Package 'bvartools'

Add Priors to Bayesian Models A generic function used to generate prior specifications for a list of models. The function invokes particular methods which depend on the class of the first argument.

Description

Add Priors to Bayesian Models

A generic function used to generate prior specifications for a list of models. The function invokes particular methods which depend on the class of the first argument.

Usage

add_priors(object, ...)

Arguments

object	an object of class "bvarmodel" or "bvecmodel".
...	arguments passed forward to method.

Examples

# Load data
data("e1")
e1 <- diff(log(e1)) * 100

# Obtain data matrices
model <- gen_var(e1, p = 2, deterministic = 2,
                 iterations = 100, burnin = 10)
# Chosen number of iterations and burn-in draws should be much higher.

# Add prior specifications
model <- add_priors(model)

---

Add Priors for a Vector Autoregressive Models

Description

Adds prior specifications to a list of models, which was produced by function gen_var.

Usage

## S3 method for class 'bvarmodel'
add_priors(
  object,
  coef = list(v_i = 1, v_i_det = 0.1, shape = 3, rate = 1e-04, rate_det = 0.01),
  sigma = list(df = "k", scale = 1, mu = 0, v_i = 0.01, sigma_h = 0.05, constant = 1e-04),
  ssvs = NULL,
  bvs = NULL,
  ...
)

Arguments

object	a list, usually, the output of a call to gen_var.
coef	a named list of prior specifications for the coefficients of the models. For the default specification all prior means are set to zero and the diagonal elements of the inverse prior variance-covariance matrix are set to 1 for coefficients corresponding to non-deterministic and structural terms. For deterministic coefficients the prior variances are set to 10 via v_i_det = 0.1. The variances need to be specified as precisions, i.e. as inverses of the variances. For further specifications such as the Minnesota prior see 'Details'.
sigma	a named list of prior specifications for the error variance-covariance matrix of the models. For the default specification of an inverse Wishart distribution the prior degrees of freedom are set to the number of endogenous variables and the prior variances to 1. See 'Details'.
ssvs	optional; a named list of prior specifications for the SSVS algorithm. Not allowed for TVP models. See 'Details'.
bvs	optional; a named list of prior specifications for the BVS algorithm. See 'Details'.
...	further arguments passed to or from other methods.

Details

The arguments of the function require named lists. Possible specifications are described in the following. Note that it is important to specify the priors in the correct list. Otherwise, the provided specification will be disregarded and default values will be used.

Argument coef can contain the following elements

v_i

a numeric specifying the prior precision of the coefficients. Default is 1.

v_i_det

a numeric specifying the prior precision of coefficients corresponding to deterministic terms. Default is 0.1.

coint_var

a logical specifying whether the prior mean of the first own lag of an endogenous variable should be set to 1. Default is FALSE.

const

a numeric or character specifying the prior mean of coefficients, which correspond to the intercept. If a numeric is provided, all prior means are set to this value. If const = "mean", the mean of the respective endogenous variable is used as prior mean. If const = "first", the first values of the respective endogenous variable is used as prior mean.

minnesota

a list of length 4 containing parameters for the calculation of the Minnesota prior, where the element names must be kappa0, kappa1, kappa2 and kappa3. For the endogenous variable $i$ the prior variance of the $l$th lag of regressor $j$ is obtained as

$\frac{\kappa_{0}}{l^2} \textrm{ for own lags of endogenous variables,}$

$\frac{\kappa_{0} \kappa_{1}}{l^2} \frac{\sigma_{i}^2}{\sigma_{j}^2} \textrm{ for endogenous variables other than own lags,}$

$\frac{\kappa_{0} \kappa_{2}}{(l+1)^2} \frac{\sigma_{i}^2}{\sigma_{j}^2} \textrm{ for exogenous variables,}$

$\kappa_{0} \kappa_{3} \sigma_{i}^2 \textrm{ for deterministic terms,}$

where $\sigma_{i}$ is the residual standard deviation of variable $i$ of an unrestricted LS estimate. For exogenous variables $\sigma_{i}$ is the sample standard deviation.

max_var

a numeric specifying the maximum prior variance that is allowed for non-deterministic coefficients.

shape

a numeric specifying the prior shape parameter of the error term of the state equation. Only used for models with time varying parameters. Default is 3.

rate

a numeric specifying the prior rate parameter of the error term of the state equation. Only used for models with time varying parameters. Default is 0.0001.

rate_det

a numeric specifying the prior rate parameter of the error term of the state equation for coefficients, which correspond to deterministic terms. Only used for models with time varying parameters. Default is 0.01.

If minnesota is specified, v_i and v_i_det are ignored.

Argument sigma can contain the following elements:

df

an integer or character specifying the prior degrees of freedom of the error term. Only used, if the prior is inverse Wishart. Default is "k", which indicates the amount of endogenous variables in the respective model. "k + 3" can be used to set the prior to the amount of endogenous variables plus 3. If an integer is provided, the degrees of freedom are set to this value in all models.

scale

a numeric specifying the prior error variance of endogenous variables. Default is 1.

shape

a numeric or character specifying the prior shape parameter of the error term. Only used, if the prior is inverse gamma or if time varying volatilities are estimated. For models with constant volatility the default is "k", which indicates the amount of endogenous variables in the respective country model. "k + 3" can be used to set the prior to the amount of endogenous variables plus 3. If a numeric is provided, the shape parameters are set to this value in all models. For models with stochastic volatility this prior refers to the error variance of the state equation.

rate

a numeric specifying the prior rate parameter of the error term. Only used, if the prior is inverse gamma or if time varying volatilities are estimated. For models with stochastic volatility this prior refers to the error variance of the state equation.

mu

numeric of the prior mean of the initial state of the log-volatilities. Only used for models with time varying volatility.

v_i

numeric of the prior precision of the initial state of the log-volatilities. Only used for models with time varying volatility.

sigma_h

numeric of the initial draw for the variance of the log-volatilities. Only used for models with time varying volatility.

constant

numeric of the constant, which is added before taking the log of the squared errors. Only used for models with time varying volatility.

covar

logical indicating whether error covariances should be estimated. Only used in combination with an inverse gamma prior or stochastic volatility, for which shape and rate must be specified.

df and scale must be specified for an inverse Wishart prior. shape and rate are required for an inverse gamma prior. For structural models or models with stochastic volatility only a gamma prior specification is allowed.

Argument ssvs can contain the following elements:

inprior

a numeric between 0 and 1 specifying the prior probability of a variable to be included in the model.

tau

a numeric vector of two elements containing the prior standard errors of restricted variables ($\tau_0$) as its first element and unrestricted variables ($\tau_1$) as its second.

semiautomatic

an numeric vector of two elements containing the factors by which the standard errors associated with an unconstrained least squares estimate of the model are multiplied to obtain the prior standard errors of restricted ($\tau_0$) and unrestricted ($\tau_1$) variables, respectively. This is the semiautomatic approach described in George et al. (2008).

covar

logical indicating if SSVS should also be applied to the error covariance matrix as in George et al. (2008).

exclude_det

logical indicating if deterministic terms should be excluded from the SSVS algorithm.

minnesota

a numeric vector of length 4 containing parameters for the calculation of the Minnesota-like inclusion priors. See below.

Either tau or semiautomatic must be specified.

The argument bvs can contain the following elements

inprior

a numeric between 0 and 1 specifying the prior probability of a variable to be included in the model.

covar

logical indicating if BVS should also be applied to the error covariance matrix.

exclude_det

logical indicating if deterministic terms should be excluded from the BVS algorithm.

minnesota

a numeric vector of length 4 containing parameters for the calculation of the Minnesota-like inclusion priors. See below.

If either ssvs$minnesota or bvs$minnesota is specified, prior inclusion probabilities are calculated in a Minnesota-like fashion as

$\frac{\kappa_1}{l}$	for own lags of endogenous variables,
$\frac{\kappa_2}{l}$	for other endogenous variables,
$\frac{\kappa_3}{1 + l}$	for exogenous variables,
$\kappa_{4}$	for deterministic variables,

for lag $l$ with $\kappa_1$, $\kappa_2$, $\kappa_3$, $\kappa_4$ as the first, second, third and forth element in ssvs$minnesota or bvs$minnesota, respectively.

Value

A list of country models.

References

Chan, J., Koop, G., Poirier, D. J., & Tobias J. L. (2019). Bayesian econometric methods (2nd ed.). Cambridge: Cambridge University Press.

George, E. I., Sun, D., & Ni, S. (2008). Bayesian stochastic search for VAR model restrictions. Journal of Econometrics, 142(1), 553–580. doi:10.1016/j.jeconom.2007.08.017

Korobilis, D. (2013). VAR forecasting using Bayesian variable selection. Journal of Applied Econometrics, 28(2), 204–230. doi:10.1002/jae.1271

Lütkepohl, H. (2006). New introduction to multiple time series analysis (2nd ed.). Berlin: Springer.

Examples

data("e1")
e1 <- diff(log(e1)) * 100

model <- gen_var(e1, p = 2, deterministic = 2,
                 iterations = 100, burnin = 10)

model <- add_priors(model)

---

Add Priors for Vector Error Correction Models

Description

Adds prior specifications to a list of models, which was produced by function gen_vec.

Usage

## S3 method for class 'bvecmodel'
add_priors(
  object,
  coef = list(v_i = 1, v_i_det = 0.1, shape = 3, rate = 1e-04, rate_det = 0.01),
  coint = list(v_i = 0, p_tau_i = 1, shape = 3, rate = 1e-04, rho = 0.999),
  sigma = list(df = "k", scale = 1, mu = 0, v_i = 0.01, sigma_h = 0.05, constant = 1e-04),
  ssvs = NULL,
  bvs = NULL,
  ...
)

Arguments

object	a list, usually, the output of a call to gen_vec.
coef	a named list of prior specifications for coefficients that do not determine the cointegration space. For the default specification all prior means are set to zero and the diagonal elements of the inverse prior variance-covariance matrix are set to 1 for coefficients corresponding to non-deterministic terms. For deterministic coefficients the prior variances are set to 10 by v_i_det = 0.1. The variances need to be specified as precisions, i.e. as inverses of the variances. For further specifications such as the Minnesota prior see 'Details'.
coint	a named list of prior specifications for coefficients determining the cointegration space of VEC models. See 'Details'.
sigma	a named list of prior specifications for the error variance-covariance matrix of the models. For the default specification of an inverse Wishart distribution the prior degrees of freedom are set to the number of endogenous variables plus the rank of the cointegration matrix. The prior variance is to 1. See 'Details'.
ssvs	optional; a named list of prior specifications for the SSVS algorithm. Not allowed for TVP models. See 'Details'.
bvs	optional; a named list of prior specifications for the BVS algorithm. See 'Details'.
...	further arguments passed to or from other methods.

