This paper considers the problem of estimation and testing
for ARCH models under the assumption of conditional correlation.
For a bivariate model with unknown volatility parameter vector,
we construct an estimator for this parameter vector using
the conditional least squares estimator given by Tj$\o$stheim (1986).
Such an estimation procedure is applied to more general ARCH models.
Next we turn to discuss the testing problem for the multivariate model
in the setting of Taniguchi and Kakizawa (2000), based on a quasi-Gaussian
maximum likelihood estimator. The results show that the tests such as
Gaussian likelihood ratio (GLR), Wald (W), and Langrange multiplier (LM)
provide asymptotic equivalent procedures for testing a general linear
hypothesis. For a composite hypothesis, the limiting distribution of
such tests is derived in a parametric form. The W test is used for
constructing approximate confidence intervals. As an example,
the local power property is illustrated.