216 Recent Developments in Density Forecasting
zt|t−h. For empirical examples and references, see Clements and Smith (2000),
Clements (2004) and Hall and Mitchell (2004).
By taking the inverse normal cumulative density function (c.d.f.) transforma-
tion ofzt|t−hto give, say,zt∗|t−h, the test for uniformity can be considered to
be equivalent to one for normality ofz∗t|t−h; see Berkowitz (2001). For Gaussian
forecast densities with mean given by the point forecast,zt∗|t−his simply the
standardized forecast error (outturn minus point forecast divided by the stan-
dard error of the Gaussian density forecast). Testing normality is convenient
as normality tests are widely seen to be more powerful than uniformity tests.
However, testing is complicated by the fact that the impact of dependence on
the tests for uniformity/normality is unknown, as is the impact of non-unifor-
mity/non-normality on tests for dependence.
Consequently, various single and joint tests of uniformity/normality and
independence have been employed in empirical studies.^5 These include
Kolmogorov–Smirnov, Anderson–Darling and Doornik and Hansen (1994) tests
for uniformity/normality, Ljung–Box tests and Lagrange multiplier (LM) tests for
independence, and Hong (2002), Thompson (2002) and Berkowitz (2001) LR tests
for both uniformity/normality and independence. Using Monte Carlo techniques,
Noceti, Smith and Hodges (2003) found the Anderson–Darling test to have more
power to detect misspecification than the Kolmogorov–Smirnov test (and related
distributional tests). However, they maintained an assumption of a random sample
and did not consider the effect dependence may have on the performance of the
tests. Many of the popular distributional tests, such as the Kolmogorov–Smirnov
and Anderson–Darling tests, are not robust to dependence, their properties having
been developed under independence.
Parameter uncertainty and dependence Testing uniformity is complicated by both
parameter uncertainty and possible dependence in thezt|t−h. For a review and
derivation of out-of-sample versions of the tests we consider below, see Corradi
and Swanson (2006c).
Parameter uncertainty is a concern when the density forecast is model-based and
depends on estimated parameters. This is because when parameters are estimated
the Kolmogorov test is no longer asymptotically distribution free, meaning that
critical values cannot be tabulated as they are dependent on the null hypothesis and
the parameter values. Bai (2003) therefore developed a modified Kolmogorov-type
test, based on a martingale transformation, which is asymptotically distribu-
tion free. While this test has power against violations of uniformity, it does not
necessarily have power against violations of independence in thezt|t−h.
This means that these Kolmogorov tests require the density forecast not just to
capture the distribution ofytcorrectly but to be correctly specified dynamically.
Following Corradi and Swanson (2006c), let us illustrate what this means via a
simple example. Let the true (conditional) density bef(yt|!t− 1 )=N(α 1 yt− 1 +
α 2 yt− 2 ,σ 2 ), but the density forecast, while normal, be misspecified in terms of
its dynamics:g(yt|t− 1 )=N(α 1 ∗yt− 1 ,σ 1 ), whereα∗ 1 =α 1. In this casezt|t− 1 is
no longer independent but remains uniform. To test the null hypothesis that the