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# Intro to the Tidyverse by Colleen O'Briant
# Koan #18: first differences
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# In order to progress:
# 1. Read all instructions carefully.
# 2. When you come to an exercise, fill in the blank, un-comment the line
# (Ctrl/Cmd Shift C), and execute the code in the console (Ctrl/Cmd Return).
# If the piece of code spans multiple lines, highlight the whole chunk or
# simply put your cursor at the end of the last line.
# 3. Save (Ctrl/Cmd S).
# 4. Test that your answers are correct (Ctrl/Cmd Shift T).
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library(tidyverse)## ── Attaching packages ─────────────────────────────────────── tidyverse 1.3.1 ──## ✔ ggplot2 3.3.6 ✔ purrr 0.3.4
## ✔ tibble 3.1.7 ✔ dplyr 1.0.9
## ✔ tidyr 1.2.0 ✔ stringr 1.4.0
## ✔ readr 2.1.2 ✔ forcats 0.5.1## ── Conflicts ────────────────────────────────────────── tidyverse_conflicts() ──
## ✖ dplyr::filter() masks stats::filter()
## ✖ dplyr::lag() masks stats::lag()library(gapminder)
# In class, we talked about first differences as a way to side-step the
# multicollinearity issue of including lags of an independent variable:
# If the true model is this:
# y_t = \beta_0 + \beta_1 x_t + \beta_2 x_{t-1} + u_t
# Then you can also estimate this and recover \beta_1 and \beta_2 without
# getting a large standard error on the coefficient for x_t:
# y_t = \alpha_0 + \alpha_1 x_t + \alpha_2 (x_t - x_{t-1}) + u_t
# Where \alpha_0 = \beta_0,
# \alpha_1 = \beta_1 + \beta_2, and
# \alpha_2 = -\beta_2.
# In the next few problems, we'll verify this is true using `gapminder`.
# 1. Create a new variable `gdpPercap_lag` that is the first lag of ------------
# `gdpPercap`. Make sure to `group_by` country so that we don't compare GDP's
# of different countries.
#01@
# gapminder %>%
# group_by(__) %>%
# mutate(gdpPercap_lag = __)
#@01
# 2. Lagging a variable is a transformation you can do inside `mutate` or ------
# inside lm() itself. Fill in the blanks below to estimate what we'll call
# model A:
# lifeExp_t = \beta_0 + \beta_1 log(gdpPercap)_t +
# \beta_2 log(gdpPercap)_{t-1} + u_t
# Write down your estimates for \beta_0, \beta_1, and \beta_2. It doesn't matter
# whether you log() and then lag(), or lag() and then log().
#02@
# gapminder %>%
# group_by(__) %>%
# lm(__, data = .) %>%
# broom::tidy()
#
# beta_0 <- __
# beta_1 <- __
# beta_2 <- __
#@02
# 3. Now estimate what we'll call model B, which includes a first --------------
# difference:
# lifeExp_t = \alpha_0 + \alpha_1 log(gdpPercap)_t +
# \alpha_2 I(log(gdpPercap)_t - log(gdpPercap)_{t-1}) + u_t.
# Write down your estimates for \alpha_0, \alpha_1, and \alpha_2.
#03@
# gapminder %>%
# group_by(__) %>%
# lm(__ ~ __ + I(__), data = .) %>%
# broom::tidy()
#
# alpha_0 <- __
# alpha_1 <- __
# alpha_2 <- __
#@03
# 4. Verify that the algebra we did in class works by running this code: -------
# Does estimating model B allow us to recover the estimates from model A with a
# smaller standard error for the coefficient on log(gdpPercap)_t?
# You should get TRUE's for all of the following:
#04@
# near(alpha_0, beta_0)
# near(alpha_1, beta_1 + beta_2)
# near(alpha_2, -beta_2)
#@04
# Note: we use near() instead of `==` because of something called "floating
# point imprecision" in computers. This demonstrates how `==` won't work, but
# near() will:
.1 + .2 == .3## [1] FALSEnear(.1 + .2, .3)## [1] TRUEMore info about floating point imprecision here.
