Question

Your time series data are as follows.
0.513 -0.396 -0.85 -1.73 0.127 0.403 0.959 1.498 1.349 -0.186 1.082 -0.92 -1.504 0.135 -0.271 0.726 0.702 0.769 0.854 1.498
For these data, plot the times series, plot the acf, and get the acf function numerically. Consider this as the training set.
Please answer the following.

The serial correlations of lags 1,2,3 are

lag 1:

lag 2:

lag:3

Answer 1

Auto correlation is the simple correlation of a variable (or data) with itself over successive periods of time. For the given time series data, the time series plot and the ACF function (using excel) for the given dataset is shown below:

The serial correlation (or auto correlation) for lags 1, 2 and 3, using excel, are calculated simply by using the CORREL command.

For example, in this dataset there are 20 observations. For lag 1, the serial correlation is calculated by selecting the 1st 19 observations in the 1st array, and next 19 observations (leaving the 1st one) in the 2nd array. It is displayed in the image below:

Therefore, the serial correlation coefficients are:

1) For lag 1: 0.4327

2) For lag 2: 0.1941

3) For lag 3: -0.0930

This can also be obtaine numerically using the formula:

$r_{k}=\frac{s_k}{s_0}$

where r_k is the ACF at lag k, s₀ is the variance of the time series data, and s_k is the auto-covariance function at lag k which is calculated as:

$s_k=\frac{1}{n}\sum_{i=k+1}^{n}(y_i-\bar{y})(y_{i-k}-\bar{y})$;

n being the number of observations.