Time Series Models with an EGB2 Conditional Distribution

Abstract

A time series model in which the signal is buried in noise that is non-Gaussian may throw up observations that, when judged by the Gaussian yardstick, are outliers. We describe an observation driven model, based on an exponential generalized beta distribution of the second kind (EGB2), in which the signal is a linear function of past values of the score of the conditional distribution. This speci…cation produces a model that is not only easy to implement, but which also facilitates the development of a comprehensive and relatively straight-1 forward theory for the asymptotic distribution of the maximum likelihood estimator. Score driven models of this kind can also be based on conditional t-distributions, but whereas these models carry out what, in the robustness literature, is called a soft form of trimming, the EGB2 distribution leads to a soft form of Winsorizing. An EGARCH model based on the EGB2 distribution is also developed. This model complements the score driven EGARCH model with a conditional t-distribution. Finally dynamic location and scale models are combined and applied to data on the UK rate of in ‡ation.

Time‐series models with an EGB2 conditional distribution

“…The data are seasonally adjusted and taken from the Federal Reserve Economic Data (FRED) database of the Federal Reserve of St. Louis. Table I shows the results; corresponding results for GDP and industrial production can be found in our original discussion article, Caivano and Harvey (2013). The asymptotic and numerical standard errors, the latter computed from the estimated Hessian, are reasonably close given that the sample size is only T D 268.…”

“…/. Unfortunately, the presence of h D h. ; & / complicates the information matrix, as shown in Caivano and Harvey (2013…”

Two EGARCH Models and One Fat Tail

“…The above criteria are related to the behavior of the conditional score and whether or not it discounts large observations. This, in turn, connects to robustness, as shown in Caivano and Harvey (2013). More speci…cally, consider a power law PDF, (3), with y divided by a scale parameter; ', so that F (y=') = cL(y=')(y=') and f (y) cL(y)' 1 (y=') 1 : Then…”

“…The associated dynamic scale model is described in Section 4. (The dynamic EGB2 location model is discussed in Caivano and Harvey, 2013).…”

“…1 it tends to a normal distribution. When = & = 0 in the EGB2, the distribution is double exponential or Laplace; see Caivano and Harvey (2013). The following results will be used in a number of places when = & : (i) h 2 = 2 as !…”