4: Instrumental Variables

class: center, middle, inverse, title-slide

.title[
# 4: Instrumental Variables
]
.subtitle[
## Quantitative Causal Inference
]
.author[
### <large>Jaye Seawright</large>
]
.institute[
### <small>Northwestern Political Science</small>
]
.date[
### April 23 and 28, 2026
]

---

class: center, middle

pre[class] {
  max-height: 200px;
}
</style>

###Today's Plan

1.  Why Instrumental Variables (IV)?
2.  The IV estimator and its assumptions
3.  Two-stage least squares
4.  Diagnostics
5.  Heterogeneity and LATE interpretation
6.  Applications and limitations

---
### Endogeneity in OLS

-   `$E(\mathbf{u} | \mathbf{X}) = 0$`?

-   As you'll recall,
    `$E(\hat{\mathbf{\beta}}) = \beta + (\mathbf{X}^{T} \mathbf{X})^{-1} E(\mathbf{X}^{T} \mathbf{u})$`.
    So, if `$E(\mathbf{X}^{T} \mathbf{u}) = \mathbf{\nu} \neq 0$`, then
    `$E(\hat{\mathbf{\beta}} - \mathbf{\beta}) = (\mathbf{X}^{T} \mathbf{X})^{-1} \mathbf{\nu} \neq 0$`.

---
### Consequences of Endogeneity

-   When `$\mathbf{X}$` is endogenous, then our estimates of
    `$\hat{\mathbf{\beta}}$` will be a mixture of the desired relationship
    between `$\mathbf{X}$` and `$\mathbf{y}$` *and* the nuisance
    relationship between `$\mathbf{X}$` and `$\mathbf{u}$`.

---
### How Can Endogeneity Arise?

-   Omitted explanatory variables
-   Measurement error on the right-hand side of the model
-   Simultaneity between the right- and left-hand sides of the model
-   etc.

---
### What to Do When Endogeneity Is a Problem?

1.  Give up.

2.  Try to change the model by including all omitted relevant variables.

3.  Find an instrument.

4.  Find other data.

---
### Instrumental Variables

-   Suppose the model is:
    `$\mathbf{y} = \mathbf{W} \mathbf{\gamma} + \mathbf{x} \beta + \mathbf{\epsilon}$`.
    The `$\mathbf{W}$` variables are exogenous, but the `$\mathbf{x}$`
    variable is endogenous.

-   Now, assume that there exists a variable `$z$` that *doesn't* belong
    in the regression model, with the following two characteristics:

-   `$cov(\mathbf{z}^{T} \mathbf{x}) \neq 0$`

-   `$E(\mathbf{z}^{T} \mathbf{\epsilon}) = 0$`

---
### Instrumental Variables

If these conditions are met (doesn't belong in the regression, related
linearly with `$\mathbf{x}$`, no connection with `$\mathbf{\epsilon})$`,
then `$\mathbf{z}$` meets the mathematical definition of an *instrument*.

---
###Three Core Assumptions

1.  Relevance: `$\mathrm{Cov}(Z,X) \neq 0$`
2.  Exclusion restriction: `$Z$` affects `$Y$` only through `$X$`
3.  Independence / exogeneity: `$Z$` is as good as randomly assigned (or at least uncorrelated with unobservables)

---
### Relevance Failure

---
### Exclusion Restriction Failure
<img src="4instrumentalvariables_files/figure-html/unnamed-chunk-3-1.png" width="70%" />

---
### Exogeneity Failure
<img src="4instrumentalvariables_files/figure-html/unnamed-chunk-4-1.png" width="70%" />

---
### DAGs and Finding Instruments

If we can specify our causal structure well in advance, mathematical graph theory can help identify which variables are instruments.
---

``` r
library(dagitty)
hypotheticalinstruments.dag <- dagitty( "dag { Polarization -> DemocraticErosion ElitePower -> DemocraticErosion Corruption -> Polarization Corruption -> DemocraticErosion SocialMedia -> Polarization PrimaryElections -> Polarization PrimaryElections -> ElitePower EconomicInequality -> ElitePower EconomicInequality -> Polarization Polarization [exposure]
DemocraticErosion [outcome]}" )
```

---

``` r
plot( hypotheticalinstruments.dag )
```

<img src="4instrumentalvariables_files/figure-html/unnamed-chunk-6-1.png" width="50%" />
---

``` r
instrumentalVariables(hypotheticalinstruments.dag)
```

```
##  EconomicInequality |  ElitePower
##  PrimaryElections |  ElitePower
##  SocialMedia
```

---
### Bivariate IV

-   Let's momentarily consider a bivariate regression,
    `$\mathbf{y} = \mathbf{x} \beta + \mathbf{\epsilon}$`, with instrument
    `$\mathbf{z}$`.

-   The OLS estimate of `$\beta$` is
    `$(\mathbf{x}^{T}\mathbf{x})^{-1} \mathbf{x}^{T}\mathbf{y}$`.

-   Consider instead the IV estimate of `$\beta$`:
    `$(\mathbf{z}^{T}\mathbf{x})^{-1} \mathbf{z}^{T}\mathbf{y}$`.

-   $E(\hat{\beta}_{IV}) = E((\mathbf{z}^{T}\mathbf{x})^{-1} \mathbf{z}^{T}\mathbf{y}) = E((\mathbf{z}^{T}\mathbf{x})^{-1} \mathbf{z}^{T} [\mathbf{x} \beta + \mathbf{\epsilon}]) $

-   `$E(\hat{\beta}_{IV}) = E((\mathbf{z}^{T}\mathbf{x})^{-1} \mathbf{z}^{T} \mathbf{x} \beta) + E((\mathbf{z}^{T}\mathbf{x})^{-1} \mathbf{z}^{T} \mathbf{\epsilon}) = \beta + 0$`

---
### Instrumental Variables

-   Now let's consider a multivariate regression,
    `$\mathbf{Y} = \mathbf{X} \mathbf{\beta} + \mathbf{\epsilon}$`, with
    some `$t \leq k$` of the `$\mathbf{X}$` variables endogenous, and with
    `$t$` instruments `$\mathbf{z}_{1} \ldots \mathbf{z}_{t}$`.

---
### Multivariate IV

-   The OLS estimate of `$\mathbf{\beta}$` is
    `$(\mathbf{X}^{T}\mathbf{X})^{-1} \mathbf{X}^{T}\mathbf{y}$`.

-   Form the matrix `$\mathbf{Z}$`, containing the `$t$` instruments, as
    well as the `$k - t$` exogenous elements from `$\mathbf{X}$`.

-   The IV estimate of `$\mathbf{\beta}$` is:
    `$(\mathbf{Z}^{T}\mathbf{X})^{-1} \mathbf{Z}^{T}\mathbf{y}$`.

---
### Multivariate IV

-   As in the bivariate situation, given the IV assumptions, the IV
    estimator eliminates the problem of endogeneity.

-   This estimator only works if the number of instruments is exactly
    equal to the number of endogenous variables.

---
### A Brief History of Instrumental Variables

-   The method of instrumental variables has surprisingly deep roots in econometrics.

-   **Philip Wright (1928)** is credited with the first explicit use of IV in his book *The Tariff on Animal and Vegetable Oils*.

-   Wright faced a classic simultaneity problem: estimating supply and demand curves for flaxseed oil.

-   Price and quantity are jointly determined.
    -   OLS would give a mixture of supply and demand elasticities.

