GAUSS vs R vs BEAR: Side-by-Side#

This page compares the same BVAR analysis implemented in GAUSS, R, and MATLAB (ECB BEAR Toolbox). All code is copy-paste runnable.

The Task#

Estimate a Bayesian VAR(4) on 200 quarters of US macroeconomic data (GDP growth, CPI inflation, federal funds rate), compute impulse responses, and forecast 8 quarters ahead.

GAUSS#

library timeseries;

fname = getGAUSSHome("pkgs/timeseries/examples/data/us_macro_quarterly.csv");
data = loadd(fname);

// BVAR(4)
result = bvarFit(data, p=4, ar=0);

// IRF
rv = varFit(data, 4, quiet=1);
irf = irfCompute(rv, 20);

// Forecast
fc = bvarForecast(result, 8);

Estimation, IRF, and forecast in one script using the timeseries library.

R (vars + BVAR packages)#

library(vars)
library(BVAR)

y <- as.matrix(read.csv("us_macro_quarterly.csv"))

# BVAR(4)
set.seed(42)
bv <- bvar(y, lags = 4, n_draw = 6000, n_burn = 1000,
           n_thin = 1, verbose = FALSE)

# IRF (from OLS VAR — BVAR package doesn't do Cholesky IRF directly)
v <- VAR(y, p = 4, type = "const")
ir <- irf(v, n.ahead = 20, boot = FALSE)

# Forecast
fc <- predict(bv, horizon = 8, conf_bands = 0.68)

Requires two packages: vars for OLS/IRF and BVAR for Bayesian estimation. The BVAR package uses Gibbs sampling with hierarchical hyperparameter tuning — a different algorithm than both GAUSS and BEAR.

MATLAB (ECB BEAR Toolbox)#

addpath(genpath('tbx'));

opts = BEARsettings(2);
opts.frequency  = 2;
opts.startdate  = '1970q2';
opts.enddate    = '2019q4';
opts.varendo    = 'YER HICSA STN';
opts.lags       = 4;
opts.prior      = 21;
opts.It         = 10000;
opts.Bu         = 5000;
opts.IRF        = 1;
opts.IRFperiods = 20;
opts.F          = 1;
opts.Fendsmpl   = 1;
opts.Fstartdate = '2020q1';
opts.Fenddate   = '2021q4';

BEARmain(opts);

Data path, date range, and variable names are configured as strings. Output is saved to Excel and .mat files rather than returned to the workspace. Applications (IRF, forecast) must be enabled via flags. Each BEARmain() call re-estimates the model from scratch.

Timing Comparison#

All timings on the same 200-quarter, 3-variable dataset. GAUSS and R timed on Apple M-series. BEAR on MATLAB R2025b.

Task

GAUSS

R

BEAR

OLS VAR(4)

0.003s

0.004s

0.058s

BVAR(4), 5K draws

0.08s

0.66s

3.69s

8-step forecast

0.02s

0.12s

4.34s 1

Cholesky IRF (20h)

0.001s

0.003s

4.23s 1

1 BEAR re-estimates the full model for each application. Marginal IRF/forecast time is ~0.5s after subtracting re-estimation.

Implementation Differences#

GAUSS vs R: GAUSS uses the conjugate Normal-Inverse-Wishart posterior, which produces exact draws without MCMC. R’s BVAR package uses a Gibbs sampler. Both are correct Bayesian inference — the conjugate form avoids iterative sampling.

GAUSS vs BEAR: BEAR uses an independent Normal-Wishart prior with Gibbs sampling (MATLAB interpreted loops). GAUSS uses conjugate posterior draws (compiled backend). The difference compounds with draw count: at 50K draws, BEAR takes approximately 4 minutes vs approximately 1 second for GAUSS.

Numerical Agreement#

GAUSS matches BEAR to \(10^{-8}\) on OLS coefficients (same data, same model). BVAR posterior means agree within 0.06 with matched hyperparameters (conjugate vs independent NW prior form). R’s BVAR package uses a different prior (hierarchical hyperparameter tuning), so posterior means differ by design.

Full verification details: Verification and Cross-Validation.

What You Get With Each Platform#

GAUSS

R

BEAR

OLS VAR

varFit()

vars::VAR

VARtype=1

Bayesian VAR

bvarFit()

BVAR::bvar

VARtype=2

Stochastic volatility

bvarSvFit()

bsvars

VARtype=5

Cholesky IRF

irfCompute()

vars::irf

opts.IRF=1

Sign restrictions

svarIdentify()

bayesianVARs

opts.IRFt=4

Conditional forecast

condForecast()

(manual)

opts.CF=1

Density forecast

bvarSvForecast()

bayesianVARs

(manual)

Hyperparameter opt

bvarHyperopt()

Built-in

opts.hogs=1

MCMC diagnostics

varDiagnostics()

coda

(manual)

Forecast evaluation

dmTest() pitTest()

forecast

(not included)

FRED data access

fred_load

fredr

(not included)

Verified against

R + BEAR (428 tests)

—

—

Multi-Run Timing (5 runs, median)#

GAUSS timing variability is minimal — the compiled Rust backend produces deterministic execution times:

Task

Median

Min

Max

IQR

OLS VAR(4)

0.0001s

0.0001s

0.0023s

0.0000s

BVAR(4), 5K draws

0.077s

0.076s

0.082s

0.000s

SV-BVAR(4), 10K draws

1.161s

1.154s

1.166s

0.006s

Cholesky IRF

0.0005s

0.0005s

0.0011s

0.0000s

BVAR Forecast

0.024s

0.024s

0.024s

0.000s

All measurements on 3-variable, 200-quarter data. 5 runs each, Apple M-series. GAUSS runs under Rosetta 2 (x86_64 on ARM) — native arm64 GAUSS will be faster.

Scaling: Large Systems#

GAUSS handles large BVAR systems efficiently. Memory scales with \(n_{draws} \times K \times m\):

m

p

K

BVAR (5K draws)

SV (2K draws)

Memory (B draws)

3

4

13

0.08s

0.34s

1.5 MB

5

4

21

0.19s

0.66s

4.0 MB

10

4

41

0.71s

2.0s

15.6 MB

20

4

81

2.9s

7.8s

61.8 MB

50

4

201

18.0s

—

383 MB

For systems above m=10, use bvarSvFit() with sv_keep = "online" to reduce memory from O(n_draws * T * m) to O(reservoir_size * m). .. seealso:: Guides Getting Started, Choosing a VAR Model, Verification and Cross-Validation

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