bvartools: Bayesian Inference of Vector Autoregressive and Error Correction Models

Description

Assists in the set-up of algorithms for Bayesian inference of vector autoregressive (VAR) and error correction (VEC) models. Functions for posterior simulation, forecasting, impulse response analysis and forecast error variance decomposition are largely based on the introductory texts of Chan, Koop, Poirier and Tobias (2019, ISBN: 9781108437493), Koop and Korobilis (2010) doi:10.1561/0800000013 and Luetkepohl (2006, ISBN: 9783540262398).

Author(s)

Maintainer: Franz X. Mohr franz.x.mohr@outlook.com (0009-0003-8890-7781)

See Also

Useful links:

• https://github.com/franzmohr/bvartools

• Report bugs at https://github.com/franzmohr/bvartools/issues

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

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

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 lth 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

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

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 lth 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

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

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

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:

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

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

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:

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

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

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

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)

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

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:

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

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

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

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

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

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

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

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 ith 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 jth 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 jth 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

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:

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)

Vector Autoregressive Model Input

Description

gen_var produces the input for the estimation of a vector autoregressive (VAR) model.

Usage

gen_var(
  data,
  p = 2,
  exogen = NULL,
  s = NULL,
  deterministic = "const",
  seasonal = FALSE,
  structural = FALSE,
  tvp = FALSE,
  sv = FALSE,
  fcst = NULL,
  iterations = 50000,
  burnin = 5000
)

Arguments

Details

The function produces the data matrices for vector autoregressive (VAR) models, which can also include unmodelled, non-deterministic variables:

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,

where y_t is a K-dimensional vector of endogenous variables, A_0 is a K \times K coefficient matrix of contemporaneous endogenous variables, A_i is a K \times K coefficient matrix of endogenous variables, x_t is an M-dimensional vector of exogenous regressors and B_i its corresponding K \times M coefficient matrix. D_t is an N-dimensional vector of deterministic terms and C its corresponding K \times N coefficient matrix. p is the lag order of endogenous variables, s is the lag order of exogenous variables, and u_t is an error term.

If an integer vector is provided as argument p or s, the function will produce a distinct model for all possible combinations of those specifications.

If tvp is TRUE, the respective coefficients of the above model are assumed to be time varying. If sv is TRUE, the error covariance matrix is assumed to be time varying.

Value

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

References

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

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

Examples

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

# Generate model data
data <- gen_var(e1, p = 0:2, deterministic = "const")

Vector Error Correction Model Input

Description

gen_vec produces the input for the estimation of a vector error correction (VEC) model.

Usage

gen_vec(
  data,
  p = 2,
  exogen = NULL,
  s = 2,
  r = NULL,
  const = NULL,
  trend = NULL,
  seasonal = NULL,
  structural = FALSE,
  tvp = FALSE,
  sv = FALSE,
  fcst = NULL,
  iterations = 50000,
  burnin = 5000
)

Arguments

Details

The function produces the variable matrices of vector error correction (VEC) models, which can also include exogenous variables:

\Delta y_t = \Pi w_t + \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,

where \Delta y_t is a K \times 1 vector of differenced endogenous variables, w_t is a (K + M + N^{R}) \times 1 vector of cointegration variables, \Pi is a K \times (K + M + N^{R}) matrix of cointegration parameters, \Gamma_i is a K \times K coefficient matrix of endogenous variables, \Delta x_t is a M \times 1 vector of differenced exogenous regressors, \Upsilon_i is a K \times M coefficient matrix of exogenous regressors, d^{UR}_t is a N \times 1 vector of deterministic terms, and C^{UR} is a K \times N^{UR} coefficient matrix of deterministic terms that do not enter the cointegration term. p is the lag order of endogenous variables and s is the lag order of exogenous variables of the corresponding VAR model. u_t is a K \times 1 error term.

If an integer vector is provided as argument p, s or r, the function will produce a distinct model for all possible combinations of those specifications.

If tvp is TRUE, the respective coefficients of the above model are assumed to be time varying. If sv is TRUE, the error covariance matrix is assumed to be time varying.

Value

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

References

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

Examples

# Load data
data("e6")

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

Prior Inclusion Probabilities

Description

Prior inclusion probabilities as required for stochastic search variable selection (SSVS) à la George et al. (2008) and Bayesian variable selection (BVS) à la Korobilis (2013).

Usage

inclusion_prior(
  object,
  prob = 0.5,
  exclude_deterministics = TRUE,
  minnesota_like = FALSE,
  kappa = c(0.8, 0.5, 0.5, 0.8)
)

Arguments

Details

If minnesota_like = TRUE, prior inclusion probabilities \underline{\pi}_1 are calculated as

for lag r with \kappa_1, \kappa_2, \kappa_3, \kappa_4 as the first, second, third and forth element in kappa, respectively.

