Ralph Bradley and Robert McClelland (1996)
"A Kernel Test for Neglected Nonlinearity."
This paper develops a new kernel test for neglected
nonlinearity in the conditional expectation function, and
compares this test to the Ramsey RESET test (1969) and the
Neural Net test of Lee, White, and Granger (1993). Like the
Neural Test and the Ramsey Reset Test, this Kernel test is a
Lagrange Multiplier test based on the R-Square statistic from
a regression of the estimated residuals on the regressors,
and an additional nonlinear function of the regressors.
Unlike the other tests, our test has a two stage approach
where in the first stage we estimate the structure of the
misspecification and in the second stage we test for whether
or not the estimate of the misspecification can better
predict the residuals than their mean. This two stage
approach can give the researcher guidance on the nature of
the misspecification, and should improve the power of the
test since the added function in the regression of the
residuals is itself an estimate of the conditional
expectation of the residuals given the independent variables.
In addition, because it uses simple, well known estimation
methods it can be easily implemented by researchers when
using linear models.
Our test uses a cross-validated kernel regression estimate of
the conditional mean of the residuals given the explanatory
variables, and from the results of the Rao-Blackwell Theorem
this should asymptotically be the nonlinear function of these
variables whose correlation with the estimated residuals has
the highest absolute value. As a result our test should have
greater power at detecting nonlinearity than either the RESET
or the Neural Net tests, even though we use an established
estimation technique in a standard Lagrange Multiplier
Because the kernel regression is an estimate of the
conditional mean of the residuals, it contains information
about the form of the nonlinearity. For example a research
may plot the confidence interval and estimated conditional
mean of the residuals, noting where and how the estimated
mean diverges significantly from zero. One may also calculate
the derivatives of the estimate at the mean for an indication
of the direction of the misspecification.
Last Modified Date: July 19, 2008