Lecture 19: Instrumental Variables

Economics 326 — Introduction to Econometrics II

Author

Vadim Marmer, UBC

Endogeneity

In the linear regression model, Y_{i}=\beta _{0}+\beta _{1}X_{i}+U_{i}, the condition for consistent estimation of \beta _{1} by OLS is that X is exogenous: \mathrm{Cov}\left(X_{i},U_{i}\right) =0.
When \mathrm{Cov}\left(X_{i},U_{i}\right) \neq 0, we say that the regressor X is endogenous.
When the regressor is endogenous, the OLS estimator is inconsistent: \begin{align*} \hat{\beta}_{1,n}-\beta _{1} &= \frac{\frac{1}{n}\sum_{i=1}^{n}\left( X_{i}-\bar{X}_{n}\right) U_{i}}{\frac{1}{n}\sum_{i=1}^{n}\left( X_{i}-\bar{X}_{n}\right) ^{2}} \\ &\rightarrow _{p}\frac{{\color{red}\mathrm{Cov}\left(X_{i},U_{i}\right)}}{\mathrm{Var}\left(X_{i}\right)} \neq 0. \end{align*}

Consequences of endogeneity

The causal effect of X on Y is not estimated consistently: \hat{\beta}_{1,n}\rightarrow _{p}\beta _{1}+\frac{\mathrm{Cov}\left(X_{i},U_{i}\right)}{\mathrm{Var}\left(X_{i}\right)}. The effect can be over- or underestimated depending on the sign of \mathrm{Cov}\left(X_{i},U_{i}\right).
Tests and confidence intervals are invalid.

Sources of endogeneity

Several possible sources of endogeneity:
1. Omitted explanatory variables.
2. Simultaneity.
3. Errors in variables.
All result in regressors correlated with the errors.

Omitted explanatory variables

Suppose that the true model is \ln \text{Wage}_{i}=\beta _{0}+\beta _{1}\text{Education}_{i}+\beta _{2}\text{Ability}_{i}+V_{i}, where V_{i} is uncorrelated with \text{Education} and \text{Ability}.
Since \text{Ability} is unobservable, the econometrician regresses \ln \text{Wage} against \text{Education}, and \beta _{2}\text{Ability} goes into the error: \begin{aligned} \ln \text{Wage}_{i} &= \beta _{0}+\beta _{1}\text{Education}_{i}+U_{i}, \\ U_{i} &= \beta _{2}\text{Ability}_{i}+V_{i}. \end{aligned}
\text{Education} is correlated with \text{Ability}: we can expect that \mathrm{Cov}\left(\text{Education}_{i},\text{Ability}_{i}\right) >0, \beta _{2}>0, and therefore \mathrm{Cov}\left(\text{Education}_{i},U_{i}\right) >0. Thus, OLS will overestimate the return to education.

Simultaneity

Consider a demand-supply system: \begin{array}{ll} \text{Demand:} & Q^{d}=\beta _{0}^{d}+\beta _{1}^{d}P+U^{d}, \\ \text{Supply:} & Q^{s}=\beta _{0}^{s}+\beta _{1}^{s}P+U^{s}, \end{array} where Q^{d} = quantity demanded, Q^{s} = quantity supplied, P = price.
The quantity and price are determined simultaneously in the equilibrium: Q^{d}=Q^{s}=Q.
Q^{d} and Q^{s} are not observed separately; we observe only the equilibrium values Q.

Simultaneity

Solving for P: \begin{aligned} Q^{d} &= Q^{s} \\ \beta _{0}^{d}+\beta _{1}^{d}P+U^{d} &= \beta _{0}^{s}+\beta _{1}^{s}P+U^{s}, \end{aligned} so P=-\frac{\beta _{0}^{d}-\beta _{0}^{s}}{\beta _{1}^{d}-\beta _{1}^{s}}-\frac{U^{d}-U^{s}}{\beta _{1}^{d}-\beta _{1}^{s}}.
Thus, \mathrm{Cov}\left(P,U^{d}\right) \neq 0 \text{ and } \mathrm{Cov}\left(P,U^{s}\right) \neq 0. The demand-supply equations cannot be estimated by OLS.

