Exercise

# Calculating autocorrelations

Autocorrelations or lagged correlations are used to assess whether a time series is dependent on its past. For a time series `x`

of length `n`

we consider the `n-1`

pairs of observations one time unit apart. The first such pair is `(x[2],x[1])`

, and the next is `(x[3],x[2])`

. Each such pair is of the form `(x[t],x[t-1])`

where t is the observation index, which we vary from 2 to n in this case. The lag-1 autocorrelation of `x`

can be estimated as the sample correlation of these `(x[t], x[t-1])`

pairs.

In general, we can manually create these pairs of observations. First, create two vectors, `x_t0`

and `x_t1`

, each with length `n-1`

, such that the rows correspond to `(x[t], x[t-1])`

pairs. Then apply the `cor()`

function to estimate the lag-1 autocorrelation.

Luckily, the acf() command provides a shortcut. Applying `acf(..., lag.max = 1, plot = FALSE)`

to a series `x`

automatically calculates the lag-1 autocorrelation.

Finally, note that the two estimates differ slightly as they use slightly different scalings in their calculation of sample covariance, `1/(n-1)`

versus `1/n`

. Although the latter would provide a biased estimate, it is preferred in time series analysis, and the resulting autocorrelation estimates only differ by a factor of `(n-1)/n`

.

In this exercise, you'll practice both the manual and automatic calculation of a lag-1 autocorrelation. The time series `x`

and its length `n`

(150) have already been loaded. The series is shown in the plot on the right.

Instructions

**100 XP**

- Create two vectors,
`x_t0`

and`x_t1`

, each with length`n-1`

such that the rows correspond to the`(x[t], x[t-1])`

pairs. - Confirm that
`x_t0`

and`x_t1`

are`(x[t], x[t-1])`

pairs using the pre-written code. - Use
`plot()`

to view the scatterplot of`x_t0`

and`x_t1`

. - Use
`cor()`

to view the correlation between`x_t0`

and`x_t1`

. - Use
`acf()`

with`x`

to automatically calculate the lag-1 autocorrelation. Set the`lag.max`

argument to`1`

to produce a single lag period and set the`plot`

argument to`FALSE`

. - Confirm that the difference factor is
`(n-1)/n`

using the pre-written code.