Need Clarification of Terms: Innovation v.s. Disturbance

Question

I have been reading some econometrics paper and came across terms "innovation" and "disturbance" of regression models. Can someone please explain to me what they are? I have searched on the internet and got some confused answers.

For example, here (https://stats.stackexchange.com/questions/130900/error-terms-vs-innovations) says that innovations are the same as errors in regression model.

Then the answer here (https://stats.stackexchange.com/questions/221891/difference-between-residual-and-disturbance-epsilon) says that disturbances are the same as errors in regression model.

Does that mean the two terms (innovation and disturbance) mean the same thing (i.e. the error term (epsilon) in the regression model)? Or if one of the terms actually means residuals (e or epsilon_hat) in the regression model?

Any help is much appreciated!

Answer 1

They are virtually the same thing but you should not use them completely interchangeably.

Disturbance Term

Disturbance term is a synonym for an error term. For example, as explained in Verbeek's A Guide to Modern Econometrics pp 14:

$ε_i$ is unobserved and referred to as an error term or disturbance term

So these two words can be interchanged.

Innovations

Innovations, as the Cross-Validated answer you linked are also error terms, but we use the term innovations almost exclusively only in time series (or sometimes panel data) analysis.

This is because technically innovations are supposed to be the difference between the observed value of a variable at time $t$ and the expectation of that value based on information available prior to time $t$, but in practice this boils down to the difference between $y_t-\hat{y}_t$ in context of most econometric models which is by definition the error term (however, as mentioned in the comments innovations require the term to be iid, which is common assumption for error term but in principle errors might not always be iid).

For example, have a look at Verbeek A Guide to Modern Econometrics. In chapters 1-6, which deal with cross-section the textbook always uses the term error term or disturbance term to describe errors. However, from chapter 7 on (chapters that deal with time series and panel data) textbook starts calling error term either error term or innovations.

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Consequently, to sum up disturbance term is just a synonym for an error term and you can use disturbance term and error term interchangeably. The word innovations should only be applied in the context of time series analysis, as it is the difference between actual value and expected value conditional on given information, but in practice this boils down just to error term/white noise so virtually always in time series analysis when people talk about innovations they refer to the error term(s).