This problem and the next use CRSP daily returns. First, get the data and plot the ACF in two ways — Explain what lag means plots Why does differ between
General StudiesGeneralWorked Solution
This problem and the next use CRSP daily returns. First, get the data and plot the ACF in two ways:
library(Ecdat)
data(CRSPday)
crsp=CRSPday[,7]
acf(crsp)
acf(as.numeric(crsp))
(a) Explain what \lag" means in the two ACF plots. Why does lag differ between the plots?
(b) At what values of lag are there significant autocorrelations in the CRSP returns? For which of these values do you think the statistical significance might be due to chance?
SOLUTION
Plots of the output from the acf function for each of the two calls are given in Figure 7.
Figure 7: Left: The acf plot for the crsp data (as a time series object). Right: The acf plot for the crsp data (treated as a numeric array).
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Part (a):
Note that variable crsp is a ts (time series) object and as such has some auxiliary data associated with it. If we try to display the variable crsp in the command window we get the following (partial output)
When we plot the acf of the ts object the lag is in units of time since the time series object has a notion of how much time is represented between each data point. For the output above we see that the frequency is 365 (corresponding to the number of days in a year) thus a lag of one day corresponds to 1/365 = 0.002739726 yearly time units which is the spacing between the acf values in the leftmost plot. If we look at the rightmost acf plot in Figure 7 we see a small downward spike at lag seven. In units of time this is located at 7/365 = 0.01917808 in the leftmost plot.
Part (b):
We see that the three significant autocorrelations (with the smallest lags) are found at lags 1, 7, and 16. I would expect the autocorrelation at lag 1 to be statistically significant but the others are most likely due to change. Autocorrelations with larger lags that seem “significant” (in that they are above the 2σ standard error bars) are also probably due to chance.