Two Steps from Hell - Two-Stage Least Squares

In this article, we delve into a method used in econometrics and statistics to estimate causal relationships: the Two-Stage Least Squares (2SLS) approach. This is a popular method used to address issues of endogeneity in regression models.

Two-Stage Least Squares: An Overview

The Two-Stage Least Squares (2SLS) is a form of instrumental variable (IV) estimation used to estimate the causal effect of an explanatory variable on the dependent variable when the explanatory variable is potentially endogenous, i.e., correlated with the error term.

Assumptions

The 2SLS method relies on several key assumptions:

1. Relevance: The instrument is correlated with the endogenous explanatory variable.

2. Exogeneity: The instrument is not correlated with the error term in the main equation.

Method

The 2SLS involves two steps:

1. First, the endogenous explanatory variable is regressed on all exogenous variables in the model, including the instruments.

2. Second, the predicted values from the first stage are used to replace the endogenous explanatory variable in the main equation.

Pros and Cons

The 2SLS has its limitations. It requires a valid instrument, which is not always available. It also relies heavily on the assumptions of relevance and exogeneity.

However, the 2SLS provides a method to estimate causal effects in the presence of endogeneity, which is a common issue in econometrics. It's widely used in economics, political science, and other fields.

Two-Stage Least Squares Explained with Math

The 2SLS method involves the following two regression models:

1. First Stage:

   $$ X = AZ + u $$

   where $ X $ is the endogenous variable, $Z$ is the instrument, $A$ is the parameter to be estimated, and $u$ is the error term.

2. Second Stage:

   $$Y = B\hat{X} + e$$

   where $Y$ is the dependent variable, $\hat{X}$ is the predicted values from the first stage, $B$ is the parameter to be estimated, and $e$ is the error term.

The indirect effect of $Z$ on $Y$ through $X$ is given by $BA$.

Relation between Two-Stage Least Squares and Other Causal Inference Methods

The 2SLS is a type of instrumental variable (IV) estimation. It's used when the explanatory variable is endogenous, i.e., correlated with the error term. The 2SLS, along with other IV methods, provides a way to obtain consistent and unbiased estimates of the causal effect in such situations.

Regression Discontinuity Design (RDD) and Regression Kink Design (RKD) are other causal inference methods that can be used to estimate local treatment effects at a certain cutoff or kink point. However, these methods require a clear cutoff or kink point, which is not always available.

In contrast, the 2SLS can be used in a broader set of contexts, as it doesn't require a specific cutoff or kink point, but instead requires a valid instrument. The 2SLS and RDD/RKD can be seen as complementary tools in the causal inference toolbox, with each method being more suitable for certain types of research questions and data situations.

Conclusion

The Two-Stage Least Squares method provides a valuable tool for estimating causal effects in the presence of endogeneity. It's widely used in economics, political science, and other fields. While it comes with its own set of assumptions and limitations, when used correctly, the 2SLS can provide important insights into causal relationships.