day_3.Rmd

---
title: |
 | Uncertainty, consistency 
 | \& hypothesis testing
bibliography:
 - 'BIB/MasterBibliography.bib'
---

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# Variance of Difference-in-Means

::: frame
### Variance of Difference-in-Means estimator

-   Yesterday we showed that an example based on the village
    heads study, the Difference-in-Means estimator is distributed as\
    ![image](images/cra_est_dist_plot.pdf){width="0.75\\linewidth"}
    
-   \pause What is the variance of this estimator? \pause (And why do we care?)
:::

### Variance of Difference-in-Means estimator {.build}

-   Variance is average squared distance of estimator from its expected
    value:
    $$\underbrace{\mathop{\mathrm{\rm{E}}}\left[\underbrace{\left(\hat{\tau}\left(\bm{Z}, \bm{Y}\right) - \underbrace{\mathop{\mathrm{\rm{E}}}\left[\hat{\tau}\left(\bm{Z}, \bm{Y}\right)\right]}_{\color{red}{\text{Expected value}}}\right)^2}_{\color{blue}{\text{Squared distance from expected value}}}\right]}_{\color{green}{\text{Average (or expected) squared distance}}}$$

-   Diff-in-Means unbiased for $\tau$, so write variance as
    $$\mathop{\mathrm{\rm{E}}}\left[\left(\hat{\tau}\left(\bm{Z}, \bm{Y}\right) - \tau\right)^2\right]$$

-   In "village heads" example with 21 possible assignments
    $$\begin{aligned}
    \left(\hat{\tau}\left(\bm{z}_1, \bm{y}_1\right) - \tau\right)^2 \Pr\left(\bm{Z} = \bm{z}_1\right) + \ldots + \left(\hat{\tau}\left(\bm{z}_{21}, \bm{y}_{21}\right) - \tau\right)^2 \Pr\left(\bm{Z} = \bm{z}_{21}\right)
    \end{aligned}$$

-   Variance is $\approx 21.19$

::: frame
### Variance of Difference-in-Means estimator

-   Neyman (1923) derived expression for variance of Diff-in-Means under
    complete random assignment
    $$\mathop{\mathrm{\rm{Var}}}\left[\hat{\tau}\left(\bm{Z}, \bm{Y}\right)\right] = \frac{1}{N - 1}\left(\frac{n_1 {\color{magenta}{\sigma^2_{\bm{y}(\bm{0})}}}}{n_0} + \frac{n_0 {\color{magenta}{\sigma^2_{\bm{y}(\bm{1})}}}}{n_1} + 2{\color{magenta}{\sigma_{\bm{y}(\bm{0}), \bm{y}(\bm{1})}}}\right)$$
    where

-   $\sigma^2_{\bm{y}(\bm{0})} = \frac{1}{N} \sum \limits_{i = 1}^N \left(y_i(0) - \frac{1}{N} \sum \limits_{i = 1}^N y_i(0)\right)^2$
    is var of control POs

-   $\sigma^2_{\bm{y}(\bm{1})} = \frac{1}{N} \sum \limits_{i = 1}^N \left(y_i(1) - \frac{1}{N} \sum \limits_{i = 1}^N y_i(1)\right)^2$
    is var of treated POs

-   $\sigma_{\bm{y}(\bm{0}), \bm{y}(\bm{1})} = \frac{1}{N} \sum \limits_{i = 1}^N \left(y_i(0) - \frac{1}{N} \sum \limits_{i = 1}^N y_i(0)\right)\left(y_i(1) - \frac{1}{N} \sum \limits_{i = 1}^N y_i(1)\right)$
    is cov of POs

$\star$ Note that ${\color{magenta}{\sigma^2_{\bm{y}(\bm{0})}}}$,
${\color{magenta}{\sigma^2_{\bm{y}(\bm{1})}}}$ and
${\color{magenta}{\sigma_{\bm{y}(\bm{0}), \bm{y}(\bm{1})}}}$ depend on
unknown potential outcomes
:::

::: frame
### Variance of Difference-in-Means estimator

-   Sometimes you might see equivalent expression (Imbens and
    Rubin 2015)
    $$\dfrac{{\color{magenta}{S_{\bm{y}(\bm{0})}}}}{n_0} + \dfrac{{\color{magenta}{S_{\bm{y}(\bm{1})}}}}{n_1} - \dfrac{{\color{magenta}{S_{\tau}}}}{N},$$
    where

