This lecture covers the bread-and-butter tool of applied econometrics and data science: regression analysis. This adapted from work by Grant McDermott. Today is focused on practical skills of using regression tools with R to do some analysis. I am not working through major econometric theory of regression analysis and inference. We’re instead focusing on the practical skills of producing regression tables, interpreting coefficients, standardizing variables, and understanding the different types of standard errors.
There are also many R packages I’m leaving out! There are tons of ways to do regressions in R. I’ll present just a few.
Software requirements
R packages
It’s important to note that “base” R already provides all of the tools we need for basic regression analysis. However, we’ll be using several additional packages today, because they will make our lives easier and offer increased power for some more sophisticated analyses.
- New: fixest, broom, modelsummary, vtable
- Already used: tidyverse, hrbrthemes, listviewer, tidycensus
A convenient way to install (if necessary) and load everything is by running the below code chunk.
## Load and install the packages that we'll be using today
if (!require("pacman")) install.packages("pacman")
pacman::p_load(tidyverse, fixest, vtable, broom, modelsummary,tidycensus)
## My preferred ggplot2 plotting theme (optional)
theme_set(theme_minimal())While we’ve already loaded all of the required packages for today, I’ll try to be as explicit about where a particular function is coming from, whenever I use it below.
Something else to mention up front is that we are using the
Opportunity Atlas today for our regressions. Some of these regressions
will be a bit simple, but the point is to show you how to run them in R
with data that are meaningful. We’ll also use the
fips_codes data from the tidycensus
package to merge in county names and state abbreviations.
The Opportunity Atlas file lives on GitHub, so we’ll download it from
there (Note: I am using the githack URL. You could also sync
your fork, pull to your cloned repo, and navigate to
lectures/10-regression/ and the file is there too.). I’ve
also amended the file to make it smaller so it can be easily pushed to
GitHub. Note: This is bad practice generally, do not store data
files on GitHub unless you have a really good reason to do so.
# Create opp atlas object by reading in a CSV off the internet.
opp_atlas <- read_csv("https://raw.githubusercontent.com/big-data-and-economics/big-data-class-materials/main/lectures/10-regression/opp_atlas_amended.csv") ## Rows: 3219 Columns: 9
## ── Column specification ────────────────────────────────────────────────────────
## Delimiter: ","
## chr (1): czname
## dbl (8): state, county, cz, kfr_pooled_pooled_p25, kfr_white_pooled_p25, poo...
##
## ℹ Use `spec()` to retrieve the full column specification for this data.
## ℹ Specify the column types or set `show_col_types = FALSE` to quiet this message.#Quickly renaming fips data to make it easier to merge
fips <- fips_codes %>%
rename(state_abb=state,
state=state_code,
county_name=county,
county=county_code) %>%
mutate(across(c(state,county),as.numeric))opp_atlas <- opp_atlas %>%
left_join(fips) %>% # Join together the data
rename(kfr_p25=kfr_pooled_pooled_p25,
kfr_white_p25=kfr_white_pooled_p25) # Rename so you have to type less later.## Joining with `by = join_by(state, county)`Regression basics
The lm() function
R’s base workhorse command for running regression models is the
built-in lm() function. The “lm” stands
for “linear models” and the syntax is
very intuitive. The syntax is straight-forward to run a simple
regression, but it can be a bit cumbersome to run more complex
models.
Where y is the dependent variable, x1,
x2, x3, etc. are the independent variables,
and df is the data frame that contains these variables.
You’ll note that the lm() call includes a reference to the
data source. We covered
this in our earlier lecture on R language basics and
object-orientated programming, but the reason is that many objects
(e.g. data frames) can exist in your R environment at the same time. So
we need to be specific about where our regression variables are coming
from — even if opp_atlas is the only data frame in our
global environment at the time.
Here’s an example using the Opportunity Atlas data. We’ll run a simple bivariate regression of the 1990 poverty rate on income mobility for children in the 25th percentile.
##
## Call:
## lm(formula = kfr_p25 ~ poor_share1990, data = opp_atlas)
##
## Coefficients:
## (Intercept) poor_share1990
## 5.623064 -0.006886You might immediately notice that the coefficient seems wrong. A higher share that are poor in 1990 is associated with greater income mobility? We’ll get there in a second.
First, let’s note some limitations of lm(). If you want
to throw in fixed effects, non-standard errors, or any other fancy
stuff, you’ll need to bring in other packages. So why not just teach you
the same syntax, but with a more flexible package? That sounds better.
Let’s do that.
The fixest package and feols()
function
The fixest package (link) is a powerful and flexible package for running linear models in R created by Laurent Bergé. It’s particularly well-suited for high-dimensional fixed effects models, but it also has a bunch of other neat features. It is extremely fast too because it leverages some more complicated econometric theorems to speed up computation, reduce memory usage, and parallelize the computation. (We’ll do parallelization later.)
fixest has many different linear models built in,
but feols() is what is used for ordinary least squares
(OLS) regression. The syntax is very similar to lm(), but
with a few extra bells and whistles.