For vector error correction models the function generates prior inclusion probabilities for differenced variables and unrestricted deterministc terms as described above. For variables in the error correction term prior inclusion probabilites are calculated as

Value

A list containing a matrix of prior inclusion probabilities and an integer vector specifying the positions of variables, which should be included in the variable selction algorithm.

References

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

# Prepare data
data("e1")

# Generate model input
object <- gen_var(e1)

# Obtain inclusion prior
pi_prior <- inclusion_prior(object)

Impulse Response Function A generic function used to calculate impulse response functions.

Description

A plot function for objects of class "bvarirf".

Usage

irf(x, ...)

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

Arguments

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)
# Number of iterations and burnin should be much higher.

# Add prior specifications
model <- add_priors(model)

# Optain posterior draws
object <- draw_posterior(model)

# Calculate IR
ir <- irf(object, impulse = "invest", response = "cons")

# Plot IR
plot(ir)

Impulse Response Function

Description

Computes the impulse response coefficients of an object of class "bvar" for n.ahead steps.

Usage

## S3 method for class 'bvar'
irf(
  x,
  impulse = NULL,
  response = NULL,
  n.ahead = 5,
  ci = 0.95,
  shock = 1,
  type = "feir",
  cumulative = FALSE,
  keep_draws = FALSE,
  period = NULL,
  ...
)

Arguments

Details

The function produces different types of impulse responses 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).

Forecast error impulse responses \Phi_i are obtained by recursions

\Phi_i = \sum_{j = 1}^{i} \Phi_{i-j} A_j, i = 1, 2,...,h

with \Phi_0 = I_K.

Orthogonalised impulse responses \Theta^o_i are calculated as \Theta^o_i = \Phi_i P, where P is the lower triangular Choleski decomposition of \Sigma.

Structural impulse responses \Theta^s_i are calculated as \Theta^s_i = \Phi_i A_0^{-1}.

(Structural) Generalised impulse responses for variable j, i.e. \Theta^g_ji are calculated as \Theta^g_{ji} = \sigma_{jj}^{-1/2} \Phi_i A_0^{-1} \Sigma e_j, where \sigma_{jj} is the variance of the j^{th} diagonal element of \Sigma and e_i is a selection vector containing one in its j^{th} element and zero otherwise. If the "bvar" object does not contain draws of A_0, it is assumed to be an identity matrix.

Value

A time-series object of class "bvarirf" and if keep_draws = TRUE a simple matrix.

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 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 IR
ir <- irf(object, impulse = "invest", response = "cons")

# Plot IR
plot(ir)

Durbin and Koopman Simulation Smoother

Description

An implementation of the Kalman filter and backward smoothing algorithm proposed by Durbin and Koopman (2002).

Usage

kalman_dk(y, z, sigma_u, sigma_v, B, a_init, P_init)

Arguments

Details

The function uses algorithm 2 from Durbin and Koopman (2002) to produce a draw of the state vector a_t for t = 1,...,T for a state space model with measurement equation

y_t = Z_t a_t + u_t

and transition equation

a_{t + 1} = B_t a_{t} + v_t,

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

The algorithm takes into account Jarociński (2015), where a possible missunderstanding in the implementation of the algorithm of Durbin and Koopman (2002) is pointed out. Following that note the function sets the mean of the initial state to zero in the first step of the algorithm.

Value

A M \times T+1 matrix of state vector draws.

References

Durbin, J., & Koopman, S. J. (2002). A simple and efficient simulation smoother for state space time series analysis. Biometrika, 89(3), 603–615.

Jarociński, M. (2015). A note on implementing the Durbin and Koopman simulation smoother. Computational Statistics and Data Analysis, 91, 1–3. doi:10.1016/j.csda.2015.05.001

Examples

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

# Generate model data
temp <- gen_var(data, p = 2, deterministic = "const")
y <- t(temp$data$Y)
z <- temp$data$SUR
k <- nrow(y)
tt <- ncol(y)
m <- ncol(z)

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

a_Q <- diag(.0001, m)

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

# Initial values for Kalman filter
y_init <- y * 0
a_filter <- matrix(0, m, tt + 1)

# Initialise the Kalman filter
for (i in 1:tt) {
  y_init[, i] <- y[, i] - z[(i - 1) * k + 1:k,] %*% a_filter[, i]
}
a_init <- post_normal_sur(y = y_init, z = z, sigma_i = sigma_i,
                          a_prior = a_mu_prior, v_i_prior = a_v_i_prior)
y_filter <- matrix(y) - z %*% a_init
y_filter <- matrix(y_filter, k) # Reshape

# Kalman filter and backward smoother
a_filter <- kalman_dk(y = y_filter, z = z, sigma_u = sigma,
                      sigma_v = a_Q, B = diag(1, m),
                      a_init = matrix(0, m), P_init = a_Q)
                      
a <- a_filter + matrix(a_init, m, tt + 1)

Calculates the log-likelihood of a multivariate normal distribution.