Simultaneity

Consider a labour supply model for married women: \text{Hours}_{i}=\beta _{0}+\beta _{1}\text{Children}_{i}+\text{Other Factors}+U_{i}, where \text{Hours} = hours of work, \text{Children} = number of children.
It is reasonable to assume that women decide simultaneously how much time to devote to career and family.
Thus, while we may be mainly interested in the effect of family size on labour supply, there is another equation: \text{Children}_{i}=\gamma _{0}+\gamma _{1}\text{Hours}_{i}+\text{Other Factors}+V_{i}, and \text{Children} and \text{Hours} are determined simultaneously in an equilibrium.
As a result, \mathrm{Cov}\left(\text{Children}_{i},U_{i}\right) \neq 0, and the effect of family size cannot be estimated by OLS.

Errors in variables

Consider a model: Y_{i}=\beta _{0}+\beta _{1}X_{i}^{\ast }+V_{i}, where X_{i}^{\ast } is the “true” regressor.
Suppose that X_{i}^{\ast } is not directly observable. Instead, we observe X_{i} that measures X_{i}^{\ast } with an error \varepsilon_{i}: X_{i}=X_{i}^{\ast }+\varepsilon _{i}.
Since X_{i}^{\ast } is unobservable, the econometrician has to regress Y_{i} against X_{i}.

Errors in variables

The model for Y_{i} as a function of X_{i} can be written as \begin{aligned} Y_{i} &= \beta _{0}+\beta _{1}\left( X_{i}-\varepsilon _{i}\right) +V_{i} \\ &= \beta _{0}+\beta _{1}X_{i}+V_{i}-\beta _{1}\varepsilon _{i}, \end{aligned} or \begin{aligned} Y_{i} &= \beta _{0}+\beta _{1}X_{i}+U_{i}, \\ U_{i} &= V_{i}-\beta _{1}\varepsilon _{i}. \end{aligned}

Errors in variables

We can assume that \mathrm{Cov}\left(X_{i}^{\ast },V_{i}\right) =\mathrm{Cov}\left(X_{i}^{\ast },\varepsilon_{i}\right) =\mathrm{Cov}\left(\varepsilon _{i},V_{i}\right) =0.
However, \begin{aligned} \mathrm{Cov}\left(X_{i},U_{i}\right) &= \mathrm{Cov}\left(X_{i}^{\ast }+\varepsilon _{i},\; V_{i}-\beta _{1}\varepsilon_{i}\right) \\ &= \mathrm{Cov}\left(X_{i}^{\ast },V_{i}\right) -\beta _{1}\mathrm{Cov}\left(X_{i}^{\ast },\varepsilon _{i}\right) \\ &\quad +\mathrm{Cov}\left(\varepsilon _{i},V_{i}\right) -\beta _{1}{\color{red}\mathrm{Cov}\left(\varepsilon _{i},\varepsilon _{i}\right)} \\ &= -\beta _{1}\mathrm{Var}\left(\varepsilon _{i}\right) \neq 0. \end{aligned}
Thus, X_{i} is endogenous and \beta _{1} cannot be estimated by OLS.

Instrumental variable (IV)

Consider \begin{aligned} Y_{i} &= \beta _{0}+\beta _{1}X_{i}+U_{i}, \\ \mathrm{Cov}\left(X_{i},U_{i}\right) &\neq 0. \end{aligned}
Suppose that the econometrician observes another variable Z_{i}, called the instrumental variable, that satisfies the following conditions:
1. The IV is exogenous: \mathrm{Cov}\left(Z_{i},U_{i}\right) =0.
2. The IV determines the endogenous regressor: \mathrm{Cov}\left(Z_{i},X_{i}\right) \neq 0.
When an IV satisfying those conditions is available, it allows us to estimate the effect of X on Y consistently.

IV regression

Consider the IV estimator of \beta _{1}: {\color{blue}\hat{\beta}_{1,n}^{IV}=\frac{\sum_{i=1}^{n}\left( Z_{i}-\bar{Z}_{n}\right) Y_{i}}{\sum_{i=1}^{n}\left( Z_{i}-\bar{Z}_{n}\right) X_{i}}}.
Substituting Y_{i}=\beta _{0}+\beta _{1}X_{i}+U_{i}: \begin{aligned} \hat{\beta}_{1,n}^{IV} &= \frac{\sum_{i=1}^{n}\left( Z_{i}-\bar{Z}_{n}\right) \left( \beta _{0}+\beta _{1}X_{i}+U_{i}\right)}{\sum_{i=1}^{n}\left( Z_{i}-\bar{Z}_{n}\right) X_{i}} \\ &= \beta _{1}+\frac{\sum_{i=1}^{n}\left( Z_{i}-\bar{Z}_{n}\right) U_{i}}{\sum_{i=1}^{n}\left( Z_{i}-\bar{Z}_{n}\right) X_{i}}. \end{aligned}

