One practical interpretation of time series models is that they represent the impact of new information on the variables we are modeling. For example, a p-order autoregressive model contains p lags to the dependent variable that represents the impact of past information, and the model error term represents the impact of new information.

In a covariance stationary model, the impact of past information diminishes over time. In contrast, the impact of past information does not diminish over time in a random walk model, and an information shock remains in the time series forever. Some asset price series may be characterized by a random walk model -- does it make sense to you that an information shock 50 years ago is still reflected in the asset price? Is this a realistic feature of the statistical model?