Description

Calculates the log-likelihood of a multivariate normal distribution.

Usage

loglik_normal(u, sigma)

Arguments

Details

The log-likelihood is calculated for each vector in period t as

-\frac{K}{2} \ln 2\pi - \frac{1}{2} \ln |\Sigma_t| -\frac{1}{2} u_t^\prime \Sigma_t^{-1} u_t

, where u_t = y_t - \mu_t.

Examples

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

# Generate VAR model
data <- gen_var(e1, p = 2, deterministic = "const")
y <- t(data$data$Y)
x <- t(data$data$Z)

# LS estimate
ols <- tcrossprod(y, x) %*% solve(tcrossprod(x))

# Residuals
u <- y - ols %*% x # Residuals

# Covariance matrix
sigma <- tcrossprod(u) / ncol(u)

# Log-likelihood
loglik_normal(u = u, sigma = sigma)

Minnesota Prior

Description

Calculates the Minnesota prior for a VAR model.

Usage

minnesota_prior(
  object,
  kappa0 = 2,
  kappa1 = 0.5,
  kappa2 = NULL,
  kappa3 = 5,
  max_var = NULL,
  coint_var = FALSE,
  sigma = "AR"
)

Arguments

Details

The function calculates the Minnesota prior of a VAR model. For the endogenous variable i the prior variance of the lth 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.

For VEC models the function only provides priors for the non-cointegration part of the model. 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.

Value

A list containing a matrix of prior means and the precision matrix of the cofficients and the inverse variance-covariance matrix of the error term, which was obtained by an LS estimation.

References

Chan, J., Koop, G., Poirier, D. J., & Tobias, J. L. (2020). Bayesian Econometric Methods (2nd ed.). Cambridge: University Press.

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

Examples

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

# Generate model input
object <- gen_var(data)

# Obtain Minnesota prior
prior <- minnesota_prior(object)

Plotting Posterior Draws of Bayesian VAR or VEC Models

Description

A plot function for objects of class "bvarlist".

Usage

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

Arguments

Plotting Forecasts of BVAR Models

Description

A plot function for objects of class "bvarprd".

Usage

## S3 method for class 'bvarprd'
plot(x, n.pre = NULL, ...)

Arguments

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)

# Add prior specifications
model <- add_priors(model)

# Obtain posterior draws
object <- draw_posterior(model)

# Calculate forecasts
pred <- predict(object, new_d = rep(1, 10))

# Plot forecasts
plot(pred)

Posterior Draw for Cointegration Models

Description

Produces a draw of coefficients for cointegration models with a prior on the cointegration space as proposed in Koop et al. (2010) and a draw of non-cointegration coefficients from a normal density.

Usage

post_coint_kls(
  y,
  beta,
  w,
  sigma_i,
  v_i,
  p_tau_i,
  g_i,
  x = NULL,
  gamma_mu_prior = NULL,
  gamma_v_i_prior = NULL
)

Arguments

Details

The function produces posterior draws of the coefficient matrices \alpha, \beta and \Gamma for the model

y_{t} = \alpha \beta^{\prime} w_{t-1} + \Gamma z_{t} + u_{t},

where y_{t} is a K-dimensional vector of differenced endogenous variables. w_{t} is an M \times 1 vector of variables in the cointegration term, which include lagged values of endogenous and exogenous variables in levels and restricted deterministic terms. z_{t} is an N-dimensional vector of differenced endogenous and exogenous explanatory variabes as well as unrestricted deterministic terms. The error term is u_t \sim \Sigma.

Draws of the loading matrix \alpha are obtained using the prior on the cointegration space as proposed in Koop et al. (2010). The posterior covariance matrix is

\overline{V}_{\alpha} = \left[\left(v^{-1} (\beta^{\prime} P_{\tau}^{-1} \beta) \otimes G_{-1}\right) + \left(ZZ^{\prime} \otimes \Sigma^{-1} \right) \right]^{-1}

and the posterior mean by

\overline{\alpha} = \overline{V}_{\alpha} + vec(\Sigma^{-1} Y Z^{\prime}),

where Y is a K \times T matrix of differenced endogenous variables and Z = \beta^{\prime} W with W as an M \times T matrix of variables in the cointegration term.

