Class 16 Instrumental Variables and Two-Stage Least Squares

Author

Affiliation

Dr Wei Miao

UCL School of Management

Published

November 18, 2026

1 Instrumental Variable

1.1 Class Objectives

The requirements of a valid instrumental variable and how to find good instruments
Intuition of why instrumental variables solve endogeneity problems
Apply the two-stage least squares method to estimate the causal effects using instrumental variables

1.2 Causal Inference from OLS

From non-experimental secondary data, it is impossible to control all confounding factors, which means we can never obtain causal effects from OLS regressions.
Can we still obtain causal inference from secondary data?

1.3 What is an Instrumental Variable

1.3.1 Instrumental Variable

An instrumental variable is a set of variables $Z$ that satisfies the following requirements:

$z$ is exogenous and uncorrelated with $\epsilon$; that is, $cov(Z,\epsilon) = 0$
$z$ only affects $Y$ through $X$, but not directly affect $Y$
$z$ affects $x$ to some extent, that is, $cov(Z,x) \neq 0$

Point 1 is the exogeneity requirement: the instrumental variable should be beyond an individual’s control, such that the instrumental variables are uncorrelated with any of the individual’s unobserved confounding factors.
- Potential IVs: government policy; natural disasters; randomised experiment; birthdays; etc.
Point 2 is the exclusion restriction: the instrumental variable should only affect $Y$ through $X$, and not affect $Y$ directly.
Point 3 is the relevance requirement: although beyond an individual’s control, the instrumental variable should still affect the individual’s $X$, causing some exogenous changes in $X$ that are beyond individual control.
- If the correlation between $z$ and $x$ is too small, we have a weak IV problem.

1.4 Graphical Illustration of IV

1.5 A Classic Example of Instrumental Variable

Return of Military Service to Lifetime Income¹

\[ Income = \beta_0 + \beta_1MilitaryService + \epsilon \]

OLS suffers from endogeneity problems. What are the potential endogeneity issues?
A lottery was used to determine whether soldiers with certain birthdays were drafted to the frontline.

1.6 A Classic Example of Instrumental Variable

The date of birth ($z$) or zodiacs can be an instrumental variable for military service ($x$) in this case.
- Relevance requirement: Affects years of military service: $cov(z,x) \neq 0$
- Exogeneity requirement: Randomly drawn and thus uncorrelated with any confounders: $cov(z,\epsilon) = 0$
- Exclusion restriction: $z$ only affects $Y$ through $X$, but not directly affect $Y$.

1.7 More Examples of IVs

Can you come up with IV candidates for the following causal questions?

Number of restaurants on UberEat => Number of orders on UberEat
- temporary closure of restaurants due to government inspections
- cost shifters for running restaurants: rent changes; wage changes; ingredient costs
Retail price => Sales
- cost shifters: wholesale price; costs of raw materials; COGS
- BLP instruments: prices of other products in the same category
- Hausman instruments: the prices of the same product in other markets

2 Two-Stage Least Squares

2.1 Solving Endogeneity Using IV

Given an endogenous OLS regression,

\[ y_{i}=X_{i} \beta+\varepsilon_{i}, \quad \operatorname{cov}\left(X_{i}, \varepsilon_{i}\right) \neq 0 \]

Find instrumental variables $Z_i$ that do not (directly) influence $y_i$ , but are correlated with $X_i$

2.2 Two-Stage Least Squares: Stage 1

Run a regression of X on Z. The predicted value $\hat X$ is predicted by Z, and so should be uncorrelated with the error term $\epsilon$.
- $\hat{X}$ (the part of the changes in $X$ that is due to $Z$) is exogenous, because $Z$ is exogenous
- All endogenous parts are now left over in the error term in the first-stage regression $\epsilon_{i}$

\[ X_{i}=Z_{i}\eta+\epsilon_{i} \]

2.3 Two-Stage Least Squares: Stage 2

Run a regression of $Y$ on $\hat{X}$: now $\hat{X}$ is uncorrelated with the error term, and thus we can get a causal inference from the second-stage regression.

\[ y_{i}=\hat{X} \beta+\varepsilon_{i}, \quad \operatorname{cov}\left(\hat{X}_{i}, \varepsilon_{i}\right) = 0 \]

2.4 After-Class Readings

(optional) Econometrics with R: Instrumental Variables Regression

Footnotes

Angrist, Joshua D., Stacey H. Chen, and Jae Song. “Long-term consequences of Vietnam-era conscription: New estimates using social security data.” American Economic Review 101, no. 3 (2011): 334-38.↩︎