Consistency of the IV estimator

The IV conditions:
1. Exogeneity: {\color{red}\mathrm{Cov}\left(Z_{i},U_{i}\right) =0}.
2. Relevance: {\color{blue}\mathrm{Cov}\left(Z_{i},X_{i}\right) \neq 0}.
By the LLN: \begin{aligned} \frac{1}{n}\sum_{i=1}^{n}\left( Z_{i}-\bar{Z}_{n}\right) U_{i} &\rightarrow _{p} {\color{red}\mathrm{Cov}\left(Z_{i},U_{i}\right)}, \\ \frac{1}{n}\sum_{i=1}^{n}\left( Z_{i}-\bar{Z}_{n}\right) X_{i} &\rightarrow _{p} {\color{blue}\mathrm{Cov}\left(Z_{i},X_{i}\right)}. \end{aligned}
The IV estimator is consistent: \begin{align*} \hat{\beta}_{1,n}^{IV} &= \beta _{1}+\frac{\frac{1}{n}\sum_{i=1}^{n}\left( Z_{i}-\bar{Z}_{n}\right) U_{i}}{\frac{1}{n}\sum_{i=1}^{n}\left( Z_{i}-\bar{Z}_{n}\right) X_{i}} \\ &\rightarrow _{p}\beta _{1}+\frac{{\color{red}0}}{{\color{blue}\mathrm{Cov}\left(Z_{i},X_{i}\right)}} =\beta _{1}. \end{align*}

Natural experiments

Theoretically, the causal effect can be estimated from controlled experiments:
- To estimate the return to education, select a random sample of children, randomly assign how many years of education they should have, and measure their income several years after graduation.
- To estimate the effect of family size on labor supply, select a random sample of parents and randomly assign how many children they should have, and measure their labor market outcomes.
Such an approach is infeasible due to high cost and/or ethical reasons.
Natural experiments: use the random variation in the variable of interest to estimate the causal effect.

Example: Compulsory schooling laws

Angrist and Krueger (1991, QJE) suggested using school start age policy to estimate \beta _{1} in \ln \text{Wage}_{i}=\beta _{0}+\beta _{1}\text{Education}_{i}+\beta _{2}\text{Ability}_{i}+V_{i}.
We need an IV Z such that \mathrm{Cov}\left(\text{Ability}_{i},Z_{i}\right) =0 and \mathrm{Cov}\left(\text{Education}_{i},Z_{i}\right) \neq 0.
They argue that due to compulsory schooling laws, the season of birth satisfies the IV conditions:
- A child must attend school until reaching a certain drop-out age.
- Students born in the first quarter reach the legal drop-out age before classmates born later in the year.
- The quarter-of-birth dummy is correlated with education.
- The quarter of birth is uncorrelated with ability.

Example: Sibling-sex composition

Angrist and Evans (1998, AER) argue that parents’ preferences for a mixed sibling-sex composition can be used to estimate \beta _{1} in \text{Hours}_{i}=\beta _{0}+\beta _{1}\text{Children}_{i}+\ldots +U_{i}.
We need an IV Z such that \mathrm{Cov}\left(U_{i},Z_{i}\right) =0 and \mathrm{Cov}\left(\text{Children}_{i},Z_{i}\right) \neq 0.
Consider a dummy variable equal to one if the sex of the second child matches the sex of the first child.
If parents prefer a mixed sibling-sex composition, they are more likely to have another child if their first two children are of the same sex.
The same-sex dummy is correlated with the number of children.
Since sex mix is randomly determined, the same-sex dummy is exogenous.