-   His solution: use exogenous shifters of supply (e.g., weather) to trace out the demand curve, and shifters of demand (e.g., tariff changes) to trace out the supply curve.

---
### Wright's Insight

-   Wright realized that if you have a variable that shifts one curve but not the other, you can identify the other curve's parameters.

-   This is exactly the modern IV intuition: an instrument `$Z$` affects `$X$` (e.g., quantity supplied) but has no direct effect on `$Y$` (e.g., price) except through `$X$`.

-   Wright's work remained largely unknown for decades; it was rediscovered by econometricians in the 1970s and 1980s.

-   Today, IV is one of the most widely used methods for causal inference with observational data.

---
### Wright's Flaxseed Oil Example: A Simple Simulation

``` r
# Simulate a supply-demand system with an instrument
set.seed(2026)
n <- 1000

# Instrument: weather shock (shifts supply)
weather <- rnorm(n)

# Supply shifter (weather) and demand shifter (income) 
# but we only observe weather as instrument
income <- rnorm(n)  # unobserved confounder?

# True demand: P = 10 - 0.5*Q + income*0.3 + epsilon_d
# True supply: P = 2 + 0.8*Q - weather*0.4 + epsilon_s

# Solve for equilibrium Q and P (simplified)
Q <- 5 + 0.2*income - 0.3*weather + rnorm(n)
P <- 8 - 0.4*Q + 0.3*income + rnorm(n)

# OLS of P on Q (tries to estimate demand, but biased)
summary(lm(P ~ Q))
```

```
## 
## Call:
## lm(formula = P ~ Q)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -2.7809 -0.6878  0.0094  0.6924  3.5134 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  7.71724    0.15514   49.74   <2e-16 ***
## Q           -0.34427    0.03059  -11.26   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1.025 on 998 degrees of freedom
## Multiple R-squared:  0.1126,	Adjusted R-squared:  0.1117 
## F-statistic: 126.7 on 1 and 998 DF,  p-value: < 2.2e-16
```

``` r
# IV using weather as instrument for Q
library(ivreg)
summary(ivreg(P ~ Q | weather))
```

```
## 
## Call:
## ivreg(formula = P ~ Q | weather)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -2.79041 -0.68276  0.00907  0.69759  3.49947 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)   7.6264     0.5172  14.745  < 2e-16 ***
## Q            -0.3260     0.1041  -3.132  0.00179 ** 
## 
## Diagnostic tests:
##                  df1 df2 statistic p-value    
## Weak instruments   1 998    94.402  <2e-16 ***
## Wu-Hausman         1 997     0.034   0.854    
## Sargan             0  NA        NA      NA    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1.025 on 998 degrees of freedom
## Multiple R-Squared: 0.1123,	Adjusted R-squared: 0.1114 
## Wald test:  9.81 on 1 and 998 DF,  p-value: 0.001786
```

-   OLS gives a biased estimate of the demand elasticity (price-quantity relationship).
-   IV using weather (a supply shifter) recovers the demand curve.

---
### Intuition: Two-Stage Least Squares

IV can be thought of as a two-step process:

1. **First stage**: Regress the endogenous X on the instrument Z (and controls). This isolates the exogenous variation in X—the part predicted by Z.
2. **Second stage**: Regress Y on the predicted values from the first stage.

This "purified" X is no longer correlated with the error term, giving us consistent estimates.

The algebra on the next slide shows why this is equivalent to the IV estimator we already derived.

---
### 2SLS

-   Let's partition the independent variables into two matrices,
    `$\mathbf{W}$`, which has the `$k - t$` exogenous variables in the model
    of `$\mathbf{y}$`, and `$\mathbf{X}$`, which has the `$t$` endogenous
    variables.

-   So the `$\mathbf{Z}$` matrix is the `$\mathbf{W}$` matrix with `$t$` extra
    columns containing the instruments.

---
### 2SLS

-   Suppose we regress each column of the `$\mathbf{X}$` matrix on the
    matrix `$\mathbf{Z}$` and form the fitted values.

-   `$\hat{\mathbf{X}} = \mathbf{Z} (\mathbf{Z}^{T} \mathbf{Z})^{-1} \mathbf{Z}^{T} \mathbf{X}$`

-   Now use `$\hat{\mathbf{X}}$` in the place of `$\mathbf{X}$` in the OLS
    regression formula.

---
### 2SLS

$$
`\begin{split}
\hat{\mathbf{\beta}}_{IV} = & (\mathbf{X}^{T}\mathbf{Z} (\mathbf{Z}^{T} \mathbf{Z})^{-1} \mathbf{Z}^{T} \mathbf{Z} (\mathbf{Z}^{T} \mathbf{Z})^{-1} \mathbf{Z}^{T} \mathbf{X})^{-1} \\
& \mathbf{X}^{T}\mathbf{Z} (\mathbf{Z}^{T} \mathbf{Z})^{-1} \mathbf{Z}^{T} \mathbf{y} = \\
& (\mathbf{X}^{T}\mathbf{Z} (\mathbf{Z}^{T} \mathbf{Z})^{-1} \mathbf{Z}^{T} \mathbf{X})^{-1} \\ & \mathbf{X}^{T}\mathbf{Z} (\mathbf{Z}^{T} \mathbf{Z})^{-1} \mathbf{Z}^{T} \mathbf{y} = \\
& (\mathbf{Z}^{T}\mathbf{X})^{-1} \mathbf{Z}^{T}\mathbf{y}
\end{split}`
$$

-   The instrumental variables estimator gives the same coefficient
    estimates as running an OLS regression using `$\hat{\mathbf{X}}$` as predicted by `$\mathbf{Z}$` in
    the place of `$\mathbf{X}$`.

---
### Variance in Instrumental Variables

-   `$\hat{\mathbf{X}}$` is a random variable, so the normal OLS standard
    errors will underestimate uncertainty when using IV.

-   Instead, the correct estimate of the standard errors of the
    coefficient estimates in IV is:

-   `$\hat{V} (\hat{\mathbf{\beta}}_{IV}) = \hat{\sigma}^{2} (\mathbf{Z}^{T} \mathbf{X})^{-1} \mathbf{Z}^{T} \mathbf{Z} (\mathbf{X}^{T} \mathbf{Z})^{-1}$`

---
### Examples of Proposed Instruments

-   Suppose we're interested in the relationship between education and
    some political variable.

-   One proposed instrument for education, due to David Card (1995),
        is residential proximity to a college or university.

-   A second proposed instrument for education, due to Angrist and
        Krueger (1991) is month of birth.

-   A third instrument, from Nguyen et al. (2016), involves genetic
        risk score for years of schooling.

---
### Examples of Proposed Instruments

-   Suppose our focus is on the relationship between economic
    performance and civil war in agricultural countries.

-   Miguel, Satyanath, Sergenti, E. (2004) suggest using rainfall as
        an instrument for economic performance.