--- aliases: - /Week8-Lecture2.html date: "`r (first_date+lubridate::dweeks(7))`" title: "Class 16 Instrumental Variables and Two-Stage Least Squares" --- # Instrumental Variable ## Class Objectives - The requirements of a valid instrumental variable and how to find good instruments - Intuition of why instrumental variables solve endogeneity problems - Apply the two-stage least squares method to estimate the causal effects using instrumental variables ## Causal Inference from OLS - From non-experimental secondary data, it is impossible to control all confounding factors, which means we can never obtain causal effects from OLS regressions. - Can we still obtain causal inference from secondary data? ## What is an Instrumental Variable ::: block ### Instrumental Variable An instrumental variable is a set of variables $Z$ that satisfies the following requirements: 1. $z$ is exogenous and uncorrelated with $\epsilon$; that is, $cov(Z,\epsilon) = 0$ 2. $z$ only affects $Y$ through $X$, but not directly affect $Y$ 3. $z$ affects $x$ to some extent, that is, $cov(Z,x) \neq 0$ ::: \small - Point 1 is the **exogeneity** requirement: the instrumental variable should be beyond an individual's control, such that the instrumental variables are uncorrelated with any of the individual's unobserved confounding factors. - Potential IVs: government policy; natural disasters; randomised experiment; birthdays; etc. - Point 2 is the **exclusion restriction**: the instrumental variable should only affect $Y$ through $X$, and not affect $Y$ directly. - Point 3 is the **relevance requirement**: although beyond an individual's control, the instrumental variable should still affect the individual's $X$, causing some exogenous changes in $X$ that are beyond individual control. - If the correlation between $z$ and $x$ is too small, we have a [**weak IV** problem]((https://www.econometrics-with-r.org/12.3-civ.html)). ## Graphical Illustration of IV ```{r} #| echo: false #| fig-align: 'center' knitr::include_graphics("../../images/Week 8/InstrumentalVariable.png") ``` ## A Classic Example of Instrumental Variable **Return of Military Service to Lifetime Income**[^1] [^1]: Angrist, Joshua D., Stacey H. Chen, and Jae Song. "Long-term consequences of Vietnam-era conscription: New estimates using social security data." *American Economic Review* 101, no. 3 (2011): 334-38. $$ Income = \beta_0 + \beta_1MilitaryService + \epsilon $$ - OLS suffers from endogeneity problems. What are the potential endogeneity issues? - A lottery was used to determine whether soldiers with certain birthdays were drafted to the frontline. ## A Classic Example of Instrumental Variable - The date of birth ($z$) or zodiacs can be an **instrumental variable** for military service ($x$) in this case. - Relevance requirement: Affects years of military service: $cov(z,x) \neq 0$ - Exogeneity requirement: Randomly drawn and thus uncorrelated with any confounders: $cov(z,\epsilon) = 0$ - Exclusion restriction: $z$ only affects $Y$ through $X$, but not directly affect $Y$. ```{r} #| echo: false #| fig-align: 'center' knitr::include_graphics("../../images/Week 8/zodiac.jpg") ``` ## More Examples of IVs ::: {.content-visible when-format="beamer"} **Can you come up with IV candidates for the following causal questions?**[^2] - Number of restaurants on UberEat =\> Number of orders on UberEat - Retail price =\> Sales ::: [^2]: See html version for answers. ::: {.content-visible when-format="html"} **Can you come up with IV candidates for the following causal questions?** - Number of restaurants on UberEat =\> Number of orders on UberEat - temporary closure of restaurants due to government inspections - cost shifters for running restaurants: rent changes; wage changes; ingredient costs - Retail price =\> Sales - cost shifters: wholesale price; costs of raw materials; COGS - BLP instruments: prices of other products in the same category - Hausman instruments: the prices of the same product in other markets ::: # Two-Stage Least Squares ## Solving Endogeneity Using IV - Given an endogenous OLS regression, $$ y_{i}=X_{i} \beta+\varepsilon_{i}, \quad \operatorname{cov}\left(X_{i}, \varepsilon_{i}\right) \neq 0 $$ - Find instrumental variables $Z_i$ that do not (directly) influence $y_i$ , but are correlated with $X_i$ ## Two-Stage Least Squares: Stage 1 1. Run a regression of `X` on `Z`. The predicted value $\hat X$ is predicted by Z, and so should be uncorrelated with the error term $\epsilon$. - $\hat{X}$ (the part of the changes in $X$ that is due to $Z$) is exogenous, because $Z$ is exogenous - All endogenous parts are now left over in the error term in the first-stage regression $\epsilon_{i}$ $$ X_{i}=Z_{i}\eta+\epsilon_{i} $$ ## Two-Stage Least Squares: Stage 2 2. Run a regression of $Y$ on $\hat{X}$: now $\hat{X}$ is uncorrelated with the error term, and thus we can get a causal inference from the second-stage regression. $$ y_{i}=\hat{X} \beta+\varepsilon_{i}, \quad \operatorname{cov}\left(\hat{X}_{i}, \varepsilon_{i}\right) = 0 $$ ## After-Class Readings - (optional) [Econometrics with R: Instrumental Variables Regression](https://www.econometrics-with-r.org/12-ivr.html)