Asymptotic distribution of the IV estimator

Write \begin{align*} \sqrt{n}\left( \hat{\beta}_{1,n}^{IV}-\beta _{1}\right) &= \frac{\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\left( Z_{i}-\bar{Z}_{n}\right) U_{i}}{\frac{1}{n}\sum_{i=1}^{n}\left( Z_{i}-\bar{Z}_{n}\right) X_{i}} \\ &\rightarrow _{d}\frac{N\left( 0,\;\mathrm{E}\left[\left( Z_{i}-\mathrm{E}\left[Z_{i}\right]\right)^{2}U_{i}^{2}\right]\right)}{\mathrm{Cov}\left(Z_{i},X_{i}\right)}. \end{align*}
Thus, \begin{aligned} \sqrt{n}\left( \hat{\beta}_{1,n}^{IV}-\beta _{1}\right) &\rightarrow_{d} N\left( 0,V^{IV}\right), \text{ where} \\ V^{IV} &= \frac{\mathrm{E}\left[\left( Z_{i}-\mathrm{E}\left[Z_{i}\right]\right) ^{2}U_{i}^{2}\right]}{\left(\mathrm{Cov}\left(Z_{i},X_{i}\right)\right) ^{2}}. \end{aligned}

Variance estimation

Let \hat{\beta}_{0,n}^{IV}=\bar{Y}_{n}-\hat{\beta}_{1,n}^{IV}\cdot \bar{X}_{n}.
Let \hat{U}_{i}=Y_{i}-\hat{\beta}_{0,n}^{IV}-\hat{\beta}_{1,n}^{IV}X_{i}.
Estimate V^{IV} by \hat{V}_{n}^{IV}=\frac{\frac{1}{n}\sum_{i=1}^{n}\left( Z_{i}-\bar{Z}_{n}\right) ^{2}\hat{U}_{i}^{2}}{\left( \frac{1}{n}\sum_{i=1}^{n}\left( Z_{i}-\bar{Z}_{n}\right) X_{i}\right) ^{2}}.
In finite samples, we use the approximation: \hat{\beta}_{1,n}^{IV}\overset{a}{\sim }N\left( \beta _{1},\frac{\hat{V}_{n}^{IV}}{n}\right).

