Appendix A: Mathematical Notation

This appendix provides a reference for the mathematical notation used throughout the book.

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General Conventions

Symbol	Meaning
Lowercase letters ($x, y, z$)	Scalars or realized values of random variables
Uppercase letters ($X, Y, Z$)	Random variables
Bold lowercase ($\mathbf{x}, \mathbf{y}$)	Vectors
Bold uppercase ($\mathbf{X}, \mathbf{Y}$)	Matrices
Greek letters ($\alpha, \beta, \theta$)	Parameters
Hat ($\hat{\theta}$)	Estimator or estimate
Bar ($\bar{X}$)	Sample mean
Tilde ($\tilde{X}$)	Transformed or residualized variable

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Subscripts and Superscripts

Notation	Meaning
$Y_i$	Outcome for unit $i$
$Y_{it}$	Outcome for unit $i$ at time $t$
$X_k$	The $k$-th covariate
$\beta_j$	Coefficient on variable $j$
$Y^{(1)}, Y^{(2)}$	Different outcomes or transformations
$X'$	Transpose of matrix/vector $X$
$X^{-1}$	Inverse of matrix $X$

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Treatment and Potential Outcomes

Symbol	Meaning
$D$ or $D_i$	Treatment indicator (1 = treated, 0 = control)
$Y(1)$	Potential outcome under treatment
$Y(0)$	Potential outcome under control
$Y(d)$	Potential outcome under treatment status $d$
$\tau_i = Y_i(1) - Y_i(0)$	Individual treatment effect
$Z$	Instrument

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Treatment Effect Estimands

Symbol	Definition	Name
$\text{ATE}$	$E[Y(1) - Y(0)]$	Average Treatment Effect
$\text{ATT}$	$E[Y(1) - Y(0) \mid D = 1]$	Average Treatment Effect on the Treated
$\text{ATU}$	$E[Y(1) - Y(0) \mid D = 0]$	Average Treatment Effect on the Untreated
$\text{LATE}$	$E[Y(1) - Y(0) \mid \text{Compliers}]$	Local Average Treatment Effect
$\text{CATE}(x)$	$E[Y(1) - Y(0) \mid X = x]$	Conditional Average Treatment Effect
$\tau(x)$	Shorthand for $\text{CATE}(x)$	Treatment effect function

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Probability and Expectation

Symbol	Meaning
$P(A)$	Probability of event $A$
$P(A \mid B)$	Conditional probability of $A$ given $B$
$E[X]$	Expected value of $X$
$E[X \mid Y]$	Conditional expectation of $X$ given $Y$
$\text{Var}(X)$	Variance of $X$
$\text{Cov}(X, Y)$	Covariance of $X$ and $Y$
$\text{Corr}(X, Y)$	Correlation of $X$ and $Y$
$\sigma^2$	Variance (population)
$s^2$	Variance (sample)
$\sigma_{XY}$	Covariance between $X$ and $Y$

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Distributions

Notation	Meaning
$X \sim F$	$X$ is distributed according to $F$
$X \sim N(\mu, \sigma^2)$	Normal distribution with mean $\mu$, variance $\sigma^2$
$X \sim \text{Bernoulli}(p)$	Bernoulli with probability $p$
$X \sim \text{Binomial}(n, p)$	Binomial distribution
$X \sim \chi^2_k$	Chi-squared with $k$ degrees of freedom
$X \sim t_k$	Student's t with $k$ degrees of freedom
$X \sim F_{k_1, k_2}$	F-distribution
$\Phi(\cdot)$	Standard normal CDF
$\phi(\cdot)$	Standard normal PDF

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Convergence

Notation	Meaning
$\xrightarrow{p}$	Convergence in probability
$\xrightarrow{d}$	Convergence in distribution
$\xrightarrow{a.s.}$	Almost sure convergence
$= O_p(1)$	Bounded in probability
$= o_p(1)$	Converges to zero in probability
$\approx$	Approximately equal
$\propto$	Proportional to

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Regression and Estimation

Symbol	Meaning
$\beta$	Coefficient vector
$\hat{\beta}$	OLS (or other) estimator
$\hat{\beta}_{OLS}$	OLS estimator specifically
$\hat{\beta}_{IV}$ or $\hat{\beta}_{2SLS}$	IV/2SLS estimator
$\varepsilon$, $u$, $e$	Error terms (various contexts)
$\hat{\varepsilon}$, $\hat{u}$, $\hat{e}$	Residuals
$R^2$	R-squared (coefficient of determination)
$\bar{R}^2$	Adjusted R-squared
$\text{SE}(\hat{\beta})$	Standard error of estimator

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Propensity Scores and Weights

