General overview of BVARs:

• structural vs reduced form BVARs

Structural: \(Ay_t = B_0+ B_1 y_{t-1} +\ldots + B_p y_{t-p} + u_t\)

Reduced form: \(y_t =\Phi_{const}+ \Phi_1 y_{t-1} + \Phi_2 y_{t-2} +\ldots + \Phi_p y_{t-p} + \varepsilon_t\)

Link: \(B_i = A \Phi_i\).

• time varying parameters vs classic BVARs

The parameters \(\Phi_i\) may change over time.

Classical BVARs in reduced form

\(y_t =\Phi_{const}+ \Phi_1 y_{t-1} + \Phi_2 y_{t-2} +\ldots + \Phi_p y_{t-p} + \varepsilon_t\)

We should impose prior on \(\Sigma\), \(\Phi_{const}\), \(\Phi_1\), \(\Phi_2\), …

Here \(y_t\) is multivariate: \(m\times 1\).

For \(m=10\) variables and \(p=4\) lags we have more than 400 parameters.

Grimpen Mire of priors

1. Confusing names for priors

2. No clear classification of priors

3. Contradictory notation

Results in:

1. Coding mistakes

2. You should struggle a lot to understand

3. Underuse of BVARs

Everything is simple:

\[ \begin{cases} y_t =\Phi_{const}+ \Phi_1 y_{t-1} + \Phi_2 y_{t-2} +\ldots + \Phi_p y_{t-p} + \varepsilon_t \\ \varepsilon_t\sim \mathcal{N}(0,\Sigma) \end{cases} \]

We should place prior on:

1. Covariance matrix \(\Sigma\)

2. Coefficients \(\Phi_i\)

Prior for \(\Sigma\) and \(\Phi_i\):

1. Independent \(\Sigma\) and \(\Phi_i\)

Prior \(p(\Sigma)\) and prior \(p(\Phi_{const}, \Phi_1, \ldots, \Phi_p)\)

1. Conjugate \(\Sigma\) and \(\Phi_i\)

Prior \(p(\Sigma)\) and prior \(p(\Phi_{const}, \Phi_1, \ldots, \Phi_p | \Sigma)\)

Sharp classification:

1. Independent \(\Sigma\) and \(\Phi_i\):
   • Independent Normal-Inverse Wishart prior
   • Independent Normal-Jeffreys prior
   • Minnesota prior
2. Conjugate \(\Sigma\) and \(\Phi_i\):
   • Conjugate Normal-Inverse Wishart prior
   • Conjugate Normal-Jeffreys prior
   • Kronecker-subcase of Minnesota prior

Technical details:

• Independent Normal-Inverse Wishart prior

\[ \begin{cases} \Sigma \sim \mathcal{IW}(\underline S,\underline \nu) \\ \phi \sim \mathcal{N}(\underline \phi, \underline \Xi) \\ p(\phi, \Sigma) = p(\phi)\cdot p(\Sigma) \end{cases} \]

• Minnesota prior

\[ \begin{cases} \Sigma = const\\ \phi \sim \mathcal{N}(\underline \phi, \underline \Xi) \end{cases} \]

• Conjugate Normal-Inverse Wishart prior

\[ \begin{cases} \Sigma \sim \mathcal{IW}(\underline S,\underline \nu) \\ \phi | \Sigma \sim \mathcal{N}(\underline \phi, \Sigma \otimes \underline \Omega) \end{cases} \]

Here be dragons

Conjugate Normal-Inverse Wishart prior

\[ \begin{cases} \Sigma \sim \mathcal{IW}(\underline S,\underline \nu) \\ \phi | \Sigma \sim \mathcal{N}(\underline \phi, \Sigma \otimes \underline \Omega) \end{cases} \]

• Explicit formula for posterior distribution

• You can sample directly from posterior

• Posterior distribution can be calculated in a second way:

  1. Add artificial observations

  2. Use OLS with augmented dataset

Two ways to specify a prior

1. Explicit use of the formula

\[ \begin{cases} \Sigma \sim \mathcal{IW}(\underline S,\underline \nu) \\ \phi | \Sigma \sim \mathcal{N}(\underline \phi, \Sigma \otimes \underline \Omega) \end{cases} \]

2. Implicit approach with artificial observations:
   • Augment dataset: obtain \(X^*\) and \(Y^*\)
   • \(\overline \Phi\) — estimates of the regression of \(Y^*\) on \(X^*\)
   • \(\overline S\) — cross-products of residuals
   • \(\overline \Omega^{-1}\) — cross-products of of regressors

Current situation:

• a bunch of separated MATLAB functions. Poor documentation, poor stability

• Eviews point and click interface. Poor documentation. Poor stability.

