Adaptive Models and Heavy Tails

Abstract

This paper proposes a novel and ‡exible framework to estimate autoregressive models with time-varying parameters. Our setup nests various adaptive algorithms that are commonly used in the macroeconometric literature, such as learning-expectations and forgetting-factor algorithms. These are generalized along several directions: speci…cally, we allow for both Student-t distributed innovations as well as time-varying volatility.Meaningful restrictions are imposed to the model parameters, so as to attain local stationarity and bounded mean values. The model is applied to the analysis of in ‡ation dynamics. Allowing for heavy-tails leads to a signi…cant improvement in terms of …t and forecast. Moreover, it proves to be crucial in order to obtain well-calibrated density forecasts.JEL classi…cation: C22, C51, C53, E31.

Feasible Invertibility Conditions and Maximum Likelihood Estimation for Observation-Driven Models

“…The invertibility conditions of Straumann and Mikosch (2006) often fail to be guaranteed in empirical studies. In Section 2 and 6 we illustrate this issue through some empirical examples featuring the Beta-t-GARCH(1, 1) model of Harvey (2013) and Creal et al (2013), the dynamic autoregressive model of Blasques et al (2014b) and Delle Monache and Petrella (2016), and the fat-tailed location model of Harvey and Luati (2014). The main problem is due to the conditions themselves since they depend on the unknown data generating process.…”

Feasible invertibility conditions and maximum likelihood estimation for observation-driven models

“…This problem is also not specific for the class of conditional heteroscedastic models. We illustrate this point considering the autoregressive model of Blasques et al (2014b) and Delle Monache and Petrella (2016) and the location model of Harvey and Luati (2014). We find that, in general, the typical invertibility conditions needed to ensure the consistency of the ML estimator, which are considered for instance in Straumann and Mikosch (2006), Straumann (2005) and Blasques et al (2014a), lead often to a parameter region that is too small for practical purposes.…”

“…The usefulness of our theoretical results is further illustrated considering two examples in the context of dynamic location model. In particular, we discuss the implications of our theoretical results considering the dynamic autoregressive model of Blasques et al (2014b) and Delle Monache and Petrella (2016) and the fat-tailed location model of Harvey and Luati (2014).…”

“…On the contrary, it applies to the general class of observation driven models. Consider the first-order autoregressive model with a time-varying autoregressive coefficient and with a fat-tailed distribution as discussed in Blasques et al (2014b) and Delle Monache and Petrella (2016). This model is specified by the equations…”

“…In Section 2 and 6 we illustrate this issue through some empirical examples featuring the Beta-t-GARCH(1, 1) model of Harvey (2013) and Creal et al (2013), the dynamic autoregressive model of Blasques et al (2014b) and Delle Monache and Petrella (2016), and the fat-tailed location model of Harvey and Luati (2014). The main problem is due to the conditions themselves since they depend on the unknown data generating process.…”