ECON 371 Class Recording 11/03

Overview

The class develops continuous-by-dummy interaction models, showing how to interpret group-specific slopes and intercepts, test interaction significance, calculate predicted outcomes, and find where group predictions cross. Examples use log wages and beauty scores, mate height, and 135,000 NBA shot observations, with Stata interaction syntax and cautions about extrapolating beyond observed data.

Key takeaways

In a continuous-by-dummy regression, the D=0 slope is the continuous-variable coefficient, and the D=1 slope is that coefficient plus the interaction coefficient.
A positive interaction makes the D=1 slope larger than the D=0 slope, while a negative interaction makes it smaller; the interaction’s p-value determines whether the estimated slope difference is statistically significant.
In the log-wage example, a beauty coefficient of 0.04 implies about a 4% wage change per score unit, while adding a 0.046 female interaction yields a female slope near 0.086, or 8.6%.
The beauty-score wage crossing is estimated around 13, outside the observed 0-to-10 range, illustrating why calculated thresholds must be checked against the data’s support.
For the NBA model, a tall-player-by-distance interaction changes the shot-distance slope for players over 80 inches; the estimated crossing distance of about -24 is infeasible and should be interpreted with possible selection effects in mind.
Adding regressors always weakly increases ordinary R-squared, but adjusted R-squared can move either way; the gap between the two measures grows as the adjusted R-squared penalty increases.

Chapters

0:00 Problem Set 4: Omitted-Variable Bias and Variance

Problem Set 4 is due by the end of the day and covers material through nonlinearities.
For the variance question, use the known effect of the charter-school variable to recover the true coefficient and then back out the variance; the answer key will also explain the shortcut using Stata’s sum command.
Practice problems for the next quiz are planned for Canvas.

2:30 Why Interact a Continuous Variable with a Dummy

Without an interaction, a regression of wages on education and a white dummy estimates a shared education effect across white and non-white individuals.
If the data are 80% white, the pooled estimate can be influenced more by the larger group, even if education’s effect differs between groups.
In a model with education, a group dummy, and their product, the education slope for the dummy-equals-one group is the base education coefficient plus the interaction coefficient.

7:12 Interaction Signs and Group-Specific Slopes

For a model with continuous variable X and dummy D, the slope for D=0 is the coefficient on X, while the slope for D=1 is the X coefficient plus the interaction coefficient.
A positive interaction means the D=1 group has a larger X slope than the D=0 group; a negative interaction means its slope is smaller.
A pooled model without the product term can conceal effects that differ in size or even sign between groups.

11:14 Testing Interactions and Comparing R-Squared Measures

An estimated difference between group slopes is not necessarily statistically significant; test the interaction coefficient’s p-value.
Adding an interaction cannot lower ordinary R-squared, but adjusted R-squared can rise or fall depending on the added explanatory power relative to the penalty for another regressor.
Adding a regressor increases the gap between ordinary and adjusted R-squared because the adjusted measure’s scaling factor grows.

16:25 Mapping Interaction Coefficients to Intercepts and Slopes

The dummy coefficient shifts the predicted starting point for the D=1 group when the continuous variable is zero.
The continuous-variable coefficient gives the D=0 slope; adding the interaction coefficient gives the D=1 slope.
The class compares ordinary and interaction regressions, where R-squared changes from about 0.4071 to 0.4077.

19:00 Interpreting Beauty-by-Female Effects on Log Wages

For the log-wage model, the male group’s beauty slope is about 0.04, which corresponds to roughly a 4% wage increase per one-unit increase in the beauty score.
The female group’s slope adds an interaction estimate of about 0.046, giving a total slope near 0.086, or about an 8.6% wage increase per beauty-score unit.
At beauty equal to zero, the female prediction adds the female dummy coefficient, about -0.53, to the assumed constant of 0.5.

26:20 Finding the Beauty Score Where Wage Predictions Cross

The predicted female-minus-male wage difference is the female dummy coefficient plus the beauty interaction coefficient multiplied by beauty.
Setting that difference to zero gives a crossing point of roughly 13 using the class’s rounded coefficients, above the observed beauty scale of 0 to 10.
A calculated crossing point should be checked against the variable’s observed minimum and maximum; a threshold outside the data range is an extrapolation, not an observed comparison.
The class connects the example to economist Daniel Hamermesh’s research on beauty and labor-market outcomes, including interview requests.

