ECON 371 Class Recording 10/22
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Overview
Professor Lantis develops nonlinear regression methods for ECON 371, showing how quadratic terms make marginal effects depend on X and how to calculate and assess the turning point, including with Stata. The class then introduces logarithmic models and their coefficient interpretations, compares model fit, and closes with exam guidance and attendance-related evidence.
Key takeaways
- In a quadratic regression, the marginal effect is beta one plus two beta two X, and the slope changes sign at negative beta one divided by two beta two.
- A turning point is useful only if it falls within a plausible, observed range; for example, the study-effort estimate of 5.7 was relevant because observed values ranged from 0 to 11.
- Omitted-variable bias affects nonlinear terms as well as linear terms: adding game time as a control changed the free-throw squared-term p-value from about 0.18 to 0.001.
- In a linear-log model, a 1% increase in X changes predicted Y by beta one divided by 100 units; in a log-linear model, a one-unit increase in X changes Y by approximately 100 beta one percent.
- For the high-school-rank GPA example, the linear-log model had R-squared about 0.16 compared with about 0.11 for the linear model, while the SAT example favored the linear-linear specification.
- The attendance analysis found an average association of about four fewer in-class quiz points per missed class, and quiz times below 10 minutes were negatively correlated with in-class exam performance.
Chapters
- The midterm totaled 104 possible points and was graded out of 100, effectively including a four-point curve; the class average was just under 72.
- The midterm and final together count for 60% of the course grade, and the final will not include a take-home portion.
- Quiz 4 will be assigned after Tuesday's class and due the following Friday; a new problem set will be posted the next day.
- For a regression with X and X squared, the marginal effect is beta one plus two times beta two times X.
- Unlike a linear model's constant slope, a quadratic model's marginal effect changes with the value of X.
- In the population and COVID-cases example, an additional 10,000 residents predicts about 1,200 extra cases for a city of 80,000 and about 1,300 for a city of 2 million.
- A quadratic relationship can have a positive slope at lower X values and a negative slope at higher values.
- Used cars illustrate the pattern: rising income may initially increase used-car purchases, while sufficiently high income shifts buyers toward new cars.
- Setting beta one plus two times beta two times X equal to zero gives the turning point: negative beta one divided by two beta two.
- A calculated turning point may lie outside the observed or plausible range, so it does not necessarily describe a real-world reversal.
- The Yerkes-Dodson idea predicts that performance may first improve with stress, then worsen once stress becomes excessive.
- Free-throw win-probability added serves as a measure of shot importance: players may focus more as a shot matters more, but extreme pressure can impair performance.
- Adding game time as a control changes the squared-term p-value from about 0.18 to 0.001, illustrating omitted-variable bias in nonlinear regressions.
- Stata's i. notation creates category indicators, such as basketball-position dummies, while omitting a reference category to avoid perfect multicollinearity.
- The free-throw example uses the estimated linear and squared coefficients to locate where the marginal effect changes sign.
- A turning point below zero cannot be meaningful for variables such as age or income, which cannot take negative values.
- A turning point above the sample's observed range may be mathematically calculable but irrelevant to the data.
- A quadratic term cannot be added for a binary dummy variable because squaring a zero-one variable reproduces the original variable and creates perfect multicollinearity.
- Professor Lantis demonstrates two Stata approaches: create an effort-squared variable directly or use factor-variable syntax such as c.effort##c.effort.
- The turning point is calculated from the estimated coefficients as negative beta one divided by two beta two.
- For the test-score data, the estimated turning point is about 5.7 on the effort measure, and observed effort values range from 0 to 11.
- Checking the observed range establishes that the 5.7 turning point is represented in the sample rather than lying beyond the data.
- The effort marginal effect is beta one plus two times beta two times the effort value.
- Using the Stata estimates, an additional effort unit at effort level 2 predicts improved math performance.
- At effort level 6, just above the 5.7 turning point, an additional effort unit predicts lower math performance.
- Marginal effects can be calculated at selected X values or evaluated across a range and plotted.
- In a linear-log model, Y stays in its original units while the regression uses the natural log of X.
- The coefficient interpretation is that a 1% increase in X changes predicted Y by beta one divided by 100 units.
- Income illustrates why percentage changes can matter more than equal dollar increases: an extra $10,000 is proportionally larger for someone earning $40,000 than for someone earning $250,000.
- Because the natural log of zero is undefined, linear-log models cannot use a zero-valued X observation or a dummy variable as the logged predictor.
- The GPA example compares a linear model using high-school rank with a linear-log model using the natural log of rank.
- In the linear-log specification, a 1% increase in rank predicts a GPA change of beta one divided by 100, not beta one full units.
- The linear-log model has an R-squared of about 0.16 versus about 0.11 for the linear model, and its test statistic has greater absolute magnitude.
- Stata creates the logged predictor with the ln() function; the GPA data are added to the course's week-nine materials.
- Professor Lantis compares linear-linear and linear-log specifications for predicting college GPA from SAT scores.
- The linear-linear model has the higher R-squared, showing that a logarithmic specification is not automatically more appropriate.
- The linear-linear model also has a larger absolute test statistic, supporting it over the linear-log alternative for this example.
- In a log-linear model, the dependent variable is logged and X remains in its original units.
- A one-unit increase in X changes predicted Y by approximately 100 times beta one percent; a 2% effect corresponds to a coefficient near 0.02.
- A log-linear specification cannot use a dummy variable as the logged dependent variable because taking the natural log of zero is undefined.
- In a log-log model, both variables are logged, so a 1% increase in X predicts a beta-one-percent change in Y.
- For exam scores predicted by attendance, quadratic and cubic terms are possible, as is a log-linear model, but a log-log model fails if attendance includes zero.
- The class previews adjusted R-squared and coefficient significance for comparing models, while noting that some comparisons require further discussion.
- Professor Lantis reports that quiz completion times below about 10 minutes were negatively correlated with in-class exam performance.
- The attendance analysis suggests each missed class corresponded to about a four-point decline in the in-class quiz grade on average; students can review exams after class or during office hours.
Transcript excerpt
0:01 All right. So, before we get started, everyone should see I posted yesterday the exam scores. So, I uploaded two things. So, you already see that kind of multiple choice take-home portion of the grade. I uploaded the in-class kind of number of points you got. I then added those together to get your midterm grade. So, 104 points possible, so I graded out of 100. So, it's already kind of like a four-point curve on that. And I now have it being weighted into your grade. So, the final and midterm add up to 60%. So, it's kind of like the grade you see now is like if you did the exact same on the midterm, that's what your score in the class would be. Right?
0:48 Um obviously, like if you do better on the final, that weight then goes to the final, that could change. But, just making it 30% the way that Canvas calculated so that it's not as high of a weight as the exams will be in your final grade. So, it kind of gives you like an overinflated idea of where you're at. This way it doesn't. Right? Um so, [clears throat] I might have been pessimistic before I actually calculated the grades last class. They weren't quite as bad as I thought once I factored in the four additional points since it was like a four-point curve. And kind of the you know, knew the take-home portion would likely be relatively high, and it was. So, I think the average was like just under a 72. So, usually I like to have my exams be right around that kind of 75
1:37 range. It's kind of the way I construct the course. So, not far off. Um and you know, some extremely low grades might have pulled that down a
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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.