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Course Outline

• Focus: Data, descriptive statistics, and causality
• Topics include:
  • Introduction to data
  • Samples and descriptive statistics
  • Conditional descriptive statistics
  • Causality and randomization
  • Statistical inference
  • Revealed preferences in observed data
• Methods: Quasi-experimental

Learning Objectives

• Characterizing conditional distributions
• Linear relationships between variables are characterized.
• Application of knowledge when interpreting data on income distributions.

Kernel Density Graphs

• Kernel density estimator is a weighted average for each value x.
• Bandwidth (h): measurement of the amount of data used around x.
• Kernel Function (Kh): weighting of observations within the bandwidth, and whether further observations from x get lower weight.

Conditional Descriptive Statistics

• Conditional descriptives are statistics of a variable, conditional on other variables.
• Conditional expectation is the most important.
  • Denoted as E[Y|X = x], representing the expectation of a random variable Y when another random variable X holds the value x.
• Conditional sample average is the empirical counterpart.
• Joint Distribution: All conditional descriptive statistics follow the joint distribution of two or more variables.

Joint Distribution (review)

• Cross Tabulation: Displays small data sets of two variables efficiently
  • Rows indicate the number of values that Y can take.
  • Columns indicate the number of values that X can take.
  • Cells report the number of observations with value (y, x).
• Alternatively, cross tabulation cells can report the share of observations with value (y, x).
• Joint Density Function: An empirical counterpart of the joint distribution, defined as fxy(x, y) = P(X = x, Y = y).

Marginal Distribution

• The marginal distribution of Y is defined as fy(y) = ∑ fxy(x,y)
• Value of X is not factored into the probability of Y.
• The marginal distribution of X is fx(x) = ∑ fxy(x,y)

Conditional Distribution

• A conditional distribution of Y: fy\x(y\x) = fxy(x,y) / fx(x)
• i.e., Probability that Y takes value y conditional that X takes value x
• One can estimate P(Y = b\X = w) where the "hats" indicate estimates.

Conditional Expectation

• The conditional expectation function (CEF), when Y is discrete is E[Y|X = x] = ∑ tfy\x(t\X = x).
• Population average of Y holding X fixed.
• Weight average of Y is used.
• The weight used for each value of Y is the share of sub-population.
• X can be a vector.

Income Distribution in the U.S.

• Basic results of income distribution literature.
  • Group averages are reviewed.
  • Changes over the entire distribution is researched.
  • Top percent shares, social mobility extras are reviewed.
• Researched based on the tax data that is available over long time periods in many countries.
  • Earlier periods are limited to the top.
  • Tax records do not capture all income.
• Surveys completed based on surveys.
  • Particularly the labor force survey.
• In the periods of 1946 - 1980, there was roughly 2% annual income growth across the distribution among "the 99%"
• From 1980-2018 income growth was faster among the more wealthy even among "the 99%", the very different from the rest
• CDFs for Income Distribution: The U.S. income distribution, 1962–2014, bottom 95 percentiles
  • 90/10 percentile ratio.
• CDF comparisons over time: the CDFs show that horizontal distance btw the CDFs = dollar change for each percentile.
  • Are not the same people, and percentiles are being compared.
• With very skewed distributions, CDFs are uninformative, and CDF changes are made visible.

Correlation

• Conditional expectation can detect variable associations.
• Scatter plots: show variable relationship observations.
• Persistence of income over time can be graphed in a scatter plot
• Correlation indicates how two variables' values aligned.
  • A correlation of 0.92 was observed.

Covariance

• Used to determine correlation.
• Formula: Cov(X, Y) = E[X – E(X)]E[Y – E(Y)].
• Empirical Counterpart: Cov(X, Y) = 1 / n ∑(xi – x) (yi – ỹ)
• For the example shown, the covariance is 256.6.
• Measurement unit is equal to the unit of X times the unit of Y.

Pearson Correlation Coefficient

• Scaled covariance that varies between -1 ≤ Cor(X, Y) ≤ 1, making the number easier to interpret.
• Cor(X, Y) = PX,Y = Cov(X, Y) / SD(X)SD(Y)
• Additional Examples
  • correlation = 1 observed in the presentation
  • correlation = 0.009 observed in the presentation
  • correlation 0 observed in the presentation
• Correlation measures a linear dependence.
  • It's possible to have dependence and zero correlation.

Regression

• A regression model is a closely associated approach for assessing linear dependence.
• A bivariate regression model: Y = βο + β₁X + €
  • Y = dependent variable
  • X = independent variable
  • € = residual or error term
  • βο = constant parameter
  • β₁ = regression coefficient parameter
• In a regression, the data was described best.
• In Ordinary Least Squares the variables can equal arg min βο,βι Σ[Υ; - (βο + β₁X;)]²
• Minimizes difference between observed data and the regression model's prediction.

Regression vs Correlation

• Correlation and bivariate regression are related.
• β1 = Cov(X, Y) / Var(X)
• Pearson correlation coefficient formula: Px,y = Cov(X, Y) / ✓ Var(X) Var(Y)✓
• Y ≈ B1 because Var(X) ≈ Var(Y)

Intergenerational Mobility

• A complementary way to think about inequality.
• Involves questions of equality of opportunity.
• Compares the extent to which people can inherit position vs compete of level playing field.
• Powerful, but incomplete measurement: E[pc|Pp = pp] pc is child's position pp is her parent's position

Regression & Expectation

• Conditional expectation function where Y is linear in X: E[Y|X = x] = βο + β₁×.
• If CEF isn't linear, regression provides an approximation.
• Regression is the minimum mean squared error linear CEF approximation.
• The approximation is often good enough, especially with flexible multivariate regression.

Multivariate Regression Model

• The formula is: Y = βο + β₁X + β2X² + €.
• In one example, this means the estimates that fit the data best are: βο = -37,549, β₁ = 2.857, β₂ = -31
• This is generally good for approx E[Y|X = x] within certain age range.
• This analysis found multivariate regressions are pretty good for approximating the figure.
• Less good for new born incomes
• Looking at the data through the lenses of several methods is a good idea.

End of Lecture Summary

• Marginal and conditional distribution. Conditional expectation function Scatter plots. Covariance and correlation Regression, ordinary least square (OLS)
• Stata code for those things is on MyCourses/More Materials/. Homework: In-class worksheet 2 due on MyCourses before next lecture.
• Aim measures the actual state by trying to determine, " "what is joint distribution of X and Y?"

Up Next: Causal Questions

• A need to be able to to evaluate the impact of X on Y.
  • Impacts: of education on earnings; marketing campaigns on sales; carbon tax on emissions; R&D subsidy on innovation; fiscal stimulus on unemployment.
• These are CAUSAL questions.
• Aim compare states.
• Aim " how would Y change if we changed X?"

Explore conditional descriptive statistics, focusing on characterizing conditional distributions and linear relationships between variables. Learn about kernel density estimators, bandwidth, and kernel functions. Apply knowledge to interpret income distribution data and quasi-experimental methods.