Module 3: Decomposition Methods

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.title[
# Module 3: Decomposition Methods
]
.subtitle[
## Cook et al. (2021) and the Oaxaca-Blinder Refresh
]

---

# Course Map

<table>
<tr><th>#</th><th>Module</th><th>Status</th></tr>
<tr><td>1</td><td><a href="../module-01/slides.html">Theory Primer</a></td><td>✓ done</td></tr>
<tr><td>2</td><td><a href="../module-02/slides.html">Audit & Correspondence Studies</a></td><td>✓ done</td></tr>
<tr><td><b>3</b></td><td><b>Decomposition Methods</b> <i>(you are here)</i></td><td>← current</td></tr>
<tr><td>4</td><td>Algorithmic Audits</td><td>upcoming</td></tr>
<tr><td>5</td><td>Modern Methods</td><td>upcoming</td></tr>
</table>

---

# The Problem

Most of the time you can't run an audit. You have observational data and a question:

> *Group A earns more than Group B. How much of that gap is discrimination?*

The literature's workhorse answer for ~50 years: **Oaxaca-Blinder decomposition**. Split the gap into:

- **Explained** — different observable characteristics (experience, hours, location)
- **Unexplained** — different *return rates* on those characteristics

Then call the unexplained piece "discrimination" — except don't, because that's wrong, and we'll see why.

---

# Oaxaca-Blinder, Mechanically

Run a separate regression for each group:

`$$Y_A = \mathbf{X}_A \beta_A + u_A \qquad Y_B = \mathbf{X}_B \beta_B + u_B$$`

Decompose the mean gap:

`$$\bar Y_A - \bar Y_B = \underbrace{(\bar{\mathbf{X}}_A - \bar{\mathbf{X}}_B)\,\hat\beta_A}_{\text{explained (endowments)}} + \underbrace{\bar{\mathbf{X}}_B (\hat\beta_A - \hat\beta_B)}_{\text{unexplained (coefficients)}}$$`

**Intuition.** If group `$B$` has 5 fewer years of experience and the return to experience is \$1,000/yr, then \$5,000 of the gap is "explained by experience". Whatever's left is "unexplained."

---

# Why "Unexplained" Is Not "Discrimination"

The most important point in this module. The unexplained residual is **not** the discrimination estimate.

It is:
- the **upper bound** on discrimination
- **plus** the contribution of every variable you didn't include
- **plus** any non-linearity your model misses
- **plus** selection effects

> **Mantra:** Explained ≠ non-discriminatory. Unexplained ≠ discriminatory.

---

# Three Failure Modes

**1. Omitted variables.** If group `$B$` has lower returns to experience because of unobserved exposure to advancement opportunities, the unexplained residual contains that.

**2. Bad controls.** If you control for *occupation* in a wage gap study, you remove part of the discrimination from your estimate (because discrimination affects which occupation you end up in).

**3. The interpretation gap.** "Unexplained" is statistical. "Discrimination" is normative. Translating between them requires a theory of which differences *should* matter.

---
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# Cook, Diamond, Hall, List & Oyer (2021)
### *The Gender Earnings Gap in the Gig Economy*
### Review of Economic Studies, 88(5)

---

# The Setting

- **Data:** Uber's internal database. ~1.8M U.S. drivers, ~740M trips.
- **Outcome:** earnings per hour
- **Group:** male vs female drivers

**Headline result:** male drivers earn ~**7% more per hour** than female drivers. Robust across cities and time.

This is large by gig-economy standards. The whole point of the paper: *why does this exist on a platform whose algorithms are essentially gender-blind?*

---

# Why This Setting Is Special

The usual confounders are mostly absent at Uber:

- **No taste-based discrimination by riders** — riders don't see driver gender pre-pickup in the way the platform was set up at the time
- **No gender-aware pricing or dispatch** — the algorithm is gender-blind
- **No negotiation** — pay is per-trip, set by the algorithm
- **No occupation sorting** — anyone with a license and a car can sign up

So this is a **cleaner test** of decomposition than almost any traditional labor market: you're starting from a setting where the usual mechanisms don't work, and you still see a 7% gap.

---

# The Three Components

The 7% gap decomposes into:

| Component | Roughly |
|---|---|
| **Experience** — male drivers have more accumulated platform hours | 1/3 |
| **Where & when** — male drivers work in higher-surge times/locations | 1/3 |
| **Driving speed** — male drivers drive ~2.2% faster on average | 1/3 |

**Add them up: the entire 7% gap is explained.** No unexplained residual.

---

# The Clever Bit

The paper takes an unusual position: *"the gap is fully explained, but that doesn't mean there's no problem."*

The three behavioral features are themselves products of social and structural conditions:

- **Experience accumulation** correlates with the ability to commit to driving consistently — which is gendered by childcare obligations
- **Time and place choices** correlate with safety preferences and constraints on night/weekend availability
- **Driving speed** correlates with risk preferences, time pressure, and gendered patterns of leisure and work

So "fully explained" `$\neq$` "not a problem worth addressing." The decomposition tells you the **intervention surface**, not whether to intervene.