For a given prior mean vector \underline{\Gamma} and prior covariance matrix \underline{V_{\Gamma}} the posterior covariance matrix of non-cointegration coefficients in \Gamma is obtained by

\overline{V}_{\Gamma} = \left[ \underline{V}_{\Gamma}^{-1} + \left(X X^{\prime} \otimes \Sigma^{-1} \right) \right]^{-1}

and the posterior mean by

\overline{\Gamma} = \overline{V}_{\Gamma} \left[ \underline{V}_{\Gamma}^{-1} \underline{\Gamma} + vec(\Sigma^{-1} Y X^{\prime}) \right],

where X is an M \times T matrix of explanatory variables, which do not enter the cointegration term.

Draws of the cointegration matrix \beta are obtained using the prior on the cointegration space as proposed in Koop et al. (2010). The posterior covariance matrix of the unrestricted cointegration matrix B is

\overline{V}_{B} = \left[\left(A^{\prime} G^{-1} A \otimes v^{-1} P_{\tau}^{-1} \right) + \left(A^{\prime} \Sigma^{-1} A \otimes WW^{\prime} \right) \right]^{-1}

and the posterior mean by

\overline{B} = \overline{V}_{B} + vec(W Y_{B}^{-1} \Sigma^{-1} A),

where Y_{B} = Y - \Gamma X and A = \alpha (\alpha^{\prime} \alpha)^{-\frac{1}{2}}.

The final draws of \alpha and \beta are calculated using \beta = B (B^{\prime} B)^{-\frac{1}{2}} and \alpha = A (B^{\prime} B)^{\frac{1}{2}}.

Value

A named list containing the following elements:

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

Examples

# Load data
data("e6")

# Generate model data
temp <- gen_vec(e6, p = 1, r = 1)
y <- t(temp$data$Y)
ect <- t(temp$data$W)

k <- nrow(y) # Endogenous variables
tt <- ncol(y) # Number of observations

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

# Initial values of beta
beta <- matrix(c(1, -4), k)

# Draw parameters
coint <- post_coint_kls(y = y, beta = beta, w = ect, sigma_i = sigma_i,
                        v_i = 0, p_tau_i = diag(1, k), g_i = sigma_i)

Posterior Draw for Cointegration Models

Description

Produces a draw of coefficients for cointegration models in SUR form with a prior on the cointegration space as proposed in Koop et al. (2010) and a draw of non-cointegration coefficients from a normal density.

Usage

post_coint_kls_sur(
  y,
  beta,
  w,
  sigma_i,
  v_i,
  p_tau_i,
  g_i,
  x = NULL,
  gamma_mu_prior = NULL,
  gamma_v_i_prior = NULL,
  svd = FALSE
)

Arguments

Details

The function produces posterior draws of the coefficient matrices \alpha, \beta and \Gamma for the model

y_{t} = \alpha \beta^{\prime} w_{t-1} + \Gamma z_{t} + u_{t},

where y_{t} is a K-dimensional vector of differenced endogenous variables. w_{t} is an M \times 1 vector of variables in the cointegration term, which include lagged values of endogenous and exogenous variables in levels and restricted deterministic terms. z_{t} is an N-dimensional vector of differenced endogenous and exogenous explanatory variabes as well as unrestricted deterministic terms. The error term is u_t \sim \Sigma.

Draws of the loading matrix \alpha are obtained using the prior on the cointegration space as proposed in Koop et al. (2010). The posterior covariance matrix is

\overline{V}_{\alpha} = \left[\left(v^{-1} (\beta^{\prime} P_{\tau}^{-1} \beta) \otimes G_{-1}\right) + \left(ZZ^{\prime} \otimes \Sigma^{-1} \right) \right]^{-1}

and the posterior mean by

\overline{\alpha} = \overline{V}_{\alpha} + vec(\Sigma^{-1} Y Z^{\prime}),

where Y is a K \times T matrix of differenced endogenous variables and Z = \beta^{\prime} W with W as an M \times T matrix of variables in the cointegration term.

For a given prior mean vector \underline{\Gamma} and prior covariance matrix \underline{V_{\Gamma}} the posterior covariance matrix of non-cointegration coefficients in \Gamma is obtained by

\overline{V}_{\Gamma} = \left[ \underline{V}_{\Gamma}^{-1} + \left(X X^{\prime} \otimes \Sigma^{-1} \right) \right]^{-1}

and the posterior mean by

\overline{\Gamma} = \overline{V}_{\Gamma} \left[ \underline{V}_{\Gamma}^{-1} \underline{\Gamma} + vec(\Sigma^{-1} Y X^{\prime}) \right],

where X is an M \times T matrix of explanatory variables, which do not enter the cointegration term.

