EXPECTED VALUE is the objectuve of all models. In time series, the model answers the question of "what is the long-run equilibrium value?", i.e. what is the estimated value or expected value of the series. If there is a shock upon the system, will that value change? If it changes, will it go back to its original value? If it does, how long does it take, i.e. lag 1, 2, etc.? If it does not go back to the equilibrium point, was the effect of the shock integrated into the system? If so, revise and rerun the model.
MISUSE may occur if time series is use just because of the coincidental incidence of "time" or that the data was measured in periodic fashion. Other model should also be explaored. Time series model generally consists of a family of regression, i.e. autoregressive. The data should be tested for distribution and other model alternatives should be explored. The data should preceed the model, do not put model before data. The investigation is the investigation of the data, not testing the model. The model is a short-hand explanation of the data. Time series modeling often time failed or has low explanatory power of many data.
It may be a good idea to start with a textbook on time series analysis, such as 'time series analysis by state space methods' by Durbin and Koopman.
A classical time series model is the Sun Spot model. Data, Y(t), on the number of dark spots on the Sun are collected for hundred of years now. The problem is to have a model that explain the data. It is difficult to find explanatory variables that affect Y(t). So a model that regresses Y(t) on its lagged values was developed. It was found that a lagged value of 11 years fits the data well.
An old distinction is between cross-section and time series data. Data collected for a number of cities for one period is an example of the former. See a textbook definition on them. A simple time series model is to regress GDP on Time. Also, a whole set of stochastic or random walk models for business and economics have been developed. Many techniques have been developed to smooth, balance, detrend, deseasonlaize, cointigrate etc time series data.
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