Ramsey RESET test

The RESET test according to Ramsey or the test for incorrect specification of the regression equation according to Ramsey ( English Ramsey Regression Equation Specification Error Test ) is a test proposed by James B. Ramsey in 1969 in statistics for checking the model specification in the context of linear regression . It checks whether non-linear combinations of the explanatory variables have an influence on the explained variable . If the non-linear combinations of the explanatory variables have an influence, then the linear model specification should be reconsidered. However, incorrect specifications such as disregarding relevant variables , structural breaks , homoscedasticity etc. can also be indicated by the test. One advantage of the Ramsey RESET test is that no explicit alternative model has to be specified; the disadvantage that it does not provide any indication of a “correct” specification . ${\displaystyle X_{i}}$ ${\displaystyle Y}$

Mathematical formulation

The following model specification is assumed in the linear model

${\displaystyle Y=\beta _{0}+\beta _{1}X_{1}+\ldots +\beta _{m}X_{m}+\epsilon ,}$

and one appreciates

${\displaystyle {\hat {Y}}=b_{0}+b_{1}X_{1}+\ldots +b_{m}X_{m}.}$

The test checks for a nonlinear model of the shape

${\displaystyle Y={\hat {Y}}+\gamma _{2}{\hat {Y}}^{2}+\ldots +\gamma _{k}{\hat {Y}}^{k}+\zeta }$

does not have a greater explanatory power than the linear model.

The hypotheses are

${\displaystyle H_{0}:\gamma _{2}=\ldots =\gamma _{k}=0}$vs. . ${\displaystyle H_{1}:{\text{es existiert mindestens ein}}\;\gamma _{i}\neq 0}$

The test statistic is

${\displaystyle {\frac {\frac {R_{1}^{2}-R_{0}^{2}}{k}}{\frac {1-R_{1}^{2}}{n-(m+k)-1}}}\sim F_{k-1;n-m-k}}$

With

• ${\displaystyle R_{0}^{2}}$: the coefficient of determination of the linear model,
• ${\displaystyle R_{1}^{2}}$: the coefficient of determination of the nonlinear model,
• ${\displaystyle n}$: the sample size,
• ${\displaystyle m}$: the number of explanatory variables and
• ${\displaystyle k}$: the number of additional parameters in the nonlinear model.

If the coefficients of the linear regression in the nonlinear model are also re-estimated and if they differ significantly from the estimated coefficients in the linear model, this is also an indication of an incorrect specification.

The Ramsey RESET test can also be extended to generalized linear models .

example

In the Boston Housing data , the mean purchase price of houses per district (medv) is estimated as a function of the proportion of the lower class population (lstat) using a simple linear regression. The regression line in the scatter plot clearly shows that the relationship between the two variables is non-linear.

The Ramsey RESET test (with and ) gives the following result: ${\displaystyle {\hat {y}}^{2}}$ ${\displaystyle {\hat {y}}^{3}}$

 RESET test
data:  linreg
RESET = 83.4103, df1 = 2, df2 = 502, p-value < 2.2e-16

As the graphic already suggests, the null hypothesis is rejected because the p -value is smaller than z. B. a significance level of . ${\displaystyle \alpha =5\,\%}$

Individual evidence

1. ↑ JB Ramsey: Tests for Specification Errors in Classical Linear Least-squares Regression Analysis . In: Journal of the Royal Statistical Society, Series B . tape 31 , no. 2 , 1969, p. 350-371 , JSTOR : 2984219 .
2. ↑ Peter Hackl : Introduction to Econometrics . Addison-Wesley Verlag, 2004, ISBN 978-3-8273-7118-8 .
3. ↑ Sunil Sapra: A regression error specification test (RESET) for generalized linear models . In: Economics Bulletin . tape 3 , no. 1 , 2005, p. 1–6 ( economicsbulletin.com [PDF; accessed September 9, 2012]). A regression error specification test (RESET) for generalized linear models ( Memento from December 25, 2015 in the Internet Archive )