--- title: "Lecture 19: Instrumental Variables" subtitle: "Economics 326 — Introduction to Econometrics II" author: - name: "Vadim Marmer, UBC" format: html: output-file: 326_19_IV.html toc: true toc-depth: 3 toc-location: right toc-title: "Table of Contents" theme: cosmo smooth-scroll: true html-math-method: katex embed-resources: true pdf: output-file: 326_19_IV.pdf pdf-engine: xelatex geometry: margin=0.75in fontsize: 10pt number-sections: false toc: false classoption: fleqn revealjs: output-file: 326_19_IV_slides.html theme: solarized css: slides_no_caps.css smaller: true slide-number: c/t incremental: true html-math-method: katex scrollable: true chalkboard: false self-contained: true transition: none --- ## Endogeneity ::: {.hidden} \gdef\E#1{\mathrm{E}\left[#1\right]} \gdef\Var#1{\mathrm{Var}\left(#1\right)} \gdef\Cov#1{\mathrm{Cov}\left(#1\right)} \gdef\Vhat#1{\widehat{\mathrm{Var}}\left(#1\right)} \gdef\se#1{\mathrm{se}\left(#1\right)} ::: - In the linear regression model, $$ Y_{i}=\beta _{0}+\beta _{1}X_{i}+U_{i}, $$ the condition for consistent estimation of $\beta _{1}$ by OLS is that $X$ is **exogenous**: $$ \Cov{X_{i},U_{i}} =0. $$ - When $\Cov{X_{i},U_{i}} \neq 0,$ we say that the regressor $X$ is **endogenous**. - When the regressor is **endogenous**, the OLS estimator is **inconsistent**: \begin{align*} \hat{\beta}_{1,n}-\beta _{1} &= \frac{\frac{1}{n}\sum_{i=1}^{n}\left( X_{i}-\bar{X}_{n}\right) U_{i}}{\frac{1}{n}\sum_{i=1}^{n}\left( X_{i}-\bar{X}_{n}\right) ^{2}} \\ &\rightarrow _{p}\frac{{\color{red}\Cov{X_{i},U_{i}}}}{\Var{X_{i}}} \neq 0. \end{align*} ## Consequences of endogeneity - The causal effect of $X$ on $Y$ is not estimated consistently: $$ \hat{\beta}_{1,n}\rightarrow _{p}\beta _{1}+\frac{\Cov{X_{i},U_{i}}}{\Var{X_{i}}}. $$ The effect can be over- or underestimated depending on the sign of $\Cov{X_{i},U_{i}}$. - Tests and confidence intervals are invalid. ## Sources of endogeneity - Several possible sources of endogeneity: 1. Omitted explanatory variables. 2. Simultaneity. 3. Errors in variables. - All result in regressors correlated with the errors. ## Omitted explanatory variables - Suppose that the true model is $$ \ln \text{Wage}_{i}=\beta _{0}+\beta _{1}\text{Education}_{i}+\beta _{2}\text{Ability}_{i}+V_{i}, $$ where $V_{i}$ is uncorrelated with $\text{Education}$ and $\text{Ability}$. - Since $\text{Ability}$ is unobservable, the econometrician regresses $\ln \text{Wage}$ against $\text{Education}$, and $\beta _{2}\text{Ability}$ goes into the error: $$ \begin{aligned} \ln \text{Wage}_{i} &= \beta _{0}+\beta _{1}\text{Education}_{i}+U_{i}, \\ U_{i} &= \beta _{2}\text{Ability}_{i}+V_{i}. \end{aligned} $$ - $\text{Education}$ is correlated with $\text{Ability}$: we can expect that $\Cov{\text{Education}_{i},\text{Ability}_{i}} >0$, $\beta _{2}>0$, and therefore $$ \Cov{\text{Education}_{i},U_{i}} >0. $$ Thus, OLS will overestimate the return to education. ## Simultaneity - Consider a **demand-supply** system: $$ \begin{array}{ll} \text{Demand:} & Q^{d}=\beta _{0}^{d}+\beta _{1}^{d}P+U^{d}, \\ \text{Supply:} & Q^{s}=\beta _{0}^{s}+\beta _{1}^{s}P+U^{s}, \end{array} $$ where $Q^{d}$ = quantity demanded, $Q^{s}$ = quantity supplied, $P$ = price. - The quantity and price are determined **simultaneously** in the equilibrium: $$ Q^{d}=Q^{s}=Q. $$ - $Q^{d}$ and $Q^{s}$ are not observed separately; we observe only the equilibrium values $Q$. ## Simultaneity - Solving for $P$: $$ \begin{aligned} Q^{d} &= Q^{s} \\ \beta _{0}^{d}+\beta _{1}^{d}P+U^{d} &= \beta _{0}^{s}+\beta _{1}^{s}P+U^{s}, \end{aligned} $$ so $$ P=-\frac{\beta _{0}^{d}-\beta _{0}^{s}}{\beta _{1}^{d}-\beta _{1}^{s}}-\frac{U^{d}-U^{s}}{\beta _{1}^{d}-\beta _{1}^{s}}. $$ - Thus, $$ \Cov{P,U^{d}} \neq 0 \text{ and } \Cov{P,U^{s}} \neq 0. $$ The