instrumental variables | Chen Xing

Notes on causal inference with No Overlap – Regression Discontinuity

Sat, 31 May 2025 00:00:00 +0000

Here is my notes on regression discontinuity from Prof. Ding’s textbook (2024) and Prof. Imai’s lecture notes.

Motivation

We often cannot run a randomized experiment and have to use/design observational studies to find a setting where credible causal inference is possible.

The key is the knowledge of treatment assignment mechanism.

Regression discontinuity design (RD Design):

RD Design is a simple and widely used quasi-experimental design. The term “quasi experimental” is to emphasize that these approaches are still framed using concepts from randomized experiments but require econometric innovations to compensate for the lack of random treatment assignment.

Sharp RD Design: treatment assignment is based on a deterministic rule (i.e. we have full knowledge of how treatment is assigned)
Fuzzy RD Design: encouragement to receive treatment is based on a deterministic rule

Setting

Binary treatment $Z\in \{0,1\}$
Potential outcomes $\{Y(0), Y(1)\}$
There is a running variable $X \in \R$ such that $Z=I\left(X \geq x_0\right)$, where $x_0$ is a pre-determined threshold. Note that, the treatment assignment is deterministic
The unconfoundedness assumption holds automatically $$ Z \indep \{Y(1), Y(0)\} \mid X $$
The overlap assumption does not hold

$$ e(X)=\operatorname{pr}(Z=1 \mid X)=1\left(X \geq x_0\right) = \text{1 or 0} $$

Identification

RD can identify a local average causal effect at the cutoff point $x_0$
Estimand:

$$ \tau\left(x_0\right)=E\left\{Y(1)-Y(0) \mid X=x_0\right\} . $$

Assumption 1 (continuity assumption).

1. $E\{Y(1) \mid X=x\}$ is continuous from the right at $x_0$
2. $E\{Y(0) \mid X=x\}$ is continuous from the left at $x_0$

We have $$ \begin{align} E\left\{Y(1) \mid X=x_0\right\} & =\lim _{\varepsilon \rightarrow 0+} E\left\{Y(1) \mid X=x_0+\varepsilon\right\} \tag{continuity} \\ & =\lim _{\varepsilon \rightarrow 0+} E\left\{Y(1) \mid Z=1, X=x_0+\varepsilon\right\} \tag{def of Z}\\ & =\lim _{\varepsilon \rightarrow 0+} E\left(Y \mid Z=1, X=x_0+\varepsilon\right), \end{align} $$ Similarly, $$ E\left\{Y(0) \mid X=x_0\right\}=\lim _{\varepsilon \rightarrow 0+} E\left(Y \mid Z=0, X=x_0-\varepsilon\right) $$ So the local average causal effect at $x_0$ can be identified by the difference of the two limits
Advantage: internal validity
Disadvantage: external validity

Key Theorem

Theorem 1.

Assume that the treatment is determined by $Z=I\left(X \geq x_0\right)$ where $x_0$ is a predetermined threshold. Assume that $E\{Y(1) \mid X=x\}$ is continuous from the right at $x_0$ and $E\{Y(0) \mid X=x\}$ is continuous from the left at $x_0$. Then the local average treatment effect at $X=x_0$ is identified by $$ \tau\left(x_0\right)=\lim _{\varepsilon \rightarrow 0+} E\left(Y \mid Z=1, X=x_0+\varepsilon\right)-\lim _{\varepsilon \rightarrow 0+} E\left(Y \mid Z=0, X=x_0-\varepsilon\right) $$

$\tau\left(x_0\right)$ is nonparametrically identified.

Reference

Ding, P. (2024). A First Course in Causal Inference. CRC Press.

Lecture notes: Regression Discontinuity Design

Notes on Control Function Method

Sat, 31 May 2025 00:00:00 +0000

Motivation

Endogeneity: When an explanatory variable (treatment) is correlated with the unobserved error term (e.g., due to omitted variables, measurement error, or simultaneity).
Consequence: Standard regression (e.g., OLS) yields biased estimates.
Goal: The control function (CF) approach “purges” endogeneity by modeling the correlation between the treatment and unobservables.

