1.6 Auto correlation
A sample auto correlation co-efficiency are an important guide to the properties of a time
series.
A sample auto correlation measures the correlation between observations of different time
apart.
Note. Since \(n\) pairs of observations on \(x\) and \(y\) the correlation co efficient is given by
\[r = \frac {\sum (x-\overline {x})(y-\overline {y})}{\sqrt {\sum (x-\overline {x})^2\sum (y-\overline {y})^2}}\]
Now, given n observations \(x_1,x_2,.........,x_n\). We can form \(n-1\) pairs of observations, \((x_1-x_2), (x_2-x_3),............., (x_{t-1}-x_t)\) and the correlation coefficient between \(x_t\) and \(x_{t-1}\)
\[r_1 = \frac {\sum \limits ^{n-1}_{t=1}(x_t-\overline {x_1})(x_{t+1}-\overline {x_2})}{\sqrt {\sum \limits ^{n-1}_{t=1}(x_t-\overline {x_1})\sum \limits ^{n-1}_{t=1}(x_{t+1}-\overline {x_2})}}\]
on the assumption that \(\overline {x_1}=\overline {x_2}\)
\[r_1 = \frac {\sum \limits ^{n-1}_{t=1}(x_t-\overline {x})(x_{t+1}-\overline {x})}{(n-1)\frac {\sum \limits ^n_{t=1}(x_t-\overline {x})^2}{n}}\]
for large n, we have
\[r_1 = \frac {\sum \limits ^{n-1}_{t=1}(x_t-\overline {x})(x_{t+1}-\overline {x})}{\sum \limits ^n_{t=1}(x_t-\overline {x})^2}\]
This is the formula used to compute the auto correlation coefficient.
In the similar manner we can find the correlation between observation at distance k as
\[r_k = \frac {\sum \limits ^{n-k}_{t=1}(x_t-\overline {x})(x_{t+k}-\overline {x})}{\sum \limits ^n_{t=1}(x_t-\overline {x})^2}\]
this the auto correlation coefficient at lag k.
The auto correlation coefficients are usually calculated using the auto ca variance and defined
as
\[c_k = \frac {1}{n}\sum ^{n-k}_{t=1}(x_t-\overline {x})(x_{t+1}-\overline {x})\]
This is the auto co variance at lag k.
We can the compute the auto-correlation co efficiency as
\(r_k = \frac {c_k}{c_0} \)
for k = 1,2, ......,n where k ¡ n.
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