Econometrics 25117 Lecture 9: Instrumental Variables

An instrument variable (IV) is a variable used to estimate the causal effect of one variable on another in the presence of endogeneity. It satisfies two conditions: relevance and exogeneity.

The relevance condition implies that the instrument is a strong predictor of the endogenous variable. In other words, the instrument has a significant correlation with the independent variable.

The exogeneity condition requires that the instrument is not correlated with the error term in the regression model. This ensures that the instrument's influence on the outcome variable is only through its effect on the independent variable.

Two-Stage Least Squares (2SLS) is a statistical technique to estimate the causal effect of a variable when the independent variable is endogenous. It involves two stages: regressing the independent variable on the instrument variable and then regressing the dependent variable on the predicted values from the first stage.

The first stage of 2SLS involves regressing the endogenous independent variable (X) on the instrument variable (Z) and obtaining predicted values for X. This stage is effectively using the instrument to isolate the variation in X that is not correlated with the error term.

The second stage of 2SLS involves regressing the dependent variable (Y) on the predicted values of X obtained in the first stage. This stage is essentially using the instrument-purified variation in X to estimate the causal effect on Y.

The reduced-form estimator is a way to estimate the effect of the instrument variable (Z) on the outcome variable (Y) without directly estimating the effect of the independent variable (X) on Y. This estimator is used in the context of instrumental variables.

The IV estimator is a consistent estimator of β1. It is obtained by replacing population covariances with sample covariances in the reduced-form estimator. This estimator is a crucial tool in estimating causal effects in the presence of endogeneity.

The variable that is being affected by the instrument and is correlated with the error term.

The standard errors from the second stage regression of 2SLS are not reliable because they don't account for the estimation in the first stage. Using a single command in regression software allows for calculation of the correct standard errors.

The variable that is being explained in the regression model.

We use the first-stage F-statistic to test that our instrument is relevant to the endogenous variable. The F-statistic should be greater than 10 to reject the null hypothesis of weak instruments.

The J-statistic tests whether the exclusion restriction holds. The null hypothesis is that the instrument is exogenous, meaning it doesn't affect the dependent variable directly.

The IV-estimated coefficient is only accurate when all instruments are valid. Exogeneity (J-test) and relevance (F-test) require careful scrutiny to ensure the instrument's validity.

When instruments are found to be weak, we can improve our estimation by seeking stronger instruments, dropping weak instruments in multi-instrument settings, or switching to methods robust to weak instruments.

The reduced-form estimator directly estimates the effect of the instrument on the dependent variable. It is a crucial step for understanding how the instrument influences the outcome.

Testing the assumption that the instruments are not correlated with the error term in the regression model. It involves checking if the instruments have a significant influence on the residual term of the regression, independent of the endogenous variable.

The process of estimating the residuals using the two-stage least squares (2SLS) method, which involves initial estimation of the endogenous variable and then incorporating those estimates into the final regression model.

A statistical test designed to assess the validity of the chosen instruments. It tests the null hypothesis that all the instruments are exogenous, meaning they don't have a direct effect on the dependent variable beyond their influence on the endogenous variable.

The degree of overidentification refers to the number of instruments used in the regression model that exceed the number of endogenous variables. It reflects the extra information provided by the instruments, which can be useful for testing hypotheses about the model.

A regression model used to estimate the relationship between the residuals from the first stage and the instruments, controlling for the predetermined variables. It aims to isolate the relationship between the residuals and the instruments themselves, providing valuable insights into the validity of the chosen instruments.

Indicative of a scenario where the instruments are not sufficiently strong predictors of the endogenous variable. It signifies that the chosen instruments provide limited explanatory power for the endogenous variable, undermining the efficacy of two-stage least squares (2SLS) estimation.

Visual illustration showing a strong instrument (Z) explaining a significant portion of the variation in the endogenous variable (X). The relationship between Z and X is clear and substantial, facilitating accurate estimation.

A key assumption for 2SLS estimator validity, stating that all instruments are uncorrelated with the error term in the model. In other words, instruments only influence the outcome variable through their impact on the endogenous variable.

If the instruments are correlated with the error term, the instrumental variable technique fails to isolate a component of the endogenous variable that is uncorrelated with the error, leading to biased and inconsistent estimates.

The use of more instruments than endogenous regressors allows for a potential test of instrument exogeneity. By comparing 2SLS estimates using different sets of instruments, significant discrepancies point to the potential invalidity of some instruments.

A statistical test to assess instrument exogeneity when there are more instruments than endogenous variables, comparing 2SLS estimates in a statistically precise manner.

In the context of 2SLS, if the instruments do not provide enough information about the exogenous movements in the endogenous variable, it is impossible to disentangle the effects of different variables on the outcome.

The predicted values of the endogenous variable obtained from the first stage of 2SLS using the instrument, which are then used as the regressor in the second stage to estimate the causal effect.

The Two-Stage Least Squares (2SLS) method is an estimation technique used to address endogeneity issues, where the independent variable is correlated with the error term in the regression model.

The reduced-form equation shows how the instrument variable (Z) affects the dependent variable (Y) directly, without considering the endogenous variable (X).

The first stage of 2SLS involves regressing the endogenous variable (X) on the instrument (Z) to obtain predicted values for X. This helps isolate the variation in X that is not correlated with the error term.

The second stage of 2SLS involves regressing the dependent variable (Y) on the predicted values of X from the first stage. This allows us to estimate the causal effect of X on Y without the influence of the error term.

The IV estimator, derived using sample covariances, is a consistent estimate of β1, the effect of X on Y, after dealing with endogeneity.