The control-function (CF) approach tackles this by explicitly modelling the source of endogeneity and then partialling it out. For linear models that modelling step turns out to be algebraically equivalent to 2SLS, but the real power of CF is that it extends seamlessly to nonlinear or limited‐dependent‐variable settings where 2SLS cannot be applied directly.

Isolate the problematic part of the error term correlated with treatment, then add it as a control variable to ’neutralize’ endogeneity.

How It Works

Outcome model: $Y = \beta_0 + \beta_1 T + \beta_2 X + U$

$T$: Endogenous treatment
$X$: Exogenous controls
$U$: Unobservables (correlated with $T$).

First-stage model: $ T = \gamma_0 + \gamma_1 Z + \gamma_2 X + V $

$Z$: Instrumental variable (IV)
$V$: First-stage error.

CF Insight:

If $U$ and $V$ are correlated (i.e., $\text{Cov}(U,V) \neq 0$), we can decompose $U$ into:
$$ U = \rho V + \epsilon $$ where $\epsilon$ is uncorrelated with $T$ and $V$ (by construction).

Estimation

First Stage:
- Regress $T$ on $Z$ and $X$:
  $$ \hat{T} = \hat{\gamma}_0 + \hat{\gamma}_1 Z + \hat{\gamma}_2 X $$
- Obtain the residual: $\hat{V} = T - \hat{T}$.
Second Stage:
- Add $\hat{V}$ to the outcome model:
  $$ Y = \beta_0 + \beta_1 T + \beta_2 X + \rho \hat{V} + \epsilon $$
- Estimate via OLS.

Why this works:

$\hat{V}$ “controls for” the part of $U$ correlated with $T$.
Once $\hat{V}$ is included, $T$ becomes exogenous in the modified model ($\text{Cov}(T, \epsilon) = 0$).
$\hat{\beta}_1$ is consistent for the causal effect.

Key Assumptions

Instrument Validity:
- $Z$ is relevant: $\text{Cov}(Z, T) \neq 0$ (strong first stage).
- $Z$ is exogenous: $\text{Cov}(Z, U) = 0$.
Correct Functional Form:
- Linearity in the first stage and control function (e.g., $U = \rho V + \epsilon$).
Exclusion Restriction: $Z$ affects $Y$ only through $T$.

CF vs. Other Methods

Method	Key Difference
2SLS	Uses fitted values ($\hat{T}$); efficient but inconsistent under heteroscedasticity/nonlinearity.
Control Function	Uses residuals ($\hat{V}$); more flexible for nonlinear models (e.g., probit).

Advantage of CF:

Directly models the endogeneity structure (via $\hat{V}$).
Extends to non-additive errors, discrete outcomes, and heteroscedastic settings.

Example

Problem: Estimate returns to education ($T$) on wages ($Y$), where ability ($U$) is unobserved and correlated with education.

IV: Distance to college ($Z$).
Steps:
1. Regress: education $\sim$ distance + controls → get residuals $\hat{V}$.
2. Regress: wages $\sim$ education + controls + $\hat{V}$.
Result: Coefficient on education is causal.

Generalization

Setting	Control-function term	Typical reference
Binary/probit $Y$ with endogenous $T$	Include $h(\hat v)$ where $h(\cdot)$ is the generalized residual (Smith & Blundell, 1986)	Rivers & Vuong (1988)
Heckman sample selection	Inverse Mills ratio $\lambda(\hat v)$ controls for non-random sample entry	Heckman (1979)
Count models (Poisson, NB)	Nonparametric sieve $h(\hat v)$ or parametric polynomial	Wooldridge (2015)
Semiparametric/ML partially-linear	Learnt nuisance $m(X)$; CF gives Neyman-orthogonal moment for Double ML	Chernozhukov et al. (2018)

The principle is identical: obtain a residual that captures unobserved heterogeneity driving $T$, then include it (or a flexible transformation) in the structural equation.

