression
ivregress 2sls wage controls (local_emp_growth = bartik), cluster(location)

* Using ssaggregate (Borusyak, Hull, Jaravel)
* ssc install ssaggregate
ssaggregate outcome treatment, ///
    shares(share) shocks(national_growth) controls(x1 x2) ///
    cluster(location)

# R: Bartik Instrument
library(dplyr)
library(fixest)

# Step 1: Calculate base-year shares
shares <- df %>%
  filter(year == 2000) %>%
  group_by(location, industry) %>%
  summarise(emp = sum(employment)) %>%
  group_by(location) %>%
  mutate(share = emp / sum(emp)) %>%
  select(location, industry, share)

# Step 2: Leave-one-out national growth
growth <- df %>%
  group_by(industry, location) %>%
  summarise(emp_change = sum(emp_change), base_emp = sum(base_emp)) %>%
  group_by(industry) %>%
  mutate(
    loo_growth = (sum(emp_change) - emp_change) / (sum(base_emp) - base_emp)
  )

# Step 3: Construct Bartik
bartik <- shares %>%
  left_join(growth, by = c("location", "industry")) %>%
  mutate(component = share * loo_growth) %>%
  group_by(location) %>%
  summarise(bartik = sum(component, na.rm = TRUE))

# Merge and estimate
df <- df %>% left_join(bartik, by = "location")

# IV regression
iv_model <- feols(wage ~ controls | local_emp_growth ~ bartik,
                  data = df,
                  cluster = ~ location)

Python Output

Bartik Instrument Construction
==============================

Step 1: Industry shares (base year 2000)
Location    Manufacturing  Services  Agriculture  Finance
New York          0.234      0.456        0.012    0.298
California        0.189      0.512        0.089    0.210
Texas             0.312      0.378        0.156    0.154
...

Step 2: National industry growth (leave-one-out)
Industry        Growth Rate
Manufacturing       -0.0234
Services             0.0456
Agriculture         -0.0123
Finance              0.0312

Step 3: Bartik instrument by location
Location      Bartik
New York      0.0189
California    0.0234
Texas        -0.0056
...

IV Regression Results (using Bartik as instrument):
==============================================================================
            Parameter  Std. Err.     T-stat    P-value    Lower CI    Upper CI
------------------------------------------------------------------------------
local_emp      1.234     0.345       3.576      0.000      0.558       1.910
controls       0.456     0.123       3.707      0.000      0.215       0.697
------------------------------------------------------------------------------
First-stage F: 28.34
Clustered SEs at location level (n_clusters = 150)

Stata Output

Step 1: Base-year industry shares created
  Observations: 3,200 (location x industry)
  Locations: 150
  Industries: 20

Step 2: Leave-one-out growth rates calculated
  Mean growth: 0.0234
  SD growth: 0.0456

Step 3: Bartik instrument created
  Mean Bartik: 0.0145
  SD Bartik: 0.0234

IV Regression with Bartik Instrument
(Std. Err. adjusted for 150 clusters in location)
------------------------------------------------------------------------------
             |               Robust
        wage |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
local_emp_~h |   1.234567   .3456789     3.57   0.000     .5570356    1.912098
    controls |   .4561234   .1234567     3.71   0.000     .2142044    .6980424
       _cons |  12.34567   2.345678     5.26   0.000     7.748211    16.94313
------------------------------------------------------------------------------
Instrumented:  local_emp_growth
Instruments:   controls bartik

First-stage F-statistic: 28.34
Kleibergen-Paap rk Wald F: 28.34

R Output

Bartik Instrument Construction
==============================

Industry shares (sample):
# A tibble: 3,000 x 3
  location   industry       share
                  
1 New York   Manufacturing  0.234
2 New York   Services       0.456
3 California Manufacturing  0.189
...

Bartik instrument summary:
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max.
-0.0456 -0.0123  0.0145  0.0178  0.0312  0.0567

IV Regression with Clustered Standard Errors
============================================
TSLS estimation, Pair(s): 1, Observations: 4,500
Cluster: location, Number of clusters: 150

                 Estimate Std. Error   t value   Pr(>|t|)
local_emp_growth   1.2346     0.3457     3.572   0.000489 ***
controls           0.4561     0.1235     3.707   0.000312 ***

First-stage statistics:
                    F-stat   p-value
local_emp_growth     28.34   < 2e-16

Partial R2 of excluded instruments: 0.0892

Recent Advances in Shift-Share Identification

Recent papers provide new understanding of when Bartik instruments are valid:

Goldsmith-Pinkham, Sorkin, Swift (2020): Identification comes from the shares; requires shares to be exogenous
Borusyak, Hull, Jaravel (2022): Identification comes from the shocks; requires shocks to be as-good-as-randomly assigned
Adão, Kolesár, Morales (2019): Standard errors need adjustment for exposure-weighted structure

Choose your identification argument and conduct appropriate diagnostics.

Multiple Instruments

With multiple instruments, you can test overidentifying restrictions, but be careful of overfitting and weak-IV bias magnification.