#library(fixest) # already loaded
ols_fixest = feols(kfr_p25 ~ poor_share1990, data = opp_atlas)
ols_fixest## OLS estimation, Dep. Var.: kfr_p25
## Observations: 3,219
## Standard-errors: IID
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 5.623064 0.402329 13.976276 < 2.2e-16 ***
## poor_share1990 -0.006886 0.014945 -0.460733 0.64502
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## RMSE: 21.9 Adj. R2: -2.448e-4The output of ols_lm and ols_fixest differs
a little, but check – are the coefficients and standard errors the same?
Always check if a new tool gives the same results as the simpler tool
before diving into further.
What’s in these objects?
First, let’s talk about the object we have. The resulting
fixest (and lm) object is pretty
terse, but that’s only because it buries most of its valuable
information — of which there is a lot — within an internal list
structure. If you’re in RStudio, you can inspect this structure by
typing View(ols_fixest) or simply clicking on the
“ols_fixest” object in your environment pane. Doing so will prompt an
interactive panel to pop up for you to play around with. That approach
won’t work for this knitted R Markdown document, however, so I’ll use
the listviewer::jsonedit() function (used in the APIs
lecture) instead.
# View(ols_fixest) ## Run this instead if you're in a live session
listviewer::jsonedit(ols_fixest, mode="view") ## Better for R MarkdownAs we can see, this ols_fixest object has a bunch of
important slots… containing everything from the regression coefficients,
to vectors of the residuals and fitted (i.e. predicted) values, to the
rank of the design matrix, to the input data, etc. etc. To summarise the
key pieces of information, we can use the — wait for it —
generic summary() function. This will look pretty similar
to the default regression output from Stata that many of you will be
used to.
## OLS estimation, Dep. Var.: kfr_p25
## Observations: 3,219
## Standard-errors: IID
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 5.623064 0.402329 13.976276 < 2.2e-16 ***
## poor_share1990 -0.006886 0.014945 -0.460733 0.64502
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## RMSE: 21.9 Adj. R2: -2.448e-4We can then dig down further by extracting a summary of the regression coefficients:
## (Intercept) poor_share1990
## 5.623064305 -0.006885866Multiple regression
The feols() function can handle multiple regression
models unlike lm(). It can split regressions by different
dependent variables, independent variables, fixed effects, and subsets
all in one. I need to briefly create an indicator for region to show you
multiple subset regression.
# Indicator for south created.
opp_atlas <- opp_atlas %>%
mutate(insouth= ifelse(state_abb %in% c("AL", "AR", "FL", "GA", "KY", "LA", "MS", "NC", "OK", "SC", "TN", "TX", "VA", "WV"), "South", "North"))
ols_multi <- feols(c(kfr_p25, kfr_white_p25) ~
sw(poor_share1990,ann_avg_job_growth_2004_2013),
fsplit=~insouth, # split by region
data = opp_atlas)
etable(ols_multi) # explained below## ols_multi.1 ols_multi.2
## Sample (insouth) Full sample Full sample
## Dependent Var.: kfr_p25 kfr_p25
##
## Constant 5.623*** (0.4023) 5.688*** (0.3943)
## poor_share1990 -0.0069 (0.0149)
## ann_avg_job_growth_2004_2013 36.74 (25.58)
## ____________________________ _________________ _________________
## S.E. type IID IID
## Observations 3,219 3,214
## R2 6.6e-5 0.00064
## Adj. R2 -0.00024 0.00033
##
## ols_multi.3 ols_multi.4
## Sample (insouth) Full sample Full sample
## Dependent Var.: kfr_white_p25 kfr_white_p25
##
## Constant 0.4620*** (0.0010) 0.4636*** (0.0010)
## poor_share1990 -1.49e-5 (4.65e-5)
## ann_avg_job_growth_2004_2013 0.5987*** (0.0676)
## ____________________________ __________________ __________________
## S.E. type IID IID
## Observations 3,131 3,126
## R2 3.28e-5 0.02450
## Adj. R2 -0.00029 0.02419
##
## ols_multi.5 ols_multi.6
## Sample (insouth) North North
## Dependent Var.: kfr_p25 kfr_p25
##
## Constant 5.594*** (0.5449) 5.848*** (0.5363)
## poor_share1990 0.0172 (0.0188)
## ann_avg_job_growth_2004_2013 25.63 (35.33)
## ____________________________ _________________ _________________
## S.E. type IID IID
## Observations 1,825 1,820
## R2 0.00046 0.00029
## Adj. R2 -8.67e-5 -0.00026
##
## ols_multi.7 ols_multi.8
## Sample (insouth) North North
## Dependent Var.: kfr_white_p25 kfr_white_p25
##
## Constant 0.4816*** (0.0014) 0.4846*** (0.0014)
## poor_share1990 4.31e-5 (6.73e-5)
## ann_avg_job_growth_2004_2013 0.9318*** (0.0950)
## ____________________________ __________________ __________________
## S.E. type IID IID
## Observations 1,742 1,737
## R2 0.00024 0.05253
## Adj. R2 -0.00034 0.05199
##
## ols_multi.9 ols_multi.10
## Sample (insouth) South South
## Dependent Var.: kfr_p25 kfr_p25
##
## Constant 5.665*** (0.5949) 5.459*** (0.5825)
## poor_share1990 -0.0535* (0.0249)
## ann_avg_job_growth_2004_2013 51.23 (37.07)
## ____________________________ _________________ _________________
## S.E. type IID IID
## Observations 1,394 1,394
## R2 0.00329 0.00137
## Adj. R2 0.00258 0.00065
##
## ols_multi.11 ols_multi.12
## Sample (insouth) South South
## Dependent Var.: kfr_white_p25 kfr_white_p25
##
## Constant 0.4371*** (0.0012) 0.4377*** (0.0012)
## poor_share1990 -1.01e-5 (5.08e-5)
## ann_avg_job_growth_2004_2013 0.3043*** (0.0754)
## ____________________________ __________________ __________________
## S.E. type IID IID
## Observations 1,389 1,389
## R2 2.83e-5 0.01160
## Adj. R2 -0.00069 0.01088
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1One function call produced 12 regressions and you could technically do thousands or even millions at once. This is a very powerful tool for running many regressions at once. Use it wisely.
