Special Topics in R

Effects of Measurement Error

Effects of Measurement Error

In the panel on the left, you’ll see that the points spread out horizontally but do not change their vertical positions. Notice how the regression line, depicted in solid black, seems to quickly flatten out as the measurement error increases. This is what I mean by saying measurement error in \(X\) pushes the slope coefficient towards zero.

In the panel on the right, the opposite movement is occurring for the points. Here, they are spreading out vertically, but not changing their position horizontally. In this example, the regression line becomes more vertical over time (i.e., the slope coefficient is pushed away from zero), but this is due to the particularities of this specific simulation. With measurement error in \(Y\), the slope coefficient is as likely to increase as it is to decrease.

See below for the second set of simulations with similar results.

Effects of Measurement Error

Standard Errors

Now, what do the parameter estimates and standard errors look like from these regressions? Remember, the slope should be about equal to \(1\).

Column (a) in this table is the homoskedastic version, and should be thought of as a baseline. Since the errors are different, the estimates will also be a bit different, so I want you to focus just on the standard errors. Heteroskedasticity can both artificially inflate or deflate standard errors, though it’s not always clear how the standard errors will be impacted. In general, though, the coefficients are all close to \(1\), though some of the standard errors are larger and some are smaller than the homoskedastic one.

Standard Errors

To address the issue of heteroskedasticity, we can use heteroskedasticity-robust standard errors. Essentially, without getting too deep into the math, we are going to correct for the non-constant variance of the errors. This will give us better (more accurate) standard errors for testing our hypotheses.

To apply this correction, we are going to use an R package called fixest.

At this point, the main benefits of this package are two fold. First, fixed effects are estimated extremely quickly. We’ll do some “speed tests” in just a bit. Second, we can apply different standard error corrections to our models fairly easily.

feols

library("fixest")
df <- read.csv("https://vincentarelbundock.github.io/Rdatasets/csv/openintro/satgpa.csv")
df <- df[df$hs_gpa <= 4,]

r <- feols(sat_sum ~ hs_gpa, df, fsplit = ~sex)

# Note that r is now a "list" of three regressions.
regz <- list(`All Students` = r[[1]],
             `Female` = r[[2]],
             `Male` = r[[3]])
coefz <- c("hs_gpa" = "High School GPA",
           "(Intercept)" = "Constant")
gofz <- c("nobs", "r.squared")
modelsummary(regz,
             title = "Effect of GPA on SAT Scores",
             estimate = "{estimate}{stars}",
             coef_map = coefz,
             gof_map = gofz)

feols

For speed comparisons between lm and feols, we are going to simulate data of varying sizes with a proportional number of fixed effects (e.g. 1,000 observations and 10 FEs; 10,000 obs and 100 FEs) and see how long each function takes.

lm_time <- c()
feols_time <- c()
nz <- c(1000, 5000, 10000, 20000, 35000, 50000, 75000, 100000)
for(n in nz){
  
  x <- rnorm(n)
  fe <- sample(1:(n/100), n, TRUE)
  y <- x + rnorm(n)
  df <- data.frame(y = y, x = x, fe = fe)
  lm_time <- c(lm_time, system.time(lm(y ~ x + as.factor(fe), data = df))[3])
  feols_time <- c(feols_time, system.time(feols(y ~ x | fe, df))[3])
}

plot(nz, lm_time, pch = 19, col = "tomato", type = "b",
     xlab = "Sample Size", ylab = "Time in Seconds", las = 1)
lines(nz, feols_time, pch = 19, col = "dodgerblue", type = "b")
legend("topleft", legend = c("lm()", "feols()"), bty = "n",
       col = c("tomato", "dodgerblue"), lty = 1)
cat("Maximum lm() time:", max(lm_time), "\n")
cat("Maximum feols() time:", max(feols_time))

Maximum lm() time: 164.28 
Maximum feols() time: 0.04

With the largest dataset, lm() needed almost six minutes to estimate while feols() did not even take a tenth of a second. When datasets are small, this is not necessarily an issue, but hopefully you can see the advantage of using a faster function.

feols

Finally, let’s apply those standard error corrections. To do this, we are going to use vcov = "hetero" in our function call. The table of the new estimates can be found below:

In some cases, the standard errors increased. In other cases, the standard errors decreased. So what? When standard errors grow larger, it means that we would have been failing to reject the null too often, making our relationship seem weaker than it really is. The same is true in reverse, of course. Sometimes, students get caught up in wanting significant results, but this is not the be all, end all. It is more important that you come to the correct conclusion, even that is inconclusiveness, rather than an incorrect conclusion.

Dates in R

ymd(), dmy(), mdy(): when you read data into R and one of the columns has entries like "January 2 2014", we can use these functions to help us out. Note that y stands for year, d stands for day, and m stands for month, so just choose the function that makes sense for your situation.

Output from these functions appear in year-month-day format, but are actually numeric under the hood. We can see this by executing as.numeric(mdy("January 2 2014")): 16072.

Dates in R

Once your data is converted to well-behaved date objects, we can begin to manipulate them. We can extract the year, month, day, etc. using lubridate’s helpfully named year(), month(), and day() functions. We can also grab the weekday by using wday(df$date, label = TRUE).

Next, we can use floor_date(df$date, "quarter") “round” our dates to the nearest quarter, month, year, whatever.

Finally, we can calculate differences in dates by using simple subtraction. However, often times we want to know the number of, say, months between two dates. For this, we can use the following code: interval(date1, date2) %% months(1).

Text as Data

It’s also often the case that quantifiable information is recorded as textual data. There is a robust set of tools for handling text in R, but these are outside the scope of this course, and you will not be tested on them. However, I wanted to include some information just in case it’s needed for a project or perhaps some students are just interested. See below.

Text as Data

Date data is a very particular type of text data. However, working with text more generally is equally important when programming. We will not use any new packages for this, as base R has plenty of functions for us.

The first function we’ll take a look at is paste(), which has a sibling function paste0().

paste() takes whatever vectors you give it and smushes them together. paste() will recycle the shorter of the vector(s) until it matches the length of the longest vector. paste() will separate text with spaces by default, but this can be changed via the sep argument. paste0() defaults to sep = "", or nothing inbetween text. Some examples are below:

paste("alex", "cardazzi"); cat("\n")
paste("alex", c("cardazzi", "trebek", "hamilton")); cat("\n")
paste(c("alexander", "alex"), c("cardazzi", "trebek", "hamilton")); cat("\n")
paste(c("alex"), "middle-name", c("cardazzi", "trebek")); cat("\n")
paste("alex", c("cardazzi", "trebek", "hamilton"), sep = "-"); cat("\n")
paste0("alex", c("cardazzi", "trebek", "hamilton")); cat("\n")
paste(c("alex", "christine"), c("cardazzi", "strong"), sep = " abcxyz ")

[1] "alex cardazzi"

[1] "alex cardazzi" "alex trebek"   "alex hamilton"

[1] "alexander cardazzi" "alex trebek"        "alexander hamilton"

[1] "alex middle-name cardazzi" "alex middle-name trebek"  

[1] "alex-cardazzi" "alex-trebek"   "alex-hamilton"

[1] "alexcardazzi" "alextrebek"   "alexhamilton"

[1] "alex abcxyz cardazzi"    "christine abcxyz strong"