Unit 6 Practical Autocorrelation

6.1 Estimation!

6.1.1 \(\rho_k \rightarrow 0\)

For a stationary time series, if the autocorrelation approaches zero, then :

A Single realization lets us estimate mean, variance, and autovvariance!

Remember that \(\rho_k = \frac{\gamma_k}{\gamma_0}\), and we use n-k pairs to calculate it (the summation)

6.1.1.1 Mean

Just calculate the mean normally for this case.

6.1.1.2 Variance

\[\mathrm{Var}\left(\bar{X}\right) = \frac{\sigma^2}{n} \sum^{n-1}_{k = -(n-1)} \left( 1 - \frac{\mid{k}\mid}{n} \right)\rho_k\]

\(\sigma^2\) is calculated as normal, we will see rhok next!

remember !

Now it is time for some code!

library(glue)

## 
## Attaching package: 'glue'

## The following object is masked from 'package:dplyr':
## 
##     collapse

xbar <- function(xs) {
    mean(xs)
}

ghat_zero <- function(xs) {
    summand <- (xs - xbar(xs))^2
    mean(summand)
}

ghat_one <- function(xs) {
    lhs <- xs[1:(length(xs) - 1)] - xbar(xs)
    rhs <- xs[2:length(xs)] - xbar(xs)
    summand <- lhs * rhs
    summate <- sum(summand)
    summate/length(xs)
    
}
rhohat_zero <- 1
rhohat_one <- function(xs) {
    ghat_one(xs)/ghat_zero(xs)
}
v <- c(76, 70, 66, 60, 70, 72, 76, 80)
xbar(v)

## [1] 71.25

ghat_zero(v)

## [1] 34.9375

ghat_one(v)

## [1] 14.74219

rhohat_one(v)

## [1] 0.4219589