HC3 is the finite-sample variant of the heteroskedasticity-consistent covariance estimator proposed by MacKinnon and White (1985). It weights each squared OLS residual by the inverse squared leverage, $$\widehat{V}_{HC3}=(X^\top X)^{-1}\left(\sum_i \frac{\hat e_i^2}{(1-h_{ii})^2}\, x_i x_i^\top\right)(X^\top X)^{-1}$$, where $$h_{ii}$$ is the i-th diagonal of the hat matrix $$H=X(X^\top X)^{-1}X^\top$$. The leverage correction approximates a jackknife variance and inflates the contribution of high-leverage points, yielding better Type I error control than HC0/HC1 in small samples. Long and Ervin (2000) recommend HC3 as the default when $$N<250$$.

HC3 Heteroskedasticity-Consistent Standard Errors

Heteroskedasticity-consistent (HC) standard errors are 'sandwich' variance estimators for regression coefficients that remain valid when the error variance is not constant across observations. Introduced by White (1980), the basic HC0 estimator replaces the usual homoskedastic variance with $$\widehat{V}=(X^\top X)^{-1}\left(\sum_i \hat e_i^2\, x_i x_i^\top\right)(X^\top X)^{-1}$$, using squared OLS residuals $$\hat e_i$$. This delivers asymptotically correct standard errors under arbitrary heteroskedasticity without requiring the analyst to model the variance structure.

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Causal inference refers to making conclusions about finding the reasonings behind the relationship of two or more variables. After coming to some conclusions about statistical data, we ask why such relationships exist between variables.

Causal Inference

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A machine passes this test if a human could perceive their interactions with the machine as human, indicating that the machine has successfully emulated human reasoning. 

Turing Test

- Causal Inference Books
- Causal Inference Papers

Causal Inference References

The calculus of causation is formed of:
1. Causal diagrams (the language of knowledge)
2. Causal symbolic language (the language of queries)

The calculus of causation

Organisms, as well as ML, can be divided into 3 levels of causation corresponding to three cognitive abilities: seeing, doing, and imagining.
1. Association
2. Intervention
3. Counterfactuals


Ladder of Causation 

Bayes Theorem is about predicting the probability of an event A occurring given an event B has already occurred. We can use the inverse probability of this statment and the probability of each event to find this value. 
P(A) is called prior probability, P(B|A) is Likelihood, P(B) is evidence, and P(A|B) is posterior probability.


Bayes Theorem Overview

Most of the statistical methods are entirely objective, which means reasoning exclusively from data and experiments. But causal analysis and Bayesian analysis are different in that they involves some subjectivity.

Causal analysis requires some subjective commitments based on one's knowledge in the field before the causal digram is drawn and data is explored.

Similarly, Bayesian analysis combines objective evidence with the prior subjective belief.

From objectivity to subjectivity

Both Induction and Deduction are necessary within casual inference because they are two ways of approaching a problem and often go hand in hand. Induction goes from evidence -> hypothesis while deduction works in the forward direction; cause (hypothesis) -> effect (theory, conclusion). Artificial Intelligence has started automating Induction and Deduction, an example is Bayesian networks. 

Stages of Casual Inference: Induction and Deduction

Logical reasoning is the process of using a rational, systematic series of steps based on sound certain mathematical procedures and statements to arrive at a conclusion. 

There are three types of the reasoning:
- Abductive Reasoning 
- Deductive Reasoning 
- Inductive Reasoning

Reasoning

Hill's Criteria is a collection of nine informal guidelines ("viewpoints" as supposed to requirements) used to determine whether a causal relationship exists within an observed association.  Initially, five out of the nine viewpoints were proposed by the surgeon general's advisory committee, to which Bradford Hill later added four more.  Hill's Criteria is not meant to be the necessary conditions to support or refute a cause-and-effect hypothesis.  The debate over the causality between cigarette and lung cancer prompted the criteria, which were used for the proponent of the causal relationship to justify their claim that cigarette smoking is indeed causally related to lung cancer.  

Hill's Criteria

1. necessary causation (but-for causation)
2. sufficient causation
3. necessary-and-sufficient causation

Three different kinds of causation

1. The Law of Counterfactuals
2. The Law of Conditional Independence

The Two Fundamental Laws of Causal Inference

A Randomized Controlled Trial (RCT) or a Controlled Experiment is a method to eliminate unmeasurable confounders effect on causal inference by implementing an intervention on a randomly selected group of individuals (treatment/intervened group) while keeping other individuals unchanged (controlled group, placebo, not-intervened group) to observe the changes in outcomes due to the intervention effect. The RCTs are intended to "disarm" confounders (confounding variables).

Randomized Controlled Trial (RCT) = Controlled Experiment

List of techniques that help solve intractable inference problems in machine learning that usually arise from interactions between latent variables in a structured graphical model.

