Economics 20: Instrumental Variables and 2SLS

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Questions and Answers

What is the primary reason for using Instrumental Variables estimation in a model?

  • To reduce the impact of multicollinearity
  • To solve the issue of omitted variable bias (correct)
  • To ensure that the errors in the model are normally distributed
  • To address the problem of heteroskedasticity

What is a necessary condition for a variable to be a valid Instrumental Variable?

  • The instrument must have a significant partial effect on the dependent variable
  • The instrument must be correlated with the dependent variable
  • The instrument must be a proxy for the unobserved variable
  • The instrument must be correlated with the endogenous variable (correct)

What is the difference between using an Instrumental Variable and a proxy variable in a model?

  • A proxy variable is used to address the omitted variable bias, while an Instrumental Variable is used to remove the unobserved variable from the error term
  • A proxy variable is used to remove the unobserved variable from the error term, while an Instrumental Variable is used to address the omitted variable bias (correct)
  • An Instrumental Variable is used to remove the unobserved variable from the error term, while a proxy variable is used to address the omitted variable bias
  • A proxy variable is used to address the errors-in-variables problem, while an Instrumental Variable is used to address the omitted variable bias

What is the purpose of the first-stage regression in Instrumental Variables estimation?

<p>To test if the Instrumental Variable is correlated with the endogenous variable (B)</p> Signup and view all the answers

What is an advantage of using Instrumental Variables estimation over using a proxy variable?

<p>IV estimation can provide a more accurate estimate of the coefficient of the endogenous variable (A)</p> Signup and view all the answers

Why can't we test if the Instrumental Variable is uncorrelated with the error term?

<p>Because it is a population assumption (C)</p> Signup and view all the answers

What is a characteristic of a valid Instrumental Variable?

<p>It is uncorrelated with the error term and has no partial effect on the dependent variable (C)</p> Signup and view all the answers

What is the purpose of using Instrumental Variables estimation in the context of omitted variable bias?

<p>To address the bias caused by the omitted variable (A)</p> Signup and view all the answers

What is the condition required for the instrument to provide consistent estimates of the parameters in the simple regression case?

<p>Cov(z,u) = 0 (C)</p> Signup and view all the answers

What is the expression for the IV estimator for β1 in the simple regression case?

<p>βˆ1 = ∑ (z i − z )( y i − y ) / ∑ (z i − z )( x i − x ) (A)</p> Signup and view all the answers

What is the assumption required to get estimates of the standard errors in IV estimation?

<p>E(u2|z) = σ2 (C)</p> Signup and view all the answers

What is the difference between the standard error in IV case and OLS?

<p>R2 from regressing z on x (C)</p> Signup and view all the answers

What is the expression for the asymptotic variance of β1-hat in IV estimation?

<p>Var (βˆ1 ) = σ2 / nσx2ρxz2 (D)</p> Signup and view all the answers

Why are IV standard errors larger than OLS?

<p>Because R2 &lt; 1 (B)</p> Signup and view all the answers

What is the estimated variance of β1-hat in IV estimation?

<p>Var (βˆ1 ) = σˆ2 / SSTxR2x,z (C)</p> Signup and view all the answers

What is the distribution of the IV estimator in large samples?

<p>Normal distribution (C)</p> Signup and view all the answers

What happens to the IV standard errors as the correlation between z and x increases?

<p>They decrease (C)</p> Signup and view all the answers

What is the condition required for the IV estimator to be the same as the OLS estimator?

<p>z = x (D)</p> Signup and view all the answers

What is the condition under which IV is preferred over OLS?

<p>If Corr(z,u)/Corr(z,x) &lt; Corr(x,u) (A)</p> Signup and view all the answers

What happens to the inconsistency in IV estimation if Corr(z,u) is small but Corr(z,x) is also very small?

<p>It increases (A)</p> Signup and view all the answers

What is the population variance of x denoted by in the expression for the asymptotic variance of β1-hat?

<p>σx2 (A)</p> Signup and view all the answers

What is the problem with using R2 in IV estimation?

