Interpretation Under Heterogeneity

Recall that in the model

$$Y=X^{\prime} \beta+U$$

the effect of a change in $X$ (say, from $X=x$ to $X=x^{\prime}$ ) is the same for everybody.

However, the effect of a change in $X$ on $Y$ can be different for different people. To capture this: allow for $\beta$ to be random. When $\beta$ is random, we may absorb $U$ into the intercept and simply write

$$Y=X^{\prime} \beta .$$

Notation

1. With a random sample where variables are indexed by $i$, we would write $Y_i=X_i^{\prime} \beta_i$, which makes it explicit that every individual has a unique effect $\beta_i$.

2. Assume $k=1$ and write $D$ in place of $X_1$, which is assumed to take values in $\{0,1\}$, i.e. $D$ is binary. Then,

   $$Y=\beta_0+\beta_1 D .$$
3. We interpret $\beta_0$ as $Y(0)$ and $\beta_1$ as $Y(1)-Y(0)$, where $Y(1)$ and $Y(0)$ are potential or counterfactual outcomes. Using this notation, we may rewrite the equation as Potential Outcome Model:

   $$Y=D Y(1)+(1-D) Y(0)\\\text{or}\\Y= \begin{cases}Y(1) & \text { if } D=1 \\ Y(0) & \text { if } D= 0 .\end{cases}$$

   1. $Y(0)$ denotes the value of the outcome that would have been observed if (possibly counter-to-fact) $D$ were 0 ;

   2. $Y (1)$ denotes the value of the outcome that would have been observed if (possibly counter-to-fact) $D$ were 1.

4. The variable $D$ is typically called the treatment

5. $Y(1)-Y(0)$ is called the treatment effect. The quantity $E[Y(1)-Y(0)]$ is usually referred to as the average treatment effect (ATE). This denotes the "treatment effect on the overall population".

6. The quantity $E[Y(1)-Y(0) \mid D=1]$ is called the average treatment effect on treatment group (ATET). This denotes the "treatment effect on populations who are treated".

Note that:

The treatment effect is for one individual, however, we cannot observe both outcomes ($Y(0) \& Y(1)$) from one individual at the same time.

Four Types of People in treatment

1. Always Taker: $D(1)=1 \quad D(0)=1$

2. Complier: $D(1)=1 \quad D(0)=0$

3. Never Taker: $D(1)=0 \quad D(0)=0$

4. Defier: $D(1)=0 \quad D(0)=1$

Random Assignment

If $D$ were randomly assigned (e.g., by the flip of a coin), then

$$(Y(0), Y(1)) \perp D$$

In this case, under mild assumptions, the slope coefficient from OLS regression of $Y$ on a constant and $D$ yields a consistent estimate of the average treatment effect.

Note that if $D$ is randomly assigned or the treated group is randomly drawn from sample (Treated group represents for whole population). Then ATE = ATET.

If $D$ is binary, we have that: $Y=\beta_0+\beta_1 D+U$

$$\beta_1=\frac{\operatorname{Cov} ( Y, D)}{\operatorname{Var}(D)}=E[Y \mid D=1]-\mathbb{E}[Y \mid D=0]=\mathbb{E}[Y(1) \mid D=1]-\mathbb{E}[Y(0) \mid D=0]$$

As $(Y(0), Y(1)) \perp D$, we can have that $\beta_1 = E[Y(1)]-E[Y(0)]$ = ATE / ATET

The estimator of $\beta_1$ is

$$\hat{\beta}_1=\frac{1}{n} \sum_{i=1}^n Y_i D_i-\frac{1}{n} \sum_{i=1}^n Y_i \left(1-D_i\right)=\bar{Y}_{D=1}-\bar{Y}_{D=0} .$$

Therefore, we got $plim_{n \rightarrow \infin} \hat{\beta_1}=\beta_1$, which is ATE / ATET.

However, if $D$ is not randomly assigned, $\beta_1 =$ ATET + Bias. Since under this case,

$$E[Y \mid D=1]-\mathbb{E}[Y \mid D=0] \\= \underbrace{E\left[\mathrm{Y}_{1 i} \mid \mathrm{D}_i=1\right]-E\left[\mathrm{Y}_{0 i} \mid \mathrm{D}_i=1\right]}_{\text {average treatment effect on the treated }}+\underbrace{E\left[\mathrm{Y}_{0 i} \mid \mathrm{D}_i=1\right]-E\left[\mathrm{Y}_{0 i} \mid \mathrm{D}_i=0\right]}_{\text {selection bias }}$$

This can also be reformed to $\beta_1 =$ ATE + Bias.