Get “tidy” regression coefficients with the broom
package
While it’s easy to extract regression coefficients via the
summary() function, in practice I often use the
broom package (link) to do so.
broom has a bunch of neat features to convert
statistical objects like regressions into “tidy” data frames. This is
especially useful because regression output is so often used as an input
to something else, e.g. a plot of coefficients or marginal effects.
Here, I’ll use broom::tidy(..., conf.int = TRUE) to coerce
the ols_fixest regression object into a tidy data frame of
coefficient values and key statistics.
## # A tibble: 2 × 7
## term estimate std.error statistic p.value conf.low conf.high
## <chr> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 (Intercept) 5.62 0.402 14.0 3.88e-43 4.83 6.41
## 2 poor_share1990 -0.00689 0.0149 -0.461 6.45e- 1 -0.0362 0.0224Again, I could now pipe this tidied coefficients data frame to a
ggplot2 call, using saying
geom_pointrange() to plot the error bars. Feel free to
practice doing this yourself now, but we’ll get to some explicit
examples further below.
broom has a couple of other useful functions too.
For example, broom::glance() summarises the model “meta”
data (R2, AIC, etc.) in a data frame.
## # A tibble: 1 × 9
## r.squared adj.r.squared within.r.squared pseudo.r.squared sigma nobs AIC
## <dbl> <dbl> <dbl> <dbl> <dbl> <int> <dbl>
## 1 0.0000660 -0.000245 NA NA 21.9 3219 29017.
## # ℹ 2 more variables: BIC <dbl>, logLik <dbl>By the way, if you’re wondering how to export regression results to other formats (e.g. LaTeX tables), don’t worry: We’ll get to that at the end of the lecture.
Regressing on subsetted data
Our simple model isn’t particularly good; the R2 is only 0. Let’s check for outliers by plotting the data.
It looks like NAs were replaced with 99 by someone… ahem… someone who wants to remind you to plot your data and check for quirks like this. Maybe we should exclude outliers from our regression? You can do this in two ways:
- Subset (
filter()) the original data frame directly in thefeols()call. - Create a new data frame and then regress
1) Subset directly in the lm() call
Running a regression directly on a subsetted data frame is equally easy.
ols_fixest_sub = feols(kfr_p25 ~ poor_share1990, data = opp_atlas %>% filter(kfr_p25!=99 & poor_share1990!=99))
summary(ols_fixest_sub)## OLS estimation, Dep. Var.: kfr_p25
## Observations: 2,824
## Standard-errors: IID
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.464994 0.002621 177.3889 < 2.2e-16 ***
## poor_share1990 -0.211911 0.014356 -14.7616 < 2.2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## RMSE: 0.060377 Adj. R2: 0.0713522) Create a new data frame
Recall that we can keep multiple objects in memory in R. So we can easily create a new data frame that excludes Jabba using, say, dplyr (lecture) or data.table (lecture. For these lecture notes, I’ll stick with dplyr commands. But it would take just a little elbow grease (with help from ChatGPT or CoPilot) to switch to data.table if you prefer.
opp_atlas =
opp_atlas %>%
filter(kfr_p25!=99 & poor_share1990!=99)
ols_fixest_filter = feols(kfr_p25 ~ poor_share1990, data = opp_atlas)
summary(ols_fixest_filter)## OLS estimation, Dep. Var.: kfr_p25
## Observations: 2,824
## Standard-errors: IID
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.464994 0.002621 177.3889 < 2.2e-16 ***
## poor_share1990 -0.211911 0.014356 -14.7616 < 2.2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## RMSE: 0.060377 Adj. R2: 0.071352The overall model fit is much improved by the exclusion of this outlier, with R2 increasing to 0.072. Still, we should be cautious about throwing out data. Another approach is to handle or account for outliers with statistical methods. Which provides a nice segue to nonstandard errors.
Nonstandard errors
Dealing with statistical irregularities (heteroskedasticity, clustering, etc.) is a fact of life for empirical researchers. However, it says something about the economics profession that a random stranger could walk uninvited into a live seminar and ask, “How did you cluster your standard errors?”, and it would likely draw approving nods from audience members.