# Python: Multiple Instruments
from linearmodels.iv import IV2SLS

# Multiple instruments for one endogenous variable
model = IV2SLS.from_formula(
    'wage ~ 1 + experience + [education ~ z1 + z2 + z3]',
    data=df
)
results = model.fit(cov_type='robust')

# Overidentification test (Sargan-Hansen)
print("Sargan J-statistic:", results.sargan.stat)
print("p-value:", results.sargan.pval)

# Multiple endogenous variables
model = IV2SLS.from_formula(
    'wage ~ 1 + age + [education + experience ~ z1 + z2 + z3 + z4]',
    data=df
)

* Stata: Multiple Instruments

* Multiple instruments
ivreg2 wage experience (education = z1 z2 z3), robust first

* Check overidentification
estat overid

* Compare specifications with different instruments
estimates store full
ivreg2 wage experience (education = z1 z2), robust
estimates store partial
estimates table full partial

* Multiple endogenous variables
ivregress 2sls wage age (education experience = z1 z2 z3 z4), robust

# R: Multiple Instruments
library(AER)
library(fixest)

# Multiple instruments
iv_model <- ivreg(wage ~ education + experience |
                  z1 + z2 + z3 + experience,
                  data = df)
summary(iv_model, diagnostics = TRUE)

# Sargan test for overidentification
summary(iv_model, diagnostics = TRUE)$diagnostics["Sargan", ]

# With fixest: check stability across instrument sets
iv1 <- feols(wage ~ experience | education ~ z1, data = df)
iv2 <- feols(wage ~ experience | education ~ z1 + z2, data = df)
iv3 <- feols(wage ~ experience | education ~ z1 + z2 + z3, data = df)

etable(iv1, iv2, iv3)

Python Output

                          IV-2SLS Estimation Summary
==============================================================================
Dep. Variable:                   wage   R-squared:                      0.2356
Estimator:                    IV-2SLS   Adj. R-squared:                 0.2352
No. Observations:                3010   F-statistic:                    162.45
Date:                Mon, Jan 27 2026   P-value (F-stat)                0.0000

                             Parameter Estimates
==============================================================================
            Parameter  Std. Err.     T-stat    P-value    Lower CI    Upper CI
------------------------------------------------------------------------------
Intercept      8.1234     1.4123      5.752      0.000      5.355      10.892
experience     0.4534     0.0231     19.627      0.000      0.408       0.499
education      0.8823     0.1089      8.102      0.000      0.669       1.096
==============================================================================

Sargan-Hansen J-statistic: 2.345
Degrees of freedom: 2 (3 instruments - 1 endogenous)
P-value: 0.310

Interpretation: Fail to reject null hypothesis.
Overidentifying restrictions are valid - instruments are consistent.

First-stage F-statistic (joint): 32.56
Individual instrument F-stats:
  z1: 28.34
  z2: 18.92
  z3: 12.45

Stata Output

IV (2SLS) estimation
--------------------
                                                     Number of obs =     3,010
                                                     F(2, 3007)    =    162.45
                                                     Prob > F      =    0.0000

------------------------------------------------------------------------------
             |               Robust
        wage |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
   education |   .8823456   .1089123     8.10   0.000     .6688234    1.095868
  experience |   .4534123   .0231234    19.63   0.000     .4080709    .4987537
       _cons |   8.123456   1.412345     5.75   0.000     5.354178    10.89273
------------------------------------------------------------------------------

Hansen J statistic (overidentification test of all instruments):
  Chi-sq(2) = 2.345      P-val = 0.310
  Instruments appear valid

Comparing instrument specifications:
------------------------------------------------------------------------------
    Variable |      full          partial
-------------+--------------------------------
   education |   .8823456        .8912345
             |  (.1089123)      (.1234567)
  experience |   .4534123        .4523456
             |  (.0231234)      (.0245678)
------------------------------------------------------------------------------
Note: Estimates are stable across instrument sets

R Output

Diagnostic tests:
                    df1  df2 statistic p-value
Weak instruments      3 3006     32.56  <2e-16 ***
Wu-Hausman            1 3006     11.89  0.0006 ***
Sargan                2   NA      2.34  0.3100
---

Sargan Test for Overidentifying Restrictions:
  J-statistic: 2.345 on 2 DF
  p-value: 0.310
  Conclusion: Fail to reject - instruments appear valid

Comparison across instrument sets (etable):
                                 iv1           iv2           iv3
Dependent Var.:                 wage          wage          wage

education                   0.9123***     0.8912***     0.8823***
                           (0.1456)      (0.1234)      (0.1089)
experience                  0.4512***     0.4523***     0.4534***
                           (0.0256)      (0.0245)      (0.0231)
Constant                    8.0123***     8.0567***     8.1234***
                           (1.5234)      (1.4678)      (1.4123)

First-stage F               18.92         25.67         32.56
Sargan J (p-val)              -           0.456         0.310
---

Estimates stable across specifications: reassuring for validity.

Key IV References

Angrist, J. & Krueger, A. (2001). "Instrumental Variables and the Search for Identification." JEP.
Stock, J. & Yogo, M. (2005). "Testing for Weak Instruments." In Identification and Inference.
Goldsmith-Pinkham, P., Sorkin, I., & Swift, H. (2020). "Bartik Instruments: What, When, Why, and How." AER.
Borusyak, K., Hull, P., & Jaravel, X. (2022). "Quasi-Experimental Shift-Share Research Designs." ReStud.

ProTools ER1

Course Modules

6D Instrumental Variables

Table of Contents

Two-Stage Least Squares

IV Diagnostics

Weak Instruments

Bartik/Shift-Share Instruments

Multiple Instruments