ECON 371 Class Recording 10/08

Overview

Professor Lantis reviews ECON 371 regression concepts and explains the midterm format: a written in-class exam of six multi-part problems, plus a 10–15-question Canvas quiz. Using Stata output and examples from North Carolina education, college-student, and powerlifting data, the class covers adjusted R-squared, joint F-tests, heteroskedasticity tests, omitted-variable bias, dummy-variable coding, coefficient interpretation, and how sample size or rescaling affects estimates and significance.

Key takeaways

Adding regressors never lowers ordinary R-squared, but adjusted R-squared can rise or fall because its fit improvement competes with a penalty for model complexity.
In joint F-tests, larger sample sizes tend to increase the test statistic and lower the p-value, making rejection more likely when the observed fit difference is held constant.
Omitted-variable bias has the sign of the omitted variable’s effect on the outcome multiplied by its covariance with the included regressor; in the credit-load example, two negative relationships produce positive bias.
For categorical predictors with an intercept, include indicators for all but one category; the excluded category becomes the reference group and prevents perfect multicollinearity.
Rescaling a predictor changes its coefficient’s numerical units but not its statistical significance or model R-squared; rescaling the outcome changes coefficient units and likewise preserves significance.
With a binary dependent variable in a linear regression, coefficients represent changes in predicted probability, and predictions require the intercept plus each coefficient multiplied by its corresponding regressor value.

Chapters

0:00 Midterm Format, Formula Sheet, and Study Strategy

The in-class exam is one week later and consists of six written, multi-part problems—about 25 response parts—with no multiple choice or fill-in-the-blank.
A separate Canvas portion has about 10–15 questions, uses LockDown Browser, and is due the following Monday; the in-class portion will make up most of the exam grade.
Professor Lantis recommends attempting practice exams and old quizzes without notes first, then reworking uncertain or incorrect answers and explaining the reasoning.
Students do not need to write Stata code, but should understand code shown with regression output and be able to answer problem-set follow-up questions.

7:22 Dummy-Variable Interpretation in North Carolina Education Data

A regression using North Carolina education data predicts student math scores from grade and a male indicator.
The male coefficient represents the predicted score difference between male and female students, with the omitted category serving as the comparison group.
Adding a gifted-student indicator creates an unrestricted model with additional regressors.

9:00 Why R-Squared Rises but Adjusted R-Squared Can Move Either Way

Ordinary R-squared cannot decrease when regressors are added; even a tiny improvement in fit raises it.
Adjusted R-squared penalizes unexplained variation by a factor involving sample size and the number of regressors.
Adding variables increases the penalty but may also reduce unexplained variation, so adjusted R-squared can rise or fall when comparing different models.
Within a single regression, adjusted R-squared is no greater than ordinary R-squared.

12:40 How Sample Size Affects Adjusted R-Squared

Holding the expected R-squared and number of regressors constant, a larger sample makes the adjusted R-squared scaling factor closer to 1.
The example contrasts n = 3 and k = 1, where the factor is 2, with n = 10 and k = 1, where it is 9/8.
A larger sample reduces the adjustment penalty and raises expected adjusted R-squared, even though expected ordinary R-squared is unchanged.

19:36 Joint F-Tests: Restricted Models and Sample-Size Effects

The North Carolina example compares restricted and unrestricted models using their R-squared values and the number of added variables, q.
The unrestricted model must have at least as high an R-squared as the restricted model.
With the R-squared comparison held constant, increasing n raises the F-statistic by increasing its effective precision.
A higher test statistic generally lowers the p-value, making rejection of the null more likely.

24:33 Interpreting Joint Significance and the Heteroskedasticity Test

A joint-significance test asks whether added regressors provide explanatory power together; its null is that their coefficients are jointly zero.
An example with a p-value near zero rejects the null, while a joint test with p = 0.17 does not reject at the 10%, 5%, or 1% levels.
For the student GPA example, Professor Lantis uses the Breusch–Pagan/Cook–Weisberg-style IM test to assess heteroskedasticity.
With an IM-test p-value of 0.29, the null of homoskedastic errors is not rejected; failure to reject is not proof that errors are homoskedastic.