Why This Matters

CF is essential when:
- You have a valid IV and suspect omitted variable bias.
- You work with nonlinear models (e.g., binary/duration outcomes).
Software Implementation:
- Stata: ivregress 2sls (equivalent to CF in linear cases) or cmp for nonlinear.
- R: ivreg (linear), controlfunction package.

Critical Caveats

Weak Instruments: If $Z$ is weak, $\hat{V}$ is noisy → bias.
Functional Form Misspecification: If $U \neq \rho V + \epsilon$, CF fails.
No Magic Bullet: Validity of $Z$ is untestable and must be justified theoretically.

Bottom Line

The CF approach harnesses IV residuals to “control” for endogeneity, converting an endogenous variable into a conditionally exogenous one. It’s a blend of IV intuition and regression control — powerful when assumptions hold.

Notes on IV Free Methods

Fri, 30 May 2025 00:00:00 +0000

Here’s a overview of instrument-free methods:

Latent Instrumental Variable (LIV)
Gaussian Copula (GC)

The goal is for dealing with endogeneity when no external instruments are available.

Motivation

Finding good instruments is hard. Around 2010, marketing researchers recognized that the “IV cure can be worse than the endogeneity disease,” and began seeking ways to exploit features of the data (rather than external IVs) to identify causal effects when valid instruments aren’t available .

Intuition

LIV treats the endogenous regressor as driven by a discrete, unobserved “latent instrument” that captures its exogenous variation, while the remaining variation is deemed endogenous. One then estimates this latent class structure (akin to a mixture model) alongside the main outcome equation under normal-error assumptions.
GC builds a joint distribution of the endogenous regressor and the structural error via a Gaussian copula. By assuming the regressor is non-normal and the error is normal, the copula decomposition isolates the exogenous component and yields consistent estimates without explicit instruments .

Data Structure

Both methods were originally developed for cross-sectional datasets.
LIV has been extended to dynamic settings (e.g., panel data with time-varying latent classes) and to nonlinear models (e.g., binary logit) .
Applying GC in panel contexts typically requires a first-difference or within transformation to purge fixed effects, which alters the error covariance and complicates copula estimation.

How It Works

LIV Approach
- Model setup: Decompose the regressor $X$ into two parts: a discrete latent instrument $Z^*$ (exogenous) and residual $u$ (endogenous).
- Assumptions:
  - $Z^*$ has a finite number of states
  - Structural errors are normally distributed
  - $X$ is non-normal
- Estimation: Use an EM algorithm (or maximum likelihood) to jointly recover $Z^*$, the mixture probabilities, and the outcome regression parameters.
Gaussian Copula
- Model setup: Specify a joint distribution of $(X,;\varepsilon)$ via a Gaussian copula linking marginal distributions.
- Assumptions:
  - $X$ is non-normal
  - $\varepsilon$ is normal
  - Dependence captured entirely by the copula correlation parameter
- Estimation: Fit marginal distributions and copula correlation by maximization of the joint likelihood, then recover the causal effect from the structural equation conditional on the estimated copula.

Pros and Cons

Method	Pros	Cons
LIV	• Avoids need for external IVs • Leverages latent class structure • Extensions to dynamic and nonlinear models exist	• Relies on an untestable assumption that exogenous variation is discrete • Normal-error and non-normal-regressor assumptions • Model complexity and identification hinge on number of latent states
GC	• Simple implementation (standard likelihood techniques) • No need for instruments beyond distributional assumptions	• Sensitive to skewness in $X$ and to deviations from normality in $\varepsilon$ • Requires large samples for stable estimates • Panel data require transformations that complicate the copula structure

Key takeaways

Instrument-free methods can be powerful when valid external instruments are unavailable, but they substitute one set of strong (often untestable) assumptions for another.
Careful diagnostic checks—testing residual normality, examining the distribution of $X$, and sensitivity analyses—are essential to ensure credible inference.

Reference

Park, Sungho and Sachin Gupta (2012), “Handling Endogenous Regressors by Joint Estimation Using Copulas,” Marketing Science, 31 (4), 567–86.

