Learn about instrumental variables for causal inference.

Instrumental Variables

Performing 2SLS in R is easy if we use the `ivreg()` function from the `AER` package.

The key difference in syntax between `ivreg()` and other regression functions is that the `formula` argument of the `ivreg()` function must include the instrument. If we wanted to perform 2SLS regression with variables `outcome` as the outcome, `treatment` as the treatment, and `instrument` as the instrument, the model formula would be `outcome ~ treatment | instrument`.

To fit the 2SLS regression using the recycling data, we would use the following code:

```r
# import library
library(AER)

# run 2SLS regression
iv_mod <- ivreg(
 #outcome ~ treatment | instrument
 formula = recycled ~ rebate | distance, 
 data = recycle_df
 )
```

To view the coefficients and standard errors, we can use `summary(iv_mod)$coefficients`, which gives the following output (you may need to make this section of the screen wider to view the full table):

```
 Estimate Std. Error t value Pr(>|t|)
(Intercept) 129.36463 0.8683141 148.98368 0.000000e+00
rebate 31.25452 1.4629239 21.36442 5.118885e-68
```

The results of 2SLS regression show that the estimate of the effect of the rebate program is 31.25, meaning participation in the rebate program led to an average increase in recycling of 31.25 kilograms/person. This only applies to compliers: those individuals who participated in the rebate program because they lived within 5 miles of a recycling center, but who would not have participated otherwise.

You may be wondering why we couldn't just fit the two separate regression models described in the previous exercise using `lm()` or `glm()` functions. The `ivreg()` function is preferred because it automatically corrects standard errors to account for the fact that the second stage regression model uses predicted values of the treatment. 

If we use incorrect standard errors, we could make incorrect conclusions about the treatment effect:

* Lower standard errors correspond with more precise treatment effect estimates and a greater likelihood that the treatment coefficient will be found to be significantly different from zero. 
* Higher standard errors correspond with less precise treatment effect estimates and a lesser likelihood that the treatment coefficient will be found to be significantly different from zero.

IV Estimation in R

In observational studies, when randomization is not possible, balance of measured and unmeasured confounding variables is not guaranteed. Without taking appropriate measures, the causal estimate of the effect of a treatment on an outcome of interest will be biased.

Instrumental variable (IV) estimation is one causal inference method that uses _instruments_ to help reduce bias from both measured AND unmeasured confounding variables.

If you started this lesson hoping to learn about pianos and guitars, you may be disappointed. In IV estimation, an _instrument_ (or _instrumental variable_) is a variable that is causally related to an outcome variable ONLY through another variable &mdash; typically the treatment variable of interest.

An instrumental variable would be depicted in a causal diagram as follows:

![IV diagram where Treatment, Outcome, Confounders (both measured and unmeasured), and Instruments are all represented. Treatment feeds the Outcome. Instruments feeds the treatment. The Confounders feed both the Outcome and the Treatment.](https://static-assets.codecademy.com/Courses/causal-inference/rdd-and-iv/iv-e2-narr.svg)

In this diagram, the arrows signify the presence and direction of a causal relationship. For example, there is no arrow directly from the instrument to the outcome because the instrument only impacts the outcome through its causal relationship with the treatment.

Not That Kind of Instrument

Compliance with the treatment assignment influences which causal estimand we calculate:

 - When compliance is perfect, the treatment assignment and treatment received can be used interchangeably to get an accurate estimate of the causal effect.
 - When compliance is NOT perfect, we cannot estimate the average treatment effect (ATE) because treatment assignment and treatment compliance are NOT interchangeable. Instead, the average effect of treatment assignment can be thought of as the effect of the _intention to treat_, or ITT effect.

IV estimation allows us to approximate the ATE by estimating a local average treatment effect (LATE) just among the compliers. This is known as the _compliers' average causal effect_ (CACE). To estimate the CACE, we must make four assumptions about the sample:

 - _Relevance_: the instrument has a causal effect on the treatment received.
 - _Exclusion_: the instrument affects the outcome ONLY indirectly through the treatment received.
 - _Exchangeability_: there are no confounders that influence both the instrument AND the outcome.
 - _Monotonicity_: there are no "defiers" in the sample.

