vector of maturities (J x 1). Maturities are in multiples of the discrete interval used in the model

risk-neutral feedback matrix (N x N)

string-vector containing the label of the model to be estimated

state loadings for the one-period rate (1xN). Default is a vector of ones

the long run risk neutral mean of the short rate (scalar)

the covariance matrix of the errors (N x N)

string-vector containing the names of the economies which are part of the economic system

String-vector containing the label of the model to be estimated

List to store the bootstrap set

List of model estimated parameters

String-vector containing the names of the economies which are part of the economic system

Number of the bootstrap draw

available options are "IRF" and "GIRF"

available options are "Factors" and "Yields"

desired path to save the graphs

Economies of the economic system

Desired estimated model type

available options are "IRF" and "GIRF"

available options are "Factors" and "Yields"

desired path to save the graphs

Economies of the economic system

Desired estimated model type

time series of bond yields (CJ x T1 )

time series of risk factors (F x T2)

string-vector containing the names of the economies of the system.

Type of constraint (Full Name)

Constrained parameter value

Type of constraint

string-list based which contains the labels of all the variables present in the model

string-vector containing the names of the economies which are part of the economic system

Constrained parameter value

Type of constraint

lower bound

upper bound

Constrained parameter value

Type of constraint

string-list based which contains the labels of all the variables present in the model

string-vector containing the names of the economies which are part of the economic system

list of necessary inputs for the estimation of JLL-based models (see "JLL" function)

Constrained parameter

Type of constraint

string-list based which contains the labels of all the variables present in the model

string-vector containing the names of the economies which are part of the economic system

list of necessary inputs for the estimation of JLL-based models (see "JLL" function)

Constrained parameter value

Type of constraint

number of global unspanned factors

number of domestic unspanned factors

number of domestic spanned factors

number of economies of the economic system

number of country-specific observed bond yields

B that accomodates only the map to the spanned factors only

number of global unspanned factors

number of domestic unspanned factors per country

list of model parameter estimates (See the "Optimization" function)

complet set of economies of the economic system

specific economy under study

a string-vector containing the label of the model to be estimated

number of global unspanned factors

number of country-specific domestic unspanned factors

list of model parameter estimates (see the "Optimization" function) after a bootstrap draw

string-vector containing the names of the economies which are part of the economic system

string-vector containing the names of the economy under study

string-vector containing the label of the model to be estimated

number of the bootstrap draw

A character vector indicating the model type to be estimated.

A list containing the necessary inputs for the BRW model estimation:

1. flag_mean: Logical. Determines whether mean- (TRUE) or median- (FALSE) unbiased estimation is desired. Default is TRUE.

2. gamma: Numeric. Adjustment parameter between 0 and 1. Default is 0.5.

3. N_iter: Integer. Number of iterations for the stochastic approximation algorithm after burn-in. Default is 5,000.

4. N_burn: Integer. Number of burn-in iterations. Default is 15

5. B: Integer. Number of bootstrap samples per iteration for calculating the noisy measure of the OLS estimator's mean or median. Default is 50.

6. check: Logical. Indicates whether to perform a closeness check. Default is TRUE.

7. B_check: Integer. Number of bootstrap samples for the closeness check. Default is 100,000.

A numeric matrix (T x F) representing the time series of risk factors.

Integer. Number of country-specific spanned factors.

A character vector containing the names of the economies included in the system.

A list of character vectors with labels for all variables in the model.

List. Inputs for GVAR model estimation (see GVAR function). Default is NULL.

List. Inputs for JLL model estimation (see JLL function). Default is NULL.

Numeric. Restriction on the largest eigenvalue under the P-measure. Default is 1.

Integer. Number of elements in the output list. Default is 4.

numerical output set from the bootstrap analysis

numerical output set from the point estimate analysis

index associated with the shock variable

Label of the response variable

Output types "IRF", "GIRF" and "IRF Ortho"

desired model type

dimension from the P-dynamics

horizon of analysis

name of economies forming the economic system

Option for orthogonal outputs, for JLL models. Default is FALSE.

numerical output set from the bootstrap analysis

numerical output set from the point estimate analysis

index associated with the shock variable

Label of the response variable

Output types "IRF" or "FEVD"

desired model type

dimension from bond yield set

dimension from the P-dynamics

horizon of analysis

name of economies forming the economic system

Option for orthogonal outputs, for JLL models. Default is FALSE.

numerical output set from the bootstrap analysis

numerical output set from the point estimate analysis

desired model type

dimension of the risk factor set

horizon of numerical outputs

Economies that are part of the economic system

Path to save the desired graphs

available options are "IRF" and "FEVD"

Binary variable reflecting the graphs of interest

Binary variable reflecting the graphs of interest (orthogonalized version). Default is NULL

numerical output set from the bootstrap analysis

numerical output set from the point estimate analysis

desired model type

dimension of the risk factor set

horizon of numerical outputs

Economies that are part of the economic system

Path to save the desired graphs

Available option are "IRF" or "FEVD"

Binary variable reflecting the graphs of interest

Binary variable reflecting the graphs of interest (orthogonalized version). Default is NULL

data-frame containing the necessary data for building the grahs

string containing the name of the response variable

A character vector indicating the model type to be estimated.

A list containing the point estimates of the model parameters. For details, refer to the outputs from the Optimization function.

The point estimate derived from numerical outputs. See the outputs from the NumOutputs function for further information.

A character vector containing the names of the economies included in the system.

A list containing the necessary inputs for generating IRFs, GIRFs, FEVDs, GFEVDs and Term Premia.

A list of character vectors with labels for all variables in the model.

List. Inputs for JLL model estimation (see JLL function). Default is NULL.

List. Inputs for GVAR model estimation (see GVAR function). Default is NULL.

Whether to estimate the physical parameter model with bias correction, based on the method by Bauer, Rudebusch and Wu (2012) (see Bias_Correc_VAR function). Default is set to 0.

List of necessary inputs for performing the bias-corrected estimation (see Bias_Correc_VAR function).

string-vector containing the label of the model to be estimated

list containing the complete set of model parameters after the bootstrap estimation procedure

point estimate from the numerical outputs (see the outputs of the "NumOutputs" function)

list containing the desired inputs for the construction of IRFs, GIRFs, FEVDs, and GFEVDs

string-vector containing the names of the economies which are part of the economic system

Sample starting date

Sample last date

matrix (J x T), where J - the number of maturities and T - time series length

time series of the global macroeconomic risk factors (G x T)

time series of the country-specific macroeconomic risk factors for all C countries (CM x T)

string-vector containing the names of the economies which are part of the economic system

string-list based which contains the labels of all variables present in the model

string-vector containing the label of the model to be estimated

adjustment of global series for sepQ model in the Bootstrap setting. Default is set to FALSE.

numerical output set from the bootstrap analysis

numerical output set from the point estimate analysis

Label of interest. available options are "IRF" and "FEVD"

desired model type

name of the economies forming the economic system

index associated with the response variable

index associated with the shock variable

Option for orthogonal outputs, for JLL models. Default is FALSE.

list of computed outputs containing the model fit, IRFs, FEVDs, GIRFs, GFEVDs and Term Premia

Economies of the economic system

Desired model type

time-horizon of the FEVD and GFEVD

dimension of the risk factor vector

dimension of the model set of yields

available option are 'FEVD' and 'GFEVD'

Set of VARX model parameters

Set of marginal model parameters

List of inputs for GVAR-based models

string-based vector containing label of the domestic risk factors

string-based vector containing label of the global risk factors

number of country-specific spanned factors (scalar)

list of computed outputs containing the model fit, IRFs, FEVDs, GIRFs, GFEVDs and Term Premia

Economies of the economic system

Desired model type

time-horizon of the IRF and GIRF

dimension of the risk factor vector

available option are 'IRF' and 'GIRF'

List of inputs for GVAR-based models

number of country-specific spanned factors (scalar)

number of country-specific unspanned factors (scalar)

List of point estimates of the model parameter

Time-series of the residuals from the P-dynamics equation (T x F)

Time-series of the residuals from the observational equation (T x J or T x CJ)

List containing the desired inputs for the construction of the numerical outputs.