A Review of Copula Correction Methods to Address Regressor–Error Correlation

Notes on Instrumental Variables

Thu, 29 May 2025 00:00:00 +0000

Here are my notes on instrumental variables from Stefan’s lecture materials.

Motivation

How can we identify causal eﬀects $W \rightarrow Y$ when we are in the presence of unobserved confounding $U$?

One popular way is to ﬁnd and use instrumental variables.

Partially Linear IV Models

When instrumental variables are available, it becomes possible to point identify causal effects in partially linear models and certain types of causal effects in nonlinear models.

The keyword in this section is linear, constant effect IV

Here we begin with partially linear models.

Assumption 1 (partially linear).

Assume the structural equation for $Y$ is linear: $$ Y=f_Y\left(W, U, \varepsilon_Y\right)=\alpha+W \tau+\varepsilon \tag{PLM}, $$ where $\epsilon$ is an error term that captures the contribution of both $U$ and $\epsilon_Y$.

This is a semiparametric specification, in that we impose a linear relationship between $W$ and $Y$ but let the rest be non-parametric. Notice that, besides the linearity assumption, we also assume a constant treatment effect (i.e. $\tau$), which is also a strong assumption.

Remark 1 (partialling out X).

I ignore the role of covariates $X$ and the possible extension like $$ \varepsilon_i \indep Z_i \mid X_i $$ because we can recover the same DAG after partialling out observed confounder $X$. As an illustration, we can use $\tilde{Y}, \tilde{W}, \tilde{Z}$, where $$\tilde{V} = V - \E[V \mid X]$$

Remark 2 (relax linearity assumption).

The key limitation here is that we assume the linear structure and constant treatment effect. Later, we will consider IV estimator without the linearity assumption.

When the constant treatment eﬀect model (PLM) doesn’t hold, the average treatment eﬀect $\tau_{ATE} = \E [Y_i (1) − Y_i (0)]$ is NOT identified without more data, because we don’t have any observations on treated never takers, etc. Without linearity, the estimator $\tau_{IV}$ still converges to a large-sample limit $\tau_{LATE}$, the local average treatment effect (LATE).

Identifying assumptions

There are 3 identification assumptions. Or let’s say there are three main assumptions that must be satisfied for a variable $Z$ to be considered an instrument.

Assumption 2 (exogeneity).

The instrument $Z_i$ must be exogenous, which here means $\epsilon_i \indep Z_i$

Assumption 3 (relevance).

The instrument $Z_i$ must be relevant, such that $\Cov[W_i, Z_i] \neq 0$

Graphically, the relevance assumption corresponds to the existence of an active edge from $Z$ to $W$ in the causal graph.

Assumption 4 (exclusion restriction).

The instrument $Z_i$ must satisfy exclusion restriction, meaning that any eﬀect of $Z_i$ on $Y_i$ must be fully mediated via the treatment $W_i$

Graphically, this means that we’ve excluded enough potential edges between variables in the causal graph so that all causal paths from $Z$ to $Y$ go through $W$.

Case I (Easiest)

Consider the fully linear version,

$$ \begin{align} & Y=\alpha+W \tau+\varepsilon, \quad \varepsilon \indep Z \tag{C1} \\ & W=Z \gamma+\eta \end{align} $$

Then,

$$\operatorname{Cov}[Y, Z]=\operatorname{Cov}[\tau W+\varepsilon, Z]=\tau \operatorname{Cov}[W, Z] $$ $$\tau= \frac{\operatorname{Cov}[Y, Z]}{\operatorname{Cov}[W, Z]}$$

It implies,

$$ \hat{\tau}_{IV}= \frac{\widehat{\operatorname{Cov}}[Y_i, Z_i]}{\widehat{\operatorname{Cov}}[W_i, Z_i]} $$

Case II (More general: optimal instruments)

Case I assumes (1) linear relationship between $Y$ and $W$ and (2) linear relationship between $W$ and $Z$. This may be too restrictive. How to extend to more general specification? What should we do if

we have multiple instruments
or we believe that the instrument may act non-linearly

