Bartlett’s paradox has been taken to imply that using improper priors results in Bayes factors that are not well defined, preventing model comparison in this case. We use well understood principles underlying what is already common practice, to demonstrate that this implication is not true for some improper priors, such as the Shrinkage prior due to Stein (1956). While this result would appear to expand the class of priors that may be used for computing posterior odds, we warn against the straightforward use of these priors. Highlighting the role of the prior measure in the behaviour of Bayes factors, we demonstrate pathologies in the prior measures for these improper priors. Using this discussion, we then propose a method of employing such priors by setting rules on the rate of diffusion of prior certainty.
In the paper, we document how conditional dependencies observed in the FOREX market change during a trading day. The analysis is performed for the pairs (GBP/EUR, USD/EUR) and (GBP/USD, EUR/USD) of exchange rates. We consider daily returns calculated using the exchange rates quoted at different hours of a day. The dynamics of the dependencies is modeled by means of 3-regime Markov regime switching copula models, and the strength of the linkages is described using dynamic Spearman’s rho and the dynamic coefficients of tail dependence. The established approach allows us to monitor the changes in the dependence structure.
If the most parsimonious behavioral model between an observed behavior, Y, and some factors, X, can be defined as f(Y|X1, X2), then fx1 will measure the impact in behavior of a change in factor X1. Additionally, if fx1x2 ≠ 0, then the impact in behavior of a change in factor X1 is qualified, or moderated by X2. If this is the case, X2 is said to be a moderating variable and fx1x2 is said to be the moderating effect. When Y is modeled via a logistic regression, the moderation effect will exist regardless of whether the index function of the logit specification includes a moderation term or not. Thus, including a moderation terms in the index function will help the researcher more precisely qualify the moderation effect between X1 and X2. The question that naturally arises is whether the researcher must include the moderation term or not. In this document, we provide the conditions in which moderation terms will naturally arise in a logistic regression and introduce some modeling guidelines. We do so by introducing a general framework that nests models with no moderation terms in three scenarios for the independent variables, commonly found in applied research.
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