17: Robust Standard Errors.

class: center, middle, inverse, title-slide

.title[
# 17: Robust Standard Errors.
]
.subtitle[
## Linear Models
]
.author[
### <large>Jaye Seawright</large>
]
.institute[
### <small>Northwestern Political Science</small>
]
.date[
### March 4, 2026
]

---

class: center, middle

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

### The Problem: Why We Need Robust SEs

---

### **Classical OLS Assumption:**
$$
\text{Var}(\epsilon_i | X) = \sigma^2
$$
Constant variance (homoskedasticity)

---

### **Reality:**
- Some groups are more predictable than others
- Richer districts have more variation in outcomes
- Survey error varies by respondent type
- Time-series volatility changes over time

**This is heteroskedasticity**

---

![](RobustStandardErrors_files/figure-html/hetero-example-1.png)

---

**Consequence:** Standard errors are wrong → confidence intervals wrong → p-values wrong

---
### **What Happens?**

1. **OLS coefficients remain unbiased**
2. **But variance estimates are biased:**
   - Can be too small (common) → false significance
   - Can be too large → missed findings

---
### **Examples from Political Science:**
- **Wealth and voting:** Richer districts show different patterns of variation in turnout than poor districts (e.g., Avery 2015)
- **Media effects:** Highly attentive individuals are differently predictable compared to less attentive people (e.g., Walgrave et al., 2017)
- **Policy adoption:** Effects may differ across space and time (e.g. Matisoff 2008)
- **Conflict studies:** Actors have different distributions of information, motivations, and resources (e.g., Bas 2012)

---

```
## Average SE (homoskedastic): 0.101
```

```
## Average SE (heteroskedastic): 0.143
```

```
## 
## Typical underestimation: -29 %
```

---

![](RobustStandardErrors_files/figure-html/consequence-plot-1.png)

---
### **Huber-White (Sandwich) Estimator**
Developed by Huber (1967) and White (1980)

**Regular OLS variance:**
$$
\text{Var}(\hat{\beta}) = (X'X)^{-1} \sigma^2
$$

**Robust (HC) variance:**
$$
\text{Var}(\hat{\beta}) = (X'X)^{-1} X' \Omega X (X'X)^{-1}
$$
where `$\Omega = \text{diag}(e_i^2)$` for HC0

---
### **Intuition:**
- Weight observations by their residual size
- Bigger residuals → more uncertainty in that region
- Adapts to heteroskedasticity

---
### **Flavors of HC Estimators**
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<table class="gt_table" data-quarto-disable-processing="false" data-quarto-bootstrap="false">
  <thead>
    <tr class="gt_heading">
      <td colspan="3" class="gt_heading gt_title gt_font_normal gt_bottom_border" style>Common HC Estimators</td>
    </tr>
    
    <tr class="gt_col_headings">
      <th class="gt_col_heading gt_columns_bottom_border gt_left" rowspan="1" colspan="1" scope="col" id="Type">Type</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_left" rowspan="1" colspan="1" scope="col" id="Formula">Weight</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_left" rowspan="1" colspan="1" scope="col" id="Description">Description</th>
    </tr>
  </thead>
  <tbody class="gt_table_body">
    <tr><td headers="Type" class="gt_row gt_left" style="font-size: 16px;">HC0</td>
<td headers="Formula" class="gt_row gt_left" style="font-size: 16px;">$e_i^2$</td>
<td headers="Description" class="gt_row gt_left" style="font-size: 16px;">Basic White</td></tr>
    <tr><td headers="Type" class="gt_row gt_left" style="font-size: 16px;">HC1</td>
<td headers="Formula" class="gt_row gt_left" style="font-size: 16px;">$\frac{n}{n-k}e_i^2$</td>
<td headers="Description" class="gt_row gt_left" style="font-size: 16px;">Incomplete small sample correction</td></tr>
    <tr><td headers="Type" class="gt_row gt_left" style="font-size: 16px;">HC2</td>
<td headers="Formula" class="gt_row gt_left" style="font-size: 16px;">$\frac{e_i^2}{1-h_{ii}}$</td>
<td headers="Description" class="gt_row gt_left" style="font-size: 16px;">Leverage adjustment</td></tr>
    <tr><td headers="Type" class="gt_row gt_left" style="font-size: 16px;">HC3</td>
<td headers="Formula" class="gt_row gt_left" style="font-size: 16px;">$\frac{e_i^2}{(1-h_{ii})^2}$</td>
<td headers="Description" class="gt_row gt_left" style="font-size: 16px;">Better small sample correction</td></tr>
    <tr><td headers="Type" class="gt_row gt_left" style="font-size: 16px;">HC4</td>
<td headers="Formula" class="gt_row gt_left" style="font-size: 16px;">$\frac{e_i^2}{(1-h_{ii})^{\delta_i}}$</td>
<td headers="Description" class="gt_row gt_left" style="font-size: 16px;">Designed for outliers</td></tr>
  </tbody>
  