-   $S_{\bm{y}(\bm{0})} = \frac{1}{N - 1} \sum \limits_{i = 1}^N \left(y_i(0) - \frac{1}{N} \sum \limits_{i = 1}^N y_i(0)\right)^2$

-   $S_{\bm{y}(\bm{1})} = \frac{1}{N - 1} \sum \limits_{i = 1}^N \left(y_i(1) - \frac{1}{N} \sum \limits_{i = 1}^N y_i(1)\right)^2$

-   $S_{\tau} = \frac{1}{N - 1} \sum \limits_{i = 1}^N \left(\tau_i - \frac{1}{N} \sum \limits_{i = 1}^N \tau_i\right)^2$

\pause$\star$ Note that ${\color{magenta}{S_{\bm{y}(\bm{0})}}}$,
${\color{magenta}{S_{\bm{y}(\bm{1})}}}$ and
${\color{magenta}{S_{\tau}}}$ depend on unknown potential outcomes
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::: frame
### Variance of Difference-in-Means estimator
\fontsize{10pt}{10pt}\selectfont

-   Example: "Village heads" study (Gerber & Green 2012, Ch. 2):

-   ::: center
      --------- ----------------------- ----------------------- -------------
                       Budget share (%)                         
      Village     $\bm{\bm{y}(\bm{0})}$   $\bm{\bm{y}(\bm{1})}$   $\bm{\tau}$
      1                              10                      15             5
      2                              15                      15             0
      3                              20                      30            10
      4                              20                      15            -5
      5                              10                      20            10
      6                              15                      15             0
      7                              15                      30            15
      Average                        15                      20             5
      --------- ----------------------- ----------------------- -------------
    :::

-   With access to true POs, we can directly calculate
    $\mathop{\mathrm{\rm{Var}}}\left[\hat{\tau}\left(\bm{Z}, \bm{Y}\right)\right]$:
    ${\color{magenta}{\sigma^2_{\bm{y}(\bm{0})}}} \approx 14.29$,
    ${\color{magenta}{\sigma^2_{\bm{y}(\bm{1})}}} \approx 42.86$,
    ${\color{magenta}{\sigma_{\bm{y}(\bm{0}), \bm{y}(\bm{1})}}} \approx 7.14$\pause

-   So,
    $\mathop{\mathrm{\rm{Var}}}\left[\hat{\tau}\left(\bm{Z}, \bm{Y}\right)\right] = \frac{1}{N - 1}\bigg(\frac{n_1 {\color{magenta}{\sigma^2_{\bm{y}(\bm{0})}}}}{n_0} + \frac{n_0 {\color{magenta}{\sigma^2_{\bm{y}(\bm{1})}}}}{n_1} + 2{\color{magenta}{\sigma_{\bm{y}(\bm{0}), \bm{y}(\bm{1})}}}\bigg) \approx 21.19$\pause

-   In practice, ${\color{magenta}{\sigma^2_{\bm{y}(\bm{0})}}}$,
    ${\color{magenta}{\sigma^2_{\bm{y}(\bm{1})}}}$ and
    ${\color{magenta}{\sigma_{\bm{y}(\bm{0}), \bm{y}(\bm{1})}}}$
    unknown, so we estimate
    $\mathop{\mathrm{\rm{Var}}}\left[\hat{\tau}\left(\bm{Z}, \bm{Y}\right)\right]$
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# Variance estimation

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### Variance estimation
\fontsize{10pt}{10pt}\selectfont
We showed that the Diference-in-Means estimator's
variance is
$$\mathop{\mathrm{\rm{Var}}}\left[\hat{\tau}\left(\bm{Z}, \bm{Y}\right)\right] = \frac{1}{N - 1}\left(\frac{n_1 {\color{blue}{\sigma^2_{\bm{y}(\bm{0})}}}}{n_0} + \frac{n_0 {\color{blue}{\sigma^2_{\bm{y}(\bm{1})}}}}{n_1} + 2{\color{magenta}{\sigma_{\bm{y}(\bm{0}), \bm{y}(\bm{1})}}}\right)$$

