by

• Andrea Carriero (Queen Mary University of London, and University of Bologna)
• Todd Clark (Federal Reserve Bank of Cleveland)
• Massimiliano Marcellino (Bocconi University, Bocconi University, CEPR, IGIER, BIDSA, and BAFFI)
• Elmar Mertens (Deutsche Bundesbank)

Accepted for publication in the Review of Economics and Statistics

The usual disclaimers apply; the views and results conveyed in our research are solely those of the authors and do not necessarily reflect the views of the Federal Reserve Bank of Cleveland, the Federal Reserve System, the Eurosystem, or the Deutsche Bundesbank.

• Pre-publication versions of paper and supplementary material can be found here: https://www.elmarmertens.com/research/publications
• The replication code is also posted at https://github.com/elmarmertens/CCMMoutlierVAR-code

This repository provides replication codes and raw input data to replicate (and update) the results shown in our paper and its supplementary appendix. All core scripts are in the main directory. In addition, there are the following subdirectories:

• data scripts for data construction based on input files obtained from FRED-MD, available at https://research.stlouisfed.org/econ/mccracken/fred-databases/
• matlabtoolbox for general utilities (also available at https://github.com/elmarmertens/em-matlabbox)

The code requires a recent version of Matlab (we used Matlab versions 2019a-2021a) including access to Matlab’s Statistics and Machine Learning Toolbox. The codes employ parfor loops that are executed in parallel when a parpool has been created in Matlab, which requires availability of the Matlab Parallel Computing Toolbox (otherwise the loops will be executed sequentially).

• All scripts set the MATLAB path to point to toolboxes in matlabtoolbox.

• In addition, most scripts collect output (in form of LaTeX tables and figures) in a temporary directory, which is by default created as subfolder foo within the main directory. Figures and tables are also inserted in a LaTeX file, which is compiled at the end of each script (provided a LaTeX installation can be found on the path). -- Edit localtemp.m to change the location of the temp directory that contains the LaTeX output -- To control whether to compile this output with LaTeX (or not), please edit finishwrap.m. -- To avoid collecting output files, comment out the call to initwrap in each script (and make sure to define instead a variable called wrap that is set to empty).

• Model names in paper and code:

-- VARconstvcv is the constant-variance VAR, labeled CONST in the paper -- VARSV is the VAR with (standard) SV, labeled SV in the paper -- VARSVO is the SVO specification -- VARSVOt combines t and SVO, and is labeled SVO-t in the paper -- VARSVnanOutlier is the VAR that treats (pre-specified) outliers as missing data, labeled SV-OutMiss in the paper

• Additional model variants considered in the supplementary material:

-- VARSVt is SV with student-t distributed errors, called SV-t -- VARSVdummy is the VAR with SV and separate dummies for each month of COVID -- VARSVobar is the common-outlier variant of the SVO model -- VARSV*ar1* are model variants with AR(1) processes (instead of RW) for SV

The folder data contains code to transform raw data, obtained from FRED-MD, into data files for subsequent use by the analytical routines in the repository's main folder.

The raw data file that serves as input is 2021-04.csv, which reflects the 2021-04 vintage of FRED-MD and is available under that name from https://research.stlouisfed.org/econ/mccracken/fred-databases/.

The script generate_freddata.m selects the appropriate contents of 2021-04.csv and prepares an output file called fredMD16-2021-04.csv that contains the 16-variable data inputs (after all necessary transformations) as described in Table 1 of the paper. The script also prepares LaTeX output to produce Table 1 of the paper.

The supplementary appendix also contains an alternative specification of our VAR models using variables in levels rather than differences. To prepare data for this specification, use generate_freddataLEVELS.m which generates fredMD16levels-2021-04.csv.

Copies of fredMD16-2021-04.csv and fredMD16levels-2021-04.csv have been stored in the main folder of this repository for further use as described below. (In case of data updates, these copies are not created automatically but must be made manually by the user.)

Plots of the transformed input data, as shown in Section I of the supplementary appendix, are produced by calling chartData16.m contained in the main folder of this repository.

• To execute individual models for a given sample, use doVAR.m and select the desired model by setting the variable modeltype in the preamble to the script.

• To launch out-of-sample runs for a given model, consider scripts called goXXX.m, where XXX refers to one of the model labels listed above. After computing the out-of-sample runs each of these goXXX.m scripts stores results for further post-processing in a AAA-BBB-p12.mat file where AAA reflects the name of the input data file (default: fredMD16-2021-4) and BBB refers to the model name and additional estimation options. By default, these mat files are stored in the current folder, but the user can also chose to move these manually to a different location. Based on these mat files, figures and tables for the paper can be created as described further below. Each of these driver files calls MCMC sampling routines named mcmcXXX.m where XXX corresponds to the model labels listed above.

• To use data files other than fredMD16-2021-04.csv as input for estimation, edit the variables datalabel that is set in the preamble of each Matlab script to point to the name of the data file to be used.

• The main directory contains a bash script gobatch.sh that can be used to launch a sequence of multiple Matlab scripts from the shell. The Matlab scripts are executed in sequence and in separate Matlab sessions. Each Matlab session opens a parallel pool. For example, the shell command sh gobatch.sh goVARSV.m goVARSVO.m will launch a command line session of Matlab, start a parallel pool, and then execute goVARSV.m; once goVARSV.m has been executed, the Matlab sessions closes, a new one is reopened for execution of goVARSVO.m. (The shell script supports as many command line arguments as supported by bash and has been written for use on macOS and Linux.) Alternatively, Matlab scripts can, of course, also be called interactively on the Matlab GUI’s command line.

Below we describe various scripts that generate figures and tables from previously stored model estimates (obtained from calls to the various goVARXXX.m routines described above). The preamble to each of these scripts specifies a variable resultsdir that is expected to point to a directory containing the mat files containing those estimation results.

Specifically:

• To collect predictive scores as compiled in Table 2, use oosMVlogscores.m (oosMVlogscoresLevels.m and oosMVlogscoresSVar1 produce corresponding tables for VAR specifications in levels and AR(1) processes for SV, respectively, as shown in the supplementary appendix).

• To report relative RMSE and CRPS, as in Table 3, call oosEvaluationTables.m (and oosEvaluationTablesSVAR1.m for the case of SV-AR(1) models as reported in the supplementary appendix).

• To produce the panels of Figure 1 (and additional output for other variables) use compareSVOt.m.

• Figure 2 is produced by compareSVpaths.m.

• The panels of Figures 3 and 4 are obtained from plotPredictiveDensitiesPaper2021.m. To obtain additional results for other variables (than those shown in the paper), set the variable doFullYlist to true in the preamble of the script.