String-vector containing the names of the economies which are part of the economic system

Desired model to be estimated

String-list based which contains the labels of all the variables present in the model

List of necessary inputs for the estimation of GVAR-based models

List of necessary inputs for the estimation of JLL-based models

Whether the user wishes to estimate the physical parameter model with the Bias correction model from BRW (2012) (see "Bias_Correc_VAR" function). Default is set to 0.

List of necessary inputs for performing the bias-corrected estimation (see "Bias_Correc_VAR" function)

Number of lags in the P-dynamics. Default is set to 1.

List of point estimates of the model parameter

String-vector containing the label of the model to be estimated

Time-series of the risk factors (F x T)

B matrix of loadings

Set of bootstrap inputs

String-vector containing the names of the economies which are part of the economic system

variable inputs used in the optimization (see "optimization" function)

string-vector containing the names of the economies which are part of the economic system

list of necessary inputs for the estimation of JLL-based models (see JLL function)

Necessary inputs for the estimation of the JLL-based models

Sample starting date

Sample last date

string-vector containing the names of the economies of the system.

time series of the country-specific macroeconomic risk factors for all C countries (CM x T)

time series of the global macroeconomic risk factors (G x T)

(i) "Month": if maturity of yields are expressed in months or (ii) "Year": if maturity of yields are expressed in years

single element character-based vector. Available options are: "Daily All Days",
"Daily Business Days", "Weekly", "Monthly", "Quarterly", "Annually"

string-vector containing the names of the single country setups

string-vector containing the names of the multicountry setups

string-list based which contains the labels of all variables present in the model

list of necessary inputs for the estimation of GVAR-based models (see "GVAR" function)

string-vector containing the label of the model to be estimated

List of inputs for GVAR-based models

number of country-specific spanned factors (scalar)

Risk factors before the orthogonalization (FxT)

List of necessary inputs to estimate JLL-based setups

feedback matrix after the bias correction procedure

feedback matrix obtained by OLS estimation (without bias correction)

number of bootstrap samples used in the closeness check

time series of the risk factors (F x T)

number of country-specific spanned factors (scalar)

inputs used in the estimation of the GVAR-based models (see "GVAR" function). Default is set to NULL

inputs used in the estimation of the JLL-based models (see "JLL" function). Default is set to NULL

string-list based which contains the labels of all variables present in the model

string-vector containing the names of the economies which are part of the economic system

string-vector containing the label of the model to be estimated

string-vector containing the names of the economies which are part of the economic system

time series of the country-specific domestic risk factors (C(M+N) x T)

time series of the global risk factors (G X T)

string-vector containing the names of the desired domestic risk factors

string-vector containing the names of the desired global risk factors

Index of the last set of observations in the information set at the first forecasting round

Index of the first set of observations in the information set at the first forecasting round

Number of forecasting sets generated

Forecast type. Available options are "Rolling" and "Expanding".

Time-series dimension of the model

unrestricted variance-covariance matrix (K X K)

number of domestic unspanned factors per country (scalar)

number of global unspanned factors (scalar)

number of country-specific spanned factors (scalar)

string-vector containing the names of the economies which are part of the economic system

Name of the economy which is assigned as the dominant unit.
If no dominant unit is assigned, then this variable is defined as "none"

List of outputs to store bootstrap draws

Total number of bootstrap draws

A character vector indicating the model type to be estimated

numerical output set from the bootstrap analysis

quantile of the confidence bounds

dimension of the risk factor set

dimension of the bond yield set

desired model type

Economies that are part of the economic system

number of draws

horizon of numerical outputs

numerical output set from the bootstrap analysis

quantile of the confidence bounds

dimension of the risk factor set

dimension of the bond yield set

Desired model type

Economies that are part of the economic system

number of draws selected

horizon of numerical outputs

Variance-covariance matrix

Loading As

contemporaneous terms

Loading Bs

List containing the label of factors

Dimension of the P-dynamics

Length of the maturity vector

Horizon of the analysis

Label of bond yields

Desired model type

specific economy under study

Cholesky factorization term from JLL models

matrix PI for JLL-based models

allows for the orthogonalized version in the case of JLL-based models

Variance-covariance matrix

Loading As

contemporaneous terms

Loading Bs

List containing the label of factors

Dimension of the P-dynamics

Length of the maturity vector

Horizon of the analysis

Label of bond yields

Desired model type

specific economy under study

matrix PI for JLL-based models

allows for the orthogonalized version in the case of JLL-based models

Variance-covariance matrix

Feedback matrix

Loading Bs

matrix of contemporaneous terms

List containing the labels of the factors

Dimension of the P-dynamics

Length of the maturity vector

Horizon of the analysis

Label o yields

desired Model type

Economy under study

matrix PI for JLL-based models

allows for the orthogonalized version in the case of JLL-based models

#' @references

• This function is a partially based on the version of the "irf" function from Smith, L.V. and A. Galesi (2014). GVAR Toolbox 2.0, available at https://sites.google.com/site/gvarmodelling/gvar-toolbox.

• Pesaran and Shin, 1998. "Generalized impulse response analysis in linear multivariate models" (Economics Letters)

Variance-covariance matrix

Loading As

Loading Bs

List containing the label of factors

Dimension of the P-dynamics

Length of the maturity vector

Horizon of the analysis

Label of bond yields

Desired model type

specific economy under study

matrix PI for JLL-based models

allows for the orthogonalized version in the case of JLL-based models

vector of maturities (J x 1). Maturities are in multiples of the discrete interval used in the model

number of country-specific spanned factors

risk neutral feedback matrix (N x N)

the long run risk neutral mean of the short rate (scalar)

state loadings for the one-period rate (1xN). Default is a vector of ones

the covariance matrix of the errors (N x N)

string-vector containing the names of the economies which are part of the economic system

string-vector containing the label of the model to be estimated

string-vector containing the labels of the models estimated on a country-by-country basis

list of model parameter estimates

Desired model type

Available options: "Month" and "Year"

Adjusted vector of matutities

number of country-specific spanned factors

List of risk-neutral parameters

string-vector containing the names of the economies which are part of the economic system

List of factor labels

If users wants to compute the forward premia. Default is FALSE.

For the forward premia, the shortest maturity of the remium of interest

For the forward premia, the longest maturity of the remium of interest

squared matrix in non-Jordan form

Start date of the sample period in the format yyyy-mm-dd.

End date of the sample period in the format yyyy-mm-dd.

A character vector containing the names of the economies included in the system.

Integer. Number of country-specific spanned factors.