-Expectation Maximization

-MAP Inference and Sparse Coding

-Variational Inference and Learning

-Learned Approximate Inference



Approximate Inference 

"The estimand is the target quantity—the purpose of the statistical analysis ... the thing we are estimating" " that connects statistical evidence to theory."

It refers "to both the theoretical and the empirical estimands."

Estimand

1- "Set a **theoretical estimand**, clearly connecting this quantity to theory"

2- "Link to an **empirical estimand**, which is informative about the theoretical estimand under some identification assumptions"

3- Use "an **estimation strategy** to learn the empirical estimand (e.g., a regression model) ... from data."

Three Critical Choices in Causal Inference

- Correlation does not imply causation.
- Causation does not imply correlation either.

Correlation vs. Causation

Establishing causality in economics is a significant challenge for two main reasons. First, unlike in many natural sciences, it is often impossible to conduct controlled experiments to isolate a single variable. The economy is a dynamic system where numerous factors change simultaneously, creating confounding variables that can lead to misleading correlations. Second, the economy's fundamental complexity, arising from the interactions of millions of people, makes it impossible to measure and understand every variable. This inherent complexity means economists must be skeptical of simple claims that one complex phenomenon, like capitalism, is the direct cause of another, such as a rise in living standards. To address these challenges, economists use methods like natural experiments and advanced statistical techniques to infer causal relationships from observational data.

The Challenge of Establishing Causality in Economics

Instrumental Variables (IV) estimation is a causal-inference method used when a regressor of interest is endogenous, i.e., correlated with the error term. An instrument is a variable that (i) is correlated with the endogenous regressor (relevance) and (ii) affects the outcome only through that regressor (exclusion restriction). Under these conditions, IV recovers a consistent estimate of the causal effect that ordinary least squares cannot.

Instrumental Variables Estimation

An encouragement design is a randomized experiment in which units are randomly assigned an encouragement to take a treatment rather than the treatment itself. Because compliance is partial, the random encouragement serves as an instrument for the realized exposure: it satisfies random assignment, is associated with uptake (relevance), and is assumed to affect the outcome only through that uptake (exclusion). The encouragement-design framework links the intention-to-treat effect of encouragement to the causal effect of the treatment among compliers, providing the standard bridge between randomization and instrumental-variables identification.

Encouragement Design (Randomized Encouragement)

Heteroskedasticity-Consistent (HC) Standard Errors

The Intent-to-Treat (ITT) effect is the average causal effect of being **assigned** to a treatment condition, regardless of whether the assigned unit actually complied with or received the treatment. In a randomized experiment with outcome $$Y$$ and binary assignment $$Z\in\{0,1\}$$, the ITT effect is $$\mathrm{ITT}=\mathbb{E}[Y\mid Z=1]-\mathbb{E}[Y\mid Z=0]$$. Because $$Z$$ is randomized, ITT is unbiased for the effect of the **offer/assignment** itself, but under partial compliance it underestimates the effect of the treatment when actually received.

Intent-to-Treat (ITT) Effect

The Treatment-on-the-Treated (TOT) effect, also called the average treatment effect on the treated (ATET), is the average causal effect of **actually receiving** the treatment among the subpopulation that took it up. Under one-sided non-compliance (no-shows) with random assignment $$Z$$ to treatment $$D$$, Bloom (1984) shows it can be estimated as $$\widehat{\mathrm{TOT}}=\dfrac{\mathbb{E}[Y\mid Z=1]-\mathbb{E}[Y\mid Z=0]}{\Pr(D=1\mid Z=1)}$$, i.e., the ITT scaled by the compliance rate. With two-sided non-compliance and a monotonicity assumption, this is generalized to the Local Average Treatment Effect (LATE) among compliers (Imbens & Angrist, 1994).

Treatment-on-the-Treated (TOT) Effect

When a randomized experiment has imperfect compliance, two distinct causal estimands separate: the **Intent-to-Treat (ITT)** effect captures the impact of **assignment**, while the **Treatment-on-the-Treated (TOT)** effect captures the impact of **actually receiving** the treatment among takers. Under randomization, ITT is unbiased for the effect of the offer but is attenuated toward zero relative to the effect of exposure as compliance shrinks. TOT can be recovered from ITT by scaling by the compliance rate (Bloom, 1984) or, more generally, by instrumental-variables estimation using assignment as an instrument for realized exposure, which identifies the LATE among compliers under monotonicity (Imbens & Angrist, 1994; Angrist, Imbens & Rubin, 1996). Reporting ITT alone hides the effect of exposure; reporting TOT/IV alone hides what happens at scale when many assigned users do not comply. The distinction is what justifies running an IV on a randomized default rather than just reporting the assignment-level contrast.

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