Does the following qualify as a univariate regression?
I fully comprehend the implications of adding regressors and need no background information - a "yes" or "no" will do :-)
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Yes. "Multivariate" and "univariate", when they're used to describe models, refer to the number of dependent variables, not the number of independent variables. A multivariate linear regression model, for example, predicts several different variables, and the residuals are multivariate normal rather than univariate normal. See, for example, the Wikipedia article "Linear regression":
For more than one explanatory variable, the process is called multiple linear regression. (This term should be distinguished from multivariate linear regression, where multiple correlated dependent variables are predicted, rather than a single scalar variable.)
Hence, your model is a multiple linear regression model, but also a univariate linear regression model.