Details

The arguments of the function require named lists. Possible specifications are described in the following. Note that it is important to specify the priors in the correct list. Otherwise, the provided specification will be disregarded and default values will be used.

Argument coef contains the following elements

v_i

a numeric specifying the prior precision of the coefficients. Default is 1.

v_i_det

a numeric specifying the prior precision of coefficients that correspond to deterministic terms. Default is 0.1.

const

a character specifying the prior mean of coefficients, which correspond to the intercept. If const = "mean", the means of the series of endogenous variables are used as prior means. If const = "first", the first values of the series of endogenous variables are used as prior means.

minnesota

a list of length 4 containing parameters for the calculation of the Minnesota prior, where the element names must be kappa0, kappa1, kappa2 and kappa3. For the endogenous variable $i$ the prior variance of the $l$th lag of regressor $j$ is obtained as

$\frac{\kappa_{0}}{l^2} \textrm{ for own lags of endogenous variables,}$

$\frac{\kappa_{0} \kappa_{1}}{l^2} \frac{\sigma_{i}^2}{\sigma_{j}^2} \textrm{ for endogenous variables other than own lags,}$

$\frac{\kappa_{0} \kappa_{2}}{(l+1)^2} \frac{\sigma_{i}^2}{\sigma_{j}^2} \textrm{ for exogenous variables,}$

$\kappa_{0} \kappa_{3} \sigma_{i}^2 \textrm{ for deterministic terms,}$

where $\sigma_{i}$ is the residual standard deviation of variable $i$ of an unrestricted LS estimate. For exogenous variables $\sigma_{i}$ is the sample standard deviation.

The function only provides priors for the non-cointegration part of the model. However, the residual standard errors $\sigma_i$ are based on an unrestricted LS regression of the endogenous variables on the error correction term and the non-cointegration regressors.

max_var

a numeric specifying the maximum prior variance that is allowed for non-deterministic coefficients.

shape

a numeric specifying the prior shape parameter of the error term of the state equation. Only used for models with time varying parameters. Default is 3.

rate

a numeric specifying the prior rate parameter of the error term of the state equation. Only used for models with time varying parameters. Default is 0.0001.

rate_det

a numeric specifying the prior rate parameter of the error term of the state equation for coefficients, which correspond to deterministic terms. Only used for models with time varying parameters. Default is 0.01.

If minnesota is specified, elements v_i and v_i_det are ignored.

Argument coint can contain the following elements:

v_i

numeric between 0 and 1 specifying the shrinkage of the cointegration space prior. Default is 0.

p_tau_i

numeric of the diagonal elements of the inverse prior matrix of the central location of the cointegration space $sp(\beta)$. Default is 1.

shape

an integer specifying the prior degrees of freedom of the error term of the state equation of the loading matrix $\alpha$. Default is 3.

rate

a numeric specifying the prior variance of error term of the state equation of the loading matrix $\alpha$. Default is 0.0001.

rho

a numeric specifying the autocorrelation coefficient of the state equation of $\beta$. It must be smaller than 1. Default is 0.999. Note that in contrast to Koop et al. (2011) $\rho$ is not drawn in the Gibbs sampler of this package yet.

Argument sigma can contain the following elements:

df

an integer or character specifying the prior degrees of freedom of the error term. Only used, if the prior is inverse Wishart. Default is "k", which indicates the amount of endogenous variables in the respective model. "k + 3" can be used to set the prior to the amount of endogenous variables plus 3. If an integer is provided, the degrees of freedom are set to this value in all models. In all cases the rank $r$ of the cointegration matrix is automatically added.

scale

a numeric specifying the prior error variance of endogenous variables. Default is 1.

shape

a numeric or character specifying the prior shape parameter of the error term. Only used, if the prior is inverse gamma or if time varying volatilities are estimated. For models with constant volatility the default is "k", which indicates the amount of endogenous variables in the respective country model. "k + 3" can be used to set the prior to the amount of endogenous variables plus 3. If a numeric is provided, the shape parameters are set to this value in all models. For models with stochastic volatility this prior refers to the error variance of the state equation.

rate

a numeric specifying the prior rate parameter of the error term. Only used, if the prior is inverse gamma or if time varying volatilities are estimated. For models with stochastic volatility this prior refers to the error variance of the state equation.

mu

numeric of the prior mean of the initial state of the log-volatilities. Only used for models with time varying volatility.

v_i

numeric of the prior precision of the initial state of the log-volatilities. Only used for models with time varying volatility.

sigma_h

numeric of the initial draw for the variance of the log-volatilities. Only used for models with time varying volatility.

constant

numeric of the constant, which is added before taking the log of the squared errors. Only used for models with time varying volatility.

covar

logical indicating whether error covariances should be estimated. Only used in combination with an inverse gamma prior or stochastic volatility, for which shape and rate must be specified.

df and scale must be specified for an inverse Wishart prior. shape and rate are required for an inverse gamma prior. For structural models or models with stochastic volatility only a gamma prior specification is allowed.

Argument ssvs can contain the following elements:

inprior

a numeric between 0 and 1 specifying the prior probability of a variable to be included in the model.

tau

a numeric vector of two elements containing the prior standard errors of restricted variables ($\tau_0$) as its first element and unrestricted variables ($\tau_1$) as its second.

semiautomatic

an numeric vector of two elements containing the factors by which the standard errors associated with an unconstrained least squares estimate of the model are multiplied to obtain the prior standard errors of restricted ($\tau_0$) and unrestricted ($\tau_1$) variables, respectively. This is the semiautomatic approach described in George et al. (2008).

covar

logical indicating if SSVS should also be applied to the error covariance matrix as in George et al. (2008).

exclude_det

logical indicating if deterministic terms should be excluded from the SSVS algorithm.

minnesota

a numeric vector of length 4 containing parameters for the calculation of the Minnesota-like inclusion priors. See below.

Either tau or semiautomatic must be specified.

The argument bvs can contain the following elements

inprior

a numeric between 0 and 1 specifying the prior probability of a variable to be included in the model.

covar

logical indicating if BVS should also be applied to the error covariance matrix.

exclude_det

logical indicating if deterministic terms should be excluded from the BVS algorithm.

minnesota

a numeric vector of length 4 containing parameters for the calculation of the Minnesota-like inclusion priors. See below.

If either ssvs$minnesota or bvs$minnesota is specified, prior inclusion probabilities are calculated in a Minnesota-like fashion as

$\frac{\kappa_1}{l}$	for own lags of endogenous variables,
$\frac{\kappa_2}{l}$	for other endogenous variables,
$\frac{\kappa_3}{1 + l}$	for exogenous variables,
$\kappa_{4}$	for deterministic variables,

for lag $l$ with $\kappa_1$, $\kappa_2$, $\kappa_3$, $\kappa_4$ as the first, second, third and forth element in ssvs$minnesota or bvs$minnesota, respectively.

Value

A list of models.

References

Chan, J., Koop, G., Poirier, D. J., & Tobias J. L. (2019). Bayesian econometric methods (2nd ed.). Cambridge: Cambridge University Press.

George, E. I., Sun, D., & Ni, S. (2008). Bayesian stochastic search for VAR model restrictions. Journal of Econometrics, 142(1), 553–580. doi:10.1016/j.jeconom.2007.08.017

Koop, G., León-González, R., & Strachan R. W. (2010). Efficient posterior simulation for cointegrated models with priors on the cointegration space. Econometric Reviews, 29(2), 224–242. doi:10.1080/07474930903382208

Koop, G., León-González, R., & Strachan R. W. (2011). Bayesian inference in a time varying cointegration model. Journal of Econometrics, 165(2), 210–220. doi:10.1016/j.jeconom.2011.07.007

Korobilis, D. (2013). VAR forecasting using Bayesian variable selection. Journal of Applied Econometrics, 28(2), 204–230. doi:10.1002/jae.1271

Lütkepohl, H. (2006). New introduction to multiple time series analysis (2nd ed.). Berlin: Springer.

Examples

# Get data
data("e6")

# Create model
model <- gen_vec(e6, p = 4, r = 1,
                 const = "unrestricted", seasonal = "unrestricted",
                 iterations = 100, burnin = 10)
# Chosen number of iterations and burnin should be much higher.

# Add priors
model <- add_priors(model)

---

Add Priors to Dynamic Factor Model

Description

Adds prior specifications to a list of models, which was produced by function gen_dfm.

Usage

## S3 method for class 'dfmodel'
add_priors(
  object,
  lambda = list(v_i = 0.01),
  sigma_u = list(shape = 5, rate = 4),
  a = list(v_i = 0.01),
  sigma_v = list(shape = 5, rate = 4),
  ...
)

Arguments

object	a list, usually, the output of a call to gen_dfm.
lambda	a named list of prior specifications for the factor loadings in the measurement equation. For the default specification the diagonal elements of the inverse prior variance-covariance matrix are set to 0.01. The variances need to be specified as precisions, i.e. as inverses of the variances.
sigma_u	a named list of prior specifications for the error variance-covariance matrix. See 'Details'.
a	a named list of prior specifications for the coefficients of the transition equation. For the default specification the diagonal elements of the inverse prior variance-covariance matrix are set to 0.01. The variances need to be specified as precisions, i.e. as inverses of the variances.
sigma_v	a named list of prior specifications for the error variance-covariance matrix. See 'Details'.
...	further arguments passed to or from other methods.

Details

Argument lambda can only contain the element v_i, which is a numeric specifying the prior precision of the loading factors of the measurement equation. Default is 0.01.

