Autocorrelation and Autocovariance: Calculation, Examples, and More – Part II

See Part I for an overview of autocovariance.

Calculation of the autocovariance with an example

You might have been thinking up to now:
Why are the autocovariance and autocorrelation defined with an “s” subscript?
Great question!

Let us explain: Actually, the autocovariance formula defined above is a function which allows the calculation of the autocovariance for different lags. The same for the autocorrelation function.

Confused? Don’t worry! We got you covered!

Let’s see an example to make the concept clear to your thoughts! We are going to make an example of how to calculate the autocovariance of the Microsoft price returns at lag 1. We are going to use the autocovariance function shown above.

Imagine we have the following returns for Microsoft prices:

Day 1	Day 2	Day 3	Day 4	Day 5	Day 6	Day 7	Day 8	Day 9	Day 10
5%	1%	-2%	3%	-4%	6%	2%	-1%	-3%	4%

Let’s suppose we want to compute the autocovariance at lag 1. You will need the returns up to day 10, and the 1-period lagged returns up to day 9.

Thus, you have the following data structure for returns on days 10 and 9:

Variable	Day 1	Day 2	Day 3	Day 4	Day 5	Day 6	Day 7	Day 8	Day 9	Day 10
rt	5%	1%	-2%	3%	-4%	6%	2%	-1%	-3%	4%
rt−1		5%	1%	-2%	3%	-4%	6%	2%	-1%	-3%

Do you get to see the difference between the 2 variables?
The second one is the first lag of Xt.

Now, since the 2 variables have different dimensions (the first one has 10 observations, while the second one has 9), we are going to use data from day 2 onwards.

Consequently, our data is as follows:

Variable	Day 2	Day 3	Day 4	Day 5	Day 6	Day 7	Day 8	Day 9	Day 10
rt	1%	-2%	3%	-4%	6%	2%	-1%	-3%	4%
rt−1	5%	1%	-2%	3%	-4%	6%	2%	-1%	-3%

The covariance between these 2 variables will be the autocovariance of the returns at lag 1.

You can do this, right?
Check an example we give in our previous article.

Before you get ready to use a pencil and a piece of paper, let us tell you something important.

Remember the autocovariance formula:

If you paid attention to details, you could see that the average return is the same for both returns, in our case, for returns up to day 10 and up to day 9. As we explained before, autocovariance and autocorrelation functions are applied only to stationary time series.

Consequently, not only the variance but also the mean is a unique value for the whole span. That’s why the mean is the same for any lag of the price returns.

The mean of the Microsoft price returns is 1.1%. Let’s follow the procedure to compute the autocovariance:

Variable	rt	rt−1	(rt−r)	(rt−1−r)	(rt−r)(rt−1−r)
Day 2	1%	5%	-0.100%	3.900%	-0.004%
Day 3	-2%	1%	-3.100%	-0.100%	0.003%
Day 4	3%	-2%	1.900%	-3.100%	-0.059%
Day 5	-4%	3%	-5.100%	1.900%	-0.097%
Day 6	6%	-4%	4.900%	-5.100%	-0.250%
Day 7	2%	6%	0.900%	4.900%	0.044%
Day 8	-1%	2%	-2.100%	0.900%	-0.019%
Day 9	-3%	-1%	-4.100%	-2.100%	0.086%
Day 10	4%	-3%	2.900%	-4.100%	-0.119%

The autocovariance is just the sum of the last column values divided by (N-1, equal to 8), which results in -0.046%.

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Calculation of the autocorrelation with an example

Let’s follow the same exercise and compute the autocorrelation of the Microsoft price returns up to day 10 at lag 1. The autocorrelation is the autocovariance divided by the variance. We give you the exact hint you need: The variance of Microsoft price returns up to day 10 is 0.121%.

Let’s follow the algebraic formulas and use the numbers to compute the autocorrelation:

Stay tuned for the next installment in this series to learn about computation of autocovariance and autocorrelation in Python

Visit QuantInsti for additional insight on this topic: https://blog.quantinsti.com/autocorrelation-autocovariance/.