The good news is that there are lots of ways to get
nonstandard errors in R. For many years, these have been based on the
excellent sandwich package (link)
and estimatr package (link).
However, I’ll demonstrate using the fixest package,
which has a built-in vcov argument that allows you to
easily specify different types of standard errors.
Robust standard errors
One of the primary reasons that you might want to use robust standard errors is to account for heteroskedasticity. What is heteroskedasticity, well it is when the variance of the error term is not constant across observations. This is a problem because it violates one of the key assumptions of OLS regression, namely that the error term is homoskedastic (i.e. constant variance).1 I present an example below with fake data (cause it is easier than forcing it out of real data.)
# Create an example of heteroskedasticity
# This creates a simple regression, but with variance that increases with x
hetero_df <- tibble(
x = rnorm(1000),
y = 1 + 2 * x + rnorm(1000, sd = 1 + 5 * abs(x))
)
# Plot the data
hetero_df %>%
ggplot(aes(x = x, y = y)) +
geom_point(alpha = 0.5) +
geom_smooth(method = "lm", se = TRUE) +
labs(
title = "Heteroskedasticity",
subtitle = "The variance of the error term is not constant across observations",
caption = "Source: Gen AI (GitHub CoPilot) helped me write this example, but I still used my brain too."
)## `geom_smooth()` using formula = 'y ~ x'
There are many ways to deal with heteroskedasticity, but one of the
most common is to use robust standard errors or “HC1”
heteroskedasticity-consistent standard errors. fixest
and feols() has an argument, vcov (for
variance-covariance matrix) that you can set as “HC1” or “hetero” to
generate standard errors. I’ll illustrate below!
ols_robust = feols(kfr_p25 ~ poor_share1990, data = opp_atlas, vcov = "HC1")
ols_robust = feols(kfr_p25 ~ poor_share1990, data = opp_atlas, vcov = "hetero")
# tidy(ols1_robust, conf.int = TRUE) ## Could tidy too
ols_robust## OLS estimation, Dep. Var.: kfr_p25
## Observations: 2,824
## Standard-errors: Heteroskedasticity-robust
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.464994 0.002385 194.9884 < 2.2e-16 ***
## poor_share1990 -0.211911 0.012915 -16.4083 < 2.2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## RMSE: 0.060377 Adj. R2: 0.071352If you use the package estimatr, there is a function
lm_robust(), which defaults to using Eicker-Huber-White
robust standard errors, commonly referred to as “HC2” standard errors.
You can easily specify alternate methods using the
se_type = and vcov argument.2 For example, you can
specify Stata robust standard errors (reg y x, robust) if
you want to replicate code or results from that language. (See here
for more details on why this isn’t the default and why Stata’s robust
standard errors differ from those in R and Python. tl;dr: A few years
ago several people realized Stata was reporting different SEs than they
expected.) See Grant’s notes for more advanced treatment of
robustness.
Clustered standard errors
Another way that standard errors can violate homoskedasticity is for the error to be “clustered” by groups in the data. A classic example is students in the same classroom will all have the same teacher and so their learning outcomes will be correlated. For example, all of you have the same teacher (me), so you will all learn the same coding habits. This is a problem because it violates the assumption that the error term is independent across observations. Here’s an example with the Opportunity Atlas data.
# Let's look at just a few states, so we can easily see clusters
filter(opp_atlas, state_abb %in% c('FL','NY','TX','ME')) %>%
ggplot(aes(x=poor_share1990,y=kfr_p25)) +
geom_point(aes(col=state_abb)) +
geom_smooth(method='lm') +
labs(
col = 'State',
title = "Error clusters",
subtitle = "The error term is correlated within clusters",
caption = "Source: GitHub CoPilot helped me write this example, but I still used my brain"
)## `geom_smooth()` using formula = 'y ~ x'
Clustered standard errors is an issue that most commonly affects
panel data. As such, I’m going to hold off discussing clustering until
we get to the panel data
section below. But here’s a quick example of clustering with
fixest::feols:
Manipulating variables
Standardize variables
Regression coefficients can be tricky to understand! The units may be strange or the coefficients may be hard to interpret. One way to make coefficients easier to interpret is to manipulate the variables to be more informative. For example, you might want to standardize the variables to return a correlation coefficient.
The statistical equation for a correlation coefficient is:
\[r = \frac{\sum_{i=1}^{n} (x_i - \bar{x})(y_i - \bar{y})}{\hat{\sigma}^2_x,\hat{\sigma}^2_y} \]
Where \(x_i\) and \(y_i\) are the individual observations, and
\(\bar{x}\) and \(\bar{y}\) are the means of the variables.
The \(\hat{\sigma}^2_x\) and \(\hat{\sigma}^2_y\) are the sample variances
of the variables. Let’s change the variables kfr_p25 and
poor_share1990 to be standardized and then run a
regression.
opp_atlas <- opp_atlas %>%
mutate(
kfr_p25_std = (kfr_p25-mean(kfr_p25))/sd(kfr_p25),
poor_share1990_std = (poor_share1990-mean(poor_share1990))/sd(poor_share1990)
)Take a look at the variables to see if they are standardized.
