Mitigating the choice of the duration in DDMS models through a parametric link

TL;DR

This paper tackles the persistent challenge of selecting the duration parameter in duration-dependent Markov-switching (DDMS) models by introducing a flexible parametric Aranda-Ordaz link to govern transition probabilities. The model replaces the conventional fixed logit link with $g(y;\lambda)=\log\left(\frac{(1-y)^{-\lambda}-1}{\lambda}\right)$, yielding $P(S_t=i|S_{t-1}=i,D(S_{t-1})=d)=g^{-1}(\gamma_1(i)+\gamma_2(i)(d\wedge\tau);\lambda)$, with $\lambda$ estimated from data; as $\lambda\to0^+$, the link approaches the cloglog form. Through two Monte Carlo experiments—in-sample bull/bear identification and out-of-sample volatility forecasting—and an empirical S&P 500 volatility forecast using realized measures and robust loss functions, the Aranda-Ordaz DDMS demonstrates superior forecasting performance and higher likelihoods under duration misspecification, while remaining competitive when duration is correctly specified. The results suggest that incorporating a flexible link reduces sensitivity to duration choices and broadens the applicability of DDMS models for volatility forecasting and risk management, with potential extensions to VaR and trading strategies. The work provides a practical, data-driven method to mitigate a long-standing modeling limitation in DDMS frameworks.

Abstract

One of the most important hyper-parameters in duration-dependent Markov-switching (DDMS) models is the duration of the hidden states. Because there is currently no procedure for estimating this duration or testing whether a given duration is appropriate for a given data set, an ad hoc duration choice must be heuristically justified. In this paper, we propose and examine a methodology that mitigates the choice of duration in DDMS models when forecasting is the goal. Two Monte Carlo simulations, based on classical applications of DDMS models, are employed to evaluate the methodology. In addition, an empirical investigation is carried out to forecast the volatility of the S\&P 500, which showcases the capabilities of the proposed model.

Figures (5)

• Figure 1: Transition probabilities shape for different values of $\lambda$ and $d$.
• Figure 2: Bear market filtered probabilities, for $\tau=4$.
• Figure 3: Time series plot of the S&P500 (a) index and (b) log-returns.
• Figure 4: Estimated values of $\lambda$ for the DDMS A-O model with $\tau=5$.
• Figure 5: Fluctuation test between the DDMS A-O and Garch (top row) and DDMS A-O and MS-Garch (bottom) for different loss functions.