Draws of the cointegration matrix \beta are obtained using the prior on the cointegration space as proposed in Koop et al. (2010). The posterior covariance matrix of the unrestricted cointegration matrix B is

\overline{V}_{B} = \left[\left(A^{\prime} G^{-1} A \otimes v^{-1} P_{\tau}^{-1} \right) + \left(A^{\prime} \Sigma^{-1} A \otimes WW^{\prime} \right) \right]^{-1}

and the posterior mean by

\overline{B} = \overline{V}_{B} + vec(W Y_{B}^{-1} \Sigma^{-1} A),

where Y_{B} = Y - \Gamma X and A = \alpha (\alpha^{\prime} \alpha)^{-\frac{1}{2}}.

The final draws of \alpha and \beta are calculated using \beta = B (B^{\prime} B)^{-\frac{1}{2}} and \alpha = A (B^{\prime} B)^{\frac{1}{2}}.

Value

A named list containing the following elements:

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

Examples

# Load data
data("e6")

# Generate model data
temp <- gen_vec(e6, p = 1, r = 1)
y <- t(temp$data$Y)
ect <- t(temp$data$W)

k <- nrow(y) # Endogenous variables
tt <- ncol(y) # Number of observations

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

# Initial values of beta
beta <- matrix(c(1, -4), k)

# Draw parameters
coint <- post_coint_kls_sur(y = y, beta = beta, w = ect,
                            sigma_i = sigma_i, v_i = 0, p_tau_i = diag(1, nrow(ect)),
                            g_i = sigma_i)

Posterior Draw from a Normal Distribution

Description

Produces a draw of coefficients from a normal posterior density.

Usage

post_normal(y, x, sigma_i, a_prior, v_i_prior)

Arguments

Details

The function produces a vectorised posterior draw a of the K \times M coefficient matrix A for the model

y_{t} = A x_{t} + u_{t},

where y_{t} is a K-dimensional vector of endogenous variables, x_{t} is an M-dimensional vector of explanatory variabes and the error term is u_t \sim \Sigma.

For a given prior mean vector \underline{a} and prior covariance matrix \underline{V} the posterior covariance matrix is obtained by

\overline{V} = \left[ \underline{V}^{-1} + \left(X X^{\prime} \otimes \Sigma^{-1} \right) \right]^{-1}

and the posterior mean by

\overline{a} = \overline{V} \left[ \underline{V}^{-1} \underline{a} + vec(\Sigma^{-1} Y X^{\prime}) \right],

where Y is a K \times T matrix of the endogenous variables and X is an M \times T matrix of the explanatory variables.

Value

A vector.

References

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

Examples

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

# Generate model data
temp <- gen_var(data, p = 2, deterministic = "const")
y <- t(temp$data$Y)
x <- t(temp$data$Z)
k <- nrow(y)
tt <- ncol(y)
m <- k * nrow(x)

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

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

# Draw parameters
a <- post_normal(y = y, x = x, sigma_i = sigma_i,
                 a_prior = a_mu_prior, v_i_prior = a_v_i_prior)

Posterior Simulation of Error Covariance Coefficients

Description

Produces posterior draws of constant error covariance coefficients.

Usage

post_normal_covar_const(y, u_omega_i, prior_mean, prior_covariance_i)

Arguments

Details

For the multivariate model A_0 y_t = u_t with u_t \sim N(0, \Omega_t) the function produces a draw of the lower triangular part of A_0 similar as in Primiceri (2005), i.e., using

y_t = Z_t \psi + u_t,

where

Z_{t} = \begin{bmatrix} 0 & \dotsm & \dotsm & 0 \\ -y_{1, t} & 0 & \dotsm & 0 \\ 0 & -y_{[1,2], t} & \ddots & \vdots \\ \vdots & \ddots & \ddots & 0 \\ 0 & \dotsm & 0 & -y_{[1,...,K-1], t} \end{bmatrix}

and y_{[1,...,K-1], t} denotes the first to (K-1)th elements of the vector y_t.

Value

A matrix.

References

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

# Load example data
data("e1")
y <- log(t(e1))

# Generate artificial draws of other matrices
u_omega_i <- diag(1, 3)
prior_mean <- matrix(0, 3)
prior_covariance_i <- diag(0, 3)

# Obtain posterior draw
post_normal_covar_const(y, u_omega_i, prior_mean, prior_covariance_i)

Posterior Simulation of Error Covariance Coefficients

Description

Produces posterior draws of time varying error covariance coefficients.

