I have a simple question. I want to know the difference between the residuals obtained from a model and idiosyncratic errors.
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The biggest difference is that residuals is a term used in statistics and "idiosyncratic errors" isn't.
In statistics there is a distinction between error and residuals, where in the model
$$ y = f(x) + \varepsilon $$
$\varepsilon$ is the error term, while
$$ r = y - \hat y $$
where $\hat y$ is the fitted value, is the residual.
I did a quick online search and appears in publications be econometricians, like those slides
The basic panel data model is given by:
$$ y_{it} = \beta_0 + \beta_1 x_{1,it} + \beta_2 x_{2,it} + \dots + \beta_K x_{K,it} + a_i + u_{it}, \qquad (18) $$
with idiosyncratic errors $u_{it}$ [...]
where clearly they mean the error term. Same here where the author also has econometrics background, or discussed on /r/econometrics.
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$\begingroup$ Great explanation! If the errors in the model are allowed to follow and AR(1) process, does it mean that residuals can also follow an AR(1) process? $\endgroup$ Jan 12, 2022 at 8:03
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3$\begingroup$ If $\hat y$ is a prediction, I would say $r=y-\hat y$ is a prediction error, not a residual. It would be a residual if $\hat y$ were a fitted value. $\endgroup$ Jan 12, 2022 at 8:05
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