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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.
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