Jonas Dovern and Hans Manner, "Order Invariant Tests for Proper
Calibration of Multivariate Density Forecasts", Journal of Applied
Econometrics, Vol. 35, No. 4, 2020, pp. 440-456.

The material provided along with this readme file enables readers to
replicate the two empirical applications in Section 4 of the paper and
to implement all of the used tests. Code for replicating the
simulations can be obtained from the authors upon request. We note
that the computations are time consuming and are best done using
parallel computing on fairly powerful desktop computers. 

Note that both the estimation of the TVP-BVAR used in the macro
application and the non-parametric estimation of marginal and
conditional predictive densities were done in R (using the packages
'bvarsv' and 'np' respectively). Especially the latter task is very
time-consuming and we, therefore, do not include the corresponding
files in this package. Intermediate results are stored in the files
'PITs_macro.mat' (PITs for all marginal and conditional predictive
densities, both for non-parameteric case and for Gaussian case),
'VCV_h_t.txt' (Covariance matrices of predictive densities under
assumption of normality), and 'SimPITs_h_t.txt' (distribution of PITs
of marginal and conditional predictive densities in the non-parametric
case); these are called by 'application_macro.m' to reproduce the
results in Section 4.2 of the paper.

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Data files: 

- Macro_data.txt - The raw data for the macro application (data are
from FRED (St. Louis Fed): UNRATE, GDPDEF, TB3MS. 264
observations/rows; first column indicates the quarter. Inflation is
calculated as the q/q change of the GDP deflator in percent.)

- PITs_macro.mat - The macro data after the PIT for applying our tests

- MSCI_weekly.txt - The raw data for the finance application (data are
from Datastream: MSUSAML, MSJPANL, MSUTDKL, MSAUSTL, MSGERML. 2505
observations/rows; first column indicates the week.) )

- MSCI_data.mat - The finance data in Matlab format 

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Files for replication of the applications:

- application_macro.m - Replicates the macro application of the paper

- application_finance.m - Replicates the finance application for the
  Gaussian DCC GARCH

- application_finance_dyn_copula.m - Replicates the finance
  application for the time-varying t-copula model and t-GARCH margins

- application_finance_dyn_copula_skewt.m - Replicates the finance
  application for the time-varying t-copula model with skew-t GJR-Garch
  margins

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Functions needed to implement the various tests: 

- CS_PIT.m - Probability integral transform based on the product of
  iid uniforms as in Clements and Smith (2000) for any dimension d

- KP_PIT.m - Probability integral transform based on the demeaned
  product of iid uniforms as in Ko and Park (2013) for any dimension d

- Neyman_ST.m - Standard version of Neyman's smooth test

- NST_EU.m - Neyman's smooth test accounting for estimation uncertainty

- NST_DM.m - Neyman's smooth test accounting for dynamic misspecification

- NST_EU_DM.m - Neyman's smooth test accounting for estimation
  uncertainty and dynamic misspecification

- PIT_Z2S.m - Z2 transformation into uniform under normality

- PIT_Z2dag.m - Z2_dagger transformation into uniform under normality

- PIT_Z2star.m - Z2_star transformation into uniform under normality

- PIT_Z2_student.m - Z2 transformation for data from a multivariate
  t-distribution

- PIT_Z2dag_Student.m - Z2_dagger transformation into uniform for
  multivariate t-distribution

- PIT_Z2star_Student.m - Z2_star transformation into uniform for
  multivariate t-distribution

- PIT_chi2_mix.m - PIT for data coming from a mixture of chi-square
  variables

- Z2dag_Student.m and Z2star_Student.m - Transformations Z2star and
  Z2dagger based on a multivariate t-distribution with time varying
  correlation matrix and time-varying degrees-of-freedom

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Matlab toolboxes required:

- Dynamic Copula Toolbox
- ucsd_garch toolbox
- MFE toolbox 
- LeSage toolbox