FD vs FE and new tidyr commands

For two periods, first differences is numerically equivalent to a fixed effect model. The following code reproduces that result, using the (new) pivot_longer and pivot_wider commands from tidyr.

library(tidyverse)
library(lfe)
library(broom)

# Create some fake data

data <- as_tibble(expand.grid(group = c("A", "B"),
                              period = 0:1))

data <- data %>% 
    mutate(x = rnorm(nrow(.)))  %>% 
    mutate(y = 3*x + period * 1.5 + rnorm(nrow(.)))

# Long differences ----

# Reshape long-then-wide so that variable names are row IDs, take differences, reshape wide again to get differences
ld <- data %>% 
    pivot_longer(c(x, y),
                 names_to = "var") %>%
    pivot_wider(id_cols = c(group, var),
                names_from = period,
                names_prefix = "period",
                values_from = "value") %>%
    mutate(diff = period1 - period0) %>%
    pivot_wider(id_cols = c(group), 
                names_from = var,
                names_prefix = "diff_",
                values_from = diff)

# LD
tidy(lm(diff_y ~ diff_x - 1, data = ld))

# FE
tidy(lm(y ~ x + group, data = data))