The function assumes an inverse gamma prior for the errors of the measurement equation. Argument sigma_u can contain the following elements:

shape

a numeric or character specifying the prior shape parameter of the error terms of the measurement equation. Default is 5.

rate

a numeric specifying the prior rate parameter of the error terms of the measurement equation. Default is 4.

Argument a can only contain the element v_i, which is a numeric specifying the prior precision of the coefficients of the transition equation. Default is 0.01.

The function assumes an inverse gamma prior for the errors of the transition equation. Argument sigma_v can contain the following elements:

shape

a numeric or character specifying the prior shape parameter of the error terms of the transition equation. Default is 5.

rate

a numeric specifying the prior rate parameter of the error terms of the transition equation. Default is 4.

Value

A list of models.

References

Chan, J., Koop, G., Poirier, D. J., & Tobias J. L. (2019). Bayesian econometric methods (2nd ed.). Cambridge: Cambridge University Press.

Lütkepohl, H. (2007). New introduction to multiple time series analysis (2nd ed.). Berlin: Springer.

Examples

# Load data
data("bem_dfmdata")

# Generate model data
model <- gen_dfm(x = bem_dfmdata, p = 1, n = 1,
                 iterations = 5000, burnin = 1000)
# Number of iterations and burnin should be much higher.

# Add prior specifications
model <- add_priors(model,
                    lambda = list(v_i = .01),
                    sigma_u = list(shape = 5, rate = 4),
                    a = list(v_i = .01),
                    sigma_v = list(shape = 5, rate = 4))

---

FRED-QD data

Description

The data set contains quarterly time series for 196 US macroeconomic variables from 1959Q3 to 2015Q3. It was produced from file "freddata_Q.csv" of the data sets associated with Chan, Koop, Poirier and Tobias (2019). Raw data are available at https://web.ics.purdue.edu/~jltobias/second_edition/Chapter18/code_for_exercise_4/freddata_Q.csv.

Usage

data("bem_dfmdata")

Format

A named time-series object with 225 rows and 196 variables. A detailed explanation of the variables can be found in McCracken and Ng (2016).

References

Chan, J., Koop, G., Poirier, D. J., & Tobias J. L. (2019). Bayesian econometric methods (2nd ed.). Cambridge: Cambridge University Press.

McCracken, M. W., & Ng, S. (2016). FRED-MD: A monthly database for macroeconomic research. Journal of Business & Economic Statistics 34(4), 574-589. doi:10.1080/07350015.2015.1086655

---

Bayesian Vector Autoregression Objects

Description

bvar is used to create objects of class "bvar".

A plot function for objects of class "bvar".

Forecasting a Bayesian VAR object of class "bvar" with credible bands.

Usage

bvar(
  data = NULL,
  exogen = NULL,
  y,
  x = NULL,
  A0 = NULL,
  A = NULL,
  B = NULL,
  C = NULL,
  Sigma = NULL
)

## S3 method for class 'bvar'
plot(x, ci = 0.95, type = "hist", ...)

## S3 method for class 'bvar'
predict(object, ..., n.ahead = 10, new_x = NULL, new_d = NULL, ci = 0.95)

Arguments

data	the original time-series object of endogenous variables.
exogen	the original time-series object of unmodelled variables.
y	a time-series object of endogenous variables with $T$ observations, usually, a result of a call to gen_var.
x	an object of class "bvar", usually, a result of a call to draw_posterior.
A0	either a $K^2 \times S$ matrix of MCMC coefficient draws of structural parameters or a named list, where element coeffs contains a $K^2 \times S$ matrix of MCMC coefficient draws of structural parameters and element lambda contains the corresponding draws of inclusion parameters in case variable selection algorithms were employed. For time varying parameter models the coefficient matrix must be $TK^2 \times S$. Draws of the error covariance matrix of the state equation can be provided as a $K^2 \times S$ matrix in an additional list element.
A	either a $pK^2 \times S$ matrix of MCMC coefficient draws of lagged endogenous variables or a named list, where element coeffs contains a $pK^2 \times S$ matrix of MCMC coefficient draws of lagged endogenous variables and element lambda contains the corresponding draws of inclusion parameters in case variable selection algorithms were employed. For time varying parameter models the coefficient matrix must be $pTK^2 \times S$. Draws of the error covariance matrix of the state equation can be provided as a $pK^2 \times S$ matrix in an additional list element.
B	either a $((1 + s)MK) \times S$ matrix of MCMC coefficient draws of unmodelled, non-deterministic variables or a named list, where element coeffs contains a $((1 + s)MK) \times S$ matrix of MCMC coefficient draws of unmodelled, non-deterministic variables and element lambda contains the corresponding draws of inclusion parameters in case variable selection algorithms were employed. For time varying parameter models the coefficient matrix must be $(1 + s)TMK \times S$. Draws of the error covariance matrix of the state equation can be provided as a $(1 + s)MK \times S$ matrix in an additional list element.
C	either a $KN \times S$ matrix of MCMC coefficient draws of deterministic terms or a named list, where element coeffs contains a $KN \times S$ matrix of MCMC coefficient draws of deterministic terms and element lambda contains the corresponding draws of inclusion parameters in case variable selection algorithms were employed. For time varying parameter models the coefficient matrix must be $TKN \times S$. Draws of the error covariance matrix of the state equation can be provided as a $KN \times S$ matrix in an additional list element.
Sigma	a $K^2 \times S$ matrix of MCMC draws for the error variance-covariance matrix or a named list, where element coeffs contains a $K^2 \times S$ matrix of MCMC draws for the error variance-covariance matrix and element lambda contains the corresponding draws of inclusion parameters in case variable selection algorithms were employed to the covariances. For time varying parameter models the coefficient matrix must be $TK^2 \times S$. Draws of the error covariance matrix of the state equation can be provided as a $K^2 \times S$ matrix in an additional list element.
ci	a numeric between 0 and 1 specifying the probability mass covered by the credible intervals. Defaults to 0.95.
type	either "hist" (default) for histograms, "trace" for a trace plot or "boxplot" for a boxplot. Only used for parameter draws of constant coefficients.
...	additional arguments.
object	an object of class "bvar", usually, a result of a call to bvar or bvec_to_bvar.
n.ahead	number of steps ahead at which to predict.
new_x	an object of class ts of new non-deterministic, exogenous variables. The object must have the same frequency as the time series in object[["x"]] and must contain at least all necessary observations for the predicted period.
new_d	a matrix of new deterministic variables. Must have n.ahead rows.

Details

For the VARX model

$A_0 y_t = \sum_{i = 1}^{p} A_i y_{t-i} + \sum_{i = 0}^{s} B_i x_{t - i} + C d_t + u_t$

the function collects the S draws of a Gibbs sampler (after the burn-in phase) in a standardised object, where $y_t$ is a K-dimensional vector of endogenous variables, $A_0$ is a $K \times K$ matrix of structural coefficients. $A_i$ is a $K \times K$ coefficient matrix of lagged endogenous variabels. $x_t$ is an M-dimensional vector of unmodelled, non-deterministic variables and $B_i$ its corresponding coefficient matrix. $d_t$ is an N-dimensional vector of deterministic terms and $C$ its corresponding coefficient matrix. $u_t$ is an error term with $u_t \sim N(0, \Sigma_u)$.

For time varying parameter and stochastic volatility models the respective coefficients and error covariance matrix of the above model are assumed to be time varying, respectively.

The draws of the different coefficient matrices provided in A0, A, B, C and Sigma have to correspond to the same MCMC iterations.

For the VAR model

$A_0 y_t = \sum_{i = 1}^{p} A_{i} y_{t-i} + \sum_{i = 0}^{s} B_{i} x_{t-i} + C D_t + u_t,$

with $u_t \sim N(0, \Sigma)$ the function produces n.ahead forecasts.

Value

An object of class "bvar" containing the following components, if specified:

data	the original time-series object of endogenous variables.
exogen	the original time-series object of unmodelled variables.
y	a $K \times T$ matrix of endogenous variables.
x	a $(pK + (1+s)M + N) \times T$ matrix of regressor variables.
A0	an $S \times K^2$ "mcmc" object of coefficient draws of structural parameters. In case of time varying parameters a list of such objects.
A0_lambda	an $S \times K^2$ "mcmc" object of inclusion parameters for structural parameters.
A0_sigma	an $S \times K^2$ "mcmc" object of the error covariance matrices of the structural parameters in a model with time varying parameters.
A	an $S \times pK^2$ "mcmc" object of coefficient draws of lagged endogenous variables. In case of time varying parameters a list of such objects.
A_lambda	an $S \times pK^2$ "mcmc" object of inclusion parameters for lagged endogenous variables.
A_sigma	an $S \times pK^2$ "mcmc" object of the error covariance matrices of coefficients of lagged endogenous variables in a model with time varying parameters.
B	an $S \times ((1 + s)MK)$ "mcmc" object of coefficient draws of unmodelled, non-deterministic variables. In case of time varying parameters a list of such objects.
B_lambda	an $S \times ((1 + s)MK)$ "mcmc" object of inclusion parameters for unmodelled, non-deterministic variables.
B_sigma	an $S \times ((1 + s)MK)$ "mcmc" object of the error covariance matrices of coefficients of unmodelled, non-deterministic variables in a model with time varying parameters.
C	an $S \times NK$ "mcmc" object of coefficient draws of deterministic terms. In case of time varying parameters a list of such objects.
C_lambda	an $S \times NK$ "mcmc" object of inclusion parameters for deterministic terms.
C_sigma	an $S \times NK$ "mcmc" object of the error covariance matrices of coefficients of deterministic terms in a model with time varying parameters.
Sigma	an $S \times K^2$ "mcmc" object of variance-covariance draws. In case of time varying parameters a list of such objects.
Sigma_lambda	an $S \times K^2$ "mcmc" object of inclusion parameters for error covariances.
Sigma_sigma	an $S \times K^2$ "mcmc" object of the error covariance matrices of the coefficients of the error covariance matrix of the measurement equation of a model with time varying parameters.
specifications	a list containing information on the model specification.

