Two-Stage Least Squares (2SLS) regression is a powerful econometric tool used when you suspect a problem of endogeneity—meaning one or more of your independent variables are correlated with the error term. This issue, common in non-experimental data (e.g., economics, finance, or social sciences), violates a core assumption of Ordinary Least Squares (OLS) and leads to biased and inconsistent coefficient estimates.
2SLS solves this by using instrumental variables (IVs) to isolate the exogenous variation in the endogenous variable. Interpreting the results, however, requires careful attention to both the statistical output and the validity of your instruments.
Step 1: Confirming the Need for 2SLS (Endogeneity Test)
Before interpreting the 2SLS results, you should statistically confirm that OLS would indeed be problematic.
- The Test: Use a test like the Hausman Test or a regression-based equivalent (like a Durbin-Wu-Hausman test).
- Interpretation:
- Null Hypothesis (): There is no systematic difference between the OLS and 2SLS coefficients (i.e., OLS is consistent and endogeneity is not a problem).
- If you reject (typically when the p-value is below ), you confirm that endogeneity is present. This justifies the use of 2SLS, and its coefficients are the correct ones to interpret. If you fail to reject, you should generally revert to the simpler OLS estimates.
Step 2: Validating the Instrumental Variables (IVs)
The quality of your instruments is paramount. If your IVs are weak or invalid, the 2SLS estimates will be just as, or even more, biased than OLS.
A. The First Stage: Relevance (Weak Instruments Test)
The first stage of 2SLS regresses the endogenous variable on all exogenous variables and the instruments.
- Requirement: The instruments must be highly correlated with the endogenous variable.
- Test: Examine the F-statistic on the instruments in the first-stage regression (often called the “First-Stage F-Statistic”).
- Interpretation:
- A common rule of thumb, introduced by Staiger and Stock, suggests an F-statistic of 10 or higher indicates the instruments are sufficiently relevant (not weak).
- If the F-statistic is low (e.g., below 10), the instruments are weak, and the 2SLS coefficients and standard errors will be unreliable.
B. The Second Requirement: Exogeneity (Over-Identification Test)
The instruments must be uncorrelated with the second-stage error term. This means they should only affect the dependent variable through the endogenous variable.
- The Test: If you have more instruments than endogenous variables (an over-identified model), you can perform a test like the Sargan or Hansen J-Test.
- Interpretation:
- Null Hypothesis (): The instruments are valid (i.e., uncorrelated with the error term).
- If you fail to reject (p-value is above), your instruments are considered valid, which is the desired result.
- If you reject (p-value below), at least one of your instruments is invalid, and your 2SLS results are likely biased.
Step 3: Interpreting the Second-Stage Coefficients
Assuming you have justified the use of 2SLS and validated your instruments, you can now interpret the main results.
The second stage of 2SLS is structurally identical to an OLS regression, except that the problematic endogenous variable is replaced by the fitted values () from the first stage.
1. Coefficient Magnitude
The primary coefficient of interest is the one on the endogenous variable (now).
- The Interpretation: The coefficient represents the causal effect of the endogenous variable on the dependent variable, holding all other variables constant.
- Example: If the dependent variable is income (in thousands of dollars) and the 2SLS coefficient on the endogenous variable (education in years) is, you would interpret this as: “A one-year increase in education causes an average increase in income, after accounting for endogeneity.”
2. Significance (P-Values and T-Statistics)
Check the p-values or t-statistics for your coefficients to determine if the relationship is statistically significant.
- The Interpretation: The standard interpretation applies: if the p-value is below your chosen significance level (e.g.,), you conclude that the coefficient is statistically different from zero.
3. Comparing to OLS
It is highly informative to compare your 2SLS coefficient for the endogenous variable to the coefficient estimated by OLS.
- If 2SLS OLS: The original OLS estimate likely suffered from negative bias (e.g., omitted variable bias was pulling the coefficient toward zero).
- If 2SLS OLS: The original OLS estimate likely suffered from positive bias (e.g., measurement error or reverse causality was inflating the coefficient).
The difference between the two confirms the magnitude of the bias introduced by endogeneity and highlights the necessity of using 2SLS. The 2SLS estimate is the consistent and reliable causal effect.