Usage

post_normal_covar_tvp(y, u_omega_i, v_sigma_i, psi_init)

Arguments

Details

For the multivariate model A_{0,t} y_t = u_t with u_t \sim N(0, \Omega_t) the function produces a draw of the lower triangular part of A_{0,t} similar as in Primiceri (2005), i.e., using

y_t = Z_t \psi_t + u_t,

where

Z_{t} = \begin{bmatrix} 0 & \dotsm & \dotsm & 0 \\ -y_{1, t} & 0 & \dotsm & 0 \\ 0 & -y_{[1,2], t} & \ddots & \vdots \\ \vdots & \ddots & \ddots & 0 \\ 0 & \dotsm & 0 & -y_{[1,...,K-1], t} \end{bmatrix}

and y_{[1,...,K-1], t} denotes the first to (K-1)th elements of the vector y_t.

The algorithm of Chan and Jeliazkov (2009) is used to obtain time varying coefficients.

Value

A matrix.

References

Chan, J., & Jeliazkov, I. (2009). Efficient simulation and integrated likelihood estimation in state space models. International Journal of Mathematical Modelling and Numerical Optimisation, 1(1/2), 101–120. doi:10.1504/IJMMNO.2009.030090

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

# Load example data
data("e1")
y <- log(t(e1))

# Generate artificial draws of other matrices
u_omega_i <- diag(1, 3)
v_sigma_i <- diag(1000, 3)
psi_init <- matrix(0, 3)

# Obtain posterior draw
post_normal_covar_tvp(y, u_omega_i, v_sigma_i, psi_init)

Posterior Draw from a Normal Distribution

Description

Produces a draw of coefficients from a normal posterior density for a model with seemingly unrelated regresssions (SUR).

Usage

post_normal_sur(y, z, sigma_i, a_prior, v_i_prior, svd = FALSE)

Arguments

Details

The function produces a posterior draw of the coefficient vector a for the model

y_{t} = Z_{t} a + u_{t},

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

For a given prior mean vector \underline{a} and prior covariance matrix \underline{V} the posterior covariance matrix is obtained by

\overline{V} = \left[ \underline{V}^{-1} + \sum_{t=1}^{T} Z_{t}^{\prime} \Sigma_{t}^{-1} Z_{t} \right]^{-1}

and the posterior mean by

\overline{a} = \overline{V} \left[ \underline{V}^{-1} \underline{a} + \sum_{t=1}^{T} Z_{t}^{\prime} \Sigma_{t}^{-1} y_{t} \right].

Value

A vector.

Examples

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

# Generate model data
temp <- gen_var(data, p = 2, deterministic = "const")
y <- t(temp$data$Y)
z <- temp$data$SUR
k <- nrow(y)
tt <- ncol(y)
m <- ncol(z)

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

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

# Draw parameters
a <- post_normal_sur(y = y, z = z, sigma_i = sigma_i,
                     a_prior = a_mu_prior, v_i_prior = a_v_i_prior)

Stochastic Search Variable Selection

Description

ssvs employs stochastic search variable selection as proposed by George et al. (2008) to produce a draw of the precision matrix of the coefficients in a VAR model.

Usage

ssvs(a, tau0, tau1, prob_prior, include = NULL)

Arguments

Details

The function employs stochastic search variable selection (SSVS) as proposed by George et al. (2008) to produce a draw of the diagonal inverse prior covariance matrix \underline{V}^{-1} and the corresponding vector of inclusion parameters \lambda of the vectorised coefficient matrix a = vec(A) for the VAR model

y_t = A x_t + u_t,

where y_{t} is a K-dimensional vector of endogenous variables, x_{t} is a vector of explanatory variabes and the error term is u_t \sim \Sigma.

Value

A named list containing two components:

References

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

Examples

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

# Generate model data
temp <- gen_var(data, p = 2, deterministic = "const")
y <- t(temp$data$Y)
x <- t(temp$data$Z)
k <- nrow(y)
tt <- ncol(y)
m <- k * nrow(x)

# Obtain SSVS priors using the semiautomatic approach
priors <- ssvs_prior(temp, semiautomatic = c(0.1, 10))
tau0 <- priors$tau0
tau1 <- priors$tau1

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

# Priors
a_mu_prior <- matrix(0, m)
a_v_i_prior <- diag(c(tau1^2), m)

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

# Draw parameters
a <- post_normal(y = y, x = x, sigma_i = sigma_i,
                 a_prior = a_mu_prior, v_i_prior = a_v_i_prior)

# Run SSVS
lambda <- ssvs(a = a, tau0 = tau0, tau1 = tau1,
               prob_prior = prob_prior)

Stochastic Search Variable Selection Prior

Description

Calculates the priors for a Bayesian VAR model, which employs stochastic search variable selection (SSVS).