-   We have unbiased estimators for
    ${\color{blue}{\sigma^2_{\bm{y}(\bm{0})}}}$ and
    ${\color{blue}{\sigma^2_{\bm{y}(\bm{1})}}}$, but not
    ${\color{magenta}{\sigma_{\bm{y}(\bm{0}), \bm{y}(\bm{1})}}}$

-   So what do we do?\pause 

-   We use **conservative** "plug-in" estimator (Neyman 1923)

-   **Conservative** means that
    $$\mathop{\mathrm{\rm{E}}}\left[\widehat{\mathop{\mathrm{\rm{Var}}}}\left[\hat{\tau}\left(\bm{Z}, \bm{Y}\right)\right]\right] \geq \mathop{\mathrm{\rm{Var}}}\left[\hat{\tau}\left(\bm{Z}, \bm{Y}\right)\right]$$\pause

-   When potential outcomes perfectly positively correlated
    $$\mathop{\mathrm{\rm{E}}}\left[\widehat{\mathop{\mathrm{\rm{Var}}}}\left[\hat{\tau}\left(\bm{Z}, \bm{Y}\right)\right]\right] = \mathop{\mathrm{\rm{Var}}}\left[\hat{\tau}\left(\bm{Z}, \bm{Y}\right)\right]$$

-   Otherwise,
    $\widehat{\mathop{\mathrm{\rm{Var}}}}\left[\hat{\tau}\left(\bm{Z}, \bm{Y}\right)\right]$
    is positively biased (conservative)\pause

-   Potential outcomes will be perfectly positively correlated if and
    only if\
    $\tau_i$ is the same for all $i = 1, \ldots , N$ units
:::

::: frame
### Conservative variance estimator

$$\mathop{\mathrm{\rm{Var}}}\left[\hat{\tau}\left(\bm{Z}, \bm{Y}\right)\right] = \frac{1}{N - 1}\left(\frac{n_1 {\color{blue}{\sigma^2_{\bm{y}(\bm{0})}}}}{n_0} + \frac{n_0 {\color{blue}{\sigma^2_{\bm{y}(\bm{1})}}}}{n_1} + 2{\color{magenta}{\sigma_{\bm{y}(\bm{0}), \bm{y}(\bm{1})}}}\right)$$

-   The maximum possible value of
    $2{\color{magenta}{\sigma_{\bm{y}(\bm{0}), \bm{y}(\bm{1})}}}$ is
    ${\color{blue}{\sigma^2_{\bm{y}(\bm{0})}}} + {\color{blue}{\sigma^2_{\bm{y}(\bm{1})}}}$\pause

-   So substitute
    ${\color{blue}{\sigma^2_{\bm{y}(\bm{0})}}} + {\color{blue}{\sigma^2_{\bm{y}(\bm{1})}}}$
    for $2{\color{magenta}{\sigma_{\bm{y}(\bm{0}), \bm{y}(\bm{1})}}}$:
    $$\begin{aligned}
    \mathop{\mathrm{\rm{Var}}}\left[\hat{\tau}\left(\bm{Z}, \bm{Y}\right)\right] & = \frac{1}{N - 1}\left(\frac{n_1 {\color{blue}{\sigma^2_{\bm{y}(\bm{0})}}}}{n_0} + \frac{n_0 {\color{blue}{\sigma^2_{\bm{y}(\bm{1})}}}}{n_1} + \left({\color{blue}{\sigma^2_{\bm{y}(\bm{0})}}} + {\color{blue}{\sigma^2_{\bm{y}(\bm{1})}}}\right)\right) \\
    & = \frac{N}{N - 1}\left(\frac{{\color{blue}{\sigma^2_{\bm{y}(\bm{0})}}}}{n_0} + \frac{{\color{blue}{\sigma^2_{\bm{y}(\bm{1})}}}}{n_1}\right)
    \end{aligned}$$

-   Now all quantities can be estimated!\pause

-   So, just "plug-in" estimators $\hat{\sigma}^2_{\bm{y}(\bm{0})}$ and
    $\hat{\sigma}^2_{\bm{y}(\bm{1})}$ for
    ${\color{blue}{\sigma^2_{\bm{y}(\bm{0})}}}$ and
    ${\color{blue}{\sigma^2_{\bm{y}(\bm{1})}}}$
    $$\widehat{\mathop{\mathrm{\rm{Var}}}}\left[\hat{\tau}\left(\bm{Z}, \bm{Y}\right)\right] = \frac{N}{N - 1}\left(\frac{\hat{\sigma}^2_{\bm{y}(\bm{0})}}{n_0} + \frac{\hat{\sigma}^2_{\bm{y}(\bm{1})}}{n_1} \right)$$

