Explain instrumental variables with an example

Question

Explain instrumental variables in simple terms and provide a concrete example.

Codemia · Accepted Answer

The analytical question we are addressing is: "How can we determine the causal impact of a new feature on ride frequency for Lyft users?" To answer this, we need to collect data on user ride frequency before and after the feature’s implementation, as well as demographic information about the users (age, location, ride preferences). The central challenge is that simply observing increased ride frequency after the feature launch does not imply causation, as other factors (e.g., seasonality, promotions) could also influence this change.

To address the confounding variables affecting our analysis, we will use instrumental variables (IV). An instrumental variable is a third variable that is correlated with the independent variable (the new feature) but not directly with the dependent variable (ride frequency), except through its correlation with the independent variable. In this case, let's use 'user engagement score' (a derived metric from app usage patterns) as our instrumental variable. We will employ Two-Stage Least Squares (2SLS) regression. In the first stage, we will regress the new feature adoption on the user engagement score. In the second stage, we will regress ride frequency on the predicted values of feature adoption from the first stage.

Assume we have the following data for 100 Lyft users:

- **User Engagement Score (0-10)**: [2, 3, 4, 5, 6, 7, 8, 9, 10] (repeated across 10 users each)
- **Feature Adoption (1 = adopted, 0 = not adopted)**: [0, 0, 1, 1, 1, 1, 1, 1, 1]
- **Ride Frequency (trips per month)**: [5, 6, 8, 10, 12, 15, 20, 22, 25]

In the first stage, we would run a regression of feature adoption on user engagement score. Let's say the regression output indicates a coefficient of 0.5 indicating that for every point increase in engagement score, the probability of feature adoption increases by 0.5.

In the second stage, we would use the predicted values of feature adoption to regress on ride frequency, yielding a coefficient of 3. This means that adopting the new feature increases the ride frequency by approximately 3 rides per month, after accounting for the engagement score.

The result of our analysis indicates that the new feature leads to an increase in ride frequency by 3 rides per month when controlled for user engagement. This suggests that the feature is likely a positive contributor to user behavior and that the initial correlation observed between feature adoption and ride frequency can be interpreted as causal. However, it is crucial to recognize that our findings are specific to the users in our sample, and further validation may be necessary across different user segments.

Common mistakes in this type of analysis include:
- **Choosing a weak instrumental variable**: The instrumental variable must be strongly correlated with the independent variable (feature adoption). If the correlation is weak, the results will be biased.
- **Ignoring confounding variables**: Even with IV, it’s essential to ensure that other confounding factors (e.g., external promotions or market changes) are accounted for. Failing to do so can lead to misleading conclusions.
- **Misinterpreting the results**: It’s important to remember that while IV can suggest causation, it does not eliminate all biases. The analysis should be interpreted in the context of the data, and not as absolute proof of causation.

Explain instrumental variables with an example

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