The relevance assumption implies that there are no "never-takers" and "always-takers" in the sample. In these subgroups, the treatment received doesn't depend on the value of the assigned treatment. And in order to estimate the treatment effect among the "compliers," we must also assume that there are no "defiers" in the sample.

Assumptions of IV Estimation

Suppose that a city is interested in increasing the per-capita rate of recycling among its citizens while decreasing the city’s operating costs.

To encourage this, the city allows individuals to discontinue curbside recycling pickup and instead opt into a rebate program. The city wants to evaluate whether or not this rebate program increases the amount of recycling in the city.
 
The city compiles the following variables in a dataset to evaluate the success of the rebate program:  
 
`recycled`: amount recycled (kg/person).  
`rebate`: participation in rebate program (curbside vs. rebate).  
`distance`: distance from recycling center (5 miles vs. > 5 miles). An individual who lives less than five miles from a recycling center might be more likely to opt into the rebate program than someone who lives more than five miles from a center. However, distance to a recycling center should not directly impact the amount of waste each person recycles. 

We are going to use Instrumental Variables to answer the city's question, but before we dive into that, we need to establish some more tools.

The first is **conditional exchangeability.**

One key assumption made in causal inference methods like weighting or stratification is _conditional exchangeability_. This assumption states that there are no unmeasured confounding variables that have a causal effect on both the treatment assignment and the outcome.

Randomization of the treatment assignment ensures that both measured and unmeasured confounding variables are evenly balanced between treatment groups. In a non-randomized setting, balance is NOT guaranteed. The assumption of conditional exchangeability cannot be tested or verified &mdash; in most cases, the best we can do without randomization is to identify and measure as many potential confounding variables as possible.

Instrumental variable (IV) estimation is a causal inference technique that helps us estimate the causal effect of the treatment even in the presence of unmeasured confounding variables. In this lesson, we will learn about the assumptions and potential applications of IV estimation.

Diagram of causal relationships made of three boxes and arrows between them. The treatment box has an arrow pointing to the outcome box. The  "measured and unmeasured confounders" box points to both the treatment and the outcome boxes.

Introduction to Instrumental Variables

Congratulations on finishing this lesson on instrumental variable estimation! We covered a lot of topics:

 - Instrumental variable (IV) estimation is a causal inference technique that can be used to estimate a causal treatment effect even in the presence of unobserved confounding variables.
 - IV estimation can be used in non-randomized studies when compliance with the assigned or encouraged treatment is not perfect.
 - An instrument is a variable that is related to an outcome of interest ONLY through the treatment variable.
 - The four assumptions of IV estimation are relevance, exclusion, exchangeability, and monotonicity.
 - IV estimation is performed via two-stage least squares (2SLS) regression.
 - The `ivreg()` function in the `AER` package performs 2SLS regression and automatically provides corrected standard errors of the treatment effect.

Conclusion

While IV estimation can be effective in certain circumstances, it also has limitations. One limitation is that the causal estimand (CACE) is not generalizable. The CACE only describes the effect of those who comply with treatment.

Another limitation is that it is difficult to find a suitable instrument that has a strong relationship with the treatment. If an instrument is only weakly related to the treatment, 2SLS regression will produce inaccurate estimates of the CACE.

To illustrate this, suppose that instead of using distance as an instrument for participation in the rebate program, we used another variable, `children`. The variable indicates whether or not an individual has children. Having children wouldn't directly cause a change in recycling, but individuals with children might be less likely to participate in the rebate program. The rebate program requires individuals to take the time to drop off recycling &mdash; time that people with children might not have.

Performing the 2SLS regression again with the `ivreg()` function and the `children` variable as an instrument highlights the effect of using a weak instrument:

```r
iv_mod_weak <- ivreg(
 formula = recycled ~ rebate | children, #new weak instrument
 data = recycle_df
 )
```

Using `summary(iv_mod_weak)$coefficients` we can view just the coefficient table from the results summary.

```
 Estimate Std. Error t value Pr(>|t|)
(Intercept) 126.10140 8.930242 14.120715 5.477652e-37
rebate 37.84692 18.018181 2.100485 3.631487e-02
```

The estimate of 37.85 is neither accurate nor precise, as highlighted by the large standard error. This estimate is similar to the estimate from OLS regression, demonstrating that this is a weak instrument.