</table>
</div>

---
### **Social Science Practice:**
- **HC1**, **HC2**, and **HC3** converge for larger samples (above about 500)
- **HC3** performs best for smaller samples and works well down to samples of about 25 
- **Stata default:** HC1, in spite of its limitations
- **R common practice:** HC3, which is probably the best choice (Long and Ervin 2000)
]

---
### Implementation in R: The `sandwich` Package

---
### **Basic Implementation**

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

la_electoral <- read.csv("https://raw.githubusercontent.com/jnseawright/ps405/refs/heads/main/Data/laelectoral.csv")

# After fitting model
# Fit regular model
model <- lm(voteswoninc ~ gdpgrowthlag + votesinclag, data = la_electoral)
```

---

``` r
# Regular SEs
summary(model)
```

```
## 
## Call:
## lm(formula = voteswoninc ~ gdpgrowthlag + votesinclag, data = la_electoral)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -49.632  -9.563   6.100  13.505  29.843 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)   
## (Intercept)   -3.3713    12.0876  -0.279  0.78099   
## gdpgrowthlag   1.0473     0.4600   2.277  0.02528 * 
## votesinclag    0.7166     0.2361   3.036  0.00318 **
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 19.01 on 86 degrees of freedom
## Multiple R-squared:  0.1439,	Adjusted R-squared:  0.124 
## F-statistic:  7.23 on 2 and 86 DF,  p-value: 0.001252
```
---

``` r
# Robust SEs (HC3 default)
coeftest(model, vcov = vcovHC)
```

```
## 
## t test of coefficients:
## 
##              Estimate Std. Error t value Pr(>|t|)   
## (Intercept)  -3.37130   12.54174 -0.2688 0.788722   
## gdpgrowthlag  1.04729    0.44261  2.3661 0.020221 * 
## votesinclag   0.71660    0.25701  2.7882 0.006523 **
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
```

---

``` r
# Different HC types
coeftest(model, vcov = vcovHC(model, type = "HC0"))
```

```
## 
## t test of coefficients:
## 
##              Estimate Std. Error t value Pr(>|t|)   
## (Intercept)  -3.37130   11.67991 -0.2886 0.773551   
## gdpgrowthlag  1.04729    0.40190  2.6058 0.010800 * 
## votesinclag   0.71660    0.24007  2.9850 0.003692 **
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
```

---

``` r
coeftest(model, vcov = vcovHC(model, type = "HC1"))
```

```
## 
## t test of coefficients:
## 
##              Estimate Std. Error t value Pr(>|t|)   
## (Intercept)  -3.37130   11.88188 -0.2837 0.777296   
## gdpgrowthlag  1.04729    0.40885  2.5615 0.012165 * 
## votesinclag   0.71660    0.24422  2.9342 0.004286 **
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
```

---

``` r
coeftest(model, vcov = vcovHC(model, type = "HC3"))
```

---
### **Using `modelsummary`**

``` r
library(modelsummary)

