We introduce a score-driven volatility model for the t distribution, the Beta-t-QVAR (quasi-vector autoregressive) model, in which the scale and degrees of freedom parameters interact through a multivariate score-driven filter. Both components of the filter influence the conditional volatility of returns. This paper aims to improve the statistical and density forecasting performances of Beta-t-EGARCH (exponential generalized AR conditional heteroscedasticity). We present the Beta-t-QVAR model and the conditions of its maximum likelihood (ML) estimation. We apply Beta-t-QVAR to 15 international stock indices using daily data from December 1997 to April 2024. We compare the in-sample statistical and out-of-sample density forecasting performances of Beta-t-QVAR, normal-GARCH (NGARCH), asymmetric power-ARCH (APARCH), and Beta-$t$-EGARCH. We find that Beta-t-QVAR is superior to NGARCH, APARCH, and Beta-t-EGARCH, motivating its practical use for financial forecasting.