Spurious Regression in R

spuriousRegression.r

library(tidyverse)
library(ggplot2)

set.seed(123)
sample_num <- 1000
sigma <- 1

# Generate Two Unit Root Processes
x <- c(0,cumsum(rnorm(sample_num-1, mean=0, sd=sigma)))
y <- c(0,cumsum(rnorm(sample_num-1, mean=0, sd=sigma)))

# Plot
df %>%
    gather(-t, key=var, value=val) %>%
    ggplot(aes(x=t, y=val, fill=var)) +
    geom_line(aes(col=var), size = 1) +
    theme_minimal() +
    scale_color_brewer(palette="Paired") +
    geom_label(aes(x = 10, y = 30, label = "Here is a line"), fill = "#dddddd") +
    labs(title = "SPURIOUS REGRESSION EXAMPLE") +
    annotate("text", x = 60, y = 40, hjust=0.08, size=4,
    label = "R^2 = 0.25 \n p-value < 2.2e-16", color="#333333", fontface="bold")

# Regression
df <- data.frame(t=1:sample_num, x, y)
summary(lm(y~x,df))

# White Noise
white_noise <- data.frame(t=1:100,x=rnorm(100, mean=0, sd=1))
white_noise %>%
	gather(-t, key=var, value=val) %>%
	ggplot(aes(x=t, y=val, fill=var)) +
	geom_line(aes(col=var), size = 1) +
	theme_minimal() +
	scale_color_brewer(palette="Paired") +
    labs(title = "White Noise (Mean 0, Variance 1)")

# Random Walk
randomWalk <- function(size, d, rho, sigma)
{
    return(Reduce(function(x, noise){d+rho*x+noise}, rnorm(size-1, mean=0, sd=sigma), init=0, accumulate=TRUE))
}

rw <- data.frame(t=1:100, x=randomWalk(100, 0, 1, 1))
rw %>%
    gather(-t, key=var, value=val) %>%
    ggplot(aes(x=t, y=val, fill=var)) +
    geom_line(aes(col=var), size = 1) +
    theme_minimal() +
    scale_color_brewer(palette="Paired") +
    labs(title = "Random Walk without Drift")

rwd <- data.frame(t=1:100, x=randomWalk(100, 0.25, 1, 1))
rwd %>%
    gather(-t, key=var, value=val) %>%
    ggplot(aes(x=t, y=val, fill=var)) +
    geom_line(aes(col=var), size = 1) +
    theme_minimal() +
    scale_color_brewer(palette="Paired") +
    labs(title = "Random Walk with Positive Drift")