## OLS estimation, Dep. Var.: kfr_p25_std
## Observations: 2,824
## Standard-errors: IID
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -1.990000e-16 0.018134 -1.100000e-14 1
## poor_share1990_std -2.677334e-01 0.018137 -1.476157e+01 < 2.2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## RMSE: 0.963322 Adj. R2: 0.071352Is that a correlation coefficient? Maybe? That’d be neat! How can we
find out? Well use the cor() function and see if the
coefficients are the same.
## [1] -0.2677334They are! This is a handy trick to standardized two radically different variables to tell you about the overall correlation when you’re less interested in the individual coefficients. This regression trick also returns standard errors which are really nice to have.
Convert variables to logs
Another useful trick is to log variables! You can do this directly or
use the log() function.
## # A tibble: 2 × 5
## term estimate std.error statistic p.value
## <chr> <dbl> <dbl> <dbl> <dbl>
## 1 (Intercept) -1.00 0.0107 -93.7 0
## 2 log(poor_share1990) -0.0766 0.00541 -14.2 4.67e-44Dummy variables and interaction terms
For the next section, we’ll need to create a dummy variable. We’ll create two: one for whether a county is in the South and one for having positive job growth.
# Get the regions and do minor cleaning
opp_atlas <- mutate(opp_atlas,
in_south = ifelse(state_abb %in% c("AL", "AR",
"FL", "GA", "KY", "LA", "MS", "NC", "OK",
"SC", "TN", "TX", "VA", "WV"), "South", "North"))Dummy variables as factors
Dummy variables are a core component of many regression models. However, these can be a pain to create in some statistical languages, since you first have to tabulate a whole new matrix of binary variables and then append it to the original data frame. In contrast, R has a very convenient framework for creating and evaluating dummy variables in a regression: Simply specify the variable of interest as a factor.3
Here’s an example where we explicitly tell R that “in_south” is a factor. Since I don’t plan on reusing this model, I’m just going to print the results to screen rather than saving it to my global environment.
## OLS estimation, Dep. Var.: kfr_p25
## Observations: 2,824
## Standard-errors: IID
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.467104 0.002368 197.24602 < 2.2e-16 ***
## poor_share1990 -0.072648 0.014080 -5.15974 2.6445e-07 ***
## as.factor(in_south)South -0.056854 0.002246 -25.31793 < 2.2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## RMSE: 0.054501 Adj. R2: 0.243025Okay, I should tell you that I’m actually making things more complicated than they need to be with the heavy-handed emphasis on factors. R is “friendly” and tries to help whenever it thinks you have misspecified a function or variable. While this is something to be aware of, normally It Just WorksTM. A case in point is that we don’t actually need to specify a string (i.e. character) variable as a factor in a regression. R will automatically do this for you regardless, since it’s the only sensible way to include string variables in a regression.
## Use the non-factored version of "in_south" instead; R knows it must be ordered
## for it to be included as a regression variable
summary(feols(kfr_p25 ~ poor_share1990 + in_south, data = opp_atlas))## OLS estimation, Dep. Var.: kfr_p25
## Observations: 2,824
## Standard-errors: IID
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.467104 0.002368 197.24602 < 2.2e-16 ***
## poor_share1990 -0.072648 0.014080 -5.15974 2.6445e-07 ***
## in_southSouth -0.056854 0.002246 -25.31793 < 2.2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## RMSE: 0.054501 Adj. R2: 0.243025What happens if I use job growth quartile as a dummy variable? First, let’s check the values of this value. It is uneven because we removed missing values before (a bit haphazardly, I might add.)
##
## 1 2 3 4
## 678 715 710 716## NOTE: 5 observations removed because of NA values (RHS: 5).## OLS estimation, Dep. Var.: kfr_p25
## Observations: 2,819
## Standard-errors: IID
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.455969 0.004145 110.00006 < 2.2e-16 ***
## poor_share1990 -0.203346 0.014739 -13.79654 < 2.2e-16 ***
## job_growth_quartile 0.003074 0.001048 2.93406 0.0033727 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## RMSE: 0.060123 Adj. R2: 0.075183We can fix this by telling R that job_growth_quartile is a factor.
## NOTE: 5 observations removed because of NA values (RHS: 5).## OLS estimation, Dep. Var.: kfr_p25
## Observations: 2,819
## Standard-errors: IID
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.459600 0.003791 121.241972 < 2.2e-16 ***
## poor_share1990 -0.204437 0.014890 -13.729452 < 2.2e-16 ***
## as.factor(job_growth_quartile)2 0.002975 0.003279 0.907244 0.364355
## as.factor(job_growth_quartile)3 0.004225 0.003333 1.267562 0.205059
## as.factor(job_growth_quartile)4 0.009747 0.003315 2.940454 0.003304 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## RMSE: 0.060116 Adj. R2: 0.074744Interaction effects
Like dummy variables, R provides a convenient syntax for specifying interaction terms directly in the regression model without having to create them manually beforehand.4 You can use any of the following expansion operators:
x1:x2“crosses” the variables (equivalent to including only the x1 × x2 interaction term)x1/x2“nests” the second variable within the first (equivalent tox1 + x1:x2; more on this later)x1*x2includes all parent and interaction terms (equivalent tox1 + x2 + x1:x2)
As a rule of thumb, if not always, it is
generally advisable to include all of the parent terms alongside their
interactions. This makes the * option a good default.