-   For exact expressions of $\hat{\sigma}^2_{\bm{y}(\bm{0})}$ and
    $\hat{\sigma}^2_{\bm{y}(\bm{1})}$, see [](#Variance estimators)
:::

::: frame
### Variance estimation
\fontsize{9pt}{9pt}\selectfont

-   Here is the conservative variance estimator for "village heads":\
    ![image](images/cra_var_est_dist_plot.pdf){width="0.8\\linewidth"}

-   **Solid** line is true variance of Diff-in-Means,
    $\mathop{\mathrm{\rm{Var}}}\left[\hat{\tau}\left(\bm{Z}, \bm{Y}\right)\right]$

-   **Dashed** line is expected value of the conservative variance estimator, 
    i.e. $\mathop{\mathrm{\rm{E}}}\left[\widehat{\mathop{\mathrm{\rm{Var}}}}\left[\hat{\tau}\left(\bm{Z}, \bm{Y}\right)\right]\right]$
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# Asymptotic properties

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Asymptotics

-   So far, we have derived

    1.  unbiased Difference-in-Means estimator of ATE

    2.  variance of Difference-in-Means estimator

    3.  conservative estimator of Difference-in-Means estimator's
        variance\pause

-   All of these derivations were for a fixed $N$, either small or large

-   Now let's see what happens to our estimator when $N$ grows large,
    $N \to \infty$

-   But first, why do we care?\pause

-   $N$ never goes to $\infty$ in an actual experiment

-   But properties as $N \to \infty$ **approximate** properties with
    fixed, but large $N$
:::

# Consistent estimation

::: frame
### Consistent estimation

-   Difference-in-Means estimator is consistent:
    $$\lim \limits_{N \to \infty} \Pr\left(\left\lvert \hat{\tau}\left(\bm{Z}, \bm{Y}\right) - \tau \right\rvert < \epsilon \right) = 1 \text{ for all } \epsilon > 0$$

-   In words:

    ::: center
    Pick any $\epsilon$ you want, no matter how small. There will be
    some value $N^{*}$ such that, if size of experiment is greater than
    $N^*$, the probability of an estimate within a distance of
    $\epsilon$ from the truth is equal to $1$.
    :::

-   Intuitively, with large experiment, our estimate will be close to
    true ATE!
:::

::: frame
### Consistent estimation

-   "Village heads" example:

    ![image](images/asymp_ests_plot.pdf){width="\\linewidth"}
:::

# Hypothesis testing

::: frame
Hypothesis tests of the weak null

-   The finite population CLT tells us that $$\begin{aligned}
    \cfrac{\hat{\tau}\left(\bm{Z}, \bm{Y}\right) - \mathop{\mathrm{\rm{E}}}\left[\hat{\tau}\left(\bm{Z}, \bm{Y}\right)\right]}{\sqrt{\mathop{\mathrm{\rm{Var}}}\left[\hat{\tau}\left(\bm{Z}, \bm{Y}\right)\right]}} & \overset{d}{\to} \mathcal{N}\left(0, 1\right)
    \end{aligned}$$\pause

-   Diff-in-Means is unbiased, so write $$\begin{aligned}
    \cfrac{\hat{\tau}\left(\bm{Z}, \bm{Y}\right) - \tau}{\sqrt{\mathop{\mathrm{\rm{Var}}}\left[\hat{\tau}\left(\bm{Z}, \bm{Y}\right)\right]}} & \overset{d}{\to} \mathcal{N}\left(0, 1\right)
    \end{aligned}$$\pause

-   The CLT is an asymptotic results as $N \to \infty$

-   But we can bound error of Normal approximation for fixed $N$

-   Thus, with experiments of at least moderate size and outcomes that
    aren't too skewed or have extreme outliers, $$\begin{aligned}
    \cfrac{\hat{\tau}\left(\bm{Z}, \bm{Y}\right) - \tau}{\sqrt{\mathop{\mathrm{\rm{Var}}}\left[\hat{\tau}\left(\bm{Z}, \bm{Y}\right)\right]}} & \overset{\text{approx.}}{\sim} \mathcal{N}\left(0, 1\right)
    \end{aligned}$$