A time-series object of class "bvarprd".

References

Lütkepohl, H. (2006). New introduction to multiple time series analysis (2nd ed.). Berlin: Springer.

Examples

# Get data
data("e1")
e1 <- diff(log(e1))
e1 <- window(e1, end = c(1978, 4))

# Generate model data
data <- gen_var(e1, p = 2, deterministic = "const")

# Add priors
model <- add_priors(data,
                    coef = list(v_i = 0, v_i_det = 0),
                    sigma = list(df = 0, scale = .00001))

# Set RNG seed for reproducibility 
set.seed(1234567)

iterations <- 400 # Number of iterations of the Gibbs sampler
# Chosen number of iterations and burnin should be much higher.
burnin <- 100 # Number of burn-in draws
draws <- iterations + burnin # Total number of MCMC draws

y <- t(model$data$Y)
x <- t(model$data$Z)
tt <- ncol(y) # Number of observations
k <- nrow(y) # Number of endogenous variables
m <- k * nrow(x) # Number of estimated coefficients

# Priors
a_mu_prior <- model$priors$coefficients$mu # Vector of prior parameter means
a_v_i_prior <- model$priors$coefficients$v_i # Inverse of the prior covariance matrix

u_sigma_df_prior <- model$priors$sigma$df # Prior degrees of freedom
u_sigma_scale_prior <- model$priors$sigma$scale # Prior covariance matrix
u_sigma_df_post <- tt + u_sigma_df_prior # Posterior degrees of freedom

# Initial values
u_sigma_i <- diag(1 / .00001, k)

# Data containers for posterior draws
draws_a <- matrix(NA, m, iterations)
draws_sigma <- matrix(NA, k^2, iterations)

# Start Gibbs sampler
for (draw in 1:draws) {
 # Draw conditional mean parameters
 a <- post_normal(y, x, u_sigma_i, a_mu_prior, a_v_i_prior)

 # Draw variance-covariance matrix
 u <- y - matrix(a, k) %*% x # Obtain residuals
 u_sigma_scale_post <- solve(u_sigma_scale_prior + tcrossprod(u))
 u_sigma_i <- matrix(rWishart(1, u_sigma_df_post, u_sigma_scale_post)[,, 1], k)

 # Store draws
 if (draw > burnin) {
  draws_a[, draw - burnin] <- a
  draws_sigma[, draw - burnin] <- solve(u_sigma_i)
 }
}

# Generate bvar object
bvar_est <- bvar(y = model$data$Y, x = model$data$Z,
                 A = draws_a[1:18,], C = draws_a[19:21, ],
                 Sigma = draws_sigma)
                 

# Load data 
data("e1")
e1 <- diff(log(e1)) * 100

# Generate model
model <- gen_var(e1, p = 1, deterministic = 2,
                 iterations = 100, burnin = 10)
# Chosen number of iterations and burn-in should be much higher.

# Add priors
model <- add_priors(model)

# Obtain posterior draws
object <- draw_posterior(model)

# Plot draws
plot(object)


# Load data
data("e1")
e1 <- diff(log(e1)) * 100
e1 <- window(e1, end = c(1978, 4))

# Generate model data
model <- gen_var(e1, p = 0, deterministic = "const",
                 iterations = 100, burnin = 10)
# Chosen number of iterations and burnin should be much higher.

# Add prior specifications
model <- add_priors(model)

# Obtain posterior draws
object <- draw_posterior(model)

# Generate forecasts
bvar_pred <- predict(object, n.ahead = 10, new_d = rep(1, 10))

# Plot forecasts
plot(bvar_pred)

---

Posterior Simulation for BVAR Models

Description

Produces draws from the posterior distributions of Bayesian VAR models.

Usage

bvarpost(object)

Arguments

object

an object of class "bvarmodel", usually, a result of a call to gen_var in combination with add_priors.

Details

The function implements commonly used posterior simulation algorithms for Bayesian VAR models with both constant and time varying parameters (TVP) as well as stochastic volatility. It can produce posterior draws for standard BVAR models with independent normal-Wishart priors, which can be augmented by stochastic search variable selection (SSVS) as proposed by Geroge et al. (2008) or Bayesian variable selection (BVS) as proposed in Korobilis (2013). Both SSVS or BVS can also be applied to the covariances of the error term.

The implementation follows the descriptions in Chan et al. (2019), George et al. (2008) and Korobilis (2013). For all approaches the SUR form of a VAR model is used to obtain posterior draws. The algorithm is implemented in C++ to reduce calculation time.

The function also supports structural BVAR models, where the structural coefficients are estimated from contemporary endogenous variables, which corresponds to the so-called (A-model). Currently, only specifications are supported, where the structural matrix contains ones on its diagonal and all lower triangular elements are freely estimated. Since posterior draws are obtained based on the SUR form of the VAR model, the structural coefficients are drawn jointly with the other coefficients.

Value

An object of class "bvar".

References

Chan, J., Koop, G., Poirier, D. J., & Tobias J. L. (2019). Bayesian econometric methods (2nd ed.). Cambridge: Cambridge University Press.

George, E. I., Sun, D., & Ni, S. (2008). Bayesian stochastic search for VAR model restrictions. Journal of Econometrics, 142(1), 553–580. doi:10.1016/j.jeconom.2007.08.017

Korobilis, D. (2013). VAR forecasting using Bayesian variable selection. Journal of Applied Econometrics, 28(2), 204–230. doi:10.1002/jae.1271

Examples

# Get data
data("e1")
e1 <- diff(log(e1)) * 100

# Create model
model <- gen_var(e1, p = 2, deterministic = "const",
                 iterations = 50, burnin = 10)
# Number of iterations and burnin should be much higher.

# Add priors
model <- add_priors(model)

# Obtain posterior draws 
object <- bvarpost(model)

---

Bayesian Vector Error Correction Objects

Description

'bvec' is used to create objects of class "bvec".

A plot function for objects of class "bvec".

Usage

bvec(
  y,
  alpha = NULL,
  beta = NULL,
  beta_x = NULL,
  beta_d = NULL,
  r = NULL,
  Pi = NULL,
  Pi_x = NULL,
  Pi_d = NULL,
  w = NULL,
  w_x = NULL,
  w_d = NULL,
  Gamma = NULL,
  Upsilon = NULL,
  C = NULL,
  x = NULL,
  x_x = NULL,
  x_d = NULL,
  A0 = NULL,
  Sigma = NULL,
  data = NULL,
  exogen = NULL
)

## S3 method for class 'bvec'
plot(x, ci = 0.95, type = "hist", ...)

Arguments

y	a time-series object of differenced endogenous variables, usually, a result of a call to gen_vec.
alpha	a $Kr \times S$ matrix of MCMC coefficient draws of the loading matrix $\alpha$.
beta	a $Kr \times S$ matrix of MCMC coefficient draws of cointegration matrix $\beta$ corresponding to the endogenous variables of the model.
beta_x	a $Mr \times S$ matrix of MCMC coefficient draws of cointegration matrix $\beta$ corresponding to unmodelled, non-deterministic variables.
beta_d	a $N^{R}r \times S$ matrix of MCMC coefficient draws of cointegration matrix $\beta$ corresponding to restricted deterministic terms.
r	an integer of the rank of the cointegration matrix.
Pi	a $K^2 \times S$ matrix of MCMC coefficient draws of endogenous varaibles in the cointegration matrix.
Pi_x	a $KM \times S$ matrix of MCMC coefficient draws of unmodelled, non-deterministic variables in the cointegration matrix.
Pi_d	a $KN^{R} \times S$ matrix of MCMC coefficient draws of restricted deterministic terms.
w	a time-series object of lagged endogenous variables in levels, which enter the cointegration term, usually, a result of a call to gen_vec.
w_x	a time-series object of lagged unmodelled, non-deterministic variables in levels, which enter the cointegration term, usually, a result of a call to gen_vec.
w_d	a time-series object of deterministic terms, which enter the cointegration term, usually, a result of a call to gen_vec.
Gamma	a $(p-1)K^2 \times S$ matrix of MCMC coefficient draws of differenced lagged endogenous variables or a named list, where element coeffs contains a $(p - 1)K^2 \times S$ matrix of MCMC coefficient draws of lagged differenced endogenous variables and element lambda contains the corresponding draws of inclusion parameters in case variable selection algorithms were employed.
Upsilon	an $sMK \times S$ matrix of MCMC coefficient draws of differenced unmodelled, non-deterministic variables or a named list, where element coeffs contains a $sMK \times S$ matrix of MCMC coefficient draws of unmodelled, non-deterministic variables and element lambda contains the corresponding draws of inclusion parameters in case variable selection algorithms were employed.
C	an $KN^{UR} \times S$ matrix of MCMC coefficient draws of unrestricted deterministic terms or a named list, where element coeffs contains a $KN^{UR} \times S$ matrix of MCMC coefficient draws of deterministic terms and element lambda contains the corresponding draws of inclusion parameters in case variable selection algorithms were employed.
x	an object of class "bvec", usually, a result of a call to draw_posterior.
x_x	a time-series object of $Ms$ differenced unmodelled regressors.
x_d	a time-series object of $N^{UR}$ deterministic terms that do not enter the cointegration term.
A0	either a $K^2 \times S$ matrix of MCMC coefficient draws of structural parameters or a named list, where element coeffs contains a $K^2 \times S$ matrix of MCMC coefficient draws of structural parameters and element lambda contains the corresponding draws of inclusion parameters in case variable selection algorithms were employed.
Sigma	a $K^2 \times S$ matrix of MCMC draws for the error variance-covariance matrix or a named list, where element coeffs contains a $K^2 \times S$ matrix of MCMC draws for the error variance-covariance matrix and element lambda contains the corresponding draws of inclusion parameters in case variable selection algorithms were employed to the covariances.
data	the original time-series object of endogenous variables.
exogen	the original time-series object of unmodelled variables.
ci	interval used to calculate credible bands for time-varying parameters.
type	either "hist" (default) for histograms, "trace" for a trace plot or "boxplot" for a boxplot. Only used for parameter draws of constant coefficients.
...	further graphical parameters.

