1. Estimating serial correlations
I hope you could spot the volatility clustering in the Dow Jones log-return series in the previous exercise. This phenomenon is evidence that there are serial dependencies in the log-returns.
If serial dependencies are present, they can be exploited for prediction. For example, when there is volatility clustering, and you know you are in the middle of a particularly volatile period, then you know that the probability of another extreme market move is increased.
2. Sample autocorrelations
So how can serial dependencies be quantified? A classic tool from time series analysis is the sample autocorrelation function or sample acf.
Sample autocorrelations measure the correlations between variables in the series that are separated by different amounts of time. The separation in time is known as the lag.
When you estimate sample autocorrelations, you are making the implicit assumption that the series has a property known as stationarity.
Consider a daily log-return series. Stationarity would mean that the expected daily return is always the same; the variance of daily returns is always the same, and the correlation between returns on days the same distance apart is always the same.
The sample autocorrelation for lag k will be denoted in this course by rho hat k.
This is the estimated correlation between returns on days which are k apart. For example, rho hat one is the estimated correlation between returns on successive days.
And, if you have equity return data since equities are traded on weekdays, rho hat five is the estimated correlation between returns on days, which are one week apart.
3. The sample acf plot or correlogram
The function acf calculates the sample autocorrelations and plots them in a well-known picture, which is called a sample acf plot or also a correlogram. Let's go ahead and make the picture and then discuss it.
For example, for the FTSE log-returns in the years 2008-9, you can obtain the sample acf plot by entering acf(ftse).
The x-axis shows the lag, which is k in the formula, and the y-axis shows the estimated correlation value. Instead of plotting points, it is usual to plot vertical bars. Note that the bar of length 1 when the lag k is zero simply reflects the trivial observation that return on the same day are 100% correlated.
The blue dashed lines are 95% confidence bounds. If the data were, in fact, from an iid process, it would be expected that 95% of the correlation estimates (excluding the lag zero value) lie between these bounds. About 1 in every 20 would be expected to protrude outside.
The picture on the slide is, in fact, a rather dull acf plot. The estimated autocorrelations are very small in magnitude and mostly within the confidence bands, although a few more than 1 in 20 protrude. At first sight, this picture is a disappointment as it might have been expected that the volatility clustering would lead to stronger evidence of serial dependence.
4. The sample acf plot or correlogram
In fact it does, but you need to view the data through a different lense to see the serial dependence. Suppose instead you made the acf plot of the absolute values of the FTSE returns as shown on the slide.
Suddenly there are many estimated correlations outside the confidence bounds. This is very typical for a financial return series. Because of the frequent sign changes the strong serial dependence is hidden when you look at the acf plot. But it is revealed when you remove the sign changes with the absolute function.
One way of interpreting these pictures is that there is information in the data that would allow the prediction of the magnitude of returns - that is the volatility - but not much information about the direction of returns - that is whether the market is going to go up or down.
5. Let's practice!
Now it's your turn to estimate sample autocorrelations and experiment with the acf plot.