Why manual autocorrelation does not match acf() results?

Question

I'm trying to understand acf and pacf. But do not understand why acf() results do not match simple cor() with lag1

I have simulated a time series

set.seed(100)
ar_sim <- arima.sim(list(order = c(1,0,0), ar = 0.4), n = 100)

ar_sim_t <- ar_sim[1:99]
ar_sim_t1 <- ar_sim[2:100]

cor(ar_sim_t, ar_sim_t1)   ## 0.1438489
acf(ar_sim)[[1]][2]        ## 0.1432205

Could you please explain why the first lag correlation in acf() does not exactly match the manual cor() between the series and lag1?

Answer 1

下面是估计具有已知均值和方差的离散过程的自相关的正确方法。参见例如Wikipedia。

n <- length(ar_sim)
l <- 1
mu <- mean(ar_sim)
s <- sd(ar_sim)

sum((ar_sim_t - mu)*(ar_sim_t1 - mu))/((n - l)*s^2)
#[1] 0.1432205

此值不是identical由内置计算得出的值，stats::acf但非常接近它。

a.stats <- acf(ar_sim)[[1]][2]
a.manual <- sum((ar_sim_t - mu)*(ar_sim_t1 - mu))/((n - l)*sd(ar_sim)^2)

all.equal(a.stats, a.manual)  # TRUE
identical(a.stats, a.manual)  # FALSE

a.stats - a.manual
#[1] 1.110223e-16