Selction

In general, we expect $D$ to depend on $(Y(1), Y(0))$. Under this condition, OLS does not yield a consistent estimate of the average treatment effect.

Note that as treatment $D$ is not randomly assigned, which means it is for a selected group, we have that ATE $\neq$ ATET.

To proceed further, we therefore assume, as usual, that there is an instrument $Z$. Let $Z \in{0,1}$.

Consider the slope coefficient from TSLS/IV regression of $Y$ on $D$ with $Z$ as an instrument, $Z$ is also binary:

$$\frac{\operatorname{Cov}[Y, Z]}{\operatorname{Cov}[D, Z]}=\frac{E[Y \mid Z=1]-E[Y \mid Z=0]}{E[D \mid Z=1]-E[D \mid Z=0]}$$

Expressing in Terms of Expected Values:

• The numerator, $E[Y \mid Z=1]-E[Y \mid Z=0]$, represents the difference in the expected value of $\mathrm{Y}$ when $\mathrm{Z}$ changes from 0 to 1 .

• The denominator, $E[D \mid Z=1]-E[D \mid Z=0]$, represents the difference in the expected value of $\mathrm{D}$ (take-up of the treatment) when $\mathrm{Z}$ changes from 0 to 1.

Take-Up Ratio: The ratio $\frac{E[D \mid Z=1]-E[D \mid Z=0]}{\operatorname{Var}[Z]}$ can be interpreted as the take-up ratio. It indicates the percentage of people who will take the treatment when offered (i.e., when $\mathrm{Z}$ changes from 0 to 1 ). It's a measure of the effectiveness of the instrument in inducing changes in the treatment variable.

Proof:

Standard Regression Formula: In a simple linear regression, without considering endogeneity, the coefficient $\beta_1$ is given by:

$$\beta_1=\frac{\operatorname{Cov}(Y, D)}{\operatorname{Var}(D)}$$

IV Regression Setup: However, in the presence of endogeneity (where D is correlated with the error term), this formula doesn't yield a consistent estimate. Here's where IV regression using an instrument $Z$ comes into play.

Assumptions:

1. Relevance: $Z$ is correlated with $\mathrm{D}$ (i.e., $\operatorname{Cov}(D, Z) \neq 0$ ).

2. Exogeneity: $Z$ is not correlated with the error term in the $Y$ regression.

Two-Stage Least Squares (TSLS) Process:

1. First Stage: Regress $\mathrm{D}$ on $\mathrm{Z}$ and get the fitted values $\hat{D}$ :

$$\begin{aligned} & D=\pi_0+\pi_1 Z+\epsilon \\ & \hat{D}=\pi_0+\pi_1 Z \end{aligned}$$

1. Second Stage: Regress Y on $\hat{D}$ :

$$Y=\alpha_0+\beta_1 \hat{D}+u$$

Now, let's derive the IV estimate of $\beta_1$ :

1. Covariance of $Y$ and $\hat{D}$ : The covariance of $Y$ with $\hat{D}$ is given by:

$$\operatorname{Cov}(Y, \hat{D})=\operatorname{Cov}\left(Y, \pi_0+\pi_1 Z\right)$$

Since $\pi_0$ is a constant, it drops out in the covariance, leaving:

$$\operatorname{Cov}(Y, \hat{D})=\pi_1 \operatorname{Cov}(Y, Z)$$

1. Variance of $\hat{D}$ : Similarly, the variance of $\hat{D}$ is:

$$\operatorname{Var}(\hat{D})=\operatorname{Var}\left(\pi_0+\pi_1 Z\right)$$

Again, $\pi_0$ being constant, it drops out, leaving:

$$\operatorname{Var}(\hat{D})=\pi_1^2 \operatorname{Var}(Z)$$

1. Substituting in the Second Stage: In the second stage, $\beta_1$ is estimated as the coefficient of $\hat{D}$ in the regression of $Y$ on $\hat{D}$ :

$$\beta_1=\frac{\operatorname{Cov}(Y, \hat{D})}{\operatorname{Var}(\hat{D})}$$

Substituting our expressions for $\operatorname{Cov}(Y, \hat{D})$ and $\operatorname{Var}(\hat{D})$ :

$$\begin{aligned} & \beta_1=\frac{\pi_1 \operatorname{Cov}(Y, Z)}{\pi_1^2 \operatorname{Var}(Z)} \\ & \beta_1=\frac{\operatorname{Cov}(Y, Z)}{\pi_1 \operatorname{Var}(Z)} \end{aligned}$$