Symbol	Meaning
$e(X)$ or $p(X)$	Propensity score: $P(D = 1 \mid X)$
$\hat{e}(X)$	Estimated propensity score
$w_i$	Weight for observation $i$
$\text{IPW}$	Inverse probability weighting
$\text{AIPW}$	Augmented inverse probability weighting

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Panel Data and Time Series

Symbol	Meaning
$i$	Cross-sectional unit index
$t$	Time period index
$T$	Number of time periods
$N$	Number of cross-sectional units
$\alpha_i$	Unit fixed effect
$\gamma_t$ or $\lambda_t$	Time fixed effect
$L$	Lag operator: $LY_t = Y_{t-1}$
$\Delta$	First difference operator: $\Delta Y_t = Y_t - Y_{t-1}$

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Difference-in-Differences

Symbol	Meaning
$\text{Post}_t$	Post-treatment indicator
$\text{Treat}_i$	Treatment group indicator
$\delta$ or $\tau$	DiD treatment effect
$Y_{it}(0)$, $Y_{it}(1)$	Potential outcomes in panel setting

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Regression Discontinuity

Symbol	Meaning
$X$ or $R$	Running variable
$c$	Cutoff value
$h$	Bandwidth
$\tau_{RD}$	RD treatment effect at cutoff

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Instrumental Variables

Symbol	Meaning
$Z$	Instrument(s)
$\pi$	First-stage coefficient
$\rho$	Reduced-form coefficient
$\beta = \rho / \pi$	Wald estimator

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Matrix Notation

Symbol	Meaning
$\mathbf{X}$	$n \times k$ matrix of regressors
$\mathbf{y}$	$n \times 1$ outcome vector
$\mathbf{I}_n$	$n \times n$ identity matrix
$\mathbf{0}$	Zero vector or matrix
$\mathbf{1}$	Vector of ones
$\text{tr}(\mathbf{A})$	Trace of matrix $\mathbf{A}$
$$	\mathbf{A}
$\text{rank}(\mathbf{A})$	Rank of matrix

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Independence and Conditional Independence

Notation	Meaning
$X \perp Y$	$X$ is independent of $Y$
$X \perp Y \mid Z$	$X$ is independent of $Y$ conditional on $Z$
$(Y(0), Y(1)) \perp D \mid X$	Conditional independence assumption

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Indicators and Sets

Notation	Meaning
$\mathbf{1}\{A\}$ or $\mathbb{I}(A)$	Indicator function (1 if $A$ true, 0 otherwise)
$\{x : \text{condition}\}$	Set of $x$ satisfying condition
$\in$	Element of
$\subset$	Subset of
$\cup$, $\cap$	Union, intersection
$\emptyset$	Empty set
$\mathbb{R}$	Real numbers
$\mathbb{R}^k$	$k$-dimensional real space

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Summation and Products

Notation	Meaning
$\sum_{i=1}^n x_i$	Sum from $i = 1$ to $n$
$\prod_{i=1}^n x_i$	Product from $i = 1$ to $n$
$\bar{x} = \frac{1}{n}\sum_{i=1}^n x_i$	Sample mean

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Limits and Asymptotics

Notation	Meaning
$\lim_{n \to \infty}$	Limit as $n$ approaches infinity
$\text{plim}$	Probability limit
$n \to \infty$	Sample size goes to infinity
$\sqrt{n}(\hat{\theta} - \theta)$	Scaled deviation (for CLT)

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Calculus and Optimization

Notation	Meaning
$\frac{\partial f}{\partial x}$	Partial derivative
$\nabla f$	Gradient
$\frac{\partial^2 f}{\partial x^2}$	Second derivative
$\mathbf{H}$	Hessian matrix
$\arg\max_x f(x)$	Value of $x$ that maximizes $f$
$\arg\min_x f(x)$	Value of $x$ that minimizes $f$

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Chapter-Specific Notation

DAGs (Chapter 9)

• $\to$: Direct causal effect

• $\leftarrow$: Reverse causal direction

• $X \to M \to Y$: Chain (mediation)

• $X \leftarrow C \to Y$: Fork (confounding)

• $X \to C \leftarrow Y$: Collider

Meta-Analysis (Chapter 24)

• $\theta_i$: True effect in study $i$

• $\hat{\theta}_i$: Estimated effect in study $i$

• $\tau^2$: Between-study variance

• $I^2$: Heterogeneity measure

Machine Learning (Chapter 21)

• $\eta$: Nuisance parameters

• $\psi$: Influence function/moment condition

• $\hat{\mu}(x)$: Estimated outcome function

• $\hat{g}(x)$: Estimated propensity score

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For notation not listed here, consult the chapter where the symbol is first introduced.