library(tidyverse)Recall how you can use the function tibble() to generate
a dataset from scratch, like this:
tibble(
x = c(1, 7, 3, 7),
y = 1:4,
z = sample(1:10, size = 4, replace = TRUE)
)## # A tibble: 4 × 3
## x y z
## <dbl> <int> <int>
## 1 1 1 1
## 2 7 2 3
## 3 3 3 9
## 4 7 4 3You can even have variables be functions of other variables. Note that w is a function of x and y:
tibble(
x = c(1, 7, 3, 7),
y = 1:4,
z = sample(1:10, size = 4, replace = TRUE),
w = x + 2*y - 10
)## # A tibble: 4 × 4
## x y z w
## <dbl> <int> <int> <dbl>
## 1 1 1 1 -7
## 2 7 2 5 1
## 3 3 3 9 -1
## 4 7 4 1 5And you can introduce noise from a random normal distribution using
the function rnorm(). Just like rt(), it
generates a vector of random numbers from its distribution (the normal
distribution instead of the t distribution like with
rt()).
tibble(
x = c(1, 7, 3, 7),
y = 1:4,
z = sample(1:10, size = 4, replace = TRUE),
w = x + 2*y - 10 + rnorm(n = 4, mean = 0, sd = 1)
)## # A tibble: 4 × 4
## x y z w
## <dbl> <int> <int> <dbl>
## 1 1 1 3 -7.32
## 2 7 2 9 0.977
## 3 3 3 6 0.290
## 4 7 4 3 5.22Every time you run the code above, R will generate slightly different numbers for z and w because of the random elements in z and w.
Take a look at your answer to classwork 5, #1e. It asked you to be creative and to think of an example of omitted variable bias yourself. In particular, it asked you to think of three variables: \(x\), \(y\), and \(u\). \(x\) and \(u\) should both effect \(y\), \(x\) and \(u\) should also be correlated, and \(u\) should be hard to or not commonly measured (unobservable).
In this classwork, you’ll simulate the omitted variable bias issue that you explained in classwork 5 #1e.
First, generate an artificial dataset that has the 3 variables you talked about in 1e. There should be 100 observations (rows) in your tibble. Make sure that \(y\) is determined by a linear function of \(x\) and \(u\), along with additive noise. Also make sure that \(x\) and \(u\) correlate, but are not perfectly collinear (there are many ways to do this).
Don’t use the names “x”, “y”, and “u” in your tibble; use the names from your answer to 1e. Try to generate those variables so that they have reasonable values that are not too far away from the values they might take on in the real world.
# dataset <- tibble(
# u = __,
# x = __,
# y = __
# )questions about your dataset:
2.1 What is the median value for x in your dataset?
# __2.2 What are the median values for x among people with above-average y? What about the median values for x among people with below-average y?
# __2.3 What is the value of u and x for the single person with the highest y?
# __You should use ggplot, geom_point or
geom_jitter, and geom_smooth to verify that
x covaries with u and that y
covaries both with x and with u.
# __4.1) As the econometricians, x and y will
be observable to us, but u will not be observable. Fit the
model y ~ x, copy-pasting the way you originally defined
your dataset using the function tibble(). That way, each
time you run the piece of code, it will generate slightly different
estimates for the model parameters. Do you seem to get biased or
unbiased estimates of the true effect of x on y?
# tibble(
# __
# ) %>%
# lm(y ~ x, data = .) %>%
# broom::tidy()4.2) Make a change to your simulation
Take the code from above and change your dataset so that x and u no longer covary; make them independent. Fit the same model as before. If x is independent of u, but u is still an important determinant of y, when we omit u from the model, do we bias the estimate for the impact of x on y?
# __4.3) Make another change to your simulation
Take the code from 4.1) and change your dataset again so that u is no longer a direct determinant of y. x should still covary with both u and y. When u no longer determines y, does omitting u bias the impact of x on y?
# __