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.

This exercise is part of the course

Time Series Analysis in R

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Exercise instructions

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.

Hands-on interactive exercise

Have a go at this exercise by completing this sample code.

# Define x_t0 as x[-1]
x_t0 <- 

# Define x_t1 as x[-n]
x_t1 <- 

# Confirm that x_t0 and x_t1 are (x[t], x[t-1]) pairs  
head(cbind(x_t0, x_t1))
  
# Plot x_t0 and x_t1
plot(___, ___)

# View the correlation between x_t0 and x_t1
cor(___, ___)

# Use acf with x
acf(___, lag.max = ___, plot = ___)

# Confirm that difference factor is (n-1)/n
cor(x_t1, x_t0) * (n-1)/n

Edit and Run Code