Sum the Magnitude of an Effect from 2SLS

[dolcampb]

In assessing the aggregate effect of the China shock, Acemoglu, Autor, Dorn, Hanson and Price, the authors use the increase in import penetration in a few other countries as an IV to predict import penetration increases in the US, and thus US job losses.

To get jobs lost in one sector, of course, they multiply the beta from the 2nd stage by the Instrument (page S160). And then they sum. Interestingly, they then multiply by the first-stage R-squared.

My question is, is it necessary to multiply the magnitude in an IV regression by the first stage R-square? Is it only necessary when summing the results across sectors, or also when interpreting the magnitude of an individual beta? (The answer to the latter question seems like clearly not.) If so, why?

[dolcampb]

So, I am answering my own question. From Perplexity:

In instrumental variable (IV) regression, the first-stage R-squared measures the proportion of variation in the endogenous variable (e.g., import penetration) explained by the instrument (e.g., import shocks in other countries). When aggregating results across sectors, multiplying by the first-stage R-squared adjusts for the fact that the instrument only captures a fraction of the endogenous variable’s variation. Here’s why: 1. Interpreting Individual Coefficients

• The second-stage coefficient (β) from IV regression already represents the causal effect of the endogenous variable on the outcome, conditional on the instrument’s strength.
• No adjustment by R-squared is needed for individual coefficients, as β inherently reflects the local average treatment effect (LATE) identified by the instrument14. The R-squared does not alter the magnitude of β itself but indicates the instrument’s explanatory power.

2. Aggregating Across Sectors

• When summing effects (e.g., total job losses), the raw second-stage coefficient multiplied by the instrument’s value assumes the instrument explains all variation in the endogenous variable. This overstates the effect if the instrument is weak (low R-squared).
• Multiplying by the first-stage R-squared scales the aggregate effect to account for the proportion of variation in the endogenous variable actually explained by the instrument. For example:
  • If the first-stage R² = 0.3, only 30% of the endogenous variable’s variation is instrument-driven. Summing β × instrument × R² ensures the total reflects only this exogenous variation35.

3. Why It Matters

• Avoiding Overestimation: Without R-squared adjustment, the aggregate effect would incorrectly treat the instrument as explaining 100% of the endogenous variation, leading to inflated estimates5.
• Heterogeneity in Instrument Strength: If sectors vary in how well the instrument explains their endogenous variables (heterogeneous R²), sector-specific adjustments improve accuracy5.

Example from the Study

The authors multiply β × instrument × R² when summing across sectors to ensure their aggregate job-loss estimate reflects only the portion of import penetration changes explained by the instrument. This guards against attributing non-instrument-driven variation to the causal effect5. Key Takeaway

• Individual β: No R-squared adjustment needed (β is already conditional on the instrument’s strength).
• Aggregate effects: Multiply by R-squared to isolate the instrument-driven component of the endogenous variable’s variation345.