Positive serial correlation Blows are the pictures from my course lecture. The lecture states only the second picture shows a positive serial correlation, and the first picture requires time to be added as a predictor while second picture requires to be moved to time series modeling. I'm a little bit confused. Why it is not Positive serial correlation in the first picture or why we can't move it to time series modeling as well?  How to tell there is "positive serial correlation" among the error terms?


 A: In the first picture you have residuals that are trend stationary, that means that you have a regression like $y_{i,t} = \alpha + \beta x_{i,t} + \epsilon_{i,t}$ where $\epsilon_{i,t} = \delta t + \eta_{i,t}$ and $\eta_{i,t}$ is iid. So when you include a trend in your regression the residuals becomes iid: $y_{i,t} = \alpha + \delta t + \beta x_{i,t} + \eta_{i,t}$. You solved the problem without including serial correlation in your model.
In the second picture you have residuals that are not trend stationary, but they not seem to be iid too. If you just include a trend you will not make them iid. To solve this issue you need to include serial correlation of the residuals and estimate a model like this: $y_{i,t} = \alpha + \beta x_{i,t} + \epsilon_{i,t}$ where $\epsilon_{i,t} = \omega + \phi \epsilon_{i,t-1} + \eta_{i,t}$, with $\eta_{i,t}$ iid. Well, that is an AR(1) model for the residuals.
To tell if there is positive serial correlation you may check the autocorrelation function of the residuals or even perform statistical tests such as the Ljung-Box test, but before doing that you may want to remove a possible trend in the residuals. You see, the difference between those two pictures is that in the first one, residuals are not autocorrelated after removing the linear trend, while in the second one they are.