<p>R2 can be negative (A)</p> Signup and view all the answers

Why is it not important to use R2 in IV estimation?

<p>Because we prefer consistent estimates of the coefficients (C)</p> Signup and view all the answers

What is the best approach when it comes to dealing with poor instruments?

<p>Test for correlation in the first stage regression (C)</p> Signup and view all the answers

What is the primary reason why we cannot use z1 as an instrument for y2 in the multiple regression case?

<p>z1 appears in the structural model (A)</p> Signup and view all the answers

What is the condition required for an instrument z2 to be valid in the multiple regression case?

<p>z2 is correlated with y2 and uncorrelated with u1 (D)</p> Signup and view all the answers

What is the advantage of using Two Stage Least Squares (2SLS) in the multiple regression case?

<p>It allows for the use of multiple instruments (D)</p> Signup and view all the answers

What is the best instrument for y2 in the multiple regression case?

<p>A linear combination of the exogenous variables (D)</p> Signup and view all the answers

What is the condition required for the model to be identified in the multiple regression case?

<p>z1 is uncorrelated with u1 and y2 is correlated with z2 (A)</p> Signup and view all the answers

What is the purpose of the reduced form equation in the multiple regression case?

<p>To partial out the effect of z1 on y2 (B)</p> Signup and view all the answers

What is the implication of having multiple instruments in the multiple regression case?

<p>Any linear combination of the instruments is also a valid instrument (B)</p> Signup and view all the answers

What is the relationship between y2 and y2* in the multiple regression case?

<p>y2 = y2* + v2 (D)</p> Signup and view all the answers

What is the primary goal of testing for endogeneity in OLS regression?

<p>To ensure the consistency of OLS estimates (D)</p> Signup and view all the answers

In the Hausman test, what is the null hypothesis?

<p>y2 is exogenous (B)</p> Signup and view all the answers

What is the assumption required for the z's in the first stage equation of the regression test?

<p>They are uncorrelated with the error term (A)</p> Signup and view all the answers

What is the consequence of finding statistically significant differences in the coefficients in the Hausman test?

<p>We conclude that y2 is endogenous (A)</p> Signup and view all the answers

What is required to test the overidentifying restrictions in the instrumental variable estimation?

<p>Multiple instruments for the endogenous variable (A)</p> Signup and view all the answers

What is the purpose of saving the residuals from the first stage equation in the regression test?

<p>To include them in the structural equation (C)</p> Signup and view all the answers

What is the null hypothesis in the regression test for endogeneity?

<p>The coefficient on the residual is statistically not different from zero (A)</p> Signup and view all the answers

Why can't we test whether an instrument is uncorrelated with the error term if we only have one instrument for the endogenous variable?

<p>Because the error term is unobserved (D)</p> Signup and view all the answers

Flashcards

Instrumental Variables (IV)

A method used when variables in a model are endogenous (correlated with the error term).

Problems IV Addresses

Omitted variable bias and errors-in-variables.

Instrumental Variable

A variable used in IV estimation that is correlated with the endogenous variable but not with the error term.

Exogeneity Condition

The instrument must not directly affect the outcome variable (y) and uncorrelated with the error term 'u'.

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Correlation Condition

The instrument must be correlated with the endogenous variable.

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IV vs. Proxy Variables

IV leaves the unobserved variable in the error term, while proxy variables attempt to remove it.

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Testing Validity of Instruments

We rely on common sense and economic theory.

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IV Estimator Formula

The IV estimator is βˆ1 = ∑(zi−z)(yi−y) / ∑(zi−z)(xi−x).

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Distribution of IV Estimator

The IV estimator has an approximate normal distribution in large samples.

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Homoskedasticity in IV

We need to assume E(u2|z) = σ2 = Var(u), conditioning on z.

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IV vs. OLS Standard Errors

IV standard errors are larger due to the weaker correlation between 'z' and 'x'.

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Effect of Invalid Instrument

If Cov(z,u) = 0 is false, the IV estimator will be inconsistent.

