  Score tests have the advantage of requiring estimation alone of the model
restricted by the null hypothesis, which often is much simpler than models
defined under the alternative hypothesis. This is typically so when the
alternative hypothesis involves inequality constraints. However, existing score
tests address only jointly testing all parameters of interest; a leading
example is testing all ARCH parameters or variances of random coefficients
being zero or not. In such testing problems rejection of the null hypothesis
does not provide evidence on rejection of specific elements of parameter of
interest. This paper proposes a class of one-sided score tests for testing a
model parameter that is subject to inequality constraints. Proposed tests are
constructed based on the minimum of a set of $p$-values. The minimand includes
the $p$-values for testing individual elements of parameter of interest using
individual scores. It may be extended to include a $p$-value of existing score
tests. We show that our tests perform better than/or perform as good as
existing score tests in terms of joint testing, and has furthermore the added
benefit of allowing for simultaneously testing individual elements of parameter
of interest. The added benefit is appealing in the sense that it can identify a
model without estimating it. We illustrate our tests in linear regression
models, ARCH and random coefficient models. A detailed simulation study is
provided to examine the finite sample performance of the proposed tests and we
find that our tests perform well as expected.
