We develop a general methodology to account for entry into and exit from a system modeled as a Markov process estimated using macro aggregate-frequency data. By adding a closure state that does not exist in reality, we straightforwardly account for entities that may come into or go out of existence. When collecting macro data to estimate such a process, we cannot see the closure state explicitly, so it is necessary to posit some value for the number (or proportion) of entities in it. By means of extensive designed simulation experiments, we determine the effect of the size of the closure state on estimate, hypothesis-test, and prediction performance. For best point-estimate performance, results indicate that the size of the closure state should be the smallest that it can be theoretically. An application is presented in a model of advertising media-share shifts over the last twenty years. Here, the real states represent specific advertising media among which a firm allocates its advertising budget, while the closure state would hold that portion of the firm's budget either allocated to other media or not allocated to advertising at all. Additional advertising expenditures over time would be modeled as a transition out of the closure state and into a real state for a particular medium, while a fall in advertising expenditures would represent the opposite type of transition. Implications of our results extend beyond marketing to many fields in which systems are modeled as Markov processes.
展开▼