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# It's also commonplace to see time series variables transformed into their
# first difference. This can be useful because it lets us focus on how the
# period-to-period change in a variable may effect the period-to-period change
# in another variable. It's a different question than asking how the level of a
# variable may effect the level of another variable. We'll explore this in the
# next few problems.
# 5. Use `ggplot()` to plot the lifeExp for the United States over time. -------
#05@
# gapminder %>%
# filter(__) %>%
# ggplot(aes(__)) +
# geom_point() +
# geom_line()
#@05
# Notice that lifeExp has a strong upward trend in the US. It will therefore
# correlate strongly with any other variable that has an upward time trend, and
# a regression between that variable and lifeExp will yield spurious results.
# We'll talk about this more next week in class, but for now, notice what
# happens to the time trend when we first difference lifeExp:
# 6. Take the plot above, but instead of plotting US lifeExp over time, --------
# now plot the first difference of US lifeExp (lifeExp_t - lifeExp_{t-1}) over
# time. You should see that the time trend disappears:
#06@
# gapminder %>%
# filter(__) %>%
# mutate(lifeExp_diff = __) %>%
# ggplot(aes(__)) +
# geom_point() +
# geom_line()
#@06
# 7. Next we'll take the plot above, but add the time series for the first -----
# difference of gdpPercap. We'll also focus on Afghanistan rather than the
# United States.
#07@
# gapminder %>%
# filter(__) %>%
# mutate(
# lifeExp_diff = __,
# gdpPercap_diff = __,
# ) %>%
# ggplot() +
# geom_line(aes(__)) +
# geom_line(aes(__))
#@07
# 8. The plot above isn't very good at comparing the two time series -----------
# because changes in gdpPercap are so much larger than changes in lifeExp.
# Transform gdpPercap through multiplication, division, addition, or subtraction
# until patterns in both the variables are visually clear. I've also added an
# aesthetic mapping for linetype so that we can differentiate between the two
# series.
#08@
# gapminder %>%
# filter(__) %>%
# mutate(
# lifeExp_diff = __,
# gdpPercap_diff = __,
# ) %>%
# ggplot() +
# geom_line(aes(x = year, y = lifeExp_diff, linetype = "Life Expectancy")) +
# geom_line(
# aes(x = year,
# y = (gdpPercap_diff/__) + __,
# linetype = "GDP per cap")
# )
#@08
# 9. Finally, we'll add two vertical axis labels and a title to make the -------
# plot look really professional. The second axis should reverse the algebraic
# transformation you made to gdpPercap_diff. So if you divided and then added,
# the second axis should subtract and then multiply: the syntax is:
# ~ (. - 5)*50
#09@
# gapminder %>%
# filter(__) %>%
# mutate(
# lifeExp_diff = __,
# gdpPercap_diff = __)
# ) %>%
# ggplot() +
# geom_line(aes(x = year, y = lifeExp_diff, linetype = "Life Expectancy")) +
# geom_line(
# aes(x = year,
# y = (gdpPercap_diff/__) + __,
# linetype = "GDP per cap")
# ) +
# scale_y_continuous(
# "Change in Life Expectancy (Years)",
# sec.axis = sec_axis(
# ~ (. - __)*__,
# name = "Change in GDP per Capita ($)"
# )
# ) +
# labs(title = "GDP and Life Expectancy in Afghanistan")
#@09
# A plot like the one you made in 9 lets us visually inspect whether a bump to
# one variable (GDP per capita in this case) seems to cause a bump to another
# (Life Expectancy) in the same period, or in one or two periods ahead. This is
# the intuition behind a time series test called# It's plain to see in this plot that increases in GDP per capita does not
# clearly cause a boost to life expectancy in Afghanistan: the relationship
# between the two variables may be a little more complicated, or it might be
# completely spurious!
# 10. Consider another country that isn't the US or Afghanistan, and draw a-----
# plot just like the plot in 9 for that country. Do increases in gdpPercap seem
# to Granger Cause increases in life expectancy there?
#10@
# gapminder %>%
# filter(__) %>%
# mutate(
# lifeExp_diff = __,
# gdpPercap_diff = __)
# ) %>%
# ggplot() +
# geom_line(aes(x = year, y = lifeExp_diff, linetype = "Life Expectancy")) +
# geom_line(
# aes(x = year,
# y = (gdpPercap_diff/__) + __,
# linetype = "GDP per cap")
# ) +
# scale_y_continuous(
# "Change in Life Expectancy (Years)",
# sec.axis = sec_axis(
# ~ (. - __)*__,
# name = "Change in GDP per Capita ($)"
# )
# ) +
# labs(title = "GDP and Life Expectancy in __")
#@10
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# Great work! You're one step closer to tidyverse enlightenment. Make sure to
# return to this topic to meditate on it later.