30:53 Omitted-Variable Bias: Signs from Credit Loads and Course Difficulty

The example predicts GPA using math and science credits while leaving out total credit hours.
Total credits are expected to covary negatively with math and science credits if students taking heavier loads avoid as many difficult courses.
The effect of total credits on GPA is expected to be negative because heavier course loads may lower GPA.
The omitted-variable-bias sign is the product of the omitted variable’s effect and its covariance with the included regressor, divided by the included regressor’s variance; two negative terms imply positive bias.

37:23 Calculating Omitted-Variable Bias from Regression and Covariance Output

The GPA regression illustrates continuous-variable interpretation: holding other regressors constant, one additional year of age predicts about a 0.06-point GPA decrease.
The intercept is predicted GPA when all included regressors equal zero.
For the math-and-science-credit coefficient, the bias calculation uses the omitted variable’s coefficient (about −0.244), the covariance with the included variable (0.44), and the included variable’s variance (1.48).
Including an omitted variable removes its bias from the affected coefficient; a positive omitted-variable bias makes the corrected estimate lower.

44:06 Using Confidence Intervals to Test Regression Coefficients

In the powerlifting regression, the outcome is total weight lifted in kilograms, with body weight and a male indicator as predictors.
A 95% confidence interval that excludes zero implies rejection of the null hypothesis that the coefficient equals zero at the 5% level.
Confidence intervals can therefore reveal statistical significance even when standard errors, test statistics, or p-values are hidden.

49:23 Age-Category Dummies and the Perfect Multicollinearity Trap

The powerlifting example creates young, middle-aged, and old indicators using age cutoffs of under 30, 30–39, and 40 or older.
Including all three indicators alongside an intercept creates perfect multicollinearity because the categories exhaust the sample and sum to one.
Leave out one category; its members become the reference group, so the young coefficient compares young lifters with the omitted old group.
To estimate the young-versus-middle-aged difference directly, change the omitted category and include the other two indicators; the resulting coefficient is about 10 kilograms.

52:23 Sample Size and Predictor Rescaling Change Different Things

A smaller sample raises standard errors, lowers test statistics, raises p-values, and makes coefficient significance harder to establish, while the coefficient estimate itself is not expected to shift systematically.
Expressing body weight in 10-kilogram units instead of kilograms changes the coefficient’s numerical scale.
Rescaling an independent variable changes its coefficient and standard error proportionally, leaving its test statistic and statistical significance unchanged.
The model’s R-squared also remains unchanged when only the units of a predictor are rescaled.

57:00 Rescaling the Outcome Changes Coefficient Units

When total weight lifted is expressed in tens of kilograms rather than kilograms, the dependent variable is divided by 10.
The estimated slopes and intercept are correspondingly divided by 10; the example’s body-weight coefficient changes from about 3.03 to about 0.3.
Changing outcome units does not change the coefficients’ statistical significance because estimates and standard errors scale together.

1:03:29 Binary Outcomes, Probability Effects, and Prediction Calculations

When the dependent variable is a 0/1 indicator, a linear-regression coefficient describes a change in predicted probability, not a percentage-point interpretation stated without regard to units.
For example, a coefficient of 0.02 means a one-kilogram increase in body weight raises the predicted probability of being middle-aged by 0.02.
To calculate a prediction, multiply every coefficient by its corresponding hypothetical regressor value and add the intercept.
For a female observation, a male dummy equals zero, so the male coefficient contributes nothing to the prediction.

1:07:05 Final Review Priorities, Problem-Set Deadline, and Exam Q&A

Professor Lantis asks students to identify topics for Tuesday’s review, including omitted-variable bias, categorical dummies, heteroskedasticity, multicollinearity, joint tests, sample size, and scaling.
The third problem set is due by the end of the next day, with no late work accepted; old quizzes and problem sets are recommended exam practice.
Students should know how to interpret Stata output and answer related questions, but do not need to reproduce the code.
The in-class exam is entirely free response and requires work or explanations for most questions; the Canvas take-home portion is smaller.

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.