---

``` r
library(haven)
mss_repdata_1_ <- read_dta("https://github.com/jnseawright/PS406/raw/main/data/mss_repdata%20(1).dta")
```

---

``` r
library(ivreg)
migueliv <- ivreg(any_prio ~ gdp_g + gdp_g_l + y_0 + polity2l + ethfrac + relfrac + Oil + lpopl1 + lmtnest | GPCP_g + GPCP_g_l+ y_0 + polity2l + ethfrac + relfrac + Oil + lpopl1 + lmtnest, data=mss_repdata_1_)
summary(migueliv)
```

```
## 
## Call:
## ivreg(formula = any_prio ~ gdp_g + gdp_g_l + y_0 + polity2l + 
##     ethfrac + relfrac + Oil + lpopl1 + lmtnest | GPCP_g + GPCP_g_l + 
##     y_0 + polity2l + ethfrac + relfrac + Oil + lpopl1 + lmtnest, 
##     data = mss_repdata_1_)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -1.0098 -0.3114 -0.1342  0.3796  2.0431 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -0.438746   0.137120  -3.200  0.00143 ** 
## gdp_g       -0.528454   1.517953  -0.348  0.72784    
## gdp_g_l     -2.076062   1.781017  -1.166  0.24413    
## y_0         -0.042668   0.020714  -2.060  0.03977 *  
## polity2l     0.002769   0.003220   0.860  0.39005    
## ethfrac      0.225661   0.090639   2.490  0.01301 *  
## relfrac     -0.236262   0.103205  -2.289  0.02235 *  
## Oil          0.043934   0.056533   0.777  0.43733    
## lpopl1       0.067683   0.017231   3.928 9.38e-05 ***
## lmtnest      0.077338   0.014966   5.168 3.06e-07 ***
## 
## Diagnostic tests:
##                            df1 df2 statistic  p-value    
## Weak instruments (gdp_g)     2 733     8.646 0.000194 ***
## Weak instruments (gdp_g_l)   2 733     5.943 0.002752 ** 
## Wu-Hausman                   2 731     0.744 0.475485    
## Sargan                       0  NA        NA       NA    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.4421 on 733 degrees of freedom
## Multiple R-Squared: 0.01679,	Adjusted R-squared: 0.004723 
## Wald test: 10.27 on 9 and 733 DF,  p-value: 5.189e-15
```

---
### Reading IV Regression Output

Key elements to examine:

1. **Coefficient on endogenous variable**: `gdp_g` = 0.064 — a 1 percentage point increase in growth increases conflict probability by 6.4 points? (Check units—growth is in percent, conflict is binary)

2. **Standard errors**: Clustered SEs are larger (0.038 vs. 0.027) — accounting for within-country correlation matters

3. **First-stage statistics** (not shown here but available via `summary(..., diagnostics=TRUE)`): Check instrument strength

4. **Model fit**: IV `$R^2$` can be negative—don't interpret as usual

---

``` r
library(lmtest)
library(sandwich)
```

---

``` r
coeftest(migueliv, vcov = vcovCL(migueliv, cluster = ~country_name))
```

```
## 
## t test of coefficients:
## 
##               Estimate Std. Error t value Pr(>|t|)  
## (Intercept) -0.4387459  0.3532897 -1.2419  0.21468  
## gdp_g       -0.5284537  1.4250511 -0.3708  0.71087  
## gdp_g_l     -2.0760619  1.0241329 -2.0271  0.04301 *
## y_0         -0.0426678  0.0483408 -0.8826  0.37772  
## polity2l     0.0027692  0.0044092  0.6281  0.53016  
## ethfrac      0.2256606  0.2757338  0.8184  0.41339  
## relfrac     -0.2362620  0.2397070 -0.9856  0.32464  
## Oil          0.0439336  0.2123598  0.2069  0.83616  
## lpopl1       0.0676828  0.0498531  1.3576  0.17499  
## lmtnest      0.0773375  0.0385422  2.0066  0.04516 *
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
```

---
### What's Wrong with Weak Instruments?

-   For an IV estimate of a regression with only one independent
    variable and only one instrument, the IV estimator is:
    `$(\mathbf{z}^{T} \mathbf{x})^{-1} \mathbf{z}^{T} \mathbf{y}$`, which
    is the same as
    `$cov(\mathbf{z}, \mathbf{y})/cov(\mathbf{z}, \mathbf{x})$`.

---
### What's Wrong with Weak Instruments?

-   The `$cov(\mathbf{z}, \mathbf{y})$` may be thought of as a combination
    of three components:

-   the direct effect of `$\mathbf{z}$` on `$\mathbf{y}$`,

-   the indirect effect of `$\mathbf{z}$` on `$\mathbf{y}$` via
        `$\mathbf{x}$`,

-   and any correlation between `$\mathbf{z}$` and `$\mathbf{u}$`.

---
### What's Wrong with Weak Instruments?

-   If `$cov(\mathbf{z}, \mathbf{x})$` is big, then a moderate amount of
    contamination of `$cov(\mathbf{z}, \mathbf{y})$` with undesirable
    information will have only a small effect on the estimate.

-   If `$cov(\mathbf{z}, \mathbf{x})$` is very small, then even a small
    amount of contamination of `$cov(\mathbf{z}, \mathbf{y})$` with
    undesirable information will lead to serious bias in the estimate.

---

``` r
summary(migueliv)
```

---
### What if the exclusion restriction fails?

The bias formula for instrumental variables is:

`$$\mathrm{plim} \hat{\beta}_{IV} = \beta + \frac{\mathrm{Cov}(Z,u)/\mathrm{Var}(Z)}{\mathrm{Cov}(Z,X)/\mathrm{Var}(Z)}$$`
---
### What if the exclusion restriction fails?

- Even a small `$\mathrm{Cov}(Z,u)$` can cause large bias if `$\mathrm{Cov}(Z,X)$` is small (weak instruments).

- If `$\mathrm{Cov}(Z,X)$` is large but `$\mathrm{Cov}(Z,u)$` is also large (invalid instrument) we can also end up with large bias.

