class: center middle main-title section-title-1 # Estimating Causal Effects .class-info[ **Session 15** .light[STA 379/679: Causal Inference <br> Lucy D'Agostino McGowan ] ] --- class: title title-1 # Weighted Estimator .box-1[ `$$\hat{\tau} = \frac{\sum_{i=1}^n Y_i X_i w_i}{\sum_{i=1}^nX_iw_i}-\frac{\sum_{i=1}^n Y_i (1-X_i) w_i}{\sum_{i=1}^n(1-X_i)w_i}$$` ] --- class: title title-1 # Weighted Estimator .box-1[ `$$\hat{\tau}_{ATE} = \frac{1}{n}\sum_{i=1}^n\left\{\frac{X_iY_i}{\hat{e}(\mathbf{Z_i})}-\frac{(1-X_i)Y_i}{1-\hat{e}(\mathbf{Z_i})}\right\}$$` ] -- <br> .box-1[ Is this an unbiased estimator for `\(\tau_{ATE}\)`? ] --- class: title title-1 # Using the propensity score .footer[Imbens and Rubin (2015) Causal Inference] .box-1[We can show that] .box-inv-1[ `$$E\left[\frac{Y_i^{obs}X_i}{e(Z_i)}\right] = E[Y_i(1)]$$` ] -- <br> .box-inv-1[ `$$E\left[\frac{Y_i^{obs}(1 - X_i)}{1 - e(Z_i)}\right] = E[Y_i(0)]$$` ] --- class: title title-1 # Using the propensity score .footer[Imbens and Rubin (2015) Causal Inference] .box-1[ `$$E\left[\frac{Y_i^{obs}X_i}{e(Z_i)}\right]=E\left[E\left[\frac{Y_i^{obs}X_i}{e(Z_i)}\Big|Z_i\right]\right]$$` ] -- .box-inv-1.medium[Why?] -- .box-1[Law of iterated expectations] --- class: title title-1 # Using the propensity score .footer[Imbens and Rubin (2015) Causal Inference] .box-1[ `$$E\left[E\left[\frac{Y_i^{obs}X_i}{e(Z_i)}\Big|Z_i\right]\right] = E\left[E\left[\frac{Y_i(1)X_i}{e(Z_i)}\Big|Z_i\right]\right]$$` ] --- class: title title-1 # Using the propensity score .footer[Imbens and Rubin (2015) Causal Inference] .box-1[ `$$E\left[E\left[\frac{\color{orange}{Y_i^{obs}}X_i}{e(Z_i)}\Big|Z_i\right]\right] = E\left[E\left[\frac{\color{orange}{Y_i(1)}X_i}{e(Z_i)}\Big|Z_i\right]\right]$$` ] -- .box-inv-1.medium[Why?] -- .box-1[We're only looking at the observations where `\(X_i = 1\)`] --- class: title title-1 # Using the propensity score .footer[Imbens and Rubin (2015) Causal Inference] .box-1[ `$$=E\left[E\left[\frac{Y_i(1)X_i}{e(Z_i)}\Big|Z_i\right]\right]=E\left[\frac{E[Y_i(1)|Z_i]E_X[X_i|Z_i]}{e(Z_i)}\right]$$` ] --- class: title title-1 # Using the propensity score .footer[Imbens and Rubin (2015) Causal Inference] .box-1[ `$$=E\left[E\left[\frac{\color{orange}{Y_i(1)X_i}}{e(Z_i)}\Big|\color{orange}{Z_i}\right]\right]=E\left[\frac{\color{orange}{E[Y_i(1)|Z_i]E_X[X_i|Z_i]}}{e(Z_i)}\right]$$` ] -- .box-inv-1.medium[Why?] -- .box-1[ Unconfoundedness (we can split `\(E[Y(1)X]=E[Y(1)]E[X]\)`) ] --- class: title title-1 # Using the propensity score .footer[Imbens and Rubin (2015) Causal Inference] .box-1[ `$$E\left[\frac{E[Y_i(1)|Z_i]E_X[X_i|Z_i]}{e(Z_i)}\right]=E[E[Y_i(1)|Z_i]]$$` ] --- class: title title-1 # Using the propensity score .footer[Imbens and Rubin (2015) Causal Inference] .box-1[ `$$E\left[\frac{E[Y_i(1)|Z_i]\color{orange}{E_X[X_i|Z_i]}}{\color{orange}{e(Z_i)}}\right]=E[E[Y_i(1)|Z_i]]$$` ] -- .box-inv-1.medium[Why?] -- .box-1[ propensity score definition: `\(e(Z_i)=E_X[X_i|Z_i]\)` ] --- class: title title-1 # Using the propensity score .footer[Imbens and Rubin (2015) Causal Inference] .box-1[ `$$E[E[Y_i(1)|Z_i]]=E[Y_i(1)]$$` ] -- .box-inv-1.medium[Why?] -- .box-1[ Law of iterated expectation ] --- class: title title-1 # <svg aria-hidden="true" role="img" viewBox="0 0 640 512" style="height:1em;width:1.25em;vertical-align:-0.125em;margin-left:auto;margin-right:auto;font-size:inherit;fill:currentColor;overflow:visible;position:relative;"><path d="M624 416H381.54c-.74 19.81-14.71 32-32.74 32H288c-18.69 0-33.02-17.47-32.77-32H16c-8.8 0-16 7.2-16 16v16c0 35.2 28.8 64 64 64h512c35.2 0 64-28.8 64-64v-16c0-8.8-7.2-16-16-16zM576 48c0-26.4-21.6-48-48-48H112C85.6 0 64 21.6 64 48v336h512V48zm-64 272H128V64h384v256z"/></svg> Application Exercise .box-1[Open `appex-10`] .box-1[Calculate the ATE and ATO estimates for the effect of `satisfied_customer_service` on `next_spend`] .box-1[Knit, commit, push to GitHub] <div class="countdown" id="timer_623fa91a" style="right:0;bottom:0;" data-warnwhen="0"> <code class="countdown-time"><span class="countdown-digits minutes">10</span><span class="countdown-digits colon">:</span><span class="countdown-digits seconds">00</span></code> </div>