Interpretation and Considerations of IV Estimation

In a standard ordinary least squares (OLS) regression model, the outcome is predicted directly from the treatment assignment and other confounding variables. Such an OLS model would take the form:

Outcome = &#946;0 + &#946;1 * Treatment Assignment.

In OLS regression, we would use &#946;1 as an estimate of the treatment effect. To fit the OLS regression model using the recycling data, we would use the following code:

```r
lm(recycled ~ rebate, #outcome ~ treatment
 data = recycle_df #dataset
)

# Output:

Call:
lm(formula = recycled ~ rebate, data = recycle_df)

Coefficients:
(Intercept) rebate
 126.00 38.04
```

The OLS estimate suggests that participation in the rebate program leads to an average increase in recycling of 38.04 kilograms/person. However, this estimate is biased because OLS regression does not control bias introduced by unmeasured confounding variables or imperfect compliance.

In IV estimation, we account for unmeasured confounding variables and imperfect compliance via _two-stage least squares_ (2SLS) regression. This type of regression predicts the outcome in two separate steps:

 - In the first stage, treatment received is predicted by the instrument (treatment assignment): 
 
 Predicted Treatment Received = &#945;0 + &#945;1 * Treatment Assignment

 - In the second stage, the outcome is predicted as a function of the predicted treatment received from the first stage:

 Outcome = &#946;0 + &#946;1 * Predicted Treatment Received

&#946;1 from the second stage of the 2SLS regression model is used as the estimate of the CACE.

In this lesson, we focus on 2SLS regression with a continuous outcome, binary instrument, and binary treatment to keep things simple. When this is the case, the first stage uses logistic regression, while the second stage uses linear regression.

Two-Stage Least Squares Regression

When treatment group assignments cannot be randomized due to ethical or practical reasons, the best we can do is to encourage compliance. However, encouragement does not guarantee compliance.

Compliance with the treatment assignment can be influenced by many factors, only some of which may be measurable. To account for unmeasured confounders of treatment compliance and the outcome, we could use IV estimation with treatment assignment as the instrument:

![IV diagram where Treatment Compliance, Treatment Assignment, Outcome, and Confounders (both measured and unmeasured are represented. Treatment Assignment feeds Treatment Compliance. Treatment Compliance feeds the Outcome, and the Confounders feed both Treatment Compliance and the Outcome variables. ](https://static-assets.codecademy.com/Courses/causal-inference/rdd-and-iv/iv-e3-narr.svg)

When the instrument AND treatment are both binary variables, we can define four types of "compliers":

 1. _Always takers_: takes the treatment regardless of treatment assignment.
 2. _Never takers_: never takes the treatment regardless of treatment assignment.
 3. _Compliers_: takes the assigned treatment.
 4. _Defiers_: takes the opposite of the assigned treatment.

In the context of the recycling program example, the four types of compliers would be defined as follows:

<table class = "table table-condensed">
 <thead>
 <tr class="header">
 <th></th><th>Instrument Value</th>
 </tr>
 <tr>
 <th></th><th>&#8804; 5 miles</th><th>&#062; 5 miles</th>
 </tr>
 </thead>
 <tbody>
 <tr>
 <td>Always takers</td> <td>Rebate</td><td>Rebate</td>
 </tr>
 <tr>
 <td>Never takers</td> <td>Curbside</td><td>Curbside</td>
 </tr>
 <tr>
 <td>Compliers</td><td>Rebate</td><td>Curbside</td>
 </tr>
 <tr>
 <td>Defiers</td><td>Curbside</td><td>Rebate</td>
 </tr>
 </tbody>
</table>

The issue of compliance is related to the fundamental problem of causal inference, which states that we can never observe both potential outcomes. In an IV estimation, we can never observe both potential treatments received for an individual. Thus, we cannot know with certainty what kind of complier an individual is.

Fortunately, we can get around this issue by making additional assumptions about our sample.