Explain instrumental variables with an example

Question

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

Codemia · Accepted Answer

In the context of Postmates, suppose we want to analyze the effect of a marketing campaign on order volume. We hypothesize that the campaign increases order volume (H1), while the null hypothesis (H0) is that the campaign has no effect. However, we suspect that there might be an unobserved variable, such as seasonal demand, that influences both the campaign effectiveness and order volume, leading to potential confounding. To address this, we can use instrumental variables (IV) to isolate the effect of the marketing campaign on order volume.

1. **Identify an Instrumental Variable**: We need a variable that is correlated with the marketing campaign but not directly with the order volume, except through the campaign. For instance, let’s consider the amount spent on social media ads as our instrument.  
2. **Check Relevance**: We verify that the social media ad spend affects the campaign's reach.  
3. **Check Exogeneity**: We ensure that the social media ad spend does not directly affect the order volume outside its influence on the campaign.  
4. **Two-Stage Least Squares (2SLS)**: We perform 2SLS regression. In the first stage, we regress the campaign variable on the instrument (social media ad spend). In the second stage, we regress order volume on the predicted values from the first stage.

Suppose we have the following data for a month:  
- Campaign reach (in thousands): [10, 20, 30, 40, 50]  
- Social media ad spend (in thousands): [2, 4, 6, 8, 10]  
- Order volume (in units): [100, 150, 200, 250, 300]  
1. First, we regress campaign reach on social media ad spend and find a strong correlation (R² = 0.9).  
2. Next, we use the predicted values from this regression in a second regression with order volume as the dependent variable.  
3. Suppose the result shows a coefficient of 5 for the predicted campaign reach. This indicates that for every additional thousand reach from the campaign, order volume increases by 5 units.

The final analysis suggests that the marketing campaign has a statistically significant effect on order volume, as indicated by the positive coefficient of 5. However, we must be cautious in interpreting this as practical significance. While the marketing campaign does increase orders, we should assess the cost-effectiveness of the campaign relative to the increase in order volume and profits. We should also consider the confidence intervals around our estimates and potential biases that may still be present.

The use of instrumental variables connects to broader statistical concepts such as confounding factors and causal inference. Additionally, this approach relates to hypothesis testing techniques, where we evaluate the significance of our findings (p-values) and how to adjust for multiple testing if we test various marketing strategies concurrently. In future discussions, the interviewer may ask about other techniques to address confounding, such as propensity score matching or regression discontinuity, as well as the implications of sample size on our power analysis.

Explain instrumental variables with an example

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