String-list based which contains the labels of all the variables present in the model

String-vector containing the label of the model to be estimated

Character-based-vector. Available options are: "Daily All Days", "Daily Business Days", "Weekly", "Monthly", "Quarterly", "Annually"

Three possibilities:

• Full Sample: if one wishes ALL weight matrices of each year from which data is available (it may extrapolate the sample period);

• Sample Mean: if one wishes a SINGLE weight matrix containing the average of weights over of the entire sample period;

• Some year in particular (e.g. "1998", "2005" ...).

Sample starting date (year)

Sample last date (year)

Path of the Excel file containing the macroeconomic data (if any). The default is linked to the excel file present in the package.

Path of the Excel file containing the yields data (if any). The default is linked to the excel file present in the package.

Path of the Excel file containing the trade data (if any). The default is linked to the excel file present in the package.

A character vector containing the label of the model to be estimated.

A matrix of the complete set of risk factors (K x T).

A transition matrix from GVAR models (C x C).

A character vector containing the names of the economies included in the system.

A list of character vectors with labels for all variables in the model.

Start date of the sample period in the format yyyy-mm-dd.

End date of the sample period in the format yyyy-mm-dd.

A character vector containing the names of the economies included in the system.

Integer. Number of country-specific spanned factors.

A list of character vectors with labels for all variables in the model.

A character vector indicating the model type to be estimated.

GVAR transition matrix of size C x C, applicable if a GVAR-type model is selected. Default is NULL.

File path to the Excel file containing macroeconomic data, if provided. The default path points to the Excel file available within the package.

File path to the Excel file containing yields data, if provided. The default path points to the Excel file available within the package

Unrestricted variance-covariance matrix (K x K)

residuals from the VAR of a GVAR model (K x T)

index of the variable that is selected as strictly exogenous

unrestricted variance-covariance matrix (K x K)

residuals from the VAR of the JLL model (K x T)

number of domestic unspanned factors per country (scalar)

number of global unspanned factors (scalar)

string-vector containing the names of the economies which are part of the economic system

Name of the economy which is assigned as the dominant unit.
If no dominant unit is assigned, then this variable is defined as "none"

list of model parameter estimates

list containing the desired horizon of analysis for the model fit, IRFs, GIRFs, FEVDs, GFEVDs, and risk premia decomposition

desired model type

string-vector containing the names of the economies which are part of the economic system

string-list based which contains all the labels of all the variables present in the model

If users wants to compute the forward premia. Default is FALSE.

string-vector containing the label of the model to be estimated

list of model parameter estimates (see the "Optimization" function)

single numerical vector containing the desired horizon of analysis for the FEVDs and GFEVDs

string-list based which contains all the labels of all the variables present in the model

string-vector containing the names of the economies which are part of the economic system

string-vector containing the label of the model to be estimated

list of model parameter estimates (see the "Optimization" function) after a bootstrap draw

single numerical vector containing the desired horizon of analysis for the FEVDs

string-list based which contains all the labels of all the variables present in the model

string-vector containing the names of the economies which are part of the economic system

string-vector containing the label of the model to be estimated

list containing the complete set of model parameters after bootstrap estimation procedure

list of model parameter point estimates

list containing the desired inputs for the construction of the outputs of interest

string-vector containing the names of the economies which are part of the economic system

path of the folder in which the graphs will be saved

a string-vector containing the label of the model to be estimated

list of computed outputs containing the model fit, IRFs, FEVDs, GIRFs, and GFEVDs

binary variable: set 1, if the user wishes graphs to be generated; or set 0, otherwise

binary variable: set 1, if the user wishes graphs to be generated; or set 0, otherwise

single numerical vector containing the desired horizon of analysis for the FEVDs

Path of the folder in which the graphs will be saved

Available options are 'FEVD' and 'GFEVD'

a string-vector containing the names of the economies which are part of the economic system

available options are "FEVD", "GFEVD", "FEVD Ortho" and "GFEVD Ortho"

list of FEVD and GFEVD outputs

name of variable of interest. Options: "Factors" and "Yields"

label of the model factors

Path to save graphs

vector of parameters to be estimated numerically

log-likelihood function

temporary vector of parameters to be estimated numerically

scaling factor

log-likelihood function

list of model parameters as described below

matrix (N x N)

vector (N x 1). Optional. Default: vector of zeros.

Draw-specific set

vector containing the labels "IRF" and "GIRF"

number of draws

quantile of the confidence bounds

horizon of numerical outputs

dimension of the risk factor set

labels of the factor set

desired model type

Orthogonolized version for the JLL models. Default is FALSE.

numerical output set from the bootstrap analysis

quantile of the confidence bounds

desired model type

number of draws

horizon of numerical outputs

dimension of the risk factor set

vector containing the labels "FEVD" and "GFEVD"

Economies that are part of the economic system

numerical output set from the bootstrap analysis

quantile of the confidence bounds

desired model type

number of draws

horizon of numerical outputs

dimension of the risk factor set

vector containing the labels "IRF" and "GIRF"

Economies that are part of the economic system

Risk factors before the orthogonalization (FxT)

List of necessary inputs to estimate JLL-based setups

Variable labels from JLL-based models

number of country-specific spanned factors (scalar)

String-vector containing the label of the model to be estimated

Time-series of risk factors of the bootstrap (F x T)

String-list based which contains the labels of all the variables present in the model

String-vector containing the names of the economies which are part of the economic system

List of necessary inputs for the estimation of GVAR-based models

List of necessary inputs for the estimation of JLL-based models

Whether the user wishes to estimate the physical parameter model with the Bias correction model from BRW (2012) (see "Bias_Correc_VAR" function). Default is set to 0.

List of necessary inputs for performing the bias-corrected estimation (see Bias_Correc_VAR function)

Name of the economy which is assigned as the dominant unit.
If no dominant unit is assigned, then this variable is defined as "none"

Total number of risk factors of the economic system (scalar)

Number of global unspanned factors (scalar)

Number of country-specific unspanned factors (scalar)

Number of country-specific spanned factors (scalar)

Time series of bond yields

Time series of fitted bond yields

Time series of model-implied bond yields

number of country-specific maturities

vector of maturities

Economies of the economic system

Desired estimated model

Path to save the graphs

a string-vector containing the label of the model to be estimated

binary variable: set 1, if the user wishes graphs to be generated; or set 0, otherwise

List of model parameter estimates (See the "Optimization" function)

list of computed outputs containing the model fit, IRFs, FEVDs, GIRFs, and GFEVDs

a string-vector containing the names of the economies which are part of the economic system

Path of the folder in which the graphs will be saved

a string-vector containing the label of the model to be estimated

a string-vector containing the names of the economies which are part of the economic system

A character vector indicating the model type to be estimated.

A character vector containing the names of the economies included in the system.

Desired model type

Output types "IRF", "GIRF" and "IRF Ortho"

economies of the economic system

Available option "Factors" or "Yields

Option for orthogonal outputs, for JLL models. Default is FALSE.

available options are "IRF", "GIRF", "IRF Ortho" and "GIRF Ortho"

binary variable specifing whether the user whishes IRFs and/or GIRFs of risk factors

binary variable specifing whether the user whishes IRFs and/or GIRFs of bond yields

Set of economies that are part of the economic system

Desired modem type

available options are "IRF", "GIRF", "IRF Ortho" and "GIRF Ortho"

binary variable specifying whether the user whishes IRFs and/or GIRFs of risk factors

binary variable specifying whether the user whishes IRFs and/or GIRFs of bond yields

Set of economies that are part of the economic system

Desired modem type

A character vector indicating the model type to be estimated.