ms.display <- modelsummary(
  list("OLS" = model, 
       "Robust SE" = model),
  vcov = list("iid", "HC3"),
  stars = TRUE, output = "kableExtra"
)
```

---

<table style="NAborder-bottom: 0; width: auto !important; margin-left: auto; margin-right: auto;" class="table">
 <thead>
  <tr>
   <th style="text-align:left;">   </th>
   <th style="text-align:center;"> OLS </th>
   <th style="text-align:center;"> Robust SE </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> (Intercept) </td>
   <td style="text-align:center;"> −3.371 </td>
   <td style="text-align:center;"> −3.371 </td>
  </tr>
  <tr>
   <td style="text-align:left;">  </td>
   <td style="text-align:center;"> (12.088) </td>
   <td style="text-align:center;"> (12.542) </td>
  </tr>
  <tr>
   <td style="text-align:left;"> gdpgrowthlag </td>
   <td style="text-align:center;"> 1.047* </td>
   <td style="text-align:center;"> 1.047* </td>
  </tr>
  <tr>
   <td style="text-align:left;">  </td>
   <td style="text-align:center;"> (0.460) </td>
   <td style="text-align:center;"> (0.443) </td>
  </tr>
  <tr>
   <td style="text-align:left;"> votesinclag </td>
   <td style="text-align:center;"> 0.717** </td>
   <td style="text-align:center;"> 0.717** </td>
  </tr>
  <tr>
   <td style="text-align:left;box-shadow: 0px 1.5px">  </td>
   <td style="text-align:center;box-shadow: 0px 1.5px"> (0.236) </td>
   <td style="text-align:center;box-shadow: 0px 1.5px"> (0.257) </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Num.Obs. </td>
   <td style="text-align:center;"> 89 </td>
   <td style="text-align:center;"> 89 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> R2 </td>
   <td style="text-align:center;"> 0.144 </td>
   <td style="text-align:center;"> 0.144 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> R2 Adj. </td>
   <td style="text-align:center;"> 0.124 </td>
   <td style="text-align:center;"> 0.124 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> AIC </td>
   <td style="text-align:center;"> 781.7 </td>
   <td style="text-align:center;"> 781.7 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> BIC </td>
   <td style="text-align:center;"> 791.7 </td>
   <td style="text-align:center;"> 791.7 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Log.Lik. </td>
   <td style="text-align:center;"> −386.874 </td>
   <td style="text-align:center;"> −386.874 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> F </td>
   <td style="text-align:center;"> 7.230 </td>
   <td style="text-align:center;"> 7.018 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> RMSE </td>
   <td style="text-align:center;"> 18.69 </td>
   <td style="text-align:center;"> 18.69 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Std.Errors </td>
   <td style="text-align:center;"> IID </td>
   <td style="text-align:center;"> HC3 </td>
  </tr>
</tbody>
<tfoot><tr><td style="padding: 0; " colspan="100%">
<sup></sup> + p &lt; 0.1, * p &lt; 0.05, ** p &lt; 0.01, *** p &lt; 0.001</td></tr></tfoot>
</table>

---
### **Complete Example**

``` r
# Load data
data("CASchools", package = "AER")
# Simulate heteroskedasticity
CASchools$testscore <- 
  CASchools$read + CASchools$math + 
  rnorm(nrow(CASchools), 0, 
        abs(CASchools$income)/1000)

# Fit model
model <- lm(testscore ~ income + english, 
            data = CASchools)

# Compare SEs
library(modelsummary)
models <- list(
  "Regular" = model,
  "HC0" = model,
  "HC1" = model,
  "HC2" = model,
  "HC3" = model
)