-   This justifies use of standard Normal distribution for hyp.
    tests
:::

::: frame
Hypothesis tests of the weak null

-   "Village heads" example:

    ![image](images/asymp_stand_ests_plot.pdf){width="\\linewidth"}
:::

::: frame
### Hypothesis tests of the weak null
\fontsize{10pt}{10pt}\selectfont

-   To test null hypothesis relative to alternative $$\begin{aligned}
    H_0: & \tau = \tau_0 \text{ versus either } \\
    H_A: & \tau > \tau_0, \, H_A: \tau < \tau_0 \text{ or } H_A: \left\lvert \tau \right\rvert > \left\lvert \tau_0 \right \rvert
    \end{aligned}$$

-   Calculate upper(u), lower(l) or two-sided(t) p-value as
    $$\begin{aligned}
    p_u & =  1 - \Phi\left(\frac{\hat{\tau}\left(\bm{Z}, \bm{Y}\right) - \tau_0}{\sqrt{\widehat{\mathop{\mathrm{\rm{Var}}}}\left[\hat{\tau}\left(\bm{Z}, \bm{Y}\right)\right]}}\right) \\
    p_l & =  \Phi\left(\frac{\hat{\tau}\left(\bm{Z}, \bm{Y}\right) - \tau_0}{\sqrt{\widehat{\mathop{\mathrm{\rm{Var}}}}\left[\hat{\tau}\left(\bm{Z}, \bm{Y}\right)\right]}}\right) \\
    p_t & = 2\left(1 - \Phi\left(\frac{\left\lvert\hat{\tau}\left(\bm{Z}, \bm{Y}\right) - \tau_0\right\rvert}{\sqrt{\widehat{\mathop{\mathrm{\rm{Var}}}}\left[\hat{\tau}\left(\bm{Z}, \bm{Y}\right)\right]}}\right)\right)
    \end{aligned}$$

-   If p-value is less than size $\alpha$-level of test, reject.
    Otherwise, don't

-   Note that, since we don't know
    $\mathop{\mathrm{\rm{Var}}}\left[\hat{\tau}\left(\bm{Z}, \bm{Y}\right)\right]$,\
    we have used its conservative estimator instead,
    $\widehat{\mathop{\mathrm{\rm{Var}}}}\left[\hat{\tau}\left(\bm{Z}, \bm{Y}\right)\right]$
:::

::: frame
### Hypothesis tests of the weak null

\bh{Hypothesis tests susceptible to two errors}:

-   Type I error: Rejecting null hypothesis when it is true

-   Type II error: *Failing* to reject null hypothesis when it is false

\bh{A good test} controls these errors:

1.  Type I error probability is less than or equal to size
    ($\alpha$-level) of test

2.  Power (1 - type II error probability) is at least as great as
    $\alpha$-level

3.  Power tends to $1$ as $N \to \infty$
:::

::: frame
### Hypothesis tests of the weak null

-   We can prove that tests of weak null satisfy (1) -- (3) as
    $N \to \infty$

-   Thus, when experiments are large, we can often safely use such tests

-   But (1) -- (3) may not always be satisfied when experiments are
    small, have skewed outcome distributions or extreme outliers
:::