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Two-Stage Least Squares (2SLS)

Two-Stage Least Squares can be used with multiple instruments.

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Choosing the Best Instrument

The best instrument is the one that is most highly correlated with y2.

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Reduced Form Equation

y2 = π0 + π1 z1 + π2 z2 + π3 z3 + v2.

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Hausman Test

It tests whether an explanatory variable is actually endogenous.

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Regression Test (Endogeneity)

Include the residual from the first stage equation in the structural equation.

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Testing Overidentifying Restrictions

Used when we have multiple instruments to check if some are correlated with the error.

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Endogeneity

When standard OLS assumptions are violated, leading to biased or inconsistent estimates.

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What is a valid IV?

A variable related to the explanatory variable but unaffected by outcome.

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Causal effect in IV

IV estimates the causal effect of a treatment or policy.

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Strategy for finding IV

Find an instrument that strongly predicts treatment and is unrelated to confounders.

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Drawbacks of IV

IV relies on strong assumptions; violations can cause more bias than OLS.

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How to Verify IV

We examine the relationship between the instrument and both explanatory and outcome variables.

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Use cases for IV

Used to fix multicollinearity and measurement error in X.

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Data Needed to Fit an IV Model

Requires data on instruments, explanatory variable, observed outcomes.

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Two stages of 2SLS

First-stage: regress endogenous variable on instrument to get predicted values. Second-stage: use predicted values as instrument for endogenous variable in the structural equation.

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The first-stage regression R-squared.

In the first stage, the r-squared must be high.

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How Does an Instrument Work?

The instrument is measured in the independent variable.

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Be like the journals

In real application of IV, you need to read journals and follow the theory of application.

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Study Notes

Instrumental Variables (IV)

  • IV estimation is used when there are endogenous variables in the model, i.e., Cov(x,u) ≠ 0.
  • IV can be used to address the problem of omitted variable bias and errors-in-variables problem.

What is an Instrumental Variable?

  • For a variable z to be a valid instrument for x, it must be:
    • Exogenous: Cov(z,u) = 0, meaning z has no partial effect on y and is uncorrelated with u.
    • Correlated with the endogenous variable x: Cov(z,x) ≠ 0.

Difference between IV and Proxy

  • IV estimation leaves the unobserved variable in the error term, but uses an estimation method that recognizes the presence of the omitted variable.
  • Proxy variables, on the other hand, try to remove the unobserved variable from the error term.

Valid Instruments

  • We can't test if Cov(z,u) = 0, as this is a population assumption.
  • We rely on common sense and economic theory to decide if an instrument is valid.
  • We can test if Cov(z,x) ≠ 0 using a random sample.

IV Estimation in the Simple Regression Case

  • If we have a valid instrument, we can get consistent estimates of the parameters.
  • The IV estimator for β1 is βˆ1 = ∑(zi−z)(yi−y) / ∑(zi−z)(xi−x).

Inference with IV Estimation

  • The IV estimator has an approximate normal distribution in large samples.
  • To get estimates of the standard errors, we need a slightly different homoskedasticity assumption: E(u2|z) = σ2 = Var(u), conditioning on z.

IV versus OLS Estimation

  • The standard error in IV case differs from OLS only in the R2 from regressing x on z.
  • IV standard errors are larger due to the weaker correlation between z and x.

The Effect of Poor Instruments

  • If the assumption that Cov(z,u) = 0 is false, the IV estimator will be inconsistent.
  • The asymptotic bias in IV can be compared to that in OLS.

Two-Stage Least Squares (2SLS)

  • 2SLS can be used with multiple instruments.
  • The best instrument is the one that is most highly correlated with y2.
  • The reduced form equation is: y2 = π0 + π1 z1 + π2 z2 + π3 z3 + v2.

Testing for Endogeneity

  • The Hausman test can be used to determine if y2 is endogenous.
  • The regression test can also be used by including the residual from the first stage equation in the structural equation.

Testing Overidentifying Restrictions

  • If we have multiple instruments, we can test the overidentifying restrictions to see if some of the instruments are correlated with the error.

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