caschools.msummary <- msummary(models,
         vcov = c("iid", "HC0", "HC1", "HC2", "HC3"),
         fmt = 2,
         stars = TRUE,
         output = "kableExtra") %>%
  kable_styling(font_size = 10)
```

---

<table style="NAborder-bottom: 0; width: auto !important; margin-left: auto; margin-right: auto; font-size: 10px; margin-left: auto; margin-right: auto;" class="table table">
 <thead>
  <tr>
   <th style="text-align:left;">   </th>
   <th style="text-align:center;"> Regular </th>
   <th style="text-align:center;"> HC0 </th>
   <th style="text-align:center;"> HC1 </th>
   <th style="text-align:center;"> &amp;nbsp;HC2 </th>
   <th style="text-align:center;"> &amp;nbsp;HC3 </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> (Intercept) </td>
   <td style="text-align:center;"> 1277.85*** </td>
   <td style="text-align:center;"> 1277.85*** </td>
   <td style="text-align:center;"> 1277.85*** </td>
   <td style="text-align:center;"> 1277.85*** </td>
   <td style="text-align:center;"> 1277.85*** </td>
  </tr>
  <tr>
   <td style="text-align:left;">  </td>
   <td style="text-align:center;"> (2.86) </td>
   <td style="text-align:center;"> (3.41) </td>
   <td style="text-align:center;"> (3.42) </td>
   <td style="text-align:center;"> (3.46) </td>
   <td style="text-align:center;"> (3.52) </td>
  </tr>
  <tr>
   <td style="text-align:left;"> income </td>
   <td style="text-align:center;"> 3.00*** </td>
   <td style="text-align:center;"> 3.00*** </td>
   <td style="text-align:center;"> 3.00*** </td>
   <td style="text-align:center;"> 3.00*** </td>
   <td style="text-align:center;"> 3.00*** </td>
  </tr>
  <tr>
   <td style="text-align:left;">  </td>
   <td style="text-align:center;"> (0.15) </td>
   <td style="text-align:center;"> (0.18) </td>
   <td style="text-align:center;"> (0.18) </td>
   <td style="text-align:center;"> (0.19) </td>
   <td style="text-align:center;"> (0.19) </td>
  </tr>
  <tr>
   <td style="text-align:left;"> english </td>
   <td style="text-align:center;"> −0.98*** </td>
   <td style="text-align:center;"> −0.98*** </td>
   <td style="text-align:center;"> −0.98*** </td>
   <td style="text-align:center;"> −0.98*** </td>
   <td style="text-align:center;"> −0.98*** </td>
  </tr>
  <tr>
   <td style="text-align:left;box-shadow: 0px 1.5px">  </td>
   <td style="text-align:center;box-shadow: 0px 1.5px"> (0.06) </td>
   <td style="text-align:center;box-shadow: 0px 1.5px"> (0.06) </td>
   <td style="text-align:center;box-shadow: 0px 1.5px"> (0.06) </td>
   <td style="text-align:center;box-shadow: 0px 1.5px"> (0.06) </td>
   <td style="text-align:center;box-shadow: 0px 1.5px"> (0.06) </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Num.Obs. </td>
   <td style="text-align:center;"> 420 </td>
   <td style="text-align:center;"> 420 </td>
   <td style="text-align:center;"> 420 </td>
   <td style="text-align:center;"> 420 </td>
   <td style="text-align:center;"> 420 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> R2 </td>
   <td style="text-align:center;"> 0.707 </td>
   <td style="text-align:center;"> 0.707 </td>
   <td style="text-align:center;"> 0.707 </td>
   <td style="text-align:center;"> 0.707 </td>
   <td style="text-align:center;"> 0.707 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> R2 Adj. </td>
   <td style="text-align:center;"> 0.706 </td>
   <td style="text-align:center;"> 0.706 </td>
   <td style="text-align:center;"> 0.706 </td>
   <td style="text-align:center;"> 0.706 </td>
   <td style="text-align:center;"> 0.706 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> AIC </td>
   <td style="text-align:center;"> 3741.1 </td>
   <td style="text-align:center;"> 3741.1 </td>
   <td style="text-align:center;"> 3741.1 </td>
   <td style="text-align:center;"> 3741.1 </td>
   <td style="text-align:center;"> 3741.1 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> BIC </td>
   <td style="text-align:center;"> 3757.2 </td>
   <td style="text-align:center;"> 3757.2 </td>
   <td style="text-align:center;"> 3757.2 </td>
   <td style="text-align:center;"> 3757.2 </td>
   <td style="text-align:center;"> 3757.2 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Log.Lik. </td>
   <td style="text-align:center;"> −1866.540 </td>
   <td style="text-align:center;"> −1866.540 </td>
   <td style="text-align:center;"> −1866.540 </td>
   <td style="text-align:center;"> −1866.540 </td>
   <td style="text-align:center;"> −1866.540 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> F </td>
   <td style="text-align:center;"> 503.396 </td>
   <td style="text-align:center;"> 445.674 </td>
   <td style="text-align:center;"> 442.491 </td>
   <td style="text-align:center;"> 432.888 </td>
   <td style="text-align:center;"> 420.409 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> RMSE </td>
   <td style="text-align:center;"> 20.60 </td>
   <td style="text-align:center;"> 20.60 </td>
   <td style="text-align:center;"> 20.60 </td>
   <td style="text-align:center;"> 20.60 </td>
   <td style="text-align:center;"> 20.60 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Std.Errors </td>
   <td style="text-align:center;"> IID </td>
   <td style="text-align:center;"> HC0 </td>
   <td style="text-align:center;"> HC1 </td>
   <td style="text-align:center;"> HC2 </td>
   <td style="text-align:center;"> HC3 </td>
  </tr>
</tbody>
<tfoot><tr><td style="padding: 0; " colspan="100%">
<sup></sup> + p &lt; 0.1, * p &lt; 0.05, ** p &lt; 0.01, *** p &lt; 0.001</td></tr></tfoot>
</table>

---

### **Key Insight:**
Robust SEs often (but not always) increase standard errors, making findings **less significant**. This is more honest given real-world data.

---
# Clustered Standard Errors

### **When Observations Are Not Independent**
Common in political science:
1. **Panel data:** Multiple observations per country
2. **Survey data:** Respondents within states
3. **Experimental data:** Treatments assigned to groups
4. **Spatial data:** Nearby locations correlated

---
### **The Problem:**
Within-cluster correlation → underestimation of variance

**Example:** If all respondents in Texas share unobserved factors, we don't have 100 independent observations, but 1 cluster of 100 correlated observations

---
### **Clustering Formula**
$$
\text{Var}(\hat{\beta}) = (X'X)^{-1} \left( \sum_{c=1}^C X_c' e_c e_c' X_c \right) (X'X)^{-1}
$$
where `$c$` indexes clusters

---
### **Implementation in R:**

