MOD3

CEF (Conditional Expectation Function)

A function that gives the expected value of some random variable $Y$ given the value of another random variable $X$.

Gender Pay Gap Variables

In the application of CEF to the gender pay gap, the $Y$ variable represents earnings and the $X$ variable represents gender.

Variance Formula

The expression $E\{[X - E(X)]^2\}$, which is used to define the variance of a random variable.

Covariance Formula

The expression $E\{[X - E(X)][Y - E(Y)]\}$, which defines the covariance between random variables $X$ and $Y$.

Relationship Estimation

To estimate the covariance, sample means for $E(X)$ and $E(Y)$ are plugged in, and the outer expectation is replaced with another sample mean.

Covariance

A measurement that indicates the direction of a relationship between variables but not the strength of that relationship.

Earnings and Age Correlation

The estimated correlation between earnings and age among 23-62 year-olds using the March 2009 CPS is $0.13$.

$E(\text{earnings} | \text{age})$ Estimation

The simplest estimation method is to plug in the sample mean earnings for each specific value of age.

Career Earnings Pattern

According to Figure 6, earnings tend to increase early in a career and plateau after roughly age 40.

Linear Career Earnings Model

A model that assumes the difference in earnings from one age to the next remains constant throughout a career.

Quadratic Career Earnings Model

A model that captures the concave shape of the earnings-age relationship, where the difference in earnings from one age to the next varies with age.

Earnings Peak Prediction

Using March 2009 CPS data, a quadratic model of $E(\text{earnings} | \text{age})$ predicts that earnings increase until approximately age 50.

Human Capital Theory

The theory that provides the justification for using a quadratic model to fit earnings and age data.

Variance

A measure of how a random variable $X$ deviates from its mean, defined as $\text{var}(X) = E\{[X - E(X)]^2\}$.

Covariance

A measure of the direction of the relationship between two random variables, defined as $\text{cov}(X,Y) = E\{[X - E(X)][Y - E(Y)]\}$.

Correlation coefficient

A normalized version of covariance that describes the strength of a linear relationship between two variables, calculated as $\text{corr}(X, Y) = \frac{\text{cov}(X,Y)}{\sqrt{\text{var}(X)} \times \sqrt{\text{var}(Y)}}$.

Standard deviation

The square root of the variance, shorthandly referred to as $\text{sd}$.

Degrees-of-freedom adjustment

The practice of dividing by $N - 1$ instead of $N$ when estimating variance and covariance to ensure the estimator is not biased.

Conditional Expectation Function (CEF)

The framework used for analysis represented as $E(Y | X)$, which focuses on average values of a dependent variable $Y$ conditional on the values of an independent variable $X$.

Hamilton Project Career Definition

A definition that treats a career as lasting 40 years, typically restricting samples to individuals between the ages of 23 and 62 inclusive.

Top-coding

An effect in the March CPS data where earnings are capped at a specific limit, creating a visible horizontal line of high-density points at the top of a scatterplot.

Linear model (Age-Earnings)

A model that assumes earnings are a linear function of age, specified as $E(\text{earnings} | \text{age}) = \beta_0 + \beta_1 \text{age}$, implying a constant earnings difference between ages.

Quadratic model (Age-Earnings)

A polynomial relationship of order 2 specified as $E(\text{earnings} | \text{age}) = \beta_0 + \beta_1 \text{age} + \beta_2 \text{age}^2$, which accommodates a concave earnings profile.

Concave relationship

The shape of the age-earnings profile suggested by human capital theory, where earnings increases are larger early in a career and decline toward the end.

Human capital theory

A theory that models education as an investment, predicting that individuals concentrate investments early in their careers when the payoff horizon is long, leading to a concave age-earnings profile.

Log earnings

A transformation of the earnings variable, denoted as $\ln(\text{earnings})$, used to model the relationship in terms of rates of return.

Returns to experience

The rate at which earnings change as labor-market experience grows, calculated as the derivative of the quadratic log-earnings CEF: $\frac{d E(\text{learnings} | \text{age})}{d \text{age}} = \beta_1 + 2\beta_2(\text{age} - 23)$.

Gender earnings gap

The difference between male and female average earnings; in the March 2009 23-62 age group, males had a mean of $\$64,189.77$ compared to females at $\$44,827.78$.

March 2009 CPS Sample

A dataset containing 50,742 individuals who worked at least 36 hours per week for at least 48 weeks, excluding those in the military.