A list containing the point estimates of the model parameters. For details, refer to the outputs from the Optimization function.

A list containing the necessary inputs for generating IRFs, GIRFs, FEVDs, GFEVDs and Term Premia.

A list of character vectors with labels for all variables in the model.

A character vector containing the names of the economies included in the system.

A list of necessary inputs for the estimation of JLL-based models (see the JLL function).

A list containing the necessary inputs for the estimation of GVAR-based models (see the GVAR function).

Whether to estimate the physical parameter model with bias correction, based on the method by Bauer, Rudebusch and Wu (2012) (see Bias_Correc_VAR function). Default is set to 0.

List of necessary inputs for performing the bias-corrected estimation (see Bias_Correc_VAR function).

list of model parameter estimates

list containing the country-specific expected component of the forward period

desired model type

List of factor labels

list containing the desired horizon of analysis for the model fit, IRFs, GIRFs, FEVDs, GFEVDs, and risk premia decomposition

string-vector containing the names of the economies which are part of the economic system

Set of inputs that are necessary to the log-likelihood function

string-vector containing the names of the economies which are part of the economic system

character-based vector: "Daily All Days", "Daily Business Days", "Weekly", "Monthly", "Quarterly", "Annually"

string-list based which contains the labels of all the variables present in the model

string-vector containing the label of the model to be estimated

Generates simplified output list in the bootstrap setting. Default is set to FALSE.

vector containing all the vectorized auxiliary parameters

matrix (6x2) containing the size information of all parameters

vector-valued objective function (function)

if con = 'concentration', then set the value of the parameter whose name contains @ to empty

variable inputs used in the optimization (see inputs from "optimization" function)

string-vector containing the label of the model to be estimated

string-list based which contains the labels of all the variables present in the model

string-vector containing the names of the economies which are part of the economic system

Set of necessary inputs used in the estimation of the JLL-based models (see "JLL" function)

Set of necessary inputs used in the estimation of the GVAR-based models (see "GVAR" function)

if TRUE, returns only the values of the likelihood function, else generates the entire set of outputs

List of inputs for GVAR model estimation:

1. Economies: A character vector containing the names of the economies included in the system.

2. GVARFactors: A list of all variables used in the estimation of the VARX model
   (see e.g. CM_Factors_GVAR file for details);

3. VARXtype: A character vector with three possible options:

   • 'unconstrained': model is estimated without constraints (each equation is estimated individually by ordinary least square);

   • 'constrained: Spanned Factors': The model is estimated with the restriction that foreign pricing factors do NOT affect (i) domestic economic variables and (ii) domestic pricing factors.
     (Equations are estimated using restricted least squares)

   • 'constrained : [factor_name]': The model is estimated with the restriction that the specified risk factor is influenced only by its own lagged values and the lagged values of its corresponding star variables. (Equations are estimated using restricted least squares.)

4. Wgvar: The GVAR transition matrix (C x C) used in the model solution.
   (See the output from the Transition_Matrix function.).

Integer. Number of country-specific spanned factors.

A logical flag to indicate whether to perform a prior consistency check on the inputs provided in GVARinputs. The default is set to FALSE

List of inputs for GVAR-based models

string-based vector containing label of the domestic risk factors

string-based vector containing label of the star domestic risk factors

string-based vector containing label of the global risk factors

number of country-specific spanned factors (scalar)

Bond yield forecasts for a single date

Bond yield forecasts for multiple dates

String-vector containing the names of the economies which are part of the economic system

String-vector containing the label of the model to be estimated

matrix of residuals (N x T)

covariance matrice or array of covariance matrices (N x N) or (N x N x T)

Inverse of SS (N x N) or (N x N x T) - optional input

log(abs(|SS|)) (1 x T) - optional input

List of point estimates of the model parameter

Time-series of the residuals from the P-dynamics equation (T x F)

Time-series of the residuals from the observational equation (T x J or T x CJ)

Desired model to be estimated

Matrix B of loadings (CJ x F or J x F)

List containing the desired inputs for the construction

String-vector containing the names of the economies which are part of the economic system

String-list based which contains the labels of all the variables present in the model

List of necessary inputs for the estimation of GVAR-based models

List of necessary inputs for the estimation of JLL-based models

Whether the user wishes to estimate the physical parameter model with the Bias correction model from BRW (2012) (see "Bias_Correc_VAR" function). Default is set to 0.

List of necessary inputs for performing the bias-corrected estimation (see "Bias_Correc_VAR" function)

Number of lags in the P-dynamics. Default is set to 1.

Intercept from the P-dynamics (F x 1)

Feedback matrix from the P-dynamics (F x F)

Intercept of model-implied yields model (J x 1)

Slope of model-implied yields model (J x N or CJ x CN)

Sub-sample of risk factors (F x t)

Number of countries in the economic cohort (scalar)

Number of country-specific bond yields

Labels of bond yields

Forecast labels (string-based vector)

Forecast horizon (scalar)

A string-vector containing the label of the model to be estimated

matrix (CJ x T) or a list containing those matrices, where C is the number of countries, J - the number of maturities and T - time series length

time series of risk factors (F x T). Could be stored in a list depending on the model

string-list based which contains the labels of all variables present in the model

vector of maturities (in years) used in the estimation

single element character-based vector. Available options are: "Daily All Days",
"Daily Business Days", "Weekly", "Monthly", "Quarterly", "Annually"

string-vector containing the names of the multicountry setups

string-vector containing the names of the economies which are part of the economic system

string-vector containing the label of the model to be estimated

Constrained parameter

character-based vector that describes the constraints. Constraints are:

• 'Jordan';

• 'Jordan; stationary'

• 'Jordan MultiCountry'

• 'Jordan MultiCountry; stationary'

• 'stationary'

• 'psd'

• 'BlockDiag'

• 'bounded'

• 'diag'

• 'JLLstructure'

lower bounds (a scalar or a matrix) of ParaValue (for the bounded case).

upper bounds (a scalar or a matrix) of ParaValue (for the bounded case).

string-vector containing the names of the economies which are part of the economic system

string-list based which contains the labels of all the variables present in the model

list of necessary inputs for the estimation of JLL-based models (see "JLL" function)

Risk factors before the orthogonalization (FxT)

List of necessary inputs to estimate JLL-based setups

number of global unspanned factors (scalar)

string containing the names of the economy to be estimated

string-vector containing the label of the model to be estimated

list containing the factor labels

time series of risk factors (F x T). Could be stored in a list depending on the model

string-list based which contains the labels of all variables present in the model

string-vector containing the names of the economies which are part of the economic system

string-vector containing the label of the model to be estimated

list of necessary inputs for performing the bias-corrected estimation (see "Bias_Correc_VAR" function)

list of necessary inputs for the estimation of GVAR-based models (see "GVAR" function)

list of necessary inputs for the estimation of JLL-based models (see "JLL" function)

Logical. Whether to perform a prior check on the consistency of the provided input list. Default is FALSE.