For example, we might wonder whether the relationship between a location’s income mobility for children in the 25th percentile and its 1990 poverty rate differs by region. That is, we want to run a regression of the form,
\[KFR_{P25} = \beta_0 + \beta_1 D_{South} + \beta_2 + \beta_3 D_{South} \times 1990 Poverty Share\]
To implement this in R, we simply run the following,
## OLS estimation, Dep. Var.: kfr_p25
## Observations: 2,824
## Standard-errors: IID
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.468163 0.003317 141.120464 < 2.2e-16 ***
## in_southSouth -0.058977 0.005170 -11.406464 < 2.2e-16 ***
## poor_share1990 -0.080372 0.022029 -3.648380 0.00026868 ***
## in_southSouth:poor_share1990 0.013060 0.028646 0.455913 0.64848794
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## RMSE: 0.054499 Adj. R2: 0.242813Presentation
Tables
Regression tables
There are loads of different options here.
Fixest actually provides its own table function,
etable() (for “estimation table”), which is a bit like
Stata’s esttab or outreg command. It’s a bit
more flexible than the default summary() function, but it
only works with fixest list objects.
## ols_fixest ols_fixest_filter
## Dependent Var.: kfr_p25 kfr_p25
##
## Constant 5.623*** (0.4023) 0.4650*** (0.0026)
## poor_share1990 -0.0069 (0.0149) -0.2119*** (0.0144)
## poor_share1990_std
## in_southSouth
## in_southSouth x poor_share1990
## ______________________________ _________________ ___________________
## S.E. type IID IID
## Observations 3,219 2,824
## R2 6.6e-5 0.07168
## Adj. R2 -0.00024 0.07135
##
## ols_robust fixest_cluster
## Dependent Var.: kfr_p25 kfr_p25
##
## Constant 0.4650*** (0.0024) 0.4650*** (0.0108)
## poor_share1990 -0.2119*** (0.0129) -0.2119*** (0.0464)
## poor_share1990_std
## in_southSouth
## in_southSouth x poor_share1990
## ______________________________ ___________________ ___________________
## S.E. type Heteroskedast.-rob. by: state
## Observations 2,824 2,824
## R2 0.07168 0.07168
## Adj. R2 0.07135 0.07135
##
## fixest_corr ols_ie
## Dependent Var.: kfr_p25_std kfr_p25
##
## Constant -1.99e-16 (0.0181) 0.4682*** (0.0033)
## poor_share1990 -0.0804*** (0.0220)
## poor_share1990_std -0.2677*** (0.0181)
## in_southSouth -0.0590*** (0.0052)
## in_southSouth x poor_share1990 0.0131 (0.0286)
## ______________________________ ___________________ ___________________
## S.E. type IID IID
## Observations 2,824 2,824
## R2 0.07168 0.24362
## Adj. R2 0.07135 0.24281
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Hmm… how can I clean that up a bit? Well etable() has a
number of options including dict where you can assign
variable names.
dict = c("poor_share1990"="Poverty Rate (1990)",
'poor_share1990_std'='Std. Poverty Rate (1990)',
'kfr_p25'='Income Mobility (25th Percentile)',
'kfr_p25_std'='Std. Income Mobility (25th Percentile)',
'in_south'='In South')
etable(ols_fixest,ols_fixest_filter,ols_robust,fixest_cluster,fixest_corr, dict = dict)## ols_fixest
## Dependent Var.: Income Mobility (25th Percentile)
##
## Constant 5.623*** (0.4023)
## Poverty Rate (1990) -0.0069 (0.0149)
## Std. Poverty Rate (1990)
## ________________________ ______________________________
## S.E. type IID
## Observations 3,219
## R2 6.6e-5
## Adj. R2 -0.00024
##
## ols_fixest_filter
## Dependent Var.: Income Mobility (25th Percentile)
##
## Constant 0.4650*** (0.0026)
## Poverty Rate (1990) -0.2119*** (0.0144)
## Std. Poverty Rate (1990)
## ________________________ ______________________________
## S.E. type IID
## Observations 2,824
## R2 0.07168
## Adj. R2 0.07135
##
## ols_robust
## Dependent Var.: Income Mobility (25th Percentile)
##
## Constant 0.4650*** (0.0024)
## Poverty Rate (1990) -0.2119*** (0.0129)
## Std. Poverty Rate (1990)
## ________________________ ______________________________
## S.E. type Heteroskedasticity-robust
## Observations 2,824
## R2 0.07168
## Adj. R2 0.07135
##
## fixest_cluster
## Dependent Var.: Income Mobility (25th Percentile)
##
## Constant 0.4650*** (0.0108)
## Poverty Rate (1990) -0.2119*** (0.0464)
## Std. Poverty Rate (1990)
## ________________________ ______________________________
## S.E. type by: state
## Observations 2,824
## R2 0.07168
## Adj. R2 0.07135
##
## fixest_corr
## Dependent Var.: Std. Income Mobility (25th Percentile)
##
## Constant -1.99e-16 (0.0181)
## Poverty Rate (1990)
## Std. Poverty Rate (1990) -0.2677*** (0.0181)
## ________________________ ______________________________
## S.E. type IID
## Observations 2,824
## R2 0.07168
## Adj. R2 0.07135
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1There are tons of other tools to use too!
