Data with multivariate, longitudual categorical responses often occur in applications. It can be difficult to analyze and model such data while simultaneously taking into account explanatory variables and correlations between the responses over time. We take a generalized linear model approach to this problem in analyzing panel data from the Health and Retirement Survey (HRS) that includes older Americans’ mobility over several years as a response. We provide a general formula for the likelihood of such data and apply it to the case when there are three binary responses. This approach can be taken, with computational limits, for data with multivariate, categorical responses with any number of categories. We consider, simultaneously, interpretations of coefficients, dependence of responses and goodness-of-fit in reduced models for parsimony while taking into account explanatory data. The gradient of the objective function is provided for use in gradient descent and the coded optimization algorithm is tested with a Monte Carlo simulation. Dependence of responses in mobility is shown before taking explanatory variables into account, and dependence is shown in a Markov logistic regression model and in the generalized linear model taking into account race, age, gender and interactions between them.