string-vector containing the label of the model to be estimated

list of necessary inputs for performing the bias-corrected estimation (see "Bias_Correc_VAR" function)

time series of risk factors (F x T). Could be stored in a list depending on the model

string-vector containing the names of the economies which are part of the economic system

string-list based which contains the labels of all variables present in the model

list of necessary inputs for the estimation of GVAR-based models (see "GVAR" function)

list of necessary inputs for the estimation of JLL-based models (see "JLL" function)

string-vector containing the label of the model to be estimated

time series of risk factors (F x T). Could be stored in a list depending on the model

string-vector containing the names of the economies which are part of the economic system

number of country-specific spanned factors

list of necessary inputs for the estimation of GVAR-based models (see "GVAR" function)

list of necessary inputs for the estimation of JLL-based models (see "JLL" function)

unconstrained auxiliary parameter

One of the following options:

• 'Jordan'

• 'Jordan; stationary'

• 'Jordan MultiCountry'

• 'Jordan MultiCountry; stationary'

• 'psd';

• 'BlockDiag'

• 'bounded'

• 'diag'

• 'JLLstructure'

lower bounds of b (if option 'bounded' is chosen)

upper bounds of b (if option 'bounded' is chosen)

string-list based which contains the labels of all the variables present in the model

string-vector containing the names of the economies which are part of the economic system

Inputs used in the estimation of the JLL-based models

Inputs used in the estimation of the GVAR-based models

List of model parameter estimates

String-vector containing the names of the economies which are part of the economic system

A string-vector containing the label of the model to be estimated

list containing the B loadings

matrix of weights of the portfolios observed without errors (N x J or CN x J)

long-run interest rate (scalar or vector with length C)

time interval unit of the model (scalar). For instance, if data is (i) monthly, dt <- 12; (ii) quarterly, dt <- 4; (iii) yearly, dt <- 1

string-vector containing the names of the economies which are part of the economic system

string-vector containing the label of the model to be estimated

List of point estimates of the model parameter

String-list based which contains the labels of all the variables present in the model

Vector of bond yield maturities

String-vector containing the names of the economies which are part of the economic system

A character vector indicating the model type to be estimated

vector of maturities (in years) of yields used in estimation (J x 1)

time interval unit of the model (scalar). For instance, if data is (i) monthly, dt <- 12; (ii) quarterly, dt <- 4; (iii) yearly, dt <- 1

risk-neutral feedback matrix (N x N or CN x CN)

variance-covariance matrix (F x F)

matrix of weights of the portfolios observed without errors (N x J or CN x J)

string-list based which contains the labels of all the variables present in the model

string-vector containing the names of the economies which are part of the economic system

string-vector containing the label of the model to be estimated

Set of VARX model parameters

List of inputs for GVAR-based models

list containing the country-specific intercepts

list containing the country-specific feedback matrices

list containing the country-specific link matrices

List containing the set of VARX model parameters

List of inputs for GVAR-based models

feedback matrix from a GVAR without global variables

feedback matrix from a marginal model

matrix of yields used in estimation (J x T or CJ x T)

number of country-specific spanned factors

vector of maturities (in years) of yields used in estimation (J x 1)

composite matrix of weights the portfolios observed with and without errors

Variance of the portfolio of yields observed with error (scalar). Default is set to NULL

string-vector containing the label of the model to be estimated

List of necessary inputs to estimate JLL-based setups

Set of factors used in the estimation of JLL-based setups

Set of parameters obtained from the JLL regressions

Set of parameters obtained from the VAR(1) of JLL-based models

number of country-specific spanned factors (scalar)

List of model parameter estimates

A string-list based which contains all the labels of all the variables present in the model

String-vector containing the names of the economies which are part of the economic system

A string-vector containing the label of the model to be estimated

Feedback matrix resulting from the bias-correction procedure

number of country-specific spanned factors (scalar)

time series of the risk factors (T x F)

inputs used in the estimation of the GVAR-based models (see "GVAR" function). Default is set to NULL

inputs used in the estimation of the JLL-based models (see "JLL" function). Default is set to NULL

string-list based which contains the labels of all variables present in the model

string-vector containing the label of the model to be estimated

List of inputs for GVAR-based models

List containing the set of VARX model parameters

time-series of spanned factors (N x T or CN x T)

time-series of yields (J x T or CJ x T)

time-series of risk factors (F x T)

number of country-specific spanned factors

vector of maturities (in years) of yields used in estimation (J x 1)

matrix of weights of the portfolios observed with errors ((J-N) x J or C(J-N) x CJ)

matrix of weights of the portfolios observed without errors (N x J or CN x CJ)

matrix of intercepts (P-dynamics)

feedback matrix (P-dynamics)

variance-covariance matrix (P-dynamics)

list containing the B loadings

list containing the A loadings

string-vector containing the label of the model to be estimated

matrix of yields used in estimation (J x T or CJ x T)

complete set of spanned factors (N x T or CN x T)

number of country-specific spanned factors

vector of maturities (in years) of yields used in estimation (J x 1)

time interval unit of the model (scalar). For instance, if data is (i) monthly, dt <- 12; (ii) quarterly, dt <- 4; (iii) yearly, dt <- 1.

list containing the B loadings

matrix of weights of the portfolios observed without errors (N x J or CN x J)

matrix of weights of the portfolios observed with errors ((J-N) x J or C(J-N) x CJ)

string-vector containing the names of the economies which are part of the economic system

string-vector containing the label of the model to be estimated

single element character-based vector. Available options are: "Daily All Days",
"Daily Business Days", "Weekly", "Monthly", "Quarterly", "Annually"

A character vector indicating the model type to be estimated.

List of model parameter estimates (See the "Optimization" function)

list of computed outputs containing the model fit, IRFs, FEVDs, GIRFs, GFEVDs and Term Premia

list containing the desired inputs for the construction of the desired output

A character vector containing the names of the economies included in the system.

A list of character vectors with labels for all variables in the model.

number of country-specific unspanned factors (scalar)

number of country-specific spanned factors (scalar)

number of global unspanned factors (scalar)

Set of economies that are part of the economic system (string-vector)

Name of the economy which is assigned as the dominant unit.
If no dominant unit is assigned, then this variable is defined as "None"

string-vector containing the label of the model to be estimated

list of model parameter estimates (See the "Optimization" function)

single numerical vector containing the desired horizon of analysis for the IRFs

string-list based which contains the labels of all the variables present in the model

string-vector containing the names of the economies which are part of the economic system

string-vector containing the label of the model to be estimated

list of model parameter estimates (see the "Optimization" function) after a bootstrap draw

single numerical vector containing the desired horizon of analysis for the IRFs

string-list based which contains all the labels of all the variables present in the model

string-vector containing the names of the economies which are part of the economic system

string-vector containing the label of the model to be estimated

list containing the complete set of model parameters after bootstrap estimation procedure

list of model parameter point estimates

list containing the desired inputs for the construction of the outputs of interest

string-vector containing the names of the economies which are part of the economic system

path of the folder in which the graphs will be saved

a string-vector containing the label of the model to be estimated

list of computed outputs containing the model fit, IRFs, FEVDs, GIRFs, and GFEVDs

binary variable: set 1, if the user wishes graphs to be generated; or set 0, otherwise

binary variable: set 1, if the user wishes graphs to be generated; or set 0, otherwise

single numerical vector containing the desired horizon of analysis for the IRFs

Path of the folder in which the graphs will be saved

Available options are 'IRF' and 'GIRF'

a string-vector containing the names of the economies which are part of the economic system

