MOD2

Random sampling

The key idea that one draw from a population does not depend on another.

Infer

The process of learning about the underlying data-generating process from the data we observe.

Frequentist approach to probability

A definition of probability as the number of times an event occurs out of an infinite number of random trials.

Law of Large Numbers (Relative Frequency)

The idea of relative frequency converging to the true probability.

Lognormal distribution

A model often used for earnings data because earnings distributions tend to be skewed right.

Estimand

The thing we want to learn about.

Estimator

The thing we compute to learn about the estimand.

Estimate

The result given by an estimator for a given set of data.

Unbiased

A state where $E(\text{estimator})$ equals the thing we want to learn about.

Bias

A potential result of sample selection if the data does not represent the population we want to learn about.

Conditional expectation

The method used to compare earnings of women and men by estimating earnings given gender.

Expected value

A weighted average of all the random variable's possible outcomes.

Law of Large Numbers (Sample Size)

The principle that sample averages tend to the population mean as sample size grows.

Indicator variable expected value

For a variable taking values $1$ and $0$, this is equivalent to the probability that the random variable equals $1$.

LIE (Law of Iterated Expectations)

The law stating that the expected value of the CEF of $Y$ given $X$ is the expected value of $Y$.

Frequentism

An approach to probability defined as the relative frequency of an event $A$ in the limit of $N$ random trials, expressed as $P(A) = \text{lim}_{N \rightarrow \text{∞}} \frac{S_N}{N}$, where $S_N$ is the number of occurrences.

Law of Large Numbers (LLN)

A fundamental result in statistics stating that as the number of trials increases, the sample average converges to the population expected value, such that $\bar{Y} \rightarrow E(Y)$ as $N \rightarrow \text{∞}$.

Estimand

The theoretical concept or population parameter we want to learn about, such as the true probability P(\text{earnings} \req 100,000).

Estimator

A rule or mathematical formula applied to a sample to infer the value of an estimand, denoted by a "hat" symbol, such as \bar{Y} = \frac{1}{N} \text{∑}_{i=1}^{N} Y_i.

Estimate

The specific numerical value observed when an estimator is applied to a particular set of data.

Unbiased Estimator

An estimator that is equal to the estimand on average, defined by the condition $E(\text{estimator}) = \text{estimand}$.

Topcoding

A technique used in data collection to protect respondent privacy by clustering the highest observed values at a specific limit, such as earnings capped at $400,000$ in the March 2009 CPS.

Grammar of Graphics

A framework implemented in the ggplot2 package for building visualizations from components including data, aesthetic mappings, geometric objects, scales, facet specifications, statistical transformations, and coordinate systems.

Full-time employed (CPS 2009 Definition)

Individuals in the March 2009 CPS extract who worked at least $36$ hours per week for at least $48$ weeks during the previous year.

Expected Value ($E(Y)$)

The weighted average of all possible outcomes of a random variable. For a discrete variable $Y$, it is calculated as E(Y) = \text{∑}_{j=1}^{J} y_j f(y_j), where $f(y_j)$ represents the weights from the PDF.

Conditional Expectations Function (CEF)

Known as the "workhorse of data science," it defines the probabilities of a random variable $Y$'s values conditional on the value of another variable $X$, denoted as $E(Y | X)$ or $μ(X)$.

Plug-in Principle

The justification for estimating population expectations by substituting them with their corresponding sample averages under random sampling.

Expected Value of an Indicator Variable

The outcome where the expected value of a variable taking values represented by $1$ and $0$ equals the probability that the indicator is $1$, expressed as $E(Y) = P(Y = 1)$.

Law of Iterated Expectations (LIE)

A theorem stating that the unconditional expectation of a random variable is equal to the weighted average of its conditional expectations: $E(Y) = E[E(Y|X)]$, or specifically $E(\text{earnings}) = E[E(\text{earnings}|\text{gender})]$.

Difference-in-logs Approximation

A method where a difference in natural logs is used to approximate percentage change: $\text{log}(Y_2) - \text{log}(Y_1) \text{≈} \frac{Y_2 - Y_1}{Y_1}$. The approximation is more accurate for smaller percentage changes.

Jensen's Inequality

A theorem relating the log of an average to the average of logs; for a concave function $g$, it states g(E[x]) \req E[g(x)], meaning the log of the average is greater than or equal to the average of the logs.

Censoring

A source of bias where data values are limited or obscured at a certain threshold, such as the topcoding of high earnings.

Sample Selection Bias

A bias occurring when the collected data does not accurately represent the population of interest or when the process of selecting the sample is non-random.

Measurement Error

A source of bias that occurs when a variable's empirical measurement fails to accurately capture the theoretical concept of interest.