Consider the following,

$$ Y=\tau W+\varepsilon, \quad \varepsilon \indep Z, \quad Y, W \in \mathbb{R}, \quad Z \in \mathcal{Z}, \tag{C2} $$

where $\mathcal{Z}$ can be a high-dimensional space. Define function $w$ that maps $Z_i$ to the real line $$w: \mathcal{Z} \rightarrow \R$$

Then by the same argument as the Case I (note: we can regard $w(Z)$ as a “pseudo $Z$”),

$$ \tau=\frac{\operatorname{Cov}[Y, w(Z)]}{\operatorname{Cov}[W, w(Z)]} $$

provided the denominator is non-zero, resulting a feasible estimator

$$ \hat{\tau}_{I V}=\frac{\widehat{\operatorname{Cov}}\left[Y_i, w\left(Z_i\right)\right]}{\widehat{\operatorname{Cov}}\left[W_i, w\left(Z_i\right)\right]}=\frac{\frac{1}{n} \sum_{i=1}^n\left(Y_i-\bar{Y}\right)\left(w\left(Z_i\right)-\overline{w(Z)}\right)}{\frac{1}{n} \sum_{i=1}^n\left(W_i-\bar{W}\right)\left(w\left(Z_i\right)-\overline{w(Z)}\right)} $$

Theorem 1.

Suppose $\left(X_i, W_i, Y_i, Z_i\right)$ are IID draws from a distribution satisfying (C2), and let $w: \mathcal{Z} \rightarrow \mathbb{R}$ be such that $\operatorname{Cov}[W, w(Z)] \neq 0$. Then, $\hat{\tau}_{I V}$ as given above is consistent for $\tau$, and $$ \sqrt{n}\left(\hat{\tau}_{I V}-\tau\right) \Rightarrow \mathcal{N}\left(0, V_w\right), \quad V_w=\frac{\operatorname{Var}\left[\varepsilon_i\right] \operatorname{Var}\left[w\left(Z_i\right)\right]}{\operatorname{Cov}\left[W_i, w\left(Z_i\right)\right]^2} $$

What is the best function $w$, say $w^*(z)$, that minimizes the variance $V_w$? It turns out that the optimal instrument is the best prediction of $W_i$ from $Z_i$.

$$ w^*(z)=\mathbb{E}\left[W_i \mid Z_i=z\right] $$

How to do we estimate? Do cross-fitting! Why? Because

$$ \epsilon_i \rightarrow W_i \rightarrow \hat{w}(Z_i) $$

We no longer have $\hat{w}(Z_i) \indep \epsilon_i$. Therefore, we need cross-ﬁtting to address this issue.

Case III (Much more general: non-parametric IV regression)

The more general version than Case II is the following,

$$ Y_i=\alpha+g\left(W_i\right)+\varepsilon_i, \quad Z_i \indep \varepsilon_i, \quad Y_i, W_i \in \mathbb{R}, \quad Z_i \in \mathcal{Z} \tag{C3} $$

where $g(\cdot)$ is some generic smooth function we want to estimate. Note that:

(C3) still requires the eﬀect of $W_i$ on $Y_i$ to be additive
however, unlike (C2), it now allows this additive eﬀect to be modified by a non-linearity $g(\cdot)$

Now,

$$ \begin{aligned} \mathbb{E}\left[Y_i \mid Z_i=z\right] & =\mathbb{E}\left[\alpha+g\left(W_i\right)+\varepsilon_i \mid Z_i=z\right] \\ & =\alpha+\mathbb{E}\left[g\left(W_i\right) \mid Z_i=z\right] \\ & =\alpha+\int_{\mathbb{R}} g(w) f(w \mid z) d w, \end{aligned} $$

There are two steps for learning $g(\cdot)$:

non-parametric model $\hat{f}(w \mid z)$ using cross-fitting
estimate $g(\cdot)$ using empirical minimization

For more details, see Section 9.2 in Stefan’s lecture.