Data-frame with basic features of a single IRF for risk factors

Data-frame with basic features of a single IRF for yields

string-vector containing the label of the model to be estimated

string-list based which contains the labels of all the variables present in the model

string-vector containing the names of the economies which are part of the economic system

number of global unspanned factors (scalar)

number of domestic unspanned factors per country (scalar)

number of domestic spanned factors per country (scalar)

number of countries of the economic system (scalar)

Time series of risk factors for the jointly estimated models (CJ x T)

A string-list based which contains all the labels of all the variables present in the model

String-vector containing the names of the economies which are part of the economic system

numerical vector before imposing stationary constraint

parameter of interest (scalar or matrix)

risk-neutral feedback matrix

Start date of the sample period in the format "dd-mm-yyyy"

End date of the sample period in the format "dd-mm-yyyy"

A character vector indicating the model type to be estimated. Available options: "JPS original", "JPS global", "GVAR single", "JPS multi", "GVAR multi", "JLL original", "JLL No DomUnit", "JLL joint Sigma".

A numerical matrix with time series of yields (JxT or CJ x T)

A numerical matrix with time series of the global risk factors (G x T)

A numerical matrix with time series of the country-specific risk factors for all C countries (CM x T)

A list of character vectors with labels for all variables in the model.

A character vector containing the names of the economies included in the system.

A character vector specifying the frequency of the data. Available options are: "Daily All Days", "Daily Business Days", "Weekly", "Monthly", "Quarterly", or "Annually".

A list containing the necessary inputs for the estimation of GVAR-based models

A list of necessary inputs for the estimation of JLL-based models. If the chosen model is "JLL original" or "JLL joint Sigma", then a dominant unit economy must be chosen. Otherwise, this list must be set as 'None'.

Logical. Whether to estimate the physical parameter model with bias correction, based on the method by Bauer, Rudebusch and Wu (2012). Default is FALSE.

List of necessary inputs for performing the bias-corrected estimation.

A character string indicating the maturity unit of yields. Options are: "Month" for yields expressed in months, or "Year" for yields expressed in years. Default is "Month".

Logical. Whether to perform a prior check on the consistency of the provided input list. Default is TRUE.

Logical. Whether to adjust the global series for the sepQ models in the Bootstrap setting. Default is FALSE.

A character vector indicating the model type to be estimated.

A numeric scalar specifying the desired analysis horizon for the outputs.

A list of desired graphical outputs. Available options are: "Fit", "IRF", "FEVD", "GIRF", "GFEVD", "TermPremia".

A string for the name of the output label to be stored.

A binary variable (1 or 0) indicating whether to impose that the largest eigenvalue under Q is strictly smaller than 1. Set to 1 to impose the restriction, or 0 otherwise.

A character vector specifying the data frequency. Available options: "Daily All Days", "Daily Business Days", "Weekly", "Monthly", "Quarterly", "Annually".

A binary variable (1 or 0) indicating whether the user wishes to generate graphs for yields. Default is 0.

A binary variable (1 or 0) indicating whether the user wishes to generate graphs for risk factors. Default is 0.

A binary variable (1 or 0) indicating whether the user wishes to generate orthogonalized JLL-based graphs. Default is 0.

A binary variable (1 or 0) indicating whether the user wishes to generate forward premia graphs. Default is 0.

A numeric vector containing the maturities associated with the start and end dates of the loan.

A binary variable (1 or 0) indicating whether the user wishes to perform bootstrap-based estimation. Default is 0.

A List containing the following four elements:

1. methodBS: Desired bootstrap method: (a) 'bs' for standard residual bootstrap, (b) 'wild' for wild bootstrap, or (c) 'block' for block bootstrap.

2. BlockLength: If block bootstrap is chosen, specify the block length (numeric scalar).

3. ndraws: Number of bootstrap draws.

4. pctg: Confidence level expressed in basis points (numeric vector).

A binary variable (1 or 0) indicating whether the user wishes to generate forecasts. Default is 0.

A list containing the following three elements:

1. ForHoriz: forecast horizon;

2. t0Sample: Index of the first variable in the information set.

3. t0Forecast: Index of the first forecast cut-off date.

4. ForType: A string specifying the desired forecast type. Available options are: "Rolling" or "Expanding".

A character string indicating the maturity unit of yields. Options are: (i) "Month" for yields expressed in months, or (ii) "Year" for yields expressed in years. Default is "Month".

A numeric matrix (F x T) representing the time series of risk factors before the orthogonalization process.

Integer. Number of country-specific spanned factors.

List of necessary inputs to estimate JLL models:

1. Economies: set of economies that are part of the economic system (string-vector)

2. DomUnit: A string specifying the name of the economy assigned as the dominant unit.
   If no dominant unit is assigned, set this variable to "None".

3. WishSigmas: Set to "1" if the user wishes to estimate the variance-covariance matrices and Cholesky factorizations
   (this can take a long time). Set to "0" if not.

4. SigmaNonOrtho: A NULL value or an F x F matrix from the non-orthogonalized dynamics.

5. JLLModelType: A string specifying the type of JLL model. Available options are: "JLL original", "JLL joint Sigma", or "JLL No DomUnit".

A logical flag to indicate whether to perform a prior consistency check on the inputs provided in JLLinputs. The default is set to FALSE

Constrained parameter value

number of countries of the economic system

number of country-specific spanned factors

Binary variable: set "1" if the user wishes the largest eigenvalue to be strictly smaller than 1. Set "0", otherwise

Integer. Number of country-specific spanned factors.

A character vector containing the names of the domestic variables.

A character vector containing the names of the global variables.

A character vector containing the names of the economies included in the system.

A character vector indicating the model type to be estimated.

number of spanned factors

Factor labels

Available options are BR_2017 (Bauer and Rudebusch, 2017) , CM_2023 (Candelon and Moura, 2023), CM_2024 (Candelon and Moura, 2024)

risk-neutral feedback matrix (N x N or CN x CN)

long-run interest rate (scalar or vector with length C)

variance-covariance matrix (F x F)

intercept from the P-dynamics (F x 1)

feedback matrix from the P-dynamics (F x F)

Variance of the portfolio of yields observed with error (scalar). Default is set to NULL.

matrix of contemporaneous terms from the P-dynamics (F x F)

vector of maturities (in years) of yields used in estimation (J x 1)

matrix of yields used in estimation (J x T or CJ x T)

complete set of spanned and unspanned factors (F x T)

complete set of spanned factors (N x T or CN x T)

matrix of weights of the portfolios observed without errors (N x J or CN x J)

matrix of weights of the portfolios observed with errors ((J-N) x J or C(J-N) x CJ)

composite matrix of weights the portfolios observed with and without errors

time interval unit of the model (scalar). For instance, if data is (i) monthly, dt <- 12; (ii) quarterly, dt <- 4; (iii) yearly, dt <- 1.

string-vector containing the names of the economies which are part of the economic system

string-list based which contains the labels of all the variables present in the model

string-vector containing the label of the model to be estimated

if the model chosen is the "GVAR single" or "GVAR multi", the "GVARinputs" should be specified (see "GVAR" function)

if the model chosen is JLL-based. "JLLinputs" should contain (i) DomUnit, (ii) WishSigmas, (iii) SigmaNonOrtho, (iv) JLLModelType (See JLL function)

Generates simplified output list in the bootstrap setting. Default is set to FALSE.

if nargout == 1: provides only the values of the likelihood; if nargout == 2: complete ATSM outputs

List of inputs for GVAR-based models

vector of maturities (J x 1)

Available options: "Month" and "Year"

matrix containing all yields of the system (JxT,if the model is single-country-based or CJxT if the model is multy-country-based)

vector containing names of all the economies of the system

(i) "Month": if maturity of yields are expressed in months or (ii) "Year": if maturity of yields are expressed in years

Set of parameters obtained from the JLL regressions

Set of parameters obtained from the VAR(1) of JLL-based models

A character vector indicating the model type to be estimated.

