  This paper develops tests for the correct specification of the conditional
variance function in GARCH models when the true parameter may lie on the
boundary of the parameter space. The test statistics considered are of
Kolmogorov-Smirnov and Cram\'{e}r-von Mises type, and are based on a certain
empirical process marked by centered squared residuals. The limiting
distributions of the test statistics are not free from (unknown) nuisance
parameters, and hence critical values cannot be tabulated. A novel bootstrap
procedure is proposed to implement the tests; it is shown to be asymptotically
valid under general conditions, irrespective of the presence of nuisance
parameters on the boundary. The proposed bootstrap approach is based on
shrinking of the parameter estimates used to generate the bootstrap sample
toward the boundary of the parameter space at a proper rate. It is simple to
implement and fast in applications, as the associated test statistics have
simple closed form expressions. A simulation study demonstrates that the new
tests: (i) have excellent finite sample behavior in terms of empirical
rejection probabilities under the null as well as under the alternative; (ii)
provide a useful complement to existing procedures based on Ljung-Box type
approaches. Two data examples are considered to illustrate the tests.