Details

For the vector error correction model with unmodelled exogenous variables (VECX)

$A_0 \Delta y_t = \Pi^{+} \begin{pmatrix} y_{t-1} \\ x_{t-1} \\ d^{R}_{t-1} \end{pmatrix} + \sum_{i = 1}^{p-1} \Gamma_i \Delta y_{t-i} + \sum_{i = 0}^{s-1} \Upsilon_i \Delta x_{t-i} + C^{UR} d^{UR}_t + u_t$

the function collects the $S$ draws of a Gibbs sampler in a standardised object, where $\Delta y_t$ is a K-dimensional vector of differenced endogenous variables and $A_0$ is a $K \times K$ matrix of structural coefficients. $\Pi^{+} = \left[ \Pi, \Pi^{x}, \Pi^{d} \right]$ is the coefficient matrix of the error correction term, where $y_{t-1}$, $x_{t-1}$ and $d^{R}_{t-1}$ are the first lags of endogenous, exogenous variables in levels and restricted deterministic terms, respectively. $\Pi$, $\Pi^{x}$, and $\Pi^{d}$ are the corresponding coefficient matrices, respectively. $\Gamma_i$ is a coefficient matrix of lagged differenced endogenous variabels. $\Delta x_t$ is an M-dimensional vector of unmodelled, non-deterministic variables and $\Upsilon_i$ its corresponding coefficient matrix. $d_t$ is an $N^{UR}$-dimensional vector of unrestricted deterministics and $C^{UR}$ the corresponding coefficient matrix. $u_t$ is an error term with $u_t \sim N(0, \Sigma_u)$.

For time varying parameter and stochastic volatility models the respective coefficients and error covariance matrix of the above model are assumed to be time varying, respectively.

The draws of the different coefficient matrices provided in alpha, beta, Pi, Pi_x, Pi_d, A0, Gamma, Ypsilon, C and Sigma have to correspond to the same MCMC iteration.

Value

An object of class "gvec" containing the following components, if specified:

data	the original time-series object of endogenous variables.
exogen	the original time-series object of unmodelled variables.
y	a time-series object of differenced endogenous variables.
w	a time-series object of lagged endogenous variables in levels, which enter the cointegration term.
w_x	a time-series object of lagged unmodelled, non-deterministic variables in levels, which enter the cointegration term.
w_d	a time-series object of deterministic terms, which enter the cointegration term.
x	a time-series object of $K(p - 1)$ differenced endogenous variables
x_x	a time-series object of $Ms$ differenced unmodelled regressors.
x_d	a time-series object of $N^{UR}$ deterministic terms that do not enter the cointegration term.
A0	an $S \times K^2$ "mcmc" object of coefficient draws of structural parameters. In case of time varying parameters a list of such objects.
A0_lambda	an $S \times K^2$ "mcmc" object of inclusion parameters for coefficients corresponding to structural parameters.
A0_sigma	an $S \times K^2$ "mcmc" object of the error covariance matrices of the structural parameters in a model with time varying parameters.
alpha	an $S \times Kr$ "mcmc" object of coefficient draws of loading parameters. In case of time varying parameters a list of such objects.
beta	an $S \times ((K + M + N^{R})r)$ "mcmc" object of coefficient draws of cointegration parameters corresponding to the endogenous variables of the model. In case of time varying parameters a list of such objects.
beta_x	an $S \times KM$ "mcmc" object of coefficient draws of cointegration parameters corresponding to unmodelled, non-deterministic variables. In case of time varying parameters a list of such objects.
beta_d	an $S \times KN^{R}$ "mcmc" object of coefficient draws of cointegration parameters corresponding to restricted deterministic variables. In case of time varying parameters a list of such objects.
Pi	an $S \times K^2$ "mcmc" object of coefficient draws of endogenous variables in the cointegration matrix. In case of time varying parameters a list of such objects.
Pi_x	an $S \times KM$ "mcmc" object of coefficient draws of unmodelled, non-deterministic variables in the cointegration matrix. In case of time varying parameters a list of such objects.
Pi_d	an $S \times KN^{R}$ "mcmc" object of coefficient draws of restricted deterministic variables in the cointegration matrix. In case of time varying parameters a list of such objects.
Gamma	an $S \times (p-1)K^2$ "mcmc" object of coefficient draws of differenced lagged endogenous variables. In case of time varying parameters a list of such objects.
Gamma_lamba	an $S \times (p-1)K^2$ "mcmc" object of inclusion parameters for coefficients corresponding to differenced lagged endogenous variables.
Gamma_sigma	an $S \times (p - 1)K^2$ "mcmc" object of the error covariance matrices of the coefficients of lagged endogenous variables in a model with time varying parameters.
Upsilon	an $S \times sMK$ "mcmc" object of coefficient draws of differenced unmodelled, non-deterministic variables. In case of time varying parameters a list of such objects.
Upsilon_lambda	an $S \times sMK$ "mcmc" object of inclusion parameters for coefficients corresponding to differenced unmodelled, non-deterministic variables.
Upsilon_sigma	an $S \times sMK$ "mcmc" object of the error covariance matrices of the coefficients of unmodelled, non-deterministic variables in a model with time varying parameters.
C	an $S \times KN^{UR}$ "mcmc" object of coefficient draws of deterministic terms that do not enter the cointegration term. In case of time varying parameters a list of such objects.
C_lambda	an $S \times KN^{UR}$ "mcmc" object of inclusion parameters for coefficients corresponding to deterministic terms, that do not enter the conintegration term.
C_sigma	an $S \times KN^{UR}$ "mcmc" object of the error covariance matrices of the coefficients of deterministic terms, which do not enter the cointegration term, in a model with time varying parameters.
Sigma	an $S \times K^2$ "mcmc" object of variance-covariance draws. In case of time varying parameters a list of such objects.
Sigma_lambda	an $S \times K^2$ "mcmc" object inclusion parameters for the variance-covariance matrix.
Sigma_sigma	an $S \times K^2$ "mcmc" object of the error covariance matrices of the coefficients of the error covariance matrix of the measurement equation of a model with time varying parameters.
specifications	a list containing information on the model specification.

Examples

# Load data
data("e6")
# Generate model
data <- gen_vec(e6, p = 4, r = 1, const = "unrestricted", season = "unrestricted")
# Obtain data matrices
y <- t(data$data$Y)
w <- t(data$data$W)
x <- t(data$data$X)

# Reset random number generator for reproducibility
set.seed(1234567)

iterations <- 400 # Number of iterations of the Gibbs sampler
# Chosen number of iterations should be much higher, e.g. 30000.

burnin <- 100 # Number of burn-in draws
draws <- iterations + burnin

r <- 1 # Set rank

tt <- ncol(y) # Number of observations
k <- nrow(y) # Number of endogenous variables
k_w <- nrow(w) # Number of regressors in error correction term
k_x <- nrow(x) # Number of differenced regressors and unrestrictec deterministic terms

k_alpha <- k * r # Number of elements in alpha
k_beta <- k_w * r # Number of elements in beta
k_gamma <- k * k_x

# Set uninformative priors
a_mu_prior <- matrix(0, k_x * k) # Vector of prior parameter means
a_v_i_prior <- diag(0, k_x * k) # Inverse of the prior covariance matrix

v_i <- 0
p_tau_i <- diag(1, k_w)

u_sigma_df_prior <- r # Prior degrees of freedom
u_sigma_scale_prior <- diag(0, k) # Prior covariance matrix
u_sigma_df_post <- tt + u_sigma_df_prior # Posterior degrees of freedom

# Initial values
beta <- matrix(c(1, -4), k_w, r)
u_sigma_i <- diag(1 / .0001, k)
g_i <- u_sigma_i

# Data containers
draws_alpha <- matrix(NA, k_alpha, iterations)
draws_beta <- matrix(NA, k_beta, iterations)
draws_pi <- matrix(NA, k * k_w, iterations)
draws_gamma <- matrix(NA, k_gamma, iterations)
draws_sigma <- matrix(NA, k^2, iterations)

# Start Gibbs sampler
for (draw in 1:draws) {
  # Draw conditional mean parameters
  temp <- post_coint_kls(y = y, beta = beta, w = w, x = x, sigma_i = u_sigma_i,
                         v_i = v_i, p_tau_i = p_tau_i, g_i = g_i,
                         gamma_mu_prior = a_mu_prior,
                         gamma_v_i_prior = a_v_i_prior)
  alpha <- temp$alpha
  beta <- temp$beta
  Pi <- temp$Pi
  gamma <- temp$Gamma
  
  # Draw variance-covariance matrix
  u <- y - Pi %*% w - matrix(gamma, k) %*% x
  u_sigma_scale_post <- solve(tcrossprod(u) +
     v_i * alpha %*% tcrossprod(crossprod(beta, p_tau_i) %*% beta, alpha))
  u_sigma_i <- matrix(rWishart(1, u_sigma_df_post, u_sigma_scale_post)[,, 1], k)
  u_sigma <- solve(u_sigma_i)
  
  # Update g_i
  g_i <- u_sigma_i
  
  # Store draws
  if (draw > burnin) {
    draws_alpha[, draw - burnin] <- alpha
    draws_beta[, draw - burnin] <- beta
    draws_pi[, draw - burnin] <- Pi
    draws_gamma[, draw - burnin] <- gamma
    draws_sigma[, draw - burnin] <- u_sigma
  }
}

# Number of non-deterministic coefficients
k_nondet <- (k_x - 4) * k

# Generate bvec object
bvec_est <- bvec(y = data$data$Y, w = data$data$W,
                 x = data$data$X[, 1:6],
                 x_d = data$data$X[, 7:10],
                 Pi = draws_pi,
                 Gamma = draws_gamma[1:k_nondet,],
                 C = draws_gamma[(k_nondet + 1):nrow(draws_gamma),],
                 Sigma = draws_sigma)


# Load data 
data("e6")

# Generate model
model <- gen_vec(data = e6, p = 2, r = 1, const = "unrestricted",
                 iterations = 20, burnin = 10)
# Chosen number of iterations and burn-in should be much higher.

