Plan: Lecture 14 — Treatment Effects and Difference-in-Differences
File: 326_14_treatment_did.qmd Outputs: 326_14_treatment_did.html, 326_14_treatment_did.pdf, 326_14_treatment_did_slides.html
Context
New lecture on causal inference for Economics 326 (undergraduate, scalar notation only, no vectors). Based on handwritten notes: Dropbox/Notability/527/DID causal Nov 13, 2025.pdf. Datasets: jtrain2 and kielmc from the wooldridge R package.
Lecture Structure (~36 slides)
Part 1: Potential Outcomes and Treatment Effects (Slides 1–12)
| # | ## Title | Content |
|---|
| 1 | Motivation: causal questions | Job training → earnings? Drug → health? Incinerator → house prices? Fundamental challenge: can’t observe same person both ways. |
| 2 | Potential outcomes | $Y_i(1)$, $Y_i(0)$; individual effect $Y_i(1)-Y_i(0)$. |
| 3 | The fundamental problem | $D_i\in{0,1}$; $Y_i = D_i Y_i(1)+(1-D_i)Y_i(0)$; only one PO observed. |
| 4 | Average treatment effects | ATE $=\mathrm{E}[Y_i(1)-Y_i(0)]$; ATT $=\mathrm{E}[Y_i(1)-Y_i(0)\mid D_i=1]$. |
| 5 | Selection bias | Naive diff = ATT + selection bias; add/subtract derivation. |
| 6 | Random assignment | Under randomization, selection bias = 0; simple diff = ATE = ATT. |
| 7 | Regression with a treatment dummy | $Y_i=\alpha+\tau D_i+U_i$; $\hat\tau=\bar Y_1-\bar Y_0$; equals ATE if random. |
| 8 | Example: Lalonde data | data(jtrain2); NSW randomized experiment; head(...). |
| 9 | Estimating the ATE | lm(re78 ~ train); estimated ATE ≈ $1,794. |
| 10 | Observational studies | Self-selection → selection bias; need covariates. |
| 11 | Potential outcomes with a covariate | $Y_i(0)=\alpha_0+\beta_0 X_i+U_i(0)$; $Y_i(1)=\alpha_1+\beta_1 X_i+U_i(1)$; conditional mean independence. |
| 12 | The ATE with a covariate | ATE $=(\alpha_1-\alpha_0)+(\beta_1-\beta_0)\mathrm{E}[X_i]$. |
Part 2: Regression with Covariates and Interactions (Slides 13–20)
| # | ## Title | Content |
|---|
| 13 | Two separate regressions | OLS for each group; compute ATE from $(\hat\alpha_1-\hat\alpha_0)+(\hat\beta_1-\hat\beta_0)\bar X$. |
| 14 | Combined regression with interactions | $Y_i=\alpha_0+(\alpha_1-\alpha_0)D_i+\beta_0 X_i+(\beta_1-\beta_0)X_i D_i+\tilde U_i$; coefficient on $D_i$ is NOT ATE. |
| 15 | The demeaning trick | Substitute $X_i=\mathrm{E}[X_i]+(X_i-\mathrm{E}[X_i])$; get $\tau=\text{ATE}$ as coefficient on $D_i$. |
| 16 | Estimating the ATE with covariates | Replace $\mathrm{E}[X_i]$ by $\bar X$; regress on $D_i$, $X_i$, $(X_i-\bar X)D_i$; coeff on $D_i$ = ATE. |
| 17 | Why not regress $Y_i$ on $D_i$ and $X_i$? | Valid if $\beta_1=\beta_0$; interaction nests simpler model. |
| 18 | Example: separate regressions | lm(re78 ~ educ, subset=train==0/1); compute ATE manually. |
| 19 | Example: demeaned regression | lm(re78 ~ train + educ + I(train*educ_dm)); coeff on train = ATE. |
| 20 | Example: regression lines | Base R figure: two lines, vertical at $\bar X$, ATE arrow. |
Part 3: Difference-in-Differences (Slides 21–36)
| # | ## Title | Content |
|---|
| 21 | From cross-sections to panel data | Cross-section needed “selection on observables”; panel → exploit within-unit changes. |
| 22 | DID setup | Two periods $t\in{0,1}$; two groups $D_i\in{0,1}$; treatment between periods. |
| 23 | DID regression model | $Y_{it}=\alpha+\delta t+\gamma D_i+\beta(t\cdot D_i)+U_{it}$; 2×2 table of conditional means. |
| 24 | Interpreting the coefficients | $\alpha$=baseline; $\delta$=time effect; $\gamma$=group diff at $t=0$; $\beta$=DID. |
| 25 | Deriving the DID estimand | Full derivation: $\beta=\mathrm{E}[Y_{i1}-Y_{i0}\mid D_i=1]-\mathrm{E}[Y_{i1}-Y_{i0}\mid D_i=0]$. |
| 26 | DID diagram | Base R: 4-point diagram, counterfactual dashed line, $\beta$ arrow. |
| 27 | Example: incinerator and house prices | data(kielmc); Kiel & McClain (1995); head(...). |
| 28 | The 2×2 table of means | tapply(rprice, list(y81, nearinc), mean); DID by hand. |
| 29 | DID regression | lm(rprice ~ y81 + nearinc + y81nrinc); matches 2×2 table; p≈0.11. |
| 30 | DID with covariates | Add age + I(age^2); DID becomes significant. |
| 31 | DID and potential outcomes | Panel POs $Y_{it}(d)$; what’s observed for each group; ATT definition. |
| 32 | DID as a treatment effect | Express $\beta$ in terms of POs via add/subtract derivation. |
| 33 | Assumption 1: no anticipation | $\mathrm{E}[Y_{i0}(1)\mid D_i=1]=\mathrm{E}[Y_{i0}(0)\mid D_i=1]$; incinerator interpretation. |
| 34 | Assumption 2: parallel trends | $\mathrm{E}[Y_{i1}(0)-Y_{i0}(0)\mid D_i=0]=\mathrm{E}[Y_{i1}(0)-Y_{i0}(0)\mid D_i=1]$; $\Rightarrow\beta=\text{ATT}$. |
| 35 | Parallel trends diagram | Base R: annotated diagram, equal $\delta$ for both groups, $\beta$=ATT arrow. |
| 36 | Summary | Recap: POs → ATE/ATT → regression + demeaning → DID → assumptions → ATT. |
Datasets
jtrain2: Lalonde NSW randomized job training. Variables: train, re78, educ, age, black, married.kielmc: Kiel & McClain housing prices near incinerator. Variables: year, rprice, nearinc, y81, y81nrinc, age.
- YAML: standard (matching
326_13_dummy.qmd), outputs named 326_14_treatment_did.* - Hidden macro div:
\E, \Var, \Cov, \Vhat, \se immediately after first ## - Scalar notation throughout (no vectors)
- All display math and R chunks indented 2 spaces under a parent bullet
- Base R figures (
#| fig-width: 8, #| fig-height: 5.5, #| fig-align: center) \text{ATE}, \text{ATT} for multi-letter names in math
Implementation steps
Write scratch file (done as scratch_lecture21.qmd)- Fix numbering (21 → 14) and apply to
326_14_treatment_did.qmd quarto render 326_14_treatment_did.qmd- Grep for
katex-error in HTML output - Run
proofread-326 agent - Move TODO item to DONE.md
- Update course page
_teaching/2026-01-Econ-326.md - Update
SESSION_NOTES.md
