<p>They are virtually the same thing but you should not use them completely interchangeably.</p>
<p><strong>Disturbance Term</strong></p>
<p>Disturbance term is a synonym for an error term. For example, as explained in Verbeek's  A Guide to Modern Econometrics pp 14:</p>
<blockquote>
<p><span class="math-container">$ε_i$</span> is unobserved and referred to as an
error term or disturbance term</p>
</blockquote>
<p>So these two words can be interchanged.</p>
<p><strong>Innovations</strong></p>
<p>Innovations, as the <a href="https://stats.stackexchange.com/questions/130900/error-terms-vs-innovations">Cross-Validated</a> answer you linked are also error terms, but we use the term innovations almost exclusively only in <strong>time series</strong> (or sometimes panel data) analysis.</p>
<p>This is because technically innovations are supposed to be the difference between the observed value of a variable at time <span class="math-container">$t$</span> and the expectation of that value based on information available prior to time <span class="math-container">$t$</span>, but in practice this boils down to the difference between <span class="math-container">$y_t-\hat{y}_t$</span> 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).</p>
<p>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.</p>
<hr />
<p>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).</p>

CC BY-SA 4.0
2021-08-12T15:11:47.130Z
Thank you SO much! That is clear to me now!
Judy Zhang
CC BY-SA 4.0
2021-08-12T18:45:30.333Z
I would just add that 'innovations' is somewhat more of a loaded term since it typically implies that the errors / residuals are independent random variables, whereas the other terms don't necessarily impose this assumption. See also Duarte and Hoover (2012) for an interesting discussion of how model residuals are often interpreted as "shocks" and some of the historical ideas behind this: http://public.econ.duke.edu/~kdh9/Source%20Materials/Research/Observing%20Shocks.pdf
Andrew M
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2021-08-12T18:52:47.987Z
@AndrewM thanks for the addition, I edited my answer
1muflon1
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2021-08-12T00:52:27.997Z
2021-08-12T18:52:33.513Z
2021-08-12T18:52:33.513Z
1muflon1
<p>I have been reading some econometrics paper and came across terms &quot;innovation&quot; and &quot;disturbance&quot; of regression models. Can someone please explain to me what they are? I have searched on the internet and got some confused answers.</p>
<p>For example, here (<a href="https://stats.stackexchange.com/questions/130900/error-terms-vs-innovations">https://stats.stackexchange.com/questions/130900/error-terms-vs-innovations</a>) says that innovations are the same as errors in regression model.</p>
<p>Then the answer here (<a href="https://stats.stackexchange.com/questions/221891/difference-between-residual-and-disturbance-epsilon">https://stats.stackexchange.com/questions/221891/difference-between-residual-and-disturbance-epsilon</a>) says that disturbances are the same as errors in regression model.</p>
<p>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?</p>
<p>Any help is much appreciated!</p>

<econometrics><regression><statistics><autoregressive>
Need Clarification of Terms: Innovation v.s. Disturbance
Judy Zhang
# Need Clarification of Terms: Innovation v.s. Disturbance

(Asked by: username_0 on 2021-08-12)

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!

**Tags:** econometrics, regression, statistics, autoregressive

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## 1 answers

### Answer by username_1 on 2021-08-12. Score: 3 | Accepted answer

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](https://stats.stackexchange.com/questions/130900/error-terms-vs-innovations) 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.

---

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).

> **Comments:**
> **username_0**: Thank you SO much! That is clear to me now!  
> **username_2**: I would just add that 'innovations' is somewhat more of a loaded term since it typically implies that the errors / residuals are independent random variables, whereas the other terms don't necessarily impose this assumption. See also Duarte and Hoover (2012) for an interesting discussion of how model residuals are often interpreted as "shocks" and some of the historical ideas behind this: http://public.econ.duke.edu/~kdh9/Source%20Materials/Research/Observing%20Shocks.pdf  
> **username_1**: @AndrewM thanks for the addition, I edited my answer  

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