# Add priors
model <- add_priors(model)

# Obtain posterior draws
object <- draw_posterior(model)

# Plot draws
plot(object)

---

Transform a VEC Model to a VAR in Levels

Description

An object of class "bvec" is transformed to a VAR in level representation.

Usage

bvec_to_bvar(object)

Arguments

object

an object of class "bvec".

Value

An object of class "bvar".

References

Lütkepohl, H. (2006). New introduction to multiple time series analysis (2nd ed.). Berlin: Springer.

Examples

# Load data
data("e6")

# Generate model
data <- gen_vec(e6, p = 4, r = 1, const = "unrestricted", season = "unrestricted")

# Obtain data matrices
y <- t(data$data$Y)
w <- t(data$data$W)
x <- t(data$data$X)

# Reset random number generator for reproducibility
set.seed(1234567)

iterations <- 100 # Number of iterations of the Gibbs sampler
# Chosen number of iterations should be much higher, e.g. 30000.

burnin <- 100 # Number of burn-in draws
draws <- iterations + burnin

r <- 1 # Set rank

tt <- ncol(y) # Number of observations
k <- nrow(y) # Number of endogenous variables
k_w <- nrow(w) # Number of regressors in error correction term
k_x <- nrow(x) # Number of differenced regressors and unrestrictec deterministic terms

k_alpha <- k * r # Number of elements in alpha
k_beta <- k_w * r # Number of elements in beta
k_gamma <- k * k_x

# Set uninformative priors
a_mu_prior <- matrix(0, k_x * k) # Vector of prior parameter means
a_v_i_prior <- diag(0, k_x * k) # Inverse of the prior covariance matrix

v_i <- 0
p_tau_i <- diag(1, k_w)

u_sigma_df_prior <- r # Prior degrees of freedom
u_sigma_scale_prior <- diag(0, k) # Prior covariance matrix
u_sigma_df_post <- tt + u_sigma_df_prior # Posterior degrees of freedom

# Initial values
beta <- matrix(c(1, -4), k_w, r)
u_sigma_i <- diag(1 / .0001, k)
g_i <- u_sigma_i

# Data containers
draws_alpha <- matrix(NA, k_alpha, iterations)
draws_beta <- matrix(NA, k_beta, iterations)
draws_pi <- matrix(NA, k * k_w, iterations)
draws_gamma <- matrix(NA, k_gamma, iterations)
draws_sigma <- matrix(NA, k^2, iterations)

# Start Gibbs sampler
for (draw in 1:draws) {
  # Draw conditional mean parameters
  temp <- post_coint_kls(y = y, beta = beta, w = w, x = x, sigma_i = u_sigma_i,
                         v_i = v_i, p_tau_i = p_tau_i, g_i = g_i,
                         gamma_mu_prior = a_mu_prior,
                         gamma_v_i_prior = a_v_i_prior)
  alpha <- temp$alpha
  beta <- temp$beta
  Pi <- temp$Pi
  gamma <- temp$Gamma
  
  # Draw variance-covariance matrix
  u <- y - Pi %*% w - matrix(gamma, k) %*% x
  u_sigma_scale_post <- solve(tcrossprod(u) +
     v_i * alpha %*% tcrossprod(crossprod(beta, p_tau_i) %*% beta, alpha))
  u_sigma_i <- matrix(rWishart(1, u_sigma_df_post, u_sigma_scale_post)[,, 1], k)
  u_sigma <- solve(u_sigma_i)
  
  # Update g_i
  g_i <- u_sigma_i
  
  # Store draws
  if (draw > burnin) {
    draws_alpha[, draw - burnin] <- alpha
    draws_beta[, draw - burnin] <- beta
    draws_pi[, draw - burnin] <- Pi
    draws_gamma[, draw - burnin] <- gamma
    draws_sigma[, draw - burnin] <- u_sigma
  }
}

# Number of non-deterministic coefficients
k_nondet <- (k_x - 4) * k

# Generate bvec object
bvec_est <- bvec(y = data$data$Y, w = data$data$W,
                 x = data$data$X[, 1:6],
                 x_d = data$data$X[, 7:10],
                 Pi = draws_pi,
                 Gamma = draws_gamma[1:k_nondet,],
                 C = draws_gamma[(k_nondet + 1):nrow(draws_gamma),],
                 Sigma = draws_sigma)

# Thin posterior draws
bvec_est <- thin(bvec_est, thin = 5)

# Transfrom VEC output to VAR output
bvar_form <- bvec_to_bvar(bvec_est)

---

Posterior Simulation for BVEC Models

Description

Produces draws from the posterior distributions of Bayesian VEC models.

Usage

bvecpost(object)

Arguments

object

an object of class "bvecmodel", usually, a result of a call to gen_vec in combination with add_priors.

Details

The function implements posterior simulation algorithms proposed in Koop et al. (2010) and Koop et al. (2011), which place identifying restrictions on the cointegration space. Both algorithms are able to employ Bayesian variable selection (BVS) as proposed in Korobilis (2013). The algorithm of Koop et al. (2010) is also able to employ stochastic search variable selection (SSVS) as proposed by Geroge et al. (2008). Both SSVS and BVS can also be applied to the covariances of the error term. However, the algorithms cannot be applied to cointegration related coefficients, i.e. to the loading matrix $\alpha$ or the cointegration matrix $beta$.

The implementation primarily follows the description in Koop et al. (2010). Chan et al. (2019), George et al. (2008) and Korobilis (2013) were used to implement the variable selection algorithms. For all approaches the SUR form of a VEC model is used to obtain posterior draws. The algorithm is implemented in C++ to reduce calculation time.

The function also supports structural BVEC models, where the structural coefficients are estimated from contemporary endogenous variables, which corresponds to the so-called (A-model). Currently, only specifications are supported, where the structural matrix contains ones on its diagonal and all lower triangular elements are freely estimated. Since posterior draws are obtained based on the SUR form of the VEC model, the structural coefficients are drawn jointly with the other coefficients. No identifying restrictions are made regarding the cointegration matrix.

Value

An object of class "bvec".

References

Chan, J., Koop, G., Poirier, D. J., & Tobias J. L. (2019). Bayesian econometric methods (2nd ed.). Cambridge: Cambridge University Press.

George, E. I., Sun, D., & Ni, S. (2008). Bayesian stochastic search for VAR model restrictions. Journal of Econometrics, 142(1), 553–580. doi:10.1016/j.jeconom.2007.08.017

Koop, G., León-González, R., & Strachan R. W. (2010). Efficient posterior simulation for cointegrated models with priors on the cointegration space. Econometric Reviews, 29(2), 224–242. doi:10.1080/07474930903382208

Koop, G., León-González, R., & Strachan R. W. (2011). Bayesian inference in a time varying cointegration model. Journal of Econometrics, 165(2), 210–220. doi:10.1016/j.jeconom.2011.07.007

Korobilis, D. (2013). VAR forecasting using Bayesian variable selection. Journal of Applied Econometrics, 28(2), 204–230. doi:10.1002/jae.1271

Examples

# Get data
data("e6")

# Create model
model <- gen_vec(e6, p = 4, r = 1,
                 const = "unrestricted", seasonal = "unrestricted",
                 iterations = 100, burnin = 10)
# Chosen number of iterations and burnin should be much higher.

# Add priors
model <- add_priors(model)

# Obtain posterior draws 
object <- bvecpost(model)

---

Bayesian Variable Selection

Description

bvs employs Bayesian variable selection as proposed by Korobilis (2013) to produce a vector of inclusion parameters for the coefficient matrix of a VAR model.

Usage

bvs(y, z, a, lambda, sigma_i, prob_prior, include = NULL)

Arguments

y	a $K \times T$ matrix of the endogenous variables.
z	a $KT \times M$ matrix of explanatory variables.
a	an M-dimensional vector of parameter draws. If time varying parameters are used, an $M \times T$ coefficient matrix can be provided.
lambda	an $M \times M$ inclusion matrix that should be updated.
sigma_i	the inverse variance-covariance matrix. If the variance-covariance matrix is time varying, a $KT \times K$ matrix can be provided.
prob_prior	an M-dimensional vector of prior inclusion probabilities.
include	an integer vector specifying the positions of variables, which should be included in the BVS algorithm. If NULL (default), BVS will be applied to all variables.

Details

The function employs Bayesian variable selection as proposed by Korobilis (2013) to produce a vector of inclusion parameters, which are the diagonal elements of the inclusion matrix $\Lambda$ for the VAR model

$y_t = Z_t \Lambda a_t + u_t,$

where $u_t \sim N(0, \Sigma_{t})$. $y_t$ is a K-dimensional vector of endogenous variables and $Z_t = x_t^{\prime} \otimes I_K$ is a $K \times M$ matrix of regressors with $x_t$ as a vector of regressors.

Value

A matrix of inclusion parameters on its diagonal.

References

Korobilis, D. (2013). VAR forecasting using Bayesian variable selection. Journal of Applied Econometrics, 28(2), 204–230. doi:10.1002/jae.1271

Examples

# Load data
data("e1")
data <- diff(log(e1)) * 100

# Generate model data
temp <- gen_var(data, p = 2, deterministic = "const")

y <- t(temp$data$Y)
z <- temp$data$SUR

tt <- ncol(y)
m <- ncol(z)

# Priors
a_mu_prior <- matrix(0, m)
a_v_i_prior <- diag(0.1, m)

# Prior for inclusion parameter
prob_prior <- matrix(0.5, m)

# Initial value of Sigma
sigma <- tcrossprod(y) / tt
sigma_i <- solve(sigma)

lambda <- diag(1, m)

z_bvs <- z %*% lambda

a <- post_normal_sur(y = y, z = z_bvs, sigma_i = sigma_i,
                     a_prior = a_mu_prior, v_i_prior = a_v_i_prior)

lambda <- bvs(y = y, z = z, a = a, lambda = lambda,
              sigma_i = sigma_i, prob_prior = prob_prior)

---

Covariance: Data Preparation

Description

Convenience function, which generates the input data for posterior simulation of covariance parameters.

