Note: The red line is the normalized BTC price. At first glance, it looks like a correlation is present but I need concrete proof.

While there are several ways to conduct correlation analysis, I am going to use the Pearson correlation coefficient as it is a simple and well-known method. The coefficient tells whether two-time series are correlated or not. The value ranges from -1 to 1. A value close to 1 means that two data sets have a positive relationship, whereas a value close to -1, implies a negative relationship.

Since the price data has a trend, first-order differences are used to calculate the coefficient.

The coefficient is shown below:

According to the coefficient, BTC and ETH have a strong positive relationship, 0.76. As a rule of thumb, we say that two variables have a strong positive relationship when their coefficient is 0.6 or greater. The strong positive relationship indicates that when the price of BTC goes up, so will the price of ETH with a great likelihood. The coefficient between BTC and XRP is 0.55-smaller than 0.76 yet still relatively high value. We can conclude that all three coins have a strong positive relationship one to another.

The previous correlation was done for whole 2019 data. But if we can determine the correlations on shorter time scales and track that correlation it will offer more insight into the correlation.

Let’s compute the correlation again with a rolling window. A rolling window means that sub-datasets of the full data set are used.

*In time series analysis, too often recent lags have higher predictive power than old ones. Therefore, a model that is applied to the whole series (or on a ‘training’ subset) might hide several characteristics of the whole series.*

*The rolling window, therefore, consists of the selection of given window size (by analyzing the Root Mean Square Error). For example, if the best window size is 24, we will be applying the model on the sample that is comprised of the first 24 observations; then on the second window, comprised of observations 2–25; then observations 3–26; and so forth.*

*The rolling window helps you to assess whether the parameters estimated by linear methods (intercept and slopes) are time-invariant (an important assumption of the model).*

Source — https://www.mathworks.com/help/econ/rolling-window-estimation-of-state-space-models.html