E371 Class Recording

Overview

Professor Lantis develops simple linear regression from least-squares fitting through coefficient interpretation, hypothesis tests, confidence intervals, and Stata output. Examples using marijuana use and income, county COVID data, and student-athlete GPAs show how to interpret continuous and dummy variables, while repeated-sample simulations illustrate how sample size affects precision and how confidence intervals capture population coefficients.

Key takeaways

Ordinary least squares estimates the regression line by minimizing the sum of squared residuals, and the simple-regression slope equals the sample covariance of x and y divided by the sample variance of x.
A regression slope always describes the predicted outcome change for a one-unit predictor change; rescaling income from dollars to thousands changes the coefficient’s units and makes its interpretation more useful.
For a zero-slope null hypothesis, the regression test statistic is the estimated slope divided by its standard error; a confidence interval excluding zero supports rejection at the corresponding confidence level.
Larger samples make coefficient estimates more precise: repeated samples of 100 counties produce a tighter slope distribution than samples of 30, while smaller samples increase standard errors.
With a zero-one predictor, the slope is the difference between the predicted averages of the one and zero groups; with a zero-one outcome, predicted changes are probability changes.
A statistically significant coefficient does not by itself establish a large practical effect: the COVID-income example motivates expressing household income in $1,000 units rather than interpreting a per-dollar change.

Chapters

0:00 Least Squares: Choosing the Regression Line

The fitted line predicts each observation’s outcome as ŷ = β̂₀ + β̂₁x; the residual is the difference between actual y and predicted ŷ.
Ordinary least squares chooses the intercept and slope to minimize the sum of squared residuals, penalizing large prediction errors more heavily.
The estimated slope is the sample statistic used to approximate the unknown population slope; its formula is the covariance of x and y divided by the variance of x.

5:00 Interpreting the Intercept and Slope in Income Data

The slope β̂₁ predicts the change in y for a one-unit increase in x; the intercept β̂₀ predicts y when x equals zero.
In the example relating monthly marijuana-use days to income, the intercept predicts income for someone reporting zero use days.
The estimated slope indicates about $253 lower predicted income for each additional reported marijuana-use day per month; the fitted line slopes downward.

12:00 Scaling a COVID Vaccination Effect

In the county example, the outcome is COVID cases per 1,000 people and the predictor is the county vaccination percentage.
The intercept predicts about 50.9 cases per 1,000 people when the vaccination percentage is zero.
The slope is roughly −0.25 cases per 1,000 for a one-percentage-point increase; a 10-point increase corresponds to about 2.5 fewer cases per 1,000.

15:00 Sampling Variation and Regression Standard Errors

A regression coefficient estimated from one sample can differ from the population coefficient, just as a sample mean can differ from the population mean.
Across repeated samples, coefficient estimates are treated as a sampling distribution centered around the true population slope.
A coefficient test statistic uses the estimate minus the hypothesized value, divided by its standard error; Stata calculates the more complex regression standard error.

21:00 Testing Whether Vaccination Has a Regression Effect

For the usual test of whether x and y are related, the null hypothesis sets the slope to zero; the test statistic is therefore the estimated slope divided by its standard error.
The county vaccination regression reports a test statistic near −11 and a p-value rounded to zero, providing strong evidence against a zero slope.
Professor Lantis reviews two-sided standard-normal critical values of about 1.65, 1.96, and 2.58 for 90%, 95%, and 99% confidence levels.

25:00 Confidence Intervals and Alternative Slope Hypotheses

A slope confidence interval is formed by adding and subtracting a critical value times the coefficient’s standard error.
The income example’s approximate 95% interval runs from −$350 to −$150 per additional marijuana-use day; because it excludes zero, the zero-slope null is rejected.
If zero falls inside a confidence interval, the corresponding zero-slope null cannot be rejected at that confidence level.
The hypothesized slope need not be zero: the marijuana example’s interval also excludes a slope of −$100, so that value can be tested and rejected.

32:00 Reading Weak Evidence from Test Statistics and Intervals

For an oil-well births example, a standard error much larger than the estimated slope produces a test statistic with absolute value below one.
The reported p-value is about 0.96, and the confidence interval includes zero, so the data do not support rejecting a zero-slope null.
A confidence interval’s midpoint recovers the coefficient estimate: the example interval’s midpoint is close to the reported estimate of −3.9.
Professor Lantis notes that quiz questions may omit parts of Stata output and require students to reconstruct results from the remaining information.

37:00 Stata Examples: Implausible Intercepts and Income Units

A regression of county COVID cases on population has an intercept near −16, which predicts cases for a county with zero residents—a value outside the observed data range.
The population coefficient is highly significant, with a test statistic around 250, consistent with larger counties having more cases.
When household income is measured in dollars, its slope can look numerically small; dividing income by 1,000 expresses the predictor in thousands of dollars.
The scaled income regression gives a slope around −0.4 per $1,000, an interpretation more useful than the effect per dollar.

44:00 Simulating the Sampling Distribution and Interval Coverage

A Stata program repeatedly samples 30 counties from the full county dataset, runs the vaccination regression, and stores 1,000 slope estimates.
The mean of those 1,000 estimates is close to the population-data slope of roughly −0.25, and their distribution is approximately normal.
Repeating the exercise with samples of 100 produces a tighter distribution of estimates than samples of 30.
Professor Lantis uses simulated intervals to illustrate coverage: a nominal 90% interval, built with a critical value of 1.65, contains the population slope in roughly 90% of repeated samples.

53:00 Dummy Predictors: Comparing Student-Athlete GPAs

A dummy or indicator predictor takes only the values zero and one; its regression slope is the difference in predicted group averages.
For the GPA example, zero identifies non-athletes and one identifies student-athletes, so the intercept is the predicted GPA for non-athletes.
The athlete coefficient indicates a predicted GPA about 28 points lower for student-athletes than for non-athletes, with a p-value near zero.
The same hypothesis-testing and confidence-interval procedures apply to dummy predictors as to continuous predictors.

58:00 Binary Outcomes: Interpreting Coefficients as Probability Changes

When the dependent variable is a zero-one indicator, changes in its predicted value represent changes in probability.
In the county example, the outcome indicates whether a county is metropolitan and the predictor is the unemployment rate.
A slope near −0.045 means a one-percentage-point increase in unemployment is associated with about a 0.045 decrease in predicted metro probability, or 4.5 percentage points.

1:02:00 South Indicator Tests and the Effect of Smaller Samples

For a regression of changes in county COVID cases on a South-region indicator, the p-value is about 0.01: the null is rejected at 10% and 5% levels, but narrowly not at 1%.
A result significant at 99% confidence would also be significant at lower confidence levels; significance cannot occur only at the highest level.
Reducing sample size increases the standard error, which lowers the test statistic’s magnitude and generally raises the p-value.
Professor Lantis postpones the comparison of R-squared and adjusted R-squared until the next class.

1:06:00 Quiz Access, Assignment Timing, and Office Hours

The in-class quiz access code is 1020; Professor Lantis says incomplete questions will be revisited after the next class covers the remaining material.
The next problem set is posted, and students are advised to begin its practice problems early because the quiz window runs from Tuesday through the following Friday.
Additional virtual office hours are announced for the next day from 9 to 11, with adjusted Monday office-hour times to be posted.

Summary, takeaways, and chapters were generated by AI from the video's transcript and may contain errors. The video belongs to its creator, Professor Lantis.