Solved – Real life examples of difference between independence and correlation

correlationindependenceintuition

It is well known that independence of random variables implies zero correlation but
zero correlation need not imply independence.

I came across plenty of mathematical examples demonstrating dependence despite zero correlation. Are there any real life examples to support this fact?

Best Answer

Stock returns are a decent real-life example of what you're asking for. There's very close to zero correlation between today's and yesterday's S&P 500 return. However, there is clear dependence: squared returns are positively autocorrelated; periods of high volatility are clustered in time.

R code:

library(ggplot2)
library(grid)
library(quantmod)

symbols   <- new.env()
date_from <- as.Date("1960-01-01")
date_to   <- as.Date("2016-02-01")
getSymbols("^GSPC", env=symbols, src="yahoo", from=date_from, to=date_to)  # S&P500

df <- data.frame(close=as.numeric(symbols$GSPC$GSPC.Close),
                 date=index(symbols$GSPC))
df$log_return     <- c(NA, diff(log(df$close)))
df$log_return_lag <- c(NA, head(df$log_return, nrow(df) - 1))

cor(df$log_return,   df$log_return_lag,   use="pairwise.complete.obs")  # 0.02
cor(df$log_return^2, df$log_return_lag^2, use="pairwise.complete.obs")  # 0.14

acf(df$log_return,     na.action=na.pass)  # Basically zero autocorrelation
acf((df$log_return^2), na.action=na.pass)  # Squared returns positively autocorrelated

p <- (ggplot(df, aes(x=date, y=log_return)) +
      geom_point(alpha=0.5) +
      theme_bw() + theme(panel.border=element_blank()))
p
ggsave("log_returns_s&p.png", p, width=10, height=8)

The timeseries of log returns on the S&P 500:

log return timeseries

If returns were independent through time (and stationary), it would be very unlikely to see those patterns of clustered volatility, and you wouldn't see autocorrelation in squared log returns.

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