::: frame
### Confidence intervals
\fontsize{8pt}{8pt}\selectfont

-   Equivalence between hypothesis testing and confidence intervals

-   Confidence interval is set of null hypotheses we fail to reject

Consider two-sided confidence interval, $\mathcal{C}_t$: $$\begin{split}
\mathcal{C}_t & = \left\{\tau_0 : \left\lvert \cfrac{\hat{\tau}\left(\bm{Z}, \bm{Y}\right)- \tau_0}{\sqrt{\widehat{\mathop{\mathrm{\rm{Var}}}}\left[\hat{\tau}\left(\bm{Z}, \bm{Y}\right)\right]}} \right\rvert \leq z_{1 - \alpha/2}\right\} \\
& = \left\{\tau_0 : - z_{1 - \alpha/2} \leq \cfrac{\hat{\tau}\left(\bm{Z}, \bm{Y}\right)- \tau_0}{\sqrt{\widehat{\mathop{\mathrm{\rm{Var}}}}\left[\hat{\tau}\left(\bm{Z}, \bm{Y}\right)\right]}} \leq z_{1 - \alpha/2} \right\} \\
& = \left\{\tau_0 : - z_{1 - \alpha/2} \sqrt{\widehat{\mathop{\mathrm{\rm{Var}}}}\left[\hat{\tau}\left(\bm{Z}, \bm{Y}\right)\right] }\leq \hat{\tau}\left(\bm{Z}, \bm{Y}\right)- \tau_0 \leq z_{1 - \alpha/2} \sqrt{\widehat{\mathop{\mathrm{\rm{Var}}}}\left[\hat{\tau}\left(\bm{Z}, \bm{Y}\right)\right] }\right\} \\
& = \left\{\tau_0 : -\hat{\tau}\left(\bm{Z}, \bm{Y}\right)- z_{1 - \alpha/2} \sqrt{\widehat{\mathop{\mathrm{\rm{Var}}}}\left[\hat{\tau}\left(\bm{Z}, \bm{Y}\right)\right] }\leq - \tau_0 \leq - \hat{\tau}\left(\bm{Z}, \bm{Y}\right)+ z_{1 - \alpha/2} \sqrt{\widehat{\mathop{\mathrm{\rm{Var}}}}\left[\hat{\tau}\left(\bm{Z}, \bm{Y}\right)\right] }\right\} \\
& = \left\{\tau_0 : \hat{\tau}\left(\bm{Z}, \bm{Y}\right)+ z_{1 - \alpha/2} \sqrt{\widehat{\mathop{\mathrm{\rm{Var}}}}\left[\hat{\tau}\left(\bm{Z}, \bm{Y}\right)\right] }\geq \tau_0 \geq  \hat{\tau}\left(\bm{Z}, \bm{Y}\right)- z_{1 - \alpha/2} \sqrt{\widehat{\mathop{\mathrm{\rm{Var}}}}\left[\hat{\tau}\left(\bm{Z}, \bm{Y}\right)\right] }\right\} \\
& = \left\{\tau_0 : \hat{\tau}\left(\bm{Z}, \bm{Y}\right)- z_{1 - \alpha/2} \sqrt{\widehat{\mathop{\mathrm{\rm{Var}}}}\left[\hat{\tau}\left(\bm{Z}, \bm{Y}\right)\right] }\leq \tau_0 \leq \hat{\tau}\left(\bm{Z}, \bm{Y}\right)+ z_{1 - \alpha/2} \sqrt{\widehat{\mathop{\mathrm{\rm{Var}}}}\left[\hat{\tau}\left(\bm{Z}, \bm{Y}\right)\right] }\right\}
\end{split}$$
:::

::: frame
### Appendix: Estimator of Difference-in-Means estimator's variance

Neyman's conservative estimator of the Difference-in-Means estimator's
variance is
$$\widehat{\mathop{\mathrm{\rm{Var}}}}\left[\hat{\tau}\left(\bm{Z}, \bm{Y}\right)\right] = \frac{N}{N - 1}\left(\frac{\hat{\sigma}^2_{\bm{y}(\bm{0})}}{n_0} + \frac{\hat{\sigma}^2_{\bm{y}(\bm{1})}}{n_1} \right),$$
where $$\begin{aligned}
\hat{\sigma}^2_{\bm{y}(\bm{0})} & = \left(\frac{N - 1}{N\left(n_0 - 1\right)}\right)\sum \limits_{i: Z_i = 0}^N \left(y_i(0) - \hat{\mu}_{\bm{y}(\bm{0})}\right)^2 \\
\hat{\sigma}^2_{\bm{y}(\bm{1})} & = \left(\frac{N - 1}{N\left(n_1 - 1\right)}\right)\sum \limits_{i: Z_i = 1}^{n} \left(y_i(1) - \hat{\mu}_{\bm{y}(\bm{1})}\right)^2 \\
\hat{\mu}_{\bm{y}(\bm{0})} & = \left(\frac{1}{n_0}\right) \sum \limits_{i = 1}^N \left(1 - Z_i\right)y_i(0) \\
\hat{\mu}_{\bm{y}(\bm{1})} & = \left(\frac{1}{n_1}\right) Z_i y_i(1)
\end{aligned}$$
:::