Usage

covar_prepare_data(y, omega_i, k, tt, tvp)

Arguments

y	a $KT \times 1$ vector of input data.
omega_i	a $K \times K$ or $KT \times KT$ matrix of error variances. The matrix must be sparse.
k	an integer of the number of endogenous variables.
tt	an integer of the number of observations.
tvp	logical indicating if the SUR matrix with the values of regressors should be prepared for the estimation of constant or time varying parameters.

Details

For the model

$y_t = Z_{t} a_t + u_t$

with $u_t \sim N(0, \Psi \Omega_{t} \Psi^{\prime})$ and $\Omega_{t}$ as a diagonal matrix of error variances, the function produces the input data for the posterior simulation of the lower triangular covariance coefficients of $\Psi$ as presented in Primiceri (2005).

Value

A list with three elements:

y	The prepared vector of endogenous variables.
z	The prepared matrix of regressors.
omega_i	The prepared diagonal matrix of measurement error variances.

All matrices are returned as sparse matrices.

References

Chan, J., Koop, G., Poirier, D. J., & Tobias J. L. (2019). Bayesian econometric methods (2nd ed.). Cambridge: Cambridge University Press.

Primiceri, G. E. (2005). Time varying structural vector autoregressions and monetary policy. The Review of Economic Studies 72(3), 821–852. doi:10.1111/j.1467-937X.2005.00353.x

Examples

# Create artificial data
k <- 3
tt <- 4
u <- matrix(1:(k * tt))
omega_i <- Matrix(diag(1:3, k))

# Generate input data (constant parameters)
covar_prepare_data(u, omega_i, k, tt, FALSE)

# Generate input data (time varying parameters)
covar_prepare_data(u, omega_i, k, tt, TRUE)

---

Covariance: Vector to Matrix

Description

Convenience function, which takes the vector of draws of lower triangular covariance coefficients and transforms it into a matrix with ones on the main diagonal. In case of time varying parameters the resulting matrix will be block diagonal.

Usage

covar_vector_to_matrix(psi, k, tt)

Arguments

psi	a $K (K - 1) / 2 \times 1$ or $T K (K - 1) / 2 \times 1$ vector of input data.
k	the number $K$ of endogenous variables.
tt	the number $T$ of observations.

Value

A sparse, block diagonal matrix.

References

Chan, J., Koop, G., Poirier, D. J., & Tobias J. L. (2019). Bayesian econometric methods (2nd ed.). Cambridge: Cambridge University Press.

Primiceri, G. E. (2005). Time varying structural vector autoregressions and monetary policy. The Review of Economic Studies 72(3), 821–852. doi:10.1111/j.1467-937X.2005.00353.x

Examples

# Create artificial data
k <- 5
tt <- 4
n_covar <- (k - 1) * k / 2

# Constant parameters
psi <- matrix(1:(n_covar))
covar_vector_to_matrix(psi, k, tt)

# Time varying parameters
psi <- matrix(1:(n_covar * tt))
covar_vector_to_matrix(psi, k, tt)

---

Bayesian Dynamic Factor Model Objects

Description

dfm is used to create objects of class "dfm".

A plot function for objects of class "dfm".

Usage

dfm(x, lambda = NULL, fac, sigma_u = NULL, a = NULL, sigma_v = NULL)

## S3 method for class 'dfm'
plot(x, ci = 0.95, ...)

Arguments

x	an object of class "dfm", usually, a result of a call to dfm.
lambda	an $MN \times S$ matrix of MCMC coefficient draws of factor loadings of the measurement equation.
fac	an $NT \times S$ matrix of MCMC draws of the factors in the transition equation, where the first N rows correspond to the N factors in period 1 and the next N rows to the factors in period 2 etc.
sigma_u	an $M \times S$ matrix of MCMC draws for the error variances of the measurement equation.
a	a $pN^2 \times S$ matrix of MCMC coefficient draws of the transition equation.
sigma_v	an $N \times S$ matrix of MCMC draws for the error variances of the transition equation.
ci	interval used to calculate credible bands.
...	further graphical parameters.

Details

The function produces a standardised object from S draws of a Gibbs sampler (after the burn-in phase) for the dynamic factor model (DFM) with measurement equation

$x_t = \lambda f_t + u_t,$

where $x_t$ is an $M \times 1$ vector of observed variables, $f_t$ is an $N \times 1$ vector of unobserved factors and $\lambda$ is the corresponding $M \times N$ matrix of factor loadings. $u_t$ is an $M \times 1$ error term.

The transition equation is

$f_t = \sum_{i=1}^{p} A_i f_{t - i} + v_t,$

where $A_i$ is an $N \times N$ coefficient matrix and $v_t$ is an $N \times 1$ error term.

Value

An object of class "dfm" containing the following components, if specified:

x	the standardised time-series object of observable variables.
lambda	an $S \times MN$ "mcmc" object of draws of factor loadings of the measurement equation.
factor	an $S \times NT$ "mcmc" object of draws of factors.
sigma_u	an $S \times M$ "mcmc" object of variance draws of the measurement equation.
a	an $S \times pN^2$ "mcmc" object of coefficient draws of the transition equation.
sigma_v	an $S \times N$ "mcmc" object of variance draws of the transition equation.
specifications	a list containing information on the model specification.

Examples

# Load data
data("bem_dfmdata")

# Generate model data
model <- gen_dfm(x = bem_dfmdata, p = 1, n = 1,
                 iterations = 20, burnin = 10)
# Number of iterations and burnin should be much higher.

# Add prior specifications
model <- add_priors(model,
                    lambda = list(v_i = .01),
                    sigma_u = list(shape = 5, rate = 4),
                    a = list(v_i = .01),
                    sigma_v = list(shape = 5, rate = 4))

# Obtain posterior draws
object <- dfmpost(model)


# Load data
data("bem_dfmdata")

# Generate model data
model <- gen_dfm(x = bem_dfmdata, p = 1, n = 1,
                 iterations = 20, burnin = 10)
# Number of iterations and burnin should be much higher.

# Add prior specifications
model <- add_priors(model,
                    lambda = list(v_i = .01),
                    sigma_u = list(shape = 5, rate = 4),
                    a = list(v_i = .01),
                    sigma_v = list(shape = 5, rate = 4))

# Obtain posterior draws
object <- draw_posterior(model)

# Plot factors
plot(object)

---

Posterior Simulation for Dynamic Factor Models

Description

Produces draws from the posterior distributions of Bayesian dynamic factor models.

Usage

dfmpost(object)

Arguments

object

an object of class "dfmodel", usually, a result of a call to gen_dfm in combination with add_priors.

Details

The function implements the posterior simulation algorithm for Bayesian dynamic factor models.

The implementation follows the description in Chan et al. (2019) and C++ is used to reduce calculation time.

Value

An object of class "dfm".

References

Chan, J., Koop, G., Poirier, D. J., & Tobias J. L. (2019). Bayesian econometric methods (2nd ed.). Cambridge: Cambridge University Press.

Examples

# Load data
data("bem_dfmdata")

# Generate model data
model <- gen_dfm(x = bem_dfmdata, p = 1, n = 1,
                 iterations = 20, burnin = 10)
# Number of iterations and burnin should be much higher.

# Add prior specifications
model <- add_priors(model,
                    lambda = list(v_i = .01),
                    sigma_u = list(shape = 5, rate = 4),
                    a = list(v_i = .01),
                    sigma_v = list(shape = 5, rate = 4))

# Obtain posterior draws
object <- dfmpost(model)

---

Posterior Simulation

Description

Forwards model input to posterior simulation functions. This is a generic function.

Usage

draw_posterior(object, ...)

Arguments

object	a list of model specifications. Usually, the output of a call to gen_var, gen_vec or gen_dfm in combination with add_priors.
...	arguments passed forward to method.

---

Posterior Simulation

Description

Forwards model input to posterior simulation functions.

Usage

## S3 method for class 'bvarmodel'
draw_posterior(object, FUN = NULL, mc.cores = NULL, ...)

Arguments

object	a list of model specifications, which should be passed on to function FUN. Usually, the output of a call to gen_var in combination with add_priors.
FUN	the function to be applied to each model in argument object. If NULL (default), the internal functions bvarpost is used.
mc.cores	the number of cores to use, i.e. at most how many child processes will be run simultaneously. The option is initialized from environment variable MC_CORES if set. Must be at least one, and parallelization requires at least two cores.
...	further arguments passed to or from other methods.

Value

For multiple models a list of objects of class bvarlist. For a single model the object has the class of the output of the applied posterior simulation function. In case the package's own functions are used, this will result in an object of class "bvar".

Examples

# Load data 
data("e1")
e1 <- diff(log(e1)) * 100

# Generate model
model <- gen_var(e1, p = 1:2, deterministic = 2,
                 iterations = 100, burnin = 10)
# Chosen number of iterations and burn-in should be much higher.

# Add priors
model <- add_priors(model)

# Obtain posterior draws
object <- draw_posterior(model)

---

Posterior Simulation for Vector Error Correction Models

Description

Forwards model input to posterior simulation functions for vector error correction models.

Usage

## S3 method for class 'bvecmodel'
draw_posterior(object, FUN = NULL, mc.cores = NULL, ...)

Arguments

object	a list of model specifications, which should be passed on to function FUN. Usually, the output of a call to gen_vec in combination with add_priors.
FUN	the function to be applied to each list element in argument object. If NULL (default), the internal function bvecpost is used.
mc.cores	the number of cores to use, i.e. at most how many child processes will be run simultaneously. The option is initialized from environment variable MC_CORES if set. Must be at least one, and parallelization requires at least two cores.
...	further arguments passed to or from other methods.

Value

For multiple models a list of objects of class bvarlist. For a single model the object has the class of the output of the applied posterior simulation function. In case the package's own functions are used, this will be "bvec".

References

Koop, G., León-González, R., & Strachan R. W. (2010). Efficient posterior simulation for cointegrated models with priors on the cointegration space. Econometric Reviews, 29(2), 224–242. doi:10.1080/07474930903382208

Koop, G., León-González, R., & Strachan R. W. (2011). Bayesian inference in a time varying cointegration model. Journal of Econometrics, 165(2), 210–220. doi:10.1016/j.jeconom.2011.07.007

Examples

# Load data 
data("e6")
e6 <- e6 * 100

# Generate model
model <- gen_vec(e6, p = 1, r = 1, const = "restricted",
                 iterations = 10, burnin = 10)
# Chosen number of iterations and burn-in should be much higher.