As a note, here’s what you need to do in Rmarkdown to get it to work
in a PDF format. You must specify to output as a latex file, then you
can report “asis” and ask etable() to output a tex file.
(See PDF for that.)
Alternatively, you can save this file to a tex file and then include
it in your LaTeX document using input() or
include().
etable(ols_fixest,ols_fixest_filter,ols_robust,fixest_cluster,fixest_corr, dict = dict, tex=TRUE,file="regression_table.tex")Other neat tricks with etable() is that you can specify
a “note” and “label” for the table. Also, you can change standard error
calculations for all the models at once.
etable(ols_fixest,ols_fixest_filter,ols_robust,fixest_cluster,fixest_corr, dict = dict, notes="This is a caption", label="tab:regression_table",se='IID')## ols_fixest
## Dependent Var.: Income Mobility (25th Percentile)
##
## Constant 5.623*** (0.4023)
## Poverty Rate (1990) -0.0069 (0.0149)
## Std. Poverty Rate (1990)
## ________________________ ______________________________
## S.E. type IID
## Observations 3,219
## R2 6.6e-5
## Adj. R2 -0.00024
##
## ols_fixest_filter
## Dependent Var.: Income Mobility (25th Percentile)
##
## Constant 0.4650*** (0.0026)
## Poverty Rate (1990) -0.2119*** (0.0144)
## Std. Poverty Rate (1990)
## ________________________ ______________________________
## S.E. type IID
## Observations 2,824
## R2 0.07168
## Adj. R2 0.07135
##
## ols_robust
## Dependent Var.: Income Mobility (25th Percentile)
##
## Constant 0.4650*** (0.0026)
## Poverty Rate (1990) -0.2119*** (0.0144)
## Std. Poverty Rate (1990)
## ________________________ ______________________________
## S.E. type IID
## Observations 2,824
## R2 0.07168
## Adj. R2 0.07135
##
## fixest_cluster
## Dependent Var.: Income Mobility (25th Percentile)
##
## Constant 0.4650*** (0.0026)
## Poverty Rate (1990) -0.2119*** (0.0144)
## Std. Poverty Rate (1990)
## ________________________ ______________________________
## S.E. type IID
## Observations 2,824
## R2 0.07168
## Adj. R2 0.07135
##
## fixest_corr
## Dependent Var.: Std. Income Mobility (25th Percentile)
##
## Constant -1.99e-16 (0.0181)
## Poverty Rate (1990)
## Std. Poverty Rate (1990) -0.2677*** (0.0181)
## ________________________ ______________________________
## S.E. type IID
## Observations 2,824
## R2 0.07168
## Adj. R2 0.07135
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Other tools
Another great tool for creating and exporting regression tables is the modelsummary package (link). It is extremely flexible and handles all manner of models and output formats. modelsummary also supports automated coefficient plots and data summary tables, which I’ll get back to in a moment. The documentation is outstanding and you should read it, but here is a bare-boned example just to demonstrate.
# library(modelsummary) ## Already loaded
## Note: msummary() is an alias for modelsummary() add variable names
msummary(list("lm"=ols_lm, "fixest"=ols_fixest,"filter"=ols_fixest_filter,"robust"=ols_robust,"cluster"=fixest_cluster),
stars=TRUE, ## Output type
coef_map = c("poor_share1990"="Poverty Rate (1990)",
"(Intercept)"="Constant"),
title="Relationship between Average Income Percentile in 2015 of Children born in 25th percentile and 1990 poverty rate") ## Rename coefficients| lm | fixest | filter | robust | cluster | |
|---|---|---|---|---|---|
| Poverty Rate (1990) | −0.007 | −0.007 | −0.212*** | −0.212*** | −0.212*** |
| (0.015) | (0.015) | (0.014) | (0.013) | (0.046) | |
| Constant | 5.623*** | 5.623*** | 0.465*** | 0.465*** | 0.465*** |
| (0.402) | (0.402) | (0.003) | (0.002) | (0.011) | |
| Num.Obs. | 3219 | 3219 | 2824 | 2824 | 2824 |
| R2 | 0.000 | 0.000 | 0.072 | 0.072 | 0.072 |
| R2 Adj. | 0.000 | 0.000 | 0.071 | 0.071 | 0.071 |
| AIC | 29018.7 | 29016.7 | −7836.6 | −7836.6 | −7836.6 |
| BIC | 29036.9 | 29028.8 | −7824.7 | −7824.7 | −7824.7 |
| Log.Lik. | −14506.342 | ||||
| RMSE | 21.92 | 21.92 | 0.06 | 0.06 | 0.06 |
| Std.Errors | IID | IID | Heteroskedasticity-robust | by: state | |
| + p < 0.1, * p < 0.05, ** p < 0.01, *** p < 0.001 |
One nice thing about modelsummary is that it plays very well with R Markdown and will automatically coerce your tables to the format that matches your document output: HTML, LaTeX/PDF, RTF, etc. Of course, you can also specify the output type if you aren’t using R Markdown and want to export a table for later use. Finally, you can even specify special table formats like threepartable for LaTeX and, provided that you have called the necessary packages in your preamble, it will render correctly (see example here).
