# Difference between residuals and idiosyncratic errors?

I have a simple question. I want to know the difference between the residuals obtained from a model and idiosyncratic errors.

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.

• 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? Jan 12, 2022 at 8:03
• 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. Jan 12, 2022 at 8:05
• @RichardHardy yeah, corrected.
– Tim
Jan 12, 2022 at 8:06