Error terms vs Innovations I noticed that we sometimes call the error terms "innovations". I do not understand if this is in special situations or if these terms can be used one for another. Then, another question is "why do we call error terms "innovations"? thanks
 A: The innovations are used in the time series the same way as errors in cross-sectional analysis (such as OLS). For instance if you data generating process is $$y_t=0.9y_{t-1}+\varepsilon_t$$, then we estimated it as $$y_t=0.85y_{t-1}+e_t$$, we call $\varepsilon_t$ innovations (or errors), and $e_t$ - residuals.
For instance, take a look at this MATLAB help page on ARIMA class, where they always refer to innovations in the place where you'd expect to see errors in cross-sectional analysis such as in this MATLAB help page for LinearModel class. In cross-sectional context the model could look like $$y_i=0.9x_i+\varepsilon_i$$
In this MATLAB help page for arima.infer() method, which estimates innovations, the estimated errors are called residuals as usual.
So, I conclude that innovations are ok to interchange with errors. It's called innovations because in time series context the errors bring new information to the system. In cross-sectional context it doesn't make a sense to call them new, as the observations come not in time-ordered sequence. So, observation number 10 is not newer or older than observation number 9. In time series, 10 comes after 9, so in this regard the error/innovation can be seen as a new information from the point of view of the observer who hold the information set up to time 9.
A: *

*See Lawrence Christiano''s "Brief Review of VAR's" notes. He distinguishes between u(t)'s which are new errors that occur at each time in the time series (t,t+1) but do not necessarily mean anything, with underlying economic shocks which MAY persist on an Impulse response graph or show a pattern at some level "C", e.g. ut=C*et.

*If there is no impact in the Long run, C=0, and ut=0 means random Gaussian white noise with E[ut]=0 mean. But if for example supply shocks are persistent, they will show some impulse response pattern long run where C has some value or level, like may C=.90% of the error ut will impact demand each time.

*At least that's how I read it. But agree sometimes used interchangeably.