Usage

ssvs_prior(object, tau = c(0.05, 10), semiautomatic = NULL)

Arguments

Value

A list containing the vectors of prior standard deviations for restricted and unrestricted variables, respectively.

References

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

Examples

# Prepare data
data("e1")
data <- diff(log(e1))

# Generate model input
object <- gen_var(data)

# Obtain SSVS prior
prior <- ssvs_prior(object, semiautomatic = c(.1, 10))

Stochastic Volatility

Description

Produces a draw of log-volatilities.

Usage

stoch_vol(y, h, sigma, h_init, constant)

Arguments

Details

The function is a wrapper for function stochvol_ksc1998.

Value

A vector of log-volatility draws.

References

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

Kim, S., Shephard, N., & Chib, S. (1998). Stochastic volatility. Likelihood inference and comparison with ARCH models. Review of Economic Studies 65(3), 361–393. doi:10.1111/1467-937X.00050

Examples

data("us_macrodata")
y <- matrix(us_macrodata[, "r"])

# Initialise log-volatilites
h_init <- matrix(log(var(y)))
h <- matrix(rep(h_init, length(y)))

# Obtain draw
stoch_vol(y - mean(y), h, matrix(.05), h_init, matrix(0.0001))

Stochastic Volatility

Description

Produces a draw of log-volatilities.

Usage

stochvol_ksc1998(y, h, sigma, h_init, constant)

Arguments

Details

For each column in y the function produces a posterior draw of the log-volatility h for the model

y_{t} = e^{\frac{1}{2}h_t} \epsilon_{t},

where \epsilon_t \sim N(0, 1) and h_t is assumed to evolve according to a random walk

h_t = h_{t - 1} + u_t,

with u_t \sim N(0, \sigma^2).

The implementation is based on the algorithm of Kim, Shephard and Chip (1998) and performs the following steps:

1. Perform the transformation y_t^* = ln(y_t^2 + constant).

2. Obtain a sample from the seven-component normal mixture for approximating the log-\chi_1^2 distribution.

3. Obtain a draw of log-volatilities.

The implementation follows the code provided on the website to the textbook by Chan, Koop, Poirier, and Tobias (2019).

Value

A vector of log-volatility draws.

References

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

Kim, S., Shephard, N., & Chib, S. (1998). Stochastic volatility. Likelihood inference and comparison with ARCH models. Review of Economic Studies 65(3), 361–393. doi:10.1111/1467-937X.00050

Examples

data("us_macrodata")
y <- matrix(us_macrodata[, "r"])

# Initialise log-volatilites
h_init <- matrix(log(var(y)))
h <- matrix(rep(h_init, length(y)))

# Obtain draw
stochvol_ksc1998(y - mean(y), h, matrix(.05), h_init, matrix(0.0001))

Stochastic Volatility

Description

Produces a draw of log-volatilities based on Omori, Chib, Shephard and Nakajima (2007).

Usage

stochvol_ocsn2007(y, h, sigma, h_init, constant)

Arguments

Details

For each column in y the function produces a posterior draw of the log-volatility h for the model

y_{t} = e^{\frac{1}{2}h_t} \epsilon_{t},

where \epsilon_t \sim N(0, 1) and h_t is assumed to evolve according to a random walk

h_t = h_{t - 1} + u_t,

with u_t \sim N(0, \sigma^2).

The implementation follows the algorithm of Omori, Chib, Shephard and Nakajima (2007) and performs the following steps:

1. Perform the transformation y_t^* = ln(y_t^2 + constant).

2. Obtain a sample from the ten-component normal mixture for approximating the log-\chi_1^2 distribution.

3. Obtain a draw of log-volatilities.

The implementation is an adaption of the code provided on the website to the textbook by Chan, Koop, Poirier, and Tobias (2019).

Value

A vector of log-volatility draws.

References

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

Omori, Y., Chib, S., Shephard, N., & Nakajima, J. (2007). Stochastic volatiltiy with leverage. Fast and efficient likelihood inference. Journal of Econometrics 140(2), 425–449. doi:10.1016/j.jeconom.2006.07.008

Examples

data("us_macrodata")
y <- matrix(us_macrodata[, "r"])

# Initialise log-volatilites
h_init <- matrix(log(var(y)))
h <- matrix(rep(h_init, length(y)))

# Obtain draw
stochvol_ocsn2007(y - mean(y), h, matrix(.05), h_init, matrix(0.0001))

Summarising Bayesian VAR Coefficients

Description

summary method for class "bvar".

