ECON 371 Class Recording 9/24

Overview

Professor Lantis reviews heteroskedasticity, standard errors, hypothesis tests, confidence intervals, and Stata tools before introducing multiple linear regression. The class develops the “holding other variables constant” interpretation, shows how adding predictors changes coefficients and R-squared, explains adjusted R-squared’s penalty for extra variables, and previews omitted-variable bias and correlation versus causation.

Key takeaways

When heteroskedasticity is ignored, standard errors can be too small, inflating test statistics and reducing p-values; Stata’s robust option adjusts the standard errors.
A confidence interval containing zero corresponds to failing to reject a zero coefficient at that confidence level, while smaller samples generally produce wider intervals and lower statistical power.
Stata’s predict command can create fitted values, and its residuals option produces actual-minus-fitted errors for residual plots and heteroskedasticity checks.
In multiple regression, a coefficient describes a one-unit change in its predictor while holding the other included predictors constant—not all other influences in the world.
Adding population to the police-and-crime regression raises R-squared from about 0.74 to 0.82 and reduces the police coefficient by more than half, revealing confounding by city size.
Ordinary R-squared never falls when predictors are added, but adjusted R-squared penalizes model size and can decline—or become negative with small samples and many predictors.

Chapters

0:00 Class Deadlines and How Heteroskedasticity Distorts Tests

The second quiz covers simple linear regression and is due by the end of the following day; problem set 2 is due Wednesday.
Homoskedastic errors have the same distribution across values of X, while heteroskedastic errors vary with X.
Ignoring heteroskedasticity can make standard errors too small, inflate the test statistic, shrink the p-value, and increase false rejections.

5:25 Nine-Minute Simple Regression Quiz

Professor Lantis opens a three-question in-class quiz reviewing simple linear regression.
Students receive nine minutes to work, are encouraged to discuss questions with classmates, and must submit before the stated deadline.
The quiz uses the course site’s Week 5 materials and the all-caps access code LINEAR.

13:55 Reading Confidence Intervals and Recovering a Regression Coefficient

A confidence interval that contains zero means the sample does not provide enough evidence to reject a zero slope coefficient.
Given a slope’s standard error of about 308 and a test statistic of 4.04, the coefficient can be recovered by multiplying them.
When only the confidence interval is supplied, its midpoint gives the coefficient estimate; the example bounds are approximately 444 and 2,051.

17:37 Smaller Samples Widen Intervals and Reduce Rejection Power

Reducing a sample from 200 observations to 50 raises standard errors because less data provides less precise estimates.
A larger standard error lowers the test statistic, raises the p-value, and makes rejection of the null less likely.
Confidence intervals widen because their margins of error are based on standard errors.

23:55 Heteroskedasticity Definitions and Stata’s Robust Standard Errors

For a regression of savings on age, heteroskedasticity means error variance depends on age; homoskedasticity means it does not.
Adding predictors does not itself correct heteroskedasticity; Professor Lantis identifies Stata’s robust option as the correction used in class.
If robust standard errors rise, the test statistic falls and the p-value rises, making rejection less likely.

27:00 Interpreting Slopes and Generating Predicted Values in Stata

In a regression of COVID cases on county vaccination percentage, a slope near −0.25 means a one-percentage-point increase in vaccination predicts about 0.25 fewer cases, in the outcome’s stated units.
Stata’s predict command generates fitted Y-values for every observation after a regression is run.
The predicted value is the intercept plus the slope times X; a county with 50% vaccination is evaluated by plugging in 50.

30:39 Residuals and the Breusch–Pagan Heteroskedasticity Test

A residual is the actual outcome minus its fitted value; Stata’s predict command with the residuals option creates residuals for each observation.
A scatterplot of residuals against predicted cases helps inspect whether error spread changes across fitted values.
The heteroskedasticity test reports p = 0.07, so homoskedasticity is rejected at the 10% level but not at 5% or 1%; robust standard errors are then appropriate.

35:34 Why Multiple Regression Adds Predictors

A single predictor often misses important influences: test scores may depend on prior coursework as well as study habits.
Demand can depend on price, income, and substitute prices, so multiple regression can include several explanatory variables.
Police-officer counts may track crime partly because large cities have more residents, making population an important additional predictor.

38:14 Interpreting Multiple Regression as “Holding Constant” Other Factors

A police-officer coefficient in a crime regression describes the predicted change associated with one more officer while population is held constant.
The comparison is a statistical thought experiment: cities with the same population but differing by one officer.
Adding controls such as income, education, or demographics makes the coefficient’s interpretation more specific to the variables included.

43:26 IQ, Experience, Age, and the Limits of Regression Intercepts

In a wage regression with IQ, experience, and age, each slope measures a one-unit change in that predictor while the other included predictors stay fixed.
The example assigns experience a negative coefficient of about $1.40 per additional year, which may reflect omitted factors rather than a simple causal effect.
The intercept is predicted wage when IQ, experience, and age all equal zero, so it may be economically meaningless and outside the observed data range.
Age and experience can be correlated without being perfectly related; perfect multicollinearity, not ordinary correlation, prevents separate coefficient estimation.

50:37 Making Predictions and Testing Coefficients in Multiple Regression

For a hypothetical person with IQ 105, 10 years of experience, and age 45, the prediction sums the intercept and each coefficient multiplied by its corresponding value.
Multiple regression uses the same t-statistics, p-value-versus-alpha decisions, and confidence-interval logic as simple regression.
The null hypothesis is that each coefficient equals zero, considered separately for the predictors in the model.

55:18 Adding Population Changes the Police-Crime Estimate

R-squared is the proportion of variation in Y explained by the regression’s predictors; it ranges from zero to one in ordinary regression.
Adding population to a police-and-crime model raises R-squared from about 0.74 to 0.82.
The police coefficient falls by more than half after controlling for population, showing that the simpler model attributed some population-related association to police counts.
Both police coefficients remain statistically significant, but the smaller test statistic in the expanded model implies a larger p-value.

59:45 Adjusted R-Squared Penalizes Unhelpful Predictors

Adjusted R-squared uses the unexplained share of variation and scales it by (n − 1)/(n − k − 1), where n is sample size and k is the number of predictors.
Ordinary R-squared cannot decrease when predictors are added, even if a new variable contributes almost no explanatory value.
The adjusted measure penalizes extra predictors, so it can fall when the added variables do not explain enough additional variation.

1:03:24 How Sample Size and Predictor Count Affect Adjusted R-Squared

Increasing k shrinks the denominator n − k − 1 in the adjustment factor, increasing the penalty applied to the unexplained share.
As n grows relative to k, the penalty becomes less severe; Professor Lantis illustrates this with n changing from 3 to 4 when k = 1.
With a small sample and many predictors, adjusted R-squared can be negative even though ordinary R-squared cannot.
A falling adjusted R-squared indicates that added predictors are not improving explanatory power enough to justify their number.

1:11:43 Omitted-Variable Bias and the Correlation-versus-Causation Preview

Leaving population out of a crime regression lets police counts absorb some association attributable to city size.
A variable correlated with both the included predictor and the outcome can make an included coefficient reflect more than its own relationship with Y.
Professor Lantis previews cases where relevant data—such as students’ socioeconomic circumstances—may be unavailable, complicating causal interpretation of attendance and performance.
The second quiz is due the following day and problem set 2 is due Wednesday; Monday and Wednesday office hours are available for questions.

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.