We consider a degenerate system of stochastic differential equations. The first component of the system has a parameter
θ
1
\theta _1
in a non-degenerate diffusion coefficient and a parameter
θ
2
\theta _2
in the drift term. The second component has a drift term with a parameter
θ
3
\theta _3
and no diffusion term. Parametric estimation of the degenerate diffusion system is discussed under a sampling scheme. We investigate the asymptotic behavior of the joint quasi-maximum likelihood estimator for
(
θ
1
,
θ
2
,
θ
3
)
(\theta _1,\theta _2,\theta _3)
. The estimation scheme is non-adaptive. The estimator incorporates information of the increments of both components, and under this construction, we show that the asymptotic variance of the estimator for
θ
1
\theta _1
is smaller than the one for standard estimator based on the first component only, and that the convergence of the estimator for
θ
3
\theta _3
is much faster than for the other parameters. By simulation studies, we compare the performance of the joint quasi-maximum likelihood estimator with the adaptive and one-step estimators investigated in Gloter and Yoshida [Electron. J. Statist 15 (2021), no. 1, 1424–1472].