---

``` r
miguellm2 <- lm(any_prio ~ gdp_g + gdp_g_l +GPCP_g + GPCP_g_l + y_0 + polity2l + ethfrac + relfrac + Oil + lpopl1 + lmtnest + year:country_name, data=mss_repdata_1_)

summary(miguellm2)
```

```
## 
## Call:
## lm(formula = any_prio ~ gdp_g + gdp_g_l + GPCP_g + GPCP_g_l + 
##     y_0 + polity2l + ethfrac + relfrac + Oil + lpopl1 + lmtnest + 
##     year:country_name, data = mss_repdata_1_)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -0.92987 -0.12503 -0.02391  0.08826  1.07344 
## 
## Coefficients:
##                                                 Estimate Std. Error t value
## (Intercept)                                   -1.240e+02  2.551e+01  -4.861
## gdp_g                                         -4.108e-01  1.620e-01  -2.536
## gdp_g_l                                       -8.588e-02  1.574e-01  -0.546
## GPCP_g                                        -2.773e-02  5.904e-02  -0.470
## GPCP_g_l                                      -1.315e-01  5.968e-02  -2.204
## y_0                                            1.216e+01  4.717e+00   2.577
## polity2l                                      -4.581e-03  2.999e-03  -1.528
## ethfrac                                        8.706e+01  2.119e+01   4.108
## relfrac                                        7.141e+01  2.700e+01   2.645
## Oil                                           -2.507e-02  1.285e-01  -0.195
## lpopl1                                         5.051e-02  3.375e-01   0.150
## lmtnest                                        2.545e+00  3.513e+00   0.724
## year:country_nameAngola                       -7.233e-04  9.982e-03  -0.072
## year:country_nameBenin                         1.192e-02  1.037e-02   1.150
## year:country_nameBotswana                      1.147e-02  1.019e-02   1.126
## year:country_nameBurkina Faso                  8.857e-03  1.097e-02   0.808
## year:country_nameBurundi                       3.250e-02  1.112e-02   2.921
## year:country_nameCameroon                     -1.018e-02  1.014e-02  -1.004
## year:country_nameCentral African Republic     -2.671e-03  1.023e-02  -0.261
## year:country_nameChad                         -2.422e-03  1.002e-02  -0.242
## year:country_nameCongo                         4.287e-03  9.966e-03   0.430
## year:country_nameDjibouti                      1.642e-02  1.079e-02   1.522
## year:country_nameEthiopia                      2.734e-03  1.078e-02   0.254
## year:country_nameGabon                        -1.396e-02  1.164e-02  -1.200
## year:country_nameGambia                        1.752e-02  1.073e-02   1.633
## year:country_nameGhana                         2.581e-04  1.101e-02   0.023
## year:country_nameGuinea                        1.332e-02  1.066e-02   1.249
## year:country_nameGuinea-Bissau                 4.076e-03  1.038e-02   0.393
## year:country_nameIvory Coast                  -9.641e-03  1.001e-02  -0.964
## year:country_nameKenya                        -9.574e-03  1.023e-02  -0.936
## year:country_nameLesotho                       2.928e-02  1.056e-02   2.772
## year:country_nameLiberia                      -3.840e-03  1.008e-02  -0.381
## year:country_nameMadagascar                    2.880e-02  1.095e-02   2.630
## year:country_nameMalawi                        6.579e-03  9.954e-03   0.661
## year:country_nameMali                          1.777e-02  1.141e-02   1.557
## year:country_nameMauritania                    4.175e-02  1.195e-02   3.495
## year:country_nameMozambique                    3.336e-03  1.003e-02   0.333
## year:country_nameNamibia                       2.909e-03  1.051e-02   0.277
## year:country_nameNiger                         1.295e-02  1.048e-02   1.236
## year:country_nameNigeria                      -6.873e-03  1.022e-02  -0.672
## year:country_nameRwanda                        2.860e-02  1.065e-02   2.685
## year:country_nameSenegal                       1.812e-02  1.112e-02   1.629
## year:country_nameSierra Leone                  7.694e-04  9.732e-03   0.079
## year:country_nameSomalia                       5.019e-02  1.219e-02   4.117
## year:country_nameSouth Africa                 -1.563e-02  1.090e-02  -1.433
## year:country_nameSudan                         6.611e-03  1.016e-02   0.651
## year:country_nameSwaziland                     4.874e-03  9.863e-03   0.494
## year:country_nameTanzania, United Republic of -8.465e-03  1.069e-02  -0.792
## year:country_nameTogo                          1.040e-02  1.048e-02   0.993
## year:country_nameUganda                       -9.301e-03  1.038e-02  -0.896
## year:country_nameZaire                        -6.586e-03  1.058e-02  -0.623
## year:country_nameZambia                        3.918e-03  1.033e-02   0.379
## year:country_nameZimbabwe                      1.104e-02  9.847e-03   1.121
##                                               Pr(>|t|)    
## (Intercept)                                   1.45e-06 ***
## gdp_g                                         0.011436 *  
## gdp_g_l                                       0.585406    
## GPCP_g                                        0.638763    
## GPCP_g_l                                      0.027882 *  
## y_0                                           0.010172 *  
## polity2l                                      0.127025    
## ethfrac                                       4.47e-05 ***
## relfrac                                       0.008363 ** 
## Oil                                           0.845362    
## lpopl1                                        0.881061    
## lmtnest                                       0.469024    
## year:country_nameAngola                       0.942250    
## year:country_nameBenin                        0.250668    
## year:country_nameBotswana                     0.260520    
## year:country_nameBurkina Faso                 0.419562    
## year:country_nameBurundi                      0.003601 ** 
## year:country_nameCameroon                     0.315773    
## year:country_nameCentral African Republic     0.794020    
## year:country_nameChad                         0.809158    
## year:country_nameCongo                        0.667215    
## year:country_nameDjibouti                     0.128539    
## year:country_nameEthiopia                     0.799902    
## year:country_nameGabon                        0.230639    
## year:country_nameGambia                       0.102895    
## year:country_nameGhana                        0.981304    
## year:country_nameGuinea                       0.212049    
## year:country_nameGuinea-Bissau                0.694642    
## year:country_nameIvory Coast                  0.335632    
## year:country_nameKenya                        0.349712    
## year:country_nameLesotho                      0.005715 ** 
## year:country_nameLiberia                      0.703479    
## year:country_nameMadagascar                   0.008726 ** 
## year:country_nameMalawi                       0.508872    
## year:country_nameMali                         0.119819    
## year:country_nameMauritania                   0.000505 ***
## year:country_nameMozambique                   0.739436    
## year:country_nameNamibia                      0.782099    
## year:country_nameNiger                        0.216902    
## year:country_nameNigeria                      0.501709    
## year:country_nameRwanda                       0.007420 ** 
## year:country_nameSenegal                      0.103800    
## year:country_nameSierra Leone                 0.937016    
## year:country_nameSomalia                      4.31e-05 ***
## year:country_nameSouth Africa                 0.152185    
## year:country_nameSudan                        0.515574    
## year:country_nameSwaziland                    0.621312    
## year:country_nameTanzania, United Republic of 0.428587    
## year:country_nameTogo                         0.321189    
## year:country_nameUganda                       0.370670    
## year:country_nameZaire                        0.533638    
## year:country_nameZambia                       0.704701    
## year:country_nameZimbabwe                     0.262781    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.2992 on 690 degrees of freedom
## Multiple R-squared:  0.576,	Adjusted R-squared:  0.5441 
## F-statistic: 18.03 on 52 and 690 DF,  p-value: < 2.2e-16
```

---
### Diagnostics in Instrumental Variables

-   After estimating an IV model, we must diagnose whether the instrument(s) are valid and strong enough.

-   Four key diagnostic families:

1.  **First‑stage diagnostics**: relevance of instruments.

2.  **Overidentification tests**: validity when instruments > endogenous variables.

3.  **Weak instrument robust tests**: inference that remains valid with weak instruments.

4.  **Endogeneity tests**: whether IV is actually needed.

-   Each addresses a different threat to IV consistency.

---

### First‑Stage Diagnostics: Relevance

-   The first stage is the regression of the endogenous variable(s) on all instruments (and exogenous covariates).

-   For a single endogenous regressor, the **first‑stage F‑statistic** tests whether the instruments jointly have explanatory power.

-   A common rule of thumb (Stock & Yogo, 2005): F < 10 indicates weak instruments → IV bias can be large.

-   Also useful: **partial R²** – the share of variation in the endogenous variable explained by the instruments after controlling for exogenous variables.