A list containing the point estimates of the model parameters. For details, refer to the outputs from the Optimization function.

A list containing the necessary inputs for generating IRFs, GIRFs, FEVDs, GFEVDs and Term Premia.

A list of character vectors with labels for all variables in the model.

A character vector containing the names of the economies included in the system.

A character vector indicating the model type to be estimated.

A list of model parameter estimates (see the "Optimization" function) after a bootstrap draw

A list containing the necessary inputs for generating IRFs, GIRFs, FEVDs, GFEVDs and Term Premia.

A list of character vectors with labels for all variables in the model.

A character vector containing the names of the economies included in the system.

Forecast horizon-ahead (scalar)

Index of the last set of observations in the information set at a given forecasting round

List of model parameter estimates

A string-list based which contains all the labels of all the variables present in the model

Time-series of bond yields, complete set (J x T or CJ x T)

String-vector containing the names of the economies which are part of the economic system

A string-vector containing the label of the model to be estimated

matrix of yields used in estimation (J x T or CJ x T)

complete set of spanned and unspanned factors (F x T)

vector of maturities (in years) of yields used in estimation (J x 1)

number of country-specific spanned factors

time interval unit of the model (scalar)

matrix of weights of the portfolios observed without errors (N x J or CN x J)

risk-neutral feedback matrix (N x N or CN x CN)

variance-covariance matrix (F x F)

list containing the A loadings

list containing the B loadings

long-run interest rate (scalar or vector with length C)

Variance of the portfolio of yields observed with error (scalar).

intercept from the P-dynamics (F x 1)

feedback matrix from the P-dynamics (F x F)

matrix of contemporaneous terms from the P-dynamics (F x F)

variance-covariance matrix of the bond yields

likelihood of each time series (Tx1)

List of inputs from GVAR models

List of inputs from JLL models

string containing the names of the economy to be estimated

string-vector containing the label of the model to be estimated

Bootstrap output. Default is FALSE.

A list containing the necessary inputs for building the log-likelihood function (see InputsForOpt function).

A binary variable (1 or 0) indicating whether to impose that the largest eigenvalue under Q is strictly smaller than 1. Set to 1 to impose the restriction, or 0 otherwise.

A character vector specifying the data frequency. Available options: "Daily All Days", "Daily Business Days", "Weekly", "Monthly", "Quarterly", "Annually".

A list of character vectors with labels for all variables in the model.

A character vector containing the names of the economies included in the system.

A character vector indicating the model type to be estimated.

Convergence tolerance (scalar). The default is 1e-4.

Logical. If TRUE, computes the time required for model estimation. Default is TRUE.

Logical. If TRUE, generates a simplified output list in the bootstrap setting. Default is FALSE.

List containing features for estimation of the risk-neutral parameters.

Log-likelihood function

Estimation type

convergence tolerance (scalar). Default value is set as 1e-4.

vector-valued objective function (function)

list contain starting values and constraints: for each input argument K (of f), we need four inputs that look like:

1. a starting value: K0

2. a variable label ('K0') followed by a ':' followed by a type of constraint. The constraint can be:

   • 'bounded': bounded matrix;

   • 'Jordan' or 'Jordan MultiCountry': a matrix of Jordan type;

   • 'psd': psd matrix;

   • 'stationary': largest eigenvalue of the risk-neutral feedback matrix is strictly smaller than 1;

   • 'diag' or 'BlockDiag': a diagonal or block diagonal matrix.

   • 'JLLstructure': to impose the zero-restrictions on the variance-variance matrix along the lines of the JLL models

3. a lower bound lb (lb <- NULL -> no lower bound)

4. an upper bound ub (ub <- NULL -> no upper bound)

5. Specification of the optimization settings:

   • 'iter off': hide the printouts of the numerical optimization routines;

   • 'fminunc only': only uses fminunc for the optimization;

   • ”fminsearch only': only uses fminsearch for the optimization.

A list of character vectors with labels for all variables in the model.

A character vector containing the names of the economies included in the system.

A character vector indicating the model type to be estimated.

List. Inputs for JLL model estimation (see JLL). Default is NULL.

List. Inputs for GVAR model estimation (see GVAR). Default is NULL.

convergence tolerance (scalar). Default value is 1e-4.

computes the required time for estimation of the model. Default is TRUE.

Starting time

List of necessary inputs to estimate JLL-based setups

number of country-specific spanned factors (scalar)

Set of factors used in the estimation of JLL-based setups

Variable labels of the non-orthogonalized risk factors

Variable labels of the orthogonalized risk factors

Risk factors before the orthogonalization (FxT)

List of necessary inputs to estimate JLL-based setups

Set of orthogonalized risk factors

Variable labels of the orthogonalized risk factors

number of country-specific spanned factors (scalar)

string-vector containing the label of the model to be estimated

list of model parameter estimates (See the "Optimization" function)

list containing the desired horizon of analysis for the model fit, IRFs, GIRFs, FEVDs, GFEVDs, and risk premia decomposition

string-list based which contains all the labels of all the variables present in the model

string-vector containing the names of the economies which are part of the economic system

string-vector containing the names of the multicountry setups

string-vector containing the names of the economies which are part of the economic system

time series of risk factors (F x T). Could be stored in a list depending on the model

matrix (CJ x T) or a list containing those matrices, where C is the number of countries, J - the number of maturities and T - time series length

vector of maturities (in years) used in the estimation

List of generic inputs

List of specific inputs

Model parameters estimated in the P-dynamics the

string-vector containing the label of the model to be estimated

List of model parameter features updated

llk after optimization

List containing features for estimation of several parameters after optimization

Several variable labels

a string-vector containing the label of the model to be estimated

User must set "1" is she wishes to impose the largest eigenvalue under the Q to be strictly smaller than 1. Otherwise set "0"

Set of inputs that are necessary to the log-likelihood function

Generates simplified output list in the bootstrap setting. Default is set to FALSE.

A character vector containing the label of the model to be estimated

A character vector containing the names of the economies included in the system.

Point estimate from the model parameters

List of country-specific forecasts

matrix of yields used in estimation (J x T)

vector of maturities (in years) of yields used in estimation (J x 1)

pricing factors (can be yields-based or non-yields/macro variables) (N x T)

time unit of the model (scalar). For instance, if data is (i) monthly, dt <- 12; (ii) quarterly, dt <- 4; (iii) yearly, dt <- 1.

