# Linearity in Parameters: Ramsey RESET Test

Last Update: February 21, 2022

Linearity in Parameters within linear regression requires that model equation has correct functional form specification. This can be evaluated through Ramsey RESET test [1] which evaluates whether linear regression fitted values non-linear combinations explain dependent variable. If linear regression fitted values non-linear combinations explain dependent variable, then model equation has incorrect functional form specification.

As example, we can fit a three-variable multiple linear regression with formula $\hat{y}_{(1)i}=\hat{\beta}_{0}+\hat{\beta}_{1}x_{1i}+\hat{\beta}_{2}x_{2i}\;(1)$ . Then, as example again, we can fit Ramsey RESET test augmented multiple linear regression by adding multiple linear regression (1) squared fitted values $\hat{y}_{(1)i}^{2}$ as independent variable with formula $\hat{y}_{(2)i}=\hat{\beta}_{0}+\hat{\beta}_{1}x_{1i}+\hat{\beta}_{2}x_{2i}+\hat{\delta}_{1}\hat{y}_{(1)i}^{2}\;(2)$. Next, we can do an F-test with null hypothesis that $\hat{\delta}_{1}$ coefficient is equal to zero with formula $H_{0}:\;\hat{\delta}_{1}=0\;(3)$. If null hypothesis is rejected, then multiple linear regression (1) has incorrect functional form specification. Notice that Ramsey RESET test augmented multiple linear regression (2) can include more fitted values powers, independent variables powers or independent variables first principal component powers. Also, notice that null hypothesis test (3) can be done with a chi-square test.

Below, we find an example of Ramsey RESET test augmented multiple linear regression coefficients and F-test from original multiple linear regression of house price explained by its lot size and number of bedrooms [2].

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References

[1] Ramsey, J. B. (1969). “Tests for Specification Errors in Classical Linear Least Squares Regression Analysis”. Journal of the Royal Statistical Society, Series B. 31 (2): 350–371.

[2] 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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