8.2 Holding other things equal
We add a second explanatory variable. What does \(\beta_1\) now measure?
The motivation for multiple regression was straightforward. We wanted to compare students who differed in attendance but were otherwise similar.
To do that we introduce additional explanatory variables.
With two explanatory variables, the population model is
\[y = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + u\]
and with \(k\) explanatory variables,
\[y = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + \cdots + \beta_k x_k + u\]
Nothing fundamental has changed. There is still a population regression function that we cannot observe, an error term containing everything left out of the model, and a sample from which we estimate the unknown coefficients.
What has changed is the meaning of a coefficient.
A different comparison
In the simple regression of Unit 7, the coefficient on attendance compared students who attended different numbers of classes.
Now the comparison is narrower.
The coefficient on attendance compares students who attended different numbers of classes but who had the same values of the other explanatory variables.
That is the meaning of holding other things equal.
Mathematically,
\[\frac{\partial E(Y \mid X_1, X_2)}{\partial X_1} = \beta_1\]
The notation looks new, but the idea is not. A partial derivative asks how the conditional mean changes when one explanatory variable changes while the others remain fixed.
\(\beta_1\) is the change in the average value of \(Y\) associated with a one-unit increase in \(X_1\), holding all the other explanatory variables constant.
This ceteris paribus interpretation is the entire reason for using multiple regression.
An example
Adding explanatory variables in R is as simple as separating them with +.
full <- lm(sem_gpa ~ attendance + cum_gpa + admission_score,
data = students)
round(coef(summary(full)), 4)#> Estimate Std. Error t value Pr(>|t|)
#> (Intercept) -7.6603 1.2944 -5.918 0
#> attendance 0.1360 0.0104 13.040 0
#> cum_gpa 0.5748 0.0441 13.026 0
#> admission_score 0.0835 0.0158 5.299 0Compare the coefficient on attendance with the simple regression.
#> simple.attendance multiple.attendance
#> 0.189 0.136The estimated association falls from 0.189 to 0.136.
The interpretation has changed. In the simple regression, the comparison was between students who differed in attendance. In the multiple regression, the comparison is between students who have the same prior GPA and the same admission score, but who differ in attendance.
Some of the relationship that the simple regression attributed to attendance is now explained by differences in students’ previous academic performance. The coefficient on attendance therefore becomes smaller.
Notice that nothing has happened to the data. The students are the same. Only the comparison has changed.
Reading the other coefficients
Every coefficient is interpreted in exactly the same way.
The coefficient on cumulative GPA says that among students with the same attendance and the same admission score, a one-point increase in prior GPA is associated with an increase of about 0.575 points in semester GPA.
The coefficient on admission score compares students with the same attendance and the same prior GPA.
Each coefficient asks what happens when one explanatory variable changes while all the others remain fixed.
Another example
The same interpretation applies outside education. This is a synthetic dataset of 20,000 Indian wage earners, built to resemble the structure of household survey data.
wages <- read.csv("data/wages-india-synthetic.csv")
wages$male <- as.integer(wages$sex == "Male")
wages$urban <- as.integer(wages$residence == "Urban")
w_full <- lm(annual_earnings ~ education + age + male + urban,
data = wages)
round(coef(summary(w_full)), 1)#> Estimate Std. Error t value Pr(>|t|)
#> (Intercept) -28175.8 2612.7 -10.8 0
#> education 4281.2 134.7 31.8 0
#> age 559.8 57.4 9.8 0
#> male 31999.1 1442.8 22.2 0
#> urban 51287.4 1517.5 33.8 0The coefficient on education is interpreted as the association between an additional year of schooling and annual earnings holding age, sex and place of residence constant.
The variables male and urban take only the values 0 and 1, so their
coefficients compare two groups rather than describing the effect of a one-unit
increase. We return to indicator variables in the next unit.
What “holding fixed” does not mean
The phrase holding other things fixed often gives the wrong impression.
Regression does not perform an experiment.
Nobody took the same student, kept their prior GPA unchanged and forced them to attend one more class. Nor did anyone move a rural worker to an urban area while leaving everything else unchanged.
The comparison is entirely statistical. The regression compares observations that already exist and are similar in the variables included in the model.
Whether that comparison is enough to recover a causal effect depends on whether the important differences between individuals have been measured.
That question returns in Section 8.7.