35:48 In-Class Quiz 13 and the Mate-Height Example

The class opens In-Class Quiz 13 with access code 10.09 and reviews a regression predicting mate height from a person’s height, a male dummy, and their interaction.
The male coefficient is about 11, shifting the predicted starting point upward, while the interaction is about -0.28, making the male height slope flatter.
The regression’s approximate coefficients are a 27.71 intercept and a 0.67 own-height slope; for a 69-inch male, substituting all terms gives a predicted mate height of about 65.62.

51:49 NBA Shot Data: Tall Players and Shot Distance

The NBA example defines a tall-player dummy as one for players over 80 inches, or 6 feet 8 inches, and uses shot success as a binary outcome.
The initial model controls for factors including shot distance, time remaining, player age, score gap, and closest-defender distance.
Without an interaction, the shot-distance coefficient represents an average relationship across tall and non-tall players.

55:13 Interacting Tall Status with Shot Distance

The expanded NBA regression interacts tall status with shot distance to allow distance’s effect on shot success to differ between the two player groups.
With a positive interaction and a negative base distance effect, the tall group’s distance slope is less negative than the non-tall group’s slope.
The data contain about 135,000 observations, so even small coefficient estimates may be statistically significant; the point-gap variable is an example discussed as not statistically significant.

59:37 Stata Syntax for Interaction Coefficients and Marginal Effects

Stata factor-variable notation can specify a dummy-by-continuous interaction, conventionally with a form such as i.tall##c.shot_distance.
The interaction coefficient name appears in the regression output and is needed alongside the shot-distance coefficient to calculate the tall group’s slope.
The class works through uncertainty about the exact post-estimation syntax for displaying marginal effects and returns to the calculation using a manually created product term.

1:05:09 Manual Interaction Terms and the NBA Distance Threshold

Creating a product variable manually and including it with both main effects makes the group-specific slope calculation explicit: shot-distance coefficient plus interaction coefficient for tall players.
The group-prediction difference is the tall dummy coefficient plus the interaction coefficient times shot distance; setting it to zero gives a crossing distance of about -24.
Because shot distance cannot be negative, the estimated crossing point lies outside the feasible range, implying the fitted model predicts a tall-player advantage at all nonnegative distances.
Selection may complicate the result because tall players may be less likely to take long shots; the class flags selection issues for later study.

1:10:06 Next Topic: Continuous-by-Continuous Interactions

The next class will introduce continuous-by-continuous interactions, whose relationships are harder to visualize than the two-group lines used in this lecture.
Practice problems will be posted on Canvas, and an interaction-term problem set is tentatively planned after Thursday’s class.
The timing of the homework and quiz may shift if interaction material extends into the following week.

Transcript excerpt

0:00 Okay. So, a couple things before we get started here. Um >> [clears throat] >> that fourth problem set is due by the end of the day today. So, it should just kind of go up through nonlinearities. Um One thing I'll mention, and I think we'll probably talk about this after it's due a little bit, but I think it's in this question. What I intended was there's a way in which you can solve it without simply putting the summarized command and looking for the variance, um which is to think about if I can obtain So, I I forgot that I left those variables in the data set for you, so you could just do that. But, if you want to know the bias, it's going

0:47 to be the coefficient of the variable that you knew its true effect should be. So, what I had intended for you to do >> [clears throat] >> is kind of use that um square variable of the charter schools uh or what was the omitted variable? Yeah, charter school to kind of obtain what that true coefficient should be, and then kind of plug that in and back out the variance. So, I'll post the uh answer key to this, and it'll have kind of like how you could solve it. If you simply use the sum command, I'll give you kind of credit for that. Um but, when you're looking at the answer key, you can back that out if you didn't have the data just based off of what you kind of have from this earlier problem. So, kind of worry about that one, that second part, finding the variance.

1:34 Should be pretty simple since I since I you have that that charter school variable in the set.

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