# Add priors
model <- add_priors(model)

# Obtain posterior draws
object <- draw_posterior(model)

---

Posterior Simulation

Description

Forwards model input to posterior simulation functions.

Usage

## S3 method for class 'dfmodel'
draw_posterior(object, FUN = NULL, mc.cores = NULL, ...)

Arguments

object	a list of model specifications, which should be passed on to function FUN. Usually, the output of a call to gen_var, gen_vec or gen_dfm in combination with add_priors.
FUN	the function to be applied to each list element in argument object. If NULL (default), the internal functions bvarpost is used for VAR model, bvecpost for VEC models and dfmpost for dynamic factor models.
mc.cores	the number of cores to use, i.e. at most how many child processes will be run simultaneously. The option is initialized from environment variable MC_CORES if set. Must be at least one, and parallelization requires at least two cores.
...	further arguments passed to or from other methods.

Value

For multiple models a list of objects of class bvarlist. For a single model the object has the class of the output of the applied posterior simulation function. In case the package's own functions are used, this will be "bvar", "bvec" or "dfm".

Examples

# Load data 
data("e1")
e1 <- diff(log(e1)) * 100

# Generate model
model <- gen_var(e1, p = 1:2, deterministic = 2,
                 iterations = 100, burnin = 10)
# Chosen number of iterations and burn-in should be much higher.

# Add priors
model <- add_priors(model)

# Obtain posterior draws
object <- draw_posterior(model)

---

West German economic time series data

Description

The data set contains quarterly, seasonally adjusted time series for West German fixed investment, disposable income, and consumption expenditures in billions of DM from 1960Q1 to 1982Q4. It was produced from file E1 of the data sets associated with Lütkepohl (2007). Raw data are available at http://www.jmulti.de/download/datasets/e1.dat and were originally obtained from Deutsche Bundesbank.

Usage

data("e1")

Format

A named time-series object with 92 rows and 3 variables:

invest

fixed investment.

income

disposable income.

cons

consumption expenditures.

References

Lütkepohl, H. (2006). New introduction to multiple time series analysis (2nd ed.). Berlin: Springer.

---

German interest and inflation rate data

Description

The data set contains quarterly, seasonally unadjusted time series for German long-term interest and inflation rates from 1972Q2 to 1998Q4. It was produced from file E6 of the data sets associated with Lütkepohl (2007). Raw data are available at http://www.jmulti.de/download/datasets/e6.dat and were originally obtained from Deutsche Bundesbank and Deutsches Institut für Wirtschaftsforschung.

Usage

data("e6")

Format

A named time-series object with 107 rows and 2 variables:

R

nominal long-term interest rate (Umlaufsrendite).

Dp

$\Delta$ log of GDP deflator.

Details

The data cover West Germany until 1990Q2 and all of Germany aferwards. The values refer to the last month of a quarter.

References

Lütkepohl, H. (2006). New introduction to multiple time series analysis (2nd ed.). Berlin: Springer.

---

Forecast Error Variance Decomposition A generic function used to calculate forecast error varianc decompositions.

Description

A plot function for objects of class "bvarfevd".

Usage

fevd(object, ...)

## S3 method for class 'bvarfevd'
plot(x, ...)

Arguments

object	an object of class "bvar".
...	further graphical parameters.
x	an object of class "bvarfevd", usually, a result of a call to fevd.

Examples

# Load data
data("e1")
e1 <- diff(log(e1)) * 100

# Generate model data
model <- gen_var(e1, p = 2, deterministic = 2,
                 iterations = 100, burnin = 10)
# Chosen number of iterations and burnin should be much higher.

# Add prior specifications
model <- add_priors(model)

# Obtain posterior draws
object <- draw_posterior(model)

# Obtain FEVD
vd <- fevd(object, response = "cons")

# Plot
plot(vd)

---

Forecast Error Variance Decomposition

Description

Produces the forecast error variance decomposition of a Bayesian VAR model.

Usage

## S3 method for class 'bvar'
fevd(
  object,
  response = NULL,
  n.ahead = 5,
  type = "oir",
  normalise_gir = FALSE,
  period = NULL,
  ...
)

Arguments

object	an object of class "bvar", usually, a result of a call to bvar or bvec_to_bvar.
response	name of the response variable.
n.ahead	number of steps ahead.
type	type of the impulse responses used to calculate forecast error variable decompositions. Possible choices are orthogonalised oir (default) and generalised gir impulse responses.
normalise_gir	logical. Should the GIR-based FEVD be normalised?
period	integer. Index of the period, for which the variance decomposition should be generated. Only used for TVP or SV models. Default is NULL, so that the posterior draws of the last time period are used.
...	further arguments passed to or from other methods.

Details

The function produces forecast error variance decompositions (FEVD) for the VAR model

$A_0 y_t = \sum_{i = 1}^{p} A_{i} y_{t-i} + u_t,$

with $u_t \sim N(0, \Sigma)$. For non-structural models matrix $A_0$ is set to the identiy matrix and can therefore be omitted, where not relevant.

If the FEVD is based on the orthogonalised impulse resonse (OIR), the FEVD will be calculated as

$\omega^{OIR}_{jk, h} = \frac{\sum_{i = 0}^{h-1} (e_j^{\prime} \Phi_i P e_k )^2}{\sum_{i = 0}^{h-1} (e_j^{\prime} \Phi_i \Sigma \Phi_i^{\prime} e_j )},$

where $\Phi_i$ is the forecast error impulse response for the $i$th period, $P$ is the lower triangular Choleski decomposition of the variance-covariance matrix $\Sigma$, $e_j$ is a selection vector for the response variable and $e_k$ a selection vector for the impulse variable.

If type = "sir", the structural FEVD will be calculated as

$\omega^{SIR}_{jk, h} = \frac{\sum_{i = 0}^{h-1} (e_j^{\prime} \Phi_i A_0^{-1} e_k )^2}{\sum_{i = 0}^{h-1} (e_j^{\prime} \Phi_i A_0^{-1} A_0^{-1\prime} \Phi_i^{\prime} e_j )},$

where $\sigma_{jj}$ is the diagonal element of the $j$th variable of the variance covariance matrix.

If type = "gir", the generalised FEVD will be calculated as

$\omega^{GIR}_{jk, h} = \frac{\sigma^{-1}_{jj} \sum_{i = 0}^{h-1} (e_j^{\prime} \Phi_i \Sigma e_k )^2}{\sum_{i = 0}^{h-1} (e_j^{\prime} \Phi_i \Sigma \Phi_i^{\prime} e_j )},$

where $\sigma_{jj}$ is the diagonal element of the $j$th variable of the variance covariance matrix.

If type = "sgir", the structural generalised FEVD will be calculated as

$\omega^{SGIR}_{jk, h} = \frac{\sigma^{-1}_{jj} \sum_{i = 0}^{h-1} (e_j^{\prime} \Phi_i A_0^{-1} \Sigma e_k )^2}{\sum_{i = 0}^{h-1} (e_j^{\prime} \Phi_i A_0^{-1} \Sigma A_0^{-1\prime} \Phi_i^{\prime} e_j )}$

.

Since GIR-based FEVDs do not add up to unity, they can be normalised by setting normalise_gir = TRUE.

Value

A time-series object of class "bvarfevd".

References

Lütkepohl, H. (2006). New introduction to multiple time series analysis (2nd ed.). Berlin: Springer.

Pesaran, H. H., & Shin, Y. (1998). Generalized impulse response analysis in linear multivariate models. Economics Letters, 58, 17-29.

Examples

# Load data
data("e1")
e1 <- diff(log(e1)) * 100

# Generate models
model <- gen_var(e1, p = 2, deterministic = 2,
                 iterations = 100, burnin = 10)

# Add priors
model <- add_priors(model)

# Obtain posterior draws
object <- draw_posterior(model)

# Obtain FEVD
vd <- fevd(object, response = "cons")

# Plot FEVD
plot(vd)

---

Dynamic Factor Model Input

Description

gen_dfm produces the input for the estimation of a dynamic factor model (DFM).

Usage

gen_dfm(x, p = 2, n = 1, iterations = 50000, burnin = 5000)

Arguments

x	a time-series object of stationary endogenous variables.
p	an integer vector of the lag order of the measurement equation. See 'Details'.
n	an integer vector of the number of factors. See 'Details'.
iterations	an integer of MCMC draws excluding burn-in draws (defaults to 50000).
burnin	an integer of MCMC draws used to initialize the sampler (defaults to 5000). These draws do not enter the computation of posterior moments, forecasts etc.

Details

The function produces the variable matrices of dynamic factor models (DFM) with measurement equation

$x_t = \lambda f_t + u_t,$

where $x_t$ is an $M \times 1$ vector of observed variables, $f_t$ is an $N \times 1$ vector of unobserved factors and $\lambda$ is the corresponding $M \times N$ matrix of factor loadings. $u_t$ is an $M \times 1$ error term.

The transition equation is

$f_t = \sum_{i=1}^{p} A_i f_{t - i} + v_t,$

where $A_i$ is an $N \times N$ coefficient matrix and $v_t$ is an $N \times 1$ error term.

If integer vectors are provided as arguments p or n, the function will produce a distinct model for all possible combinations of those specifications.

Value

An object of class 'dfmodel', which contains the following elements:

data	A list of data objects, which can be used for posterior simulation. Element X is a time-series object of normalised observable variables, i.e. each column has zero mean and unity variance.
model	A list of model specifications.

References

Chan, J., Koop, G., Poirier, D. J., & Tobias, J. L. (2019). Bayesian Econometric Methods (2nd ed.). Cambridge: University Press.

Lütkepohl, H. (2007). New introduction to multiple time series analysis (2nd ed.). Berlin: Springer.

Examples

# Load data
data("bem_dfmdata")

# Generate model data
model <- gen_dfm(x = bem_dfmdata, p = 1, n = 1,
                 iterations = 5000, burnin = 1000)

---