Econometrics 25117 Lecture 9: Instrumental Variables

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Questions and Answers

What does the parameter β1 represent in the demand equation for butter?

The quantity demanded of butter

The total demand for butter

The price-elasticity of butter (correct)

The supply elasticity of butter

In the Two Stages Least Squares (2SLS) method, what is the relationship between an exogenous change in Z and its effect on Y?

Y will decrease for any change in Z

The effect on Y is given by the ratio γ1 / π1 (correct)

It has no effect on Y

It leads to an increase in Y by the amount of π1

What issue does the OLS regression of ln(Qibutter) on ln(Pibutter) face?

Overfitting the model

Simultaneity (reverse causality) bias (correct)

Inconsistency in price measurement

Lack of sufficient data

What role does Z play in the 2SLS context as described?

It acts as an instrument that is exogenous to the error terms Signup and view all the answers

Who developed IV regressions to estimate supply and demand elasticities for agricultural goods?

Philip G. Wright Signup and view all the answers

Which condition must an instrumental variable Z satisfy to be considered valid?

Z must be uncorrelated with the error term u Signup and view all the answers

What is the first step in the Two Stages Least Squares (2SLS) estimation process?

Obtain predicted values X̂i by regressing Xi on Zi Signup and view all the answers

Which formula represents the reduced-form estimator, β1RF?

cov(Zi, Yi) / var(Zi) Signup and view all the answers

What does the second stage of the 2SLS regression involve?

Regressing Yi on the predicted values X̂i Signup and view all the answers

What is the condition termed 'instrument relevance' in relation to Z?

Z must strongly influence X (corr(Z, X) ≠ 0) Signup and view all the answers

What is the primary goal of using an instrumental variable in regression analysis?

To provide bias adjustments in the estimation of β1 Signup and view all the answers

How is the IV estimator viewed in terms of estimation consistency?

It is a consistent estimator of β1 Signup and view all the answers

What indicates that Z is an appropriate instrument for estimating the effect of X on Y?

Z is uncorrelated with u Signup and view all the answers

What is the requirement for a model to be considered underidentified in the context of the General IV Regression Model?

There are more regressors than instruments. Signup and view all the answers

What must be done in the first stage of 2SLS with a single endogenous regressor?

Regress X1 on all exogenous regressors and instruments. Signup and view all the answers

Why are the standard errors from the second stage regression in 2SLS considered incorrect?

They do not account for the estimation in the first stage. Signup and view all the answers

What does it mean for a model to be exactly identified?

As many instruments as regressors. Signup and view all the answers

For the correct computation of standard errors in a 2SLS regression, what is recommended?

Using heteroskedasticity-robust standard errors. Signup and view all the answers

What is one of the main characteristics of an overidentified model?

Less regressors than instruments. Signup and view all the answers

What is the sequence followed in the second stage of the 2SLS method?

Predict values of X1 and then regress Y on those. Signup and view all the answers

In the context of the General IV Regression Model, what must be true about the coefficients for them to be considered underidentified?

There must be more regressors than instruments. Signup and view all the answers

What determines if an instrument is considered relevant in the first-stage regression?

At least one of π1, ..., πm is non-zero Signup and view all the answers

In the context of 2SLS, what is indicated if all π1, ..., πm are either zero or nearly zero?

The instruments are considered weak. Signup and view all the answers

What does the homoskedasticity-only F-statistic test for in the context of 2SLS?

The null hypothesis that the instruments are exogenous Signup and view all the answers

What distribution does the test statistic J follow under the null hypothesis when ei is homoskedastic?

Chi-squared distribution Signup and view all the answers

What impact do weak instruments have on the 2SLS sampling distribution?

They may distort the normality of the distribution. Signup and view all the answers

In what scenario can causal inference from 2SLS be drawn confidently?

When at least one instrument is strong. Signup and view all the answers

What does the notation J = mF represent in the context of 2SLS?

The test statistic for overidentifying restrictions. Signup and view all the answers

What is indicated if cov(Zi, Xi) approaches zero when instruments are weak?

Statistical inference cannot be conducted reliably. Signup and view all the answers

What is the F-statistic threshold proposed by Staiger and Stock for rejecting weak instruments?

F ≥ 10 Signup and view all the answers

When using Stock and Yogo's method, what does the notation J10(m) represent?

A specific F-statistic threshold Signup and view all the answers

What should a researcher do if all instruments are weak?

Switch to a robust method that can handle weak instruments Signup and view all the answers

What does it mean for an instrument to be valid in IV regression?

It isolates variation in X that is uncorrelated with u Signup and view all the answers

What is implied by the critical requirement of at least m valid instruments?

Researchers need to rely on sound judgment since this cannot be quantitatively tested. Signup and view all the answers

What is the relationship between dropping weak instruments and the first-stage F-statistic?

Dropping weak instruments increases the first-stage F-statistic. Signup and view all the answers

What is the importance of testing overidentifying restrictions in IV regression?

To validate the exogeneity of the instruments Signup and view all the answers

Which textbook is NOT referenced for weak instrument discussion?

Advanced Econometrics by Greene Signup and view all the answers

What is the implication of instrument exogeneity in the context of 2SLS estimation?

