Is multivariate ordinal regression suitable for my reserach? I want to test what leads to the support for two policies (Likert scale ß disagree to strongly agree). There are independent variables, some of which are demographic and two are of interest (continuous and binary variables).
Can I use multivariate ordinal regression? Which statistical model would also be suitable?
Some more context:

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*dependent variables - support for two policies 1) subsidies for electric vehicles, 2) CO2 tax

*independent variables - 1) ownership of an electric vehicle, 2) environmental ideology (voting for the green party), 3-6) some demographic variables like gender, age, income, and education level.

*research question: Who supports the environmental policies encouraging the purchase of electric vehicles - the greens or the owners of electric vehicles?

 A: This seems like an appropriate application of multivariate (meaning multiple-outcome) ordinal multiple regression (meaning multiple predictors in a regression model). With a multivariate analysis you can allow for different associations of the predictors with each of the two outcomes while handling correlations of responses to the two questions within individuals.
Be careful, however, as many use the word "multivariate" in a different way, to mean multiple predictors rather than multiple outcomes. Some references you find to "multivariate ordinal regression" thus might not be what you need, as the comment from @JohnMadden implies.
It's best to code your Likert items to point in the same direction (both increasingly "environmentally friendly" here) so that responses tend to be non-negatively correlated within individuals. See Agresti's discussion of a similar context of binary responses to a set of related questions in Section 12.3.2 of Categorical Data Analysis, second edition.
Given the high likely correlation between the predictors "greens" and "the owners of electric vehicles" I'm not sure that you will be able to resolve your research question cleanly, but you won't know until you try.
