Stephen G. Hall and James Mitchell 217density forecasts are optimal with dynamic misspecification under both the null
and alternative hypotheses therefore requires a test for uniformity that is robust to
dependence. Use of the traditional Kolomogorov-type tests, including the Bai test,
will lead to invalid inference as the critical values are invalid.
Accordingly, Hong (2002) and Corradi and Swanson (2005a) have developed
uniformity tests robust to dependence. The Hong test is based on the generalized
cross-spectrum; see also Hong, Li and Zhao (2004). Corradi and Swanson suggest
a Kolmogorov-type test. At the expense of an assumption of strict stationarity
for
{
yt}
and having to use the block bootstrap, which they prove can be used
to construct valid critical values, the advantage of the Corradi and Swanson test
relative to Hong’s is that it converges at a parametric rather than nonparametric
rate. In addition, it directly accounts for parameter estimation uncertainty; Hong
assumes parameter estimation error vanishes asymptotically.
Multi-step-ahead density forecasts A distinct form of dependence to that caused by
dynamic misspecification can be induced in thepit’s when forecasting more than
one step ahead (h>1). Even when the density forecasts are correctly conditionally
calibrated we should expect dependence of order (h−1) because consecutive obser-
vations are subject to common shocks. This complicates further the task of evalu-
ating multi-step-ahead (h>1) density forecasts since one risks confoundinggood
dependence, explained byh>1, withbaddependence, due to dynamic misspeci-
fication. Distributional tests applied to thepit’s, which are designed to be robust to
dependence, will ideally distinguish betweengoodandbaddependence. Otherwise,
whenh>1, one risks declaring incorrect density forecasts “correct,” on the basis
that thepit’s are uniform, with the dependence in thepit’s dismissed on the grounds
that it is not a symptom of dynamic misspecification but attributable toh>1.
Distributional tests designed to accommodate dependence of order (h−1), i.e.,
gooddependence, have been considered. Most simply, as suggested by Diebold,
Gunther and Tay (1998), thepit’s have been partitioned into (h−1) blocks for which
we expect uniformity and independence when the density forecasts are condition-
ally well-calibrated. For further discussion see Clements and Smith (2000). Dowd
(2007) compares, using simulation experiments, alternative methods of dealing
with the dependence and finds that it is best to carry out tests on a bootstrapped
resample of thepit’s designed to be independent. This remains an active area for
research, since applied studies continue to employ different evaluation methods
in similar contexts.
Joint tests Another option to overcome the deleterious effects of dependence is to
consider a joint test for uniformity and independence ofzt|t−h(see Hong, 2002).
Berkowitz (2001) also presents a parametric, LR, test for the null of standard normal-
ity against autoregressive alternatives. While obviously not robust under the null
hypothesis to dynamic misspecification, these joint tests do, at least in principle,
have power against violations of both uniformity/normality and independence.
Forh=1 Berkowitz (2001) proposes a three degrees-of-freedom LR test of the
joint null hypothesis of a zero mean, unit variance and independentz∗t|t− 1 against