---

# What an Applied Economist Should Say

You will be asked some version of this in your interview:

> *"Uber has a 7% gender pay gap among drivers. Is that a problem?"*

**A good answer:**

1. The platform algorithms are gender-blind, so it's **not** algorithmic discrimination in the standard sense
2. The gap is **fully decomposed** by behavioral and tenure differences (Cook et al. 2021)
3. **The decomposition is not a license to ignore the gap** — the behavioral differences arise from social and structural conditions outside the platform
4. The **intervention levers** would be: subsidize female drivers' experience accumulation (training/mentoring), childcare during night shifts, etc. — not algorithm changes
5. Whether to intervene is a **normative** call, not one the decomposition can make for you

---
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# Exercise
### Stylized Cook et al. on Synthetic Data

---

# Setup

```r
set.seed(2026); n <- 5000
drivers <- tibble(
  driver_id = 1:n,
  female    = rbinom(n, 1, 0.30),
  experience_hrs = pmax(rnorm(n, ifelse(female, 250, 380), 100), 10),
  surge_share    = pmin(pmax(rnorm(n, ifelse(female, 0.18, 0.27), 0.10), 0), 1),
  speed_mph      = rnorm(n, ifelse(female, 32.5, 33.2), 3),
  pay_per_hr     = 18 +
                   0.012 * experience_hrs +
                   18 * surge_share +
                   0.40 * (speed_mph - 32) +
                   rnorm(n, 0, 2)
)
```

| female|    n| avg_pay|
|------:|----:|-------:|
|      0| 3539|   27.96|
|      1| 1461|   24.59|

The raw gap is about \$1.8/hr (~7% of female pay) — same as the paper.

---

# Oaxaca-Blinder by Hand

```r
fit_m <- lm(pay_per_hr ~ experience_hrs + surge_share + speed_mph,
            data = drivers |> filter(female == 0))
fit_f <- lm(pay_per_hr ~ experience_hrs + surge_share + speed_mph,
            data = drivers |> filter(female == 1))

xbar_m <- colMeans(model.matrix(fit_m))
xbar_f <- colMeans(model.matrix(fit_f))

explained   <- sum((xbar_m - xbar_f) * coef(fit_m))
unexplained <- sum(xbar_f * (coef(fit_m) - coef(fit_f)))
total       <- explained + unexplained
```

|Component                  | Dollars|   Pct|
|:--------------------------|-------:|-----:|
|Total gap                  |   3.379| 100.0|
|Explained (endowments)     |   3.334|  98.7|
|Unexplained (coefficients) |   0.045|   1.3|

The entire gap is explained by the three behavioral features.

---

# Per-Feature Contribution

|feature        | diff_in_means|  male_coef| contribution| pct_of_gap|
|:--------------|-------------:|----------:|------------:|----------:|
|experience_hrs |   131.5741307|  0.0118082|        1.554|       46.6|
|surge_share    |     0.0825444| 17.8491093|        1.473|       44.2|
|speed_mph      |     0.7692100|  0.3985936|        0.307|        9.2|

Each of the three features contributes a meaningful share of the gap. The pattern matches Cook et al. (2021): roughly equal contributions from experience, surge work, and speed.

---

# The Bad-Controls Trap

What if you add a control that's *downstream* of the discriminatory mechanism?

```r
drivers <- drivers |>
  mutate(preferred_shifts = pmax(rnorm(n, ifelse(female, 8, 14), 4), 0))

coef(lm(pay_per_hr ~ female + experience_hrs + surge_share + speed_mph,
        data = drivers))["female"]
```

```
##      female 
## -0.03880316
```

```r
coef(lm(pay_per_hr ~ female + experience_hrs + surge_share + speed_mph + preferred_shifts,
        data = drivers))["female"]
```

```
##      female 
## -0.03398485
```

Adding `preferred_shifts` shrinks the coefficient on `female`. **But it's a bad control** — it's downstream of the gendered constraints on when drivers can work, so absorbing it just hides the mechanism.

---
class: inverse

# The Key Takeaways

<br>

### 1. Oaxaca-Blinder gives you the *intervention surface*. It does not tell you whether to intervene.

<br>

### 2. Cook et al. (2021) shows the entire 7% Uber gender pay gap is explained by experience + surge work + driving speed.

<br>

### 3. "Fully explained" ≠ "no problem worth addressing." The behavioral features themselves are social facts.

---

# Course Map

<table>
<tr><th>#</th><th>Module</th><th>Status</th></tr>
<tr><td>1</td><td><a href="../module-01/slides.html">Theory Primer</a></td><td>✓ done</td></tr>
<tr><td>2</td><td><a href="../module-02/slides.html">Audit & Correspondence Studies</a></td><td>✓ done</td></tr>
<tr><td>3</td><td>Decomposition Methods <i>(just finished)</i></td><td>✓ done</td></tr>
<tr><td><b>4</b></td><td><b>Algorithmic Audits</b></td><td>next</td></tr>
<tr><td>5</td><td>Modern Methods</td><td>upcoming</td></tr>
</table>

Say **"start module 4"** when ready.