demand-supply equations cannot be estimated by OLS. ## Simultaneity - Consider a labour supply model for married women: $$ \text{Hours}_{i}=\beta _{0}+\beta _{1}\text{Children}_{i}+\text{Other Factors}+U_{i}, $$ where $\text{Hours}$ = hours of work, $\text{Children}$ = number of children. - It is reasonable to assume that women decide **simultaneously** how much time to devote to career and family. - Thus, while we may be mainly interested in the effect of family size on labour supply, there is another equation: $$ \text{Children}_{i}=\gamma _{0}+\gamma _{1}\text{Hours}_{i}+\text{Other Factors}+V_{i}, $$ and $\text{Children}$ and $\text{Hours}$ are determined **simultaneously** in an equilibrium. - As a result, $\Cov{\text{Children}_{i},U_{i}} \neq 0,$ and the effect of family size cannot be estimated by OLS. ## Errors in variables - Consider a model: $$ Y_{i}=\beta _{0}+\beta _{1}X_{i}^{\ast }+V_{i}, $$ where $X_{i}^{\ast }$ is the "true" regressor. - Suppose that $X_{i}^{\ast }$ is not directly observable. Instead, we observe $X_{i}$ that measures $X_{i}^{\ast }$ with an error $\varepsilon_{i}$: $$ X_{i}=X_{i}^{\ast }+\varepsilon _{i}. $$ - Since $X_{i}^{\ast }$ is unobservable, the econometrician has to regress $Y_{i}$ against $X_{i}$. ## Errors in variables - The model for $Y_{i}$ as a function of $X_{i}$ can be written as $$ \begin{aligned} Y_{i} &= \beta _{0}+\beta _{1}\left( X_{i}-\varepsilon _{i}\right) +V_{i} \\ &= \beta _{0}+\beta _{1}X_{i}+V_{i}-\beta _{1}\varepsilon _{i}, \end{aligned} $$ or $$ \begin{aligned} Y_{i} &= \beta _{0}+\beta _{1}X_{i}+U_{i}, \\ U_{i} &= V_{i}-\beta _{1}\varepsilon _{i}. \end{aligned} $$ ## Errors in variables - We can assume that $$ \Cov{X_{i}^{\ast },V_{i}} =\Cov{X_{i}^{\ast },\varepsilon_{i}} =\Cov{\varepsilon _{i},V_{i}} =0. $$ - However, $$ \begin{aligned} \Cov{X_{i},U_{i}} &= \Cov{X_{i}^{\ast }+\varepsilon _{i},\; V_{i}-\beta _{1}\varepsilon_{i}} \\ &= \Cov{X_{i}^{\ast },V_{i}} -\beta _{1}\Cov{X_{i}^{\ast },\varepsilon _{i}} \\ &\quad +\Cov{\varepsilon _{i},V_{i}} -\beta _{1}{\color{red}\Cov{\varepsilon _{i},\varepsilon _{i}}} \\ &= -\beta _{1}\Var{\varepsilon _{i}} \neq 0. \end{aligned} $$ - Thus, $X_{i}$ is **endogenous** and $\beta _{1}$ cannot be estimated by OLS. ## Instrumental variable (IV) - Consider $$ \begin{aligned} Y_{i} &= \beta _{0}+\beta _{1}X_{i}+U_{i}, \\ \Cov{X_{i},U_{i}} &\neq 0. \end{aligned} $$ - Suppose that the econometrician observes another variable $Z_{i}$, called the **instrumental variable**, that satisfies the following conditions: 1. The IV is **exogenous**: $\Cov{Z_{i},U_{i}} =0.$ 2. The IV **determines** the endogenous regressor: $\Cov{Z_{i},X_{i}} \neq 0.$ - When an IV satisfying those conditions is available, it allows us to estimate the effect of $X$ on $Y$ consistently. ## IV regression - Consider the **IV estimator** of $\beta _{1}$: $$ {\color{blue}\hat{\beta}_{1,n}^{IV}=\frac{\sum_{i=1}^{n}\left( Z_{i}-\bar{Z}_{n}\right) Y_{i}}{\sum_{i=1}^{n}\left( Z_{i}-\bar{Z}_{n}\right) X_{i}}}. $$ - Substituting $Y_{i}=\beta _{0}+\beta _{1}X_{i}+U_{i}$: $$ \begin{aligned} \hat{\beta}_{1,n}^{IV} &= \frac{\sum_{i=1}^{n}\left( Z_{i}-\bar{Z}_{n}\right) \left( \beta _{0}+\beta _{1}X_{i}+U_{i}\right)}{\sum_{i=1}^{n}\left( Z_{i}-\bar{Z}_{n}\right) X_{i}} \\ &= \beta _{1}+\frac{\sum_{i=1}^{n}\left( Z_{i}-\bar{Z}_{n}\right) U_{i}}{\sum_{i=1}^{n}\left( Z_{i}-\bar{Z}_{n}\right) X_{i}}. \end{aligned} $$ ## Consistency of the IV estimator - The IV conditions: 1. **Exogeneity**: ${\color{red}\Cov{Z_{i},U_{i}} =0}.$ 2. **Relevance**: ${\color{blue}\Cov{Z_{i},X_{i}} \neq 0}.