``` r
# Using sandwich and lmtest
coeftest(model, 
         vcov = vcovCL(model, 
                       cluster = ~state))

# Using fixest package (fast!)
library(fixest)
model_fe <- feols(vote_share ~ income + education,
                  data = election_data,
                  cluster = ~state)

# Using clubSandwich for multi-way clustering
library(clubSandwich)
coef_test(model, 
          vcov = "CR2", 
          cluster = election_data$state)
```

---
# Multi-Way Clustering

### **Complex Dependencies**
Sometimes observations cluster in multiple ways:
1. **States and years** (Cameron, Gelbach & Miller, 2011)
2. **Individual and time** (two-way FE)
3. **Industry and region**

---

### **R Implementation:**

``` r
# Using clubSandwich
vcov_multi <- vcovCR(model,
                     type = "CR2",
                     cluster = election_data[, c("state", "year")])

coeftest(model, vcov = vcov_multi)

# Using lfe (older but works)
library(lfe)
model_multi <- felm(vote_share ~ income + education |
                      state + year,
                    data = election_data)
summary(model_multi, robust = TRUE)
```

---
### **When to Cluster?**

<table class="table" style="font-size: 10px; margin-left: auto; margin-right: auto;">
<caption style="font-size: initial !important;">Clustering Guidelines for Political Data</caption>
 <thead>
  <tr>
   <th style="text-align:left;"> Data_Structure </th>
   <th style="text-align:left;"> Clustering </th>
   <th style="text-align:left;"> R_Package </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> Cross-section </td>
   <td style="text-align:left;"> No clustering or heteroskedasticity-robust </td>
   <td style="text-align:left;"> sandwich </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Panel (country-year) </td>
   <td style="text-align:left;"> Cluster by country </td>
   <td style="text-align:left;"> plm/sandwich </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Survey (individuals in states) </td>
   <td style="text-align:left;"> Cluster by state (or primary sampling unit) </td>
   <td style="text-align:left;"> survey/sandwich </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Experimental (treatments in groups) </td>
   <td style="text-align:left;"> Cluster by treatment group </td>
   <td style="text-align:left;"> clubSandwich </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Time series </td>
   <td style="text-align:left;"> Newey-West (HAC) errors </td>
   <td style="text-align:left;"> sandwich </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Network/spatial </td>
   <td style="text-align:left;"> Spatial HAC or Conley errors </td>
   <td style="text-align:left;"> spatialreg </td>
  </tr>
</tbody>
</table>

---

**Rule of thumb:** Cluster at the level of treatment assignment or at the level where you suspect correlation.

---
# Limitations and When NOT to Use Robust SEs

### **1. Very Small Samples**
- HC estimators biased in tiny samples
- N < 25: Consider alternatives
- Better: Use exact methods or permutation tests

### **2. Weighted Data**
- Survey weights already account for heterogeneity
- Don't double-adjust

---

### **3. Nonlinear Models**
- GLMs (logit, probit) have different variance structure

### **4. Model Misspecification**
Robust SEs don't fix:
- Omitted variable bias
- Functional form misspecification
- Measurement error
- Simultaneity

---

### **5. Efficiency Loss**
- Robust SEs are less efficient when homoskedasticity holds

Use robust SEs unless you have strong reason not to

---

![Robust1](Images/Xu1.png)

---

![Robust2](Images/Xu2.png)