Usage

## S3 method for class 'bvar'
summary(object, ci = 0.95, period = NULL, ...)

## S3 method for class 'summary.bvar'
print(x, digits = max(3L, getOption("digits") - 3L), ...)

Arguments

Value

summary.bvar returns a list of class "summary.bvar", which contains the following components:

Summarising Bayesian VAR or VEC Models

Description

summary method for class "bvarlist".

Usage

## S3 method for class 'bvarlist'
summary(object, ...)

Arguments

Details

The log-likelihood for the calculation of the information criteria is obtained by

LL = \frac{1}{R} \sum_{i = 1}^{R} \left( \sum_{t = 1}^{T} -\frac{K}{2} \ln 2\pi - \frac{1}{2} \ln |\Sigma_t^{(i)}| -\frac{1}{2} (u_t^{{(i)}\prime} (\Sigma_t^{(i)})^{-1} u_t^{(i)} \right)

, where u_t = y_t - \mu_t. The Akaike, Bayesian and Hannan–Quinn (HQ) information criteria are calculated as

AIC = 2 (Kp + Ms + N) - 2 LL

,

BIC = (Kp + Ms + N) ln(T) - 2 LL

and

HQ = 2 (Kp + Ms + N) ln(ln(T)) - 2 LL

, respectively, where K is the number of endogenous variables, p the number of lags of endogenous variables, M the number of exogenous variables, s the number of lags of exogenous variables, N the number of deterministic terms and T the number of observations.

Value

summary.bvarlist returns a table of class "summary.bvarlist".

Summarising Bayesian VEC Coefficients

Description

summary method for class "bvec".

Usage

## S3 method for class 'bvec'
summary(object, ci = 0.95, period = NULL, ...)

## S3 method for class 'summary.bvec'
print(x, digits = max(3L, getOption("digits") - 3L), ...)

Arguments

Value

summary.bvec returns a list of class "summary.bvec", which contains the following components:

Summarising Bayesian Dynamic Factor Models

Description

summary method for class "dfm".

Usage

## S3 method for class 'dfm'
summary(object, ci = 0.95, ...)

Arguments

Value

summary.dfm returns a list of class "summary.dfm", which contains the following components:

Thinning Posterior Draws

Description

Thins the MCMC posterior draws in an object of class "bvar".

Usage

## S3 method for class 'bvar'
thin(x, thin = 10, ...)

Arguments

Value

An object of class "bvar".

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)

# Obtain posterior draws
object <- draw_posterior(model)

object <- thin(object)

Thinning Posterior Draws

Description

Thins the MCMC posterior draws in an object of class "bvarlist".

Usage

## S3 method for class 'bvarlist'
thin(x, thin = 10, ...)

Arguments

Value

An object of class "bvarlist".

Examples

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

# Generate multiple model matrices
model <- gen_var(e1, p = 1:2, deterministic = 2,
                 iterations = 100, burnin = 10)

# Add prior specifications
model <- add_priors(model)

# Obtain posterior draws
object <- draw_posterior(model)

# Thin
object <- thin(object)

Thinning Posterior Draws

Description

Thins the MCMC posterior draws in an object of class "bvec".

Usage

## S3 method for class 'bvec'
thin(x, thin = 10, ...)

Arguments

Value

An object of class "bvec".

Examples

# Load data
data("e6")

# Generate model data
model <- gen_vec(e6, p = 2, r = 1,
                 const = "unrestricted", seasonal = "unrestricted",
                 iterations = 100, burnin = 10)

# Add prior specifications
model <- add_priors(model)

# Obtain posterior draws
object <- draw_posterior(model)

# Thin
object <- thin(object)

Thinning Posterior Draws

Description

Thins the MCMC posterior draws in an object of class "dfm".

Usage

## S3 method for class 'dfm'
thin(x, thin = 10, ...)

Arguments

Value

An object of class "dfm".

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 <- draw_posterior(model)

# Plot factors
object <- thin(object, thin = 2)

US macroeconomic data

Description

The data set contains quarterly time series for the US CPI inflation rate, unemployment rate, and Fed Funds rate from 1959Q2 to 2007Q4. It was produced from file "US_macrodata.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/Chapter20/code_for_exercise_1/US_macrodata.csv.

Usage

data("us_macrodata")

Format

A named time-series object with 195 rows and 3 variables:

Dp

CPI inflation rate.

u

unemployment rate.

r

Fed Funds rate.

References

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