# Linear Regression: Residual Standard Error

Last Update: February 21, 2022

Linear Regression: Residual Standard Error is used to evaluate linear regression goodness of fit by estimating its residual standard deviation.

As example, we can fit a three-variable multiple linear regression with formula $\hat{y}_{i}=\hat{\beta}_{0}+\hat{\beta}_{1}x_{1i}+\hat{\beta}_{2}x_{2i}\;(1)$. Then, we can estimate its residual standard error with formula $se_{res}=\sqrt{ms_{res}}\;(2)$. Residual mean squared error $ms_{res}$ with formula $ms_{res}=\frac{ss_{res}}{df_{res}}\;(3)$ is estimated as residual sum of squares $ss_{res}$ divided by residual degrees of freedom $df_{res}$. Residual sum of squares $ss_{res}$ with formula $ss_{res}=\sum_{i=1}^{n}\hat{e}_{i}^{2}\;(4)$ is estimated as the sum of squared regression residuals $\hat{e}_{i}$. Regression residuals $\hat{e}_{i}$ with formula $\hat{e}_{i}=y_{i}-\hat{y}_{i}\;(5)$ are estimated as differences between actual $y_{i}$ and fitted $\hat{y}_{i}$ values. Residual degrees of freedom $df_{res}$ with formula $df_{res}=n-p-1\;(6)$ are the number of observations $n$ minus number of independent variables $p$ minus constant term.

Below, we find an example of estimated residual standard error from multiple linear regression of house price explained by its lot size and number of bedrooms [1].

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References

[1] Data Description: Sales prices of houses sold in the city of Windsor, Canada, during July, August and September, 1987.

Original Source: Anglin, P., and Gencay, R. (1996). Semiparametric Estimation of a Hedonic Price Function. Journal of Applied Econometrics, 11, 633–648.

Source: AER R Package HousePrices Object. Christian Kleiber and Achim Zeileis. (2008). Applied Econometrics with R. Springer-Verlag, New York.

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