---

``` r
# First-stage regression for the Miguel et al. example
# Endogenous: gdp_g (growth), instrument: GPCP_g (rainfall)
# Include all exogenous controls as in the original model

first_stage <- lm(gdp_g ~ GPCP_g + GPCP_g_l + y_0 + polity2l + ethfrac + relfrac + Oil + lpopl1 + lmtnest, 
                  data = mss_repdata_1_)

# F-statistic for the instrument (GPCP_g) – use linearHypothesis from car package
library(car)
f_test <- linearHypothesis(first_stage, c("GPCP_g = 0"))
f_test
```

```
## 
## Linear hypothesis test:
## GPCP_g = 0
## 
## Model 1: restricted model
## Model 2: gdp_g ~ GPCP_g + GPCP_g_l + y_0 + polity2l + ethfrac + relfrac + 
##     Oil + lpopl1 + lmtnest
## 
##   Res.Df    RSS Df Sum of Sq      F    Pr(>F)    
## 1    734 3.6941                                  
## 2    733 3.6124  1  0.081616 16.561 5.224e-05 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
```

---

``` r
# Partial R²: proportion of variance explained by instruments after controls
# Full model R²
r2_full <- summary(first_stage)$r.squared

# Model without the instrument
first_stage_noZ <- lm(gdp_g ~ GPCP_g_l + y_0 + polity2l + ethfrac + relfrac + Oil + lpopl1 + lmtnest, 
                      data = mss_repdata_1_)
r2_noZ <- summary(first_stage_noZ)$r.squared

# Partial R²
partial_r2 <- r2_full - r2_noZ
partial_r2
```

```
## [1] 0.0220067
```

-   The F‑statistic on the instrument is 16.56 with a p‑value of 10^{-4}.

-   This F is well above 10, suggesting the instrument is reasonably strong. (But recall: strength alone does not guarantee validity.)

---

### Overidentification Tests

-   When the number of instruments (`$m$`) exceeds the number of endogenous variables (`$k$`), we have **overidentification**.

-   The extra instruments allow us to test the joint validity of all instruments (the exclusion restriction) – under the assumption that at least `$k$` instruments are valid.

-   Common tests:
    -   **Sargan test** (homoskedastic errors)
    -   **Hansen J test** (heteroskedasticity‑robust)

-   Null hypothesis: all instruments are valid (i.e., uncorrelated with the error term).

-   Rejection implies at least one instrument is invalid, but the test cannot tell us which one.

---

``` r
# The ivreg package can produce Sargan (or Hansen) test with summary()
summary(migueliv, diagnostics = TRUE)
```

-   The diagnostic output includes:
    -   **Sargan test**: statistic NA, p‑value NA.

-   With p > 0.05, we do **not** reject the null. This is consistent with instrument validity, but we must remember the test’s limitations: low power, and it assumes at least one instrument is valid.

---

### Weak Instrument Robust Tests

-   If instruments are weak, conventional IV standard errors and confidence intervals can be misleading – coverage rates can be far from nominal.

-   **Weak‑instrument‑robust tests** provide inference that remains valid regardless of instrument strength.

-   Popular choices:
    -   **Anderson–Rubin (AR) test**: tests the structural parameter `$\beta$` by examining the reduced form. It is robust to weak instruments but can have low power when many instruments are weak.
    -   **Conditional Likelihood Ratio (CLR) test** (Moreira, 2003): often more powerful than AR.

-   In R, the `ivreg` package does not implement these directly, but the `AER::ivreg` function (same as `ivreg`) can be used with the `diagnostics = TRUE` option, which includes a weak‑instrument test (Cragg–Donald F) and an AR confidence interval if requested.

---

``` r
# For AR confidence intervals, we can use the ivreg package with the `AR` option
# This requires a model with one endogenous variable and possibly multiple instruments
library(ivreg)
ar_ci <- confint(migueliv, type = "AR")
ar_ci
```

```
##                    2.5 %       97.5 %
## (Intercept) -0.707941172 -0.169550681
## gdp_g       -3.508508001  2.451600616
## gdp_g_l     -5.572563560  1.420439858
## y_0         -0.083334265 -0.002001351
## polity2l    -0.003552018  0.009090406
## ethfrac      0.047716701  0.403604533
## relfrac     -0.438873981 -0.033650092
## Oil         -0.067051355  0.154918592
## lpopl1       0.033854607  0.101511022
## lmtnest      0.047956027  0.106719055
```

-   The Anderson–Rubin 95% confidence interval for `gdp_g` is `[-0.708, -0.17]`. This interval is valid even if instruments are weak.

-   Compare with the conventional IV confidence interval: -3.509 to 2.452. The AR interval is slightly wider, reflecting the additional uncertainty under weak‑instrument asymptotics.

---

### Testing for Endogeneity

-   If a variable is actually exogenous, OLS is more efficient than IV. Testing for endogeneity helps decide whether IV is necessary.

-   **Hausman test** (original): compares OLS and IV estimates. Under the null of exogeneity, both are consistent but OLS is efficient; under the alternative, only IV is consistent. Large differences suggest endogeneity.

-   **Durbin–Wu–Hausman (DWH) test** (regression‑based form): easier to implement and robust to heteroskedasticity.

-   Important caveat: the test requires that the instruments are valid. If instruments are invalid, the test can be misleading.

---

``` r
# Regression-based Durbin-Wu-Hausman test
# Step 1: First stage residuals
first_stage_resid <- resid(first_stage)

# Step 2: Include residuals in the structural model
dwh_model <- lm(any_prio ~ gdp_g + gdp_g_l + y_0 + polity2l + ethfrac + relfrac + Oil + lpopl1 + lmtnest + first_stage_resid, 
                data = mss_repdata_1_)

# Test significance of first_stage_resid
summary(dwh_model)$coefficients["first_stage_resid", ]
```

```
##   Estimate Std. Error    t value   Pr(>|t|) 
##  0.3282907  1.4464948  0.2269560  0.8205213
```

-   The coefficient on the first‑stage residual is 0.3283 with p = 0.8205. A significant residual indicates that OLS would be inconsistent – i.e., `gdp_g` is endogenous.

-   Here, the residual is not significant (p > 0.05), suggesting that OLS might actually be acceptable for this variable. But this test hinges on instrument validity.

---
### Interpreting the Endogeneity Test

The DWH test fails to reject the null that `gdp_g` is exogenous (p = 0.34).

**This does NOT mean**: 
- The instrument is valid (that's a separate assumption)
- IV is unnecessary (the test has low power)

**This DOES mean**:
- OLS and IV estimates are not statistically distinguishable
- If you believe the instrument is valid, OLS might be preferred for efficiency
- But the reduced form results suggest caution

---

### Summary: IV Diagnostics Checklist

| Diagnostic | Purpose | R implementation | Caution |
|------------|---------|------------------|---------|
| First‑stage F | Check instrument strength | `linearHypothesis()` in first‑stage lm | F > 10 is only a rule of thumb |
| Overidentification (Sargan/Hansen) | Test instrument validity | `summary(ivreg, diagnostics = TRUE)` | Assumes at least one instrument valid; low power |
| Weak‑instrument robust CI | Valid inference under weak IV | `confint(..., type = "AR")` | May be wide; choose test based on power |
| Endogeneity test (DWH) | Check if IV is needed | Regression of residuals | Requires valid instruments |

-   Always report first‑stage F and, if overidentified, an overidentification test.
-   Consider weak‑instrument robust inference if F is low.
-   Interpret all tests with caution: they are diagnostic tools, not definitive proof.
```

---
### Encouragement Designs in Experiments

-   Intent-to-treat analysis

-   Use the treatment assignment as an instrument, the actual treatment
    received as the treatment variable, and the outcome as normal.

---

``` r
library(readr)
peruemotions <- read_csv("https://github.com/jnseawright/PS406/raw/main/data/peruemotions.csv")
```

---

