Ramsey RESET test

In statistics, the Ramsey Regression Equation Specification Error Test (RESET) test (Ramsey, 1969) is a general specification test for the linear regression model. More specifically, it tests whether non-linear combinations of the fitted values help explain the response variable. The intuition behind the test is that if non-linear combinations of the explanatory variables have any power in explaining the response variable, the model is mis-specified.

Technical summary

Consider the model

$\hat{y}=E\{y|x\}=\beta x.$

The Ramsey test then tests whether (β₁x)²,(β₂x)³...,(β_{k − 1}x)^k has any power in explaining y. This is executed by estimating the following linear regression

$y=\beta x + \beta_1\hat{y}^2+...+\beta_{k-1}\hat{y}^k+\epsilon$,

and then testing, by a means of a F-test whether $\beta_1~$ through $~\beta_{k-1}$ are zero. If the null-hypothesis that all regression coefficients of the non-linear terms are zero is rejected, then the model suffers from mis-specification.

For a univariate x the test can also be performed by regressing on the truncated power series of the explanatory variable and using an F-Test for

$~H_0:\beta_i=0 \quad \forall i=1,\ldots,k-1.$

Test rejection implies the same insight as the first version mentioned above.

$y=\beta x + \beta_1x^2+...+\beta_{k-1}x^k+u. \,$

The F-test compares both regressions, the original one and the Ramsey's auxiliary one, as done with the evaluation of linear restrictions. The original model is the restricted one opposed to the Ramsey's unrestricted model.

$\frac{\frac{R_1^2-R_0^2}{k-1}}{\frac{1-R_1^2}{n-k}}$ ~ F_{(k − 1,n − k)}, where:

$R_0^2$ is the determination coefficient of the original linear model regression;

$R_1^2$ is the determination coefficient of the Ramsey's auxiliary regression;

n is the sample size;

k is number of parameters in the Ramsey's model.

Furthermore, the linear model

$\hat{y}=\beta_0 + \beta_1 x_1 + ... + \beta_m x_m$

and the model with the non-linear power terms

$y=\beta_0 + \beta_1 x_1 + ... + \beta_m x_m + \beta_{m+1}\hat{y}^2+...+\beta_{m+k-1}\hat{y}^k+\epsilon$,

are sujected to the F-test, similarly as before:

$\frac{\frac{R_1^2-R_0^2}{k-1}}{\frac{1-R_1^2}{n-(m+k)}}$ ~ F_{(k − 1,n − m − k)} ,

where m + k is number of parameters in the Ramsey's model, which are k − 1 variables in the Ramsey group (non-linear $\hat{y}$) plus m + 1 the number of parameters in the original model.

The critical (rejection) region is on the right side of the F distribution, thus

$p-value=P(F_{(k-1, n-m-k)} > \frac{\frac{R_1^2-R_0^2}{k-1}}{\frac{1-R_1^2}{n-(m+k)}} )$.

References

• 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. JSTOR 2984219

• Thursby, J.G., Schmidt, P. (1977) "Some Properties of Tests for Specification Error in a Linear Regression Model", Journal of the American Statistical Association, 72, 635–641. JSTOR 2286231

• Murteira, Bento. (2008) Introdução à Estatística, McGraw Hill.

• Wooldridge, Jeffrey M. (2006) Introductory Econometrics - A Modern Approach, Thomson South-Western, International Student Edition.