Independence

The plot below shows data from response variable \(y\) and explanatory variable \(x\), a measure of time. The right hand side plot shows a fitted regression line (from a simple linear regression).

Assessing model fit, from the simple linear regression, we can see that the residuals do appear normally distributed with mean zero and constant variance

However, if we check independence of residuals, we can see that from the right hand side plot above that the residuals are not independent. There is a clear positive correlation between a residual (shown on y-axis) and the residual at the previous time point (residuals lag 1 shown on the x-axis). In this case, the independence assumption is violated.

Essentially, what we provided evidence to suggest that the residuals are not independent. The right hand side plot shows a relationship between the residuals \(\epsilon_1, \ldots, \epsilon_{n-1}\) plotted against residuals \(\epsilon_2,\ldots,\epsilon_n\).