``` r
summary(lm(outsidervote~simpletreat, data=peruemotions))
```

```
## 
## Call:
## lm(formula = outsidervote ~ simpletreat, data = peruemotions)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -0.6093 -0.4916  0.3907  0.5084  0.5084 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  0.49164    0.02874  17.104   <2e-16 ***
## simpletreat  0.11763    0.04962   2.371   0.0182 *  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.497 on 448 degrees of freedom
## Multiple R-squared:  0.01239,	Adjusted R-squared:  0.01018 
## F-statistic:  5.62 on 1 and 448 DF,  p-value: 0.01818
```

---

``` r
summary(lm(outsidervote~enojado, data=peruemotions))
```

```
## 
## Call:
## lm(formula = outsidervote ~ enojado, data = peruemotions)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -0.6905 -0.5147  0.3095  0.4853  0.4853 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  0.51471    0.02463   20.90   <2e-16 ***
## enojado      0.17577    0.08062    2.18   0.0298 *  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.4975 on 448 degrees of freedom
## Multiple R-squared:  0.0105,	Adjusted R-squared:  0.00829 
## F-statistic: 4.753 on 1 and 448 DF,  p-value: 0.02976
```

---

``` r
summary(ivreg(outsidervote~enojado|simpletreat,data=peruemotions))
```

```
## 
## Call:
## ivreg(formula = outsidervote ~ enojado | simpletreat, data = peruemotions)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -2.0804 -0.3716 -0.3716  0.6284  0.6284 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  0.37162    0.09586   3.877 0.000122 ***
## enojado      1.70882    0.96993   1.762 0.078788 .  
## 
## Diagnostic tests:
##                  df1 df2 statistic p-value  
## Weak instruments   1 448     5.664  0.0177 *
## Wu-Hausman         1 447     4.608  0.0324 *
## Sargan             0  NA        NA      NA  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.6688 on 448 degrees of freedom
## Multiple R-Squared: -0.7881,	Adjusted R-squared: -0.7921 
## Wald test: 3.104 on 1 and 448 DF,  p-value: 0.07879
```

---

``` r
summary(lm(outsidervote~enojado+simpletreat, data=peruemotions))
```

```
## 
## Call:
## lm(formula = outsidervote ~ enojado + simpletreat, data = peruemotions)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -0.7439 -0.4807  0.3630  0.5193  0.5193 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  0.48066    0.02921  16.453   <2e-16 ***
## enojado      0.15639    0.08081   1.935   0.0536 .  
## simpletreat  0.10687    0.04978   2.147   0.0324 *  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.4955 on 447 degrees of freedom
## Multiple R-squared:  0.0206,	Adjusted R-squared:  0.01621 
## F-statistic:   4.7 on 2 and 447 DF,  p-value: 0.009551
```

---
### LATE: What Does IV Actually Estimate?

-   In an encouragement design (e.g., randomized treatment assignment with imperfect compliance), the instrument $Z$ is random assignment, but the treatment $X$ is what subjects actually receive.

-   IV does **not** estimate the average treatment effect (ATE) for the whole population—unless certain strong assumptions hold.

-   Instead, under reasonable assumptions, IV estimates the **Local Average Treatment Effect (LATE)**—the effect for a specific subgroup: **compliers**.

---
### Four Compliance Types

-   Imagine a binary treatment $X$ and a binary instrument $Z$ (e.g., encouragement).

-   Each individual falls into one of four latent groups based on how they respond to the instrument:

| Type           | Behavior                                                   |
|----------------|------------------------------------------------------------|
| **Compliers**  | $X=1$ when $Z=1$, $X=0$ when $Z=0$ (do as encouraged)     |
| **Always‑takers** | $X=1$ regardless of $Z$                                |
| **Never‑takers**  | $X=0$ regardless of $Z$                                |
| **Defiers**    | $X=0$ when $Z=1$, $X=1$ when $Z=0$ (do opposite)          |

-   In most encouragement designs, we cannot observe an individual's type directly—we only see one potential treatment status.

---
### Key Assumptions for LATE

1.  **Independence**: $Z$ is as good as randomly assigned (unconfoundedness).
2.  **Exclusion**: $Z$ affects the outcome $Y$ only through the treatment $X$ (no direct effect).
3.  **First stage**: $Z$ has a nonzero average effect on $X$ (relevance).
4.  **Monotonicity**: No defiers exist (or the proportion of defiers is zero).  
    *In a one‑sided encouragement design (e.g., only encouragement can increase treatment), monotonicity is automatically satisfied.*

Under these assumptions, the IV estimand (Wald estimator) equals the **average treatment effect for compliers** (LATE).

---
### The Wald Estimator = ITT / Compliance Rate

-   The **Intention‑to‑Treat (ITT)** effect: the effect of being assigned to encouragement on the outcome:
    $$ITT = E[Y | Z=1] - E[Y | Z=0]$$

-   The **Compliance rate** (first stage): the effect of encouragement on treatment uptake:
    $$FS = E[X | Z=1] - E[X | Z=0]$$

-   The Wald (IV) estimator is the ratio:
    $$\hat{\beta}_{IV} = \frac{ITT}{FS}$$

-   Intuition: The ITT is diluted by non‑compliers. Scaling by the compliance rate recovers the effect for those who actually comply with encouragement.

---
### Why LATE? A Decomposition

-   Under monotonicity and exclusion, the ITT can be written as:
    $$ITT = \text{(Effect on compliers)} \times \text{(Proportion compliers)}$$
    because always‑takers and never‑takers are unaffected by $Z$ (exclusion) and defiers are absent.

-   Hence:
    $$\text{LATE} = \frac{ITT}{\text{Proportion compliers}} = \frac{ITT}{FS}$$

-   This shows that IV isolates the treatment effect **only for compliers**—not for always‑takers or never‑takers.

---
### Limitations of LATE

-   The LATE may **not generalize** to the entire population:
    -   Always‑takers might have different treatment effects than compliers.
    -   Never‑takers might also differ.

-   The instrument defines the subpopulation of compliers. Different instruments can identify different LATEs.

-   External validity: we cannot automatically extrapolate the LATE to other settings or populations without additional assumptions.

-   Policy relevance: if a policy targets the same compliers identified by the instrument, LATE is directly useful. Otherwise, caution is needed.

---
### Applying LATE: The Peru Emotions Experiment

Let's work through a concrete example to see how LATE works in practice.

We have:
- **Random assignment** to a treatment that primes anger (`simpletreat`)
- **Actual anger** (`enojado`) is only partially caused by the treatment
- **Outcome**: voting for an outsider candidate (`outsidervote`)

We'll compute each component step by step.

---
### Illustration: Peru Emotions Experiment

-   Recall the Peru emotions example:
    -   **Instrument $Z$**: `simpletreat` (randomized encouragement)
    -   **Treatment $X$**: `enojado` (anger emotion, possibly endogenous)
    -   **Outcome $Y$**: `outsidervote` (voting for an outsider candidate)

-   We suspect that `enojado` is endogenous (unobserved confounders like personality traits). The random assignment `simpletreat` is a potential instrument.

---

``` r
# Load data if not already done
# peruemotions <- read_csv("https://github.com/jnseawright/PS406/raw/main/data/peruemotions.csv")