Local Average Treatment Effects

Motivation

IV without the linearity assumption: One may doubt the validity the linearity and constant treatment eﬀect assumption in previous section. What about non-parametric identification using IV?
Encouragement Design and Noncompliance: Noncompliance is a common problem in encouragement designs involving human beings as experimental units. In those cases, the experimenters cannot force the units to take the treatment but rather only encourage them to do so. Heterogeneous effects should be allowed.

Setup

Consider an randomized experiment,

Let $Z_i \in \{0, 1\}$ be the treatment assigned
Let $W_i \in \{0, 1\}$ be the treatment received
When $Z_i \neq W_i$, the noncompliance problem arises
Potential outcome $\{W_i(1), W_i(0)\}$ s.t. $W_i = W_i(Z_i)$
Potential outcome $\{Y_i(w,z)\}_{(w,z) \in \{0,1\}}$ s.t. $Y_i = Y_i(W_i, Z_i)$

Identifying assumptions

LATE Theorem

Idea of proof:

Start with $\Cov(Y, Z) = \E[YZ] - \E[Y]\E[Z]$, then apply LIE by conditioning on $Z$
Derive $\Cov(W, Z)$ similarly as above, then get the ratio
Decompose the ATE on $Y$, $\E[Y(1) - Y(0)]$, into four terms (always-taker, compiler, defier, never-taker) using law of total probability
By exclusion restriction and monotonicity assumption, only “compiler” remains

Multiple instruments

We may have access to data from multiple randomized trials that can be used to study a treatment effect via a non-compliance analysis.

Marketing example:

goal: study the effect of subscription to a loyalty program ($W_i$) on long-term customer CLV ($Y_i$)
randomized trial 1: offering discounts for joining the loyalty program $\left(Z_i=\1(\{\right.$ customer received a discount $\left.\})\right)$
randomized trial 2: showing advertisements $\left(Z_i=\1(\{\right.$ customer was shown an ad for the program $\left.\})\right)$

Previously, under the linear treatment effect model, multiple instruments could be combined into a single optimal instrument, and the optimal instrument corresponds to the summary of all the instruments that best predicts the treatment.

Without the linear treatment effect model, however, we caution that no such result is available. Diﬀerent instruments may induce diﬀerence compliance patterns, and so the LATEs identiﬁed diﬀerent instruments may not be the same.

In the marketing example, the ATE for customers who respond to a discount may be diﬀerent from the ATE for customers who respond to an advertisement.

Reference

Wager, S. (2024). Causal inference: A statistical learning approach. https://web.stanford.edu/~swager/causal_inf_book.pdf

Hausman-Taylor estimator Notes

Wed, 27 Mar 2024 00:00:00 +0000

Hausman-Taylor in R

The following example is from here.

Example: The fixed effects model, however, does not allow time-invariant variables such as educ or black. Since the problem of the random effects model is endogeneity, one can use instrumental variables methods when time-invariant regressors must be in the model. The Hausman-Taylor estimator uses instrumental variables in a random effects model; it assumes four categories of regressors: time-varying exogenous, time-varying endogenous, time-invariant exogenous, and time-invariant endogenous. The number of time-varying variables must be at least equal to the number of time-invariant ones. In our wage model, suppose exper, tenure and union are time-varying exogenous, south is time-varying endogenous, black is time-invariant exogenous, and educ is time-invariant endogenous. The same plm() function allows carrying out Hausman-Taylor estimation by setting model= “ht”.

wage.HT <- plm(lwage~educ+exper+I(exper^2)+
      tenure+I(tenure^2)+black+south+union |
      exper+I(exper^2)+tenure+I(tenure^2)+union+black,
      data=nlspd, model="ht")
kable(tidy(wage.HT), digits=5, caption=
     "Hausman-Taylor estimates for the wage equation")

Note that, the instruments are specified at the end of the formula after a | sign (pipe). For more details, check

library(plm)
browseVignettes("plm")

Reference

check the plm 📦 in R.

plm 📦 github repo: https://github.com/ycroissant/plm?tab=readme-ov-file
plm 📦 cran: https://cran.r-project.org/web/packages/plm/index.html

Additional Resources for Panel Data Analysis

计量经济学笔记（三）：面板数据分析