• three R-packages:
  • BMR. Good documentation. Poor stability.
  • MSBVAR. Poor documentation. Poor stability.
  • bvarsv. Good documentation. Complex model.

Almost no integration at all.

Ideal package for BVARs:

1. Open source. Only R/python/Julia/gretl. MATLAB/Stata/Eviews are almost dead.

2. Well documented. Consistent notation with major articles.

3. Robust. Sever multicollinearity problem. Should not throw errors!

4. Easy to use. Reasonable defaults to plug and play!

5. Well integrated. Do not reinvent the weel.

6. Fast. Probably Julia/C++ implementation.

Developing R package bvarr

• Open source. OK

• Well documented. Almost OK. Article «BVAR mapping» should soon appear in Prikladnaya ekonometrika.

• Easy to use. Almost OK. Some defaults are imposed. More testing is on the way.

• Robust. Almost OK. We use SVD and other tricks of the trade.

We never use the formula \(\hat\beta = (X'X)^{-1}X'y\) :)

Package bvarr todo list:

1. More priors. Only conjugate-inverse normal Wishart for the moment

2. Visualise priors. Almost no one really understands priors.

3. Well integrated. Integration with forecast, vars and BMR R-packages is planned.

4. Fast. Irrelevant for conjugate-inverse normal Wishart.

Some random plot

library(ggplot2)
qplot(rnorm(100)) + xlab("Привет, Ира!")

Plug-and-play!

1. Install

install.packages(c("sophisthse", "devtools", "dplyr"))
devtools::install_github("bdemeshev/bvarr")

2. Attach

library("dplyr")
library("sophisthse")
library("bvarr")

3. Load and clean data

y <- sophisthse(c("UNEMPL_Y", "GDPVA_T_Y_DIRI"))
y <- y[, c("UNEMPL_Y", "GDPVA_T_Y_DIRI")] %>% na.omit

4. Estimate and predict

setup <- bvar_conj_setup(y, p = 4)
model <- bvar_conj_estimate(setup)
y_hat <- bvar_conj_forecast(model, h = 2)

What can we do?

1. If you are a probability/econometrics lecturer. Just introduce MCMC in the course!

2. If you are a student. Ask your lecturer about Bayesian approach and MCMC. It’s modern, stylish and ye-ye!

How to introduce MCMC?

1. Don’t panic!

2. Basic course. Use MCMC as a black box for the conditional probability formula.

3. Advanced course. Use MCMC to illustrate the convergence in distribution.

Software to use

1. Plug-and-play bayesian solutions for a wide range of classic models.
   • R + MCMCpack.
2. Make students build they own hierarchical models:
   • R/python/Julia + STAN/JAGS
   • Julia + Lora/Mamba
   • python + PyMC

Our questions:

1. An analog of M3-competition for multivariate time series?

2. A good multivariate time series plot? For 20+ series at once?

3. How to obtain \((X'X)^{-1}\) if SVD of \(X\) fails?

4. How to visualise Inverse-Wishart prior? Or prior for 400+ coefficients?

5. Any experience with MCMC in bachelor courses?

Contacts:

Boris Demeshev: boris.demeshev@gmail.com

Oxana Malakhovskaya: oxana.malakhovskaya@gmail.com

Project web-page: bdemeshev.github.io/bvarr

Quote for the BVAR paradox:

Я вам, ребята, на мозги не капаю, но вот он, перегиб и парадокс: кого-то выбирают римским папою, кого-то запирают в тесный бокс.

Владимир Высоцкий