$ - By the LLN: $$ \begin{aligned} \frac{1}{n}\sum_{i=1}^{n}\left( Z_{i}-\bar{Z}_{n}\right) U_{i} &\rightarrow _{p} {\color{red}\Cov{Z_{i},U_{i}}}, \\ \frac{1}{n}\sum_{i=1}^{n}\left( Z_{i}-\bar{Z}_{n}\right) X_{i} &\rightarrow _{p} {\color{blue}\Cov{Z_{i},X_{i}}}. \end{aligned} $$ - The IV estimator is consistent: \begin{align*} \hat{\beta}_{1,n}^{IV} &= \beta _{1}+\frac{\frac{1}{n}\sum_{i=1}^{n}\left( Z_{i}-\bar{Z}_{n}\right) U_{i}}{\frac{1}{n}\sum_{i=1}^{n}\left( Z_{i}-\bar{Z}_{n}\right) X_{i}} \\ &\rightarrow _{p}\beta _{1}+\frac{{\color{red}0}}{{\color{blue}\Cov{Z_{i},X_{i}}}} =\beta _{1}. \end{align*} ## Natural experiments - Theoretically, the causal effect can be estimated from **controlled experiments**: - To estimate the return to education, select a random sample of children, **randomly** assign how many years of education they should have, and measure their income several years after graduation. - To estimate the effect of family size on labor supply, select a random sample of parents and **randomly** assign how many children they should have, and measure their labor market outcomes. - Such an approach is infeasible due to high cost and/or ethical reasons. - **Natural experiments**: use the random variation in the variable of interest to estimate the causal effect. ## Example: Compulsory schooling laws - Angrist and Krueger (1991, *QJE*) suggested using school start age policy to estimate $\beta _{1}$ in $$ \ln \text{Wage}_{i}=\beta _{0}+\beta _{1}\text{Education}_{i}+\beta _{2}\text{Ability}_{i}+V_{i}. $$ - We need an IV $Z$ such that $\Cov{\text{Ability}_{i},Z_{i}} =0$ and $\Cov{\text{Education}_{i},Z_{i}} \neq 0.$ - They argue that due to compulsory schooling laws, the **season of birth** satisfies the IV conditions: - A child must attend school until reaching a certain drop-out age. - Students born in the first quarter reach the legal drop-out age before classmates born later in the year. - The quarter-of-birth dummy is correlated with education. - The quarter of birth is uncorrelated with ability. ## Example: Sibling-sex composition - Angrist and Evans (1998, *AER*) argue that parents' preferences for a mixed sibling-sex composition can be used to estimate $\beta _{1}$ in $$ \text{Hours}_{i}=\beta _{0}+\beta _{1}\text{Children}_{i}+\ldots +U_{i}. $$ - We need an IV $Z$ such that $\Cov{U_{i},Z_{i}} =0$ and $\Cov{\text{Children}_{i},Z_{i}} \neq 0$. - Consider a dummy variable equal to one if the sex of the second child matches the sex of the first child. - If parents prefer a mixed sibling-sex composition, they are more likely to have another child if their first two children are of the same sex. - The same-sex dummy is correlated with the number of children. - Since sex mix is randomly determined, the same-sex dummy is exogenous. ## Asymptotic distribution of the IV estimator - Write \begin{align*} \sqrt{n}\left( \hat{\beta}_{1,n}^{IV}-\beta _{1}\right) &= \frac{\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\left( Z_{i}-\bar{Z}_{n}\right) U_{i}}{\frac{1}{n}\sum_{i=1}^{n}\left( Z_{i}-\bar{Z}_{n}\right) X_{i}} \\ &\rightarrow _{d}\frac{N\left( 0,\;\E{\left( Z_{i}-\E{Z_{i}}\right)^{2}U_{i}^{2}}\right)}{\Cov{Z_{i},X_{i}}}. \end{align*} - Thus, $$ \begin{aligned} \sqrt{n}\left( \hat{\beta}_{1,n}^{IV}-\beta _{1}\right) &\rightarrow_{d} N\left( 0,V^{IV}\right), \text{ where} \\ V^{IV} &= \frac{\E{\left( Z_{i}-\E{Z_{i}}\right) ^{2}U_{i}^{2}}}{\left(\Cov{Z_{i},X_{i}}\right) ^{2}}. \end{aligned} $$ ## Variance estimation - Let $\hat{\beta}_{0,n}^{IV}=\bar{Y}_{n}-\hat{\beta}_{1,n}^{IV}\cdot \bar{X}_{n}.$ - Let $\hat{U}_{i}=Y_{i}-\hat{\beta}_{0,n}^{IV}-\hat{\beta}_{1,n}^{IV}X_{i}.$ - Estimate $V^{IV}$ by $$ \hat{V}_{n}^{IV}=\frac{\frac{1}{n}\sum_{i=1}^{n}\left( Z_{i}-\bar{Z}_{n}\right) ^{2}\hat{U}_{i}^{2}}{\left( \frac{1}{n}\sum_{i=1}^{n}\left( Z_{i}-\bar{Z}_{n}\right) X_{i}\right) ^{2}}. $$ - In finite samples, we use the approximation: $$ \hat{\beta}_{1,n}^{IV}\overset{a}{\sim }N\left( \beta _{1},\frac{\hat{V}_{n}^{IV}}{n}\right). $$