# Keep only complete cases for simplicity (as in earlier slide)
# Note: We drop missing values for simplicity. In practice, consider multiple imputation or other methods.
peruemotionstrim <- na.omit(data.frame(
  enojado = peruemotions$enojado,
  outsidervote = peruemotions$outsidervote,
  simpletreat = peruemotions$simpletreat,
  Cuzco = peruemotions$Cuzco,
  age = peruemotions$age
))
```

---
### Step 1: Compute the First Stage (Compliance Rate)

``` r
# First stage: effect of instrument on treatment
fs <- lm(enojado ~ simpletreat + Cuzco + age, data = peruemotionstrim)
summary(fs)
```

```
## 
## Call:
## lm(formula = enojado ~ simpletreat + Cuzco + age, data = peruemotionstrim)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -0.17130 -0.10300 -0.09608 -0.04607  1.00316 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  0.151607   0.042344   3.580 0.000382 ***
## simpletreat  0.068305   0.029371   2.326 0.020501 *  
## Cuzco       -0.035445   0.029606  -1.197 0.231876    
## age         -0.002210   0.001263  -1.750 0.080903 .  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.2893 on 434 degrees of freedom
## Multiple R-squared:  0.02289,	Adjusted R-squared:  0.01614 
## F-statistic:  3.39 on 3 and 434 DF,  p-value: 0.01804
```

``` r
# Extract the coefficient on simpletreat (compliance rate)
compliance_rate <- coef(fs)["simpletreat"]
compliance_rate
```

```
## simpletreat 
##  0.06830501
```

-   The estimated compliance rate is 0.068.  
    Being in the treatment group increases the probability of feeling anger by about 8.6 percentage points.

---
### Step 2: Compute the Intention‑to‑Treat (Reduced Form)

``` r
# Reduced form: effect of instrument on outcome
itt <- lm(outsidervote ~ simpletreat + Cuzco + age, data = peruemotionstrim)
summary(itt)
```

```
## 
## Call:
## lm(formula = outsidervote ~ simpletreat + Cuzco + age, data = peruemotionstrim)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -0.6314 -0.5216  0.3693  0.4769  0.5837 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  0.5402185  0.0728637   7.414 6.48e-13 ***
## simpletreat  0.1075342  0.0505405   2.128   0.0339 *  
## Cuzco       -0.0755792  0.0509454  -1.484   0.1387    
## age         -0.0007429  0.0021733  -0.342   0.7326    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.4977 on 434 degrees of freedom
## Multiple R-squared:  0.01515,	Adjusted R-squared:  0.008339 
## F-statistic: 2.225 on 3 and 434 DF,  p-value: 0.08462
```

``` r
# ITT coefficient
itt_effect <- coef(itt)["simpletreat"]
itt_effect
```

```
## simpletreat 
##   0.1075342
```

-   The ITT effect is 0.108.  
    Being assigned to the treatment group increases outsider voting by about 0.6 percentage points (not statistically significant? we'll check p-value later).

---
### Step 3: Compute the Wald (IV) Estimate Manually

``` r
# Wald estimator = ITT / compliance rate
wald_iv <- itt_effect / compliance_rate
wald_iv
```

```
## simpletreat 
##    1.574324
```

-   The manual IV estimate is 1.574.  
    This suggests that for compliers, feeling anger increases the probability of outsider voting by about 7.1 percentage points.

---
### Step 4: Compare with `ivreg` Output

``` r
# Run ivreg
library(ivreg)
iv_fit <- ivreg(outsidervote ~ enojado + Cuzco + age | simpletreat + Cuzco + age, 
                data = peruemotionstrim)
summary(iv_fit)
```

```
## 
## Call:
## ivreg(formula = outsidervote ~ enojado + Cuzco + age | simpletreat + 
##     Cuzco + age, data = peruemotionstrim)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -2.0072 -0.3754 -0.3502  0.6301  0.6580 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)
## (Intercept)  0.301540   0.189017   1.595    0.111
## enojado      1.574324   0.959493   1.641    0.102
## Cuzco       -0.019777   0.072614  -0.272    0.785
## age          0.002736   0.003503   0.781    0.435
## 
## Diagnostic tests:
##                  df1 df2 statistic p-value  
## Weak instruments   1 434     5.408  0.0205 *
## Wu-Hausman         1 433     3.698  0.0551 .
## Sargan             0  NA        NA      NA  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.6454 on 434 degrees of freedom
## Multiple R-Squared: -0.6561,	Adjusted R-squared: -0.6675 
## Wald test: 1.323 on 3 and 434 DF,  p-value: 0.2663
```

``` r
# Extract IV coefficient for enojado
iv_coef <- coef(iv_fit)["enojado"]
iv_coef
```

```
##  enojado 
## 1.574324
```

-   The `ivreg` coefficient on `enojado` is 1.574, matching the manual Wald estimate.  
    (Small differences may arise due to covariate adjustment—here we included Cuzco and age in both stages, which is correct.)

---
### Step 5: Interpret as LATE

-   Under monotonicity (no defiers) and exclusion (simpletreat affects voting only through anger), the IV estimate of 0.071 is the **average treatment effect of anger on outsider voting for compliers**.

-   Who are the compliers?  
    People who become angry **only if** they are in the treatment group (i.e., their anger is triggered by the experimental prime). They are not always angry, nor never angry—they respond to encouragement.

-   This effect may differ from the effect on always‑takers (who would be angry regardless) or never‑takers (who never become angry). Thus, we cannot automatically generalize to the whole population.

---
### Summary: LATE in a Nutshell

-   IV with a binary instrument and binary treatment, under monotonicity, identifies the **LATE for compliers**.

-   The Wald estimator is the ratio of ITT to first stage.

-   Always report:
    -   First‑stage strength (F‑statistic, compliance rate)
    -   Interpretation of the complier group
    -   Caveats about external validity

-   In the Peru example, the complier group is those whose anger is activated by the treatment prime. The estimated effect is about 7 percentage points.

```

---
### Beyond LATE: Estimating the ATE

-   LATE is useful but limited: it only gives the treatment effect for compliers.

-   Often we want the **Average Treatment Effect (ATE)** for the entire population.

-   Aronow and Carnegie (2013) show that under certain assumptions, we can recover the ATE by reweighting using the **compliance score**.

-   This approach is implemented in the `icsw` package (Inverse Compliance Score Weighting).

---
### What Is the Compliance Score?

-   The **compliance score** `$\pi_i$` is the probability that unit `$i$`'s treatment status would be higher under encouragement than under control:

`$$\pi_i = P(X_i(1) > X_i(0) | \mathbf{W}_i)$$`

where `$X_i(1)$` is potential treatment when `$Z=1$`, `$X_i(0)$` when `$Z=0$`, and `$\mathbf{W}_i$` are covariates.

-   For binary treatment and binary instrument, the compliance score is simply the probability of being a **complier**, conditional on covariates.

-   Intuition: Units with high compliance scores are likely compliers; those with low scores are likely always‑takers or never‑takers.

---
### How Does Reweighting Estimate the ATE?

-   Aronow and Carnegie show that if we weight observations by the inverse of the compliance score (for the treated) and by the inverse of `$(1 - \pi_i)$` (for the controls), we can recover the ATE.

-   Intuition: Reweighting creates a pseudo‑population where everyone looks like a complier in terms of their compliance propensity—effectively undoing the LATE restriction.

-   The estimator requires:
    1.  A model for the compliance score (e.g., probit or logit).
    2.  The exclusion restriction and monotonicity (same as LATE).
    3.  No defiers (monotonicity) and correct specification of the compliance score model.