'Jordan' -> K1Q will be of the Jordan type

left hand side variables (M x T)

regressors (i.e. N-1 variables + the intercept) (N x T)

constraints matrix (M x N). If Bcon(i,j) = nan –> B(i,j) is a free parameter

weighting matrix (psd) - (M x M). Default is set to be identity

List of country-specific bond yields

List of country-specific and global economic variables

string-vector containing the names of the economies which are part of the economic system

Time-series of the residuals from the P-dynamics equation (T x F)

Time-series of the residuals from the observational equation (T x J or T x CJ)

List containing the desired inputs for the construction of the numerical outputs.

A character vector indicating the model type to be estimated

Number of lags in the P-dynamics. Default is set to 1.

string-vector containing the label of the model to be estimated

list of model parameter estimates (see the "Optimization" function)

string-vector containing the names of the economies which are part of the economic system

string-list based which contains the labels of all the variables present in the model

Factor set list (see e.g. "CM_Factors_GVAR" data file)

A character vector containing the names of the economies included in the system.

A list of character vectors with labels for all variables in the model.

Start date of the sample period in the format yyyy-mm-dd

End date of the sample period in the format yyyy-mm-dd

A character vector specifying the data frequency. Available options: "Daily All Days", "Daily Business Days", "Weekly", "Monthly", "Quarterly", "Annually".

A matrix (J x T), where J is the number of maturities and T is the length of the time series.

A character vector containing the names of the economies included in the system.

Scalar representing the desired number of country-specific spanned factors (maximum allowed is N = J).

string-vector containing the label of the model to be estimated

string-vector containing the names of the economies of the system

time series of risk factors (F x T)

string-list based which contains the labels of all the variables present in the model

A list of required inputs to estimate the GVAR-based setups:

1. VARXtype string-vector containing the VARX feature (see "GVAR" function) (GVAR-based models)

2. t_First_Wgvar Sample starting date (year) (GVAR-based models)

3. t_Last_Wgvar Sample last date (year) (GVAR-based models)

4. W_type Criterion used in the computation of the star variables (see "Transition_Matrix" function) (GVAR-based models)

A list of required inputs to estimate the JLL-based setups:

1. DomUnit name of the economy which is assigned as the dominant unit (JLL-based models)

2. WishSigmas equal to "1" if one wishes the variance-covariance matrices and the Cholesky factorizations (JLL-based models)

3. SigmaNonOrtho NULL or some F x F matrix from the non-orthogonalized dynamics (JLL-based models)

Whether the user wishes to estimate the physical parameter model with the Bias correction model from BRW (2012) (see "Bias_Correc_VAR" function).
Default is set to 0.

A list of required inputs to estimate the bias corrected setups of the type of BRW:

1. BiasCorrection binary variable. it takes value equal to 1 if the user whishes the estimates to be bias-corrected and 0, otherwise. (BRW model)

2. flag_mean flag whether mean- (TRUE) or median- (FALSE) unbiased estimation is desired

3. gamma adjustment parameter (BRW model)

4. N_iter number of iterations (BRW model)

5. N_burn number of burn-in iterations (BRW model)

6. B number of bootstrap samples (BRW model)

7. checkBRW flag whether the user wishes to perform the closeness check (BRW model)

8. B_check number of bootstrap samples for closeness check

path of the Excel file containing the data (if any)

time series of the risk factors (F x T)

string-vector containing the names of the economies which are part of the economic system

GVAR transition matrix (C x C)

a string-vector containing the label of the model to be estimated

list of computed outputs containing the model fit, IRFs, FEVDs, GIRFs, GFEVDs and Risk premia

list of model parameter estimates (See the "Optimization" function)

binary variable: set 1, if the user wishes graphs to be generated; or set 0, otherwise

(i) "Month": if maturity of yields are expressed in months or (ii) "Year": if maturity of yields are expressed in years

a string-vector containing the names of the economies which are part of the economic system

Path of the folder in which the graphs will be saved

list of model parameter estimates

list containing the country-specific pure expected component

desired model type

string-vector containing the names of the economies which are part of the economic system

list of model parameter estimates (see the "Optimization" function)

string-list based which contains all the labels of all the variables present in the model

string-vector containing the label of the model to be estimated

list containing the desired horizon of analysis for the model fit, IRFs, GIRFs, FEVDs, GFEVDs, and risk premia decomposition

string-vector containing the names of the economies which are part of the economic system

A matrix of the complete set of risk factors (F x T).

A character vector containing the names of the economies included in the system.

List of domestic risk factors (both spanned and unspanned)

List of risk factors

List of transition matrices

A list of character vectors with labels for all variables in the model.

Sample starting date (in the format: yyyy).

Sample ending date (in the format: yyyy).

A character vector containing the names of the economies included in the system.

A character string indicating the method for computing interdependence. Possible options include:

• Time-varying: Computes time-varying interdependence and returns the weight matrices for each year based on available data (may extrapolate the sample period).

• Sample Mean: Returns a single weight matrix containing the average weights over the entire sample period, suitable for time-invariant interdependence.

• A specific year (e.g., "1998", "2005"): Used to compute time-invariant interdependence for the specified year.

Data used to compute the transition matrix. Default is set to NULL.

Path to the Excel file containing the data (if applicable). The default is linked to the Excel file available in the package.

Constrained parameter

Type of constraint

string-list based which contains the labels of all the variables present in the model

string-vector containing the names of the economies which are part of the economic system

Inputs used in the estimation of the GVAR-based models

Constrained parameter value

Type of constraint

lower bound

upper bound

Constrained parameter value

Type of constraint

string-list based which contains the labels of all the variables present in the model

string-vector containing the names of the economies which are part of the economic system

Inputs used in the estimation of the JLL-based models

Constrained parameter

Type of constraint

string-list based which contains the labels of all the variables present in the model

string-vector containing the names of the economies which are part of the economic system

Constrained parameter value

Type of constraint

vector containing all the vectorized auxiliary parameters

matrix (6x2) containing the size information of all parameters

if con = 'concentration', then set the value of the parameter whose name contains @ to empty

string-list based which contains the labels of all the variables present in the model

string-vector containing the names of the economies which are part of the economic system

Set of necessary inputs used in the estimation of the JLL-based models

Set of necessary inputs used in the estimation of the GVAR-based models

variable inputs used in the optimization (see "Optimization" function)

Variance-covariance matrix from JLL model

complete set of spanned and unspanned factors (F x T)

number of country-specific spanned factors

List of inputs from JLL models

A numeric matrix (FTx T) representing the time series of risk factors.

String vector with two possible values: 'unconstrained' or 'constrained'.

Constraints matrix (F+1 x N), which includes an intercept. If Bcon(i,j) = NA, then B(i,j) is treated as a free parameter.
Default is set to NULL.

List of inputs for GVAR-based models

list containing the set of eisk factors used in the estimation of the VARX models

string-based vector containing label of the domestic risk factors

string-based vector containing label of the star domestic risk factors

string-based vector containing label of the global risk factors

number of country-specific spanned factors (scalar)

string-vector containing the label of the model to be estimated

List of model parameter estimates (see the "Optimization" function)

string-list based which contains all the labels of all the variables present in the model

string-vector containing the names of the economies which are part of the economic system

list containing the desired inputs for the construction of IRFs, GIRFs, FEVDs, and GFEVDs

desired model type

list containing the desired inputs for the construction of IRFs, GIRFs, FEVDs, and GFEVDs

desired model type

List of inputs for outputs

Output types "FEVD", "GFEVD" and "FEVD Ortho"

desired model type