Summary tables
A variety of summary tables — balance, correlation, etc. — can be
produced by the companion set of
modelsummary::datasummary*() functions. Again, you should
read the documentation
to see all of the options. But here’s an example of a very simple
balance table using a subset of our “humans” data frame.
datasummary_balance(~ in_south,
data = opp_atlas %>%
dplyr::select(kfr_p25:ann_avg_job_growth_2004_2013,in_south))| Mean | Std. Dev. | Mean | Std. Dev. | Diff. in Means | Std. Error | |
|---|---|---|---|---|---|---|
| kfr_p25 | 0.5 | 0.1 | 0.4 | 0.0 | -0.1 | 0.0 |
| kfr_white_p25 | 0.5 | 0.1 | 0.4 | 0.0 | 0.0 | 0.0 |
| poor_share1990 | 0.1 | 0.1 | 0.2 | 0.1 | 0.1 | 0.0 |
| ann_avg_job_growth_2004_2013 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
Another package that I like a lot in this regard is vtable (link). Not only can it be used to construct descriptive labels like you’d find in Stata’s “Variables” pane, but it is also very good at producing the type of “out of the box” summary tables that economists like. For example, here’s the equivalent version of the above balance table.
# library(vtable) ## Already loaded
## An additional argument just for formatting across different output types of
## this .Rmd document
otype = ifelse(knitr::is_latex_output(), 'return', 'kable')
## vtable::st() is an alias for sumtable()
vtable::st(opp_atlas %>%
dplyr::select(kfr_p25:ann_avg_job_growth_2004_2013, in_south),
group = 'in_south',
out = otype)| Variable | N | Mean | SD | N | Mean | SD |
|---|---|---|---|---|---|---|
| kfr_p25 | 1581 | 0.46 | 0.062 | 1243 | 0.4 | 0.044 |
| kfr_white_p25 | 1577 | 0.48 | 0.058 | 1240 | 0.44 | 0.044 |
| poor_share1990 | 1581 | 0.14 | 0.062 | 1243 | 0.2 | 0.085 |
| ann_avg_job_growth_2004_2013 | 1577 | -0.003 | 0.014 | 1243 | -0.0021 | 0.015 |
Lastly, Stata users in particular might like the qsu()
and descr() functions from the lightning-fast
collapse package (link).
Figures
Coefficient plots
We’ve already worked through an example of how to extract and compare
model coefficients here.
I use this “manual” approach to visualizing coefficient estimates all
the time. However, our focus on modelsummary in the
preceding section provides a nice segue to another one of the package’s
features: modelplot().
Consider the following, which shows both the degree to which
modelplot() automates everything and the fact that it
readily accepts regular ggplot2 syntax.
# library(modelsummary) ## Already loaded
mods = list('No clustering' = summary(ols_fixest, se = 'standard'))
modelplot(mods) +
## You can further modify with normal ggplot2 commands...
coord_flip() +
labs(
title = "Relationship between Pov Rate (1990) and Income Mobility",
subtitle = "Comparing fixed effect models"
)
Further resources
- Ed Rubin has outstanding teaching notes for econometrics with R on his website. This includes both undergrad- and graduate-level courses. Seriously, check them out.
- Several introductory texts are freely available, including Introduction to Econometrics with R (Christoph Hanck et al.), Using R for Introductory Econometrics (Florian Heiss), and Modern Dive (Chester Ismay and Albert Kim).
- Tyler Ransom has a nice cheat sheet for common regression tasks and specifications.
- Itamar Caspi has
written a neat unofficial appendix to this lecture, recipes
for Dummies. The title might be a little inscrutable if you
haven’t heard of the
recipespackage before, but basically it handles “tidy” data preprocessing, which is an especially important topic for machine learning methods. We’ll get to that later in course, but check out Itamar’s post for a good introduction. - I promised to provide some links to time series analysis. The good news is that R’s support for time series is very, very good. The Time Series Analysis task view on CRAN offers an excellent overview of available packages and their functionality.
- Lastly, for more on visualizing regression output, I highly encourage you to look over Chapter 6 of Kieran Healy’s Data Visualization: A Practical Guide. Not only will learn how to produce beautiful and effective model visualizations, but you’ll also pick up a variety of technical tips.
See Causal Inference: The Mixtape for more details.↩︎
See the package documentation for a full list of options.↩︎
Factors are variables that have distinct qualitative levels, e.g. “male”, “female”, “non-binary”, etc.↩︎
Although there are very good reasons that you might want to modify your parent variables before doing so (e.g. centering them). As it happens, Grant McDermott has strong feelings that interaction effects are most widely misunderstood and misapplied concept in econometrics. However, that’s a topic for another day.↩︎