Treatment Effect¶
HTE = Heterogeneous Treatment Effect = ATE with high dimensional \(s\)
If \(x\) is binary $$ \begin{aligned} x &\in { 0, 1 } \ \implies \text{ATE}(x, s) &= E[ y \vert \text{do}(x=1) ] - E[ y \vert \text{do}(x=0) ] \ &= E_s \Bigg[ E[y \vert x=1, s ] - E[ y \vert x=0, s] \Bigg] \ \end{aligned} $$
| Treatment | Model | \(\widehat{\text{ATE}}(x)\) |
|---|---|---|
| Binary | Linear \(\hat \beta_0 + \hat \beta_1 x + \hat \beta_2 s\) | Constant \(\hat \beta_1\) |
| Multi-Level/ Continuous | Non-linear | Functional |
Heterogeneous Treatment Effects¶
If we consider effect modifier \(s\) $$ \begin{aligned} \text{ATE} &= \dfrac{\partial}{\partial x} E[ y \vert \text{do}(x), s] \ &= \int \limits_s E[y^1 - y^0 | s] \cdot P(s) \cdot ds \ \end{aligned} $$
where \(P(s)\) is the distribution of effect modifier.
Then, the result of the randomized test actually gives us \(E[\tilde \tau]\), which may not be equal to \(E[\tau]\)
Example¶
For example, the yield depends on the season and the crop for which was grown previous year. Let’s take the example of a a randomized test of a fertilizer used in the summer.
If no crop was grown the previous year, then we
| Crop Grown in Field Last Year | Result of Randomized Test \(E[\tilde \tau]\) |
|---|---|
| No crop | $E[\tau |
| Rice only | $E[\tau |
| 50% Barley, 50% Rice | $0.5 \cdot E[\tau |
Regression Discontinuity Design¶
Causal effect cannot be obtained directly due to lack of overlap for different \(s\), hence we take the neighborhood
When a quasi-experiment partially determines the treatment status, the “as if” randomness can be used as an instrument for identifying the causal effect of interest
Eg: Evaluating the treatment effect of college on students in the neighborhood of college acceptance cutoff, where the selection of students is random
Limitations
- Poor generalizability: The validity of the results is usually restricted to this region
- Throws away the lot of information in the non-random parts
- Doesn’t allow building structural causal model
Differences-in-Differences¶
Let
- Control: \(y_0\) be the time series with \(x=0\)
- Treated: \(y_1\) be the time series with \(x=1\)
- \(D_t\) be the difference of the 2 series
Assumptions
- parallel trends: confirmed by evaluating regions without the treatment
- absence treatment: no other variables
- Hence, the difference between the treatment & the control group is time-invariant, any difference in their difference must be due to the treatment effect.
$$ \begin{aligned} E[\Delta D] &= \Big( E[y_{1 t}] - E[y_{0 t}] \Big) - \Big( E[y_{1 t_0}] - E[y_{0 t_0}] \Big) \ &= \hat \beta_1(x=1) \ &= E(y \vert x=1) - E(y \vert x=1) \ \implies \text{ATE}(x) &\approx \text{ATT}(x) \ &= E[\Delta D] \end{aligned} $$ Technically, \(\hat \beta_1\) is an ATT because the parallel trend assumption assumes what the treated series would be like in the absence of the treatment, not what the control series would be like given the treatment
Didn’t understand
Technically, α is an ATT because the parallel trend assumption assumes what the treated cities would be like in the absence of the program, not what the control cities would be like given the program