Instruments should not be correlated with the error term. Signup and view all the answers

Which condition must be met for the 2SLS estimator to be consistent?

There should be no correlation between instruments and the error term. Signup and view all the answers

What does the J-test of overidentifying restrictions help to assess?

It compares different 2SLS estimates from valid instruments. Signup and view all the answers

Which statement accurately describes the concept of instrument relevance?

Instruments provide sufficient information about exogenous movements in the endogenous variables. Signup and view all the answers

When testing for instrument exogeneity, what does it mean if two separate 2SLS estimates are very different?

One or both instruments may be invalid. Signup and view all the answers

What does it indicate if the error term in the model is correlated with the instruments?

The 2SLS estimates will be inconsistent. Signup and view all the answers

Which assumption is violated if large outliers are present in the data?

The variables have nonzero finite fourth moments. Signup and view all the answers

What is the consequence of having more instruments than endogenous regressors?

It allows for partial testing of instrument exogeneity. Signup and view all the answers

Study Notes

Lecture 9: An Introduction to Instrumental Variables

Lecture is for Econometrics 25117, Universitat Pompeu Fabra, November 18, 2024.
Statistical studies are evaluated by internal and external validity.
A study is internally valid if statistical inferences about causal effects are accurate for the population studied. Regression estimation of causal effects is threatened by correlated regressors and error terms, leading to biased and inconsistent OLS estimators. Incorrect standard errors also invalidate confidence intervals and hypothesis tests.
A study is externally valid if inferences and conclusions can be generalized from the studied population and setting to other populations and settings.

Instrumental Variables

OVB, simultaneous causality bias, and errors-in-variables bias cause E(u | X) ≠ 0 in a regression model.
Instrumental variables (IV) regression can eliminate bias if E(u | X) ≠ 0, by instrumenting with variable Z. Intuitively, IV splits variable X into two parts: exogenous part which is not correlated with error term u, and endogenous part correlated with u.
The part of X not correlated with an error term u is estimate for B1.

Instrumental Variables (IVs) Conditions

Instrument relevance: A strong determinant of X, (corr(Z, X) ≠ 0).
Instrument exogeneity (exclusion restriction): Unrelated to the error term (u), corr(Z, u) = 0.

Two-Stage Least Squares (2SLS)

To obtain estimator β1, first regress X on Z (including the intercept) to get predicted values X1.
Then regress Y on X1 (including the intercept), where coefficient on X1 is the 2SLS estimator β2SLS

A Buttery Example

IV regressions estimate supply and demand elasticities, starting in 1928.
Demand equation: ln(Qbutter) = β₀ + β₁ln(Pbutter) + u
Problem: Price (P) is determined by supply and demand, making OLS (Ordinary Least Squares) regression of ln(Qbutter) on ln(Pbutter) susceptible to simultaneity bias.
Equilibrium results from supply and demand intersection.
Price and Quantity are not informative about specific demand/supply elasticities.
Solutions: find instruments that shift supply curve only while demand curves remain constant.

A Potential IV Example

Drought shocks to dairy regions are a potential instrument. Below-average rainfall reduces butter production (changes supply) without directly influencing demand.

Large-Sample Properties

As sample size (n) gets very large, 2SLS estimator (β2SLS) distribution approaches normal distribution with mean β1 and variance σ2/n[Σ(cov(Zi,Xi)/var(Zi))]2
Standard errors (SE) from 2SLS need to account for the estimation of first stage regression. Employ specialized commands.

The General IV Regression Model

Extends the IV framework to multiple endogenous regressors, exogenous regressors/control variables, and exogenous instruments.
Underidentified: More regressors than instruments (m < k).
Exactly identified: Equivalent regressors & instruments (m = k)
Overidentified: More instruments than regressors (m > k).

2SLS with a Single Endogenous Regressor

Steps to implement a 2SLS with single endogenous variable regression
First Stage regresses X on all exogenous regressors and instruments. Compute predicted X1 values.
Second Stage regresses Y on X1 and exogenous regressors to arrive at 2SLS estimators.

Checking Assumption 4.1 – Instrument Exogeneity

Instrument exogeneity implies that instruments only influence Y through X, thereby not directly influencing the error term.
If instruments are correlated with the error term, then first stage of 2SLS cannot disentangle an X component that is unrelated to the error term, causing a 2SLS estimator to be inconsistent.
If there are more instruments than endogenous regressors, then partial testing for instrument exogeneity is possible.

Checking Assumption 4.2 – Instrument Relevance

Instruments are relevant if at least one of the coefficients in the first-stage regression is non-zero/ non-trivial.
Weak instruments are highly problematic in that they do not explain much of the variation in the endogenous regressor. With weak instruments, the sampling distribution is no longer normal.
Validating instrument relevance is vital; methods for formally evaluating instrument relevance are recommended.

Summary

A valid instrument isolates a portion of X that is independent of the error term 'u', permitting estimation of the causal effect of X on Y.
IV regression relies on valid instruments. Exogeneity & relevance conditions must be checked via J-statistic and first-stage F-statistic.

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Description

This lecture introduces instrumental variables in the context of econometrics, focusing on internal and external validity of statistical studies. It discusses the threats to causal inference and the role of instrumental variables in mitigating biases in regression models.