UNCERTAINTY AND DENSITY FORECASTS OF
ARMA MODELS: COMPARISON OF ASYMPTOTIC,
BAYESIAN, AND BOOTSTRAP PROCEDURES
ao Henrique Gonc¸alves Mazzeu and Esther Ruiz*
Universidad Carlos III de Madrid
Universidad Carlos III de Madrid and BRU-Unide,
Instituto Universitario de Lisboa
The objective of this paper is to analyze the effects of uncertainty on density forecasts
of stationary linear univariate ARMA models. We consider three speciﬁc sources of uncertainty:
parameter estimation, error distribution, and lag order. Depending on the estimation sample size
and the forecast horizon, each of these sources may have different effects. We consider asymptotic,
Bayesian, and bootstrap procedures proposed to deal with uncertainty and compare their ﬁnite sample
properties. The results are illustrated constructing fan charts for UK inﬂation.
Bayesian forecast; Bootstrap; Fan charts; Model misspeciﬁcation; Parameter uncertainty
Forecasting economic time series has traditionally focused on point forecasts. However, many aspects of
the decision-making process require making forecasts of an uncertain future and, consequently, forecasts
ought to be probabilistic in nature; see Granger and Machina (2006). Analytic construction of density
forecasts has historically required restrictive and sometimes dubious assumptions, such as no parameter
and/or model uncertainty and Gaussian innovations. However, in practice, any forecast model is an
approximation to the data-generating process (DGP). As stated by Box (1979), all models are wrong,
but some are useful. Even if the model is correctly speciﬁed, its parameters need to be estimated and/or
assumptions about the error distribution might not be good approximations to the true distribution; see
Clements and Hendry (1998) for a detailed taxonomy of uncertainty applied to forecast errors in economic
time series. In the context of economic problems, Draper (1995) shows that ignoring model uncertainty
when forecasting oil prices can lead to forecast intervals that are too narrow. Onatski and Stock (2002)
and Onatski and Williams (2003) show that monetary policy may perform poorly when faced with a
wrong error distribution or with slight variations of the model. Brock et al. (2007) also explore ways to
integrate model uncertainty into monetary policy evaluation. Finally, some spectacular failures in risk
management have also emphasized the consequences of neglecting model uncertainty in the context of
ﬁnancial models; see, for example, Schrimpf (2010) and Boucher et al. (2014).
Corresponding author contact email: email@example.com; Tel: 34 91 624 9851.
Journal of Economic Surveys (2018) Vol. 32, No. 2, pp. 388–419
2017 John Wiley & Sons Ltd.