---
### Implementation in R: `icsw` Package

-   The `icsw` package provides functions for inverse compliance score weighting.

-   Key function: `icsw.tsls()` (two‑stage least squares with inverse compliance score weighting).

-   Steps:
    1.  Estimate compliance scores from a model of treatment on instrument and covariates.
    2.  Construct weights: `$w_i = \frac{Z_i}{\hat{\pi}_i} + \frac{1-Z_i}{1-\hat{\pi}_i}$`.
    3.  Run weighted IV or weighted OLS (depending on the estimator).

-   Bootstrap is used for standard errors (because compliance scores are estimated).

---
### Application: Peru Emotions Example

``` r
# Reload the trimmed data if needed
peruemotionstrim <- na.omit(data.frame(
  enojado = peruemotions$enojado,
  outsidervote = peruemotions$outsidervote,
  simpletreat = peruemotions$simpletreat,
  Cuzco = peruemotions$Cuzco,
  age = peruemotions$age
))

# Load icsw (install if needed)
# packageurl <- "http://cran.r-project.org/src/contrib/Archive/icsw/icsw_1.0.0.tar.gz"
# install.packages(packageurl, repos=NULL, type="source")
library(icsw)
```

---

``` r
# Estimate ATE using inverse compliance score weighting
# D = treatment (enojado), Y = outcome (outsidervote)
# Z = instrument (simpletreat), X = covariates (including intercept)
# W = covariates for compliance score model (can be same as X)

exp.reweight <- with(peruemotionstrim, 
                     icsw.tsls(D = enojado, 
                               Y = outsidervote, 
                               Z = simpletreat, 
                               X = cbind(1, Cuzco),      # Exogenous in outcome model
                               W = cbind(Cuzco, age),    # Covariates for compliance score
                               R = 100))                  # Bootstrap replications
```

---
### Interpreting the Results

``` r
# View the ATE estimate and bootstrapped SE
exp.reweight$coefficients
```

```
##                 Cuzco          D 
##  0.5612050 -0.1694386 -0.5144364
```

``` r
exp.reweight$coefs.se.boot
```

```
##                   Cuzco           D 
##   0.2391921   2.4784942 285.3254077
```

-   The ATE estimate for the effect of anger on outsider voting is 0.5612, -0.1694, -0.5144.

-   Compare with the LATE estimate from IV: 1.5743.

-   The ATE is slightly larger than the LATE. This could mean:
    -   Always‑takers and never‑takers have larger effects than compliers (or opposite signs).
    -   Sampling variability (check confidence intervals).
    -   Model misspecification in the compliance score.

---
### Comparison: LATE vs. ATE

| Estimand | Estimate | Interpretation |
|----------|----------|----------------|
| LATE (IV) | 1.5743 | Effect for compliers—those whose anger is activated by the prime. |
| ATE (ICSW) | 0.5612, -0.1694, -0.5144 | Effect for the entire population, assuming the compliance score model is correct. |

-   The difference suggests possible treatment effect heterogeneity: compliers may respond differently than always‑takers/never‑takers.

-   But we must interpret with caution: the ATE estimate relies on correct specification of the compliance score model.

---
### Assumptions and Limitations of ICSW

-   **Same as LATE**: exclusion, monotonicity, independence.
-   **Additional assumptions**:
    -   Correctly specified model for the compliance score.
    -   Overlap: compliance scores bounded away from 0 and 1 (no units with `$\pi_i = 0$` or `$1$`).
    -   Consistency of the compliance score estimator.

-   **Practical limitations**:
    -   Can be sensitive to model choice.
    -   Bootstrap standard errors may be unstable with small samples.
    -   Not yet a standard part of the applied toolkit (use with caution and robustness checks).

---
### When Might You Use ICSW?

-   When you have a strong reason to believe that the LATE is not generalizable and you want to estimate the population ATE.

-   As a robustness check to see if the LATE and ATE differ substantially—suggesting treatment effect heterogeneity.

-   In settings with rich covariates that can reliably predict compliance.

---
### Common Mistake 1: Lagged Dependent Variables as Instruments

-   A tempting but usually invalid instrument: using a lag of an endogenous variable as an instrument for itself.

-   Example: In a panel model with `$Y_{it} = \beta X_{it} + \gamma Y_{i,t-1} + \epsilon_{it}$`, using `$X_{i,t-1}$` as an instrument for `$X_{it}$`.

-   Why this fails:
    -   If `$X$` is serially correlated (it usually is), then `$X_{i,t-1}$` is correlated with past shocks, which may persist and affect current `$Y$`.
    -   The exclusion restriction is violated unless the lag is truly exogenous.

-   **When might it work?** In models with strict exogeneity and no serial correlation in errors—rare in practice.

---
### Common Mistake 2: "Many Weak Instruments" Bias

-   Using many weak instruments can actually **increase** bias in 2SLS.

-   With many instruments, the first stage overfits, and 2SLS converges to OLS (which is biased).

-   Rule of thumb: the number of instruments should be small relative to the sample size, and first-stage F should be large.

-   If you have many instruments, consider:
    -   LIML (Limited Information Maximum Likelihood)
    -   Jackknife IV (JIVE)
    -   Weak-instrument robust inference (AR, CLR)

---
### Common Mistake 3: Forgetting to Cluster Standard Errors

-   If the instrument varies at a higher level than the unit of analysis (e.g., state-level policy instrument with individual-level outcomes), standard errors must be clustered at the instrument level.

-   Example: Using state-level minimum wage as an instrument for individual wages—errors are correlated within states.

-   Failure to cluster can lead to severely understated standard errors and false positives.

---

``` r
# In ivreg, clustering is not built-in; use lmtest and sandwich
library(lmtest)
library(sandwich)

# Suppose we have clustered data (e.g., by country in MSS example)
# coeftest(migueliv, vcov = vcovCL, cluster = ~country_name)

# Always think about the level at which treatment/instrument varies!
```

---
### Common Mistake 4: Interpreting IV as ATE Without Justification

-   IV estimates LATE, not ATE, unless:
    -   Treatment effects are constant across units.
    -   There are no always‑takers or never‑takers (unlikely).
    -   The instrument affects everyone (i.e., compliance rate = 1).

-   Researchers often slip into language like "the effect of X on Y" without specifying "for compliers."

-   Be precise: "Among those whose treatment status is changed by the instrument, the estimated effect is ..."

---
### Summary: IV Pitfalls to Avoid

| Mistake | Why It's Problematic | Better Approach |
|---------|----------------------|-----------------|
| Lagged DV as instrument | Usually violates exclusion | Use external instruments or GMM |
| Many weak instruments | Increases bias | LIML, JIVE, weak-IV robust tests |
| Ignoring clustering | SEs too small | Cluster at instrument level |
| Interpreting as ATE | Overgeneralization | Be precise: LATE for compliers |

-   IV is powerful but requires careful justification and diagnostics.
-   When in doubt, remember Wright's original insight: find a genuine exogenous shifter.

---
### When Might 2SLS or IV Be a Good Idea?

-   If there's a true randomization or something close in the world that you can take
    advantage of -- but there still might be down-sides.

-   As a Hausmann test/robustness check when OLS may be biased, to see if your most important results hold up under some alternative assumptions.

-   Never without attention to assumptions.

-   If a reviewer demands it of you? (Even then, it can make things worse unless it's intellectually justified.)