1. Relation to $\operatorname{Cov}(D, Z)$ : From the first stage, we know that $\pi_1=\frac{\operatorname{Cov}(D, Z)}{\operatorname{Var}(Z)}$. Substituting $\pi_1$ :

$$\beta_1=\frac{\operatorname{Cov}(Y, Z)}{\frac{\operatorname{Cov}(D, Z)}{\operatorname{Var}(Z)} \cdot \operatorname{Var}(Z)}$$

Simplifying, we get:

$$\beta_1=\frac{\operatorname{Cov}(Y, Z)}{\operatorname{Cov}(D, Z)}$$

WALD estimator is the estimator used for this TSLS regression $\beta_1$, it can be represented as:

$$\hat{\beta}_1=\frac{\hat{E[Y \mid Z=1]}-\hat{E[Y \mid Z=0]}}{\hat{E[D \mid Z=1]}-\hat{E[D \mid Z=0]}}$$

Potential Treatments

Now, we want to express this quantity

$$\beta_1 = \frac{\operatorname{Cov}[Y, Z]}{\operatorname{Cov}[D, Z]}=\frac{E[Y \mid Z=1]-E[Y \mid Z=0]}{E[D \mid Z=1]-E[D \mid Z=0]}$$

in terms of the treatment effect $Y(1)-Y(0)$ somehow.

Towards our goal, it is useful to also introduce the following equation for $D$:

$$\begin{aligned} D &= Z D(1)+(1-Z) D(0) \\ & =D(0)+(D(1)-D(0)) Z \\ & =\quad \pi_0+\pi_1 Z \end{aligned}$$

where $\pi_0=D(0), \pi_1=D(1)-D(0)$, and $D(1)$ and $D(0)$ are potential or counterfactual treatments.

We impose the following versions of instrument exogeneity and instrument relevance, respectively:

$$(Y(1), Y(0), D(1), D(0)) \perp Z$$

Note that, this assumption is actually stronger than the exogeneity.

$$P\{D(1) \neq D(0)\}=P\left\{\pi_1 \neq 0\right\}>0$$

We further assume the following Monotonicity Condition:

$$P\{D(1) \geq D(0)\}=P\left\{\pi_1 \geq 0\right\}=1$$

This monotonicity condition eliminates the defiers. If the monotonicity does not hold, we can have that under this condition, $D(0)=0 \quad (Z=1) < D(0)=1\quad(Z=0)$, which will include the defiers.

TSLS Estimator to Form Y(1) - Y(0)

Since the potential outcome is $E[Y \mid Z=1]-E[Y \mid Z=0]$

as we have

$$\left\{\begin{array}{l} Y=D Y(1)+(1-D) Y(0) \\ D=Z D(1)+(1-Z) D(0) \end{array}\right.$$

We got

$$\begin{aligned} &E[Y \mid Z=1]-E[Y \mid Z=0]\\ & =\mathbb{E}[D Y(1)+(1-D) Y(0) \mid Z=1]-\mathbb{E}[D Y(1)-(1-D) Y(0) \mid Z=0] . \\ & =\mathbb{E}[D(1) Y(1)+(1-D(1)) Y(0) \mid Z=1]-\mathbb{E}[D( 0) Y(1)-(1-D(0)) Y(0) \mid Z=0] \end{aligned}$$

since the instrument $Z$ here is exogenous

$$\begin{gathered} =\mathbb{E}\left[D(1) Y(1)+(1-D(1)) Y_0\right]-\mathbb{E}[D(0) Y(1)-(1-D(0)) Y(0)] \\ =\mathbb{E}\{[D(1)-D(0)] Y(1)-[D(1)-D(0)] Y(0)\} \\ =\mathbb{E}\{(D(1)-D(0))(Y(1)-Y(0))\} \end{gathered}$$

As $D(1)-D(0)$ is always 0 for Always Taker and Never Taker, and by monotonicity condition, we ruled out the Defiers.

$$= \mathbb{E}\left[( D ( 1 ) - D ( 0 ) ) \left(Y(1)-Y\left(0\right)\right) \mid D(1)-D(0)=1\right] P(D(1)-D(0)=1)$$

$$= \mathbb{E}\left[( D ( 1 ) - D ( 0 ) ) \left(Y(1)-Y\left(0\right)\right) \mid D(1)-D(0)>1\right]$$

Which only focuses on the Compliers.

