Ramsey RESET test: Information from Answers.com

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 estimated values help explain the endogenous variable. The intuition behind the test is that, if non-linear combinations of the explanatory variables have any power in explaining the endogenous variable, then the model is mis-specified.

Technical summary

Consider the model

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

The Ramsey test then tests whether $(β 1 x) 2,(β 2 x) 3 ...,(β k - 1 x) k$ 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 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", J. Roy. Statist. Soc. B., 31(2), 350–371.

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.

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

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

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