LATE

The TSLS/IV estimand equals

$$\frac{\operatorname{Cov}[Y, Z]}{\operatorname{Cov}[D, Z]}=E[\underbrace{Y(1)-Y(0)}_{\mathrm{TE}} \mid \underbrace{D(1)>D(0)}_{\text {local }}] \equiv \mathrm{LATE}$$

This is called the local average treatment effects. Which denotes the Average treatment effect of the subpopulation of people for whom a change in the value of the instrument switched them from being non-treated to treated: the so-called compliers.

Since $\operatorname{Cov}[Y, Z] = E[Y \mid Z=1]-E[Y \mid Z=0] = E\left[Y(1)-Y{(0)} \mid D(1)>D(0)\right] P(D(1)-D(0)=1)$

$\operatorname{Cov}[D, Z] = E[D \mid Z=1]-E[D \mid Z=0]=P\left(D(1)>D(0)\right)$

This is because $D=Z D(1)+(1-Z) D(0)$, then

$$\begin{aligned} E[D \mid Z=1]-E[D \mid Z=0] = & E[D(1) \mid Z=1]-E[D(0) \mid Z=0]\\ = & E[D(1)-D(0)] \\ = & 1 \cdot P(D(1)-D(0)=1)+(-1) P(D(1)-D(0)=-1) \\ & +0 \cdot P(D(1)-D(0)=0)\\ =&P(D(1)-D(0)=1) \end{aligned}$$

Because we ruled out the defiers.

Monotonicity

As before, we have shown that monotonicity will rule out the defiers.

Monotonicity: while the instrument may have no effect on some people, all those who are affected are affected in the same way. Without monotonicity, we would have

$$\begin{gathered} E[Y \mid Z=1]-E[Y \mid Z=0]=E[Y(1)-Y(0) \mid D(1)>D(0)] P\{D(1)> \\ D(0)\}-E[Y(1)-Y(0) \mid D(1)<D(0)] P\{D(1)<D(0)\} . \end{gathered}$$

Treatment effects may be positive for everyone (i.e., $Y(1)-Y(0)>0$ ) yet the reduced form is zero because effects on compliers are canceled out by effects on defiers, i.e., those individuals for which the instrument pushes them out of treatment $(D(1)=0$ and $D(0)=1)$.

This doesn't come up in a constant effect model where $\beta=Y(1)-Y(0)$ is constant, as in such case

$$\begin{aligned} E[Y \mid Z=1]-E[Y \mid Z=0] & =\beta\{P\{D(1)>D(0)\}-P\{D(1)<D(0)\}\} \\ & =\beta E[D(1)-D(0)], \end{aligned}$$

and so a zero reduced-form effect means either the first stage is zero or $\beta=0$.

Monotonicity with one-sided Compliance

Randomized trial with non-compliance: the treatment assignment as an "offer of treatment" $Z$ (the instrument) and the actual treatment $D$ determines whether the subject actually had the treatment.

Assume no one in the control group has access to the treatment: $D(0)=0$ while $D(1) \in\{0,1\}$, in this case, Monotonicity automatically holds: $D(1) \geq D(0)$

Since $D(1)$ is a choice, a comparison between those actually treated $(D=1)$ and the control $(D=0)$ group is misleading. Two alternatives are frequently used.

Intention to Treat Effect: a comparison between those who were offered treatment $(Z=1)$ and the control $(Z=0)$ group.

In this case: LATE = ATT: IV using $Z$ as an instrumental variable for $D$, which leads to LATE. Since $D(0)=0$, LATE returns the effect of treatment on the treated (ATT).

LATE: $E[Y(1)-Y(0) \mid D(1)>D(0)]$

ATT: $E[Y(1)-Y(0) \mid D=1]$